## Abstract

The layered antiferromagnetic MnBi_{2}Te_{4} films have been proposed to be an intrinsic quantum anomalous Hall (QAH) insulator with a large gap. It is crucial to open a magnetic gap of surface states. However, recent experiments have observed gapless surface states, indicating the absence of out-of-plane surface magnetism, and thus, the quantized Hall resistance can only be achieved at the magnetic field above 6 T. We propose to induce out-of-plane surface magnetism of MnBi_{2}Te_{4} films via the magnetic proximity with magnetic insulator CrI_{3}. A strong exchange bias of ∼40 meV originates from the long Cr-*e _{g}* orbital tails that hybridize strongly with Te

*p*orbitals. By stabilizing surface magnetism, the QAH effect can be realized in the MnBi

_{2}Te

_{4}/CrI

_{3}heterostructure. Moreover, the high–Chern number QAH state can be achieved by controlling external electric gates. Thus, the MnBi

_{2}Te

_{4}/CrI

_{3}heterostructure provides a promising platform to realize the electrically tunable zero-field QAH effect.

## INTRODUCTION

The quantum anomalous Hall (QAH) effect is a topological phenomenon characterized by quantized Hall resistance and zero longitudinal resistance (*1*–*4*). Different from the conventional quantum Hall effect, the QAH effect is induced by the interplay between spin-orbit coupling (SOC) and magnetic exchange coupling and thus can occur in certain ferromagnetic (FM) materials at zero external magnetic field. Owing to its topological and dissipation-free properties, the QAH insulator is an outstanding quantum-coherent material platform for the next-generation quantum-based technologies, including spintronics and topological quantum computations. Following the early theoretical predictions (*5*–*7*), the QAH effect was first demonstrated in magnetically (Cr or V) doped (Bi,Sb)_{2}Te_{3} (*8*–*11*), in which magnetic doping provides the required exchange coupling between magnetic moments and electron spins and thus is essential for the occurrence of the QAH state. However, magnetic doping inevitably degrades sample quality with the presence of massive disorders and thus limits the critical temperature of the QAH state below 2 K (*11*). Therefore, it is desirable to realize the QAH effect in intrinsic magnetic materials with stoichiometric crystals.

Recently, a tetradymite-type layered compound, MnBi_{2}Te_{4}, was proposed to be a promising topological material platform (*12*–*15*), with intrinsic A-type anti-FM (AFM) order, in which the magnetic moments of Mn atoms are ferromagnetically coupled within one septuple layer (SL) and anti-ferromagnetically coupled between the adjacent SLs, for the realization of the QAH effect, as well as other magnetic topological phases (*16*–*18*). Early first-principles calculations show that the QAH state can be realized in the MnBi_{2}Te_{4} films with odd numbers of SLs at zero magnetic field for the ideal AFM order (*13*–*19*). The A-type AFM order was demonstrated via magnetization measurements for bulk MnBi_{2}Te_{4} as the typical spin-flop transition was observed when the external magnetic field perpendicular to the SL plane was increased above 3.5 T (*15*, *20*–*24*). However, the magnetotransport experiments in the MnBi_{2}Te_{4} films only revealed a quantized Hall resistance for the magnetic field above 6 T (*21*, *25*, *26*), larger than the critical field of spin-flop transition. Therefore, the thin film has already become FM under this magnetic field. The predicted zero-field QAH state induced by the ideal AFM order has yet been demonstrated experimentally. The early angular-resolved photon emission spectroscopy (ARPES) measurements observed a band gap, ranging from 50 meV to hundreds of meVs (*15*, *20*, *27*, *28*), of topological surface states (TSSs) in MnBi_{2}Te_{4}. However, this gap is shown to persist well above the Néel temperature and could be observed even at room temperature (*20*, *27*, *15*), making it unlikely originated from the AFM order. More recent high-resolution ARPES studies based on synchrotron and laser light sources show that the TSS remains gapless below the Néel temperature (*29*–*32*). The negligible magnetic gap of TSS is consistent with the absence of the zero-field QAH effect in magnetotransport measurements (*20*, *21*, *23*, *25*, *26*). The absence of magnetic gap of TSSs suggests that the surface magnetism may not be well developed and different from the bulk AFM order. Physically, this is not unexpected since more complex magnetic interactions, including dipole-dipole interaction and Dzyaloshinskii-Moriya interaction, may play an important role for the surface magnetic mechanism. Consequently, the surface Mn magnetic moments may be canted, or lie in the SL plane, or become disordered, all of which may lead to a gapless TSS. Furthermore, magnetic domains ubiquitously exist in AFM materials and cannot be easily eliminated even by field cooling. All these problems hamper the realization of zero-field QAH state in the MnBi_{2}Te_{4} films.

In this work, we propose to overcome the challenge of surface magnetism by coupling the MnBi_{2}Te_{4} films to a two-dimensional (2D) FM insulator with the example of CrI_{3} via exchange bias. Our density functional theory (DFT) calculations on the MnBi_{2}Te_{4}/CrI_{3} heterostructure show a FM exchange bias around 40 meV, much larger than the Néel temperature of MnBi_{2}Te_{4} [24 K (*15*)] and the Curie temperature of CrI_{3} [61 K for bulk (*33*) and 45 K for monolayer (*34*)]. Moreover, CrI_{3} has little influence on electronic band structure of MnBi_{2}Te_{4} films, and thus, the QAH state with the Chern number (CN) = 1 can exist in 3- and 5-SL-thick MnBi_{2}Te_{4}, consistent with the early studies on pure MnBi_{2}Te_{4} films. We also studied the electric gating effect and the CrI_{3}/MnBi_{2}Te_{4}/CrI_{3} heterostructures. Our results show that (i) the high-CN QAH state with CN = 3 can be achieved by tuning gate voltages and (ii) the strong exchange bias can always align the magnetization of both surfaces of MnBi_{2}Te_{4} films, thus driving even SL MnBi_{2}Te_{4} into the QAH state in the CrI_{3}/MnBi_{2}Te_{4}/CrI_{3} heterostructure.

## RESULTS

### FM exchange bias at the MnBi_{2}Te_{4}/CrI_{3} interface

The required exchange bias material should provide strong magnetic coupling at the interface but not change the electronic states near the Fermi energy. Therefore, we choose a magnetic insulator, CrI_{3} (*34*). Its monolayer is FM and can couple with MnBi_{2}Te_{4} through the van der Waals interface, which may weakly disturb the band structure of MnBi_{2}Te_{4}. Because the interaction is determined by the interface layer, we only choose a monolayer of CrI_{3} for the interface model.

We construct interface models with one layer of CrI_{3} and different layers of MnBi_{2}Te_{4} on its top, as shown in Fig. 1. Both materials share the same triangular lattice but different in-plane lattice parameters, 7.04 Å for CrI_{3} and 4.36 Å for MnBi_{2}Te_{4} from our DFT calculations, which is consistent with recent works (*13*, *14*, *35*). A 2 × 2 supercell of CrI_{3} can match well with a 3 × 3 supercell of MnBi_{2}Te_{4}. Alternatively, the primitive unit cell of CrI_{3} can also match a _{2}Te_{4} with a mismatch of 7%. Because we a5re mostly interested in the band structure of MnBi_{2}Te_{4}, we stretch the CrI_{3} lattice to match the _{2}Te_{4} supercell. We fully optimized the atomic structures by including the van der Waals interactions in DFT calculations within the generalized gradient approximation (GGA) and the Hubbard U. We have tested both models and found that they give similar results in the exchange coupling and band structure (see figs. S1 and S3). Thus, we choose the smaller model, _{2}Te_{4}/1 × 1 CrI_{3}, for further investigations in the following.

At the interface, MnBi_{2}Te_{4} exhibits strong FM coupling with CrI_{3}. For 1-SL MnBi_{2}Te_{4} on top of CrI_{3}, the energy difference between the FM and AFM coupling is about 40 meV. We note that different ways of stacking between two materials give very similar strength of exchange coupling, which is also true for the 3 × 3 MnBi_{2}Te_{4}/2 × 2 CrI_{3} case (fig. S3). When increasing the MnBi_{2}Te_{4} layer to 2 SLs and more, the interface FM coupling remains with the same exchange energy and the two SLs still couple in the AFM way (fig. S2). Therefore, the CrI_{3} layer couples only with the neighboring MnBi_{2}Te_{4} layer and does not affect the AFM order between different MnBi_{2}Te_{4} layers. We point out that such an exchange coupling is much stronger than the magnitude of the exchange interactions between two MnBi_{2}Te_{4} layers (∼3 meV for _{3} layers [∼10 meV for 1 × 1 unit cell (*36*)]. Therefore, CrI_{3} can stably pin the FM order of the proximity MnBi_{2}Te_{4} layer and act as an effective exchange bias. In addition, we find that the SOC weakly affect the magnetic coupling strength (see figs. S1 to S4) and the magnetic moments prefer the out-of-plane direction (see fig. S8).

The strong exchange coupling originates in the orbital feature at the interface. The Mn site has *d*^{5} configuration as *d*^{3} as *e _{g}* to Mn-

*t*

_{2g}states through the intermediate I, Te, Bi, and Te atoms, which is beyond the simple superexchange interaction. In the localized Wannier orbitals, we observe a crucial feature in the Cr-

*e*states. Tails of the Cr-

_{g}*e*Wannier functions extend beyond the van der Waals gap and strongly overlap with the neighboring Te

_{g}*p*orbitals (see Fig. 1C). This strong orbital overlap rationalizes the strong coupling between two materials. We also notice that the exchange channels from Cr-

*e*to Mn-

_{g}*t*

_{2g}and Cr-

*e*to Mn-

_{g}*e*are both of FM type, further enhancing the overall exchange coupling strength. This is in sharp contrast to the exchange coupling between two CrI

_{g}_{3}layers, which is of FM type for the channel from Cr-

*e*to Cr-

_{g}*t*

_{2g}and of AFM type from Cr-

*t*

_{2g}to Cr-

*t*

_{2g}(

*36*). In addition, AFM-type coupling at the interface can also play a role of the exchange bias, although the present specific interface structure exhibits the FM coupling.

### QAH effect

We next investigate the electronic band structure and discuss its topological properties. Figure 2 shows band structures for 1 to 6 MnBi_{2}Te_{4} SL(s) on top of CrI_{3}. As discussed above, there is FM coupling between CrI_{3} and neighboring the MnBi_{2}Te_{4} SL and AFM coupling between MnBi_{2}Te_{4} SLs. The interface band structure can be approximately regarded as an overlap of two different materials. An essential feature is the existence of an energy gap in these band structures, which is crucial for the realization of QAH effect. The occupied Cr-*t*_{2g} bands are far below the valence bands of MnBi_{2}Te_{4}. The Cr-*e _{g}* states overlap with the conduction band bottom of MnBi

_{2}Te

_{4}and remain unoccupied. This means that there is no charge transfer through the van der Waals junction. The calculated Cr-

*t*

_{2g}and Cr-

*e*gap is about 1 eV, which is consistent with previous GGA calculations and can be corrected to about 1.5 eV by hybrid functionals (

_{g}*37*). Although some Cr-

*e*bands appear as the lowest conduction bands at the interface for thinner MnBi

_{g}_{2}Te

_{4}films (1 to 4 SLs), they will be pushed to even higher energy by the self-energy correction and do not affect our understanding of the band structure topology. When the MnBi

_{2}Te

_{4}layer is thicker (e.g., 5 to 6 SLs), the MnBi

_{2}Te

_{4}states become the lowest conduction bands in the GGA band structure. Thus, CrI

_{3}serves an ideal proximity exchange bias without destroying the MnBi

_{2}Te

_{4}band structure.

We find that isolate MnBi_{2}Te_{4} layers are trivial magnetic insulators for 1, 2, 4, and 6 SLs thick and QAH effect insulators for 3 and 5 SLs, which is consistent with recent theoretical studies (*13*, *19*). Here, the QAH insulator has the CN = 1, as showed by our Berry phase calculations using the Wilson loop method (*38*, *39*) and the Berry curvature distribution in the 2D Brillouin zone. In proximity to the CrI_{3} layer, MnBi_{2}Te_{4} band structures are modified weakly without changing their topological nature. For example, the isolated MnBi_{2}Te_{4} layer of 2, 4, or 6 SLs thick exhibits the double degeneracy in the band structure caused by the symmetry combining spatial inversion and time reversal. The existence of the CrI_{3} layer weakly breaks this symmetry and splits the degenerate bands. We verify the topological character of the interface structures by observing the band gap evolution with respect to the SOC strength. For 3- and 5-SL-thick MnBi_{2}Te_{4}/CrI_{3}, the band gap closes at about 90% of the normal SOC strength but reopens an energy gap with increasing SOC, showing a topological phase transition (TPT) (see figs. S5 and S6). The QAH insulator gaps are 49 and 14 meV for the 3- and 5-SL interface, respectively. For 1-, 2-, 4-, and 6-SL-thick MnBi_{2}Te_{4}/CrI_{3}, however, the bandgap remains open as varying SOC from 0 to 100%.

### Electrically tunable high-CN QAH effect

The 2D layered structure offers an opportunity to tune the band structure topology by applying a vertical electric field. The electric field induces different potential variation at different layers and subsequently modifies the overall band structure and its topological nature. For the interface structure, an electric field (ϵ) along the −*z* direction can push the Cr-*e _{g}* states up in the conduction band, as illustrated in Fig. 1B, leaving only MnBi

_{2}Te

_{4}states right above and below the energy gap. Further increasing the electric field can induce an inversion between the occupied and unoccupied bands, giving rise to the TPT. Since the CrI

_{3}brings little modifications to the low-energy band structure of MnBi

_{2}Te

_{4}, we only consider isolated MnBi

_{2}Te

_{4}models when applying an electric field in following discussions.

The electric field can induce the high-CN QAH state. In a simple two-band model (*5*), a band inversion at the Γ point usually leads to a change of the CN by ±1. If the band inversion occurs at generic *k*-points, then it can induce a jump of the CN by the number of the transition points. The MnBi_{2}Te_{4} film under an electric field exhibits two important symmetries, the threefold rotation (denoted as *C*_{3}) and a combined symmetry between the time reversal and mirror reflection (denoted as *TM*). Since the mirror plane crosses the Γ – *M* line in the Brillouin zone and perpendicular to the layer plane, the Γ – *K* line is invariant under the *TM* symmetry. Therefore, if a transition happens at a generic *k*-point away from the Γ – *K* line, then the gapless points must exist at six different *k*-points related by the *C*_{3} and *TM* symmetries. If a transition happens along the Γ – *K* line that is invariant under *TM*, then the gapless points must simultaneously appear at three different *k*-points related by *C*_{3} (see the inset of Fig. 3A). If a transition appears at Γ that is invariant under both *C*_{3} and *TM* symmetries, then a single Dirac point transition can occur.

To verify this scenario, we carried out band structure calculations on 3-SL-thick MnBi_{2}Te_{4} and demonstrate that the CN can jump by both 1 and 3 via applying a small electric field, as shown in Fig. 3. At zero electric field, the 3-SL-thick MnBi_{2}Te_{4} is a QAH state with CN = 1 and changes to a trivial insulator for ϵ = 0.005 V/Å. This transition is through a gap-closing point at Γ for ϵ = 0.002/Å. For a larger electric field (ϵ = 0.015 V/Å), another transition occurs with three gap-closing points along the Γ – *K* lines, leading to a QAH state with CN = 3. The gap-closing and reopening points can be recognized as hot spots of the Berry curvature in Fig. 3C. Furthermore, the electric field can also drive the MnBi_{2}Te_{4} film with even numbers of SLs from a trivial magnetic insulator with zero CN to the QAH state. For instance, red ϵ = 0.025 V/Å induces a TPT with three gapless points along the Γ – *K* lines in the 2-SL-thick MnBi_{2}Te_{4} film at ϵ = 0.0223 V/Å, resulting in the QAH state with CN = −3 (see fig. S7).

### Sandwiched MnBi_{2}Te_{4} structures

Given the short range nature of exchange bias, the CrI_{3} is expected to align the magnetization of the bottom MnBi_{2}Te_{4} layer in the MnBi_{2}Te_{4}/CrI_{3} heterostructure but may have little influence on the top MnBi_{2}Te_{4} layer when the film thickness is large. This issue can be resolved by considering a sandwiched structure CrI_{3}/MnBi_{2}Te_{4}/CrI_{3}. For the MnBi_{2}Te_{4} films with an odd number of SLs, the AFM order in MnBi_{2}Te_{4} is compatible with the FM orders in the top and bottom CrI_{3} monolayers. In contrast, for the MnBi_{2}Te_{4} films with an even number of SLs, the compensated AFM ordering between MnBi_{2}Te_{4} layers can be changed by CrI_{3}. As an example of the 4-SL case in Fig. 4, the magnetization of the top MnBi_{2}Te_{4} SL feels frustration from the upper CrI_{3} layer and the lower MnBi_{2}Te_{4} layer. Because the MnBi_{2}Te_{4}/CrI_{3} coupling is much stronger than the MnBi_{2}Te_{4}/MnBi_{2}Te_{4} coupling, magnetic moments of the top MnBi_{2}Te_{4} SL aligns parallel to those of the CrI_{3} layer. Such a rearrangement of magnetic moments in the MnBi_{2}Te_{4} SL leads to a net magnetization for the MnBi_{2}Te_{4} film. As verified by our band structure calculations, reversing magnetic moments of the top MnBi_{2}Te_{4} SL layer is indeed energetically favored by 30 to 40 meV (fig. S4). Subsequently, the system becomes a QAH state with an energy gap of 34 meV. As shown by the Wilson loop calculations, it exhibits a nontrivial CN = − 1. Therefore, the sandwich configuration may always provide a QAH insulator for either odd or even numbers of MnBi_{2}Te_{4} SLs.

## DISCUSSION

In summary, the magnetic order of MnBi_{2}Te_{4} thin film can be pinned and also manipulated by a strong exchange bias in proximity to CrI_{3}. Thus, the heterostructures with MnBi_{2}Te_{4} and CrI_{3} provide an experimentally feasible platform to realize the QAH effect. An external electric field can further modify the thin-film band structure and induce QAH effect with large CNs. Since the magnetic insulator CrI_{3} weakly disturbs the electronic states of MnBi_{2}Te_{4}, it can also be used to pin the surface magnetic order of the bulk MnBi_{2}Te_{4} and assist the observation of the axion insulator phase (*16*, *17*) in ARPES. In addition, it is worth noting that other magnetic insulators with out-of-plane magnetization, such as Tm_{3}Fe_{5}O_{12} (TmIG) and Cr_{2}Ge_{2}Te_{6}, may also play the same role of exchange bias as CrI_{3}.

In the proof stage of our manuscript, we were aware of the recent experimental report on the zero-field QAH effect in MnBi_{2}Te_{4} thin layers (*41*).

## METHODS

DFT calculations were performed using the Vienna ab initio simulation package (*40*), with core electrons represented by the projector augmented wave potential. The Perdew-Burke-Ernzerhof exchange-correlation functional with GGA + *U* method was used in the DFT calculations. The parameter *U* = 2.9 and 3.0 eV was chosen to describe the localized *d* orbitals of Cr and Mn, respectively. Plane waves with a kinetic energy cutoff of 270 eV were used as the basis set. Geometry optimization was carried out until the residual force on each atom was less than 0.01 eV/Å. The DFT-D3 correction method was considered to treat the van der Waals interactions between the CrI_{3} and MnBi_{2}Te_{4} slabs. We projected the Wannier functions of the bulk MnBi_{2}Te_{4} in the AFM phase. On the basis of the bulk tight-binding parameters of Wannier functions, we constructed the slab model for MnBi_{2}Te_{4} thin films and evaluated their band structures and Berry phases.

## SUPPLEMENTARY MATERIALS

Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/6/10/eaaz0948/DC1

Section S1. Structure models and exchange coupling

Section S2. QAH effect and electrically tunable high-CN QAH effect

Section S3. Magnetic crystalline anisotropy

Table S1. The gap at Γ from MnBi_{2}Te_{4} bands of the heterostructures.

Fig. S1. The magnetic coupling energy ∆*E* for different stacking ways of the interface for 1-SL MnBi_{2}Te_{4} on CrI_{3}.

Fig. S2. The total energy ∆*E* for different magnetic structures of the interface for 2-SL MnBi_{2}Te_{4} on CrI_{3}.

Fig. S3. The total energy ∆*E* for different coupling of the interface for 1-SL 3 × 3 MnBi_{2}Te_{4} on 2 × 2 CrI_{3}.

Fig. S4. Energies of different magnetic structures and band structure for 4-SL MnBi_{2}Te_{4} sandwiched between two CrI_{3} layers.

Fig. S5. Band structure evolution for 3-SL-thick MnBi_{2}Te_{4}/CrI_{3} heterostructure with varying SOC strength from 0 to 110%.

Fig. S6. Band structure evolution for 5-SL-thick MnBi_{2}Te_{4}/CrI_{3} heterostructure with varying SOC strength from 0 to 110%.

Fig. S7. Band structure evolution for 2-SL-thick MnBi_{2}Te_{4} at different electric field ϵ.

Fig. S8. The relative energy ∆*E* for different magnetic structures of the interface for 1-SL MnBi_{2}Te_{4} on CrI_{3} with SOC included.

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## REFERENCES AND NOTES

**Acknowledgments:**We acknowledge helpful discussions with C.-z. Chang at the Penn State University and X. Xu at the University of Washington.

**Funding:**Work at Penn State (C.-X.L.) was primarily supported by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES) under Award DE-SC0019064. C.-X.L. also acknowledges the support from the Office of Naval Research (grant no. N00014-18-1-2793) and Kaufman New Initiative research grant KA2018-98553 of the Pittsburgh Foundation. B.Y. acknowledges the financial support by the Willner Family Leadership Institute for the Weizmann Institute of Science, the Benoziyo Endowment Fund for the Advancement of Science, Ruth and Herman Albert Scholars Program for New Scientists, and the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant no. 815869).

**Author contributions:**B.Y. and C.-X.L. conceived the project. H.F. performed DFT calculations. All authors performed the Berry phase calculations, analyzed results, and wrote the manuscript.

**Competing interests:**The authors declare that they have no competing interests.

**Data and materials availability:**All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.

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