Exchange Bias and Quantum Anomalous Hall Effect in the MnBi2Te4-CrI3 Heterostructure

TL;DR

This paper proposes a heterostructure of MnBi2Te4 and CrI3 to induce out-of-plane surface magnetism, enabling the realization of the quantum anomalous Hall effect at zero magnetic field through strong exchange bias and electric control.

Contribution

It introduces a novel MnBi2Te4/CrI3 heterostructure that stabilizes surface magnetism and achieves electrically tunable quantum anomalous Hall states, addressing previous limitations of gapless surface states.

Findings

Strong exchange bias (~40 meV) from CrI3 stabilizes surface magnetism.

Realization of the quantum anomalous Hall effect in the heterostructure.

High Chern number QAH states can be electrically controlled.

Abstract

The layered antiferromagnetic MnBi2Te4 films have been proposed to be an intrinsic quantum anomalous Hall (QAH) insulator with a large gap. To realize this proposal, 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. In this work, we propose to induce out-of-plane surface magnetism of MnBi2Te4 films via the magnetic proximity with magnetic insulator CrI3. Our calculations have revealed a strong exchange bias ~ 40 meV, originating 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. Our calculations also demonstrate 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.

Tables

Table 1. Table S1: The gap at Γ Γ \Gamma from MnBi 2 Te 4  bands of the heterostructures. When placing MnBi 2 Te 4  on top of CrI 3 , the whole system remains to be insulating. Here, we show the band gap of the MnBi 2 Te 4  thin film as shown in Fig.  2 .

Thickness (SL)	1	2	3	4	5	6
Gap (meV)	299	59	49	42	14	22

Full text

*Introduction: * The quantum anomalous Hall (QAH) effect is a topological phenomenon characterized by quantized Hall resistance and zero longitudinal resistance Haldane (1988); Liu et al. (2016); Chang and Li (2016); Wang et al. (2015). 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  Zutic et al. (2004); Moore (2010); Šmejkal et al. (2018) and topological quantum computations Nayak et al. (2008). Following the early theoretical predictions Qi et al. (2006); Liu et al. (2008); Yu et al. (2010), the QAH effect was first demonstrated in magnetically (Cr or V) doped (Bi,Sb)2Te3 Chang et al. (2013); Checkelsky et al. (2014); Kou et al. (2014); Chang et al. (2015), 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 2K Chang et al. (2015). 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 platformGong et al. (2019); Li et al. (2019b); Zhang et al. (2019); Otrokov et al. (2018) with intrinsic A-type anti-ferromagnetic (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 phasesQi et al. (2008); Mong et al. (2010); Wan et al. (2011). Early first principles calculations show that the QAH state can be realized in the MnBi2Te4 films with odd number of SLs at zero magnetic field for the ideal AFM order Li et al. (2019b); Zhang et al. (2019); Otrokov et al. (2019). 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 3.5T Otrokov et al. (2018); Lee et al. (2018); Deng et al. (2019); Yan et al. (2019); Wu et al. (2019); Chen et al. (2019b). However, the magneto-transport experiments in the MnBi2Te4 films only revealed a quantized Hall resistance for the magnetic field above 6T Deng et al. (2019); Liu et al. (2019); Ge et al. (2019), larger than the critical field of spin-flop transition. Therefore, the thin film has already become ferromagnetic 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 Otrokov et al. (2018); Lee et al. (2018); Vidal et al. (2019); Chen et al. (2019c), 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 Lee et al. (2018); Vidal et al. (2019); Otrokov et al. (2018), making it unlikely originated from the AFM order. Indeed, more recent high-resolution ARPES studies based on synchrotron and laser light sources show that the TSS remains gapless below the Néel temperature Hao et al. (2019); Chen et al. (2019a); Li et al. (2019a); Swatek et al. (2019). The negligible magnetic gap of TSS is consistent with the absence of the zero-field QAH effect in magneto-transport measurementsLee et al. (2018); Deng et al. (2019); Liu et al. (2019); Wu et al. (2019); Ge et al. (2019). 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 surprising 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) ferromagnetic insulator with the example of CrI3 via exchange bias. Our density-functional theory (DFT) calculations on the MnBi2Te4 /CrI3 heterostructure show a ferromagnetic exchange bias around $40$ meV, much larger than the Néel temperature of MnBi2Te4 (24 K Otrokov et al. (2018)) and the Curie temperature of CrI3(61 K for bulk McGuire et al. (2015) and 45 K for monolayerHuang et al. (2017)) . 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 (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.

*Ferromagnetic 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, CrI3Huang et al. (2017). Its monolayer is ferromagnetic (FM) and can couple with MnBi2Te4 through the van der Waals interface, which may disturb the band structure of MnBi2Te4 weakly. 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, $7.04~{}\text{\AA}$ for CrI3 and $4.36~{}\text{\AA}$ for MnBi2Te4 from our DFT calculations, which is consistent with recent worksHou et al. (2019); Li et al. (2019b); Zhang et al. (2018). A $2\times 2$ supercell of CrI3 can match well with a $3\times 3$ supercell of MnBi2Te4. Alternatively, the primitive unitcell of CrI3 can also match a $\sqrt{3}\times\sqrt{3}$ supercell of MnBi2Te4 with a mismatch of 7%. Because we are mostly interested in the band structure of MnBi2Te4, we stretch the CrI3 lattice to match the $\sqrt{3}\times\sqrt{3}$ MnBi2Te4 supercell. We fully optimized the atomic structures by including the van der Waals interactions in DFT calculationsKresse and Furthmüller (1996) within the generalized-gradient approximation (GGA)Perdew et al. (1996). We have tested both models and found that they give similar results in the exchange coupling and band structure (see more information in the Supplementary Materials (SM) SMs ). Thus, we choose the smaller model, $\sqrt{3}\times\sqrt{3}$ MnBi2Te4/$1\times 1$ CrI3, for further investigations in the following.

At the interface, MnBi2Te4 exhibits strong the 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 $3\times 3$ MnBi2Te4/$2\times 2$ CrI3 case SMs . When increasing the MnBi2Te4 layer to 2 SLs and more, the interface FM coupling remains with the same exchange energy and the two SLs couples still in the AFM way. Therefore, 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 ($\sim$ 3 meV for $\sqrt{3}\times\sqrt{3}$ supercell) or two CrI3 layers ($\sim$ 10 meV for $1\times 1$ unitcellSivadas et al. (2018)). 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 SOC does not affect the magnetic coupling at the interface.

The strong exchange coupling originates in the orbital feature at the interface. The Mn site has $d^{5}$ configuration as $t_{2g}^{3}e_{g}^{2}$ and the Cr site has $d^{3}$ as $t_{2g}^{3}e_{g}^{0}$. There is a long exchange pathway from Cr-$e_{g}$ to Mn-$t_{2g}$ states through the intermediate I, Te, Bi and Te atoms, which is clearly beyond the simple super-exchange interaction. In the localized Wannier orbitals Mostofi et al. (2008), we observe a crucial feature in the Cr-$e_{g}$ states. Tails of the Cr-$e_{g}$ 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. In addition, we note that 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 1-6 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-$t_{2g}$ bands are far below the valence bands of MnBi2Te4. The Cr-$e_{g}$ states overlap with the conduction band bottom of MnBi2Te4 and remain unoccupied. This means there is no charge transfer through the van der Waals junction. The calculated Cr-$t_{2g}$ and Cr-$e_{g}$ gap is about 1 eV, which is consistent with previous GGA calculations and can be corrected to about 1.5 eV by hybrid-functionals Zhang et al. (2015). Although some Cr-$e_{g}$ bands appear as the lowest conduction bands at the interface for thinner MnBi2Te4 films (1-4 SLs), they do not affect our understanding of the band structure topology. When the MnBi2Te4 layer is thicker (e.g. 5-6 SLs), the MnBi2Te4 states become the lowest conduction bands. Thus, CrI3 serves an ideal proximity exchange bias without destroying the MnBi2Te4 band structure.

We find that isolate MnBi2Te4 layers are trivial magnetic insulators for 1, 2, 4 and 6 SLs thick and QAHE insulators for 3 and 5 SLs, which is consistent with recent theoretical studies Li et al. (2019b); Otrokov et al. (2019). Here, the QAH insulator possesses the CN = 1, as showed by our Berry phase calculations using the Wilson loop method Yu et al. (2011); Soluyanov and Vanderbilt (2011) 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 breaks weakly 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 SLs thick MnBi2Te4 /CrI3, the band gap closes at about 90% of the normal SOC strength but re-opens an energy gap as increasing SOC, showing a topological phase transition (TPT)SMs . The QAH insulator gaps are 49 and 14 meV for the 3 and 5 SLs interface, respectively. For 1, 2, 4 and 6 SLs thick MnBi2Te4 / CrI3, however, the band gap remains open as varying SOC from 0 to 100%. (see more information in Ref. SMs ).

*Electrically tunable high-Chern-number 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 ($\epsilon$) along the $-z$ direction can push the Cr-$e_{g}$ 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 modelQi et al. (2006), a band inversion at the $\Gamma$ point usually leads to a change of the CN by $\pm 1$. If the band inversion occurs at generic k-points, 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 three-fold 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 $\Gamma-M$ line in the Brillouin zone and perpendicular to the layer plane, the $\Gamma-K$ line is invariant under the $TM$ symmetry. Therefore, if a transition happens at a generic k-point away from the $\Gamma-K$ line, the gapless points must exist at six different k-points related by the $C_{3}$ and $TM$ symmetries. If a transition happens along the $\Gamma-K$ line that is invariant under $TM$, 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 $\Gamma$ that is invariant under both $C_{3}$ and $TM$ symmetries, a single Dirac point transition can occur.

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 $\epsilon=0.005~{}V/\text{\AA}$. This transition is through a gap-closing point at $\Gamma$ for $\epsilon=0.002/\text{\AA}$. For a larger electric field ($\epsilon=0.015$ V/Å), another transition occurs with three gap-closing points along the $\Gamma-K$ lines, leading to a QAH state with CN = 3. Furthermore, the electric field can also drive the MnBi2Te4 film with even number of SLs from a trivial magnetic insulator with zero CN to the QAH state. For instance, $\epsilon=0.03~{}V/\text{\AA}$ induces a TPT with three gapless points along the $\Gamma-K$ lines in the 2-SL-thick MnBi2Te4 film at $\epsilon=0.023~{}V/\text{\AA}$, resulting in the QAH state with CN = 3 (See more information in SM SMs ).

*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 re-arrangement 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 $\sim$ 40 meV. 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 number of MnBi2Te4 SLs.

Discussion and Conclusion: * 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 disturbs weakly 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 phaseQi et al. (2008); Mong et al. (2010) in ARPES. Additionally, it is worth noting that other magnetic insulators with out-of-plane magnetization, such as Tm3Fe5O12* (TmIG)Avci et al. (2017) and Cr2Ge2Te6Gong et al. (2017), may also play the same role of exchange bias as CrI3.

*Acknowledgement: * We acknowledge helpful discussions with Cui-zu Chang and Xiaodong Xu. Work at Penn State (CXL) was primarily supported by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES) under Award DE-SC0019064. CXL 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, the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant No. 815869).

Appendix 1 Calculation Methods

Density functional theory (DFT) calculations were performed using the Vienna ab initio simulation package (VASP) with core electrons represented by the projector-augmented-wave (PAW) potential. 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. We projected the Wannier functions of the bulk MnBi2Te4 in the AFM phase. Based on 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.

Appendix 2 Structure models and exchange coupling

In the interface model, we can overlap the CrI3 layer and the MnBi2Te4 layer with different sliding shift in the plane. We have tested different ways of stacking and found they give quantitatively very similar results in energy and band structure for the $\sqrt{3}\times\sqrt{3}$ MnBi2Te4/$1\times 1$ CrI3 interface. At the same FM phase, the stacking way that preserves the three-fold rotation ($C_{3}$) exhibits slightly lower energy (by 20$\sim$30 meV for 1 to 3 MnBi2Te4 SLs) than other low-symmetry stacking ways. Therefore, we used the $C_{3}$-symmetric model in the main text.

We show the exchange coupling $\Delta_{E}=E_{FM}-E_{AFM}$ for 1-2 SL MnBi2Te4 on a CrI3 layer in Figs.S1 and S2. Two different stacking ways give similar exchange coupling strength. One can also deduce that the MnBi2Te4 - MnBi2Te4 interlayer coupling ($\sim$ 4 meV) is much weaker than the MnBi2Te4 - CrI3 coupling ($\sim$ 40 meV) from Fig.S2.

For the much larger supercell, $3\times 3$ MnBi2Te4/$2\times 2$ CrI3, it still prefers the FM coupling at the interface with an exchange coupling energy $\sim$ 40 meV. The system remains as an insulator, as shown in Fig. S3.

The band gaps for different SLs of MnBi2Te4 films (Fig.2) are shown in the table S1.

Figure S4 show the energies of different magnetic configurations of 4-SL of MnBi2Te4 sandwiched between two CrI3 layers. The structure 1, in which the top MnBi2Te4 SL flips its spin direction due to the top CrI3 layer, has the lowest energy. In the band structure, Cr-$e_{g}$ states appear as the conduction bands. There is still an energy gap for the QAH state with CN $=-1$, as shown in Fig. 4.

Figures S5 and S6 show the band evolution as increasing SOC strength for 3 and 5 SL MnBi2Te4 , respectively. The band gap closing is found when the SOC strength is increased to 90% (98%) of the full SOC strength for 3 (5) SL MnBi2Te4 , reflecting the occurrence of TPT and the change of CN.