Abstract
Topological insulators are characterized by a nontrivial band topology driven by the spinorbit coupling. To fully explore the fundamental science and application of topological insulators, material realization is indispensable. Here we predict, based on tightbinding modelling and firstprinciples calculations, that bilayers of perovskitetype transitionmetal oxides grown along the [111] crystallographic axis are potential candidates for twodimensional topological insulators. The topological band structure of these materials can be finetuned by changing dopant ions, substrates and external gate voltages. We predict that LaAuO_{3} bilayers have a topologically nontrivial energy gap of about 0.15 eV, which is sufficiently large to realize the quantum spin Hall effect at room temperature. Intriguing phenomena, such as fractional quantum Hall effect, associated with the nearly flat topologically nontrivial bands found in e_{g} systems are also discussed.
Introduction
Since the discovery^{1,2} of the quantum Hall effect (QHE), the quest for topologically ordered states of matter has become a major subject of interest in condensed matter physics. Haldane^{3} first proposed that electrons hopping on a honeycomb lattice could realize the QHE in the absence of Landau levels, pointing out the possibility of nontrivial topology in simple band insulators. Along this direction, recent efforts have culminated in the theoretical prediction^{4,5,6,7} and subsequent experimental realization^{8,9,10} of the socalled topological insulators (TIs) in materials with strong spinorbit coupling (SOC). Many interesting phenomena, including giant magnetoelectric effects^{11} and the appearance of Majorana fermions^{12}, have been predicted. Once realized in real materials, these phenomena could lead to entirely new device paradigms for spintronics and quantum computing.
However, so far, the material realization of TIs has been limited to narrow bandgap semiconductors based on Hg or Bi, in which the electronic properties are dominated by s and p orbitals. Here, we report our theoretical investigation of topological insulating behaviour in a completely different materials class—heterostructures of transitionmetal oxides (TMOs) involving d electrons. Our motivation is twofold. First, artificial heterostructures of TMOs are becoming available owing to the recent development^{13,14,15} in the fields of oxide superlattices and oxide electronics^{16}. In particular, layered structures of TMOs can be now prepared with atomic precision, thus offering a high degree of control over important material properties, such as lattice constant, carrier concentration, SOC and correlation strength. As we show below, these advantages can be readily exploited in the design of TIs. Second, TMOs constitute a wide class of compounds that exhibit a variety of intriguing properties and electronic states associated with the electron–electron interactions, encompassing superconductivity, magnetism, ferroelectricity and Mott insulators. Combined with the TI phase, TMO heterostructures provide a very promising platform to explore various topological effects.
Our main results are summarized below. We first demonstrate the design principle for realizing twodimensional (2D) TIs in bilayers of perovskitetype TMOs grown along the [111] crystallographic axis by using phenomenological tightbinding (TB) modelling. Based on this design principle and firstprinciples calculations, a number of candidate materials are identified. The topological band structure of these materials can be finetuned by changing dopant ions, substrates and external gate voltages, which will enable also the control of the topological quantum phase transition. In particular, we predict that LaAuO_{3} bilayer has a topologically nontrivial energy gap about 0.15 eV, which is sufficiently large to realize the quantum spin Hall effect at room temperature. When electron–electron interaction is included, our system with topologically nontrivial band structure could have far more interesting physics. Here, we demonstrate this possibility by focusing on the TMO bilayers of e_{g} systems, which are characterized by nearly flat topologically nontrivial Z_{2} bands. We argue that when these bands are partially filled, electron correlation could give rise to the quantum anomalous Hall (QAH) effect and the fractional QHE. Our results may open new directions focusing on topological phenomena in the rapidly growing field of oxide electronics.
Results
Design principle
To demonstrate the design principle for engineering TIs in TMO heterostructures, we consider perovskitetype TMOs as our prototype system. These compounds are very common and have the chemical formula ABO_{3}, where O is oxygen and B is a transitionmetal (TM) ion. The key idea is to start with a band structure that possesses 'Dirac points' in the Brillouin zone without the SOC, and then examine whether an energy gap can be opened at those points with the SOC turned on. If an energy gap does open, combined with proper filling the resulting state could be a TI. In an ideal perovskite structure, the TM ions sit on a simple cubic lattice, with the octahedral crystalline field splitting the TM d orbitals into twofold degenerate and threefold degenerate levels, well separated by socalled 10Dq on the order of 3 eV. Such a lattice geometry usually does not support Dirac points. Instead, we consider bilayers of the perovskite structure grown in the [111] direction. As shown in Figure 1, the TM ions in the (111) bilayer are located on a honeycomb lattice consisting of two trigonal sublattices on different layers. This lattice geometry has three consequences: firstly, it is well known from the study of graphene that electrons hopping on a honeycomb lattice generally give rise to Dirac points in the band structure; secondly, a layer potential difference can be easily created by applying a perpendicular electric field or by sandwiching the bilayer between two different substrates, which allows experimental control of the band topology; and, thirdly, the honeycomb lattice further reduces the symmetry of the crystalline field from octahedral (O_{h}) to trigonal (C_{3v}), and introduces additional level splitting of the d orbitals. The last point turns out to be crucial for realizing the topologically insulating phase.
We first consider the t_{2g} manifold, in which the onsite SOC is active. In our modelling, only nearestneighbour (NN) hopping of d electrons between the TM sites via oxygen p orbital is included. As we are interested in the band topology, which is robust against small perturbations as long as the band gap remains open, our model is justified and allows us to capture the essential ingredients with minimal parameterization. The TB Hamiltonian is given by
where r and τ label the lattice sites and the t_{2g} orbitals, respectively. The first term is the hopping term represented by a single amplitude t and the dimensionless structural factor . The second term is the onsite SOC, which splits the t_{2g} levels into a j=1/2 doublet with energy λ and a j=3/2 quadruplet with energy −λ/2. l_{r} and s_{r} are the angular momentum and spin operators. The third term is the trigonal crystalline field which splits the t_{2g} manifold into a_{1g} and e_{g}′ manifolds with their level separation given by 3Δ/2. V in the last term is the layer potential difference, and ξr=1 when r is in the top or bottom layer. The explicit form of the Hamiltonian is presented in the Methods.
The large number of orbitals (six per TM site) involved in our model give rise to a very rich behaviour of the topological band structure in the parameter space. Depending on the strength of the SOC, the system falls into two different phases. In the strong SOC limit (λ/t>8/3 when Δ=0), bands originating from the j=1/2 and j=3/2 orbitals are completely separated. The trigonal crystal field then opens up an energy gap within each manifold, and a nontrivial Z_{2} topology can be realized. This is similar to the results reported on Iridium compounds^{17,18}. In the weak SOC limit, bands from j=1/2 and j=3/2 orbitals become mixed away from the Γ point. Again, we find topologically nontrivial energy gaps that can be opened by Δ. The Z_{2} topological invariant is determined using two different methods. One can either evaluate it directly from the bulk band structure^{19} or count the number of edge states. The calculated band structure for both cases together with the Z_{2} index is shown in Figure 2a–d. By inspection, we find that t_{2g}^{1}, t_{2g}^{2} , t_{2g}^{3} and t_{2g}^{5} TMOs are all possible candidates for TIs in the strong SOC limit, and t_{2g}^{2} , t_{2g}^{4} and t_{2g}^{5} in the weak SOC. The dependence of the band topology on the layer potential difference V is rather interesting. While increasing V will eventually destroy the Z_{2} nontrivial phase, under moderate values of V, the t_{2g}^{3} system remains TI in the strong SOC limit, and the t_{2g}^{4} system in the weak SOC limit.
Next, we consider the e_{g} manifold. It is well known that the SOC is quenched within the e_{g} manifold so it seems that the resulting band topology should be trivial. However, similar to graphene^{20,21} and some TM ions^{22,23}, the SOC can still take place through the virtual excitation of electrons between e_{g} and t_{2g} levels. According to the secondorder perturbation theory, the effective SOC is given by
where ɛ labels the e_{g} orbitals. Hence, the Hamiltonian can be written
where the SOC magnitude is given by , with Δ_{E} roughly being the energy difference between e_{g} and t_{2g} levels. (The explicit form of the Hamiltonian is presented in the Methods and in Supplementary Note 1.) We find that opens up an energy gap at the Γ point and also the Dirac point located at K. From the inspection of the Z_{2} topological invariant and counting the number of edge states, we found that e_{g}^{1}, e_{g}^{2} and e_{g}^{3} systems become TIs (Fig. 2e,f). Here, the trigonal crystalline field is also important—if all t_{2g} levels are degenerated, even the secondorder SOC vanishes. A layer potential difference comparable to closes the gap at the Dirac point, turning the e_{g}^{2} system into a trivial insulator. On the other hand, gaps at the Γ point are stable against this perturbation. Instead, these gaps close when the local potential difference between and is comparable to . Thus, the TI state and the Jahn–Teller effect^{24} compete in real materials with the e_{g}^{1} or e_{g}^{3} configuration.
Materials consideration
Having established that the TIs can be realized in (111)bilayer TMO for both t_{2g} and e_{g} configurations, we now turn to real materials. We aim to realize the integer fillings established above using TM B ions with the formal valence +3 or +4. For B^{3+(4+)}, we choose La (Sr) for the Asite element in both the target TMO and the insulating substrate AB′O_{3}, and Al (Ti) for the B′site element in the insulating substrate. Controlling the strain effects and the layer potential difference is possible by replacing A and/or B′ with their isovalent elements. It is well known that some of the TMOs are insulating due to strong correlations^{25}. Therefore, if the corresponding bulk system is heavily insulating, bilayering may not be useful. Even if the corresponding bulk system is metallic, the low dimensionality in (111) bilayers may drive the system into a Mott insulator^{26}. Further, the correlation effects are expected to reduce the effective band width and increase the splitting between occupied levels and unoccupied levels. While this effect does not change the band topology in e_{g} electron systems, this could influence the topology in t_{2g} systems by modifying the crystal field splitting between a_{1g} and e^{′}_{g} levels. In addition, in a system with an integer number of electrons per site, local moments could be induced by the correlation effects resulting in the magnetic ordering. If the symmetry breaking by the magnetic ordering is strong, the system could become a trivial insulator. We do not consider such complexities by focusing on rather itinerant 4d and 5d electrons of TM ion, yet t_{2g} electron systems are more susceptible for magnetic orderings than e_{g} electron systems because of the smaller hopping intensity. These considerations somewhat limit the choice of TM and substrate material. Our candidate materials for TIs are, therefore, LaRe^{3+}O_{3} as a t_{2g}^{4} electron system, LaRu^{3+}O_{3}, LaOs^{3+}O_{3}, SrRh^{4+}O_{3} and SrIr^{4+}O_{3} as t_{2g}^{5} systems, and LaAg^{3+}O_{3} and LaAu^{3+}O_{3} as e_{g}^{2} electron systems. Most of these materials have been synthesized and their references are summarized in Table 1. LaReO_{3}, LaOsO_{3} and LaAgO_{3} have yet to be synthesized. According to Ralle and Jansen^{35}, bulk LaAuO_{3} has CaF_{2} structure rather than the perovskite. We expect this material shapes the perovskite structure by, for example, highpressure synthesis and grown on a substrate with the perovskite structure. If properly synthesized, perovskite LaAuO_{3} is expected to be metallic as LaAgO_{3} predicted by the density functional theory (DFT) calculation^{33}. We also note that growing thin films of perovskite TMOs along the [111] direction has already started^{36,37}. While t_{2g}^{2} systems are also candidates for TIs, the TI state is hard to realize because of the band overlap. For e_{g}^{1} and e_{g}^{3} systems, additional effects such as longer range transfer and the Jahn–Teller effect can easily modify the dispersion relations.
We first performed the DFT calculations for the bilayers of t_{2g} systems LaReO_{3}, LaRuO_{3}, LaOsO_{3}, SrRhO_{3} and SrIrO_{3} (details are presented in the Methods section.) Their dispersion relations are shown in Figure 3a–d. We notice the remarkable agreement between the DFT results and the TB result, Figure 2c, especially for LaReO_{3} and LaOsO_{3}. For SrRhO_{3}, LaReO_{3} and LaRuO_{3} (not shown), the Fermi level crosses several bands. Thus, these systems are classified as topological metals rather than TIs. In LaOsO_{3} and SrIrO_{3}, the Fermi level is located inside the gap. Therefore, from the analogy to the TB model, (111) bilayers of LaOsO_{3} and SrIrO_{3} are TIs. From our DFT calculations, it is noted that the material dependence of the dispersion relations is rather large for t_{2g} systems. This is because a large number of band parameters are involved in the band structure including the local crystalline field.
We now move to e_{g}^{2} electron systems, LaAgO_{3} and LaAuO_{3}. Our DFT results for these systems are shown in Figure 3e,f. As in the t_{2g} case, the DFT reproduces the TB result fairly well. In both undoped systems, the Fermi level is inside the gap at the K point, and from the analogy to the TB result, these systems are TIs. The gap amplitude is found to be about 150 meV for LaAuO_{3} and 40 meV for LaAgO_{3}, so these systems should remain TIs at room temperature. The band gap and topological property can be controlled by breaking the symmetry between top and bottom layers. As shown in Figure 3g, the asymmetric bilayer with LaScO_{3} has larger band gap ~300 meV, and from the inspection of the symmetry of the wave function at the K point, this bilayer is a trivial insulator. On the other hand, the asymmetric bilayer with YAlO_{3} has smaller band gap ~50 meV and remains to be a TI (Fig. 3h). Such a small band gap TI is especially useful to control the topological property by using the gate voltage. Similar to LaAgO_{3} and LaAuO_{3}, we also performed the DFT calculations for a 3d system LaCuO_{3}. We found that this system develops an instability towards magnetic ordering because the itinerancy is reduced compared with 4d and 5d systems.
Nearly flat Z_{2} topologically nontrivial bands
One of the appealing aspect of realizing nontrivial band topology in TMOs is the rich possibilities of novel phenomena that could emerge when electron correlation is considered. Here, we demonstrate one of the possibilities in the e_{g} systems with nearly flat Z_{2} topologically nontrivial bands (see Fig. 2e). Obviously, when the chemical potential is tuned into the nearly flat Z_{2} bands, we are in a novel regime of quantum manybody physics. From the inspection of dispersion relations shown in Figure 3e and f, the upper flat band appears to be more stable than the lower flat band. Tuning the chemical potential in the upper flat band corresponds to removing electrons from d^{10} systems or adding electrons to d^{9} systems. When such a situation is realized, kinetic energy is suppressed and physics is controlled mainly by interaction, whose energy scale is typically 1–2 eV, much larger than the width of flat bands W~0.2 eV. What would be the ground state of the system? Here, we propose several natural candidate states. One very likely consequence of the shortrange repulsion U, when the doping of the nearly flat band is not too small, is to drive the system ferromagnetic because of Stoner's criteria U/W»1. On the other hand, when doping is too small, Wigner crystal phase should naturally occur. In the following, we assume that spontaneous ferromagnetic ordering occurs. The magnetic order should be viewed as a breaking of a discrete symmetry, due to the SOC, instead of the breaking of a continuous spin rotation symmetry, which is not realized in a 2D system at any finite temperature. Our discussion is not limited to spontaneous ordering, because magnetism can be also introduced externally by using a magnetic insulator as a substrate.
We model the effect of ferromagnetic ordering by adding a Zeeman spin splitting to the Hamiltonian:
Resulting dispersion relations for e_{g} system with two characteristic Zeeman fields are presented in Figure 4. From the inspection of the Chern number, the QAHinsulating state^{38} is realized in e_{g}^{0.5} and e_{g}^{3.5} systems, and also e_{g}^{1}, e_{g}^{1.5}, e_{g}^{2.5} and e_{g}^{3} systems when the Zeeman field is large. The e_{g}^{1} configuration with the large Zeeman splitting is realized in undoped perovskite manganites. More interesting physics occurs when the nearly flat 'Chern' band is partially filled (see Figs 2e and 4). In this case, as pointed out recently, fractional quantum Hall (FQH) states are likely to be realized^{39,40,41,42,43,44}. To elaborate this possibility, we perform exact diagonalization calculation after projecting onsite repulsion and NN repulsion into the 1/3 filled highest Chern band (that is, at e_{g}^{3.5+1/6}). Indeed, signatures of a ν=1/3 FQH state are observed (details of the exact diagonalization and the numerical result are presented in the Supplementary Note 2). We find that, when the NN repulsive interaction is larger than the width of the flat band, the ground state degeneracy is threefold on a torus, and the Chern number (excluding the integer Chern number from the filled bands) of the ground state wave function is ~1/3 up to finite size correction.
The aforementioned FQH effects and QAH effects are both hightemperature effects and fundamentally different from the quantum Hall states realized in GaAs 2D electron gas in a magnetic field, where quasiparticle energy gap is controlled by the longrange Coulomb repulsion (ɛ is the dielectric constant and l_{B} is the magnetic length), typically around a few Kelvin. In the present systems, quasiparticle energy gap is determined by the shortrange repulsion ~1–2 eV. This indicates that room temperature FQH effects may be realized. FQH states, in particular the nonAbelian states, have been shown to be very useful as building blocks of a quantum computer^{45,46}. A hightemperature nonAbelian quantum Hall states in the TMO heterostructures, for example, at ν=1/2 filling where natural candidate states are in the same universality class of Pfaffian states^{47} or antiPfaffian states^{48,49}, if realized experimentally, would have strong impacts on both fundamental physics and its applications, including the efforts of realizing topological quantum computation.
Discussion
Before closing, we make few remarks on TIs in the TMO bilayers. The direct confirmation of the TI state is possible by measuring the conductance. As in Bernevig et al.^{5} and König et al.^{8}, the conductance should be quantized as σ=2e^{2}/h per (111) bilayer in the twoterminal measurement. The conductance can be controlled by using the gate voltage. In our DFT calculations, the (111) bilayers are repeated along the [111] axis. With the nonzero interbilayer coupling, the helical edge channels on the surface of the sample will turn to the two Dirac fermions at k_{[111]}=0 and 1/2 in the unit of the reciprocal lattice vector along the [111] direction. Thus, strictly speaking, the TI is classified as a 'weak' TI and the backward scattering between the two Dirac fermions, if it exists, causes the localization. In order to avoid this, one needs to make the single (111) bilayer or keep the neighbouring (111) bilayers far apart so that the interbilayer coupling becomes exponentially small. When fabricating the (111) bilayer, a small number of defects would have minor effects because the edge modes surrounding them are disconnected. However, as the defect density increases, two surfaces are eventually connected through the edge modes belonging to the islands. As a result, the backward scattering takes place. Another source of the localization is islands of thicker (111) TMO layers. The suppressed trigonal field inside such thick islands is expected to create the gapless bulk modes. In further thicker islands of (111) TMO layers, metallic state would be realized inside the sample. In addition to (111) bilayers, we have studied model (111) trilayers and found that the TI states are robust without the even–odd oscillation between TI and trivial insulator, as predicted for the bismuth thin films^{50}. Structurally, a (111) trilayer forms a socalled dice lattice, which could also bring about interesting quantum effects characterized by the Chern number C=±2 (ref. 51). Of course, if the layer structure is too thick then the bulk cubic symmetry is restored and the system is no longer a TI. So far, we did not mention the correlation effects and competing ground states except for the previous section. For limiting cases, we have performed unrestricted Hartree–Fock calculations for multiorbital Hubbard models defined on (111) bilayers. We found that the TI states are rather robust for e_{g}^{2} systems and become unstable against antiferromagnetic insulating states when the interaction strength is comparable to the full bandwidth as in the 2D Hubbard model on the honeycomb lattice^{52}. For e_{g}^{1} or e_{g}^{3} systems, the QAHinsulating states could be generated dynamically by correlation effects without the SOC^{53,54}, yet trivialinsulating states due to the Jahn–Teller effect would also be stabilized depending on the relative balance between the Coulomb interaction and the Jahn–Teller coupling. In this paper, we focused on the perovskitetype TMOs. Thus, our design principle for the TI state works only for the [111] plane because other planes such as [001] and [110] do not support a honeycomb lattice. However, this approach is not limited to the perovskite systems. For example, the [0001] plane of corundum Al_{2}O_{3}, that is, sapphire, involves a honeycomb lattice formed by Al atoms. Such a system could also be utilized as the substrate material to artificially create the TI state.
Methods
TB models in the real space
First, we consider a general multiband TB model on a cubic lattice given by
where r labels the TM sites, σ spin and μ orbitals. is a transfer matrix, which depends on the pair of orbitals but not on the spin; its detail will be presented shortly.
For t_{2g} electron systems, the trigonal crystal field directly couples with the local t_{2g} level. In addition, the angular momentum is not quenched, and therefore the SOC is active. Including these two effects, a TB model for t_{2g} systems is written as with H_{SO} and H_{tri} given by the second and the third terms of equation (1), respectively. The explicit form of H_{SO} for the t_{2g}alone model is given by
with the use of the following convention for the orbital index: , and . σ^{τ} with τ=a,b,c is the Pauli matrix, and is the Levi–Civita antisymmetric tensor.
The dependence of transfer matrices on the orbital and direction is given by the Slater–Koster formula^{55} as follows:
for the NN hopping and
for the secondneighbour (SN) hopping. Here, are the unit vector along the x, y and z direction, respectively. Although it is via weak π hybridization t_{pd}^{π} between a TM ion and an oxygen ion, the NN hopping is the largest parameter in this model, thus, taken as the unit of energy t. Δ_{pd} is the level difference between TM d orbitals and oxygen p orbitals. The ratio between and t_{π} is the dimensionless parameter t_{δ′} is also the NN hopping, but it is via weak direct overlap and, therefore, is expected to be small. t_{σ′′} and t_{π′} are the SN hoppings due to the higherorder processes involving the transfer between two oxygen ions as As involves relatively strong (weak) σ(π) hybridization between two oxygen ions we expect Typical transfer intensities are shown in Supplementary Fig. S1(a).
For e_{g} electron systems, linear coupling with the trigonal crystal field is absent. Therefore, the C_{3} lattice symmetry of a (111) bilayer does not influence the onsite e_{g} level. On the other hand, e_{g} degeneracy can be lifted by the distortion of an O_{6} cage surrounding a TM ion, that is, the Jahn–Teller effect. Focusing on the metallic regime, we neglect this effect. The angular momentum is quenched unless the coupling between e_{g} and t_{2g} orbitals are considered. We also neglect this effect at the moment but reconsider it later. Thus, for the e_{g}alone model, The dependence of transfer matrices on the orbital and direction is again given by the Slater–Koster formula^{55}. For the NN hopping, we have
and for the SN hopping
Here, ɛ(=α, β) labels the e_{g} orbitals as and For e_{g} electron systems, the NN hopping is via strong σ hybridization between a TM ion and an oxygen ion t_{pd}^{σ} and, therefore, largest. This hopping integral is taken as the unit of energy t. Again, the dimensionless parameter is defined by the ratio between and t_{σ} as The NN hopping t_{δ} is due mainly to the direct overlap between two TM ions and, therefore, expected to be small as t_{δ′} in the t_{2g} orbital model. t_{σ′} is the SN hopping due to the higherorder processes involving the transfer between two oxygen ions as . Typical transfer intensities are shown in Supplementary Fig. S1(b).
TB models on the (111) bilayer
By constraining the atomic coordinate r within the (111) bilayer R=(X,Y), it is straightforward to derive the TB Hamiltonian as a function of 2D momentum k=(k_{x}, k_{y}). We use the convention in which the projection of the NN bond into the (111) plane is taken as the unit of the length scale. This is a factor smaller than the lattice constant of the cubic perovskite, and the size of the new unit cell is Taking the primitive lattice vectors as and the first Brillouin zone is a hexagon with six corners located at
For the t_{2g} orbital model, we obtain
with
Here, the spin indices are suppressed for simplicity. ∓V/2 is the sublatticedependent potential, which breaks the symmetry between the top (labelled 1) and bottom (labelled 2) layers.
For the e_{g} orbital model, we obtain
with
DFT calculations
DFT calculations were carried out using the projector augmented wave method^{56} with the generalized gradient approximation in the parametrization of Perdew, Burke and Enzerhof^{57} for exchange correlation as implemented in the Vienna Ab Initio Simulation Package^{58}. The default planewave energy cutoff for O, 400.0 eV, was consistently used in all the calculations. The optimized crystal parameter is 3.81 Å for bulk LaAlO_{3}, and 3.95 Å for bulk SrTiO_{3}. These values are in consistent with the experimental values of 3.79 Å (Berkstresser et al.^{59}) and 3.91 Å (Hellwege and Hellwege^{60}). The TM bilayer structures were simulated by a supercell consisting of 12 AO_{3} and 12 B layers along the [111] direction with (A,B)=(La,Al) or (Sr,Ti) with two adjacent B layers replaced by TM ions. In the (111) plane, the supercell contains a 1×1 unit cell. A 6×6×1 special kpoint mesh including the Γ point (0,0,0) was used for integration over the Brillouin zone. Optimized atomic structures were achieved when forces on all the atoms were <0.01 eV/Å.
Additional information
How to cite this article: Xiao, D. et al. Interface engineering of quantum Hall effects in digital transitionmetal oxide heterostructures. Nat. Commun. 2:596 doi: 10.1038/ncomms1602 (2011).
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Acknowledgements
We thank H. Christen, K. Yamaura, Wanxiang Feng and A. Rüegg for their stimulating discussions. Work by W.Z. and S.O. was supported by the US Department of Energy, Office of Basic Energy Sciences, Materials Sciences and Engineering Division. D.X. was supported by the Laboratory Directed Research and Development Program of ORNL. N.N. acknowledges support from MEXT GrantinAid for Scientific Research (19048008, 19048015 and 21244053). Computational support was provided by NERSC of US Department of Energy.
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S.O. conceived the idea for the (111) bilayer and constructed the theoretical models. Model calculations were performed by D.X. and S.O. (noninteracting models), and Y.R. (interacting models). The density functional theory calculations were performed by W.Z. N.N. provided further theoretical inputs. S.O. and D.X. were responsible for overall project direction, planning and management.
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Supplementary Figures S1S2, Supplementary Notes 12 and Supplementary References (PDF 297 kb)
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Xiao, D., Zhu, W., Ran, Y. et al. Interface engineering of quantum Hall effects in digital transition metal oxide heterostructures. Nat Commun 2, 596 (2011). https://doi.org/10.1038/ncomms1602
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