Abstract
Chern insulators are twodimensional magnetic topological materials that conduct electricity along their edges via the onedimensional chiral modes. The number of these modes is a topological invariant called the first Chern number C that defines the quantized Hall conductance as S_{xy} = Ce^{2}/h. Increasing C is pivotal for the realization of lowpowerconsumption topological electronics, but there has been no clearcut solution to this problem so far, with the majority of existing Chern insulators showing C = 1. Here, by using stateoftheart theoretical methods, we propose an efficient approach for the realization of the highC state in MnBi_{2}Te_{4}/hBN van der Waals multilayer heterostructures. We show that a stack of n MnBi_{2}Te_{4} films with C = 1 intercalated by hBN monolayers gives rise to a high Chern number state with C = n, characterized by n chiral edge modes. This state can be achieved both under the external magnetic field and without it, both cases leading to the quantized Hall conductance S_{xy} = Ce^{2}/h. Our results, therefore, pave the way to practical highC quantized Hall systems.
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Introduction
The Chern insulator (CI) state is a quantum phase of twodimensional (2D) gapped materials with broken timereversal invariance and nontrivial electronic band topology^{1,2,3,4}. It is most straightforwardly probed via Hall measurements, the hallmark being a vanishing longitudinal conductance S_{xx} along with a transversal conductance S_{xy} quantized to integer multiples of the conductance quantum, Ce^{2}/h^{5,6}. Here, e is the electron charge, h is Planck’s constant, and C is a dimensionless integer called the first Chern number, corresponding to the number of the 1D gapless chiral modes residing at the CI film’s edge. The existence of these modes is guaranteed by the nontrivial band topology of CI.
The edge modes of a CI conduct electricity without dissipation, which could be useful for the construction of novel highly efficient chiral interconnects for lowpowerconsumption electronics^{7,8}. However, the contact resistance between a metal electrode and CI in the envisioned interconnect devices is a bottleneck limiting their performance. To reduce this resistance as much as possible, the number of chiral edge modes, i.e., the Chern number C, should be as large as possible^{7,8}. Therefore, it is of great interest and importance to engineer CIs with high Chern numbers.
Historically, the CI state was first observed in 2D electron gases in 1980 in a transport phenomenon that is now known as the quantum Hall effect (QHE)^{5}. The QHE in this system stems from the formation of Landau levels under the external magnetic field, which drives the system into a topologically nontrivial state. However, welldefined Landau levels are only possible in systems with high carrier mobility under strong external magnetic fields, which prevents this QHE from a widely applied use.
Notwithstanding, the developments in the research field of magnetic topological insulators (TIs) in the last decade have allowed a qualitative leap toward QHE without Landau levels. In particular, the quantum anomalous Hall effect (QAHE), a special kind of QHE that occurs without the external magnetic field, has been observed^{9}. It is mainly realized in the thin films of TIs of the (Bi,Sb)_{2}Te_{3} family doped by Cr or/and V atoms^{9,10,11,12,13,14}. In these systems, C = 1, and although it is theoretically possible to increase C by increasing the dopant concentration and the film thickness^{15,16}, this has not been experimentally realized to date.
Instead, complex materials engineering has been resorted to in order to achieve C = n > 1 state based on the magnetically doped TIs using the following idea. Rather than seeking a highC state in a particular system, it can be realized by stacking n CI layers with C = 1 each. In this case, it is necessary, however, that the adjacent CI layers are efficiently decoupled from each other by a normal, i.e., topologically trivial, insulator layer. In this way, C up to 4 and 5 have been achieved in (Cr,V)_{x}(Bi,Sb)_{2−x}Te_{3}/CdSe^{17} and heavily Crdoped (Bi,Sb)_{2}Te_{3}/Cr_{x}(Bi,Sb)_{2−x}Te_{3} multilayers^{18}, respectively. Incidentally, such a multilayer heterostructure design also allows the realization of one of the very first proposals to achieve a Weyl semimetal phase^{19}. Although the studies^{17,18} represent a proofofconcept of the C enhancement approach, it is well known that the potential of magnetically doped TIs for QAHbased applications is quite limited. Namely, due to a strong disorder in the pnictogen sublattice, which is randomly occupied by Bi, Sb, and magnetic dopants, both electronic^{20,21} and magnetic^{22,23} properties of such materials are strongly inhomogeneous. Therefore, the observation of the QAHE in these systems appears to be limited to about 2 K at best^{12,24}, with no further improvements achieved over the last several years^{12}.
Recently, new systems showing the C = 1 QAHE have emerged, such as the intrinsic magnetic topological insulators of the MnBi_{2}Te_{4} (MBT) family^{25,26,27}, the twisted bilayer graphene^{28}, and the transition metal dichalcogenide moiré superlattices^{29} opening new opportunities for C engineering. MBT, shown in Fig. 1a, appears as particularly promising due to its van der Waals (vdW) nature, the intrinsic combination of nontrivial band topology and longrange antiferromagnetic order (T_{Néel} = 25 K), as well as large predicted surface band gap^{25,26,30,31,32,33,34,35,36,37,38,39,40,41,42}. The QAHE in thin MBT flakes has been achieved up to about 1.4 K, leaving a large room for the observation temperature enhancement. Indeed, a recent study^{36} demonstrates the actual potential of this material by registering the C = 1 (C = 2) QHE up to 30 K (13 K) in its thin flakes, ferromagnetically polarized by an external magnetic field. Remarkably, the quantization, in this case, appears without the Landau levels, in contrast to the conventional QHE observed in 2D electron gas^{5}.
Here, inspired by the recent progress on the Q(A)HE in MBT, we propose a novel MBTbased high Chern number material. Namely, we design a multilayer vdW heterostructure, in which thin MBT CI films are stacked on top of each other, interlayed by hexagonal boron nitride (hBN) monolayers that decouple and insulate them from one another (Fig. 1b). As an inert wide band gap insulator, hBN is an ideal material for such a decoupling, widely used in vdW heterostructure devices as an encapsulating layer or substrate for the stacked 2D materials^{43,44,45,46}. Using the stateoftheart density functional theory and tightbinding calculations, we show that the weak vdW bonding between MBT and hBN essentially preserves the C = 1 CI state in the individual MBT layers. This state can correspond to either (1) the QH insulator phase achieved in thin MBT films under the external magnetic field but without the Landau levels or (2) the QAH insulator phase at zero field if the MBT films are made of the odd number of septuple layer blocks. In either case, stacking n MBT films with C = 1 interlayed by (n − 1) hBN monolayers gives rise to a C = n state, with n as large as allowed by the vdW heterostructures growth technology. Our results provide an excellent platform for the realization of highC materials.
Results
Crystal and electronic structure of MBT/hBN interface
MnBi_{2}Te_{4} crystallizes in the trigonal \(R\bar{3}m\)group structure^{47,48,49}, made up of septuple layer (SL) blocks, in which hexagonal atomic layers are stacked in the TeBiTeMnTeBiTe sequence, as shown in Fig. 1a. Neighboring SLs are bound by vdW forces. Below T_{Néel} = 25 K, MnBi_{2}Te_{4} orders antiferromagnetically due to the antiparallel alignment between the alternating ferromagnetically ordered Mn layers^{25,34}, with the local moments pointing outofplane (Fig. 1a).
We start our study with the consideration of structural, magnetic, and electronic properties of the MBT/hBN bilayer made of the 2 SL thick MBT film (MBT_{2SL}) and hBN monolayer. This can be considered a minimal system because it contains all of the essential characteristics of MBT, such as intra and interlayer exchange couplings as well as the nontrivial topology in the forced FM state^{30}, so it can be used to test whether they are affected by hBN.
The MBT and hBN basal planes are symmetry compatible and show a good lattice parameter matching in the MBT (1 × 1)/hBN(\(\sqrt{3}\times \sqrt{3}\)) configuration, with a mismatch of only about 0.6%. The optimal hBN adsorption geometry was then determined by the comparison of total energies of structurally optimized MBT_{2SL}/hBN in four highsymmetry registries of such configuration, shown in Supplementary Fig. 3a–d. These energies were calculated for both interlayer FM and AFM spin configurations of MBT_{2SL}, assuming the outofplane magnetic moment direction, to determine a possible influence of hBN on the interlayer exchange coupling, which is in MBT significantly weaker than the intralayer one (the latter is addressed below, as well). For more details, see Supplementary Note 2.A.1.
The relevant numerical results are listed in Table 1. It can be seen that for all adsorption registries, the interlayer distance d between MBT and hBN is about 3.5–3.6 Å, and the AFM spin configuration is lower than the FM configuration by at least ΔE_{A/F} = E_{AFM} − E_{FM} ≃ 1 meV per Mn pair. The lowest energy is obtained for the hollow site geometry, shown in Fig. 2a and Supplementary Fig. 3a. The large d on one hand and the overall similarity of the energy differences ΔE_{A/F} in MBT_{2SL}/hBN and MBT_{2SL}^{30} on the other suggest the vdW bonding of MBT and hBN.
Having determined the MBT/hBN interface geometry, we can explore the effect of hBN on the MBT_{2SL} electronic structure and topology. Since we are interested in the CI state, the FM interlayer spin alignment is considered here, for which MBT_{2SL} has been predicted to have C = 1^{30}. Noteworthy, in experiments, the FM alignment in MBT is achieved by the external magnetic field application^{26,35,36,50}.
Comparison of the band structure along the \(\overline{{{{\rm{K}}}}}\overline{{{\Gamma }}}\overline{{{{\rm{M}}}}}\) path for MBT_{2SL}/hBN in the hollow site registry and pure MBT_{2SL} is shown in Fig. 2b. Their similarity near the Fermi level is immediately obvious, and the band gaps, 63.5 and 62.7 meV, respectively, are very close. The hBN states can be found at about 1.5 eV below the Fermi level (and deeper) as well as over 3.2 eV above it. Furthermore, Fig. 2c shows that the Fermi level dependencies of the anomalous Hall conductance, S_{xy}(E_{F}), of MBT_{2SL}/hBN and MBT_{2SL} are very well matched, too. In particular, in both cases, S_{xy} is constant inside the band gap, where the actual Fermi level is, and it is equal to one conductance quantum e^{2}/h, indicating the CI state with C = 1. Accordingly, the edge spectral function of MBT_{2SL}/hBN in Fig. 2d features a single chiral edge mode traversing the band gap, similar to MBT_{2SL}. Finally, a Wilson loop method calculation for MBT_{2SL}/hBN yields ∣C∣ = 1 as well, with the sign depending on the magnetization direction, as expected for a CI^{4}.
The calculations of the band structures, S_{xy}(E_{F}) and C for the other three highsymmetry registries show that the same results hold for all of them (see Table 1 and Supplementary Fig. 3e–k for visualization). These findings clearly confirm the vdW nature of the bond between hBN and MBT, which preserves the CI state in the forced FM phase of the MBT_{2SL} film.
We note that recently it has been reported elsewhere^{51} that while hBN and MBT are bound by vdW interaction, the interlayer FM configuration in MBT_{2SL}/hBN becomes significantly lower in energy than the AFM one (by up to 45 meV) for any adsorption registry. We have attempted to reproduce those results by retracing the steps outlined in ref. ^{51} (see Supplementary Note 2.A.2), but arrived at the same results that we present here. We believe that our result is correct on the physical basis that the vdW interaction, along with the insulating character of hBN, should not produce such a drastic effect on magnetism as it was found in ref. ^{51}.
Finally, we confirm the implicit assumptions of preference for the FM intralayer spin configuration and the outofplane easy axis direction by total energy calculations on MBT_{1SL}/hBN (see Supplementary Notes 2.A.3 and 2.A.4, respectively). The former calculation reveals that the FM configuration is 16.9 meV (per Mn pair) lower than the AFM configuration, while the latter yields a positive magnetic anisotropy energy of 0.07 meV per Mn atom (vs. 0.074 meV in pure MBT_{1SL}), meaning that the easy axis indeed stays outofplane. Thus, neither the intralayer magnetic order of MBT nor its magnetic anisotropy are changed by interfacing with hBN. The above results concerning the insensitivity of magnetic, electronic, and topological properties of MBT to hBN should hold for thicker MBT films as well because of the vdW nature of the bond. Thus, we conclude that hBN can be efficiently used to decouple MBT CI layers from each other without altering their properties.
HighC state in the forced FM phase
We can now proceed with the study of the topological properties of the MBT/hBN multilayer heterostructures. We first note that in experiments, the CI state in the forced FM phase, achieved through the application of the external magnetic field, is observed in thin MBT flakes made of both even and odd numbers of SLs^{26,35,36,37,38,39,40,41,42,52}. According to the previous density functional theory calculations^{30}, the minimal MBT film thickness required for the realization of such a C = 1 state is two SL blocks. Let us therefore consider nMBT_{2SL}/hBN, n > 1, heterostructures in which n films of MBT_{2SL} are interlayed by (n − 1) hBN monolayers, as schematically depicted in Fig. 1b. We will assume that all MBT_{2SL} films are FMpolarized in the +z direction by the external magnetic field. These heterostructures were constructed based on the structure of the MBT/hBN/MBT system that was determined after a series of calculations outlined in Supplementary Note 2.B.1.
Figure 3 shows the calculated lowenergy band structures along the \(\overline{{{{\rm{K}}}}}\overline{{{\Gamma }}}\overline{{{{\rm{M}}}}}\) path together with the respective S_{xy}(E_{F}) for nMBT_{2SL}/hBN with n = 2, 3, 4, and 5. The band structures basically correspond to that of a freestanding MBT_{2SL} repeated n times, slightly shifted in energy due to a slight variation of the electrostatic potential across the multilayer. The bands stemming from the spatial inversion equivalent MBT_{2SL} layers come in pairs, as it is seen in the insets to Fig. 3. In the corresponding S_{xy}(E_{F}) dependencies, shown in Fig. 4a, there are plateaus in the band gap that are equal to an integer number of conductance quanta, the integer being equal to n, suggesting C = n state in the respective multilayers. Accordingly, in the plots of the calculated edge spectral functions, shown in Fig. 4b, two (three) edge modes can be seen for the n = 2 (n = 3) system.
From these results, we can conclude that C = n in the examined heterostructures and infer by induction that the same will hold for heterostructures with greater n.
HighC QAH state: odd number of SLs
In the thin films made of an odd number of SLs, MBT has been predicted^{30,32} and subsequently experimentally confirmed^{26} to show the intrinsic C = 1 QAHE. We therefore now turn to the topological properties of the MBT/hBN heterostructures based on the thinnest possible oddSL MBT film, i.e., MBT_{3SL}^{30}. A comparison of S_{xy}(E_{F}) calculated for MBT_{3SL}/hBN and MBT_{3SL} is shown in Fig. 5. A good matching between the two can be observed, especially for the flat portion in the band gap where S_{xy} = e^{2}/h for both, confirming the insensitivity of the C = 1 QAH state of MBT_{3SL} to interfacing with hBN.
As we are seeking the highC QAH state realized at zero external field, the question arises whether it is supported by the magnetic ground state of nMBT_{3SL}/hBN, n > 1. While we have shown above that the magnetic ordering inside MBT film is not affected by hBN, the interlayer exchange coupling between the local Mn moments through hBN should now be studied. Our totalenergy calculations show that the AFM configuration is ~0.03 meV (per 2 Mn atoms) lower in energy than the FM one (see Supplementary Note 3). Such a small energy difference is due to the large separation between neighboring Mn planes (about 17.8 Å) and the vdW coupling between MBT and hBN. Although this number is at the limit of our calculation accuracy, assuming that its sign is correct yields a zero (nonzero) net magnetization in nMBT_{3SL}/hBN with even (odd) n. As the net C is a sum of C’s of individual MBT_{3SL} films, with the sign of C changing with the magnetization direction, our result predicts that an nMBT_{3SL}/hBN heterostructure should have C equal to 0 (1) in its ground state.
However, the weakness of the interlayer exchange coupling through hBN makes a reliable prediction of its sign hardly possible using the density functional theory, so it should be defined in future experiments. If the FM coupling through hBN occurs experimentally, the Chern numbers of the individual MBT films will sum up to create the highC state. However, even if the experiment would show the worstcase scenario, i.e., the AFM coupling through hBN, a metamagnetic state that is appropriate for achieving the zerofield highC state in the nMBT_{3SL}/hBN (n > 1) multilayers can nevertheless be “prepared” by the external magnetic field, in the manner we will describe below. In this state, the alignment of the MBT SLs across hBN would be FM, while the interlayer alignment inside MBT films stays AFM, thus guaranteeing an overall nonzero net magnetization increasing with n, in turn yielding the desired C = n state. Crucial for the realization of such a state is the fact that the magnetic anisotropy energy of MBT_{3SL} is ~0.22 meV^{30}, several times larger than the energy difference of AFM and FM configurations across hBN. Note that the magnetic anisotropy energy of MBT_{3SL} should not change appreciably upon its interfacing with hBN, as shown in the section “Crystal and electronic structure of MBT/hBN interface”.
We, therefore, propose the following procedure to induce the abovedescribed metamagnetic state. First, the external magnetic field of about 10 T^{26,35,36} should be applied to overcome the AFM coupling inside individual MBT_{3SL} films and polarize all SLs along the same direction. Next, the external magnetic field should be gradually reduced to zero leading to the recovery of the uncompensated interlayer AFM state with nonzero magnetization in each individual MBT_{3SL} film^{26,30}. However, their net magnetizations will remain parallel to each other even if the exchange coupling across hBN is AFM because the energy barrier due to anisotropy prevents the magnetization relaxation into the only slightly more favorable AFM configuration across hBN. Indeed, a similar metamagnetic state has been recently experimentally observed in bulk MnBi_{4}Te_{7} and MnBi_{6}Te_{10}^{53,54,55,56,57}, where the uniaxial anisotropy dominates over the AFM interlayer exchange coupling^{57}. The magnetic anisotropy energy of MBT_{3SL} is slightly larger, while the exchange coupling across hBN is significantly weaker than those in MnBi_{4}Te_{7} and MnBi_{6}Te_{10} (c.f. ref. ^{56}), making the proposed realization of the metamagnetic state in nMBT_{3SL}/hBN feasible.
We now have a firm basis to claim a possibility of the highC QAH state realization in nMBT_{3SL}/hBN, n > 1. As in the preceding section, the highC state can be demonstrated by the S_{xy}(E_{F}) and the edge electronic structure calculations, the results of which are shown in Fig. 5. Analogously to the nMBT_{2SL}/hBN multilayers, the S_{xy}(E_{F}) for nMBT_{3SL}/hBN with n = 2 shows a clear plateau within the band gap, where the conductance is equal to two conductance quanta, 2e^{2}/h, i.e., C = n = 2. Moreover, the edge electronic structure shows two edge modes traversing the band gap.
From these results, it can be inferred that the nMBT_{3SL}/hBN, n > 1, heterostructures have C = n either in the metamagnetic state or even intrinsically, depending on the actual interlayer coupling over hBN. Similar results are expected for MBT/hBN multilayers based on MBT_{5SL} and MBT_{7SL} films. As a side note, the weakness of the exchange interactions through hBN implies that the magnetic domain dynamics of the individual MBT subsystems should be decoupled, which is one particular difference between the here proposed highC multilayer and nonsynthetic highC CIs^{58,59}.
Discussion
Using ab initio and tightbinding calculations, we studied novel MnBi_{2}Te_{4}/hBN multilayer heterostructures in which thin MnBi_{2}Te_{4} films are interlayed by hexagonal boron nitride monolayers. The van der Waals bonding between hBN and MnBi_{2}Te_{4} preserves the magnetic and electronic properties of the latter, in particular, the C = 1 Chern insulator state. Taking advantage of this, we have shown that a stack of n MnBi_{2}Te_{4} films with C = 1 in the MnBi_{2}Te_{4}/hBN multilayer gives rise to a high Chern number state, C = n, characterized by n chiral edge modes.
There are two ways to achieve this state in the proposed heterostructures. The first way is to use the external magnetic field to drive MnBi_{2}Te_{4} films into a forced ferromagnetic state, which is nowadays widely used to observe a new kind of the quantum Hall effect that does not require the formation of Landau levels^{26,35,36,37,38,39,40,41,42,52}. In this case, both even and odd septuple layer MnBi_{2}Te_{4} films can be used. One may well expect the high Chern number state to persist up to temperatures as high as 20–30 K, as previously observed for the C = 1 state in MnBi_{2}Te_{4} thin flakes^{36,40}. The second way relies on the use of the odd septuple layer films as they realize the quantum anomalous Hall state intrinsically^{26}. Although a prior application of the external field might be needed to align the net magnetizations of the individual MnBi_{2}Te_{4} slabs (as it is done in the Crdoped (Bi,Sb)_{2}Te_{3}^{9,60}), the high Chern number quantum anomalous Hall state can later be observed in remanence, i.e., at zero field. Currently, the observation temperature of the C = 1 quantum anomalous Hall effect in MnBi_{2}Te_{4} is about 1.4 K. Improving the structural quality of MnBi_{2}Te_{4} should allow the resolution of the intensely debated issue of its Dirac point gap size^{25,61,62,63,64,65,66} and push the effect observation temperature toward MnBi_{2}Te_{4}’s Néel point. The steps in this direction are currently underway^{38,52,67,68}.
MnBi_{2}Te_{4} thin films currently used in the quantized transport measurements are mostly obtained by exfoliation^{26,35,36,37,38,39,40,41,42}. Using exfoliation and transfer, which is a standard technique for the construction of the van der Waals heterostructures^{45,46}, should be suitable to realize the here proposed MnBi_{2}Te_{4}/hBN multilayers as well. However, this approach will not necessarily favor the formation of the interface structure with the lowest energy, but it could rather result in an arbitrary nonsymmetric alignment between MnBi_{2}Te_{4} and hBN. Fortunately, our results show that the Chern insulator state of individual MnBi_{2}Te_{4} layers is not sensitive to the MnBi_{2}Te_{4}/hBN interface registry. Therefore the exfoliation and transfer strategy should be appropriate to synthesize the here proposed MnBi_{2}Te_{4}/hBN multilayers. The accumulated worldwide experience in the synthesis of twodimensional van der Waals heterostructures is expected to greatly facilitate a prompt realization of the high Chern number state in the MnBi_{2}Te_{4}/hBN system.
In conclusion, we presented a novel concept of realization of a high Chern number magnetic topological insulator state, which relies on the use of fundamentally different but highly compatible van der Waals materials assembled in a multilayer heterostructure. While there is hardly a good alternative to hBN as an inert wide band gap monolayer insulator, other choices of the Chern insulators are in principle possible, such as transition metal dichalcogenide moiré superlattices^{29} or compounds of the MnBi_{2}Te_{4} family^{27,69,70,71,72,73,74}. The design proposed here allows a widerange tuning of the Chern number, its upper limit being restricted only by the van der Waals heterostructures growth technology, making such multilayers interesting for future fundamental research and efficient interconnect technologies.
Methods
Density functional theory calculations
The firstprinciples calculations were carried out on the level of density functional theory (DFT) as implemented in the Vienna Abinitio Simulation Package (VASP)^{75,76,77,78,79}. All calculations shared several common parameters: the VASP’s PAW datasets^{79} were used, the plane wave cutoff was set to 500 eV, the spinorbit coupling (SOC) was enabled, the exchangecorrelation functional was that of Perdew, Burke, and Ernzerhof (PBE)^{80}, while the vdW interaction was taken into account through the Grimme D3 model with the Becke–Jonson cutoff^{81,82}. The Mn 3dstates were treated employing the GGA+U approach^{83} within the Dudarev scheme^{84}. The U_{eff} = U − J value for the Mn 3dstates was chosen to be equal to 5.34 eV, as in previous works^{25,30,69,85}.
The DFT calculations were carried out with 11 × 11 × 1 Monkhorst–Pack grid used to sample the Brillouin zone (BZ) for all but n = 5 system, where 9 × 9 × 1 Monkhorst–Pack grid was used instead, and the electronic convergence threshold was 10^{−6} eV. For structural relaxations, VASP’s conjugate gradient algorithm was employed until the force on each atom decreased under 0.01 eV/Å, and the electronic occupations were smeared by the Gaussian function of the width of 0.01 eV. For the static calculations, the tetrahedron method with Blöchl corrections was used to treat the electronic occupations instead. The noncollinear intralayer AFM configuration was calculated using hexagonal cells containing three atoms per layer [\((\sqrt{3}\times \sqrt{3})R3{0}^{\circ }\) inplane periodicity] (see Supplementary Note 2.A.3) and a 7 × 7 × 1 BZ sampling. For the purpose of electronic structure visualization, the band energies for selected systems were calculated along the \(\overline{{{{\rm{K}}}}}\overline{{{\Gamma }}}\overline{{{{\rm{M}}}}}\) path by a nonselfconsistent DFT calculation, again with Gaussian smearing of 0.01 meV. The magnetic anisotropy energy was calculated as explained in ref. ^{25}, using the 25 × 25 × 1 Monkhorst–Pack grid and electronic convergence threshold of 10^{−8} eV.
Wannierbased calculations
The obtained Kohn–Sham functions were used to construct the maximally localized Wannier functions by the Wannier90 code^{86,87}. We refer the reader to Supplementary Note 1 for further details regarding the wannierization procedure. The Wannier functions were, in turn, used to calculate anomalous Hall conductances of the proposed heterostructures through the Kubo formula as implemented in the WannierBerri code^{88}. In WannierBerri calculations, the broadening of 10K was used. For hBNcovered MBT_{2SL} film, as well as MBT_{2SL}/hBN/MBT_{2SL} heterostructure, the Chern number was also calculated by the Wilson loop method as implemented in Z2Pack^{89,90}. The edge spectral functions were calculated by the Green function method^{91} from the tightbinding Wannier Hamiltonian as implemented in WannierTools^{92}. The number of principal layers in these calculations was set to four, while the details of the cell used can be found in Supplementary Note 4.
Data availability
Inputs and results are available from the corresponding authors upon a reasonable request.
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Acknowledgements
The authors thank J. IbañezAzpiroz, S.S. Tsirkin, I. Souza, and M. Garnica for stimulating discussions. M.B. and M.M.O. acknowledge the support of the Spanish Ministerio de Ciencia e Innovacion (Grant no. PID2019103910GBI00) and the University of the Basque Country (Grant no. IT152722). A.Y.V. and E.K.P. acknowledge the Ministry of Science and Higher Education of the Russian Federation (state task № FSWM20200033). E.V.C. acknowledges support from Saint Petersburg State University (Grant ID 94031444). The calculations were performed using computational resources of Donostia International Physics Center (http://dipc.ehu.es/cc/) and Research Park of Saint Petersburg State University “Computing Center” (http://www.cc.spbu.ru/).
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The problem was conceptualized by M.M.O. and E.V.C. Density functional theory calculations were performed by M.B., A.Y.V., E.K.P., and M.M.O. Wannier90 and surface state (WannierTools) calculations were performed by M.B. and E.K.P. Z2Pack Chern number calculations were performed by M.B. and A.Y.V. Hall conductance calculations were performed by M.B. Analysis of results was done by M.B. and M.M.O. Figures were produced by M.B. The manuscript was written by M.M.O. and M.B., with input from the other authors.
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Bosnar, M., Vyazovskaya, A.Y., Petrov, E.K. et al. High Chern number van der Waals magnetic topological multilayers MnBi_{2}Te_{4}/hBN. npj 2D Mater Appl 7, 33 (2023). https://doi.org/10.1038/s4169902300396y
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DOI: https://doi.org/10.1038/s4169902300396y
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