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 *AM*O_{3} thin film along (111) direction [3], 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 [4] and thus a quantum spin Hall state may emerge. However, (111) terminations of perovskite oxide *AM*O_{3} are polar and it is very difficult to synthesize such films with precise control of their thickness in experiments [5]. By contrast, (001) perovskite oxide heterostructures have been routinely synthesized [6,7]. In particular, those oxides with non-polar terminations [*A*^{2+}O^{2-}] and [*M*^{4+}(O^{2-})_{2}] can be accurately controlled on the atomic scale in a layer-by-layer manner in an oxide superlattice [8].

Therefore, we wonder whether non-trivial topological properties may emerge in (001) oxide superlattices. We start from the simplest configuration: a (001) (*AM*O_{3})_{1}/(*AM’*O_{3})_{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.

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**Fig. 1 a**: The crystal structure of a (001) (*AM*O_{3})_{1}/(*AM’*O_{3})_{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) (SrTaO_{3})_{1}/(SrIrO_{3})_{1} superlattice. In this case, the *d* occupancy of Ta^{4+} and Ir^{4+} ions are *d*^{1 }and *d*^{5}, 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 (SrTaO_{3})_{1}/(SrIrO_{3})_{1} superlattice is a strong topological insulator (TI) with (1;001) *Z*_{2} 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) (SrMoO_{3})_{1}/(SrIrO_{3})_{1} superlattice. Different from *d*^{1 }+ *d*^{5}, the total *d* occupancy of Mo^{4+} and Ir^{4+} are *d*^{2} and *d*^{5}, 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 (SrMoO_{3})_{1}/(SrIrO_{3})_{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 [11] or electric-field gating [12].

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**Fig. 2 a**: DFT+SOC band structure of the (SrTaO_{3})_{1}/(SrIrO_{3})_{1} superlattice. **b**: Band structure of (100) semi-infinite slab of the (SrTaO_{3})_{1}/(SrIrO_{3})_{1} superlattice, calculated by the Green-function method. **c**: DFT+SOC band structure of (SrMoO_{3})_{1}/(SrIrO_{3})_{1} superlattice along G to Z path. **d**: Band structure of (010) semi-infinite slab of the (SrMoO_{3})_{1}/(SrIrO_{3})_{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.

**Reference**

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[8] Ramesh, R. & Schlom, D. G. Creating emergent phenomena in oxide superlattices. *Nat.** **Rev. Mater.* **4**, 257-268 (2019).

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