**Pressure-Induced Phase Transformations (Volume II)**

Editors

**Daniel Errandonea Enrico Bandiello**

Basel • Beijing • Wuhan • Barcelona • Belgrade • Novi Sad • Cluj • Manchester

*Editors* Daniel Errandonea Departamento de F´ısica Aplicada-ICMUV, MALTA Consolider Team, Universidad de Valencia, Valencia, Spain `

Enrico Bandiello Instituto de Diseno para La ˜ Fabricacion y ´ Produccion Automatizada ´ MALTA Consolider Team Universitat Politecnica ` de Valencia ` Valencia, Spain

*Editorial Office* MDPI St. Alban-Anlage 66 4052 Basel, Switzerland

This is a reprint of articles from the Special Issue published online in the open access journal *Crystals* (ISSN 2073-4352) (available at: https://www.mdpi.com/journal/crystals/special issues/Pressure Induced Phase Transformations VolumeII).

For citation purposes, cite each article independently as indicated on the article page online and as indicated below:

Lastname, A.A.; Lastname, B.B. Article Title. *Journal Name* **Year**, *Volume Number*, Page Range.

**ISBN 978-3-0365-8564-2 (Hbk) ISBN 978-3-0365-8565-9 (PDF) doi.org/10.3390/books978-3-0365-8565-9**

© 2023 by the authors. Articles in this book are Open Access and distributed under the Creative Commons Attribution (CC BY) license. The book as a whole is distributed by MDPI under the terms and conditions of the Creative Commons Attribution-NonCommercial-NoDerivs (CC BY-NC-ND) license.

## **Contents**



## **About the Editors**

#### **Daniel Errandonea**

Daniel Errandonea, Professor Dr., is a full professor with the Department of Applied Physics of University of Valencia (Spain). He is an Argentinean-born physicist (married with two sons) who received an M.S. from the University of Buenos Aires (1992) and a Ph.D. from the University of Valencia (1998). He has authored/co-authored over 350 research articles in refereed scientific journals including *Prog. Mat. Sci.*, *Nat. Commun.*, *Phys. Rev. Lett.*, and *Advanced Science*, which have attracted 11000+ citations. His work on materials under extreme conditions of pressure and temperature has implications for fundamental and applied research. Among other subjects, during the last decade, Prof. Errandonea has comprehensively explored phase transitions in ternary oxides, semiconductors, metals, and related materials. Some of his accomplishments include the determination of high-pressure phase transitions, high-pressure and high-temperature phase diagrams (including melting curves), and the study of their implications in the physical properties of materials. Prof. Errandonea is a fellow of the Alexander von Humboldt Foundation, winning the Van Valkenburg Award and IDEA Prize, among others. He is presently a member of the MALTA Consolider and GREENMAT teams, the Editorial Board of *Crystals*, and the executive committee of the International Association for the Advancement of High Pressure Science and Technology (AIRAPT).

#### **Enrico Bandiello**

Enrico Bandiello, Ph.D., is a researcher at the Technical University of Valencia—UPV (Spain). He is an Italian-born physicist (married with one son) who received an M.S. (2012) and a Ph.D. (2017) from the University of Valencia. He has authored/co-authored over 38 research articles in refereed scientific journals including *Materials Today Advances*, *JACS*, and *Advanced Materials*, which have attracted 2000+ citations. His work on materials under extreme conditions of pressure and temperature has implications for fundamental and applied research. Among other subjects, during the last decade, Dr. Bandiello has comprehensively explored phase transitions in ternary oxides, semiconductors, and related materials. Some of his recent accomplishments include the determination of the high-pressure phase transitions of topological semiconductors like GaGeTe. Dr. Bandiello performed postdoctoral stays at the University of Valencia and the University of Verona. He is presently a member of the MALTA Consolider Team.

## **Preface**

The study of phase transitions in solids and liquids under high-pressure and high-temperature conditions is a very active and vigorous research field. In recent decades, thanks to the development of experimental techniques and computer simulation methods, a plethora of important discoveries have been made. Many of the achievements accomplished in recent years affect various research fields, from solid-state physics to chemistry, materials science, and geophysics. They not only contribute to a deeper understanding of solid–solid phase transitions but also to a better understanding of the properties of matter. This Special Issue presents and discusses these recent achievements, and is addressed to physicists, chemists, materials scientists, and researchers working in geophysics and Earth and planetary sciences.

#### **Daniel Errandonea and Enrico Bandiello**

*Editors*

## *Review* **Recent Progress in Phase Stability and Elastic Anomalies of Group VB Transition Metals**

**Yixian Wang 1,2,\*, Hao Wu 1, Yingying Liu 1, Hao Wang 2,3, Xiangrong Chen <sup>3</sup> and Huayun Geng 2,\***


**Abstract:** Recently discovered phase transition and elastic anomaly of compression-induced softening and heating-induced hardening (CISHIH) in group VB transition metals at high-pressure and hightemperature (HPHT) conditions are unique and interesting among typical metals. This article reviews recent progress in the understanding of the structural and elastic properties of these important metals under HPHT conditions. Previous investigations unveiled the close connection of the remarkable structural stability and elastic anomalies to the Fermi surface nesting (FSN), Jahn–Teller effect, and electronic topological transition (ETT) in vanadium, niobium, and tantalum. We elaborate that two competing scenarios are emerging from these advancements. The first one focuses on phase transition and phase diagram, in which a soft-mode driven structural transformation of BCC→RH1→RH2→BCC under compression and an RH→BCC reverse transition under heating in vanadium were established by experiments and theories. Similar phase transitions in niobium and tantalum were also proposed. The concomitant elastic anomalies were considered to be due to the phase transition. However, we also showed that there exist some experimental and theoretical facts that are incompatible with this scenario. A second scenario is required to accomplish a physically consistent interpretation. In this alternative scenario, the electronic structure and associated elastic anomaly are fundamental, whereas phase transition is just an outcome of the mechanical instability. We note that this second scenario is promising to reconcile all known discrepancies but caution that the phase transition in group VB metals is elusive and is still an open question. A general consensus on the relationship between the possible phase transitions and the mechanical elasticity (especially the resultant CISHIH dual anomaly, which has a much wider impact), is still unreached.

**Keywords:** phase stability; elastic anomalies; structural transition; group VB transition metals; high pressure and high temperature

#### **1. Introduction**

The group VB transition metals (vanadium, niobium, and tantalum) have extensive applications because of their excellent mechanical and physical properties. Vanadium is well known as a kind of metallic "vitamin", which is commonly added to iron or steel to increase their toughness, strength, and abrasion resistance. Moreover, vanadium plays an excellent role in titanium alloys, which greatly promotes the development of the aerospace industry. With the progression in science and technology, there are higher requirements for advanced materials, with the application of vanadium becoming more and more extensive, which covers batteries, pharmaceuticals, optics, and many other fields. Niobium and tantalum belong to refractory metals because of their melting point is higher than 2700 K [1]. One can increase a material's strength at high temperature and improve the processing performance by adding niobium or tantalum as alloying elements. On the

**Citation:** Wang, Y.; Wu, H.; Liu, Y.; Wang, H.; Chen, X.; Geng, H. Recent Progress in Phase Stability and Elastic Anomalies of Group VB Transition Metals. *Crystals* **2022**, *12*, 1762. https://doi.org/10.3390/ cryst12121762

Academic Editors: Daniel Errandonea and Enrico Bandiello

Received: 14 November 2022 Accepted: 30 November 2022 Published: 5 December 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

other hand, tantalum is also a widely used pressure standard and high technology material due to its strong stability in chemistry and mechanics, as well as the very high melting point (3269 K) [2].

Due to their important applications, group VB transition metals have always attracted much attention. However, most studies in early days mainly focused on their superconducting properties [3–10] because their superconducting transition temperature is quite high at ambient conditions. Theoretical studies show that the superconducting transition temperature of vanadium and niobium depends on the pressure; and the density of electron states at Fermi level has a significant impact on the relationship [11]. In 1997, Struzhkin et al. [10] measured the superconducting *Tc* of niobium and tantalum up to 100 GPa through a highly sensitive magnetic susceptibility technique. The results showed that *Tc* in tantalum remains nearly constant at 4.38 K in the range of 0–45 GPa. However, they observed *Tc* anomalies in niobium around 5–6 GPa and 60–70 GPa, where *Tc* increases by 0.7 K and decreases by about 1.0 K, respectively. Struzhkin et al. suggested that the anomalies in niobium arise from stress-sensitive ETT. Later, Tse et al. [12] calculated the Fermi surface of niobium by the density functional theory (DFT) method and found that the Fermi surface does undergo a topological transformation under pressure, confirming the results of Struzhkin et al. [10].

Unlike niobium and tantalum, vanadium has a large positive pressure coefficient of *Tc*. Smith [13] measured the *Tc* of vanadium under pressure up to 2.4 GPa and found a linear increase in *Tc* with d*Tc*/d*P* = 0.062 K/GPa. Subsequently, Brandt et al. [14] also observed a monotonic increase in *Tc* up to 18 GPa using ice bombs and mechanical presses. After that, Akahama et al. [15] carried out electrical resistance measurements of vanadium up to 49 GPa to investigate the upper bound of the monotonically increased *Tc* with pressure. They found that *Tc* increases linearly with a coefficient of d*Tc*/d*P* = 0.096 K/GPa and reaches a value of 9.6 K at about 18 GPa, the maximum pressure in their experiment. This value of *T*<sup>c</sup> is comparable to that of niobium at ambient pressure. With the development of experimental technology, Ishizuka et al. [16] studied the superconducting properties of vanadium under higher pressure by using a vibrating coil magnetometer. The results showed that *Tc* increases from 5.3 K to 17.2 K from zero pressure to 120 GPa, and the superconducting transition temperature increases almost linearly.

In order to understand these experimental results, Suzuki et al. [17] further studied the superconducting properties of vanadium under pressure by first-principles calculations. It was found that the *Tc* of vanadium shows an obvious upward trend with the change of pressure, and the increasing rate of *Tc* decreases gradually at about 80 GPa, which is qualitatively consistent with the experimental results. Moreover, they interpreted such characteristic behavior of *Tc* under pressure by attributing it to a significant frequency softening of the transverse mode near the Γ-H line as pressure increased. According to their results, when the pressure is above 130 GPa, the transverse acoustic mode (TA) even has imaginary frequency, indicating that the BCC phase of vanadium becomes dynamically unstable and there is an opportunity of structural phase transformation. However, at that time, the research on vanadium was limited to its superconducting properties and did not pay much attention to its structural instability under high pressure. This softening was subsequently confirmed by calculated elastic constants [18,19], in which *C*<sup>44</sup> continuously decreased to negative values, indicating mechanical instability of the BCC structure, but the crystalline structure of the high-pressure phase was not proposed at that time.

Until 2007, Ding et al. [20] performed X-ray diffraction (XRD) experiments on vanadium using diamond anvil cell (DAC) up to 150 GPa. They discovered a novel rhombohedral (RH) phase appearing around 63–69 GPa. This novel high-pressure structural phase transition had not been detected in any of the earlier experiments. After that, the focus of investigation on group VB transition metals has been gradually shifted to study their structural stability and phase transition.

In addition to the structural transformation, the group VB transition elements also exhibit striking anomalous elastic softening under pressure, which is quite different from other transition metals. According to existing literature reports [18,21,22], the pressureinduced shear elastic softening of vanadium, niobium, and tantalum originates from the electronic structure, which is closely related to the FSN and ETT. Furthermore, the elastic softening in vanadium and niobium were discovered to gradually diminish with increased temperature, effectively giving rise to a heating-induced *hardening* phenomenon, which is very rare (if any) to our knowledge. These elastic anomalies might be taken as due to the phase transition. However, some experimental and theoretical facts that are incompatible with this scenario.

During the past two decades, the structural stability and elastic anomalies of the group VB transition elements have been the subject of numerous theoretical and experimental studies [23–76]. The progress is tremendous; regardless, there are still many controversies on some key issues. In this paper, we review the current understanding of the structural and mechanical anomalies of these important metals under high temperature and high pressure. It is hoped that this paper will significantly promote the understanding of the physical properties for more broad types of metals under extreme conditions.

#### **2. Phase Stability and Elastic Anomalies in V, Nb, Ta**

#### *2.1. Pressure Effect on Structure Stability*

The first direct evidence of a phase transition in vanadium came from the static compression experiment by Ding et al. [20]. They observed a new type of high-pressure structural transition from BCC to an RH phase at 63–69 GPa, which once was thought of as a second-order transition and was not found in any of the earlier experiments with elements or compounds. In general, the phase transition sequence of transition metals under pressure is hexagonal close packed (HCP)→body centered cubic (BCC)→HCP→face centered cubic (FCC). Based on this sequence, the stability of the BCC phase has long been predicted to be very high [77]. Therefore, the discovery of a phase transition in vanadium below 70 GPa is very remarkable.

Soon after, Lee et al. [36] confirmed this phase transition with DFT calculations and showed that a metastable RH structure is formed at 73 GPa and becomes the ground state at 84 GPa. This low-pressure RH phase is termed as "RH1", which has an angle of *α* = 110.25◦. Furthermore, Lee et al. predicted two other transformations that were not detected in Ding et al.'s experiment: the second transformation to a high-pressure structure "RH2" with an angle *α* = 108.14◦ at 120 GPa, and the third transformation back to the high-symmetric BCC structure (*α* = 109.47◦) at 280 GPa. As the pressure continues to increase, the BCC phase becomes the only stable structure at 315 GPa. Since the latent heat of BCC→RH transition is much smaller than the thermal fluctuation at room temperature, Lee et al. [36] suggested that this transformation is first-order, which contradicts the second-order transition proposed by Ding et al. [20]. To verify this result, Lee et al. [37] further studied the elastic constants and volume changes associated with two high-pressure RH phase transitions in vanadium. The results shown are that there were small discontinuities in shear modulus and other elastic properties in the phase transitions even at zero temperature, indicating that the phase transitions should be first-order.

The prediction of RH1 and RH2 phases in vanadium was supported by phonon calculations of Luo et al. [35]. They found that the lattice dynamical instability of vanadium starts at 62 GPa and phonon softening leads to a phase transition of BCC→RH1 (*α* = 110.5◦). At about 130 GPa, the angle of RH1 phase changes to 108.2◦, and the electronic structure changes drastically. At a pressure of 250 GPa, lattice dynamics calculations show that the stability of BCC structure is restored. Luo et al. [35] suggested that the dramatic change in the electronic structures of vanadium under pressure are the driving force behind the structural phase transitions. Later, Verma et al. [38] and Qiu et al. [39] further investigated the structural stability of vanadium under high pressure through first-principles calculations. Both confirmed the existence of the RH1 and RH2 phases, as well as the pressure-induced structural transition sequence of BCC→RH1→RH2→BCC reported by Lee et al. [36]. Meanwhile, detailed electronic structure analysis by Verma et al. showed that the phase transition of BCC→RH1 is caused by the Jahn–Teller mechanism. Although different theoretical studies

have reached a consensus on the phase transition sequence of vanadium under pressure, there is still disagreement on the exact value of the phase transition pressure. For example, Qiu et al. reported a BCC→RH1 transition pressure of 32 GPa, while all other theoretical transition pressures were located between 60 and 84 GPa.

To trace the possible origin of the discrepancy between these calculations, two different methods were exploited by Wang et al. [54] to evaluate the phase transition pressures of vanadium. The first method is to choose RH1 and RH2 as the initial structure, and directly optimize them under different pressures without any symmetry constraints. The calculated enthalpy difference with respect to the BCC phase as a function of pressure is shown in Figure 1 (taken from [54]). It can be seen clearly that the BCC→RH1 transition is not at 30 GPa, where RH1 is dynamically instable and spontaneously collapses to BCC phase. According to Wang et al. [54], the relaxation of the RH1 phase to the BCC phase is far from being perfect, and the angle is about 109.51◦ at 20–40 GPa, which reflects a possibility that non-hydrostatic loading can easily drive vanadium towards RH-like deformations. It is worth noting that another RH phase (RH2) transforms to a similar twisted BCC structure with *α* = 109.39◦ when below 110 GPa. In addition, when the pressure is about 98 GPa, the RH1 phase becomes the ground state with an angle of *α* = 110.17◦. With the further increase of pressure, the RH1 phase transforms into another RH phase (RH2) at about 128 GPa. At the pressure of 211 GPa, RH2 reaches the maximum stability, with *α* = 108.23◦. As compression increases, the RH2 phase eventually collapses to the BCC phase at about 300 GPa.

**Figure 1.** (**a**) Enthalpy difference of vanadium in RH1 and RH2 structures at zero Kelvin with respect to the BCC phase as a function of pressure. (**b**) Variation of angle α in RH1 and RH2 structures as a function of pressure at zero Kelvin. Note that *α* = 109.47◦ corresponds to the perfect BCC structure. (By the courtesy of Ref. [54]).

In the second method, Wang et al. [54] adopted the same method as Lee et al. [36], that is, twisting the BCC structure along a predetermined path. Note that the unit volume is conserved in this approach. Qiu et al. [39] argued that such a treatment would result in a higher phase transition pressure. According to Lee et al., the error caused by fixed volume can be corrected by the following formula

$$H(\delta, P\_0) \approx lI(\delta, V\_0) + P\_0V\_0 - \frac{1}{2B(\delta, V\_0)} \Delta P(\delta, V\_0)^2 V\_0. \tag{1}$$

It is worth noting that only the first term was used in Lee et al.'s calculations [36]. After careful examination, Wang et al. [54] found that the correction of the third term is indeed small, which means that the contribution of volume relaxation can be safely ignored when studying the relative phase stability. This supports the assessment of Lee et al. [36]. In addition, the transition pressures of 'unrelaxed' calculations (method II) by Wang et al. [54]

are in good accordance with the full structural relaxation calculations (method I), therefore, Qiu et al.'s comment on Lee et al.'s results is inappropriate.

Wang et al. [54] further studied the influence of changing the position of Fermi level on the structural stability of vanadium by using a partial jellium model, see Figure 2 (taken from [54]). Here, the Fermi level is shifted by charge transfer (chemical doping) changing the orbital occupations. Since the RH2 phase reaches the maximal stability at 211 GPa, it should have the optimum orbital occupation. Thus, raising (adding electrons) or lowering (removing electrons/adding holes) the Fermi level pushes the system away from the optimum occupation, so the stability of RH2 phase will be weakened in both cases. However, Wang et al.'s calculations showed that moving the Fermi level down further stabilized the RH2 phase, while moving the Fermi level up greatly destabilized the RH2 phase. This is consistent with Landa et al.'s band-filling argument when alloying vanadium with the same transition series [41]; however, it is incompatible with Jahn–Teller mechanism.

**Figure 2.** Calculated enthalpy difference with respect to BCC phase as a function of the RH deformation parameter *δ* when the Fermi level is shifted up or down at a pressure of 211 GPa. Inset: Calculated differential charge density between Δ = −0.77% and Δ = 0. (By the courtesy of Ref. [54]).

According to the calculation of Wang et al. [54], RH2 phase reaches the maximal stability at 211 GPa when Δ = −2.15% (Δ represents the percentage of total charge added to /removed from the system). Further shifting down the Fermi level reduces the stability of RH phase. When Δ < −4.85%, BCC becomes stable again. In the inset of Figure 2, Wang et al. plotted the differential charge density between Δ = −0.77% and Δ = 0. As shown, the removed electrons/added holes are distributed around the nucleus and mainly exhibit *d* orbital characteristics. Since the *d* orbitals' delocalization results in lower electronegativity of the RH2 phase relative to the BCC phase, the localization (or delocalization) of *d* electrons has an important effect on the stability of the RH phase.

In addition, Wang et al. [54] found that even if the absolute convergence is achieved, the DFT method still has the problem of insufficient accuracy when exploring the phase transition of vanadium. Consider that the semi-local functional PBE may not be able to handle strong electron correlations in narrow-band systems, so they thought that the quality of the exchange-correlation (XC) functional may be an important factor affecting the results. For this reason, an evaluation of the advanced hybrid functionals in vanadium and niobium was further carried out by Wang et al. [66]. The results show that the common HSE06, PBE0, and B3LYP hybrid functionals are complete failures in describing the mechanical properties of these metals. The unexpected failure is due to the very rare localization error in these functionals, which is further supported by a similar failure of the DFT + *U* method. To solve this problem, Wang et al. [66] proposed a DFT + *J* method to promote on-site electron exchange, which well reproduces the experimental shear modulus under ambient conditions. However, Wang et al. found that the PBE + *J* increases the BCC→RH transition pressure, and the correction of localization error weakens (or even eliminates) the RH phases. They concluded that RH phase could be unstable under more accurate calculation methods, which is a striking prediction and challenges the previously reported structural transition in vanadium.

Following this prediction, an independent XRD experiment was performed by Akahama et al. [73] to study the structural stability of vanadium. The results showed that BCC vanadium is stable up to 189 GPa at room temperature, while the RH phase (*α* > 109.47◦) reported in previous studies should be a metastable phase induced by non-hydrostatic pressure. This supports the prediction of Wang et al. [66] that RH phase may be unstable in vanadium. Furthermore, they observed a new high-pressure phase after annealing at 242 GPa, which was also confirmed from a different experiment at room temperature. Akahama et al. interpreted the phase as RH phase with *α* < 109.47◦. However, the pressure range does not agree with the previous theoretical [35–37,54] and experimental results [21,47,51,55].

At the same time, Stevenson et al. [74] re-performed the XRD experiment at pressures up to 154 GPa using polycrystalline (powder) and single crystal samples with various pressure transfer media (PTM). It was found that only the single-crystal samples reveal two RH phases, and the distortions from cubic symmetry are much smaller than previous results. That is to say, the observed RH phase is far from being perfect, and should be interpreted as a kind of lattice distortion rather than a phase transition. Moreover, Wang et al. [72] measured the sound velocity of vanadium through shock wave experiment recently. They found that when the pressure was above 79 GPa, the sound velocity of the shocked vanadium was closer to the RH phase rather than the BCC phase. The unexpected high-pressure phases along the Hugoniot can be seen as slight distortions of the BCC structure, which may be caused by the dynamic, nonequilibrium, and nonhydrostatic nature of planar shock waves. The above two experimental signatures are compatible with the experimental results of Akahama et al. [73] and further confirm the theoretical assessment of Wang et al. made in Ref. [66].

In other experimental research, Jenei et al. [47] performed DAC experimental studies and found that the BCC structure transformed to RH1 phase at about 30 GPa in non-pressure medium, while it was around 60 GPa when Ne pressure medium was used. In addition, the transition can occur at a much lower pressure if under nonhydrostatic conditions. Nonetheless, in Ding et al.'s experiments [20], the transition pressure of BCC→RH in nonpressure medium and He pressure medium was 69 GPa and 63 GPa, respectively. Thus, the deviation in transition pressure of BCC→RH might not be due to the non-hydrostatic condition. In addition, Antonangeli et al. [55] used inelastic X-ray scattering to detect the phonon dispersion of single crystal vanadium under pressure up to 45 GPa. Their results showed that the transverse acoustic mode has abnormally high-pressure behavior along (100) direction, and the softening of *C*<sup>44</sup> causes the RH distortion around 34–39 GPa. It is obvious that the transition pressure is consistent with the diffraction results of 30 GPa in nonhydrostatic conditions by Jenei et al. [47]. It should be noticed that Antonangeli et al. [55] performed the measurements on relatively large single crystals, which are more susceptible to non-hydrostatic stress than powders. Moreover, according to the experimental study of Stevenson et al. [74], an RH high-pressure phase was indeed observed when using single crystal samples; but the high-pressure diffraction profiles from the polycrystalline samples is not suitable for RH lattice, regardless of the PTM used. Why there is such a big difference between the experimental data of powder and single crystal is still an open question that needs further study.

On the other hand, Yu et al. [51] recently measured the sound velocities and yield strength of vanadium through reverse impact experiments. They found an indication of the shock induced BCC→RH transition at about 60.5 GPa by the discontinuity of longitudinal sound velocity against shock pressure, which disagrees with the results of Jenei et al. [47] and Antonangeli et al. [55] but remains consistent with both DAC measurements by Ding et al. [20]. The aforementioned experiment studies cautioned that the phase transition in group VB metals is elusive, which remains an open question and no general consensus on this issue has been achieved.

#### *2.2. Temperature Effect on Structure Stability*

In 2014, Landa et al. [78] explored the phase stability of vanadium at high temperatures and pressures by using the self-consistent ab initio lattice dynamics (SCAILD) approach combined with DFT. In this study, the phonon-phonon interactions at elevated temperatures were considered. In Figure 3 (taken from [78]), Landa et al. showed the calculated phonon frequencies for vanadium under different temperatures at 182 GPa. The maximum stability of RH phase was measured by analyzing the variation in phonon dispersion of BCC phase. Their results showed that temperature promoting the phonon frequencies from being imaginary to being real along the Γ→H and Γ→N lines. When the temperature is above 8000 K, BCC phase becomes stable again. Since this temperature is significantly higher than the shock melting temperature of 6800 ± 800 K at 182 GPa [79], Landa et al. [78] concluded that the BCC phase is actually never stable at this density. Namely, high-pressure RH phases of compressed vanadium should have a very broad pressure-temperature stability region.

**Figure 3.** Calculated phonon dispersions for vanadium under different temperatures at a pressure of 182 GPa. (By the courtesy of Ref. [78]).

Nonetheless, lattice dynamics stability is not the unequivocal criterion for the thermodynamic stability of a phase. For the latter purpose, one should resort to free energy difference. To this end, based on Landa et al.'s initial results [78], Wang et al. further investigated the effect of thermo-electrons on the structure stability of vanadium by calculating the free energy using finite temperature DFT method [54]. Since the phase transitions of vanadium are closely related to the changes in the electronic structure, Wang et al. estimated that the contribution of thermo-electrons may be greater than that of lattice dynamics. As shown in Figure 4 (taken from [54]), the electronic temperature significantly reduces the stability of RH phases, and RH1 and RH2 transform back to BCC at around 1440 K (at 140 GPa) and 1915 K (at 211 GPa), respectively. Compared with the results of Landa et al. [78], this new transition temperature is much lower. It clearly demonstrated that compressed vanadium should transition back to BCC structure in the solid state; and the transition temperature is much lower than the melting temperature. Wang et al. analyzed the impact of lattice dynamics by including both thermo-electronic and phonon corrections in their assessment. It revealed that phonon correction will further reduce the transition temperature by about 260 K at 200 GPa [54], comparable to the usual expectation. Further study by Wang et al. indicated that this heating-induced reentrant transition in vanadium was mainly driven by electronic entropy.

**Figure 4.** Effect of thermo-electrons on the phase stability of vanadium at 211 GPa. (By the courtesy of Ref. [54]).

Based on these new theoretical data, Wang et al. [54] constructed a comprehensive phase diagram for vanadium under high pressure and temperature for the first time. According to their diagram (see Figure 5 taken from [54]), RH1 phase stabilizes at 100–126 GPa with a maximum transition temperature of 1440 K at 140 GPa, while RH2 phase stabilizes at 126–280 GPa with a maximum transition temperature of 1915 K at 211 GPa. In addition, the stability of RH1 and RH2 phases decreases with increasing temperature, and both transform back to BCC before melting. This picture completely changes our understanding about vanadium which insisted that RH phases could stand up to the melting temperature [79,80]. Meanwhile, Wang et al. identified a triple point at about 1440 K and 140 GPa [54], where there may be spectacular physical properties due to the structural frustration.

**Figure 5.** High-pressure and high-temperature phase diagram of vanadium. (By the courtesy of Ref. [54]).

Following Wang et al.'s prediction, Errandonea et al. [62] performed powder XRD experiments on vanadium up to 120 GPa and 4000 K. Under compression, the BCC vanadium was observed up to 53 GPa at room temperature. At higher pressure of 64 GPa, the measured XRD spectra at room temperature belonged to the RH structure of the *R*-3m space group. This observation supports the previous report by Ding et al. [20]. According

to Errandonea et al. [62], the RH phase could be observed at temperatures up to 1560 K at 64 GPa and up to 1700 K at 120 GPa. Under the higher temperature of 1840 K at 64 GPa, the existence of BCC phase can be seen through the measured XRD pattern. This result is in accordance with the prediction of Wang et al. [54] but is less than one quarter of the initial estimate of Landa et al. [78]. Moreover, Errandonea et al. [62] interpreted their observations as the RH lattice distortion in vanadium that is triggered by phonon anomalies at high pressure and can be eliminated by phonon-phonon scattering effects at high temperatures.

According to the experimental results, Errandonea et al. further presented a phase diagram for vanadium under HPHT, as shown in Figure 6 (taken from [62]). The phase boundary of BCC→RH (dashed blue line) is tentatively drawn, which is qualitatively similar to the phase boundary given by Wang et al. [54] and consistent with other available results [19–21,47,78,81]. In addition, Wang et al. [54] also predicted the reentrance of the BCC phase at room temperature around 280 GPa, but this exceeded the pressure limit in Errandonea et al.'s experiments. This ultra-high pressure prediction still requires experimental verification.

**Figure 6.** Pressure–temperature phase diagram of vanadium. (By the courtesy of Ref. [62]).

Most recently, Zhang et al. [69] also determined the phase stability of vanadium at 0–4400 K and 20–100 GPa by using synchrotron XRD independently. The results showed that BCC vanadium stabilized below 44 GPa at room temperature. With increasing the pressure above 52 GPa, a BCC→RH phase transition occurs, which is consistent with the observations of Errandonea et al. using NaCl as the pressure medium at 53 GPa [62]. Moreover, Zhang et al.'s experimental results showed that the RH vanadium stabilized between 50 and 100 GPa at room temperature [69]. At the pressure of 52 GPa, the RH phase transformed back to BCC when increasing the temperature to 1881 K. This supports the prediction by Wang et al. [54] that electronic temperatures will reduce the stability of RH phase in vanadium, as well as the experiment of Errandonea et al. [62].

In contrast to vanadium, only few studies [65,82] on the structure stability of niobium and tantalum have been carried out in the literature. In 2018, Haskins et al. [82] examined possible HPHT polymorphism in tantalum with complementary DFT-based model generalized pseudopotential theory (MGPT) multi-ion interatomic potentials. Their results showed that four orthorhombic structures of Pnma, Fddd, Pmma, and *α*-U are similarly energetically favorable. Moreover, the MGPT-MD simulations of them further revealed possible spontaneous heatinginduced Pnma→BCC and Fddd→BCC transitions at modest temperatures. Nevertheless, neither unequivocal experimental (DAC or shock wave) nor direct DFT calculation evidence exists for these proposed phase transitions in tantalum by far.

As for niobium, Errandonea et al. [65] recently reported a result of static laser-heated DAC experiments up to 120 GPa, as well as ab initio quantum molecular dynamics simulations. They found that niobium undergoes a BCC→Pnma phase transition at high temperatures, which can be seen from their experimental XRD data. Errandonea et al.'s finding could provide evidence for the topological similarity of the phase diagrams of niobium and tantalum. All of them may undergo orthogonal phase transitions, and the HPHT phase is Pnma. However, this phase transition in niobium is the only report available so far; and no other relevant studies have been reported. Errandonea et al.'s experiments and theoretical calculations seem to be self-consistent, suggesting that such a type of phase transition may exist. However, more accurate experimental and theoretical studies are still needed for final confirmation.

#### *2.3. Elastic Anomalies in V, Nb, Ta*

In addition to the investigations on structural stability, the anomaly in the elastic constants of the group VB transition elements also attracts a lot of attention. Early theoretical and experimental studies [17,21] have shown that the transverse acoustic phonon mode of vanadium and niobium all exhibit softening. Since the shear elastic constant *C*<sup>44</sup> is directly related to the transverse acoustic mode in the limit of short *q*-vector lengths, anomalous softening in this phonon mode implies that vanadium and niobium should have extraordinary elastic moduli. Landa et al. [18,19] confirmed this anomaly in the elastic constants of vanadium and niobium through first-principles calculations. The results show that vanadium is mechanical instability in *C*<sup>44</sup> at pressures between 120 and 245 GPa. The results also show a softening in niobium at around 50 GPa. Their further study suggested that the pressure-induced shear instability (softening) in vanadium (niobium) is mainly due to the electronic structure with FSN.

Later, Koˇci et al. [22] confirmed the mechanical instability of *C*<sup>44</sup> in vanadium through first-principles calculations. Meanwhile, they found that the elastic constants of group VB elements (V, Nb, Ta) exhibit anomalous behaviors, while those of group VIB elements (Mo, W) increase monotonically with pressure. According to Koˇci et al. [22], the calculated *C*<sup>44</sup> of both vanadium and niobium is significantly underestimated by comparing to experimental data, while *C*<sup>11</sup> and *C*<sup>12</sup> are consistent with the experimental results. They further analyzed these metals by Fermi surface calculations and found that the nesting vectors of vanadium, niobium, and tantalum contracted with increasing pressure. This phenomenon, however, was not observed in molybdenum and tungsten. To explore the reason why *C*<sup>44</sup> was underestimated in theoretical calculation, Liu et al. [48] further calculated the Fermi surface of these metals. The results suggested that the underestimation of *C*<sup>44</sup> is mainly caused by the FSN.

To verify the theoretically predicted shear modulus anomalies, Jing et al. conducted XRD experiments on niobium powders under pressures up to 61 GPa at room temperature using DAC technique [60]. They observed an obvious softening in the yield strength of niobium between 42 and 47 GPa, which unexpectedly follows the trend of abnormal softening in the shear modulus predicted by recent theoretical studies [56]. Therefore, Jing et al. predicted that there should be a close relationship between the abnormal strength softening of niobium and the abnormal shear modulus softening [60]; however, this needs to be confirmed by further experimental evidence.

Furthermore, Li et al. [75] experimentally investigated the sound velocities of niobium up to 69 GPa and 1100 K under shock compression. It was found that both the compressional and shear sound velocities soften significantly between 50 and 60 GPa. Li et al. [75] suggested this anomalous behavior might be due to a pressure-induced ETT at 50–60 GPa. However, their data deviate significantly from all theoretical calculations [18,19,22,48]. Especially, by extrapolating their sound velocity to an impact pressure above 100 GPa, the longitudinal sound velocity of niobium would be much higher than that of vanadium; that is unphysical. In this regard, it is still an open question, and more experimental data with much higher precision are needed to pin down the predicted elastic anomalous softening in niobium.

It should also be mentioned that a similar softening in *C*<sup>44</sup> was predicted by calculations for tantalum [83,84]. According to Gülseren et al. [84], the *C*<sup>44</sup> of tantalum shows softening at pressures of 100–200 GPa. They also found that the *C*<sup>44</sup> softens with temperature at low pressures, but then it becomes rather flat at higher pressures. Soon after, Landa et al. [41] further predicted a similar softening of *C*<sup>44</sup> in tantalum between 50 and 80 GPa through first-principles calculations. Later, Antonangeli et al. [43] studied the elasticity of tantalum under pressures over 100 GPa. Their experiments showed that the shear velocity softened between 90 and 100 GPa, and with a pressure dependence above 120 GPa. They suggested that this abnormal behavior may be due to the intraband FSN that causes an ETT and a concomitant transverse acoustic phonon mode softening, which is consistent with the other theoretical predictions [22,48].

Jing et al. thought that there was a potential physical relationship between yield strength and shear modulus, so they further measured the yield strength of tantalum up to 101 GPa at room temperature by XRD-DAC experiment [53]. They detected a yield strength softening at 52–84 GPa, which is in accordance with one of the previous calculations [41] that suggested a significant softening in the shear modulus of tantalum between 50 and 80 GPa, but not others [43,84]. In addition, their measurements showed that the softening trend of the yield stress is roughly the same as that of the shear modulus given by the first-principles calculations [41]. To verify this result, Zhang et al. [85] re-studied the elastic properties of tantalum at high pressures through first-principles calculations. Their calculations showed elastic softening for both the *C*<sup>11</sup> and *C*<sup>44</sup> at pressures above 100 GPa, rather than at 50 GPa. The softening in *C*<sup>44</sup> overall agrees with previous powder tantalum IXS data by Antonangeli et al. [43], but is different from Jing et al.'s results of 52–84 GPa [53].

In order to have a comprehensive understanding about the elastic anomalies of compression-induced softening, Wang et al. [54,56] revisited the elastic properties of the group VB transition metals, by noticing that DFT already correctly predicted many metals that also having Fermi surface nesting or Van Hove singularities. They first calculated the *C*<sup>44</sup> and *C*' of BCC vanadium under different pressures [54], as shown in Figure 7 (taken from [54]). The obtained *C*<sup>44</sup> is negative between 125 and 260 GPa, which is in agreement with that of Landa et al.'s full-potential linear muffin-tin orbitals (FP-LMTO) [18]. However, the calculated *C*<sup>44</sup> from first-principles by Qiu et al. [39] shows that the first mechanical instability pressure for vanadium occurs at about 60 GPa, which is markedly different from the results of Landa et al. [18,19]. Qiu et al. suggested that this difference is mainly due to the fact that Landa et al.'s work ignored the pressure correction. However, Wang et al. [54] explicitly included the same pressure correction as Qiu et al. [39] when calculating the elastic modulus *C*44. The obtained results perfectly match with that of Landa et al. [19], demonstrating that the pressure correction is not the reason for the difference between them. By using DFT, Wang et al. [56] further studied the elastic properties of niobium under pressure through first-principles calculations. The results demonstrated that the *C*<sup>44</sup> and *C*' of niobium soften significantly in the range of 20–150 GPa. In addition, a new softening range for *C*<sup>44</sup> at 275–400 GPa was also discovered. They suggested that the first anomaly was directly related to the underlying RH distortion, whereas the latter originated in an ETT.

In addition, the shear modulus *C*<sup>44</sup> calculated by Wang et al. [66] through GGA (in PBE) is underestimated by about −40% (−30%) for vanadium (niobium) compared with the experimental results [86–89], which is consistent with the evaluation of Liu et al. [48] and Koˇci et al. [22]. Since the semi-local functional (such as PBE) may not be able to handle strong electron correlations in narrow-band systems, Wang et al. [66] further thoroughly evaluated the accuracy of different XC functionals in describing the *C*<sup>44</sup> of vanadium and niobium. The results unexpectedly showed that *C*<sup>44</sup> calculated by the common hybrid functionals (PBE0, HSE06, and B3LYP) are negative value at 0 GPa, which means that these functionals incorrectly predict the mechanical stability of these metals. Through systematic analysis, Wang et al. suggested that this unexpected failure is mainly caused by the localization error of these functionals.

**Figure 7.** Calculated elastic moduli of BCC vanadium as a function of pressure. (By the courtesy of Ref. [54]).

Subsequently, Wang et al. [66] tentatively proposed a DFT + *J* method to correct the localization error, which corrected the *C*<sup>44</sup> from 25.5 to 37.34 GPa for vanadium, and from 19.77 to 22.07 GPa for niobium at 0 GPa, respectively. To explore the influence of this correction on the high-pressure properties, Wang et al. further used the PBE + *J* method to study the elastic properties of vanadium under pressure, as shown in Figure 8 (taken from [66]). Obviously, the *C*<sup>11</sup> and *C*<sup>12</sup> calculated by different methods are consistent with each other, but the *C*<sup>44</sup> calculated by different methods as a function of pressure is quite different. Among them, the PBE + *J* method has the largest correction, especially in the softening pressure range, which means that the correction of localization error could slightly weaken the elastic anomaly of compression-induced softening in group VB transition metals.

**Figure 8.** *Cont*.

**Figure 8.** Calculated elastic moduli of BCC vanadium as a function of pressure by using different XC functionals and the PBE + *J* method (with *J* = 2 eV): (**a**) *C*<sup>11</sup> and *C*12, and (**b**) *C*44. (By the courtesy of Ref. [66]).

Although vanadium, niobium, and tantalum have nearly identical valence electronic configuration, their compression-induced softening ranges of *C*<sup>44</sup> are not the same. To explore the potential connection between them, Wang et al. [56] carefully calculated and compared the band structures of these metals under pressure of 0–400 GPa (see Figure 9 taken from [56]). They discovered two electronic topological transitions in the Γ→H direction for vanadium, niobium, and tantalum. The first ETT occurs around 300, 110, and 280 GPa in vanadium, niobium, and tantalum, respectively. The second ETT occurs at 300 GPa in niobium and at about 600 GPa in both vanadium and tantalum. According to Wang et al. [56], the second pressure-induced ETT should be the cause of the abnormal softening of *C*<sup>44</sup> in niobium at 275–400 GPa, thus they predicted that vanadium and tantalum should have the same elastic modulus *C*<sup>44</sup> softening at about 600 GPa. This is a quite interesting prediction that needs to be further confirmed by more precise experimental and theoretical studies.

In addition to the elastic anomaly of compression-induced softening, the heatinginduced *hardening* in group VB metals was further predicted by Wang et al. [54]. They evaluated the effect of electronic temperature on the *C*<sup>44</sup> of BCC vanadium and found that *C*<sup>44</sup> increases with the temperature (see Figure 10 taken from [54]). This means that there is a heating-induced *hardening* in this metal, which is against our empirical intuition. In addition, as shown in the inset of Figure 10, at selected pressures of 50 GPa and 300 GPa, when the temperature increases from 0 to 3000 K, *C*<sup>44</sup> increases by about 75% and 53%, respectively. At higher temperatures, however, the thermal motion of the nucleus will inevitably soften the metal. Therefore, as the temperature increases, the strength and shear modulus of vanadium will rise to a maximum and then drop to zero.

**Figure 9.** Band structures of vanadium, niobium, and tantalum in BCC phase at various pressures. (By the courtesy of Ref. [56]).

**Figure 10.** Calculated *C*<sup>44</sup> of BCC vanadium as a function of pressure under different electronic temperatures. (By the courtesy of Ref. [54]).

Further investigation by Wang et al. showed that thermo-electrons also have a significant effect on the elastic anomalies of niobium [56]. Like its light neighbor vanadium, the elastic softening in niobium is gradually diminished with increased electronic temperature, effectively causing a heating-induced *hardening* phenomenon. Wang et al. [56] noticed that this thermo-electron effect only presents in the softening pressure ranges for niobium. At the pressure of 75 GPa, the *C*<sup>44</sup> of niobium increases by about 135% when the temperature increases from 0 to 2000 K. Further research by Wang et al. [56] showed that the inclusion of phonon contribution could slightly soften the metal; however, it cannot change the conclusion qualitatively.

Subsequently, Keuter et al. [63] studied the anomalous thermoelastic properties of vanadium, niobium, and tantalum based on a DFT model, which could calculate the *C*<sup>44</sup> under different temperatures. The results showed that the calculated elastic constants of cuprum and molybdenum decrease monotonically with the increase of temperature, which accords with the general thermoelastic behavior. However, for vanadium and niobium, the *C*<sup>44</sup> falls to a minimum at about 500 K and then increases at higher temperatures, which is consistent with the heating-induced *hardening* observed by Wang et al. [56] in these elements.

As a rule of thumb, solids usually harden under compression and soften at high temperatures. Therefore, the theoretical predicted compression-induced softening and heating-induced *hardening* in group VB metals is quite remarkable. Nevertheless, direct experimental evidence has long been lacking. Until recently, Wang et al. [72] measured the HPHT sound velocities at Hugoniot states generated by shock waves and reported the first evidence for the CISHIH counterintuitive phenomenon in group VB metals. They observed that the shock vanadium not only had a significant reduction in sound velocity due to compression, but also had a strong increase in sound velocity due to heating. The former reflects the softening of the shear modulus by compression, while the latter corresponds to the reverse hardening by heat. Their experimental study further highlights the CISHIH dual anomaly behavior in group VB metals and provided inspiration for further theoretical and experimental research on this outstanding problem. The conceptual advancement also might be inspirational in understanding the general exotic behavior of matter, such as electrides [90,91], under extreme conditions.

#### **3. Conclusions**

In summary, it is a basic topic in condensed matter physics to reveal and elucidate the trend and mechanism of structural transformation and elastic anomaly of elemental metals. This review covers a large number of theoretical and experimental research on the phase stability and elastic anomalies in group VB transition metals over the last two decades. Two quite different scenarios are emerging from this progress. The first one is focused on phase transition and is represented by the phase diagram shown in Figures 5 and 6. As the basis of this scenario, a tentative theoretical "consensus" has been established on the phase transition sequence of BCC→RH1→RH2→BCC in vanadium under pressure, and RH→BCC under high temperature. Meanwhile, we also showed that some experimental and theoretical facts are incompatible with this scenario. They suggested that the resultant RH is far from being perfect and should be interpreted as lattice distortion rather than a phase transition. The same issue exists for niobium and tantalum. The above discussions cautioned that the phase transition in group VB metals is elusive, and the main problem of this scenario is that it cannot provide a unified and general description for all group VB elements.

The second scenario emphasizes on electronic structure and the resultant elastic anomalies. It viewed the phase transition in vanadium as the natural outcome of the anomaly, rather than the cause of it. In this direction, recent investigations revealed that different from other groups of transition metals, the group VB transition elements exhibit striking anomalous elastic softening under pressure. It is known that vanadium, niobium, and tantalum have nearly identical valence electronic configuration, but the variation of the *C*<sup>44</sup> under pressure for them is not the same. According to existing literature reports, the pressure-induced softening of *C*<sup>44</sup> in vanadium, niobium, and tantalum originates from the

electronic structure, which is closely related to the FSN and ETT. In addition, the elastic softening in vanadium and niobium are gradually diminished with increased temperature, effectively giving rise to a heating-induced *hardening* phenomenon. The most striking feature of this scenario is that there actually could be *no phase transition* in vanadium at all. We have increasing confidence on this picture, especially after the reported facts that accurate DFT + *J* method completely eliminates RH phases from the thermodynamic equilibrium phase diagram [66], the absence/imperfect RH phase in Stevenson et al.'s experiment [74], the stability of BCC and meta-stability of RH as reported in Akahama et al.'s experiment [73], and the unexpected appearance of RH in shock experiment as reported by Wang et al. [72]. All of them cannot be consistently interpreted in the first scenario. Nonetheless, they can be reconciled in the second scenario, in which the "imperfect" RH as reported in various DAC experiments is viewed as lattice distortions caused by deviatoric stress rather than a new phase. These distortions lead to prominent change in modulus and sound velocity, and are modulated by compression and temperature, leading to CISHIH dual anomaly in both vanadium and niobium. We conclude that it seems the second scenario is more promising, but a lot of investigation is still required to achieve a general consensus.

**Author Contributions:** Y.W. and H.G. conceived the research. H.W. (Hao Wu), Y.L., H.W. (Hao Wang), X.C. and H.G. analyzed the results. Y.W. wrote the manuscript and all the authors commented on it. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by National Key R&D Program of China under Grant No. 2021YFB3802300; the NSAF under Grant Nos. U1730248 and U1830101; the National Natural Science Foundation of China under Grant Nos. 11672274, 11602251, 11872056, and 11904282; the CAEP Research Project under Grant No. CX2019002; and the Science Challenge Project Tz2016001.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


**Vinod Panchal 1, Catalin Popescu <sup>2</sup> and Daniel Errandonea 3,\***


**Abstract:** The structural behavior of nanocrystalline α-CuMoO4 was studied at ambient temperature up to 2 GPa using in situ synchrotron X-ray powder diffraction. We found that nanocrystalline α-CuMoO4 undergoes a structural phase transition into γ-CuMoO4 at 0.5 GPa. The structural sequence is analogous to the behavior of its bulk counterpart, but the transition pressure is doubled. A coexistence of both phases was observed till 1.2 GPa. The phase transition gives rise to a change in the copper coordination from square-pyramidal to octahedral coordination. The transition involves a volume reduction of 13% indicating a first-order nature of the phase transition. This transformation was observed to be irreversible in nature. The pressure dependence of the unit-cell parameters was obtained and is discussed, and the compressibility analyzed.

**Keywords:** high pressure; phase transition; synchrotron radiation; X-ray diffraction

#### **1. Introduction**

Nanomaterials play an important role in catalysis, optics, electricity, and other research fields due to their exclusive and typical characteristics. In the past decades, compared to bulk materials, their nanocrystalline counterparts have gained massive attention due to their novel properties such as thermal, electronic, and magnetic, owing to their specific shape, size, and surface to volume ratio. In addition to that, nanomaterials also find their application in the fields of biology, medicine, and chemical industry [1–4]. Metal molybdates (AMO4) are important inorganic materials which have gained enormous scientific importance due to their wide range of applicability, such as industrial catalysts, photoluminescence, microwave applications, optical fibers, humidity sensors, scintillator materials and due to their magnetic and electrochemical properties [5–8].

Among molybdates, CuMoO4 is the compound with a very complex polymorphism [9–12]. Till date, six different polymorphs of CuMoO4 have been identified in the literature: namely ambient condition α-CuMoO4 [9], high temperature β-CuMoO4 [10], low temperature γ-CuMoO4 [11], high pressure (HP) CuMoO4-II [12], distorted wolframite CuMoO4-III [13] and monoclinic ε-CuMoO4 [14]. Earlier, single-crystal high-pressure and low-temperature X-ray diffraction studies carried by Wiesmann et al. [15] found that phase α-CuMoO4 undergoes a structural phase transition to γ-CuMoO4. This occurs during cooling at 190 K at ambient pressure as well as under increasing pressure at 0.2 GPa and room temperature. Congruently, high pressure and low temperature optical-absorption measurements carried out by Rodriguez et al. [16] found that α-CuMoO4 undergoes a first-order transition at 0.25 GPa to γ-CuMoO4. They found the same transition by cooling α-CuMoO4 to 200 K at ambient pressure. The α to γ phase transition involves a change of part of the copper coordination polyhedra, from square-pyramidal (*C*4*<sup>v</sup>*) CuO5 in α-CuMoO4 to octahedral elongated (*D*4*h*) CuO6 in γ-CuMoO4. The transition also involves a volume drop of 13%.

**Citation:** Panchal, V.; Popescu, C.; Errandonea, D. An Investigation of the Pressure-Induced Structural Phase Transition of Nanocrystalline α-CuMoO4. *Crystals* **2022**, *12*, 365. https://doi.org/10.3390/ cryst12030365

Academic Editor: Heinz-Günter Brokmeier

Received: 1 March 2022 Accepted: 8 March 2022 Published: 9 March 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

Many of the applications of CuMoO4 involve its use in the form of nanoparticles [17–19]. However, the information available on the structural and mechanical properties of nanocrystalline CuMoO4 is scarce. It is known that the high-pressure (HP) behavior of materials could be different for nanoparticles than for bulk materials [20–23]. In particular, transition pressures and compressibilities could be affected when the particle size is reduced [20–23]. To explore the structural and mechanical properties of nanocrystalline CuMoO4 under compression, the performance of an HP X-ray diffraction (XRD) investigation is needed. However, such studies have not been carried out yet in nanocrystalline α-CuMoO4. In the present work we have investigated nanocrystalline α-CuMoO4 by high pressure powder X-ray diffraction up to a pressure of 2 GPa under hydrostatic conditions.

#### **2. Experiment**

CuMoO4 nanoparticles were prepared following the method reported by Hassani et al. [20]. The chemical composition of the obtained CuMoO4 nanoparticles was determined by energy-dispersive X-ray spectrometry (EDXS), which was recorded with a JEOL JEM-6700F device. A EDXS spectrum is shown in Figure 1. The analysis showed that the composition of the nanoparticles was 67 (3) atomic % oxygen, 16 (1) atomic % copper, and 17 (1) atomic % molybdenum, which agrees with the stoichiometry of CuMoO4.

**Figure 1.** Energy dispersive X-ray analysis (EDXS) spectrum of nanocrystalline CuMoO4.

Ambient conditions powder XRD (λ = 0.4246 Å) performed in the same set up as highpressure measurements confirmed that CuMoO4 nanoparticles crystallize in the triclinic α polymorph (space group *P*1). The average particle size was estimated from powder XRD using the Scherrer equation [24], to be 43 (7) nm. This value was confirmed using MAUD [25], which gave an average particle size of 40 (9) nm. Angle-dispersive powder HP-XRD measurements were performed at the BL04-MSPD beamline of ALBA synchrotron [26]. An incident monochromatic X-ray beam with a wavelength of 0.4246 Å was focused down to a spot size of 20 μm × 20 μm (FWHM). Two-dimensional (2D) XRD data were recorded using a Rayonix SX165 CCD detector. The 2D images were integrated into one-dimensional intensity versus 2θ patterns using Dioptas [27]. The sample-to-detector distance along with detector parameters were determined using a LaB6 calibrant. A pellet of CuMoO4 nanocrystalline powder was loaded into a Boehler-Almax diamond-anvil cell (DAC) with diamond culets of 500 μm using a T301 stainless-steel gasket, pre-indented to a thickness of 50 μm and with a sample chamber of 200 μm diameter in the center. As pressuretransmitting medium, we used a 4:1 methanol–ethanol to allow a direct comparison with previous HP studies performed in microscopic samples [15]. The pressure medium can be considered nearly hydrostatic within the pressure limit of the experiments [28]. Special attention was paid to occupy only a minor fraction on the pressure chamber with sample to avoid sample bridging between diamonds [29]. The pressure was determined using ruby fluorescence [30]. Small ruby spheres with a grain size of ~1 μm and concentration of 3000 ppm Cr3+ [31] were used for pressure calibration [31] based on the wavelength shift of the R1 fluorescence band of the trivalent Cr3+ [32,33]. The estimated relative error for all measured pressures was better than 1% [34]. The unit-cell parameters were initially determined using the Le Bail analysis incorporated in the GSAS software [35]. Using the same software, Rietveld analysis [36] was also carried out for the initial α-CuMoO4 phase and high pressure γ-CuMoO4 phase.

#### **3. Results and Discussion**

At ambient conditions, CuMoO4 crystallizes in a triclinic α-CuMoO4 structure (space group *P*1, Z = 6). The α-CuMoO4 structure can be described by MoO4 tetrahedra, CuO5 square pyramidal polyhedra, and CuO6 distorted octahedra, interconnected via common corners and edges forming layers as shown in Figure 2.

**Figure 2.** Crystal structure of α-CuMoO4. Blue solid spheres correspond to Cu atoms, gray solid spheres correspond to Mo atoms, and red solid spheres correspond to O atoms. The different coordination polyhedra are illustrated in the figure.

Figure 3 shows a selection of HP powder X-ray diffraction profiles of nanocrystalline α-CuMoO4 at representative pressures. There were no noticeable changes (other than typical peak shifts due to the changes induced by pressure in the unit-cell parameters) in the diffraction patterns up to 0.3 GPa and all the diffraction peaks could be indexed to α-CuMoO4 phase. At 0.5 GPa, the appearance of many weak extra diffraction peaks was observed at 2θ of 4.2, 4.7, 6.8 and 9.5◦ along with the diffraction peaks of the α-CuMoO4 phase. The red arrows in Figure 3 identify some of the additional peaks observed at 0.5 GPa. The red dashed lines show how these peaks evolve with pressure until the highest pressure corresponding to peaks of the γ-phase. In the pattern at 2.0 GPa (γ-phase) they are identified with asterisks.

**Figure 3.** Evolution of the X-ray diffraction patterns of α-CuMoO4 as a function of pressure. Pressures are provided in the figure. Red arrows indicate the appearance of extra diffraction peaks evidencing the onset of the transition to the γ-CuMoO4 phase. Red dashed lines show how these peaks evolve under compression. The asterisks identify the peaks in the γ-phase. Ticks are the position of Bragg peaks for different structures.

These modifications in the diffraction pattern are indicative of the onset of a structural phase transition in α-CuMoO4 at 0.5 GPa. The transition pressure observed in this study is almost double the transition pressure determined from earlier single-crystal high-pressure XRD measurements and optical studies on bulk samples, which identified the phase transition at 0.2 and 0.25 GPa, respectively [15,16]. The increase of the transition pressure could be explained in terms of an increase in surface energy in the newly formed highpressure phase crystallites [37]. The coexistence of both the phases has been observed till 1.2 GPa. Beyond this pressure, all the diffraction peaks could be assigned to the triclinic γ-CuMoO4 (space group *P*1). On further increase of pressure, the γ-CuMoO4 phase has

been found to remain stable up to 2 GPa, which is the highest pressure reached in our XRD measurements. On release of pressure, the γ-CuMoO4 phase has been observed at ambient conditions, thus indicating the irreversible nature of the phase transition. In previous studies [15,16] the reversibility of the HP α-γ transition was not evaluated. However, it was shown that the γ-phase has a smaller band gap and different optical properties than the α-phase [16]. In particular, the band gap of γ-CuMoO4 is close to 2 eV, making it ideal for hydrogen production via photocatalytic water splitting [38]. Given the low-pressure requested to obtain γ-CuMoO4 as an ambient-conditions metastable polymorph, HP could be an efficient way to synthesize γ-CuMoO4 for water splitting applications. The crystal structure of the HP γ-CuMoO4 phase is represented in Figure 4. In this structure, all Cu atoms display an octahedral elongated coordination, and all Mo atoms have a distorted octahedral coordination. Rietveld refinements supporting the structural assignments are provided in Figure 5.

**Figure 4.** Crystal structure of γ-CuMoO4. Blue solid spheres correspond to Cu atoms, gray solid spheres correspond to Mo atoms, and red solid spheres correspond to O atoms. The different coordination polyhedra are illustrated in the figure.

**Figure 5.** *Cont*.

**Figure 5.** (**a**) XRD pattern and Rietveld refinement of α-CuMoO4 at ambient pressure. (Rwp = 13.4%, Rp = 9.2%). (**b**) XRD pattern and Rietveld refinement of γ-CuMoO4 at 1.6 GPa (Rwp = 11.4%, Rp = 6.8%). (**c**) XRD pattern and Rietveld refinement of γ-CuMoO4 at ambient pressure after pressure release (Rwp = 12.4%, Rp = 8.0%), denoted by (r).

At ambient pressure, the lattice parameters for α-CuMoO4 were refined as *a* = 9.870 (1) Å, b = 6.764 (1) Å, *c* = 8.337 (1) Å, α = 101.13(5)◦, β = 96.90 (5)◦, γ = 107.03 (5)◦. The lattice parameters and fractional coordinates of the α-CuMoO4 are given in Table 1. They are in good agreement with those reported in previous single-crystal high-pressure XRD measurements by Wiesmann et al., *a* = 9.901 (3) Å, b = 6.786 (2) Å, *c* = 8.369 (3) Å, α = 101.13◦, β = 96.88◦, γ = 107.01◦ [15]. On the other hand, the lattice parameters of γ-CuMoO4 phase at 1.6 GPa were found to be *a* = 9.608 (9)Å, b = 6.219 (7) Å, *c* = 7.875 (5) Å, α = 94.91 (1)◦, β = 103.10 (1)◦, γ = 102.48 (9)◦. The lattice parameters and fractional coordinates are summarized in Table 2. They are in good agreement with those reported in previous single crystal high-pressure XRD measurements by Wiesmann et al., *a* = 9.708 (3) Å, b = 6.302 (7) Å, *c* = 7.977 (2) Å, α = 94.76◦, β = 103.35◦, γ = 103.26◦ [15]. After release of pressure, the lattice parameters for γ-CuMoO4 were found to be *a* = 9.6605 Å, b = 6.2751 Å, *c* = 7.9334 Å, α = 94.76◦, β = 103.29◦, γ = 103.14◦. The unit-cell volume of the γ-phase was 11.5% smaller than the volume of the α-phase.

**Table 1.** Lattice parameters and fractional coordinates of α-CuMoO4 at ambient conditions. Space group *P*1, Z = 6, *a* = 9.870 (1) Å, b = 6.764 (1) Å, *c* = 8.337 (1) Å, α = 101.13 (5)◦, β = 96.90 (5)◦, γ = 107.03 (5)◦.


**Table 2.** Lattice parameters and fractional coordinates of γ-CuMoO4 at 1.6 GPa. Space group *P*1, Z = 6, *a* = 9.608 (9) Å, b = 6.219 (7) Å, *c* = 7.875 (5) Å, α = 94.91 (1)◦, β = 103.10 (1)◦, γ = 102.48 (9)◦.


From Rietveld refinements, we obtained the pressure evolution of the lattice parameters and equation of state of α-CuMoO4 and γ-CuMoO4. For the HP phase, they were obtained only for 0.7 GPa and higher pressures because at 0.3 and 0.5 GPa the fraction of γ-CuMoO4 was too small to allow an accurate determination of unit-cell parameters (The peaks of γ-CuMoO4 are too weak at these two pressures). For the low-pressure phase they were obtained up to 1.2 GPa. The pressure dependence of the lattice parameters and angles are plotted in Figure 6. From the figure, it can be seen the behavior under compression is anisotropic.

**Figure 6.** Pressure dependence of the lattice parameters and angles. Filled circles correspond to the α-CuMoO4 phase and solid triangles corresponds to γ-CuMoO4. Errors are smaller than symbol's size. The solid lines in the lattice-parameter plots are linear fits of the data. Vertical dashed lines delimit the region of phase coexistence.

For any triclinic structure, the description of the deformation of the unit cell under compression is provided by the compressibility tensor [39]; in particular, by its eigenvectors (*e*λi) and eigenvalues (λi). They give the principal axes of the compressibility and the compressibilities along them. We have obtained *e*λ<sup>i</sup> and λ<sup>i</sup> for α-CuMoO4 using PASCal [40]. The results are given in Table 3.

**Table 3.** Eigenvalues, λi, and Eigenvectors, *e*λi, for the isothermal compressibility tensor of α-CuMoO4.


The pressure dependence of volume of both the phases is plotted in Figure 7. From the plot, it can be seen that the phase transition from α-CuMoO4 to γ-CuMoO4 exhibits a volume collapse of ~13% at 0.7 GPa which is also in agreement with earlier XRD measurements [15]. The results of Figure 7 were fitted using second- and third-order Birch– Murnaghan equations of state. The fits were carried out using EOSFIT [41]. A fit of the volume vs. pressure data of the α-CuMoO4 phase to a second-order Birch–Murnaghan equation of state (*B* <sup>0</sup> = 4) gives a bulk modulus of 67 (2) GPa, if we use third order Birch-Murnaghan equation of state it gives bulk modulus of 62 (4) GPa with *B* <sup>0</sup> = 5.1 (8) which is also in agreement with earlier XRD measurements [15]. A fit of the volume vs. pressure data of the γ-CuMoO4 phase to a second-order Birch-Murnaghan equation of state (*B* <sup>0</sup> = 4) gives bulk modulus of 62 (6) GPa, if we use third-order order Birch-Murnaghan equation of state it gives bulk modulus of 68 (9) GPa with *B* <sup>0</sup> = 3.1 (9). In the previous study, *B*<sup>0</sup> = 44.0 (3.4) GPa and *B* <sup>0</sup> = 15.3 (3.0) were reported for γ-CuMoO4. Such a large pressure derivative for the bulk modulus is unusual in oxides [42]. On the other hand, it is not reasonable to have a smaller bulk modulus for γ-CuMoO4 than for α-CuMoO4, because the first phase is denser than the second one. The mismatch of *B*<sup>0</sup> and *B* <sup>0</sup> in the previous work could be related to the fact that volume at zero pressure (V0) was unknown, being extrapolated from HP results. In our case, V0 has been experimentally determined after decompression. V0, *B*0, and *B* <sup>0</sup> are correlated parameters [43] and therefore an overestimation of V0 (466 Å<sup>3</sup> extrapolated in Ref. [15] and 454 Å3 here measured) could easily lead to inaccurate values of *B*<sup>0</sup> and *B* <sup>0</sup>. Interestingly, our results indicate, that despite the volume collapse of the transition, the two phases of CuMoO4 have comparable bulk moduli.

**Figure 7.** Pressure dependence of the volume. Filled circles correspond to the α-CuMoO4 phase and solid triangles corresponds to γ-CuMoO4. Errors are smaller than symbol size. The solid lines are linear fits of the data.

Interestingly, the two phases of CuMoO4 have considerably smaller bulk moduli than CuWO4 (*B*<sup>0</sup> = 139 GPa) [44]. This is related to the fact that CuMoO4 has a much more open structure than CuWO4. This is a direct consequence of the way in which polyhedra are linked in CuMoO4, which gives to the structure of the two polymorphs a pseudo-layered characteristic. This feature makes the reduction of the "empty" space between layers easier and consequently leads to a smaller bulk modulus in CuMoO4. The second fact that facilitates the reduction of volume in CuMoO4 is that it can be achieved by tilting of the polyhedra. It should be noted that with respect to its polyhedral constitution, CuMoO4 is more similar to Cu3V2O8 and Cu2V2O7 [45,46] than to typical orthomolybdates [47,48]. This is consistent in that the two vanadates have bulk moduli of 52 and 64 GPa, respectively; values that are comparable to the bulk modulus of α- and γ-CuMoO4.

#### **4. Conclusions**

X-ray diffraction studies on nanocrystalline α-CuMoO4 up to 2 GPa suggest that the low pressure α-phase undergoes an irreversible α–γ phase transition. The onset of phase transition occurs at 0.5 GPa, the coexistence of both phases was observed between 0.5 to 1.2 GPa, and the transition completes at 1.6 GPa, indicating a sluggish nature of the phase transition. The α to γ phase transition involves a change of part of the copper coordination polyhedra, from square-pyramidal (*C*4*<sup>v</sup>*) CuO5 in α-CuMoO4 to octahedral elongated (*D*4*h*) CuO6 in γ-CuMoO4 along with a reduction in volume by 13%, indicative of first-order phase transition. The transition pressure observed in this case is almost double compared to its bulk counterpart which can be attributed to an increase in surface energy in the newly formed high-pressure phase. In addition, we found that in nanocrystalline CuMoO4 the high-pressure γ-polymorph can be recovered as a metastable phase upon decompression. Since the γ-phase has a smaller band-gap energy than the α-phase, our findings suggest that high pressure can be used to synthesize a metastable polymorph with a tailored band-gap energy for water splitting applications. Finally, we have also discussed the compressibility of both polymorphs and determined their equations of state.

**Author Contributions:** D.E. conceived the experiments. D.E. and C.P. conducted the experiments. V.P. analyzed the results. The manuscript was written through contributions of all authors. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by the Spanish Research Agency (AEI) and Spanish Ministry of Science and Investigation (MCIN) under project PID2019-106383GB-C41 (DOI:10.13039/501100011033), co-financed by EU funds. The authors also acknowledge financial support from the MALTA Consolider Team network, under project RED2018-102612-T.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** All relevant data that support the findings of this study are available from the corresponding authors upon request.

**Acknowledgments:** The authors acknowledge the ALBA synchrotron facilities for provision of beamtime on the beamline MSPD-BL04 (Proposal 2016081779). They also thanks S. Agouram from Servicio Central de Soporte a la Investigación Experimental (SCSIE) at Universidad de Valencia for technical assistance in EXDS measurements.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


## *Article* **Computational Investigation of the Stability of Di-***p***-Tolyl Disulfide "Hidden" and "Conventional" Polymorphs at High Pressures**

**Valeriya Yu. Smirnova 1,2, Anna A. Iurchenkova 1,3 and Denis A. Rychkov 1,2,\***


**Abstract:** The investigation of molecular crystals at high pressure is a sought-after trend in crystallography, pharmaceutics, solid state chemistry, and materials sciences. The di-*p*-tolyl disulfide (CH3−C6H4−S−)2 system is a bright example of high-pressure polymorphism. It contains "conventional" solid–solid transition and a "hidden" form which may be obtained only from solution at elevated pressure. In this work, we apply force field and periodic DFT computational techniques to evaluate the thermodynamic stability of three di-*p*-tolyl disulfide polymorphs as a function of pressure. Theoretical pressures and driving forces for polymorphic transitions are defined, showing that the compressibility of the γ phase is the key point for higher stability at elevated pressures. Transition state energies are also estimated for α → β and α → γ transitions from thermodynamic characteristics of crystal structures, not exceeding 5 kJ/mol. The β → γ transition does not occur experimentally in the 0.0–2.8 GPa pressure range because transition state energy is greater than 18 kJ/mol. Relations between free Gibbs energy (in assumption of enthalpy) of phases α, β, and γ, as a function of pressure, are suggested to supplement and refine experimental data. A brief discussion of the computational techniques used for high-pressure phase transitions is provided.

**Keywords:** high-pressure polymorph; relative stability; di-*p*-tolyl disulfide; high-pressure DFT; molecular crystals; hidden polymorph

#### **1. Introduction**

High-pressure research of molecular crystals is an important direction of modern crystallography [1–11]. A lot of organic systems have polymorphic modifications at ambient pressure, while other organic crystals undergo phase transitions at extreme conditions [12,13]. These new high-pressure structures may be the result of direct solid-solid transformation and may be called "conventional". Nevertheless, the formation of new polymorphs may also be hindered by the high transition state energies and crystalize only from liquid phase. These solid–liquid–solid phase transitions (which can also undergo through mobile fluid phase, e.g., partial dissolution) are called "hidden" polymorphs [14–16]. The comparison of the thermodynamic stability of both "hidden" and "conventional" phases is a complicated question, especially if several phases exist at the same conditions. Highpressure experiments could only show the structural stability of such phases, but not the thermodynamic stability. If there is no direct phase transition between polymorphs, it becomes impossible to investigate their thermodynamic stability. Both computational and experimental approaches should be used in such cases [13,17].

**Citation:** Smirnova, V.Y.; Iurchenkova, A.A.; Rychkov, D.A. Computational Investigation of the Stability of Di-*p*-Tolyl Disulfide "Hidden" and "Conventional" Polymorphs at High Pressures. *Crystals* **2022**, *12*, 1157. https:// doi.org/10.3390/cryst12081157

Academic Editors: Daniel Errandonea and Enrico Bandiello

Received: 27 July 2022 Accepted: 13 August 2022 Published: 17 August 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

Di-*p*-tolyl disulfide (*p*-Tol2S2) is a demonstrative organic system that contains a "conventional" form and a "hidden" high-pressure form (Scheme 1). The "hidden" form recrystallizes from solution at elevated pressures, while the "conventional" form is obtained via solid–solid transition [18].

**Scheme 1.** Molecular structure of di-*p*-tolyl disulfide (*p*-Tol2S2).

Szymon Sobczak and Andrzej Katrusiak previously found that the α phase is the most stable phase at ambient conditions, while, at higher pressures, two phase transitions occur with the formation of β and γ polymorphs [18]. On the one hand, the reversible α → β phase transition occurs at a pressure near 1.6 GPa and the β phase remains stable up to 2.8 GPa (glycerin was used as a pressure transmitting media). On the other hand, α transforms to the γ-phase form near 0.45 GPa via recrystallization (methanol, ethanol, and isopropanol solvents were simultaneously used as hydrostatic fluids). The γ-phase form is also stable up to 2.8 GPa, similarly to the β one. Based on those experiments, the authors suggested a scheme of phase transitions and schematic relations between the free Gibbs energy of the α, β, and γ phases as a function of pressure for *p*-Tol2S2 (Figure 1).

**Figure 1.** (**a**) Principal scheme of the experimentally observed phase transitions in a di-*p*-tolyl disulfide system. The α → β phase transition is given by two arrows due to formation of two different molecular conformations. (**b**) Suggested relations between the free Gibbs energy of phases α, β, and γ as a function of pressure based on experimental study (adapted from [18]).

It is impossible to measure the relative stability of β and γ polymorphs in different pressure regions due to the absence of a direct phase transition between them. Thus, the aim of this study is to calculate the thermodynamic stability of all three polymorphs of the *p*-Tol2S2 system and explain the driving force for experimentally observed structural reorganization at pressures of 0.45 GPa and 1.6 GPa. Based on several previous works [19–24] Force Field (FF) and periodic DFT methods in conjunction with equations of states (EoS), the enthalpies of all polymorphs are calculated as a first rough approximation of the Gibbs energy for each.

#### **2. Computational Details**

Lattice energies were calculated using different approaches. The force field approach, as implemented in the CrystalExplorer 21.5 (CE21) software package, calculates pair-wise energies, summing up the energy of all intermolecular energies in the experimental crystal structure in terms of kJ/mol [25–27]. CE21 uses data calculated via DFT (e.g., electron

density mapped on a Hirshfeld surface at B3LYP/6-31G (d,p)) and should be strictly named as an FF-based method, but, following major principles, it is further referred to as an FF method. Periodic DFT, as implemented in VASP 5.4.4 [28–31], calculates an overall (electronic) energy of the optimized system, which may be presented as a sum of inter- and intra-molecular interactions (which are of course interrelated from molecular geometry). Thus, periodic DFT (apart from obvious differences in principles in comparison to FF methods) considers conformational changes in crystal structures, whereas the CE21 algorithm does not. Enthalpies were calculated by manual addition of the PV term at pressure points, where volume was known experimentally (for FF calculations) or from EoS calculations (for DFT calculations). Zero-point energy was not considered, as well as temperature (T = 0 K). An overall workflow is presented in Scheme S1.

#### *2.1. FF Method Calculations*

CrystalExplorer 21.5 [25] was used with a Gaussian09 [32] backend at D2-B3LYP/6-31G (d,p) level of theory (LOT) with four further terms (electrostatic, polarization, dispersion, and exchange-repulsion) separation. Corresponding default scaling factors for this LOT were used to calculate total energy, which were further interpreted as lattice energy.

#### *2.2. Periodic DFT Calculations*

All periodic DFT calculations were performed using the VASP 5.4.4 package using a PBE functional [33] plane-wave basis set with a kinetic energy cut-off of 500 eV and PAW atomic pseudopotentials [34,35]. The integrals in reciprocal space were calculated on a kpoint Monkhorst-Pack mesh of 4 × 4 × 2 [36]. Grimme D3 dispersion correction with Becke– Johnson damping was used for better simulation of van der Waals (VdW) interactions in crystal structures [37]. Convergence criteria were applied, where the maximum change in system energy was 10−<sup>5</sup> eV and the norms of all forces were smaller than 10−4. The volumes of unit cells in all periodic DFT calculations were fixed at calculated values from corresponding EoS (which are respectively calculated from experimental data), while cell shapes and atom positions were fully relaxed during the optimization procedure (ISIF = 4). Crystal structures for all calculations were prepared manually by editing the experimental lowest pressure structure (cif) file cell parameters. This editing considered the Vp/Vo ratio (preserving same angles as in the initial structures), where Vp is the calculated volume of the crystal structure at pressure P from EoS. All unit cell parameters including angles were fully relaxed further during DFT optimization procedure, keeping volume fixed only. Experimental structure (.cif) files were obtained from the CCDC database [38].

#### *2.3. Equations of States*

Experimentally obtained pressure dependencies of unit cell volumes of the polymorphs (Table S1 in Supplementary Materials) were fitted by the EoSFit7-GUI software [39,40], using the third order Birch–Murnaghan equation of state [40,41]. Standard deviations of the volumes and pressures were considered.

#### **3. Results and Discussion**

The thermodynamic stability of different polymorphs is described as the enthalpy of these forms in the current study. The authors are aware of possible significant changes in the entropy term due to change in lattice vibrations under pressure. Nevertheless, we suggest such simplification reasonable for this system while taking into account the isothermal conditions of the experiments, minor changes in the molecular arrangement of the α and β forms, which results in very small lattice changes, and the preserved symmetry (P)2 point group of the α and γ polymorphs (Figure S1 and Table S1). Thus, the entropy difference should be less significant than that which is expected for major materials and may be neglected as a rough approximation of the Gibbs energies to save computational resources. Enthalpy may be divided into several terms to find a driving force for phase transitions (Equation (1)).

$$\text{pH} = \text{U}\_{\text{cryptal}} + \text{P} \times \text{V} = \text{U}\_{\text{irter}} + \text{U}\_{\text{irtrra}} + \text{P} \times \text{V} \tag{1}$$

Lattice energy is presented as an Ucrystal and presents a sum of inter- and intramolecular interactions. Considering the computational techniques, Equation (1) transforms to Equation (2) in the case of FF calculations as implemented in CrystalExplorer 21.5 software (where only intermolecular interaction energies are parametrized).

$$\text{pH} = \text{U}\_{\text{crystal}} + \text{P} \times \text{V} = \text{U}\_{\text{inter}} + \text{P} \times \text{V} \tag{2}$$

Uintra, being conformational energy, was calculated in the work [18] separately using several gas-phase DFT methods, and is reproduced in Figure S2, showing γ-phase conformations are 6–8 kJ/mol less favorable than the α and γ forms at the same pressures. Thus, Uintra may be added to Uinter manually, which in fact does not change the overall picture for used FF method (Figure S3).

Following this logic, an overall behavior of *p*-Tol2S2 system may be assessed regarding Ucrystal lattice energies (mostly intermolecular interactions) and PV term (compressibility) to find a driving force for phase transitions and evaluate relative stability of polymorphs using different computational techniques.

#### *3.1. Force Field Calculations*

Lattice energies of all experimentally founded crystal structures were calculated using the CrystalExplorer 21.5 software package at corresponding experimental pressure values (Figure 2). The function of lattice energies on the pressure shows relative stability of all three phases in the regions of their structural stability. A significant decrease in the lattice energy of α phase in the interval 0.0–1.5 GPa is probably an artifact of CE21 parametrization, which was already discussed in the work [24]. It was shown that decrease of lattice energies with an initial increase in pressure is similar for all phases/interactions and does not affect results of polymorphs relative energies.

**Figure 2.** Lattice energies (Ucrystal = Uinter) of phases α, β, and γ as a function of pressure calculated based on experimental data. Regions of structural stability according to experimental data [18] are shown with dashed lines.

According to the lower lattice energy of the γ phase (Figure 2), it has higher stability in the pressure range of 0.45–2.5 GPa with relatively reversed stability for the β and γ polymorphs at 2.8 GPa. The addition of conformational energies does not change this tendency (Figure S3). Thus, based only on the lattice energies, one can expect a γ to β phase transition at higher pressures. Nevertheless, compressibility of different phases should be considered, which may affect PV term and consequently H drastically [42].

The enthalpies of all phases were calculated at experimental pressures by manual addition of a PV term (Figure S4). Higher lattice energy of the γ polymorph was fully compensated by PV term (compressibility of γ phase) and the γ phase preserved relative stability at 2.8 GPa. It is impossible to compare the enthalpies of all phases at the same pressures in a whole pressure interval (0–3 GPa) due to the limitation of lacking a region where the phases are all structurally stable. This is one of the key limitations for the FF method, because atom coordinates for other pressures cannot be obtained without expensive DFT calculations or new labor-intensive experiments. Linearization of enthalpy data may be successfully used in such cases [17] and be applied for a *p*-Tol2S2 system (Figure 3).

**Figure 3.** Calculated enthalpy differences between *p*-Tol2S2 polymorph α and γ (green), α and β (purple), and β and γ (orange) as a function of pressure (i.e., ΔHα − β = Hα − Hβ, ΔHα − γ = Hα − Hγ, and ΔHα − β = Hα − Hβ). Phase transition between the involved polymorphs can be expected when their enthalpy difference is equal to zero, marked with circles.

The phase transitions from α → γ and α → β can be expected when the free energies (in the approximation of the calculations used here—the enthalpies) of the corresponding phases become equal. These calculations predict this to happen at 0.44 GPa and 1.45 GPa, respectively (Figure 3). The obtained theoretical values are extremely close to the experimental values of 0.45 and 1.6 GPa for the α → γ and α → β phase transitions, respectively (Table S2). Nevertheless, the applied approximation of linear enthalpy change (due to high PV energies) at higher pressures is a significant simplification and is recommended for use only with short pressure intervals. The periodic DFT calculations, in conjunction with EoS, were applied to estimate the possible margins of the used approach.

#### *3.2. Periodic DFT Calculations with EoS*

The prediction of structure change at finite pressure points is mandatory to overcome the limitations of experimental data at different pressures. A previously reported [21] technique of fully automated structure change prediction at any pressure using the PSTRESS tag in VASP turned out to be inaccurate for small pressure ranges, at least for a *p*-Tol2S2 system. Thus, a concept similar to those reported in the work [43] was applied. Equations of state were calculated based on experimental data for all three polymorphs to predict the cell volume of each phase at the 0.0–3.0 GPa pressure range with a 0.5 GPa step size (Figure S5 and Table S3). At each pressure point, the full electronic energy was calculated using DFT optimization with a fixed cell volume (Figure 4).

**Figure 4.** Relative lattice energies of phases α, β, and γ as a function of pressure calculated based on EoS predicted data. Full electronic structure energy contains both inter and intramolecular interaction energies. The α phase energy at ambient pressure is taken as zero.

Lattice energies show a similar trend as calculated using the FF approach, predicting that the relative stability of β to γ changes at elevated pressures. The γ phase is slightly less stable than the α phase in the whole pressure range, which may be explained by less favorable conformations. What is more important, however, is that these calculations predict γ and β phase instability at ambient pressure, which fully coincides with the experimental study.

It is not only possible to estimate phase stability, but also to compare separate enthalpy terms and find a prevailing term for phase transition. Pairwise comparisons of lattice energies with PV in terms of polymorphs are presented in Figure 5a,b.

**Figure 5.** Calculated (**a**) lattice energy ΔUcrystal, (**b**) ΔPV term and (**c**) ΔH differences between *p*-Tol2S2 polymorphs α and γ (green), α and β (purple), β and γ (orange) as a function of pressure (i.e., ΔUα − β = Uα − Uβ, ΔPVα–β = PVα − PVβ, ΔHα − β = Hα − Hβ, etc.). Phase transition between the involved polymorphs can be expected when their enthalpy difference is equal to zero.

Phase transition α → γ is justified by higher compressibility (the PV component of the enthalpy, rather than the specific pairwise energies) of γ polymorph (what also shown with K' coefficients from EoS data (Table S3)), whereas α → β transition cannot be unequivocally defined by any term. This also coincides well with structure data, showing β and α phases being very similar in terms of geometry (Figure S1).

Calculated cell volumes from EoS helps to exclude simplification of enthalpy linearization it is possible to calculate enthalpies at each point by direct sum of Ucrystal (Uinter + Uintra) and PV term (Figure S6). Enthalpy difference is plotted in Figure 5c (and Figure S7 and Table S5 with linearization).

Based on this more sophisticated approach, one can predict phase transitions at 0.34 GPa and 0.36 GPa for α → γ and α → β transitions in comparison to experimental 0.45 GPa and 1.6 GPa. Such deviations may be explained by several factors, apart from the absence of calculated thermal effects. It is mandatory to understand that experimental techniques show a range of pressures where transition occurs, due to features of the highpressure devices (e.g., a diamond anvil cell and the limitations of a single crystal diffraction technique at elevated pressure). The experimental data are limited, so the exact pressure of phase transition is not reported: α → β transition occurred in the interval 1.52–1.60 GPa, whereas the γ phase was crystallized at 0.45 GPa and other pressures were not checked. On the other hand, transition state energy during phase change may be significant and lead to hysteresis and higher pressures of experimental phase transition in comparison to thermodynamic data [21,44,45]. Finally, the entropy term may also affect transitions, changing the Gibbs energies of different polymorphs. Nevertheless, it is possible to estimate energy limits of the transition state from the suggested enthalpy functions.

The differences in polymorph enthalpies, when at pressure of experimental phase transition, denote the upper limit for the transition state energy. This is because the initial phase (α in this case) stores "excessive" energy in relation to the final phase. Following this logic, the formation of the "hidden" γ polymorph has an energy barrier less than 2 kJ/mol, which is reasonable for solid–liquid–solid phase transition. The "conventional" β polymorph also has very small transition state energy less than 3 kJ/mol, which may be explained by extremely small structure changes during this transition. The β → γ phase transition has a TS energy barrier of no less than 18 kJ/mol. So, the "excessive" energy of the β phase is not enough to result in solid–solid crystal structure rearrangement.

Finally, it is possible to refine the phase diagram for a *p*-Tol2S2 system based on the enthalpy calculations, where the α form is the most stable in the pressure range of 0.0–0.34 GPa, the γ phase is the most stable at a pressure range from 0.34 GPa to 3.0 GPa, and the β phase is metastable in the whole pressure range from 0.0 GPa to 3.0 GPa (nevertheless, being more stable than the γ phase in the pressure range of 0.0–0.32 GPa) (Table S4). Figure 6 illustrates the thermodynamic relation between *p*-Tol2S2 polymorphs, which turned out to be more complicated than suggested from experimental data (Figure 1b, Table S6).

**Figure 6.** Suggested relations between the free Gibbs energy of phases α, β, and γ as a function of pressure based on a periodic DFT study where only enthalpies were calculated. Table S4 shows the relative phase stability according to calculated enthalpies in different pressure ranges.

#### **4. Conclusions**

In this study, we show how different computational techniques supplement previous experimental studies to evaluate thermodynamic stability of *p*-Tol2S2 polymorphs in the approximation of enthalpy changes. Force field methods (as implemented in CE21) show reliable stability ranking of polymorphs at pressure ranges where phases coexist, if enthalpies are calculated. The "hidden" γ form was predicted to be more stable than the α form in the 0.45–1.52 GPa experimental pressure range, and more stable than the "conventional" β phase in the interval of 1.60–2.8 GPa. Periodic DFT methods, in conjunction with EoS, help to answer questions regarding polymorph stability in the whole pressure (not only experimental) range, even if a specific form does not exist experimentally. It was shown that the α form is the most stable form when in the pressure range 0.0–0.34 GPa, while the γ phase is the most stable from 0.34–3.0 GPa due to higher compressibility (PV term). Upper limits for transition state energies for both experimental phase transitions were estimated from the suggested relations between the free Gibbs energy values of phases α, β, and γ as a function of pressure. Both methods proved to reveal valuable information about relative polymorph stability (FF for experimental and periodic DFT for the whole pressure range). A refined phase diagram (in a rough assumption of calculated enthalpies) over pressure shows the necessity of computational methods for high-pressure crystallography.

**Supplementary Materials:** The following supporting information can be downloaded at: https: //www.mdpi.com/article/10.3390/cryst12081157/s1. **Scheme S1**: An overall workflow of computational procedures. **Table S1**: Summarized structural data for *p*-Tol2S2 polymorphs used in CE21 and EoSFit7 calculations. **Figure S1**: Crystal structures of (**a**) alpha, (**b**) beta and (**c**) gamma polymorphs of *p*-Tol2S2. Colors according to symmetry equivalence in corresponding crystal structures. **Figure S2**: Potential energy changes (ΔEp) calculated by Gaussian (Experimental Section of [14]) for the isolated molecule in its conformation, experimentally determined in the *p*-Tol2S2 structures of phases α, β, and γ. **Figure S3**: Lattice energies (the sum of Uinter and Uintra) of phases α, β, and γ as a function of pressure calculated based on experimental data. Regions of structural stability according to experimental data are shown with dashed lines. Uintra is reproduced from work [14]. **Figure S4**: Enthalpies of phases α, β, and γ as a function of pressure calculated based on experimental data using CE21. **Table S2**: Predicted by FF and experimental phase transition pressures. **Figure S5**: *p*-Tol2S2 volumepressure dependence based on calculated EoS. **Table S3**: EoS Birch-Murnaghan 3rd order coefficients. V0—reference pressure volume at ambient pressure, K0—bulk modulus, Kp—derivative of Bulk Modulus (dK/dP). **Figure S6**: Enthalpies of phases α, β, and γ as a function of pressure calculated using periodic DFT and EoS. **Table S4**: Relative stability of *p*-Tol2S2 polymorphs in different pressure ranges calculated by periodic DFT. **Figure S7**: ΔH differences between *p*-Tol2S2 polymorph α and γ (green), α and β (purple), β and γ (orange) as a function of pressure (i.e., ΔUα − β = Uα − Uβ, ΔPVα − β = PVα − PVβ, ΔHα − β = Hα − Hβ, etc.). **Table S6**: Phase transitions of *p*-Tol2S2 polymorphs under pressure according to various experimental and computational techniques.

**Author Contributions:** Conceptualization, D.A.R.; methodology, D.A.R.; software, V.Y.S. and A.A.I.; validation, A.A.I. and D.A.R.; formal analysis, V.Y.S. and A.A.I.; investigation, V.Y.S., A.A.I. and D.A.R.; resources, D.A.R.; data curation, V.Y.S. and A.A.I.; writing—original draft preparation, D.A.R.; writing—review and editing, V.Y.S., A.A.I. and D.A.R.; visualization, V.Y.S., A.A.I. and D.A.R.; supervision, D.A.R.; project administration, D.A.R.; funding acquisition, D.A.R. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by Ministry of Higher Education and Science, grant number FWUS-2021-0005 (FF and periodic DFT calculations, data processing and analysis) and the Russian Science Foundation, grant number 18-73-00154 (system choice, database search, preliminary calculations).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Data available on request from corresponding author.

**Acknowledgments:** The Siberian Branch of the Russian Academy of Sciences (SB RAS) Siberian Supercomputer Center (http://www.sscc.icmmg.nsc.ru/, accessed on 12 August 2022) is gratefully acknowledged for providing access to their supercomputer facilities. The authors also acknowledge the Supercomputing Center of the Novosibirsk State University (http://nusc.nsu.ru, accessed on 12 August 2022) for providing computational resources.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.

#### **References**


### *Article* **On the Definition of Phase Diagram**

**Kamil Filip Dziubek**

European Laboratory for Non-Linear Spectroscopy (LENS), Via Nello Carrara 1, 50019 Sesto Fiorentino, Italy; dziubek@lens.unifi.it

**Abstract:** A phase diagram, which is understood as a graphical representation of the physical states of materials under varied temperature and pressure conditions, is one of the basic concepts employed in high-pressure research. Its general definition refers to the equilibrium state and stability limits of particular phases, which set the stage for its terms of use. In the literature, however, a phase diagram often appears as an umbrella category for any pressure–temperature chart that presents not only equilibrium phases, but also metastable states. The current situation is confusing and may lead to severe misunderstandings. This opinion paper reviews the use of the "phase diagram" term in many aspects of scientific research and suggests some further clarifications. Moreover, this article can serve as a starting point for a discussion on the refined definition of the phase diagram, which is required in view of the paradigm shift driven by recent results obtained using emerging experimental techniques.

**Keywords:** phase diagram; phase transitions; high pressure; metastability; phase boundaries; kinetic lines; nonequilibrium conditions

#### **1. Introduction**

In their pioneering work on semantics, semiotics, and the science of symbolism, published almost a century ago, Ogden and Richards recounted: "*In all discussions we shall find that what is said is only in part determined by the things to which the speaker is referring. Often without a clear consciousness of the fact, people have preoccupations which determine their use of words. Unless we are aware of their purposes and interests at the moment, we shall not know what they are talking about and whether their referents are the same as ours or not*" [1]. Today, this statement is still very true, even in the scientific language, where precise definitions are to be generally agreed upon. One of the frequently used and, at the same time, elusive physico-chemical terms that defies a single accepted definition is *phase diagram*. This expression is not reported in the IUPAC Gold Book [2], nor in the IUCr Online Dictionary of Crystallography [3]. In a recent review on pressure-induced phase transitions, Grochala noted that "*the phase diagram is in its most common meaning a graph in which a property or a state of matter is shown in the function of pressure and temperature,*" adding a caveat about the described system being in equilibrium [4]. Therefore, I decided to use the following general Oxford English Dictionary (OED) definition [5] as a reference point:

**phase diagram** *n*. Chemistry a diagram which represents the limits of stability of the various phases of a chemical system at equilibrium, with respect to two or more variables (commonly composition and temperature); an equilibrium diagram.

In the high-pressure research, phase diagrams are usually constructed for systems of constant composition (empirical formula), with two variables: temperature (*T*) and pressure (*P*). Hence, the OED definition applied to the *P*–*T* space assumes that: (1) the limits of the stability of the phases are represented in a *P*–*T* chart as separate fields that correspond to each phase and are delimited by phase-boundary lines, also known as coexistence curves (representing the *P–T* conditions at which phase transitions occur), or the abscissa and ordinate axes (representing the *P* = 0 isobar and *T* = 0 isotherm, respectively); (2) phase diagrams pertain only to the equilibrium state. Note that the stability conditions may be

**Citation:** Dziubek, K.F. On the Definition of Phase Diagram. *Crystals* **2022**, *12*, 1186. https://doi.org/ 10.3390/cryst12091186

Academic Editor: Daniel Errandonea

Received: 25 July 2022 Accepted: 20 August 2022 Published: 23 August 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

ambiguous, as there are various definitions of stability (this will be discussed in more detail later). The OED definition therefore describes an *ideal phase diagram,* and it will be referred to as such from here onwards.

The phase diagrams published in the literature are determined in experimental studies or are predicted using various theoretical approaches. Both methods have their intrinsic limitations. In predicted phase diagrams, the boundary lines between the crystalline phases and melting curves are usually constructed by comparing the Gibbs free energies of individual phases. Because they can be computed using different models and approximations that also include temperature effects, the plots reported by various authors may differ significantly. On the contrary, the disagreements between experimental phase diagrams depend not only on the laboratory protocols, but also on the character of the phase transition.

To aid the reader and for clarity, a glossary of key terms related to the concept of the phase diagram is provided below.

#### **2. Glossary of the Key Terms Related to Phase Diagrams**


states displayed on a diagram involve chemical reactions and the rearrangement of the bonding pattern (e.g., transformations from a molecular to an extended solid), although this additional specification seems redundant given the previously defined types of *P–T* diagrams.

#### **3. Classification of Phase Transitions and Definitions of Phase Stability**

In the most basic terms, phase transitions are usually related to the transformations of the physical properties of matter, which can be quantified by a change in the state parameters. The concept of a phase-transition order was introduced by Paul Ehrenfest in 1933 [9–11]. Ehrenfest's approach is purely thermodynamic: Assuming that the thermodynamic potential (usually the Gibbs free energy (*G*) or chemical potential (*μ*)) is continuous across the phase boundary of any equilibrium phase transition, one can investigate its particular derivatives concerning an external thermodynamic variable, such as temperature or pressure. Hence, in the general case, for an *n*th-order phase transition:

$$
\Delta \left( \frac{\partial^n G}{\partial T^n} \right)\_P \neq 0; \quad \Delta \left( \frac{\partial^{n-1} G}{\partial T^{n-1}} \right)\_P = 0 \qquad \text{at } T = T\_{\text{trans}}.\tag{1}
$$

$$
\Delta \left( \frac{\partial^n G}{\partial P^n} \right)\_T \neq 0; \quad \Delta \left( \frac{\partial^{n-1} G}{\partial P^{n-1}} \right)\_T = 0 \qquad \text{at} \quad P = P\_{\text{trans}}.\tag{2}
$$

where *G* represents the Gibbs free energy, and *Ttrans* and *Ptrans* are the equilibrium transition temperature and pressure, respectively. Thus, the order of the transition is determined by the lowest nonzero derivative of the thermodynamic potential. At a first-order transition, the first derivatives of the *G* (namely, entropy (*S*) and volume (*V*)) are discontinuous; at a second-order transition, the *S* and *V* are continuous, but the second derivatives of the *G* (for instance, the constant-pressure heat capacity (*CP*), the isothermal compressibility (*βT*), and the isobaric thermal expansion coefficient (α)) are discontinuous, and so on. This simple view, however, is often inadequate for the characterization of some more complex transitions beyond the first order, which cannot be effectively described by a jump in any *n*th derivative of the thermodynamic potential, but by the nonanalytic behavior of these functions. In the modern classification scheme, which is essentially a generalization of the Ehrenfest one, phase transitions are classified into two broad categories: (1) "discontinuous" first-order transitions, which are distinguished by the presence of the latent heat and discontinuity in the order parameter defined on the grounds of the phenomenological theory formulated by Lev Landau [12]; (2) "continuous" second-order transitions (also comprising higher-order transitions, as defined in the Ehrenfest scheme), which are marked by a steady evolution of the order parameters, which are fully reversible across the phase boundary. This claim has significant consequences for the experimental determination of phase diagrams.

Consider a multidimensional space where the thermodynamic potential is a function of different variables. In a system of constant composition, each phase can be represented as a potential-energy surface in terms of pressure and temperature. The equilibriumtransformation conditions between two phases are defined by the intersection of surfaces, corresponding to the phase-boundary lines. In the example shown in Figure 1, if an isothermal compression is considered, then one can expect that the thermodynamic potential of the system should vary along the path 1–2–3–4–5 upon compression (5–4–3–2–1 upon decompression), with the sharp transformation of the entire sample occurring at Point 3, corresponding to the pressure (*Ptrans*) at which the potentials of Phases I and II become equal. In practice, however, during the course of first-order transitions, the potential often follows the path 1–2–3–7–4–5 upon compression (5–4–3–8–2–1 upon decompression). Phase I, which exists for *Ptrans* < *P* < *P*\*\* during compression, is not in equilibrium because it is not the minimum potential configuration but a "superpressurized" metastable state. Likewise, Phase II, which persists upon decompression for *P*\* < *P* < *Ptrans*, corresponds to a "superdepressurized" state (named by an analogy to superheated and supercooled

states). Hence, the experimental onset of the transition upon the pressure increase can be regarded as an upper bound of the metastability range, while the onset of the reverse transition marks a lower bound. The pressure range Δ*P* = *P*\*\* − *P*\* of this hysteresis (and the values of *P*\* and *P*\*\*) depend on many factors, such as the pressure-change rate and the purity of the substance. Moreover, note that the intervals (*P*\*\*, *Ptrans*) and (*Ptrans*, *P*\*) are not necessarily equal, and so the center of the hysteresis loop does not need to coincide with the transition pressure. Therefore, the actual value of the *Ptrans* is difficult, if not virtually impossible, to locate from the experiment.

**Figure 1.** Thermodynamic-potential surfaces for two phases, I and II, as functions of pressure and temperature. The projection onto the *P*–*T* plane represents the phase diagram, whereas the line of intersection of the two surfaces corresponds to the I–II-phase-transition boundary. Black dots represent the stages of experimental isothermal compression or decompression routes. *P*\* and *P*\*\* define the limits of the metastability range, while *Ptrans* denotes the equilibrium transition pressure.

One of the most common examples is the first-order solid–liquid phase transitions and the determination of the melting line. In the isothermal runs, crystallization often occurs from the superpressurized liquid, while in standard experimental settings, virtually all the solids melt immediately under the equilibrium solid–liquid coexistence conditions without entering the underpressurized (superdepressurized) state. The reason behind this is that, in the case of heterogeneous melting, the crystal surfaces and grain boundaries suppress the energy barrier required for melt nucleation [13–15].

Furthermore, it has to be stressed that the solid-state first-order transitions proceed through the nucleation and growth of interfaces. If the kinetics of nucleation is much slower than the rate of the pressure (or temperature) change, then one can observe the phase coexistence, where the mass fraction between the two phases changes across the transformation [16]. If the crystal structures are not similar, and the change from one structure to the other involves the significant reorganization of the bonding scheme at the atomic level, then the energy barrier of nucleation may be considerable and the transformation rate sluggish. Moreover, many other factors (such as the pressure- or temperature-variation rate, sample microstructure, or sample environment) may affect this process. Figure 2 demonstrates the fraction of the transformed phase as a function of pressure in a pressure-induced first-order transition for different kinetic rates.

Moreover, many of the reported high-pressure datasets are collected on compression only. The reasons are varied but are often quite mundane: limited beamtime or technical obstacles (jammed diamond-anvil cell, plastic deformation of the gasket, failure of the diamonds). Therefore, the observed conditions at which the new phase begins to appear may be far away from the point where the Gibbs free energies of the two phases are equal. This discrepancy is even more severe in the ultrahigh-pressure regime.

**Figure 2.** The fraction of the transformed phase as a function of pressure representing a relatively fast (**a**) and slow (**b**) rate of interconversion. *P*\* and *P*\*\* mark the lower and upper bounds of the phase coexistence range.

To sum up, various generic features are associated with first-order transitions, such as metastability, phase coexistence, hysteresis, and their dependence on the path followed in the pressure–temperature space. All these factors can affect the shape of the experimentally determined phase-boundary lines and, consequently, the phase diagram.

Finally, one should be aware that the phase stability has different definitions. The phase is *thermodynamically stable* if it corresponds to the minimum of the thermodynamic potential at a given pressure and temperature with respect to the other phases. It is roughly consistent with the IUPAC definition of "stable" regarding the chemical stability: "*As applied to chemical species, the term expresses a thermodynamic property, which is quantitatively measured by relative molar standard Gibbs energies. A chemical species* A *is more stable than its isomer* B *if* Δr*G*◦ > 0 *for the (real or hypothetical) reaction* A → B *under standard conditions*" [17]. An immediate conclusion from this definition is that, at the given pressure and temperature, there is only one thermodynamically stable phase. At the same time, the other structures are *metastable* (i.e., thermodynamically unstable states that can be identified with the local minima on the potential-energy hypersurface and persist due to kinetic hindrance). A good general definition of *metastability* was provided by Tschoegl [18]: "*When a thermodynamic system is in* stable equilibrium, *a perturbation will result only in small (virtual) departures from its original conditions and these will be restored upon removal of the cause of the perturbation. A thermodynamic system is in* unstable equilibrium *if even a small perturbation will result in large, irreversible changes in its conditions. A system in* metastable equilibrium *will act as one in stable equilibrium if perturbed by a small perturbation but will not return to its initial conditions upon a large perturbation.*"

Under equilibrium conditions, a crystal is *dynamically stable* (or *vibrationally stable*; these terms are synonymous) if its potential energy always increases for any combination of atomic displacements; hence, all of the atoms are trapped within the local energy minima. In the harmonic approximation, this corresponds to the real and positive phonon frequencies of all the phonon modes [19]. The presence of imaginary (negative) frequencies of vibration at reciprocal lattice vectors in the Brillouin zone testifies to the dynamic instability.

A crystal is *mechanically stable* (or *elastically stable*; synonymous terms) if it fulfills Born's stability criteria, which were formulated for the first time in his 1940 paper [20]. In the harmonic approximation (i.e., for an unstressed material at zero pressure), the second-order elastic stiffness tensor (**C**) is defined as:

$$\mathbb{C}\_{ij} = \frac{1}{V\_0} \left( \frac{\partial^2 E}{\partial \varepsilon\_i \partial \varepsilon\_j} \right) ,\tag{3}$$

where *E* is the elastic energy of the crystal, *V*<sup>0</sup> is its equilibrium volume, *εi*, and *ε<sup>j</sup>* are the components of the strain tensor, and *Cij* are the elastic constants (i.e., the elements of the elastic tensor). The elastic energy of a crystal is quadratic with applied strains, as follows:

$$E = E\_0 + \frac{V\_0}{2} \sum\_{i=1}^{6} \sum\_{j=1}^{6} \mathbb{C}\_{ij} \varepsilon\_i \varepsilon\_j. \tag{4}$$

According to the Born *mechanical* (*elastic*)*-stability* criterion, the crystal is *mechanically* (*elastically*) *stable* if the elastic energy is always positive. This corresponds to the requirement that the tensor (**C**) is positive definite, or equivalently, that all the eigenvalues of the **C** are positive. In a general case, the symmetric 6 × 6 s-order elastic stiffness tensor (**C**) contains 21 independent elements. While a more detailed analysis is beyond the scope of this paper, the interested reader is referred to the comprehensive article of Mouhat and Coudert [21], who generalized the necessary and sufficient Born mechanical-stability conditions for all crystal classes. It should be emphasized that the *mechanical* (*elastic*)-*stability* conditions can be generalized to systems under an arbitrary external load (*σ*) by introducing an elastic stiffness tensor (**B**) under load. The applied external stress is often a game-changing factor for the phase stability. If a system is mechanically unstable but dynamically stable under given conditions, then this may indicate a thermodynamically metastable state.

The definitions of *dynamic* and *mechanical* stability refer to different physical variables and are therefore independent of each other. For example, scandium carbide (ScC) was found to be *mechanically stable*, but *dynamically unstable* [22], while the bcc titanium structure is *mechanically unstable,* but *dynamically stable* [23].

#### **4. Case Studies**

The following examples will illustrate a variety of issues and concerns regarding the literature phase diagrams. Questions about their conformity to the universally accepted *ideal-phase-diagram* definition will be discussed.

#### *4.1. Stable or Metastable, That Is the Question*

#### 4.1.1. Chemical Elements

Chemical elements have been extensively studied under pressure since the pioneering works of Percy Bridgman at the beginning of the last century. Several monographs are entirely devoted to the phase diagrams of elements [24,25]. The systematic behavior of chemical elements over a range of pressures and temperatures was also illustrated graphically in a specific form of the periodic table [26]. Despite the expected simplicity, designing phase diagrams of elements can pose a real challenge.

Carbon is arguably the element with the highest known number of chemical compounds due to its ability to bind covalently with other carbon atoms and form extended structures. It is also a chemical element with an overwhelming number of predicted crystal forms; the most recent release of the comprehensive Samara Carbon Allotrope Database (SACADA) contains 524 allotropes [27,28]. This complexity is not reflected in the experimental phase diagram of carbon [8,29–32], which contains only two experimentally determined solid phases, graphite and diamond, and other predicted phases, still not confirmed in experiments (see Figure 3). The first-order graphite diamond phase transition has a high energy barrier, and the room-temperature compression of graphite does not yield diamond, but another three-dimensional metastable structure, in which chemical bonds between the neighboring graphene sheets are formed: the monoclinic *M*-carbon [33]. As expected, the kinetic barrier between the phases is lowered with the increase in temperature. On the one hand, synthetic laboratory-grown diamonds are produced at the pressure of ~5 GPa and a temperature of ~1800 K. On the other hand, the back transformation of diamond to graphite is kinetically hindered at room temperature and atmospheric pressure; hence, "diamonds are forever." Fullerenes and nanotubes, which are famous families of carbon allotropes, are not found in the phase diagram of carbon because they are metastable under

any *P*,*T* conditions with respect to graphite or diamond. Another well-known carbon allotrope, lonsdaleite, or the so-called "hexagonal diamond," is not a separate phase but actually adopts a stacking-disordered diamond structure [34,35]. Hypothetical superdense carbon allotropes, some of them calculated to be thermodynamically stable in the terapascal regime, have been included in predicted phase diagrams of carbon [31,36,37]. However, a recent study on the ramp compression of diamond to 2 TPa demonstrates that the transition to stable dense allotropes is kinetically arrested due to a high energy barrier, similar to the hindered graphite diamond transition in the modest-pressure range [32].

**Figure 3.** The simplified *transitional P–T diagram* of carbon, after [8,29–32]. Note the logarithmic scale for pressure. The left and right hashed areas correspond to the regions of metastability of graphite and diamond, with metastability boundaries marked as dashed lines. Dash-dotted contours delineate the calculated melting curve of diamond and stability fields of the predicted BC8, simple cubic (sc), and simple hexagonal (sh) phases [31], which have been never confirmed experimentally. The horizontal hashed area demonstrates the persistence of diamond in a ramp-compression experiment up to 2 TPa [32] within the predicted thermodynamic stability field of the BC8 phase. While the kinetics of the graphite diamond transition can be very slow (at room temperature and ambient pressure, the diamond → graphite conversion timescale is longer than the age of the universe), the proposed metastability of diamond above 1 TPa is reported in the order of magnitude of nanoseconds.

As mentioned above, the experimental determination of phase boundaries may be affected by numerous factors. A striking example is the phase transitions in lithium, where quantum and isotope effects play a crucial role in shaping its phase diagram [38]. It was known from earlier studies [39–41] that, upon cooling below ~77 K at atmospheric pressure, bcc 7Li transforms to a rhombohedral close-packed *α*-Sm-like 9R structure (space group *R*3*m*), which, for a long time, was regarded as the ground state of lithium [40,41]. Careful selection of the pressure–temperature pathway (pressure-induced bcc → fcc transition at room temperature, followed by cooling to 20 K and subsequent decompression along the T = 20 K isotherm) helped to circumvent the bcc → 9R transition and facilitated the establishment of the actual ground state of the element, which is the fcc phase. By a prudent choice of words, the authors prevented ambiguities: while presenting the metastable 9R state on *P*–*T* charts, they entitled the figure "*Observed stable and metastable crystal structures of 6Li and 7Li measured along the identified P–T paths*", avoiding the term "phase diagram" [38].

Another interesting case is phosphorous, where two textbook examples of allotropes, white and red phosphorous, are thermodynamically metastable under any *P*–*T* conditions and are missing in the phase diagram [42]. Black phosphorous, a ground-state structure built of corrugated layers of covalently bonded atoms, was synthesized for the first time by Percy Bridgman in 1914 by heating the white modification at high pressure [43]. It was one of the first spectacular achievements of high-pressure research. This form exhibits an interesting phenomenon of layer destabilization upon heating [44], although no other stable crystal phase was revealed in this experiment.

As already mentioned, the mechanical stability of the crystal can be tuned by the external stress. For example, first-principles calculations and stability analysis suggest that diamond becomes mechanically unstable when the difference between the radial- and axial-stress components exceeds 200 GPa [45]. In his review article, Levitas distinguished three types of phase transitions at high pressure: pressure-induced, stress-induced, and strain-induced transformations [46], and he described the differences between them in the following manner: "*In most cases nucleation of the product phase occurs heterogeneously at some defects (dislocations, grain and twin boundaries), which produce a concentration of the stress tensor and/or provide some initial surface energy.* Temperature-induced transformations *nucleate predominantly at pre-existing defects without stresses at the specimen surface. Similarly,* pressure-induced transformations *occur mostly by nucleation at the same pre-existing defects under action of external hydrostatic pressure.* Stress-induced transformations *occur at the same defects when external nonhydrostatic stresses do not exceed the macroscopic yield strength in compression σy. If the phase transformations take place during plastic deformations they are classified as* strain-induced transformations. *They occur by nucleation at new defects generated during plastic deformation*". The effects of uniaxial stress on the phase-transition onset, hysteresis, and phase-coexistence range were demonstrated experimentally for the *α ε* transition of iron [47] and the *α ω* transition of titanium [48], employing various pressure-transmitting media.

#### 4.1.2. Chemical Compounds

Chemical compounds exhibit a higher degree of complexity. The *P–T* phase diagrams are determined for all the phases with the same net stoichiometry (constant empirical formula). Even for the simplest cases, there are many contradictory results, which have stirred major controversies. One of the examples is carbon dioxide (CO2), which, despite the simple stoichiometry, has quite a complex phase diagram that consists of several crystalline molecular polymorphs below ~35 GPa. Above this pressure, in laser-heated diamond-anvil-cell experiments, they transform to a polymeric tetragonal crystal phase, which is named CO2-V, which can be described as a network of fourfold coordinated carbon atoms interconnected by oxygen bridges [49,50]. The substantial kinetic barrier that results from the reconstruction of the bonding scheme may preclude the formation of this phase, leading to the quenching of metastable states in the stability field of CO2-V. Indeed, other extended covalent (CO2-VI [51]) and even ionic (*i*-CO2 [52]) structures of CO2 have been suggested in experimental studies of CO2 in the nonmolecular *P*–*T* regime. Recent reports that extend the pressure range beyond 100 GPa [53] and that employ in situ highpressure–high-temperature (*HP*–*HT*) X-ray diffraction [54] confirmed that CO2-V is the only stable modification of carbon dioxide at *P* > 35 GPa. Moreover, a previously reported dissociation of CO2 to elements [55–57] was not observed, even in long laser-heating cycles up to a pressure above 100 GPa and T = 6200 K [54]. The *P*–*T* diagram of CO2 showing the relationships between the equilibrium crystal phases and nonequilibrium states is illustrated in Figure 4. Please note that the position of the so-called kinetic transition lines may depend strongly on the pressure- and temperature-change rate and the *P*–*T* pathway.

**Figure 4.** The *P*–*T* diagram of carbon dioxide compiled from literature data. Black solid lines represent the phase boundaries of thermodynamically stable phases delineating the stability range of CO2-I and the melting curves of molecular phases [58,59], as well as the coexistence line between CO2-II and CO2-IV [60]. The black dashed line is the extrapolated CO2-IV melting curve. Black dash-dotted lines correspond to the computed boundary line between molecular phases and CO2-V [61] and the calculated CO2-V melting curve [62], used in the absence of reliable experimental data. Metastable states are highlighted in blue: CO2-III is most probably a metastable manifestation of CO2-VII, and both have the same structure [61], while CO2-V appears to be the only thermodynamically stable crystal phase of carbon dioxide above ~35 GPa [53,54,61,63]. Blue dotted traces are the kinetic transition lines involving the metastable states: between CO2-IV and CO2-VII (the extrapolation indicated by the dashed line) [59], between CO2-II and CO2-III [51], between amorphous CO2 and CO2-VI [51], between CO2-V and CO2-VI [51], and between CO2-V and ionic *i*-CO2 [52]. Red dotted lines denote the thresholds of the dissociation of CO2 into chemical elements (indicated by an arrow), which was reported in three independent studies [55–57]. This reaction, however, was not confirmed by the subsequent in situ *HP*–*HT* study [54].

The above example refers to a simple system that consists of two chemical elements. In principle, the vast majority of the materials are metastable, including almost all the organic compounds, as reflected, for example, in the description of the methodology of the constrained-evolutionary-algorithm calculations by Zhu et al. [64]: "*Most of the molecular compounds are thermodynamically less stable than the simpler molecular compounds from which they can be obtained (such as H2O, CO2, CH4, NH3). This means that a fully unconstrained global optimization approach in many cases will produce a mixture of these simple molecules*". This statement falls squarely into the category of chemical stability and supports the expectation that the decomposition of complex molecules into a mixture of simpler structures is thermodynamically preferred; hence, such compounds are chemically metastable. Because the rate of transition is very often so slow (the energetic barrier is so high) that the transition to the thermodynamic ground state is practically not observed, and the shelf lives of many molecular compounds are in a timescale of years, the *P–T* diagrams of most of them (such as, e.g., paracetamol [65]) do not conform to the *ideal-phase-diagram* definition. Indeed, as Brazhkin noted in his viewpoint article [66]: "*the objects of the physics of condensed media are primarily the equilibrium states of substances with metastable phases viewed as an exception, while in chemistry the overwhelming majority of organic substances under investigation are metastable. It turns out that at normal pressure many simple molecular compounds based on light elements (* ... *) are metastable substances too,* i.e., *they do not match the Gibbs free energy minimum for a given atomic chemical composition. (* ... *) Actually almost all so-called 'phase diagrams' of molecular substances in the reference books are in fact transitional diagrams (* ... *)*". Such *transitional*

*diagrams* are useful tools for analyzing molecular systems. In many organic compounds, the upper *P–T* limit of the metastability range of the molecular state is not a melting curve but a chemical-stability boundary, above which decomposition, polymerization, or pyrolysis is observed [67,68].

It is worth mentioning that the transformations between constrained metastable states have most of the features of phase transitions. In particular, the transition point in molecular solids can depend on factors such as the pressure-transmitting medium [69], nonhydrostatic conditions [70], and so on, affecting the shape of the associated transitional diagrams.

Lastly, it should be noted that, in the computational structure predictions of chemical compounds, the idea of the so-called convex hull is widely employed to predict stable phases. In the simplest case, for a binary AB system at zero temperature and zero pressure, it shows the enthalpy of formation (Δ*HF*), plotted against the A:B molar ratio, represented on a linear segment between the pure A and B elements (unary phases) as the endmembers (composition–energy hull) [71]. Convex hulls contain only thermodynamically stable phases with the lowest possible formation energy for a given composition. Most frequently, these phases correspond to stoichiometric compounds that are characterized by the simplest whole-number A:B ratios. This idea can be extended to more complex systems and more dimensions: ternary systems are represented graphically in an equilateral triangle, quaternary systems in a tetrahedron, and so forth. Pressure can be included by adding volume as a parameter (composition–volume–energy hull) [71]. This concept can be helpful in the quest for new materials under high pressure [72], and even ones as complicated as the H–C–N–O quaternary system [73]. While the convex hulls cannot be directly rendered into *P–T* phase diagrams (mostly due to the difficulties related to accounting for temperature effects and creating a composition–volume–temperature–energy hull), a quick glimpse at a hull allows us to answer the question of whether the phase is stable towards decomposition into the constituent elements or other compounds with different stoichiometry by comparing their Δ*HF* under the given thermodynamic conditions.

#### *4.2. Pressure-Induced Amorphization, Glassy States, and Liquid–Liquid Transitions*

Pressure-induced amorphization is a ubiquitous phenomenon that is observed as crystalline solids are compressed beyond their stability range under specific conditions (fast compression rate, low temperature) that hamper the expected crystal-to-crystal phase transition [74,75]. Amorphization can also occur upon the decompression of high-density phases. It can be associated with a chemical reaction; for example, at a moderate temperature (up to 680 K), molecular CO2 starts to polymerize, forming amorphous carbonia glass [76], which is a chemically distinct species that contains carbon in threefold and fourfold coordination (see Figure 4). This transformation occurs when the temperature is too low to initiate and complete the phase transition to the thermodynamically stable CO2-V crystal phase.

As McMillan and Wilding noted [75]: "Polyamorphic transformations between different amorphous "phases" that appear to mimic crystalline phase transitions also occur as a function of variables such as pressure (P) or temperature (T). Because glasses and other amorphous materials are not in internal thermodynamic equilibrium, such analogies must be made with care, however". Indeed, glasses are not phases but are kinetically trapped metastable states (even if one can plot boundaries between the glassy states), and they shall not be present in the phase diagrams. For instance, the formation of amorphous ice results from the sluggish transition kinetics and can be overcome if the compression rate is very slow [77]. In this regard, one can plot a phase diagram of ice reporting the crystalline phases and a transitional P–T diagram revealing amorphous states (with boundaries between low-density amorphous, high-density amorphous, and very-high-density amorphous ice). Presenting "mixed" plots that comprise both stable and metastable states is also possible [78], but they have to be presented with appropriate comments to avoid any ambiguities.

Phase transitions in liquids are a different subject because they pertain to the state, which corresponds to the energy minimum in the *P–T* range beyond the melting curve. First-order liquid–liquid phase transitions usually occur between the states that are characterized by different local structures and short-range bonding schemes [75]. Examples are transitions in phosphorus between the low-density-liquid (LDL) phase consisting of discrete P4 molecules and the polymeric high-density liquid (HDL) [79], or between the LDL and HDL phases of sulfur [80]. In the latter case, the phase-boundary line between the LDL and HDL ends with a critical point, which was a first experimental observation of a critical point in liquid–liquid phase transitions.

#### *4.3. Critical Points and Supercritical States*

As per the definition, in a phase diagram of a pure substance, liquid and gas are undistinguishable beyond the liquid–vapor critical point and constitute a single fluid phase. However, boundary lines that delimit the gas-like and liquid-like states of supercritical fluid can be noted in some instances. These boundaries do not correspond to classical phase transitions but can be revealed in experimental studies (e.g., investigating sound velocity in supercritical fluid argon reveals the so-called Widom line) [81].

Critical points can also be observed at the end of the phase-boundary lines between the liquid states (e.g., in sulfur [80]). In solid phases, they are rare, primarily due to symmetry requirements. Usually a solid–solid phase-boundary line is intercepted by other phase boundaries or runs to infinity. One of the notable exceptions is the *α γ* phase transition in cerium discovered by Percy Bridgman in 1927 [82], which has been intensively studied ever since [83]. Both phases share the same space-group symmetry (*Fm*-3*m*), and the observed first-order transition of electronic origin [84] is associated with a substantial volume collapse, which decreases with the pressure and temperature rise along the phase-boundary line, and eventually vanishes at the critical point. Interestingly, upon the extension of the *α–γ* phase boundary, no evidence of a second-order phase transition is found, but a minimum in the isothermal bulk modulus in the function of pressure is observed to the highest experimentally achieved temperature [85]. The *P–V* compression data revealed in the radiography study indicated that, at the extrapolation of the *α–γ* coexistence line to the liquid-state regime, the *P*(*V*) isotherm shows an inflection point, which indicates a plausible liquid–liquid phase transition [86] (see Figure 5).

**Figure 5.** The phase diagram of cerium adapted from the literature data [86,87]. Solid traces denote the phase-boundary lines, and CP marks the critical point. The dashed line corresponds to the minimum observed in the isothermal-bulk-modulus plots [85]. The dash-dotted line refers to the liquid–liquid transition [86]. The phase denoted in the literature as *α*" (*mC*4 in Pearson notation) and claimed to coexist with *α*' in part of its *P–T* field was not confirmed in the recent experimental study [87]. The stability of the *α*' and *α*" structures is a matter of ongoing debate.

#### *4.4. Nonequilibrium Studies and Dynamic P–T Diagrams*

The advent of dynamic-compression techniques has enabled experimental studies of matter under *P–T* conditions that are unattainable in static studies, and that allow for a reinvestigation of the previous reports with a new method. In shock-compressed silicon, the observed onset of phase transitions is lower than in similar static experiments [88]. In shock-compressed antimony, a similar lowering of the transition pressure is observed and an additional structure, not existing under "static" (i.e., very slow) compression, is revealed [89]. Some authors report the "*dynamic phase diagrams*" and speculate about the "modification of the phase diagram" of studied materials. However, it has to be emphasized in the strongest possible terms that dynamic *P–T* charts correspond to nonequilibrium conditions and cannot be regarded as phase diagrams in light of the *ideal-phase-diagram* definition. A perfect illustration of this fact is the accurate determination of phase-transition pressures in bismuth in a broad compression-rate range (10<sup>−</sup>2–103 GPa/s) using a dynamic diamond-anvil cell [90]. The variation in the transition pressure of the Bi-III - Bi-V transition can actually be determined as a function of the compression rate. Similar relations can be observed in compression-rate-dependent transitional diagrams of organic compounds (e.g., L-serine) [91]. Such phenomena are therefore relatively frequent, and there is no single "*dynamic phase diagram*" for a given system, but a continuum of *P–T* diagrams, where the compression rate is an additional variable. One should also consider that, in dynamic experimental studies, many other factors can effectively tune the observed phase boundaries (such as the sample microstructure and environment (see, e.g., the differences between the runs on powder and foil bismuth, or the experiments on bismuth surrounded by neon versus those performed without a pressure-transmitting medium [90])). Finally, it must be taken into account that nonequilibrium processes can lead to the kinetic stabilization of metastable states. This refers to the definition of a *transitional P–T diagram* and the difference between *dynamic* and *transitional diagrams*, as formulated in the glossary of terms (Section 2 of this article). As the timescale is the principal and perhaps even the only basis for distinguishing these two terms, with no quantitative threshold, the criterion of discrimination is therefore purely discretionary. Hence, I am inclined to believe that only one of these terms should be officially recommended. In this regard, I would prefer to use the term *transitional* rather than *dynamic*, as virtually any variable-pressure study is de facto *dynamic*, although some experiments can appear as quasistatic.

#### **5. Conclusions and Call for Actions**

Phase diagrams are simple visual representations that communicate key data related to the state of matter under varied thermodynamic conditions. Their focus and clarity make them understandable and accessible, even at the undergraduate level [92]. However, the notion of a phase diagram needs to be better defined. After running through all these examples, it is time to go back to the opening quotation from Ogden and Richards' "The Meaning of Meaning" and reflect on the status quo. I think the community should settle the terms and find a common language to avoid further misunderstandings. For many, "phase diagram" is a buzzword that automatically springs to mind when referring to any pressure–temperature chart. Do not worry, there is no scientific police that hounds for the misuse of the *phase-diagram* definition. Let me conclude with some closing remarks, which may serve as a starting point for a further discussion. It would be perfect if this debate resulted in the reaching of a consensus on the most significant working definitions and the issuing of official recommendations by international scientific organizations, such as IUPAC, IUCr, and AIRAPT [93]. To catalyze and facilitate the exchange of ideas, in Table 1, I present some preliminary suggestions and measures to tackle the fundamental problems in the current presentation of *phase diagrams*, providing insight into issues and potential solutions.


**Table 1.** The issues associated with the presentation of phase diagrams and suggestions for improvements.

<sup>1</sup> For example, a *transitional P–T diagram* of buckminsterfullerene (C60) is not a phase diagram of carbon, and a *transitional P–T diagram* of ozone (O3) is not a phase diagram of oxygen, and so on.

**Funding:** This research received no external funding.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** Some of the thoughts included in this article were presented by the author in the 56th course of the International School of Crystallography in Erice, "Crystallography under Extreme Conditions: The Future is Bright and very Compressed", 3–11 June 2022. The author gratefully acknowledges Agnès Dewaele for the valuable comments on the manuscript and fruitful discussions. The help of Demetrio Scelta with designing the carbon dioxide diagram is greatly appreciated. The anonymous reviewer is kindly thanked for the constructive comments, which helped to improve the quality and presentation of this article.

**Conflicts of Interest:** The author declares no conflict of interest.

#### **References**


## *Article* **Chinese Colorless HPHT Synthetic Diamond Inclusion Features and Identification**

**Ying Ma 1, Zhili Qiu 2,\*, Xiaoqin Deng 2, Ting Ding 1, Huihuang Li 1, Taijin Lu 3, Zhonghua Song 3, Wenfang Zhu <sup>1</sup> and Jinlin Wu <sup>1</sup>**


**Abstract:** Chinese HPHT diamonds have improved dramatically in recent years. However, this brings a challenge in identifying type IIa colorless diamonds. In this study, eleven HPHT and three natural, colorless, gem-quality IIa diamonds were analyzed using magnified observation, Raman, PL and chemical element analysis. The results show that only HPHT samples possessed kite-like inclusions and lichenoid inclusions, as verified by their complex Raman spectra (100–750 cm−1). Through PL mapping, HPHT and natural IIa diamonds were distinguished by their growth environments, which were reflected by PL peaks at 503, 505, 575, 637, 693, 694 and 737 nm. The chemical components of HPHT IIa diamond carbide inclusions are mainly Fe, Co, Ni and Mn, but those of Natural IIa are mainly Fe and Ni. As a result, the chemical components can be used to distinguish a natural colorless IIa diamond from a synthetic diamond.

**Keywords:** HPHT diamond; inclusion; PL mapping; chemical components

#### **1. Introduction**

HPHT synthetic diamonds are fundamental for high-pressure research (Crystals 2021, 11, 1185). Thanks to advancements in experimental techniques and computer simulations, the rate of new, important discoveries has significantly increased. Chinese HPHT synthetic diamonds have improved dramatically since the first was produced in China in 1963 [1–3]. After 2016, manufacturers began to produce high-quality IIa diamonds which have been made available in the Chinese jewelry market in large quantities [2–5] (see Figure 1). Nevertheless, HPHT synthetic diamonds are often sold as natural diamonds, or natural diamond parcels are intentionally mixed with HPHT synthetics [4]. There have been reports of cases of fraud in the global diamond business [2–4]. Therefore, there is a demand for distinguishing between natural and synthetic diamonds in the jewelry market.

Luminescence is currently used to identify IIa colorless diamonds. DeBeers company developed the DiamondView fluorescence imaging microscope using an ultra-shortwave excitation of 225 nm in the 1990s to distinguish HPHT-growth diamonds from natural diamonds [6]. They have developed a variety of methods since then, including cathodoluminescence (CL) images, X-ray images, PL spectrums, IR spectrums and inclusion tests [2,3,7,8]. DiamondView fluorescence imaging, PL spectrums and IR spectrums are the most commonly used methods for identifying diamonds. However, under DiamondView fluorescence imaging, many synthetic diamonds have a similar fluorescence color or IR spectrum to natural diamonds, especially IIa diamonds. There are also difficulties distinguishing between natural and synthetic diamonds using only fluorescence imaging, PL and IR spectrums with the progress of synthetic diamond technology. Gemologists face a challenge in identifying IIa colorless diamonds.

Inclusion characteristics, which reflect the diamond's formation environment, are also necessary for identifying diamonds [3,5]. This article discusses eleven HPHT-growth IIa

**Citation:** Ma, Y.; Qiu, Z.; Deng, X.; Ding, T.; Li, H.; Lu, T.; Song, Z.; Zhu, W.; Wu, J. Chinese Colorless HPHT Synthetic Diamond Inclusion Features and Identification. *Crystals* **2022**, *12*, 1266. https://doi.org/ 10.3390/cryst12091266

Academic Editor: Giuseppe Prestopino

Received: 10 August 2022 Accepted: 30 August 2022 Published: 6 September 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

colorless diamonds from three producers (HH-HuangHe, J-HuaJing and Z-ZhongNan) and three natural ones from natural primary diamond deposits (Guizhou). With the addition of previous corresponding research results, this article reports that special inclusions, their Raman features and chemical components are useful for identifying natural diamonds and HPHT synthetic colorless diamonds.

**Figure 1.** Jewelry made from Chinese HPHT synthetic diamonds (weights ranging from 0.01 to 5.00 ct), 2019. Images courtesy of two companies (Zhengzhou Sino-Crystal Diamond Co., Ltd. Shenzhen Multi-Win Business Co., Ltd.-VCK growth Diamond, China).

#### **2. Materials and methods**

#### *2.1. Materials*

In this study, 14 samples were tested, including 11 HPHT type IIa colorless diamonds from three Chinese factories (HH, J, Z) and three natural type IIa colorless diamonds selected from the primary diamond mine in Guizhou (China), which are presented in Table 1 and Figure 2. Eleven HPHT type IIa colorless diamonds were cut and polished to expose inclusions near the surface, with the cuts oriented by the {111}, {100} and {110} planes. We also cut the three rough Guizhou diamonds with irregular shapes into plates. We cleaned all of the plates without inclusions on the diamond face in acetic acid for 1 h at 100 ◦C. The detailed inclusions and PL peak characteristics of these samples are described in Table 1.

**Figure 2.** Diamond samples.


#### **Table 1.** Characteristic of Chinese HPHT synthetic diamonds in recent years.

#### *2.2. Methods*

Photomicrographs of samples were tested at the School of Earth Sciences and Engineering (SESE), Sun Yat-sen University, using digital (VHX-5000, Keyence, Japan) and polarizing (BX51, Olympus, Japan) microscopic image technology.

A Renishaw inVia Raman micro-spectrometer was used to collect Raman and photoluminescence (PL) spectra with four laser excitations (325, 473, 532 and 785 nm) at room temperature and liquid nitrogen temperature (77 K), at 1 cm−<sup>1</sup> resolution, for which the laser power was 20 mw × 24%–20 mw × 100%.

We performed PL mapping with a Thermo Scientific DXR 2xi Raman imaging microscope equipped with an Olympus optical microscope and an EMCCD detector at Sun Yat-sen University's School of Earth Sciences and Engineering, with an accuracy of 100 nm. We conducted mapping with 455 and 532 nm laser excitation wavelengths. An Olympus <sup>10</sup> × 0.25 numerical aperture objective lens was used, 1–3 scans were conducted at 2–4 cm−<sup>1</sup> resolution and the laser power was 10 mw × 20%–10 mw × 100%. In addition, the sample was mounted on a liquid-nitrogen-cooled Linkam cold stage for analysis at 123 K. The

data were processed using Thermo Scientific's OMNIC 2xi analysis software package and baseline-corrected peak area profiles were used to produce the observed PL maps.

LA–ICP–MS analyses were performed at the Metallurgical Geology Bureau (MGB) Shandong Bureau, China. The ThermoFisher ThermoX2 laser ablation–inductively coupled plasma–mass spectrometer, coupled with 193 nm LA system was used to analyze trace elements of inclusions in nine samples employed a 10 Hz pulse rate, 30 μm spot size and ablation time of 30–45 s. The US National Institute of Standards and Technology (NIST) standard reference materials 610 and 612 were used for internal calibration. The uncertainty was 1σ.

#### **3. Results and Discussion**

#### *3.1. Inclusion Feature*

Using microphotographs, inclusions in Chinese type IIa colorless HPHT diamonds appear transparent to opaque, silver or black in color, and range in size from a few hundred to a few microns. Our samples contained rod-like, fine point-like, lichenoid (tree-like), tubular (cone-like), kite-like, pear-shaped and water-drop-shaped inclusions, as shown in Figure 3. All diamonds from the three factories displayed black point-like, rod-like and irregular clusters. Inclusions of a lichenoid (tree-like) nature were mostly found in diamonds from factories Z and HH, see Figure 3. Tubular or kite-like inclusions were only found in factory J, and these were transparent or filled with black materials, reflecting the differences in production methods, see Figure 3J-2. Rod-like, tree-like, pear-shaped and water drop inclusions were metal alloys or carbides, with minor sulfides and phosphides (see Sections 3.2 and 3.3). Some contained graphite coatings and methane jackets (in Raman). As the tubular and kite-like inclusions were too small to analyze, their chemical composition is unknown.

**Figure 3.** The inclusions in Chinese HPHT synthetic diamonds collected from factories Z, J and HH. **The first row**: Rod-like, lichenoid (tree-like) and fine-point-like inclusions from factory Z, {100}; **The second row**: Disc shape, tubular and kite-like inclusions, {111}, from factory J; **The third row**: Pear or water-drop inclusions, fine-point-like and rod-like inclusions, {111}, from factory HH. These samples were produced in 2017–2020. Photographs by Ying Ma and Xu Ye.

The tubular inclusions (Figure 3J-2) are similar to earlier HPHT diamond cone-like inclusions (the temperature was from 1290 to 1250 ◦C, the pressure was 6 Gpa) from Frumkin Institute of Physical Chemistry and Electrochemistry RAS [9]. Differing from cone-like inclusions, our samples contained opaque black materials. According to previous research, small-angle grain boundaries (GBs) and associated dislocation bundles decorated with microscopic oxide inclusions are the most plausible explanation for the cone-like defects. Black materials fill the kite-like inclusions in HPHT IIa diamonds (Figure 3J-2). Although there appeared to be networks of planar cracks in natural pink diamond [10], the networks of planar cracks in the natural pink diamonds had multiple planar due to the slip of the crystal plane. However, HPHT synthetic diamonds do not have linear slips in the same way as natural diamonds. The cause of kite-like inclusions appearing in HPHT diamonds is still unclear; however, they can help to distinguish HPHT IIa diamonds from natural diamonds.

#### *3.2. Inclusion Raman Spectroscopy*

The Raman method was used to determine inclusions in diamonds using non-destructive testing. In this study, all samples displayed approximately a 1332 cm−<sup>1</sup> diamond intrinsic Raman line. The three HPHT manufacturers differed. Factory Z samples had a wider HFWT, while J factory samples had a peak at 1400 cm−1, representing graphite D bands. The Raman line of the Guizhou sample showed graphite inclusion and contained the sample's intrinsic peak. Sample differences are not representative of the specific production quality of the three manufacturers and may be caused by sampling. We observed Raman peaks around 2912 cm−<sup>1</sup> (Figure 4), which corresponds to CH4 [11,12], see Figure 4J-5. Some HPHT samples contain cohenite inclusions, pear-shaped inclusions and were opaque, silver or black in color, with a graphite coating on the surface, as can be seen in Figures 3 and 4 for HH-20. Furthermore, the Raman spectrum contained many impurity emission peaks, including 4382 cm−<sup>1</sup> and 4382 cm−1, equal to 692 nm and 693 nm, respectively, which corresponded to Ni (see Figure 4Z-3, right side). Our HPHT sample Z-3 lichenoid inclusions (tree-like inclusions) displayed a complex Raman feature (see Figure 4Z-3, left side). For lichenoid inclusions in HPHT synthetic diamonds, the fit analysis ranged from 100 cm−<sup>1</sup> to 750 cm−<sup>1</sup> with software PeakFit V4.12 and Origin 2015, R2 > 94%. By fitting and dividing the original Raman peak position, Raman emission peaks (100–750 cm−1) of metal oxide were observed. Ni emission peaks can be seen in the lichenoid inclusion complex's Raman feature at 535–554 nm. The 547 nm (546.6 nm), 540nm (539.4 nm) and 535 nm (535.2 nm) emission peaks are related to Ni (see Figure 4Z-3, left side). More detailed electronic activities remain to be studied in the future. To the best of the authors' knowledge, Raman spectroscopy of lichenoid inclusions in HPHT IIa diamonds has yet to be found in natural diamonds.

Our research results were compared with those of previous studies to reveal the differences between type IIa natural diamonds and synthetic diamond inclusions (see Table 2). Perovskites and other mineral crystals are common inclusions in natural type IIa diamonds, but amorphous metal compounds, carbides and others can also occur. Inclusions in natural IIa diamonds and HPHT share various similarities, as shown in Table 2. There are inclusions such as metal alloys, carbides, H2 and CH4 in both natural and HPHT IIa diamonds, with some inclusions grouped in <111> chains [13–15]. Since HPHT samples do not undergo as many geological processes as natural diamonds, they do not contain inclusions associated with healed cracks. Crystal minerals appear only in natural IIa diamonds, but not in every diamond. It is important to distinguish metal carbides in natural and HPHT synthetic IIa diamonds when there are no crystalline mineral inclusions. Thus, the morphology and Raman spectroscopy of inclusions in IIa diamonds are useful indicators.

**Figure 4.** Raman spectroscopy of diamonds and inclusions in Chinese HPHT synthetic diamonds. **The red frame:** Raman Spectroscopy of inclusions in Chinese HPHT synthetic diamond. In sample Z-3, lichenoid inclusions demonstrated complex Raman and electron emission peaks; two red arrows point to the fit and division of the sample Z-3 original Raman lichenoid inclusions. The Cohenite in sample HH-20 are coated with graphite, which is often observed in our HPHT samples. Sample J-5 contains methane gas inclusions encircling metal alloys. Diamond Raman features are marked "D". **The blue frame:** Raman spectroscopy of samples.


**Table 2.** Summary of inclusions observed in type IIa colorless diamonds.

#### *3.3. PL Spectroscopy and Mapping*

For testing purposes, we directionally sectioned the synthetic samples in this study. There was a smooth plane thrown out for the sample without a crystal row in the Guizhou diamond samples. Diamond inclusions are invisible microscopic defects that can reflect the conditions under which diamonds are formed. The position of PL peaks is often used to identify diamonds. HH, J and Z factory HPHT diamonds (total eight grains) had peaks at 693, 694, 883 and 884 nm, which were attributed to an interstitial Ni<sup>+</sup> atom that was distorted [16,17]. As shown in Figure 5, sample Z-3 exhibited an emission multiplet with lines at 484 nm (2.56 eV), which was Ni-related. The negatively charged silicon-vacancy doublet (SiV−, 737 nm, 1.68 eV) was only detected for J factory in four of the samples. We did not observe some PL peaks features which were previously observed in products of ZhongNan (peaks at 494, 503, 658 nm) [18], Power (489 nm) [2] and ZhongWu (peaks at 450, 659, 670, 707 and 714 nm) [19] which were HPHT diamonds factories of China. The peaks at 406, 612, 676, 710, 741, 745 and 945 nm were only observed in the Guizhou natural type IIa diamonds. We found that the greater the number of inclusions, the more impurity defect peaks there were. For more details, see Table 1 and Figure 5.

**Figure 5.** PL Spectroscopy of typical samples (GZ samples in blue color, HPHT samples in black color).

Previous studies indicated which PL peaks are unique to natural diamonds and which are unique to synthetic diamonds [16]. However, with the improvement of HPHT type IIa synthesis technology, impurity defects are decreasing. Some only appeared at 505 and 637 peak positions, or only showed the same PL peaks (693, 694, 737, 883 and 884 nm) as natural diamonds. In this study, both the HPHT and the Guizhou IIa diamonds possessed the H3, NV<sup>0</sup> and NV<sup>−</sup> centers with peaks at 505, 575 and 637 nm, respectively. Additionally, a 737 nm peak (SiV−) was found in both the HPHT and natural samples [2,20]. Additionally, PL peaks at 693 and 694 nm were found in synthetic diamonds from IIa HPHT in AOTC [17], as well as natural ultra-deep mantle diamonds [21]. Prior studies distinguished between natural and synthetic colorless diamonds by different types (Ia, Ib, IIa) of diamonds, mineral inclusions, fluorescence and spectral characteristics caused by defects from impurities entering the crystal lattice [22]. Similar impurity defects (fluorescence and spectral characteristics) make it increasingly difficult to distinguish between colorless HPHT synthetic and natural diamonds.

We performed PL mapping at the same peak position in HPHT and natural diamonds, with the aim of distinguishing between them. We mapped each sample using a smooth plane's PL, with peak area distribution peaks at 637, 575 and 496 nm. Figure 6 demonstrates the typical samples. HPHT synthetics have an interior morphological structure quite different from that of naturally grown IIa diamonds according to PL mapping (637 nm). An HPHT IIa diamond revealed growth sector distribution, whereas natural diamonds had multiple lines and a cloud-like growth morphology. This underlying mechanism was also useful in studying the difference between HPHT-grown and Natural IIa diamonds during PL mapping (Figure 6A,C). The PL peak intensity is essential for this method; otherwise it is ineffective.

**Figure 6.** PL mapping images of samples. The baseline-corrected peak area intensity for each laser excitation wavelength. (**A**) Sample GZ-25, the peak area distribution of the 637 nm peak (532 nm excitation). (**B**) Sample GZ-42, the peak area distribution of the 575 nm peak (455 nm excitation). (**C**) Sample J-2, The peak area distribution of the 637 nm peak (532 nm excitation). (**D**) Sample GZ-42, The peak area distribution of the 496 nm peak (455 nm excitation).

#### *3.4. Chemical Composition of Inclusions*

Solvents and catalysts were added to decrease the pressure and temperature of Chinese synthetic HPHT diamonds. Among the Chinese catalysts used are the Ni-Mn-Co, Fe-Al Fe-Ni, Fe-Ni-Si, Fe, Fe-S, Fe-Ni–Co, Ni–Mn–Co–Si, and Fe–Ni–B systems that include metal -Fe, Co, Mn, Al, Ni, Pt, Ru, Rh, Pd, Ir, Os, Ta and Cr- and non-metal-N, and B [1,3]. Catalyst materials can cause inclusions in samples. Aside from catalysts, pressure transmitting media and solvents also contribute to HPHT diamond inclusions [1]. As previously found, the inclusions in HPHT diamonds have mainly solid mineral and gas inclusions including graphite, magnetite, carbide Fe3C, (FeNi)23C6, NiC, FeSi2, FeNi, FeS, SiC, native metals (e.g., F, Co, Ni, Mn); trace elements (e.g., Ca, Al, Si, S, Cr, Cu, Na, Cl, K, Na, Ti, Mg, P, B, and N) and CH4 and H2 (gases) [3,11,23,24]. There is evidence that various elements (N, B, H, O, Ni, Co, S, Ti, P, Si, Ge and Sn, etc) can enter the diamond crystal lattice. Nevertheless, Fe3C and (Fe, Ni)3C carbide inclusions were also found in type IIa super-deep diamonds from Juina Rio Soriso, Brazil; Fe–Ni alloy, Fe-alloy and pyrrhotite also occur in these diamonds. [13–15]. Metallic Fe–Ni–C–S melt systems in natural diamonds have been described previously [14].

For inclusions protruding from diamond faces, we performed a chemical analysis with LA–ICP–MS. Using four colorless typical samples, we tested their inclusions, resulting in 26 points of data. The chemical compositions are shown in the Appendix A, with major elements of Fe-Co-Ni and Co-Mn-Ni distributions (see Figure 7), and different products shown as blue, green and gray balls, respectively, in Chinese gem-quality HPHT IIa diamonds. Trace elements in inclusions in HPHT diamonds include B, Ti, Cu, Zn, Ca, S, P, Cr, etc. (see Appendix A).

A previous study demonstrated that Ni, Mn and Co are the predominant elements in Chinese small abrasive diamond inclusion samples (50–250 m) [24]. The Ni content in inclusions of type Ib small abrasive diamonds has obvious advantages. Iron-carbide inclusions in deep-mantle IIa natural diamonds mainly contain Fe elements, as well as small amounts of Ni and lack Co and Mn [15]. The elements B, Ti and Cu are found in colorless HPHT diamonds at a higher quantity than in yellow diamonds. In recent years, Fe-Co and Fe-Co-Ni systems have been mainly used in Chinese colorless HPHT diamonds; B, Ti and Cu may be nitrogen collectors in HPHT diamonds. We compared the composition of the carbide inclusions in natural diamonds and synthetic diamonds. The major difference between natural and synthetic diamonds is the absence of Co and Mn elements in the natural carbide inclusions. Carbide inclusions in natural diamonds were rare and little information was available about them. Ideally, more trace elements from

natural IIa diamond carbide inclusions should be examined in the future to compare the differences between HPHT diamonds and natural IIa diamonds.

**Figure 7.** (**a**) Fe–Co–Ni, (**b**) Co–Mn–Ni elements plots in inclusions of diamonds, data did normalization operation. Sample numbers are HH-5, HH-20, Z-1, Z-3, J-3 in this study; SG1 are yellow abrasive diamonds, data from Litasov et al. [24]; Julia-iron carbide is a natural super deep diamond, data from Kaminsky et al. [15]; SYN- Synthetic diamond, NAT- Natural diamond; colorless HPHT diamonds characteristics in the blue circle.

#### **4. Conclusions**

This study indicates that it is difficult to identify colorless type IIa diamonds, but some details can provide support. The main conclusions are as follows:


Thus, the morphology of inclusions and complex Raman spectroscopy of lichenoid (tree-like) inclusion, PL perks, particular peak position PL mapping and the chemical composition of the iron-carbide inclusion can help to distinguish natural diamonds from HPHT IIa colorless diamonds. These tests provided more information than can be determined from a DiamondView image, IR and PL alone. However, for diamonds with no characteristic inclusions and no PL peaks, we suggest that other images such as CL images and X-ray images (XRT) are needed for further analysis.

**Author Contributions:** Conceptualization, Y.M. and Z.Q.; data curation, Z.S. and T.L.; formal analysis, Y.M.; funding acquisition, X.D. and Z.Q.; investigation, Y.M., T.D., Z.S. and H.L.; methodology, Y.M. and Z.Q.; software, Y.M.; supervision, T.D., H.L. and Z.S.; validation, Y.M., J.W. and W.Z.; visualization, Y.M.; writing—original draft, Y.M.; writing—review and editing, Y.M. and Z.Q. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by NSFC (Nos. 41473030 and 42073008).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** Thanks to General Manager Weizhang Liang of the Guangzhou Diamond Exchange; and Jianguo Liu, Jingru Shao and Dufu Wang for arranging our factory visits during 2017– 2020. The authors also thank anonymous contributors for generously donating diamond samples, and the analysts at the Shandong Bureau of China Metallurgical Geology Bureau. We also would like to thank Thiering Gerg and J.W. Harris for their assistance during the writing process.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Appendix A**


**Table A1.** Trace element compositions of the inclusions in HPHT diamonds.

#### **References**


## *Article* **Understanding the Semiconducting-to-Metallic Transition in the CF2Si Monolayer under Shear Tensile Strain**

**Tarik Ouahrani 1,2,***<sup>∗</sup>* **and Reda M. Boufatah <sup>1</sup>**


**Abstract:** With the ever-increasing interest in low-dimensional materials, it is urgent to understand the effect of strain on these kinds of structures. In this study, taking the CF2Si monolayer as an example, a computational study was carried out to investigate the effect of tensile shear strain on this compound. The structure was dynamically and thermodynamically stable under ambient conditions. By applying tensile shear, the structure showed a strain-driven transition from a semiconducting to a metallic behavior. This electronic transition's nature was studied by means of the electron localization function index and an analysis of the noncovalent interactions. The result showed that the elongation of covalent bonds was not responsible for this metallization but rather noncovalent interactions governing the nonbonded bonds of the structure. This strain-tuned behavior might be capable of developing new devices with multiple properties involving the change in the nature of chemical bonding in low-dimensional structures.

**Keywords:** ab initio calculations; tensile strain; electronic transition; topological analysis of bonds

#### **1. Introduction**

Strain is ubiquitous in solid-state materials; it can be caused by the melt growth of the material [1], due to the quantum effect driven by the structure [2], or simply by an external application. One fundamental transformation induced by this strain is the change in bonding and structural pattern [3]. Strain applied to a versatile low-dimensional material and also a bulk counterpart could combine several intriguing mechanisms that are still in their infancy. For example, we could obtain the conjunction of metallic (a compound with a vanishing electronic band gap) and semiconducting regions on a monolithic catalytic MX2 nanosheet that could be used to make electronic devices [4,5]. Furthermore, as is the way with pressure, the application of external strains on such structures can become radically different from what they are at ambient pressure, as shown in recent examples with the giant piezoelectricity induced by the mean of pressure in monolayer tellurene [6] and the experimental study showing evidence of an electronic phase transition in molybdenum disulfide [7].

Two-dimensional silicon carbide, a famous member of the 2D family, has been studied extensively owing to its distinctive electronic, optical, and mechanical properties. This quality pushed an increase in the synthesis of new structures based on this material. For instance, graphitic SiC [8], planar SiC2 silagraphene with tetracoordinate Si [9], planar graphitic SiC2 [10], carbon-rich SiC3 [11], g-SiC2 [10], pt-SiC2 [9], SiC6-SW, SiC2-b, SiC2-p [12], quasi-planar tetragonal SiC and SiC2 [13], penta-SiC2 [14], a series of silagraphyne [13], silicon-rich Si3C [12], and the recently reported tetrahex SiC [15] have been experimentally synthesized. All these structures show covalent bonding with strong C–Si bonds. In this contribution, we are aiming to analyze the effect of strain on a buckled geometry, namely the CF2Si structure. Its network also has fluorine atoms on its surface, which functionalizes the C–F bonds and improves structural integrity, surface activity,

**Citation:** Ouahrani, T.; Boufatah, R.M. Understanding the Semiconducting-to-Metallic Transition in the CF2Si Monolayer under Shear Tensile Strain. *Crystals* **2022**, *12*, 1476. https://doi.org/ 10.3390/cryst12101476

Academic Editor: Dmitri Donetski

Received: 17 September 2022 Accepted: 16 October 2022 Published: 18 October 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

and processability, opening up new opportunities for catalytic applications. Such a functionalization not only inherits the C–F bonds but also brings about a promise to alter the planar SiC structure, transforming the hybridization from sp<sup>2</sup> to sp3, and enhancing dispersion, orientation, interaction, and electronic properties [16].

An in-depth understanding of the relationship between chemical structure and macroscopic behavior holds the key to rationalizing the design of new synthetic routes addressing a certain property [17–19]. One of the best possible strategies to unravel these transformations is to look at the bonding patterns, especially in the electronic population [20,21]. Such a link can be analyzed by the so-called localization index as the localization tensor (LT), the localized-electrons detector (LED) [22,23], and the electron localization function (ELF) [24]. With this in mind, in this paper, we look at how the bonding pattern changes from the ambient condition to the highly strained structure. The main goal is to see if the buckled structure of CF2Si behaves like its planar or bulk SiC counterpart when subjected to no hydrostatic strain/stress deformations. In this manner, the band gap of the title structure can be effectively manipulated through various strategies, including mechanical strain application. The task requires, however, access to the more stable ambient structure. For this purpose, a brief analysis of dynamical, thermal, and thermodynamic stability is done by the density functional theory method. The effect of tensile strain is also analyzed within this strategy. The result gives us insight into whether the modulation vanishing of the electronic band gap is caused by the change in the structure or also a result of the change in the bonding trend.

#### **2. Computational Details**

The study of the bonding pattern of ground or strained material required first an analysis of its stability. This task was carried out by means of the density functional theory (DFT), as implemented in the Vienna ab initio simulation package (VASP) [20,25,26]. The projector augmented-wave (PAW) [27] method was employed. Furthermore, the kinetic energy cutoff for the plane-wave basis was converged at 500 eV. We used the exchange potential of Perdew–Burke–Ernzerhof (PBE) [28] weighted by the DFT-D3/BJ approach of the Becke–Jonson damping [29,30], which simulates the van der Waals (vdW) interactions between the adjacent layers. Here, we used a 2 × 2 × 1 supercell with a 20 Å vacuum along the **c**-direction. A Monkhorst–Pack grid of 12 × 12 × 1 k-points was considered. Structural optimizations were deemed to be converged when all the forces reached their convergence below F*tol* = 10−<sup>6</sup> eV/Å. The mechanical properties were estimated within the stress–strain methodology carried through the VASPKIT toolkit [31].

For the calculations of the phonon dispersion of a system, we utilized the direct method as implemented in VASP and interfaced in the Phonopy open-source package [32]. The phonon frequencies were investigated in a 3 × 3 × 1 supercell within a self-consistent way by alternating between the DFT calculations of the Hellmann–Feynman forces acting on atoms displaced from their equilibrium positions and the calculations of improved phonon frequencies and atomic displacement vectors. To achieve calculation convergence, we selected displacements of 0.05 Åas appropriate values. All the topological analyses were carried out with the TOPCHEM2 package [33] using very fine grids of size 300 × 300 × 100.

#### **3. Results and Discussion**

#### *3.1. Optimized Structure and Its Stability*

The structure under investigation has a space group P3m1 with a trigonal shape (T1), see Figure 1, which means that the lattice parameters *a* and *b* are equal. As a result, the optimized lattice parameter was limited to the *a* parameter, which was calculated to be 3.16 Å. Unfortunately, no experimental data were available to compare with it. As a result, it was necessary to examine the structure's stability as well as its growth conditions. In terms of thermodynamic stability, the chemical potentials of the constituent atoms of

the CF2Si low-dimensional structure obeyed several restrictions based on the energetic equilibrium as follows:

$$\begin{array}{ll} \Delta\mu\_{\text{C}} + 2\Delta\mu\_{F} + \Delta\mu\_{Si} = \Delta E(\text{CF}\_{2}\text{Si}) = -1.44\,\text{eV} \\ 2\Delta\mu\_{Si} \le \Delta E(\text{Si}\_{2}) = 0.64\,\text{eV} \\ 2\Delta\mu\_{\text{C}} \le \Delta E(\text{C}\_{2}) = -0.01\,\text{eV} \\ 2\Delta\mu\_{\text{C}} + 2\Delta\mu\_{F} \le \Delta E(\text{C}\_{2}\text{F}\_{2}) = -0.91\,\text{eV} \\ \Delta\mu\_{\text{C}} + 2\Delta\mu\_{F} \le \Delta E(\text{CF}\_{2}) = -1.39\,\text{eV} \\ 2\Delta\mu\_{F} + 2\Delta\mu\_{Si} \le \Delta E(\text{F}\_{2}\text{Si}\_{2}) = -1.77\,\text{eV} \end{array} \tag{1}$$

where Δ*μC*, Δ*μF*, Δ*μSi*, Δ*E*(*Si*2), Δ*E*(*C*2), Δ*E*(*CF*2), Δ*E*(*F*2*Si*2), and Δ*E*(*C*2*F*2) are the chemical potentials for the computing elements, and Δ*E*(*CF*2*Si*) corresponds to the investigated low-dimensional structure

We present the resulting limiting conditions and the intersection points bounding the stability region in Table 1. Figure 2 shows its corresponding diagram. According to this diagram, the title structure competes mainly with the F2Si2 and C2F2 monolayers. We can stipulate this conclusion by taking a number of equations into consideration, considering the constraints to building this structure.

**Figure 1.** (**a**) The top and (**b**) the side views of CF2Si in its low-dimensional structure.

**Figure 2.** Chemical potential's phase diagram, showing the region of stability of CF2Si in terms of the excess chemical potentials. (Color online) Variation in Δ*μ<sup>F</sup>* formation enthalpy as a function of chemical potential, shown within the stability region for the formation of low dimensional CF2Si. Points A, B, C, and D are shown only in order to delimit the zone of stability.


**Table 1.** The limiting inequalities applied to the independent variables Δ*μ<sup>C</sup>* and Δ*μSi*. The point column lists the intersection points bonding the stability region. The corresponding values of the dependent variable Δ*μ<sup>F</sup>* and the relevant competing phases are also given. All energies are in eV.

According to all of the above equations and the diagram in Figure 2, the CF2Si growth is preferred for intersection points bounding along points A (Si-poor, C-rich, F-rich) where the CF2Si competes. We also analyzed the dynamical stability of the low-dimensional structure in the title. This task was done via the calculation of the phonon dispersion spectrum displayed in Figure 3a. Due to the absence of an imaginary frequency, we can clearly see that the structure is mechanically stable. The shape of the plot seems to share the general scheme of 2D chalcogenide structures. We can note some anomalies in the form of longitudinal acoustic branches, the signature of weak interplanar interactions. According to [34], these acoustic modes have a linear dependence in the **q**-space near the Γ point. The modes at the center of the Γ point, according to the crystal point group, obey the formula Γ = 4*A*<sup>1</sup> + 4*E*, where Γ*acoustic* = *A*<sup>1</sup> + *E* and Γ*optic* = 3*A*<sup>1</sup> + 3*E*. The *E* mode denotes the double degenerated one. The analysis of the activity of each mode shows that they are both Raman and infrared ones. The calculated wavenumbers for each mode are gathered in Table 2.


**Table 2.** Calculated modes of 2D CF2Si. Raman (R) and infrared (IR) modes are indicated.

At the same time, the phonon density of state gathered in Figure 3 shows that the acoustic branches were mainly formed from the lighter atoms, namely the F one, and the branches at high frequency originated from the carbon one. To assess more information on the mechanical properties of the studied structure, we also evaluated the elastic constants of CF2Si. The results are shown in Table 3. The mechanical stability of the 2D CF2Si monolayer can be examined by using the elastic constants C*ij*. For symmetry reasons, we had only two elastic stiffness components, C11 and C12, the C66 = (C11 − C12)/2. Because CF2Si is bidimensional, the Born criteria of mechanical stability should satisfy the conditions C11 > 0 and C66 > 0 [35–37]. According to the calculated results of C*ij*, the mechanical stability was satisfied, implying that the investigated structure would eventually be mechanically stable in its low-dimensional structure.

**Table 3.** Calculated lattice parameters, elastic components, and elastic modulus of CF2Si monolayer.


According to the results of this table, and due to its buckled nature, the compound was rather rigid and had comparable properties to its SiC counterpart in the 2D honeycomb structure [21]. The application of strain on the 2D-SiC honeycomb structure showed a stable structure until 17% of compression. CF2Si in fact manifested comparable dynamical properties as TH-SiC2 and TH-SiC structures. C11 (119.7 N/m) and C12 (26.5 N/m) had values that were close to TH-SiC2 ((T) and (H) stand, respectively, for tetragonal and hexagonal structures). The main small difference was mainly due to the F atom bonded in the out-plane direction of the Si–C bonds. As a result, the structure was less covalent than a TH-SiC2, TH-SiC, or h-SiC monolayer and then more ductile along the shear direction. Furthermore, the shear modulus of CF2Si was 17% lower than that of TH-SiC2 [38]. This allowed a precise application of strain along the shear direction.

**Figure 3.** (**a**) Phonon dispersion plot and its corresponding (**b**) phonon density of states of CF2Si monolayer in cm<sup>−</sup>1.

#### *3.2. Low-Dimensional CF*2*Si Structure under Shear Tensile Strain*

Starting with the fully relaxed 2D crystal structure CF2Si, we simulated the shear tensile strain by decreasing the value of the *γ* angle from 120◦ to 98◦ (see Figure 4a,b). The tensile strain was defined as *ε* = (*θ* − *θ*0)/*θ*<sup>0</sup> × 100%, where *θ* and *θ*<sup>0</sup> are the lattice constants of the strained and relaxed structure, respectively. After the application of such a strain, the structure was fully relaxed, keeping the volume and lattice parameters constant, but allowing the relaxation of the internal atomic coordinates. The shear strain was applied step by step until *ε* reached a value of 98◦ of the initial *γ* = 120◦ angle.

To gain a better understanding of the structure's trend under shear deformation, we plotted the evolution of the C–Si bonds as a function of *θ* in three directions in Figure 4b. The linear compressibility of the d1 and d2 axial bonds was correlated. They gave a value of *<sup>κ</sup>* = 1.6 × <sup>10</sup>−<sup>3</sup> GPa−1, where the lateral direction d3 had a linear compressibility of *<sup>κ</sup>* = 3.7 × <sup>10</sup>−<sup>3</sup> GPa−1. According to this result, we believe that the enlargement of the d2 C–Si bonds under shear compression was responsible for the increase of most highfrequency modes and was a consequence of the change in the band-gap nature. Qualitatively, under the harmonic approximation, the stretching mode was proportional to frequency *ω*−2/3 [39]. Thus, according to our mode assignment, the strongest Raman active mode related to the carbon atoms, and located at a wavenumber of 970.1 cm−1, increased to 1354.86 cm−<sup>1</sup> under a shear deformation of 18%. This result implied that the structure was rather compressible under shear deformation.

**Figure 4.** (**a**) Schematic representation of the pristine and strained (deformed) structure of CF2Si monolayer. (**b**) Calculated strain dependence of C–Si bond distances. d1, d2 represent the axial bonds and d3 the lateral bonds.

In a subsequent step, we analyzed the effect of an imposed strain on the electronic band gap of the investigated structure. The result is gathered in Figure 5. The calculation of the band gap of the unstrained structure within the Heyd–Scuseria–Ernzerhof hybrid density functional (HSE06) [40] gave a value of 3.31 eV, whereas the PBE gave a band gap of 1.94 eV. The plot shows that with the increase of tensile strain, the band gap decreased and vanished for a strain of 18% on the structure. The new Si–C–Si reached a value of 98◦, and this behavior (semiconductor-to-metal transition) was also shown in the application of a zigzag strain on the 2D–SiC [21]. As a result, we suspect a corresponding electronic transformation on the CF2Si structure.

**Figure 5.** Evolution of the band gap as a function of gamma angle. The inset shows the 3D-ELF isosurface of the structure at (**a**) (unstrained) 0% and (**b**) (strained) 18% of the tensile strain.

In order to understand this electronic transformation, we analyzed the nature of both Si–C and C–F bonds for strained and pristine structures. For this task, we performed the electron localization function (ELF) designed by Becke and Edgecombe [24] according to the method proposed by Savin et al. [41]. The ELF topology gives a partition into localized electronic domains known as basins. They are used to rationalize the bonding schemes. The synaptic order of a valence ELF basin is determined by the number of core basins with which they share a common boundary. The basins' spatial locations are very close to the valence-shell electron-pair repulsion domains [42]. According to Savin et al. [41], in the DFT framework, the ELF can be understood as a local measure of the excess of local kinetic energy of electrons, *tp*(*r*). This quantity is computed by subtracting the bosonic contribution, |∇*ρ*(*r*)| 2/8, from the kinetic energy density of the system, *t*(*r*). A rescaling of it with respect to the homogeneous electrons gas provides the core of the ELF, *χ*(*r*):

$$\chi(\vec{r}) = \frac{t\_p(\vec{r})}{t\_w(\vec{r})} = \frac{t(\vec{r}) - \frac{1}{8}\frac{|\nabla\rho(\vec{r})|^2}{\rho}}{c\_F \rho^{5/3}(\vec{r})} \tag{2}$$

where *cF* is the Fermi constant. Accordingly, the regions of electron pairing would have a small *χ* value. In order to inverse this relationship and map it in a closed interval, the final function was defined as follows:

$$\eta(\vec{r}) = \frac{1}{1 + \chi^2(\vec{r})} \tag{3}$$

The ELF partition shown in the inset of Figure 5 depicts two cases: the top for pristine structure, which depicts three disynaptic basins ELF–V(Si, C) built at isosurface = 0.95.

These basins had similar shapes and populations (see Table 4), while both basins V(Si,C) and V(C,Si) had *η*(*r*) = 0.96 and their population was approximately equal to two electrons, indicating their covalent nature. Table 4 also indicates the existence of a monosynaptic basin V(F) with an ionic nature (*η*(*r*) = 0.79), plus a small disynaptic V(C, F) basin (*η*(*r*) = 0.85), with a weak electronic population. This behavior was a result of the strong polarization between the in-plane Si–C and C–F bonds. While the charge density (*ρ*) in the covalent bonds was weak, the ionic one in V(F) was strong. On the other hand, the strained structure showed deformed electronic basins. In the strained structure, the covalent disynaptic V(C, Si) basins also appeared, as well as new monosynaptic ones, for instance, a V(Si) basin with a small *η*(*r*) = 0.5, as well as three polarized ionic V(F) basins, where we could show an increase in the electron population. We could also show an increase in the metallicity of the bonds and an extension of the charge density. The asymmetry in the bond population and the electronic volume of the emerged basins were attributed to the strong distortion of the strained structure.

A deeper analysis can untangle this behavior with both the 1D ELF profile and the NCI index. In Figure 6a,b, we displayed the ELF profile of Si–C–F, of unstrained and strained structures in black, the deformed Si–C–F bond of a strained structure in red, and also the lateral direction of Si–C–F of both structures in blue. The 1D ELF profile confirmed that the Si–C bond was covalent and C–F ionic, but it also confirmed the existence of a metallic plateau in the lateral direction of Si–C–F showing a region of very low ELF values with a small hill around *η*(*r*) = 0.5. The same trend was noticed whilst studying the bulk SiC under strain [20]. In fact, in the application of tensile strain, the Si–C bond length built from the unit cell of the CF2Si structure became different. This distortion enhanced the noncovalent interactions in the nonbonded Si and C direction. Given the relevance of this result, we used the same tool as in [20]. The noncovalent interactions' (NCI) isosurface is presented in Figure 6c,d. The noncovalent domain seemed to increase from the unstrained to the strained structure in the nonbonded Si–C–F direction. This region, in fact, did not play any meaningful role in the cohesion of low-dimensional CF2Si, but rather in the stability and the enhancement of the polarization of bonds. This could be shown in the region of blue surrounding the distorted Si–C bonds of the strained structure, which explained the increase in ionicity and metallicity.


**Table 4.** Electron localization analysis of a basin of bonding of the CF2Si monolayer. V(Ω), *q*, bond metallicity (*ρ*/∇(*ρ*)), and *ρ* stand, respectively, for basin volumes of disynaptic V(X,Y) and monosynaptic V(X) bonds, electronic charges of the bond, calculated bond metallicity and electronic density charge (*ρ*), and ELF (3, −3) basin.

**Figure 6. Left**, one-dimensional ELF profile of bonds for (**a**) pristine and (**b**) strained CF2Si monolayer. The black, blue, and red correspond, respectively, to Si–C–F, deformed Si–C–Si for strained structure, and Si–C–F lateral direction. Right, the noncovalent interactions index NCI = 0.3 isosurface of (**c**) pristine and (**d**) strained structures.

Finally, to investigate the relationship between conductivity and shear effect on the low-dimensional CF2Si structure, we used the semiclassical Boltzmann transport theory with a fixed relaxation time approximation to predict the electrical conductivity (*σ*). This task was done by the use of the BoltzTraP code [43]. We note that, as the procedure was related to the band structure calculation, we took care to increase the grid of the calculation by the use of 37 × 37 × 1 k-points. Figure 7 gives the *σ*/*τ* (*τ* being the relaxation time) at 300 K as a function of the carrier concentration for both pristine (0%) and strained (18%) structures. We can see that the tensile shear strain increased in both the **p**- and **n**-type doped systems. In fact, the increase in *σ* coincided with that of the band gap. Therefore, the electrical conductivity was efficiently tuned by the strain via band structure engineering. It was noted that the p-type doped region was more affected due to the closing of the band gap.

**Figure 7.** Calculated electrical conductivity of pristine (0%) and strained (18%) structures at ambient temperature. The electrical conductivity was calculated with respect to the relaxation time *τ*. The transporting directions are not distinguished here.

#### **4. Summary**

We showed in this contribution that when the CF2Si low-dimensional structure was submitted to the application of tensile strain along the shear direction, an electronic transition occurred from the semiconducting phase with a wide electronic band gap to a metallic one with a vanishing band gap. A first attempt to understand this transformation was made within the density functional method calculation, by calculating the stability and mechanical properties of the investigated structure. Furthermore, to understand this mechanism, we used the inherent topological bonding. This task allowed us to understand that this transition was a response to the new repartition of the electronic population and an enhancement of weak noncovalent interactions along the nonbonded directions of the unit cell. Overall, such interactions could be responsible for the relative stability of the structure and also play an important role in controlling the nature of the electronic wave function describing the computed structure. The increase of electric conductivity as a function of strain stipulated the poor performance of the CF2Si low-dimensional structure as a thermoelectric system.

**Author Contributions:** Conceptualization, methodology, validation, formal analysis, investigation, writing—original draft preparation, writing—review and editing, supervision, T.O.; DFT calculation, review and editing, R.M.B. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Data are available upon request from the corresponding author.

**Acknowledgments:** T.O. acknowledges the financial support given by the PRFU, no. B00L02EP13022-0230001.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


## *Article* **Effects of Physical and Chemical Pressure on Charge Density Wave Transitions in LaAg1***−x***Au***x***Sb2 Single Crystals**

**Li Xiang 1,†, Dominic H. Ryan 1,2, Paul C. Canfield <sup>1</sup> and Sergey L. Bud'ko 1,\***


**Abstract:** The structural characterization and electrical transport measurements at ambient and applied pressures of the compounds of the LaAg1−*x*Au*x*Sb2 family are presented. Up to two charge density wave (CDW) transitions could be detected upon cooling from room temperature and an equivalence of the effects of chemical and physical pressure on the CDW ordering temperatures was observed with the unit cell volume being a salient structural parameter. As such LaAg1−*x*Au*x*Sb2 is a rare example of a non-cubic system that exhibits good agreement between the effects of applied, physical, pressure and changes in unit cell volume from steric changes induced by isovalent substitution. Additionally, for LaAg0.54Au0.46Sb2 anomalies in low temperature electrical transport were observed in the pressure range where the lower charge density wave is completely suppressed.

**Keywords:** charge density wave; pressure; chemical substitution; resistivity; LaAgSb2; LaAuSb2

#### **1. Introduction**

Physical (hydrostatic) pressure and chemical substitution are two common ways to tune the physical properties of materials. Whereas hydrostatic pressure is considered to be a clean parameter that does not introduce additional disorder, as well as changes in band filling in many cases, the experimental techniques available under pressure are more limited. Chemical substitution necessarily involves some additional disorder. In the case of aliovalent substitutions the corresponding electron- or hole- doping effects are often dominant. For isovalent substitution the primary effect is thought to be steric and the comparison with physical pressure can be more relevant. Whereas such isovalent substitutions can be referred to as "chemical pressure" differences in how pressure and substitution affect a compound, especially a non-cubic one, can be greater than similarities in some cases. It is of particular importance when there is a desire to stabilize a particular high pressure phase/state (like high temperature superconductivity [1]) using chemical and/or physical pressure. Additionally, observation of an apparent equivalence [2–4] or non-equivalence [5] of chemical and physical pressure can help in understanding of structure—property relations and in recognizing relevant structural motifs.

The members of the family of compounds chosen for this study, LaAg1−*x*Au*x*Sb2, demonstrate charge density wave (CDW) transitions, or spontaneous superstructures formed by electrons [6]. Decades ago Peierls showed the instability of a (one-dimensional) metal interacting with the lattice towards a lattice distortion and the opening of a gap in the electronic spectrum [7]. This concept is often applied to CDW formation in low-dimensional materials, although alternatives are widely discussed [8–10]. Studies of CDW phenomena in solids and competition of CDW with other collective phenomena remain one of the active subfields of quantum materials research.

**Citation:** Xiang, L.; Ryan, D.H.; Canfield, P.C.; Bud'ko, S.L. Effects of Physical and Chemical Pressure on Charge Density Wave Transitions in LaAg1−*x*Au*x*Sb2 Single Crystals. *Crystals* **2022**, *12*, 1693. https://doi.org/10.3390/ cryst12121693

Academic Editors: Daniel Errandonea and Enrico Bandiello

Received: 1 November 2022 Accepted: 21 November 2022 Published: 23 November 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

In this work, we study effects of pressure on single crystals of selected members of the LaAg1−*x*Au*x*Sb2 family which are then compared to those of chemical pressure implemented via Ag ↔ Au substitution. The end-compounds, LaAgSb2 and LaAuSb2 were first synthesized almost three decades ago [11,12] and were reported to crystallize in the same tetragonal ZrCuSi2-type structure (*P*4/*nmm*, No. 129). CDW transitions were observed in electrical transport at ∼210 K and ∼100 K for LaAgSb2 and LaAuSb2, respectively, refs. [13,14]. Synchrotron X-ray scattering study [15] and further thermodynamic measurements [16] identified second, lower temperature, CDW transition at ∼185 K in LaAgSb2. Similarly, in addition to *TCDW*<sup>1</sup> ≈ 110 K, a second CDW transition at *TCDW*<sup>2</sup> ∼90 K was detected by electrical transport measurements in near stoichiometric LaAuSb2 [17]. The synthesis and evolution of the higher temperature, *TCDW*<sup>1</sup> in the LaAg1−*x*Au*x*Sb2 series was reported in Ref. [18] but without any measurements of *TCDW*<sup>2</sup> or companion applied pressure studies.

For LaAgSb2, pressure reportedly suppressed *TCDW*<sup>1</sup> [19–23] as well as *TCDW*<sup>2</sup> [21,23] with the results being consistent in majority of publications [19–21]. Qualitatively similar behavior under pressure was also observed for LaAuSb2 [17,24,25]. Additionally, in LaAgSb2 and LaAuSb2, low temperature superconductivity was discovered and studied under pressure [23,24]. It has to be noted that the exact Au stoichiometry in LaAu*x*Sb2 depends on details of the synthesis and affects both the ambient pressure values of *TCDW*<sup>1</sup> and *TCDW*<sup>2</sup> and their pressure derivatives [17].

Earlier comparison of the effects of pressure and chemical substitution in this family of materials [19] was based on a study of the La1−*xRx*AgSb2 series (*R* = Y, Ce, Nd, Gd) and a significant contribution of disorder prevailing in the case of substitution was found, thus resulting in a significant difference between physical and chemical pressure. In this work, we address the same question in the different, transition metal site, substitution series.

#### **2. Materials and Methods**

Single crystals of LaAg1−*x*Au*x*Sb2 were grown from an antimony-rich self-flux following the method described in Refs. [13,17,26]. Pure elements were loaded into an alumina Canfield crucible set [27] which was placed into an amorphous silica tube and sealed in partial atmosphere of argon. The sealed tubes were heated to 1050 ◦C over 10 h, held for 8 h, then cooled to 800 ◦C over a period of 10 h prior to starting the crystal growth. Crystal growth occurred during the 100 h cooling from 800 ◦C to 670 ◦C, after which the excess flux was decanted with the aid of a centrifuge.

In this work, crystals of LaAg1−*x*Au*x*Sb2 with nominal compositions *x* = 0, 0.25, 0.5, 0.75 were grown with the initial La:T:Sb (T = Ag1−*x*Au*x*) growth compositions: 1:2:20 (T2). To investigate whether reported Au deficiency [14,17,18] is relevant and can be tuned for the intermediate Au concentrations in LaAg1−*x*Au*x*Sb2, for *x* = 0.25, 0.75 the growth composition of 1:6:20 (T6) was used as well. For the end compound, LaAu*x*Sb2, the data from the recent Ref. [17] are used when appropriate.

Cu-K*α* X-ray diffraction patterns were taken using a Rigaku Miniflex-II diffractometer. The crystals were ground and the powder was mounted on a low-background single-crystal silicon plate using a trace amount of Dow Corning silicone vacuum grease. The mount was spun during data collection to reduce possible effects of texture. Data taken for Rietveld refinement were collected in two overlapping blocks: 10◦ ≤ 2*θ* ≤ 48◦ and 38◦ ≤ 2*θ* ≤ 100◦, with the second block counted for 4–5 times longer than the first to compensate for the loss of scattered intensity at higher angles due to the X-ray form factors. The two data blocks for each sample were co-refined within GSAS [28,29] using a single set of structural and instrumental parameters but with independent scale factors to allow for the different counting times used. Parameters for both the primary phase and any impurity were refined. We found that the materials were easy to grind into a random powder and no texture or preferential orientation effects were observed in the residuals. The diffractometer and analysis procedures were checked using Al2O3 (SRM 676a [30]); our fitted values of *a* = 4.7586(2) Å and *<sup>c</sup>* = 12.9903(7) Å were both 1.6(4) × <sup>10</sup>−<sup>4</sup> Å smaller than the values on

the certificate [30], suggesting a small but statistically significant mis-calibration of the instrument. The fitted lattice parameters given in the analysis that follows do not include this correction.

Chemical analysis of the crystals was performed using an Oxford Instruments energydispersive X-ray spectroscopy (EDS) system on a Thermo Scientific Teneo scanning electron microscope. The measurements were performed on polished *ab* surfaces of single crystals with four to eight points taken for every sample.

Standard, linear 4-probe ac resistivity was measured on bar—shaped samples of LaAg1−*x*Au*x*Sb2 in two arrangements: *I*||*ab* and, when needed, *I*||*c*. The size of the samples was 1.5–2 mm length, 0.2–0.4 mm width and about 0.1 mm thickness. The frequency used was 17 Hz, typical current values were 3 mA for in-plane electrical transport and 5 mA for the *c*-axis measurements. The contact resistances between the leads and the samples were below 1 Ω. Based on our experience with the LaAu*x*Sb2 samples with similar size and contact resistance [17], we do not expect heating effects to be observed either at ambient pressure or in the pressure cell environment. The measurements were performed using the ACT option of a Quantum Design Physical Property Measurement System (PPMS).

For selected samples, resistivity measurements under pressure were performed in a hybrid, BeCu/NiCrAl piston-cylinder pressure cell (modified version of the one used in Ref. [31]) in the temperature environment provided by a PPMS instrument. A 40:60 mixture of light mineral oil and n-pentane was used as a pressure-transmitting medium. This medium solidifies at room temperature in the pressure range of 30–40 kbar, [4,31,32] which is above the maximum pressure in this work. Elemental Pb was used as a low temperature pressure gauge [33]. It has been shown that in piston-cylinder pressure cells the value of pressure depends on temperature (see Ref. [34] for mineral oil:n-pentane pressure medium and this particular design of the cell). Below we use the Pb gauge pressure value. Given that the upper transition for LaAgSb2, highest in the series, is at ambient pressure at ∼200 K, this may give rise to pressure differences with the values determined by Pb gauge by at most 2 kbar.

#### **3. Results**

#### *3.1. Structure and Substitution*

The X-ray diffraction patterns for all LaAg1−*x*Au*x*Sb2 samples were fitted using the GSAS/EXPGUI packages [28,29]. Small amounts of residual flux were generally observed as impurity phases and were included in the fits as necessary. Figure 1 shows a typical X-ray diffraction data set for the T2 growth of LaAg0.75Au0.25Sb2 with ∼1 wt.% Sb as impuritiy. In the fit, the occupations of the La, Sb1, and Sb2 sites as well as the total occupation of the T = Ag/Au 2*b* site were fixed as 1, whereas the Ag/Au ratio was allowed to vary. As the parameter that actually contributes to the scattering from a given site in the structure is the average scattering length for that site, it was not meaningful to refine both the Au/Ag ratio and a possible vacancy level using a single measurement (our X-ray diffraction patterns) of the average scattering length. The same (reduced) average scattering could be constructed from Au-only + some level of vacancy, a fully occupied site with Au + some Ag, or some appropriate, and continuously variable combination of Au + Ag + vacancy. The results from Rietveld analysis of the powder X-ray data for LaAgSb2 and five LaAg1−*x*Au*x*Sb2 samples are listed in Tables A1 and A2 in the Appendix A. The EDS results for LaAgSb2 and four LaAg1−*x*Au*x*Sb2 samples are presented in Table A3 in the Appendix A. The values in the table are the average of the measurements taken at between four and eight different places on the samples' surfaces, standard deviations are listed in the parentheses.

**Figure 1.** Cu-K*<sup>α</sup>* X-ray diffraction patterns for the T2 growth of LaAg0.75Au0.25Sb2 showing the two overlapping data blocks that were co-fitted using the GSAS/EXPGUI packages [28,29]. The red points are the data and the green lines show the fits with the residuals shown below each fitted pattern. The Bragg markers show the positions of the reflections from (top) Sb, and (bottom) LaAg1−*x*Au*x*Sb2.

Analysis of the X-ray diffraction as well as EDS results show (Figure 2) that the measured Ag/Au ratio deviates from the nominal with the experimental points for *xmeas* being slightly below the *xmeas* = *xnom* line with *xmeas*/*xnom* = 0.88 ± 0.02 and 0.90 ± 0.03 for X-ray diffraction and EDS results, respectively. In the rest of the text we will use *x*-values determined from the X-ray diffraction.

**Figure 2.** Measured vs. nominal values of Au concentration *x* in LaAg1−*x*Au*x*Sb2 (filled symbols). Double filled red circles for *xnom* = 0.25 (not clearly discerned on the plot) and 0.75, as well as double filled black rhombi for *xnom* = 0.25 correspond to T2 and T6 growths, see Section 2. These data are presented in the Appendix A in a tabular form (Tables A2 and A3). For example, of two red circles at *xnom* = 0.75, higher corresponds to T2 and lower to T6 growth. Data for LaAu*x*Sb2 [17] (open symbols) are added for the reference, here again multiple symbols correspond to different initial La:Au:Sb growth compositions [17]. Dashed line corresponds to *xmeas* = *xnom*.

The lattice parameters, unit cell volume and the *c*/*a* ratio as a function of Au substitution are presented in Figure 3. All these quantities have an approximately linear dependence of *x*, in good agreement with Ref. [18].

**Figure 3.** (**a**) Lattice parameters, (**b**) unit cell volume and *c*/*a* ratio vs. Au concentration in LaAg1−*x*Au*x*Sb2 determined from Rietveld refinement (filled symbols). Data for LaAu*x*Sb2 [17] (open symbols) are added for the reference. Dashed lines are guide to the eye.

#### *3.2. CDW at Ambient Pressure*

Whereas in the case of LaAu*x*Sb2 the CDW temperatures were significantly affected by initial growth compositions [17], this appears to be not so critical for LaAg1−*x*Au*x*Sb2 with the intermediate Au compositions. For the nominal LaAg0.75Au0.25Sb2 and LaAg0.25Au0.75Sb2 samples the difference between T2 and T6 initial compositions in the XRD-refined Au concentrations is 0.01–0.04 (5–6%) (Table A2) and in the CDW ordering temperatures 3–6 K (2–5%), with the difference, not surprisingly, being larger for the latter samples with higher Au concentration. The normalized in-plane resistivity and the CDW ordering temperatures for T2 and T6 samples of LaAg0.75Au0.25Sb2 and LaAg0.25Au0.75Sb2 are shown in Figure 4a,b, respectively.

**Figure 4.** Normalized in-plane resistivity, *ρab*/*ρab*(300 K) and the ordering temperatures CDW1 and CDW2 for T2 and T6 samples with nominal compositions (**a**) LaAg0.75Au0.25Sb2 and (**b**) LaAg0.25Au0.75Sb2.

The overall evolution of the in-plane resistivity of LaAg1−*x*Au*x*Sb2 is shown in Figure 5. CDW transition temperatures decrease with Au substitution. The suppression of *TCDW*<sup>1</sup> is

in fair agreement with the prior results [18]. The ambient pressure *x* − *T* phase diagram based on the data of Figures 4 and 5 is presented in Figure 6. The *TCDW* values were determined from extrema in the *dρab*/*dT* data; an example of which is shown in the inset to Figure 5.

**Figure 5.** Normalized in-plane resistivity for LaAg1−*x*Au*x*Sb2. Data are vertically shifted for clarity. The curve for LaAu0.991Sb2 [17] is added for the reference. The inset shows an example of *dρab*/*dT* for LaAgSb2 with two CDW transitions marked.

**Figure 6.** Transition temperatures, CDW1 and CDW2, as a function of *x* in LaAg1−*x*Au*x*Sb2. Data are vertically shifted for clarity. Dashed lines are guide for the eye. The inset shows residual resistivity ratio, *RRR* = *ρab*(300 K)/*ρab*(1.8 K) as a function of *x*-Au.

On going from LaAgSb2 to LaAuSb2, based on electrical transport data, both CDW transitions persist, but both of them are suppressed by ∼100 K without a significant change of the value of *TCDW*<sup>1</sup> − *TCDW*2. The *TCDW*1(*x*) and *TCDW*2(*x*) behavior has an upward curvature. Most likely the disorder induced by substitution contributes to additional suppression of CDW transition temperatures, although to the extent significantly smaller than, e.g., in 2H—TaSe2−*x*S*<sup>x</sup>* [35]. The presence of substitutional disorder is, expectedly, seen in the evolution of residual resistivity ratio (*RRR* = *ρab*(300 K)/*ρab*(1.8 K)) with Au substitution (Figure 6, inset), which shows a broad local minimum for intermediate substitution values. Similar moderate but visible effect of substitutional disorder was observed in studies of superconducting transition temperature in Y*x*Lu1−*<sup>x</sup>*Ni2B2C [36]. A clearer example can be found in a similar, isoelectronic substitution in the Mn(Pt1−*x*Pd+*x*)5P series [37].

#### *3.3. CDW under Pressure*

Note that often (see above) both CDW1 and CDW2 are reasonably well discerned in the derivatives of in-plane resistivity, with the feature associated with CDW2 being less pronounced. Having in mind that (at least in LaAgSb2 [15]) the CDW2 wavevector is along the *c*-axis, the *ρ<sup>c</sup>* measurements provide better identification of the *TCDW*<sup>2</sup> with *TCDW*<sup>1</sup> still being strong. Therefore the measurements for LaAgSb2 under pressure were performed in *Ic*, *Hab* geometry.

Main panel of Figure 7 presents *c*-axis resistivity data taken for LaAgSb2 at different pressures. The overall resistivity is suppressed as pressure increases. The CDW transitions are moving down in temperature. The insets help to quantify above statement. The relative change of the c-axis resisitivity under pressure, is similar for both temperatures presented, 300 K and 50 K, 1/*ρc*(0) · *<sup>d</sup>ρc*/*dP* = −(0.012 − 0.014) kbar−1. Both CDW temperatures decrease under in a close to linear fashion, with the derivatives *dTCDW*1/*dP* = −4.6 ± 0.2 K/kbar and *dTCDW*2/*dP* = −7.0 ± 0.3 K/kbar. Simple, linear, extrapolation suggests that CDW2 will be suppressed to 0 K at ∼25 kbar and CDW1 at ∼43 kbar. The observed *TCDW* derivatives are consistent with the published values of −(4.3–5.1) K/kbar for CDW1 [19–21] and −8.0 K/kbar for CDW2 [21].

**Figure 7.** Temperature-dependent *c*-axis resistivity of LaAgSb2 measured at different applied pressures. Upper inset: CDW transition temperatures, *TCDW*<sup>1</sup> and *TCDW*2, as a function of pressure. Lower inset: normalized resistivity at 300 K and 50 K as a function of pressure.

In order to extend the pressure dependence of *TCDW*<sup>1</sup> and *TCDW*<sup>2</sup> across the substitutional series, a similar data set (but for *ρab*) for LaAg0.54Au0.46Sb2 (nominal LaAg0.5Au0.5Sb2) is presented in Figure 8. The relative change in the in-plane resistivity at 300 K and 200 K is 1/*ρab*(0) · *<sup>d</sup>ρab*/*dP* <sup>=</sup> −(0.009–0.01) kbar−1, the same as that in LaAgSb2 [20]. The initial pressure derivatives of the CDW transitions are *dTCDW*1/*dP* = −4.9 ± 0.1 K/kbar and *dTCDW*2/*dP* = −9.4 ± 0.1 K/kbar. For CDW2 simple, linear, extrapolation yields ∼12.5 kbar as a critical pressure of complete suppression of CDW2. Since we cannot observe any distinguishable feature in *dρab*/*dT* data at 9.3 kbar below 50 K (see Figure A1 in

the Appendix B), it is possible that the *TCDW*2(*P*) behavior is super-linear and the critical pressure for CDW2 is lower than ∼12.5 kbar obtained from the linear extrapolation. Alternatively, the feature associated with CDW2 could be suppressed so much, that it cannot be detectable within our signal-to-noise ratio and digital differentiation. The data in Figure 9 potentially favors the former possibility. *TCDW*1(*P*) dependence has some curvature, the data extrapolate to the value of the critical pressure of ∼26 kbar.

**Figure 8.** Temperature-dependent in-plane resistivity of LaAg0.54Au0.46Sb2 measured at different applied pressures. Upper inset: CDW transition temperatures, *TCDW*<sup>1</sup> and *TCDW*2, as a function of pressure. Lower inset: normalized resistivity at 300 K and 200 K as a function of pressure. Note, some minor corrections of pressure values, following Ref. [17] were applied.

**Figure 9.** Pressure dependence of (**a**) in-plane resistivity at 2 K (dashed lines are guide to the eye); (**b**) exponent *α* in *ρ* = *ρ*<sup>0</sup> + *AT<sup>α</sup>* fit of low temperature resistivity (fit performed between 1.8 K and 20 K) for LaAg0.54Au0.46Sb2.

Since for LaAg0.54Au0.46Sb2 CDW2 appears to be suppressed to 0 K within our pressure range, we examine if this suppression has any bearing on the low temperature electrical transport. Indeed, the zero applied field data in Figure 9a,b show changes in behavior near 9.3 kbar with the power law exponent, *α*, having the clearest signature of a possible transition near the 9.3 kbar. So most probably the value of the critical pressure for CDW2 is around 9.3 kbar. The changes observed are rather subtle, however the features associated with CDW suppression in LaAgSb2 [21] and LaAu*x*Sb2 [17] were subtle as well.

#### **4. Discussion**

The pressure dependence of the CDW transitions of LaAgSb2, LaAg0.54Au0.46Sb2 and LaAu*x*Sb2 samples is shown in Figure 10. For different members of the family the behavior is very similar. It is noteworthy that the CDW2 suppression rates are almost a factor of 2 higher than those for CDW1. This is possibly due to different effect of pressure on the nesting features along *a*- and *c*-axes (note that for LaAgSb2 the CDW wave-vectors were found to be (0.026 0 0) and (0 0 0.16) for CDW1 and CDW2, respectively, [15]).

**Figure 10.** Pressure dependence of (**a**) *TCDW*<sup>1</sup> and (**b**) *TCDW*<sup>2</sup> for different LaAg1−*x*Au*x*Sb2 shown on the same plot. Data for LaAu*x*Sb2 are taken from Ref. [17]. Insets: the same data positioned on universal lines by horizontal shifts, Δ*P*.

The *TCDW*(*P*) data could be combined on the same universal line by horizontal shift of the data (as shown in the insets to Figure 10). This universal behavior suggest equivalence of the chemical and physical pressure. The approximate scaling is 2 kbar∼0.1 *x*-Au for CDW1 and slightly smaller pressure shift per 0.1 *x*-Au for CDW2. We recall that in a similar way the *P* − *T* phase diagrams for Ba(Fe1−*x*Ru*x*)2As2 with different value of *x* were combined to form a universal phase diagram by Δ*P* shifts with 30 kbar∼0.1 *x*-Ru [4]. In contrast, the pressure and substitution data in the La1−*xRx*AgSb2 (*R* = Ce, Nd) series [20] cannot be combined on the same line by Δ*P* shifts. Apparently rare earth and transition metal substitutions in LaAgSb2 affect the pressure derivatives of CDW transition temperatures in different manner, with *R*-substituted compounds having higher (and *R*dependent) suppression rates. Of course, whereas both Ag/Au and La/R substitutions are isoelectronic, substitution of Ce or Nd for La brings local moment magnetism that subsitution of Au for Ag does not.

To gain some further insight on which structural parameter could be of importance for change of the CDW temperature under pressure and with Au substitution we plot *TCDW*<sup>1</sup> as a function of basic structural parameters, *a*, *c*, *c*/*a* and *V* in Figure 11. For ambient pressure data the structural parameters obtained from the Rietveld refinement are used. For the high pressure data the structural parameters were obtained from the *P* = 0 values using LaAgSb2 elastic constants from Ref. [19] and assuming that their change within the LaAg1−*x*Au*x*Sb2 series is insignificant.

**Figure 11.** CDW1 transition temperature for LaAg1−*x*Au*x*Sb2 and LaAu*x*Sb2 [17] at ambient and high pressure as a function of (**a**) *a*; (**b**) *c*; (**c**) *c*/*a*; and (**d**) *V* structural parameters. Encircled points discussed in the text.

The data in Figure 11 clearly show that whereas using *a*, *c*, and *c*/*a* as a structural parameter results in distinctly different trends for chemical and physical pressure, all the data on *TCDW*<sup>1</sup> vs. *V* plot fall fairly well on the same line. To check if this holds for CDW2 as well, we plotted *TCDW*<sup>2</sup> vs. *V* in Figure 12 as well. The *TCDW*<sup>2</sup> data also scale with the unit cell volume well. We have few outlier points encircled in Figures 11d and 12. These point correspond to measurably *off-stoichiometric* LaAu*x*Sb2 [17], whereas the rest of the data are for the compounds with stiochiometry very close to 1:1:2. That is possibly the reason for these few point being outliers. The *TCDW* vs. *V* scaling could be even better if the elastic constants measured for each compound were used, however to address this, further elastic properties measurements should be performed. It is of a surprise, that despite LaAg1−*x*Au*x*Sb2 being tetragonal, anisotropic compounds, the chemical and physical pressure appear to be equivalent with a salient structural parameter being the unit cell volume (that lacks any anisotropic information). Hopefully further band structure calculations will be able to address this issue.

**Figure 12.** CDW2 transition temperature for LaAg1−*x*Au*x*Sb2 and LaAu*x*Sb2 [17] at ambient and high pressure as a function of the unit cell volume, *V*. Encircled points discussed in the text.

#### **5. Summary**

Study of compounds of the LaAg1−*x*Au*x*Sb2 family at ambient and high pressure show that both CDW transitions are suppressed with Au substitution and under pressure in a manner that indicate equivalence of chemical and physical pressure in this series with the unit cell volume being a suitable structural control parameter and with suppression rates being different for CDW1 and CDW2. Such equivalence of physical and chemical pressure was not observed in the La1−*xRx*AgSb2 series [20]. Different CDW suppression rates probably reflect the fact that (at least for LaAgSb2) the CDW wave-vectors are orthogonal, along *a*- and *c*-axis for CDW1 and CDW2, respectively.

Additionally, for LaAg0.54Au0.46Sb2 anomalies in low temperature electrical transport were observed in the pressure range where CDW2 is completely suppressed.

**Author Contributions:** Conceptualization, S.L.B.; formal analysis, L.X., D.H.R. and S.L.B.; investigation, L.X. and D.H.R.; data curation, L.X. and D.H.R.; writing—original draft preparation, S.L.B.; writing—review and editing, L.X., D.H.R. and P.C.C.; visualization, L.X., D.H.R. and S.L.B.; supervision, P.C.C. and S.L.B.; funding acquisition, P.C.C. All authors have read and agreed to the published version of the manuscript.

**Funding:** Work at the Ames National Laboratory was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division. The Ames National Laboratory is operated for the U.S. Department of Energy by Iowa State University under contract No. DE-AC02-07CH11358. L.X. was supported, in part, by the W. M. Keck Foundation. Much of this work was carried out while D.H.R. was on sabbatical at Iowa State University and Ames Laboratory and their generous support (again under under contract No. DE-AC02-07CH11358) during this visit is gratefully acknowledged. D.H.R. was supported as well by Fonds Québécois de la Recherche sur la Nature et les Technologies.

**Data Availability Statement:** The data presented in this study are available on reasonable request from the corresponding author.

**Acknowledgments:** We thank Warren E. Straszheim for help with the EDS measurements, and Elena Gati and Raquel A. Ribeiro for useful discussions. PCC would like to acknowledge Paul Delvaux for ongoing inspiration.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Appendix A. Rietveld Refinement and EDS Results**

This Appendix A contains tables with the results of Rietveld refinements and EDS chemical analysis of the LaAg1−*x*Au*x*Sb2 samples. Data for LaAu*x*Sb2 [17] in Table A1 are added for comparison.

**Table A1.** Lattice parameters of LaAg1−*x*Au*x*Sb2 samples (labels in parentheses indicate initial growth compositions, see Experimental details section for more details).



**Table A2.** Atomic coordinates, occupancy, and isotropic displacement parameters of LaAg1−*x*Au*x*Sb2 samples (labels in parentheses indicate initial growth compositions, see Experimental details section for more details).

**Table A3.** EDS results for LaAg1−*x*Au*x*Sb2 samples.


#### **Appendix B. LaAg0.54Au0.46Sb2 under Pressure**

Figure A1 presents the derivatives of the resistivity data taken at 5.2 kbar and 9.3 kbar for LaAg0.54Au0.46Sb2 sample.

**Figure A1.** Resistivity derivatives, *dρab*/*dT*, data at 5.2 kbar and 9.3 kbar for LaAg0.54Au0.46Sb2. Arrows point to CDW transition temperatures. The 5.5 kbar data are shifted vertically by 10 for clarity.

#### **References**


**Oliver Tschauner**

Department of Geoscience, University of Nevada Las Vegas, Las Vegas, NV 89154, USA; oliver.tschauner@unlv.edu

**Abstract:** Many non-molecular elemental solids exhibit common features in their structures over the range of 0 to 0.5 TPa that have been correlated with equivalent valence electron configurations. Here, it is shown that the pressures and volumes at polymorphic transitions obey corresponding states given by a single, empirical universal step-function Vtr/L = −0.0208(3) · Ptr + Ni, where Vtr is the atomic volume in Å<sup>3</sup> at a given transformation pressure Ptr in GPa, and L is the principal quantum number. Ni assumes discrete values of approximately 20, 30, 40, etc. times the cube of the Bohr radius, thus separating all 113 examined polymorphic elements into five discrete sets. The separation into these sets is not along L. Instead, strongly contractive polymorphic transformations of a given elemental solid involve changes to different sets. The rule of corresponding states allows for predicting atomic volumes of elemental polymorphs of hitherto unknown structures and the transitions from molecular into non-molecular phases such as for hydrogen. Though not an equation of state, this relation establishes a basic principle ruling over a vast range of simple and complex solid structures that confirms that effective single-electron-based calculations are good approximations for these materials and pressures The relation between transformation pressures and volumes paves the way to a quantitative assessment of the state of very dense matter intermediate between the terrestrial pressure regime and stellar matter.

**Keywords:** corresponding states; high pressure; Wigner–Seitz radii; elemental solids; phase transformation

#### **1. Introduction**

In difference to gases whose states can be described by the ideal gas- or the van der Waals equations of state, solids seem to defy the concept of a general equation of state. The range of structures and properties as well as their changes upon compression appear too large to be comprised within a single formula. Over a range of pressure of 0 to 500 GPa, many elements that are molecular solids at ambient conditions transform into atomic metals [1], while others that are normal metals at ambient conditions become semiconductors [2–4], and some monatomic elemental solids assume complex structures under compression that do not exist at ambient pressure [5,6].

These changes are not arbitrary but reflect general trends in the effect of pressure on the valence electron structure of solid matter over large pressure intervals [1–12]. The changes in the valence electron configuration reflect the different response of different orbital states to the increase in electron density upon compression [2,7,10–15]. The atomic volume often is markedly reduced upon such transitions. The transition from the bcc to the hcp structure in many transition metal elements is correlated with an increase in the hybridization of filled d-states with p-states of the next higher principal quantum number [16,17]. Similarly, the pressure-induced transformations of lanthanides from hcp to the Sm type to dhcp to fcc is explained through the increased occupancy of valence d-states [11–13], whereas the marked volume collapse and transitions to low symmetric phases in lanthanides are assigned to delocalization of the 4f states [12,13,15]. A different type of delocalization occurs in the low-Z elemental alkaline metals, which at high pressure transform into electrides in which inner

**Citation:** Tschauner, O.

Corresponding States for Volumes of Elemental Solids at Their Pressures of Polymorphic Transformations. *Crystals* **2022**, *12*, 1698. https:// doi.org/10.3390/cryst12121698

Academic Editors: Daniel Errandonea and Enrico Bandiello

Received: 1 November 2022 Accepted: 16 November 2022 Published: 23 November 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

shell electron states overlap with valence s-states and non-bonding p-states are occupied on expenses of the bonding s- and p-hybrid states [2,4,14]. As a consequence, elements such as Li and Na, which are normal metals at ambient conditions, become semiconductors at pressures between 100 and 200 GPa [2–4]. Changes in the electronic valence states have been studied quantitatively for some alkaline metals, transition group elements, and lanthanides through X-ray spectroscopy [10,14–17]. These changes are correlated with structural transformations. It has been noted that these pressure-induced valence changes follow general correlations of the nuclear charge number with the pressure, volume, and structure type [5,8–10]. In addition, there are correlations between the atomic volumes of equivalent polymorphs of different elements [8,18].

Here, it is shown that all known non-molecular elemental solids follow a rule of corresponding states if only the volumes and pressures at the polymorphic transitions are considered. This observation extends to high-pressure atomic polymorphs of elements that are molecular at ambient conditions, such as O, N, S, P, Cl, Br, and I. Only experimental data are considered here (see Section 2), hence the relation presented is without any kind of simplifying assumptions or theory.

#### **2. Materials and Methods**

Only published volumes and transformation pressures of polymorphs of elemental solids are used in this paper. Recently published data and work conducted under hydrostatic or nearly hydrostatic conditions are given preference. This includes data obtained from samples embedded in He or Ne as pressure media or data from crystals grown from melt at a high pressure. Data from compression experiments without pressure media are only included if the average pressure has been below 10 GPa. However, the induced transformations in Ir, Pb, and Th under non-hydrostatic stresses are included for reference. Similarly, compression data of some lanthanides are included due to their principal interest, although most of these data were not and possibly could not be acquired with the use of hydrostatic media. The plotted volumes Vtr and pressures Ptr at the polymorphic transitions are generally those of the first observation of a high-pressure polymorph upon compression at 300 K. In a few cases in which the transformations exhibits hystereses larger than 5 GPa, the arithmetic middle pressure has been taken as Ptr. In all other cases, the hysteresis has been added to the uncertainty. For some elemental high-pressure polymorphs, the structures have been redetermined in more recent studies but without reassessment of the 300 K isotherms. In such cases, the atomic volume of the correct structure has been used to reassess the atomic volume at the pressure of the transformation that has been reported in the earlier studies. The complete data are given in Table 1 at the end of the paper.

#### **3. Results**

For all of the examined 113 elemental solids, the volume at the phase transition Vtr = V(Ptr) (Ptr = transition pressure) divided by the principal quantum number L exhibits a universal linear pressure dependence of −0.0208(3) · Ptr (Ptr in GPa; atomic volume in Å3; the adjusted R<sup>2</sup> of the linear fit was 0.9985; see Supplemental Figure S1). This is shown in Figure 1. Furthermore, within the uncertainties, the constant term of this equation provides a volume that for all of the examined elemental solids is nearly equal to n · 10 · rB 3, where n is an integer number between 2 and 6, and rB is the Bohr radius in Å. This is shown in Figure 2. In other words:

$$\rm{N}\_{\rm tr}/\rm{L} + 0.0208(3) \cdot \rm{P}\_{\rm tr} = \rm{N}\_{\rm i} \approx \rm{n} \cdot 10 \cdot \rm{r}\_{\rm B}^{\rm 3} \tag{1}$$

**Figure 1.** Volume of elemental polymorphs in Å3/at at 300 K divided by the principal quantum number L as a function of pressure in GPa. The volumes are those that are observed at the pressure of their formation through direct phase transformation from lower pressure polymorphs; the plotted pressures are the pressures of the transformation. The linear relation between these volumes and pressures and the existence of five distinct sets of pressure–volume relations is clearly visible.

**Figure 2.** This figure shows the data from Figure 1 with the volume corrected for the linear pressure dependence from Equation (1). The remaining constant term of each set is approximately equal to integer multiples of 2, 3, 4, 5, and 6 of 10 times the cube of the Bohr radius.

The actual values of the five distinct volumes for Ni are 2.88(8), 4.58(4), 5.51(5), 7.14(5), 8.53(6) Å3/at, which is 19.5(4), 30.9(4), 37.2(8), 48.2(1), and 57.6(5) times the cube of the Bohr radius rB <sup>3</sup> in Å3, respectively. Therefore, Figures 1 and 2 show that the volumes of elemental solids at their polymorphic transitions assume corresponding states. It is important to recall that this universal relation of corresponding states only holds for the pressures and volumes at the phase transitions: Within the stability field of each polymorph, the pressure–volume relation is controlled by its specific compressibility that itself depends on pressure and temperature and that, in general, is far from being linear, universal, and discrete. Equation (1) is therefore not an equation of state.

#### **4. Discussion**

The 'collective quantization' shown in Figure 1 may, upon first glance, be assigned to the factorization by the principal quantum number L. However, the series of corresponding states Ni does not follow the rows of the periodic table. Rather, many elements with subsequent polymorphic transitions change between states Ni. For instance, the highpressure polymorphs fcc, cl16, and oc120 of Na reside in N4 = 5.51(5) Å3/at ≈ <sup>40</sup> · rB <sup>3</sup> along with K-IV, Rb-IV, -V, Cs-IV, and Ca-III to -IV; whereas the h4p-type Na resides in N5 = 7.14(5) Å3/at ≈ <sup>50</sup> · rB <sup>3</sup> along with Rb-II and -III. In some cases, for instance Si, all polymorphs remain within one set Ni; however, for most elements at least some polymorphs fall into different sets. For instance, Cs-II and Cs-III are in N6 but Cs-IV and -V belong to N4 and Cs-VI to N3. A sequence of stepwise transitions to sets with a lower state Ni along with an increasing transformation pressure is found for many elements: the high-pressure polymorphs of Rb fall into the sets N5 (Rb-II and -III), N4 (Rb-IV, -V, -VIII), and N3 (Rb-VI, -VII); Li into N6 (Li-II), N4 (Li-III, -IV,-VI), and N3 (Li-V); and the various polymorphs of the alkaline earths Ca, Sr, and Ba are found in the sets N3 to N6 (Figures 1 and 2), where the higher pressure polymorphs belong to sets with a smaller Ni. Lower sets Ni have markedly smaller volumes at a given pressure than the corresponding higher sets Ni+1 (Figure 1), and the polymorphic transitions that involve changes Ni → Ni−<sup>1</sup> are accompanied by marked volume contractions (VLPtr − VHPtr)/V0 where VLPtr and VHPtr are the atomic volumes of the low- and the high-pressure polymorphs, respectively, at the pressure of the phase transitions; and V0 is the volume of the ambient pressure phase at standard conditions. A smaller number of elements also exhibit transitions into higher sets Ni+1 at higher pressure. In particular, this holds for the polymorphic transitions to Sc-V [19], bcc-Ti [20,21], hp4-Na [3], and Rb-VIII (Figure 1), but also for the elastic anomaly of Os around 290 GPa [22], which is taken here for an isostructural transition. There is a simple reason for these 'upward transitions' Ni → Ni+1: at the given transformation pressure, the corresponding volumes of the next lower set Ni−<sup>1</sup> would be negative. It is to be expected that for all elements at sufficiently high pressures, the cascade of successive, markedly contractive transitions Ni → Ni−<sup>1</sup> is followed by transitions Ni → Ni+1 to polymorphs in higher sets with a less pronounced contraction once Ni → Ni−<sup>1</sup> reaches the limit of negative volumes. Transitions to higher sets come with smaller contractions (VLPtr − <sup>V</sup>HPtr)/V0 than those into lower sets.

#### *4.1. Theoretical Explanation*

'Upward' and 'downward' transitions between different states Ni can be correlated to changes in the valence electron configuration wherever such information has been obtained experimentally [10,15,17] or through calculation [14].The fact that the volumes Ni are nearly equal to integer multiples of rB <sup>3</sup> has a straightforward theoretical basis: It implies that the volumes that occur at polymorphic transitions of non-molecular elements are controlled by the number of valence electrons rather than by their interactions. Besides the limited experimental accuracy the electron exchange- and Coulomb interactions account for part of the deviations from Equation (1) and from integer Ni values; thus, the interaction terms are of the second order with respect to the volumes. In other words, Equation (1) is the equivalent to an ideal equation of state for elemental solids, only that it does not describe continuous pressure–volume changes but discrete changes that occur at the phase transitions. This finding can be further quantified by considering the effect of the different valence orbital states as based on the experimentally determined pressure-induced hybridization of d-states of the alkaline metals Rb, Cs, and K (the latter element exhibits no measurable d-state occupancy up to 40 GPa) [10]. Figure 1 may be re-parametrized in terms of pressure-induced changes in the d-state occupancy at the valence level. In particular, the d-state occupancy of the elemental polymorphs can be correlated with the five sets N2 to N6 shown in Figure 1 through their 0 GPa intercepts. This is shown in Figure 3a. For instance, the d-state occupancy of Cs increases for phase Cs-II to -VI from ~0.0 to ~0.6, and for Rb from ~0.2 to 0.4 electrons between Rb-II and -VI [10]. The high-P polymorphs of Li and Na have little or no occupied d-states in this diagram and according to earlier computational work [2,4,14]. The correlation between Δn(d) and Ni is linear, at least for the polymorphs K-II, Rb-II,-IV, Cs-II, and Cs-III (black squares in Figure 3a): Within uncertainty, the slopes are equal for all sets. In terms of multiples of the electron elemental charge, this linear correlation between d-state occupancy, Vtr, and L is:

$$\text{N}\_{\text{i}} = \text{N}\_{\text{i}}^{\text{0}} - 6.11(14) \cdot 10^{-11} \cdot \text{n(d)} \cdot \text{e} \tag{2}$$

where Ni <sup>0</sup> = Ni at n(d) = 0, the constant factor is given in m3/(C · at), and e is the electron elemental charge. However, the correlation between Ni and n(d) is not continuous: Equations (1) and (2) can be combined to give:

$$\frac{\text{V}\_{\text{tr}}}{\text{L}} = -2.08(3) \cdot 10^{-32} \cdot \text{P} - 6.11(14) \cdot 10^{-11} \cdot \sum\_{i=0}^{\text{m}-1} \text{i} \cdot \text{n}(\text{d}) \cdot \text{e} + \text{N}\_{\text{m}} \tag{3}$$

where Nm is the value assumed at n(d) = 0, P is in GPa, V is in m3/at, and e is the electron elemental charge. The effect of the d → p transfer on shifts into sets with a higher Ni has been discussed above. Overall, there are fewer data on pressure-induced changes in the valence p-state occupancy (summarized in Supplementary Figure S2, see refs. [10,16,17]), but within uncertainties, the relation between Vtr/L and n(p) has the same slope as in (3) but with an opposed sign (consistent with the transfer into higher sets Ni, whereas changes in the d-state occupancy generally result in changes to a lower Ni). Implementing n(p) into (3), dividing through rB 3, and using the approximation Ni rB <sup>3</sup> ~ 10·m gives:

$$\begin{array}{l} \frac{\text{V}\_{\text{F}}}{\text{L} \cdot \text{r}\_{\text{B}}^{\text{r}}} = -0.1405(20) \cdot \text{P} + 10 \cdot \text{m} - 66.1(15) \cdot \sum\_{i=0}^{\text{m}-1} \text{i} \cdot 0.15(4) + 66(2) \cdot \sum\_{j=\text{P}}^{\text{m}} \text{j} \cdot \\\ 0.09(21) = -0.1405(20) \cdot \text{P} + 10 \cdot \text{m} - 10.0(6) \cdot \sum\_{i=0}^{\text{m}-1} \text{i} + 6(1) \cdot \sum\_{j=\text{P}}^{\text{m}} \text{j} \end{array} \tag{4}$$

where the pressure is given in GPa, the volumes Ni are taken as integer multiples 10 · i ∈ IN (the set of natural numbers) of rB <sup>3</sup> per atom, p ≤ n(d), and the increments <sup>Δ</sup>n(d) are taken as 0.151; hence: 66.1 · 0.151 = 10 d-electrons and accordingly 66(2). 0.09(1) = 6(1) for the p-electrons. Thus, within uncertainties, the factors of the sum terms equal the number of p- and d-states and an additional term for f-electrons with factor −14 may be added. The actual values Ni may be substituted for 10m.

The volumes of elemental atomic solids at ambient conditions are not bound to Equation (1) because they do not represent volumes at phase transitions. However, the correlation V = Ni · L is within ±10% of the observed volumes at ambient conditions for 28 and within 20% for 38 out of the 42 non-molecular solid elements (Figure 3b). Considering that the atomic volumes of elemental metals are equal to the cube of the Wigner–Seitz radii, their approximate relation to Ni · L is equivalent to a relation recently found for the ionic radii [23]. However, in the present case of elemental solids the relation does not rely on any approximate model of bonding.

**Figure 3.** (**a**) Correlation between the normalized transformation volumes Vtr from Equation (1) and d-state occupancy. The volumes Ni from Equation (1) were used to define the Vtr/L sets of the alkaline metals [10] and of a selection of other elements as a function of d-state occupancy. Filled symbols: measured d-state occupancies; hollow symbols: estimated occupancies based on Equations (1) and (2). Solid lines: fitted correlation for KII, Rb-II and -IV, and Cs-II and -III. The same slope has then been applied to all data. Dashed lines give the values Ni and are guides for the eye. (**b**) Comparison of the atomic volumes of elemental non-molecular solids at ambient pressure and temperature with the calculated volumes Ni · L. The correlation is linear with an adjusted R2 = 0.9928 and a slope of 1.015(13), which reflects a general slight contraction of the elemental volumes relative to Ni · L.

#### *4.2. Molecular Gap and Forbidden Zone*

There are two regions in Figure 1 that contain no values Vtr/L: (1) There is no observed Vtr/L equal to or smaller than 1.48 Å3/at at any pressure; 1.48 Å3/at equals 10 times the cube of the Bohr radius (per atom; this limiting value 10 rB 3/at is defined here as N1). Upon compression, elemental solids such as Au and Re reach this limit (see Figure 4) but without further known transformations [24,25]. There is no known elemental solid that undergoes a polymorphic transition between 0 and 300 GPa where the volume at the transition is equal to or smaller than 10 rB 3.

(2) The area between P ~ 20 GPa for Vtr/L ~ 8 Å3/at and 200–300 GPa for ~3 Å3/at contains no observations. This empty range has a nearly parabolic shape in Figure 1. It corresponds to volumes and pressures of molecular elemental solids (Figure 4).

For instance, the volumes Vtr of the molecular polymorphs β, γ, δ, and ε of oxygen as well as the metallic molecular phase ζ-O2 [26], are not compliant with Equation (1). The isotherm of ε-O2 intersects Vtr/L + N6 at 24 GPa without any observed transition ([26]; Figure 4). The isotherm of H2 intersects the Vtr/L + N6 at 26 GPa and Vtr/L + N5 at 54 and 135 GPa and again Vtr/L + N6 around 250 GPa, which is at the pressure of the transition to H2-IV ([27], Figure 4), but neither of these intersections corresponds to the transition to an atomic metal [27]. However, the transition from molecular N2 to the polymeric, semiconducting cg-phase of nitrogen [28] falls onto the N4 trajectory (Figure 1, Table 1). Cl2 intersects Vtr/L + N6 at 260 GPa, where it has been reported to become an atomic metal with an In-type structure [29] and thus follows the behavior predicted by Equation (1). Hence, the gap between 50–180 GPa for Vtr/L + N5 and N6 indirectly reflects the strength of the molecular bonds of these materials under compression and should be called a 'molecular gap'. The intersection of the 300 K isotherm of the chlorine phase III (mc8; [29]) with Vtr/L + N5 at 180 GPa may be coincidental because this phase is probably still molecular

and the Vtr of the intermediate phase IV (i-oF3, [29]) falls in between Vtr/L + N5 and Vtr/L + N6. The isotherms of N, S, P, Br, and I all intersect the trajectories Vtr/L + Ni at the pressures of the observed transitions to non-molecular phases [8,27,30,31] and fall mostly below the 'molecular gap' because their transitions to non-molecular solids occur at lower pressure.

**Figure 4.** Data from Figure 1 with the equations of state for gold [24], chlorine [29], hydrogen [27], and oxygen [26] shown as yellow, green, red, and blue lines, respectively. Dashed lines indicate the distinct sets that defined Vtr/L versus P as defined by Equation (1). The additional data points for hydrogen [32] and the second elastic anomaly of osmium [22] have been added. Hachured areas mark the molecular gap and the forbidden zone below Vtr/L = 10 · rB 3.

Interestingly, the extrapolated pressure of the direct gap closure of H2 that has been assessed through inelastic X-ray spectroscopy of H2 [32] matches the intersection of the extrapolated isotherm of H2 [27] and a trajectory Vtr/L + N9 (of 13.0 to 13.3 Å3/at = 90 rB 3 by using the intersection of the trajectory N9 in Figures 3a and 4) at around 0.5 TPa, which suggests that the transition from molecular to atomic hydrogen occurs at that pressure at 300 K.

Finally, it is noticed that Equation (1) could be used to assess the atomic volumes of polymorphs with extant structure analyses such as Ca-VII, Sc-III and -IV, Ga-IV, and metastable Ir-II to 8.0(5), 12.08(8), 10.3(11), 9.2(4), and 9.78(12) Å3/at at 170, 26, 35, 120, and 59 GPa, respectively (Table 1). The volume of hydrogen at the predicted transition at 490–530 GPa is estimated to 2.3(2) Å3/at. This and above values (marked with an asterisk) and the entire list of volumes Vtr and pressures Ptr are given in Table 1.

**Table 1.** Columns from left to right: Name of polymorph, pressure of phase transition at 300 K, volume of phase transition divided by the principal quantum number L, volume of phase transition at 300 K, and reference(s).


#### *Crystals* **2022**, *12*, 1698

**Table 1.** *Cont.*


**Table 1.** *Cont.*


\* Parameters calculated using Equation (1).

#### **5. Conclusions**

The volume and pressure of phase transformations of non-molecular elemental solids obeyed a universal relation of discrete corresponding states. These states, as defined in Equation (1), were close to integer multiples of the cube of the Bohr radius. Therefore, the volumes and pressures of polymorphic transitions of these solids were controlled by the number of valence electrons as specified in Equation (4). This finding was in accordance with the commonly used effective single-electron-based computational approaches for modeling elemental solids within the range of 0 to 0.5 TPa. However, Equation (1) can be used for constraining elemental metal volumes for first-principal- and empirical-potentialbased calculations, thus removing the weakest point in these approaches (the proper assessment of volumes). The volumes of polymorphs at the phase transitions exhibited a general linear dependence on the pressure of the transition (see Equation (1)). Though not an equation of state, this relation establishes a very simple principle that rules over a vast range of simple and complex solid structures and a range of pressure of 0.5 TPa.

**Funding:** This research received no external funding.

**Supplementary Materials:** The following supporting information can be downloaded at: https:// www.mdpi.com/article/10.3390/cryst12121698/s1, Figure S1: Volume of elemental polymorphs in Å3/at divided by the principal quantum number L as function of pressure in GPa and mapped onto the set N4; Figure S2: Correlation between the normalized transformation volumes from Equation (1) and p-state occupancy.

**Data Availability Statement:** All data are provided in the paper.

**Acknowledgments:** Comments and suggestions by S. Huang, H.-k. Mao, A. Zerr, and by two anonymous reviewers are gratefully acknowledged.

**Conflicts of Interest:** The author declares no conflict of interest.

#### **References**


**Pricila Betbirai Romero-Vázquez 1, Sinhué López-Moreno 2,3,\* and Daniel Errandonea <sup>4</sup>**


**Abstract:** In this work, we report first-principles calculations to study FeVO4 in the CrVO4-type (phase **II**) structure under pressure. Total-energy calculations were performed in order to analyze the structural parameters, the electronic, elastic, mechanical, and vibrational properties of FeVO4-**II** up to 9.6 GPa for the first time. We found a good agreement in the structural parameters with the experimental results available in the literature. The electronic structure analysis was complemented with results obtained from the Laplacian of the charge density at the bond critical points within the Quantum Theory of Atoms in Molecules methodology. Our findings from the elastic, mechanic, and vibrational properties were correlated to determine the elastic and dynamic stability of FeVO4-**II** under pressure. Calculations suggest that beyond the maximum pressure covered by our study, this phase could undergo a phase transition to a wolframite-type structure, such as in CrVO4 and InVO4.

**Keywords:** FeVO4 under pressure; CrVO4-type structure; first-principles; mechanical properties; vibrational properties; electronic properties

#### **1. Introduction**

High-pressure studies on *AB*O4 compounds have increased in the last two decades due to the evolution of the diamond-anvil cell (DAC) and its impact on other relevant areas, such as materials sciences [1,2]. Among *AB*O4 compounds, those with a CrVO4-type structure (Space Group, SG, *Cmcm*, No. 63, *Z* = 4, with [6 − 4] coordination for the [*A*-*B* cations]) have received less attention. CrVO4-type compounds are a family formed mainly by phosphates, vanadates, chromates, and sulfates [3]. Given their position in the Bastide diagram [4], it has been reported that they could have a structural phase transition driven by pressure to the wolframite [5,6] or zircon [7,8] structure. This implies a change in the coordination to [6 − 4 + 2] and [8 − 4] in the high-pressure phases, respectively.

According to the literature, *A*VO4 vanadates with CrVO4-type structures have been used in applications as photocatalyst materials [1,9,10], cathodoluminescent materials [11,12], and scintillators [13,14], among others. The properties of these compounds are reported to be highly dependent on the occupation of the *d* valence electrons of the transition metal cations [9–14]. Until now, only four vanadates with a CrVO4-type structure have been reported: CrVO4 [6], FeVO4 [15], InVO4 [16], and the less studied TlVO4 [3]. Unlike the other vanadates, FeVO4 presents a great polymorphism, crystallizing at ambient conditions in the triclinic AlVO4-type [17] structure (SG: *P*1¯, No. 2, *Z* = 6) [15], while the CrVO4-type phase was obtained from high-pressure-high-temperature studies [18] and synthesized at ambient conditions by hydrothermal method [19]. Further, the antiferromagnetic order in FeVO4-**II** was observed after the characterization of its magnetic properties [19]. However, until now, no more experimental or theoretical studies have been reported to characterize this phase. It

**Citation:** Romero-Vázquez, P. B.; López-Moreno, S.; Errandonea, D. Stability of FeVO4-II under Pressure: A First-Principles Study. *Crystals* **2022**, *12*, 1835. https://doi.org/ 10.3390/cryst12121835

Academic Editor: Artem R. Oganov

Received: 18 November 2022 Accepted: 13 December 2022 Published: 15 December 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

is worth mentioning that most of the studies carried out at high pressure on this compound have focused on characterizing the crystal structure of the different polymorphs [15,18,20–22].

In this work, we performed a first-principles characterization of FeVO4-**II** up to 9.6 GPa. The main goal is to determine the elastic and dynamic stability of the studied polymorph at ambient conditions and under pressure. In this sense, a complete report of the elastic constants, mechanical properties, elastic anisotropy, and vibrational properties is presented. Furthermore, given the relevance of the electronic structure of this compound in materials science, we computed the density of states and the band structure together with an analysis of the charge density with the QTAIM methodology [23,24].

#### **2. Computational Details**

Total energy calculations were performed within the framework of the density functional theory (DFT) [25] and the projector-augmented wave (PAW) [26,27] method as implemented in the Vienna *Ab initio* Simulation Package (VASP) [28–31]. We used a plane-wave energy cutoff of 520 eV to ensure high precision in our calculations. The exchange-correlation energy was described within the generalized gradient approximation (GGA) in the Perdew–Burke–Ernzerhof for solids (PBEsol) formulation [32]. The GGA+*U* was used to account for the strong correlation between the electrons in the *d* shell, on the basis of Dudarev's method [33]. In this method, the Coulomb interaction *U* and the onsite exchange interaction *J*<sup>H</sup> are treated together as *U*eff = *U* - *J*H. For our GGA+*U* calculations, we choose *U* = 6 eV and *J*<sup>H</sup> = 0.95 eV. Similar values were previously used with relative success in the study of other iron and vanadate compounds [15,20,34–37].

The Monkhorst-Pack scheme was employed for the Brillouin-zone (BZ) integrations [38] with a mesh 4 × 3 × 3, which corresponds to a set of 8 special *k*-points in the irreducible BZ. In the relaxed equilibrium configuration, the forces are less than 0.3 meV/Å per atom in each Cartesian direction. The highly converged results on forces are required for the calculations of the dynamical matrix using the direct force constant approach [39], which allows us to identify the irreducible representation and the character of the phonon modes at the zone center (Γ point). The phonon density of states (PDOS) were obtained from the calculation of the phonons in the whole BZ with a supercell 2 × 2 × 2 times the conventional unit cell by using the PHONON software. The calculations of elastic constants were made following the methodology validated in Ref. [40] with a *k*-points mesh of 8 × 6 × 6, a plane-wave energy cutoff of 570 eV, and a POTIM parameter of 0.016. The elastic tensor is determined by performing six finite distortions of the lattice and deriving the elastic constants from the strain–stress relationship [41]. The electron density was computed with a refined grid to be analyzed within the fundaments of Quantum Theory of Atoms in Molecules (QTAIM) [42] with CRITIC [43,44], and AIM [45] codes.

#### **3. Results and Discussion**

#### *3.1. Crystal Structure*

According to the literature, FeVO4 has been successfully synthesized as a metastable polymorph at ambient conditions in the orthorhombic CrVO4-type structure (SG: *Cmcm*, No. 63, *Z* = 4) [19], also called the FeVO4-**II** phase, see Figure 1a. In this structure, there are four non-equivalent Wyckoff positions (WP): Fe occupies the 4*a* (0, 0, 0), V the 4*c* (0, *y*, 1/4), and O the 8 *f* (0, *y*, *z*), and 8*g* (*x*, *y*, 1/4). Internally, the structure consists of edge-sharing FeO6 distorted octahedra along the *c* direction. The –FeO6–FeO6– chains are linked together by means of the VO4 distorted tetrahedra, while the VO4 tetrahedra are not linked to each other, as can be seen in Figure 1b. As shown in Figure 1c, there are two non-equivalent interatomic distances in the FeO6 octahedra, the apical Fe-O distance (*d*Fe−O1) and the equatorial Fe-O distance (*d*Fe−O2). In the VO4 tetrahedra, there are two V–O interatomic distances (*d*V−O1 and *d*V−O2). In both polyhedra, the respective interatomic distance *dX*−O2 is larger than *dX*−O1 (*X* = Fe, V). Inside the polyhedrons, there are important angles to mention that are relevant for the structural changes in the polyhedra under pressure. The *α*

angle is formed between the equatorial plane of the FeO6 octahedra and the polar bond, while the *β* (*γ*) angle is between the two *d*<sup>1</sup> (*d*2) interatomic distances of the VO4 tetrahedra.

**Figure 1.** (**a**) Crystal structure of FeVO4-**II** polymporph, (**b**) polyhedral representation of the structure, and (**c**) irregular polyhedra of FeO6 and VO4 units. The different bond distances mentioned in the text are labeled as well as the bond angles. We used the VESTA software [46] to build the structures.

The equilibrium volume and unit-cell parameters at *P* ≈ 0 GPa were computed by minimizing the total crystal energy for different volumes ranging from −4 to 10 GPa while allowing the internal atomic positions and the lattice parameters to relax. We used the thirdorder Birch–Murnaghan [47] equation-of-state (EOS) to fit the volume–energy curve. Our results showed that the lowest energy configuration is obtained when an antiferromagnetic coupling between Fe ions is obtained, in accordance with the experimental results from Mössbauer effect [19]. The optimized lattice parameters (*a*, *b*, *c*), volume (*V*), interatomic distances (*d*Fe−O1, *d*Fe−O2, *d*V−O1, *d*V−O2), internal angles (*α*, *β*, *γ*), bulk modulus (*B*) and pressure derivative of *B* (*B*') appear in Table 1 together with experimental results from the literature [19]. Furthermore, we have included the reported parameters for other CrVO4 type structure vanadates for comparison. According to Table 1, the equilibrium volume at ambient pressure of *A*VO4 (*A* = Cr, Fe, In, Tl) compounds increases with the atomic mass of cation *A*, while the opposite occurs for bulk modulus. The obtained results for FeVO4–**II** are in excellent agreement with experiments, with a difference of less than 1% in the lattice parameters. For this volume, the WPs are 4*a* (0, 0, 0) for Fe, 4*c* (0, 0.35788, 1/4) for V, 8 *f* (0, 0.76239, 0.96753), and 8*g* (0.26481, 0.47778, 1/4) for O. Following Table 1, the equilibrium volume at ambient pressure of *A*VO4 (*A* = Cr, Fe, In, Tl) compounds increases with the atomic mass of cation *A*. In contrast, the opposite occurs for bulk modulus.

**Table 1.** Calculated structural parameters for FeVO4-**II** phase. *a*, *b*, and *c* are the lattice parameters, *V* is the volume, *dX*−O*<sup>n</sup>* (*X* = Fe, V; *n* = 1, 2) and *α*, *β*, and *γ*, are the interatomic distances and angles inside the crystal structure, respectively (see Figure 1). *B* is the bulk modulus and *B* is the pressure derivative of *B*. Results from this work are denoted with . We also include experimental results from the literature of *A*VO4 compounds for comparison.


According to the most recent high-pressure experiments carried out on FeVO4 [15,20], this compound undergoes the following phase transition sequence, *<sup>P</sup>*1¯ <sup>→</sup> *<sup>P</sup>*1¯ <sup>→</sup> *<sup>α</sup>*-MnMoO4-type, driven by pressure. However, previous high-pressure studies reported

the next phase transition sequence *<sup>P</sup>*1¯ <sup>→</sup> CrVO4-type <sup>→</sup> wolframite in a small range of pressure [49]. Therefore, it is unknown if the CrVO4-type structure of FeVO4 would transition to wolframite when high-pressure experiments are carried out starting from the CrVO4-type structure, as occurs with CrVO4 [6] and InVO4 [5,16], and was predicted through firstprinciples calculations [15]. Therefore, from now on, we will only deal with studying the behavior of FeVO4-**II** under pressure without considering whether the CrVO4 → wolframite transition occurs. Figure 2 shows the pressure evolution of the structural parameters for the FeVO4-**II** phase. According to Figure 2a, a volume reduction of Δ*V* = −9 % is observed for a range of pressure of ≈10 GPa. Considering the anisotropy of the orthorhombic crystal structure, Δ*V* is more reflected in the [010] and [001] directions, while the change in [100] is practically negligible, see Figure 2b. Indeed the compressibility of the [001] direction is comparable to the linear compressibility of diamond, making the studied material a near zero linear compressibility material along [001]. The anisotropy can also be understood with the help of the other internal parameters from Figure 2c–f. As seen in Figure 2c, the interatomic distances *d*Fe−O*<sup>n</sup>* of FeO6 undergo more changes than the *d*V−O*<sup>n</sup>* distances of VO4, which is reflected in the larger compressibility of *V*FeO4 octahedra against the minimal change in *V*VO4 , see Figure 2d. Hence, the large compressibility of the *y*-axis is due primarily to the compressibility of octahedra due to the reduction in *d*Fe−O2. Whereas, despite the reduction in *d*V−O2, the *β* and *γ* angles adjust in such a way that *V*VO4 remains almost constant with pressure. On the other hand, the distortion parameter Δ*<sup>d</sup>* presents an important change at 6.2 GPa, which will be discussed in detail in Section 3.3.

**Figure 2.** Structural parameters for FeVO4-**II** as a function of pressure: (**a**) volume, (**b**) lattice parameters, (**c**) interatomic distances, (**d**) polyhedral volumes, (**e**) angles, and (**f**) polyhedral distortion parameter.

#### *3.2. Electronic Structure*

Figure 3a,b displays our calculated density of states (DOS) and band structures along the high-symmetry path Z–Γ–S–R–Z–T–Y–Γ of the FeVO4-**II** phase at ambient pressure and 9.6 GPa, respectively. Red dashed lines at 0 eV indicate the Fermi level. At ambient pressure, the valence band maximum (VBM) is located at an H ( <sup>1</sup> 2 , 1 <sup>2</sup> , −*u*) point (between T and Y), mainly occupied by 2*p* (*px*) states from oxygen. In contrast, the conduction band minima (CBM) is located at the Γ point with contributions of the 3*d* states from Fe, mostly from *dyz* sub-orbital. Therefore, FeVO4-**II** phase presents an indirect energy band-gap *Eg* = 2.11 eV. In comparison, the direct Γ–Γ energy band-gap is 2.17 eV. In this case, the maximum at Γ below the Fermi level is occupied by 2*p* (*px* and *py*) states from oxygen, as happens in other *AB*O4 compounds [40]. It has been reported that CBM of many rare-earth *A*VO4 vanadates is dominated by vanadium 3*d* states, which makes these compounds present energy band-gaps close to 3.8 eV [50,51]. However, in the case of vanadates InVO4 (phase **II**), CrVO4 (phase **II**), and FeVO4 (phase **I**), the energy band-gap lowers to 3.2, 2.6, and 2.1 eV, respectively, due to contribution from cation *A* to the valence and conduction band [5,6,50,52]. Hence, our *Eg* value follows the observed results for those vanadates. According to partial electronic DOS of Figure 3a, the states below the Fermi level mainly belong to the O atoms. In contrast, the states above the conduction bands are due to Fe, followed by V and O.

**Figure 3.** Band structure and electronic density of states at (**a**) ambient pressure and (**b**) 9.6 GPa of FeVO4-**II** phase.

As pressure increases up to 9.6 GPa, the VBM shifts to the high-symmetry point Y while the CBM is still at the Γ point with the same band occupation observed at ambient pressure, see Figure 3b. Hence, at this pressure, the band-gap is still indirect with *Eg* = 2.05 eV, which gives a pressure coefficient of *dEg*/*dp* = −7.1 meV/GPa. In contrast, the pressure coefficients for InVO4 and CrVO4 are positive, with values of 13.5 [52] (8.9 [5]) meV for InVO4 and 1.9 (4.1) meV for CrVO4 [6], obtained by optical absorption measurements (DFT calculations). On the other hand, the forbidden direct Γ–Γ band-gap increases with pressure up to 2.21 eV at 9.6 GPa, which gives a value of *dEg*/*dP* = 4.2 meV, similar to the obtained values for the other *A*VO4 vanadates.

We performed a topological analysis of the charge density (*ρ*) with the QTAIM methodology to complement the study of the electronic structure of FeVO4-**II**. Figure 4a,b show the gradient vector field of the electron density (∇*ρ*) of FeO6 and VO4 polyhedra, respectively (gold for Fe, purple for V, and red for O). Blue points indicate the atomic nucleus, corresponding to the maximum critical points (MCP). The minimal CP (mCP) are in green, and the bond CPs (BCPs) are in black and are located in the bond path (blue dashed lines) between Fe (V) and O. The distance between Fe (V) and the BCP (*d*BCP) is plotted as a function of pressure in Figure 4c. As seen, the distances *d*BCP follow the same trend observed in the interatomic distances of Figure 2c. Therefore, the crystal structure can be interpreted in terms of charge density. In this sense, the Laplacian of the charge density (∇2*ρ*BCP) at the BCP gives an idea of the ionicity (∇2*ρ*BCP > 0) or covalency (∇2*ρ*BCP < 0) of a system [53]. According to Figure 4d, the Fe–O and V–O bonds have an ionic nature. The obtained values of ∇2*ρ*BCP (0.214–0.295 a.u.) for Fe–O bonds in the distorted octahedra are larger than those reported for *A*–O bonds in *A*WO4 (0.086–0.147 a.u.) [54], *A*MoO4 (0.085–0.22 a.u.) [53], and

*A*TcO4 (0.045–0.153 a.u.) [40] compounds. In the case of the VO4 distorted tetrahedra, there is a significative difference in the values of ∇2*ρ*BCP1 and ∇2*ρ*BCP2 , which is related to the interatomic distances *d*V−O*<sup>n</sup>* (see Table 1) being significantly larger than those obtained for the FeO6 distorted octahedra. This is the reason for the low compressibility of the VO4 polyhedra. As the pressure grows, the Laplacian values increase, strengthening the Fe–O and V–O bonds and the resistance of the polyhedrons to compression.

**Figure 4.** Gradient vector field of the electron density (∇*ρ*) for (**a**) FeO6 and (**b**) VO4 polyhedra. (**c**) Evolution of the interatomic distances from the cations to the critical bond points and (**d**) Laplacian of the charge density at the critical bond point as a function of pressure for FeVO4.

#### *3.3. Elastic constants and mechanical properties*

The elastic constants (*cij*) determine the resistance to deformation in the directions and planes of a crystal, making them highly dependent on the system's symmetry. For the orthorhombic crystalline system, there are nine independent elastic constants: *c*11, *c*<sup>22</sup> and *c*<sup>33</sup> for longitudinal compression, *c*12, *c*<sup>13</sup> and *c*<sup>23</sup> for transverse expansion, *c*44, *c*55, and *c*<sup>66</sup> for pure shear [55]. Table 2 shows the elastic constants *cij* of FeVO4-**II** at 0 GPa and the respective pressure coefficients *dcij*/*dP*. The necessary and sufficient Born stability criteria for orthorhombic systems are [56]: *c*<sup>11</sup> > 0, *c*11*c*<sup>22</sup> > *c*<sup>2</sup> <sup>12</sup>, *c*11*c*22*c*<sup>33</sup> + <sup>2</sup>*c*12*c*13*c*<sup>33</sup> + <sup>2</sup>*c*12*c*13*c*<sup>23</sup> − *<sup>c</sup>*11*c*<sup>2</sup> <sup>23</sup> − *<sup>c</sup>*22*c*<sup>2</sup> <sup>13</sup> − *<sup>c</sup>*33*c*<sup>2</sup> <sup>12</sup> > 0, *c*<sup>44</sup> > 0, *c*<sup>55</sup> > 0, and *c*<sup>66</sup> > 0, which are fulfilled for FeVO4-**II**. Under hydrostatic pressure (*P*), the stability criteria are [57]: *cii* − *P* > 0 (*i* = 1 to 6), *c*<sup>11</sup> + *c*<sup>22</sup> + *c*<sup>33</sup> + 2*c*<sup>12</sup> + 2*c*<sup>13</sup> + 2*c*<sup>23</sup> + 3*P* > 0, *c*<sup>11</sup> + *c*<sup>22</sup> − 2*c*<sup>12</sup> − 4*P* > 0, *c*<sup>11</sup> + *c*<sup>33</sup> − 2*c*<sup>13</sup> − 4*P* > 0, and *c*<sup>22</sup> + *c*<sup>33</sup> − 2*c*<sup>23</sup> − 4*P* > 0, which are also meet. According to Table 2, the elastic constant *c*<sup>11</sup> presents the higher value, followed by *c*<sup>33</sup> and *c*22, indicating that longitudinally the crystal is more resistant to deformation in the *a*-axis; this was also observed in the evolution of the lattice parameters under pressure in Figure 2b. As would be expected, the largest transverse expansion constant is *c*12, followed by *c*<sup>13</sup> and *c*23, *c*<sup>33</sup> being the smallest of the nine elastic constants. At the same time, the shear elastic constants are generally small against those from longitudinal compression and transverse expansion. We found that elastic constants follow a linear trend with pressure, where only *c*<sup>22</sup> presents

a negative pressure coefficient related to the trends observed in the structural parameters with pressure.

**Table 2.** Elastic constants *cij* (in GPa) at 0 GPa and pressure coefficients.


The elastic constants and compliances were used to compute the bulk (*B*) and shear (*G*) modulus with the Hill procedure [58], which is the average of the Voigt (V) [59] and Reuss (R) [60] methods. The bulk modulus in the Voigt and Reuss methods is computed as follows: 9*B*<sup>V</sup> = (*c*<sup>11</sup> + *c*<sup>22</sup> + *c*<sup>33</sup> + 2*c*<sup>12</sup> + 2*c*<sup>13</sup> + 2*c*23), and *B*−<sup>1</sup> <sup>R</sup> = (*s*<sup>11</sup> + *s*<sup>22</sup> + *s*<sup>33</sup> + 2*s*<sup>12</sup> + 2*s*<sup>13</sup> + 2*s*23), respectively. While the shear modulus is obtained with the following equations: <sup>15</sup>*G*<sup>V</sup> = (*c*<sup>11</sup> <sup>+</sup> *<sup>c</sup>*<sup>22</sup> <sup>+</sup> *<sup>c</sup>*<sup>33</sup> <sup>−</sup> *<sup>c</sup>*<sup>12</sup> <sup>−</sup> *<sup>c</sup>*<sup>23</sup> <sup>−</sup> *<sup>c</sup>*<sup>13</sup> + 3*c*<sup>44</sup> + 3*c*<sup>55</sup> + 3*c*66), and 15*G*−<sup>1</sup> <sup>R</sup> = [4(*s*<sup>11</sup> + *s*<sup>22</sup> + *s*33) − 4(*s*<sup>12</sup> + *s*<sup>13</sup> + *s*23) + 3(*s*<sup>44</sup> + *s*<sup>55</sup> + *s*66)], respectively. *B* and *G* in the Hill average are obtained by means of *B* = (*B*<sup>V</sup> + *B*R)/2 and *G* = (*G*<sup>V</sup> + *G*R)/2, respectively. On the other hand, Young's modulus (*E*) and Poisson's ratio (*ν*) are given by *E* = 9*BG*/(3*B* + *G*) and *ν* = (3*B* − 2*G*)/(6*B* + 2*G*), respectively. The Vickers hardness was calculated with the approximation of Tian et al. [61]: *Hv* = 0.92 *k*1.137*G*0.708, *k* = *G*/*B*.

The mechanical properties and their evolution with pressure are provided in Figure 5. As seen, *B* increases almost linearly with pressure, which means that the resistance to deformation increases for this pressure range. Whereas *G* undergoes a change in the linear trend above 3 GPa, reaching a maximum value of 53.9 at 6.2 GPa and falling down to 53.3 GPa at 9.6 GPa. There is no significant change in *G* for this pressure range, but the change in trend will substantially affect the other properties that depend on *G*. Interestingly, in other materials, such change in the slope of the pressure dependence of the shear modulus *G* has been identified as a precursor effect of a phase transition observed at higher pressures [62]. According to the equations of Voigt and Reuss for *G*, the behavior of *G* is due to the negative pressure coefficient for *c*<sup>22</sup> and the significant values of *dcij*/*dP* for the transverse expansion elastic constants. On the other hand, *B* behaves linearly because *B*<sup>V</sup> (*B*G) only sums the longitudinal and transverse elastic constants (compliances), while *c*<sup>22</sup> (*s*22) decreases with pressure, so the value of *c*<sup>22</sup> affects *B* by a small constant factor.

The behavior of Young's modulus is similar to *G* due to its relation with this mechanical property. According to Figure 5c, *E* increases by 4.3 % in a pressure range of 7.86 GPa; above this pressure, it decreases from 138.1 to 137.6 GPa. This behavior indicates that FeVO4 could undergo a phase transition from a CrVO4-type phase to another one at a relatively small pressure since it has been observed that FeVO4 undergoes several phase transitions below 10 GPa, see Refs. [15,20] and references therein. The Poisson's ratio follows an increasing trend, taking values between 0.26 and 0.29, which are in the range for ceramic semiconductors (0.25–0.42) [63] and are similar for other *AB*O4 compounds [40,54,64]. These values indicate that ionic contribution dominates in FeVO4-**II**. The Pugh relation, Figure 5e, represents the ductility or brittleness of the system. A value below 1.75 [11] reveals a tendency to brittleness, while a value greater than this indicates that the material behaves in a ductile manner. FeVO4-**II** presents a value of 1.745 at ambient pressure and increases to 2.06 at 9.6 GPa, which means that ductility increases with pressure. As was expected, FeVO4-**II**, as other *AB*O4 compounds, presents a small Vickers hardness due to the relatively small values of *B* and *G*. Given the relation between *H*<sup>V</sup> and *B*/*G*, it is natural that the Vickers hardness decreases with pressure, Figure 5f.

**Figure 5.** Pressure dependence of mechanical properties obtained from elastic constants with the Hill average: (**a**) bulk modulus, *B*, (**b**) shear modulus, *G*, (**c**) Young's modulus, *E*, (**d**) Poisson's ratio, *ν*, (**e**) Pugh's ratio, *B*/*G*, and (**f**) Vickers hardness, *H*V, for FeVO4-**II**.

On the other hand, elastic anisotropy has important implications in engineering science and crystal physics because it is commonly responsible for the formation of microcracks in materials [65]. In this work, we analyzed several anisotropic factors discussed below. The anisotropic shear factors provide a measure of the degree of anisotropy in the bonding between atoms in different crystallographic planes. The shear anisotropic factor for the {100} shear planes between the <sup>011</sup> and <sup>010</sup> directions is *<sup>A</sup>*{100} = <sup>4</sup>*c*44/(*c*<sup>11</sup> + *<sup>c</sup>*<sup>33</sup> − <sup>2</sup>*c*13). For the {010} shear planes between <sup>101</sup> and <sup>001</sup> directions it is *<sup>A</sup>*{010} = <sup>4</sup>*c*55/(*c*<sup>22</sup> + *<sup>c</sup>*<sup>33</sup> − <sup>2</sup>*c*23). For the {001} shear planes between <sup>110</sup> and <sup>010</sup> directions, it is *<sup>A</sup>*{001} = <sup>4</sup>*c*66/(*c*<sup>11</sup> + *<sup>c</sup>*<sup>22</sup> − <sup>2</sup>*c*12). The factors *<sup>A</sup>*{100}, *<sup>A</sup>*{010}, and *<sup>A</sup>*{001} must be one for isotropic crystals, and any value greater or smaller than one measures the degree of elastic anisotropy. According to Figure 6a, *A*{100} presents the smallest deviation from one, followed by *A*{010} and *A*{001}. However, once pressure is applied, *A*{010} is more affected than the others. Factor *A*{010} is related to the equatorial interatomic distances of the distorted octahedra [*d*Fe−O2, see Figure 2c], which makes quite a bit of sense since it is *d*Fe−O2 that undergoes the greatest change with pressure. While *A*{001} is more related to *d*Fe−O1, with this interatomic distance being the second one most affected by pressure.

Other important factors are the universal anisotropy *AU* = 5*G*V/*G*<sup>R</sup> + *B*V/*B*<sup>R</sup> − 6, the percentage of elastic anisotropy in bulk [*AB* = (*B*<sup>V</sup> − *B*R)/(*B*<sup>V</sup> + *B*R)] and shear [*AG* = (*G*<sup>V</sup> − *G*R)/(*G*<sup>V</sup> + *G*R)] modulus, where a value of zero is associated with isotropic elastic constants, which means that *B*<sup>V</sup> = *B*<sup>R</sup> and *G*<sup>V</sup> = *G*R. For the entire range of pressure, *AB* is larger than *AG* because the difference between *B*<sup>V</sup> and *B*<sup>R</sup> is bigger than between *G*<sup>V</sup> and *G*R. However, the pressure affects more *AG* than *AB* as happens with *G*. This difference increases above 6.2 GPa as *G*<sup>R</sup> presents the same behavior of *G*, while *G*<sup>V</sup> is always increasing. *AU* is larger than *AG* and *AB* since the anisotropies of both modules are added.

**Figure 6.** (**a**) Pressure dependence of anisotropy factors: (**a**) by crystallographic direction, (**b**) percentage bulk and shear anisotropy, and (**c**) universal anisotropy for FeVO4-**II**.

#### *3.4. Vibrational Properties*

According to the group theory, the space group *Cmcm* has the following Raman, infrared (IR), and silent phonon modes at the zone center Γ = 5*Ag* + 4*B*1*<sup>g</sup>* + 2*B*2*<sup>g</sup>* + 4*B*3*g*, Γ = 6*B*1*<sup>u</sup>* + 7*B*2*<sup>u</sup>* + 5*B*3*u*, and Γ = 3*Au*, respectively. These modes can be further classified into internal (symmetric stretching, *ν*1, symmetric bending, *ν*2, asymmetric stretching, *ν*3, and asymmetric bending, *ν*4) and external (translational, *T*, and rotational, *R*) modes of VO4 units. The calculated Raman, IR, and silent phonon frequencies, the pressure coefficients (*dω*/*dP*), and the Grüneisen parameters (*γ* = -*∂*ln(*ω*)/*∂*ln*V*) at the Γ point appear in Table 3.

**Table 3.** Calculated Raman, infrared, and silent phonon frequencies (*ω* in cm<sup>−</sup>1), pressure coefficients (*dω*/*dp*), and Grüneisen parameters (*γ*) of FeVO4-**II**.


According to Table 3, one Raman, *R*(*B*1*g*), and two IR modes, *R*(*B*3*u*) and *ν*4(*B*2*u*), have negative pressure coefficients and Grüneisen parameters (soften with pressure), which are related to the instability of the CrVO4-type structure under pressure [2]. Such behavior was already observed in InVO4 [5] and CrVO4 [20]. The soft translational Raman *T*(*B*1*g*) mode involves a translation of the VO4 polyhedra accompanied by a rotation of the FeO6 units around the *z*-axis as rigid units, see Figure 7, whose freedom of movement is affected by the decrease in the lattice parameter *b*, which is the most compressible under pressure. In contrast, the remaining Raman modes harden with pressure. The IR soft *R*(*B*3*u*) mode is quite similar to the *R*(*B*1*g*), but in this case, the FeO6 vibrates asymmetrically. While the other soft *B*2*<sup>u</sup>* mode implies the asymmetric bending in the VO4 polyhedra with a translation of the Fe cation in the *c* direction, which produces a stretching of the apical *d*Fe−O1 interatomic distances. Therefore, the softening of R and IR modes is related to the compression of the *y*-axis. Interestingly, phosphates with CrVO4-type structure, such

as InPO4 and TiPO4, only present some IR soft modes. The main difference being that Ti(In)PO4 is more compressible in the *c*(*b*)-axis [7].

**Figure 7.** Eigenvectors of the phonon frequencies with negative pressure coefficients of FeVO4-**II**. We used the VESTA software [46] to build the structures.

The phonon dispersion relation and the phonon density of states (DOS) are displayed in Figure 8 at (a) ambient pressure and (b) 9.6 GPa. As can be seen in Figure 8a, the CrVO4-type structure of FeVO4 complains with the dynamic stability criteria at *P* = 0, which implies that *ω*<sup>2</sup> (**q**, *s*)> 0 for all wave vectors **q** and polarization *s* (longitudinal and transverse modes) [66]. Therefore, FeVO4-**II** complains with the elastic and dynamic stability criteria to be experimentally synthesized [19], such as InVO4 [5] and CrVO4 [20]. We can see no significant differences in the phonon spectrum at 9.6 GPa, which ensures that the FeVO4-**II** phase would be stable in a pressure range where the isostructural vanadates InVO4 and CrVO4 undergo a phase transition to the wolframite-type structure [5,20].

**Figure 8.** Phonon spectrum and phonon DOS at (**a**) 0 GPa, and (**b**) 9.6 GPa of FeVO4-**II**.

The phonon DOS of FeVO4-**II** is more similar to CrVO4 than InVO4 because In is more than double the mass of Cr and Fe. The partial phonon DOS of the first lower frequency zone (up to ≈ 505 cm−1) is due to external (*<sup>R</sup>* and *<sup>T</sup>*) and internal (*ν*<sup>2</sup> and *<sup>ν</sup>*4) phonon modes, with Fe and V similar contributions. In contrast, the two high-frequency zones are due to pure internal vibrations *ν*<sup>1</sup> and *ν*3, which indicates that phonon DOS is only due to O and V.

#### **4. Conclusions**

First-principles calculations were performed to study FeVO4 in the CrVO4-type structure up to 9.6 GPa. Our results indicate that this phase is elastically and dynamically stable for this pressure range. FeVO4 presents an indirect band-gap *Eg* = 2.11 eV, similar to the value reported for the triclinic phase of FeVO4 at ambient pressure. The Laplacian charge density at BCPs in both polyhedra is larger than those reported for *A*MoO4 and

*A*WO4 compounds and increases with pressure. It was observed that the *y*-axis is the most compressible, related to the reduction in the interatomic distances of the FeO6 polyhedra. These changes in the structural parameters have strong effects on the phonon modes with negative pressure coefficients. We found that the shear modulus *G* softens with pressure, which suggests that the FeVO4-**II** phase could undergo a phase transition below 10 GPa as was predicted in a previous study and happen in CrVO4 and InVO4, where these compounds have a phase transition from CrVO4-type to the wolframite structure. We hope this work encourages other research groups to perform high-pressure characterization studies with the diamond-anvil cell device in order to increase the knowledge of less studied orthovanadates with CrVO4-type structure, such as FeVO4 and TlVO4.

**Author Contributions:** Methodology, P.B.R.-V., S.L.-M.; calculations, P.B.R.-V., S.L.-M.; writing original draft preparation, P.B.R.-V., S.L.-M.; supervision, S.L.-M.; discussion, S.L.-M., D.E.; writing review and editing, P.B.R.-V., S.L.-M., D.E. All authors have read and agreed to the published version of the manuscript.

**Funding:** S.L.-M. thanks CONACYT of Mexico for financial support through the program "Programa de Investigadoras e Investigadores por México". D.E. thanks the financial support from the Spanish Ministerio de Ciencia e Investigación ( 10.13039/501100011033) under Project PID2019- 106383GB-41, as well as through the MALTA Consolider Team research network (RED2018-102612-T) and from Generalitat Valenciana under Grants PROMETEO CIPROM/2021/075-GREENMAT and MFA/2022/007.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** All relevant data that support the findings of this study are available from the corresponding authors upon request.

**Acknowledgments:** The authors gratefully acknowledge the computing time granted by LANCAD and CONACYT on the supercomputer Miztli at LSVP DGTIC UNAM. Furthermore, the IPICYT Supercomputing National Center for Education and Research, grant TKII-R2022-SLM1/PBRV1.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


## *Article* **Phase Relations of Ni2In-Type and CaC2-Type Structures Relative to Fe2P-Type Structure of Titania at High Pressure: A Comparative Study**

**Khaldoun Tarawneh \* and Yahya Al-Khatatbeh**

Department of Basic Sciences, Princess Sumaya University for Technology, Amman 11941, Jordan **\*** Correspondence: khaldoun@psut.edu.jo

**Abstract:** Density functional theory (DFT) based on *first-principles* calculations was used to study the high-pressure phase stability of various phases of titanium dioxide (TiO2) at extreme pressures. We explored the phase relations among the following phases: the experimentally identified nine-fold hexagonal Fe2P-type phase, the previously predicted ten-fold tetragonal CaC2-type phase of TiO2, and the recently proposed eleven-fold hexagonal Ni2In-type phase of the similar dioxides zirconia (ZrO2) and hafnia (HfO2). Our calculations, using the generalized gradient approximation (GGA), predicted the Fe2P → Ni2In transition to occur at 564 GPa and Fe2P → CaC2 at 664 GPa. These transitions were deeply investigated with reference to the volume reduction, coordination number decrease, and band gap narrowing to better determine the favorable post-Fe2P phase. Furthermore, it was found that both transitions are mostly driven by the volume reduction across transitions in comparison with the small contribution of the electronic energy gain. Additionally, our computed Birch–Murnaghan equation of state for the three phases reveals that CaC2 is the densest phase, while Ni2In is the most compressible phase.

**Keywords:** phase transitions; enthalpy difference components; equation of state; first principles; phase relations; band gap

#### **1. Introduction**

The nature of bonding in titania (TiO2) has attracted great interest over the last few decades due to its interesting industrial applications such as photocatalysts, energy generation and storage, environmental protection, and many more [1,2]. One of the important and promising research directions in studying this dioxide is investigating the high-pressure behavior of TiO2 polymorphs, both experimentally and theoretically, due to their interesting properties [3–39]. However, much attention, using measurements and calculations, has been given to exploring the high-pressure phase stability of TiO2 phases and the transition pressures between different phases and their equation of state parameters as well as to searching for new possible phases (e.g., [3,8,11,15,18,19,21,22,29,31–33,37]). Over the past decades, density functional theory has become the workhorse theory for the identification of the pressure-driven phase transitions [40,41].

As mentioned above, titania can be found in many structural forms with increasing pressure. In this regard, the well-known transition sequence that is experimentally observed and theoretically predicted is as follows: Orthorhombic OI → orthorhombic cotunnite OII → Fe2P-type [3,15,29,31]. Thus, the hexagonal Fe2P-type structure (Figure 1a) is the highest-pressure phase experimentally observed for TiO2 which has been found to be stable at (210 GPa, 4000 K) using diamond-anvil experiments [31]. Recently, the tetragonal CaC2-type structure (Figure 1c) has been theoretically predicted as a post-Fe2P phase of titanium dioxide at pressures beyond 647–689 GPa [33,36]. Additionally, recent density functional theory (DFT) calculations have predicted the hexagonal Ni2In-type structure

**Citation:** Tarawneh, K.; Al-Khatatbeh, Y. Phase Relations of Ni2In-Type and CaC2-Type Structures Relative to Fe2P-Type Structure of Titania at High Pressure: A Comparative Study. *Crystals* **2023**, *13*, 9. https://doi.org/10.3390/ cryst13010009

Academic Editors: Daniel Errandonea and Enrico Bandiello

Received: 27 November 2022 Revised: 17 December 2022 Accepted: 18 December 2022 Published: 21 December 2022

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

(Figure 1b) to be the post-Fe2P phase in the similar dioxides ZrO2 and HfO2 at pressures that exceed 300 GPa [42–44].

**Figure 1.** Crystal structures of TiO2 phases (generated using XCrySDen software [45]). The green spheres represent the titanium atom, while the alloy orange spheres represent the oxygen atom. (**a**) Fe2P-type: crystal structure: hexagonal, space group: *P*62*m*, coordination number: 9, lattice parameters: *a* = *b* = 5.3274 Å and *c* = 3.1234 Å, atomic coordinates: Ti1 (1/3, 2/3, 1/2), Ti2 (0, 0, 0), O1 (0.262, 0, 1/2), O2 (0.601, 0, 0). (**b**) Ni2In-type: crystal structure: hexagonal, space group: *P*63/*mmc*, coordination number: 11, lattice parameters: *a* = *b* = 3.2632 Å and *c* = 6.0335 Å, atomic coordinates: Ti (1/3, 2/3, 1/4), O1 (0, 0, 0), O2 (1/3, 2/3, 3/4). (**c**) CaC2-type: crystal structure: tetragonal, space group: *I4*/*mmm*, coordination number: 10, lattice parameters: *a* = *b* = 2.7407 Å and *c* = 6.7179 Å, atomic coordinates: Ti (1/2, −1/2, 0), O (1, −1, 0.152).

However, it is important to note that the three dioxides (TiO2, ZrO2, and HfO2) share obvious similarities, especially in their high-pressure behavior (e.g., see Ref. [26] and references therein). In this regard, the overlapping high-pressure phase transition sequence, experimentally observed and theoretically confirmed in the three dioxides is as follows: Monoclinic baddeleyite MI → OI → OII → Fe2P [3,15,26,29,31,46–50].

The obvious and close similarities of the high-pressure behavior of titania, zirconia, and hafnia have motivated us to test the stability of Fe2P-TiO2 with respect to CaC2 and Ni2In phases. In detail, until recently, the theoretical work performed on TiO2 predicted the Fe2P → CaC2 transition [33,36], while the Fe2P → Ni2In transition was predicted in ZrO2 and HfO2 [42–44]. Therefore, to better understand the upper part of the phase transition sequence in TiO2, we have performed DFT calculations to investigate the phase relations among Fe2P, CaC2, and Ni2In phases at megabar pressures. Consequently, the long-range target of this study is to draw the similarities and differences between TiO2 and other similar transition-metal dioxides in terms of predicting the ultrahigh-pressure phase transition sequence, which will hopefully lead to a better understanding of the high-pressure behavior of such dioxides.

#### **2. Computational Details**

In order to investigate the phase stability and the equations of state (EOSs) of all tested phases of TiO2, we used static *first-principles* computations performed within the framework of density functional theory (DFT) [51]. The projector-augmented wave (PAW) formalism [52,53] was used to treat the interactions between the titanium (Ti) and oxygen (O) atoms with the valence configuration of 3*s*23*p*63*d*24*s*<sup>2</sup> for Ti and 2*s*22*p*<sup>4</sup> for O. Following previous theoretical high-pressure studies carried out on TiO2 and similar dioxides [29,43,44,46–48], the electronic exchange and correlation effects were treated within the GGA [54]. We performed our calculations using the Quantum ESPRESSO package [55] with an energy cutoff of 80 Ry and Γ-centered *k*-point meshes [56]. Our calculations yielded sufficient convergence to better than 10−<sup>5</sup> Ry in the total energies for both phases, and pressures were converged to better than 0.1 GPa. The Brillouin zone integration was performed using the following *k*-point meshes for the ZrO2 phases: 8 × 8 × 12 for Fe2P, 12 × 12 × 8 for CaC2, and 20 × 20 × 16 for Ni2In. For a fixed volume, all internal degrees of freedom and unit-cell parameters of the structure were optimized simultaneously during the geometry optimizations. The ground-state energy for each phase was determined for 13–16 volumes, and the EOS parameters for each phase were obtained by fitting the total energy as a function of volume to a second-order Birch–Murnaghan equation of state (BM-EOS) [57] (Table 1). Phonon dispersion calculations have not been performed and will be the scope of future studies.

**Table 1.** Calculated equations of state for the Fe2P, Ni2In, and CaC2 phases of TiO2. Our EOS is determined from GGA calculations using the second-order BM-EOS [57]. For comparison, we list other calculated EOSs for TiO2 and the similar dioxides ZrO2 and HfO2. 1σ uncertainties are given in parentheses.


#### **3. Results and Discussion**

#### *3.1. Phase Stability and Equation of State*

We computed our EOS parameters for each phase by fitting the energy-volume (*E*-*V*) data to the following second-order BM-EOS [57] for which the first pressure derivative of the bulk modulus at zero pressure (*K*0') was fixed to 4 and the zero-pressure volume (*V*0) and the zero-pressure bulk modulus (*K*0) were used as fitting parameters. The determination of the EOS for the three phases allowed us to investigate their compressibilities and phase stability at high pressure as well as the volume change across different transitions.

The EOS parameters for all phases are summarized in Table 1 along with results from previous calculations [31,32]. We note that our calculated EOS for the Fe2P phase agrees well with previous studies [31,32]. On the other hand, the EOS of either Ni2In or CaC2 has not been previously obtained. In this regard, we should emphasize that the Ni2In-type structure has not been tested for TiO2, while CaC2 has been proposed for TiO2 [33,36]. Therefore, to our knowledge, we provide the EOS of the Ni2In and CaC2 phases of TiO2 for the first time (Table 1). When comparing the EOS for the three phases, we notice that Fe2P has the highest bulk modulus and thus is the most incompressible phase, followed by CaC2 and finally Ni2In that exhibits the most compressibility among all tested phases. In detail, a small bulk modus decrease (~7%) is predicted across Fe2P → CaC2, while we note an obviously large decrease (~39%) across Fe2P → Ni2In (Table 1). Furthermore, the density of CaC2 at zero pressure is higher than that of Fe2P, as expected, which is not the case for Ni2In where the density is less than that of Fe2P by ~9% (Figure 2). We should note that such predicted large changes in *K*<sup>0</sup> and *V*<sup>0</sup> across Fe2P → Ni2In are likely to be unexpected across transitions to higher-pressure phases, whereas the corresponding changes across Fe2P → CaC2 are much more reasonable.

**Figure 2.** Pressure versus volume of TiO2 phases as determined by GGA calculations using BM-EOS [57]. The dashed circles show the large volume reduction across the Fe2P → Ni2In and Fe2P → CaC2 transitions.

To obtain the transition pressures, we calculated the enthalpy change relative to the Fe2P phase (Figure 3). First, we note that the transition from Ni2In to CaC2 or vice versa is not possible at ultrahigh pressures as their enthalpy curves are unlikely to intersect. Furthermore, although the enthalpy curves of the two phases seem to cross at lower pressures, this transition is not expected to occur within the stability field of the Fe2P phase. Our enthalpy calculations indicate that the transition pressure across the Fe2P → Ni2In transition is 564 GPa. On the other hand, our calculated transition pressure across the Fe2P → CaC2 transition is 664 GPa, in agreement with previous findings (Table 2) [33,36]. We should note that although the Ni2In phase has a lower enthalpy than the CaC2 phase, this does not necessarily mean that Fe2P → Ni2In is the favorable transition when compared to the Fe2P → CaC2 transition. Consequently, there are two possible scenarios for the transition from the Fe2P phase, and thus one must be eliminated from the transition sequence of TiO2. Therefore, in the next sections, we further explore the two transitions to predict the favored post-Fe2P phase of TiO2.

#### *3.2. Enthalpy Difference and Volume Collapse across Phase Transitions*

In this section, we discuss the enthalpy difference across the two transitions and the contribution of the volume change in this difference as well as the correlation between the volume decrease and the coordination number increase. We note that the Fe2P → Ni2In and Fe2P → CaC2 transitions are associated with a large enthalpy difference (Figure 3, Table 3) across the phase transition. Our calculations reveal that such noticeable enthalpy difference is mainly due to the obvious volume reduction across both transitions (Figure 2, Table 3), while the contribution of electronic energy gain has a minimal effect.

Although both transitions are driven by a large enthalpy change as discussed above, it has been found that the Fe2P → Ni2In transition requires a larger enthalpy change compared to the Fe2P → CaC2 transition. This result is not unexpected and can be explained in view of the coordination number change across transitions; in this regard, we notice that the Fe2P → Ni2In is related to a 2-coordination-number increase (from 9 to 11) whereas Fe2P → CaC2 is driven by a 1-coordination-number increase (from 9 to 10).

**Figure 3.** Change in enthalpy with respect to Fe2P phase versus pressure of one formula unit as determined by GGA calculations for TiO2. The transition pressures across transitions from Fe2P to Ni2In and CaC2 phases are shown.

**Table 2.** Calculated transition pressures across the Fe2P → Ni2In and Fe2P → CaC2 phase transitions for TiO2. For comparison, we list other calculated results.


**Table 3.** Enthalpy difference, band gap difference, volume change, and coordination number change across the Fe2P → Ni2In and Fe2P → CaC2 phase transitions for TiO2.


However, regardless of the coordination number increase (either 1 or 2), the calculated volume collapse across both transitions does not show a large difference (3.7% for CaC2 vs. 5.6% for Ni2In), where Fe2P → Ni2In is expected to show a much larger volume reduction due to the large coordination increase across this transition. In detail, it has been previously found that for TiO2, the 2-coordination increase (from 7 to 9 in OI → OII transition) results in an ~ 8% volume reduction [29] compared to a 5.6% reduction for the same coordination increase in Fe2P → Ni2In. On the other hand, the 3.7% volume decrease in Fe2P → CaC2 transition looks more appropriate for a 1-coordination increase and agrees well with previous studies that reported values of 3.3–3.4% [33,36]. Therefore, based on the interplay between the volume reduction and coordination number change, the Fe2P → CaC2 transition is likely favorable in TiO2 when compared to the Fe2P → Ni2In transition.

#### *3.3. Band Gap Calculations at the Transition Pressures*

To further investigate the proposed ultrahigh-pressure phases of titania, we explored the pressure dependence of the band gap by analyzing the band structure of each phase at different pressures (Figures 4 and 5). Often, DFT calculations systematically underestimate band gaps; however, they accurately describe the pressure dependence [58,59]. We speculate that such analysis might be helpful in gaining a deeper insight into the Fe2P → Ni2In and Fe2P → CaC2 transitions to better determine the favorable phase transition. Our band gap calculations show that the band gap of the Fe2P-type structure at zero pressure is ~0.94 eV and decreases as pressure increases, where the drop in the band gap becomes more obvious at megabar pressures (Figure 4). However, regardless of such band-gap collapse, we should note that the Fe2P phase remains a semiconductor up to multi-megabar pressures before its metallization begins at pressures greater than 650 GPa (Figure 4), in good agreement with previous predictions obtained for Fe2P-TiO2 [33,36].

**Figure 4.** Calculated band gap of the Fe2P phase as a function of pressure. The vertical dashed lines indicate the transition pressures of the Fe2P → Ni2In and Fe2P → CaC2 transitions.

Additionally, our analysis of the band structures of the Ni2In and CaC2 phases reveals that both structures have metallic characteristics (band gap = 0 eV) at their predicted transition pressures from the Fe2P phase (Figures 4 and 5, Table 2). In this regard, we note that the band gap difference (Table 2, Figure 4) across Fe2P → CaC2 transition is zero (0 eV → 0 eV) compared to a ~0.04 eV difference across Fe2P → Ni2In transition (0.04 eV → 0 eV). Consequently, our band gap calculations likely provide further evidence that supports the Fe2P → CaC2 transition being the favorable transition when compared to the Fe2P → CaC2 transition. Thus, we confirm previous theoretical findings that predict CaC2 to be the post-Fe2P phase for TiO2 [33,36]. Finally, it should be noted that such confirmation is important since the Ni2In phase was a highly possible scenario for TiO2 as it has been predicted to be the post-Fe2P phase in the similar dioxides ZrO2 and HfO2 [42–44]. In this regard, we should emphasize that the three dioxides undergo the same transition sequence at high pressures (MI → OI → OII → Fe2P), but the post-Fe2P phase in TiO2 (i.e., CaC2-type) is predicted to be different from that of ZrO2 and HfO2 (i.e., Ni2In-type).

**Figure 5.** Electronic band structure for: (**Top**) the Fe2P phase at 0 GPa, 564 GPa, and 664 GPa; (**Bottom**) the Ni2In and CaC2 phases at 564 GPa and 664 GPa, respectively.

#### **4. Conclusions**

In conclusion, we employed DFT computations to investigate the ultrahigh-pressure phase stability of the Ni2In and CaC2 structures with respect to the Fe2P structure of titania. We explored the Fe2P → Ni2In and Fe2P → CaC2 transitions to better predict the favored post-Fe2P of TiO2. These transitions were thoroughly studied in terms of the volume decrease, the coordination number increase, and the band gap reduction. Our analysis favored the Fe2P → CaC2 transition, and thus we predict that the CaC2-type structure is likely the post-Fe2P phase of TiO2, and therefore the most stable phase of titania at pressures that exceed 664 GPa. This result is also evidenced by a similar conclusion we recently obtained for ZrO2, where the Fe2P → CaC2 transition was predicted to be more favorable than the Fe2P → Ni2In transition (unpublished), thus emphasizing the close similarities of the high-pressure behavior of titania and zirconia. Finally, our equation of state determination shows that CaC2 is the densest phase with a high bulk modulus that is comparable to that of the experimentally observed Fe2P phase, while the Ni2In phase reveals a low density and high compressibility.

**Author Contributions:** Conceptualization, Y.A.-K. and K.T.; methodology, Y.A.-K.; software, K.T.; formal analysis, Y.A.-K. and K.T.; writing—original draft preparation, K.T. and Y.A.-K.; writing—review and editing, Y.A.-K. and K.T. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Data Availability Statement:** The data presented in this study are available in the article.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


**Disclaimer/Publisher's Note:** The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

## *Article* **First-Principle Study of Ca3Y2Ge3O12 Garnet: Dynamical, Elastic Properties and Stability under Pressure**

**Alfonso Muñoz \* and Plácida Rodríguez-Hernández**

Departamento de Física, Instituto de Materiales y Nanotecnología, Malta Consolider Team, Universidad de La Laguna, 38200 La Laguna, Tenerife, Spain

**\*** Correspondence: amunoz@ull.edu.es; Tel.: +34-922-318-275

**Abstract:** We present here an ab initio study under the framework of the Density Functional Theory of the Ca3Y2Ge3O12 garnet. Our study focuses on the analysis of the structural, electronic, dynamic, and elastic properties of this material under hydrostatic pressure. We report information regarding the equation of state, the compressibility, and the structural evolution of this compound. The dynamical properties and the evolution under pressure of infrared, silent, and Raman frequencies and their pressure coefficients are also presented. The dependence on the pressure of the elastic constants and the mechanical and elastic properties are analyzed. From our results, we conclude that this garnet becomes mechanically unstable at 45.7 GPa; moreover, we also find evidence of soft phonons at 34.4 GPa, showing the dynamical instability of this compound above this pressure.

**Keywords:** ab initio; garnet; equation of state; elasticity; phonons; Raman

#### **1. Introduction**

The study of garnet minerals under high pressure is an essential topic in Earth Sciences; garnets are among the major components of Earth's deep interior. The physical properties of these materials at extreme conditions have an important effect on the dynamics of the mantle transition zone [1].

In technological applications, garnets are potential candidates for many devices. Their excellent thermal conductivity, hardness, mechanical and chemical stability, and high optical transparency suggest that these materials can be used as laser materials. Doping garnets with Rare Earth (RE3+) elements allows the use of their great luminescence properties. For example, YAG, a Y3Al5O3 garnet, used as a solid-state laser doped with Nd, is well-known in medical and commercial applications [2]. Under pressure, these garnets can be employed as sensors for high-pressure experiments [3,4], and can also be used with adequate doping as scintillators or light-emitting diodes [5]. Recently, a Ca3Y2Ge3O12 garnet doped with Cr+3 was studied as a near-infrared LED combined with a 455 nm LED chip [6]. In addition, the near-infrared phosphor Ca3Y2Ge3O12: Cr+3 garnet has been synthesized, showing a broadband NIR emission of 700–1100 nm, peaking around 800 nm [7].

The oxide garnets have the general formula A3B2C3O12, where A denotes the dodecahedral, B the octahedral, and C the tetrahedral sites with different coordination. They usually crystallize in the body-centered cubic (bcc) structure (*Ia3d* space group), with a unit cell of 160 atoms and a primitive cell of 80 atoms. In our case, Ca atoms (A) occupy the 24 *c* positions with coordination 8, Y atoms (B) occupy the 16 *a* positions with coordination 6 and octahedral point symmetry (C3i), and Ge atoms (C) occupy the 24 *d* positions with coordination 4 and tetrahedral point symmetry (S4). Finally, the O atoms occupy the 96 *h* positions (Figure 1).

#### **Citation:** Muñoz, A.;

Rodríguez-Hernández, P. First-Principle Study of Ca3Y2Ge3O12 Garnet: Dynamical, Elastic Properties and Stability under Pressure. *Crystals* **2023**, *13*, 29. https://doi.org/ 10.3390/cryst13010029

Academic Editor: Claudio Cazorla

Received: 5 December 2022 Revised: 19 December 2022 Accepted: 21 December 2022 Published: 24 December 2022

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

**Figure 1.** The conventional unit cell with polyhedral units CaO8 (green), YO6 (blue), and GeO4 (orange).

Due to the large number of atoms in the unit cell and hence the complexity of the simulation, few ab initio studies of the structural, electronic, dynamical, or mechanical properties both at room and high pressures are available for garnets [8,9].

In the last decades, the use of Density Functional Theory (DFT) simulations of material under high pressure has become a very well-established technique. The quality and precision of these studies make possible the prediction of many properties of materials under extreme conditions, providing information on the stability of these compounds.

The application of pressure allows modifications of the bonds and interatomic distances of material and, therefore, changes in their elastic and vibrational properties. On the other hand, the optical properties of Re3+ doped oxides garnets are affected by changes in the host structure due to the reduction in interatomic distances and possible coordination changes. Thus, it is worth studying garnets' structural and electronic properties under pressure to help understand their luminescence properties and improve their practical applications. A study of this garnet under pressure is required to explore potential applications of the Ca3Y2Ge3O12 garnet, for example, as a pressure optical sensor. The knowledge of its mechanical properties under pressure is also essential in geological studies due to the abundance of this mineral phase in the upper mantle and Earth's transition zone [10].

Since it is important to study how the properties of garnets evolve under high pressure, in this work, we report an extensive first-principle study of the electronic, structural, dynamic, and elastic properties of the Ca3Y2Ge3O12 garnet under pressure. To the best of our knowledge, there are no reported ab initio studies of this compound under high pressure. We hope the information provided in this work will stimulate further experimental studies on this garnet.

#### **2. Simulation Details**

Ab initio total energy simulations at zero temperature were performed in the framework of the Density Functional Theory, DFT, with the plane-wave method and the pseudopotential theory implemented in the Vienna ab initio simulations package [11,12]. The Projector Augmented-Wave (PAW) [13] pseudopotentials provided in this package were employed to take into account the full nodal character of the all-electron charge density in the core region. For the Ca atoms, 8 valence electrons (3p64s2) were included, for Y atoms

11 valence electrons (4s24p65s24d1), and for Ge atoms 14 valence electrons (3d104s24p2), whereas for O atoms 6 valence electrons (2s22p4) were used. The basis set included plane waves up to a cut-off energy of 520 eV in order to achieve highly converged results. The exchange-correlation energy was described with the generalized gradient approximation, GGA, using the PBEsol functional [14]. The integrations over the Brillouin zone, BZ, were performed with a 4 × 4 × 4 Monkorts-Pack grid [15] k-points sampling to obtain accurate converged energies and forces. At a set of selected volumes, the structure was fully optimized to obtain its equilibrium configuration through the calculations of forces and stress tensors. The convergence criteria were to achieve forces smaller than 0.004 eV/Å and deviations of stress tensor diagonal lower than 0.1 GPa. The data set of volume (V) and energy (E) obtained as result of our simulations was fitted with a standard equation of state, EOS, to determine the bulk modulus and its pressure derivative.

Lattice-dynamic calculations of the phonon modes were carried out at the zone center (Γ point) of the BZ with the direct force-constant approach [16]. These calculations provide the frequency of the normal modes, their symmetry, and their polarization vectors. This allowed to identify the irreducible representations and the character of the phonon modes at the Γ-point. The study of the dynamical stability was performed through the phonon density of state and the phonon dispersion calculated using a 2 × 2 × 2 supercell.

The mechanical and elastic properties were studied by obtaining the elastic constants with the strain–stress method implemented in the VASP code following Le Page method [17].

#### **3. Results and Discussion**

#### *3.1. Structural Properties under Pressure*

From our ab initio simulations, the obtained unit cell volume at zero pressure is 2095.5 Å<sup>3</sup> with a lattice constant of a0 = 12.7966 Å, and the O 96 *h* positions are defined by x = 0.9646, y = 0.0573, and z = 0.1610 (experiment a = 12.8059 Å, O 96h positions: 0.9637, 0.0567, and 0.1609) [18]. Our results are in excellent agreement with the available experimental data, and they differ by less than 0.3%. The Ca, Y, and Ge atoms occupy fixed Wyckoff positions already described in a previous section. In Table 1, we report the calculated structural parameter, the bulk modulus, and its pressure derivative. The bulk modulus, B0 = 111.24 GPa, with a pressure derivative B0' of 4.75, was obtained from the theoretical energy and volume data using a Birch–Murnaghan EOS [19]. This bulk modulus is smaller than those of most silicate garnets (between 150 and 190 GPa) [10,20], due to the smaller ionic radius of Ca.

**Table 1.** The lattice constant, a0, the volume, V0, at cero pressure, and the bulk modulus, B0, and its pressure derivative B0'.


<sup>1</sup> Reference [18].

The volume versus pressure evolution of the Ca3Y2Ge3O12 garnet is presented in Figure 2, and the inset shows the lattice constant evolution. The linear axial compressibility, ka = 0.255 × <sup>10</sup>−<sup>3</sup> GPa<sup>−</sup>1, was obtained from these data.

**Figure 2.** Pressure dependence of the volume. The inset shows the pressure dependence of the lattice constant.

The structural changes of Ca3Y2Ge3O12 under pressure are illustrated in Figures 3 and 4. The pressure dependence of the interatomic distances is shown in Figure 3. It can be observed that even at zero pressure, CaO8 dodecahedra are slightly distorted with two different Cadod-O distances (2.4587 and 2.5489 Å), while Yoct-O and Getet-O distances are 2.2327 and 1.7739 Å, respectively. When compared to distances in Sr3Y2Ge3O12, a garnet of the same family, the main difference is that Cadod-O distances are slightly lower than Srdod-O ones (5-3 %), while the Yoct-O and Getet-O distances are approximately the same in both compounds [21]. All the interatomic distances decrease with increasing pressure. However, whereas Yoct–O and Getet-O distances decrease at similar rates, −3.8 × <sup>10</sup>−<sup>3</sup> Å/GPa and −2.1 × <sup>10</sup>−<sup>3</sup> Å/GPa, respectively, one of the distances between Ca and O atoms varies more significantly than the other. The smaller distance (Cadod-O)2 changes at a rate of −4.95 × <sup>10</sup>−<sup>3</sup> Å/GPa, while the larger (Cadod-O)1 distance changes faster, at about −11.7 × <sup>10</sup>−<sup>3</sup> Å/GPa. Both become equal at around 15.9 GPa. This effect is common among garnets, e.g., the Lu-O distances become equal in Lu3Ga5O12 [22] or the Y-O distances in Y3Al5O12 [9].

The evolution with pressure of the polyhedral units of Ca3Y2Ge3O12 are plotted in Figure 4. The CaO8 dodecahedra are the polyhedra with the major reduction in volume: 23.4% in the pressure range under study. In the case of the GeO4 tetrahedra, the volume is almost constant along the pressure range with a small change of −0.09%. Finally, the YO6 octahedra volume decreases by 13.9%.

The volume changes of the CaO8 dodecahedra seem to be responsible for most of the volume reduction induced by pressure in Ca3Y2Ge3O12. In fact, the GeO4 tetrahedra can be treated as rigid units within the structure.

**Figure 3.** Evolution of interatomic distances as a function of pressure. Cadod stands for Ca atoms with dodecahedral coordination, Yoct for Y atoms with octahedral coordination, and Getet for Ge atoms with tetrahedral coordination.

**Figure 4.** Pressure evolution of the polyhedral units: CaO8 (green), YO6 (blue), and GeO4 (orange).

#### *3.2. Electronic Structure*

The electronic structure of Ca3Y2Ge3O12 has been scarcely investigated. Even though Baklanova et al. [23] have studied the electronic structure of this garnet at ambient pressure, the electronic properties under pressure are still unknown. Therefore, we performed band structure calculations at zero and different pressures of the Ca3Y2Ge3O12 garnet.

Figure 5 shows the electronic band structure at zero pressure. The valence band maxima and the conduction band minima are located at the Γ point, with a direct band gap of Eg = 3.33 eV. This value is in good agreement with the DFT result of 3.32 eV given in reference [23]. In the same reference, the experimental value obtained for the band gap is 5.56 eV, larger than the calculated value. As it is well known, the energy gap is underestimated by DFT compared with the experiments, although the order of magnitude of the pressure coefficients is in agreement.

**Figure 5.** The band structure of Ca3Y2Ge3O12 along high symmetry directions in the Brillouin zone, at zero pressure.

As observed, the dispersion of the upper valence bands, VB, is small, similar to other garnets [9,22]. On the other hand, at the Γ point, the band at the bottom of the conduction bands, CB, is well separated with respect to the other conduction bands. To understand the nature of the electronic structure of Ca3Y2Ge3O12, we analyzed the total density of states (DOS) and the projected density of state (PDOS). The PDOS corresponding to the top of the VBs and bottom of CBs, from −2 to 7eV (Figure 6), agrees with previous results [23]. O-2p states dominate the top of the valence bands. The bottom of the CBs is dominated by Y-4d states, with a contribution of O-2p states and a minor contribution of O-2s states near 5.5 eV.

As stated above, although the DFT calculations underestimate the band gap energy, they provide a good description of the pressure dependence of the band gap. To study the effect of pressure on electronic properties, we obtained the band structure of this compound under pressure. According to our calculations, the band gap remains direct in all the stability ranges and increases with a pressure derivative of 29.5 meV/GPa (Figure 7).

**Figure 6.** The total (DOS) and partial (PDOS) density of states of Ca3Y2Ge3O12, at zero pressure. The black curve represents the DOS.

**Figure 7.** The pressure dependence of the band gap energy of Ca3Y2Ge3O12.

#### *3.3. Elastic Properties*

Since Ca3Y2Ge3O12 has a cubic structure, it has, therefore, only three independent elastic constants: C11, C12, and C44. The elastic constants allow us to obtain the elastic moduli, bulk modulus (B), shear modulus (G), Young's modulus, and Poisson's ratio (ν) that describe the significant elastic properties of a compound along with the B/G ratio.

Cubic crystals the bulk modulus, B, are defined as the following:

$$\mathcal{B} = \frac{\mathcal{C}\_{11} + 2\mathcal{C}\_{12}}{3}$$

The shear modulus in the scheme of Hill [24], GH, can be expressed as the average between the upper bond, in the scheme of Voigt, Gv, and the lower bond, in scheme of Reuss, GR, [25,26]. In the present case [8,27]:

$$\mathbf{G\_{H}} = \frac{1}{2} \left[ \frac{\mathbf{C\_{11}} - \mathbf{C\_{12}} + 3\mathbf{C\_{44}}}{5} + \frac{5\mathbf{C\_{44}}(\mathbf{C\_{11}} - \mathbf{C\_{12}})}{4\mathbf{C\_{44}} + 3(\mathbf{C\_{11}} - \mathbf{C\_{12}})} \right]$$

The Young's modulus and the Poisson's ratio can be defined as the following:

$$\mathbf{E\_{x}} = \frac{9\mathbf{B\_{x}}\,\mathbf{G\_{x}}}{3\mathbf{B\_{x}} + \mathbf{G\_{x}}}$$

and

$$\mathbf{v}\_{\mathbf{x}} = \frac{\mathbf{E}\_{\mathbf{x}} - 2\mathbf{G}\_{\mathbf{x}}}{2\mathbf{G}\_{\mathbf{x}}}$$

where the subscript x refers to the symbols V, R, and H.

The computed elastic constants, Cij, and the elastic moduli at 0 GPa are presented in Tables 2 and 3. The bulk modulus, the inverse of the compressibility, is related to the material resistance to hydrostatic pressure. The bulk modulus computed through the elastic constants, 112.430 GPa, is in excellent agreement with the value obtained with the EOS using total energy calculations, 111.24 GPa. This is an indication of the quality and consistency of our simulations. The Young modulus, 129.94 GPa, provides a measure of the material stiffness. Young's modulus, Poisson's ratio, and the B/G ratio are essential for engineering applications. B/G is 2.26; therefore, according to the Pugh criterion (B/G > 1.75), Ca3Y2Ge3O12 behaves like a ductile material at 0 GPa [28]. The Poisson's ratio is ν = 0.31; hence, the ionic bonding is predominant against the covalent bonding, and the interatomic forces can still be considered predominantly central [29,30].

**Table 2.** Elastic constants in GPa of Ca3Y2Ge3O12 at 0 Gpa.


**Table 3.** Elastic moduli B, E, and G, the B/G and the Poisson's ratio, ν, in the Voigt, Reuss, and Hill approximations, at 0 GPa.


This compound is stable at zero pressure; hence, the elastic constants fulfill the Born stability criteria [31]:

$$\mathbb{C}\_{11} - \mathbb{C}\_{12} > 0, \mathbb{C}\_{11} + 2\mathbb{C}\_{12} > 0, \text{ and } \mathbb{C}\_{44} > 0$$

However, when hydrostatic pressure is applied to the crystal, the above criteria to describe the stability limits of a cubic crystal are not adequate. Moreover, the relevant magnitudes that describe the elastic properties are the elastic stiffness coefficients, Bij, related to the elastic constants through the following relationships [32,33]:

$$\mathbf{B}\_{11} = \mathbf{C}\_{11} - \mathbf{P}\_{\prime} \ \mathbf{B}\_{12} = \mathbf{C}\_{12} + \mathbf{P}\_{\prime} \ \text{and} \ \mathbf{B}\_{44} = \mathbf{C}\_{44} - \mathbf{P}\_{\prime}$$

The evolution of the Bij with pressure is plotted in Figure 8. The diagonal coefficient B11 and the B12 coefficient increase with pressure in the studied pressure range. However, the diagonal coefficient B44 decreases rapidly, becoming negative at about 45 GPa. B11 is higher than B44; this indicates that the resistance to a shear deformation is weaker than the resistance to compression.

**Figure 8.** Pressure evolution of the elastic stiffness coefficients, Bij.

The elastic stiffness coefficients enable us to study the evolution of the elastic properties under pressure. Replacing the Cij elastic constants with the Bij elastic stiffness coefficients in the above expressions, we obtain the analytical expressions for the elastic moduli, valid under pressure [31,34]. The pressure dependence of the elastic moduli with pressure up to 50 GPa is displayed in Figure 9. The bulk modulus increases almost linearly with increasing pressure. Both the shear modulus and the Young modulus decrease, and their values converge at high pressure. The Poisson's ratio becomes larger, and this can be interpreted as an increase in metallization. The B/G ratio indicates that the material is ductile until approximately 45 GPa; above this pressure, there is a dramatic change in the slope (Figure 10a). This could indicate a possible mechanical instability at higher pressures.

To study the mechanical stability of a crystal under pressure, the generalized Born stability criteria must be employed [31,35]. In the present case, as stated above, these criteria can be written as the following:

$$\mathbf{M}\_1 = \mathbf{B}\_{11} - \mathbf{B}\_{12} > 0, \; \mathbf{M}\_2 = \mathbf{B}\_{11} + 2\mathbf{B}\_{12} > 0, \; \mathbf{M}\_3 = \mathbf{B}\_{44} > 0 \tag{1}$$

These generalized stability criteria are plotted as a function of pressure in Figure 10b. The M3 stability criterion is violated at 45.7 GPa. This result indicates that the Ca3Y2Ge3O12 garnet is mechanically unstable above this pressure. Experimental studies on garnets report that an amorphous phase appears at high pressure [8,22,36]. However, in the present case, as we will see in the next section, the Ca3Y2Ge3O12 garnet becomes dynamically unstable at pressure over 34.4 GPa.

**Figure 9.** (**a**) Pressure evolution of the elastic moduli: bulk modulus, B, Young Modulus, E, and shear modulus, G. The blue line represents the moduli in the Voigt approximation, the green line in the Reuss approximation, and the red one in the Hill approximation. (**b**) Pressure evolution of the Poisson's ratio in the three approximations: Voigt, Reuss, and Hill.

**Figure 10.** (**a**) Pressure evolution of the B/G ratio in the approximations of Voigt, Reuss, and Hill. (**b**) Generalized stability criteria as a function of pressure.

#### *3.4. Vibrational Properties*

The Ca3Y2Ge3O12 garnet belongs to the *Ia-3d* space group and, according to group theory, has 98 vibrational modes. At the zone center, they can be classified as 25 Raman active modes (3A1g + 8Eg +14T2g), 17 infrared active modes (17T1u), and 55 silent modes (16T2u + 14T1g + 5A2u + 5A2g + 10Eu + 5A1u), which are optically inactive.

Figure 11 shows the phonon dispersion. It can be observed that the phonon spectrum could be divided into three regions. The low-frequency region ranges up to approximately 400 cm−<sup>1</sup> and is separated by a small gap from the medium-frequency region, which is

very narrow and reaches up to 510 cm <sup>−</sup>1. The high-frequency region extends from 625 to 780 cm<sup>−</sup>1.

**Figure 11.** Phonon dispersion of Ca3Y2Ge3O12 garnet at 0 GPa along the high symmetry directions.

The Raman modes are connected with the vibration of the different polyhedral units strongly coupled to each other; therefore, the attribution of each mode to a single unit is not straightforward. To analyze the dynamical contribution of each atom, the total phonon density of states (DOS) and the partial density of states (PDOS) were calculated (see Figure 12). As shown, in the case of Ca3Y2Ge3O12, the PDOS is very compact at low and medium frequencies, between 100 and 510 cm−1, and also in the high-frequency region between 625 and 780 cm−1. The contributions of Y atoms are located at the lowand medium-frequency regions, mainly up to 200cm<sup>−</sup>1. Ca atoms also contribute in these regions, although the contribution is small in the region of very low frequencies. Ge atoms have a small contribution in the region from 200 to 500 cm−<sup>1</sup> and a larger one in the highfrequency region. Finally, the O atoms have more significant contributions, spread over all frequencies, mainly in the high-frequency region.

**Figure 12.** Total and partial phonon density of states at zero pressure. The black line represents the total DOS, and the contributions of O, Ca, Ge, and Y atoms are represented by the red, blue, magenta, and green lines, respectively.

The pressure dependence of the calculated frequencies has been fitted with a secondorder polynomial. All theoretical Raman active modes are reported at zero pressure in Table 4, along with the pressure derivative coefficients, dω/dP, and the Grüneisen parameters, γ = (dω/dP)/(B0/ωo), where B0 is the bulk modulus and ω<sup>o</sup> the frequency of the corresponding mode at zero pressure.

**Table 4.** Wavenumbers of Raman active vibrational frequencies at zero pressure, ω0, their pressure coefficients, dω/dP, and Grüneisen parameters, γ, of Ca3Y2Ge3O12 garnet.


The evolution of the Raman modes of this garnet with pressure is presented in Figure 13. Most modes harden under compression; however, the frequencies of the lowest Raman modes change slightly. The modes on the low- and medium-frequency regions are related to movements of Ca, Ge, Y, and O atoms and exhibit very different pressure coefficients, since they involve several types of bonds with diverse compressibility. Due to the significant differences between the pressure dependence of these modes, crossing and anti-crossing phenomena are observed in the medium-frequency region. The highest modes, however, increase with pressure with similar coefficients and exhibit similar and low Grüneisen parameters.

**Figure 13.** Theoretical evolution with pressure of Raman modes of the Ca3Y2Ge3O12 garnet.

Additionally, the pressure evolution of the infrared modes is plotted in Figure 14. As in the case of Raman modes, there is clearly a large gap between the medium- and high-frequency region. The phonons in the medium- and high-frequency regions harden under pressure, and there are several anti-crossing phonons in the medium region. Except for the lowest mode (ω<sup>0</sup> = 67.96 cm<sup>−</sup>1), all the phonons in the low-frequency region are very slightly affected by pressure. The frequency values, pressure derivative, and Grüneisen parameters for the infrared modes are reported in Table 5.

**Figure 14.** Theoretical evolution with the pressure of infrared modes of the Ca3Y2Ge3O12 garnet.


**Table 5.** Infrared active vibrational modes, ω0, pressure coefficients, dω/dP, and Grüneisen parameters, γ, of Ca3Y2Ge3O12 garnet.

Finally, the pressure evolution of the 55 silent modes is presented in Figure 15. The most crucial point is the softening of the lower silent mode (T2u).

**Figure 15.** Theoretical evolution with the pressure of silent modes of Ca3Y2Ge3O12 garnet.

At 34.4 Gpa, this T2u mode becomes imaginary, as shown in Figure 16. It is important to note that this indicates a dynamical instability at 34.4 GPa. Therefore, our results evidence that, under hydrostatic pressure, this compound becomes dynamically unstable.

**Figure 16.** Theoretical evolution with pressure of silent mode T2u of the Ca3Y2Ge3O12 garnet.

We analyzed this soft phonon and obtained a triclinic *P-1* structure with 80 atoms in the cell at positions *1a* (x y z). By studying the stability of this new structure, we obtained that it is also dynamically unstable, so we discard it as a possible high-pressure phase of this garnet.

#### **4. Conclusions**

An ab initio study of the structural, electronic, elastic, and vibrational properties of the Ca3Y2Ge3O12 garnet was performed both at ambient and under hydrostatic pressure.

The lattice parameter compares nicely with the experimental value at 0 GPa. The evolution of the lattice parameter and interatomic distances under compression, the equation of state, and the compressibility are reported. The dodecahedral units (CaO8 in the present case) that account for most of the volume changes are distorted and behave under pressure in a common way among garnets.

The electronic band structure agrees with previous theoretical data and shows a direct band gap at Γ that increases with increasing pressure.

The elastic constants and elastic stiffness coefficients were accurately determined and the elastic behavior of the Ca3Y2Ge3O12 garnet was analyzed. The pressure evolution of B, G, and E elastic moduli and the Poisson's ratio were reported. The B/G relation indicates that the material is ductile in the whole pressure stability range and more sensitive to volume compression than to shear deformation (B>G). The study of the stability criteria indicates that the garnet is mechanically unstable above 45.7 GPa.

The total and partial phonon density of states were analyzed to determine each atom's contribution to the different vibrational modes. The pressure evolutions of the Raman, infrared, and silent modes were presented. At 34.4 GPa, the lower silent mode (T2u) softens and becomes imaginary; this means that the garnet structure is unstable at a pressure above 34.4 GPa.

**Author Contributions:** Both authors contributed equally to this work: simulations, analysis and interpretation of results, and manuscript writing, A.M., P.R.-H. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the Spanish Research Agency (AEI) and the Spanish Ministry of Science and Investigation (MCIN) under grant PID2019-106383GB-43 (DOI: 10.13039/501100011033) and RED2018-102612-T (MALTA Consolider-Team Network).

**Data Availability Statement:** All relevant data that support the findings of this study are available from the corresponding authors upon request.

**Acknowledgments:** The authors thank the financial support from the Spanish Research Agency (AEI) and Spanish Ministry of Science and Investigation (MCIN) under grant PID2019-106383GB-43 (DOI: 10.13039/501100011033) and RED2018-102612-T (MALTA Consolider-Team Network).

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


**Disclaimer/Publisher's Note:** The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

## *Article* **Structural and Luminescence Properties of Cu(I)X-Quinoxaline under High Pressure (X = Br, I)**

**Javier Gonzalez-Platas 1,\*, Ulises R. Rodriguez-Mendoza 2, Amagoia Aguirrechu-Comeron 3, Rita R. Hernandez-Molina 4, Robin Turnbull 5, Placida Rodriguez-Hernandez <sup>2</sup> and Alfonso Muñoz <sup>2</sup>**


**Abstract:** A study of high-pressure single-crystal X-ray diffraction and luminescence experiments together with ab initio simulations based on the density functional theory has been performed for two isomorphous copper(I) halide compounds with the empirical formula [C8H6Cu2*X*2N2] (*X* = Br, I) up to 4.62(4) and 7.00(4) GPa for X-ray diffraction and 6.3(4) and 11.6(4) GPa for luminescence, respectively. An exhaustive study of compressibility has been completed by means of determination of the isothermal equations of state and structural changes with pressure at room temperature, giving bulk moduli of *K*<sup>0</sup> = 14.4(5) GPa and *K* <sup>0</sup> = 7.7(6) for the bromide compound and *K*<sup>0</sup> = 13.0(2) GPa and *K* <sup>0</sup> = 7.4(2) for the iodide compound. Both cases exhibited a phase transition of second order around 3.3 GPa that was also detected in luminescence experiments under the same high-pressure conditions, wherein redshifts of the emission bands with increasing pressure were observed due to shortening of the Cu–Cu distances. Additionally, ab initio studies were carried out which confirmed the results obtained experimentally, although unfortunately, the phase transition was not predicted.

**Keywords:** equation of state; luminescence; high pressure; phase transition; copper halides

#### **1. Introduction**

Coordination complexes based on copper(I) halides with N-donor ligands have been some of the most attractive and widely studied complexes over the last three decades [1–8]. This has been due to the variety of physical and chemical properties exhibited by the copper(I) halide complexes, such as photo- and electroluminescence, nonlinear optics, and electrical conductivity. In particular, the electrical conductivity properties lead to many potential technological applications [9–17], such as light-emitting diodes (LEDs) [18], organic light-emitting diodes (OLEDs) [19–22], simpler light-emitting electrochemical cells (LECs) [23–27], biosensors [28], or solar energy conversion [29–31].

Systems based on these copper(I) halide complexes are very interesting because copper (Cu) is more abundant and less expensive than noble or rare earth metals. Copper halides also exhibit remarkable structural diversity [2,4,32]. Specifically, the *d*<sup>10</sup> electron configuration of Cu<sup>I</sup> does not lead to a pre-defined spatial configuration of the ligands; rather, the coordination sphere is determined by molecular mechanics and electrostatic factors. The halide ions exhibit four pairs of electrons in the outer shell and can therefore coordinate with four Cu<sup>I</sup> ions. When a nitrogen-centered Lewis base is added to a CuI halide, the

**Citation:** Gonzalez-Platas, J.; Rodriguez-Mendoza, U.R.; Aguirrechu-Comeron, A.; Hernandez-Molina, R.R.; Turnbull, R.; Rodriguez-Hernandez, P.; Muñoz, A. Structural and Luminescence Properties of Cu(I)X-Quinoxaline under High Pressure (X = Br, I). *Crystals* **2023**, *13*, 100. https:// doi.org/10.3390/cryst13010100

Academic Editors: Alessandra Toncelli, Daniel Errandonea and Enrico Bandiello

Received: 14 December 2022 Revised: 29 December 2022 Accepted: 3 January 2023 Published: 5 January 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

coordination number of the halide ions is reduced, resulting in a cluster of Cu<sup>I</sup> salt (CuX)n. This can give rise to interesting geometries such as mononuclear copper(I) complexes [33], square dimers [34], cubane tetramers [35], or polymers in one or two dimensions [36–39].

Regarding the photo-physical properties of metal complexes, they are intimately related to the electronic configuration of the metal center and the ligands around it. Cu<sup>I</sup> with a *d*<sup>10</sup> electronic configuration has a completely filled *d*-subshell, which excludes the possibility of *d–d* electronic transitions. Therefore, the luminescence of these *d*<sup>10</sup> copper(I) complexes arises from other types of electronic transitions, including metal-to-ligand charge transfer (MLCT), ligand-to-metal charge transfer (LMCT), and ligand-to-ligand charge transfer (LLCT). The presence of other types of interactions is also possible. For example, transitions involving the CuX cluster core, whereby a combination of halideto-metal charge transfer (XMCT) and metal-cluster-centered d10 → <sup>d</sup>9s1 (MCC) charge transfer occurs due to the weak attractive *d*10–*d*<sup>10</sup> interactions between the CuI ions in the cluster. In this latter case, known as cuprophilic interaction, the intensity of the interaction is directly connected with the Cu–Cu distances, which play an important role. Cuprophilic interaction has been observed experimentally in systems which exhibit luminescence for Cu–Cu distances shorter than twice the Cu van der Waals radius (2.8 Å). In addition, the *d*<sup>10</sup> configuration favors tetrahedral four-coordinate environments of the ligands around the metal center Cu<sup>I</sup> [40]. Pressure is a valuable thermodynamic variable which can be used as a tool to induce, in a continuous way, changes in the crystal structure and the immediate coordination complex environment, thereby provoking changes in the emission spectra [41,42]. Therefore, high-pressure techniques allow us to relate structural changes with optical properties.

Despite the large number of studies into copper(I) halides, there has historically been a deficit of research into these systems under extreme conditions of temperature or pressure, with still fewer publications utilizing ab initio studies for comparison with experimental results. Pressure and temperature are external stimuli which both affect intermolecular interactions, molecular packing, and structural parameters, thereby affecting emergent physical and chemical properties. Knowledge of how the properties of copper(I) halide complexes are affected by these different external stimuli is essential for technological applications [15,43]. In recent years, this situation seems to be changing with the publication of studies into how temperature [12,44,45] and pressure [31,36–39,46,47] affect the luminescent properties of CuI iodide complexes.

In this work, we present the pressure-induced changes in the crystal structure and luminescence properties of the 2D isomorphous coordination polymers [(CuX)2(Quin)] (X = Br, I and Quin = quinoxaline) at room temperature.

#### **2. Materials and Methods**

#### *2.1. Synthesis of [(CuX)2(Quin)] with X = Br, I*

The compound [(CuBr)2(Quin)] was prepared by mixing an equimolar amount of CuBr (0.1 g, 6.9 mmol) dissolved in 20 mL of acetonitrile with quinoxaline (0.09 g, 6.9 mmol) dissolved in 10 mL of acetonitrile. A red-orange precipitate immediately formed. The precipitate was filtered off. This powder was recrystallized from acetonitrile, and orange single crystals suitable for X-ray diffraction were obtained by slow evaporation of the solvent over 1 week. The compound [(CuI)2(Quin)] was prepared from CuI (0.2g, 1.05mmol) and quinoxaline (0.13g, 1mmol) following the same procedure as described above for the equivalent bromine compound [(CuBr)2(Quin)].

Elemental analysis. [(CuBr)2(Quin)] Found: C, 22.9; H, 1.4; N, 6.6%. Calcd: C, 23.0; H, 1.5; N, 6.7%. [(CuI)2(Quin)] Found: C, 18.5; H, 1.2; N, 5.4%. Calcd: C, 18.8; H, 1.2; N, 5.5%.

#### *2.2. X-ray Diffraction Measurements*

At ambient conditions, we used an Agilent SuperNOVA diffractometer equipped with an EOS detector (CCD) and a Mo radiation micro-source. Data were collected and processed using CrysAlisPro software [48]. The structures for both compounds were solved and refined using the SHELXT and SHELXL programs [49,50]. The PLATON program [51] was used for geometric calculations.

For high-pressure (HP) X-ray measurements, we used the Agilent SuperNOVA diffractometer for the bromine compound [(CuBr)2(Quin)]. The equivalent iodine compound [(CuI)2(Quin)] was measured at the MSPD beamline at ALBA synchrotron with a focused beam of 15 × <sup>15</sup> <sup>μ</sup>m<sup>2</sup> (FWHM) and a Rayonix SX165 CCD detector (Figure S1 in Supporting Information). The synchrotron X-ray energy used was 38.9 keV (λ = 0.3185 Å), selected from the La absorption K-edge. The sample-to-detector distance (240 mm), beam center, and detector tilt were calibrated from LaB6 (NIST) diffraction data measured under the same experimental conditions as the HP experiments. For each pressure, data collection was performed by rotating the DAC around the omega axis in small steps (0.2◦) from −30 to 30◦. The data were processed using CrysAlisPro software through Esperanto conversion.

For both samples, a single crystal of the compound was placed into a Diacell Bragg-Mini diamond anvil cell (DAC) from Almax-EasyLab (Figure S2 in Supporting Information), with an opening angle of 90◦ and anvil culets of 500 μm in diameter. The DAC was fitted with a stainless gasket of 60 μm thickness containing a hole of 200 μm in diameter. A small ruby sphere was placed next to the sample on one of the diamond anvils (diffraction side) as a pressure sensor and measured using the ruby *R*<sup>1</sup> fluorescence line [52]. A methanol–ethanol–water mixture (16:3:1) was used as the pressure-transmitting medium (PTM). This PTM remains hydrostatic in the range of pressures used in our experiments [53,54], thereby minimizing deviatoric stresses which can cause incorrect values for the bulk modulus to be determined [55].

Structures were refined using results from the previous pressure as a starting point, and they were refined on *F*<sup>2</sup> by full-matrix least-squares refinement using the SHELXL program. Due to limitations of the opening angle of our DAC, it was only possible to observe about 30% of the reflections which were observed outside of the DAC at ambient conditions. In general, structure refinements were performed with isotropic displacement parameters for all atoms except for the heavy atoms (Cu, Br, or I) that were refined with anisotropic displacement parameters whenever they did not become non-positive. In the case of the [(CuBr)2(Quin)] compound, constraints were introduced to maintain the planarity of the organic ligand as well as to adjust certain thermal factors influenced by the quality of the intensities measured during the HP experiment, which significantly decreased after the phase transition. Hydrogen atoms were included in the final procedure in the same way as for ambient conditions.

#### *2.3. High-Pressure Optical Measurements*

Two different setups were used to obtain the high-pressure emission spectra. The first setup consisted of a 375-nanometer continuum diode laser as the excitation source and a 0.75-m single grating monochromator (Spex 750 M) equipped with a cooled photomultiplier (PMT) (Hamamatsu 928b) for detection. The second setup consisted of a 532-nanometer diode laser as the excitation course and a commercial scanning confocal Raman instrument (Renishaw InVia) with a cooled CCD for detection, where a 20× SLWD objective was used to achieve a laser-spot diameter of less than 5 μm. The luminescence of the [(CuBr)2(Quin)] compound was obtained with the second setup, while for the [(CuI)2(Quin)] compound, both setups were used. All the spectra have been corrected for the instrument response. A miniature DAC (Figure S2 in Supplementary Information) designed at the University of Paderborn (Germany) was used for high-pressure experiments using the same hydrostatic pressure-transmitting medium and pressure sensor as for the high-pressure X-ray diffraction experiments. Lifetime measurements were carried out by exciting with an Optical Parametric Oscillator (OPO) (EKSPLA/NT342/3UVE). The emission was focused onto a 0.32-m monochromator (Jobin Yvon Triax 320) coupled with a cooled PMT (Hamamatsu 928P) and recorded and averaged using a digital storage oscilloscope (LeCroy WS424).

#### *2.4. Computer Simulations*

Combining ab initio simulations with experimental studies of materials under high pressure has proven to be a very powerful technique [56]. In this work, we have performed first principles simulations in the framework of density functional theory (DFT) using the Vienna ab initio simulation package (VASP) [57–59]. A generalized gradient approximation (GGA) with the Perdew–Burke–Ernzerhof (PBE) functional [60] was used to describe the exchange-correlation energy. We have also included the van der Waals dispersion energy correction employing the Grimme D3 method [61] to take into account the weak interactions. Interactions among the core and the valence electrons were treated with pseudopotentials through the projector-augmented wave scheme (PAW) [62] to solve the Schrödinger equation. A plane wave basis with an energy cut-off of 520 eV was used, which assures high accuracy in the results. The integration over the Brillouin zone (BZ) was performed with a Monkhorst–Pack scheme [63] grid of 8 × 2 × 2, which ensures high accuracy in the results since we are working with a primitive cell of 80 atoms in general positions. The structural parameters of the crystalline structures and the atomic positions have been relaxed at selected volumes. During the process of relaxation and optimization of the structures, it was required that the forces on the atoms were less than 0.003 eV/Å, and the stress tensor was diagonal with differences below 0.1 GPa, to ensure hydrostaticity.

#### **3. Results and Discussion**

#### *3.1. Structural Analysis*

The main single-crystal X-ray diffraction data and structure refinement parameters at ambient conditions for [(CuBr)2(Quin)] and [(CuI)2(Quin)] are reported in Table 1. Only the structure of [(CuI)2(Quin)] has been previously reported [2,32]. Both compounds are mutually isomorphous and can therefore be described using the iodide compound as the reference. The asymmetric unit is shown in Figure 1. When expanded, the asymmetric unit results in polymeric 2D layers with a staircase motif as copper bromide/iodide is situated parallel to the *a*-axis (Figure 2).

**Figure 1.** Thermal ellipsoids plot (50%) and labeling scheme for compound [(CuI)2(Quin)] at ambient conditions.


**Table 1.** Summary of single-crystal X-ray diffraction crystal data and structure refinement parameters for [(CuBr)2(Quin)] and [(CuI)2(Quin)] at ambient conditions.

**Figure 2.** *Cont*.

**Figure 2.** View of the stacking 2D layers along the *c*-axis (**upper**) and view of the staircase motif parallel to the *a*-axis (**lower**) for the complex [(CuI)2(Quin)].

Each Cu<sup>I</sup> atom is coordinated to three adjacent halogen ions over a range of 2.4584(8) Å to 2.5942(11) Å for [(CuBr)2(Quin)] and 2.6275(10) Å to 2.7030(10) Å for [(CuI)2(Quin)]. In both cases, the Cu<sup>I</sup> atom is also coordinated by one of the nitrogen donors from the quinoxaline ligand, which bridges the adjacent Cu-X staircase, resulting in the formation of 2D sheets. Typically, a single geometric index [64] called τ4 is used to describe the shape of this four-fold coordination environment, which takes a range of values from 1.0 for perfect tetrahedral configuration to 0.0 in the case of perfect square planar configuration. In this study, the τ4 values were 0.88 (for Cu1) and 0.93 (for Cu2) in [(CuBr)2(Quin)] and 0.87 (for Cu1) and 0.92 (for Cu2) in [(CuI)2(Quin)], indicating a distorted trigonal pyramidal configuration. The quinoxaline adopts an arrangement in order to maximize the π–π interactions. Neighboring 2D sheets are related by inversion symmetry and are separated by half the *c*-axis.

The geometric parameters for the [(CuX)2(Quin)] at ambient conditions agree with the values obtained in a search in the CSD (v5. 43) [65] for similar structures. The average values found in the CSD for Cu-Br, Cu-I, and Cu-N were 2.50(8) Å, 2.63(5) Å, and 2.06(7) Å, respectively. Similarly, the Cu–Cu distance was 2.9(3)Å when bromine atoms were involved and 2.8(3) Å when iodide atoms were involved. According to the structural analysis of the present work, all values fall within the expected ranges: 2.53 Å (Cu-Br), 2.02 Å (Cu-N), and 2.83 Å (Cu–Cu) for [(CuBr)2(Quin)], and 2.68 Å (Cu-I), 2.05 Å (Cu-N), and 2.79 Å (Cu–Cu) for [(CuI)2(Quin)].

The bond angles determined in this work are also consistent with those reported in the literature. The average value for the I-Cu-N angle calculated from the CSD was 110(7)◦. In this work, it was found to be 111.3(2)◦ and 112.2(2)◦ for [(CuBr)2(Quin)] and [(CuI)2(Quin)], respectively. In the case of X-Cu-X (with X = Br, I), the range found in the CSD is between 93 and 120◦, and all values for both compounds fall within this range.

With respect to the intermolecular interactions, the perpendicular stacking distance between quinoxaline layers was around 3.33 Å (4.1080(3) Å for [(CuBr)2(Quin)] and 4.3722(2) Å for [(CuI)2(Quin)] Cg–Cg distance), indicating weak π–π interactions.

High-pressure experiments were conducted at room temperature. The dependence of the unit cell parameters on pressure is shown in Figure 3. Although the behavior of the volume does not show any discontinuity, there was a significant change in the behavior of the angles in the unit cell parameters, where the *α* and *γ* angles differed from their original values of 90◦ at around 3.5 GPa for [(CuBr)2(Quin)] and 3.2 GPa for [(CuI)2(Quin)], with the *β* angle increasing smoothly with pressure. Therefore, it can be concluded that there

**Figure 3.** Pressure dependence of the cell parameters and volume for [(CuBr)2(Quin)] (**top**) and [(CuI)2(Quin)] (**bottom**). The dashed magenta line marks the phase transition zone. Error bars are smaller than their respective symbols.

Cu–Cu interactions are shown in Figure 4. In the low-pressure monoclinic phase (*P21*/*n*), we observed three different Cu–Cu interactions imposed by the symmetry (red, black, and dashed blue lines). In the high-pressure triclinic phase (*P-1*), the number of different Cu–Cu interactions increased up to five (red, black, green, magenta, and dashed blue lines). A similar situation occurred with the other bond distances (Cu-I and Cu-N, Figures S3 and S4 in Supplementary Information). Regarding the bond angles, the N-Cu-X (X = Br, I) angles presented the most significant changes after the phase transition (Figure 5). The other parameters produced only a slight distortion (gliding movement) of the Cu-X staircases which is associated with a slight decrease in the space

between the quinoxaline ligands, which maintained their planarity and orientation, with an additional small displacement of the ligands relative to one another (Figure S5 in Supplementary Information).

**Figure 4.** Variation in Cu–Cu distances with pressure for [(CuX)2(Quin)] (X = Br, I). Upper plots correspond with the scheme of distance variation before and after the phase transition: monoclinic (**left**) and triclinic (**right**). Different colored arrows correspond to different Cu–Cu interactions. The bottom part of the figure shows the Cu–Cu distances for [(CuBr)2(Quin)] (**left**) and for [(CuI)2(Quin)] (**right**) as a function of pressure. Error bars are smaller than respective symbols. Colored lines correspond with the different Cu–Cu interactions shown in the upper part of the figure.

**Figure 5.** Variation in N-Cu-X (X = Br, I) angles with pressure for [(CuBr2(Quin)] (**left**) and [(CuI2(Quin)] (**right**). In the right-hand figure, the label 'Staircase (I)' corresponds to the Cu1 atoms, while 'Staircase (II)' corresponds to the Cu2 atoms.

#### *3.2. The Equation of State Analysis*

The determination of the variation in volume with hydrostatic pressure at a fixed temperature is known as isothermal equation of state (EoS) analysis, whereby the volume variation is characterized by the bulk modulus (*K*) and its pressure derivatives (*K* , *K*", . . . ). We calculated such parameters for both compounds using a Birch–Murnaghan (BM) EoS model using EoSFit7-GUI software [66,67]. In order to assign the correct order to be used in the fitting, we have to represent the normalized pressure (*F*) against finite strain (*f*) (i.e., *f-F* plot). In both cases, we can observe a linear behavior with a positive slope, indicating that volume can be fitted with a third-order BM EoS, as can be seen in Figure 6.

The bulk modulus (*K*0) and its first derivative (*K* 0) were 14.4(5) GPa and 7.7(6) for [(CuBr)2(Quin)] and 13.0(2) GPa and 7.4(2) for [(CuI)2(Quin)], respectively.

**Figure 6.** *Cont*.

**Figure 6.** *f-F* plots and EoS BM3 fit for volume variation for [(CuBr)2(Quin)] (**upper**) and [(CuI)2(Quin)] (**lower**). Dashed red curves represent the EoS fitting results. Orange curves are the DFT simulations using the PBE + D3 model.

In general, this type of compound has a very anisotropic compression, especially when the compounds crystallize in monoclinic or triclinic systems. From Figure 3, we can observe this phenomenon, whereby the *c*-axis is substantially softer than the *b-* and *a*-axes. The bulk modulus obtained for both compounds falls in the range of 10–20 GPa, which is typical for organometallic compounds [68]. A quite similar case is the 2D coordination polymer with the formula Cu2I2(2-aminopyrazine) [36], where the bulk modulus (*K*0) and its first derivative (*K* 0) were 14.1(3) GPa and 7.4(2), respectively, using the same EoS model (BM3) in the fitting calculations. For similar staircase polymers of CuI iodides (1D), the bulk moduli were slightly smaller at around 10 GPa [37,46], and the exceptionally small bulk modulus of 7.5(4) GPa for CuI(3,5-dichloropyridine) [38] was probably due to the existence of a phase transition of the first order at 6 GPa. For 0D cases, such as a Cu4I4 cluster with different organic ligands [47], the trend is that the bulk modulus is even smaller (*K*<sup>0</sup> around 9 GPa or less). Therefore, we can conclude that the coordination dimensionality in this type of compound influences the compressibility values. In cases where the compressibility is high, this can be attributed to the deformability of the intermolecular interactions.

Unfortunately, the simulations carried out in this investigation (not only with the PBE + D3 functional but also with other commonly used functionals such as PBESol) have not reproduced the phase transition that was experimentally observed in both compounds. The simulation study shows us that in both compounds, the most favorable energy configuration is the monoclinic one rather than the triclinic one, as can be seen in Figure 7.

**Figure 7.** Energy curves of DFT simulations for [(CuBr)2(Quin)] (**left**) and [(CuI)2(Quin)] (**right**) considering monoclinic or triclinic phases.

#### *3.3. Luminescence Properties under Pressure*

In both samples, asymmetric emission bands were observed, exhibiting their multicomponent character. To follow the pressure-induced evolution of the peak maxima of the emission bands, the center of gravity (centroid) (N1) was used, defined as follows:

$$N\_1 = \frac{\int\_{-\infty}^{+\infty} \lambda I(\lambda) d\lambda}{\int\_{-\infty}^{+\infty} I(\lambda) d\lambda}. \tag{1}$$

The [(CuBr)2(Quin)] emission spectra with increasing pressure were recorded up to 6.3 GPa, with an excitation source wavelength of 532 nm (Figure 8 left). The spectra consist of a single asymmetric band that can be deconvoluted into three Gaussians (see Figure S5 in Supplementary Information).

**Figure 8.** Pressure dependence of luminescence spectra of [(CuBr)2(Quin)] compound obtained at RT under 532 nm laser excitation (**left**), and pressure evolution of the centroid (**right**).

At ambient conditions, the centroid of the band was around 720 nm, and when the pressure was increased, two different behaviors were observed (Figure 8 right) in the pressure interval from ambient to 6.3 GPa. In the first interval (0–3.2 GPa), a linear redshift was obtained with a slope of around −89 cm−1/GPa. This tendency drastically changed in the interval of 3.2–6.3 GPa with an opposite behavior, giving rise to a linear blue shift around 149.2 cm<sup>−</sup>1/GPa. Upon releasing pressure, the original peak positions were almost recovered.

A similar behavior was observed in the pressure evolution of the emission intensity, with a pressure-induced decrease in intensity observed at pressures higher than 3.2 GPa up to 6.0 GPa. The initial intensity was almost recovered after releasing pressure (see Figure 9).

In the [(CuI)2(Quin)] compound, two different excitation wavelengths were used to obtain the emission spectra, resulting in similar observed behaviors.

Under 375 nm excitation, the spectra consisted of two bands: one asymmetric band centered at 750 nm, denoted as the low-energy band (LE), and another more symmetric band centered at 510 nm, denoted as the high-energy band (HE). The LE band could be deconvoluted into two Gaussian profiles (Figure S5). It is important to point out that in this setup, the limit imposed by the response of the photomultiplier was around 850 nm, which precluded measurements at higher wavelengths.

**Figure 9.** Integrated intensity pressure dependence of the [(CuBr)2(Quin)] compound.

Upon compressing the copper(I) halide coordination complexes, two different regimes could be distinguished. The first regime is in the pressure interval between ambient and 2.9 GPa, where both the integrated intensity and the centroid of the band only exhibit slight changes of 20% and −49 cm−1/GPa, respectively. The second regime is in the pressure interval between 3.0 and 5.4 GPa, where a progressive decrease in the integrated intensity of the band occurs until it is no longer observable beyond 6 GPa along with an enhancement of the redshift rate up to −209 cm<sup>−</sup>1/GPa. These pressure-induced behaviors are depicted in Figure 10, where clear changes can be seen around 3.0 GPa in the slopes of the peak positions and in the integrated intensities (Figure 11). Upon decompression, the emission behavior was almost fully recovered.

**Figure 10.** Pressure dependence of luminescence spectra of [(CuI)2(Quin)] compound obtained at ambient temperature under 375 nm laser excitation.

**Figure 11.** Integrated intensity pressure dependence of the [(CuI)2(Quin)] compound.

Concerning the luminescence measurements obtained using an excitation wavelength of 532 nm, similar pressure behavior was found, with bands that can be deconvoluted into two Gaussian profiles (see Figure S5 in Supplementary Information). In this case, a change in tendency was observed around 4.0 GPa. Initially, a linear redshift of around −66 cm<sup>−</sup>1/GPa was found up to 4.0 GPa; then, up to 11.6 GPa, an increase in the redshift to −112 cm<sup>−</sup>1/GPa was observed (Figure 12). Due to the 532 nm excitation, it was impossible to observe the HE band appearing with 375 nm excitation.

**Figure 12.** Pressure dependence of luminescence spectra of [(CuI)2(Quin)] compound obtained at ambient temperature under 532 nm laser excitation.

Additionally, the decay curves of [(CuBr)2(Quin)] and [(CuI)2(Quin)] at ambient conditions with an excitation wavelength of 375 nm were also obtained, showing nonexponential behaviors, as expected (Figure 13), that can be related to the multicomponent character of the emission bands. For comparison purposes, an average lifetime (<τ>) was

considered according to Equation (2), and values of 0.7 μs and 0.3 μs for [(CuBr)2(Quin)] and [(CuI)2(Quin)] were found, respectively:

$$
\langle \tau \rangle = \frac{\int\_{-\infty}^{+\infty} t I(t) dt}{\int\_{-\infty}^{+\infty} I(t) dt}. \tag{2}
$$

**Figure 13.** Decay curves of [(CuBr)2(Quin)] (red) and [(CuI)2(Quin)] (black) obtained at ambient conditions.

The average lifetime obtained for the emission bands, around the order of microseconds, allows us to consider the luminescence process as phosphorescence, which reflects the triplet character of the emitter state [69].

In previous studies, the correlation between the structure of Cu<sup>I</sup> halide complexes and their luminescent properties has been demonstrated [31,36–39,41–43,46,70].

As commented on in the Introduction section, the electronic configuration for Cu<sup>I</sup> corresponds to *d*10. Therefore, since the *d*–*d* electronic transitions are excluded, charge transfer transitions can be the only mechanisms responsible for the electronic transitions. On the basis of DFT calculations, the HE band observed in the CuI -based complexes is usually ascribed to a mixed triplet metal-to-ligand charge transfer and triplet iodide-toligand charge transfer transition 3(M + X)LCT. Moreover, the LE band is connected with transitions in the CuX cluster core, exhibiting a combination of halide-to-metal 3XMCT and metal-cluster-centered (3MCC) *<sup>d</sup>*<sup>10</sup> → *<sup>d</sup>*9*s*1, which is independent of the nature of the ligands.

The presence of structure in the emission bands reveals their multicomponent character and clearly indicates the contribution of different excited states involved in this mechanism. From the high-pressure X-ray experiments, some important information on luminescent properties can be inferred. Firstly, at around 3.5 GPa and 3.0 GPa, a second-order phase transition from monoclinic (*P21*/*n*) to triclinic phase (*P-1*) took place for the [(CuBr)2(Quin)] and [(CuI)2(Quin)], respectively. In the low-pressure *P21*/*n* phase, three different Cu–Cu distances were observed, of which only one was equal to or shorter than the minimum distance required for so-called cuprophilic interactions, established as 2.8 Å for the Cu(I) ions [70]. This is depicted by solid red diamond symbols in Figure 4, with distances at ambient conditions of 2.8293 Å and 2.7869 Å for [(CuBr)2(Quin)] and [(CuI)2(Quin)], respectively. The pressure evolution of the Cu–Cu distances consisted of shortening at different rates depending on the sample and the pressure range. For [(CuBr)2(Quin)], the change in Cu–Cu distance was −0.0396 Å/GPa for the low-pressure *P21*/*n* phase (0–3.5 GPa), which then stabilized or showed a slight increase in the high-pressure *P-1* phase (Figure 4, left). Unfortunately, the X-ray diffraction pressure measurements were limited to 4.5 GPa. For the [(CuI)2(Quin)] sample, the change in Cu–Cu distance in the

*P21*/*n* phase (0–3.0 GPa) was around −0.0289 Å/GPa and was around −0.0134 Å/GPa in the *P-1* phase (3.2–7.0 GPa), thereby revealing the same tendency over the whole pressure range studied.

The redshift of the emission bands with the shortening of the Cu–Cu distances is one of the main features of the cuprophilic interaction, since from molecular orbital theory, it is known that the Cu–Cu distances in the lowest unoccupied molecular orbital (LUMO) have a bonding character [71] which increases with the shortening of the Cu–Cu distances. In this context, some correlation with the observed experimental results can be drawn. For example, the [(CuBr)2(Quin)] complex in the low-pressure *P21*/*n* phase showed a redshift of the emission centroid concurrently with shortening of the Cu–Cu distances of around 5% in the 0–3.5 GPa pressure interval. In the high-pressure *P-1* phase, the opposite tendency was observed—that is, a blueshift occurred while the Cu–Cu distances stabilized or even increased. The latter behavior can be connected with the rigidochromism effect, which is observed when the environment becomes more rigid and is connected to metal-to-ligand charge transfer (MLCT) excited states [72] and consists of a blue shift of the emission band. It seems that in the [(CuBr)2(Quin)] compound, at least two independent excited states are competing: the metal-cluster-centered (3MCC) *<sup>d</sup>*<sup>10</sup> → *<sup>d</sup>*9s1 transition and the 3MLCT. Considering the fact that the emission data have been interpreted using the centroid, which is itself an average value, it can be interpreted that in the *P21*/*n* phase, the 3MCC state predominates over the 3MLCT state, and that in the *P-1* phase, the roles are reversed.

For the other complex, [(CuI)2(Quin)], the same tendency was observed over the whole pressure interval (0 GPa to 11.5 GPa). In the *P21*/*n* phase, as can be seen in Figure 4, the variation in the Cu–Cu distances was less than 5% in the 0.0–3.0 GPa interval. Accordingly, small differences in the shifts of the centroids of the emission bands were observed (−49 cm<sup>−</sup>1/GPa and −70 cm<sup>−</sup>1/GPa) in this interval for excitations at 375 nm and 532 nm, respectively, that were more pronounced in the triclinic (*P-1*) phase (−209 cm<sup>−</sup>1/GPa and −100 cm−1/GPa) in the 4.0–7.0 GPa interval. Unfortunately, beyond 7.0 GPa, there is no information about the Cu–Cu distances since X-ray diffraction data were not acquired at these pressures, although the evolution of the emission spectra characterized by redshifts suggests a progressive shortening of the Cu–Cu distances. These distances correspond to the blue closed and open square symbols in Figure 4, which suggest that the Cu–Cu distances decrease below 2.8 Å for pressures over 7.0 GPa since decreases of around 14%, from 3.5105 Å (AC) to 3.0306 Å (7 GPa), were already observed. Therefore, in this complex, the metal-cluster-centered (3MCC) d10 → <sup>d</sup>9s1 transition is expected to be the dominant charge transfer mechanism.

#### **4. Conclusions**

We have performed a structural and luminescence study of a 2D polymeric staircase copper(I) halogen (halogen = bromine or iodine) with quinoxaline as an organic ligand. The high-pressure X-ray diffraction showed a phase transition of second order at around 3.3 GPa for both compounds whereby the cell parameters changed from the monoclinic to the triclinic crystal system. The compressibility study determined similar values in both compounds for the bulk modulus and its first derivate. The geometric changes revealed that the main structural variations occurred in the distortion of the ladder, with predominantly gliding versus folding movements affecting the Cu–Cu interactions. In general, the luminescence spectra showed asymmetric bands that reflect their multicomponent origin relating to the excited states involved in the transitions. The non-exponential decay curves confirmed this hypothesis, with averaged decay time data on the order of microseconds indicating the triplet character of the emitter states. Different behaviors of the pressure evolutions were observed for both compounds. With regard to [(CuBr)2(Quin)], a change in the tendency of the emission centroid from redshift to blueshift was observed around 3.5 GPa, which agrees with the phase transition pressure from the monoclinic *P21*/*n* phase to the triclinic *P-1* phase. From this, a combination of metal-cluster-centered (3MCC) *<sup>d</sup>*<sup>10</sup> → *<sup>d</sup>*9s1 (cuprophilic) and 3MLCT excited states were considered competing emitter states. In the 0–3.5 GPa

interval, the former dominated, but after the phase transition, the rigidochromism effect became more important. In the [(CuI)2(Quin)] compound, the same tendency (redshift) was observed over the whole pressure range studied (0–11.5 GPa), although different rates were observed around the phase transition. In this case, metal-cluster-centered (3MCC) *<sup>d</sup>*<sup>10</sup> → *<sup>d</sup>*9s1 transition seemed to be the dominant mechanism.

**Supplementary Materials:** The following supporting information can be downloaded at: https: //www.mdpi.com/article/10.3390/cryst13010100/s1, Figure S1: Instrumental setup used for HP experiments. On the left, a SuperNOVA diffractometer with EoS CCD detector. On the right, HP Station in MSPD-BL04 (ALBA Synchrotron) with Rayonix SX165 CCD Detector; Figure S2: Bragg-Mini DAC (Almax EasyLab company) with 500 μm diameter in the culets of diamonds (top). Paderborn Mini DAC with 400 μm diameter in the culets of diamonds (bottom); Figure S3: Normalized distances for Cu-N for (CuX)2-Quin (X = Br, I) compounds; Figure S4: Normalized distances for Cu-X for (CuX)2-Quin (X = Br, I) compounds; Figure S5: Room temperature emission spectrum fitted to three Gaussian profiles for (CuBr)2-Quin (top) and (CuIr)2-Quin (bottom) compounds; Table S1: Crystal data and structure refinements for (CuBr)2-Quin sample at different pressures (0.00–4.67GPa) at room temperature; Table S2: Crystal data and structure refinements for (CuI)2-Quin sample at different pressures (0.00–7.00GPa) at room temperature.

**Author Contributions:** J.G.-P. worked on the X-ray diffraction and EoS calculations; U.R.R.-M. performed luminescence measurements; R.R.H.-M. and A.A.-C. conducted the synthesis of compounds; P.R.-H. and A.M. performed the simulations calculations; R.T. conducted the discussion and writing. All authors have read and agreed to the published version of the manuscript.

**Funding:** The authors acknowledge financial support from the Spanish Research Agency (AEI) and the Spanish Ministry of Science and Investigation (MCIN) under project PID2019-106383GB-C43/C44 (DOI:10.13039/5011000011033) and the MALTA Consolider Team network under project RED2018- 102612-T. J.G.-P. thanks the Servicios Generales de Apoyo a la Investigación (SEGAI) at La Laguna University. U.R.R.-M. thanks the Gobierno de Canarias and EU-FEDER for grants ProID2020010067 and ProID2021010102.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** CCDC 2226239-2226264 contains the supplementary crystallographic data for this paper. The data can be obtained free of charge from The Cambridge Crystallographic Data Centre via www.ccdc.cam.ac.uk/structures.

**Acknowledgments:** Some parts of the HP diffraction experiments were performed with the BL04-MSPD beamline at ALBA Synchrotron with the collaboration of ALBA staff. J.G.-P. thanks Servicio de Difracción de Rayos X (SIDIX) at La Laguna University for complementary support with the HP diffraction experiments. R.T. acknowledges funding from the Generalitat Valenciana through the APOSTD postdoctoral Fellowship No. CIAPOS/2021/20.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


**Disclaimer/Publisher's Note:** The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

## *Article* **Ab Initio Theoretical Study of DyScO3 at High Pressure**

**Enrique Zanardi, Silvana Radescu, Andrés Mujica, Plácida Rodríguez-Hernández and Alfonso Muñoz \***

Departamento de Física and Instituto de Materiales y Nanotecnología, MALTA Consolider Team, Universidad de La Laguna, E38200 San Cristóbal de La Laguna, Tenerife, Spain

**\*** Correspondence: amunoz@ull.edu.es

**Abstract:** DyScO3 is a member of a family of compounds (the rare-earth scandates) with exceptional properties and prospective applications in key technological areas. In this paper, we study theoretically the behavior of DyScO3 perovskite under pressures up to about 65 GPa, including its structural and vibrational properties (with an analysis of the Raman and infrared activity), elastic response, and stability. We have worked within the ab initio framework of the density functional theory, using projector-augmented wave potentials and a generalized gradient approximation form to the exchange-correlation functional, including dispersive corrections. We compare our results with existing theoretical and experimental published data and extend the range of previous studies. We also propose a candidate high-pressure phase for this material.

**Keywords:** DyScO3; high pressure; density functional theory; stability

#### **1. Introduction**

On account of their dielectric and optical properties, as well as their optimal chemical stability, the rare-earth (RE) scandates with formula REScO3, and among them DyScO3, have been researched for the past two decades with a view mainly to replace SiO2 in nextgeneration MOSFETs [1–3]. Other usages of RE-scandates include their employment as optimal substrates for the epitaxial growth and fine-tuning of high-quality perovskite-type thin films, as well as their application in the terahertz range when embedded in perovskite heterostructures [4–10]. The enticing properties and promising scope of applications of DyScO3 suggest extending the fundamental research on this material to include as broad a range of pressures as possible, and very recently Bura and co-workers have experimentally undertaken this goal in the range from ambient conditions up to 40 GPa [11]. In the present work we aim to provide an ab initio theoretical study, performed within the framework of the density functional theory (DFT), of the so far only observed phase of DyScO3, in the range up to 65 GPa, with a focus on its structural, elastic, and vibrational response, to compare with the experimental results of Bura et al. [11] as well as with other previously existing works and to provide a bound to its local stability. Our results indicate the existence of local instabilities, both elastic and dynamic, above 60–65 GPa. We also provide a possible candidate high-pressure phase, a well-known post-perovskite structure. We hope that these results will stimulate future experiments which, combined with theoretical studies, may lead to a better understanding of the properties of this technologically relevant, emerging material under high pressure.

#### **2. Theoretical Method and Details of the Calculations**

All the calculations were performed within the ab initio framework of the density functional theory (DFT) [12,13], using a projector augmented wave scheme (PAW) [14,15] and the VASP computational package [16–19]. The outermost, valence electrons of each species were explicitly considered in the calculations whereas their innermost, closedshell core electrons were treated at the PAW level. For dysprosium we have used the recommended VASP Dy\_3 PAW potential, which represents Dy with partially frozen 4f

**Citation:** Zanardi, E.; Radescu, S.; Mujica, A.; Rodríguez-Hernández, P.; Muñoz, A. Ab Initio Theoretical Study of DyScO3 at High Pressure. *Crystals* **2023**, *13*, 165. https://doi.org/10.3390/ cryst13020165

Academic Editor: Artem R. Oganov

Received: 21 December 2022 Revised: 12 January 2023 Accepted: 14 January 2023 Published: 17 January 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

electrons and formal valency 3, a standard way to cope with localized f electrons in DFT calculations. The cutoff in the kinetic energy of the basis's plane waves was 540 eV and the required Brillouin-zone integrations were performed with a 4 × 4 × 4 grid [20]. Several levels of approximations for the exchange-correlation functional were considered in our preliminary calculations, with optimal results obtained for the PBE generalized gradient approximation form [21] including dispersive corrections within the so-called Grimme's D3 scheme [22], which was accordingly the level of approximation adopted for the production of all the results shown in this paper. The material was structurally relaxed at different volumes, sampling the region between negative pressures just below 0 GPa to about 60 GPa. Hydrostatic conditions were assumed in all cases and the criteria for stopping the structural relaxations was set to stress anisotropies less than 0.1 GPa and residual forces on atoms less than 5 meV/Å.

Further to the total energy calculations just described, which provide information on the pressure dependence of the structural properties of DyScO3, we also performed a general study of its dynamical and elastic properties and their variation with pressure, and draw conclusions on the stability of the system based on them, as described in Sections 3.2 and 3.3. For the study of the lattice vibrations, we used the small-displacement method as implemented in the phonopy package [23], employing a 2 × 2 × 2 supercells for sampling the phonons across the whole Brillouin zone. At the zone center, the symmetry of the calculated modes was analyzed to determine their possible spectroscopical activity, Raman or infra-red (IR). For the calculation of the elastic constants, we used the Le Page stress-strain method as implemented in the VASP code [24].

#### **3. Results and Discussion**

#### *3.1. Crystallographic Description of DyScO*<sup>3</sup> *Perovskite and Evolution of Its Structural Properties with Pressure*

Like many other ABO3 oxide compounds, DyScO3 crystallizes in the perovskitetype structure. The ideal cubic perovskite structure, space group (SG) *Pm-3m* No 221, is characterized by corner-sharing BO6 octahedra and 12-fold coordinated A sites. For nonideal or distorted perovskites, octahedron tilting lowers the space group symmetry of the structure [25]. For DyScO3 that means an orthorhombic phase with SG *Pbnm* No 62 and four formula units on the conventional unit cell (see Figure 1).

Our calculated lattice parameters *a*, *b*, *c*, and unit cell volume *V* at 0 GPa are in good agreement with the experimental results, as shown in Table 1. The calculated variation with the pressure of the lattice parameters (Figure 2) also reproduces the experimental results within a wide range of pressures up to approximately 10–12 GPa, although for larger pressures the calculated values of *a* and *c* are somewhat lower than the experimental values while *b* still follows the experimental trend. These deviations may be related to the loss of hydrostaticity above 10–12 GPa [26] in the methanol/ethanol pressure-transmitting medium used in the experiments [11] but more experimental and theoretical research would be needed to shed light on this point. The calculated linear axial compressibilities are 1.469 × <sup>10</sup>−<sup>3</sup> GPa−<sup>1</sup> for *<sup>a</sup>*, 0.731 × <sup>10</sup>−<sup>3</sup> GPa−<sup>1</sup> for *<sup>b</sup>*, and 1.411 × <sup>10</sup>−<sup>3</sup> GPa−<sup>1</sup> for *<sup>c</sup>*, and the value for the bulk modulus *B*<sup>0</sup> of 164.7 GPa obtained by fitting the calculated energy-volume curve to a third-order Birch-Murnaghan equation of state [27] is a bit below the experimental one in Ref. [11] (see Table 2) whereas it is in good agreement with the calculated value of 165 GPa retrieved from the Materials Project database [28,29].

**Figure 1.** The structure of DyScO3 is of the orthorhombic perovskite type. In this plot the ScO6 octahedra of the structure are shown in magenta, with oxygen sites (in red) at their corners and Sc sites at the centers. The Dy cation sites are shown in blue.

**Table 1.** Calculated and experimental lattice parameters and unit cell volume at 0 GPa. Also provided are the calculated values at 60 GPa, for reference.


**Figure 2.** (**left**) Variation of the lattice parameters *a*, *b*, *c* of DyScO3 perovskite with pressure. (**right**) Variation of the unit cell volume with pressure. In both panels, the curves represent our calculated values and the symbols represent the experimental values from Ref. [11].


**Table 2.** Comparison of the calculated and experimental unit cell volume, bulk modulus, and pressure derivative of the bulk modulus at zero pressure.

The physical properties of the ABO3 perovskites and the instabilities associated with soft modes found near the phase transitions are related to the distortions from the ideal cubic structure. Those distortions can be broken up into three components: two independent tilts of the BO6 octahedra (with tilt angles respectively *θ* and *φ*), the distortion of those octahedra, and the displacement of the A cations [31–34]. In Table 3 we summarize the calculated values of the atomic positions in DyScO3, showing, in particular, the displacement of the Dy cation, the tilt angles, and the octahedra distortion index, compared to the values of the ideal cubic perovskite structure, and the experimental values published in Ref. [25]. (The tilt angles were calculated using Equations (A1) and (A2) in Ref. [33].)

**Table 3.** Summary of the atomic sites of DyScO3 perovskite, showing the values of the internal parameters and tilt angles. The values for the ideal cubic perovskite structure are also provided, for reference.


The variation of those values with pressure is shown in Figure 3. We observe that as the applied pressure increases the Dy cations move further away from the cubic perovskite positions, as the octahedra rotate further away around the [001] axes.

In Figure 4 we show the calculated pressure-evolution of near-neighbor distances in DyScO3. Our calculated values for the Sc-O interatomic distances, which are related to the ScO6 octahedra, are typical for octahedrally coordinated scandium, and the octahedra show little distortion over the studied pressure range. The Dy-O distances show a larger variation and there is a gap between the first eight and the last four such distances, which makes it possible to distinguish between a first and second shell for the Dy sites and has led some authors to consider Dy as being 8-fold coordinated [25,35].

**Figure 3.** (**left**) Evolution of the internal parameters of the 4c Wickoff positions of Dy with pressure. (The values for the ideal cubic perovskite are ½, ½) (**right**) Tilt angles change with pressure.

**Figure 4.** Evolution of interatomic distances with pressure in DyScO3 perovskite. The O1-Sc1, O2-Sc1, and Dy1-O2 distances have all a multiplicity of two (each represented curve corresponds to two symmetrically equivalent but different bonds).

Related to the observed behavior for the cell volumes and the lattice parameters, we find that the calculated values of the polyhedra volumes stray somewhat away from the experimental ones at large pressures, but nonetheless the volume ratio of DyO12 to ScO6 stays above the value of four as suggested in Ref. [34] for a stable material. The ScO6 octahedra are less compressible than the DyO12 polyhedra, so the latter are the main contributors to the volume change of the material, see Figure 5.

**Figure 5.** Variation of the polyhedral volumes of DyScO3 perovskite with pressure. Solid curves represent the calculated values and symbols represent the experimental values from Ref. [11].

#### *3.2. Elastic Properties*

As described in Section 3.1, DyScO3 has an orthorhombic structure with SG *Pbnm* No 62 and therefore there are nine independent elastic constants: *C*11, *C*22, *C*33, *C*44, *C*55, *C*66, *C*<sup>12</sup> *C*13, and *C*<sup>23</sup> [36]. The study of elastic constants is useful to understand both the mechanical properties and the structural stability of the material. Our calculated values at zero pressure of these constants for DyScO3 are compared with other ab initio and experimental results in Table 4.

**Table 4.** Elastic constants (in GPa) of DyScO3 perovskite at 0 GPa. (The *Pbnm* setting is used.)


The *C*11, *C*22, and *C*<sup>33</sup> elastic constants, which are related to unidirectional compressions along the principal crystallographic directions, are much larger than the *C*44, *C*55, and *C*<sup>66</sup> constants that are related to resistance against shear deformations. Consequently, there is a larger resistance to unidirectional compression than to shear deformation. The structure is stable at 0 GPa as it fulfills the Born stability criteria [38] given by the following expressions [39]:

$$\begin{aligned} \mathsf{C}\_{11} &> 0 \; ; \; \mathsf{C}\_{11} \mathsf{C}\_{22} - \mathsf{C}\_{12}^2 > 0 \\\\ \mathsf{C}\_{11} \mathsf{C}\_{22} \mathsf{C}\_{33} &+ 2 \mathsf{C}\_{12} \mathsf{C}\_{13} \mathsf{C}\_{23} - \mathsf{C}\_{11} \mathsf{C}\_{23}^2 - \mathsf{C}\_{22} \mathsf{C}\_{13}^2 - \mathsf{C}\_{33} \mathsf{C}\_{12}^2 > 0 \\\\ \mathsf{C}\_{44} &> 0 \; ; \; \mathsf{C}\_{55} > 0 \; ; \; \mathsf{C}\_{66} > 0 \end{aligned}$$

When stress is applied to the crystal, the elastic properties of the material are no longer described by the elastic constants *Cij*, and the elastic stiffness coefficients *Bij* must be employed instead. If the external stress corresponds to a hydrostatic pressure *p*, the elastic stiffness coefficients for a crystal are expressed as [40,41]:

$$B\_{ii} = \mathbb{C}\_{ii} - p \text{ , } i = 1, \dots, 6 \text{ ; } B\_{ij} = \mathbb{C}\_{ij} + p \text{ , } i \neq j \text{ , } i \text{, } j = 1, 2, 3$$

where the *Cij* are the elastic constants evaluated at the current stressed state. (As can be seen, the *Bij*, coefficients reduce to the *Cij* when no pressure is applied.) The evolution of the elastic stiffness coefficients *Bij* under pressure is depicted in Figure 6. The coefficients *B*22, *B*33, *B*12, *B*23, and *B*<sup>13</sup> increase with pressure in the whole pressure range studied. On the other hand, *B*44, *B*55, and *B*<sup>66</sup> decrease slowly. The *B*<sup>11</sup> coefficient increases slightly up to 30 GPa and then decreases smoothly.

**Figure 6.** Calculated pressure evolution of the elastic stiffness coefficients *Bij* of DyScO3 perovskite.

The stiffness coefficients allow us to obtain the elastic moduli that describe the major elastic properties of the material at any hydrostatic pressure. Employing standard relations for orthorhombic crystals [42], we have obtained the bulk modulus *B*, the shear modulus *G*, the young modulus *E*, the Poison's ratio *ν*, and the B/G ratio. The values of the abovementioned elastic moduli at zero pressure as calculated within the standard schemes of Voigt, Reuss, and Hill (with the latter one defined as the arithmetic average of the other two models [43–45]) are displayed in Table 5.


**Table 5.** Elastic moduli *B*0, *E*<sup>0</sup> and *G*0; *B*0/*G*<sup>0</sup> ratio; and Poisson's ratio *ν*0, calculated using the Voigt, Reuss, and Hill approximations from the calculated elastic coefficients at 0 GPa of DyScO3 perovskite.

It is worth noting that the bulk modulus, an important parameter related to the resistance of a material to compression, yields a similar value in the Hill formalism (164.9 GPa) than the one obtained from the EOS fitting (164.7 GPa), which reflects the consistency of the calculations. As the shear modulus *G* is associated with the resistance to plastic deformation, it is also interesting to analyze the *B/G* ratio. According to Pugh's criterion, if *B/G* > 1.75 the material is ductile, otherwise the material is brittle [46]. In the present case, our results indicate that DyScO3 is a ductile material at zero pressure. The Poisson's coefficient *ν* is related to volume changes during uniaxial deformation. Lower coefficients indicate large volume changes during deformation; if *ν* = 0.5 no volume changes occur during elastic deformation [40]. This parameter also provides information about the characteristics of the bonding forces. A value of *ν* = 0.25 is a lower limit for central forces while *ν* = 0.5 is an upper limit [40]. For DyScO3, the calculated value of *ν* suggests that the forces in this material are predominantly central.

The calculated pressure evolution of the elastic moduli of DyScO3 is presented in Figure 7. The bulk modulus *B* as calculated using Voigt's prescription increases in the whole pressure range under study. However, using Reuss' scheme the bulk modulus decreases dramatically around 50 GPa. This could be related to mechanical instability, as we will show in the next paragraph. As for the shear and Young moduli (*G* and *E*), *E* reaches a shallow maximum at about 14 GPa, then both moduli decrease with pressure and tend to get closer in values.

**Figure 7.** Pressure evolution of the elastic moduli: bulk modulus *B*, Young modulus *E*, and Shear modulus *G* of DyScO3 perovskite. The blue line represents the moduli in the Voigt approximation, the green line in the Reuss approximation, and the red one in the Hill approximation.

in the Reuss approximation, and the red one in the Hill approximation.

0.45 in the Hill expression at 60 GPa, see Figure 8.

**Figure 8.** (**left**) Pressure evolution of the *B/G* ratio of DyScO3 perovskite. (**right**) Pressure evolution of the Poisson's ratio. The blue line represents the moduli in the Voigt approximation, the green line

The *B/G* ratio becomes larger under pressure and so the material becomes more ductile under compression. The Poisson's ratio increases rapidly with pressure, taking a value of

To conclude, we analyze the mechanical stability of DyScO3 perovskite under pressure. As stated before, when hydrostatic pressure is applied, the elastic stiffness coefficients must be employed instead of the elastic constants [38]. Therefore, the stability conditions are modified, and one needs to use the following generalized stability conditions for an orthorhombic crystal [38,47]:

$$M\_1 = B\_{11} > 0 \; ; \; M\_2 = B\_{11}B\_{22} - B\_{12}^2 > 0$$

$$M\_3 = B\_{11}B\_{22}B\_{33} + 2B\_{12}B\_{13}B\_{23} - B\_{11}B\_{23}^2 - B\_{22}B\_{13}^2 - B\_{33}B\_{12}^2 > 0$$

$$M\_4 = B\_{44} > 0 \; ; \; M\_5 = B\_{55} > 0 \; ; \; M\_6 = B\_{66} > 0$$

The crystalline structure is mechanically stable when all the above criteria are fulfilled. The evolution of these generalized Born stability criteria *Mi* (*i* = 1 to 6) as functions of pressure is depicted in Figure 9 where it can be observed that the *M*<sup>3</sup> criterion is violated above 60 GPa, indicating that DyScO3 would become mechanically unstable at such pressure. Above this pressure an amorphization or a pressure-driven phase transition can occur, a result that may entice further experimental and theoretical research along this line.

181

**Figure 9.** Generalized stability criteria as a function of pressure for DyScO3 perovskite.

#### *3.3. Vibrational Properties*

We have also explored the vibrational properties of DyScO3 perovskite at high pressure using the small-displacements method. The mechanical representation of DyScO3 yields the following multiplicities for the irreducible representations of the modes at the zone center: M = 7A*<sup>g</sup>* (R) + 8A*<sup>u</sup>* (S) + 5B1*<sup>g</sup>* (R) + 10B1*<sup>u</sup>* (IR) + 7B2*<sup>g</sup>* (R) + 8B2*<sup>u</sup>* (IR) + 5B3*<sup>g</sup>* (R) + 10B3*<sup>u</sup>* (IR); of which the A*g*, B1*g*, B2*g*, and B3*<sup>g</sup>* modes are active in Raman (R); the B1*u*, B2*u*, and B3*<sup>u</sup>* modes are infrared-active (IR); and the A*<sup>u</sup>* modes are silent (S). In Figure 10 we display the behavior of the phonon modes at the Γ point (Raman, infrared and silent modes) under applied hydrostatic pressure.

**Figure 10.** (**left**) Pressure dependence of the calculated Raman-active modes of DyScO3 perovskite; (**right**) idem for the infrared and silent modes.

The present calculated values of the frequencies are in overall agreement with previous theoretical values obtained by Delugas et al. [48]. The pressure behavior in the very large region investigated in our work displays some curvature as evidenced in Figure 10 but up to about 15 GPa it can be approximated fairly well by a linear relation. The corresponding fit to the theoretical data yields the values of the pressure coefficients and calculated mode Grüneisen parameters shown in Table 6. As can be seen, they are in overall agreement with the experimental data obtained by Bura et al. [11].

The calculated phonon-dispersion curves along selected paths within the Brillouin zone with endpoints at the high symmetry points for SG 62 are plotted in Figure 11. The phonon dispersion curves at ambient pressure in these figures correspond to a dynamically stable behavior where all the frequencies are positive whereas above around 63 GPa the structure becomes dynamically unstable, displaying phonon branches with negative (imaginary) frequencies around the zone-boundary X point. This instability becomes more accentuated as the pressure is further increased, a pressure regime that we have not considered in this paper.

**Figure 11.** (**left**) Phonon dispersion curves of DyScO3 perovskite at ambient pressure and (**right**) at 66 GPa.

To provide more detail about the nature of the lattice vibrations, the spectral distribution of the phonon density of states (phDOS) was obtained along with the partial phonon density of states, to extract the contribution of each atomic species in the local lattice configuration. Figure 12 shows phDOS plots at ambient pressure. The shape of the phDOS curve above 150 cm−<sup>1</sup> is dominated by the oxygen atoms, which are present in larger numbers. At low frequencies (up to 150 cm<sup>−</sup>1) the phonon peaks correspond mainly to the heavy Dy atoms, whereas the contribution of these atoms is very small at higher frequencies. In the intermediate frequency region (from 150 cm−<sup>1</sup> to 450 cm−1) there are significant contributions from both the Sc and O atoms.

**Figure 12.** Total phonon density of states and contribution per species, as calculated at ambient pressure for DyScO3 perovskite.

**Table 6.** Raman (R), infrared (IR), and silent (S) mode frequencies as calculated in this work for DyScO3 perovskite, and their respective pressure coefficients obtained by fitting in the range up to 15 GPa with a linear equation *ω*(*p*) = *ω<sup>o</sup>* + *αp*. Mode Grüneisen parameters (*γ*) are also provided using the calculated bulk modulus *Bo* (164.7 GPa). Experimental values from Ref. [11], where available, are given in parenthesis.



**Table 6.** *Cont.*

#### *3.4. Prospective High-Pressure Post-Perovskite Phase*

The recent experimental work of Bura et al. [11] shows that the DyScO3 perovskite has a large stability range, up to at least 40 GPa, which are the highest pressures for which experimental results on this compound exist. For this reason, up to now we have been concerned with the local stability of the perovskite phase of DyScO3, the only one so far observed. However, other perovskites, like MgSiO3, undergo a pressure-induced transition to a post-perovskite phase with a *Cmcm*-type structure [49–51]. We have performed simulations on this post-perovskite structure for DyScO3 (see Table 7) and our results show that this new phase has lower enthalpy above 14.4 GPa, see Figure 13. The study of the phonon dispersion curves at 15 GPa (Figure 14) shows that this phase is also dynamically stable. The existence of large kinetic barriers (and the enhanced local stability of the perovskite phase) might explain why this high-pressure phase was not observed in the experimental study of Ref. [11], but more experimental work, involving heating at high pressure to overcome barriers, would be needed to test this hypothesis. In the Supplementary Material we provide some extra results for this proposed high-pressure phase of DyScO3, but its full study will not be pursued here.

**Figure 13.** Energy-volume curves and (inset) enthalpy-presssure curves for the *Pbnm* (perovskite) and *Cmcm* (post-perovskite) phases of DyScO3 (black and blue curves, respectively). The enthalpies are given with respect to that of the perovskite phase.

**Figure 14.** Calculated phonon dispersion curves of the *Cmcm* phase at 15 GPa.

**Table 7.** Crystallographic description of *Cmcm* at 15 GPa.


#### *3.5. Summary and Conclusions*

We have performed an ab initio study of DySCO3 perovskite both at ambient and under hydrostatic pressure up to just above 60 GPa. We have analyzed the evolution of its structural, dynamic, and elastic properties. The calculated lattice parameters compare well with experimental values up to 10–15 GPa, above which pressures the experiments are probably affected by the loss of hydrostatic conditions of the pressure transmitting medium. The equation of state, the evolution of interatomic distances with pressure, and the compressibility were calculated. The DyO12 polyhedra account for most of the volume reduction under compression. The phDOS and partial phDOS were analyzed to determine the contribution of each atomic species to the vibrational spectrum. The evolution of all the Raman, infrared and silent phonon modes was obtained along with their corresponding Grüneisen parameters. At about 63 GPa, imaginary frequencies appear around the X point of the Brillouin zone and therefore the structure becomes dynamically unstable. The elastic constants and the elastic stiffness coefficients were also determined. This allowed us to study the major elastic moduli (*B*, *G*, *E*, *ν*). The value of the *B/G* ratio points out that this material is ductile and that its ductility increases under pressure. Above 60 GPa the phase becomes mechanically unstable. We have also considered the possibility of a high-pressure post-perovskite phase with *Cmcm* symmetry. Our calculations suggest that it is favored over the perovskite phase above 14.4 GPa, which is a pressure within the range of observation of the perovskite phase and well below its predicted local (i.e., dynamical and elastic) instabilities. This prediction should thus entice further experimental and theoretical studies on DyScO3.

**Supplementary Materials:** The following supporting information can be downloaded at: https: //www.mdpi.com/article/10.3390/cryst13020165/s1, Figure S1. Variation of the lattice parameters *a*, *b*, *c* with pressure for the proposed *Cmcm* phase. Table S1. Elastic moduli *B*, *E* and *G*; *B*/*G* ratio; and Poisson's ratio *ν* for the proposed *Cmcm* phase., calculated using the Voigt, Reuss, and Hill approximations from the calculated elastic coefficients at 15 GPa. Figure S2. Calculated pressure evolution of the elastic stiffness coefficients for the proposed *Cmcm* phase. Figure S3. M3 generalized stability criteria as a function of pressure for the proposed *Cmcm* phase. Figure S4. Pressure dependence of the

calculated Raman-active modes of the *Cmcm* phase. Figure S5. Pressure dependence of the calculated infrared modes of the *Cmcm* phase. Figure S6. Pressure dependence of the calculated silent modes of the *Cmcm* phase. Table S2. Calculated Raman (R), infrared (IR), and silent (S) mode frequencies for the proposed *Cmcm* phase as calculated in this work, and their respective pressure coefficients obtained by fitting in the range up to 60 GPa with a cuadratic equation *ω*(*p*) = *ω<sup>o</sup>* + *αp* + *βp*2. Figure S7. Phonon dispersion curves at 65.7 GPa for the proposed *Cmcm* phase.

**Author Contributions:** All authors contributed to this work doing simulations, analysis, interpretation of results, and/or manuscript writing. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the Spanish Research Agency (AEI) and the Spanish Ministry of Science and Investigation (MCIN) under grant PID2019-106383GB-43 (DOI: 10.13039/501100011033) and RED2018-102612-T (MALTA Consolider-Team Network).

**Data Availability Statement:** All relevant data that support the findings of this study are available from the corresponding author upon request.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.

#### **References**


**Disclaimer/Publisher's Note:** The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

**Meera Varma 1, Markus Krottenmüller 1, H. K. Poswal 2,\* and C. A. Kuntscher 1,\***


**Abstract:** This study reports high-pressure structural and spectroscopic studies on polycrystalline cubic chromium spinel compound LiInCr4O8. According to pressure-dependent X-ray diffraction measurements, three structural phase transitions occur at ∼14 GPa, ∼19 GPa, and ∼36 GPa. The first high-pressure phase is indexed to the low-temperature tetragonal phase of the system which coexists with the ambient phase before transforming to the second high-pressure phase at ∼19 GPa. The pressure-dependent Raman and infrared spectroscopic measurements show a blue-shift of the phonon modes and the crystal field excitations and an increase in the bandgap under compression. During pressure release, the sample reverts to its ambient cubic phase, even after undergoing multiple structural transitions at high pressures. The experimental findings are compared to the results of first principles based structural and phonon calculations.

**Keywords:** high-pressure studies; Raman spectroscopy; infrared spectroscopy; X-ray diffraction

#### **1. Introduction**

The chromium spinels *A*Cr2O4 belong to the widely studied geometrically frustrated systems, owing to their varied magnetic couplings, magnetostructural transitions, and exotic ground states. With the *A*-site occupied by a non-magnetic ion, the magnetic chromium ions (Cr3+) at the *B*-site form a network of corner-linked Cr4 tetrahedra, i.e., the pyrochlore lattice. The dominant antiferromagnetic (AFM) interactions between the Cr3<sup>+</sup> ions in the pyrochlore network leads to strong magnetic frustrations, which result in Jahn– Teller-driven structural distortion and antiferromagnetic ordering at low temperatures. Due to their varied response to external parameters, the chromium-based spinels could be used in magnetic sensing devices, data storage or spintronic devices [1]. There have also been studies reporting their use as a potential electrode support material [2]. A study also suggested cobalt containing chromium spinels as possible candidates for catalytic combustion [3].

An archetype derived by substituting two different ions at the *A*-site of *A*Cr2O4 was first reported by Joubert and Durif in 1966 [4]. The difference in ionic radii at the *A*-site led to an alternate arrangement of small and large Cr4 tetrahedra at the *B*-sites, known as the breathing pyrochlore lattice (see Figure 1) [5–8]. This new family of Cr spinels, namely Li*M*Cr4O8 (with *M* = In, Ga, Fe), are being actively investigated due to the geometrical frustration and the Cr–Cr bond alternations in these materials. The substitution of different ions at the *A*-site leads to the loss of inversion symmetry found in the conventional spinels, and the crystal symmetry is reduced to F4¯3m. The alternating smaller and larger Cr–Cr bonds between Cr4 tetrahedra also cause a difference in the nearest-neighbour magnetic interactions, without relieving the frustration in the system. The magnitude of AFM interactions between neighbouring Cr ions is denoted as J and J for small and large tetrahedra, respectively. The ratio of J and J is defined as the breathing factor B*<sup>f</sup>* , i.e., B*<sup>f</sup>* = J /J, and determines the degree of frustration in the system [5].

**Citation:** Varma, M.; Krottenmüller, M.; Poswal, H.K.; Kuntscher, C.A. Pressure-Induced Structural Phase Transitions in the Chromium Spinel LiInCr4O8 with Breathing Pyrochlore Lattice. *Crystals* **2023**, *13*, 170. https://doi.org/10.3390/ cryst13020170

Academic Editors: Daniel Errandonea and Enrico Bandiello

Received: 20 December 2022 Revised: 12 January 2023 Accepted: 13 January 2023 Published: 18 January 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

The physical properties of this family of frustrated breathing pyrochlores Li*M*Cr4O8 (with *M* = In, Ga, Fe) are highly dependent on the cationic radii at the *A*-site, which define the breathing factor and in turn have an overwhelming influence on the response of the material to different thermodynamic conditions. Recent studies have been conducted to improving the understanding of the structural and magnetic response of these materials at low temperatures. A temperature-dependent neutron diffraction experiment on LiInCr4O8 indicated the opening of a spin gap below 65 K, which is followed by a long-range magnetic ordering at ∼15.9 K [5]. An NMR study on LiInCr4O8 suggested a singlet ground state with a gap at 18 K and a structural transition at 16 K followed by a second-order AFM transition at 13 K [6]. However, another study reported a structural transition at 18 K followed by magnetic ordering at 12 K [7]. The related compound LiGaCr4O8 has been reported to show short-range AFM ordering close to ∼50 K, which is followed by a firstorder magnetostructural transition around ∼15 K [5,6]. However, another study reported it as two consecutive events of magnetic and structural transitions at 14.1 K and 14.5 K, respectively, [8]. As opposed to LiInCr4O8 and LiGaCr4O8, the compound LiFeCr4O8 has been reported to undergo a ferrimagnetic transition at 94 K. Furthermore, the opening of a spin gap at ∼60 K and a magnetostructural transition at ∼23 K were found [9]. It is to be noted that all three compounds, having different degrees of frustration, show a structural instability at low temperature.

Although low-temperature-induced structural and magnetic responses have been explored and reported for this class of compounds, high-pressure studies have not been reported to the best of our knowledge. In this work, we study the structural phase transitions in LiInCr4O8 induced by high pressure using synchrotron-based X-ray diffraction (XRD) and Raman and infrared (IR) spectroscopic measurements supplemented by density functional theory (DFT)-based simulations.

**Figure 1.** Sketch of the breathing chromium spinel LiInCr4O8 crystal structure with LiO4 and InO4 tetrahedra and CrO6 octahedra as structural units.

#### **2. Materials and Methods**

Polycrystalline LiInCr4O8 was synthesized using the solid-state reaction method reported previously [5,8], where the stoichiometric amounts of In2O3 and Cr2O3 and 10% excess of Li2CO3 were thoroughly ground, pelleted, and heated in a furnace at 1100 ◦C for 48 h in an alumina crucible with intermittent grinding. As lithium is volatile, a slow rate of cooling was maintained. The phase purity of the synthesized compound LiInCr4O8 was characterized using XRD measurements performed in the angle-dispersive mode at the ECXRD beamline (BL-11), Indus-2, RRCAT. X-rays with energies of 19.7 keV (*λ* = 0.6280 Å) were incident on the synthesized polycrystalline sample. NIST standard LaB6 was used to calibrate the distance from sample to detector.

High-pressure XRD (HPXRD) measurements on LiInCr4O8 were performed at the XPRESS beamline of the Elettra synchrotron radiation source, Italy. Powdered sample was loaded into a Mao–Bell type diamond anvil (DAC) along with gold (Au) as a marker to determine the pressure with an accuracy of ∼0.1 GPa. [10,11]. The diamonds had a culet diameter of ∼400 μm. A tungsten gasket of thickness ∼180 μm was pre-indented to a thickness of ∼50 μm before drilling a hole of 150 μm at the centre of the gasket. A methanol–ethanol mixture in a 4:1 ratio served as a quasi-hydrostatic pressure transmitting medium (PTM) [12]. Monochromatic X-rays of energy 25 keV (*λ* = 0.4957 Å) were incident on the sample. A MAR345 detector was used to record the diffraction patterns. Standard LaB6 loaded in the DAC was used to calibrate the experimental setup. The diffraction images were reduced into 2*θ*-intensity patterns using the program Fit2D [13], and the refinement of the XRD patterns obtained were performed using GSAS [14].

Raman spectroscopy measurements were performed with a confocal micro-Raman setup (Jobin–Yvon T64000 spectrograph, single stage mode, 1800 groves/mm grating, and a resolution of 2 cm−1) using a 50× objective in the back scattering geometry. A 488 nm argon ion laser was used as excitation source. The pressure-dependent Raman scattering measurements were carried out using a gas membrane type DAC, with diamonds with ∼500 μm culet diameter. The pressure inside the DAC was monitored using the well-known ruby fluorescence shift with an accuracy of 0.1 GPa [15]. A methanol–ethanol mixture in a 4:1 ratio was used as PTM.

The infrared spectroscopic measurements were performed using a Bruker Vertex FTIR spectrometer coupled to an IR microscope (Bruker Hyperion). The ambient pressure infrared reflectivity measurements were carried out in the spectral range 100–25,000 cm−<sup>1</sup> on a polycrystalline sample pressed into a pellet of thickness 72 μm. The resolution for the various frequency ranges, viz., FIR, MIR, NIR-VIS, amount to 2 cm<sup>−</sup>1, 4 cm−1, and 8 cm−1, respectively. Reflection from an aluminium mirror was used for the reference measurement, for normalizing the sample spectrum. The optical conductivity *σ*<sup>1</sup> was obtained by Kramer– Kronig (KK) transformation. For the KK transformation, the reflectivity spectrum was fitted with the Lorentz model and extrapolated to zero frequency based on the fitting model; in the high-energy range, a constant extrapolation up to 105 cm−<sup>1</sup> was used, and beyond this, an extrapolation following a 1/w4 dependency was chosen.

For the pressure-dependent infrared transmittance measurements in the spectral range 500–20,500 cm−<sup>1</sup> with a resolution of 4 cm−1, the powder sample was diluted with CsI in the ratio 1:20 and pressed into a pellet of thickness 64 μm. A small piece of this diluted pellet was loaded into a membrane type DAC (500 μm culets) along with ruby spheres and well-ground CsI powder serving as reference for normalizing the sample spectrum while at the same time acting as a quasihydrostatic PTM [16]. The absorption spectrum was calculated using the formula A = −log10(T), where A is the absorbance and T is the measured transmittance spectrum.

*Ab initio*-based simulations for structural relaxations were performed using DFT within the framework of the projected augmented wave (PAW) method [17], as implemented in the Quantum Espresso 6.4.1 package [18]. Calculations were performed using Perdew, Burke, and Ernzerhof (PBE) [19] generalized gradient approximations (GGA) for exchange and correlation functional. Structural relaxations were performed on a primitive cell of LiInCr4O8 with fourteen atoms of four different types in a non-magnetic configuration. Simulations were performed by considering one valence electron of Li (2s1), thirteen valence electrons of In (5s2 5p<sup>1</sup> 4d10), six valence electrons of Cr (4s2 3d4), and six valence electrons of O (2s<sup>2</sup> 2p4). A plane wave cut-off energy of 160 Ry was used for expanding the basis set. The Brillouin zone integration was performed at the zone centre on a 6 × 6 × 6 Monkhorst– Pack k-point mesh [20]. The Hellman–Feynman forces were converged until the largest force component was less than 1 × <sup>10</sup>−5. For the purpose of phonon-mode assignments, density functional perturbation theory (DFPT) [21] calculations were performed using Martins–Trouilier [22] pseudopotentials with local density approximation (LDA) on the primitive lattice of LiInCr4O8 at ambient volume.

#### **3. Results**

#### *3.1. Ambient Pressure Results*

LiInCr4O8 crystallizes in the cubic space group F4¯3m with four formula units per unit cell [5]. The lattice parameter, shape profile parameters, and asymmetry corrections were refined during the Rietveld refinement of the ambient pressure XRD pattern, shown in Figure 2. The background was fitted using a Chebyschev polynomial. The synthesized sample contains less than 2% unreacted Cr2O3 as observed from Figure 2. The refined lattice parameter is *a* = 8.4038(1) Å and the unit cell volume amounts to V = 593.52(2) Å3, with R*<sup>p</sup>* = 0.075 and R*wp* = 0.109. These values are in good agreement with the reported values of *a* = 8.4205 Å and V = 597.05 Å3 [4,5]. The fractional coordinates, occupations, and Wyckoff sites as reported by an earlier study [5] and the refined thermal parameter obtained from the Rietveld analysis are given in Table 1.

**Figure 2.** Rietveld refinement of the X-ray diffraction pattern of LiInCr4O8 at ambient pressure.

**Table 1.** The Wyckoff sites, fractional coordinates, and occupations as reported in an earlier study [5] and thermal parameter *U* as obtained from Rietveld refinement of LiInCr4O8 at ambient conditions. Space group F43m, Z = 4, ¯ *a* = 8.4038(1) Å, V = 593.52(2) Å3.


DFT-based simulations were performed using LDA and GGA pseudopotentials and the results for ambient volume calculations are compared in Table 2. The *A*-site cations Li1<sup>+</sup> and In3<sup>+</sup> at 4a and 4d crystallographic sites form LiO4 and InO4 tetrahedra, respectively, and share corners with CrO6 at the B-site (see Figure 1). The CrO6 octahedral units at the general 16e Wyckoff positions have shared edges between them, and the Cr4 tetrahedra form the breathing pyrochlore lattice. The Cr–Cr distances are usually short enough to facilitate electron hopping in some of the conventional spinels (Mott insulators). The quality of synthesized sample was verified by determining the degree of distortion (d /d) due to the two different Cr–Cr bond lengths of Cr4 tetrahedra from the Rietveld refinement, where the ratio d /d is found to be 1.051, which matches with the reported value [4,5,8]. This ratio of dissimilar Cr–Cr distances can be directly associated with the breathing factor B*<sup>f</sup>* in LiInCr4O8.

**Table 2.** Comparison of experimental and theoretical primitive cell volume together with the distortion parameter and the reported values from the literature.


For a primitive cell containing one formula unit (14 atoms), the factor group analysis gives the following irreducible representations:

$$
\Gamma\_{total} = 3\mathbf{A}\_1 + 3\mathbf{E} + 3\mathbf{T}\_1 + 8\mathbf{T}\_2.
$$

This can further be classified as:

$$
\Gamma\_{\text{acoustic}} = \text{T}\_2$$

$$
\Gamma\_{IR} = \text{TT}\_2$$

$$
\Gamma\_{\text{Raman}} = 3\text{A}\_1 + 3\text{E} + \text{TT}\_2$$

From the expected 13 Raman-active modes at ambient conditions, we experimentally observe nine modes, which are denoted as M1, M2,...M9 in Figure 3. Due to the loss of inversion symmetry in comparison with conventional spinels, LiInCr4O8 has T2 modes which are both IR- and Raman-active. The weak mode at ∼530 cm−<sup>1</sup> has been assigned to Cr2O3 [23,24]. The peak observed at ∼220 cm−<sup>1</sup> is a plasma line from the excitation source used, which served as an internal calibrant. The ambient pressure Raman spectrum of LiInCr4O8 in Figure 3 matches well with a recently reported study [25], except for the low-energy Raman mode M1, which is not observed in the recorded spectral range of Ref. [25]. Because there has been no detailed report on the vibrational properties of LiInCr4O8, DFPT calculations were carried out for explicit assignment of Raman and IR modes. These calculations were performed on a primitive cell consisting of 14 atoms using the Martins–Trouilier pseudopotentials with LDA approximations. The phonon modes were assigned with the help of Molden, a visualization software [26]. Simulations were also performed using PAW potentials with GGA approximations at ambient volume. The theoretical results are compared with experimental findings, summarized in Table 3.

Predominantly, the observed Raman modes can be classified as the internal vibrations of polyhedral units (viz., LiO4, InO4 and CrO6). Due to the covalent nature of In–O and Cr–O bonds, they are expected to have stronger contributions in the Raman spectrum. The symmetry of a free CrO6 ion is reduced inside the crystal site. Table 4 shows the changes in the internal modes of a free CrO6 ion at the crystal site (C3*v*) of the ambient pressure cubic phase (T*d*). Of these CrO6 modes, only the A1 and T2 modes are Raman active in LiInCr4O8.

From the DFPT calculations, the Raman-active T2 mode observed at 164 cm−<sup>1</sup> is assigned to the translation motion of In. The modes M2 and M3 at 310 cm−<sup>1</sup> and 443 cm<sup>−</sup>1, respectively, are both assigned to the translation motion of Li ions. The Raman modes at 460 cm−<sup>1</sup> and 491 cm−<sup>1</sup> (M4 & M5) are associated with asymmetric and symmetric O–Li– O bending vibrations, respectively. The strongest Raman mode observed at ∼590 cm−<sup>1</sup> is related to the O–Cr–O symmetric bending and O–In symmetric stretching vibrations, whereas the adjacent T2 mode at ∼582 cm−<sup>1</sup> is related to the O–Cr–O asymmetric bending and In-O asymmetric stretching vibrations. In a recent study, the DFT-based calculations for LiGaCr4S8 show large differences in the calculated Cr–Cr distances in magnetic and non-magnetic configurations of the system [27]. This could explain the large difference in the calculated and observed M6 and M7 modes, as all the calculations are performed in a non-magnetic configuration in the present study. The Raman modes at 718 cm−<sup>1</sup> and 739 cm−<sup>1</sup> (M8 and M9) are assigned to asymmetric and symmetric stretching vibrations of Li–O. A study on LiFeCr4O8 has assigned the strongest observed mode to Cr–O stretching vibration [9]. However, as the CrO6 octahedra have shared edges, they restrict the Cr–O stretching motion in this structure, and hence only the bending modes are prominent.

**Figure 3.** Ambient pressure Raman spectrum of LiInCr4O8, together with the fitting curve. Shown also are the theoretical mode frequencies (GGA) and the Raman mode frequencies for Cr2O3 [23,24].

**Table 3.** Mode assignment and comparison of experimental and theoretical mode frequencies (in cm−1). (w), (m), and (s) denote the strength of active mode, viz., weak, medium and strong, respectively.



**Table 3.** *Cont.*

**Table 4.** Internal modes of CrO6 octahedra.


Figure 4a depicts the ambient pressure reflectivity spectrum of LiInCr4O8 over a broad frequency range, together with the Lorentz fitting. The inset shows the low-frequency range up to 800 cm−1, where the phonon modes are located. The corresponding optical conductivity *σ*<sup>1</sup> obtained from the KK transformation is depicted in Figure 4b. The *σ*<sup>1</sup> spectrum shows strong phonon contributions in the far-infrared range, which is followed by the onset of electronic excitations around ∼0.1 eV, which gradually increases to two prominent absorption bands centred at ∼1.65 and 2.4 eV. Based on previous studies, the absorption bands can be assigned to intra-atomic *d*-*d* excitations, i.e., crystal field (CF) excitations, of the Cr3<sup>+</sup> ions in an octahedral environment. Electronic excitations from the <sup>4</sup>*A*2*<sup>g</sup>* ground state to the <sup>4</sup>*T*2*<sup>g</sup>* and <sup>4</sup>*A*1*<sup>g</sup>* excited states are expected in the spectral ranges of ∼13,000 to 17,000 cm−<sup>1</sup> and ∼13,000 to 17,000 cm−1, respectively, and the spin-forbidden transitions from <sup>4</sup>*A*2*<sup>g</sup>* to <sup>2</sup>*Eg* and <sup>2</sup>*E*2*<sup>g</sup>* are expected in the spectral ranges of 13,000 to 14,400 cm−<sup>1</sup> and 18,000 to 19,200 cm<sup>−</sup>1, respectively, [28–36].

The inset in Figure 4b shows the observed ten phonon modes fitted with Lorentzian oscillators. However, the group theoretical analysis predicts only seven T2 modes for LICO. Therefore, modes were assigned to the strongest modes after comparison with DFPT calculations to the closest calculated values (see Table 3). An earlier infrared spectroscopic study on LiFeCr4O8 has assigned the observed five phonon modes to the internal vibrations of the polyhedral units, viz., Li–O stretching modes in the range 400–500 cm−<sup>1</sup> and CrO6 and FeO4 vibrations around 500 cm−<sup>1</sup> and ∼640 cm−1, respectively [37]. In the present study, based on the DFPT calculations, the phonon modes above 600 cm−<sup>1</sup> are assigned to the Li–O stretching vibrations. The low-energy modes are assigned to the In–O vibrations and those observed around ∼500 cm−<sup>1</sup> belong to the CrO6 internal vibrations.

**Figure 4.** Ambient pressure (**a**) reflectance and (**b**) optical conductivity spectrum of LiInCr4O8 in the range 100–22,000 cm<sup>−</sup>1, together with the Lorentz fitting. The insets of (**a**,**b**) depict the corresponding low-frequency range (200–800 cm<sup>−</sup>1) of the reflectance and optical conductivity spectrum, respectively, where the phonon modes are located, together with the Lorentz fitting.

#### *3.2. High-Pressure Results*

#### 3.2.1. X-ray Diffraction Measurements

A few XRD patterns at selected pressures are presented in Figure 5. According to these results, the sample remains in the ambient phase up to ∼14 GPa. Above 10 GPa, i.e., beyond the hydrostatic pressure limits of the PTM used, the peaks at higher values of 2*θ* become weaker and undergo a pressure-induced broadening, thereby making it difficult to trace their behaviour precisely with pressure. However, the peaks at lower angles provide clear signatures of phase transitions. Above 10 GPa, the (200) and (220) reflections show a broadening. Beyond ∼14 GPa, they undergo a splitting, which is an indication of a first-order structural phase transition to a high-pressure phase (HP-1). The clear splitting of these off-diagonal planes into two peaks while the diagonal planes (111) remain intact is an indication of a cubic-to-tetragonal structural phase transition. Previous temperature-dependent XRD studies on Li*M*Cr4O8 (*M* = In, Ga, Fe) compounds have

reported a cubic-to-tetragonal phase transition at very low temperatures. Based on earlier reported studies, the HP-1 phase was indexed to a tetragonal structure (space group I4¯m2) with two formula units per unit cell (Z = 2). The Rietveld refinement of our XRD data collected at ∼14.3 GPa indicated the coexistence of the ambient and HP-1 phases (see Figure 6). The refined lattice parameters of the HP-1 phase at ∼14 GPa are *a* = 5.9030(26) Å, *c* = 8.0681(60) Å, and V = 281.13(22) Å3, with R*<sup>p</sup>* = 0.038 and R*wp* = 0.058. The structural information from the Rietveld analysis for the coexisting ambient and the HP-1 phases are given in Table 5 and Table 6, respectively. This information is also shown in Figure 8. For both the ambient and HP-1 phases, the atomic positions and their occupations were not refined and are used as reported in earlier studies [5,8]. The structural refinement of the diffraction patterns above 14 GPa could not be performed; hence, only one data point of the HP-1 phase is shown in Figure 8. Detailed information regarding the coordination of polyhedral units cannot be extracted from this powder diffraction as the oxygen positions remain unrefined to maintain the reliability of the Rietveld refinement. The volume per formula unit in the ambient pressure phase is 148.38 Å3 and that of the HP-1 phase is 140.57 Å3, which gives a compression of 5.3% per formula unit.

**Table 5.** The Wyckoff sites, fractional coordinates, and occupations as reported in an earlier study [5] and the thermal parameter *U* as obtained from Rietveld refinement of LiInCr4O8 in the ambient structure at <sup>∼</sup>14 GPa. Space group F43m, Z = 4, ¯ *<sup>a</sup>* = 8.2624(13) Å and V = 564.05(26) Å3.


**Table 6.** The Wyckoff sites, fractional coordinates, and occupations as reported in an earlier study [8] and the thermal parameter *U* as obtained from Rietveld refinement of LiInCr4O8 in the HP-1 phase at <sup>∼</sup>14 GPa. Space group I4m2, Z = 2, ¯ *<sup>a</sup>* = 5.9030(26) Å, *<sup>c</sup>* = 8.0681(60) Å, and V = 281.13(22) Å3.


The low-temperature XRD studies on LiInCr4O8 reported the tetragonal phase coexisting with the ambient cubic phase down to the lowest recorded temperature of ∼2K[7,8]. Consistently, our high-pressure studies also show that the sample does not completely transform to the tetragonal phase with further compression. Instead, before this cubic to tetragonal transition is completed, the reflections from the (111) set of planes undergo a broadening at the next recorded pressure (∼16 GPa) and then split into two peaks at higher pressures. The XRD pattern recorded at ∼19.7 GPa shows clear changes, indicating the existence of a new high-pressure phase (HP-2 phase). The transition from the HP-1 to HP-2 phase is a slow and sluggish first-order structural phase transition where the HP-1 coexists with the evolving HP-2 phase. This new high-pressure phase (HP-2 phase) remains stable up to ∼35 GPa. With further compression, the diffraction pattern recorded above ∼35 GPa shows the emergence of some new peaks, suggesting a possible transition to a lower-symmetry structure (HP-3 phase). The transition to the HP-3 phase was not completed at the highest pressure recorded in this experiment (i.e., ∼36.7 GPa). Due to

the broadness of the diffraction peaks, overlapping with the strong reflections from the pressure marker (Au) and the gasket (W) used in these experiments, the high-pressure phases could not be identified. During decompression, the HP-1 and HP-2 phases coexist down up to 12 GPa. Below 8 GPa, the sample slowly transforms to the parent phase and reverts to the ambient pressure crystal structure on complete release of pressure.

**Figure 5.** Pressure-dependent XRD patterns at few selected pressures. The pressure values on the right are in GPa. The red arrows indicate the emerging new reflections. The red and black ticks at the bottom are from Au (pressure marker) and W (gasket), respectively.

**Figure 6.** Rietveld refinement of the XRD pattern at ∼14 GPa, demonstrating the coexistence of the HP-1 (space group I4m2) phase with the ambient pressure cubic phase. ¯

It is well known that the diffraction peak width has its origins from instruments, particle size, and stress-induced broadening. In this study, the XRD pattern from standard CeO2 inside the DAC recorded at ambient conditions was used to characterize the instrumental broadening. Changes in the peak width under compression observed in the present study can be attributed to inhomogeneous strain and change in crystallinity due to phase transformation. For a better understanding of the deformation at high pressures, the behaviour of the diffraction peaks and X-ray peak broadening were studied by plotting the variation of the peak position and the full-width-at-half-maximum (FWHM) value for the reflections from the (111), (200), and (220) planes as a function of pressure in Figure 7a,b, respectively. Structural phase transitions to the HP-1, HP-2, and HP-3 phases at ∼14 GPa, ∼19 GPa, ∼35 GPa, respectively, can be clearly observed from Figure 7b. The FWHM value of all peaks remains almost constant up to ∼10 GPa. Above ∼10 GPa, all the peaks show a slight discontinuity and broadening, which can be attributed to the non-hydrostatic stresses [38]. However, the (hk0) and (h00) peaks show a significant change in the FWHM as compared to the (111) set of planes. This sudden change is observed in FWHM just before the structural transition at ∼14 GPa. Immediately after the phase transition, i.e., above ∼14 GPa, both (220) and (200) undergo a clear splitting due to the transition from the cubic to tetragonal phase. Above ∼16 GPa, the (111) reflection also shows a sudden change in FWHM, which is an indication of beginning of another structural transition. With further compression, the (111) reflection exhibits a clear splitting (see Figure 7a).

**Figure 7.** Behaviour of the (**a**) Bragg peaks and (**b**) FWHM with pressure. The vertical red dashed line indicates the hydrostatic limit of the PTM used. Solid black lines mark the transition pressures.

To determine the structural stability of LiInCr4O8, the lattice parameter *a* and the volume per unit cell *V* were determined for each recorded pressure from the structural refinements. Both parameters are plotted in Figure 8, where the abrupt decrease in compressibility is notable above ∼10 GPa due to nonhydrostatic conditions [39,40]. The P–V data up to ∼10 GPa (hydrostatic limit of PTM used) was fitted with a Murnaghan equation of state according to *V*(*p*) = *V*<sup>0</sup> · [(*B* <sup>0</sup>/*B*0)· *p* + 1] −1/*B* <sup>0</sup> [41], where the first-order derivative of the bulk modulus *B* <sup>0</sup> was fixed to 4 (see Figure 8). The bulk modulus B0 of the ambient phase was found to be 186.7 ± 6.1 GPa. From the DFT calculations, the ambient pressure crystal structure was relaxed at different target pressures. The energy per formula unit and the corresponding volumes were fitted with the Murnaghan equation of state to compare with the experimental findings. The calculated B0 and *B* <sup>0</sup> amount to 176.8 ± 0.1 GPa and 4.63 ± 0.02, respectively. The difference in the experimental and calculated ambient volume and the non-magnetic configuration of the calculated system could explain the discrepancy between the experimental and theoretical values of the bulk modulus.

**Figure 8.** (**a**) Experimental lattice parameters *a* and *c* and (**b**) volume per unit cell *V* of the ambient phase and high-pressure HP-1 phase (scale on the right side of the graph) as a function of pressure. The pressure dependence of the volume is fitted with a Murnaghan equation of state (EOS), as defined in the text.

LiInCr4O8 with two different Cr–Cr distances (d = 2.90 Å and d = 3.05 Å) has values that lie in the range of conventional spinels such as ZnCr2O4 with a uniform Cr–Cr

distance of 2.944 Å, and that of CdCr2O4 is 3.041 Å. A theoretical study on MgCr2O4, MnCr2O4, and ZnCr2O4 reports their bulk moduli to be 197.3 GPa, 205.8 GPa, and 215 GPa, respectively, [42], whereas experimentally, MgCr2O4 is reported to have a bulk modulus of 189 GPa [43,44] and that of ZnCr2O4 is 183.1 GPa [45]. LiCrO2 is reported to have a bulk modulus of 161 GPa [46]. The experimentally obtained value of B0 for LiInCr4O8 from the present study is comparable with that of other chromium spinels.

#### 3.2.2. Raman Spectroscopy Measurements

To investigate the pressure-induced changes in the vibrational properties of LiInCr4O8, high-pressure Raman spectra were recorded in the spectral range ∼120–850 cm<sup>−</sup>1. Figure <sup>9</sup> depicts Raman spectra at selected pressures. At the lowest recorded pressure of ∼0.4 GPa, the spectrum shows a splitting of a few Raman modes in the range of 450–600 cm−1. Because the high-pressure XRD measurements do not provide any indication of a structural transition at such low pressures, the splitting of the Raman modes can be attributed to a loss of accidental degeneracy. All observed Raman modes shift to higher frequencies under pressure due to the pressure-induced stiffening of the lattice. The intensity of the Raman mode observed at ∼164 cm−<sup>1</sup> drops drastically with compression, and this mode vanishes above ∼2.5 GPa. Mode M2 at ∼310 cm−<sup>1</sup> also undergoes a significant decrease in the intensity with pressure. Both these modes are associated with the translational motion of the *A*-site cations. The modes M4 and M6, which are related to the O–Li–O bending and O–In stretching vibrations, respectively, undergo a splitting at ∼ 0.4 GPa. The modes M3 and M4 combine into a single mode at ∼9 GPa. At ∼2.5 GPa, a further splitting of mode M6 is observed, and this new mode can be traced up to ∼10 GPa. The splitting of modes associated with internal vibrations of polyhedral units can be attributed to the distortion under pressure. No significant changes in the Raman spectrum are observed with further compression up to ∼14 GPa. Above 14 GPa, the splitting of the mode at ∼545 cm−<sup>1</sup> can be associated with the structural phase transition as observed from XRD measurements. The most intense mode (M7), associated with the O–Cr–O bending vibrations, remains the most intense peak up to the highest recorded pressure of ∼ 18 GPa.

The frequency shift of the Raman modes with pressure is summarized in Figure 10, where the emerging new modes can be clearly seen. During pressure release (see Figure 9), the sample slowly reverts to its ambient pressure phase, where the modes at ∼310 cm−<sup>1</sup> reappear at around 7 GPa, and the mode at ∼164 cm−<sup>1</sup> can be observed at ∼3 GPa. According to the decompressed spectrum at ambient conditions, the sample has transformed to its ambient phase, consistent with our pressure-dependent XRD results.

The calculated pressure coefficients of Raman modes and the corresponding Grüneisen parameters (*γ<sup>i</sup>* = (B0/*ωi*)(d*ωi*/dP)) are tabulated in Table 7. B0 is the experimentally obtained bulk modulus which was used to calculate the mode Grüneisen parameters for the ambient phase. The modes M8 and M9 show relatively larger value of the pressure coefficient, indicating an increase in the force constant under pressure. The tabulated mode Grüneisen parameter relates the vibrational properties to the crystal deformations. The macroscopic Grüneisen parameter *γ* is a weighted sum of individual *γi*'s. The relation between the two is given as [*γ* = (∑*<sup>i</sup> γiCi*)/(∑*<sup>i</sup> Ci*)], where *Ci*'s are the mode contributions to the material's specific heat [47]. Using Einstein's specific heat relation, *Ci* = *R*[*xi* <sup>2</sup>*exp*(*xi*)]/[*exp*(*xi*) − <sup>1</sup>] <sup>2</sup> (where *xi* = *h*¯ *ωi*/*kBT* and *R* is the universal gas constant) [48], the macroscopic Grüneisen parameter was calculated using the observed modes from M1 to M9 at ambient pressure and its value was found to be *γ* = 0.99. A negative contribution to thermal expansion can be associated with a negative mode Grüneisen parameter. In the ambient phase, all the modes are notably contributing to a positive thermal expansion, consistent with recent studies [27,49] stating a positive expansion observed in LiInCr4O8, whereas other isostructural compounds LiGaCr4O8 and LiInCr4S8 are reported to exhibit negative thermal expansion. A softening of the M1−*HP* mode is, however, observed in the high-pressure phase. All the other modes observed above 14 GPa still show a positive shift with compression.

**Figure 9.** Raman spectra of LiInCr4O8 at selected pressures. The numbers on the right are in GPa scale. Red arrows indicate the emergence of new Raman modes.

**Figure 10.** Frequency shift of Raman modes with increasing pressure. The black and red symbols represent the Raman modes of the ambient phase and those observed above 14 GPa, respectively. Different shadings illustrate the changes in the Raman spectrum. The red dashed lines highlight new modes emerging under pressure.


**Table 7.** Pressure at which the modes are observed, frequency *ω<sup>i</sup>* of observed Raman modes, pressure coefficients d*ωi*/dP of Raman modes, and mode Grüneisen parameter *γ<sup>i</sup>* (see text).

#### 3.2.3. Infrared Spectroscopy Measurements

The pressure dependence of the absorbance spectrum of LiInCr4O8 is depicted in Figure 11a for pressures up to ∼17.4 GPa. The features observed close to 2000 cm−<sup>1</sup> are due to the multi-phonon absorptions in diamond. In the recorded range, we observed four phonon modes at low frequencies followed by the onset of electronic transitions at around 1000 cm<sup>−</sup>1. In comparison to the ambient pressure optical conductivity (Figure 4), the onset of electronic transitions is spread out in the absorption spectrum, and the crystal field excitation appears as a broad peak in the spectrum around 16,800 cm−<sup>1</sup> (2.05 eV), corresponding to the spin-allowed intra-atomic *d*-*d* transition between the ground state 4A2*<sup>g</sup>* and the excited 4T1*<sup>g</sup>* and 4T2*<sup>g</sup>* states [28–32]. The pressure dependence of this onset is extracted by extrapolation of a linear fit for each pressure value up to 10 GPa (Figure 12b). Beyond this pressure, the change in the spectral slope leads to unrealistic values for linear extrapolation. The onset of electronic transitions shifts to higher energies with increasing pressure indicating an increase in the band gap under compression, in agreement with the pressure-induced blue-shift of the crystal field excitation (Figure 12b). The observed crystal field excitations were fitted to a Lorentzian profile to obtain the corresponding energy position. This result is consistent with an earlier reported infrared absorption study on CdCr2O4, where a similar behaviour of the crystal field excitation under pressure was observed [32]. Weak features slowly emerging close to ∼14,500 cm−<sup>1</sup> at 4.6 GPa (marked by black arrows in Figure 11a) can be ascribed to the spin-forbidden crystal field transitions that may become infrared-active as a result of lattice vibrations which locally break the centre of symmetry. The absorption spectrum at the highest pressure of ∼17.4 GPa shows a changed crystal field excitation profile, which we relate to the pressure-induced structural phase transition observed at ∼19 GPa in our XRD measurements. The slightly lower critical pressure extracted from our infrared studies can be explained by the solid PTM used, which is less hydrostatic than the PTM (alcohol mixture) of the XRD experiment [50,51].

The pressure-dependent frequencies of the infrared-active phonon modes are presented in Figure 12a. In this spectral region, the observed modes belong to the internal bending vibrations of the CrO6 octahedral unit and Li–O stretching vibrations. The observed infrared-active modes show a blueshift under pressure, consistent with the pressure behaviour of the Raman modes. The phonon modes P1 and P2 merge at 14 GPa, whereas

the mode P4 shows a deviation from a linear fit at the same pressure. At the same pressure, one observes a sudden change in the slope of the pressure-induced shift of the crystal field excitation (Figure 12b). All these changes can be associated with the structural phase transition observed from XRD measurements (discussed earlier). In Figure 12b, a sudden jump in the onset energy is also observed at 7.4 GPa; this, however, cannot be associated with any structural phase transition. The pressure coefficients (d*ω*/dP) of the observed (P1–P4) modes are 5.1, 5.24, 4.28, and 3.69 cm−1GPa−1, respectively. These values are comparable and close to the pressure coefficients of the high-energy Raman modes (M5, M7, M8, and M9) (see Table 7).

**Figure 11.** (**a**) Pressure-dependent infrared absorbance spectra in a broad frequency range 500–20,500 cm<sup>−</sup>1. The two weak features indicated by black arrows are due to spin-forbidden crystal field excitations. (**b**) Low-frequency absorbance spectra with phonon modes (labelled P1–P4) as a function of pressure, fitted with Lorentz oscillators.

**Figure 12.** (**a**) Behaviour of infrared-active phonon mode frequencies with pressure. (**b**) Onset of electronic transitions and energy of crystal field (CF) excitation as a function of pressure. The vertical line at ∼14 GPa indicates the structural phase transition.

#### **4. Conclusions**

In summary, we report multiple pressure-induced phase transitions in the chromium spinel LiInCr4O8. From the high-pressure XRD measurements, the HP-1 phase with tetragonal structure appearing at ∼14 GPa seems to be an intermediate metastable phase. The system does not completely transform to this structure and always coexists with the ambient pressure cubic phase. The high-pressure infrared and Raman spectroscopy measurements confirm the structural phase transition at ∼14 GPa. All the observed active modes show a pressure-induced blueshift indicative of the stiffening of the lattice. The estimated onset of the electronic transitions from the high-pressure infrared measurements indicates an increase in the band gap under compression. Although the breathing pyrochlore structure of LiInCr4O8 is known to be a geometrically frustrated system, it appears to be highly resilient when the system reverts to its ambient phase even after undergoing multiple pressure-induced structural changes.

**Author Contributions:** Investigation and formal analysis, M.V. and M.K.; Visualization, data curation and writing—original draft, M.V.; Writing—review, M.K.; Supervision, resources, writing—review and editing, H.K.P. and C.A.K. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Data is contained within the article.

**Acknowledgments:** We thank Matthias Schreck for providing technical support for the Raman measurements. We acknowledge Srihari Velaga, ECXRD beamline, Indus-2, RRCAT, India, and Boby Joseph, XPRESS beamline, Elettra Synchrotron, Italy, for their support in using the beamline facilities. We acknowledge the Indus-2, RRCAT, India, and Elettra Synchrotron, Italy, for the provision of beamtime. We also thank J. Ebad-Allah, F. Meggle, S. Rojewski, M. Köpf, and G. Eickerling for the help and fruitful discussions.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


**Disclaimer/Publisher's Note:** The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

## *Article* **Phase Transformation Pathway of DyPO4 to 21.5 GPa**

**Jai Sharma, Henry Q. Afful and Corinne E. Packard \***

Department of Metallurgical and Materials Engineering, Colorado School of Mines, Golden, CO 80401, USA **\*** Correspondence: cpackard@mines.edu

**Abstract:** Interest in the deformation behavior and phase transformations of rare earth orthophosphates (REPO4s) spans several fields of science—from geological impact analysis to ceramic matrix composite engineering. In this study, the phase behavior of polycrystalline, xenotime DyPO4 is studied up to 21.5(16) GPa at ambient temperature using in situ diamond anvil cell synchrotron X-ray diffraction. This experiment reveals a large xenotime–monazite phase coexistence pressure range of 7.6(15) GPa and evidence for the onset of a post-monazite transformation at 13.9(10) GPa to scheelite. The identification of scheelite as the post-monazite phase of DyPO4, though not definitive, is consistent with REPO4 phase transformation pathways reported in both the experimental and the computational literature.

**Keywords:** high pressure; rare earth orthophosphate; phase transformation; X-ray diffraction

#### **1. Introduction**

Rare earth orthophosphates (REPO4s) are highly refractory and insoluble ceramics relevant to various research areas ranging from geoscience to structural ceramics [1]. Much of the world's rare earth element supply comes from naturally occurring xenotime and monazite minerals, whose properties and formation are of great relevance in geochronology and geothermobarometry [2,3] in addition to mineral extraction. The pressure-induced phase transformations of certain REPO4 compositions have also spurred research toward their use as fiber coatings, where they can confer additional plasticity and toughening mechanisms to oxide–oxide ceramic matrix composites [4–6]. At ambient pressure (~10−<sup>4</sup> GPa), REPO4s adopt either the xenotime (tetragonal, I41/*amd*) or monazite (monoclinic, P21/*n*) structure. The xenotime structure is also referred to as "zircon" (based on ZrSiO4); however, this study employs the former name because it specifically originates from YPO4 minerals [1]. At high pressures, xenotime compositions transform into the monazite or scheelite (tetragonal, I41/*a*) structures (see Figure 1) [7]. The scheelite-type structure has also been observed in other ABO4 materials (e.g., tungstates, molybdates, vanadates, and arsenates) [8–10]. The REPO4 xenotime, monazite, and scheelite structures feature chains of alternating PO4 tetrahedra (shown in gray) and RE-O polyhedra (shown in violet) with RE-O coordination numbers of 8, 9, and 8, respectively.

**Figure 1.** [001] views of REPO4 structure in the (**a**) xenotime, (**b**) monazite, and (**c**) scheelite phases. RE-O polyhedra are shown in violet, PO4 tetrahedra are shown in grey, and the unit cell boundaries are shown as thin black boxes. The 90◦ rotation of monazite axes with respect to those of xenotime and scheelite is a result of the monoclinic cell setting of monazite, as shown in detailed transformation schemes reported by Hay et al. [5]. Structures are visualized using the VESTA software [11].

**Citation:** Sharma, J.; Afful, H.Q.; Packard, C.E. Phase Transformation Pathway of DyPO4 to 21.5 GPa. *Crystals* **2023**, *13*, 249. https:// doi.org/10.3390/cryst13020249

Academic Editor: Daniel Errandonea

Received: 30 December 2022 Revised: 23 January 2023 Accepted: 27 January 2023 Published: 1 February 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

Figure 1 shows that the REPO4 structure becomes increasingly compact when transforming from xenotime to monazite to scheelite. This compaction can be attributed to increasing rotation and displacement of the RE-O polyhedra and resultant changes in the phosphate chain linkages (edge-sharing → corner-sharing) [10]. Although the exact unit cell volume losses during these transformations are composition-dependent, the loss associated with the xenotime → monazite transformation is significantly lower than that of the monazite → scheelite transformation [12]. This disparity likely emerges from the fact that the former transformation involves an increase in RE-O coordination (8 → 9), while the latter involves a decrease (9 → 8) [10]. An intermediate anhydrite (orthorhombic, Amma) phase has also been reported in certain xenotime compositions prior to the emergence of monazite when the xenotime composition is subject to high deviatoric stresses (e.g., TbPO4 [13]) or has a composition that is extremely close to the 1 atm (~10−<sup>4</sup> GPa) xenotime–monazite phase boundary (e.g., GdxTb1-xPO4 [14]).

Prior studies and reviews have reported phase diagrams showing REPO4 transformation pressures based on a variety of computational and experimental techniques [5,15–18]. Recent advancements in in situ diamond anvil cell (DAC) X-ray diffraction (XRD) experiments require updating the high-pressure REPO4 phase map [12,18–25]. In contrast to Raman spectroscopy and ab initio calculations, XRD provides more direct, crystallographic proof of the existence of REPO4 phases and phase transformations. Figure 2 compiles experimentally observed phase data from DAC XRD studies for all non-radioactive, single-RE compositions except PrPO4, which has only been studied thus far using DAC Raman spectroscopy [26].

**Figure 2.** High-pressure phase map of the experimentally observed phases of all single-RE REPO4s except PmPO4 because Pm is both radioactive and extremely rare. The legend lists the phases with their corresponding space groups. For each composition, the upper limit of the highest bar(s) represents the highest pressure at which data are reported and does not represent a phase boundary. See text for references to the sources of phase data. All phase data are compiled from XRD studies except for PrPO4, which has only been characterized via Raman spectroscopy \* [26].

Under hydrostatic conditions, the xenotime → monazite transformation has been reported in ErPO4, HoPO4, YPO4, DyPO4, and TbPO4 with onset pressures (Ponset) of 17.3 GPa, 17.7 GPa, 14.6 GPa, 9.1(1) GPa, and 9.9 GPa, respectively [12,20–23]. We note that for any number followed by a number in parentheses, the number in parentheses represents the standard deviation of the last digit of the number before the parentheses. In the REPO4 phase transformation literature, there has long been an assumption that the xenotime → monazite Ponset varies linearly with RE3+ radius, such as many other properties of REPO4s [5]. However, the DyPO4 Ponset of 9.1(1) GPa from our 2021 study [22] disrupts this trend, suggesting instead that xenotime → monazite Ponset values fall into two clusters:

a high-pressure one around ~16 GPa (ErPO4, HoPO4, and YPO4) and another that is <10 GPa (DyPO4 and TbPO4). Alloyed compositions of GdxDy(1-x)PO4 and GdxTb(1−x)PO4 also have transformation pressures that fall within the lower-pressure group, without following a trend with average RE3+ radius [5,14,27]. Neither thermodynamic properties (e.g., enthalpies of formation) nor structural properties (e.g., bond lengths and angles) show significant discontinuities between YPO4 and DyPO4, yet these compositions' Ponset values are known to differ by at least 5.5 GPa [7,28–31].

The xenotime → monazite transformation in REPO4s has also been described as sluggish and kinetically limited due to the experimentally observed xenotime–monazite phase coexistence being inconsistent with thermodynamic expectations (i.e., Gibbs phase rule) [12,22,23]; this is shown in Figure 2 as the regions where blue and orange bars overlap. The xenotime–monazite phase coexistence ranges for ErPO4, YPO4, and TbPO4 are 6 GPa, 4.6 GPa, and 3.9 GPa, respectively [12,20,23]. In prior experiments, HoPO4 and DyPO4 were not taken to high enough pressures to capture the full xenotime–monazite coexistence range [21,22].

Other xenotime compositions with smaller RE3+ radii undergo the xenotime → scheelite transformation, which involves no change in RE-O coordination number. ScPO4, LuPO4, and YbPO4 transform directly to the scheelite structure at Ponset values of 34.2 GPa, 19 GPa, and 22 GPa, respectively [12,19]. Their respective xenotime–scheelite phase coexistence ranges (represented by overlap of green and blue bars in Figure 2) are 10.8 GPa, 8 GPa, and 1 GPa, indicating kinetic limitations similar to the xenotime → monazite transformation [12,19].

TmPO4, as described by Stavrou et al., represents a "borderline case" between xenotime compositions that transform to monazite and those that transform to scheelite [18]. In TmPO4, the xenotime → scheelite transformation begins at 20.3 GPa, and these two phases coexist over a 2.7 GPa range. Immediately after the disappearance of xenotime at 23 GPa, monazite emerges at 23.3 GPa and coexists with scheelite until 47 GPa. Then, scheelite persists through the end of the experiment. Stavrou et al. characterize monazite TmPO4 as a "metastable minority phase" and attribute the long monazite–scheelite coexistence to the stabilization of monazite grains "when embedded in a scheelite matrix" [14].

Among the compositions, which adopt the monazite structure at 1 atm (~10−<sup>4</sup> GPa), only LaPO4 has been shown to undergo a pressure-induced phase transformation. Lacomba-Perales et al. proposed that LaPO4 transforms to barite (orthorhombic, Pnma) based on powder XRD but could not confirm the barite structure due to significant peak overlap [20]. Ruiz-Fuertes et al. used single crystal XRD and second harmonic generation analysis to confirm the post-monazite structure as non-centrosymmetric "post-barite" (orthorhombic, P212121) [32]. Post-barite first emerges at 27.1 GPa and coexists with monazite through the end of the experiment at 31 GPa. This experimental data conflicts a bit with their ab initio calculations, which show a pressure range where barite is energetically preferred before the emergence of post-barite, but the authors argue large kinetic barriers may explain the lack of barite in their LaPO4 experiments and may hinder barite formation in other monazite REPO4 compositions [32]. Ruiz-Fuertes et al. project (based on ab initio calculations) a postbarite transformation to occur at 45 GPa and 35 GPa in GdPO4 and NdPO4, respectively, with barite as a possible, but unlikely, transition phase.

Our 2021 XRD study [22] reported a DyPO4 xenotime → monazite Ponset at 9.1(1) GPa under a quasi-hydrostatic loading rate but did not go to high enough pressures to resolve the xenotime–monazite phase coexistence range [22]. This XRD study aims to identify the end of the xenotime–monazite phase coexistence range by going to higher pressures. Results reveal a xenotime–monazite phase coexistence range of 7.6(15) GPa and a previously unreported phase transformation to a post-monazite phase at 13.9(10) GPa. Comparison to the experimental and the computational literature strongly suggests this post-monazite phase adopts the scheelite (tetragonal, I41/*a*) structure.

#### **2. Materials and Methods**

Phase-pure xenotime DyPO4 powder was obtained via precipitation reaction involving Dy(NO3)3 · 5H2O (≥99.9% RE oxide basis, Alfa Aesar) precursor and H3PO4 (85% *w*/*w* aqueous solution, Alfa Aesar) and subsequent calcination. These two steps are detailed elsewhere [33]. The sample powder consists of sub-micron grains, which exhibit the anisotropic, elongated crystal habit expected of tetragonal materials (see the scanning electron micrograph in Figure S1, see supplementary materials). An energy-dispersive X-ray spectrum of the powder (shown in Figure S2) shows no elemental impurities. In situ DAC XRD was conducted at room temperature at beamline 16-ID-B, HPCAT, Advanced Photon Source, Argonne National Laboratory. Two-dimensional diffraction patterns were collected with the PILATUS 1M-F detector. The X-ray wavelength was 0.42459 Å, and the beam spot size (full width at half maximum) was ∼2 μm by ~4 μm. We used a Diacell Helios DAC with a membrane (both from Almax easyLab Inc., Cambridge, MA, USA) driven by a Druck PACE 6000 pressure controller [34]. DAC preparation involved successively loading DyPO4 powder, gold powder (>99.96% metals basis, Alfa Aesar, Ward Hill, MA, USA), ruby chips (Almax easyLab Inc., Cambridge, MA, USA), and 16:3:1 methanol–ethanol–water mixture (MEW) pressure medium into the hole of the 301 stainless steel gasket. The gasket hole diameter and indented thickness were 220 μm and 80 μm, respectively. For pressure marking during initial membrane engagement, ruby was used (R1 fluorescence calibration [35]), while gold was used (third order Birch-Murnaghan EoS [36]) during diffraction data collection. Data collection started at 3.1(2) GPa due to some initial compression required to confirm membrane engagement. There are no reported DyPO4 phase transitions below this starting pressure (as corroborated by our 2021 XRD study) [22]; therefore, the initial jump does not preclude any material insight.

XRD pattern integration, masking, and background subtraction were performed using Dioptas [37]. Pattern fitting was then performed using X'Pert HighScore Plus [38]. This software fits monazite using the P21/*c* cell setting as a default. Although both the P21/*c* and P21/*n* cell settings are valid descriptions of monazite (space group No. 14), fitted lattice parameters were converted to the P21/*n* cell setting to facilitate comparison to the literature. The LeBail fitting approach [39] was used instead of traditional Rietveld structural refinement to accommodate the significant preferred orientation present in all scans. This apparent preferred orientation appears due to the small spot size of the beam with respect to the grain size of the sample (effectively sampling a finite number of grains) rather than any inherent orientation of the sample grains. The atmospheric-pressure volume of xenotime DyPO4 (289.39(2) Å3) was derived from a prior synchrotron XRD pattern of a sample from the same batch as the sample in this study [22]. The following reference structures were used in this study: xenotime DyPO4 from Milligan et al. [40], monazite DyPO4 from Heuser et al. [41], gold from Couderc et al. [42], ruby from Jephcoat et al. [43], and calculated scheelite TbPO4 from López-Solano et al. [23]. Importantly, the scheelite TbPO4 structure file was not employed in LeBail fitting—only in peak position comparison. For the computation involving derived data (e.g., unit cell volume, gold-based pressure, cell setting conversion), Python was used to propagate error with an assumed covariance of zero.

#### **3. Results**

During the experiment, the gold lattice parameter decreases steadily as shown in Figure 3a. In Figure 3b, the pressure increases steadily with time and yields an effective sample loading rate of ~20 MPa/s, an order of magnitude faster than that of our 2021 XRD study [22].

**Figure 3.** Plots showing (**a**) gold lattice parameter and (**b**) pressure against time during the experiment. Error bars represent standard deviation. The linear fit of the pressure data yields an effective loading rate of 17.2(1) MPa/s. Green, shaded areas represent the pressure range in which a third (postmonazite) phase of DyPO4 appears in the patterns.

Figure 4 shows the pressure evolution of integrated background-subtracted XRD patterns. The square root of intensity is plotted against Q to show weak peaks more clearly.

**Figure 4.** Synchrotron XRD patterns. X, M, and S(TbPO4) ticks show Bragg reflections of xenotime DyPO4, monazite DyPO4, and scheelite TbPO4, the last based on ab initio structural data reported by Lopez-Solano et al. [23]. Star symbols mark emerging monazite peaks at 9.6(7) GPa, and diamond symbols mark emerging peaks of the unidentified phase at 13.9(10) GPa. (**a**) A contour plot showing all XRD patterns. Peak positions of gold and ruby are marked with triangles and circles, respectively. (**b**) The LeBail fits of key patterns (initial scan, onset of monazite, onset of unidentified phase, and final scan).

In Figure 4a, sample XRD peaks drift to higher Q and broaden with increasing pressure due to uniform and non-uniform strain, respectively. The first scan shows peaks from xenotime DyPO4, gold (triangles), and ruby (circles). Figure 4 also illustrates the relative loss of sample signal compared to the strong gold signal as pressure increases. Ponset values are determined by visual inspection of individual XRD patterns as shown in Figure 4b, not by the coloring in Figure 4a. Figure 4b shows the LeBail fits of key patterns (initial scan, onset of monazite, onset of new phase, and final scan). The first discernible monazite peaks emerge at 9.6(7) GPa at Q = 1.60, 2.05, and 2.14 Å–1; these are the (110), (002), and

(021) reflections. At Ponset, the monazite lattice parameters are a = 6.134(1) Å, b = 6.695(1) Å, c = 6.276(1) Å, and β = 99.79(1) ◦, and the unit cell volume loss during transformation is 7.90%. Although these lattice parameter uncertainties appear quite small, several checks on the LeBail fits do not change the outcome. The xenotime → monazite Ponset is below the hydrostatic limit of the MEW pressure medium (10.5(5) GPa), meaning non-hydrostatic stresses likely do not influence the onset of this transformation. At 10.5(5) GPa, MEW undergoes a glass transition into an amorphous phase that contributes no XRD peaks [44]. Above this hydrostatic limit, the sample stress state is understood to be non-hydrostatic. Starting at 17.2(13) GPa, there are no longer any peaks uniquely attributable to xenotime (based on visual inspection of individual patterns). The disappearance of xenotime peaks by this pressure is also apparent in Figure 4a and yields a xenotime–monazite phase coexistence range of 7.6(15) GPa.

Interestingly, a new set of previously unidentified XRD peaks emerge at 13.9(10) GPa at Q = 2.42, 2.58, and 3.94 Å–1 and persist as the pressure increases (see diamonds in Figure 4). These peak positions are inconsistent with xenotime, anhydrite, monazite, ruby, gasket material, gold, or even a "monazite II" phase reported in CeVO4 [45]. Peak positions for the post-monazite phase of TbPO4, scheelite (derived from ab initio calculations at 20.5 GPa), are shown in Figure 4 as there are no available structural data on any post-monazite phases of DyPO4. The final scan at 21.5(16) GPa contains peaks corresponding to monazite, gold, ruby, and the unidentified phase.

Next, we examine the pressure-dependence of lattice parameters more closely, finding consistent lattice parameter deviations around the pressure at which the post-monazite phase emerges. Figure 5 shows the pressure evolution of DyPO4 lattice parameters for the xenotime (ax and cx) and monazite (am, bm, cm, and βm) phases.

Figure 5a shows ax decreasing fairly steadily and monotonically with two slight disruptions at pressures consistent with the xenotime → monazite Ponset (9.6(7) GPa) and with the hydrostatic limit of MEW (10.5(5) GPa). Starting at ~14 GPa, however, ax stops decreasing, and the lattice parameter uncertainty expands significantly; this change in behavior coincides with the emergence of the unidentified peaks. Figure 5b shows similar behavior in cx, except this parameter starts decreasing rapidly at ~14 GPa. The axial ratio of xenotime (shown in Figure S3) also shows a dramatic trend change at ~14 GPa. Figure 5c shows am decreasing monotonically with slight disruptions at the hydrostatic limit and ~14 GPa. After ~15 GPa, am becomes non-monotonic with pressure, and its error bar expands significantly. Figure 5d,e does not show any anomalies in bm and cm around 14 or 15 GPa—only minor disruptions around the hydrostatic limit. Figure 5f shows β<sup>m</sup> changes monotonicity around the hydrostatic limit, steadily increases with smaller error bars after 12 GPa, then remains almost constant after ~15 GPa with larger error bars. Figure 6 shows all DyPO4 lattice parameters plotted together to better illustrate their relative values and relative compressibilities. The cx, am, bm, and β<sup>m</sup> parameters stand out with the most significant changes in behavior beginning at ~14 GPa, coinciding with the emergence of the third (post-monazite) phase of DyPO4.

**Figure 5.** Pressure dependence of xenotime (x) and monazite (m) DyPO4 lattice parameters. Green, shaded areas represent the pressure range in which a third (post-monazite) phase of DyPO4 exists. The thin gray shaded area represents the hydrostatic limit of the pressure medium: (**a**) ax; (**b**) cx; (**c**) am; (**d**) bm; (**e**) cm; (**f**) βm. All parameters show some irregularity around the hydrostatic limit (10.5(5) GPa). The ax, cx, am, and β<sup>m</sup> parameters exhibit notable changes in behavior after ~14 GPa.

**Figure 6.** Pressure evolution of all DyPO4 lattice parameters for the xenotime (ax and cx) and monazite (am, bm, cm, and βm) phases. Error bars represent standard deviation. The vertical dashed line indicates the xenotime → monazite Ponset. Green, shaded areas represent the pressure range in which a third (post-monazite) phase of DyPO4 exists. The thin gray shaded area represents the hydrostatic limit of the pressure medium. The inset shows the monazite beta angle.

Axial compressibilities of the xenotime and monazite phases are obtained by linearly fitting lattice parameter data from pressures below the hydrostatic limit (10.5 GPa). These values are summarized in Table 1. The monazite beta angle (βm) was not analyzed as this parameter's non-monotonic behavior precludes a meaningful linear fit.

**Table 1.** DyPO4 axial compressibilities derived from linear fits of lattice parameter data at pressures below the hydrostatic limit. Negative values indicate compression.


#### **4. Discussion**

This experiment shows the Ponset of DyPO4 xenotime → monazite phase transformation is 9.6(7) GPa when loading at ~20 MPa/s. This pressure is nominally higher than the 9.1(1) GPa Ponset observed in our 2021 DyPO4 study under quasi-static loading (~2 MPa/s) [22]; however, the magnitude of the Ponset error unfortunately precludes any conclusions regarding rate-dependence of the xenotime → monazite transformation (e.g., thermal activation, mechanism). The xenotime axial compressibilities (see Table 1) and the monazite lattice parameters at Ponset are not notably different from those reported in our 2021 DyPO4 study [22], while the monazite axial compressibilities differ significantly in their absolute and relative values. The discrepancy in compressibilities is likely due to this study having a much smaller quasi-hydrostatic pressure regime in which monazite exists (~1 GPa); therefore, there is a much smaller range and significantly fewer monazite datapoints suitable for fitting in this study than in our previous study.

This work also provides, for the first time, an estimate of the full DyPO4 xenotime– monazite phase coexistence range. Figure 4 shows that the xenotime phase is present up to 17.2(13) GPa (also reflected in Figure 2 as the upper bound of the blue bar). This value establishes that, while there is not a trend in the experimentally observed monazite Ponset with rare-earth radius, there is a general compositional trend in which the upper pressure bound of xenotime decreases with increasing RE3+ radius (see Figure 2). The DyPO4 xenotime–monazite coexistence range is then 7.6(15) GPa, which is significantly larger than that of both neighboring compositions (4.8 GPa for YPO4 and 4.6 GPa for TbPO4) and slightly larger than that of ErPO4 (6.0 GPa) [12,20,23]. Comparison to HoPO4 is precluded by incomplete xenotime phase transition [13].

Beyond characterizing the xenotime → monazite transformation, this study provides proof of the existence of a new, post-monazite phase of DyPO4. As a reminder, it is crucial to note that no DyPO4 structures other than xenotime and monazite were used during the LeBail fitting (as no other experiment-based DyPO4 structures have been reported). As a result, the fitting process attempted to accommodate the unidentified peaks at Q = 2.42, 2.58, and 3.94 Å–1 (emerging at 13.9(10) GPa as seen in Figure 4) with the xenotime and monazite structures. This accommodation explains the anomalies in ax, cx, am, and β<sup>m</sup> after 13.9(10) GPa as shown in Figure 5. The gold lattice parameter variation with time (and its corresponding pressure profile) is smooth and has no interruption at the post-monazite transition pressure (see Figure 3), showing that anomalies in xenotime and monazite lattice parameters do not result from experiment instabilities. Indexing the unidentified peaks to a certain structure or space group is extremely difficult because of a weak sample signal at pressures >14 GPa as well as monazite peaks covering most of the Q range.

Comparison to the literature strongly suggests the post-monazite phase is scheelite (tetragonal, I41/*a*). Based on preliminary Raman spectroscopy experiments, Stavrou et al. deduce a xenotime → monazite → scheelite transformation pathway in DyPO4 with a monazite → scheelite Ponset at ~33 GPa [18]. This Ponset value may be a significant overestimation, as the Raman spectroscopy-based xenotime → monazite Ponset has also proven to be a significant overestimation when compared to XRD work [22]. Further analysis of this Raman spectroscopy-based monazite → scheelite Ponset is complicated by the fact that the underlying Raman spectra have yet to be published. Nevertheless, the pressure-induced phase transformation pathways of other xenotime REPO4s support the existence of a monazite → scheelite transformation in DyPO4. Figure 2 shows scheelite evolves from monazite (in YPO4) or directly from xenotime (in ScPO4, LuPO4, YbPO4, and TmPO4) with increasing pressure. Experimental studies of ErPO4, HoPO4, and TbPO4 neither confirm nor deny transformation to scheelite due to limited experimental pressure ranges [20,21,23]. In the case of TbPO4, Lopez-Solano et al. point to "kinetic energy barriers" possibly hindering transformation to scheelite [23]. However, ab initio calculations performed by Bose et al. and Lopez-Solano et al. show ErPO4, HoPO4, and TbPO4 are expected to follow the xenotime → monazite → scheelite phase transformation pathway. Bose et al. predict monazite → scheelite Ponset values of ~11 GPa, ~12 GPa, and ~14 GPa for ErPO4, HoPO4, and TbPO4, respectively [17]. Lopez-Solano et al. predict a slightly higher TbPO4 monazite → scheelite Ponset of 15.5 GPa [23]. Barite and post-barite also bear consideration for the post-monazite phase, but both seem unlikely given experimental and computational data on LaPO4 put the transition for this and other compositions at pressures above 26 GPa [20,32]. Additionally, Lopez-Solano et al. found the scheelite structure to be energetically favorable to the barite structure [23]. No studies to date report a similar comparison between the scheelite and post-barite structures. Given that the unidentified XRD peaks in this study emerge at a pressure consistent with the expected monazite → scheelite transformation in neighboring compositions, it is likely that these peaks belong to a scheelite DyPO4 phase.

In the absence of scheelite unit cell data for DyPO4, we use TbPO4 for comparison. Lopez-Solano et al. predicted unit cell data of scheelite TbPO4 at 20.5 GPa, which are added to the top of Figure 4 as "S(TbPO4)" peak position ticks. Although the new peaks from

DyPO4 do not exactly match these ticks in scans around 20.5 GPa, the peaks are reasonably close to the ticks given the differences in methods, RE, temperature, kinetics, and stress state, which together have confounding effects on the unit cell. If this post-monazite DyPO4 phase is further confirmed to be scheelite, 13.9(10) GPa would be the lowest pressure at which scheelite has been reported in REPO4s to date.

#### **5. Conclusions**

This work significantly extends the characterization of the high-pressure phase behavior of DyPO4, which has been limited, particularly at pressures above 15 GPa. Our 2021 XRD study reported a xenotime → monazite Ponset at ~9 GPa but did not go to high enough pressures to reveal the xenotime–monazite phase coexistence range. This XRD study goes to higher pressures, showing a xenotime–monazite phase coexistence range of 7.6(15) GPa as well as the emergence of new peaks at 13.9(10) GPa. Contextualizing these new peaks within the experimental and the computational literature provides compelling evidence that monazite DyPO4 undergoes a pressure-induced phase transformation to the scheelite structure. Our results also motivate further XRD studies of other REPO4s (e.g., ErPO4, HoPO4, and TbPO4) at higher pressures to explore possible monazite → scheelite transformations and to elucidate high-pressure phase transformation pathways more broadly.

**Supplementary Materials:** The following supporting information can be downloaded at: https: //www.mdpi.com/article/10.3390/cryst13020249/s1, Figure S1: Scanning electron micrograph showing grain size and morphology of sample powder; Figure S2: Energy-dispersive X-ray spectrum of sample powder; Figure S3: Pressure dependence of the axial ratio (cx/ax) of the xenotime unit cell. A file containing XRD pattern fit data shown in Figure 4b is also included in the Supplementary Materials.

**Author Contributions:** Conceptualization, J.S. and C.E.P.; data curation, J.S., H.Q.A. and C.E.P.; formal analysis, J.S.; writing—original draft preparation, J.S. and H.Q.A.; writing—review and editing, J.S., H.Q.A. and C.E.P.; supervision, C.E.P. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was supported by the National Science Foundation under Award No. DMR-1352499. J.S. was supported by the Department of Defense through the National Defense Science & Engineering Graduate Fellowship Program. This work was performed at HPCAT (Sector 16), Advanced Photon Source, Argonne National Laboratory. HPCAT operations are supported by DOE National Nuclear Security Administration under Award No. DE-NA0001974 and DOE Office of Basic Energy Sciences (BES) under Award No. DE-FG02-99ER45775, with partial instrumentation funding by the National Science Foundation (NSF). A.P.S. is supported by DOE-BES, under Contract No. DE-AC02- 06CH11357.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** All relevant data that support the findings of this study are available from the corresponding authors upon request.

**Acknowledgments:** The authors thank Ivar Reimanis for lending us his Diacell Helios DAC, Matthew Musselman for synthesizing the sample used in the experiment, and Jesse Smith at HPCAT for performing initial alignment and calibration. The authors acknowledge the APS synchrotron facilities for provision of beamtime on beamline 16-ID-B.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


**Disclaimer/Publisher's Note:** The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

**Nourou Amadou 1,2,\*, Abdoul Razak Ayouba Abdoulaye 2, Thibaut De Rességuier 1,\* and André Dragon <sup>1</sup>**

<sup>1</sup> Institut Pprime, CNRS, ENSMA. Université de Poitiers, F-86961 Futuroscope, France

<sup>2</sup> Département de Physique, Université Abdou Moumouni de Niamey, Niamey P.O. Box 10662, Niger

**\*** Correspondence: nourou.amadou@ensma.fr (N.A.); resseguier@ensma.fr (T.D.R.)

**Abstract:** Non-equilibrium molecular dynamics simulations have been used to investigate strainrate dependence of plasticity and phase transition in [001]-oriented single-crystal iron under ramp compression. Here, plasticity is governed by deformation twinning, in which kinetics is tightly correlated with the loading rate. Over the investigated range of strain rates, a hardening-like effect is found to shift the onset of the structural bcc-to-hcp phase transformation to a high, almost constant stress during the ramp compression regime. However, when the ramp evolves into a shock wave, the bcc–hcp transition is triggered whenever the strain rate associated with the plastic deformation reaches some critical value, which depends on the loading rate, leading to a constitutive functional dependence of the transition onset stress on the plastic deformation rate, which is in overall consistence with the experimental data under laser compression.

**Keywords:** plasticity; iron; alpha–epsilon; phase transition; molecular dynamics simulations; ramp; shock wave; hardening-like effect

#### **1. Introduction**

Over the past few decades, there has been an increasing interest in using dynamic compression at extremely high strain rates to investigate both plasticity and pressureinduced phase transition [1–3]. Usually, such studies are performed under the conditions of uniaxial strain, which involve large deviatoric stresses. High strain rates affect material behavior in both the mechanisms of plastic deformation and the kinetics of polymorphic phase transitions [4,5]. They can even affect melting temperatures at high pressures [6]. Shock-loading experiments were classically realized under planar plate impacts where the strain rates are typically about 10<sup>4</sup> to 106 s−<sup>1</sup> [7–9]. Then, iron was found to plastically yield at a stress of 0.92 to 1.3 GPa, while the ground-state body-centered cubic (bcc) structure (*α* phase) was found to transform into the high-pressure hexagonal close-packed (hcp) structure (*ε* phase) at a stress *σ<sup>T</sup>* of about 12.88 to 14.26 GPa, which is higher for single-crystal than polycrystalline iron [7,9,10]. Higher strain rates can be achieved under laser ramp compression. Indeed, strain rates from 3 to 9 × <sup>10</sup><sup>7</sup> <sup>s</sup>−<sup>1</sup> were reported by Amadou et al. [11] under nanosecond laser compression, where the phase transformation onset stress, *σT*, was found to vary from 11 to 25 GPa with a constant completion time of 1 ns, suggesting the existence of an isokinetic regime over the explored range of strain rates. Furthermore, an extended analysis of strain-rate effects, up to 108 s<sup>−</sup>1, on both plastic flow and phase transition kinetics in iron was reported by Smith et al. [12,13], where the elastic limit (*σE*) in polycrystalline iron was found to vary from ∼1 to ∼5.5 GPa, while *σ<sup>T</sup>* ranged from ∼14 to ∼40 GPa. Moreover, a constitutive functional dependence on the plastic deformation strain rate, . *εP*, was evidenced for both *σ<sup>E</sup>* and *σT*. At strain rates higher than about 5 <sup>×</sup> <sup>10</sup><sup>6</sup> <sup>s</sup><sup>−</sup>1, a sharp increase in *<sup>σ</sup><sup>E</sup>* with increasing . *ε<sup>P</sup>* was interpreted as a transition in the plastic flow regime from thermally activated to phonon drag dislocation dynamics, which affects the structural phase transition kinetics by limiting the new phase growth

**Citation:** Amadou, N.; Abdoulaye, A.R.A.; De Rességuier, T.; Dragon, A. Strain-Rate Dependence of Plasticity and Phase Transition in [001]-Oriented Single-Crystal Iron. *Crystals* **2023**, *13*, 250. https:// doi.org/10.3390/cryst13020250

Academic Editors: Daniel Errandonea and Enrico Bandiello

Received: 27 December 2022 Revised: 19 January 2023 Accepted: 19 January 2023 Published: 1 February 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

through energy dissipation [13]. In this regime, *<sup>σ</sup><sup>T</sup>* was found to scale as . *ε* 0.18 *<sup>P</sup>* . A strain rate on the order of about 10<sup>8</sup> s−<sup>1</sup> was also reported by Hawreliak et al. [14]. However, *σ<sup>T</sup>* was found to be much lower, at about 15 GPa, with no obvious correlation with the strain rate. Even higher strain rates up to about 109 s−<sup>1</sup> are currently available under picosecond laser compression [15,16]. In this ultra-fast deformation regime, Crowhurst et al. [17] and Hwang et al. [18] found *σ<sup>T</sup>* values of 25 and 34 GPa, respectively.

These experimental ultra-high strain rates are comparable to those currently available in molecular dynamic simulations. Thus, Gunkelmann et al. [19], using the modified version of the Ackland EAM potential [20] to study the behavior of polycrystalline iron at a strain rate of about 109 s−1, showed that yielding occurs through dislocation activities around the grain boundary at 10 GPa with the onset stress of a structural bcc-to-hcp phase transformation *σ<sup>T</sup>* of about 23 GPa. A strain-rate regime ranging from 109 s−<sup>1</sup> to 1011 s−<sup>1</sup> was explored by Wang et al. [21] using a modified analytic embedded atom model [22], and they found that *<sup>σ</sup><sup>T</sup>* varied from 25 to 38 GPa with an exponential dependence in . *εP*, *<sup>σ</sup><sup>T</sup>* <sup>∝</sup> . *ε* 0.196 *<sup>P</sup>* . Finally, using Voter–Chen potential [23], singularized by a lack of plasticity before the structural phase transformation [24], Shao et al. reported a linear dependence of *<sup>σ</sup><sup>T</sup>* on . *ε<sup>P</sup>* [25] over a strain rate range from 1010 s−<sup>1</sup> to 1011 s−<sup>1</sup> with *σ<sup>T</sup>* ranging from 15 to 25 GPa. Thus, despite extensive research, the dynamic response of iron, including the bcc–hcp phase transition, its kinetics, and its strain-rate dependence under high strain rates still remains an open issue. In this context, using the modified version of the Ackland iron potential [20], we have reported in previous papers that defect-free single-crystal iron at a 50 K initial temperature subjected to ramp compression along the [001] direction exhibits a hardening-like effect, which has been shown to inhibit the nucleation of the hcp phase so that the onset of the phase transformation is shifted to very high pressures (on the order of 100 GPa) [5,26–28]. This remarkable effect, never reported before, deserves extensive study because it may contribute to the strong variations in the transition onset pressure with the loading rate classically observed experimentally. Here, we go further to explore the strain-rate dependence of both plasticity and phase transition and to investigate the influence of the hardening-like effect on the *<sup>σ</sup><sup>T</sup>* <sup>−</sup> . *ε<sup>P</sup>* relationship. Because we reported that this hardening-like effect is more important when the ramp wave propagates along the [001] direction than along other low-index crystallographic directions [26], we choose to simulate the response of a [001]-oriented single-crystal iron.

#### **2. Method and Computational Details**

Samples with up to 28 million atoms, 100a0 × 100a0 cross-section, and a length up 1400a0 (with a0 = 2.87 Angstrom, the lattice constant) were simulated using the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) molecular dynamics code [29]. The interactions between atoms were modeled through the embedded atom model (EAM) formalism [30,31], where the iron model was a modified version of the Ackland potential [20], which was successfully used to address both plasticity and phase transition in iron [5,19,26]. The sample was thermalized at a 50 K initial temperature, then dynamic ramp-wave compression was realized by driving an effective infinite-mass wall piston with an imposed velocity *v(t)* along the *z*-axis, oriented along the [001] crystallographic direction, while periodic boundary conditions were used for the transverse directions. In order to investigate strain-rate effects, the piston velocity was increased linearly from 0 to 1600 m·s−<sup>1</sup> with a rise time varying from 15 to up to 150 ps. The corresponding loading rates were about 1–10 × 109 <sup>s</sup><sup>−</sup>1, which is comparable with that reported under picosecond laser dynamic compression [15–18]. The local thermodynamic and mechanical properties, such as longitudinal stress *σz*, shear stress, temperature, etc., were evaluated within a spatial planar bin (of 3-lattice constant width) perpendicular to the wave propagation direction in the same manner as our previous work [5,26,27]. Finally, local structural analysis was performed by adaptative common neighbor analysis (CNA), centrosymmetry, and DXA, as implemented in OVITO software (https://www.ovito.org, accessed on 18 January 2023) [32].

 **[001]**

#### **3. Results and Discussion**

Figure 1 shows snapshots of atom configurations for simulations where the ramp rise time was 30, 45, and 60 ps. At each loading rate, the compressed material exhibits four distinct regimes: an elastic compression of bcc-iron (yellow zone); a regime where defects associated with plastic deformation can be observed (green zone with brown micro-features identified as twins); a regime where these defects associated with plastic deformation progressively disappear (blue zone); and a regime where iron is transformed from bcc to hcp phase (violet zone). More details on wave propagation and splitting of the compression fronts can be found in ref. [5]. At each loading rate, the sample is found to yield at a stress *σ<sup>z</sup>* of about 12 GPa via the generation of micro-twins with no sizeable influence on the loading rate. The fact that twinning deformation is independent of the loading rate is consistent with the well-known existence of the twinning threshold in bcc materials, such as tantalum, reported under both experiments [1,33] and molecular dynamics simulations [34]. Thus, regardless of the loading rate, twinning is found to be the main deformation mechanism for the defect-free single-crystal iron ramp compressed along the [001] crystallographic direction.

**Figure 1.** Snapshots of the atom configurations at times 33 ps (**a**), 48 ps (**b**), and 60 ps (**c**) under ramp compression rise times of 30 ps (**a**), 45 ps (**b**), and 60 ps (**c**). The bcc atoms are colored according to their *σz* Va value, where Va is the atomic volume, and *σz* is the stress component along the wave propagation z-axis. The hcp and fcc atoms, detected by the adaptative CNA analysis, are colored in magenta and white, while twins (detected by centrosymmetry analysis) are colored in brown (regardless of the stress). A double nucleation front of the hcp phase (violet) can be observed since the transformation occurs (i) at the top of the pressure ramp propagating from left to right and (ii) where the ramp wave steepens into a shock (see text for more details). Note that a longer ramp rise time requires longer propagation distance. Thus, the sample length in the wave propagation direction was 800 a0 for (**a**), while it was 1400 a0 for both (**b**) and (**c**).

Figure 2 shows the evolution of the twin fraction during the compression for a ramp rise time of 15, 30, 45, and 60 ps. After the nucleation period, the twin fraction grows rapidly, reaching its maximum within 3–4 ps. Then, the twin fraction starts to decrease, i.e., twins formed under moderate pressure are removed upon further compression, which we refer to as a hardening-like effect (see Figure 1, blue zone). The longer the ramp rise time, the more important the twins' fraction and the duration of the subsequent receding phase. Thus, upon increasing the ramp rise time, the maximum twins' fraction increases from 28% to a value as high as ≈67%, in consistence with the 50% twins' fraction usually assumed under dynamic loading [1,33]. Furthermore, this increase in the twins' fraction upon increasing the ramp duration is consistent with both experimental observations [35] and thermo-mechanical simulations, including a model of twins' nucleation and growth [36]. In the region affected by the hardening-like effect (blue zone), where twins are almost fully removed and no dislocation activities can be detected, the shear stress increases with compression, as can be seen in Figure 3. Such elastic stiffening of the bcc matrix leads to

a confinement effect inhibiting the nucleation of the hcp phase. This confinement effect has been interpreted in the context of classical nucleation and growth theory, and it has been shown that in order to activate new phase nucleation two different scenarios are possible [5]: (i) ramp compressing must be kept at a much higher pressure by increasing the piston maximum velocity or (ii) the ramp wave, which steepens during its propagation due to the increase in sound velocity with pressure in the bcc phase, should be allowed to evolve into a shock wave. Thus, for the simulations shown in Figure 1, both piston maximum velocity and propagation distance are enough that two nucleation fronts can be observed. The first front, on the left part of the sample, corresponds to the top of the ramp, where pressure is sufficiently high for the hcp nuclei to grow and reach a critical size, despite a rigid, confining bcc matrix due to the hardening-like effect. The second nucleation front, in the middle of the sample, corresponds to the second scenario above (ii), where the so-called P1 wave evolves into a shock wave during its propagation.

**Figure 2.** Temporal evolution of the twin fraction for various ramp rise times.

**Figure 3.** Shear stress profiles along the loading direction Z for ramp rise times of 15 ps (**a**), 30 ps (**b**), and 60 ps (**c**).

Thus, the resulting instantaneous thermodynamic driving force is high enough for hcp embryos to grow and reach a critical size almost immediately behind the shock front [37]. If the ramp rise time is increased to a sufficiently high value, the P1 wave does not form a shock upon the simulation duration. Thus, the second nucleation front can no longer be observed for a ramp rise time above 150 ps.

Figure 4 shows the structural phase transition onset stress, *σT*, as a function of the ramp rise time for both scenarios (i) and (ii) mentioned above. In the first scenario, upon increasing the ramp rise time, i.e., decreasing the loading strain rate, the phase transition onset stress can be observed to remain almost constant, at about 100 GPa (red triangles), so that no loading rate dependence can be observed. This observation can be interpreted as follows: As the hardening-like effect confines the hcp embryos, only the material strength determines the maximum pressure supported by the bcc matrix. Yet, it has been shown that above a critical strain rate of 105 s−1, the material strength becomes strain-rate independent [38,39]. This result highlights the crucial role of the strength of the parent phase matrix in the dynamics of structural phase transformation due to the generation of the elastic strains necessary to accommodate the difference between the parent and daughter crystalline structures.

**Figure 4.** The structural *α* to *ε* phase transformation onset stress as a function of the ramp rise time according to scenario (i) where high compression overcomes the hardening-like effect (red triangles) and scenario (ii) where the ramp wave steepens into a shock (blue stars).

Although the transition onset stress in scenario (i) remains almost constant after the hardening-like effect, the pressure relaxation accompanying the structural phase transformation is found to be rate-dependent (Figure 5). Indeed, this relaxation is due to the coexistence of both bcc and hcp phases in the sample, which causes a drop in the sound velocity that evolves as a result of the balance between the hcp nucleation and loading rates [13]. Moreover, increasing the ramp compression duration seems to enhance both the size and the ovaloid shape of the hcp nuclei (see Figure 6).

**Figure 5.** Stress profiles along the loading direction, Z, for the ramp rise time of 15 ps (**a**), 30 ps (**b**), and 60 ps (**c**), corresponding to Figure 3.

**Figure 6.** Snapshots of atom configurations showing a cross-section of the phase transformation front once compression overcomes the hardening-like effect scenario (i) for a ramp compression rise time of 30 ps (**a**), 45 ps (**b**), and 60 ps (**c**).

Ramp compression can be seen as a succession of infinitesimal compression waves, which all propagate at the speed of sound in the upstream medium. Because this speed usually increases with pressure (above the elastic limit), these elementary waves catch up with each other so that the ramp becomes steeper with increasing propagation distance and evolves naturally into shock waves. Its rise time and the associated plastic deformation strain rate, . *εP*, is thought to be associated with the microscopic processes of dislocation multiplication and motion, twinning, vacancy production, precipitate alteration, etc. [40,41]. Figure 5 clearly shows such gradual steepening of the compressive wave beyond the elastic limit (Pz > 12 GPa). This wave, usually referred to as the P1 wave, is associated to the plastic compression of the bcc phase, up to the onset of the transition to hcp. The average plastic deformation strain rate, . *εP*, associated with the P1 wave is calculated as follows [41,42]:

$$
\dot{\varepsilon}\_P = \frac{1}{\Delta z\_{P\_1}} \int\_{\Delta z\_{P\_1}} \dot{\varepsilon}(z) dz \tag{1}
$$

where Δ*zP*<sup>1</sup> is the width of the P1 wave between the elastic front and the bcc–hcp phase transition front in the scenario (i), as can be seen in Figure 7. The evolution of . *ε<sup>P</sup>* as a function of time for various ramp rise times is presented in Figure 8. It ranges from a few times 10<sup>9</sup> s−<sup>1</sup> to about 1011 s−1, in consistence with the values reported under the picosecond laser compression experiment [16,17]. As expected, when the ramp steepens, . *ε<sup>P</sup>* increases as a function of time more rapidly for lower ramp rise time than for higher ramp rise time [43]. For each rise time, . *ε<sup>P</sup>* shows a vertical asymptotic upward variation corresponding to the shock formation during the wave propagation through the sample. The lower the loading ramp rise time, the earlier the shock formation. Due to the complexity of the plastic response involved here, i.e., twins' formation, hardening-like effect, and phase transition, including hcp phase nucleation and growth, the time taken for steepening is much lower than the ratio of ~1.4 to 1.6 times the rise time of the ramp found from the analytical models and MD simulations of simple materials such as Al and Cu [43,44].

The bcc-to-hcp phase transition is triggered in scenario (ii) whenever the plastic strain rate, . *εP*, reaches a certain critical value that depends on the loading conditions (see Figure 8). This critical . *ε c <sup>P</sup>* ranges from 6.71 × 1010 <sup>s</sup>−<sup>1</sup> to 0.88 × <sup>10</sup><sup>10</sup> <sup>s</sup>−<sup>1</sup> when the ramp loading rise time increases from 15 to 60 ps. Indeed, when these critical values are reached, the dynamic loading switches from ramp to shock compression, where the material is carried almost instantaneously from the thermodynamic stability conditions of the bcc phase to those of the hcp phase. The resulting driving force is high enough for hcp embryos to reach a stable critical volume almost instantly behind the shock front. Thus, the hcp phase starts to nucleate behind the elastic–plastic transformation front leading to a second phase transition front in the sample, referenced above as scenario (ii), as can be seen in Figure 1. In this

scenario, the transition onset stress, *σT*, can be observed to decrease from 60 to 42 GPa upon increasing the ramp rise time from 15 to 60 ps. Thus, in contrast with scenario (i), *σ<sup>T</sup>* is rate-dependent. It increases with the strain rate, which is due to the kinetics of the transformation, and is consistent with experimental observations [11,13,45].

**Figure 7.** Stress profiles at successive times showing the evolution of the ramp compression, which steepens into a shock wave during its propagation. The width of the P1 wave is used to evaluate the plastic deformation strain rate. The bcc–hcp phase transformation occurs either under ramp compression on a ~100 GPa order scenario (i) or upon an increase in the strain-rate scenario (ii).

**Figure 8.** Time evolution of the plastic strain rate (stars) during ramp wave propagation through the sample with the critical plastic strain rate, . *ε c <sup>P</sup>*, (forest-green line) evaluated at the beginning of the hcp phase nucleation behind the compression front, once the ramp wave has steepened into a shock.

Figure <sup>9</sup> shows the variation in *<sup>σ</sup>T*, as a function of . *ε c <sup>P</sup>* (red stars), in comparison with various data reported in the literature under both experiments and MD simulations. The black line is the prediction of the Swegle–Grady (SG) law *σ<sup>T</sup>* = *α* . *ε β <sup>P</sup>* [40], using the parameters reported in ref. [14]. The red curve presents the fit of our data using a similar

law where *α* = 44.88 ± 0.47 and *β* = 0.15 ± 0.01. MD simulations are from Wang et al. [21], using a MEAM potential where the plasticity is dominated by twins; Shao et al. [25], using the Voter–Chen EAM potential [23], which does not predict plastic deformation before the structural phase transformation; and Gunkelmann et al. [19], using the same potential as we did.

**Figure 9.** Variation in the onset stress, *<sup>σ</sup><sup>T</sup>* (in scenario (ii)), with the plastic strain rate, . *ε c <sup>P</sup>* (red stars), compared with various data reported in the literature in both experiments and MD simulations, see text for more details.

Experimental data come from Smith et al. [13], where plastic deformation was thought to be dominated by phonon scattering from defects, nanosecond laser ramp compression iron by Amadou et al. [11] and Hawreliak et al. [14], or picosecond laser compression by Crowhurst et al. [17] and Hwang et al. [18]. Although our data deviate from the predictions by both the original SG model and Smith et al.'s constitutive relationship [13] at an ultra-high strain rate, their extrapolation to lower strain rates using this updated SG fit is consistent with the experimental data, including those by Amadou et al. [11], Smith et al. [13], and Hwang et al. [18]. On the other hand, the magnitude of the power law exponent in our simulations is slightly different from the 0.196 value reported by Wang et al. [21] in single-crystal iron for the same crystallographic orientation under MD simulations using the MEAM interatomic potential. This discrepancy is thought to be due to the difference in the underlying plastic deformation micro-process predicted by the different potentials. Indeed, in our simulations, twinning is followed by a hardening-like effect under further compression, while such a post-twinning effect is not predicted by the MEAM potential used by Wang et al.

#### **4. Conclusions**

NEMD simulations were used to investigate the strain-rate dependence of plasticity and phase transition in the [001]-oriented single-crystal iron at 50 K under dynamic ramp compression between 10<sup>9</sup> s−<sup>1</sup> and 10<sup>10</sup> s−1. Iron was found to yield at 12 GPa, regardless of the loading rate, through the generation of micro-twins, which grew rapidly to a peak fraction before receding upon further compression. The longer the ramp rise time, the higher the maximum twins' fraction and the longer the subsequent receding regime, in consistence with both experimental and theoretical observations in iron. As reported previously, twin recession in the absence of any dislocation slip induced a hardening-like effect, which shifted the phase transition onset stress to a very high value of ≈100 GPa scenario (i) independently of the loading rate. On the other hand, the phase transition could be triggered at lower stresses when the ramp evolved into shock wave scenario (ii). Then, the onset stress *σ<sup>T</sup>* was evidenced to be strain-rate-dependent. Indeed, the transition was triggered whenever the strain rate associated with the plastic deformation reached some critical value, . *ε c <sup>P</sup>*. The higher the ramp compression time, the lower both . *ε c <sup>P</sup>* and *σT*. Thus, the onset stress, *σT*, in this scenario has been shown to follow a Swegle–Grady power law type [40] in . *ε c <sup>P</sup>* with an exponent of 0.15, i.e., *<sup>σ</sup><sup>T</sup>* <sup>∝</sup> . *ε c* 0.15 *<sup>P</sup>* , which is in overall consistence with some experimental data under laser compression.

**Author Contributions:** Conceptualization, N.A.; methodology, N.A. and T.D.R.; software, N.A.; validation, N.A. and T.D.R.; formal analysis, N.A. and A.R.A.A.; investigation, N.A.; resources, T.D.R.; data curation, N.A. and A.R.A.A.; writing—original draft preparation, N.A.; writing—review and editing, N.A., T.D.R., and A.D.; visualization, N.A. and A.R.A.A.; supervision, N.A.; project administration, N.A.; funding acquisition, T.D.R. and A.D. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by a grant allocated by the *Conseil Scientifique de l'ENSMA* on 1 December 2021.

**Data Availability Statement:** The raw/processed data required to reproduce these findings cannot be shared at this time as the data also form part of an ongoing study.

**Acknowledgments:** Computations were performed on the supercomputer facilities of the Mésocentre de calcul de Poitou Charentes (France). We thank Mikaël Gueguen and Gérald Sailly (Institut Pprime) for their kind help to set up work stations and access.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


**Disclaimer/Publisher's Note:** The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

## *Article* **Pressure-Induced Monoclinic to Tetragonal Phase Transition in** *R***TaO4 (***R* **= Nd, Sm): DFT-Based First Principles Studies**

**Saheli Banerjee 1,2, Amit Tyagi <sup>1</sup> and Alka B. Garg 1,2,\***


**Abstract:** In this manuscript, we report the density functional theory-based first principles study of the structural and vibrational properties of technologically relevant M fergusonite (*P*2/*c*)-structured NdTaO4 and SmTaO4 under compression. For NdTaO4 and SmTaO4, ambient unit cell parameters, along with constituent polyhedral volume and bond lengths, have been compared with earlier reported parameters for EuTaO4 and GdTaO4 for a better understanding of the role of lanthanide radii on the primitive unit cell. For both the compounds, our calculations show the presence of first-order monoclinic to tetragonal phase transition accompanied by nearly a 1.3% volume collapse and an increase in oxygen coordination around the tantalum (Ta) cation from ambient six to eight at phase transition. A lower bulk modulus obtained in the high-pressure tetragonal phase when compared to the ambient monoclinic phase is indicative of the more compressible unit cell under pressure. Phonon modes are calculated for the ambient and high-pressure phases with compression for both the compounds along with their pressure coefficients. One particular IR mode has been observed to show red shift in the ambient monoclinic phase, possibly leading to the instability in the compounds under compression.

**Keywords:** high pressure; rare earth tantalates; first principles calculations; Raman modes; phase transition

**Citation:** Banerjee, S.; Tyagi, A.; Garg, A.B. Pressure-Induced Monoclinic to Tetragonal Phase Transition in *R*TaO4 (*R* = Nd, Sm): DFT-Based First Principles Studies. *Crystals* **2023**, *13*, 254. https://doi.org/10.3390/ cryst13020254

Academic Editor: Juan Ángel Sans

Received: 2 January 2023 Revised: 23 January 2023 Accepted: 27 January 2023 Published: 1 February 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

#### **1. Introduction**

*RB*O4 (*R*: rare earth; *B*: a pentavalent cation such as V, W, Mo, Nb, Ta, As, and P) compounds are the subject of extensive research due to their promising applications in areas such as proton-conducting solid oxide fuel cells [1], and as a host for nuclear radioactive waste immobilization [2,3]. It has been well established that the *B* cation plays a crucial role in deciding the stable structure of *RB*O4 compounds. Depending on the ratio of *B* cationic radii to lanthanide radii, *RB*O4 compounds are reported to crystallize in either tetragonal (zircon, scheelite type) or monoclinic (fergusonite, monazite, wulframite type) structures [4]. Rare earth orthovanadate (*R*VO4) are generally synthesized in a zircon (tetragonal) structure with a *I*41/*amd* space group, while rare earth orthotungstates (*R*WO4) and molybates (*R*MoO4) have been reported to crystallize in scheelite (tetragonal) structure with *I*41/*a* crystal symmetry [5–7]. A monoclinic M fergusonite structure with space group *I*2/*a* has been established as a stable structure for rare earth orthoniobates (*R*NbO4) at ambient temperature and pressure [8]. Depending upon the atomic radii of the lanthanide cation, the crystal structure of rare earth orthophosphate is either zircon (*R* < Gd) or monazite (*R* ≥ Gd) [9]. All the structures are closely related to each other by group– subgroup relations. A tetragonal scheelite structure (*I*41/*a*) is a subgroup of a tetragonal zircon (*I*41/amd) structure and the transformation between these two structures is generally of the first-order reconstructive type. A scheelite structure is transformed to a fergusonite structure by means of another translationgleiche, which involves lowering of point group symmetry from 4/m to 2/m [10]. Among all *RB*O4 compounds, the *R*TaO4 family of

compounds exhibit polymorphism at ambient pressure and temperature conditions, which makes them of great interest from theoretical and technological point of views [11,12]. In *R*TaO4 compounds, the final stable structure not only depends on the lanthanide radii but also depends on the heating temperature. Most of the compounds belonging to the *R*TaO4 family stabilize either in a M fergusonite or M' fergusonite structure at room temperature depending on the processing parameters. The main difference between the M fergusonite and M' fergusonite structures is the oxygen coordination around the Ta atom. In a M fergusonite structure, the Ta atom is surrounded by an oxygen tetrahedra, whereas an oxygen octahedra is formed in a M' fergusonite structure. In the M fergusonite phase, doubling of the *b* axis has been observed compared to M structure, while the other two unit cell axes have almost the same value in both structures. In recent times, *R*TaO4 family compounds are being investigated extensively due to their potential applications in the field of scintillators and laser materials, owing to their high thermal stability and good chemical stability [13,14]. These compounds are also proposed as excellent alternatives to using Yttria-stabilized zirconia as ceramic thermal barrier coatings (TBCs) due to their lower thermal conductivity and better fracture toughness at high temperatures [15]. As is well known, most often it is the crystal structure that determines the properties of a material, and the structure may be altered by varying the thermodynamic parameters. Recently, by x-ray diffraction, the Raman spectroscopic technique and density functional theory (DFT) based first principles calculations, we have shown the structural instability of EuTaO4 and GdTaO4 [16,17]. In continuation to this work, to understand the compression behavior of the *R*taO4 family of compounds, we have performed density functional theory-based first-principle simulations on technologically important NdTaO4 and SmTaO4. Wenhui Xiao et al. [18] have reported that the M' fergusonite structure is more stable compared to the M fergusonite structure. Therefore, in the present work, we report the results from DFT-based first principles calculations on the M' fergusonite structure under compression. We have also calculated the equation of state for both the compounds using the third-order Birch– Murnaghan equation of state (BM-EOS). Compressibility analysis of the simulated volume of constituent polyhedral units *R*O8 and TaO6 indicates that the major contribution to the bulk modulus comes from *R*O8 polyhedra. This behavior validates Hazen and Finger's proposed empirical model for predicting the bulk modulus, with contributions from the rare earth polyhedral unit as seen in *R*VO4, *R*WO4, *R*MoO4, *R*NbO4 compounds [6,7,19–25]. We have also calculated the pressure evolution of the Raman and IR modes, which is consistent with earlier reported results for EuTaO4 and GdTaO4 for the same structure [16,17]. Further, we have compared the results from this work with previously investigated EuTaO4 and GdTaO4 to establish the role of lanthanide contraction present in the *R*TaO4 family.

#### **2. Computational Details**

DFT-based first-principle simulations were carried out as implemented in Quantum Espresso [26] for determination of stable structures at ambient pressure as well as to investigate the influence of pressure on structural and vibrational properties. This is based on density functional theory, plane waves, and pseudopotentials. The projector-augmented wave (PAW) scheme [27], which describes electron–ion interactions, was employed as pseudo potential in self-consistency field calculation. Appropriate pseudopotentials are taken from the Pslibrary [28] considering 11 valence electrons for Nd (5s26s1.55p66p0.55d1) and Sm (5s26s1.55p66p0.55d1), 27 valence electrons for Ta (4f14 5d35p66s25s2) and 6 valence electrons for oxygen (2s<sup>2</sup> 2p4). A prescribed generalized gradient approximation (GGA) based on the parametrization proposed by Perdew, Burke, and Ernzherhof (PBE) [29] has been accounted for the for calculation of exchange and correlation energy for both the compounds with the lowest-energy M' fergusonite structure. Wave functions in the Kohn–Sham equation are expanded in a plane wave basis set due to the major advantage of orthonormality and since it is easy to control the convergence with respect to the size of the basis with only one parameter Ecut. In our calculation, the plane wave basis was extended up to 70 Ry for both the compounds (NdTaO4 and SmTaO4) to achieve highly converged results in the PAW scheme after thoroughly going through the convergence test. A dense Monkhorst pack grid of 8 × 8 × 8 is used for Brillouin zone integrations. Geometric optimization of NdTaO4 and SmTaO4 structures has been achieved using the Broyden– Fletcher–Goldfarb–Shanno (BFGS) minimization algorithm [30], where the structures have been fully optimized to the equilibrium condition at ambient pressure, by minimizing the forces on the atoms and the stress tensor. Helmann–Feynman forces lower than 0.00003 eV/atom on each atom in the unit cell and maximum deviation among the diagonal components of the stress tensor on a unit cell lower than 0.1 GPa ensure a fully relaxed structure. The same steps were followed while calculating from ambient pressure to 50 GPa with an interval of 1 GPa. After obtaining the equilibrium structures at different pressures, phonon frequencies were calculated at the center of the Brillouin zone using density functional perturbation theory (DFPT) as implemented in the Quantum Espresso code [31]. Simulations were performed at zero temperature and under a hydrostatic environment. The stable structures and transition pressures were obtained by analyzing the enthalpy–pressure curve.

#### **3. Results and Discussions**

#### *3.1. Ambient Structure*

The optimized volume obtained at ambient pressure for both the compounds in the M' fergusonite structure matches very well with previously reported experimental values [32]. Unit cell parameters obtained from the geometrically relaxed structure with constituent bond lengths along with the previously reported experimental data are given in Table 1. The equilibrium volume obtained for NdTaO4 is 159.1 Å3 and 155.4 Å3 for SmTaO4, which are within 1% and 0.3% of the experimentally observed volume. Figure 1 shows the polyhedral representation of NdTaO4 and SmTaO4 belonging to a P2/c (space group no.:13, Z = 2) structure with 2/m point group symmetry. The rare earth cation (Nd/Sm) is surrounded by eight oxygen, forming dodecahedra while oxygen coordination around the Ta cation is six. The formation of oxygen octahedra around tantalum is a distinctive signature of the M' fergusonite structure that makes it different from the M fergusonite structure with space group I2/a (space group: 15). Each rare earth cation is bonded with eight oxygen with four different *R*-O bond distances while each Ta cation bonds to six oxygen with three different bond distances, making all the constituent units distorted. These distorted polyhedral units provide structural stability to these compounds against a large range of pressure/temperature, compared to zircon- or scheelite-structured compounds [6,7,19–25]. In the calculated M' fergusonite structure, the Nd/Sm and Ta atoms occupy the 2e and 2f Wyckoff positions while oxygen atoms O1 and O2 occupy the 4g position. In Figure 2, we have plotted ambient pressure lattice parameters, the unit cell volume along with the constituent polyhedral volume (*R*O8, TaO6), and *R*-O and Ta-O bond lengths for both the compounds along with previously reported data for EuTaO4 and GdTaO4 for a better understanding of the role of the rare earth cation. An increasing linear trend has been observed in all the parameters with lanthanum cationic radii except the volume of TaO6 octahedra and its constituent bond lengths. This indicates the major influence of the R cation on structural parameters. In all four RTaO4 compounds, the polyhedral volume of all distorted TaO6 octahedra remains almost the same.

#### *3.2. Structural Behavior under Compression*

#### 3.2.1. The Low-Pressure Phase

Simulated unit cell parameters for both the compounds, NdTaO4 and SmTaO4, have been plotted at different pressures in Figure 3a,b. The anisotropic compressibility of the b axis is clearly seen, which is 2 fold as compressible compared to the other two axes. This particular behavior has been observed in almost all *R*BO4 compounds irrespective of their ambient structure [5–7,9]. The axial compressibility obtained by fitting the calculated lattice parameters to third-order BM-EOS [33] for NdTaO4 is Ka = 1.89 × <sup>10</sup>−<sup>3</sup> GPa−1, Kb= 3.79 × <sup>10</sup>−<sup>3</sup> GPa−<sup>1</sup> and Kc = 1.75 × <sup>10</sup>−<sup>3</sup> GPa<sup>−</sup>1; and SmTaO4 are Ka = 1.66 × <sup>10</sup>−<sup>3</sup> GPa<sup>−</sup>1, Kb = 3.64 × <sup>10</sup>−<sup>3</sup> GPa−<sup>1</sup> and Kc = 1.60 × <sup>10</sup>−<sup>3</sup> GPa−1, respectively. The structural arrangement of relatively more compressible *R*O8 polyhedral units along the b axis could be the reason for anisotropic compression as has been reported in other rare earth metal oxides. Similar behavior has also been observed in EuTaO4 and GdTaO4 [16,17]. It is interesting to note that a decreasing trend is observed in the axial compressibility as we go from higher ionic radii to lower ionic radii of the lanthanide cation, indicating the more incompressible behavior of *R*TaO4 compounds with a lower ionic radii lanthanide cation. Figure 4 shows the pressure evolution of the unit cell volume for both the compounds. The bulk modulus obtained by fitting the simulated pressure–volume data to third-order BM-EOS is 145.1 and 147.2 GPa for NdTaO4 and SmTaO4, respectively, which is in good agreement with the bulk modulus reported for the *R*TaO4 family of compounds. The bulk modulus reported for NdVO4 [34] in the zircon and scheelite structures is 124.2 and 136 GPa, respectively, which is similar to the obtained bulk modulus of NdTaO4 in the present studies. A similar bulk modulus is observed for NdNbO4 (138.32 GPa) [18]. Similarly, for SmTaO4, the obtained bulk modulus in the present work is similar to that reported for other SmBO4 compounds [35]. This indicates that it is NdO8/SmO8 polyhedral units that mostly contribute to the bulk modulus. A similar bulk modulus for various *R*BO4 compounds has already been reported and reaffirms that it is indeed the lower valence polyhedral units (*R*O8) that mainly contribute to the bulk modulus [16,17]. This can be validated using Hazen and Finger's proposed empirical model B0 = N × Z/(dR-O) 3, where B0 is the bulk modulus, N is the dimensional-less proportional constant (610 for tantalates and niobates), Z is the formal charge of the R cation and dR-O is the average cation–anion distance, which only considers the rare earth polyhedral unit for predicting the bulk modulus [36]. For NdTaO4 with dNd-O= 2.460 Å and Z = 3, the calculated bulk modulus using Hazen and Finger's equation is 122.9 GPa, which is similar to the simulated value obtained in the present work. For the SmTaO4 compound, the bulk modulus obtained using dSm-O = 2.460 Å is 127.4 GPa.


**Table 1.** Comparison of simulated ambient pressure lattice parameters and constituent bond lengths of NdTaO4 and SmTaO4 with previously reported experimental values.

**Figure 1.** Polyhedral representation of SmTaO4 (upper) and NdTaO4 (lower) in both LP (*P*2/*c*) and HP (*P*4/*nbm*) phases. Sm, Nd, Ta, and O atoms are green, pink, blue and red, respectively.

#### 3.2.2. The High-Pressure Phase

Earlier, based on our X-ray diffraction and Raman spectroscopic measurements along with DFT-based first principles calculations on EuTaO4 and GdTaO4, an isostructural transition was reported with a sudden drop in the monoclinic angle from 96◦ to 90◦ at the transition. This isostructural monoclinic phase could also be described as the pseudo orthorhombic structure due to all unit cell angles of 90◦. Therefore, orthorhombic Pcna was tested as an alternative description against the isostructural monoclinic phase and our calculation reveals that, at high pressure, both the structures are energetically favorable. Since NdTaO4 and SmTaO4 are found to be synthesized in the same space group (*P*2/*c*) as EuTaO4 and GdTaO4, the same orthorhombic space group has been tested as the possible high-pressure phase for both the compounds in the present work. For NdTaO4, the orthorhombic structure with the *Pcna* space group becomes energetically favorable at approximately 40 GPa as seen in Figure 5a, which depicts enthalpy difference as a function of pressure (the monoclinic phase *P*2/*c* has been taken as a reference). Similar behavior has been observed at approximately 33.5 GPa for SmTaO4 (Figure 5b). This orthorhombic structure can also be alternatively described by a pseudo tetragonal structure due to almost the same value of lattice parameter a and c as seen in calculation. Therefore, the highersymmetry tetragonal structure *P*4/*nbm* with 4/mmm point group symmetry was also tested against the Pcna structure and indeed tetragonal *P*4/*nbm* is a lower-energy structure at approximately 33 GPa for SmTaO4 and approximately 40 GPa for NdTaO4 as seen in Figure 5a,b. The phase transition in both the compounds is associated with a nearly 1.3% volume collapse at transition pressure along with change in oxygen coordination around the Ta cation from ambient six to eight at phase transition. In Figure 3, we have plotted

lattice parameters of the high-pressure tetragonal phase (HP) along with unit cell parameters corresponding to the low-pressure monoclinic phase (LP). The axial compressibility obtained by fitting the calculated lattice parameters to third-order BM-EOS for NdTaO4 and SmTaO4 in the HP phases is Ka = 3.75 × <sup>10</sup>−<sup>3</sup> GPa−1, Kc = 1.02 × <sup>10</sup>−<sup>3</sup> GPa−<sup>1</sup> and Ka = 3.4 × <sup>10</sup>−<sup>3</sup> GPa−1, Kc = 1.04 × <sup>10</sup>−<sup>3</sup> GPa−1, respectively. Unit cells for both the compounds also undergo anisotropic compression in the high-pressure tetragonal structure as observed in the low-pressure monoclinic structure. The simulated pressure volume data fitted with the third-order Birch–Murnaghan (BM) equation of state (EOS) yields a bulk modulus of 123.74 GPa for NdTaO4 and 130.60 GPa for SmTaO4. It is interesting to note that the high-pressure phase has a lower bulk modulus than the low-pressure phase, indicating a more compressible high-pressure phase in spite of volume collapse at phase transition. This can be understood by analyzing the compressional behavior of the constituent polyhedral unit in both the LP phase and the HP phase. In low-pressure phase, the compounds NdTaO4 and SmTaO4 are made up of highly compressible NdO8 and SmO8 polyhedra as evident from their value of bulk modulus being 117.3 and 115 GPa respectively, while TaO6 octahedra in both the compounds is highly incompressible with incompressibility modulus being 262.6 and 258.7 GPa for NdTaO4 and SmTaO4 respectively. The modulus of incompressibility has been obtained using calculated pressure and polyhedral volume fitted to third-order BM-EOS. A large difference in the bulk modulus at the LP phase affirms that indeed lower valence rare earth polyhedra significantly contribute to the compressibility of the compound in the low-pressure monoclinic phase, which validates Hazen and Finger's proposed empirical model for predicting the bulk modulus, taking contribution from rare earth polyhedral unit as seen in RVO4, RWO4, RMoO4, RNbO4 compounds [6,7,19–25]. In Figure 6a,b, we have plotted the distortion index of bond lengths for the polyhedral units of both NdTaO4 and SmTaO4 compounds, which show that *R*O8 polyhedra become fully symmetric under compression, while that of TaO6 more distorted and distortion in the bond length increases sharply at phase transition with the formation of a TaO8 polyhedral unit. The distortion index of the bond length as defined by Baur [37] has been computed using VESTA software [38]. For the NdTaO4 compound, the bulk modulus obtained for constituent polyhedral NdO8 and TaO8 in the HP phase is 105.9 and 99.8 GPa, respectively, whereas the bulk modulus for the SmTaO4 compound obtained for constituent polyhedral SmO8 and TaO8 in the HP phase is 109.4 and 111.42 GPa, respectively. The similar bulk modulus indicates equal contribution to the compressibility of the high-pressure unit cell unlike the low-pressure phase. Incompressible TaO6 octahedra change to very compressible TaO8 at phase transition due to the increase in oxygen coordination under pressure and can be cited as among the reasons for the lower bulk modulus of the high-pressure phase. It should also be noted that the volume collapse at transition pressure is much smaller (~1.3%), which also supports the more compressible high-pressure unit cell. In Table 2, we have tabulated the atom positions along with their unit cell lattice parameters at ambient pressure and phase transition pressure for both the compounds. It can be clearly seen that all the constituent atoms show significant atomic rearrangement in position when compared to their ambient pressure. A more packed high-pressure unit cell is a consequence of the more effective packing of oxygen anions surrounding the Ta cation. To analyze pressure effects from this perspective, we calculated the pressure evolution of *R*-O and Ta-O bond distances for both the LP and HP phases. Results are summarized in Figure 7a,b, which shows calculated bond distances plotted against pressure. Figure 7a,b clearly shows that for both the compounds, at LP phase, the two largest bond distances between Nd-O decrease at a faster rate than the other six Nd-O bond lengths, whereas the Ta-O bond lengths show lower compressibility than the Nd-O bond lengths. In the high-pressure tetragonal phase, NdO8/SmO8 polyhedra become fully symmetric and the largest Ta-O bond shows most compressibility among all the constituent bonds, indicating the contribution of TaO8 towards the compressibility of the unit cell.

**Figure 2.** Dependence of lattice parameters: (**a**) *a*, (**b**) *b*, (**c**) *c*, (**d**) unit cell volume *V*, (**e**) constituent polyhedral volume and (**f**) interatomic distances with the ionic radii of the lanthanide cation (Gd, Eu, Sm, and Nd).

**Figure 3.** Pressure dependence of unit cell lattice parameters for the compounds (**a**) NdTaO4 and (**b**) SmTaO4 in the low-pressure phase (solid circle) and the high-pressure phase (empty circle). The colored region describes the high-pressure tetragonal phase.

**Figure 4.** Pressure dependence of the unit cell lattice volume for the compounds (**a**) NdTaO4 and (**b**) SmTaO4, of the low-pressure phase (solid circle) and the high-pressure phase (empty circle). The colored region describes the high-pressure tetragonal phase.

**Figure 5.** Enthalpy difference (eV) versus pressure for (**a**) NdTaO4 and (**b**) SmTaO4. The ambient pressure monoclinic *P*2/*c* phase (black) has been taken as a reference for the both compounds. Red and green lines correspond to the *Pcna* (orthorhombic) and *P*4/*nbm* (tetragonal) structures, respectively.

**Figure 6.** Pressure dependence of the distortion index of the bond length of NdO8 and SmO8 (black), TaO6 (red) and TaO8 (orange) for the compounds (**a**) NdTaO4 and (**b**) SmTaO4, of the low-pressure phase (white region) and the high-pressure phase (colored region).


**Table 2.** Calculated structural parameters along with the atomic positions of NdTaO4 and SmTaO4 at ambient pressure and transition pressure.

#### *3.3. Vibrational Properties under Compression*

The primitive unit cell of the M' fergusonite structure has two formula units, giving rise to a total of 36 phonon modes for both the compounds belonging to the *P*2/*c* space group with 2/m point group symmetry. Out of a total of 36 phonon modes, 18 modes are Raman active (8Ag + 10Bg), 15 modes are IR active (7Au + 8Bu) and 3 are low-frequency acoustic modes. The A and B modes are one-dimensional irreducible representations which are symmetric and antisymmetric with respect to the principle axis of symmetry. The assignment of Raman and IR modes is performed in accordance with DFPT as implemented in Quantum ESPRESSO. Ambient pressure Raman and IR modes have been tabulated in Tables 3 and 4, respectively. We have also included previously published Raman mode frequencies from theoretical calculations for the EuTaO4 and GdTaO4 compounds [16,17]. The frequency distribution of the Raman modes is quite similar to the Raman mode distribution in the wolframite structure, which belongs to the same space group *P*2/*c* as M' fergusonite [39,40]. A total of twelve low-frequency Raman modes are present in the frequency region 100–400 cm<sup>−</sup>1, two Raman modes are in the 400–600 cm−<sup>1</sup> range and four are in the higher-frequency side, 600–800 cm−1. It is interesting to note that we have not observed any frequency gap as observed in the Raman spectrum of the zircon or scheelite structure [41]. This can be understood by group–subgroup relationships among zircon– scheelite–fergusonite structures by virtue of the reduction in point group symmetry from 4/mmm to 4/m to 2/m, which in turn increases the allowed numbers of Raman modes and hence fills the frequency gap. All the calculated Raman modes can be categorized as internal modes or external modes. Internal modes, lying in the higher frequency region correspond to TaO6 octahedra while, modes at lower frequency are external modes describing the movement of rigid TaO6 unit against the lanthanide cation [42]. Out of a total of six internal modes of TaO6 octahedra, the 2 Ag and 2 Bg modes lie in the higher–frequency region that

is from 600 to 800 cm<sup>−</sup>1, and two with Ag symmetry appear in the 390–500 cm−<sup>1</sup> range [42]. Identified internal modes are marked by an asterisk in Table 3. The highest-frequency Ag mode, which appears at approximately 762 cm−<sup>1</sup> for NdTaO4 and 771 cm−<sup>1</sup> for SmTaO4, describes the symmetric stretching mode of TaO6 octahedra. The frequency of the majority of the Raman active modes are observed to increase with a decrease in the lanthanum ionic radii except the two Ag modes (108 cm−<sup>1</sup> for NdTaO4 and 107cm−<sup>1</sup> for SmTaO4; 220 cm−<sup>1</sup> for NdTaO4 and 219 cm−<sup>1</sup> for SmTaO4) whereas the frequency of the two Bg modes (at 119 and 138 cm<sup>−</sup>1) remains unaltered by the change in lanthanide cationic radii. This observation is valid when we extend our comparison with the published Raman mode frequencies of EuTaO4 and GdTaO4. Table 3 lists the pressure evolution of all the Raman active modes in P2/c structure obtained by quadratic fitting of data points under pressure. No Raman mode softening has been seen in both the compounds due to the absence of negative pressure coefficients. Hardening of all the Raman active modes has been associated with the monoclinic fergusonite phase of the other compound such as rare earth niobates GdNbO4 and EuNbO4, as well as rare earth tantalates GdTaO4 and EuTaO4 [16,17,22,23]. For rare earth vanadate, mode softening has been observed in the zircon or scheelite phase, but no mode softening has been reported in the fergusonite structure consistent with our current observation in the present work [41,43]. There is crossover between the Bg and Ag modes located at 380–400 cm−<sup>1</sup> due to a nearly 3-fold higher pressure coefficient of the Bg mode than the Ag mode. No other mode crossover has been observed in spite of the large difference in the pressure coefficient of the Raman modes. In the HP phase, stabilized in the tetragonal structure, calculation predicts the presence of the 11 Raman active modes (5 Eg + 2A1g + 1B1g + 3B2g). The pressure evolution of the Raman active modes are shown in Table 5. In the HP phase, all modes show positive pressure coefficients except three modes, which show nonlinear behavior under compression.

**Figure 7.** (**a**) Nd-O bond lengths versus pressure in the LP phase (solid circle) and the HP phase (empty circle), Ta-O bond lengths versus pressure in the LP phase (solid triangle) and the HP phase (empty triangle); (**b**) Sm-O bond lengths versus pressure in LP phase (solid circle) and HP phase (empty circle), Ta-O bond lengths versus pressure in the LP phase (solid triangle) and the HP phase (empty triangle). The white region describes the LP phase while the colored region is for the HP phase.


**Table 3.** Calculated Raman modes for NdTaO4 and SmTaO4 at ambient pressure along with their pressure coefficients (present work). Reported data of earlier studies on EuTaO4 and GdTaO4 have been included for comparison. Frequencies marked with an asterisk (\*) correspond to internal modes of TaO6 octahedra.

**Table 4.** Calculated IR modes for NdTaO4 and SmTaO4 at ambient pressure along with their pressure coefficients (present work).


As seen in Tables 4 and 5 out of 15 IR modes show a decreasing trend when we go from the NdTaO4 to SmTaO4 compound, whereas one Bu mode (257 cm−1) remains unaltered. The remaining nine modes show an increase in frequency with lower lanthanide radii. From Table 4, it can be clearly seen that one IR active mode (Au) shows a nonlinear response in frequency under compression. The presence of the negative pressure coefficient of the IR active mode (Bu) indicates phonon softening in the compound with pressure. Phonon softening was cited as among the crucial trigger points for pressure-induced instability in metal oxides [39]. The other 13 IR modes have a positive pressure coefficient. According to our calculation, the high-pressure phase for both the compounds have seven IR active modes (4Eu + 3A2u). IR active modes at transition pressure, along with pressure

coefficients, are summarized in Table 6. It can be clearly seen in Table 6 that all the modes show a positive pressure coefficient except the lowest-frequency A2u mode, which shows a nonlinear response under pressure.


**Table 5.** Calculated Raman modes for NdTaO4 and SmTaO4 at transition pressure along with their pressure coefficients (present work).

**Table 6.** Calculated IR modes for NdTaO4 and SmTaO4 at 40 and 33 GPa along with their pressure coefficients (present work).


#### **4. Conclusions**

To conclude, the compressional behavior of the M' fergusonite-structured NdTaO4 and SmTaO4, investigated through DFT-based first-principle simulations indicate pressureinduced first-order phase transition from monoclinic to tetragonal structure. The transition is accompanied by an increase in oxygen coordination around the Ta cation from six to eight and a nearly 1.3% volume reduction at transition pressure (40 GPa for NdTaO4 and 33 GPa for SmTaO4). In the low-pressure monoclinic phase, the compressibility of the unit cell has a major contribution from rare earth polyhedra, whereas both the rare earth polyhedra and tantalum polyhedra exhibit a similar contribution towards the compressibility of the unit cell for both the compounds at the HP tetragonal phase, which in turn explains the lower bulk modulus obtained in the HP phase. The pressure evolution of phonon modes has been evaluated in both the LP phase and the HP phase. No Raman mode softening has been seen in both the compounds, although one particular IR mode has been observed to show red shift under pressure, possibly leading to instability in the compounds. Earlier reported experimental high-pressure studies on EuTaO4 and GdTaO4 show isostructural first-order reversible phase transition at approximately 20 GPa, although theoretical calculations predict an equally probable orthorhombic Pcna as an alternative description of the HP phase near 43 and 40 GPa, respectively. The experimental and theoretical phase transition pressures differ due to the crucial dependence of a non-hydrostatic stress environment on phase transition. Therefore, a high-pressure study on NdTaO4 and SmTaO4 by experimental

techniques is desirable for a better understanding of structural and vibrational changes under compression, which is out of scope in the present work.

**Author Contributions:** Conceptualization, methodology, validation, writing—review and editing, visualization, supervision: A.B.G. (corresponding Author); Formal analysis, investigation, writing original draft preparation, visualization: S.B. (First author); Data curation, visualization: A.T. (second author). All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data are available from the corresponding author on reasonable request.

**Acknowledgments:** S.B. and A.T. acknowledge T. Sakuntala for providing motivation and encouragement.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


**Disclaimer/Publisher's Note:** The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

## *Article* **Theoretical Study of Pressure-Induced Phase Transitions in Sb2S3, Bi2S3, and Sb2Se3**

**Estelina Lora da Silva 1,†,\*, Mario C. Santos 2,3,†, Plácida Rodríguez-Hernández 4,†, Alfonso Muñoz 4,† and Francisco Javier Manjón 3,†**


**Abstract:** We report an ab initio study of Sb2S3, Sb2Se3, and Bi2S3 sesquichalcogenides at hydrostatic pressures of up to 60 GPa. We explore the possibility that the *C2/m* , *C2/c*, the disordered *Im-3m*, and the *I4/mmm* phases observed in sesquichalcogenides with heavier cations, viz. Bi2Se3, Bi2Te3, and Sb2Te3, could also be formed in Sb2S3, Sb2Se3, and Bi2S3, as suggested from recent experiments. Our calculations show that the *C2/c* phase is not energetically favorable in any of the three compounds, up to 60 GPa. The *C2/m* system is also unfavorable for Sb2S3 and Bi2S3; however, it is energetically favorable with respect to the *Pnma* phase of Sb2Se3 above 10 GPa. Finally, the *I4/mmm* and the disordered body-centered cubic-type *Im-3m* structures are competitive in energy and are energetically more stable than the *C2/m* phase at pressures beyond 30 GPa. The dynamical stabilities of the *Pnma*, *Im-3m*, *C2/m*, and *I4/mmm* structural phases at high pressures are discussed for the three compounds.

**Keywords:** density functional theory; high-pressure effects; lattice dynamics

#### **1. Introduction**

A great deal of attention has been paid to the family of A2X3 sesquichalcogenides in the last decade since the identification of the trigonal tetradymite-like *R-3m* phases of the group-15 sesquichalcogenides Sb2Te3, Bi2Se3, and Bi2Te3 as 3D topological insulators [1,2]. Such topological insulators are a state of quantum matter where the bulk is a trivial insulator, whereas the surface states are topologically-protected conducting states due to time-reversal symmetry and strong spin-orbit coupling. Respective properties allow for potential applications in the fields of spintronics as well as for quantum computing [3].

These sesquichalcogenides have also been well studied in relation to their thermoelectric properties and as phase change materials [4,5].

Stibnite (Sb2S3), bismuthinite (Bi2S3) and antimonselite (Sb2Se3) minerals belong to group-15 sesquichalcogenides. Such systems do not crystallize into the tetradymite-type structure at room conditions, but instead into the orthorhombic U2S3-type (*Pnma*) structure (Figure 1b). Several technological applications can be considered for such compounds. These include, applications for photovoltaic solar cells, X-ray computed tomography detectors, fuel cells, biomolecules and gas sensors, solid-state batteries, fiber lasers, and photoelectrochemical devices [6–16].

High-pressure (HP) studies of Sb2Te3, Bi2Se3, and Bi2Te3 have shown that they undergo pressure-induced phase transitions (PTs) to monoclinic *C2/m* and *C2/c* structures, to the disordered solid solution *Im-3m* phase, and to the body-centered tetragonal *I4/mmm*

**Citation:** da Silva, E.L.; Santos, M.C.; Rodríguez-Hernández, P.; Muñoz, A; Manjón, F.J. Theoretical Study of Pressure-Induced Phase Transitions in Sb2S3, Bi2S3, and Sb2Se3. *Crystals* **2023**, *13*, 498. https://doi.org/ 10.3390/cryst13030498

Academic Editors: Andrei Vladimirovich Shevelkov, Daniel Errandonea and Enrico Bandiello

Received: 23 February 2023 Revised: 8 March 2023 Accepted: 12 March 2023 Published: 14 March 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

structure [17–26]. These HP studies have shown a plethora of interesting properties, such as enhanced thermoelectric properties [**? ?** ] and superconductivity, which could be of topological nature [29–33]. Therefore, there is a fundamental interest in identifying if the above mentioned HP phases can be associated to group-15 sesquichalcogenides with the *Pnma* structure crystallizing at room pressure. Since the *Pnma* structure has also been identified as a possible HP post-perovskite phase of the (Mg,Fe)SiO3 and NaFeN3 compounds [34,35], the study of Sb2S3, Bi2S3 and Sb2Se3 at HPs could also provide useful information about the structures and stability of ABO3 orthorhombic minerals at pressure conditions close to those of the Earth's mantle.

**Figure 1.** Unit-cell representations of the crystal structures of the *R-3m* (Z = 1) (**a**), *Pnma* (Z = 4) (**b**), *C2/m* (Z = 2) (**c**), *C2/c* (Z = 2) (**d**), the *C2/m* 9-/10-fold structure used to model the disordered bcc-type *Im-3m* phase (Z = 2) (**e**), and the *I4/mmm* (Z = 1) (**f**), and A2X3 sesquichalcogenide structures (A = Sb, Bi; X = S, Se). The A cations and X anions are shown as brown and yellow spheres, respectively.

Initial experimental HP studies of Sb2S3, Bi2S3, and Sb2Se3 have shown that the *Pnma* structure is stable under compression, with first-order PTs occurring at around 50 GPa [36–41]. A comparative study between experimental and theoretical analysis suggested that the *Pnma* structure of the three U2S3-type sesquichalcogenides should actually

be stable up until 50 GPa [39]. Moreover, another work has reported that the crystalline *Pnma* phase for Sb2Se3 is in fact stable even at 70 GPa [**?** ]. Curiously enough, HP studies have found that *Pnma*-type Sb2Se3 becomes a topological superconductor at around 10 GPa and at a low temperature value of 2.5 K [43], exhibiting highly conducting spin-polarized surface states, and similar to what occurs for Bi2Se3 [44]. Moreover, HP superconductivity has been recently found in amorphous Sb2Se3 and attributed to the crystallization of the material in a phase with possible "metavalent" bonding as that present in tetradymite-like Sb2Te3, Bi2Se3, Bi2Te3 [45,46].

It must be noted, however, that some experimental HP studies have suggested that several first- and second-order PTs occur for Sb2S3 up to 50 GPa [47–49]. Furthermore, it has also been suggested that the HP phases of Sb2S3 could be similar to those observed for heavier sesquichalcogenides such as Bi2Se3, Bi2Te3, and Sb2Te3. Moreover, a theoretical study of Bi2S3 at HP predicts the system to be unstable under compression, and decomposing into another stoichiometric system [50]. Therefore, there remains the question for whether the different structural phases (*C2/m*, *C2/c*, and disordered *Im-3m* and *I4/mmm*) observed for heavier cation sesquichalcogenides could be observed at HP on the three U2S3-type minerals, and at different pressure conditions.

In this work, we report theoretical simulations between 0 GPa and 60 GPa of the *Pnma* and hypothetical *R-3m*, *C2/m*, and *C2/c*; and the disordered *Im-3m* and *I4/mmm* phases for Sb2S3, Sb2Se3, and Bi2S3 (Figure 1), with a view to assessing which, if any, are likely to be observed at HP. This work complements a previous study in which we examined the stability of the tetradymite-like (*R-3m*) phase with respect to the *Pnma* phase, up until 10 GPa, for the three U2S3-type minerals [51].

#### **2. Theoretical Methodology**

The six studied crystalline phases (*Pnma*, *R-3m*, *C2/c*, *C2/m*, disordered *Im-3m*, and *I4/mmm*) for Sb2S3, Bi2S3, and Sb2Se3 were simulated within the framework of densityfunctional theory (DFT) [52]. The Vienna Ab initio Simulation Package (VASP) package [53] was used within the projector augmented-wave (PAW) scheme. The datasets included six valence electrons for S[3s23p4] and Se[4s24p4], and 15 valence electrons for Sb[4d105s25p3] and Bi[5d106s26p3]. Total energy convergence was achieved with a plane-wave kineticenergy cut-off of 600 eV. The Perdew-Burke-Ernzerhof parameterization revised for solids (PBEsol), of the generalized-gradient approximation (GGA) exchange-correlation (xc) functional [54], was considered for all the calculations.

The sampling of the Brillouin-zone (BZ) was converged with Γ-centered Monkhorst-Pack [55] grids employing adequate meshes for the different structural phases of the three compounds: *Pnma*-6 × 10 × 6, *R-3m*-12 × 12 × 12, *C2/m*-6 × 12 × 6, *C2/c*-10 × 10 × 8, disordered *Im-3m* (using a *C2/m* conventional cell)-6 × 12 × 12, and *I4/mmm*-12 × 12 × 12.

The *Im-3m* phase is a body-centered cubic (bcc) disordered structure, considered as a disordered solid solution, and it has been theoretically predicted and experimentally found for Bi2Te3 [17]. For sesquichacogenides with A2X3 stoichiometry, the bcc lattice site (2a Wyckoff position) is randomly occupied by 40% of A cations and 60% of X anions. This means that such a structure is a disordered phase with a mixture of cations and anions randomly sharing the same bcc crystallographic position and forming an A-X substitutional alloy [17]. Due to the theoretical difficulty in simulating the disordered *Im-3m* structure, we have used a 9/10-fold *C2/m* structure (the formation of 9/10 chemical A-X bonds), as was previously employed for Bi2Te3 [17] and Bi2Se3 [22]. Moreover, it has been observed that the 9/10-fold *C2/m* structure presents a bcc-like structural order, in agreement with the observed XRD patterns [17,49], therefore giving support to employ the calculated intermediate bcc-like monoclinic *C2/m* phase to confirm the experimental presence of the disordered *Im-3m* system.

For the structural relaxations, the atomic positions and the unit-cell parameters were allowed to change during the ionic relaxation, for different volume values. From these relaxations, we obtained the respective external pressure for the specific isotropic volume compression. The pressure-volume (P-V) curves for all the compounds were fitted to a third-order Birch-Murnaghan equation of state [56,57] to obtain the equilibrium volume, bulk modulus, and respective pressure derivative. The enthalpy (*H*) curves were computed by considering the relation, *H* = *E* + *pV*, where *E* is the total electronic energy of the system, *p* is pressure, and *V* is the volume. The analysis and comparison of the *H* curves for the different polymorphs provides deep insight regarding the thermodynamic stability of each phase for increasing pressure values, up until the studied pressure range (60 GPa).

Lattice dynamics calculations, within the harmonic approximation, were performed at different pressure points, which were found to be energetically favorable. These were considered for the *Pnma*, the disordered *Im-3m*, and the tetragonal phases. We have also calculated the phonon band structure for the *C2/m* system of Sb2Se3 to confirm the dynamical stability at the pressure point where the energetic stability is evidenced (20 GPa). The phonon properties were computed by using the supercell finite-displacement method implemented in the Phonopy package [58], with VASP being used as the second-order force calculator. Supercells were expanded up to 2 × 4 × 2 for the *Pnma* systems, and 2 × 2 × 2 for the disordered and tetragonal phases, enabling the exact calculation of frequencies at the zone center (Γ) and inequivalent zone-boundary wavevectors, which were then interpolated to obtain phonon-dispersion curves.

Since our calculations on the *Pnma* phases for the three compounds under study were in good agreement with the overall data found in the literature [51], we have proceeded in carrying out a theoretical study of the hypothetical *R-3m*, *C2/m*, and *C2/c*; and disordered *Im-3m* and *I4/mmm* phases of Sb2S3, Bi2S3, and Sb2Se3, to probe whether such polymorphs could be energetically competitive under hydrostatic pressure.

#### **3. Results and Discussion**

#### *3.1. Energetic Stability*

Figure 2a–c shows the pressure-dependence of the enthalpy differences relative to the stable phase at ambient pressure between the six above mentioned phases of Sb2S3, Bi2S3, and Sb2Se3, respectively. The values of the predicted transition pressures between the different phases are summarized in Table 1.

**Table 1.** Theoretical estimation of the pressure-induced phase transitions of *R-3m* → *Pnma*, *Pnma* → disordered *C2/m*, *C2/m* → disordered *Im-3m*, and *Pnma* → disordered *Im-3m* for the Sb2Se3, Sb2S3, and Bi2S3 compounds (presented in units of GPa).


\* Ref. [51].

From the enthalpy plots (Figure 2), we observe that:


**Figure 2.** Calculated relative enthalpy vs. pressure curves, for the different possible phases (shown in Figure 1) of Sb2S3 (**a**), Bi2S3 (**b**), and Sb2Se3 (**c**), relative to the lowest-energy phase at ambient pressure: the *Pnma* phase for Sb2S3 and Bi2S3, and the *R-3m* phase for Sb2Se3.

With respect to the first point, referring to Bi2S3 and Sb2S3, our calculations predict that the *Pnma* structure is energetically the most stable phase, up to 30 GPa, and in good agreement with experimental evidences that show the observation of this phase, both at room and also at HP conditions [30–33,45,46]. Surprisingly, however, our simulations indicate that the Sb2Se3 *R-3m* phase is the most stable polymorph at pressures below 4.8 GPa; being both the *Pnma* and *R-3m* phases, energetically competitive between 0 and 4.8 GPa [51].

Regarding the second point, our analysis further shows that the two monoclinic *C2/c* and *C2/m* phases are never energetically competitive in the three compounds throughout the studied pressure range up to 60 GPa; with the exception of the *C2/m* phase of the Sb2Se3 close to 10 GPa. These results are consistent with experimental analysis obtained from Refs. [30–33,45,46], in which no PT had been observed for the Sb2S3 and Bi2S3 systems up until ∼50 GPa. However, the respective results are not consistent with three recent studies reporting evidences of PTs in Sb2S3 [47–49]. A PT to an unknown phase was claimed to occur at around 15 GPa [47,48], and several transitions were also reported between 10 and 25 GPa, and tentatively proposed to be the *R-3m*, *C2/c*, and *C2/m* structural phases [49]. In this context, it must be stressed that our calculations are performed for pure hydrostatic conditions; therefore, it is not expected that a complete agreement occurs with experiments if non-hydrostatic conditions are considered for the experiments.

As for the third point, from a thermodynamic point of view, our results indicate that the bcc-like disordered *Im-3m* phase, initially identified for Bi2Se3, Bi2Te3, and Sb2Te3 [17,19,22], seems to be energetically favorable at HP for our three materials of interest. These results are consistent with the observation of such a phase at around 50 GPa for Sb2Se3 [36] and above 25 GPa for Sb2S3 [47–49]. However, our results do not agree with those found for Bi2S3 [37,40], for which a disorder, attributed mostly to a pressure-induced amorphization (PIA), has been observed above 50 GPa. Notably, PIA in Bi2S3 is consistent with a recent theoretical work that claims that a respective system is unstable above 31.5 GPa, decomposing into a mixture of BiS2 and BiS compounds [50]. In this context, it must be stressed that the existence of a bcc-like structure at HP is not only expected for A2X3 sesquichalcogenides, but also for AX (A=Ge,Sn,Pb) chalcogenides [**? ?** ]. All these compounds have in common the property of being cataloged as evidencing metavalent bonding at room pressure, such as SnTe, PbS, PbSe, PbTe, Sb2Te3, Bi2Se3, or Bi2Te3 [46]; or as being compounds with p-type covalent bonds that develop metavalent bonding at HP when approaching a six-fold coordination, such as GeSe, SnSe, As2S3, and also U2S3-type sesquichalcogenides [61,62], and ultimately, they will reach eight-fold coordination typical of bcc-like metals at very HP.

With respect to the fourth point, the *I4/mmm* structure was firstly proposed for Bi2Se3 by combining experimental and theoretical data [25]. It was found that the structural phase transition pathway for Bi2Se3 followed the sequence: *R-3m* → *C2/m* → *C2/c* → *I4/mmm*, when the quasi-hydrostatic pressure was considered. The *C2/c* phase would, however, be suppressed when nonhydrostatic pressure conditions are taken into account [25]. These results are also compatible with Raman analysis and XRD data performed on Bi2Se3, where it was evidenced that the stability of the *I4/mmm* phase was up to 81.2 GPa [24]. Moreover, it has also been claimed [26], that the alloying of Bi2Se3 with Bi2Te3 enables transitions to occur from the monoclinic phases, up to the disordered *Im-3m* at ∼19 GPa, which then would be surpassed energetically by the tetragonal *I4/mmm* polymorph at ∼23 GPa. Curiously enough, the energetic competition between the *C2/m*, *C2/c*, and *I4/mmm* phases is observed at HP for all three U2S3-type sesquichalcogenides (if we discard the disordered *Im-3m* phase), and the *I4/mmm* structure would indeed be energetically the most favorable polymorph at very HP. However, the disordered *Im-3m* phase is always energetically more competitive than *I4/mmm* for the U2S3-type sesquichalcogenides, and so this latter phase is, *a priori*, not expected to be observed at HP.

Finally, we have to mention that we have also performed the enthalpy calculations by employing two further xc functionals, in order to confirm the present theoretical data (see Appendix A). The employed functionals were the PBE-D2 method of Grimme (which takes into account the dispersion correction term) [63], and LDA [64]. From the analysis of the plots (Figures A1 and A2) we can infer similar energetic trends of the six polymorphs; however, mild differences are observed between the two competing phases of the disordered *Im-3m* and *I4/mmm*—detailed discussion is found in Appendix A.

In summary, the agreement of our results regarding the observation of the disordered *Im-3m* phase for Sb2Se3 and Sb2S3, but not for Bi2S3, suggests that thermodynamic stability is not sufficient to explain the lack of the HP disordered phase for the latter compound. In the following section, we discuss the dynamical stability of the *Pnma*, *C2/m*, *Im-3m*, and *I4/mmm* phases as a function of pressure, in order to provide a deeper understanding regarding this question.

#### *3.2. Dynamical Stability*

Energetic stability is a necessary, but it is not a sufficient condition for a structural phase to be synthetically accessible. One should also probe the dynamical stability of the system, which requires the study of the phonon frequencies. If imaginary frequencies emerge (as represented by negative frequencies in the phonon dispersion curves), we can analyze the results by considering that the system is at a potential-energy maximum (transient state), undergoing a phase transition, and therefore cannot be kinetically stable at the given temperature and/or pressure conditions [65–70].

In this section, we consider the phonon properties of different phases for the three compounds, which were observed to be energetically the most favorable (Figure 2) at different pressure values, namely:


#### 3.2.1. The *C2/m* Phase of Sb2Se3 at 20 GPa

By analyzing the enthalpies as a function of pressure (Figure 2), we observe that the *C2/m* phase in Sb2Se3 is thermodynamically the most stable phase between 10 and 20 GPa. In order to confirm the dynamical stability, we have performed the phonon dispersion curves of respective system at 20 GPa (Figure 3). The dispersion curves do not evidence any imaginary modes; therefore, we may conclude that the *C2/m* phase in Sb2Se3 is dynamically stable in this pressure range and could potentially be observed after the *Pnma* phase, although it has not yet been experimentally observed [36].

**Figure 3.** Harmonic phonon dispersion curves of the *C2/m* phase of the Sb2Se3 compound at 20 GPa.

#### 3.2.2. The Disordered bcc-Type *Im-3m* Phase of the Three Compounds above 30 GPa

To assess the possibility of dynamical stability for the disordered *Im-3m* phases of the three compounds at HP, we have evaluated the phonon dispersion curves at pressure values of 30 GPa, which is close to the transition pressures observed in Figure 2; and at higher pressures of 50 (Sb2Se3) and 60 GPa (Sb2S3, Bi2S3).

As illustrated in Figure 4, all three disordered structures at 30 GPa show imaginary modes along the dispersion curves, thus indicating that these structures are dynamically unstable at this pressure range.

**Figure 4.** Harmonic phonon dispersion curves of the disordered bcc-like *Im-3m* phases of Sb2S3 (**a**), Bi2S3 (**b**), and Sb2Se3 (**c**); and calculated at 30 GPa (**left**), and 50 (Sb2Se3) or 60 GPa (Sb2S3 and Bi2S3; **right**). The BZ **q**−vector description represents the *C2/m* space−group, according to the symmetry of the employed cell.

The phonon dispersion curves of Sb2S3 and Bi2S3 still present imaginary modes at 60 GPa (Figure 4), thus indicating that neither compound is likely to adopt this phase up to this pressure range. We note, however, that the dynamical instabilities found for Sb2S3 and Bi2S3 both decrease (the imaginary, soft modes shift to higher frequency values, towards positive values) with increasing pressure, suggesting that this phase could in principle become stable at pressures above 60 GPa. In this context, we must note that Efthimiopoulos et al. [37] had observed a pressure-induced amorphization above 50 GPa for Bi2S3; however, the authors were not able to identify the phase to be the disordered *Im-3m* structure, even at 65 GPa. On the other hand, the experimental data for Sb2S3, suggests that the disordered bcc-like phase exists between 28.2 and 50.2 GPa [49]. However, it must be noted that experimental measurements detailed in Ref. [49] were carried out under non-hydrostatic behavior due to the employed pressure-transmitting medium.

Finally, our calculations suggest that the *Im-3m* phase becomes dynamically stable in Sb2Se3 already at 50 GPa; a result that is in agreement with the experimental observation of this phase at around 50 GPa [36].

In the light of the above results regarding the disordered *Im-3m* phase of Sb2S3, Bi2S3, and Sb2Se3, we can speculate that the stability of the disordered solid solution in group-15 sesquichalcogenides seems to be related to the sizes of cations and anions. Since the *Im-3m* phase is consistently being evidenced at HP for sesquichalcogenides with heavier cations and anions (Sb2Se3, Sb2Te3, Bi2Se3, and Bi2Te3), the observation of such a HP phase could be related to the radius sizes [71] of Se, Te, Sb, and Bi (atomic radii: *r*Se = 117, *r*Te = 137, *r*Sb = 141, and *r*Bi = 182 pm, respectively). Stemming on these values, we can infer that the disordered solid solutions are energetically favorable in sesquichalcogenides if the atomic radii of the cation and anion differ by less than ∼ 65 pm, or if the size ratio between them is smaller than ∼1.55 (case of Bi2Se3). In this context, Sb2S3, which shows a radius difference of between *r*Sb and *r*<sup>S</sup> of 37 pm (141 − 104 = 37 pm) and a size ratio of 1.35, meets the criteria for the observation of the disordered *Im-3m* phase at HP. However, Bi2S3, which evidences a larger radius difference (78 pm) and ratio (1.75) between B and S, does not satisfy the above criteria. Such results would therefore be consistent with the amorphous phase observed for Bi2S3 at HP [37,40].

Finally, it must be considered, and as suggested in Ref. [17], that the atomic radii between the anion and cation tend to become approximately equal at HP due to a higher probability of charge transfer from cation to anion. Therefore, HP inherently creates a favorable environment for the disordered solid solutions due to the decrease in the difference between the cation and anion atomic radii. This means that we cannot discard that the disordered solid solution in Bi2S3 could occur at very HP values, namely, when the difference between the two radii decreases below 65 pm and when the ratio decreases below 1.55.

#### 3.2.3. The *Pnma* Phase of the Three Compounds at 50 GPa

In order to study the dynamical stability of the well known low-pressure *Pnma* phase at HP, we present in Figure 5 the phonon dispersion curves of the respective phase for Sb2S3, Bi2S3, and Sb2Se3 at 50 GPa.

**Figure 5.** Harmonic phonon dispersion curves of the *Pnma* phase of Sb2S3 (**a**), Bi2S3 (**b**), and Sb2Se3 (**c**), calculated at 50 GPa.

Curiously enough, we find that the *Pnma* phase for Sb2Se3 is already unstable at 50 GPa; therefore, a PT should occur at smaller pressures to the *C2/m* phase, and ultimately, to the disordered *Im-3m* phase, as previously commented. On the other hand, the *Pnma* phase for Sb2S3 and Bi2S3 is still dynamically stable at 50 GPa, although thermodynamically, it is not the most stable phase (Figure 2). These results, together with the dynamical instability observed for the disordered phase of Sb2S3 and Bi2S3 at 50 GPa (Figure 4) and the thermodynamic instability of the *C2/m* and *C2/c* phases, suggest that only the *Pnma* structure should be observed up to 50 GPa for both compounds, unless there is a decomposition, as theoretically predicted for Bi2S3 [50].

#### 3.2.4. The *I4/mmm* Phase of the Three Compounds at HP

By considering the enthalpy curves of the three compounds (Figure 2), we can infer that the *I4/mmm* phase is very close in energy with the disordered *Im-3m* system at pressures above 30 GPa. We therefore have carried out phonon calculations to probe the dynamical stability of this phase, and for all three compounds.

With regard to the Sb2S3 system, we observe that only at 60 GPa is the *I4/mmm* structural phase dynamically stable (Figure 6). Below this pressure value, imaginary modes are observed in the vicinity of the Γ-point, which harden for increasing pressure points. This result, together with the dynamic instability of the disordered *Im-3m* phase still at 60 GPa, suggests that both the *I4/mmm* and *Im-3m* structures are not expected to be observed at pressures below 60 GPa; however, any of these systems could potentially be observed above this pressure range.

**Figure 6.** Harmonic phonon dispersion curves of the *I4/mmm* phase of the Sb2S3 compound between 30 and 60 GPa.

With respect to the Bi2S3 compound (Figure 7), we observe that the *I4/mmm* phase is already dynamically stable at 50 GPa, maintaining the stability at 60 GPa. Therefore, the *I4/mmm* phase could in fact be observed at above 40 GPa, if there was no sample decomposition, as theoretically predicted [50].

Finally, we observed that the the tetragonal *I4/mmm* phase for Sb2Se3 is only dynamically stable at 40 GPa (Figure 8), since there is a localized imaginary mode at the high-symmetry *M*-point, at any other pressure value. These results differ from the remaining two S-based compounds, and evidence that the disordered *Im-3m* phase will be the only (thermodynamically and dynamically) stable phase for the Sb2Se3 compound at very HP (since the *Pnma* is also unstable at 50 GPa, Figure 5c), and as discussed in the previous subsection.

**Figure 7.** Harmonic phonon dispersion curves of the *I4/mmm* phase of the Bi2S3 compound between 30 and 60 GPa.

**Figure 8.** Harmonic phonon dispersion curves of the *I4/mmm* phase of the Sb2Se3 compound between 30 and 60 GPa.

#### **4. Conclusions**

We have carried out a comprehensive set of total energy and lattice dynamics calculations in order to investigate the stabilities of six possible structural phases, viz. *Pnma*, *R-3m*, *C2/m*, and *C2/c*; and disordered *Im-3m* and *I4/mmm* for the Sb2S3, Bi2S3, and Sb2Se3 sesquichalcogenides under hydrostatic pressures up to 60 GPa. Our theoretical results have been commented in the light of the available experimental data.

We find that the *Pnma* phase is energetically more stable at room pressure for the Sb2S3 and Bi2S3 compounds. Curiously, the trigonal *R-3m* phase is the most energetically favorable phase for Sb2Se3 at 0 GPa, although from an experimental perspective, this compound is synthesized in the *Pnma* phase. In fact, we find that the *Pnma* phase is dynamically stable for Sb2S3 and Bi2S3 up to 50 GPa, but not for Sb2Se3 at this pressure value.

From our calculations, we observe that the monoclinic *C2/m* and *C2/c* systems for the three compounds are energetically less favorable throughout the studied pressure range, and they are not expected to be observed at HP under hydrostatic conditions, except for Sb2Se3 between 10 and 20 GPa. Moreover, the *C2/m* phase of Sb2Se3 is dynamically stable at 20 GPa, so a pressure-induced phase transition from the *Pnma* to the *C2/m* system could be observed between 10 and 20 GPa under hydrostatic conditions.

The disordered bcc-like *Im-3m* phase is predicted to be the most energetically stable phase above 35, 30, and 22 GPa for Sb2S3, Bi2S3, and Sb2Se3, respectively; however, this structure is dynamically unstable for Sb2S3 and Bi2S3 up to 60 GPa, and for Sb2Se3 up to 50 GPa. Therefore, this HP phase is expected to occur for Sb2Se3 above 50 GPa, in good agreement with experiments, and perhaps above 60 GPa for the other two compounds.

With respect to the *I4/mmm* phase, we have learned that the structure is not energetically competitive with the *Im-3m* phase, throughout the studied pressure range, and for the three compounds; however, this phase is close in energy to the *Im-3m* phase for Bi2S3 at 30 GPa, and for Sb2Se3, close to 38 GPa. On the other hand, the *I4/mmm* system is dynamically stable for Bi2S3 above 50 GPa, and for Sb2Se3, quite close to 40 GPa. Therefore, this phase could potentially be observed for the former compound at HP; however, not for Sb2Se3.

Curiously enough, we must mention that through symmetry analysis, the *I4/mmm* → *Fmmm* → *C2/m* structural transition pathway may occur as a second-order phase transition. We have, however, not considered the intermediate *Fmmm* system, since, and to the best of our knowledge, such a phase has not yet been experimentally observed for sesquichalcogenides at HP, and therefore, respective analysis would be out the scope of the present work.

We hope that this work will stimulate further investigation of the sesquichalcogenides at HP in order to clarify which pressure-induced phase transitions are observed under hydrostatic and non-hydrostatic conditions.

**Author Contributions:** E.L.d.S., M.C.S., P.R.-H., and A.M. performed the ab initio calculations. F.J.M. supervised the work and comparison with experimental data. E.L.d.S. and F.J.M. wrote the manuscript. All authors have read and agreed to the published version of the manuscript.

**Funding:** This publication is part of the Project MALTA Consolider Team network (RED2018- 102612-T), financed by MINECO/AEI/10.13039/501100003329; by I+D+i project PID2019-106383GB-42/43, financed by MCIN/AEI/10.13039/501100011033; by projects PROMETEO CIPROM/2021/075 (GREENMAT) and MFA/2022/025 (ARCANGEL), financed by Generalitat Valenciana; and by the European Union Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant, agreement No. 785789-COMEX.

**Data Availability Statement:** The data that support the work presented within this paper are available from the corresponding author upon reasonable request.

**Acknowledgments:** E.L.d.S acknowledges the High Performance Computing Chair—an R&D infrastructure (based at the University of Évora; PI: M. Avillez), endorsed by Hewlett Packard Enterprise, and involving a consortium of higher education institutions, research centers, enterprises, and public/private organizations; and the Portuguese Foundation of Science and Technology with the CEEC individual fellowship, 5th edition, with Reference 2022.00082.CEECIND.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Appendix A. Comparison of Enthalpy vs. Pressure Behavior for Different Exchange-Correlation Functionals**

In order to confirm the enthalpy variation as the pressure increases for the six studied phases presented in Figure 2, and obtained by employing the PBEsol functional, we observe that the energetic trends are functional-independent. In Figures A1 and A2, we present enthalpy results carried out by considering the PBE-D2 and the LDA functional, respectively. For PBE-D2 (Figure A1), the transition pressures are very close to those obtained with PBEsol. Moreover, for the Sb2Se3 system, the energetic trends are also comparable to those of PBEsol, namely, that the *R-3m* phase is energetically the most stable at room pressure, and that the *C2/m* phase shows a small interval range of stability before the disordered *Im-3m* phase takes over the energetic configurational space. The *I4/mmm* system is energetically competitive, with the disordered phase at a slightly lower pressure range when compared to the results obtained with PBEsol. For Sb2S3 and Bi2S3, the tetragonal *I4/mmm* phase becomes energetically the most stable at pressures close to 60 GPa. The results obtained with the LDA functional (Figure A2) also show similar trends; however, the transition pressures are slightly higher than for PBEsol. Moreover, a competition between the two monoclinic phases and *I4/mmm* seems to occur in Sb2Se3 slightly, before the latter phase becomes the most stable phase. At around 60 GPa, the disordered phase stabilizes, becoming the most stable structure for Sb2Se3. In addition, for the Bi2S3 compound, we observe a pressure range, at which the *I4/mmm* becomes energetically the most stable (∼30–45 GPa), which then is energetically taken over by the disordered *Im-3m* phase at above 45 GPa.

**Figure A1.** PBE+vdW enthalpy vs. pressure curves, for the different possible phases (shown in Figure 1) of Sb2S3 (**a**), Bi2S3(**b**), and Sb2Se3 (**c**), relative to the lowest-energy phase at ambient pressure: the *Pnma* phase for Sb2S3 and Bi2S3, and the *R-3m* phase for Sb2Se3.

**Figure A2.** LDA enthalpy vs. pressure curves, for the different possible phases (shown in Figure 1) of Sb2S3 (**a**), Bi2S3 (**b**), and Sb2Se3 (**c**), relative to the lowest-energy phase at ambient pressure: the *Pnma* phase for Sb2S3 and Bi2S3, and the *R-3m* phase for Sb2Se3.

#### **Appendix B. Evolution of the Lattice Parameters of Sb2S3, Bi2S3, and Sb2Se3 as a Function of Pressure**

In Tables A1–A3, we present the lattice parameters of the different studied compounds at pressure values where stability (thermodynamic and dynamical) is evidenced. Since the *C2/c* structure is never stable throughout the studied pressure range, we do not present the respective theoretical values for any of the compounds. As for Sb2S3 and Bi2S3, we only present the lattice parameters related to the *Pnma* phase, at 0 and 30 GPa, since this is the pressure range for which the structure is energetically more favorable; and where none of the remaining phases satisfy both of the stability conditions that we mentioned previously.

**Table A1.** Theoretical estimation (PBEsol) of the lattice parameters of the Sb2S3 compound, for the *Pnma* structural phase, for different pressure values, at which the system is both energetically and dynamically stable (presented in units of Å).


**Table A2.** Theoretical estimation (PBEsol) of the lattice parameters of the Bi2S3 compound, for the *Pnma* structural phase, for different pressure values, at which the system is both energetically and dynamically stable (presented in units of Å).


**Table A3.** Theoretical estimation (PBEsol) of the lattice parameters of the Sb2Se3 compound, for the different structural phases that are both energetically and dynamically stable within the range of studied pressures (presented in units of Å).


#### **Appendix C. Mode Mapping of the Disordered** *Im-3m* **Phase of Sb2Se3 at 30 GPa**

By observing the phonon dispersion curves of the disordered *Im-3m* phase in Sb2Se3 at 30 GPa (Figure 4c), we may notice that the instability mode is localized at a highsymmetry point, the *M*-point, which is defined as being a zone-boundary instability (or anti-ferroelectric instability, resulting in an anti-phase periodic distortion). Since the negative mode is localized, it is therefore possible to map out the anharmonic potential energy surfaces by following the eigenvectors associated with this instability/displacement, and thus, to be able to obtain a lower energy structure corresponding to the minima of the potential energy surface. For these calculations, we use the open-source MODEMAP package [72,73]. A sequence of displaced structures in a commensurate supercell (1 × 2 × 2) expansion is generated by displacing along the phonon eigenvectors over a range of amplitudes of the normal-mode coordinate *Q*. The total energies of the "frozen phonon" structures are evaluated from single-point DFT calculations. The E(*Q*) curves are then fitted to a polynomial function, with the number of terms depending on the form of the potential well.

The displacements associated with the disordered bcc high-pressure structure result from the condensation of these soft *M*-modes, leading to a lower energy system and an enlarged primitive unit-cell. The displacement involves two Se atoms at the center of the cell, namely at (0.50000, 0.75000, 0.50000) and (0.50000, 0.25000, 0.50000). Both atoms evidence mild distortions along the x- and z-directions, minimizing the energy to 3.63 meV when compared to the original disordered *Im-3m* phase at *Q* = 0 (Figure A3).

**Figure A3.** Double −well potential−energy surfaces for the phonon instabilities associated with the high-symmetry *M* −point negative modes (Figure 4). The normal-mode coordinates *Q* have been normalized so that the minima are located at *Q* = ±1, and the energy differences are those calculated in a 1 × 2 × 2 supercell. The bottom figures represent the structure at Q = 0 (**left**) and at Q = 1 (**right**).

#### **References**


**Disclaimer/Publisher's Note:** The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

## *Article* **Pressure-Induced Structural Phase Transition of Co-Doped SnO2 Nanocrystals**

**Vinod Panchal 1, Laura Pampillo 2, Sergio Ferrari 2, Vitaliy Bilovol 2,3, Catalin Popescu <sup>4</sup> and Daniel Errandonea 5,\***


**Abstract:** Co-doped SnO2 nanocrystals (with a particle size of 10 nm) with a tetragonal rutile-type (space group *P*42/*mnm*) structure have been investigated for their use in in situ high-pressure synchrotron angle dispersive powder X-ray diffraction up to 20.9 GPa and at an ambient temperature. An analysis of experimental results based on Rietveld refinements suggests that rutile-type Co-doped SnO2 undergoes a structural phase transition at 14.2 GPa to an orthorhombic CaCl2-type phase (space group *Pnnm*), with no phase coexistence during the phase transition. No further phase transition is observed until 20.9 GPa, which is the highest pressure covered by the experiments. The low-pressure and high-pressure phases are related via a group/subgroup relationship. However, a discontinuous change in the unit-cell volume is detected at the phase transition; thus, the phase transition can be classified as a first-order type. Upon decompression, the transition has been found to be reversible. The results are compared with previous high-pressure studies on doped and un-doped SnO2. The compressibility of different phases will be discussed.

**Keywords:** high pressure; phase transition; synchrotron radiation; X-ray diffraction

#### **1. Introduction**

At present, nano-scale materials are having a great impact on human life. They are changing dental medicine, healthcare, and human life more profoundly than several scientific developments of the past decades. Research on nanomaterials has also become a striking area for applied and fundamental research due to the intriguing chemical and physical properties of nanomaterials. Nanomaterials are preferred for many technological applications over their bulk counterparts due to enhancements in their catalytic, optical, magnetic, and electrical properties [1–3]. In addition, the nature of pressure-induced structural phase transitions, pressure effects on elastic properties, and transition pressures in nanomaterials could be quite different when compared to bulk materials [4–9]. Tin dioxide (SnO2) is a very technologically important material, and it is widely used as a wide bandgap semiconductor. It finds applications in the field of solar cells [10], in ultraviolet photodetectors [11], in short wavelength light-emitting diodes [12], in spintronic applications [13], as a gas sensor [14], and in lithium-ion batteries [15], among other applications. Recently, SnO2 has also attracted substantial attention due to its wide bandgap energy, large exciton binding energy, and outstanding electrical properties and optoelectronic features. Due to

**Citation:** Panchal, V.; Pampillo, L.; Ferrari, S.; Bilovol, V.; Popescu, C.; Errandonea, D. Pressure-Induced Structural Phase Transition of Co-Doped SnO2 Nanocrystals. *Crystals* **2023**, *13*, 900. https:// doi.org/10.3390/cryst13060900

Academic Editor: Borislav Angelov

Received: 13 May 2023 Revised: 25 May 2023 Accepted: 30 May 2023 Published: 31 May 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

these characteristics, SnO2 has been employed for building ultraviolet light-emitting and light-detecting devices based on a SnO2/GaN heterojunction [16].

Under ambient conditions, SnO2, which is also known as stannic oxide or by the mineralogical name Cassiterite, crystallizes into a crystal structure isomorphic to that of tetragonal rutile (space group *P*42/*mnm*, Z = 2). In this structure, the Sn atoms are in the corners and center of the tetragonal unit cell (see Figure 1a). They are six-fold coordinated by oxygen atoms, which are shared with the adjacent SnO6 octahedral units, as shown in Figure 1a. The structure is also commonly described as a distorted hcp oxide array, with half of the octahedral sites occupied by Sn atoms. The oxygen atoms have a coordination number of three, resulting in a trigonal planar coordination.

**Figure 1.** Schematic view of (**a**) the tetragonal rutile-type structure of SnO2, space group *P*42/*mnm*, and (**b**) the high-pressure orthorhombic structure of SnO2, space group *Pnnm* (Sn+4 cations are in blue and O−<sup>2</sup> anions are in red). In the figure, we include red, green, and blue arrows showing the directions of the unit-cell axes *a*, *b*, and *c*, respectively.

In the past, it was reported that SnO2 undergoes the following sequence of pressureinduced structural phase transitions: rutile-type phase → CaCl2-type phase → PbO2-type phase → fluorite-type phase [17]. Earlier high-pressure (HP) powder X-ray diffraction (XRD) measurements [18,19], theoretical ab initio calculations [19–23], and HP Raman scattering measurements [24] on bulk as well as in nanocrystalline samples provided evidence of the occurrence of a second order-structural phase transition at pressures that vary from 7 GPa to 14 GPa. The structural phase transition is from the tetragonal rutile-type to the orthorhombic CaCl2-type structure (space group *Pnnm*), which is schematically represented in Figure 1b. In the CaCl2-type structure, the Sn atoms form a body-centered orthorhombic structure, and they are also octahedrally coordinated by oxygen atoms, as in the low-pressure phase. The SnO6 octahedral units form chains of edge-sharing octahedra running along the *c*-axis of the crystal structure, while perpendicular to the *c*-axis, the

SnO6 octahedra are linked by sharing one corner with each neighboring SnO6 octahedron. However, there are a few exceptions for the rutile to CaCl2-type phase transition. For example, high-pressure powder XRD measurements carried out on nanocrystalline 5 nm sized samples [25] and 8 nm sized samples [26,27] have indicated the occurrence of a direct tetragonal rutile to cubic fluorite first-order structural phase transition beyond 18 GPa. The transition pressure has been also found to increase when decreasing the particle size [27]. On the other hand, the doping of SnO2 has been shown to cause alterations in the highpressure behavior, elastic properties, and optical properties of the material. For instance, in Fe-doped nanoparticles (size 18 nm and 10 at % doping) of SnO2, increases in the transition pressure and bulk modulus have been reported [28]. On the contrary, in V-doped nanoparticles (size 10–30 nm and 5–12.5 at % doping) of SnO2, a decrease in the bulk modulus with an increasing concentration of the dopant has been found [29]. Hence, it is very interesting to know how the particle size, doping concentration, and the type of doping ion alter the high-pressure behavior and elastic properties of SnO2, and further systematic investigations are required in this direction. In the current contribution, we report HP synchrotron powder XRD studies up to a pressure of 20.9 GPa on Co-doped nanocrystalline SnO2 (crystallite size 15 nm and 10 at % doping) to contribute to the understanding of the effect of doping on the transition pressure and elastic properties. Such a study has not been reported yet.

#### **2. Experiments**

Cobalt-doped SnO2 nanoparticles were prepared at Universidad de Buenos Aires via the wet chemical co-precipitation method, following the procedure described previously by Ferrari et al. [30]. To implement the cobalt doping, we used anhydrous cobalt chloride (CoCl2) with a purity of 99.99%, which was obtained from Sigma-Aldrich (St. Louis, MI, USA). The synthesized nanoparticles were characterized at ambient conditions via powder XRD using Cu Ka radiation. The XRD measurements confirmed the formation of singlephase rutile-type nanoparticles with unit-cell parameters of *a* = 4.732(1) Å and *c* = 3.185(1) Å. These parameters are consistent with the parameters determined from single-crystal X-ray diffraction studies on bulk SnO2 [31,32]. An average particle size of 15 nm was determined from the full-width-at-half-maximum of the XRD peaks, using the well-known Scherrer equation [33]. The cobalt concentration (10 at %) in our samples was determined via energy-dispersive X-ray spectroscopy (EDXS).

Angle-dispersive HP powder XRD measurements were carried out at the MSPD-BL04 beamline of the ALBA synchrotron, using a monochromatic beam with a wavelength of 0.4642 Å. Wavelength selection was achieved using a silicon (111) double-crystal monochromator with a resolution of 2 × <sup>10</sup>−<sup>4</sup> [34]. The X-ray beam was focused down to a <sup>20</sup> <sup>μ</sup><sup>m</sup> full-width-at-half-maximum spot using Kirkpatrick–Baez mirrors. A Rayonix chargecoupled device (CCD) detector was used to collect the XRD patterns. The CCD detector was calibrated using LaB6 as a standard. Two-dimensional diffraction rings obtained from the detector were transformed into one-dimensional diffractograms using Dioptas. The measurements were taken at room temperature under compression with a membrane diamond anvil cell (DAC) equipped with 500 μm culet diameter diamond anvils. We used a 200 mm thick stainless-steel gasket pre-indented to a thickness of 40 μm, with a centered hole 200 μm in diameter, as a pressure chamber. As a pressure-transmitting medium we used a 16:3:1 methanol–ethanol–water mixture, which is known to remain quasi-hydrostatic up to approximately 10 GPa [35]. The pressure was determined using the equation of state (EOS) of copper (Cu), as reported by Dewaelle et al. [36], with a precision of ±0.05 GPa. During the XRD measurements, a rocking of ±3◦ of the DAC was used to reduce the influence of preferred orientations and to improve the homogeneity of the Debye rings. The structural analysis was carried out by employing the Rietveld technique via the Fullprof suite [37]. In the structural refinements, the background was fitted with a Chebyshev polynomial function of first kind with six coefficients, and the peak profiles

were modeled using a pseudo-Voigt function. In addition to the unit-cell parameters, we also refined the atomic positions. The overall displacement factor was fixed to 0.5 Å.

#### **3. Results and Discussion**

Figure 2 shows the powder XRD pattern at the lowest pressure measured for the nanocrystalline Co-doped SnO2 in the DAC (pressure = 0.6 GPa). The figure includes the results of the Rietveld refinement. As shown in the figure, the refinement is good, and the diffraction pattern can be undoubtedly assigned to the tetragonal rutile-type structure. In the XRD pattern, there is an extra weak peak, denoted by the asterisk symbol (\*), which is assigned to the copper grain used as a pressure marker. It corresponds to the (111) reflection of copper. The unit-cell parameters obtained for the nanocrystalline Co-doped SnO2 at 0.6 GPa are *a* = 4.728(1) Å and *c* = 3.184(1) Å. The goodness-of-fit parameters obtained from the structural refinement are *R*wp = 6.16% and *R*p = 3.4%.

**Figure 2.** The Rietveld profile refinement for the XRD pattern of the tetragonal rutile-type phase of nanocrystalline Co-doped SnO2 at 0.6 GPa and room temperature. The asterisk (\*) is the (111) diffraction peak of copper used to determine pressure. The vertical bars indicate the calculated positions of diffraction peaks of the sample and copper (Cu). Miller indexes are shown in the figure.

Figure 3 shows a selection of XRD patterns of nanocrystalline Co-doped SnO2 at representative pressures. There are no noticeable changes in the diffraction patterns up to 13 GPa aside from the shift of peaks to higher angles due to the typical contraction of unitcell parameters under compression. All the XRD diffraction peaks up to 13 GPa could be successfully indexed considering only the low-pressure (LP) rutile-type phase and copper. In addition to the peaks assigned to SnO2, a peak due to copper (marked by \*), which was used to determine the pressure, can be identified in all the X-ray diffraction profiles. We observed a systematic shift in all the diffraction peaks to a higher 2θ due to lattice compression. At 14.2 GPa, we observed few discernible changes in the XRD diffraction profile, which we consider to be indicative of a phase transformation. In addition, we found that at 14.2 GPa, the cell metric is no longer consistent with *P*42/*mnm* symmetry, and another crystallographic phase with *Pnmm* symmetry occurs, corresponding to an orthorhombic CaCl2-type structure. In particular, at the pressure of 14.2 GPa, the broadening of the (101), (200), and (111) diffraction peaks of rutile, in addition to the splitting of the (211) diffraction peak of rutile, were observed. The XRD diffraction profiles at 14.2 GPa and higher pressures could not be well indexed to tetragonal rutile-type phase. To make the changes in the XRD patterns indicating the structural phase transition more evident for the readers, we provide in Figure 4 an enhanced view of the XRD patterns measured at 13.0, 14.2, and 20.2 GPa. In Figure 4, it is clear that the (211) peak of rutile gradually splits

into two different peaks. The same happens with peak (111). Another fact highlighted by Figure 4 that could not be explained by assuming the tetragonal rutile-type structure is the evolution of the (200) peak of rutile towards low angles as the pressure increased beyond 13.0 GPa. This can only be explained by assuming a decrease in the crystal symmetry from tetragonal to orthorhombic. Indeed, we found that the orthorhombic CaCl2-type structure provides the correct positions of all diffraction peaks, as indicated by unbiased Rietveld refinements. An example of this can be seen in Figure 5, where we show the Rietveld refinement we performed at 14.2 GPa. Furthermore, the proposed rutile-CaCl2 structural phase transition agrees with earlier investigations on rutile SnO2 [17–20].

**Figure 3.** Room-temperature X-ray powder diffraction patterns of nanocrystalline Co-doped SnO2 at representative pressures. Pressures are indicated in the figure. The asterisks (\*) denote the (111) diffraction peak of copper at each pressure. The diffractogram of the low-pressure phase at 0.6 GPa and that of the high-pressure phase at 14.2 GPa have been indexed with the tetragonal rutile-type and the orthorhombic CaCl2-type structure, respectively. Miller indexes are shown.

**Figure 4.** Enhanced view of the regions of XRD patterns measured in nanocrystalline Co-doped SnO2 at 13.0, 14.2, and 20.2 GPa, highlighting the changes that evidence the phase transition discussed in the text.

**Figure 5.** The Rietveld profile refinement of the XRD pattern measured at 14.2 GPa and room temperature for the orthorhombic CaCl2-type phase of nanocrystalline Co-doped SnO2. The vertical bars indicate the calculated positions of diffraction peaks of sample. The contributions from the gasket and copper (Cu) are also indicated.

Upon a further increase in pressure, we found that the orthorhombic CaCl2-type structure continued to be stable in the nanocrystalline Co-doped SnO2 up to 20.9 GPa, which was the highest pressure measured in this investigation. Upon the release of the pressure, we found that the phase transition was fully reversible. In the experiments, we did not observe any coexistence of phases during the compression and decompression cycles. This fact, and the group–subgroup relationship existing between space groups *P*42/*mnm* and *Pnnm*, could be an indication that the observed phase transition is secondorder in nature, which agrees with the conclusions extracted from previous powder XRD measurements on doped and un-doped nanocrystalline SnO2 [28]. However, as we will explain below, in nanocrystalline Co-doped SnO2 at the phase transition, we detected a volume discontinuity, which undoubtedly supports a first-order nature for the phase transition in Co-doped SnO2. Notice that up to now, this is the only SnO2 nanomaterial in which such a volume discontinuity, larger than experimental uncertainties, has been detected at the phase transition. We speculate that the observed volume discontinuity could be related to changes in the strong anti-ferromagnetic super-exchange interaction between the Co ions that exist in the Co-doped SnO2 [38]. Another hypothesis to explain the observed abrupt volume decrease is the influence of structural defects that are induced by Co doping [39]. Additional studies are needed to fully understand the distinctive behavior of Co-doped SnO2. The Rietveld refinement of the XRD patterns' profile measured for the HP phase at 14.2 GPa is shown in Figure 5. In addition to the peaks from the sample and Cu, we observed a weak contribution of the gasket material, which causes the peak (200) of the HP phase of Co-doped SnO2 to become asymmetric. The unit-cell parameters determined for the HP phase of the Co-doped SnO2 at 14.2 GPa are *a* = 4.675(2) Å, *b* = 4.572(3) Å, and *c* = 3.151(1) Å. The goodness-of-fit parameters of the Rietveld refinement shown in Figure 5 are *R*wp = 9.84% and *R*p = 7.79%.

The pressure evolution of the lattice parameters of the Co-doped nanocrystalline SnO2 in the tetragonal rutile-type and orthorhombic CaCl2-type phases is shown in Figure 6. The pressure dependence of the unit-cell volume is reported in Figure 7. We observed a discontinuity in the volume (−ΔV/V~1.0%) at the transition pressure. The change in the volume was larger than the experimental error (which is smaller than the symbols), indicating the first-order nature of the structural phase transition. Using the EOSFIT7 software [40], the linear compressibility values of the axes of each phase were calculated. We observed that in the rutile-type phase of SnO2, the axial compressibility values were highly anisotropic. In particular, the *a*-axis was more compressible than the *c*-axis, as

is evident from the increase in the *c*/*a* ratio from 0.673 at ambient pressure to 0.681 at 13 GPa. The linear compressibility of the *<sup>c</sup>*-axis was found to be *Kc* = 1.02(4) × <sup>10</sup>−<sup>3</sup> GPa<sup>−</sup>1, which is almost half of the compressibility of the *<sup>a</sup>*-axis, *Ka* = 1.72(6) × <sup>10</sup>−<sup>3</sup> GPa−1. This anisotropic behavior in compressibility is in quite good agreement with the earlier reported values of bulk SnO2 samples [31,41,42] and even for un-doped, Fe-doped, and V-doped nanocrystalline SnO2 [28,29]. However, the of axial compressibility vales obtained in this investigation are slightly higher compared to earlier investigations [28,29,31,41,42]. In the case of the orthorhombic CaCl2-type phase of the Co-doped SnO2, the axial compressibility values were found to be highly anisotropic as well. In particular, the *b*-axis was more compressible when compared to other two axes. The linear compressibility values of all three axes were found to be *Ka* = 1.65(4) × <sup>10</sup>−<sup>3</sup> GPa−1, *Kb* = 1.90(1) × <sup>10</sup>−<sup>3</sup> GPa−1, and *Kc* = 1.05(2) × <sup>10</sup>−<sup>3</sup> GPa<sup>−</sup>1.

**Figure 6.** Pressure dependence of lattice parameters of nanocrystalline Co-doped SnO2. Solid squares represent the tetragonal rutile-type phase and solid circles represent the orthorhombic CaCl2-type structure. Solid lines are linear fits to lattice parameters. Error bars are smaller than the size of the symbols. The unit-cell axes, *a*, *b*, and *c* are identified in the figure.

**Figure 7.** Volume versus pressure data for nanocrystalline Co-doped SnO2. Solid squares represent the tetragonal rutile-type phase and solid circles represent the orthorhombic CaCl2-type structure. Solid lines represent the second-order Birch–Murnaghan fit to the data for both phases (see text for details). Error bars are smaller than the size of the symbols.

The *P*–*V* data of the rutile-type phase of nanocrystalline Co-doped SnO2 in the tetragonal phase, fitted to the second-order Birch–Murnaghan equation of state, provides a bulk modulus at zero pressure of B0 = 213(9) GPa, with its pressure derivative provided as

*B*0 = 4. During this fit, the bulk modulus B0 and ambient pressure volume V0 were kept as free variables. This value of the bulk modulus is in good agreement with previously reported values for the same parameter in bulk SnO2 [17] and even in un-doped and in Fe-doped nanocrystalline SnO2 [28]. The values of the bulk moduli are summarized in Table 1. However, in V-doped nanocrystalline samples, the bulk modulus was found to decrease with increases in doping concentrations in the range from 142 GPa to 185 GPa [29]. We can then conclude that in contrast with V-doping, a 10% doping with Co and Fe does not affect the mechanical properties of SnO2, which is good for technological applications. A possible reason for this it could be the fact that the V atom has larger ionic radii than the Fe and Co atoms, which would necessarily drive changes in the crystal structure more easily. For the high-pressure phase of the Co-doped SnO2, the bulk modulus we obtained was B0 = 228(9) GPa. The slight increase in the bulk modulus at the phase transition is consistent with the fact that the HP phase is denser than the low-pressure phase due to the volume contraction observed at the transition. However, both bulk moduli agree within error bars: 213(9) and 228(9) GPa. This fact is consistent with the displacive mechanism proposed for the rutile-CaCl2 phase transition [43]. In this regard, the behavior of the Co-doped SnO2 is more similar to the behavior of bulk SnO2 [17], for which no changes have been detected in the bulk modulus at the transition (see Table 1), than to the behavior of the bulk modulus for Fe-doped a nanocrystalline SnO2 [29], for which a 15% increase in the bulk modulus has been reported after the phase transition. At present, it is not clear that the difference in behavior is inherent to the difference in doping or to the difference in non-hydrostatic stresses in the experiments [44]. Future studies are needed to clarify this issue. On the other hand, it should be noted that in the equation of state, the volume at zero pressure (V0) and B0 are correlated parameters. Thus, large uncertainties on V0 could affect the value of B0 [45]. This could be the case for nanocrystalline Co-doped SnO2 for which the phase transition is reversible and therefore no data are available for the HP phase at pressures close to the ambient pressure. This means that the determination of V0 comes from the extrapolation of data measured above 14.2 GPa, which could lead to large errors in the determination of V0. This fact will necessarily propagate to the determination of B0, whose value should be taken with caution.

**Table 1.** Bulk modulus determined for different phases of nanocrystalline of SnO2. We include results from this work and for the literature [17,18,25,27–29].


To conclude the discussion, we would like to mention that the Co-doping of nanocrystalline SnO2 does not affect the structural stability and mechanical properties, making Co-doped SnO2 more attractive than SnO2 doped with other metals for practical applications for which doped SnO2 is deposited as a thin film on substrates such as amorphous solar cells and cadmium telluride solar cells [46]. Thin films are usually subjected to epitaxial strain, which gives rise to misfit stress. It is therefore better to use SnO2 films in which

the structural stability and mechanical properties are not affected by doping, as is the case in Co-doped SnO2.

#### **4. Conclusions**

In summary, we performed an in situ synchrotron powder X-ray diffraction study on nanocrystalline Co-doped SnO2 at room temperature up to a pressure of 20.9 GPa. This study shows that the studied material undergoes a first-order structural phase transition from the tetragonal rutile-type to the orthorhombic CaCl2-type phase at ∼14.2 GPa. The crystal structure of the high-pressure phase is orthorhombic, and it is isomorphic to the high-pressure structure observed previously in bulk, doped, and un-doped nanocrystalline samples of SnO2. We did not observe any coexistence of phases during the phase transition under conditions of both compression and decompression. The compressibility behavior was found to be highly anisotropic in both phases, and the value of bulk modulus for the tetragonal rutile-type phase was in good agreement with earlier measurements. However, for the orthorhombic high-pressure CaCl2-type phase, the bulk modulus of the Co-doped SnO2 was 10% smaller than the same parameter in Fe-doped SnO2. The high-pressure phase was observed to remain stable up to 20.9 GPa. Upon the release of the pressure, the phase transition was found to be reversible. The reported results could be relevant for the implementation of practical applications using Co-doped thin films prepared via deposition methods.

**Author Contributions:** Conceptualization, D.E.; methodology and validation, V.P. and D.E. formal analysis, V.P., S.F. and D.E.; investigation, V.P., L.P., S.F., V.B., C.P. and D.E.; writing—original draft preparation, V.P.; writing—review and editing, V.P., L.P., S.F., V.B., C.P. and D.E.; supervision, D.E. All authors have read and agreed to the published version of the manuscript.

**Funding:** D.E. thanks the financial support from Generalitat Valenciana under the grant PROME-TEO CIPROM/2021/075-GREENMAT, the Advanced Materials Programme, supported by MCIN and GVA, with funding from the European Union, NextGenerationEU (PRTR-C17.I1), under grant MFA/2022/007, and from the Spanish Research Agency (AEI) and the Spanish Ministry of Science and Investigation (MCIN) under projects PID2019-106383GB-C41 (DOI: 10.13039/501100011033) and RED2018-102612-T (MALTA Consolider-Team Network).

**Data Availability Statement:** Data are available upon reasonable request to the corresponding author at daniel.errandonea@uv.es (D.E.).

**Acknowledgments:** The authors are grateful for financial support from the Spanish Research Agency (AEI) and the Spanish Ministry of Science and Investigation (MCIN) under grants PID2019-106383GB-41/43 (http://dx.doi.org/10.13039/501100011033) and RED2018-102612-T (MALTA Consolider Team Network) and from Generalitat Valenciana under Grant PROMETEO CIPROM/2021/075- GREENMAT. This study forms part of the Advanced Materials program and was supported by MCIN with funding from the European Union Next Generation EU (PRTR-C17.I1) and by Generalitat Valenciana, Grant MFA/2022/007. The authors thank ALBA synchrotron for providing beam time for the XRD experiments (Proposal 2021085226).

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


**Disclaimer/Publisher's Note:** The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

## *Article* **High-Pressure Vibrational and Structural Studies of the Chemically Engineered Ferroelectric Phase of Sodium Niobate**

**Sanjay Kumar Mishra 1,2,\*, Nandini Garg 2,3,\*, Smita Gohil 4, Ranjan Mittal 1,2,\* and Samrath Lal Chaplot 1,2**


**Abstract:** Pure NaNbO3 has an antiferroelectric phase at ambient pressure. The structural behaviour of the chemically engineered ferroelectric phase of sodium niobate, NNBT05: [(0.95) NaNbO3- (0.05) BaTiO3], under high-pressure has been studied using Raman scattering and angle-dispersive synchrotron X-ray diffraction techniques. At pressure > 1 GPa, noticeable changes in the Raman spectra can be seen in the low wavenumber modes (150–300 cm<sup>−</sup>1). Large changes in the positions and intensities of the Raman bands as a function of pressure provide evidence for structural phase transition. The results indicate significant changes in the bond-lengths and the orientation of the NbO6 octahedra at ~1 GPa, and a transition to the paraelectric phase at ~5 GPa, which are at lower pressures than previously found in pure NaNbO3. The powder X-ray diffraction pattern shows an appreciable change in the peak profile in terms of position and width on increasing pressure. The pressure dependences of the structural parameters show that the response of the lattice parameters to pressure is strongly anisotropic. By fitting the pressure–volume data using the Birch–Murnaghan equation of state, the isothermal bulk modulus was estimated. The experimental results suggest that on doping BaTiO3 in NaNbO3, the bulk modulus increases. The bulk modulus of NNBT05 has been estimated to be 164.5 GPa, which is fairly close to 157.5 GPa, as previously observed in NaNbO3.

**Keywords:** ferroelectric; antiferroelectric; niobate; Raman scattering; high pressure diffraction

#### **1. Introduction**

Materials with perovskite structures exhibit diverse crystal structures and physical properties such as (anti)ferroelectricity, relaxor, magnetic, multiferroic properties, etc., and remain pertinent in next generation applications [1–14]. The perovskite structure consists of two types of polyhedra, the octahedron surrounding the B-cation, and the A-cation, coordinated by 12 oxygen atoms forming a cuboctahedron. In the perovskite family, alkaline niobates and their derivatives have attracted a great deal of attention, and are promising candidates for eco-friendly lead-free piezoceramics [10,11,15–17].

The application of hydrostatic pressure strongly modifies the short-range interatomic and long-range Coulomb interactions, which are responsible for the structural stability of ferroelectric and antiferrodistortive phases. These phases are associated with the freezing of zone-centre and zone-boundary phonon instabilities. Sodium niobate (NaNbO3) has both these competing instabilities simultaneously. At ambient conditions, sodium niobate crystallizes in the antiferroelectric orthorhombic phase. Sodium niobate perovskite is a textbook example for understanding a complicated series of structural phase changes that occur with temperature and pressure [17–28]. Using high pressure Raman scattering techniques, Shiratori et al. [28] have reported transitions at around 2, 6, and 9 GPa, respectively. The pressure-dependent neutron diffraction studied showed that the antiferroelectric (*Pbcm*) phase transformed to a paraelectric (*Pbnm*) phase at 8 GPa [19]. Neutron diffraction also

**Citation:** Mishra, S.K.; Garg, N.; Gohil, S.; Mittal, R.; Chaplot, S.L. High-Pressure Vibrational and Structural Studies of the Chemically Engineered Ferroelectric Phase of Sodium Niobate. *Crystals* **2023**, *13*, 1181. https://doi.org/10.3390/ cryst13081181

Academic Editors: Daniel Errandonea and Enrico Bandiello

Received: 7 July 2023 Revised: 26 July 2023 Accepted: 26 July 2023 Published: 29 July 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

revealed changes in the baric behaviour of Nb-O-Nb bond angles as a result of complex reorientations of NbO6 octahedra at high pressures of ~2 GPa [19]. Recent, high pressure X-ray diffraction data suggest that the paraelectric (*Pbnm*) phase is stable up to 30 GPa [29].

In addition to temperature and pressure, the structure and physical properties of materials can also be tuned by chemical doping. Specific doping elements are used to produce subtle distortions in crystal structures and to regulate changes in their physical properties. For example, when doping the BaTiO3 in the NaNbO3 matrix, there is a significant enhancement in the dielectric and piezoelectric response of the material [30–34]. The structural, dielectric, ferroelectric and piezoelectric properties of NNBTx [(1 − x) NaNbO3 − xBaTiO3] ceramics have been investigated by various researchers [10,16,30,31,33–36]. This solid solution is similar to Kx Na1−xNbO3 (KNN), except that the KNbO3 has been replaced by BaTiO3. The main problem with KNN is the poor densification of the ceramic due to the high volatility of alkaline oxides, which significantly affects the functional performance of this ceramic [37]. As suggested by Zeng et al. [37], the problem of the poor densification of KNN can be resolved by doping BaTiO3 instead of KNbO3 in the NaNbO3 matrix, because both show similar structural phase transitions. It can also be seen that BaTiO3 based piezo-ceramics show very good electromechanical properties. Raveskii et al. [32] and other researchers have studied the various compositions in the full composition range. They have reported extensive electro-physical properties of this solid solution, with expected structures. It is reported that on increasing the BaTiO3 composition, the dielectric maximum shifts to the lower temperatures and reaches near room temperature for x = 0.25. They also found P-E hysteresis loops with good saturation for x = 0.10. For the same sample, they observed the coercive field EC (11 kV/cm), the spontaneous polarization PS (15 μC/cm2) and residual polarization PR (13 μC/cm2) at 50 Hz. However, the electromechanical properties of the NNBTx in the NaNbO3 are exceptionally remarkable. The literature contains very little information about the structural phase transition with composition and temperature. Most of the reports contain only knowledge of expected room temperature structures probed using X-ray diffraction studies. Raveskii et al. [32] have proposed the monoclinic structure of the compositions x~0.05–0.07 at room temperature.

Earlier, we reported the crystal structure and phase stability of the NNBTx system for the small doping of BaTiO3 (x = 0.0 to 0.15). We reported an orthorhombic antiferroelectric phase (space group: *Pbcm*) for x < 0.02, an orthorhombic ferroelectric phase (space group: *Pmc21*) for 0.02 < x < 0.10, and an orthorhombic ferroelectric phase (space group: *Amm2*) for x ≥ 0.10. We also reported the phase transitions as a function of temperature for NNBT03 and 05 using powder X-ray and neutron diffraction studies in conjunction with dielectric and Raman scattering measurements [30,31,35,36]. Apart from this, we provided evidence of the existence of a functional monoclinic phase at low temperatures. This monoclinic (*Cc*) phase is expected to provide easy polarization rotation at low temperatures for the compositions x = 0.03 (NNBT03) and x = 0.05 (NNBT05).

The application of pressure either distorts or symmetrises the structure and causes the alteration of the crystal symmetry with huge consequences on the physical properties. The present study gives insight into the unique pressure-dependent phase stability of chemically engineered ferroelectric NaNbO3, i.e., NNBT05, and aims to deepen the understanding of structural distortion in this material. It also helps in the understanding of the role of doping on pressure responses and the phase stability of the ferroelectric phase. The pressure dependence of the powder X-diffraction and Raman scattering experiments will be used to understand the microscopic origin of the structural and vibrational properties of the sample. At pressure > 1 GPa, noticeable changes in the Raman spectra are seen in the low wavenumber modes (150–300 cm−1). These observations are most likely associated with octahedral reorientations in the parent ferroelectric phase. A detailed analysis of the high-pressure X-ray diffraction data suggests that the response of the lattice parameters to pressure is strongly anisotropic. By fitting the pressure–volume data using the Birch– Murnaghan equation of state, the isothermal bulk modulus can be estimated. We found that on doping BaTiO3 in NaNbO3, the bulk modulus increases.

#### **2. Materials and Methods**

The solid solution NNBT05 was prepared with the solid-state reaction method. Dried powders of high purity Na2CO3, BaCO3, Nb2O5 and TiO2 were taken in the stoichiometric ratios and were mixed in a planetary ball mill for 12 h in acetone medium. The mixtures were calcined in the air at 1173 K. The phase purity of the solid solutions was confirmed with powder diffraction data. The calcined powders were then sintered at 1273 K for 6 h. The sintered pellets were crushed into fine powders and were used for the high-pressure X-ray diffraction and Raman scattering measurements.

Angle-dispersive X-ray powder diffraction patterns at high pressures up to 30 GPa and at ambient temperature were measured at the BL-11 beamline at INDUS-2 India. The diffraction images were collected with the wavelength λ = 0.79997 Å on the MAR345 detector located at a distance of 190 mm from the sample. The two-dimensional X-ray diffraction (XRD) images were converted to one-dimensional diffraction patterns using the FIT2D program. The data at each pressure level were analysed with the Rietveld refinement method using the FULLPROF program [38]. A Thompson–Cox–Hastings pseudo-Voigt with Axial divergence asymmetry function was used to model the peak profiles. The background was fitted using a Chebychev polynomial. Except for the occupancy parameters of the atoms, which were fixed corresponding to the nominal composition, all other parameters, i.e., scale factor, zero displacement, and isotropic profile parameters, lattice parameters, isotropic thermal parameters, and positional coordinates, were refined.

High pressure Raman experiments were performed using an in-house-made Mao-Bell type of diamond anvil cell. Raman spectra were recorded at increasing and decreasing pressures using a 514.5 nm excitation wavelength from a mixed gas laser (model: Stabilite 2018 from Spectra Physics) and a triple grating Raman spectrometer (T64000 from Horiba Jobin Yvon) equipped with a liquid nitrogen cooled charge coupled device. We used the holographic grating with 1800 g/mm and a 20-micron exit slit with a spectrometer focal length of 1 m, which resulted in 0.7 cm−<sup>1</sup> optical resolution of the spectra.

High pressure X-ray diffraction and Raman experiments were carried out using a Mao Bell kind of diamond anvil cell, which was equipped with diamonds with a culet size of ~400 microns. The sample was loaded into a hole of 100 μm and drilled in a pre-indented 50 μm thick tungsten gasket. Methanol/ethanol in the ratio of 4:1, which is known to be quasi hydrostatic upto 10 GPa [39], was used as the pressure transmitting medium, and florescence peaks from ruby sphere were used to estimate the pressure inside the sample chamber of the diamond anvil cell. Care was taken to load a very tiny sample so that the environment around the sample remained quasi hydrostatic [40]. The limitations of the present Raman spectroscopic setup made it difficult to investigate Raman spectra below 150 cm<sup>−</sup>1.

#### **3. Results and Discussion**

#### *3.1. High Pressure Raman Scattering Study*

The high symmetry phase of the ABO3 perovskite compound had a cubic structure. In these compounds, the condensation of different phonon instabilities led to phase transitions. The A atoms (Na/Ba) were located at the cubic structure's eight corners, while the B atoms (Nb/Ti) were situated in the body's centre. The oxygen (six atoms) was arranged at the face-centred site and forms an octahedron. The Raman modes, also known as translational (Tr), librational (L), bending (B), and stretching (S) modes, were produced as a result of these atoms' vibrations. The roto-vibrational motion of the oxygen octahedra, the translational motion of the A ions against the BO6 octahedra, and the bending of the BO6 octahedra linkages gave rise to these modes. For systems of the ABO3 type, these are collectively referred to as external modes. The bending and stretching motion of the oxygen octahedra's O-B-O bonds caused the internal modes of the octahedra, known as the bending (B) and stretching (S) modes.

As said earlier, the NNBT05 ambient phase structure had orthorhombic symmetry (space group: *Pmc21*; tilt system: a−a−c+) with unit cell dimensions 2aP <sup>×</sup> <sup>√</sup>2aP <sup>×</sup> <sup>√</sup>2aP

and four numbers of formula units, i.e., a total of 20 atoms (see Figure 1a), and it was ferroelectric in nature. Here, aP is the lattice constant of the parent cubic phase. The group theoretical analysis gave 57 Raman active modes, represented by ΓRaman = 16A1 + 13A2 + 12B1 + 16B2. Due to the non-centrosymmetric structure, except for the A2 mode, these Raman active modes were simultaneously IR active. The number of observed modes was lower compared to the expected number of modes due to the peaks in the experimental Raman spectrum not being properly resolved, and due to accidental degeneracy. Generally, at elevated temperatures and pressures, the cation displacements from the centre of the octahedra (which results in nonzero polarization) may become vanishingly small and the structure may transform to the centrosymmetric orthorhombic paraelectric (*Pnma*) phase. This phase also had similar cell dimensions to ferroelectric (*Pmc21*) and in total 20 atoms in the primitive cell (Figure 1b). The number of Raman active modes in this phase (in another setting of *Pbnm*) was 24, 7Ag + 5B2g + 7B2g + 5B3g.

**Figure 1.** Crystal structure of (**a**) ferroelectric phase (space group: *Pmc21*) and (**b**) paraelectric phase (space group: *Pbnm*).

Figure 2 depicts the evolution of the high-pressure Raman spectra of NNBT05 at selected pressures and room temperature. For pressure > 1 GPa, distinct changes in the low wavenumber modes (150–300 cm<sup>−</sup>1) region of the Raman spectra were observed. The peaks containing the most prominent changes were denoted ν1, ν2, ν3, and ν<sup>4</sup> (bending mode: 150–300 cm<sup>−</sup>1) and showed drastic changes in their relative intensity in the pressure range 1 to 5.4 GPa. These observations at ~1 Gpa are most likely associated with octahedral reorientation in the parent ferroelectric phase, as has also been found earlier in pure NaNbO3. The change in Raman spectra at ~4.6 GPa may be a signature of structural phase transition from the ferroelectric phase (*Pmc21*) to the paraelectric phase (*Pbnm*). We noted that the paraelectric transition was observed in pure NaNbO3 at ~8 GPa. On further increasing the pressure above 5.4 GPa, the intensities of these peaks reduced and the stretching modes (500–750 cm<sup>−</sup>1) centred around 584 cm−<sup>1</sup> became merged. Figure 2b shows the fitted positions of the Raman bands. In literature [41], it was shown that even the 4:1 methanol/ethanol transmitting medium had Raman active modes assigned to the C-C stretch at ~450 and 880 cm−1. However, we would like to mention here that these modes were very weak and were not discernible in the presence of relatively stronger Raman scatterers. In the Raman spectra depicted in Figure 2a, we can clearly see that there was no interference from these modes.

The frequency evolution showed a blue-shift trend with increasing pressure. It is evident from Figure 2 that the frequencies of Nb-O stretching modes at ~500–700 cm−<sup>1</sup> increased with the pressure, implying a shortening of the Nb–O bond length. For pressure above 5.4 GPa, the Raman modes became very broad and weak, which could be due to an onset of disorder or due to a phase with a weak Raman response. In nanoporous silicon, it has been observed that though the Raman spectra showed very weak broad Raman modes and were termed a glassy phase, it actually was a crystalline phase which had very weak Raman activity, also because it was metallic [42]. At this point, we would also like to mention that even though the sample environment was quasi-hydrostatic, even up to ~10 Gpa, we cannot rule out the presence of some small pressure inhomogeneities, which could lead to the broadening of the Raman modes. However, on the release of pressure, the Raman modes of the parent phase were observed and so the broadening and reduction in the intensity of the Raman modes could be attributed to a reversible phase transition and not to the pressure inhomogeneities. To ascertain if indeed the high-pressure phase was disordered or it was a new crystalline phase, we carried out high pressure X-ray diffraction studies.

#### *3.2. High Pressure X-ray Diffraction Study*

High pressure X-ray diffraction patterns of NNBT05 were collected at 11 pressures up to 40 GPa in a pressure-increasing cycle. Figure 3 depicts the quality of the powder X-ray diffraction patterns at selected pressure levels. It is evident from this figure that on increasing the pressure, the diffraction peaks systematically shift towards higher angles, indicating the compression of the lattice parameters. To investigate the structural parameters with pressure, we carried out a detailed Rietveld analysis of the high-pressure X-ray diffraction data.

The detailed Rietveld refinement of the powder–diffraction data showed that the synchrotron powder diffraction pattern at 300 K could be indexed using the orthorhombic structure (space group *Pmc21*) in a low pressure range. Figure 3b shows the quality of Rietveld refinement of high-pressure powder diffraction data at 4.1 GPa. For a pressure range P > 4.1, the ferroelectric (*Pmc21*) structural model in the refinements gave highly correlated atomic positions.

**Figure 3.** (Colour online) (**a**) Evolution of the powder—X-ray diffraction patterns of NNBT05 at selected pressures. calculated (continuous red line) and difference (bottom blue line) profiles obtained from Rietveld refinement using an orthorhombic ferroelectric phase (space group: *Pmc21*) at 4.1 GPa (**b**), and paraelectric (space group: *Pbnm*) at 38.5 GPa (**c**).

The high pressure neutron diffraction study on pure sodium niobate clearly revealed a structural phase transition from the orthorhombic antiferroelectric to paraelectric phase (*Pbnm*) above 8 GPa. In view of this, we refined the high-pressure synchrotron X-ray diffraction data using the paraelectric orthorhombic phase *Pbnm*, which accounted for all the reflections and gave nearly similar goodness-of-fit parameters compared with the space group *Pmc21*. However, it should be noted that the total number of variable structural parameters for *Pmc21* was doubled (20) compared to the *Pbnm* space group (10 parameters). Even doubling the variable parameters did not help to improve the fitting. Hence, the latter phase has been preferred over the ferroelectric (*Pmc21*) phase. The Rietveld refinements using the orthorhombic (*Pbnm*) phase proceeded smoothly, revealing a monotonic decrease in lattice constants and cell volume with the pressure increasing up to 38.5 GPa. Structural parameters obtained after the Rietveld refinement for the compound are given in Table 1. The X-ray powder diffraction results indicate that the broadening and weakening of Raman modes beyond 5.4 GPa cannot be attributed to a disordered phase. It is possible that this high-pressure phase is weakly Raman active.

The pressure dependences of the lattice parameters and unit cell volume obtained after the Rietveld refinement of high-pressure synchrotron diffraction data are shown in Figure 4. For easy comparison with Figure 2, we used pseudocubic lattice parameters for the orthorhombic phase using the relations ap = Ao/ <sup>√</sup>2, bp = Bo/ <sup>√</sup>2, and cp = Co/2, where ap, bp, cp, Ao, Bo, and Co are lattice parameters corresponding to the equivalent pseudocubic and orthorhombic phases, respectively. It is clear from Figure 5 that on increasing the pressure, the lattice parameters monotonically decreased in the entire range of our measurements for NNBT05. However, unlike the 'b' and 'c' lattice constants, 'a' showed a sudden decrease around 23 GPa. This could not be attributed to the nonhydrostatic stresses caused due to the freezing of the pressure transmitting medium. If we look at the structure of sodium niobate along the b and c axes, we can see that there was only one octahedra, whereas along the 'a' axes there were two corner-shared octahedra. With pressure, there is a possibility of bending of the octahedral hinge, thus drastically reducing the lattice constant along the 'a' axes.


**Table 1.** Structural parameters obtained through Rietveld refinement of high-pressure X-ray diffraction of NNBT05 using orthorhombic ferroelectric (space group: *Pmc21*) and paraelectric (space group: *Pbnm*).

Unit cell volume (V) = 232.46(9) Å3 Rp = 3.68 Rwp = 6.61; Rexp = 5.45; χ2 = 1.22

Unit cell volume (V) = 205.41(9) Å3 Rp = 5.68 Rwp = 8.03; Rexp = 7.65; χ2 = 1.10

**Figure 4.** Evolution of lattice parameters and unit cell volume of NNBT05 as functions of pressure. For comparison, the pressure dependence of the volume of pure sodium niobate is also plotted. The solid lines are fitted to the experimental data using the Birch–Murnaghan equation of state. The error bar on the pressure is the size of the symbol.

**Figure 5.** Schematic representation of A-containing cuboctahedra (**left side**) and BO6 octahedra (**right side**).

From Figure 4, it can be seen that the lattice compression of the orthorhombic phase of NNBT05 was anisotropic. The compressibility of the 'c' parameter was smaller in comparison with those of the 'a' and 'b' parameters. Compressibility (at ambient conditions (1/l (dl/dP)) along a, b, and c was 0.0028, 0.0025 and 0.0020 GPa−1, respectively. This may be compared with the values for pure NaNbO3, which showed the compressibility

along a, b, and c as 0.0027, 0.0022, and 0.0018 GPa−1, respectively. The compressibility in the high-pressure paraelectric phase in NaNbO3 was very different from that in the low-pressure phase. However, in NNBT05, we did not find any significant discontinuous change in the compressibility as a function of pressure. Fitting the pressure versus volume data with a third-order Birch–Murnaghan equation of state gave a bulk modulus B (≈164.5 ± 1.0) GPa (with B = 4 fixed) for the orthorhombic phase at room temperature. This experimental bulk modulus in NNBT05 was fairly close to that in NaNbO3 (B ≈ 157.5 GPa) at room temperature.

It is evident from the foregoing discussion that the doping of BaTiO3 in the NaNbO3 matrix results in reduced compressibility, because on doping the free space in the Acontaining cuboctahedron reduces. It was found that in perovskites in which the A cation has a lower formal charge than the B cation, the AO12 sites are more compressible than the BO6 octahedra (see Figure 5). To reduce the unit cell volume, the tilts of the BO6 octahedra increased with increasing pressure [43]. Thus, the application of pressure drives this type of perovskite structure away from the phase transition boundary to highersymmetry structures.

On the basis of the Raman experimental results, we can conclude that NNBT05 showed changes in the Raman spectra from 1 to 1.9 GPa, 1.9 to 4.6 GPa, and above 5.4 GPa. Below 5.4 GPa, the spectra evolved continuous changes. It can be seen that the intensity of phonons located at 150–350 cm−<sup>1</sup> gradually changed with increasing pressure, up to 5.4 GPa. Then, these Raman peaks merged and became broad peaks as the pressure reached 9.9 GPa. Meanwhile, BO6 octahedra showed a growing distortion under higher pressure, as a result of the volume shrinkage. Hence, additional changes to the Raman spectra occurred when the pressure held at more than 10 GPa. As shown for pure NaNbO3, the observed anomalies in Raman spectra at a low pressure of ~2 GPa were mainly attributed to significant changes in Nb-O-Nb bond angles as a result of complex reorientations of NbO6 octahedra [19,29]. On the other hand, the lattice beyond 8 GPa belonged to a paraelectric orthorhombic phase with space group *Pbnm*. Increasing the pressure resulted in a shrinking of the polyhedral volume, and the cation was displaced towards the symmetric position to avoid over bonding. In addition to this, the ab-initio calculation of the enthalpy in the various phases of NaNbO3 [19] also supported the stabilization of the paraelectric (*Pbnm*) phase at high pressure. In view of this, we may conclude that on application of pressure, the ferroelectric phase of NNBT05 may transform to the paraelectric phase at a high pressure of ~5 GPa, which is lower than the transition pressure of ~8 GPa in pure NaNbO3. Since structural changes in the structure of the dodecahedra are very small due to changes in the average ionic radii in A and B positions, at 5% substitution, the lowering of pressure could be attributed to the chemical pressure introduced due to the substitution of BaTiO3.

#### **4. Conclusions**

In summary, we have investigated the effect of pressure on the crystal structure and structural phase transition behaviour in the engineered ferroelectric phase of sodium niobate using Raman scattering and powder X-ray diffraction techniques. At pressure > 1 GPa, noticeable changes in the Raman spectra were seen in the low wavenumber modes (150–300 cm−1). These observations are most likely associated with significant changes in Nb-O-Nb bond angles as result of complex reorientations of NbO6 octahedra in the parent ferroelectric phase, as in pure NaNbO3. Further changes were observed at ~4.6 GPa, which may be due to a transition to the paraelectric phase, which has a lower pressure than that of ~8 GPa in pure NaNbO3. For pressure levels greater than 5.4 GPa, the Raman spectrum became diffusive and may be associated with an increase in inhomogeneities in the crystal. We have determined the bulk modulus of NNBT05 to be 164.5 GPa, which is fairly close to that of 157.5 GPa observed in NaNbO3.

**Author Contributions:** S.K.M., R.M., N.G. and S.L.C. formulated the problem. N.G. carried out the high-pressure Raman and X-ray diffraction experiments. S.G. performed high-pressure Raman experiments. S.K.M. and N.G. analyzed experimental data. All the authors discussed the results and wrote the manuscript. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Data Availability Statement:** The data that support the findings of this study are available from the corresponding author upon reasonable request.

**Acknowledgments:** S.L.C. thanks the Indian National Science Academy for the financial support of the INSA Senior Scientist position.

**Conflicts of Interest:** The authors declare that they have no known competing financial interest or personal relationship that could have appeared to influence the work reported in this paper.

#### **References**


**Disclaimer/Publisher's Note:** The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

MDPI St. Alban-Anlage 66 4052 Basel Switzerland www.mdpi.com

*Crystals* Editorial Office E-mail: crystals@mdpi.com www.mdpi.com/journal/crystals

Disclaimer/Publisher's Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Academic Open Access Publishing

www.mdpi.com ISBN 978-3-0365-8565-9