Searching for non-trivial topological states has been one of the most active fields in condensed matter physics and materials science. In time-reversal-invariant crystalline solids, topological insulating state and topological metallic state (with Dirac points and/or Dirac node lines) have been intensively studied in narrow-gap semiconductors whose electronic structure is dominated by s and p orbitals [1,2]. However, complex oxides with characteristic d orbitals have been less explored. An intriguing proposal is to study a bi-layer of perovskite oxide AMO3 thin film along (111) direction , in which the transition metal atom M resides on a buckled honeycomb lattice. Like graphene, the transition metal d-bands form a linear crossing at the high-symmetry K/K’ point in the Brillouin zone and spin-orbit coupling (SOC) opens a gap at the crossing  and thus a quantum spin Hall state may emerge. However, (111) terminations of perovskite oxide AMO3 are polar and it is very difficult to synthesize such films with precise control of their thickness in experiments . By contrast, (001) perovskite oxide heterostructures have been routinely synthesized [6,7]. In particular, those oxides with non-polar terminations [A2+O2-] and [M4+(O2-)2] can be accurately controlled on the atomic scale in a layer-by-layer manner in an oxide superlattice .
Therefore, we wonder whether non-trivial topological properties may emerge in (001) oxide superlattices. We start from the simplest configuration: a (001) (AMO3)1/(AM’O3)1 superlattice, in which M and M’ are two different transition metal atoms (shown in panel a of Fig. 1). Our design principles are as follows. We choose M to be an early transition metal atom and M’ to be a late transition metal atom. Due to the electronegativity difference, the d orbitals of M have higher energies than the d orbitals of M’ [9,10]. Given a proper combination of M and M’ and their d occupancy, M-d states, and M’-d states may overlap with each other around the Fermi level (shown in panel b). We conjecture that when spin-orbit coupling (SOC) is included, we can have two situations in which non-trivial topology may emerge. One is that SOC can open a full gap between the highest valence band and the lowest conduction band, and a topological insulating state may emerge, given a proper interaction between M-d and M’-d states (shown in panel c). Another situation is that a linear crossing is stabilized between an M-d band and an M’-d band around the Fermi level even in the presence of SOC (shown in panel d), then it is possible to get a topological semi-metal state with Fermi arcs.
Fig. 1 a: The crystal structure of a (001) (AMO3)1/(AM’O3)1 superlattice. b: A schematic diagram of density of states of the superlattice. c and d: A schematic band structure close to the Fermi level where an M-d band and an M’-d band invert. c: Spin-orbit coupling opens a full gap between the highest valence band and the lowest conduction band, and non-trivial topological properties such as strong topological insulator (TI) may emerge. d: An M-d band and an M’-d band form a linear crossing around the Fermi level in the presence of spin-orbit coupling, which may lead to a topological Dirac semi-metal (TDS).
To test our conjecture, we use first-principles methods and model Hamiltonian calculations to study some non-magnetic oxide superlattices, which have time-reversal-symmetry. We find that both situations are possible, given a proper combination of two transition metal atoms.
For the first situation, we choose the two transition metal atoms to be Ta and Ir. Thus, we construct a (001) (SrTaO3)1/(SrIrO3)1 superlattice. In this case, the d occupancy of Ta4+ and Ir4+ ions are d1 and d5, respectively. Due to the cell doubling that is needed to accommodate the oxygen octahedral rotation, the total d occupancy of the superlattice is (1 + 5) × 2 = 12, which can be divided by 4. Considering that the superlattice has both time-reversal and inversion symmetries, this total occupancy (a multiple of 4) implies a possible insulating ground state. Indeed, from our calculations, we find the (SrTaO3)1/(SrIrO3)1 superlattice is a strong topological insulator (TI) with (1;001) Z2 index. As a consequence, we find topologically protected surface bands on all the surfaces of this superlattice. The bulk band structure and the topological surface bands on a representative surface are shown in panels a and b of Fig. 2.
For the second situation, we choose two transition metal atoms to be Mo and Ir, i.e. to build a (001) (SrMoO3)1/(SrIrO3)1 superlattice. Different from d1 + d5, the total d occupancy of Mo4+ and Ir4+ are d2 and d5, respectively. The superlattice also has oxygen octahedral rotations, which require cell doubling. Therefore, the total occupancy is (2 + 5) × 2 = 14, which is 3 × 4 + 2. Given the same symmetry considerations, we must have a gapless system with one band half-filled when the total d occupancy is 14. Our calculations do find that the superlattice is a semi-metal, which has a pair of type-II Dirac points (see panel c of Fig. 3). The Dirac points have a non-trivial mirror Chern number, manifested as the number of surface Fermi arcs that connect the two Dirac points. What is more intriguing is that the (SrMoO3)1/(SrIrO3)1 superlattice has multiple coexisting topological insulators (TI) and topological Dirac semi-metal (TDS) states, which is shown in panel c of Fig. 3. Both TI and TDS states induce a topologically protected surface Dirac cone. We find that in the energy-momentum space, there are three of them. The two TI Dirac cones sandwich the TDS Dirac cone (see panel d of Fig. 3). Such topological properties are rare, implying that one may induce a TI-TDS-TI topological state transition via chemical doping  or electric-field gating .
Fig. 2 a: DFT+SOC band structure of the (SrTaO3)1/(SrIrO3)1 superlattice. b: Band structure of (100) semi-infinite slab of the (SrTaO3)1/(SrIrO3)1 superlattice, calculated by the Green-function method. c: DFT+SOC band structure of (SrMoO3)1/(SrIrO3)1 superlattice along G to Z path. d: Band structure of (010) semi-infinite slab of the (SrMoO3)1/(SrIrO3)1 superlattice, calculated by using the Green-function method.
While we focus on time-reversal-invariant systems in this study, the design principles can be extended to time-reversal breaking systems (i.e. magnetic systems) as well. In those oxide superlattices, other topologically non-trivial states such as quantum anomalous Hall and Weyl semi-metal may emerge via a proper materials design.
For further information, please read our published article: npj Computational Materials (2022) 8:208; https://doi.org/10.1038/s41524-022-00894-5.
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