Chern and Z2 topological insulating phases in perovskite-derived 4d and 5d oxide buckled honeycomb lattices

Based on density functional theory calculations including a Coulomb repulsion parameter U, we explore the topological properties of (LaXO3)2/(LaAlO3)4 (111) with X = 4d and 5d cations. The metastable ferromagnetic phases of LaTcO3 and LaPtO3 with preserved P321 symmetry emerge as Chern insulators (CI) with C = 2 and 1 and band gaps of 41 and 38 meV at the lateral lattice constant of LaAlO3, respectively. Berry curvatures, spin textures as well as edge states provide additional insight into the nature of the CI states. While for X = Tc the CI phase is further stabilized under tensile strain, for X = Pd and Pt a site disproportionation takes place when increasing the lateral lattice constant from aLAO to aLNO. The CI phase of X = Pt shows a strong dependence on the Hubbard U parameter with sign reversal for higher values associated with the change of band gap opening mechanism. Parallels to the previously studied (X2O3)1/(Al2O3)5 (0001) honeycomb corundum layers are discussed. Additionally, non-magnetic systems with X = Mo and W are identified as potential candidates for Z2 topological insulators at aLAO with band gaps of 26 and 60 meV, respectively. The computed edge states and Z2 invariants underpin the non-trivial topological properties.

Nevertheless, recently a strong SOC effect was encountered in (LaMnO 3 ) 2 /(LaAlO 3 ) 4 (111) 29 which emerges as a Chern insulator with a band gap of 150 meV when the symmetry of the two sublattices is constrained. Since the ground state is a Jahn-Teller distorted trivial Mott insulator, selective excitation of phonons, as recently shown to induce an insulator-to-metal transition in NdNiO 3 /LaAlO 3 (001) SLs 30 , may present a pathway to suppress the symmetry breaking and access the CI state. Alternatively, 4d and 5d systems turn out to be less sensitive to symmetry breaking and the interplay of weaker correlation and stronger SOC makes them interesting candidates. This design principle served to identify LaRuO 3 and LaOsO 3 honeycomb bilayers sandwiched in LaAlO 3 (111) as Chern insulators 31 . Both Ru 3+ (4d 5 ) and Os 3+ (5d 5 ) are in the low-spin state with a single hole in the t 2g manifold whereas the homologous Fe 3+ (3d 5 ) in LaFeO 3 is found to be in a high-spin state with an AFM ground state.
A honeycomb pattern arises also in the corundum structure, albeit with smaller buckling and different connectivity. While in the perovskite structure the octahedra are corner sharing (cf. Fig. 1), in the corundum-derived SLs the XO 6 octahedra in the X 2 O 3 layer are edge-sharing as well as alternating corner-and face-sharing to the next layer above and below. The complex electronic behavior of corundum-derived honeycomb layers (X 2 O 3 ) 1 / (Al 2 O 3 ) 5 (0001) was recently addressed in a systematic study of the 3d series 32 . Moreover, among the 4d and 5d systems the ferromagnetic cases of X = Tc, Pt were identified as Chern insulators with C = −2 and −1 and band gaps of 54 and 59 meV, respectively 33 . This motivated us to explore here the perovskite analogues X = Tc, Pd, and Pt in (111)-oriented (LaXO 3 ) 2 /(LaAlO 3 ) 4 . Although the ground state is AFM, we find that a CI phase emerges for the metastable FM cases with C = 2 and 1. Furthermore, we explore the effect of strain on the stability of the CI state, as well as the dependence on the Coulomb repulsion parameter U and compare to the corundum-type systems. Last but not least, we concentrate on TI cases where time reversal and inversion symmetry are preserved and identify the non-magnetic phases of X = Mo, W in (LaXO 3 ) 2 /(LaAlO 3 ) 4 (111) superlattices as potential candidates for Z 2 TIs.

theoretical Methods
Density functional theory calculations were performed for (LaXO 3 ) 2 /(LaAlO 3 ) 4 (111) SLs employing the projector augmented wave (PAW) method 34 as implemented in the VASP 35 code. The plane-wave cutoff energy is fixed to 600 eV. The generalized gradient approximation (GGA) in the parametrization of Perdew-Burke-Enzerhof 36 was used for the exchange-correlation functional. The static local correlation effects were accounted for in the GGA + U approach, using = − U U J eff of Dudarev et al. 37 . Hubbard U values for the 4d and 5d ions are typically lower than for the 3d cations 31,33 . We used an U = 3 eV for X = Tc, Pd, Mo and 1-2 eV for X = Pt, W and a Hund's exchange parameter of J = 0.5 eV in all cases. Additionally, U = 8 eV is used for the empty 4 f orbitals of La. The calculations were performed using a Γ centered k-point grid of 12 × 12 × 2. The lattice parameter c and the internal coordinates of the superlattice structure were optimized until the Hellman-Feynman forces were less than 1 meV/Å. SOC was considered in the second-variational method with magnetization along the (001) quantization axis. For potentially interesting cases maximally localized Wannier functions (MLWFs) were constructed in order to calculate the Berry curvatures and the anomalous Hall conductivity (AHC) on a dense k-point mesh of 144 × 144 × 12 using the wannier90 code 38 .

Results and Discussion
Our previous study 33 on corundum-derived superlattices with honeycomb pattern and = X d 4 or 5d ion showed that the metastable ferromagnetic cases of X = Tc, Pt host quantum anomalous Hall states. Using the insight gained from this investigation, we explore the perovskite-derived SLs with the above-mentioned TM ions. Although the ground states of X = Tc, Pd and Pt in (LaXO 3 ) 2 /(LaAlO 3 ) 4 (111) superlattices are AFM, (cf. Table 1) with symmetry lowering due to a dimerization, manifested in alternating X-X bond lengths (not shown here), we concentrate here on the metastable ferromagnetic phases and explore their topological properties. Moreover, we investigate the effect of strain on the Chern insulating phases by considering two in-plane lattice constants of substrates LaAlO 3 (3.79 Å) and LaNiO 3 (3.86 Å) which corresponds to a change in strain of ~1.8%. We futhermore extend our study to non-magnetic solutions leading to Z 2 TIs. In particular, X = Mo, W turn out to possess Z 2 topologically invariant phases, albeit their non-magnetic phases are higher in energy by 2.0 eV and 0.4 eV per u.c., respectively, compared to the antiferromagnetic ground states (cf.  Table 1. Structural, electronic and magnetic properties of the FM state in X = Tc, Pt, Pd SLs at a LAO and a LNO , respectively. The relative energy difference of ferromagnetic (FM) configuration with respect to the antiferromagnetic (AFM) ground state, spin and orbital moments, band gaps with and without SOC ("m" denotes a metallic state) and Chern numbers are also listed. www.nature.com/scientificreports www.nature.com/scientificreports/ GGA + U ( + Soc) results: X = tc. In the following, we discuss the electronic and topological properties of ferromagnetic X = Tc for in-plane lattice constants a LAO and a LNO . Without SOC and for U eff = 2.5 eV semi-metallic band structures emerge as depicted in Fig. 2a,b. In both cases the band structures around E F are very similar and dominated by minority Tc t 2g bands (cf. Fig. 2a,b) touching at K, that extend to ~−1.4 eV (a LAO ) and ~−1.0 eV (a LNO ) and are completely separated from the lower lying majority bands. This feature is dissimilar to the band structure in the corundum honeycomb layer (Tc 2 O 3 ) 1 /(Al 2 O 3 ) 5 (0001) 33 where the majority and minority bands are entangled around E F .
The spin densities are shown in Fig. 2a and b. In the Tc 3+ 4d 4 configuration all electrons are in the t 2g subset with two unpaired electrons, reflected in a magnetic moment of 1.96 μ B for both Tc sites. This is in contrast to the corundum honeycomb layer with X = Tc 33 where a much lower magnetic moment of 0.93 μ B is found resulting from a violation of Hund's rule due to a strong hybridization between Tc 4d and O 2p states.
We proceed with the effect of SOC on the band structures illustrated in Fig. 3a,b. The strongest influence is observed at the Fermi level around the K point where SOC induces anti-crossings and opens gaps of 41 and 53 meV for a LAO and a LNO , respectively. Calculation of the AHC (Fig. 3e,f) shows that the LaTcO 3 honeycomb layer emerges as a Chern insulator with C = 2 for both strain values. The largest contributions to the Berry curvature Ω k ( ) (Fig. 3c,d) arises along K-M. The enhanced gap for a LNO leads to a broader Hall plateau at E F (cf Fig. 3e,f). The results demonstrate that the CI phase is further stabilized under tensile strain. A similar effect of strain was observed in (Tc 2 O 3 ) 1 /(Al 2 O 3 ) 5 (0001) 33 . We note that the sign of the Chern number C = 2 for X = Tc in the (111)-oriented perovskite bilayer is reversed compared to the corundum-derived SL (C = −2) 33 . The reversal of sign is related to the specific band topology and band gap opening mechanism and the predominance of minority bands, whereas in the corundum case majority bands reside around E F 33 .
GGA + U results for isoelectronic X = Pd, Pt. We next turn to the isoelectronic X = Pd and Pt.
Experimental studies suggest paramagnetic metallic behavior for bulk LaPdO 3 39,40 . For the honeycomb layer of X = Pd and Pt at a LAO P321 symmetry is preserved and the band structures in Fig. 4a,c show two very similar sets of four majority and minority bands, the former lying about 1 eV lower than the latter. Both exhibit Dirac crossings at K, the one of the majority band being slightly above the Fermi level. Consequently, the dispersive majority and the bottom of the minority bands cross E F and lead to a metallic state. This differs from the isoelectronic LaNiO 3 analogon where for P321 symmetry the Dirac point is fixed at the Fermi level 29,41-43 . A substantial occupation of both e g orbitals and contribution from the O 2p states is visible from the spin-densities (see Fig. 4) which www.nature.com/scientificreports www.nature.com/scientificreports/ indicates a d 8 L configuration instead of the formal d 7 occupation and bears analogies to the isoelectronic LaNiO 3 43-45 . In contrast, for tensile strain (a LNO ) the P321 symmetry is lowered and a gap of ~60 meV and ~370 meV (cf. Fig. 4b,d) is opened for LaPdO 3 and LaPtO 3 , respectively. The gap opening arises due to the disproportionation of the two Pd and Pt triangular sublattices expressed in different magnetic moments: In X = Pd the two sites acquire magnetic moments of 0.85 μ B and 0.60 μ B (cf. Fig. 4b and Table 1). For X = Pt this site-disproportionation is more pronounced with magnetic moments of 1.06 μ B and 0.32 μ B on the two Pt sites (cf. Fig. 4d) resulting in a larger gap between the occupied majority and unoccupied minority pairs of bands whose dispersion is significantly reduced. The site-disporportionation at tensile strain goes hand in hand with a breathing mode expressed in a larger and a smaller PtO 6 octahedron with volumes of 13.8 and 11.8 Å 3 , respectively. As a consequence, the Pt-O bond lengths result in 2.17, 2.18 Å at the first and 2.03, 2.10 Å at the second Pt site. Such a disproportionation is common in bulk rare earth nickelates 46  with the Fermi level (cf. Fig. 5a) and hampers the formation of a quantized Hall plateau (see Fig. 5i). For 1.0 < U eff < 2.0 eV (cf. Fig. 5b,c), the Fermi level lies inside the gap arising from band inversion between bands of opposite spin and the system becomes a Chern insulator with C = 1. Increasing the Coulomb repulsion strength from 1.0 to 1.5 eV enhances the band gap (from 31 to 38 meV) and the Hall plateau which stabilizes the Chern insulating phase (see Fig. 5k). As can be seen from Fig. 5f,g positive contributions to the Berry curvature arise  www.nature.com/scientificreports www.nature.com/scientificreports/ around K. For higher values (U eff ≥ 2.5 eV) the effect of SOC changes from band inversion between bands of opposite spin to avoided crossing between two bands with the same spin with a larger gap of 66 meV, leading to a sign reversal of the Chern number from + 1 to −1 (cf. Fig. 5l). This is consistent with the large negative Berry curvature contribution around K in Fig. 5h. While for 5d systems U values beyond 2.0 eV appear to be too high, we note that the band structure for = .  Fig. 6a) reveals that non-vanishing contributions appear solely on a rounded triangular feature around K marking the anticrossing line of the majority and minority band. The surface state in Fig. 6c calculated employing the MLWF method 53 presents a single chiral edge state associated with C = 1. For X = Tc the largest contribution to Ω k ( ) emerges along K-M (cf. Fig. 6b,d) resulting in two in-gap chiral states whose features are similar to (Tc 2 O 3 ) 1 / (Al 2 O 3 ) 5 (0001) 33 .
Here we briefly address the spin texture of the highest occupied band (marked by arrows in Figs 3a and 5b) in the Chern insulating phase for X = Tc and X = Pt, respectively. For X = Pt (see Fig. 7c) the spin texture is dominated by majority (red) components in the larger part of the BZ and exhibits an orientation reversal of minority (blue) s z spin components close to K, consistent with the SOC-induced band inversion between the occupied majority and unoccupied minority band around K discussed above. The spin texture of (LaTcO 3 ) 2 /(LaAlO 3 ) 4 (111) in Fig. 7a is rather collinear and exhibits only negative s z values throughout the entire BZ. This is consistent with the fact that only bands of minority character appear around E F . Overall, even though the number of edge states of the perovskite and corundum case (Tc 2 O 3 ) 1 /(Al 2 O 3 ) 5 (0001) 33 are identical, the differences (we remind that in the corundum case a vortex arises around Γ) can be attributed to the distinct electronic structure and the effect of SOC, as discussed in Section III A. Z 2 topological invariant systems: GGA + U ( + Soc) results for isoelectronic X = Mo, W. Besides the potential CI phases in d 7 and d 4 systems studied above, we investigate the non-magnetic phases of the two d 3 systems containing the homologous elements X = Mo and W at a LAO . Despite the AFM ground state (cf. Table 2), the potentially interesting systems were identified to be non-magnetic. We note that previous theoretical studies 54 www.nature.com/scientificreports www.nature.com/scientificreports/ suggest that bulk LaMoO 3 and LaWO 3 should be non-magnetic. The band structures in Fig. 8a,b reveal that the bandwidth of the t 2g manifold amounts to ~1.8 eV for X = W and ~1.5 eV for X = Mo. The larger bandwidth correlates with the larger extension of the 5d orbitals as compared to 4d. In contrast to bulk LaMoO 3     which are metallic, the non-magnetic perovskite superlattices exhibit semiconducting behavior with gaps of 28 meV and 62 meV, that are only weakly modified by SOC, 26 meV and 60 meV, respectively. Nevertheless, the degeneracy of bands is lifted along K-Γ (cf. Fig. 8c,d).
In order to verify the topological features of these two systems, we carry out edge state calculations by constructing the MLWFs. The edge Green's function and the local density of states (LDOS) can be simulated using an iterative method 53,56,57 . In the case of X = Mo, one can clearly see a gapless Dirac cone at the Γ point (cf. Fig. 9a), whereas one topologically protected chiral edge state is obtained for X = W (cf. Fig. 9b) connecting the valence and conduction bands. We note that additional background LDOS may arise owing to hybridization with trivial edge states emerging due to the particular procedure of creating the edge interfaced with vacuum. Similar effects have been observed previously 9,25,58 . Since the investigated systems have crystal IS and TRS, Z 2 can be calculated as a product of parities of all occupied states at the TRIM points by applying the criterion of Fu and Kane 59 . In Fig. 10a-f the Wannier function center evolution (WCC) is calculated for X = Mo using the Wilson loop method 60,61 . For the k 1 and k 2 planes the Z 2 indices are 0 whereas for k 3 = 0 and k 3 = 0.5 the Z 2 indices yield 1.

Summary
In summary, we investigated the possibility to realise topologically nontrivial states in (111)-oriented perovskite-derived honeycomb LaXO 3 layers with X = 4d and 5d, separated by the band insulator LaAlO 3 . The metastable ferromagnetic phases of (LaTcO 3 ) 2 /(LaAlO 3 ) 4 (111) and (LaPtO 3 ) 2 /(LaAlO 3 ) 4 (111) emerge as CI with C = 2 and 1, respectively, at the lateral lattice constant of LaAlO 3 (3.79 Å). Thereby, the persistence of P321 symmetry, lattice strain and the inclusion of a realistic Hubbard U term turn out to be crucial. For X = Pd the CI phase appears for U beyond 3.5 eV that likely exceeds the realistic range for a 4d compound. For X = Pt the sign of the Chern number is reversed beyond U eff = 2.0 eV due to change of the spin orientation of the contributing bands. The CI phase for (LaTcO 3 ) 2 /(LaAlO 3 ) 4 (111) is further stabilized under tensile strain (at the lateral lattice constant of LaNiO 3 ), similar to the corundum-based SL (Tc 2 O 3 ) 1 /(Al 2 O 3 ) 5 (0001). In contrast, for X = Pd and Pt in (LaXO 3 ) 2 /(LaAlO 3 ) 4 (111) tensile strain lifts the P321 symmetry and induces a site-disproportionation on the two sublattices which opens a trivial band gap and bears analogies to the behavior of the isoelectronic nickelate superlattices 29,[43][44][45]48,49 . Further insight into the topological aspects is gained by analyzing the Berry curvatures, edge states and spin textures. A closer inspection of the spin texture for LaPtO 3 reveals a spin orientation reversal along the loop of band inversion around K of two bands with opposite spin character. Moreover, we explored non-magnetic perovskite SLs where TRS is preserved and identified X = Mo and W as potential candidates for Z 2 TIs. The existence of edge states and non-trivial Z 2 indices supports this outcome.
We note that most of the effects (i.e. emergence of topologically nontrivial (Chern) insulating phases) we observe are interaction-driven, i.e. they do not appear within DFT/GGA (U = 0 eV) calculations. On the other hand with increasing static correlation effects also the tendency towards stabilization of trivial Mott insulating phases connected with symmetry lowering is enhanced. This applies not only to the systems we address here which show AFM ground states but has been observed in several previous studies 28,32,33 . Although the specific systems proposed here have not yet been synthesized, recent experimental studies reported the successful growth of related (111)-oriented nickelate superlattices 62-64 as well as nickelate and manganate perovskite heterostructures 65 . Thus, we trust that our theoretical predictions will encourage further experimental efforts to realize and characterize the proposed systems.