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Exchange bias and quantum anomalous Hall effect in the MnBi2Te4/CrI3 heterostructure

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Science Advances  06 Mar 2020:
Vol. 6, no. 10, eaaz0948
DOI: 10.1126/sciadv.aaz0948

Abstract

The layered antiferromagnetic MnBi2Te4 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 MnBi2Te4 films via the magnetic proximity with magnetic insulator CrI3. A strong exchange bias of ∼40 meV originates from the long Cr-eg orbital tails that hybridize strongly with Te p orbitals. By stabilizing surface magnetism, the QAH effect can be realized in the MnBi2Te4/CrI3 heterostructure. Moreover, the high–Chern number QAH state can be achieved by controlling external electric gates. Thus, the MnBi2Te4/CrI3 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 (14). 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 (57), the QAH effect was first demonstrated in magnetically (Cr or V) doped (Bi,Sb)2Te3 (811), 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, MnBi2Te4, was proposed to be a promising topological material platform (1215), 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 (1618). Early first-principles calculations show that the QAH state can be realized in the MnBi2Te4 films with odd numbers of SLs at zero magnetic field for the ideal AFM order (1319). The A-type AFM order was demonstrated via magnetization measurements for bulk MnBi2Te4 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, 2024). However, the magnetotransport experiments in the MnBi2Te4 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 MnBi2Te4. 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 (2932). 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 MnBi2Te4 films.

In this work, we propose to overcome the challenge of surface magnetism by coupling the MnBi2Te4 films to a two-dimensional (2D) FM insulator with the example of CrI3 via exchange bias. Our density functional theory (DFT) calculations on the MnBi2Te4/CrI3 heterostructure show a FM exchange bias around 40 meV, much larger than the Néel temperature of MnBi2Te4 [24 K (15)] and the Curie temperature of CrI3 [61 K for bulk (33) and 45 K for monolayer (34)]. Moreover, CrI3 has little influence on electronic band structure of MnBi2Te4 films, and thus, the QAH state with the Chern number (CN) = 1 can exist in 3- and 5-SL-thick MnBi2Te4, consistent with the early studies on pure MnBi2Te4 films. We also studied the electric gating effect and the CrI3/MnBi2Te4/CrI3 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 MnBi2Te4 films, thus driving even SL MnBi2Te4 into the QAH state in the CrI3/MnBi2Te4/CrI3 heterostructure.

RESULTS

FM exchange bias at the MnBi2Te4/CrI3 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, CrI3 (34). Its monolayer is FM and can couple with MnBi2Te4 through the van der Waals interface, which may weakly disturb the band structure of MnBi2Te4. Because the interaction is determined by the interface layer, we only choose a monolayer of CrI3 for the interface model.

We construct interface models with one layer of CrI3 and different layers of MnBi2Te4 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 CrI3 and 4.36 Å for MnBi2Te4 from our DFT calculations, which is consistent with recent works (13, 14, 35). A 2 × 2 supercell of CrI3 can match well with a 3 × 3 supercell of MnBi2Te4. Alternatively, the primitive unit cell of CrI3 can also match a 3×3 supercell of MnBi2Te4 with a mismatch of 7%. Because we a5re mostly interested in the band structure of MnBi2Te4, we stretch the CrI3 lattice to match the 3×3 MnBi2Te4 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, 3×3 MnBi2Te4/1 × 1 CrI3, for further investigations in the following.

Fig. 1 The atomic structure of the MnBi2Te4/CrI3 heterostructure.

(A) MnBi2Te4 and CrI3 prefers the FM-type coupling, while MnBi2Te4 layers remains the AFM interaction. (B) Schematics of the exchange interaction between Cr-eg and Mn-egt2g states. (C) The Wannier function of one of Cr-eg states. Its tails reach the neighboring Te atoms by crossing the van der Waals gap. Yellow and cyan colors represent positive and negative, respectively, isovalue surfaces of the Wannier function. An external electric field (ϵ) along −z can lift the energy of Cr-eg bands.

At the interface, MnBi2Te4 exhibits strong FM coupling with CrI3. For 1-SL MnBi2Te4 on top of CrI3, 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 MnBi2Te4/2 × 2 CrI3 case (fig. S3). When increasing the MnBi2Te4 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 CrI3 layer couples only with the neighboring MnBi2Te4 layer and does not affect the AFM order between different MnBi2Te4 layers. We point out that such an exchange coupling is much stronger than the magnitude of the exchange interactions between two MnBi2Te4 layers (∼3 meV for 3×3 supercell) or two CrI3 layers [∼10 meV for 1 × 1 unit cell (36)]. Therefore, CrI3 can stably pin the FM order of the proximity MnBi2Te4 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 d5 configuration as t2g3eg2 and the Cr site has d3 as t2g3eg0. There is a long exchange pathway from Cr-eg to Mn-t2g 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-eg states. Tails of the Cr-eg Wannier functions extend beyond the van der Waals gap and strongly overlap with the neighboring Te 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-eg to Mn-t2g and Cr-eg to Mn-eg are both of FM type, further enhancing the overall exchange coupling strength. This is in sharp contrast to the exchange coupling between two CrI3 layers, which is of FM type for the channel from Cr-eg to Cr-t2g and of AFM type from Cr-t2g to Cr-t2g (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 MnBi2Te4 SL(s) on top of CrI3. As discussed above, there is FM coupling between CrI3 and neighboring the MnBi2Te4 SL and AFM coupling between MnBi2Te4 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-t2g bands are far below the valence bands of MnBi2Te4. The Cr-eg states overlap with the conduction band bottom of MnBi2Te4 and remain unoccupied. This means that there is no charge transfer through the van der Waals junction. The calculated Cr-t2g and Cr-eg gap is about 1 eV, which is consistent with previous GGA calculations and can be corrected to about 1.5 eV by hybrid functionals (37). Although some Cr-eg bands appear as the lowest conduction bands at the interface for thinner MnBi2Te4 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 MnBi2Te4 layer is thicker (e.g., 5 to 6 SLs), the MnBi2Te4 states become the lowest conduction bands in the GGA band structure. Thus, CrI3 serves an ideal proximity exchange bias without destroying the MnBi2Te4 band structure.

Fig. 2 Band structures of MnBi2Te4 thin films in proximity to a CrI3 layer.

Band structures for 1 to 6-SL-thick MnBi2Te4 are shown in A-F, respectively. Red lines indicate Cr-eg bands and blues ones for MnBi2Te4 bands. The Fermi energy is set to zero.

We find that isolate MnBi2Te4 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 CrI3 layer, MnBi2Te4 band structures are modified weakly without changing their topological nature. For example, the isolated MnBi2Te4 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 CrI3 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 MnBi2Te4/CrI3, 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 MnBi2Te4/CrI3, 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-eg states up in the conduction band, as illustrated in Fig. 1B, leaving only MnBi2Te4 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 CrI3 brings little modifications to the low-energy band structure of MnBi2Te4, we only consider isolated MnBi2Te4 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 MnBi2Te4 film under an electric field exhibits two important symmetries, the threefold rotation (denoted as C3) 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 C3 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 C3 (see the inset of Fig. 3A). If a transition appears at Γ that is invariant under both C3 and TM symmetries, then a single Dirac point transition can occur.

Fig. 3 Electrically tunable band structure topology and the QAH effect.

(A) Band structure evolution of 3-SL-thick MnBi2Te4 as the electric field (ϵ) increases. Two topological transitions occur at ϵ = 0.002 at the Γ point and 0.0141 V/Å along Γ – K lines. (B) The Berry phase θ, i.e., the Wannier charge center, accumulated along a Wilson loop (k1 ∈ [ − π, π]) as varying k2. It is topologically equivalent to the edge state spectra. Corresponding CNs are one, zero, and three for ϵ = 0.000,0.004 and 0.015 V/Å, respectively. (C) Corresponding Berry curvature (Ω) distribution in the first Brillouin zone. The integration of Ω gives the CN (multiply by 2π).

To verify this scenario, we carried out band structure calculations on 3-SL-thick MnBi2Te4 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 MnBi2Te4 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 MnBi2Te4 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 MnBi2Te4 film at ϵ = 0.0223 V/Å, resulting in the QAH state with CN = −3 (see fig. S7).

Sandwiched MnBi2Te4 structures

Given the short range nature of exchange bias, the CrI3 is expected to align the magnetization of the bottom MnBi2Te4 layer in the MnBi2Te4/CrI3 heterostructure but may have little influence on the top MnBi2Te4 layer when the film thickness is large. This issue can be resolved by considering a sandwiched structure CrI3/MnBi2Te4/CrI3. For the MnBi2Te4 films with an odd number of SLs, the AFM order in MnBi2Te4 is compatible with the FM orders in the top and bottom CrI3 monolayers. In contrast, for the MnBi2Te4 films with an even number of SLs, the compensated AFM ordering between MnBi2Te4 layers can be changed by CrI3. As an example of the 4-SL case in Fig. 4, the magnetization of the top MnBi2Te4 SL feels frustration from the upper CrI3 layer and the lower MnBi2Te4 layer. Because the MnBi2Te4/CrI3 coupling is much stronger than the MnBi2Te4/MnBi2Te4 coupling, magnetic moments of the top MnBi2Te4 SL aligns parallel to those of the CrI3 layer. Such a rearrangement of magnetic moments in the MnBi2Te4 SL leads to a net magnetization for the MnBi2Te4 film. As verified by our band structure calculations, reversing magnetic moments of the top MnBi2Te4 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 MnBi2Te4 SLs.

Fig. 4 Interface exchange field induced magnetic order and the QAH effect.

(A) Band structure of a 4-SL-thick MnBi2Te4 with a special magnetic order. When it was sandwiched between two CrI3 layers, the AFM-type coupling between MnBi2Te4 layers are changed by the top CrI3 layer, as illustrated in the inset. This band structure is calculated without CrI3 layers, in the tight-binding scheme. The CrI3-sandwiched DFT band structure can be found in fig. S4. (B) Corresponding Berry phase of the Wilson loop, with a CN = −1. (C) The Berry curvature distribution in the first Brillouin zone.

DISCUSSION

In summary, the magnetic order of MnBi2Te4 thin film can be pinned and also manipulated by a strong exchange bias in proximity to CrI3. Thus, the heterostructures with MnBi2Te4 and CrI3 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 CrI3 weakly disturbs the electronic states of MnBi2Te4, it can also be used to pin the surface magnetic order of the bulk MnBi2Te4 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 Tm3Fe5O12 (TmIG) and Cr2Ge2Te6, may also play the same role of exchange bias as CrI3.

In the proof stage of our manuscript, we were aware of the recent experimental report on the zero-field QAH effect in MnBi2Te4 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 CrI3 and MnBi2Te4 slabs. We projected the Wannier functions of the bulk MnBi2Te4 in the AFM phase. On the basis of the bulk tight-binding parameters of Wannier functions, we constructed the slab model for MnBi2Te4 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 MnBi2Te4 bands of the heterostructures.

Fig. S1. The magnetic coupling energy ∆E for different stacking ways of the interface for 1-SL MnBi2Te4 on CrI3.

Fig. S2. The total energy ∆E for different magnetic structures of the interface for 2-SL MnBi2Te4 on CrI3.

Fig. S3. The total energy ∆E for different coupling of the interface for 1-SL 3 × 3 MnBi2Te4 on 2 × 2 CrI3.

Fig. S4. Energies of different magnetic structures and band structure for 4-SL MnBi2Te4 sandwiched between two CrI3 layers.

Fig. S5. Band structure evolution for 3-SL-thick MnBi2Te4/CrI3 heterostructure with varying SOC strength from 0 to 110%.

Fig. S6. Band structure evolution for 5-SL-thick MnBi2Te4/CrI3 heterostructure with varying SOC strength from 0 to 110%.

Fig. S7. Band structure evolution for 2-SL-thick MnBi2Te4 at different electric field ϵ.

Fig. S8. The relative energy ∆E for different magnetic structures of the interface for 1-SL MnBi2Te4 on CrI3 with SOC included.

This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial license, which permits use, distribution, and reproduction in any medium, so long as the resultant use is not for commercial advantage and provided the original work is properly cited.

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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