Metamagnetism of few layer topological antiferromagnets
MMetamagnetism of few layer topological antiferromagnets
C. Lei, O. Heinonen, A.H. MacDonald, and R. J. McQueeney
3, 4 Department of Physics, The University of Texas at Austin, Austin, TX 78712 Argonne National Laboratory, Argonne, IL USA Ames Laboratory, Ames, IA, 50011, USA Department of Physics and Astronomy, Iowa State University, Ames, IA, 50011, USA (Dated: February 24, 2021)MnBi Te (MBT) is a promising antiferromagnetic topological insulator whose films provide ac-cess to novel and technologically important topological phases, including quantum anomalous Hallstates and axion insulators. MBT device behavior is expected to be sensitive to the various collinearand non-collinear magnetic phases that are accessible in applied magnetic fields. Here, we use classi-cal Monte Carlo simulations and electronic structure models to calculate the ground state magneticphase diagram as well as topological and optical properties for few layer films with thicknesses up tosix septuple layers. Using magnetic interaction parameters appropriate for MBT, we find that it ispossible to prepare a variety of different magnetic stacking sequences, some of which have sufficientsymmetry to disallow non-reciprocal optical response and Hall transport coefficients. Other stackingarrangements do yield large Faraday and Kerr signals, even when the ground state Chern numbervanishes. INTRODUCTION
MnBi Te (MBT) is a promising platform for the de-velopment of unique devices based on topological elec-tronic bands [1–28]. The utility of MBT is consequenceof its natural layered structure, which consists of stacksof ferromagnetic (FM) septuple layers (SLs) with out-of-plane magnetization and with inverted electronic bandswith non-trivial topology [1–3]. There is great inter-est in manipulating the sequence of magnetic and topo-logical layers as a means to control phenomena relatedto the band topology [9]. Bulk MBT adopts a stag-gered antiferromagnetic (AF) stacking of the FM SLs[3, 10, 12, 18], providing the first realization of an AFtopological insulator, which is predicted to host uniqueaxion electrodynamics[29, 30]. Weak interlayer magneticinteractions across the van der Waals gap and uniaxialmagnetic anisotropy allow for facile control of the mag-netic structure. For example, a bulk Weyl semimetallicphase is predicted when all FM layers in MBT are co-aligned with a small applied magnetic field [9, 15, 16].Perhaps the most exciting opportunity in MBT ma-terials is the possibility of developing thin film deviceswith precisely controlled magnetic stacking sequences.In this context, devices with even and odd numbers ofAF stacked layers offer qualitatively different topologicalphase possibilities [11, 23, 28, 31–33]. An odd numberof magnetic layers necessarily has partially compensatedmagnetization that is beneficial for the observation of thequantum anomalous Hall (QAH) effect [11]. On the otherhand, fully compensated magnetization occurs in evenlayer devices in the absence of an applied magnetic fieldand provides an ideal platform to search for axion insu-lators with quantized magnetoelectric coupling [13, 23].However, there are also much richer possibilities in bothodd or even layer samples [33], since the application of a magnetic field can result in different collinear magneticstackings with partially compensated magnetization, oreven to non-collinear (canted) magnetic phases. Here, weconsider the possibility to stabilize such phases and thepotential for realizing unique topological phases in thisscenario.The magnetization behavior of AF magnetic multi-layers displays rich metamagnetic behavior with fea-tures, such as surface spin-flop transitions, that are notobserved in bulk AFs [34, 35]. This complex behav-ior is a consequence of competition between single-ionanisotropy, interlayer magnetic exchange, and Zeemanenergy. For topological materials, whether the symme-try of different magnetic layer stacking sequences can af-fect the topological properties of the bands is an openquestion. For example, collinear metamagnetic phaseswith the same net magnetization can result from mag-netic layer stackings that may or may not break mir-ror symmetry ( M ). Here we use classical Monte Carlosimulations to show that it is generally possible to tune-in different stacking sequences with distinct symmetries.What’s more, we find that realistic values of the single-ion anisotropy and exchange place MBT close to this tun-ability regime. Finally, we discuss the topological prop-erties of accessible field-tuned states and strategies toidentify them experimentally. MONTE CARLO SIMULATIONS OF BULKMNBI TE The spin lattice of MBT consists of triangular FM lay-ers which are stacked in a close-packed fashion alongthe direction perpendicular to the layers. Interlayerinteractions are AF, resulting in the zero-field A-typeground state [see Fig. 1(a)]. Magnetization and neu-tron diffraction experiments find Mn moments oriented a r X i v : . [ c ond - m a t . s t r- e l ] F e b perpendicular to the layers consistent with uniaxial mag-netic anisotropy [18]. These features suggest that a sim-ple spin model can be used to study the magnetizationbehavior of MBT in an applied magnetic field H = J (cid:48) (cid:88) (cid:104) ij (cid:105)|| S i · S j + J (cid:88) (cid:104) ij (cid:105)⊥ S i · S j − D (cid:88) i S i,z − gµ B H · (cid:88) i S i . (1)Here, S i is the Mn spin at site i ( S = 5 / J (cid:48) < J > D is the uniaxial single-ion anisotropy.Each Mn ion has six interlayer and intralayer nearest-neighbors ( z = 6). A representative and consistent set ofmagnetic coupling parameter values for MBT have beenobtained from magnetization [18] and inelastic neutronscattering experiments [17]; SJ (cid:48) = − .
35 meV, SJ =0 .
088 meV, and SD = 0 .
07 meV.Using these nominal values, classical Monte Carlo(MC) simulations on bulk and few layer MBT systemshave been performed using both UppASD [36] and Vam-pire [37] software packages. MC simulations are first per-formed on bulk MBT with a 21 × ×
12 system size(15876 spins) with periodic boundary conditions. MCsimulations are run with the field pointed perpendicularto the layers using 50000 MC steps per field. To accountfor hysteresis and history dependence, we begin the sim-ulations with the field polarized state at H = 10 T andramp the field down in equal steps, using the final statefrom the previous field as the initial state for the currentfield.Figure 1(b) summarizes typical MC simulation resultsfor bulk MBT that reveal a N´eel temperature of 22 Kand magnetization curves with field-polarized saturationfields ( µ H satab = 10 . µ H satc = 7 . µ H SF = 3 . D/zJ determine the magnetization behavior whenthe field is applied along the c -axis. Only three phases arepossible, the AF phase, the canted spin-flop phase (SF),and the field-polarized phase (FM). For relatively weakanisotropy D/zJ < /
3, a first-order spin-flop transitionphase is expected, followed by a second-order transitionto the field polarized state (AF → SF → FM). The nom-inal MBT parameters yield
D/zJ ≈ .
13 which is withinthe spin-flop regime. For dominant single-ion anisotropy
D/zJ >
1, the virgin AF state is swept out by a weakapplied field and a FM hysteresis loop develops. This be-havior is observed in MnBi Te , in which the addition ofa non-magnetic Bi Te spacer between MnBi Te layersdramatically weakens the interlayer magnetic exchange Figure 1. (a) Schematic magnetic layer structure of MBT in thecrystallographic unit cell with close-packed stacking of septuple lay-ers. The intralayer ( J (cid:48) ) and interlayer ( J ) magnetic interactions areindicated. The gray shaded boxes indicate the full septuple layersseparated by a van der Waals gap. (b) Monte Carlo simulations ofthe bulk magnetization of MBT with field applied perpendicularto the Mn layers. Inset shows the order parameter (OP) of thestaggered A-type AF order as a function of temperature. (c)-(f)Monte Carlo simulations of the magnetization of 3, 4, 5, and 6 layerMBT, respectively, using the nominal Heisenberg parameters. For N = 4 and 6, the ” ∗ ” indicates an additional phase transition. [39, 40]. In the intermediate regime 1 / < D/zJ <
1, theAF → FM transition occurs directly (a spin- flip transi-tion), whereas the field-reversed transition goes throughthe spin-flop regime (FM → SF → AF).
FIELD-TUNED FEW-LAYER MAGNETIZATION
We now consider the behavior of the magnetization inthin film samples consisting of N magnetic layers, with N = 3 , , N -layer systems is much more complex than the bulk phasediagram because regions of stability exist (as describedin detail below) that correspond to collinear magneticphases with partially compensated magnetization (ferri-magnetic phases). In this respect, significant differencesoccur between odd or even layer systems because the netmagnetization cannot be fully compensated in odd layerfilms. Fig. 1 (c)-(f) shows the magnetization for N = 3,4, 5, and 6-layer MBT obtained from MC simulationsusing a 11 ×
11 system size for the basal layer and thenominal MBT Heisenberg parameters. Select simulationswith a 21 ×
21 basal layer produced no significant changesin the magnetization sweeps.For N = 3 and 5 layer simulations with the nominalMBT parameters, we find the expected FM hysteresisloop corresponding to the magnetization of a single un-compensated layer ( M = ± M = ± N = 4 and N = 6,magnetization curves resemble bulk MBT with a AF → SF → FM sequence of transitions. However, one noticesevidence for an additional transition [indicated with a ” ∗ ”in Fig. 1(d) and (f)] within the spin-flop phase near 3 T.As we will show below, this transition demonstrates thatthe nominal parameters of MBT are close to a criticalpoint in the phase diagram at which the M = 2 collinearphase becomes stable. We note that experimental evi-dence exists for M = 2 magnetization plateaus in the N = 4 and 6-layer films based on reflective magnetic cir-cular dichroism experiments [32, 33]. The experimentalmagnetization results in Refs. [32, 33] are analyzed us-ing numerical methods similar to the Mills model [41]described below.To explore the nature of this critical point and param-eter regimes beyond the nominal values chosen to repre-sent MBT, we calculated the phase diagrams for the 3, 4,5, and 6-layer systems as a function of field and the single-ion anisotropy parameter, as shown in Fig. 2. Gener-ally, these phase diagrams show regions of collinear mag-netism separated by non-collinear (spin-flop-like) phases.For the largest uniaxial anisotropy values, the system ap-proaches the behavior of a finite Ising chain where suc-cessive first-order transitions occur via single-layer spin-flips. We label the collinear ground state phases as; AF( M = 0, N = even only), M n ( M = n , with n − M = N , field-polarized with allAF bonds broken). Their time-reversed states are indi-cated by a bar (eg. M2) and states that have identicalground state energies within our model, when they oc-cur, are differentiated by a prime symbol (eg. M2 (cid:48) , M2 (cid:48)(cid:48) ).Metastable excited states, when they occur, are labeledwith an asterisk (eg. M0 ∗ ). METAMAGNETIC STATES
The distinct collinear ground states that occur for N = 3 and N = 4 are illustrated in Fig. 3(a) and(b). For N = 4, the MC phase diagram in Fig. 2(b)is consistent with the phase boundaries, critical points,and metastability limits obtained using the Mills model[35, 41]. As expected, slightly increasing D from thenominal MBT value of SD = 0.07 meV stabilizes the M2collinear phase, replacing the inflection in Fig. 1(d) witha magnetization plateau. Analysis of 4-layer Mills modelreveals that this critical point occurs at ( SD, µ H ) β =(0.082 meV, 3.59 T) [35], corresponding to a critical ratioof D/zJ = 0 . N = 3 and 5), the mag-netization sequence at low D/zJ reveals M1 and M1phases that form a hysteresis loop. For N = 5, larger D/J reveals two M3 states occur, labelled as M3 (cid:48) andM3 (cid:48)(cid:48) in Fig. 3(c), which have identical energies in ourmodel but are distinguished by the presence (M3 (cid:48) ) or ab-sence (M3 (cid:48)(cid:48) ) of mirror symmetry. Analysis of the sublat-tice magnetization from our simulations shows that only
Figure 2.
Phase diagrams obtained by MC simulation showingthe layer magnetization of a uniaxial layered antiferromagnet vs. magnetic field applied perpendicular to the layers and single-ionanisotropy strength ( SD ) for (a) 3 layers, (b) 4 layers, (c) 5 layersand (d) 6 layers. Simulations start in the FM phase at 10 T and thefield strength is reduced in equal steps of 0.2 T to -10 T. The solidlines are phase boundaries and metastability limits obtained fromthe Mills model, the dashed lines are guides to the eye. Collinearphases are labeled as described in the text. For N = 6, panel (d)indicates a metastability limit below which only the M2 (cid:48) phase isstabilized out of the SF phase. Figure 3.
Possible collinear magnetic ground states for N = 3, 4, 5,and 6-layer systems, labeled as either FM, AF or M n , where n is theuncompensated net layer magnetization. Red dashed lines indicatebroken AF bonds that cost an exchange energy of 3 J each. Shadedgreen rectangles enclose degenerate states with the same magne-tization, but different stacking sequences indicated by the primesymbol. Shaded blue rectangles enclose the states with odd-paritymagnetic configuration (i.e. T M symmetrized magnetic configura-tion, with T time-reversal and M mirror symmetry). The integersbelow each state denote their Chern numbers calculated from a thesimplified Dirac-cone model and (e) are the calculated band gaps vs. the magnetic stacking sequences of few-layer MBT thin films. the mirror symmetric M3 (cid:48) phase appears in the range of D/zJ studied.Much more interesting and complex behavior is ob-tained for thin films with N = 6, where collinear phasesappear that have equal uncompensated magnetizationsbut different symmetries. These phases appear beyondthe critical point, as in the N = 4 case, which is esti-mated to be ( SD, µ H ) β = (0.084 meV, 3.46 T) fromthe 6-layer Mills model [35] with D/zJ = 0 .
16. Close tothe critical point, two different M2 phases appear at posi-tive and negative fields, labeled M2 (cid:48) and M2 (cid:48)(cid:48) as shown inFig. 3(d). The mirror symmetric M2 (cid:48) phase has a singlebroken AF bond in the center of the stack, and is foundemerging out of the spin-flop phase (FM → SF → M2 (cid:48) ).The mirror symmetry broken M2 (cid:48)(cid:48) has a broken AF bondon the surface and is always stabilized out of the AF state(AF → M2 (cid:48)(cid:48) ).The preference for the AF → M2 (cid:48)(cid:48) transition over theAF → M2 (cid:48) transition can be understood from metasta-bility arguments related to the barrier height for layerflips that is determined primarily from the uniaxialanisotropy. The AF → M2 (cid:48)(cid:48) transition requires a coherentspin flip of the surface layer only, whereas AF → M2 (cid:48) re-quires three layer flips. From the FM side, both M2 (cid:48) andM2 (cid:48)(cid:48) require two layer flips and, for this reason, eitherphase is likely to appear within our simulations when D/zJ > ∼ .
2, as indicated by the purple dotted line inFig. 2(d). Due to its larger barrier height, M2 (cid:48) has alower metastability field to enter the AF phase than M2 (cid:48)(cid:48) as the field is reduced. To illustrate the metastabilitylimit, Fig. 4(a) and 4(b) show that repeated simulationsin which the field is reduced starting the FM phase willalways generate M2 (cid:48) with SD = 0 .
11 meV, whereas ei-ther M2 (cid:48) or M2 (cid:48)(cid:48) may appear with SD = 0 .
12 meV. Thismetastability limit at intermediate
D/zJ originates fromthe intervening spin-flop phase that selectively lowers thebarrier to the mirror symmetric M2 (cid:48) phase when layers 2and 5 have a large spin-flop angle, as shown in Fig. 4(c)–(e).At large
D/zJ , metastability issues with the MC sim-ulations reveal that even excited states (such as the M0 ∗ state with stacking sequence up-down-up-down-down-up) may be trapped in a local minimum. The occurrenceof the M0 ∗ phase is dependent on whether the M2 (cid:48) orM2 (cid:48)(cid:48) phase appears when lowering the field out of theFM phase. As described above, when the M2 (cid:48)(cid:48) phaseappears, the AF phase is favored since only a spin flipof the surface layer is required. When the M2 (cid:48) phaseappears, transition to the AF phase has a high barrierrequiring three layer spin flips. It is therefore more likelyfor the M2 (cid:48) phase to transition to the metastable M0 ∗ state in which the barrier height is set by a single layerflip. While this regime is not applicable for MnBi Te ,where D/zJ ≈ .
13, it may be applicable to MnBi Te in which non-magnetic Bi Te spacer layers dramaticallyreduce J . Figure 4. (a)-(b) Twelve MC simulations for N = 6 repeatedunder identical starting conditions near the metastability limit forformation of the M2 (cid:48) phase. Curves have a slight vertical offset forclarity. (a) For SD = 0 .
11 meV, only the M2 (cid:48) phase appears. (b)For SD = 0 .
12 meV, either the M2 (cid:48) or M2 (cid:48)(cid:48) phase appears. (c)-(e)show the evolution of the magnetization angle for each layer fordifferent simulations. (c) For SD = 0.11 meV, which is below themetastability limit, the angle of layers 2 and 5 in the spin-flop phaseapproaches 90 ◦ before flipping into the M2 (cid:48) phase. For SD = 0.12meV, different simulations (labeled (cid:48) with layers 2 and 5 flipping or (e) M2 (cid:48)(cid:48) with layers 3 and 5flipping. TOPOLOGICAL AND OPTICAL PROPERTIES
A very interesting question is how the stacking se-quence of the magnetic layers in MBT thin films andthe possible concomitant breaking of symmetries affectobservable electronic properties, particular those relatedto the topological classification of the electronic struc-ture. To gain insight, we used a simple model of stacked2D Dirac metals, 2 for each MBT layer, to calculate theband structure, Chern numbers, and magneto-optical re-sponses of different magnetic states. Previous work [9]has shown that with appropriate coupling between theDirac metals, models of this type provide a reasonabledescription of MBT thin films.As summarized in Fig. 3, we find that for
N >
N > N = 3 M1 state islabelled as 0 / (cid:48) andM4 (cid:48)(cid:48) states of N = 6 films, which have larger gaps than Figure 5.
Non-reciprocal optical response in 6-layer MBT thinfilms. Panels (a) and (b) show plots of optical longitudinal ( σ xx )and Hall ( σ xy ) conductivities vs. optical frequency calculated fromthe Kubo-Greenwood formula using the simplified Dirac cone elec-tronic structure model. In these plots the solid curves show thereal part of a conductivity tensor element while the dashed curvesshow the imaginary part. Different colors show results for differ-ent metamagnetic states. The correspondence between color andmetamagnetic state is repeated in (c) and (d), in which the Faradayand Kerr rotation angles in N=6 thin films are plotted vs. opticalfrequency. the M6 state.Symmetries play an important role in film electronicproperties. We define odd-parity magnetic configurationsas ones that have the property that the magnetic mo-ments reverse upon layer reversal (i.e. T M symmetrizedmagnetic configurations with T the time-reversal oper-ator). Odd-parity magnetic configurations can be readfrom the cartoons enclosed with shaded blue rectan-gles in Fig. 3, and include even– N AF configurations.Whenever the magnetic configuration has odd parity,the band Hamiltonian is invariant under the productof time-reversal symmetry T and inversion symmetry I . (Note that the spin-orbit coupling terms in cou-pled Dirac-cone model on the top and bottom surfacesof each septuple layer differ by a sign.) For many observ-ables, the consequences of T I invariance are the sameas the consequences separate T and I invariance. Forexample, T I invariance implies that the Berry curvatureΩ n ( k ) = − Ω n ( k ) = 0. The Berry curvature thereforevanishes identically and a generalized Kramer’s theoremimplies that all bands are doubly degenerate. It followsthat the Chern number vanishes for odd-parity magneticconfigurations. N = 6 layer thin films host a richer variety of metam-agnetic states. We find that all magnetic configurationswith a non-zero net spin magnetization (FM, M2 (cid:48) /M2 (cid:48)(cid:48) or M4 (cid:48) /M4 (cid:48)(cid:48) ) have total Chern number equal to 1 and are therefore QAH insulators. The states cannot be dis-tinguished by performing DC Hall effect measurements.Those properties that are not quantized are distinct foreach of these states however. For example, the Berry cur-vature has a different dependence on momentum in eachcase, although the total Chern number is always equalto one. As shown in Fig. 5, their optical conductivitiesdiffer at finite frequencies. In Fig. 5 (a) and (b) the realand imaginary part of longitudinal optical conductivi-ties σ xx ( ω ) and transvere optical conductivities σ xy ( ω ),calculated using the Kubo-Greenwood formula, [43, 44]are shown. In these plots solid curves represent the realpart of the conductivity ( (cid:60) σ xx/xy ) while dashed curvesrepresent the corresponding imaginary parts ( (cid:61) σ xx/xy ).Different colors are use to represent different metamag-netic states. The same colors are used for the impliedfrequency-dependent Faraday and Kerr rotation anglesin Fig. 5 (c) and (d). It follows that external magneticfields drives the 4-layer and 6-layer thin films from theiraxion insulator states to M2, M2 (cid:48) or M2 (cid:48)(cid:48) Chern insula-tors.In the DC limit, all optical conductivities vanish exceptfor (cid:60) σ xy in the case of a QAH insulator. (cid:60) σ xx and (cid:61) σ xy have peaks when the optical frequencies exceeds the two-dimensional band gaps of the thin film. (cid:61) σ xx and (cid:60) σ xy are, on the other hand, non-zero for frequencies in thethin-film gaps. The frequency dependence of (cid:60) σ xy and (cid:61) σ xy in the FM states differs from that of other magneticstates in that (cid:60) σ xy initially decreases with frequency and (cid:61) σ xy is negative below the band gap. This abnormal be-havior is caused by the negative Berry curvature aroundthe Γ point in the 2D-band structure.The optical conductivity tensor components can beconverted to frequency-dependent Faraday and Kerr ro-tation angles commonly measured in experiment. TheFaraday and Kerr rotation angles are the relative rota-tions of left-handed and right-handed circularly polar-ized light[45] for transmission and reflection respectively,and these can be connected to the optical conductivityby combining electromagnetic wave boundary conditionsand Maxwell equations. The Faraday and Kerr rota-tion angle vs. optical frequency for various metamagneticstates of 6-layer thin film are shown in Fig. 5(c) and (d)correspondingly, from which we see that the optical re-sponses of all metamagnetic states with the same Chernnumber are indistinguishable in the DC limit. As fre-quency increases, the Faraday and Kerr rotation anglesof different metamagnetic states differ substantially. Itfollows that different metamagnetic states can be distin-guished magneto-optically. DISCUSSION
In our studies, we have identified the metamagneticstates that can be induced in MBT thin films with upto N = 6 septuple layers by applying external magneticfields. For N = odd and larger than 3, all MBT thinfilms states are QAH insulators. For even– N , the groundstates are axion insulators in the absence of an externalmagnetic field. Metamagnetic states that are Chern in-sulators can be induced by applying external magneticfield provided that the single-ion magnetic anisotropy islarge enough compared with the interlayer exchange in-teractions. Both M2 and M4 states are Chern insulatorswith QAH gaps comparable to those of the FM state,and appear at a much smaller magnetic fields. Thesemetamagnetic states are not distinguished in transportexperiment since they all have the same Chern numberas the FM state. However, They are distinguished bytheir magneto-optical Kerr and Faraday rotation angles.Thicker films, especially for even-layer systems and forthe thin films with high Chern number FM states (that is N = 9 layers thin or thicker), await further exploration.Interesting questions arise when the thickness increases:is it possible, for example, to have a high-Chern-numberstate at a weaker magnetic field? For MBT thin films themagnetic anisotropy seems to be comparable with the in-terlayer exchange interaction, it may therefore be inter-esting to explore other intrinsic magnetic topological in-sulators that have relative larger magnetic anisotropy toinduce more metamagnetic states, or to find tools, such aselectric field, that may increase the magnetic anisotropy. ACKNOWLEDGEMENTS
RJM and OH were supported by the Center for Ad-vancement of Topological Semimetals, an Energy Fron-tier Research Center funded by the U.S. Department ofEnergy Office of Science, Office of Basic Energy Sciences,through the Ames Laboratory under Contract No. DE-AC02-07CH11358. CL and AHM were supported by theArmy Research Office under Grant Number W911NF-16-1-0472. OH gratefully acknowledges the computing re-sources provided on Bebop and Blues, high-performancecomputing clusters operated by the Laboratory Comput-ing Resource Center at Argonne National Laboratory.
APPENDIX [1] S. V. Eremeev, M. M. Otrokov, and E. V. Chulkov,“
Competing Rhombohedral and Monoclinic Crystal Struc-tures in
MnPn Ch Compounds: An Ab-Initio Study ,” J.Alloy Compd. , 172–178 (2017).[2] M M Otrokov, T V Menshchikova, M G Vergniory,I P Rusinov, A Yu Vyazovskaya, Yu M Koroteev,G Bihlmayer, A Ernst, P M Echenique, A Arnau, et al. , “
Highly-Ordered Wide Bandgap Materials for QuantizedAnomalous Hall and Magnetoelectric Effects ,” 2D Mater. , 025082 (2017).[3] Mikhail M Otrokov, Ilya I Klimovskikh, Hendrik Bent-mann, D Estyunin, Alexander Zeugner, Ziya S Aliev,S Gaß, AUB Wolter, AV Koroleva, Alexander M Shikin, et al. , “ Prediction and Observation of an Antiferromag-netic Topological Insulator ,” Nature , 416–422 (2019).[4] E. D. L. Rienks, S. Wimmer, J. S´anchez-Barriga,O. Caha, P. S. Mandal, J. R˚uˇziˇcka, A. Ney, H. Steiner,V. V. Volobuev, H. Groiss, et al. , “
Large Magnetic Gap atthe Dirac Point in Bi Te / MnBi Te Heterostructures ,”Nature , 423–428 (2019).[5] Y. J. Chen, L. X. Xu, J. H. Li, Y. W. Li, H. Y. Wang,C. F. Zhang, H. Li, Y. Wu, A. J. Liang, C. Chen, et al. , “
Topological Electronic Structure and Its Tempera-ture Evolution in Antiferromagnetic Topological Insulator
MnBi Te ,” Phys. Rev. X , 041040 (2019).[6] Yu-Jie Hao, Pengfei Liu, Yue Feng, Xiao-Ming Ma,Eike F. Schwier, Masashi Arita, Shiv Kumar, ChaoweiHu, Rui’e Lu, Meng Zeng, et al. , “ Gapless SurfaceDirac Cone in Antiferromagnetic Topological Insulator Bi Te ,” Phys. Rev. X , 041038 (2019).[7] Hang Li, Shun-Ye Gao, Shao-Feng Duan, Yuan-Feng Xu,Ke-Jia Zhu, Shang-Jie Tian, Jia-Cheng Gao, Wen-HuiFan, Zhi-Cheng Rao, Jie-Rui Huang, et al. , “ Dirac Sur-face States in Intrinsic Magnetic Topological Insulators
EuSn As and MnBi n Te n +1 ,” Phys. Rev. X , 041039(2019).[8] Przemyslaw Swatek, Yun Wu, Lin-Lin Wang, KyungchanLee, Benjamin Schrunk, Jiaqiang Yan, and AdamKaminski, “ Gapless Dirac surface states in the antifer-romagnetic topological insulator
MnBi Te ,” Phys. Rev.B , 161109 (2020).[9] Chao Lei, Shu Chen, and Allan H MacDonald, “ Mag-netized Topological Insulator Multilayers ,” P. Natl. Acad.Sci. USA , 27224–27230 (2020).[10] Seng Huat Lee, Yanglin Zhu, Yu Wang, Leixin Miao,Timothy Pillsbury, Hemian Yi, Susan Kempinger, JinHu, Colin A Heikes, P Quarterman, et al. , “
Spin Scat-tering and Noncollinear Spin Structure-Induced IntrinsicAnomalous Hall Effect in Antiferromagnetic TopologicalInsulator
MnBi Te ,” Phys. Rev. Res. , 012011 (2019).[11] Yujun Deng, Yijun Yu, Meng Zhu Shi, Zhongxun Guo,Zihan Xu, Jing Wang, Xian Hui Chen, and YuanboZhang, “ Quantum Anomalous Hall Effect in IntrinsicMagnetic Topological Insulator
MnBi Te ,” Science ,895–900 (2020).[12] Yan Gong, Jingwen Guo, Jiaheng Li, Kejing Zhu, Meng-han Liao, Xiaozhi Liu, Qinghua Zhang, Lin Gu, LinTang, Xiao Feng, et al. , “ Experimental Realization of anIntrinsic Magnetic Topological insulator ,” Chinese Phys.Lett. , 076801 (2019).[13] Dongqin Zhang, Minji Shi, Tongshuai Zhu, Dingyu Xing,Haijun Zhang, and Jing Wang, “ Topological Axion Statesin the Magnetic Insulator
MnBi Te with the QuantizedMagnetoelectric Effect ,” Phys. Rev. Lett. , 206401(2019).[14] Ilya I Klimovskikh, Mikhail M Otrokov, Dmitry Es-tyunin, Sergey V Eremeev, Sergey O Filnov, Alexan-dra Koroleva, Eugene Shevchenko, Vladimir Voroshnin,Artem G Rybkin, Igor P Rusinov, et al. , “ Tunable 3D/2DMagnetism in the (
MnBi Te )( Bi Te ) m Topological In-sulators Family ,” npj Quantum Mater. , 1–9 (2020). [15] Jiaheng Li, Yang Li, Shiqiao Du, Zun Wang, Bing-LinGu, Shou-Cheng Zhang, Ke He, Wenhui Duan, and YongXu, “ Intrinsic Magnetic Topological Insulators in van derWaals Layered
MnBi Te -Family Materials ,” Sci. Adv. ,eaaw5685 (2019).[16] Sugata Chowdhury, Kevin F. Garrity, and FrancescaTavazza, “ Prediction of Weyl semimetal and Antiferro-magnetic Topological Insulator Phases in Bi MnSe ,” npjComput. Mater. , 33 (2019).[17] Bing Li, J-Q Yan, Daniel M Pajerowski, Elijah Gordon,A-M Nedi´c, Y Sizyuk, Liqin Ke, Peter P Orth, DavidVaknin, and Robert J McQueeney, “ Competing MagneticInteractions in the Antiferromagnetic Topological nsula-tor
MnBi Te ,” Phys. Rev. Lett. , 167204 (2020).[18] J-Q Yan, Qiang Zhang, Thomas Heitmann, ZengleHuang, KY Chen, J-G Cheng, Weida Wu, David Vaknin,Brian C Sales, and Robert John McQueeney, “ Crys-tal Growth and Magnetic Structure of
MnBi Te ,” Phys.Rev. Materials , 064202 (2019).[19] Alexander Zeugner, Frederik Nietschke, Anja U. B.Wolter, Sebastian Gaß, Raphael C. Vidal, ThiagoR. F. Peixoto, Darius Pohl, Christine Damm, AxelLubk, Richard Hentrich, et al. , “ Chemical Aspects ofthe Candidate Antiferromagnetic Topological Insulator
MnBi Te ,” Chem. Mater. , 2795–2806 (2019).[20] Jiazhen Wu, Fucai Liu, Masato Sasase, Koichiro Ienaga,Yukiko Obata, Ryu Yukawa, Koji Horiba, Hiroshi Kumi-gashira, Satoshi Okuma, Takeshi Inoshita, et al. , “ Natu-ral van der Waals Heterostructural Single Crystals withboth Magnetic and Topological Properties ,” Sci. Adv. ,eaax9989 (2019).[21] Shuai Zhang, Rui Wang, Xuepeng Wang, Boyuan Wei,Bo Chen, Huaiqiang Wang, Gang Shi, Feng Wang, BinJia, Yiping Ouyang, et al. , “ Experimental Observationof the Gate-Controlled Reversal of the Anomalous HallEffect in the Intrinsic Magnetic Topological Insulator
MnBi Te Device ,” Nano Lett. , 709–714 (2019).[22] M. M. Otrokov, I. P. Rusinov, M. Blanco-Rey, M. Hoff-mann, A. Yu. Vyazovskaya, S. V. Eremeev, A. Ernst,P. M. Echenique, A. Arnau, and E. V. Chulkov, “ UniqueThickness-Dependent Properties of the van der Waals In-terlayer Antiferromagnet
MnBi Te Films ,” Phys. Rev.Lett. , 107202 (2019).[23] Chang Liu, Yongchao Wang, Hao Li, Yang Wu, YaoxinLi, Jiaheng Li, Ke He, Yong Xu, Jinsong Zhang, andYayu Wang, “
Robust Axion Insulator and Chern Insu-lator Phases in a Two-Dimensional AntiferromagneticTopological Insulator ,” Nat. Mater. , 522–527 (2020).[24] K. Y. Chen, B. S. Wang, J.-Q. Yan, D. S. Parker, J.-S.Zhou, Y. Uwatoko, and J.-G. Cheng, “ Suppression ofthe Antiferromagnetic Metallic State in the Pressurized
MnBi Te single Crystal ,” Phys. Rev. Mater. , 094201(2019).[25] Lei Ding, Chaowei Hu, Feng Ye, Erxi Feng, Ni Ni, andHuibo Cao, “ Crystal and Magnetic Structures of Mag-netic Topological Insulators
MnBi Te and MnBi Te ,”Phys. Rev. B , 020412 (2020).[26] Raphael C. Vidal, Alexander Zeugner, Jorge I. Fa-cio, Rajyavardhan Ray, M. Hossein Haghighi, AnjaU. B. Wolter, Laura T. Corredor Bohorquez, FedericoCaglieris, Simon Moser, Tim Figgemeier, et al. , “ Topo-logical Electronic Structure and Intrinsic Magnetizationin
MnBi Te : A Bi Te Derivative with a Periodic MnSublattice ,” Phys. Rev. X , 041065 (2019). [27] R. C. Vidal, H. Bentmann, T. R. F. Peixoto, A. Zeugner,S. Moser, C.-H. Min, S. Schatz, K. Kißner, M. ¨Unzel-mann, C. I. Fornari, et al. , “ Surface States andRashba-Type Spin Polarization in Antiferromagnetic
MnBi Te (0001) ,” Phys. Rev. B , 121104 (2019).[28] Jun Ge, Yanzhao Liu, Jiaheng Li, Hao Li, TianchuangLuo, Yang Wu, Yong Xu, and Jian Wang, “ High-Chern-Number and High-Temperature Quantum Hall Ef-fect Without Landau Levels ,” Natl. Sci. Rev. , 1280–1287(2020).[29] Andrew M. Essin, Joel E. Moore, and David Vander-bilt, “ Magnetoelectric Polarizability and Axion Electrody-namics in Crystalline Insulators ,” Phys. Rev. Lett. ,146805 (2009).[30] Roger S. K. Mong, Andrew M. Essin, and Joel E. Moore,“
Antiferromagnetic topological insulators ,” Phys. Rev. B , 245209 (2010).[31] Haiming Deng, Zhiyi Chen, Agnieszka Wo(cid:32)lo´s, MarcinKonczykowski, Kamil Sobczak, Joanna Sitnicka, Irina VFedorchenko, Jolanta Borysiuk, Tristan Heider, (cid:32)LukaszPluci´nski, et al. , “ High-Temperature Quantum Anoma-lous Hall Regime in a
MnBi Te / Bi Te Superlattice ,”Nat. Phys. , 36–42 (2020).[32] Dmitry Ovchinnikov, Xiong Huang, Zhong Lin, ZaiyaoFei, Jiaqi Cai, Tiancheng Song, Minhao He, Qianni Jiang,Chong Wang, Hao Li, et al. , “ Intertwined Topologicaland Magnetic Orders in Atomically Thin Chern Insulator
MnBi Te ,” arXiv preprint arXiv:2011.00555 (2020).[33] Shiqi Yang, Xiaolong Xu, Yaozheng Zhu, Ruirui Niu,Chunqiang Xu, Yuxuan Peng, Xing Cheng, XionghuiJia, Yuan Huang, Xiaofeng Xu, et al. , “ Odd-Even Layer-Number Effect and Layer-Dependent Magnetic Phase Di-agrams in
MnBi Te ,” Phys. Rev. X , 011003 (2020).[34] Olav Hellwig, Taryl L Kirk, Jeffrey B Kortright, AndreasBerger, and Eric E Fullerton, “ A New Phase Diagram forLayered Antiferromagnetic Films ,” Nat. Mater. , 112–116 (2003).[35] Ulrich K R¨oßler and Alexei N Bogdanov, “ Reorientationin Antiferromagnetic Multilayers: Spin-Flop Transitionand Surface Effects ,” Phys. Status Solidi (C) , 3297–3305 (2004).[36] Bj¨orn Skubic, Johan Hellsvik, Lars Nordstr¨om, and OlleEriksson, “ A Method for Atomistic Spin Dynamics Sim-ulations: Implementation and Examples ,” J. Phys.: Con-dens. Mat. , 315203 (2008).[37] Richard F L Evans, Weijia J Fan, PhanwadeeChureemart, Thomas A Ostler, Matthew O A Ellis, andRoy W Chantrell, “ Atomistic Spin Model Simulations ofMagnetic Nanomaterials ,” J. Phys.: Condens Mat. ,103202 (2014).[38] Paul M Sass, Jinwoong Kim, David Vanderbilt, JiaqiangYan, and Weida Wu, “ Robust A-Type Order and Spin-Flop Transition on the Surface of the AntiferromagneticTopological Insulator
MnBi Te ,” Phys. Rev. Lett. ,037201 (2020).[39] Aoyu Tan, Valentin Labracherie, Narayan Kunchur,Anja UB Wolter, Joaquin Cornejo, Joseph Dufouleur,Bernd B¨uchner, Anna Isaeva, and Romain Giraud,“ Metamagnetism of Weakly Coupled AntiferromagneticTopological Insulators ,” Phys. Rev. Lett. , 197201(2020).[40] Jiazhen Wu, Fucai Liu, Can Liu, Yong Wang, ChangcunLi, Yangfan Lu, Satoru Matsuishi, and Hideo Hosono, “ Toward 2D Magnets in the (MnBi Te )(Bi Te ) n BulkCrystal ,” Adv. Mater. , 2001815 (2020).[41] D L Mills, “ Surface Spin-Flop State in a Simple Antifer-romagnet ,” Phys. Rev. Lett. , 18 (1968).[42] Chao Lei and Allan H. MacDonald, “ Gate-Tunable Quan-tum Anomalous Hall Effects in
MnBi Te Thin Films ,”arXiv preprint arXiv:2101.07181 (2021).[43] Ryogo Kubo, “
Statistical-Mechanical Theory of Irre-versible Processes. I. General Theory and Simple Appli-cations to Magnetic and Conduction Problems ,” J.Phys. Soc. Jpn/ , 570–586 (1957).[44] D A Greenwood, “ The Boltzmann Equation in the The-ory of Electrical Conduction in Metals ,” Proc. Phys. Soc.(1958-1967) , 585 (1958).[45] Wang-Kong Tse and A. H. MacDonald, “ Magneto-opticalFaraday and Kerr Effects in Topological Insulator Filmsand in Other Layered Quantized Hall Systems ,” Phys.Rev. B84