Unveiling the Impact of 0–20 Gpa Hydrostatic Pressure on the Physical Properties of (Cs₂HfCl₆) Double Perovskite

Abstract

The current work determines the physical properties of Cs₂HfCl₆ photovoltaic compounds including their structural, electronic, and optical behavior, utilizing the DFT approach. The simulated Cs₂HfCl₆ lattice constants, cell volumes, and bond lengths decrease as the pressure increases from 0 to 20 GPa. The band structure analysis reveals that the calculated under-pressure (0–20 GPa) of Cs₂HfCl₆ is semiconducting with a flexible indirect bandgap (5.44, 2.76, 2.02, 1.45, and 0.99) eV. The electronic bandgap diminishes (0–20 GPa), transitioning the compound from the UV to the visible spectra. This alteration improves the transition from the VBM to the CBM, hence augmenting the optical effectiveness. Concurrently, the dielectric function escalates, enhancing the absorption and conductivity, and causing a red shift in the optical spectra, while diminishing the reflection in the visible spectra. Our findings on the hydraulic pressure (0–20 GPa) and the electrical and optical properties indicate that Cs₂HfCl₆ may be utilized in the development of next-generation solar cells, LEDs, UV sensors, and high-pressure optical instruments.

1. Introduction

Organic–inorganic halide perovskites (OIHPs) are well-known for their outstanding optoelectronic properties and are used in solar cells, light emitting diodes (LEDs), X-ray detectors, and sensors [1,2,3]. Despite their success, OIHPs face significant hurdles in commercialization due to their lead (Pb) content and intrinsic instability. These challenges have encouraged scientists to develop stable lead-free all-inorganic perovskites [1,2,3,4,5,6]. The substituting of Pb²⁺ with group IVA elements like tin (Sn²⁺) or germanium (Ge²⁺) are ideal. However, these ions tend to oxidize in air, compromising their stability and their device performance [7]. To replace Pb, another approach focuses on double perovskites, where monovalent (e.g., Ag⁺) and trivalent (e.g., Bi³⁺, Sb³⁺, In³⁺) cations combine in structures like A₂M⁺M³⁺X₆. The best example is Cs₂AgBiBr₆; apart from this compound, many of these double perovskites exhibit a secondary phase, such as Cs₂AgBil₆ and Cs₂AgSbI₆, which form CsI, Cs₃Bi₂I₉, and Cs₃Sb₂I₉ structures, respectively [8].

A promising alternative lies in vacancy-ordered double perovskites (A₂B⁴⁺X₆), which replace Pb²⁺ with tetravalent metals (e.g., Zr⁴⁺, Hf⁴⁺, Pt⁴⁺) paired with ordered vacancies [9,10,11,12,13,14]. These materials maintain structural stability while offering tunable optoelectronic properties. For instance, compounds like Cs₂ZrCl₆, Cs₂RuBr₆, and Cs₂PtI₆ have demonstrated potential in light-emitting diode (LEDs) and photocatalytic applications due to their efficient charge-carrier dynamics and adaptable band structures [13,14]. By addressing both environmental concerns and stability issues, vacancy-ordered perovskites represent a critical pathway toward sustainable high-performance alternatives to conventional OIHPs. The research community has recently focused on the Cs₂BX₆ family. The cubic phase of Cs₂SnI₆, with a Sn⁴⁺ oxidation state, makes it a promising for optoelectronic applications [15,16,17,18]. Due to Sn’s tetravalent nature, Cs₂SnI₆ is more air-stable than CsSnI₃ [16]. In addition, Cs₂TiX₆ and Cs₂SnX₆ PSCs have shown environmental promise [17]. Similarly, the wide-bandgap materials Cs₂ZrCl₆ and Cs₂HfCl₆ have shown promising results in luminescence [18,19,20]. These materials show potential for LEDs, photodetectors- and thermally induced delayed fluorescence due to their unique properties including high light yields, rapid decay kinetics, and tunable luminescent properties [21]. In recent years, Cs₂HfX₆ (X = Cl, Br, I) perovskite has been used as a scintillator in X-ray and gamma-radiation [22,23]. Jaroenjittichai et al. [24] theoretically investigated that Cs₂HfCl₆ is thermodynamically very stable and is not activated with H₂O and O₂. However, there has been little study on the structural stability and scintillation properties in an ambient environment. On the other hand, despite some experimental and theoretical study, the comprehensive properties for various applications have not been explored at high pressure. To understand the mechanism behind these materials, further study is required to deeply investigate these materials for optoelectronic applications [25].

Here, we used first-principles calculations based on density functional theory (DFT), which enables the study of material properties across sizes and structures, thus enabling the prediction and preemption of possible applications [26,27]. Lattice constant optimization with PBE and TB-mBJ accurately predicts the crystal bandgap, and hydraulic pressure affects the bandgap [28]. In this study, the structural, electrical, and optical properties of Cs₂HfCl₆ were investigated by varying the pressure (0–20) GPa. The aim of this work is to reduce the bandgap values with the applied hydraulic pressure, which enhances the optical performance of the compound making it suitable for various optoelectronics applications. We anticipate that our findings will serve as reference studies for preparing and applying the relevant materials in future research.

2. Results and Discussion

2.1. Structural Properties

The vacancy order (DP) Cs₂HfCl₆ has a cubic crystal structure with a space group of (225 Fm–3 m) as shown in Figure 1a,b [5]. The symmetrical structure comprises octahedra of Hf(Cl)₆, with intervening regions occupied by Cs atoms. The octahedral structures in the material are defined by chlorine atoms, each of which exhibits a twelve-fold coordination. Figure 1 demonstrates that the Cs and Hf atoms are surrounded by 12 and 6 halogen ions, respectively. In a cubic structure, Cs, Hf, and Cl atoms are positioned at the 8c, 4a, and 24e Wyckoff sites with fractional coordinates (1/4, 1/4, 1/4), (0, 0, 0), and (0.2, 0, 0), respectively. The DP Cs₂HfCl₆ was optimized to ascertain the lattice constants. Table 1 shows the optimized lattice parameter using hydraulic pressure (0–20) GPa for Cs₂HfCl₆. The computed Cs₂HfCl₆ data were compared with the previous literature as shown in Table 1.

The present work utilizes the volume optimization approach to reduce the energy of the unit cell as a function of its volume, conforming to Murnaghan’s equation [29] for Cs₂HfCl₆, as illustrated in Figure 1. As the volume diminishes, the energy of the crystal correspondingly falls until it attains a minimum. Consequently, the minimal energy is referred to as the ground state energy, whereas the optimal volume denotes the volume of the system corresponding to the minimum energy. Raising the unit cell’s volume farther will elevate energy, resulting in system instability. The lattice constant, $a_o$, the pressure derivative of the bulk modulus, B′, the bulk modulus, $B_o$, and the ground state energy, $E_o$, of these compounds are ascertained at the optimal volume and exhibited in Table 1. As the pressure of the computed material increases, it directly impacts the lattice parameter of the compounds. The lattice parameter would consistently decrease, as the hydraulic pressure increases from 0 to 20 GPa, as listed in Figure 2b. The structural stability was established by prior research, which confirms that these compounds are stable both experimentally and theoretically [24,30,31]. The optimized parameter of the current investigated Cs₂HfCl₆ is much closer to the previously calculated data displayed in Table 1.

Table 1. The optimized lattice parameter $a_o$ (Bohr), bulk, derivative of the bulk, volume, and ground state energy of Cs₂HfCl₆.

Compound	${\mathbf{a}}_{\mathbf{o}}$ (bohar)	${\mathbf{B}}_{\mathbf{o}}$	$\acute{{\mathbf{B}}_{\mathbf{o}}}$	(${\mathbf{V}}_{\mathbf{o}}$) (a.u3)	${\mathbf{E}}_{\mathbf{o}} \left(\mathbf{R}\mathbf{y}\right)$	Ref.
Cs2HfCl6	20.09	33.14	4.45	2029.56	−66,896.15	Present work
	10.74	32.57	5.00	2091.68	−66,896.16	[5]
	10.68	-	-	-	-	[31]
	10.42	-	-	-	-	[24]

Table 1. The optimized lattice parameter $a_o$ (Bohr), bulk, derivative of the bulk, volume, and ground state energy of Cs₂HfCl₆.

Compound	${\mathbf{a}}_{\mathbf{o}}$ (bohar)	${\mathbf{B}}_{\mathbf{o}}$	$\acute{{\mathbf{B}}_{\mathbf{o}}}$	(${\mathbf{V}}_{\mathbf{o}}$) (a.u3)	${\mathbf{E}}_{\mathbf{o}} \left(\mathbf{R}\mathbf{y}\right)$	Ref.
Cs2HfCl6	20.09	33.14	4.45	2029.56	−66,896.15	Present work
	10.74	32.57	5.00	2091.68	−66,896.16	[5]
	10.68	-	-	-	-	[31]
	10.42	-	-	-	-	[24]

2.2. Electronic Properties

2.2.1. Electronic Band Structure (EBS)

Electronic band structures (EBSs) provide a fundamental framework for understanding the electronic distribution and electron behavior in materials [32]. By analyzing the EBS, the intrinsic properties of a material, such as its conductivity and optical response, can be investigated by varying the pressure from 0 GPa to 20 GPa [33]. The TB-mBJ potential was used to calculate the band structure, and the results are illustrated in Figure 3a,b. The band structures were plotted along the high-symmetry directions in the Brillouin zone (BZ), specifically W → L → Γ → X → W → K, within an energy range of −4 to +12 eV. At 0 GPa pressure, the valence band (VB) maximum and conduction band (CB) minimum of Cs₂HfCl₆ were located at the Γ point, indicating a direct bandgap semiconductor. However, as the pressure increased up to 20 GPa, the material underwent a transition from a direct to an indirect bandgap semiconductor. This shift is attributed to changes in the electron–ion potential and lattice constants under pressure, which lead to a reduction in the energy bandgap. The relationship between the hydrostatic pressure and bandgap was directly proportional, as shown in Figure 2a. The bandgap decreased consistently from 5.44 eV to 0.99 eV, as the pressure increased from 0 to 20 GPa, demonstrating a trend between the bandgap pressure as summarized in Table 1, which lists the computed bandgap values using the mBJ approach. The results confirm the direct bandgap nature of Cs₂HfCl₆ at ambient pressure and its cubic structural stability under varying pressures. The progressive decrease in the bandgap with increasing pressure, as depicted in Figure 3a,b, highlights the material sensitivity to external stress and its potential for tunable electronic properties. As the pressure increases, the nature of Cs₂HfCl₆ transforms from an insulator to a semiconductor. The present work is associated with the reported theoretical investigation and experimental work both having 5.67 eV [5] and 6.1 eV [34] bandgap values, which place these compounds in an insulating nature. Due to the larger bandgap value, the present work utilized hydraulic pressure to reduce the bandgap value to make Cs₂HfCl₆ effective for solar cells and other technological applications.

2.2.2. Density of States (DOS)

To study the electronic structure and energy distribution, the density of states (DOS) and partial density of states (PDOS) are essential tools. These metrics provide detailed insights into the contributions of different atomic orbitals (s, p, d, and f states) to the overall electronic states, enabling a deeper analysis of material electronic properties. The DOS and PDOS of Cs₂HfCl₆ were investigated within specific energy ranges: −6 to 6 eV at 0 GPa and −12 to 7 eV at 20 GPa, as shown in Figure 4a,b. Similarly, the contributions of individual atomic states (s, p, d, and f) to the DOS and PDOS are presented in Figure 5a,b. At 0 GPa, the Fermi level (E_F), defined as 0 eV, aligns with the maximum of the VB. The electronic states near the E_F were predominantly influenced by the Cs-p, Hf-d, and Cl-p states, which play a significant role in shaping the compound electronic characteristics. However, as the pressure increased to 20 GPa, significant changes occurred in the DOS and PDOS trends. The Hf-f states became more prominent in the VB, while the contributions of the p and d states diminished. This shift highlights the sensitivity of the compound electronic structure to external pressure, with the f states gaining importance under high pressure conditions. For this sequence of modeling, a range of pressures was utilized, and it was found that the pressure affected the states/eV relationship.

2.3. Electron Density

The bonding characteristics were determined using the charge density and its spatial distribution, as illustrated in Figure 5a,b. Unlike metallic and covalent bonds, which involve charge sharing between atoms, ionic bonds exhibit minimal charge overlap between their components [35]. In covalent binding, the charge density is localized, while the metallic bonds show a delocalized charge distribution. Figure 5a,b illustrates the spatial charge distribution of the vacancy-based DP along the 2D diagonal plane. The charge distributions of Cs atom do not interact with the Cl atom, which exhibits a completely spherical charge distribution. The formation of an ionic link between the Cl atom and the Cs atoms was demonstrated at (0 GPa). The nature of the compound would change due to a hydraulic pressure of (20 GPa). The charge distribution of Cs and Hf ranges from perfectly spherical to deformed, resulting in covalent interaction with the Cl atom in a dumbbell configuration. The investigated under-pressure (0–20 GPa) charge density is displayed in Figure 5a,b.

2.4. Optical Properties

The interaction of a material with electromagnetic radiation can be studied through its optical properties, which are largely influenced by two key factors: the rate of electron transitions and the rate of electron–hole (e-h) recombination. Electron transitions can be categorized into two types: intraband and interband shifts. Interband shifts, which involve electrons moving between different energy bands, are particularly significant in semiconductors and play a crucial role in determining their optical behavior [36].

The optical response of a material is often described using the dielectric function, denoted as $\epsilon \left(\omega \right)={\epsilon }_1\left(\omega \right)+i{\epsilon }_2\left(\omega \right),$ where $\omega$ represents the frequency of the incident radiation [37]. This complex function consists of two components: the real part, ε₁(ω), and the imaginary part, ε₂(ω). The real part, ε₁(ω), provides insights into how light propagates and polarizes within the material, reflecting its dispersive properties. On the other hand, the imaginary part, ε₂(ω), is associated with the material’s ability to absorb electromagnetic radiation, highlighting its absorptive characteristics. Together, these components are critical for understanding the optical behavior of semiconductor materials, which are essential for applications in photonics, optoelectronics, and energy conversion technologies.

The investigated principal peaks of ${\epsilon }_1\left(\omega \right)$ are 4.91, 8.16, 8.77, 9.63, and 10.82, which correspond to the photon energy range at 5.7, 6.8, 6.8, 2.4, and 22 eV for (0, 5, 10, 15, and 20 GPa) pressure, as illustrated in Figure 6a. As the pressure increases, the curves exhibit an upward trend, transitioning from the UV to the visible spectrum or inferior energy. The decreasing tendency towards negative values of ${\epsilon }_1\left(\omega \right)$ elucidates the reflection of light’s incidence on the compound surface, resulting in the metallic behavior of the material. The spectrum range of the imaginary component, ${\epsilon }_2\left(\omega \right)$, exhibits an increasing trend in which the peaks values are 4.29, 10.40, 11.02, 11.37, and 11.83 at 6.0, 10.97, 10.92, 10.70, and 10.62 eV for (0, 5, 10, 15, and 20 GPa) pressures. Figure 6b clearly illustrates that the curves demonstrate a little shift in confinement inside the visible region due to the pressure, transitioning from higher energies to lower energies. The ${\epsilon }_2\left(\omega \right)$ exhibits decreasing trends at a higher frequency region for various pressures (5–20 GPa), as shown in Figure 6b, respectively. Figure 6a,b enumerates the sharp and main peak’s for both ${\epsilon }_1\left(\omega \right)$ and ${\epsilon }_2\left(\omega \right)$ accompanied by the corresponding pressure (0–20 GPa).

The refractive index $i\left(\omega \right)=n\left(\omega \right)+ik\left(\omega \right)$ consists of two components: the real, $n\left(\omega \right)$, and imaginary or extinction coefficient, $k\left(\omega \right)$. When EMR propagates across a substance, the $n\left(\omega \right)$ provides insights into refraction, whereas the $k\left(\omega \right)$ elucidates the absorption aspect. There is a direct proportional relationship between the density and the refractive index [38]. The equations delineate a correlation between the dielectric function and the $n\left(\omega \right)$ and $k\left(\omega \right)$ [39]. The $k\left(\omega \right)$ and $n\left(\omega \right)$ are mathematically expressed in Equations [40]. The spectral range of n(ω) includes the prominent peaks seen at (5.7, 6.89, 6.76, 6.51, and 2.05) eV, which are 2.26, 3.07, 3.14, 3.31, and 3.45 with the hydraulic pressure (0–20 GPa), respectively. Figure 6c illustrates the alteration in the curves with increasing pressure, transitioning from the UV to the visible spectrum. The significant peaks of n(ω) demonstrate fluctuations due to differing energy level transition velocities at various pressures (0–20 GPa). Conversely, k(ω) attains its minimum at 6.21 eV, with 1.31 for 0 GPa, while the refractive index trends were completely changed due to higher pressure (5–20 GPa) in which the peaks values were 2.87, 2.95, 2.99, and 3.02 at 13.56 eV, as illustrated in Figure 6d.

Optical conductivity σ(ω) is crucial for the electrical transition strength between filled and empty levels during optical transitions. A comprehensive examination was performed on the σ(ω) of the cubic structure vacancy based DP under pressure (0–20 GPa). The σ(ω) is expressed in the following manner. Figure 6e graphically depicts numerous peaks seen at pressures (0–20 GPa). The peaks values of σ (ω) at (9.3, 9.6, 9.4, 9.4, and 9.4) eV were 5065.86, 10,942.9, 12,296.8, 13,321.3, and 13,583.7 with pressure (0–20 GPa). The σ(ω) peaks at various pressure (0–20 GPa) are dominant at higher frequency (UV) limits, as listed in Figure 6e. The permanent peaks of the computed compounds at various pressures (0–20 GPa) are highly recommended for UV photon sensor, high pressure optics, and tunable optical devices [41,42].

The absorption coefficient α(ω) defines the depth to which the EMR may contact any substance. The α(ω) can be determined, utilizing the relation (1) [43]:

$$\alpha \left(\omega \right)={\displaystyle \frac{2\pi \omega }{c}}\sqrt{{\displaystyle \frac{-Re\left(\omega \right)+\left|\epsilon \right|}{2}}}.$$

(1)

The α(ω) evaluates the capacity of a photon to infiltrate a compound at a particular wavelength prior to absorption. Furthermore, it offers essential insights into the material’s efficacy in maximum solar energy retention, a vital characteristic in solar cell applications. The under-pressure (0–20 GPa) computed compound shows prominent peaks in the UV domain as shown in Figure 6f. The graphic illustration shows visual absorption at all pressures, but the strength and range decrease with the pressure. This decrease is due to lower energy gaps. The additional UV absorption region, where the absorption effectiveness improves with pressure (0–20 GPa), makes Cs₂HfCl₆ useful for optoelectronic device uses and solar cell enhancements.

Reflectivity describes how a compound surface responds to incoming photon energy. Figure 6g shows Cs₂HfCl₆ under the impact of pressure (0–20 GPa). The static reflectivity of the under-pressure Cs₂HfCl₆ shows an increasing trend of 0.06, 0.15, 0.18, 0.21, and 0.22 for 0, 5, 10, 15, and 20 GPa, as illustrated in Figure 6g. Discrete high points were noticed in the UV spectrum at pressure (0–20 GPa) impacts. The highest fluctuating peaks were observed at 8 eV to 14 eV. As the pressure rose, several notable reflectance spectra peaks switched to smaller wavelengths. This implies that electronic shifts or vibrational patterns demand more energy. Pressure often changes the gap between energy levels in a substance. The blue shift was observed in the high frequency limits (UV) region, as shown in Figure 6g. The frequency ranges associated with the plasma resonance are determined by the energy loss function L(ω). Plasma resonance occurs when the frequency of the incident photons aligns with that of the plasmon [44]. The L(ω) of the compound dictates the energy dissipated as an electron traverses it. The L(ω) function exhibits its most significant peaks within the near-UV spectrum at 0 GPa. Plasmon peaks are characterized by distinct maxima, denoted as ωp. As compared to the 5–20 GPa pressure, the 0 GPa pressure of Cs₂HfCl₆ shows higher losses, with a value of 1.25 at 6.5 eV, as illustrated in Figure 6h. As the pressure escalates from 0 to 20 GPa, these peaks consistently shift to elevated energies, as illustrated in Figure 6h. A link exists between the minimum value of the energy loss function and the maximum value of ${\epsilon }_2\left(\omega \right)$. This is an essential characteristic of semiconductors. The computed optical results of Cs₂HfCl₆ at various pressures (0–20 GPa) were compared with the vacancy-based double perovskite literature; the current work is effective for optoelectronic applications [45].

3. Experimental Section

Computational Method

The current work employed the FP-LAPW (full-potential linearized augmented plane wave) method in the WIEN2k calculation package for all analyses [46]. This method uses DFT [47], which reflects a system’s electron density as a function of its energy. Only atomic constants are used to solve the Schrödinger equation. In structural, electrical, and optical property computations, the local density approximation (LDA) [48] estimates the electron exchange energy and correlation. The FP-LAPW approach divides the crystal unit cell into muffin-tin (MT) spheres focused on atomic sites and interstitial regions. In the interstitial area, wave functions are constructed using plane waves and a cutoff parameter of RMT × Kmax = 7 for the precise energy convergence of eigenvalues. K max is the greatest k vector used for plane-wave expansion, while RMT is the muffin-tin sphere’s minimum radius. The k-point mesh in the first irreducible Brillouin zone (BZ) was optimized to 1000 points using GGA-PBEsol. The computations converged when the system energy reached 0.0001 Ry.

4. Conclusions

The present research employed the FP-LAPW approach using Wien2k code with the PBEsol-GGA and mBJ approximations to investigate the structure, electronic, and optical properties of the Cs₂HfCl₆. A wide range of pressure (0–20 GPa) was applied to investigate the vacancy-based double perovskite Cs₂HfCl₆. The escalating pressure (0–20 GPa) reduced the lattice constant, which altered the electronic band structure. At 0 GPa, Cs₂HfCl₆ exhibited a direct bandgap; however, as the pressure varied from 0 to 20 GPa, the compound transitioned from a direct to an indirect bandgap (L-X). The compound had a decreased bandgap as the pressure increased due to narrowing of the conduction band minimums (CBMs) and valence band maximums (VBMs). Additionally, the electron density and density of states (DOS) were investigated at pressures ranging from 0 to 20 GPa. When the pressure was 0 GPa, the electron density distribution showed mixed covalent bonds and ionic bonds; furthermore, with increasing pressure, the covalent bonding became stronger. The optical properties of Cs₂HfCl₆ were calculated at pressures ranging from 0 to 20 GPa and photon energies from 0 to 14 eV. The enhanced absorption and conductivity under increased pressure indicate that the compounds could be employed in a strained condition for high-frequency applications in optoelectronic devices.