Graphene analogue in (111)-oriented BaBiO3 bilayer heterostructures for topological electronics

Topological electronics is a new field that uses topological charges as current-carrying degrees of freedom. For topological electronics applications, systems should host topologically distinct phases to control the topological domain boundary through which the topological charges can flow. Due to their multiple Dirac cones and the π-Berry phase of each Dirac cone, graphene-like electronic structures constitute an ideal platform for topological electronics; graphene can provide various topological phases when incorporated with large spin-orbit coupling and mass-gap tunability via symmetry-breaking. Here, we propose that a (111)-oriented BaBiO3 bilayer (BBL) sandwiched between large-gap perovskite oxides is a promising candidate for topological electronics by realizing a gap-tunable, and consequently a topology-tunable, graphene analogue. Depending on how neighboring perovskite spacers are chosen, the inversion symmetry of the BBL heterostructure can be either conserved or broken, leading to the quantum spin Hall (QSH) and quantum valley Hall (QVH) phases, respectively. BBL sandwiched by ferroelectric compounds enables switching of the QSH and QVH phases and generates the topological domain boundary. Given the abundant order parameters of the sandwiching oxides, the BBL can serve as versatile topological building blocks in oxide heterostructures.

exhibit extra valley-dependent spin splitting at the gapped Dirac cone and introduce spin-valley coupling, which is frequently seen in the monolayer transition metal dichalcogenide series 20 . These features are summarized in Fig. 1(b), and the corresponding ab-initio calculations are shown in Fig. 1(c-h).
Electronic structures. For the symmetric BBL configuration where M I = M II = Zr, the electronic structures without and with SOC are calculated in Fig. 1(c),(d), respectively. Gapless linear band dispersion appears inside the large band-gap of the sandwiching oxide, BaZrO 3 , and the low-energy band structure is governed by the Bi 6s-orbital Dirac cone ( Fig. 1(c)). When SOC is included, a band-gap of 79 meV opens at the Dirac point ( Fig. 1(d)); opening of a band-gap by SOC in an otherwise gapless Dirac cone indicates the presence of a Kane-Mele type topological insulator. The unexpectedly large SOC-induced gap in the s-orbital system is attributed to the strong Bi 6s-6p hybridization and the large atomic SOC of the Bi 6p-orbital (for details on the tight-binding analysis and SOC, see Supplementary Information). The topologically protected edge state, shown in Fig. 1(e), of the ribbon geometry with the zigzag edge confirms the QSH phase of the symmetric BBL heterostructure via the bulk-boundary correspondence. The same QSH phase of the symmetric configuration is attainable with other sandwiching perovskite compounds, including a Sr-based heterostructure in which all of the Ba atoms are replaced by Sr atoms. These are listed in Table 1, and the band-gap ranges from 45 to 100 meV. Therefore, the QSH phase in the symmetric BBL configuration is detectable and usable at room temperature. Figure 1(f),(g) depict the electronic structures without and with SOC, respectively, for the asymmetric BBL configuration where M I = Zr and M II = Hf. The structural inversion asymmetry induced by sandwiching with different oxides produces the potential difference at each sublattice site of the buckled honeycomb structure, and results in a QVH phase with a direct band-gap of 0.92 eV at the Dirac point ( Fig. 1(f)). In the asymmetric BBL configuration, the band topology is characterized by the valley-Chern number, and the QVH phase is verified by calculating the valley-contrasting Berry curvature 10 as shown in Fig. 1(h). The Berry curvature at each K and K′ valley has the opposite sign, leading to the non-zero valley-Chern number. As SOC is turned on, extra valley-dependent spin splitting at both the conduction and valence band edges occurs by 103 and 29 meV, respectively ( Fig. 1(g)). In addition to the usefulness of the QVH phase for topological electronics, this interesting spin-valley coupling might be available for spin-valleytronics applications 20,21 ; near K and K′, both valley-dependent Zeeman-type and valley-independent Rashba-type spin splitting occur on the out-of-plane and in-plane spin components, respectively (see Supplementary Information Fig. S2).
Ferroelectric control of topological phases. Having revealed that two distinctive topological phases, QSH and QVH, take place in the same framework, the BBL heterostructure still has room for improvement in terms of topology-tunability; the QSH phase still cannot be converted into the QVH phase and vice versa within one system. Recalling that the topological phase of the BBL heterostructure is governed by its symmetry, topology-control is synonymous with symmetry-control in this system. Therefore, introducing a symmetry-breaking order parameter in the neighboring perovskite oxide makes it possible to control the topology of the BBL heterostructure. Because the corresponding symmetry that distinguishes the QSH and QVH phases is the spatial inversion, BBL sandwiched by ferroelectric perovskite oxides provides a way to control the inversion symmetry, and ultimately switch the QSH and QVH phases.
By sandwiching the well-known ferroelectric BaTiO 3 with the [111] polarization direction, we can construct a BaTiO 3 /BBL/BaTiO 3 heterostructure as a unified system hosting and controlling both the QSH and QVH phases. Depending on the relative polarization direction of the BaTiO 3 on either side of the BBL, the inversion symmetry can be either conserved or broken; if the polarization direction is the opposite (same), the inversion symmetry of the BBL heterostructure is conserved (broken). As a result, the QSH (QVH) phase with a band-gap of 108 meV (1.17 eV) arises within the anti-parallel (parallel) polarization configuration, as shown in Fig. 2(a),(b)). Moreover, by developing a ferroelectric domain boundary, a topological domain boundary is induced, in which a topologically protected edge state emerges and the topological charge flows. Suppose that we prepare a BaTiO 3 /BBL/ BaTiO 3 heterostructure that initially has a parallel polarization configuration. As the polarization of the upper BaTiO 3 layer starts to be flipped using an atomic-force microscope (AFM) tip and a ferroelectric domain boundary is generated, the topological domain boundary between the QSH and QVH phases develops simultaneously (see Fig. 2(c)). This controllability 22 enables us to manipulate the conducting path of the topological charge in a non-volatile, reconfigurable way, which is an important operation in topological electronics.  transforms into the QVH phase by spontaneously breaking the sublattice symmetry. Therefore, it is important to verify the robustness of Bi 4+ and the graphene-like electronic structure in the BBL heterostructure against the charge ordering instability. We applied a small external electric field along the [111]-direction of the BaZrO 3 / BBL/BaZrO 3 slab geometry to examine whether a small perturbation can trigger the charge ordering and induce the QVH phase abruptly. As shown in Fig. 3(a), the QSH band-gap of 75.7 meV decreases continuously, closes at ε c = 26 mV/Å, and re-opens as the field strength increases. At the gap-closing point, band inversion occurs and the topology of the system changes from the QSH to the QVH phase. Consequently, unlike the bulk BaBiO 3 , the Bi 4+ charge state and QSH phase in the symmetric configuration of the BBL heterostructure remain intact under a small external perturbation. Recent experiments examining BaBiO 3 thin films showed that the charge ordering instability is suppressed as the film thickness decreases 24 , which is consistent with our result. This stability test revealed one more appealing aspect; the Bi 4+ charge state is not only sufficiently stable to sustain multiple Dirac cones, but is also susceptible to easy manipulation of the band-gap and topological phase by a small electric field. The QSH gap is closed by an external field of ε c = 26 mV/Å and the band topology is switched to the QVH phase at greater field strengths. The large buckling angle in the BBL honeycomb lattice and the latent Bi 3+/5+ -tendency seem to be responsible for the easy tunability, and make the system a potential platform for topological electronics via the electric field. Taking advantage of the easy controllability of the band-gap and topology of the BBL heterostructure, we designed a topological electronic field-effect-transistor 25 , as shown in Fig. 3(b-g). The single-gated transistor sketched in Fig. 3(b) can develop the QSH/QVH domain boundary, possibly leading to the QSH edge state when the gate electric field ε z > ε c . Considering the finite width W of the gate electrode, the edge states located at both sides of the topological domain boundary can overlap if the band-gap of the QVH phase induced by ε z is small. Given that the penetration length of the edge state into the insulating bulk is v where v F is the Fermi velocity of the Dirac cone (3.55 × 10 5 m/s in the BBL heterostructure) and E g is the band-gap, strong edge overlap occurs when  For example, if W = 50 nm and ε c < ε z ≤ 29.2 mV/Å, the QSH edge states on both sides easily interact via the shallow-gapped bulk region, hampering its ability to carry a topological charge through the topological domain boundary. Therefore, a gate electric field well above ε c can host the spin-momentum-locked QSH edge states (see Fig. 3(c),(d)). The double-gated system sketched in Fig. 3(e) can give rise to various topological domain boundary configurations and their corresponding topological edge states. As well as the QSH/QVH boundary, the topological domain boundary between two different QVH phases, whose non-zero valley-Chern numbers have the opposite sign, also appears depending on the gate electric fields ε 1z and ε 2z (Fig. 3(f)). Again, due to the finite width of the two gate electrodes W 1 and W 2 , a strong edge-overlapped region exists, and is shown in grey in Fig. 3(f). With |ε 1z | < ε c < |ε 2z | (|ε 2z | < ε c < |ε 1z |), the double-gated transistor behaves in the same way as the single-gate transistor, and the strong edge-overlapped region is determined by  When both gate electric fields exceed ε c and are aligned in parallel, the QVH phase is formed in a region whose effective width is W 1 + W 2 ; thus, the condition for the strong edge overlap between the QSH edge states becomes  + . Interestingly, two gate electric fields exceeding ε c but with an anti-parallel alignment develop two QVH phases possessing the opposite valley-Chern numbers. As a result, the valley-momentum-locked QVH edge states appear along the inner QVH domain boundary 26,27 , together with the QSH edge states along the outer QSH/QVH boundary. The existence and position of various non-trivial edge states can be controlled by the gate electric fields in the double-gated transistor (Fig. 3(g)). In an array of multiple topological electronic transistors based on the BBL heterostructure, we can tailor the conducting path that carries two different topological charges 28 .
In conclusion, we show that the hetrostructure of a (111)-oriented BaBiO 3 bilayer sandwiched between perovskite oxides can host QSH and QVH phases with the choice of sandwiching materials, and is therefore a promising platform of topological electronics. Although the overall discussion focuses on controlling the inversion symmetry, a BBL heterostructure sandwiched by various ferroic perovskite oxides will likely serve as a general scheme for hosting many controllable topological phases. For example, by introducing a ferromagnetic sandwiching oxide, we can break the time-reversal symmetry of the BBL in two different ways. With parallel ferromagnetic moments on both sides, the BBL honeycomb lattice is subject to a constant Zeeman field, possibly generating the quantum anomalous Hall phase by breaking the time-reversal symmetry 4 . For the anti-parallel ferromagnetic moments, a sublattice-dependent staggered Zeeman field arises, leading to the so-called quantum spin-valley Hall phase by breaking both the time-reversal and inversion symmetries but conserving the composite of the two symmetries 29 . For the BBL sandwiched by multi-ferroic perovskite oxides, we can control both the spatial inversion and time-reversal symmetries simultaneously 30 . The potential to make a three-dimensional integrated superlattice composed of various topological compartments makes the BBL heterostructure even more fascinating.

Methods
First-principles calculations. The first-principles calculations were performed using the density functional theory with projector augmented wave potentials and the Perdew-Burke-Ernzerhof exchange correlation functional as implemented in VASP code 31,32 . The plane-wave energy cutoff was set at 400 eV, and 12 × 12 × 1 mesh was used for momentum space sampling. The atomic structures were relaxed with the force criteria of 0.005 eV/Å. The wannier90 code was used to construct the maximally-localized Wannier function 33 , in which four Bi s-projectors (including spin components) were used to reproduce the band structures near the Fermi level. The Berry curvature was calculated with the linear response Kubo formula, and the tight-binding Hamiltonian of strip geometry was constructed using the obtained Wannier functions and hopping parameters. Data Availability. The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.