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

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.

For example, the first B3PW calculations, dealing with polar and charged BaTiO 3 and PbTiO 3 (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 SrTiO 3 (011) surface were carried out by Bottin et al. [88].They computed the atomic as well as electronic structure of a few (1 × 1) SrTiO 3 (011) surface terminations [88].The year after that, Heifets et al. [89] carried out firstprinciples 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].
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 ABO 3 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 ABO 3 perovskite surfaces.Finally, we systematized these common systematic trends in a system, effortlessly approachable worldwide for a comprehensive audience of scientists.

Computational Details and Surface Models
We performed very comprehensive hybrid density functional theory (DFT) calculations for eight different ABO 3 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 ABO 3 perovskites.We executed the reciprocal-space integration for the ABO 3 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 ABO 3 -type perovskites and their surfaces in high symmetry, cubic structure space group Pm3m ) [114][115][116].
With the goal of simulating the neutral BO 2 -terminated (001) surfaces of ABO 3 -type perovskites [114], we selected symmetrical slabs.These slabs, in our computations, consisted of nine neutral and alternating BO 2 as well as AO layers (Figures 1 and 2).The first slab was terminated by the BO 2 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 A 4 B 5 O 14 and A 5 B 4 O 13 , respectively (Figures 1 and 2).type perovskites and their surfaces in high symmetry, cubic structure (space group m3 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 (Figures 1 and 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 (Figures 1 and 2).type perovskites and their surfaces in high symmetry, cubic structure (space group m3 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 (Figures 1 and 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 (Figures 1 and 2).Just opposite to the (001) cleavage (Figures 1 and 2) of ABO 3 , which produce nonpolar AO and BO 2 terminations, direct cleavage of ABO 3 -type perovskites, in order to generate (011) surfaces, leads to the production of polar O 2 as well as ABO surfaces (Figure 3).The ABO 3 crystal (Figure 3), alongside the [011] crystalic direction, is composed of cyclic planes of O 2 and ABO units (Figure 3).These two alternating O 2 and ABO planes (Figure 3) have ionic charges of −4e and +4e, assuming following constituents as O 2− , B 4+ and A 2+ .Therefore, modeling of the ABO 3 (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 ABO 3 perovskite (011) surface (Figure 4), when the slab is terminated by different O 2 as well as ABO planes (Figure 4) (stoichiometric slab).Infinite charge, in case when the slab is terminated by the same planes (O 2 -O 2 ) (Figure 5) or ABO-ABO (Figure 6) (nonstoichiometric slab).Such ABO 3 perovskite (011) surface terminations (Figures 5 and 6) make the (011) surface unstable [117,118].
Materials 2023, 16, x FOR PEER REVIEW 5 Just opposite to the (001) cleavage (Figures 1 and 2) of ABO3, which produce non AO and BO2 terminations, direct cleavage of ABO3-type perovskites, in order to gen (011) surfaces, leads to the production of polar O2 as well as ABO surfaces (Figure 3) ABO3 crystal (Figure 3), alongside the [011] crystalic direction, is composed of cyclic p of O2 and ABO units (Figure 3).These two alternating O2 and ABO planes (Figure 3) ionic charges of −4e and +4e, assuming following constituents as O 2− , B 4+ and A 2+ .There modeling of the ABO3 (011) surfaces (Figure 3) precisely, as they are obtained from the tine crystal cleavage, leads to the following two problematic situations: An infinite m scopic dipole moment, which is perpendicular to the ABO3 perovskite (011) surface (F 4), when the slab is terminated by different O2 as well as ABO planes (Figure 4) (stoi metric slab).Infinite charge, in case when the slab is terminated by the same planes (O (Figure 5) or ABO-ABO (Figure 6) (nonstoichiometric slab).Such ABO3 perovskite (011 face terminations (Figures 5 and 6) make the (011) surface unstable [117,118].This was the key reason why in our ABO 3 -type perovskite (011) surface computations, with the aim of obtaining the neutral (011) slab, we deleted some atoms (Figures 7-9).Namely, we deleted the O atom (Figure 9) from the upper as well lower layers of the ninelayer O-O-terminated symmetric nonstoichiometric (011) slab.Thus, we obtain a neutral O-terminated ABO 3 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 ABO 3 perovskite (011) slabs.Thus, we obtain neutral A-terminated (Figure 8) or BO-terminated (Figure 7) ABO 3 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 ABO 3 perovskite nine-layer (011) slabs consisted of supercells enclosing 19 and 20 atoms, respectively.This was the key reason why in our ABO3-type perovskite (011) surface comp tions, with the aim of obtaining the neutral (011) slab, we deleted some atoms (Figure 9).Namely, we deleted the O atom (Figure 9) from the upper as well lower layers of nine-layer O-O-terminated symmetric nonstoichiometric (011) slab.Thus, we obta neutral O-terminated ABO3 perovskite (011) slab without any dipole moment perpen ular to the slab surface (Figure 9).Similarly, we deleted both B and O atoms (Figure 8 an A atom (Figure 7) from the upper and lower layers of the ABO-terminated symme nonstoichiometric ABO3 perovskite (011) slabs.Thus, we obtain neutral A-termina (Figure 8) or BO-terminated (Figure 7) ABO3 perovskite (011) slabs without any dip moment perpendicular to their (011) surfaces.Consequently, in our computations, the terminated symmetric, nonstoichiometric (Figure 7) nine-layer (011) slab consisted       As a further action, the ABO3 perovskite polar (111) surfaces will be described by us using BZO as an example (Figures 10 and 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 (Figures 10 and 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 As a further action, the ABO 3 perovskite polar (111) surfaces will be described by us using BZO as an example (Figures 10 and 11) [107].In order to compute the polar BZO perovskite (111) surfaces, we employed symmetrical, nonstoichiometric (111) slabs containing nine alternating Zr and BaO 3 layers (Figures 10 and 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 BaO 3 planes.It consists of a supercell accommodating 24 atoms (Figure 11b).Both these Zr-and BaO 3 -terminated BZO (111) slabs are symmetrical and nonstoichiometric (Figure 11).They have the unit-cell formulas Ba 4 Zr 5 O 12 and Ba 5 Zr 4 O 15 , 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 BaO 3 -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 BaZrO 3 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 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].With the ultimate goal of computing the ABO 3 -type perovskite, for example, the PbZrO 3 (001) surface energy, we started our B3LYP computations with the cleavage energy calculations for unrelaxed PbO-as well as ZrO 2 -terminated (001) surfaces [1][2][3]94].Surfaces with both PbO and ZrO 2 (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 (Figures 1 and 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 ZrO 2 -terminated PZO (001) slab, containing 23 atoms, together contain nine bulk unit cells or 45 atoms atoms, thus: where ϑ means PbO or ZrO 2 ; E slab unr (ϑ) is the SrO-or ZrO 2 -terminated PZO (001) slab energies without relaxation; E bulk is the PZO bulk unit cell, containing five atoms, total energy; and the factor of 1 /4 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 ZrO 2 -terminated PZO (001) slabs [1][2][3]75,82], using the following equation: 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: 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: where ϑ indicates Pb or ZrO; E slab unr (ϑ) is our computed total energy for the unrelaxed Pbor ZrO-terminated PZO (011) slabs; and E bulk is our computed PbZrO 3 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: 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 ABO 3 perovskite polar (111) surface energy computation details are described by us in Refs.[5,99,107].
Our B3PW or B3LYP computed bulk effective atomic charges Q and bond populations P for all eight ABO 3 -type perovskites are collected in Table 2.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 ABO 3 -type perovskite materials (Table 2).As we can see from Table 2, our B3PW or B3LYP computed effective atomic charges Q [127 -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 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 hightemperature 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 ABO 3 -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 ABO 3 -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 ABO 3 perovskites, whereas the density functional theory based PWGGA functional underestimated them (Figure 14 and Table 3).CaZrO 3 B3LYP 5.40 [120] No data for cubic phase PbZrO 3 B3LYP 5.63 [94] No data for cubic phase SrZrO 3 B3PW 5.30 [75] No data for cubic phase B3LYP 5.31 [

ABO 3 Perovskite (001) Surface Atomic and Electronic Structure
Our hybrid exchange-correlation functional B3LYP or B3PW computation results for the (001) surface atomic relaxations for BO 2 -as well as AO-terminated ABO 3 -type perovskite upper three or two (001) surface layers are recorded in Tables 4 and 5.As it is possible to see from Tables 4 and 5, the atomic relaxation magnitudes of surface metal atoms A or B, for all eight ABO 3 perovskite (001) surface upper two layers, are almost always noticeably larger than that for the respective O atoms (Tables 4 and 5).This leads to a significant surface rumpling s for the upper-surface plane (Table 6).The only two deviations from this systematic trend are the ZrO 2 -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 BO 2 terminations of all eight ABO 3 perovskite (001) surfaces, as a rule, all atoms of the first (upper) surface layer relax inwards towards the ABO 3 perovskite bulk (Tables 4 and 5).At the same time, all atoms of the second surface layer, for both AO and BO 2 (001) surface terminations, relax upwards (Tables 4 and 5).Again, all third-layer atoms, the same as upper-layer atoms, relax inwards, towards the ABO 3 perovskite bulk (Tables 4 and 5).There are only three exceptions to this systematic trend (Tables 4 and 5).Namely, TiO 2 -terminated PTO (001) surface upper-layer O atom relaxes upwards by +0.31% of a 0 (Table 5); SrO-terminated STO (001) surface upper-layer O atom relaxes upwards by +0.84% of a 0 , whereas the second-layer O atom on the SrOterminated SrZrO 3 (001) surface relaxes inwards by a very small relaxation magnitude equal to −0.05% of a 0 (Table 4).B3PW computed [1][2][3]134,135] as well as experimental [136,137] results, dealing with ABO 3 -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 ∆d 12 and ∆d 23 [2,134] (Table 6).Moreover, our B3PW computed [2] surface rumplings s for SrO as well as TiO 2 -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 ∆d 12 and ∆d 23 fail to agree with the LEED [136] experimental measurements for the TiO 2 -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 ∆d 12 for the SrO-terminated STO (001) surface.Also, for the TiO 2 -terminated STO (001) surface, LEED [136] and RHEED [137] experiments disagree regarding the sign of the interlayer distance ∆d 23 .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 ∆d 12 and ∆d 23 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 ∆d 12 and ∆d 23 are in good agreement with each other for the ZrO 2 -terminated SZO (001) surface.The agreement between our B3LYP [94] and Wang et al.'s [138] LDA computed surface rumpling s for the ZrO 2 -terminated SZO (001) surface (−0.72% of a 0 and −0.7% of a 0 ) is almost perfect.Unfortunately, the surface rumpling s, computed by Wang et al., using the GGA exchange-correlation functional [138] for the ZrO 2 -terminated SZO (001) surface has a different sign of +0.3% of a 0 (Table 6).
As we can see from Table 7 and Figure 15, our B3PW or B3LYP computed eight ABO 3type perovskite (001) surface energies are always around 1 eV.Namely, our largest computed (001) surface energy is for the ZrO 2 -terminated CaZrO 3 (001) surface (1.33 eV) [120], whereas the smallest is for the TiO 2 -terminated PbTiO 3 (001) surface (0.74 eV) [1].The smallest energy difference, according our B3PW computations, is for the BaZrO 3 ZrO 2 -(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 CaZrO 3 perovskite ZrO 2 − (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 BO 2 -terminated (001) surface was slightly smaller for the BTO, PTO and PZO perovskites; thus, it is more stable (Table 7 and Figure 15).(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 Figures 12 and 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.As we can see from Tables 3 and 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) Our B3PW computed electronic bulk band structures for BTO, PTO as well as BZO perovskites are illustrated in Figures 12 and 13.Our B3PW computed TiO 2 -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 ZrO 2 -(b) terminated BZO (001) surfaces are illustrated in Figure 18.Our computed Γ-Γ band gap numerical values for all eight of our computed ABO 3 perovskite bulk as well as their BO 2 -and AO-terminated (001) surfaces are collected in Table 8.As we can see from Tables 3 and 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 (Tables 3 and 8).The key effect there, as we can see from Table 8 and Figure 19, is that the ABO 3 perovskite bulk Γ-Γ band gap, for all eight of our B3PW or B3LYP computed ABO 3 perovskites, is always reduced near their AO-and BO 2 -terminated (001) surfaces.For example, our B3PW computed BZO bulk Γ-Γ band gap (4.93 eV) (Figure 8) is reduced near the BZO ZrO 2 -terminated (001) surface (4.48 eV) as well as near the AO-terminated BZO (001) surface (4.82 eV) (Table 8 and Figures 18 and 19).Also, for all of our other eight computed ABO 3 perovskites, the situation is similar, regarding the reduction of the ABO 3 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 (3.49eV) (Figure 17a) and TiO 2 -terminated BTO (001) surface (2.96 eV) (Figures 16a and 19).As we can see from Table 9 and Figure 20, for all eight of our B3LYP or B3PW computed ABO 3 perovskites, we can observe the significant increase in the B-O chemical bond covalency near their BO 2 -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 TiO 2 -terminated (001) surfaces [2,3].Just opposite, the smallest B-O chemical bond population increase is observed for the PbZrO 3 perovskite [94].Namely, the PbZrO 3 perovskite Zr-O chemical bond population increased from 0.106e (bulk case) to 0.116e near the ZrO 2 -terminated PbZrO 3 (001) surface [94] (Table 9 and Figure 20).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).

ABO3 Perovskite (011) Surface Atomic and Electronic Structure
As we can see from Table 10 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).

ABO 3 Perovskite (011) Surface Atomic and Electronic Structure
As we can see from Table 10 and Figure 21, for all eight of our B3LYP or B3PW computed ABO 3 perovskites, the systematic tendency is that for all three of their BO-, Aand 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 ABO 3 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 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).It is worth noting that the biggest relaxation magnitude between all upper-layer ABO 3 perovskite (011) surface atoms, for all three possible (011) surface terminations, demonstrates the Ca-terminated surface Ca atom shifting inwards by −18.67% of a 0 (Table 10 and Figure 16) [10].It is around three times bigger than the displacement magnitudes for the Zr atom (+6.06% of a 0 ) on the ZrO-terminated as well as O atom (+5.97% of a 0 ) on the O-terminated CZO (011) surfaces (Table 10 and Figure 21).
As we can see from Table 11 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 ABO 3 perovskites (Table 11 and Figure 22).Namely, according to our B3PW or B3LYP computations for eight ABO 3 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 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 ZrO2terminated 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 BO2or AO-terminated ABO3 perovskite (001) surface energies (Figure 19 and Table 12).As we can see from Table 12 and Figure 23, our B3LYP or B3PW computed ABO 3 perovskite A-, O-or BO-terminated polar (011) surface energies are always larger than the ABO 3 perovskite neutral BO 2 -or AO-terminated (001) surface energies.According to our B3LYP computations, the largest ABO 3 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 ABO 3 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 ZrO 2terminated CaZrO 3 (001) surface energy (1.33 eV).Nevertheless, according to our B3PW or B3LYP computations, the BO-, O-or A-terminated polar ABO 3 perovskite (011) surface energies are always larger than the neutral BO 2 -or AO-terminated ABO 3 perovskite (001) surface energies (Figure 19 and Table 12).As it is possible to see from Table 13 and Figure 24, according to our B3LYP or B3PW computation results for eight ABO 3 perovskites, the B-O chemical bond population is the smallest for the ABO 3 perovskite bulk.The B-O chemical bond population is increased near the BO 2 -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) ABO 3 perovskite surface.Namely, the plane B(I)-O(I) chemical bond population for all eight of our computed ABO 3 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 ABO 3 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).

Term.
Esurf  As it is possible to see from Table 13 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).

ABO3 Perovskite (111) Surface Atomic and Electronic Structure
As it is possible to see from Table 14, 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,

ABO 3 Perovskite (111) Surface Atomic and Electronic Structure
As it is possible to see from Table 14, according to our B3LYP computation results for seven ABO 3 perovskites, all atoms on the B-terminated ABO 3 perovskite (111) surface relax inwards.The upper-layer B atom relaxation magnitudes (Table 14) are rather strong, ranging from −3.58% of a 0 for the STO perovskite to −11.19% of a 0 for the BTO perovskite.It is worth noting that almost all second-layer A atoms on the B-terminated ABO 3 perovskite (111) surface relax inwards.In general, they exhibit very large relaxation magnitudes, for example, −14.02% of a 0 for the second-layer Ca atom on the Ti-terminated CaTiO 3 (111) surface (Table 14).As we can see from Table 15, according to our B3LYP computations, most of AO 3 -terminated ABO 3 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 ABO 3 perovskite upper layer (Tables 14 and 15).As it is possible to see from Table 16, according to our B3LYP computation results for seven ABO 3 perovskites, the B-terminated ABO 3 perovskite (111) surface energies are always smaller than the respective AO 3 -terminated ABO 3 perovskite (111) surface energies.The B-terminated ABO 3 perovskite (111) surface energies (Table 16) are in the energy range of 4.18 eV for the Ti-terminated CaTiO 3 (111) surface to 8.19 eV for the Zr-terminated CaZrO 3 (111) surface.The AO 3 -terminated ABO 3 perovskite (111) surface energies are in the range of 5.86 eV for the CaO 3 -terminated CaTiO 3 (111) surface to 9.62 eV for the CaO 3 -terminated CaZrO 3 (111) surface (Table 16).

Figure 1 .
Figure 1.Profile for the BO2-terminated (001) surface of ABO3-type perovskite accommodating nine layers and containing the definition of the surface rumpling s as well as the near-surface interplane distances Δd12 and Δd23.

Figure 1 .
Figure 1.Profile for the BO 2 -terminated (001) surface of ABO 3 -type perovskite accommodating nine layers and containing the definition of the surface rumpling s as well as the near-surface interplane distances ∆d 12 and ∆d 23 .

Figure 1 .
Figure 1.Profile for the BO2-terminated (001) surface of ABO3-type perovskite accommodating nine layers and containing the definition of the surface rumpling s as well as the near-surface interplane distances Δd12 and Δd23.

Figure 3 .
Figure 3. Sketch of the cubic ABO3 perovskite construction, containing two (011) cleavage p consisting of charged O2 as well as ABO (011) surfaces.

Figure 12 .
Figure 12.Our hybrid B3PW computed [114] bulk electronic band structure for BTO (a) as well as PTO (b) perovskites.The dotted lines correspond to the bulk valence band maximum.

Figure 12 .Figure 12 .
Figure 12.Our hybrid B3PW computed [114] bulk electronic band structure for BTO (a) as well as PTO (b) perovskites.The dotted lines correspond to the bulk valence band maximum.

Figure 17 .
Figure 17.Our B3PW simulated electronic band structures for AO-terminated ABO3 perovskite surfaces of (a) BTO and (b) PTO.

Table 6 .
B3LYP or B3PW computed as well as experimentally detected surface rumpling s and respective atomic displacements ∆d 12 and ∆d 23 (% of a 0 ) for the BO 2 -and AO-terminated (001) surfaces of eight ABO 3 perovskites.

Table 9 .
B3LYP or B3PW computed B-O bond populations for eight ABO 3 perovskites bulk and also for their BO 2 -terminated (001) surfaces (in e).

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).