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Article

Comparative Hybrid Hartree-Fock-DFT Calculations of WO2-Terminated Cubic WO3 as Well as SrTiO3, BaTiO3, PbTiO3 and CaTiO3 (001) Surfaces

by
R. I. Eglitis
1,*,
Juris Purans
1 and
Ran Jia
1,2
1
Institute of Solid State Physics, University of Latvia, 8 Kengaraga Street, LV1063 Riga, Latvia
2
Laboratory of Theoretical and Computational Chemistry, Institute of Theoretical Chemistry, Jilin University, Changchun 130023, China
*
Author to whom correspondence should be addressed.
Crystals 2021, 11(4), 455; https://doi.org/10.3390/cryst11040455
Submission received: 29 March 2021 / Revised: 17 April 2021 / Accepted: 19 April 2021 / Published: 20 April 2021
(This article belongs to the Special Issue Diffusion and Degradation Phenomena in Solid Oxide Materials)
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Abstract

:
We performed, to the best of our knowledge, the world’s first first-principles calculations for the WO2-terminated cubic WO3 (001) surface and analyzed the systematic trends in the WO3, SrTiO3, BaTiO3, PbTiO3 and CaTiO3 (001) surface ab initio calculations. According to our first principles calculations, all WO2 or TiO2-terminated WO3, SrTiO3, BaTiO3, PbTiO3 and CaZrO3 (001) surface upper-layer atoms relax inwards towards the crystal bulk, while all second-layer atoms relax upwards. The only two exceptions are outward relaxations of first layer WO2 and TiO2-terminated WO3 and PbTiO3 (001) surface O atoms. The WO2 or TiO2-terminated WO3, SrTiO3, BaTiO3, PbTiO3 and CaTiO3 (001) surface-band gaps at the Γ–Γ point are smaller than their respective bulk-band gaps. The Ti–O chemical bond populations in the SrTiO3, BaTiO3, PbTiO3 and CaTiO3 bulk are smaller than those near the TiO2-terminated (001) surfaces. Conversely, the W–O chemical bond population in the WO3 bulk is larger than near the WO2-terminated WO3 (001) surface.

1. Introduction

Throughout the last 20 years the SrTiO3, BaTiO3, PbTiO3 and CaTiO3 (001) surfaces have been broadly explored theoretically and experimentally [1,2,3,4,5,6,7,8,9,10]. At the same time, to the best of our knowledge, there are no reports of ab initio calculations dealing with the atomic relaxation and electronic structure of the pristine WO2-terminated WO3 (001) surface in the cubic perovskite-like structure. Nevertheless, a large amount of experimental studies exist dealing with WO3 (001) surfaces [11,12,13,14]. Recent theoretical studies have been devoted, for example, to hydrogen adsorption on the WO3 (001) surface [15], understanding the water splitting process on the WO3 (001) surface [16], and H2O adsorption on the WO3 (001) surface [17].
BaTiO3, PbTiO3 and CaTiO3 perovskites have attracted huge fundamental interest in these materials mostly for their phase transitions. Historically the ABO3 perovskites were highly promising low-cost energy materials. They have been used for numerous optoelectronic and photonic device applications [18]. SrTiO3 perovskite thin films are important for a large amount of technologically important applications [19,20]. For example, they are used for catalysis, optical wave guides, high-capacity memory cells as well as substrates for high-temperature cuprate superconductor growth [19,20]. Barium titanate (BaTiO3) is an excellent photorefractive material [18]. Ferroelectric PbTiO3 thin films have been applied to large numbers of electronic devices, such as non-volatile memory FET [19] and Si monolithic ultrasonic sensors [18]. CaTiO3 is used worldwide in technologically important electronic ceramic materials [18]. Tungsten trioxide (WO3) and its thin films exhibit a large number of novel properties useful for high-technology applications [21]. In particular, WO3 undergoes phase transitions, which are explored for their potential in industrial applications, display systems and microelectronics [21]. It is worth noting that the predictive power of ab initio calculations makes possible the design of new materials for high-technology applications on paper. Nowadays, consumer electronics mostly use lithium-ion batteries containing LiCoO2 cathode, which was discovered in 1980 by J. Goodenough, one of the 2019 Nobel Prize winners for Chemistry [22]. The experimentally detected LiCoO2 average intercalation voltage is 4.0–4.1 V [23]. Based on ab initio calculations by Eglitis and Borstel [24,25,26], it was demonstrated that a novel Li2CoMn3O8 battery cathode material can lead to a high-energy lithium-ion battery working at the 5 V regime.
The SrTiO3, BaTiO3, PbTiO3 and CaTiO3 perovskite cubic unit cells contain five atoms. The A type atom (A = Sr, Ba, Pb or Ca) has the coordinates (0, 0, 0), and it is located in the cube corner position. The Ti atom has the coordinates (½, ½, ½), and it is located in the cube body center position. The 3 O atoms have the coordinates (½, ½, 0), (½, 0, ½), (0, ½, ½), and they are located in the cube face centered positions. All SrTiO3, BaTiO3, PbTiO3 and CaTiO3 cubic perovskites have the same space group Pm3m with the space group number equal to 221. WO3 in its cubic perovskite-like structure has exactly the same space group as ATiO3 perovskites Pm3m, and also the same space group number 221. The only striking difference between the SrTiO3, BaTiO3, PbTiO3 and CaTiO3 cubic perovskites as well as WO3 in its cubic perovskite-like structure is that WO3 has an empty A cation position. Thereby, the cubic perovskite-like unit cell of WO3 contain only four atoms.
The objective of the reported here work was to carry out first-principles calculations for WO2-terminated polar WO3 (001) surfaces in the cubic perovskite-like structure. We compared our WO2-terminated WO3 (001) surface-atomic and electronic-structure ab initio calculations with our results for the related structure TiO2-terminated SrTiO3, BaTiO3, PbTiO3 and CaTiO3 cubic perovskite (001) surfaces. We carefully compared our calculation results for all five of our calculated materials and detected systematic common trends. The results for WO2-terminated WO3 and TiO2-terminated SrTiO3, BaTiO3, PbTiO3 and CaTiO3 (001) surfaces were summarized and analysed in a way easily readable for a broad audience of scientists.

2. Computational Methods and Surface Models

In order to carry out ab initio DFT-B3LYP or DFT-B3PW calculations, we employed the CRYSTAL computer program package [27]. Unlike the plane-wave codes widely employed in many previous studies [28,29], the CRYSTAL code [27] uses localized Gaussian-type basis sets. In our calculations, we adopted the basis sets (BS) developed for SrTiO3, BaTiO3 and PbTiO3 in [30]. The Hay–Wadt small-core, effective-core pseudopotentials (ECP) were adopted for Ca and Ti atoms [31,32,33]. The small-core ECPs replaced only the inner-core orbitals, while orbitals for subvalence electrons as well as for valence electrons were calculated self-consistently. Oxygen atoms were treated with the all-electron BS. Finally, for the W atom we used BS developed by Cora et al. [34]. Our calculations were performed by means of the B3LYP [35] or B3PW [36,37,38] hybrid exchange–correlation functionals. For all WO3, SrTiO3, BaTiO3, PbTiO3 and CaTiO3 materials we performed the reciprocal space integration with an 8 × 8 × 8 and 8 × 8 × 1 extension of Pack–Monkhorst mesh for the bulk and (001) surfaces of these materials. The CRYSTAL computer program package [27] makes possible the calculation of isolated 2D slabs perpendicular to the Oz direction. In order to compare the performance of different exchange–correlation functionals and choose the best method for our calculations, we calculated the SrTiO3, SrZrO3, BaZrO3, MgF2 and CaF2 bulk Γ–Γ band gaps [30,39,40,41,42] (Table 1 and Figure 1). The experimentally detected SrTiO3, SrZrO3, BaZrO3, MgF2 and CaF2 bulk band gaps at the Γ-point are mentioned in Table 1 for comparison purposes as well as depicted in Figure 1 [43,44,45,46,47].
As can be seen in Table 1, the ab initio Hartree-Fock (HF) calculations, for all five our calculated materials, very strongly overestimate the experimental band gap at Γ-point. Namely, the HF method most strongly (3.29 times) overestimate the experimental SrTiO3 bulk Γ–Γ band gap. Even HF calculated MgF2 bulk Γ–Γ band gap overestimates the experimental value 1.51 times (Table 1 and Figure 1).
From another side, as we can see from Table 1 and Figure 1, the generalized gradient approximations (GGA) to the density functional theory (DFT) systematically and considerably underestimate the experimental Γ–Γ bulk band gap in our calculated ABO3 perovskites as well as MgF2 and CaF2. For example, the PWGGA (6.94 eV) and PBE (6.91 eV) calculated MgF2 bulk band gap at Γ-point is 1.87 and 1.88 times, respectively, smaller than the experimental MgF2 bulk Γ–Γ band gap value of 13.0 eV [37].
To obtain the best possible results, we performed our WO3, SrTiO3, BaTiO3, PbTiO3 and CaTiO3 bulk and (001) surface calculations by means of the B3PW [36,37,38] or B3LYP [35] hybrid exchange–correlation functionals. The hybrid functional incorporates a portion of exact exchange energy density from HF theory (20%) while the rest of the exchange–correlation part is a mixture of different approaches (both exchange and correlation). It is obvious, that the B3PW and B3LYP hybrid exchange–correlation functionals, since they are a superposition of HF and DFT methods as implemented in the CRYSTAL computer code [27], allowed us to achieve as good an agreement as possible between the first principles calculated and the experimentally detected Γ–Γ band gaps for WO3, SrTiO3, BaTiO3, PbTiO3 and CaTiO3 bulk and their (001) surfaces.
In our ab initio calculations we used WO2-terminated WO3 as well as TiO2-terminated SrTiO3, BaTiO3, PbTiO3 and CaTiO3 (001) slabs containing 9 alternating layers. First our calculated WO3 (001) slab was terminated by WO2 planes from both sides (WO2–O–WO2–O–WO2–O–WO2–O–WO2) from a 19-atom supercell (Figure 2). Another of our calculated SrTiO3, BaTiO3, PbTiO3 and CaTiO3 (001) slabs was terminated by TiO2 planes from both sides (TiO2–AO–TiO2–AO–TiO2–AO–TiO2–AOvTiO2) and consisted of a 23-atom supercell (Figure 3). Both our calculated slabs were non-stoichiometric and had unit-cell equations W5O14 as well as A4Ti5O14, respectively. To analyse the chemical bonds, effective atomic charges and covalency effects for WO3 and ATiO3 perovskite bulk and (001) surfaces, we used the well-known Mulliken population analysis [48,49,50,51,52].

3. Ab initio Calculation Results for WO3, SrTiO3, BaTiO3, PbTiO3 and CaTiO3 Bulk

As a starting point, by means of the hybrid B3LYP exchange–correlation functional, we calculated the cubic WO3 bulk lattice constant (3.775 Å). Our calculated cubic WO3 constant (3.775 Å) was only slightly larger than the experimental value of a0 = 3.71–3.75 Å [51] (Table 2); nevertheless, it was in almost perfect agreement with the earlier calculation result for the WO3 cubic structure bulk lattice constant calculated by the full-potential linear muffin-tin (FP-LMTO) code equal to 3.78 Å [52]. Our B3PW-calculated SrTiO3 bulk lattice constant (3.904 Å) was only slightly overestimated with respect to the experimental SrTiO3 bulk lattice constant (3.89 Å) extrapolated to 0 K [53] (Table 2). Our ab initio calculation of the BaTiO3 bulk lattice constant (4.008 Å) was in an outstanding agreement with the experimental value of 4.00 Å [53,54,55]. Our B3PW-calculated PbTiO3 bulk lattice constant (3.936 Å) [54,55,56] was only 0.86% under the experimental value of 3.97 Å [57]. Finally, our calculated CaTiO3 bulk lattice constant (3.851 Å) was 1.17% smaller than the experimentally detected (3.8967 Å) [58,59,60] (Table 2).
Our ab initio B3LYP-calculated effective atomic charges for the WO3 bulk were (+3.095e) for the W atom, and (−1.032e) for each of the three O atoms (Table 3). Our B3LYP-calculated effective W atomic charge (+3.095e) was almost two times smaller than the generally accepted classical ionic charge for the W(+6e) atom. In addition, our calculated effective atomic charge for the O (−1.032e) atom was almost two times smaller than the generally accepted O atom classical ionic charge (−2e). In addition, for the SrTiO3, BaTiO3, PbTiO3 and CaTiO3 perovskites, our calculated A atomic charges (+1.871e, +1.797e, +1.354e and +1.782e, respectively) were considerably smaller than those of the classical Sr, Ba, Pb, Ca atom ionic charges (+2e) (Table 3) [61,62,63,64,65,66]. Our B3PW-calculated O atom Mulliken charges in SrTiO3, BaTiO3, PbTiO3 and CaTiO3 perovskites (−1.407e, −1.388e, −1.232e and −1.371e, respectively) are also at least 29.65% smaller than the classical ionic O atomic charge (−2e) [67,68,69]. Finally, our ab initio-calculated SrTiO3, BaTiO3, PbTiO3 and CaTiO3 Ti atomic charges (+2.351e, +2.367e, +2.341e and 2.330e) are more than one-and-a-half times smaller than the formal Ti atom ionic charge (+4e). Our calculated chemical bond population between W and O atoms in WO3 bulk (0.142e) is approximately one-and-a-half times larger than the Ti–O atom chemical bond population in SrTiO3, BaTiO3, PbTiO3 and CaTiO3 perovskites (+0.088e, +0.098e, +0.098e and +0.084e, respectively).
Our B3LYP-calculated WO3 bulk Γ–Γ band gap (4.95 eV) overestimated by 1.21 eV the experimental direct WO3 bulk band gap value at Γ-point of 3.74 eV [70] (Table 4). Moreover, our B3PW-calculated bulk Γ–Γ band gaps for SrTiO3, BaTiO3, PbTiO3 and CaTiO3 perovskites (3.96 eV, 3.55 eV, 4.32 eV and 4.18 eV, respectively) were always slightly overestimated with respect to the experimentally measured direct band gap values at Γ-point for SrTiO3, BaTiO3, PbTiO3 and CaTiO3 perovskites (3.75 eV [43], 3.2 eV [71], 3.4 eV [72] and 3.5 eV [73], respectively) (Table 4).

4. Ab Initio Calculation Results for the WO2-Terminated WO3 as Well as TiO2-Terminated SrTiO3, BaTiO3, PbTiO3 and CaTiO3 (001) Surfaces

Our B3LYP- or B3PW-calculated atomic displacements for the WO2-terminated WO3 and the TiO2-terminated SrTiO3, BaTiO3, PbTiO3 and CaTiO3 (001) surface upper-three or two layers are presented in Table 5. According to our B3LYP or B3PW calculations, all atoms of the first (upper) surface layer relaxed inwards, while all second-layer atoms relaxed outwards (Table 5). The only two exceptions to this systematic trend were the outward relaxation of the first layer O atom of the WO2-terminated WO3 (001) surface (+0.42% of a0) and the outward relaxation of the TiO2-terminated PbTiO3 (001) surface first-layer O atom by (0.31% of a0) (Table 5). The first layer metal atom relaxation magnitudes range from −1.71% of a0 for the TiO2-terminated CaTiO3 (001) surface to −3.08% of a0 for the TiO2-terminated BaTiO3 (001) surface (Table 5). The first- and second-layer metal atom displacement magnitudes for WO2-terminated WO3 and TiO2-terminated SrTiO3, BaTiO3, PbTiO3 and CaTiO3 (001) surfaces were always considerably larger than the respective first- and second-layer O atom displacement magnitudes (Table 5).
Our B3LYP-calculated surface rumpling amplitude s (the relative displacement of an oxygen atom relative to the metal atom in the upper surface layer) for WO2-terminated WO3 (001) surface (+2.49) is in qualitative agreement with our B3PW-calculated surface rumpling amplitudes s for TiO2-terminated BaTiO3, PbTiO3, CaTiO3 and SrTiO3 (001) surfaces (+2.73, +3.12, +1.61 and +2.12, respectively) (Table 6). Our B3PW-calculated surface rumpling amplitude s for TiO2-terminated SrTiO3 (001) surface (+2.12) is in fair agreement with available RHEED (+2.6 [74]) and LEED (+2.1 ± 2 [75]) experimental data (Table 6). Unfortunately, our B3PW-calculated interlayer distance Δd12 for the TiO2-terminated SrTiO3 (001) surface (−5.80) had the opposite sign to the experimentally measured RHEED (+1.8 [74]) and LEED (+1 ± 1 [67]) interlayer distances (Table 6). Finally, our B3PW-calculated interlayer distance Δd23 for the TiO2-terminated SrTiO3 (001) surface (+3.55) is in qualitative agreement with the RHEED experiment result (+1.3), but had the opposite sign to that of the LEED experimental result (−1 ± 1). Nevertheless, it is worth noting that the RHEED (+1.3) and the LEED experiments (−1 ± 1) had opposite signs for the interlayer distance Δd23 (Table 6).
We started the discussion of the electronic structure of WO2-terminated WO3 and TiO2-terminated SrTiO3, BaTiO3, PbTiO3 and CaTiO3 (001) surfaces with an analysis of charge redistribution in the top-three surface planes (Table 7). The ab initio-calculated atomic displacements, bond populations between the nearest metal and oxygen atoms and the effective atomic charges are collected in Table 7. For example, the effective static atomic charges on WO2-terminated WO3 as well as TiO2-terminated SrTiO3, BaTiO3, PbTiO3 and CaTiO3 (001) surface upper-layer W and Ti atoms are always reduced in comparison to the bulk WO3, SrTiO3, BaTiO3, PbTiO3 and CaTiO3 crystal charges (−0.312e, −0.06e, −0.06e, −0.062e and −0.052e, respectively). We recently observed a similar effect: the reduction of surface upper-layer metal atomic charges near the ReO2-terminated ReO3 and the ZrO2-terminated SrZrO3, BaZrO3, PbZrO3 and CaZrO3 (001) surfaces [76]. According to our ab initio calculations, the largest upper-layer metal atom displacement was observed for the TiO2-terminated BaTiO3 (001) surface Ba atom (−0.123 Å). Nevertheless, the TiO2-terminated PbTiO3 (001) surface second-layer Pb atom outward displacement (+0.209 Å) was even larger.
Our B3PW-calculated SrTiO3, BaTiO3, PbTiO3 and CaTiO3 bulk Ti-O chemical bond covalency (+0.088e, +0.098e, +0.098e and +0.084e, respectively) were always smaller than near the TiO2-terminated SrTiO3, BaTiO3, PbTiO3 and CaTiO3 (001) surfaces (0.118e, 0.126e, 0.114e, 0.114e, respectively) (Table 8 and Figure 4). Just opposite situation was obtained for the WO3 crystal: the B3LYP-calculated W-O chemical bond population in the WO3 bulk (0.142e) was larger than near the WO2-terminated WO3 (001) surface (0.108e) (Table 8 and Figure 4). Nevertheless, it is worth noting that the W–O chemical bond population between the W atom on the top layer of WO2-terminated WO3 (001) surface and the O atom on the second layer (0.278e) is the largest one (Table 8), which was in agreement with our previous B3LYP calculations dealing with ReO2-terminated ReO3 (001) surfaces [76].
As can be seen in Table 9 and Figure 5, our B3LYP- or B3PW-calculated WO3, SrTiO3, BaTiO3, PbTiO3 and CaTiO3 bulk band gaps at the Γ–Γ point were always reduced near the WO2- or TiO2-terminated WO3, SrTiO3, BaTiO3, PbTiO3 and CaTiO3 (001) surfaces. The B3PW-calculated SrTiO3 bulk band gap at Γ–Γ point near the TiO2-terminated SrTiO3 (001) surface at Γ–Γ point was reduced only by 0.01 eV. At the same time, our B3LYP-calculated WO3 bulk band gap (4.95 eV) at the Γ–Γ point near the WO2-terminated WO3 (001) surface was reduced by 3.79 eV to 1.16 eV (Table 9 and Figure 5).

5. Conclusions

For the first principles-calculated WO2- or TiO2-terminated WO3, SrTiO3, BaTiO3, PbTiO3 and CaTiO3 (001) surfaces, as a rule, all first-layer surface atoms relax inwards, whereas all second-layer surface atoms relax upwards. The only two exceptions from this systematic trend are the upward relaxation of WO2- or TiO2-terminated WO3 and PbTiO3 (001) surface first-layer O atoms. As a result of our ab initio-calculated atomic relaxation, TiO2-terminated SrTiO3, BaTiO3, PbTiO3 and CaTiO3 (001) surfaces exhibited a reduction of the interlayer distance Δd12 (−5.80, −5.59, −8.13, −4.46% of a0, respectively) as well as an expansion of Δd23 (+3.55, +2.51, +5.32, +2.75% of a0, respectively). It is worth noting that after geometry optimization, it is very useful to perform ab initio molecular dynamics computations to ensure the stability of the structures over time [77].
The changes in the interlayer distances between the first and second layer (Δd12) were always larger than between the second and third layer (Δd23) for all our calculated perovskites SrTiO3, BaTiO3, PbTiO3 and CaTiO3.
The Ti–O chemical bond population in SrTiO3, BaTiO3, PbTiO3 and CaTiO3 bulk was always smaller than near their TiO2-terminated (001) surface (see Figure 4). In contrast, the W–O chemical bond population in the WO3 bulk (0.142e) was larger than near the WO2-terminated WO3 (001) surface (0.108e). Nevertheless, the largest W–O chemical bond population, according to our ab initio calculations, is between the W atom located on the WO2-terminated WO3 (001) surface and the second-layer O atom (0.278e). It was worth noting, that also for the related material ReO3, according to our calculations [76], the situation was similar. Namely, the Re–O chemical bond population in the ReO3 bulk (0.212e) was larger than near the ReO2-terminated ReO3 (001) surface (0.170e). Nevertheless, the Re–O chemical bond population between the Re atom located on the ReO2-terminated ReO3 (001) surface upper-layer and O atom located on the ReO2-terminated ReO3 (001) surface second layer was the largest (0.262e).
According to our B3LYP or B3PW calculations, the WO3, SrTiO3, BaTiO3, PbTiO3 and CaTiO3 bulk Γ–Γ band gap values (4.95, 3.96, 3.55, 4.32, 4.18 eV, respectively) were always reduced with respect to the bulk near the WO2- or TiO2-terminated WO3, SrTiO3, BaTiO3, PbTiO3 and CaTiO3 (001) surfaces (1.16, 3.95, 2.96, 3.18, 3.30 eV, respectively) (see Figure 5).

Author Contributions

All authors equally contributed to the performed ab initio calculations as well as to the preparation of the manuscript. All authors have read and agreed to the published version of the manuscript.

Funding

This study was supported by the ERAF Project No. 1.1.1.1/18/A/073.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Acknowledgments

We greatly acknowledge the financial support via the ERAF Project No. 1.1.1.1/18/A/073. Calculations were performed using Latvian Super Cluster (LASC), located in the Center of Excellence at Institute of Solid State Physics, the University of Latvia, which is supported by European Union Horizon 2020 Framework Programme H2020-WIDESPREAD-01-2016-2017-Teaming Phase 2 under Grant Agreement No. 739508, project CAMART.

Conflicts of Interest

The authors declare no conflict of interest.

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Crystals 11 00455 g001 550
Figure 1. Ab initio calculated and experimentally measured bulk Γ–Γ band gaps for SrTiO3, SrZrO3, BaZrO3, MgF2 and CaF2 obtained by means of different exchange–correlation functionals: (1) PWGGA; (2) B3LYP; (3) B3PW; (4) Experiment; (5) HF.
Figure 1. Ab initio calculated and experimentally measured bulk Γ–Γ band gaps for SrTiO3, SrZrO3, BaZrO3, MgF2 and CaF2 obtained by means of different exchange–correlation functionals: (1) PWGGA; (2) B3LYP; (3) B3PW; (4) Experiment; (5) HF.
Crystals 11 00455 g001
Crystals 11 00455 g002 550
Figure 2. Side view of the nine-layer TiO2-terminated ATiO3 perovskite (001) surface.
Figure 2. Side view of the nine-layer TiO2-terminated ATiO3 perovskite (001) surface.
Crystals 11 00455 g002
Crystals 11 00455 g003 550
Figure 3. Side view of the nine-layer WO2-terminated WO3 polar (001) surface.
Figure 3. Side view of the nine-layer WO2-terminated WO3 polar (001) surface.
Crystals 11 00455 g003
Crystals 11 00455 g004 550
Figure 4. Our ab initio-calculated bulk (1) as well as BO2-terminated (001) surface (2). B–O chemical bond populations (in e) for WO3, SrTiO3, BaTiO3, PbTiO3 and CaTiO3.
Figure 4. Our ab initio-calculated bulk (1) as well as BO2-terminated (001) surface (2). B–O chemical bond populations (in e) for WO3, SrTiO3, BaTiO3, PbTiO3 and CaTiO3.
Crystals 11 00455 g004
Crystals 11 00455 g005 550
Figure 5. Our ab initio-calculated WO2-terminated WO3 and TiO2-terminated SrTiO3, BaTiO3, PbTiO3 and CaTiO3 (001) surface Γ–Γ band gaps (line 1). Experimentally measured bulk Γ–Γ band gaps (line 2). Our ab initio-calculated WO3, SrTiO3, BaTiO3, PbTiO3 and CaTiO3 bulk Γ–Γ band gaps (line 3).
Figure 5. Our ab initio-calculated WO2-terminated WO3 and TiO2-terminated SrTiO3, BaTiO3, PbTiO3 and CaTiO3 (001) surface Γ–Γ band gaps (line 1). Experimentally measured bulk Γ–Γ band gaps (line 2). Our ab initio-calculated WO3, SrTiO3, BaTiO3, PbTiO3 and CaTiO3 bulk Γ–Γ band gaps (line 3).
Crystals 11 00455 g005
Table
Table 1. By means of different exchange–correlation functionals calculated SrTiO3, SrZrO3, BaZrO3, MgF2 and CaF2 bulk Γ–Γ band gaps (eV). Experimental bulk band gaps at the Γ-point are listed for comparison.
Table 1. By means of different exchange–correlation functionals calculated SrTiO3, SrZrO3, BaZrO3, MgF2 and CaF2 bulk Γ–Γ band gaps (eV). Experimental bulk band gaps at the Γ-point are listed for comparison.
MethodSrTiO3 [30]SrZrO3 [39]BaZrO3 [40]MgF2 [41]CaF2 [42]
Experiment3.75 [43]5.6 [44] 5.3 [45] 13.0 [46]12.1 [47]
B3PW3.965.304.939.4810.96
B3LYP3.895.314.799.4210.85
HF12.3313.5412.9619.6520.77
PWGGA2.313.533.246.948.51
PBE2.353.52-6.918.45
Table
Table 2. Our ab initio-calculated and experimentally measured WO3, SrTiO3, BaTiO3, PbTiO3 and CaTiO3 bulk lattice constants [51,52,53,54,55,56,57,58,59,60].
Table 2. Our ab initio-calculated and experimentally measured WO3, SrTiO3, BaTiO3, PbTiO3 and CaTiO3 bulk lattice constants [51,52,53,54,55,56,57,58,59,60].
MaterialMethodTheoryExperiment
WO3B3LYP3.7753.71–3.75 [51]
FP-LMTO3.78 [52]
SrTiO3B3PW3.9043.89 [53]
BaTiO3B3PW4.008 4.00 [53,54]
PbTiO3B3PW3.9363.97 [54,57]
CaTiO3B3PW3.8513.8967 [58,59,60]
Table
Table 3. Our calculated atomic charges Q(e) as well as bond populations P(e) in WO3, SrTiO3, BaTiO3, PbTiO3 and CaTiO3 bulk materials.
Table 3. Our calculated atomic charges Q(e) as well as bond populations P(e) in WO3, SrTiO3, BaTiO3, PbTiO3 and CaTiO3 bulk materials.
Bulk MaterialsWO3SrTiO3BaTiO3PbTiO3CaTiO3
Ion PropertyB3LYPB3PWB3PWB3PWB3PW
AQ-+1.871+1.797+1.354+1.782
P-−0.010−0.034+0.016+0.006
OQ−1.032−1.407−1.388−1.232−1.371
P+0.142+0.088+0.098+0.098+0.084
BQ+3.095+2.351+2.367+2.341+2.330
Table
Table 4. Ab initio-calculated WO3, SrTiO3, BaTiO3, PbTiO3 and CaTiO3 bulk band gaps for the cubic phase at the Γ–Γ point. Our B3LYP and B3PW calculations were compared with the relevant experimental data.
Table 4. Ab initio-calculated WO3, SrTiO3, BaTiO3, PbTiO3 and CaTiO3 bulk band gaps for the cubic phase at the Γ–Γ point. Our B3LYP and B3PW calculations were compared with the relevant experimental data.
CrystalMethodOptical Bulk Band Gap at Γ–Γ Point
Ab Initio CalculationsExperimental Results
WO3B3LYP4.953.74 [70]
SrTiO3B3PW3.963.75 [43]
BaTiO3B3PW3.553.2 [71]
PbTiO3B3PW4.323.4 [72]
CaTiO3B3PW4.183.5 [73]
Table
Table 5. WO2-terminated WO3 as well as TiO2-terminated SrTiO3, BaTiO3, PbTiO3 and CaTiO3 (001) surface atom relaxation for upper-three surface layers (in percent of the bulk lattice constant).
Table 5. WO2-terminated WO3 as well as TiO2-terminated SrTiO3, BaTiO3, PbTiO3 and CaTiO3 (001) surface atom relaxation for upper-three surface layers (in percent of the bulk lattice constant).
Surfaces, (001)WO3SrTiO3BaTiO3PbTiO3CaTiO3
LayerIonWO2-Term.TiO2-Term.TiO2-Term.TiO2-Term.TiO2-Term.
MethodB3LYPB3PWB3PWB3PWB3PW
1B−2.07−2.25−3.08−2.81−1.71
O+0.42−0.13−0.35+0.31−0.10
2AAbsent+3.55+2.51+5.32+2.75
O+0.11+0.57+0.38+1.28+1.05
3B−0.01----
O0.00----
Table
Table 6. Our B3LYP- or B3PW-calculated surface rumplings s and relative displacements Δdij between the three near-surface planes for the WO2-terminated WO3 and the TiO2-terminated BaTiO3, PbTiO3, CaTiO3 and SrTiO3 (001) surfaces as a percent of the bulk material lattice constant. The available experimental data are listed for comparison purposes.
Table 6. Our B3LYP- or B3PW-calculated surface rumplings s and relative displacements Δdij between the three near-surface planes for the WO2-terminated WO3 and the TiO2-terminated BaTiO3, PbTiO3, CaTiO3 and SrTiO3 (001) surfaces as a percent of the bulk material lattice constant. The available experimental data are listed for comparison purposes.
MaterialMethodWO2- or TiO2-Terminated (001) Surface
sΔd12Δd23
WO3B3LYP+2.49--
BaTiO3B3PW+2.73−5.59+2.51
PbTiO3B3PW+3.12−8.13+5.32
CaTiO3B3PW+1.61−4.46+2.75
SrTiO3B3PW+2.12−5.80+3.55
RHEED exp. [74]+2.6+1.8+1.3
LEED exp. [75]+2.1 ± 2+1 ± 1−1 ± 1
Table
Table 7. Our B3LYP- or B3PW-calculated atomic shift magnitudes D (in Å) as well as the effective atomic charges Q (in e) and nearest atomic chemical bond populations P (in e) for the WO2- and TiO2-terminated WO3, SrTiO3, BaTiO3, PbTiO3 and CaTiO3 (001) surfaces.
Table 7. Our B3LYP- or B3PW-calculated atomic shift magnitudes D (in Å) as well as the effective atomic charges Q (in e) and nearest atomic chemical bond populations P (in e) for the WO2- and TiO2-terminated WO3, SrTiO3, BaTiO3, PbTiO3 and CaTiO3 (001) surfaces.
WO2 and TiO2-Term. (001) SurfacesWO3SrTiO3BaTiO3PbTiO3CaTiO3
LayerProperty IonWO2-Ter.TiO2-Ter.TiO2-Ter.TiO2-Ter.TiO2-Ter.
1DB−0.078−0.088−0.123−0.111−0.066
Q +2.783+2.291+2.307+2.279+2.278
P +0.108+0.118+0.126+0.114+0.114
DO+0.016−0.005−0.014+0.012−0.004
Q −1.146−1.296−1.280−1.184−1.267
P −0.014−0.014−0.038+0.044+0.016
2DA -+0.139+0.101+0.209+0.106
Q -+1.850+1.767+1.275+1.754
P -−0.008−0.030+0.008+0.006
DO+0.004+0.022+0.015+0.050+0.041
Q −0.925−1.365−1.343−1.167−1.324
P +0.064+0.080+0.090+0.080+0.086
3DB−0.0004----
Q +3.001+2.348+2.365+2.335+2.326
P +0.144+0.096+0.104+0.108+0.090
DO 0.000----
Q −1.037−1.384−1.371−1.207−1.354
P −0.032−0.010−0.034+0.018+0.008
Table
Table 8. Our B3LYP or B3PW calculated W–O or Ti–O chemical bond populations for WO3, SrTiO3, BaTiO3, PbTiO3 and CaTiO3 bulk and for WO2- or TiO2-terminated (001) surfaces (in e).
Table 8. Our B3LYP or B3PW calculated W–O or Ti–O chemical bond populations for WO3, SrTiO3, BaTiO3, PbTiO3 and CaTiO3 bulk and for WO2- or TiO2-terminated (001) surfaces (in e).
Material MethodW-O or Ti-O Chemical Bond Populations
BulkWO2, TiO2-Term. (001)
WO3B3LYP+0.142+0.108 (W(I)—O(I))
+0.142+0.278 (W(I)—O(II))
SrTiO3B3PW+0.088+0.118
BaTiO3B3PW+0.098+0.126
PbTiO3B3PW+0.098+0.114
CaTiO3B3PW+0.084+0.114
Table
Table 9. B3LYP- or B3PW-calculated Γ–Γ band gaps for WO3, SrTiO3, BaTiO3, PbTiO3 and CaTiO3 bulk as well as their WO2 or TiO2-terminated (001) surfaces.
Table 9. B3LYP- or B3PW-calculated Γ–Γ band gaps for WO3, SrTiO3, BaTiO3, PbTiO3 and CaTiO3 bulk as well as their WO2 or TiO2-terminated (001) surfaces.
MaterialMethodCalculated Band Gap at Γ–Γ Point
BulkWO2, TiO2-Term. (001)
WO3B3LYP4.951.16
SrTiO3B3PW3.963.95
BaTiO3B3PW3.552.96
PbTiO3B3PW4.323.18
CaTiO3B3PW4.183.30
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Eglitis, R.I.; Purans, J.; Jia, R. Comparative Hybrid Hartree-Fock-DFT Calculations of WO2-Terminated Cubic WO3 as Well as SrTiO3, BaTiO3, PbTiO3 and CaTiO3 (001) Surfaces. Crystals 2021, 11, 455. https://doi.org/10.3390/cryst11040455

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Eglitis RI, Purans J, Jia R. Comparative Hybrid Hartree-Fock-DFT Calculations of WO2-Terminated Cubic WO3 as Well as SrTiO3, BaTiO3, PbTiO3 and CaTiO3 (001) Surfaces. Crystals. 2021; 11(4):455. https://doi.org/10.3390/cryst11040455

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Eglitis, R. I., Juris Purans, and Ran Jia. 2021. "Comparative Hybrid Hartree-Fock-DFT Calculations of WO2-Terminated Cubic WO3 as Well as SrTiO3, BaTiO3, PbTiO3 and CaTiO3 (001) Surfaces" Crystals 11, no. 4: 455. https://doi.org/10.3390/cryst11040455

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