Review of First Principles Simulations of STO/BTO, STO/PTO, and SZO/PZO (001) Heterostructures

Abstract

In this study, we review our first-principles simulations for STO/BTO, STO/PTO, and SZO/PZO (001) heterostructures. Specifically, we report ab initio B3PW calculations for STO/BTO, STO/PTO, and SZO/PZO (001) interfaces, considering non-stoichiometric heterostructures in the process. Our ab initio B3PW calculations demonstrate that charge redistribution in the (001) interface region only subtly affects electronic structures. However, changes in stoichiometry result in significant shifts in band edges. The computed band gaps for the STO/BTO, STO/PTO, and SZO/PZO (001) interfaces are primarily determined according to whether the topmost layer of the augmented (001) film has an AO or BO₂ termination. We predict an increase in the covalency of B-O bonds near the STO/BTO, STO/PTO, and SZO/PZO (001) heterostructures as compared to the BTO, PTO, and PZO bulk materials.

1. Introduction

The new and cutting-edge technologies that have appeared over the last decade allow for the growth of STO/BTO, STO/PTO, and SZO/PZO (001) superlattices as well as ultrathin films with atomic-level control. Various aspects of the (001) surface and interface phenomena, which take place in ABO₃ perovskites and their nanostructures, as well as the nature of their surfaces and interface states are very hot topics in modern theoretical solid-state physics [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27]. All our ab initio calculated PbTiO₃ (PTO)_, BaTiO₃ (BTO), SrTiO₃ (STO), PbZrO₃ (PZO), and SrZrO₃ (SZO) complex oxide materials belong to the class of so-called ABO₃ perovskites. All five of our first-principles-simulated ABO₃ complex oxide materials have the same primitive cubic lattice space group $P\mathrm{m}\overline{3}\mathrm{m}$ (group number 221). PTO complex oxide thin films have powerful spontaneous electrical polarization; accordingly, they are extensively used in numerous technologically important devices [28,29]. BTO is an important perovskite ceramic and possesses excellent dielectric, ferroelectric, and piezoelectric properties [30,31]. STO is a typical ABO₃ perovskite crystal with high photocatalytic activity and excellent electronic properties. It has numerous applications in electronics, sensors, catalysts, and energy storage devices [32,33,34]. PZO perovskite has been frequently mentioned in connection with energy storage performance [35,36,37]. SZO has huge potential for a wide variety of technologically important applications, such as high-temperature oxygen sensors, capacitors, and devices capable of violet blue light emission [38,39]. Considering this great technological importance, PTO, BTO, STO, PZO, and SZO complex oxide materials as well as their (001) surfaces were exhaustively inspected worldwide during the last quarter of a century [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62]. Nevertheless, their STO/BTO, STO/PTO, and SZO/PZO (001) interfaces are considerably less studied; therefore, additional studies and a systematic review of the existing contributions are required [63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91].

So far, Al-Aqtash et al. [63], at an ab initio level, studied the ferroelectric properties of BZO/PZO and SZO/PZO superlattices, which are related to our current work. Kim et al. [92] studied the remarkable non-linear dielectric properties of BTO/STO superlattices. Johnston et al. [93] performed ab initio computations of polarization and symmetry lowering in BTO/STO superlattices. Neaton and Rabe [94] developed a theory of polarization intensification in epitaxial BTO/STO superlattices. Dawber et al. [95] found the uncommon performance of ferroelectric polarization in PTO/STO superlattices. It is worth noting that in recent years, a great number of ab initio calculations were also performed for related ABO₃/CDO₃ interfaces. For example, Ohtomo and Hwang [96] investigated a model (001) interface amid two insulating perovskites LaAlO₃ (LAO) and SrTiO₃. Recently, Wang et al. [97] carried out ab initio calculations in order to examine the electronic and atomic structure of the LAO/STO (001) interface system under strain. Recently, Cancellieri et al. [98] experimentally detected a polaronic metal state at the LAO/STO (001) interface. Yan et al. [99] reported photoresponsive characteristics observed at the (001), (110), and (111) LAO/STO interfaces. Finally, Yan et al. [100] investigated the control of magnetism via doping at LAO/STO heterointerfaces.

It is well known that, in most cases, ABO₃ perovskites display various formative phase transitions. They start in the cubic paraelectric phase, which is stable at high temperature, occurring as a function of the lowered temperature [101,102,103,104,105,106]. For example, PTO perovskite is stable at room temperature (RT) in the tetragonal phase [107]. Namely, it has a tetragonal phase [107] by the space group P4mm. Then, at a high temperature equal to 766 K, PTO perovskite undergoes a single phase transition [107] to a cubic, high-symmetry structure $(\mathrm{space} \mathrm{group} P\mathrm{m}\overline{3}\mathrm{m}$). In contrast to PTO, BTO perovskite undergoes three phase transitions [108]. Namely, at 183 K temperature, BTO perovskite’s structure changes from a rhombohedral phase (space group R3m) to an orthorhombic phase (space group Amm2) [108]. At a temperature of 278 K, BTO perovskite’s structure changes to a tetragonal phase (P4mm) [108]. Finally, at a temperature of 403 K, BTO perovskite’s structure changes to a cubic, high-temperature structure $(P\mathrm{m}\overline{3}\mathrm{m} \mathrm{space} \mathrm{group}$) [108]. Opposite to PTO and BTO perovskites, which exhibit different phase transitions, STO is a paraelectric perovskite that at all observed temperatures maintains a centrosymmetric cubic structure with a space group equal to $(\mathrm{space} \mathrm{group} P\mathrm{m}\overline{3}\mathrm{m}$) [109,110]. PZO perovskite has three different phases. PZO perovskite has an orthorhombic antiferroelectric phase that is stable at temperatures up to 230 °C [111,112]. From 230 °C to 233 °C, PZO enters a rhombohedral ferroelectric phase [111,112]. PZO has a highly symmetrical cubic paraelectric phase when the temperature is above 233 °C [111,112]. Finally, there are three phase transitions in the SZO perovskite matrix [113,114]. They were detected through enthalphy measurements performed by Ligny et al. [113] and are as follows: orthorhombic phase (Pnma) ↔ orthorhombic phase (Cmcm) ↔ tetragonal phase (I4/mcm) ↔ cubic phase $(P\mathrm{m}\overline{3}\mathrm{m}$) [113]. These three phase transitions in the SZO perovskite matrix occur at 995 K, 1105 K, and 1140 K, respectively [113,114].

The objective of this review paper is to conduct supplementary ab initio calculations to complete a decade-long research effort focused on the ab initio analysis of STO/BTO, STO/PTO, and SZO/PZO (001) interfaces. Our review paper is organized as follows: Section 2 outlines the calculation details. Section 3, the core of this review paper, presents an overview of atomic relaxation, electronic charge distribution, and alterations in the band structure for our ab initio computed STO/BTO, STO/PTO, and SZO/PZO (001) heterostructures. The ab initio calculation results are meticulously examined, and common systematic trends among all three STO/BTO, STO/PTO, and SZO/PZO (001) interfaces are identified and organized in a manner easily accessible to a global audience of readers. Lastly, Section 4 summarizes the conclusions of our work.

2. Computational Methods and Materials

We performed ab initio computations for STO/BTO, STO/PTO, and SZO/PZO (001) interfaces by means of the hybrid exchange-correlation functional B3PW [115,116,117] as well as the CRYSTAL computer program package [118]. It is worth noting that the B3PW exchange-correlation functional is also known as (B3PW91), i.e., Becke’s 3 parameter exchange (B3) [115] combined with the non-local correlation (PW91) [116,117]. In order to determine the performance of different exchange-correlation functionals, we computed the Γ-Γ band gap for the BaZrO₃ (BZO) perovskite [29] as well as MgF₂ [119] and CaF₂ [120] (

Table 1. BaZrO₃, MgF₂, and CaF₂ bulk Γ-Γ band gaps [29,119,120] computed by employing different exchange correlation-functionals. Experimental bulk Γ-Γ band gaps [121,122,123] are listed for illustration.

Ab initio Method	BZO [29]	MgF2 [119]	CaF2 [120]
Experiment	5.3 [121]	13.0 [122]	12.1 [123]
B3LYP	4.79	9.42	10.85
B3PW	4.93	9.48	10.96
HF	12.96	19.65	20.77
PWGGA	3.24	6.94	8.51

). The experimentally detected BZO (5.3 eV) [121], MgF₂ (13.0 eV) [122], and CaF₂ (12.1 eV) [123] bulk Γ-Γ band gaps are presented in Table 1 for comparative purposes. As we can see from Table 1, the Hartree–Fock (HF) [124,125] method, for BZO, MgF₂, and CaF₂ materials computed ab initio, very highly overestimates the experimentally detected bulk Γ-Γ band gaps. At the same time, in our density functional theory simulations (DFT), we employed local density approximation (LDA) along with the Dirac–Slater [126] exchange and the Vosko Wilk Nussair correlation [127] energy functionals in addition to the package of GGA exchange-correlation functionals proposed by Perdew-Wang (PWGGA) [117]. It is worth noting that PWGGA is the keyword in the CRYSTAL17 manual [118]. It is commonly known as PW91 [117]. Unfortunately, in contrast to the HF method, the PWGGA very strongly underestimates the bulk Γ-Γ band gap at the BZO, MgF₂, and CaF₂ matrixes (Table 1). Based on Table 1, it follows that the hybrid exchange-correlation functionals, such as B3LYP [128] and B3PW [115,116,117], allows us to achieve as-good-as-possible coincidence between the ab initio simulated and the experimental bulk Γ-Γ band gaps for our three simulated materials (BZO, MgF₂, and CaF₂) Nevertheless, the agreement between the experimental Γ-Γ bulk band gaps in the BZO, MgF₂, and CaF₂ matrixes and our computed Γ-Γ bulk band gap is slightly better for B3PW than for B3LYP. Due to this tiny difference in bulk band gap computations, we chose the B3PW hybrid exchange-correlation functional for all our future STO/BTO, STO/PTO, and SZO/PZO (001) interface computations. The key reason for such perfect agreement between the experiment and theory is that the hybrid B3LYP and B3PW functionals incorporate a portion of the exact exchange energy density from the HF method (20%). At the same time, the remnant of the exchange-correlation segment is the mixture of several approaches, both exchange and correlation. Therefore, we carried out all our forthcoming STO/BTO, STO/PTO, and SZO/PZO (001) heterostructure ab initio simulations using the B3PW hybrid exchange-correlation functional (Table 1).

One of the key advantages of the CRYSTAL [118] computer program, which is of crucial importance for B3PW simulations of STO/BTO, STO/PTO, and SZO/PZO (001) heterostructures, is the possibility of using the two-dimensional (2D) isolated slab model. We calculated [2] the energy convergence as a function of 2D slab thickness. Namely, we calculated the change in energy when one STO layer was added to the slab, with the ultimate goal being the acquisition of meaningful surface energies [2,129,130]. When the energy difference is less than 1 mHa, convergence is achieved. For STO perovskite [2], the slabs, which contained seven layers, meet these criteria. In the performed B3PW simulations, reciprocal space integration [131] was executed by scanning the Brillouin zone with an 8 × 8 × 8-fold-extended Pack Monkhorst [131] mesh for the ABO₃ perovskite bulk and an 8 × 8 × 1-fold-expanded mesh for their (001) heterostructures. With the aim of attaining the highest possible accuracy of our B3PW simulations, we used sufficiently large tolerances, such as 7, 8, 7, 7, and 14, for the Coulomb overlap, Coulomb penetration, exchange overlap, first exchange pseudo-overlap, and second exchange pseudo-overlap, respectively [118]. We relaxed the STO/BTO, STO/PTO, and SZO/PZO (001) interface atom atomic coordinates via the (001) heterostructure’s total energy minimization. Regarding the (001) interface’s atomic coordinate relaxation, we used our developed computer code. This code combines the conjugated gradient optimization technique [132] with numerical calculation of derivatives [132]. To describe the chemical bonding in the STO/BTO, STO/PTO, and SZO/PZO (001) heterostructures as well as covalency effects, we used Mulliken [133,134] population analysis. For our ab initio computed Pb, Sr, Ba, Ti, and O (001) interface atoms, we employed the basis sets described in [135]. The inner core electrons for Zr, Sr, Pb, Ba, and Ti atoms were described using the small core Hay Wadt effective pseudopotentials [118,136]. We described the oxygen atoms using the all-electron basis set [118,135].

At this stage, we will explain our ab initio computation details for the BTO/STO (001) interface, which will account for the fact that we also used the same model for STO/PTO and SZO/PZO (001) heterostructures. The STO substrate, at room temperature, has a cubic, high-symmetry structure that corresponds to the space group $(P\mathrm{m}\overline{3}\mathrm{m}$). In order to save computational time and focus on the key physics, in our ab initio simulations, we modelled the STO and BTO perovskites at their cubic, high-symmetry phase. We computed the STO/BTO, STO/PTO, and SZO/PZO (001) heterostructures using the single (001)-slab model. In our first-principles simulations, a TiO₂-terminated STO (001) substrate was formed from 11 alternating SrO and TiO₂ layers (Figure 1) with an SrO layer located in the middle of the substrate. On both sides of the TiO₂-terminated 11-layer STO (001) substrate (Figure 1), we augmented from 1 to 10 BaO and TiO₂-terminated alternating BTO perovskite (001) layers.

In the performed ab initio B3PW simulations, we allowed all atoms of the STO/BTO, STO/PTO, and SZO/PZO (001) heterostructures to relax. Atomic relaxations are a result of the cubic symmetry restrictions enforced on the system, which only occur along the z axis (Figure 1). The lattice mismatch of approximately 2.5% between the BTO perovskite and STO bulk (Figure 1) lattice constants occurs during BTO’s epitaxial growth. Our ab initio B3PW-computed joint average equilibrium lattice constant, which is used in our forthcoming simulations of STO/BTO (001) heterostructures (Figure 1), is equal to 3.958 Å. We computed the joint lattice constant (3.958 Å) for the STO/BTO (001) heterostructure in the case of the thickest possible (001) interface, consisting of the 11-layer STO (001) substrate, and on both sides (Figure 1) of this substrate’s symmetrically augmented 10 BaTiO₃ (001) layers. In order to simulate the shift (Δz) of each layer for the STO/BTO (001) interface, we accounted for the displacement magnitude of the previous (Figure 1) atomic layer. Thus, the reference coordinate z for each monolayer N was calculated using the subsequent expression:

z_N^ref = ½ [z_N-1^Me + z_N-1^O),

(1)

where z_N-1^Me and z_N-1^O describe the z coordinates for a cation and an anion located in the previous atomic monolayer.

3. Ab Initio Calculation Results

3.1. Ab Initio B3PW-Computed ABO₃ Perovskite Bulk Properties

As the starting point of our ab initio simulations, we computed the bulk lattice constants of BTO, PTO, STO_, SZO, and PZO (

Table 2. Our first-principles B3PW-computed BTO, PTO, STO, SZO, and PZO bulk lattice constants. The experimentally detected respective ABO₃ perovskite bulk lattice values are listed for analytical purposes.

Material	BTO	PTO	STO	SZO	PZO
Computed	4.008	3.936	3.904	4.195	4.220
Experiment	4.004 [137]	3.97 [138]	3.898 [139]	4.154 [140]	4.1614 [141]

). It is worth noting that our ab initio B3PW-computed BTO (4.008 Å), PTO (3.936 Å), STO (3.904 Å), SZO (4.195 Å), and PZO (4.220 Å) bulk lattice constants (Table 2) are in fair agreement with the available experimentally measured bulk lattice constants [137,138,139,140,141]. The experimentally measured BTO, PTO, STO_, SZO, and PZO bulk lattice constants are equal to 4.004 Å [137], 3.97 Å [138], 3.898 Å [139], 4.154 Å [140], and 4.1614 Å [141], respectively (Table 2).

As evident in Figure 2 and

Table 3. Our ab initio B3PW-computed PTO, BaTiO₃, and SrTiO₃ perovskite bulk. Γ-Γ band gaps (in eV) and their structures obtained at RT. The experimentally detected Γ-Γ band gaps at RT (in eV) are listed for comparative purposes.

Perovskite	Structure at Room Temperature	B3PW Γ-Γ Gap	Exp. Γ-Γ Gap
PTO	Tetragonal phase	4.32 eV	3.4 eV [121]
BTO	Tetragonal ↔ orthorhombic phase at a temperature of 278 K	3.55 eV	3.38 eV (// c); 3.27 (⊥ c) [143]
STO	Cubic phase	3.96 eV	3.75 eV [144]

, our first-principles B3PW-simulated PTO perovskite bulk Γ-Γ band gap is equal to 4.32 eV [142]. The experimentally detected PTO perovskite bulk Γ-Γ band gap in the tetragonal phase is equal to 3.4 eV [121], which coincides fairly well with our ab initio B3PW simulation result of 4.32 eV [142] (Figure 2a and Table 3). The BTO perovskite Γ-Γ bulk band gap was experimentally detected at a temperature of 278 K [143], which coincides with the BTO tetragonal-to-orthorhombic phase transition. It is equal to 3.27 or 3.38 eV under different experimental conditions [143] (Table 3). Our first-principles B3PW-simulated BTO bulk Γ-Γ band gap is (3.55 eV) [142] (Table 3 and Figure 2b), which in outstanding agreement with the abovementioned experimental data [143]. Finally, our first-principles B3PW-simulated STO bulk Γ-Γ band gap (3.96 eV) is in excellent agreement with the experimentally available results for the STO bulk band gap at a Γ-point equal to 3.75 eV [144].

Our ab initio B3PW-simulated effective atomic charges and bond population values for the BTO, PTO, STO, SZO, and PZO perovskites are collected in

Table 4. Our ab initio B3PW-computed atomic effective charges Q (in e) and bond populations P (in e) in BTO, PTO, STO, SZO, and PZO perovskites.

Ion	Property	BTO	PTO	STO	SZO	PZO
A	Q	1.79	1.35	1.87	1.88	1.30
	P	−0.034	0.016	−0.010	−0.008	−0.020
O	Q	−1.39	−1.23	−1.41	−1.33	−1.12
	P	0.100	0.098	0.088	0.084	0.100
B	Q	2.36	2.34	2.35	2.13	2.07

. As we can see from Table 4, our ab initio B3PW-computed atomic charges for all five computed perovskites are considerably smaller than their perfect ionic values equal to +2e, +4e, and −2e for A, B, and O atoms, respectively (Table 4). According to our ab initio B3PW computations, the largest chemical bond populations are between the Ti and O atoms (0.100e) in the BTO perovskite and between the Zr and O atoms (0.100e) in the PZO perovskite (Table 4).

3.2. Ab Initio B3PW-Computed STO/BTO (001) Interfaces

We carried out ab initio B3PW computations of the STO/BTO (001) heterostructures [77,91] by means of the symmetrically terminated slab model (Figure 1). The TiO₂-terminated STO (001) substrate contained eleven atomic monolayers (Figure 1). As a next step, we simulated monolayer-by-monolayer epitaxial growth (Figure 1 and

Table 5. Our ab initio B3PW-computed net atomic charges Q_atom (in e), in-plane pure charges Q_plane (in e), and degrees of relative in-plane shift Δz (% of a₀) regarding the ideal bulk positions of metal atoms and band gaps δ (in eV) of BTO/STO (001) heterostructures [91]. Planes of STO (001) substrate are enumerated using Arabic numerals (Figure 1). In contrast, Roman numerals are employed to enumerate the planes of our deposited BTO (001) film. The numeral zero corresponds to the central plane of the slab (Figure 1).

Layer	Prop.	STO	BTO1	BTO2	BTO3	BTO4	BTO5	BTO6	BTO7	BTO8	BTO9	BTO10
X	QTi											2.30
	QO											−1.27
	Qplane											−0.23
	Δz											−3.92
IX	QBa										1.75	1.76
	QO										−1.48	−1.37
	Qplane										0.27	0.39
	Δz										−2.32	4.60
VIII	QTi									2.30	2.37	2.36
	QO									−1.26	−1.41	−1.36
	Qplane									−0.23	−0.44	−0.36
	Δz									−3.84	3.65	0.65
VII	QBa								1.75	1.76	1.80	1.79
	QO								−1.48	−1.37	−1.42	−1.40
	Qplane								0.26	0.39	0.38	0.39
	Δz								−2.21	4.62	0.68	2.12
VI	QTi							2.30	2.37	2.36	2.36	2.36
	QO							−1.26	−1.40	−1.36	−1.38	−1.37
	Qplane							−0.23	−0.43	−0.36	−0.40	−0.38
	Δz							−3.70	3.69	0.74	1.88	1.50
V	QBa						1.75	1.76	1.80	1.79	1.79	1.79
	QO						−1.49	−1.38	−1.42	−1.40	−1.41	−1.41
	Qplane						0.26	0.38	0.37	0.39	0.39	0.39
	Δz						−2.07	4.68	0.76	2.15	1.40	1.71
IV	QTi					2.30	2.37	2.36	2.36	2.36	2.36	2.36
	QO					−1.26	−1.40	−1.36	−1.38	−1.37	−1.38	−1.37
	Qplane					−0.22	−0.43	−0.36	−0.40	−0.38	−0.39	−0.39
	Δz					−3.55	3.73	0.83	1.92	1.57	1.57	1.63
III	QBa				1.75	1.76	1.80	1.79	1.79	1.79	1.79	1.79
	QO				−1.49	−1.39	−1.43	−1.41	−1.41	−1.41	−1.40	−1.41
	Qplane				0.25	0.38	0.37	0.38	0.39	0.38	0.39	0.38
	Δz				−1.84	4.77	0.89	2.27	1.48	1.79	1.53	1.68
II	QTi			2.29	2.37	2.35	2.36	2.36	2.36	2.36	2.36	2.36
	QO			−1.25	−1.40	−1.35	−1.38	−1.37	−1.38	−1.37	−1.38	−1.37
	Qplane			−0.21	−0.42	−0.35	−0.39	−0.37	−0.39	−0.38	−0.39	−0.39
	Δz			−3.20	3.83	1.03	2.04	1.72	1.69	1.75	1.60	1.71
I	QBa		1.75	1.76	1.80	1.79	1.79	1.79	1.79	1.79	1.79	1.79
	QO		−1.51	−1.40	−1.44	−1.43	−1.42	−1.42	−1.42	−1.42	−1.41	−1.42
	Qplane		0.23	0.36	0.35	0.36	0.37	0.37	0.38	0.37	0.38	0.37
	Δz		−1.54	4.87	1.02	2.35	1.53	1.84	1.55	1.70	1.51	1.64
5	QTi	2.29	2.37	2.35	2.36	2.36	2.36	2.36	2.36	2.36	2.36	2.36
	QO	−1.29	−1.41	−1.38	−1.40	−1.39	−1.40	−1.40	−1.40	−1.40	−1.40	−1.40
	Qplane	−0.30	−0.45	−0.40	−0.44	−0.42	−0.44	−0.43	−0.44	−0.44	−0.45	−0.44
	Δz	−5.95	1.96	−1.13	−0.11	−0.46	−0.47	−0.43	−0.56	−0.48	−0.61	−0.53
4	QSr	1.85	1.88	1.87	1.87	1.87	1.87	1.87	1.87	1.87	1.87	1.87
	QO	−1.37	−1.41	−1.39	−1.38	−1.38	−1.37	−1.38	−1.37	−1.37	−1.36	−1.37
	Qplane	0.48	0.47	0.48	0.50	0.49	0.50	0.50	0.51	0.50	0.51	0.50
	Δz	4.13	−1.32	0.60	−0.43	−0.10	−0.39	−0.26	−0.43	−0.33	−0.47	−0.37
3	QTi	2.35	2.36	2.36	2.36	2.36	2.36	2.36	2.36	2.36	2.36	2.36
	QO	−1.38	−1.42	−1.41	−1.42	−1.41	−1.42	−1.42	−1.43	−1.42	−1.43	−1.42
	Qplane	−0.42	−0.48	−0.45	−0.48	−0.47	−0.49	−0.48	−0.49	−0.48	−0.49	−0.49
	Δz	−0.96	0.27	−0.20	−0.25	−0.21	−0.38	−0.29	−0.44	−0.35	−0.48	−0.40
2	QSr	1.87	1.87	1.87	1.87	1.87	1.87	1.87	1.87	1.87	1.87	1.87
	QO	−1.42	−1.40	−1.41	−1.39	−1.40	−1.39	−1.39	−1.38	−1.39	−1.38	−1.39
	Qplane	0.45	0.47	0.46	0.48	0.47	0.49	0.48	0.49	0.48	0.49	0.49
	Δz	0.88	−0.35	0.12	−0.29	−0.15	−0.37	−0.26	−0.42	−0.32	−0.45	−0.37
1	QTi	2.35	2.36	2.36	2.36	2.36	2.36	2.36	2.36	2.36	2.36	2.36
	QO	−1.40	−1.42	−1.41	−1.42	−1.42	−1.42	−1.42	−1.43	−1.42	−1.43	−1.42
	Qplane	−0.44	−0.47	−0.46	−0.48	−0.47	−0.49	−0.48	−0.49	−0.48	−0.49	−0.49
	Δz	0.14	−0.09	−0.01	−0.27	−0.17	−0.36	−0.26	−0.42	−0.33	−0.46	−0.37
0	QSr	1.87	1.87	1.87	1.87	1.87	1.87	1.87	1.87	1.87	1.87	1.87
	QO	−1.42	−1.40	−1.41	−1.39	−1.40	−1.39	−1.39	−1.38	−1.39	−1.38	−1.39
	Qplane	0.45	0.47	0.46	0.48	0.47	0.49	0.48	0.49	0.48	0.49	0.49
	Δz	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
	δ	2.58	3.47	2.33	3.29	2.16	3.25	2.10	3.24	2.06	3.22	2.06

); namely, we added a pair of respective BTO (001) monolayers on both sides of the symmetrical 11-layer STO (001) substrate (Figure 1).

We added from one BTO (001) monolayer (Figure 1 and Table 5) up to ten BTO monolayers on both sides of the eleven-layer TiO₂-terminated STO (001) substrate. Thus, we designed ten STO/BTO (001) interfaces, which contained different thicknesses (from one to ten monolayers) of deposited BTO (001) nano-film (Figure 1 and Table 5). Considering the symmetry restrictions, in our ab initio computations, all atom atomic positions (Figure 1) were relaxed only along the z axis (Table 5). Atomic displacements (Δz) were computed regarding the averaged position (z) of the preceding atomic monolayer (Figure 1), as described in Equation (1). Our obtained ab initio calculation results are collected in Table 5. As we can see from Table 5, all our first-principles-computed displacements are less than 5% of a₀, i.e., the TiO₂-terminated STO (001) substrates become larger on average with respect to the bulk phase (Table 5). At the same time and in stark contrast, the BTO (001) thin film contracts in order to recompensate the lattice mismatch (Table 5).

As it is possible to see from Figure 3 and Table 5, our ab initio simulated TiO₂-terminated STO (001) substrate’s upper layer (x = 0 in Figure 3), which contains Ti and O atoms, very strongly shifts towards the perovskite bulk (−5.95% of a₀). According to our ab initio simulations (Table 5), upper-BTO (001)-layer atoms, which accumulate on the TiO₂-terminated eleven-layer STO (001) substrate, rather actively relax (Δz) inwards every time (x = 1–10 in Figure 3). It is very important to stress that the BTO’s augmented upper (001)-layer atoms’ inward shift numerical value Δz (Table 5) very actively depend on the number of augmented BTO (001) layers (odd or even), which describe upper augmented layer termination (BaO or TiO₂) (Figure 3 and Table 5). For example, for one augmented BTO (001) layer, which consists of Ba and O atoms, the atom shift size Δz is equal to (−1.54%) of a₀ (Table 5 and Figure 3). For the case of two augmented BTO (001) layers, where the top BTO (001) layer consists of Ti and O atoms, the respective top layer atom shift size Δz is equal to (−3.20% of a₀) (Figure 3 and Table 5). For three augmented BTO (001) layers, the top-layer Ba and O atom shift size Δz is (−1.84% of a₀) (Table 5 and Figure 3).

Table 5 presents the total Mulliken charges of the atomic planes of SrO, TiO₂, and BaO (001) and the Mulliken charges of all atoms inside these planes, namely, Q_Sr, Q_Ti, Q_O and Q_Ba, according to our computations for STO/BTO (001) heterostructures (Table 5). Since the Ti-O bond is partly covalent, according to our ab initio computations for STO/BTO (001) heterostructures, the Mulliken net charges for Ti and O atoms are different from their well-known ionic values of (+4e) and (−2e). Our ab initio computed Mulliken charges for atoms in the STO and BTO bulk matrixes are collected in Table 4. As we have demonstrated in our recent ab initio computations [6], the Ti-O bond population around the TiO₂-terminated BTO (0.126e) and STO (0.118e) (001) surfaces increases its covalency regarding the Ti-O bond population in the BTO (0.100e) and STO (0.088e) bulk matrixes. Notably, an increase in the Ti-O bond covalency, in comparison to the BTO bulk (0.126e vs. 0.100e), is observed around our ab initio computed STO/BTO (001) heterostructure. It is important to stress once more that related effects, including an increase in the chemical bond populations near the (001) and especially (011) surfaces with respect to the bulk value, were observed for all our ab initio calculated ABO₃ perovskites [2,3,4,5,6,14,19]. According to our ab initio computations collected in Table 5, the (001) surface planes of the TiO₂-terminated BTO (001) films deposited on the eleven-layer STO (001) substrate attract ~0.25e. At the same time, the BaO-terminated STO/BTO (001) heterostructures (Figure 1) become more positively charged in order to compensate for surface relaxation (Table 5). Thus, the (001) surface Ba-O chemical bond covalency is only very modestly increased. At the same time, our ab initio computed covalency of the (001) surface Ti-O chemical bond is considerably larger than in the bulk. This may compensate for the relatively small (001) surface relaxation magnitude of the BaO-terminated STO/BTO (001) heterostructures (Figure 1) regarding the TiO₂-terminated respective (001) interfaces (Table 5 and Figure 3). In both TiO₂-terminated stoichiometric and BaO-terminated non-stoichiometric STO/BTO (001) interfaces, charges on the substrate monolayer (Table 5) did not change substantially. Namely, for stoichiometric and non-stoichiometric (001) STO/BTO heterostructures, these are equal to ±0.03e for the TiO₂-terminated and BaO-terminated interfaces (Table 5). The strongest aberrations in the atomic charges for the STO/BTO (001) interfaces are in the topmost monolayer ~± 0.26e for the stoichiometric structures and at −0.23e in the non-stoichiometric structures (Table 5).

Difference electron charge density maps provide a further chance to visualize the electronic charge density reallocation in the STO/BTO (001) heterostructures (Figure 4) in comparison to the lonely STO and BTO (001) slab components. Charge density reallocation is described as the electronic density in the (001) interface minus the sum of electron densities in the sequentially isolated STO (001) substrate and BTO (001) thin film slabs (Figure 4) for both 3 and 4 UC thick STO/BTO (001) heterostructures. These difference electron charge density maps (Figure 2) display that the most significant distortions occur at the (001) interface and are caused by the re-compensation of the (001) surface effects of the (001) slabs.

Figure 5 and Figure 6 depict our ab initio B3PW-computed, DOS-expressed, layer-by-layer results for all orbitals of the Sr, Ba, O, and Ti atoms of the 3 and 4 UC thick STO/BTO (001) heterostructures (Figure 5 and Figure 6). Akin to the bulk ABO₃ perovskites, the valence band at the top (VB) consists of O 2p orbitals, but the bottom of the conduction band (CB) mostly consists of Ti 3d states (Figure 5 and Figure 6). The hybridization among the Ti and O atoms is also very well pronounced (Figure 5 and Figure 6).

Our ab initio B3PW-computed STO bulk direct band gap (3.96 eV) (Table 3) coincides very well with the respective experimental value of 3.75 eV [144]. In addition, our ab initio B3PW-computed BTO direct bulk band gap equal to 3.55 eV (Table 4) is in fair agreement with the available experimental data measured at a temperature of 278 K (3.27 eV and 3.38 eV) [143] (Table 4). For the BaO-terminated STO/BTO (001) heterostructure (Figure 5), the obtained surplus of electron density increases the occupied levels, thereby giving rise to the expanded band gap (Table 5). In contrast, the TiO₂-terminated STO/BTO (001) heterostructure (Figure 6) experiences an absence of electron density that decreases the occupied levels, thereby lowering the band gap of the stoichiometric STO/BTO (001) heterostructures (Figure 7).

3.3. Ab Initio B3PW-Computed STO/PTO (001) Interfaces

For the STO/PTO (001) interface substrate, we employed a symmetrical STO (001) slab (Figure 8) containing 11 alternating SrO and TiO₂ layers (TiO₂_SrO_TiO₂_SrO_TiO₂_SrO_TiO₂_SrO_TiO₂_SrO_TiO₂). So, in our first-principles simulations, the STO (001) substrate consisted of a supercell that contained 28 atoms (Figure 8). Thus, in our first-principles simulations, the STO (001) substrate is non-stoichiometric and consists of Sr₅Ti₆O₁₇ formula units per cell (Figure 8).

Next, we gradually and symmetrically augmented from one to ten alternating PbO and TiO₂ layers (PbO_TiO₂_PbO_TiO₂_PbO_TiO₂_PbO_TiO₂_PbO_TiO₂) on both sides of the SrTiO₃ (001) substrate. So, the STO/PTO (001) interface in our ab initio computations contained an eleven-layer STO (001) substrate, which consisted of 28 atoms, and 10 PTO (001) layers, which consisted of 25 atoms that were symmetrically augmented on both sides on the (001) substrate (Figure 8). Thus, the largest atom number in the ab initio STO/PTO (001) interface computations was 78 atoms, with a unit cell formula equal to Sr₅Pb₁₀Ti₁₆O₄₇. It is worth noting that during the epitaxial PTO (001) film growth (Figure 8), lattice mismatch between the PTO and STO bulk lattice constants arose (Table 2). Therefore, in the first-principles B3PW simulations, we optimized the joint STO/PTO (001) interface lattice constant for the thickest computed STO/PTO (001) interface, which consisted of 31 layers and 78 atoms, in order to minimize the strain effect [72,87]. Our ab initio B3PW-computed STO/PTO (001) interface joint lattice constant is equal to 3.91 Å.

In our first-principles B3PW simulations, the atomic displacements Δz along the z axis were computed in association with the averaged z position of the previous layer, as reported in Equation (1). As evident in Figure 9, for one augmented PTO (001) layer (x = 1), consisting of PbO atoms, on top of the eleven-layer STO (001) substrate, the top layer PbO atom shift size Δz is equal to −6.01% of a₀ (Figure 9). It is worth noting that in Figure 9, (x = 0) corresponds to the shift magnitude (Δz) of the STO (001) substrate’s top-layer TiO₂ atoms (Figure 9). For two augmented PTO (001) layers, the upper layer contains TiO₂ atoms (x = 2) whose shift size (Δz) is (−7.76% of a₀) (Figure 9). For three augmented PTO (001) layers (x = 3), the top layer again contains PbO atoms, and their shift size Δz is (−6.97% of a₀) (Figure 9).

Due to the significant covalency effects in the Ti-O chemical bonds of PTO perovskite (Table 4), our ab initio B3PW-computed Mulliken charges for Ti (+2.34e) and O(−1.23e) atoms are rather different from the formal ionic charges of Ti and O (equal to +4e and −2e, respectively). As a result of the performed ab initio B3PW computations, an increase in Ti-O chemical bond covalency (0.158e) with respect to the PTO bulk (0.098e) was noted near the STO/PTO (001) interface. The augmented PTO (001) film’s upper-layer TiO₂ planes, separately from their numbers of augmented layers, which are equal to two, four, six, eight, or ten, in all cases attracts only 0.08e. In contrast, the PbO-terminated STO/PTO (001) interfaces become more positive, namely, +0.13e for one augmented PTO layer, +0.14e for three layers, +0.14e (five layers), +0.15e (seven layers), and 0.15e (nine layers).

Figure 10 describes the electronic charge density redistribution at the STO/PTO (001) interfaces with respect to the pure STO and PTO (001) slabs (Figure 10). As is known, charge density reallocation is defined as the electron density (001) on the heterostructure minus the sum of electron densities in the separately isolated STO (001) substrate and PTO (001) film slabs, as illustrated in Figure 10 for both 3- and 4 UC thick STO/PTO (001) heterostructures. Figure 10 shows that the most important distortions take place at the (001) heterostructures and are created by the compensation for the surface effects of the slabs. The difference electron charge density maps (Figure 10) also show that the electronic structure of the STO (001) substrate of the non-stoichiometric STO/PTO (001) interfaces is distorted to a degree equal to that of the stoichiometric interfaces (Figure 10).

Our ab initio B3PW-computed DOS values, which were considered layer-by-layer on all orbitals of Sr, Pb, O, and Ti atoms of the three- and four-unit-cell (UC)-thick STO/PTO (001) interfaces, are depicted in Figure 11 and Figure 12. As with all bulk ABO3 perovskites, the top of the valence band for the STO/PTO (001) interface consists mainly of O2p orbitals, whereas the bottom of the CB for the STO/PTO (001) interfaces is mostly composed of Ti3d atomic orbitals.

The first-principles B3PW-simulated band gap for the PbO-terminated PTO (001) film augmented on the STO (001) substrate, which contains one layer (x = 1), is 3.45 eV (Figure 13 and Figure 14). The PbO-terminated augmented PTO (001) film band gaps for systems that contain three, five, seven, and nine layers are 3.25 eV, 3.08 eV, 2.99 eV, and 2.94 eV (Figure 13 and Figure 14). The first-principles-simulated band gaps for the TiO₂-terminated PTO (001) film, augmented on the STO (001) substrate (Figure 13 and Figure 14), which contains two, four, six, eight, and ten layers, are 3.18 eV, 3.17 eV, 3.05 eV, 2.99 eV, and 2.93 eV.

3.4. Ab Initio B3PW-Computed SZO/PZO (001) Interfaces

Additionally, our ab initio computations for the SZO/PZO (001) heterostructures, akin to those performed for the STO/BTO and STO/PTO (001) heterostructures, were carried out using the symmetrically terminated slab model [72,77,83]. The SrZrO₃ (001) substrate consisted of 11 monolayers, and it was terminated with ZrO₂ monolayers on both sides. Then, monolayer-by-monolayer epitaxial growth was modelled. Namely, we symmetrically added a pair of PZO (001) monolayers symmetrically to both sides of a SZO (001) eleven-layer substrate slab until the deposited PZO (001) film reached the thickness of ten monolayers on both sides. Thus, we constructed 10 (001) interfaces consisting of deposited PZO (001) nano-films with different thicknesses.

As a next step, in order to determine the level of electronic charge density reallocation, we analyzed the behavior of the electronic charge density on the (001) interfaces. We compared our results to the isolated BTO, PbZO, and STO and SZO (001) slabs (Figure 15). The difference electron charge density plots (Figure 15) demonstrate that the major distortions occur at the (001) heterostructure. They occur due to the compensation for the first-principles-computed surface effects of the slabs (Figure 15). They demonstrate that the electronic structure of the (001) substrate of the non-stoichiometric (001) interface is equally as gnarled as the stoichiometric ones. The situation for the (001) thin films is the exact opposite (Figure 15). This finding corresponds well with our forecasted atomic structures.

We carefully analyzed the DOS projected layer by layer onto all orbitals of the Sr, Ba, Pb, Ti, O, and Zr atoms of the STO/BTO and SZO/PZO (001) heterostructures. As in the case of the bulk ABO3 perovskites, the top of the VB was formed from O 2p orbitals. In contrast, the bottom of the CB was mainly formed from Ti 3d and Zr 4d states. In addition, Ti-O and Zr-O hybridization are clearly pronounced. In the case of the BaO-terminated STO/BTO (001) and ZrO₂-terminated SZO/PZO (001) heterostructures, an excess of electron density shifts was gained as the number of occupied levels increased, giving rise to the expanded band gap (Figure 16). In stark contrast, the PbO and TiO₂-terminated (001) heterostructures experienced a lack of electron density. This shifted the number of occupied levels down and thereby reduced the band gaps of the SZO/PZO and STO/BTO (001) interfaces (Figure 16).

4. Conclusions

In this study, we reviewed our first-principles simulations for SrTiO₃/BaTiO₃, SrTiO₃/PbTiO₃, and SrZrO₃/PbZrO₃ (001) interfaces. Namely, we presented ab initio B3PW computations of SrTiO₃/BaTiO₃, SrTiO₃/PbTiO₃, and SrZrO₃/PbZrO₃ (001) interfaces, for which non-stoichiometric heterostructures were taken into account. In our ab initio B3PW computations, we demonstrated that charge redistribution at the (001) interface region weakly affects electronic structure, while the change in stoichiometry induces significant shifts in band edges. The optical band gaps of the SrTiO₃/BaTiO₃, SrTiO₃/PbTiO₃, and SrZrO₃/PbZrO₃ (001) interfaces depend mostly on the AO- or BO2-termination of the upper layer of augmented (001) film. We predicted an enhancement of the B-O chemical bond covalency near the SrTiO₃/BaTiO₃, SrTiO₃/PbTiO₃, and SrZrO₃/PbZrO₃ (001) interfaces as compared to the BaTiO₃, PbTiO₃, and PbZrO₃ bulk. We found that the surface covalency effects in non-stoichiometric (001) films are less pronounced than those in stoichiometric ones.