Systematic Trends in Hybrid-DFT Computations of BaTiO3/SrTiO3, PbTiO3/SrTiO3 and PbZrO3/SrZrO3 (001) Hetero Structures

We performed predictive hybrid-DFT computations for PbTiO3, BaTiO3, SrTiO3, PbZrO3 and SrZrO3 (001) surfaces, as well as their BaTiO3/SrTiO3, PbTiO3/SrTiO3 and PbZrO3/SrZrO3 (001) heterostructures. According to our hybrid-DFT computations for BO2 and AO-terminated ABO3 solid (001) surfaces, in most cases, the upper layer ions relax inwards, whereas the second layer ions shift upwards. Our hybrid-DFT computed surface rumpling s for the BO2-terminated ABO3 perovskite (001) surfaces almost always is positive and is in a fair agreement with the available LEED and RHEED experiments. Computed B-O atom chemical bond population values in the ABO3 perovskite bulk are enhanced on its BO2-terminated (001) surfaces. Computed surface energies for BO2 and AO-terminated ABO3 perovskite (001) surfaces are comparable; thus, both (001) surface terminations may co-exist. Our computed ABO3 perovskite bulk Γ-Γ band gaps are in fair agreement with available experimental data. BO2 and AO-terminated (001) surface Γ-Γ band gaps are always reduced with regard to the respective bulk band gaps. For our computed BTO/STO and PTO/STO (001) interfaces, the average augmented upper-layer atom relaxation magnitudes increased by the number of augmented BTO or PTO (001) layers and always were stronger for TiO2-terminated than for BaO or PbO-terminated upper layers. Our B3PW concluded that BTO/STO, as well as SZO/PZO (001) interface Γ-Γ band gaps, very strongly depends on the upper augmented layer BO2 or AO-termination but considerably less so on the number of augmented (001) layers.


Hybrid-DFT Calculation Details
We carried out our hybrid-DFT computations for the PbTiO 3 (PTO), BaTiO 3 (BTO), SrTiO 3 (STO), PbZrO 3 (PZO) and SrZrO 3 (001) bulk, as well as their BO 2 -and AOterminated (001) surfaces, and their different (001) heterostructures, employing the computer code CRYSTAL [97]. For our hybrid-DFT computations, we employed the nowadays very popular hybrid exchange-correlation functionals B3LYP [98] or B3PW [99,100]. It is worthwhile to note that the hybrid exchange-correlation functionals, for example, B3LYP or B3PW allows achieving an outstanding agreement with the experiment for the band gaps of ABO 3 perovskites as well as related materials, like CaF 2 [101] and MgF 2 [102]. In contrast, density functional theory (DFT) strongly underestimates, while the Hartree-Fock (HF) method, as a rule, very strongly overestimates the ab initio calculated band gaps of solids. For example, the experimental CaF 2 direct band gap value at the Γ-point is equal to 12.1 eV [103]. Our, by means of Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional [104,105], extensively used in DFT, ab initio computed CaF 2 Γ-Γ band gap is very strongly underestimated with respect to the experimental value (12.1 eV) and is equal to 8.45 eV. From another side, our ab initio HF [106] computed CaF 2 Γ-Γ band gap (20.77 eV) is 1.72 times overestimated compared to the experimental CaF 2 band gap value (12.1 eV [105]). According to our hybrid-DFT computations, the hybrid exchange correlation functionals, like B3LYP (10.85 eV) and B3PW (10.96 eV) allows the achievement of the best possible agreement with the experimental CaF 2 Γ-Γ band gap equal to 12.1 eV. The experimentally measured MgF 2 Γ-Γ band gap is equal to 13.0 eV [107]. Again, our PBE exchange-correlation functional computed MgF 2 Γ-Γ band gap is considerably underestimated regarding the experimental value of (13.0 eV) and is equal to 6.9 eV. In contrast, our ab initio HF computed MgF 2 Γ-Γ band gap (19.65 eV) very strongly, namely 1.65 times, overestimates the experimentally measured MgF 2 band gap at Γ-point (13.0 eV). The best possible results, according to our hybrid-DFT computations, are possible to achieve by means of the hybrid exchange-correlation functionals B3LYP (9.42 eV) and B3PW (9.48 eV). Since, as it is possible to see from our computation results for CaF 2 and MgF 2 solids, the hybrid exchange-correlation functionals B3LYP and B3PW allows achieving the best possible results for CaF 2 and MgF 2 Γ-Γ band gaps, we performed all of our coming ABO 3 perovskite (001) surface and interface computations using the B3LYP or B3PW functionals. The B3LYP and B3PW hybrid exchange-correlation functionals make use of 20% of the Hartree-Fock method as well as 80% of the density functional Hamiltonian, when incorporated in the CRYSTAL computer code [97].
The major advantage of the CRYSTAL computer software package, which is of key importance for our hybrid-DFT computations of neutral SrZrO 3 , PbZrO 3 , PbTiO 3 , BaTiO 3 and SrTiO 3 (001) surfaces and interfaces, is the implementation of the isolated 2-dimensional (2D) slab model. In our hybrid-DFT computations, the reciprocal space integration was carried out by examining the Brillouin zone with the 8 × 8 × 8 times expanded Pack-Monkhorst net for the bulk of SrZrO 3 , PbZrO 3 , PbTiO 3 , BaTiO 3 and SrTiO 3 solids, as well as by 8 × 8 × 1 net for their (001) surfaces and interfaces. With a goal of reaching a high accuracy in our ab initio computations, we employed sufficiently large tolerances equal to 7,8,7,7,14 for the Coulomb overlap, Coulomb penetration and exchange overlap, as well as first exchange pseudo-overlap and the second exchange pseudo-overlap [97].
With the aim of computing the neutral BO 2 -terminated (001) surfaces of ABO 3 solids, we customized symmetrical slabs. They involved nine neutral as well as alternating BO 2 and AO layers ( Figure 1). All of these nine layers were perpendicular to the [001] crystal direction [108][109][110][111]. Taking into consideration the standard ionic charges equal to (+2e) for the A atom and (+4e) for the B atom, as well as (−2e) for the O atom, both alternating BO 2 and AO layers are neutral, since they have a summary slab charge equal to zero. The slab, containing nine layers, used by us for the BO 2 -terminated ABO 3 perovskite (001) surface computations (Figure 1) consisted of a supercell containing 23 atoms. Our hybrid-DFT computed BO 2 -terminated (001) slabs ( Figure 1) of ABO 3 perovskite were non-stoichiometric. They have a unit cell, used in our hybrid-DFT computations, described by the following chemical formula-A 4 B 5 O 14 [112][113][114].
Condens. Matter 2022, 7, x FOR PEER REVIEW 4 of 20 The major advantage of the CRYSTAL computer software package, which is of key importance for our hybrid-DFT computations of neutral SrZrO3, PbZrO3, PbTiO3, BaTiO3 and SrTiO3 (001) surfaces and interfaces, is the implementation of the isolated 2-dimensional (2D) slab model. In our hybrid-DFT computations, the reciprocal space integration was carried out by examining the Brillouin zone with the 8 × 8 × 8 times expanded Pack-Monkhorst net for the bulk of SrZrO3, PbZrO3, PbTiO3, BaTiO3 and SrTiO3 solids, as well as by 8 × 8 × 1 net for their (001) surfaces and interfaces. With a goal of reaching a high accuracy in our ab initio computations, we employed sufficiently large tolerances equal to 7,8,7,7,14 for the Coulomb overlap, Coulomb penetration and exchange overlap, as well as first exchange pseudo-overlap and the second exchange pseudo-overlap [97].
With the aim of computing the neutral BO2-terminated (001) surfaces of ABO3 solids, we customized symmetrical slabs. They involved nine neutral as well as alternating BO2 and AO layers ( Figure 1). All of these nine layers were perpendicular to the [001] crystal direction [108][109][110][111]. Taking into consideration the standard ionic charges equal to (+2e) for the A atom and (+4e) for the B atom, as well as (−2e) for the O atom, both alternating BO2 and AO layers are neutral, since they have a summary slab charge equal to zero. The slab, containing nine layers, used by us for the BO2-terminated ABO3 perovskite (001) surface computations (Figure 1) consisted of a supercell containing 23 atoms. Our hybrid-DFT computed BO2-terminated (001) slabs ( Figure 1) of ABO3 perovskite were non-stoichiometric. They have a unit cell, used in our hybrid-DFT computations, described by the following chemical formula-A4B5O14 [112][113][114]. The second slab in our hybdid-DFT computations was terminated from both sides by the AO planes ( Figure 2). It also contained nine alternating layers ( Figure 2) but consisted of a supercell that incorporated only 22 atoms. Furthermore, the AO-terminated ninelayer slab, used in our hybrid-DFT computations, was non-stoichiometric and has the following chemical formula-A5B4O13. For Sr, Ba, Pb, Ti and O atoms [115], we used the basis sets developed in Reference 117. The inner-core electrons for Sr, Ba, Pb, Ti and Zr atoms were described by a small core Hay-Wadt effective pseudopotentials [97,116]. We computed the number of light oxygen atoms using the all-electron basis sets [97,115]. In order to correctly define the chemical bonding in ABO3 perovskites, as well as covalency effects, The second slab in our hybdid-DFT computations was terminated from both sides by the AO planes ( Figure 2). It also contained nine alternating layers ( Figure 2) but consisted of a supercell that incorporated only 22 atoms. Furthermore, the AO-terminated nine-layer slab, used in our hybrid-DFT computations, was non-stoichiometric and has the following chemical formula-A 5 B 4 O 13 . For Sr, Ba, Pb, Ti and O atoms [115], we used the basis sets developed in [117]. The inner-core electrons for Sr, Ba, Pb, Ti and Zr atoms were described by a small core Hay-Wadt effective pseudopotentials [97,116]. We computed the number of light oxygen atoms using the all-electron basis sets [97,115]. In order to correctly define the chemical bonding in ABO 3 perovskites, as well as covalency effects, we employed a widely accepted Mulliken population [117][118][119] analysis. The Mulliken population analysis is incorporated in the CRYSTAL computer code [97], used in all of our hybrid-DFT computations for the ABO 3 solid (001) surfaces and their respective heterostructures ( Figure 3).
we employed a widely accepted Mulliken population [117][118][119] analysis. The Mulliken population analysis is incorporated in the CRYSTAL computer code [97], used in all of our hybrid-DFT computations for the ABO3 solid (001) surfaces and their respective heterostructures ( Figure 3).   we employed a widely accepted Mulliken population [117][118][119] analysis. The Mulliken population analysis is incorporated in the CRYSTAL computer code [97], used in all of our hybrid-DFT computations for the ABO3 solid (001) surfaces and their respective heterostructures ( Figure 3).   With ambition to compute the ABO 3 solid (001) surface energies, we opened our hybid-DFT computations with cleavage energies for unrelaxed AO-as well as BO 2 -terminated (001) surfaces. In our hybrid-DFT computations, the two nine-layer AO-as well as BO 2 -terminated slabs contained 22 and 23 atoms of each of them. These two AO-and BO 2 -terminated slabs represented together nine ABO 3 perovskite bulk unit cells. Each ABO 3 perovskite bulk unit cell contained five atoms. AO as well as BO 2 -terminated ABO 3 perovskite (001) surfaces arose at the same time moment under crystal cleavage. Therefore, the relevant cleavage energy is shared uniformly among created surfaces. Thereby, our ab initio computed ABO 3 perovskite (001) surface cleavage energy is equal for both AO-as well as BO 2 -terminations: where E slab unrel (AO) and E slab unrel (BO 2 ) are our hybrid-DFT computed total energies for unrelaxed AO-and BO 2 -terminated ABO 3 perovskite nine-layer slabs. E bulk is our hybrid-DFT computed total bulk unit cell energy containing five atoms. Factor 4 in Equation (1) arises from the fact that we created four surfaces during the ABO 3 perovskite cleavage event. As a next step, we computed the relaxation energies for AO-and BO 2 -terminated ABO 3 solid (001) surfaces. We relaxed both sides of our nine-layer AO-and BO 2 -terminated slabs.
E slab (Ψ) is our hybrid-DFT computed slab total energy after the geometry relaxation. In our case the symbol Ψ denotes AO-or BO 2 -terminated ABO 3 perovskite (001) surface. After all of our hybrid-DFT computed ABO 3 perovskite AO-or BO 2 -terminated (001) surface energy is defined as a sum of the cleavage as well as relaxation energies: As a next step, we will discuss our calculation details for the ABO 3 perovskite (001) heterostructures, using as an example the BaTiO 3 /SrTiO 3 (001) interface. At room temperature, the SrTiO 3 substrate has a high-symmetry cubic structure. In our hybrid-DFT computations, we computed both SrTiO 3 and BaTiO 3 perovskites at their cubic, high-symmetry phase with the space group number Pm3m. In our hybrid-DFT computations, we modelled the BaTiO 3 /SrTiO 3 (001) interface employing the single-slab model. In order to maximally apply the advantages of symmetry, our slabs were symmetrically terminated. In our hybrid-DFT computations, the SrTiO 3 (001) substrate always contained 11 alternating SrO as well as TiO 2 layers. From 1 to 10 BaO and TiO 2 alternating layers were augmented on both sides of the 11-layer TiO 2 -terminated SrTiO 3 (001) substrate ( Figure 3).
We allowed all coordinated of atoms to relax in our hybrid-DFT computed BaTiO 3 /SrTiO 3 (001) heterostructure. In our hybrid-DFT computations, due to restrictions imposed by the cubic symmetry of the system, atomic displacements are possible only along axis z. It is worthwhile to note that the mismatch equal to approximately 2.5 percent among BaTiO 3 and SrTiO 3 bulk lattice constants happens at the time of BaTiO 3 epitaxial growth. The joint equilibrium average lattice constant employed in all of our future hybrid-DFT computations for BaTiO 3 /SrTiO 3 (001) interfaces is equal to 3.958 Å. The joint lattice constant for the BaTiO 3 /SrTiO 3 (001) interface was hybrid-DFT computed by us for the thickest (001) interface, which consisted of the 11-layer SrTiO 3 substrate as well as, from both sides of this substrate, augmented 10 BaTiO 3 layers. In our hybrid-DFT calculations of the shift (∆z) for each layer of the BaTiO 3 /SrTiO 3 (001) heterostructure, we take into consideration the shift of the previous atomic layer. Therefore, the reference z coordinate for each mono-layer N is described by the following equation: where, in Equation (4), z N-1 Me as well as z N-1 O defines the z coordinates for the cation and anion from the previous atomic monolayer.
As a next step, we computed the bulk band gaps at the Γ-point for PbTiO 3 , BaTiO 3 and SrTiO 3 perovskites and plotted the band structure for BaTiO 3 and PbTiO 3 crystals. We compared our hybrid-DFT computation results for Γ-Γ band gaps with available experimental results. Our B3PW computed SrTiO 3 bulk Γ-Γ band gap is equal to 3.96 eV, which is in fair agreement with the available experimental data for SrTiO 3 direct Γ-Γ bulk band gap equal to 3.75 eV [124] ( Table 2). The direct Γ-Γ bulk band gap in the BaTiO 3 perovskite is measured experimentally ( Table 2) at the tetragonal to orthorhombic phase transition temperature equal to 278 K. The BaTiO 3 direct bulk Γ-Γ band gap in different experimental conditions is slightly different and is equal to 3.27 eV or 3.38 eV, respectively [125]. Our B3PW computed BaTiO 3 direct bulk Γ-Γ band gap (3.55 eV) ( Table 2) is in almost perfect agreement with these experiments (Figure 4a). Finally, our B3PW computed PbTiO 3 bulk band gap at Γ-point (4.32 eV) (Figure 4b) ( Table 2) is in satisfactory agreement with the experimentally detected direct PbTiO 3 bulk Γ-Γ band gap (3.4 eV) [126] at the PbTiO 3 perovskite tetragonal phase.
As we can see from Table 3, according to our hybrid-DFT computations, all upper-layer atoms for all five perovskites relax inwards in the direction towards the perovskite bulk. The only exception from this systematic trend is the upward relaxation of the TiO 2 -terminated PbTiO 3 (001) surface upper-layer O atom by 0.31 % of a 0 . Just opposite, all second-layer atoms relax upwards. For both upper layers, the metal atom relaxation magnitudes are always larger than the respective oxygen atom relaxation magnitude [127][128][129][130]. The only exception to this systematic trend is the ZrO 2 -terminated SrZrO 3 (001) surface upperlayer oxygen atom relaxation magnitude (−2.10% of a 0 ), which is slightly larger than the same-layer Sr atom relaxation magnitude (−1.38% of a 0 ) ( Table 3).   As we can see from Table 3, according to our hybrid-DFT computations, all upperlayer atoms for all five perovskites relax inwards in the direction towards the perovskite bulk. The only exception from this systematic trend is the upward relaxation of the TiO2terminated PbTiO3 (001) surface upper-layer O atom by 0.31 % of a0. Just opposite, all second-layer atoms relax upwards. For both upper layers, the metal atom relaxation magnitudes are always larger than the respective oxygen atom relaxation magnitude [127][128][129][130]. The only exception to this systematic trend is the ZrO2-terminated SrZrO3 (001) surface upper-layer oxygen atom relaxation magnitude (−2.10% of a0), which is slightly larger than the same-layer Sr atom relaxation magnitude (−1.38% of a0) ( Table 3).
In most cases, as it is possible to see from our hybrid-DFT computation results collected in Table 4 for AO-terminated PbTiO3, BaTiO3, SrTiO3, PbZrO3 and SrZrO3 (001) surfaces, all upper-layer atoms relax inwards (Table 4), whereas all second-layer atoms relax upwards. The only two exceptions from this systematic trend are the upward relaxation of O atom on the upper layer of SrO-terminated SrTiO3 (001) surface by (+0.84% of a0), as In most cases, as it is possible to see from our hybrid-DFT computation results collected in Table 4 for AO-terminated PbTiO 3 , BaTiO 3 , SrTiO 3 , PbZrO 3 and SrZrO 3 (001) surfaces, all upper-layer atoms relax inwards (Table 4) (Table 4). It is interesting to notice that for the AO-terminated PbTiO 3 , BaTiO 3 , SrTiO 3 , PbZrO 3 and SrZrO 3 (001) surfaces, the relaxation magnitudes of metal atoms in the upper, as well as the second, surface layers are always larger than the relaxation magnitudes of the respective oxygen atoms ( Table 4).
Our B3PW computed bulk band gaps ( Figure 4) for PbTiO 3 , BaTiO 3 and SrTiO 3 perovskite bulk (4.32 eV, 3.55 eV and 3.96 eV, respectively) are in fair agreement with the available experimental data [124,125,134] (Table 8). As we can see from Table 8 and Figures 5 and 6, our B3PW computed (001) surface band gaps for all three perovskites and both (001) surface terminations are always reduced with respect to the PbTiO 3 , BaTiO 3 and SrTiO 3 bulk. Namely, our hybrid-DFT computed TiO 2 -terminated PbTiO 3 , BaTiO 3 and SrTiO 3 (001) surface band gaps are equal to 3.18 eV, 2.96 eV ( Figure 5) and 3.95 eV, respectively. At the same time, our hybrid-DFT computed AO-terminated PbTiO 3 , BaTiO 3 and SrTiO 3 (001) surface band gaps are equal to 3.58 eV, 3.49 eV ( Figure 6) and 3.72 eV, respectively, and are also smaller than the respective bulk band gap values at Γ-point (Table 8).    Our B3PW computed bulk band gaps ( Figure 4) for PbTiO3, BaTiO3 and SrTiO3 perovskite bulk (4.32 eV, 3.55 eV and 3.96 eV, respectively) are in fair agreement with the available experimental data [124,125,134] (Table 8). As we can see from Table 8 (Figure 6) and 3.72 eV, respectively, and are also smaller than the respective bulk band gap values at Γ-point (Table 8).  Our B3PW computed bulk band gaps ( Figure 4) for PbTiO3, BaTiO3 and SrTiO3 perovskite bulk (4.32 eV, 3.55 eV and 3.96 eV, respectively) are in fair agreement with the available experimental data [124,125,134] (Table 8). As we can see from Table 8 (Figure 6) and 3.72 eV, respectively, and are also smaller than the respective bulk band gap values at Γ-point (Table 8).

BaTiO 3 /SrTiO 3 , PbTiO 3 /SrTiO 3 and SrZrO 3 /PbZrO 3 (001) Interfaces
In order to start our hybrid-DFT computations for BTO/STO, PTO/STO and SZO/PZO (001) heterostructures, we computed their joint lattice constants. Our hybrid-DFT computed joint lattice constant for BTO/STO (001) interface is equal to 3.958 Å. This joint BTO/STO (001) interface joint lattice constant was computed for the system containing an 11-layer thick STO substrate, as well as 10 BTO monolayers augmented on both sides of the STO substrate ( Figure 3). Thereby, our B3PW computed thickest BTO/STO (001) interface contained 31 monolayers, as well as 78 atoms (Figure 3) [135]. Using a similar model for the PTO/STO (001) thickest interface, our B3PW computed joint lattice constant for this system is equal to 3.91 Å. Finally, our hybrid-DFT computed SZO/PZO (001) interface joint lattice constant for the system containing an 11-layer thick SZO substrate, as well as 10 PZO monolayers augmented symmetrically on both sides of the substrate, is equal to 4.167 Å.
In our hybrid-DFT (001) heterostructure computations, we relaxed all atomic positions only alongside the z-axis, due to the symmetry constraints for the cubic ABO 3 perovskite matrixes. We computed the atomic shifts ∆z regarding the averaged coordinate z of the former atomic layer as described in Equation (4). As we can see from Figure 7, our B3PW computed upper SrTiO 3 substrate (001) layer, which contains Ti and O atoms (x = 0) in (Figure 7), relaxes very strongly inside (−5.95% of a 0 ). According to our B3PW computations, upper BTO (001) layer atoms, augmented on the STO 11-layer (001) substrate, always very strongly shifts (∆z) inwards (x = 1-10 in Figure 7). It is worthwhile to note that the BTO upper-layer atoms inwards-relaxation numerical value ∆z considerably depends on the number of the STO (001) 11-layer substrate augmented BTO (001) layers.

BaTiO3/SrTiO3, PbTiO3/SrTiO3 and SrZrO3/PbZrO3 (001) Interfaces
In order to start our hybrid-DFT computations for BTO/STO, PTO/STO and SZO/PZO (001) heterostructures, we computed their joint lattice constants. Our hybrid-DFT computed joint lattice constant for BTO/STO (001) interface is equal to 3.958 Å. This joint BTO/STO (001) interface joint lattice constant was computed for the system containing an 11-layer thick STO substrate, as well as 10 BTO monolayers augmented on both sides of the STO substrate ( Figure 3). Thereby, our B3PW computed thickest BTO/STO (001) interface contained 31 monolayers, as well as 78 atoms (Figure 3) [135]. Using a similar model for the PTO/STO (001) thickest interface, our B3PW computed joint lattice constant for this system is equal to 3.91 Å. Finally, our hybrid-DFT computed SZO/PZO (001) interface joint lattice constant for the system containing an 11-layer thick SZO substrate, as well as 10 PZO monolayers augmented symmetrically on both sides of the substrate, is equal to 4.167 Å.
In our hybrid-DFT (001) heterostructure computations, we relaxed all atomic positions only alongside the z-axis, due to the symmetry constraints for the cubic ABO3 perovskite matrixes. We computed the atomic shifts Δz regarding the averaged coordinate z of the former atomic layer as described in Equation (4). As we can see from Figure 7, our B3PW computed upper SrTiO3 substrate (001) layer, which contains Ti and O atoms (x = 0) in (Figure 7), relaxes very strongly inside (−5.95% of a0). According to our B3PW computations, upper BTO (001) layer atoms, augmented on the STO 11-layer (001) substrate, always very strongly shifts (Δz) inwards (x = 1-10 in Figure 7). It is worthwhile to note that the BTO upper-layer atoms inwards-relaxation numerical value Δz considerably depends on the number of the STO (001) 11-layer substrate augmented BTO (001) layers.  In the case of one augmented BTO (001) layer, consisting of the BaO atoms, the atom relaxation magnitude ∆z is equal to −1.54% of the joint lattice constant a 0 (Figures 3 and 7). For two augmented BTO (001) layers, where the upper layer contains TiO 2 atoms, the upper-layer atom relaxation magnitude ∆z is equal to (−3.20% of a 0 ) (Figure 7). For three augmented BTO (001) layers, the upper BaO layer atom relaxation magnitude ∆z is equal to (−1.84% of a 0 ), for 4 layers (−3.55% of a 0 ), for five layers (−2.07% of a 0 ), for six layers (−3.70% of a 0 ), for seven layers (−2.21% of a 0 ), for eight layers (−3.84% of a 0 ), for nine layers (−2.32% of a 0 ) and finally, for ten augmented BTO (001) layers, the upper TiO 2 layer atom relaxation magnitude ∆z is equal to (−3.92% of a 0 ) ( Figure 7).
As we can see from Figure 10a,b, our hybrid-DFT computed Γ-Γ band gap for BTO/STO, as well as SZO/PZO (001) interface Γ-Γ band gap, very strongly depends on the upper augmented layer BO2or AO-termination but considerably less so on the number of augmented (001) layers.
Our hybrid-DFT computed surface rumpling s for the BO2-terminated ABO3 perovskite (001) surfaces (Table 5) is positive for most computed ABO3 perovskites, such as
For our B3PW computed BTO/STO as well as PTO/STO (001) interfaces, the average augmented upper-layer atom relaxation magnitudes increased by the number of augmented BTO or PTO (001) layers but always independently from the number of augmented layers, which were stronger for TiO 2 -terminated than BaO-or PbO-terminated upper augmented layers. All of our B3PW computed PTO/STO (001) interface upper augmented layer average atom displacement magnitudes ∆z are between (−6.01% of a 0 ) for the first augmented layer and (−8.54% of a 0 ) for 10 augmented layers. In contrast to the PTO/STO (001) interfaces, for our hybrid-DFT computed BTO/STO (001) interfaces, the upper augmented layer average atom displacement magnitudes ∆z are considerably smaller, and they are in the range between the (−1.54% of a 0 (1 layer) to −3.92% of a 0 (10 layers), respectively). Our B3PW computed BTO/STO, as well as SZO/PZO (001) interface Γ-Γ band gaps, very strongly depends on the upper augmented layer BO 2 -or AO-termination but considerably less so on the number of augmented (001) layers [135,139].
Summing up, all of our hybrid-DFT computed BTO/STO and PTO/STO, as well as PZO/SZO (001) heterostructures, are semiconducting. In general agreement with available experimental data [84], according to our hybrid-DFT computations, the (001) interface layer does not considerably influence the electronic structure of our studied heterostructures. At the same time, the termination of the deposited BTO and PTO, as well as PZO (001), thin films atop STO or SZO (001) substrates, respectively, may shift the band edges regarding the vacuum level and thereby reduce the (001) heterostructure band gap [84].