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The enabling electronic motif for topological insulation in ABO3 perovskites and its structural stability

Stable oxide topological insulators (TIs) that could bring together the traditional oxide functionalities with the dissipationless surface states of TIs have been sought for years but none was found. Yet, heavier chalcogenides (selenides, tellurides) were readily found to be TIs. We clarify here the basic contradiction between TI-ness and stability which is maximal for oxides, and trace the basic design principles necessary to identify the window of opportunity of stable TIs. We first identify the electronic motif that can achieve topological band inversion ("topological gene") in ABO3 as being a lone-pair electron-rich B atom (e.g. Te, I, Bi) at the octahedral site. We then illustrate that poorly screened oxide systems with large inversion energies can undergo energy-lowering atomic distortions that remove the band inversion. We identify the coexistence windows of TI functionality and structure stability for different pressures and find that the common cubic ABO3 structures have inversion energies lying outside this coexistence window at zero pressure but could be moved into the coexistence window at moderate pressures. Our study demonstrates the interplay between topological band inversion and structural stability and traces the basic principles needed to design stable oxide topological insulators at ambient pressures.


Introduction
Topological insulators (TI's) are materials having an inverted order of the occupied valence and unoccupied conduction bands at time--reversal invariant (TRI) wave vectors in the Brillouin zone (BZ), and can be characterized by the topological invariant [1] Z2 = 1.
Theory then assures that in lower--dimensional forms (2D surface or 1D edge) of the topological bulk system there will be states that possess passivation--resistant, linearly dispersed and mutually crossing (metallic) energy bands [2] . The required band inversion in the parent bulk system is generally achieved by introducing high atomic number (Z) cations and anions having strong spin--orbit coupling (SOC) [3--8] . However, such heavy--atom compounds tend to pose defective crystal structures (e.g., spontaneous vacancy formation causing metallic, not insulating behavior) associated with the relatively low cohesion of the weak heavy--atom chemical bonds [9--11] . The recent quest of topological insulators in oxides [12--18] has been partially motivated by the hope that this will deliver defect--tolerant lattices, often characteristic of metal oxides [19] , while at the same time affording the integration of topological properties with the rich oxide functionalities such as transparent conductivity [20] , ferroelectricity [21] , ferromagnetism [22] , or superconductivity [23] . However, the electronic structures of common octet metal oxides, such as ABO3 perovskites or A2BO4 spinels, show that while they may be stable and have wide energy band gaps [19] Eg, they generally lack band inversion, having a trivial topological invariant Z2 = 0. For example, the common ABO3 perovskite have oxygen--derived valence bands and B--atom--derived conduction bands [see Figure 1(a,b)], hence no band inversion. Indeed, whereas numerous topological insulators have been experimentally observed in selenides and tellurides (e.g. Bi2Se3 [4] and Bi2Te3 [24] , stable oxide TI's are yet unknown. Anecdotal examples (discussed in Supplementary Section I) abound of theoretically proposed wide gap oxide TI's in assumed hypothetical crystal structures that turned out, however, to be significantly unstable when energy--lowering structural relaxation were explored in such hypothetical structures [12--18, 25--30] .
We conjecture that the conditions needed for TI--ness-depopulation of bonding valence band states and the occupation of anti--bonding conduction band states-may be contraindicated to thermodynamic stability, if carried out throughout a significant portion of the Brillouin zone. If left unscreened (as in strongly ionic systems), such destabilizing forces may drive structural deformations that alter the crystal symmetry and could undo the band inversion, as illustrated below. Although metastable structures can certainly be made [31--33] , it would be desirable to predict compounds that are TI in not--too--unstable structures, which can be synthesized without fear of producing a topologically trivial but stabler structure, or decomposing after synthesis to a combination of phases that may not be TI's. In this work we report the results of ab--initio co--evaluation of TI--ness and stability for a class of oxides perovskites, elucidating the physical origin of the hitherto mysterious difficulty to obtain simultaneously the electronic features leading to TI--ness ('topological gene') and the structural features of the corresponding lattice structure in ABO3 oxides. We identify the topological gene that can achieve topological band inversion in oxide perovskites as being a lone--pair electron--rich B atom (e.g., Te, I, Bi) rather than Ti, Nb, Y, respectively) at the octahedral site in the cubic ABO3. Oxides tend to have larger predicted inversion energies (Δi > 1 eV) than selenides or tellurides (Δi < 0.4 eV [4,24] ), and are generally less covalent than the heavier chalcogenides where lattice distortions would be partially screened by the more delocalized electronic states. We then illustrate that poorly screened oxide systems with large inversion energies can undergo energy--lowering atomic distortions that remove the topological band inversion. Favorable screening exists in oxides at higher pressures, and lead to stable oxide TI's as illustrated here for BaTeO3 and RbIO3 under moderate pressures. Covalent oxides with strong metal--oxygen orbital mixing [34] do exist and point to the direction of future search for stable oxide TI's at ambient pressures.

Topological gene and stability gene
To address this issue we introduce two constructs: We first identify an electronic motif within a group of ABO3 oxides that would generate band inversion-the "topological gene" of this group of compounds. As Fig. 1(c, d) illustrates the "topological gene" here is the octahedral BO6 motif with lone pair B atom (generated, e.g., by replacing an electron--poor Ti atom in BaTiO3 by an electron--rich Te atom in BaTeO3 that has an additional d 10 s 2 shell).
Second, we examine the stability of the crystal structure that hosts the topological gene, relative to the ground state structures that hosts the "stability gene" of this group of compounds. The key challenge is to see if structures with the topological gene can also have the stability gene. Thus, co--evaluation of the electronic structure and stability is required.
We confirm via calculated band structures in the framework of density function theory (DFT) using projector--augmented wave (PAW) pseudopotential with the exchange-correlation of Perdew--Burke--Ernzerhof (PBE) [35--37] , and the calculated topological invariant [38,39] Z2 (see Methods Section and Supplementary Section II for the details of calculation methods) that twelve ABO3 compounds (BaTeO3, SrTeO3, CaTeO3, BaSeO3, SrSeO3, CaSeO3, RbIO3, KIO3, NaIO3, RbBrO3, KBrO3, NaBrO3) in the assumed cubic perovskite structure with lone--pair B atoms at the octahedral site are in fact TI's, whereas the corresponding compounds with non--lone--pair B ions at the octahedral site (e.g., BaTiO3 and KNbO3) are found to be normal insulators. This substantiates the identity of the topological gene in ABO3 compounds. However, the crystal structure that host this topological gene is not the stablest structure at this composition: total energy relaxation calculations reveal that the cubic topological structures would relax to lower energy non--cubic perovskite phases that are not TI's. This is consistent with the above--mentioned topology vs stability conjecture. Indeed, by performing constrained DFT calculations (see Methods Section), we observe a regained stability of the cubic phase. We further identified the window of coexistence of TI--ness and stability at ambient conditions, and find that the topological cubic ABO3 phases have inversion energies lying outside this coexistence window. Building upon this identification, we find that moderate external pressure can significantly expand the above coexistence window, and eventually leads to simultaneously stable and topological cubic ABO3 phases, thus bringing the 'stability gene' into coincidence with the 'topological gene'. This study illustrates the interplay between the topological gene needed to obtain band inversion and the structural motif needed for stability in the ABO3 oxides, and traces the window of opportunity to design stable oxide TI's at ambient pressures.

Electronic requirements for the topological gene in cubic ABO3 perovskites
In conventional ABO3 compounds with electron--poor B atoms [such as illustrated in Fig.   1(b)] the occupied valence band is oxygen derived whereas the empty conduction states are B--atom derived with normal, s--below--p orbital order (thus, no inversion). To induce band inversion, one would like to change the order of B atom orbitals to p--below--s and assure that the Fermi energy is located between these occupied and unoccupied states, respectively. The idea is to replace the B atom in ABO3 in the cubic perovskite structure [see the inset of Fig. 1(c)] by an electron--rich B element of the same formal charge. Such a B atom has an occupied (lone pair) s orbital below its empty p orbital, yet above the O--p state, as shown in Fig. 1(c). The lone pair s orbital can thus become an unoccupied state above the conduction band due to (B--atom s)-(oxygen p) level repulsion, leading to band inversion (B-p below B--s) as well as to a finite excitation band gap between B--p (occupied) and B--s (unoccupied), which is illustrated in Fig. 1(d). We will illustrate via quantitative DFT calculations the above concept and its rather broad applicability for different groups of perovskite compounds.
In the IIA--IVA--O3 group of compounds exemplified by BaTiO3, we replace Ti 4+ by the electron--richer Te 4+ having additional d 10   BaTeO3 as compared to the expected Z2 = 0 in BaTiO3. That this design principle is rather general can be seen by considering other leading ABO3 perovskite prototypes.
The Supplementary Section IV provides the values of the excitation gaps in the compounds designed according to the topological gene, and discusses effects due to possible DFT errors as compared to Hybrid functional (HSE06 [42] ) calculations. The results show that the inversion energy (Δi), i.e., the gap between the inverted valence and conduction bands at TRI wave vectors, is slightly reduced (by 4%) in HSE06 as compared to DFT, indicating that the band inversion discussed here is quite robust against DFT errors.

Band inversion is often contraindicated to stability
Having identified and verified an electronic motif-the topological gene-we next ask whether the crystal structure (here, cubic ABO3 perovskite) that harbors the topology coincides with the structure that harbors stability. Figure 3 shows the total energies (in meV/atom, given in parentheses) of various ABO3 structures (denoted S1--S15 in Fig.  2, S1 being the cubic perovskite structure while S2--S15 being the lowest--energy structures of specific ABO3 compounds), relative to the lowest-energy phase. In addition to stability, Fig. 3 also denotes if according to the calculated topological invariant Z2 the compound is a TI or a normal insulator (NI). We find from such total energy minimizations that the BO6 octahedral unit with the said electron rich B atom tends to distort towards a stabler, non--cubic crystal structures [29,30] and that this distortion removes the band inversion and thus TI--ness. Therefore, the stability gene and TI gene tend to contradict each other for the ABO3 compounds at ambient conditions.
To get a deeper understanding on the interplay between structural stability and band inversion, we perform constrained DFT calculations constructed for tuning band inversion to examine its effect on the total energy. The tuning of the inversion energy can be done by using an external potential that shifts upwards the B atom p orbital energy, thus, according to Fig.  1(d) the inverted structure (p--below--s) can be tuned to be un--inverted (s--below--p).
This constraint can be implemented, for example, by adding an external potential term [43] Vp to the DFT Hamiltonian acting on the I--p orbital in RbIO3. We then monitor the total energy of the cubic perovskite structure (S1) relative to its stable R3m rhombohedral phase [S2, see Fig.  2(b)], as a function of Δi. Figure 4(a) shows that as Δi decreases, the energy of the cubic S1 phase (relative to S2) also decreases, indicating that band inversion is contraindicated with the stability of crystal structure (see Supplementary Section V for the evolution of the electronic structures with decreasing inversion energy). Significantly, however, there is a narrow window (0 < Δi < 0.8 eV) shown by red background in Fig. 4(a), where topological band inversion and the stability of the band--inverted crystal structure can coexist. It is noticed that the currently known stable TI's (e.g. Bi2Se3 [4] and Bi2Te3 [24] ) have Δi much smaller than 0.8 eV, lying inside the coexistence window (0 < Δi < 0.8 eV). Whereas the cubic ABO3 structures with short B--O bonds have Δi > 1 eV due to the strong coupling between B--s and O--p states, lying outside of the coexistence window. Instead of searching for stable oxide TI's with small Δi, we attempt to stabilize oxide TI's with large inversion energies.

Bringing the topological gene into overlap with the stability gene by applying pressure
Inspired by the fact that many ABO3 compounds (e.g. BaTiO3 [44] ) that are not cubic at ambient conditions can be stabilized under external pressure in the cubic form, we test the effect of external pressure on the interplay between band inversion and structural stability.
We perform similar calculations as Fig. 4(a), but apply external hydrostatic pressure of ≤ 20 GPa. The result of Fig. 4(b) show that the coexistence window for band inversion and structural stability (red background) is significantly extended. At 20 GPa, a stable TI structure can form while tolerating a band inversion of the magnitude of about 2.5 eV. To demonstrate the dramatic effect of pressure on TI--ness versus stability diagram [ Fig.  4(b)], we perform enthalpy calculations for BaTeO3 (from group IIA--VIB--O3) and RbIO3 (from group IA--VIIB--O3). Figure  5(a,b) shows the calculated enthalpies of the relevant structure types for BaTeO3 and RbIO3, as a function of hydrostatic pressure, which demonstrate that the cubic perovskite [S1, see red squares in Fig. 5(a,b)] tends to be stabilized by external pressure. At pressure of 15 GPa (35 GPa), the S1 phase that contains the topological gene becomes the lowest--enthalpy structure for BaTeO3 (RbIO3). We further check the effect of external pressure on the band inversions in cubic BaTeO3 and RbIO3, finding that the band inversion is not removed by external pressure [see Fig.  5(c,d)], but on the contrary, increased by the pressure from 2.3 eV at 0 GPa to 3.4 eV at 35 GPa for BaTeO3, and from 3.0 eV at 0 GPa to 3.8 eV at 35 GPa for RbIO3. It is expected because the pressure decreases the atomic bond lengths of BO6 octahedral, leading to larger s--p repulsion. The enhanced tolerance of cubic ABO3 structures to band inversions is because of the increased electronic screening effect under pressures. Similar to heavier--chalcogenide TI's (e.g. Bi2Se3), the lattice distortions would be screened by the delocalized electronic states. Here, we use external pressure (as an example of external constraints) to tune the screening effect in oxides. Strong screening could exist in oxides without external constraints, such as covalent oxides with strong metal-oxygen orbital mixing [34] , pointing to the direction of future search of stable oxide TI's with large inversion energies that is robust under perturbations such as thermal expansion.

Conclusions and discussion
Studying the interplay between TI--ness and stability in search of oxide topological insulators opens the door of several scenarios: (i) understanding the general conflicting trends between TI--ness and stability answers the question concerning the rareness of experimentally realizable topological phases, and hopefully discourage false positive predictions of topological insulators in hypothetical, highly unstable structures. (ii) These trends can be readily applied to other band inversion quantum phases, such as topological Dirac semimetals and Weyl semimetals. (iii) The results in this study suggest the directions for future search of stable oxides TI's, that are (a) search for oxides with small inversion energies (b) design large inversion--energy oxide TIs that can be stabilized by external constraints or intrinsic screening effect. This opens the way of rational design of stable oxide TI's using topological genes as pivots, and electronic screening/external constraints (e.g. pressure and strain) as levers. Methods DFT total energy and electronic structure evaluation. Total energies were calculated using the projector--augmented wave (PAW) pseudopotential [37] total energy method without spin--orbit coupling with the exchange--correlation of Perdew--Burke--Ernzerhof (PBE) [35] as implemented in the Vienna ab--initio simulation package (VASP) [36] . We use an energy--cutoff of 520 eV and reciprocal space grids with densities of 2π×0.068 Å --1 and 2π×0.051 Å --1 for relaxation and static calculation, respectively. Electronic structure and physical properties were calculated from DFT taking into account spin--orbit coupling by a perturbation ! !" !,!,! • | , ! ! , | to the pseudopotential [45] ( | , ! is the angular momentum eigenstate of atomic site i). To illustrate the band inversions, we project the calculated wave function | on spin and orbital basis of each atomic site !,!,!,! = |( ! ⨂| , ! ! , |)| and then sum !,!,!,! for a given atomic orbital. The Wigner--Seitz radii for constructing | , ! used in this study are listed in the pseudopotentials of the VASP simulation package [36] . To illustrate the interplay between band inversion and structural stability, we performed constrained DFT calculations by applying a Hubbard--like external potential term [43] (Vp) on the I--p orbital in RbIO3 in the Hamiltonian to gradually remove the band inversion between I--p and I--s states.
Z2 characterization. For centrosymmetric structures, we evaluate Z2 from the parities of the wave functions at the time reversal invariant k--points [39] . For non--centrosymmetric structures, we use the method from Refs. [38,46,47] to calculate the topological invariant Z2, which is based on the evolution of Wannier function centers. The topological invariant Z2 is expressed as the number of times mod 2 of the partner switching between the Wannier function centers during the evolution (as illustrated in Fig. S3). To directly look at the evolution of Wannier function centers, we follow the scheme proposed by Yu et al. [38] The topological nature of a 3D compound is determined by looking at two effective 2D systems with kz = 0 and kz = π. Each 2D effective system is then considered as a series of effective 1D system with fixed ky. For an effective 1D system with periodic condition, the unitary position operator is defined as, , ! is the number of real space unit cells along direction, ! denotes the position of the jth lattice site, and denotes the index of the Wannier function | . The projection operator for the occupied subspace for a fixed ky can be defined as, ; for kz = 0 or kz = π, (2) where | ! is the Bloch state and 2 is the number of occupied states.
We then consider the eigenvalue of the projected position operator, that has the following form: where !,! , !,! , … are 2 ×2 matrices. The eigenvalue of the projected position operator can be solved by the transfer matrix method. We define a product of !,! , !,! , … as, ! = !,! !,! !,! ⋯ !"!!,!"!! !"!!,! . (5) ! has 2 eigenvalues: The evolution of the Wannier function center with ky can be easily obtained by looking at the phase factor ! ! (named Wannier center for convenience). The Wannier centers are paired at ky = 0 and ky = π, and during the evolution from ky = 0 to ky = π, the Wannier center can switch partners, with an odd number of partner switching corresponding to Z2 = 1 (see Supplementary Section II).

Supporting Information
Supporting Information is available from the Wiley Online Library or from the author.
Received: ((will be filled in by the editorial staff)) Revised: ((will be filled in by the editorial staff)) Published online: ((will be filled in by the editorial staff)) FIG. 5. Enthalpies of (a) BaTeO3 and (b) RbIO3 in the ABO3 structures as functions of pressure, illustrating that the cubic perovskite structure (S1) with the topological gene becomes the lowest--enthalpy structure at moderately high pressure. For graphic clarity, the enthalpies of the S4 structures (P21/m) are chosen as the zero at each pressure. Swapping of A and B sites has been considered, and the lowest--energy configuration is selected. (c--d) Electronic structures of cubic BaTeO3 (c) and RbIO3 (d) under external pressure of 35 GPa. The band inversion is denoted in the figure by BI, with arrows pointing to the inverted states. The dotted lines with different colors denote the band projection onto different atomic orbitals.