Abstract
Based on density functional theory calculations including a Coulomb repulsion parameter U, we explore the topological properties of (LaXO_{3})_{2}/(LaAlO_{3})_{4} (111) with X = 4d and 5d cations. The metastable ferromagnetic phases of LaTcO_{3} and LaPtO_{3} 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 LaAlO_{3}, 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 a_{LAO} to a_{LNO}. 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 (X_{2}O_{3})_{1}/(Al_{2}O_{3})_{5} (0001) honeycomb corundum layers are discussed. Additionally, nonmagnetic systems with X = Mo and W are identified as potential candidates for Z_{2} topological insulators at a_{LAO} with band gaps of 26 and 60 meV, respectively. The computed edge states and Z_{2} invariants underpin the nontrivial topological properties.
Introduction
Chern insulators and Z_{2} invariant topological insulators belong to subgroups of topological insulators (TIs) with and without broken timereversal symmetry (TRS), respectively. A Chern insulator, also known as a quantum anomalous Hall insulator (QAHI) exhibits a quantized Hall conductivity without an external magnetic field^{1,2}. In this context, CI are promising as potential candidates for the realization of Majorana fermions and the application in lowpower electronics. The Chern insulator posseses chiral edge states with electrons traversing only in one direction, where the number of conducting edge states is determined by the Chern number^{3}. A Z_{2} invariant TI supports the quantum spin Hall effect (QSHE) which can be regarded as two copies of an IQH (integer quantum Hall) system with electrons forming Kramers pairs and counterpropogating helical edge states. The properties of QSHE have been addressed in conjunction with the graphene lattice^{4} and HgTe quantum well structures^{5,6,7}. Some further systems include stanene films functionalized with an organic ethynyl molecule (SnC_{2}H)^{8}, sandwiched 2D arsenene oxide (AsO) between boron nitride (BN) sheets^{9} and amidogenfunctionalized Bi/Sb(111) films (SbNH_{2} and BiNH_{2})^{10}. Lattices hosting a honeycomb pattern are of particular interest for topologically nontrivial states, as initially proposed by Haldane^{11}. QAHI have been demonstrated in TIs doped with magnetic impurities such as Mndoped HgTe or Cr, Fedoped Bi_{2}Te_{3}, Bi_{2}Se_{3}, Sb_{2}Te_{3}^{12,13,14}. Another possibility to break TRS is by placing 5d transition metals on graphene^{15,16} or OsCl_{3}^{17}, as well as SnHN/SnOH^{18}. Recently, transition metal oxides (TMO) have attracted interest due to their interplay of spin, orbital and lattice degrees of freedom. In contrast to conventional TIs whose bands near the Fermi energy are derived from s and ptype orbitals, the narrower dbands lead to larger band gaps and a tendency towards TRS breaking. QAH phases have been predicted both for rocksalt (EuO/CdO^{19} and EuO/GdO^{20}), rutilederived heterostructures^{21,22,23}, pyrochlore oxides^{24} and 2D Nb_{2}O_{3}^{25}. As noticed by Xiao et al.^{26}, a buckled honeycomb lattice can be formed from two triangular XO_{6}layers in the AXO_{3} perovskite structure grown along the [111]direction. Perovskitederived bilayers of SrIrO_{3} and LaAuO_{3} were proposed as candidates for TIs, however interaction effects in the SrIrO_{3} bilayer lead to an AFM ground state^{27,28}. 3d TM ions tend to host stronger electronic correlations and weaker spinorbit coupling (SOC). 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 JahnTeller distorted trivial Mott insulator, selective excitation of phonons, as recently shown to induce an insulatortometal 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 lowspin 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 highspin 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 corundumderived SLs the XO_{6} octahedra in the X_{2}O_{3} layer are edgesharing as well as alternating corner and facesharing to the next layer above and below. The complex electronic behavior of corundumderived 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 corundumtype systems. Last but not least, we concentrate on TI cases where time reversal and inversion symmetry are preserved and identify the nonmagnetic 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 planewave cutoff energy is fixed to 600 eV. The generalized gradient approximation (GGA) in the parametrization of PerdewBurkeEnzerhof ^{36} was used for the exchangecorrelation functional. The static local correlation effects were accounted for in the GGA + U approach, using \({U}_{{\rm{eff}}}=UJ\) 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 \(\Gamma \) centered kpoint grid of 12 × 12 × 2. The lattice parameter c and the internal coordinates of the superlattice structure were optimized until the HellmanFeynman forces were less than 1 meV/Å. SOC was considered in the secondvariational 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 kpoint mesh of 144 × 144 × 12 using the wannier90 code^{38}.
Results and Discussion
Our previous study^{33} on corundumderived superlattices with honeycomb pattern and \(X=4d\) 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 perovskitederived SLs with the abovementioned 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 inplane 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 nonmagnetic solutions leading to Z_{2} TIs. In particular, X = Mo, W turn out to possess Z_{2} topologically invariant phases, albeit their nonmagnetic 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 2).
GGA + U ( + SOC) results: X = Tc
In the following, we discuss the electronic and topological properties of ferromagnetic X = Tc for inplane lattice constants a_{LAO} and a_{LNO}. Without SOC and for U_{eff} = 2.5 eV semimetallic 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 anticrossings 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 \(\Omega (k)\) (Fig. 3c,d) arises along KM. 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 corundumderived 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^{29,41,42,43}. A substantial occupation of both e_{g} orbitals and contribution from the O 2p states is visible from the spindensities (see Fig. 4) which indicates a d^{8}L configuration instead of the formal d^{7} occupation and bears analogies to the isoelectronic LaNiO_{3}^{43,44,45}. 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 \({\mu }_{{\rm{B}}}\) (cf. Fig. 4b and Table 1). For X = Pt this sitedisproportionation is more pronounced with magnetic moments of 1.06 \({\mu }_{{\rm{B}}}\) and 0.32 \({\mu }_{{\rm{B}}}\) 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 sitedisporportionation 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 PtO 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^{46,47}, (001) or (111)oriented LaNiO_{3}/LaAlO_{3} SLs^{29,43,44,45,48,49} as well as La_{2}CuO_{4}/LaNiO_{3}(001) SLs^{50}.
X = Pt: Emergence of a CI phase as a function of U
Since at a_{LNO} both LaPdO_{3} and LaPtO_{3} result in trivial Mott insulators due to site disproportionation, we explore here the topological properties at a_{LAO}, where the P321 symmetry is preserved. Upon including SOC, for X = Pd a CI phase emerges for U values beyond \({U}_{{\rm{eff}}}^{{\rm{c}}}=3.5\,{\rm{eV}}\) (not shown here), which are likely too high for a 4d element. We concentrate here on the topological properties of (LaPtO_{3})_{2}/(LaAlO_{3})_{4}(111) as a function of Hubbard U. Up to U_{eff} = 2.0 eV SOC leads to a band inversion between the majority and minority bands around K. At U_{eff} = 0.5 eV the conduction band still overlaps 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 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 \({U}_{{\rm{eff}}}=2.5\,{\rm{eV}}\) is similar to the results from a calculation with the hybrid functional HSE06^{51,52} with standard mixing parameter of exact exchange of 0.25 (See Supplemental Material for additional information on electronic and structural properties).
Edge states and spin textures of CI phases
Unlike in (Pt_{2}O_{3})_{1}/(Al_{2}O_{3})_{5}(0001)^{33} where contributions to the Berry curvature \(\Omega (k)\) arise along M and K, the top view of the Berry curvature for U_{eff} = 1.0 eV (see Fig. 6a) reveals that nonvanishing 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 \(\Omega (k)\) emerges along KM (cf. Fig. 6b,d) resulting in two ingap 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 SOCinduced 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 \(\Gamma \)) 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 nonmagnetic 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 nonmagnetic. We note that previous theoretical studies^{54} suggest that bulk LaMoO_{3} and LaWO_{3} should be nonmagnetic. 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} and LaWO_{3}^{55} which are metallic, the nonmagnetic 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\(\Gamma \) (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 \(\Gamma \) 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 perovskitederived 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 corundumbased 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 sitedisproportionation 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 nonmagnetic 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 nontrivial 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 interactiondriven, 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^{62,63,64} 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.
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Acknowledgements
We acknowledge discussions with W. E. Pickett and D. Khomskii on related systems. This research was supported by the German Science Foundation within CRC/TRR80 (Project number 107745057), project G03 and in part by the National Science Foundation under Grant No. NSF PHY1748958. We acknowledge computational time at the Leibniz Rechenzentrum, project pr87ro.
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O.K. performed the calculations under the guidance of R.P., O.K. and R.P. analyzed and interpreted the results and wrote the manuscript.
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Köksal, O., Pentcheva, R. Chern and Z_{2} topological insulating phases in perovskitederived 4d and 5d oxide buckled honeycomb lattices. Sci Rep 9, 17306 (2019). https://doi.org/10.1038/s41598019531251
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