Review of Systematic Tendencies in (001), (011) and (111) Surfaces Using B3PW as Well as B3LYP Computations of BaTiO₃, CaTiO₃, PbTiO₃, SrTiO₃, BaZrO₃, CaZrO₃, PbZrO₃ and SrZrO₃ Perovskites

Abstract

We performed B3PW and B3LYP computations for BaTiO₃ (BTO), CaTiO₃ (CTO), PbTiO₃ (PTO), SrTiO₃ (STO), BaZrO₃ (BZO), CaZrO₃ (CZO), PbZrO₃ (PZO) and SrZrO₃ (SZO) perovskite neutral (001) along with polar (011) as well as (111) surfaces. For the neutral AO- as well as BO₂-terminated (001) surfaces, in most cases, all upper-layer atoms relax inwards, although the second-layer atoms shift outwards. On the (001) BO₂-terminated surface, the second-layer metal atoms, as a rule, exhibit larger atomic relaxations than the second-layer O atoms. For most ABO₃ perovskites, the (001) surface rumpling s is bigger for the AO- than BO₂-terminated surfaces. In contrast, the surface energies, for both (001) terminations, are practically identical. Conversely, different (011) surface terminations exhibit quite different surface energies for the O-terminated, A-terminated and BO-terminated surfaces. Our computed ABO₃ perovskite (111) surface energies are always significantly larger than the neutral (001) as well as polar (011) surface energies. Our computed ABO₃ perovskite bulk B-O chemical bond covalency increases near their neutral (001) and especially polar (011) surfaces.

1. Introduction

Surface as well as interface phenomena, taking place in the ABO₃ perovskites, including the essence of their (001), (011) and (111) surface and interface electronic properties, are very real problems in present-day 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]. BaTiO₃, CaTiO₃, PbTiO₃, SrTiO₃, BaZrO₃, CaZrO₃, PbZrO₃ and SrZrO₃ perovskites are the members of the well-known ABO₃ (A = Ba, Ca, Pb, Sr and B = Ti or Zr) perovskite family [27,28,29]. They all have a colossal amount of technologically essential applications [30,31,32,33]. For example, BaTiO₃ is an important ABO₃ perovskite ceramic material [34]. It has outstanding dielectric as well as ferroelectric and piezoelectric properties [34]. CaTiO₃ is extensively employed in electronic ceramic materials [35]. Cubic CTO is also used as a keystone element of Synroc [35]. PbTiO₃ is one of the worldwide most extensively used piezoelectric and ferroelectric materials for technologically important industrial applications [36,37,38]. SrTiO₃ is a perovskite material with a wide-ranging spectrum of functional properties as well as physical phenomena [39]. STO is an adaptable substrate for complex oxide electronics engineering [40]. STO possesses excellent superconducting properties [41] as well as impurity- and vacancy-based magnetism [42]. BaZrO₃ perovskite has several industrially important applications in many quite different technology areas [43]. BZO is extensively used, for example, in solid-oxide fuel cells [44] as well as in wireless systems for communications [45]. CaZrO₃ is used as an ionic conductor to manufacture solid electrodes for applications in various fuel cells [46]. Moreover, CZO is used also as a key element for different sensor types [47,48]. PbZrO₃ is a very fascinating perovskite, since it is the end point of the Pb(Zr,Ti)O₃ alloy system, which is very interesting for numerous important applications in industry [49]. Finally, SrZrO₃ has a lot of technological applications [50,51,52,53]. For example, in the violet–blue light emission, laser host materials as well as capacitors and oxygen sensors [50,51,52,53]. For that reason, it is clear, that during the last twenty-five years, BTO, CTO, PTO, STO, BZO, CZO, PZO and SZO perovskite (001) surfaces have been comprehensively explored theoretically as well as experimentally around the globe [1,2,3,6,7,11,13,14,16,18,21,23,24,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81]. It is worth noting that it is relatively easy to compute the neutral ABO₃ perovskite (001) surfaces, since they consist of alternating neutral AO and BO₂ slabs [82,83,84,85]. In contrast, it is very difficult to compute the polar ABO₃ perovskite (011) and (111) surfaces, since they consist of charged planes of ABO and O₂ as well as AO₃ and B, respectively. This is main reason why the ABO₃ perovskite charged and polar (011) [86,87,88,89,90,91,92,93,94,95] and (111) [96,97,98,99,100,101,102,103,104,105,106,107] surfaces are considerably less studied than their neutral and relatively simple (001) surfaces [1,2,3,6,7,11,13,14,16,18,21,23,24,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85].

For example, the first B3PW calculations, dealing with polar and charged BaTiO₃ and PbTiO₃ (011) surface structures were performed by Eglitis and Vanderbilt in 2007 [1]. Two years later, Zhang et al. [86,87], at ab initio level, computed the electronic and structural characteristics of five different terminations of cubic PTO (110) polar surface [86,87]. The first ab initio computations for the polar SrTiO₃ (011) surface were carried out by Bottin et al. [88]. They computed the atomic as well as electronic structure of a few (1 × 1) SrTiO₃ (011) surface terminations [88]. The year after that, Heifets et al. [89] carried out first-principles Hartree–Fock computations for four terminations (Sr, TiO as well as two different O terminations) of the polar STO (011) surface [89]. As the next, Eglitis and Vanderbilt [2] performed hybrid DFT computations for three different terminations (TiO, Sr and O) of the polar and charged STO (011) surface [2]. Two years later, Enterkin et al. [90] described the results for the 3 × 1 extended STO (011) surface structure derived experimentally via transmission electron diffraction [90]. Experimental results, dealing with polar STO (011) surfaces, were also confirmed theoretically using modern ab initio DFT computations as well as scanning tunneling microscopy images [90]. Finally, five years ago, Fleischer et al. [91] experimentally investigated the STO (011) surface using reflectance anisotropy spectroscopy (RAS). World-first ab initio calculations for polar CTO (011) surfaces were performed by Zhang et al. [92]. They constructed four different CTO polar (011) surface terminations and computed the cleavage as well as (011) surface energies [92]. Zhang et al. [92] also computed the CTO (011) surface grand potential as well as the (011) surface electronic and atomic structure [92]. One year later, Eglitis and Vanderbilt, using a B3PW hybrid exchange–correlation functional, investigated three different terminations (TiO, Ca and O) of polar CTO (011) surfaces [3]. They [3] computed the polar CTO (011) surface atomic relaxations, energetics as well as chemical bonding properties for three different (011) surface terminations [3].

Two world-first ab initio simulations for the polar BaZrO₃ (011) surfaces were performed independently by Heifets et al. [93] and by Eglitis [82] in 2007. Heifets et al. [93] studied the charge redistribution, atomic and electronic structure of several different terminations of BZO (011) surfaces [93]. According to the B3PW computations performed by Eglitis [82], three different terminations of the BZO (011) surface exhibit quite different surface energies. They always (for all three terminations) are considerably larger [82] than for the neutral BZO (001) surfaces. Eglitis and Rohlfing [94] computed the SZO and PZO neutral (001) and polar (011) surface rumplings, relaxations, energetics, charge redistributions as well as the Γ-Γ band gaps [94]. Four years later, Chen et al. [95] investigated the electronic properties and stabilities of SZO (110) (1 × 1) five different polar terminations [95]. Finally, the only existing B3LYP calculations, dealing with polar CZO (110) surfaces, were recently performed by Eglitis and co-workers [10].

A quarter of century ago, Hagendorf et al. [96], experimentally investigated the polar BTO (111) surface using scanning tunneling microscopy (STM), X-ray photoelectron spectroscopy (XPS) and low-energy electron diffraction (LEED) methods [96]. Recently, Chun et al. [97], explored the BTO surface (111) termination, using the theoretical ab initio DFT calculations and experimental XPS analysis [97]. First-in-the-world ab initio linearized augmented-plane-wave method (LAPW) calculations for periodic (111) BTO slabs were performed by Cohen [98]. Cohen found [98] that the polar (111) BTO slab is considerably less stable than the BTO neutral (001) slab. In 2015, Eglitis [5,60,99] performed very comprehensive B3LYP calculations for BTO, PTO, CTO, STO, PZO and SZO (111) surfaces. Eglitis found [5] that the polar BTO, PTO, CTO, STO, PZO and SZO (111) surfaces are considerably less stable than the respective neutral (001) and even polar (011) surfaces [5,60,99]. Twenty-five years ago, Haruyama et al. [100] experimentally studied the polar STO (111) surface by means of photoemission spectroscopy [100]. In 1999, Pojani et al. [101], by means of simple semi-empirical HF method, computed the polar STO (111) and (110) surfaces [101]. Recently, Torrelles et al. [102] experimentally detected the surface structure of Ti-terminated STO (111) single crystals [102]. Finally, Marks et al. [103] described the reconstructions of the polar STO (111) surface by means of experimental-transmission electron diffraction, scanning tunneling microscopy as well as theoretical first-principles DFT calculations [103]. Pang et al. [104] performed very comprehensive first-principles computations for four different (1 × 1) polar terminations of PTO (111) surfaces. The electronic and structural properties as well as stabilities of four different polar (1 × 1) PTO (111) terminations were calculated at ab initio level [104]. Liu et al. [105] constructed the stoichiometric as well as nonstoichiometric terminations for polar CTO (111) surfaces [105]. They computed the polar CTO (111) surface electronic structure, grand potential as well as the relevant surface and cleavage energies [105]. Kim et al. [106] explored twenty-two low-indexed BZO (001), (011) as well as (111) surface terminations in order to investigate the Gibbs free energy for their surfaces [106]. Finally, Eglitis [107] performed B3LYP computations for BaO₃- and Zr-terminated polar BZO (111) surface relaxations and energetics.

The objective of our review paper was to carry out necessary additional ab initio computations in order to finalize our more-than-20-year-long research work, devoted to ABO₃ perovskite surfaces. Namely, we report in this place our B3PW and B3LYP computation results for BTO, CTO, PTO, STO, BZO, CZO, PZO, SZO neutral (001) as well as polar (011) and (111) surfaces. We meticulously analyzed B3PW and B3LYP computation results and detected systematic tendencies, typical for all eight of our ab initio computed ABO₃ perovskite surfaces. Finally, we systematized these common systematic trends in a system, effortlessly approachable worldwide for a comprehensive audience of scientists.

2. Computational Details and Surface Models

We performed very comprehensive hybrid density functional theory (DFT) calculations for eight different ABO₃ perovskite (001), (011) and (111) surfaces by means of the CRYSTAL [108] computer program. The CRYSTAL computer program [108] utilizes Gaussian-type well-localized basis sets (BSs). The BSs for BTO, PTO and STO perovskites were evolved by Piskunov et al. [109]. Almost all computations in this review were executed by means of the B3PW [110,111] or B3LYP [112] hybrid exchange–correlation functionals. It is worth noting that the hybrid exchange–correlation functionals, like B3PW or B3LYP, enable us to reach an outstanding agreement with the experiment [10,75] for the Γ-Γ band gaps of different ABO₃ perovskites. We executed the reciprocal-space integration for the ABO₃ perovskite bulk and their surfaces by examining the Brillouin zone, utilizing the 8 × 8 × 8 and 8 × 8 × 1 times, respectively, enlarged Pack Monkhorst grid [113]. The trump card of the CRYSTAL computer program [108] is its ability to compute isolated, two-dimensional slabs, without any unnatural periodicity in the direction z, perpendicular to the slab surface. We performed B3PW and B3LYP computations for all eight ABO₃-type perovskites and their surfaces in high symmetry, cubic structure $(\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{e} \mathrm{g}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{p} P\mathrm{m}\overline{3}\mathrm{m}$) [114,115,116].

With the goal of simulating the neutral BO₂-terminated (001) surfaces of ABO₃-type perovskites [114], we selected symmetrical slabs. These slabs, in our computations, consisted of nine neutral and alternating BO₂ as well as AO layers (Figure 1 and Figure 2). The first slab was terminated by the BO₂ planes and was composed of a supercell which accommodated 23 atoms (Figure 1). The second slab was terminated by the AO planes and was composed of a supercell which accommodated 22 atoms (Figure 2). Both these (001) surface slabs are nonstoichiometric. They have unit cell formulas equal to A₄B₅O₁₄ and A₅B₄O₁₃, respectively (Figure 1 and Figure 2).

Just opposite to the (001) cleavage (Figure 1 and Figure 2) of ABO₃, which produce nonpolar AO and BO₂ terminations, direct cleavage of ABO₃-type perovskites, in order to generate (011) surfaces, leads to the production of polar O₂ as well as ABO surfaces (Figure 3). The ABO₃ crystal (Figure 3), alongside the [011] crystalic direction, is composed of cyclic planes of O₂ and ABO units (Figure 3). These two alternating O₂ and ABO planes (Figure 3) have ionic charges of −4e and +4e, assuming following constituents as O²⁻, B⁴⁺ and A²⁺. Therefore, modeling of the ABO₃ (011) surfaces (Figure 3) precisely, as they are obtained from the pristine crystal cleavage, leads to the following two problematic situations: An infinite macroscopic dipole moment, which is perpendicular to the ABO₃ perovskite (011) surface (Figure 4), when the slab is terminated by different O₂ as well as ABO planes (Figure 4) (stoichiometric slab). Infinite charge, in case when the slab is terminated by the same planes (O₂-O₂) (Figure 5) or ABO-ABO (Figure 6) (nonstoichiometric slab). Such ABO₃ perovskite (011) surface terminations (Figure 5 and Figure 6) make the (011) surface unstable [117,118].

This was the key reason why in our ABO₃-type perovskite (011) surface computations, with the aim of obtaining the neutral (011) slab, we deleted some atoms (Figure 7, Figure 8 and Figure 9). Namely, we deleted the O atom (Figure 9) from the upper as well lower layers of the nine-layer O-O-terminated symmetric nonstoichiometric (011) slab. Thus, we obtain a neutral O-terminated ABO₃ perovskite (011) slab without any dipole moment perpendicular to the slab surface (Figure 9). Similarly, we deleted both B and O atoms (Figure 8) or an A atom (Figure 7) from the upper and lower layers of the ABO-terminated symmetric nonstoichiometric ABO₃ perovskite (011) slabs. Thus, we obtain neutral A-terminated (Figure 8) or BO-terminated (Figure 7) ABO₃ perovskite (011) slabs without any dipole moment perpendicular to their (011) surfaces. Consequently, in our computations, the BO-terminated symmetric, nonstoichiometric (Figure 7) nine-layer (011) slab consisted of a supercell enclosing 21 atoms. The A- (Figure 8) and O- (Figure 9) terminated nonstoichiometric and symmetric ABO₃ perovskite nine-layer (011) slabs consisted of supercells enclosing 19 and 20 atoms, respectively.

As a further action, the ABO₃ perovskite polar (111) surfaces will be described by us using BZO as an example (Figure 10 and Figure 11) [107]. In order to compute the polar BZO perovskite (111) surfaces, we employed symmetrical, nonstoichiometric (111) slabs containing nine alternating Zr and BaO₃ layers (Figure 10 and Figure 11). One of two BZO (111) slabs (Figure 11a) is terminated by Zr planes from both sides. It consists of a supercell accommodating 21 atoms (Figure 11a). The second (111) slab (Figure 11b) is terminated from both sides by BaO₃ planes. It consists of a supercell accommodating 24 atoms (Figure 11b). Both these Zr- and BaO₃-terminated BZO (111) slabs are symmetrical and nonstoichiometric (Figure 11). They have the unit-cell formulas Ba₄Zr₅O₁₂ and Ba₅Zr₄O₁₅, respectively (Figure 11). As we know from studies dealing, for example, with polar STO and CTO (111) surfaces [99,101,119], a strong electron redistribution happens for such (111) terminations (Figure 11) canceling the polarity. Therefore, such calculations are possible for the Zr- or BaO₃-terminated BZO (111) surface [99,101,119]. It is worth noting that we used the basis sets for neutral Ba, Zr and O atoms in all our B3LYP computations dealing with polar BaZrO₃ perovskite (111) surfaces [5,99,107].

With the ultimate goal of computing the ABO₃-type perovskite, for example, the PbZrO₃ (001) surface energy, we started our B3LYP computations with the cleavage energy calculations for unrelaxed PbO- as well as ZrO₂-terminated (001) surfaces [1,2,3,94]. Surfaces with both PbO and ZrO₂ (001) terminations at the same time emerge under the (001) cleavage of the PZO crystal [1,2,3,94]. We suppose that the PZO perovskite cleavage energy is uniformly shared between the created (001) surfaces (Figure 1 and Figure 2) [1,2,3]. In our B3LYP computations, the nine-layer PbO-terminated PZO (001) slab with 22 atoms as well as the nine-layer ZrO₂-terminated PZO (001) slab, containing 23 atoms, together contain nine bulk unit cells or 45 atoms atoms, thus:

E_surf^unr(ϑ) = ¼ [E_slab^unr(PbO) + E_slab^unr(ZrO₂) − 9E_bulk],

(1)

where ϑ means PbO or ZrO₂; E_slab^unr(ϑ) is the SrO- or ZrO₂-terminated PZO (001) slab energies without relaxation; E_bulk is the PZO bulk unit cell, containing five atoms, total energy; and the factor of ¼ means that we created four surfaces due the PZO crystal (001) cleavage [1,2,3]. After this, we can compute the relaxation energies for both PbO- and ZrO₂-terminated PZO (001) slabs [1,2,3,75,82], using the following equation:

E_rel(ϑ) = ½ [E_slab^rel(ϑ) − E_slab^unr(ϑ)],

(2)

where E_slab^rel(ϑ) is the (001) slab total energy after geometry relaxation [1,2,3,75,82]. The surface energy is thereby described as a sum of the relevant relaxation as well as cleavage energies:

E_surf(ϑ) = E_surf^unr(ϑ) + E_rel(ϑ).

(3)

With goal of computing the PZO (011) surface energies for the ZrO- and Pb-terminated (011) surfaces, we think about the cleavage of eight PZO bulk unit cells, in order to obtain the ZrO- and Pb-terminated (011) slabs, which contain 21 and 19 atoms. Namely, we split the cleavage energy uniformly among these two surfaces and derive:

E_surf^unr(ϑ) = ¼ [E_slab^unr(Pb) + E_slab^unr(ZrO) − 8E_bulk],

(4)

where ϑ indicates Pb or ZrO; E_slab^unr(ϑ) is our computed total energy for the unrelaxed Pb- or ZrO-terminated PZO (011) slabs; and E_bulk is our computed PbZrO₃ perovskite total energy per five-atom bulk unit cell.

In the end, when we cut the PZO perovskite crystal in other way, we obtain two equal O-terminated PZO (011) surface slabs. Each of them contains 20 atoms [1,2,3,82]. This permits us to make our computations less complex, taking into account that the unit cell of the nine-plane O-terminated PZO (011) slab includes four PZO bulk unit cells [1,2,3,82]. Thereby, the O-terminated PZO perovskite (011) surface energy is described as follows:

E_surf(O) = ½ [E_slab^rel(O) − 4E_bulk],

(5)

where E_surf(O) is the O-terminated PZO (011) surface energy, and E_slab^rel(O) is the relaxed O-terminated PZO (011) slab total energy. In the end, the ABO₃ perovskite polar (111) surface energy computation details are described by us in Refs. [5,99,107].

3. Results

3.1. B3PW and B3LYP Calculations of ABO₃ Perovskite Bulk Properties

As a starting point of calculations, we computed the bulk lattice constants for all eight of our considered ABO₃-type perovskites, using two different hybrid exchange–correlation functionals, B3PW and B3LYP. We compared our theoretically computed ABO₃ perovskite bulk lattice constants with available experimental data. Namely, our computed bulk lattice constants, using the B3PW hybrid exchange–correlation functional for BTO (4.008 Å) [1], CTO (3.851 Å) [3], PTO (3.936 Å) [1], STO (3.904 Å) [2], BZO (4.234 Å) [82] and SZO (4.155 Å) [75] perovskites are listed in

Table 1. Computed ABO₃ perovskite bulk lattice constants [1,2,3,75,82,94,99,107,109,120] (in Å) by means of the B3PW or B3LYP method. The experimentally detected ABO₃ perovskite bulk lattice constants [121,122,123,124,125,126] are listed for comparison purposes.

ABO3-Type Perovskite	B3PW	B3LYP	Experiment
BaTiO3	4.008 [1]	4.04 [109]	4.00 [121]
CaTiO3	3.851 [3]	3.851 [99]	3.8967 [122]
PbTiO3	3.936 [1]	3.96 [109]	3.97 [123]
SrTiO3	3.904 [2]	3.94 [109]	3.89 [121]
BaZrO3	4.234 [82]	4.234 [107]	4.199 [124]
CaZrO3	-	4.157 [120]	-
PbZrO3	-	4.220 [94]	4.1614 [125]
SrZrO3	4.155 [75]	4.195 [94]	4.154 [126]

. Next, we computed the relevant eight ABO₃ perovskite bulk lattice constants also using the related B3LYP hybrid exchange–correlation functional. Namely, using B3LYP, we obtain the following results for BTO (4.04 Å) [109], CTO (3.851 Å) [99], PTO (3.96 Å) [109], STO (3.94 Å) [109], BZO (4.234 Å) [107], CZO (4.157 Å) [120], PZO (4.220 Å) [94] and SZO (4.195 Å) [94] (Table 1). Experimental ABO₃-type perovskite bulk lattice constants for our eight computed ABO₃ perovskites are collected in Table 1 [121,122,123,124,125,126] for comparison purposes.

As we can see from Table 1, both in our computations for ABO₃ perovskite bulk lattice constant used B3PW and B3LYP hybrid exchange–correlation functionals allows us to achieve fair agreement with the experiment [121,122,123,124,125,126]. For example, the agreement between our B3PW computed SrZrO₃ bulk lattice constant 4.155 Å [75] and the experimental value of 4.154 Å [126] is simply outstanding. Also, our B3PW computed BTO bulk lattice constant 4.008 Å [1] is in almost perfect agreement with the experimental data of 4.00 Å [121]. In addition, the agreement between our B3LYP computed PbTiO₃ bulk lattice constant (3.96 Å) [109] and the experimental PbTiO₃ bulk lattice constant (3.97 Å) [123] is fine.

Our B3PW or B3LYP computed bulk effective atomic charges Q and bond populations P for all eight ABO₃-type perovskites are collected in

Table 2. BTO, CTO, PTO, STO, BZO, CZO, PZO and SZO perovskite bulk effective atomic charges Q (in e) and bond populations P (in e) computed using the hybrid B3PW or B3LYP exchange–correlation functionals [1,2,3,82,94].

		BTO	CTO	PTO	STO	BZO	CZO	PZO	SZO
Ion	Prop.	B3PW	B3PW	B3PW	B3PW	B3PW	B3LYP	B3LYP	B3LYP
A	Q	+1.797	+1.782	+1.354	+1.871	+1.815	+1.787	+1.368	+1.880
	P	−0.034	+0.006	+0.016	−0.010	−0.012	+0.014	+0.030	+0.002
O	Q	−1.388	−1.371	−1.232	−1.407	−1.316	−1.310	−1.160	−1.351
	P	+0.098	+0.084	+0.098	+0.088	+0.108	+0.086	+0.106	+0.092
B	Q	+2.367	+2.330	+2.341	+2.351	+2.134	+2.144	+2.111	+2.174

. We used the classical Mulliken population analysis [127,128,129,130] in order to describe the effective atomic charges Q as well as chemical bond populations P for all eight of our B3PW or B3LYP computed ABO₃-type perovskite materials (Table 2). As we can see from Table 2, our B3PW or B3LYP computed effective atomic charges Q [127,128,129,130] are always smaller than those expected from the classical ionic model (+2e for A atoms, +4e for B atoms as well as −2e for O atoms). For example, the A atom effective charges (Table 2) are in the range of only +1.354e for the PTO perovskite to +1.880e for the SZO perovskite (Table 2). The B atom effective charges are in the range (Table 2) from +2.111e for PZO to +2.367e for BTO perovskite. The O atom effective charges [127,128,129,130] are between −1.160e for PZO perovskite (Table 2) and −1.407e for the STO perovskite. Finally, the smallest B-O chemical bond population P, according to our B3PW computations, is observed between the Ti-O atom in the CTO perovskite (+0.084e), whereas the largest is between the Zr-O atoms in the BZO perovskite (+0.108e) (Table 2).

Our B3PW computed bulk Γ-Γ band gap for the BTO perovskite is equal to 3.55 eV (

Table 3. B3LYP, B3PW, HF and PWGGA computed Γ-Γ band gaps (in eV) for eight ABO₃ perovskites. Experimental bulk Γ-Γ band gaps at ABO₃ perovskite cubic phase are listed for comparison purposes.

Perovskite	Theoretical Method	Bulk Band Gap	Experiment
BaTiO3	B3PW	3.55 [114]	No data for cubic phase
CaTiO3	B3PW	4.18 [75]	No data for cubic phase
	B3LYP	4.20 [99]
	PWGGA	2.34 [75]
	HF	12.63 [75]
PbTiO3	B3PW	4.32 [114]	No data for cubic phase
SrTiO3	B3PW	3.96 [114]	3.75 [132]
BaZrO3	B3PW	4.93 [75]	5.3 [133]
	B3LYP	4.79 [107]
	PWGGA	3.24 [75]
	HF	12.96 [75]
CaZrO3	B3LYP	5.40 [120]	No data for cubic phase
PbZrO3	B3LYP	5.63 [94]	No data for cubic phase
SrZrO3	B3PW	5.30 [75]	No data for cubic phase
	B3LYP	5.31 [94]
	PWGGA	3.53 [75]
	HF	13.54 [75]

and Figure 12a). No experimental data exist for the BTO bulk Γ-Γ band gap at the cubic phase. Nevertheless, the related Γ-Γ BTO bulk electronic band structure, measured in the tetragonal towards orthorhombic phase transition temperature [131], identical to 278 K, at contrasting experimental situations, is equivalent to 3.27 or 3.38 eV, respectively. Our B3PW [75] and B3LYP [99] computed CTO bulk Γ-Γ band gaps are almost identical (4.18 eV and 4.20 eV, respectively). Again, there are no experimental data available for the high-temperature cubic CTO phase [75]. It is worth noting that our PWGGA computed CTO bulk Γ-Γ band gap is very small, only 2.34 eV [75], whereas our HF computed CTO bulk Γ-Γ band gap is 5.4 times larger and is equal to 12.63 eV (Table 3). Our B3PW computed PTO bulk Γ-Γ band gap [114] is equal to 4.32 eV (Table 3 and Figure 12b). Our B3PW computed BZO bulk band structure is plotted in Figure 13. Our B3PW computed STO Γ-Γ bulk band gap 3.96 eV [114] is almost in perfect agreement with the available experimental data for the STO cubic phase at Γ-point 3.75 eV [132] (Table 3 and Figure 14). Our B3PW computed BZO bulk band gap at Γ-point is equal to 4.93 eV [75] and is in fair agreement with the relevant experimental data (5.3 eV) [133]. As we can see from Table 3, PWGGA computed BZO bulk band gap at Γ-point is considerably underestimated (3.24 eV), whereas the HF result (12.96 eV) is considerably overestimated regarding the experimental BZO bulk band gap value of 5.3 eV. Finally, for SZO perovskite, our B3PW and B3LYP computed bulk Γ-Γ band gaps almost coincide (5.30 eV and 5.31 eV, respectively) (Table 3) [75,94]. Our B3LYP computation results, dealing with eight ABO₃-type perovskite bulk Γ-Γ band gaps, are depicted in Figure 14. As we can see from Figure 14, the best possible agreement between the theory and experiment for eight ABO₃-type perovskite bulk Γ-Γ band gaps is possible to achieve by means of the hybrid exchange–correlation functionals, for example B3PW or B3LYP (Table 3 and Figure 14). The HF method hugely overestimated the Γ-Γ bulk band gaps for all eight our computed ABO₃ perovskites, whereas the density functional theory based PWGGA functional underestimated them (Figure 14 and Table 3).

3.2. ABO₃ Perovskite (001) Surface Atomic and Electronic Structure

Our hybrid exchange–correlation functional B3LYP or B3PW computation results for the (001) surface atomic relaxations for BO₂- as well as AO-terminated ABO₃-type perovskite upper three or two (001) surface layers are recorded in

Table 4. B3LYP or B3PW computed relaxations of atoms (% of a₀) for the AO-terminated (001) surfaces of eight ABO₃ perovskites [1,2,3,75,82,94,120].

ABO3 Perovsk.		BTO	CTO	PTO	STO	BZO	CZO	PZO	SZO
Termin., (001)		AO	AO	AO	AO	AO	AO	AO	AO
Layer	Ion	B3PW	B3PW	B3PW	B3PW	B3PW	B3LYP	B3LYP	B3LYP
1	A	−1.99	−8.31	−3.82	−4.84	−4.30	−10.01	−5.69	−7.63
	O	−0.63	−0.42	−0.31	+0.84	−1.23	−0.79	−2.37	−0.86
2	B	+1.74	+1.12	+3.07	+1.75	+0.47	+1.11	+0.57	+0.86
	O	+1.40	+0.01	+2.30	+0.77	+0.18	+0.01	+0.09	−0.05
3	A	-	-	-	-	−0.01	−2.60	−0.47	−1.53
	O	-	-	-	-	−0.14	−0.48	−0.47	−0.45

and

Table 5. B3LYP or B3PW computed relaxations of atoms (% of a₀) for the BO₂-terminated (001) surfaces of eight ABO₃ perovskites [1,2,3,75,82,94,120].

ABO3 Perovsk.		BTO	CTO	PTO	STO	BZO	CZO	PZO	SZO
Termin., (001)		BO2	BO2	BO2	BO2	BO2	BO2	BO2	BO2
Layer	Ion	B3PW	B3PW	B3PW	B3PW	B3PW	B3LYP	B3LYP	B3LYP
1	B	−3.08	−1.71	−2.81	−2.25	−1.79	−1.30	−2.37	−1.38
	O	−0.35	−0.10	+0.31	−0.13	−1.70	−2.31	−1.99	−2.10
2	A	+2.51	+2.75	+5.32	+3.55	+1.94	+4.23	+4.36	+2.81
	O	+0.38	+1.05	+1.28	+0.57	+0.85	+1.25	+1.04	+0.91
3	B	-	-	-	-	−0.03	−0.05	−0.47	−0.04
	O	-	-	-	-	0.00	−0.09	−0.28	−0.05

.

As it is possible to see from Table 4 and Table 5, the atomic relaxation magnitudes of surface metal atoms A or B, for all eight ABO₃ perovskite (001) surface upper two layers, are almost always noticeably larger than that for the respective O atoms (Table 4 and Table 5). This leads to a significant surface rumpling s for the upper-surface plane (

Table 6. B3LYP or B3PW computed as well as experimentally detected surface rumpling s and respective atomic displacements Δd₁₂ and Δd₂₃ (% of a₀) for the BO₂- and AO-terminated (001) surfaces of eight ABO₃ perovskites.

Material	Method	AO-Terminated (001) Surface			BO2-Terminated (001) Surface
		s	Δd12	Δd23	s	Δd12	Δd23
BTO	B3PW [1]	1.37	−3.74	1.74	2.73	−5.59	2.51
	LDA [134]		−2.8	1.1		−3.1	0.9
CTO	B3PW [3]	7.89	−9.43	1.12	1.61	−4.46	2.75
	GGA [135]	0.37	−0.44	0.22	0.13	−0.41	0.33
PTO	B3PW [1]	3.51	6.89	3.07	3.12	−8.13	5.32
	LDA [134]		−4.2	2.6		−4.4	3.1
STO	B3PW [2]	5.66	−6.58	1.75	2.12	−5.79	3.55
	LDA [134]		−3.4	1.2		−3.5	1.6
	LEED [136]	4.1 ± 2	−5 ± 1	2 ± 1	2.1 ± 2	1 ± 1	−1 ± 1
	RHEED [137]	4.1	2.6	1.3	2.6	1.8	1.3
BZO	B3PW [82]	3.07	−4.77	0.48	0.09	−3.73	1.97
CZO	B3LYP [120]	9.22	−11.12	3.71	1.01	−5.53	4.28
PZO	B3LYP [94]	3.32	−6.26	1.04	0.38	−6.73	4.83
SZO	B3LYP [94]	6.77	−8.49	2.39	−0.72	−4.19	2.85
	LDA [138]	7.9	−9.1	3.2	−0.7	−6.1	4.2
	GGA [138]	7.8	−9.3	3.3	0.3	−7.4	4.9

). The only two deviations from this systematic trend are the ZrO₂-terminated CZO and SZO (001) surface outermost layers, where the Ca as well as Sr atom inward relaxation magnitudes are smaller than the respective O atom inward relaxation magnitudes (Table 5). The second systematic trend is that for both AO and BO₂ terminations of all eight ABO₃ perovskite (001) surfaces, as a rule, all atoms of the first (upper) surface layer relax inwards towards the ABO₃ perovskite bulk (Table 4 and Table 5). At the same time, all atoms of the second surface layer, for both AO and BO₂ (001) surface terminations, relax upwards (Table 4 and Table 5). Again, all third-layer atoms, the same as upper-layer atoms, relax inwards, towards the ABO₃ perovskite bulk (Table 4 and Table 5). There are only three exceptions to this systematic trend (Table 4 and Table 5). Namely, TiO₂-terminated PTO (001) surface upper-layer O atom relaxes upwards by +0.31% of a₀ (Table 5); SrO-terminated STO (001) surface upper-layer O atom relaxes upwards by +0.84% of a₀, whereas the second-layer O atom on the SrO-terminated SrZrO₃ (001) surface relaxes inwards by a very small relaxation magnitude equal to −0.05% of a₀ (Table 4).

B3PW computed [1,2,3,134,135] as well as experimental [136,137] results, dealing with ABO₃-type perovskite titanates BTO, CTO, PTO and STO, are collected in Table 6. As we can see from Table 6, our hybrid B3PW computation results [2] for STO (001) surfaces are in fair correspondence with the earlier LDA computation results carried out by Meyer et al. [134]. Namely, both computations, our B3PW [2] as well as those LDA computations performed by Meyer et al. [134], provide the same sign for the changes in interlayer distances Δd₁₂ and Δd₂₃ [2,134] (Table 6). Moreover, our B3PW computed [2] surface rumplings s for SrO as well as TiO₂-terminated STO (001) surfaces are in fair agreement with the actual LEED [136] as well as RHEED [137] experimental measurements. Nonetheless, our B3PW [2] and LDA [134] computed interlayer distance changes Δd₁₂ and Δd₂₃ fail to agree with the LEED [136] experimental measurements for the TiO₂-terminated STO (001) surface. It is worth noting that LEED [136] and RHEED [137] (Table 6) experimental measurements fail to agree concerning the sign of Δd₁₂ for the SrO-terminated STO (001) surface. Also, for the TiO₂-terminated STO (001) surface, LEED [136] and RHEED [137] experiments disagree regarding the sign of the interlayer distance Δd₂₃. As we can see from Table 6, our B3LYP [94] as well as Wang et al.’s [138] LDA and GGA computed surface rumpling s and relative interlayer displacements Δd₁₂ and Δd₂₃ for the SrO-terminated SZO (001) surface are in fair agreement with each other. In addition, our B3LYP [94] and Wang et al.’s [138] computed interlayer distances Δd₁₂ and Δd₂₃ are in good agreement with each other for the ZrO₂-terminated SZO (001) surface. The agreement between our B3LYP [94] and Wang et al.’s [138] LDA computed surface rumpling s for the ZrO₂-terminated SZO (001) surface (−0.72% of a₀ and −0.7% of a₀) is almost perfect. Unfortunately, the surface rumpling s, computed by Wang et al., using the GGA exchange–correlation functional [138] for the ZrO₂-terminated SZO (001) surface has a different sign of +0.3% of a₀ (Table 6).

As we can see from

Table 7. B3LYP or B3PW computed BO₂- or AO-terminated (001) surface energies (in eV per surface cell) for BTO, CTO, PTO, STO, BZO, CZO, PZO, SZO perovskites.

ABO3 Perovskite	ABO3 Perovskite (001) Surface
Termination, Functional	BO2-Terminated	AO-Terminated
BTO [1], B3PW	1.07	1.19
CTO [3], B3PW	1.13	0.94
PTO [1], B3PW	0.74	0.83
STO [2], B3PW	1.23	1.15
BZO [82], B3PW	1.31	1.30
CZO [120], B3LYP	1.33	0.87
PZO [94], B3LYP	0.93	1.00
SZO [94], B3LYP	1.24	1.13

and Figure 15, our B3PW or B3LYP computed eight ABO₃-type perovskite (001) surface energies are always around 1 eV. Namely, our largest computed (001) surface energy is for the ZrO₂-terminated CaZrO₃ (001) surface (1.33 eV) [120], whereas the smallest is for the TiO₂-terminated PbTiO₃ (001) surface (0.74 eV) [1]. The smallest energy difference, according our B3PW computations, is for the BaZrO₃ ZrO₂- (1.31 eV) and BaO− (1.30 eV) terminated (001) surfaces [82]. The largest (001) surface energy difference, according to our B3LYP hybrid exchange–correlation functional computations, is for the CaZrO₃ perovskite ZrO₂− (1.33 eV) and CaO− (0.87 eV) terminated (001) surfaces (Table 7 and Figure 10) [120]. It is worth noting that according to the calculation results, the surface energies of the nonpolar BO₂-terminated (001) surface was slightly smaller for the BTO, PTO and PZO perovskites; thus, it is more stable (Table 7 and Figure 15).

Our B3PW computed electronic bulk band structures for BTO, PTO as well as BZO perovskites are illustrated in Figure 12 and Figure 13. Our B3PW computed TiO₂-terminated electronic (001) surface band structures for BTO and PTO are depicted in Figure 16a,b, whereas the AO-terminated BTO and PTO (001) surfaces are depicted in Figure 17a,b. Our B3PW computed electronic band structures for BaO- (a) and ZrO₂- (b) terminated BZO (001) surfaces are illustrated in Figure 18. Our computed Γ-Γ band gap numerical values for all eight of our computed ABO₃ perovskite bulk as well as their BO₂- and AO-terminated (001) surfaces are collected in

Table 8. B3PW, B3LYP, PWGGA or HF computed Γ-Γ band gaps for eight ABO₃ perovskite bulk as well as their BO₂- and AO-terminated (001) surfaces (in eV).

Perovskite, Method	Γ-Γ Band Gap, Bulk	BO2-Termin., (001)	AO-Termin., (001)
BTO, B3PW	3.55 [114]	2.96	3.49
CTO, B3PW	4.18 [75]	3.30	3.87
CTO, B3LYP	4.20 [99]	3.33	3.88
CTO, PWGGA	2.34 [75]	2.06	2.19
CTO, HF	12.63 [75]	11.86	12.53
PTO, B3PW	4.32 [114]	3.18	3.58
STO, B3PW	3.96 [114]	3.95	3.72
BZO, B3PW	4.93 [75]	4.48	4.82
BZO, B3LYP	4.79 [107]	4.37	4.71
BZO, PWGGA	3.24 [75]	2.76	3.08
BZO, HF	12.96 [75]	12.62	12.84
CZO, B3LYP	5.40 [120]	5.22	5.00
PZO, B3LYP	5.63 [94]	4.60	3.86
SZO, B3PW	5.30 [75]	4.98	5.01
SZO, B3LYP	5.31 [94]	4.91	5.04
SZO, PWGGA	3.53 [75]	3.17	3.20
SZO, HF	13.54 [75]	13.19	13.25

. As we can see from Table 3 and Table 8, our B3PW computed STO Γ-Γ bulk band gap (3.96 eV) [114] is in an excellent agreement with the experimentally detected STO bulk Γ-Γ band gap (3.75 eV) [133]. Also, for the BZO perovskite Γ-Γ bulk band gap, the agreement between our B3PW computation result (4.93 eV) [75] and the experiment (5.3 eV) [133] is fine (Table 3 and Table 8). The key effect there, as we can see from Table 8 and Figure 19, is that the ABO₃ perovskite bulk Γ-Γ band gap, for all eight of our B3PW or B3LYP computed ABO₃ perovskites, is always reduced near their AO- and BO₂-terminated (001) surfaces. For example, our B3PW computed BZO bulk Γ-Γ band gap (4.93 eV) (Figure 8) is reduced near the BZO ZrO₂-terminated (001) surface (4.48 eV) as well as near the AO-terminated BZO (001) surface (4.82 eV) (Table 8 and Figure 18 and Figure 19). Also, for all of our other eight computed ABO₃ perovskites, the situation is similar, regarding the reduction of the ABO₃ perovskite bulk Γ-Γ band gap near their (001) surfaces (Figure 14 and Table 8). For example, our B3PW computed BTO bulk Γ-Γ band gap (Figure 12a) (3.55 eV) (Table 8) is also reduced near the BaO-terminated BTO (001) surface (3.49 eV) (Figure 17a) and TiO₂-terminated BTO (001) surface (2.96 eV) (Figure 16a and Figure 19).

As we can see from

Table 9. B3LYP or B3PW computed B-O bond populations for eight ABO₃ perovskites bulk and also for their BO₂-terminated (001) surfaces (in e).

Perovskite	Functional	B-O Chemical Bond Populations
		ABO3, Bulk	Surface, (001)
BTO	B3PW	0.098	0.126
CTO	B3PW	0.084	0.114
PTO	B3PW	0.098	0.114
STO	B3PW	0.088	0.118
BZO	B3PW	0.108	0.132
CZO	B3LYP	0.086	0.102
PZO	B3LYP	0.106	0.116
SZO	B3LYP	0.092	0.114

and Figure 20, for all eight of our B3LYP or B3PW computed ABO₃ perovskites, we can observe the significant increase in the B-O chemical bond covalency near their BO₂-terminated (001) surfaces, in comparison with bulk. For example, the largest Ti-O chemical bond population increase by 0.30e, according to our B3PW computations, is observed for the CTO and STO perovskites, namely, from 0.084e and 0.088e, respectively, for their bulk to 0.114e and 0.118e, respectively, near their TiO₂-terminated (001) surfaces [2,3]. Just opposite, the smallest B-O chemical bond population increase is observed for the PbZrO₃ perovskite [94]. Namely, the PbZrO₃ perovskite Zr-O chemical bond population increased from 0.106e (bulk case) to 0.116e near the ZrO₂-terminated PbZrO₃ (001) surface [94] (Table 9 and Figure 20).

3.3. ABO₃ Perovskite (011) Surface Atomic and Electronic Structure

As we can see from

Table 10. B3PW or B3LYP computed upper-layer atom shifts for 8 ABO₃ perovskite BO-, A- and O-terminated (011) surfaces.

Term. (011)	Atom	CTO	STO	PTO	BTO	CZO	SZO	PZO	BZO
Method		B3PW	B3PW	B3PW	B3PW	B3LYP	B3LYP	B3LYP	B3PW
BO	B	−7.14	−7.69	−8.13	−7.86	−6.06	−6.16	−6.87	−6.61
	O	+4.67	+3.59	+3.30	+2.61	+4.96	+4.36	+4.27	+3.35
A	A	−16.05	−12.81	−11.94	−8.67	−18.67	−15.73	−15.17	−11.81
O	O	−6.10	−6.61	−7.37	−5.40	−5.97	−6.56	−6.61	−7.32

and Figure 21, for all eight of our B3LYP or B3PW computed ABO₃ perovskites, the systematic tendency is that for all three of their BO-, A- and O-terminated (011) surfaces, all upper-layer atoms relax inwards. The only exception to this systematic trend is upward relaxation of BO-terminated (011) surface upper-layer O atoms for all eight of our computed ABO₃ perovskites (Table 10 and Figure 21).

It is worth noting that the biggest relaxation magnitude between all upper-layer ABO₃ perovskite (011) surface atoms, for all three possible (011) surface terminations, demonstrates the Ca-terminated surface Ca atom shifting inwards by −18.67% of a₀ (Table 10 and Figure 16) [10]. It is around three times bigger than the displacement magnitudes for the Zr atom (+6.06% of a₀) on the ZrO-terminated as well as O atom (+5.97% of a₀) on the O-terminated CZO (011) surfaces (Table 10 and Figure 21).

As we can see from

Table 11. B3PW or B3LYP computed second-layer atom shifts for 8 ABO₃ perovskite BO-, A- and O-terminated (011) surfaces.

Term. (011)	Atom	CTO	STO	PTO	BTO	CZO	SZO	PZO	BZO
Method		B3PW	B3PW	B3PW	B3PW	B3LYP	B3LYP	B3LYP	B3PW
BO	O	−0.44	−0.51	−0.41	−1.02	−0.38	−0.38	−0.24	−0.29
A	O	+1.35	+1.02	−0.61	+0.80	+1.25	+1.24	−0.57	+0.66
O	B	−0.26	−1.02	+0.20	−0.15	+0.75	+1.45	+0.73	+0.12
	A	−2.10	−1.18	+0.18	+1.54	+1.78	−1.43	+0.73	+0.21
	O	+3.43	+1.79	+0.51	+1.95	+0.67	+4.29	+4.29	−0.78

and Figure 22, all our B3LYP computed second-layer O-, Ca- and ZrO-terminated CZO (011) surface atoms relax upwards. The only exception to this systematic trend is the second-layer O atom on the ZrO-terminated CZO (011) surface, which relax inwards (Table 11 and Figure 22). It is worth noting that such systematic trend, mainly upward shift of the second-layer atoms on the A-, O- and BO-terminated (011) surfaces, is quite common for all eight of our computed ABO₃ perovskites (Table 11 and Figure 22). Namely, according to our B3PW or B3LYP computations for eight ABO₃ perovskite (011) surface, all second-layer atoms, located on three different (011) terminations, shift upwards 23 atoms, but relax inwards only 17 atoms (Figure 22 and Table 11).

As we can see from

Table 12. B3PW or B3LYP computed (011) and (001) surface energies for CTO, STO, PTO, BTO, CZO, SZO, PZO and BZO perovskites (in eV per surface cell).

Term.	Esurf	CTO	STO	PTO	BTO	CZO	SZO	PZO	BZO
Method		B3PW	B3PW	B3PW	B3PW	B3LYP	B3LYP	B3LYP	B3PW
BO	(011)	3.13	3.06	1.36	2.04	3.46	3.61	1.89	3.09
A	(011)	1.91	2.66	2.03	3.24	1.49	2.21	1.74	2.90
O	(011)	1.86	2.04	1.72	1.72	2.08	2.23	1.85	2.32
BO2	(001)	1.13	1.23	0.74	1.07	1.33	1.24	0.93	1.31
AO	(001)	0.94	1.15	0.83	1.19	0.87	1.13	1.00	1.30

and Figure 23, our B3LYP or B3PW computed ABO₃ perovskite A-, O- or BO-terminated polar (011) surface energies are always larger than the ABO₃ perovskite neutral BO₂- or AO-terminated (001) surface energies. According to our B3LYP computations, the largest ABO₃ perovskite (011) surface energy is for the ZrO-terminated SZO (011) surface (3.61 eV) (Figure 23 and Table 12). The smallest surface energy between all BO-terminated ABO₃ perovskite (011) surfaces is for the TiO-terminated PTO (011) surface, only 1.36 eV. This energy (1.36 eV) is only slightly larger than the ZrO₂-terminated CaZrO₃ (001) surface energy (1.33 eV). Nevertheless, according to our B3PW or B3LYP computations, the BO-, O- or A-terminated polar ABO₃ perovskite (011) surface energies are always larger than the neutral BO₂- or AO-terminated ABO₃ perovskite (001) surface energies (Figure 19 and Table 12).

As it is possible to see from

Table 13. B3LYP or B3PW computed 8 ABO₃ perovskite bulk, BO₂-terminated (001) surface and BO-terminated (011) surface B-O chemical bond populations in the plane (B(I)-O(I)) as well as in the direction perpendicular to the (011) surface (B(I)-O(II)).

Chemical bond	CZO	CTO	STO	PTO	BTO	SZO	PZO	BZO
Method	B3LYP	B3PW	B3PW	B3PW	B3PW	B3LYP	B3LYP	B3PW
Bulk (B-O)	0.086	0.084	0.088	0.098	0.098	0.092	0.106	0.108
(B-O), (001)	0.102	0.114	0.118	0.114	0.126	0.114	0.116	0.132
B(I)-O(I), (011)	0.138	0.128	0.130	0.132	0.130	0.142	0.148	0.152
B(I)-O(II), (011)	0.240	0.186	0.188	0.196	0.198	0.246	0.252	0.252

and Figure 24, according to our B3LYP or B3PW computation results for eight ABO₃ perovskites, the B-O chemical bond population is the smallest for the ABO₃ perovskite bulk. The B-O chemical bond population is increased near the BO₂-terminated (001) surface regarding the bulk value and is in the range of +0.102e for the CZO perovskite to 0.132e for the BZO perovskite. An even larger B-O chemical bond population is near the BO-terminated (011) ABO₃ perovskite surface. Namely, the plane B(I)-O(I) chemical bond population for all eight of our computed ABO₃ perovskites is in the range of 0.128e for the CTO perovskite to 0.152e for the BZO perovskite (Table 13 and Figure 24). Finally, as we can see from Table 13 and Figure 24, the ultimately largest B-O chemical bond population, according to our B3LYP or B3PW computations for eight ABO₃ perovskites, is for the BO-terminated (011) surface B(I)-O(II) chemical bond population, in the direction perpendicular to the BO-terminated (011) surface. It is in the range of 0.186e for the CTO perovskite to 0.252e for the PZO and BZO perovskites (Figure 24 and Table 13).

3.4. ABO₃ Perovskite (111) Surface Atomic and Electronic Structure

As it is possible to see from

Table 14. B3LYP computed eight ABO₃ perovskite B-terminated (111) surface three upper-layer atom relaxation (% of a₀).

Layer	Ion	BTO	CTO	PTO	STO	BZO	CZO	PZO	SZO
1	B	−11.19	−6.23	−7.57	−3.58	−8.03	−11.07	−9.24	−5.72
2	A	−6.22	−14.02	−10.09	−11.24	−9.73	−11.31	+5.92	−11.92
	O	+2.74	+1.30	−0.13	+1.53	+0.78	+0.14	+2.61	+0.79
3	B	−0.25	−0.26	+0.53	+0.26	−0.02	−0.96	−2.73	+1.53

, according to our B3LYP computation results for seven ABO₃ perovskites, all atoms on the B-terminated ABO₃ perovskite (111) surface relax inwards. The upper-layer B atom relaxation magnitudes (Table 14) are rather strong, ranging from −3.58% of a₀ for the STO perovskite to −11.19% of a₀ for the BTO perovskite. It is worth noting that almost all second-layer A atoms on the B-terminated ABO₃ perovskite (111) surface relax inwards. In general, they exhibit very large relaxation magnitudes, for example, −14.02% of a₀ for the second-layer Ca atom on the Ti-terminated CaTiO₃ (111) surface (Table 14). As we can see from

Table 15. B3LYP computed eight ABO₃ perovskite AO₃-terminated (111) surface three upper-layer atom relaxation (% of a₀).

Layer	Ion	BTO	CTO	PTO	STO	BZO	CZO	PZO	SZO
1	A	−1.24	−0.52	+1.01	+1.33	+1.70	−3.61	−0.05	−0.74
	O	−3.98	−0.81	−2.52	−0.03	−0.57	−0.07	−1.26	−0.52
2	B	+2.49	+2.13	+0.02	+1.81	+0.21	+1.20	+1.18	+0.74
3	A	+1.49	+2.60	+1.26	−0.03	+0.71	−0.02	−0.02	−0.02
	O	−0.25	−0.07	+1.26	−0.26	−0.01	−0.07	−0.02	−0.18

, according to our B3LYP computations, most of AO₃-terminated ABO₃ perovskite upper-layer atoms also relax inwards. Nevertheless, their relaxation magnitudes are considerably smaller than for the upper-layer atom inward relaxation magnitudes on the B-terminated ABO₃ perovskite upper layer (Table 14 and Table 15).

As it is possible to see from

Table 16. B3LYP computed surface energies for B as well as AO₃-terminated ABO₃ perovskite (111) surfaces (in eV per surface cell).

Termination	BTO	CTO	PTO	STO	BZO	CZO	PZO	SZO
B-terminated	7.28	4.18	6.14	4.99	7.94	8.19	6.93	7.98
AO3-terminated	8.40	5.86	8.11	6.30	9.33	9.62	8.21	9.45

, according to our B3LYP computation results for seven ABO₃ perovskites, the B-terminated ABO₃ perovskite (111) surface energies are always smaller than the respective AO₃-terminated ABO₃ perovskite (111) surface energies. The B-terminated ABO₃ perovskite (111) surface energies (Table 16) are in the energy range of 4.18 eV for the Ti-terminated CaTiO₃ (111) surface to 8.19 eV for the Zr-terminated CaZrO₃ (111) surface. The AO₃-terminated ABO₃ perovskite (111) surface energies are in the range of 5.86 eV for the CaO₃-terminated CaTiO₃ (111) surface to 9.62 eV for the CaO₃-terminated CaZrO₃ (111) surface (Table 16).

4. Conclusions

We performed B3PW and B3LYP computations for BTO, CTO, PTO, STO, BZO, CZO, PZO and SZO perovskite neutral (001) along with polar (011) and (111) surfaces. For the neutral AO- as well as BO₂-terminated (001) surfaces, in most cases, all upper-layer atoms relax inwards, although the second-layer atoms shift outwards. There are only three exceptions to this systematic trend (Table 4 and Table 5). Namely, upward relaxation of TiO₂-terminated PTO (001) surface upper-layer O atom by (+0.31% of a₀) (Table 5). On the (001) BO₂-terminated surface, the second-layer metal atoms, as a rule, exhibit larger atomic relaxations than the second-layer O atoms. For most ABO₃ perovskites, the (001) surface rumpling s is bigger for the AO- than BO₂-terminated surfaces. In contrast, the surface energies, for both (001) terminations, are practically identical. Nevertheless, for the BTO, PTO and PZO perovskites, the BO₂-terminated (001) surface has a slightly smaller surface energy, and therefore, it is more stable. In contrast, for CTO, STO, CZO and SZO perovskites, the AO-terminated (001) surface has a slightly smaller surface energy, and therefore, it is slightly more stable. Conversely, different (011) surface terminations exhibit quite different surface energies for the O-terminated, A-terminated, and BO-terminated surfaces. Our computed ABO₃ perovskite (111) surface energies are always significantly larger than the neutral (001) and polar (011) surface energies. Our computed ABO₃ perovskite bulk B-O chemical bond covalency increase near their neutral (001) and especially polar (011) surfaces. It is worth noting that for the (011) surfaces, in the plane B(I)-O(I) chemical bond population for all eight of our computed ABO₃ perovskites is in the range of 0.128e for the CTO perovskite to 0.152e for the BZO perovskite (Table 13 and Figure 24). Finally, as we can see from Table 13 and Figure 24, the ultimately largest B-O chemical bond population, according to our B3LYP or B3PW computations for eight ABO₃ perovskites, is for the BO-terminated (011) surface B(I)-O(II) chemical bond population, in the direction perpendicular to the BO-terminated (011) surface. It is in the range of 0.186e for the CTO perovskite to 0.252e for the PZO and BZO perovskites (Figure 24 and Table 13).