Abstract
The interplay between band topology and magnetism can give rise to exotic states of matter. For example, magnetically doped topological insulators can realize a Chern insulator that exhibits quantized Hall resistance at zero magnetic field. While prior works have focused on ferromagnetic systems, little is known about band topology and its manipulation in antiferromagnets. Here, we report that MnBi_{2}Te_{4} is a rare platform for realizing a cantedantiferromagnetic (cAFM) Chern insulator with electrical control. We show that the Chern insulator state with Chern number C = 1 appears as the AFM to cantedAFM phase transition happens. The Chern insulator state is further confirmed by observing the unusual transition of the C = 1 state in the cAFM phase to the C = 2 orbital quantum Hall states in the magnetic field induced ferromagnetic phase. Near the cAFMAFM phase boundary, we show that the dissipationless chiral edge transport can be toggled on and off by applying an electric field alone. We attribute this switching effect to the electrical field tuning of the exchange gap alignment between the top and bottom surfaces. Our work paves the way for future studies on topological cAFM spintronics and facilitates the development of proofofconcept Chern insulator devices.
Introduction
A Chern insulator is a twodimensional topological state of matter with quantized Hall resistance of h/Ce^{2} and vanishing longitudinal resistance^{1,2}, where the Chern number C is an integer that determines the number of topologically protected chiral edge channels^{1,3,4,5}. The formation of the Chern insulator requires timereversal symmetry breaking, which is usually achieved by magnetic doping or magnetic proximity effect^{3,6,7}. A seminal example is the magneticallydoped topological insulators that realize a ferromagnetic (FM) Chern insulator with the quantum anomalous Hall effect^{2}. Although Chern insulators have now been realized in several systems, little is known about this topological phase in antiferromagnets, which may offer a platform for exploring new physics and control of band topology^{8,9}.
MnBi_{2}Te_{4}, an intrinsic topological magnet, provides a new platform for incorporating band topology with different magnetic states^{10,11,12,13,14}. MnBi_{2}Te_{4} is a layered van der Waals compound that consists of Te–Bi–Te–Mn–Te–Bi–Te septuple layers (SL) stacked along the crystallographic caxis. At zero magnetic field, it hosts an Atype antiferromagnetic (AFM) ground state: each SL of MnBi_{2}Te_{4} individually exhibits ferromagnetism with outofplane magnetization, while the adjacent SLs couple antiferromagnetically^{13,14}. By applying an external magnetic field perpendicular to the SLs, the magnetic state evolves from AFM to canted AFM (cAFM) and then to FM^{8}. The Chern insulator state has recently been demonstrated in mechanically exfoliated MnBi_{2}Te_{4} devices at both zero and high magnetic fields^{15,16,17,18}. However, the topological properties in the cAFM state have not been investigated, where the spin structure can be continuously tuned by a magnetic field. In addition, the electric field effect, which has been demonstrated in 2D magnets^{19,20}, remains to be explored as a means to control the topological states in thin MnBi_{2}Te_{4} devices.
Here, we demonstrate that MnBi_{2}Te_{4} is a Chern insulator in the cAFM state and realize the electric field control of the band topology. We employed a combined approach of polar reflective magnetic circular dichroism (RMCD) measurement to identify the magnetic states and magnetotransport measurement to probe the topological property. To distinguish between electricfield and carrier doping effects, we fabricated MnBi_{2}Te_{4} devices with dual gates. The devices with a single gate were also used for combined transport and RMCD measurements. The transport and optical measurements were carried out at T = 50 mK and 2 K, respectively, unless otherwise specified (see the “Methods” section for fabrication and measurement details).
Results
Formation of Chern insulator phase in the cAFM state
Figure 1a shows the magnetic field dependence of the RMCD signal of a 7SL MnBi_{2}Te_{4} device with a single bottom gate (Device 1). Near zero magnetic field, the RMCD signal shows a narrow hysteresis loop due to the uncompensated magnetization in odd layernumber devices^{17,21}. Upon increasing the magnetic field, the sample enters the cAFM state at the spinflop field μ_{0}H_{C1} ~ 3.8 T, manifested by the sudden jump of the RMCD signal. Remarkably, as the spin–flop transition occurs, the Hall resistivity ρ_{yx} quantizes to about h/e^{2} (Fig. 1b), and the longitudinal resistivity ρ_{xx} drops from ~100 kΩ to near zero (Fig. 1c, see Supplementary Fig. 1 for a full gatedependent transport measurement). As the magnetic field further increases, the canted spins rotate towards the outofplane direction and eventually become fully polarized at μ_{0}H_{C2} ~ 7.2 T with saturated RMCD signal (Fig. 1a). Within the field range of about 3.8–7.2 T, where MnBi_{2}Te_{4} is in the cAFM state, ρ_{yx} remains quantized with vanishing ρ_{xx}, this demonstrates the formation of C = 1 cAFM Chern insulator state.
The cAFM Chern insulator is further supported by exploring the topological phase diagram in dualgated devices over a broad range of magnetic field. Figure 2a is a 2D color map of ρ_{yx} in a 7SL dualgated MnBi_{2}Te_{4} device (Device 2) as a function of both μ_{0}H and gateinduced carrier density n_{G} at electric field D/ε_{0} = −0.2 V/nm (see corresponding ρ_{xx} map and characterization in Supplementary Figs. 2–4). Here n_{G} partly compensates for the residual carrier density in the sample and tunes the Fermi level (see the “Methods” section). The electric field D is defined to be positive when it points from top to bottom gates. Notably, in the range of n_{G} 1.0–2.0 × 10^{12} cm^{−2}, a sharp phase boundary of the C = 1 state is observed near the spinflop field  μ_{0}H_{C1}  ~ 3.6 T. This further supports that the formation of C = 1 state is coupled to the cAFM order. The cAFM Chern insulator has been observed in multiple devices. See a device summary in Supplementary Table 1 and their basic characterization in Supplementary Figs. 5–7.
This intimate relationship between the topological and magnetic phase transitions in MnBi_{2}Te_{4} can be understood as follows. Due to the easyaxis anisotropy, the transition from the AFM to the cAFM state is a firstorder phase transition: at the spin–flop field, the magnetization in each SL suddenly rotates into the cAFM state with a finite canting angle. The abrupt change of the magnetic state is accompanied by the formation of the Chern insulator gap, hence a change in the Chern number. Further increase of the magnetic field results in a continuous rotation of the canted spins, which is expected to cause an adiabatic change in the size of the Chern insulator gap until the system enters the FM state. This understanding naturally connects our observation of cAFM Chern insulator and previously reported C = 1 Chern insulator in the field–induced FM state^{15,16,17}. As the exchange gap in both cAFM and FM states are adiabatically connected, the stability of Chern number ensures a cAFM Chern insulator state as long as the exchange gap is not closed by external disorder or temperature.
When μ_{0}H is above  μ_{0}H_{C2}  ~ 7.4 T, the sample enters the magnetic fieldinduced FM state. A rich topological phase diagram is uncovered, in which the topological states with corresponding Chern numbers are identified based on ρ_{yx} ~ h/Ce^{2} and nearly vanishing ρ_{xx}. In addition to the C = 1, C = 2 and 3 states, characterized by ρ_{yx} ~ h/Ce^{2}, appear at higher n_{G}. Unlike the C = 1 phase, the contours of C = 2 and 3 phases are linearly dependent on magnetic field μ_{0}H (Fig. 2a and its derivative in Supplementary Fig. 2c). This implies that the C = 2 and 3 states are a result of the Landau level (LL) formation coexisting with edge state from band topology^{22}. For the C = 1 phase, there is only a single region present in the phase diagram. So, the C = 1 Chern insulator state in the cAFM is adiabatically connected to the same state in the fieldinduced FM phase, as discussed above. The phase space of C = 1, 2, and 3 states can also be controlled by the top gate, as shown in Supplementary Figs. 3 and 4.
Figure 2b–g plot the μ_{0}H dependence of ρ_{yx} and ρ_{yx} at three selected n_{G}. For heavy electron doping n_{1} ~ 2.23 × 10^{12} cm^{−2}, ρ_{yx} ~ 0 and ρ_{xx} ~ 0.005 h/e^{2} near zero magnetic field. As  μ_{0}H  increases, we see a kinklike feature, namely a sudden increase of both ρ_{yx} and ρ_{xx} related to the AFM to cAFM transition at H_{C1} (Fig. 2b, e). For  μ_{0}H  > 10 T, ρ_{yx} approaches 0.5 h/e^{2}, indicating a C = 2 Chern insulator state. For n_{2} ~ 1.87 × 10^{12} cm^{−2}, upon entering the cAFM phase at  μ_{0}H  ~ 3.6 T, the sample first goes into the C = 1 state with ρ_{yx} ~ h/e^{2} and ρ_{xx} = 0.05h/e^{2}_{.} At a higher magnetic field  μ_{0}H  ~ 10 T, it then switches into the C = 2 state with ρ_{yx} ~ 0.5h/e^{2} and ρ_{xx} = 0.05h/e^{2} (Fig. 2c, f). This phase transition from the C = 1 state into a higher Chern number C = 2 state as  μ_{0}H  increases at a fixed carrier density is unusual. Increasing  μ_{0}H  increases the degeneracy of LLs. Therefore, if the quantization is caused by the formation of LLs, then the quantum Hall plateau should always change from higher to lower Chern numbers as the magnetic field increases at a fixed carrier density. The opposite observation here further supports our interpretation that the C = 1 state observed in the cAFM phase is a Chern insulator originating from the intrinsic nontrivial band structure, while the C = 2 state in the FM state is the quantum Hall state due to the formation of LLs^{23,24}. For n_{3} ~ 1.43 × 10^{12} cm^{−2}, close to the charge neutral point, the transport data shows an abrupt formation of a C = 1 Chern insulator at spinflop field H_{C1}, consistent with our discussions above (Fig. 2d, g). The sharp transition exists over a finite doping range, implying that a finite magnetic exchange gap opens suddenly with the spin–flop transition from the AFM to the cAFM phase. This further validates the conclusion that the electronic structure is coupled to the magnetic order^{17}.
We note that similar effects, i.e., the transition from C = 1 to C = 2 states as magnetic field increases, have also been studied in other material systems. For example, in doped magnetic topological insulator quantum well (Mn, Hg) Te^{23}, the increase of magnetic field first leads to a transition from C = 2 to C = 1 due to linear Hall effect, then back to C = 2 state attributed to the effective nonlinear Zeeman effect of Mn ions. A recent theoretical study^{24} on MnBi_{2}Te_{4} also suggested that the lowest Landau level, stabilized by Anderson localized state, together with quantum anomalous Hall edge state, can form a C = 2 state. Thus, the increase of magnetic field can give rise to a transition from C = 1 to C = 2 state. Note that Weyl semimetal physics has also been proposed in the FM state of bulk MnBi_{2}Te_{4}^{25,26}. So, Landau level could originate either from the surface band in a standard magnetic topological insulator picture^{15}, or from the quantum well states in a confined Weyl semimetal picture^{18}. Considering that the 3D bulk crystal of MnBi_{2}Te_{4} has not been well established experimentally as a Weyl semimetal, the application of the Weyl physics to the atomically thin flakes needs extra caution. Nevertheless, future studies in the FM state are needed to distinguish these two different physical origins of the high Chern number states.
Electric field control of the cAFM Chern insulator
The association of the formation of C = 1 state with the AFM to cAFM magnetic phase transition suggests the possibility of electricfield control of the Chern number at the cAFM to AFM phase boundary where the magnetic exchange gap should be small. Figure 3a shows ρ_{yx} vs. μ_{0}H at n_{G} = 1.0 × 10^{12} cm^{−2} and D/ɛ_{0} = −0.3 V/nm for a dual gated 6SL MnBi_{2}Te_{4} device (Device 3), from which we determine the spin–flop field μ_{0}H_{C1} is about 3 T. The small hysteresis loop in the AFM state is possibly due to magnetic domain effects, as previously reported^{17}. We then map out ρ_{yx} (Fig. 3b) and ρ_{xx} (Fig. 3c) as a function of n_{G} and D at μ_{0}H_{C1} = 3 T. The droplet shapes enclosed by the dashed lines in both plots, elongated along the D axis, indicate the C = 1 Chern insulator phases. Figure 3d shows both ρ_{yx} and ρ_{xx} as a function of D, obtained from line cuts of Fig. 3b and c at n_{G} = 1.1 × 10^{12} cm^{−2}. As D varies from positive to negative values, the sample starts with small ρ_{yx}, enters C = 1 state at optimal D_{opt}/ɛ_{0} = −0.3 V/nm supported by the observation of ρ_{yx} ~ h/e^{2} and vanishing ρ_{xx}, and finally exits the C = 1 state with reduced ρ_{yx} and increased ρ_{xx}. The behavior of ρ_{yx} and ρ_{xx} suggests that the dissipationless chiral edge transport can be switched on and off by an external electric field. The same electric field effect is reproduced in Device 2 (Supplementary Fig. 8).
The sensitive dependence of the C = 1 state on D near H_{C1} may be because the electric field directly alters the electronic band structure^{27,28}, or because it affects the magnetic order (e.g., by tuning H_{C1}), which in turn affects the band structure. To distinguish these two mechanisms, we perform gatedependent RMCD measurements on a 7SL dual gated device (Device 4). We found that both RMCD signal and spin–flop field are marginally affected by the electric field, but strongly tuned by gateinduced doping (as illustrated in Fig. 4a), consistent with previous reports on 2D magnets^{19,20}. Figure 4b shows the RMCD map as a function of D and n_{G} near spin–flop field H_{C1}. As n_{G} sweeps from electron to hole doping, RMCD significantly increases, i.e., a larger outofplane magnetization of the cAFM state at hole doping than electron doping. However, the electric field effect on the magnetic state is marginal, evident by unchanged color along the vertical direction (i.e., parallel to the D axis) in Fig. 4b. The RMCD measurements exclude electric field tuning of the magnetic state as the origin of the electric control of the cAFM Chern insulator state.
Discussion
This left us with the explanation that the electric field adjusts the relative energy alignment of the magnetic exchange gaps of the two surfaces, as depicted in Fig. 4c. There is a builtin asymmetry between the top and bottom surfaces, possibly due to the different dielectrics used for top and bottom gates. As D is tuned to the optimal electric field (D_{opt}), the exchange gaps of top and bottom surfaces are aligned. When the chemical potential is tuned into the exchange gap^{29,30,31}, a welldefined edge state with C = 1 forms. However, when the deviation of D from D_{opt} is large enough to completely misalign the two magnetic exchange gaps, i.e. the magnetic exchange gap of one surface is aligned to either bulk conduction or valence bands on the other surface, energy dissipative bulk transport is dominant. In the cAFM phase, as the magnetic field increases, the magnetic exchange gap becomes larger as well due to the increase of outofplane magnetization. The electric field modulation of the Chern insulator state thus becomes weaker at high field but can be still feasible when the chemical potential is tuned near the band edge (see Supplementary Fig. 9). Our work shows that with the combination of independent control of electric field and carrier doping, as well as intimately coupled magnetic and topological orders, MnBi_{2}Te_{4} can be a model system for developing ondemand Chern insulator devices and exploring other novel quantum phenomena such as topological magnetoelectric effects.
Methods
Device fabrication
Bulk crystals of MnBi_{2}Te_{4} were grown out of a Bi–Te flux as previously reported^{32}. Scotchtape exfoliation onto 285 nmthick SiO_{2}/pSi substrates was adopted to obtain MnBi_{2}Te_{4} from 4 SL to 10 SL, distinguished by combined optical contrast, atomic force microscopy, and RMCD measurements. A sharp tip was used to disconnect thin flakes with the surrounding thick flake. Then the flakes were fabricated into singlegated or dualgated devices. Singlegated devices were fabricated by electron beam lithography with Polymethyl methacrylate (PMMA) resist and followed by thermal evaporation of Cr (5 nm) and Au (50 nm) and liftoff in anhydrous solvents. Then the devices were covered with PMMA as the capping layer. This fabrication process brought spatial charge inhomogeneity and doped the flake. The measured voltages of charge neutrality points shifted between the thermal cycle from 2 to 300 K. Dualgated devices were fabricated by stencil mask method^{33}, followed by thermal evaporation of Au (30 nm) and transfer of 30–60 nm hBN as capping layer by polydimethylsiloxane (PDMS). Standard electron beam lithography (EBL) was adopted to define outer electrodes and metal top gates. Since the highly reflective Au top gate (Device 2 and 3) forbids the RMCD measurements, we also fabricated an optical dualgated device with graphite top gate (Device 4). Before EBL we transfer a thin graphite (5–20 nm) on top of hBN with PDMS. For this particular device, a bulk flake which connects to the thin one is used as the contact. In this way, there is no metal underneath or deposited on top of hBN. Thus, Device 4 provided higher doping density than other ones. During the fabrication, MnBi_{2}Te_{4} was either capped by PMMA/hBN or kept inside an Argonfilled glovebox to avoid surface degradation.
Transport measurements
Transport measurements were conducted in a dilution refrigerator (Bluefors) with lowtemperature electronic filters and an outofplane 13 T superconductor magnet coil. Fourterminal longitudinal resistance R_{xx} and Hall resistance R_{yx} were measured using standard lockin technique with an a.c excitation of 0.5–10 nA at 13.777 Hz. The a.c. excitation was provided by the SR830 in series with a 100 MΩ resistor, flowed through the device, and was preamplified by DL1211 at 1 V/10^{−6} A sensitivity. R_{xx} and R_{yx} signals were preamplified by differentialended mode of SR560 with a 1000 times amplification. All preamplifiers were read out by SR830. A similar amplifier chain provided approximately ±3% uncertainty in the previous study^{34}. We estimated our uncertainty was of the same order. In Figs. 1–4, ρ_{yx} overshoot 2.36%, 0.92%, 1.14% and 1.7%, respectively. The sheet resistivity ρ_{xx} and ρ_{yx} were obtained by ρ_{xx} = s × R_{xx}/l and ρ_{yx} = R_{yx} where s was the width of the current path, l was the length between two voltage probes, estimated from device geometry. Magnetotransport data involved positive and negative magnetic fields was antisymmetrized/symmetrized by a standard method to avoid geometric mixing of ρ_{xx} and ρ_{yx}. As this mixing did not affect the topological transport signal, we presented raw data for fixed magnetic field study. To convert top gate voltage V_{tg} and bottom gate voltage V_{bg} into gateinduced carrier density n_{G} and electric field D, we used n_{G} = (V_{tg}C_{tg} + V_{bg}C_{bg})/e and D/ɛ_{0} = (V_{tg}C_{tg} − V_{bg}C_{bg})/2ɛ_{0}, where C_{tg} and C_{bg} are top and bottom gate capacitance obtained from device geometry, e the electron charge and ɛ_{0} the vacuum permittivity. This formula is derived from the parallelplate capacitor model. Fixing D(n_{G}) and sweeping n_{G}(D) monotonically modify the carrier density n_{2D} (external displacement field D_{ext}) and thus the chemical potential (electric field) of the device. To obtain the n_{G}D maps, we first got V_{tg} − V_{bg} maps of transport data by sweeping V_{bg} from the negative side to the positive for every fixed V_{tg}. Converting V_{tg} − V_{bg} to n_{G}−D leads to the uncovered parameter space, e.g., in Figs. 3b, c and 4b. The nμ_{0}H (V_{bg} − μ_{0}H) maps were taken by sweeping dual gates (back gate) quickly, ~4 min per data line back and forth, while sweeping magnetic field slowly, ~0.015 T/min. This method gave digitized noise or unfinished lines near the field limit, e.g., near 0 T of Fig. 2a.
Reflective magnetic circular dichroism measurements
The experiment setup follows our previous RMCD/MOKE study of magnetic order in CrI_{3}. RMCD measurements were performed in an attoDRY cryostat with attocube xyz piezo stage, the base temperature of 1.6 K and 9 T superconducting magnet. The magnetic field was applied perpendicular to the sample plane. Linearly polarized 632.8 nm He–Ne laser with 200 nW power was focused through an aspheric lens to form ~2 µm beam spot on the sample surface. The outofplane magnetization of the sample induced magnetic circular dichroism (MCD) ΔR, the amplitude difference between the reflected right and leftcircularly polarized light. To obtain the RMCD ΔR/R signal, two lockin amplifiers SR830 were used to analyze the output signals from a photomultiplier tube with the chopping frequency p = 1.377 kHz and photoelastic modulator frequency f = 50 kHz. The ratio between pcomponent signal I_{1} and fcomponent signal I_{2} is proportional to the RMCD signal: ΔR/R = I_{2}/(J_{1}(π/2) × I_{1}) where J_{1} is the firstorder Bessel function.
Data availability
Source data of Figs. 1–4 can be found at: https://doi.org/10.6084/m9.figshare.19193498.v4. All other data that support the findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
The authors thank Chaoxing Liu for insightful discussions. The electrical control of Chern number in the canted antiferromagnetic states was mainly supported by AFOSR FA95502110177. Magnetooptical measurements and theory understanding were supported as part of Programmable Quantum Materials, an Energy Frontier Research Center funded by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES), under award DESC0019443. The authors also acknowledge the use of the facilities and instrumentation supported by NSF MRSEC DMR1719797. J.Y. acknowledges support from the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division. C.Z.C. acknowledges the partial support from the Gordon and Betty Moore Foundation’s EPiQS Initiative (Grant GBMF9063). Y.T.C. acknowledge support from NSF under award DMR2004701, and the Hellman Fellowship award. X.X. and J.H.C. acknowledge the support from the State of Washington funded Clean Energy Institute.
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J.C. and D.O. fabricated the devices, assisted by Z.F., M.H., and Z.L. J.C. and D.O. performed the transport measurements, assisted by Z.F. J.C., Z.L. and T.S. performed the RMCD measurements. C.W. and D.X. provided theoretical support. J.Y. synthesized and characterized MnBi_{2}Te_{4} bulk crystals. X.X., J.Y., D.X., C.Z.C, Y.T.C. J.H.C. D.C. supervised the project. J.C., D.X., C.Z.C., X.X., Y.T.C, and D.O. wrote the manuscript with inputs from all authors. All authors discussed the results.
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Cai, J., Ovchinnikov, D., Fei, Z. et al. Electric control of a cantedantiferromagnetic Chern insulator. Nat Commun 13, 1668 (2022). https://doi.org/10.1038/s41467022292598
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DOI: https://doi.org/10.1038/s41467022292598
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