Magnetized topological insulator multilayers

Chao Leia, Shu Chena,b , and Allan H. MacDonalda,1

aDepartment of Physics, The University of Texas at Austin, Austin, TX 78712; and bDepartment of Physics, Shanghai University, Shanghai 200444, China

Contributed by Allan H. MacDonald, September 18, 2020 (sent for review July 6, 2020; reviewed by Rob McQueeney and Jian Wang)

We discuss the magnetic and topological properties of bulk crys- with theoretical predictions (9, 12), neutron scattering mea- tals and quasi–two-dimensional (quasi-2D) thin films formed by surements (17, 26, 30) show that bulk MnBi2X4 (MBX) has stacking intrinsic magnetized topological insulator (for example, A-type antiferromagnetic order with Mn ions ordered ferromag- Mn (SbxBi1−x)2X4 with X = Se,Te) septuple layers and topolog- netically within each septuple layer and antiferromagnetically ical insulator quintuple layers in arbitrary order. Our analysis between adjacent septuple layers. Because the antiferromag- makes use of a simplified model that retains only Dirac cone netic interactions between septuple layers are weak, the Mn degrees of freedom on both surfaces of each septuple or quintu- layer moments in thin films can be aligned by magnetic fields ple layer. We demonstrate the model’s applicability and estimate ∼5 T. Thin films can be obtained either by epitaxial growth its parameters by comparing with ab initio density-functional the- or by mechanical exfoliation from bulk crystals (8, 30, 31, 34– ory (DFT) calculations. We then employ the coupled Dirac cone 39). Magnetic fields establish quantum Hall effects in thin model to provide an explanation for the dependence of thin-film films not by establishing Landau quantization, but by chang- properties, particularly the presence or absence of the quantum ing the magnetic configuration from antiferromagnetic to fer- anomalous Hall effect, on film thickness, magnetic configuration, romagnetic through overcoming the weak interlayer exchange and stacking arrangement, and to comment on the design of Weyl interactions (21, 23, 29). superlattices. In this paper, we develop a simple model that can be used to address the properties of thin films and bulk crys- Sb Bi X MnBi Te | magnetized topological insulator | topological superlattices | tals formed by stacking Mn ( x 1−x )2 4 septuple layers and 2 4 Sb , Bi X quantum anomalous Hall effect | Weyl semimetal ( x 1−x )2 3 quintuple layers in arbitrary order. The model contains only Dirac cone degrees of freedom on each surface of each septuple or quintuple layer. By comparing with ab ini- opological insulator (TI) thin films in which time-reversal tio density-functional theory (DFT) calculations we are able to Tsymmetry is broken by magnetic order (1) have long been establish that this highly simplified description is usually accurate recognized as a promising platform for the interplay between and also to fix values of the model’s material-dependent parame- transport and magnetic properties that powers spintronics. ters. The advantages of the simplified model are that it facilitates Indeed the quantum anomalous Hall (QAH) effect, a high point the descriptions of crystals with complex stacking sequences and of topological spintronics characterized by dissipationless trans- thin films containing many layers and more importantly that it port, was first observed (2, 3) in magnetic topological insulator allows trends in magnetic and topological properties across the (MTI) thin films with ferromagnetic order, and strong magneto- family of materials to be recognized and understood. Important electric response properties are expected in antiferromagnetic materials variations include changes in the Sb fraction x on the TI films (4–6). MTIs were first produced simply by doping pnictogen sites; substitutions of X = S, Se, Te on the differ- (Sbx Bi1−x )2X3 TI thin films with magnetic elements. How- ent chalcogen sites; and growth-controlled defect concentrations. ever, disorder, thought to be due mainly to inhomogeneity of the magnetic dopants (7), leads to complex magnetic order in Significance these systems. As a consequence the QAH effect appears only at extremely low temperatures, for example only at ∼30 mK in Topological insulators have Dirac cone surface states pro- Cr-doped Bi2Te3, even though the Curie temperature is ∼15 K tected by time-reversal symmetry (TRS). Weak magnetic order (3). For this reason the recent identification (8) of the Mn in topological insulator thin films can lead to the quan- (Sbx Bi1−x )2X4 family of layered van der Waals materials, which tum anomalous Hall effect (QAHE), which was first real- can be viewed as MTIs that have magnetic moments on an ized by magnetic doping of topological insulators to estab- ordered lattice, is a promising advance. lish a fragile ferromagnetic state. Recent work has estab- Important progresses have been made in understanding the lished MnBi2Te4 as a structurally ordered magnetic topolog- bulk and epitaxial thin film properties of this family of materi- ical insulator. We report on a theoretical study of superlat- als, both theoretically (8–13) and experimentally (14–31). The tices formed from MnBi2Te4 and Bi2Te3 building blocks that quantum anomalous Hall effect has now been observed in the employs a simplified model validated by ab initio density- presence of a relatively weak magnetic field ∼5 T for thick- functional theory. We use it to shed light on the depen- nesses ranging from 3 to 10 septuple layers (21, 23, 29), and large dence of the quantum anomalous Hall effect on film thick- (almost quantized) anomalous Hall effects have been observed ness, magnetic configuration, the stacking sequences of the in the absence of an external magnetic field in high-quality 5- van der Waals coupled building blocks, and bulk electronic septuple-layer MnBi2Te4 films (23), all at temperatures exceed- structure. ing 1 K. The ratio of the quantum anomalous Hall temperature to the magnetic-ordering temperature, ∼20 K, is much higher Author contributions: C.L. and A.H.M. designed research; C.L., S.C., and A.H.M. per- (29) than in doped MTI films. Some (10, 16, 18, 32) [but not all formed research; C.L., S.C., and A.H.M. analyzed data; and C.L. and A.H.M. wrote the (26–28, 33)] photoemission experiments have identified the large paper.y surface state gaps ∼100 meV that are generally expected (10, 15) Reviewers: R.M., Iowa State University; and J.W., Peking University.y theoretically in MTIs. The authors declare no competing interest.y Mn (Sbx Bi1−x )2X4 is a layered material composed of seven- Published under the PNAS license.y layer X-(B, Sb)-X-Mn-X-(Bi, Sb)-X units that are coupled by 1 To whom correspondence may be addressed. Email: [email protected] weak van der Waals interactions. These septuple layers may This article contains supporting information online at https://www.pnas.org/lookup/suppl/ be viewed as (Bi, Sb)2X3 quintuple layers in which the mid- doi:10.1073/pnas.2014004117/-/DCSupplemental.y dle X layer is replaced by an X-Mn-X trilayer. In agreement First published October 19, 2020.

27224–27230 | PNAS | November 3, 2020 | vol. 117 | no. 44 www.pnas.org/cgi/doi/10.1073/pnas.2014004117 Downloaded by guest on September 27, 2021 Downloaded by guest on September 27, 2021 h o n otmo ahlyraehbiie ihrmt ia cones the Dirac in present remote moments with local hybridized layers. the with magnetic are interactions layer exchange by each altered of and nonmagnetic bottom and N top and the layers septuple (Right magnetic layers. quintuple M of composed multilayers 1. Fig. using describe to difficult are approaches. initio that that ab disor- theories perturbations fields, for other magnetic point and external starting der, fields, a electric as gating used for be account also can model Our e tal. et Lei insula- Chern are numbers number magnetic layer that Chern sizes odd with gap tors large films and thin sufficiently antiferromagnetic grow that with and Chern that thickness, film numbers (2D) with decline Chern two-dimensional with semimetal, are Weyl insulators ferromagnets find ideal nearly We thin-film a date. is to that MBX attention ferromagnetic greatest bulk received that have which Te, Se, n omgei unul aesi h tc r ordinary are stack the in layers for quintuple insulators nonmagnetic ing hr pnlbl aebe etimplicit, left been have labels spin where therefore is Hamiltonian The hopping the the between denoting if parameter surfaces, or, spin-independent different interac- arbitrary film between and exchange moments hybridization for thin local allow Mn a We the mag- crystal. form with of tions bulk to stacking a form either arbitrary to for layers repeated, allow nonmagnetic We and 1. layer of netic bottom Fig. quintuple and in nonmagnetic top illustrated and the as on layer also septuple but only magnetic 41), insulator not each (40, freedom topological layers of surface magnetized degrees the cone a on Dirac for including by model multilayer a construct We Model Cone Dirac semimetal Coupled Weyl a achieve layers. to and chance ers large a with phase is there (1,1), larger for semimetals ihihe nFg .Tems gaps mass (∆ The layers 1. adjacent Fig. between in gap (∆ highlighted Waals hybridization layer der van important same the most the across The within cones. hopping Dirac parameters, the of velocity ones tops, odd and layer bottoms layer for for reserved integers even with labels nti ae efcson focus we paper this In ope ia oemdl (Left model. cone Dirac Coupled Bi H [MnBi 2 = MnBi Se k ∆ + X M 3 ~ ⊥ , 2 ,ij sterdcdPac osat and constant, Planck reduced the is /N X Bi 2 h ij 4 Te ia oemdli hc ia oe oaie on localized cones Dirac which in model cone Dirac ) ] 2 (1 M Sb mle hnaot3 u eoeWeyl become but 3, about than smaller 4 (−) [Bi − M 3 smgei oooia nuao lay- insulator topological magnetic as i or , hsraeadthe and surface th δ /N i 2 | ij ~v X C ) 3 i D o ueltie ih(,)= (M,N) with superlattices For . | ] c 1 = (ˆ Sb N Bi k † z ⊥ 2 ueltie omdb insert- by formed superlattices × 2 i efrhrfidta ferro- that find further We . X Te antzdtplgclinsulator topological Magnetized ) c σ k 3 ⊥ 3 ) m j and · , i stplgclinsulator topological as k ⊥ fteidvda Dirac individual the of i + MnBi and j m hsraeby surface th i j S σ 2 r ia cone Dirac are n hopping and ) z X  4 δ ij ihX= X with v D D sthe is ,are ), ∆ [1] ij . tr h oe rdcstosi-eeeaebnsaogthe along bands spin-degenerate param- two to three predicts these model only the Retaining eters layer. adjacent the adding of by surface improved be can fit The and the only retaining bands parameters DFT hybridization the to approximation reasonable a = U bulk of with (43) Nonmagnetic. approximation U per- sites. + Mn we on LDA eV which 5.25 the calculations, with using structure comparing by electronic formed tested DFT be initio can degrees systems cone ab Dirac realistic only to retains freedom that of model the of relevance The Calculations DFT to Comparison magnetic associ- between layers layers. be As be TI TI would can nonmagnetic gap. thickness inserting that variable Waals by of flexibility recovered der layers the spacer van nonmagnetic of with the some ated across below, Dirac is between discuss coupling layers we the different model this on in cones system, 42, three-dimensional ref. flexible in a imagined as form to topolog- layers magnetic between insulator layer Instead ical insulating (42). normal Balents insulator a and inserting topological Burkov of is by the configuration proposed to magnetic model reduced multilayer the model and our magnetic ferromagnetic, is or layer magnetic/magnetic that When at value its heterojunctions. differ nonmagnetic/nonmagnetic allows of and to values layers different heterojunctions MTI allows and model TI the in parameters; we properties model superlattice as MTI of only surveys retain our In magnetic. same are the splitting from exchange one layers, with Mn splitting exchange nearby with two layer have can cone Dirac aeo antzto epniua otevndrWaslayers α Waals der van that the so to perpendicular magnetization of case where moments local symmetry Mn time-reversal with break and interactions exchange from result cones and energies band valence and conduction DFT at the of average the at larger is property eter bands The low-energy DFT zone. the the Brillouin at between once larger gap bands the the bulk to that of unfolded sets are two excellent bands find the we lay- Nevertheless, between layer. three single agreement after a only after repeats of calculations the instead ers, DFT arrange- model, the simplified stacking in the the used by of lattice captured details not are of that Because ments 2. Fig. in illustrated the along tum with ∆ the and E S h oe ipie hnol n yeo ae spresent. is layer of type one only when simplifies model The fi smgei,and magnetic, is it if Z Γ Γ (k 4 meV, 143 = ±|∆ ∆ ∆ iei h rloi oewt dispersions with zone Brillouin the in line mle that implies and d z D D 1 = ) α h itnebtenlyrcnesand centers layer between distance the Bi M Z o opn oteajcn ufc fteajcn layer. adjacent the of surface adjacent the to hopping for S saddt con o h ml ifrnebetween difference small the for account to added is Z salyrlbl elmtoratninhr othe to here attention our limit We label. layer a is 2 α on (k point − ± Se h oe adeege tthe at energies band model The . ∆ = q ∆ 3 S safis eto h oe,w xmn h bands the examine we model, the of test first a As ±1 D ∆ ∆ , setmtdb F.W n htw a obtain can we that find We DFT. by estimated as Γ PNAS − | S 2 ∆ to ∆ D pcfistesneo antzto nlayer on magnetization of sense the specifies x ∆ + d D , 2∆ S Z J | = = and S D 2 D 1 ie h orsodn F ad are bands DFT corresponding The line. oebr3 2020 3, November π m and , J itn oteefu nrisw obtain we energies four these to Fitting . 8 e,and meV, −280 2∆ + r,respectively, are, ) M i D ∆ ∆ = hw si i.1 fteelayers these if 1, Fig. in as shown , α S D J X 0 = J o opn ihntesm layer same the within hopping for α S S D aeopst in.Teparam- The signs. opposite have ∆ n n rmteajcn layer adjacent the from one and hc r omlydominant, normally are which , J D nnnantclyr.Each layers. nonmagnetic on ∆ i α ∆ cos D M D 1 α tmagnetic/nonmagnetic at o opn otesame the to hopping for , | k , z o.117 vol. ∆ d 2∆ + D 1 ±|∆ = Γ S k on (k point | 1mV The meV. −11 D 1 z ∆ + o 44 no. cos h momen- the D k z | z d Z 2∆ + Bi | d , 27225 2 than 0 = Se ∆ [2] [3] D 1 Γ S 3 )

PHYSICS ABlayers are extremely weak, the ferromagnetic configuration can be realized by applying a relatively modest external magnetic field. For the ferromagnetic configuration the exchange energies are the same in every layer and the model band energy dispersion along the Γ to Z line is q 2 2 E(kz ) = ± ∆S + ∆D + 2∆S ∆D cos kz d ± mF , [4]

where the ferromagnetic exchange splitting is mF = JS + JD and is the sum of the on-layer (JS ) and neighboring-layer (JD ) exchange interactions. The energies at kz d = 0 and kz d = π are

EΓ = ±(∆S + ∆D ) ± mF [5] EZ = ±(∆S − ∆D ) ± mF .

Because mF depends only on JS + JD , we must consider other magnetic configurations to fit JS and JD independently. For the Fig. 2. Band structure of Bi Se along the Γ to Z line from DFT calculations 2 3 antiferromagnetic configuration the exchange splitting mAF = and the Dirac cone model. (A) The blue line is the result from DFT calcula- JS − JD alternates in sign from layer to layer and the gap tions. Since the bulk Bi2Se3 layer stacking arrangement has three quintuple layers per unit cell, the DFT bands must be unfolded to the larger Brillouin between conduction and valence bands at the Γ point is zone. Fitting the bulk DFT band dispersion from Γ to Z yields ∆S = 143 meV, q ∆ = −280 meV, and ∆1 = −11 meV. The sign of ∆ has been fixed by AF 2 2 D D S Egap = 2( ∆D + mAF − ∆S ). [6] noting that the single-layer conduction band state at Γ has greater weight in the middle of the layer than the valence band state. The corresponding 1 By comparing these expressions with the DFT band dispersions Bi2Te3 fit yields ∆S = 74 meV, ∆D = −200 meV, and ∆D = −10 meV. (B) Com- parison between DFT and Dirac cone models for thin-film gaps at the 2D Γ of bulk MnBi2X4 (X = Se, Te), we extract the model parameters point vs. layer number. In the displayed scale, the gaps from DFT calculations summarized in Table 1. are out of the displayed range; it is around 847 meV for Bi2Se3 and around The Γ to Z band dispersions calculated with these model 488 meV for Bi2Te3. The Dirac cone models were obtained by fitting to the parameters for the ferromagnetic configuration are illustrated bulk band dispersion. In the schematic illustration of the Dirac cone bonding in Fig. 3A. Because the Γ-point gaps, around 30 meV for MBS network, t/b label the top/bottom surfaces of individual layers represented and around 44 meV for MBT, are inverted by the exchange by magenta/green ellipses and the purple and gray links represent hopping splitting mF , the spin splitting closes at a finite value of kz , as within and between layers. noted previously (12, 13), generating a simple Weyl semimetal (44) with only two Weyl points. The model bands are in good agreement overall with the DFT results, although the small Γ- corresponding fitted parameters for Bi2Te3 are ∆S = 74 meV, point gaps are even smaller in our DFT calculations, which yield 1 ∆D = −200 meV, and ∆D = −10 meV. The property that |∆D | 23 meV for MBS and 12 meV for MBT. The role of longer-range n is larger than |∆S |, which is responsible for band inversion at Γ hopping parameters ∆D , which are responsible for a velocity and hence for the nontrivial band topology, is not surprising since magnitude difference between the crossing bands at the Weyl the former hopping parameter is across a narrow van der Waals point, is addressed in SI Appendix, Fig. S2 by DFT calculations gap whereas the latter is across a wider quintuple layer. As seen but dropped in the qualitative phase-diagram discussions below in Fig. 2A the unfolded DFT bands are in excellent agreement since they are approximately several millielectronvolts and have with this simple model. little influence on the positions of Weyl points (kw ), thin-film Using these parameters estimated from the bulk bands at Γ anomalous Hall effects (AHEs), or 2D-Γ-point thin-film gap (SI and Z , we calculated the gap at the 2D Γ point for Bi2Se3 and Appendix, Fig. S3) trends. The bulk antiferromagnetic gaps calcu- Bi2Te3 thin films with thicknesses ranging from one to six quintu- lated from this model are around 85 meV for MBS and 86 meV ple layers. As illustrated in Fig. 2B, the gaps of thin films from the for MBT, compared to gaps estimated experimentally that vary simplified model are in good agreement with DFT results (shown from 50 to 200 meV (10, 16, 30, 45). with red and blue dots), except in the single-quintuple layer case In Fig. 3B we plot 2D Chern numbers obtained by integrating for Bi2Se3 and in the single- and double-quintuple layer cases for the Berry curvature over kx and ky as a function of kz , which Bi2Te3. For the latter material the Dirac point lies in the valence are nonzero between the Weyl points at kz = ±kw . The Weyl bands instead of in the bandgap. This good agreement supports a physical picture in which the bulk gap is due to the hybridization of the Dirac cones network. The thin-film gap is small because Table 1. Model parameters for MnBi2X4 (X = Se, Te) in units of the top (bottom) Dirac cone of the top (bottom) layer has no millielectronvolts hybridization partner from adjacent layers (Fig. 2 B, Inset). The Magnetic MBS MBS MBT MBT thin-film gap is due to hopping between surface layer Dirac cones configuration (model) (DFT/exp.) (model) (DFT/exp.) via hybridized, and therefore gapped, Dirac cones in the interior. 1 ∆S 190 — 84 — Because ∆D is small compared to ∆D , we include only ∆S and ∆D in the following analysis. The model’s Dirac velocity parameter ∆D −232 — −127 — is estimated from the dependence of the DFT bands on in-plane JS 32 — 36 — momenta (SI Appendix, Fig. S1 and Table S1). JD 25 — 29 — Γ Egap (AF) 85 '100 86 50–200 Γ Magnetic. We now turn to the magnetic case, starting with bulk Egap (FM) 30 '23 44 12 MnBi2Se4 (MBS) and MnBi2Te4 (MBT). Below we refer to the For the column labeled DFT/exp. we use the experimentally available magnetic configuration in which all magnetic layers are aligned value of the antiferromagnetic state gap and DFT calculation results for as ferromagnetic. The ground state is antiferromagnetic, but quantities that were not available from experiment. AF, antiferromagnetic because magnetic interactions between different van der Waals states; FM, ferromagnetic states.

27226 | www.pnas.org/cgi/doi/10.1073/pnas.2014004117 Lei et al. Downloaded by guest on September 27, 2021 Downloaded by guest on September 27, 2021 e tal. et Lei npriua,sonta o B lstettlCennumber Chern total the have, films which MBT 29) for 23, that shown (21, particular, experiments Appendix , in recent model (SI cone with Dirac limit consistent the thick-film of are predictions the semianalytic in These layers S5). Fig. of exceeded. even is numbers for septuple identical MBS odd two are for and gaps to layers configuration septuple equal antiferromagnetic eight thickness The and film MBT that critical for in a and, layers layers numbers when septuple Chern only of numbers the case, odd Here for only distinct. nonzero quite are is configurations bands, netic semimetal Weyl simply bulk in size the detailed gap as of oscillating quantization illustrated size The the numbers. as reflects Chern decrease, integer to odd for tend 3 gaps num- Fig. Chern energy the in the As S4). when increases Fig. value Appendix, ber bulk (SI the layer approaches per films normalized thick very of conductivity sfu o B n w o B.A h hcns increases for thickness one the by As increasing MBT. indefinitely, for every increase two numbers and Chern MBS the which for layers, septuple four of number is number critical ferro- the Chern For to zero up film. have thicknesses the for films in thin layers a the septuple configurations, as of magnetic number configurations the antiferromagnetic of function and ferromagnetic with is MBX ferromagnetic value quantum bulk the in to compared layer small septuple per conductivity Γ ic h xhneinteractions | exchange the Since |∆ 2D-Γ the at MBX thin-film of Gap (D) for MBT. momentum numbers for Chern layers nonzero two have and values configurations MBS no-zero antiferromagnetic to for The jump layers MBT. layers. MBX eight septuple ferromagnetic for than of of layers number larger numbers septuple the Chern is of two the thickness function and that the show MBS when which for films model thickness cone odd-septuple-layer layers Dirac the septuple from four calculated beyond MBX thin-film of numbers Chern model cone Dirac (B) MnBi Ferromagnetic 3. Fig. ∆ nFg 3C Fig. In to D S h ukHl odciiyprlyri (46) is layer per conductivity Hall bulk The |. | − | Z ∼ ukbn iprino ermgei B acltdfo h ia oemdlwt h F-xrce aaeessmaie nTbe1. Table in summarized parameters DFT-extracted the with model cone Dirac the from calculated MBX ferromagnetic of dispersion band Bulk (A) d π/ ihincreasing with ∆ hr ehv sindangtv int h gap the to sign negative a assigned have we where D, S u ut ml oprdto compared small quite but |, k /k w h eairo antiferromag- of behavior The S5 . Fig. Appendix, SI w epo h oa hr ubr fti films thin of numbers Chern total the plot we snneofor nonzero is de etpelyr ota h unie Hall quantized the that so layers septuple added 2 X k z 4 dpnet2 hr ubr o ukMXcluae rmtemdl hwn up rm1t tteWy ons (C points. Weyl the at 0 to 1 from jumps showing model, the from calculated MBX bulk for numbers Chern 2D -dependent X=S,e saWy eiea,wt h elpitlctdat located point Weyl the with semimetal, Weyl a is Se,Te) = (X BD AC m F reaching , m F > m |∆ F D e nMXaelre than larger are MBX in d π/ 2 | − | /h |∆ . ∆ when S S | | + n oe from moves and k |∆ w m d /(π/ D F h Hall the |, = |∆ )e D 2 /h | + . ufc ttswti h omgei oooia insulator topological nonmagnetic between the hybridization within stronger that states expect therefore surface We 2. around to of decrease a Decreasing set als. we when example, that For find layers. the of mag- order of layers ratio the nonmagnetic by to determined mainly netic is gap energy the that ferromag- find (MnBi of phases configuration topological and netic gaps energy the summarize (MnBi 6) (M,N 37–39), zdeprmnal,icuig(M,N real- including been experimentally, already have ized superlattices these of Several repeated. is ocnett ermgei ofiuaini eue from reduced with needed is field configuration ferromagnetic magnetic a the has ∼5 to it that convert example, experimentally to for and alternated, shown magnetic spac- simply been When are quintuple layers. layers nonmagnetic septuple nonmagnetic magnetic inserting the by between ers simply reduced between 37– can be convert arrangements 35, stacking to for ferromagnetic (30, needed and field layers antiferromagnetic allows mag- magnetic quintuple of the materials Importantly, sequence nonmagnetic 39). MBX stacking and of the septuple changing netic character by Waals variation property der van The Diagrams Phase Superlattice septuple (29) nine reaches thickness the when layers. 2 to 1 from jumps ecnie h aiyo ukcytl nwihatemplate a which in crystals bulk of family the consider We to M ∼0. k MnBi z M 8 = 22 Te ±k /N (30). T 13 (MnBi (1,2) = ) 2 w X MnBi , where , agrta rud3last ukWy semimet- Weyl bulk to leads 3 around than larger PNAS ∆ 4 S MTI etpelyr and layers septuple | 10 /∆ k oebr3 2020 3, November w ±10% Te ≈ S TI 16 50d 3π/ 2 aosWy eiea hsswith phases semimetal Weyl favors M n MnBi and , X 6 4 Te euigtecritical the reducing /N ) M 10 o B and MBS for /(Bi 3,3) n (M,N and 39), (37, ) n sls eedn nthe on dependent less is and (MnBi (1,1) = ) N 2 | X 12 Bi o.117 vol. 3 Te ) N 2 19 X k w ueltie.We superlattices. 3) nFg we 4 Fig. In (39). ) 3 ∆ ≈ unul layers quintuple | S MTI 20d 3π/ o 44 no. M ∆ = 4 Te /N (1,3– = ) on sa as point o MBT. for h 2D The ) | 7 S TI (30, ) ratio 27227 we ,

PHYSICS For the limit ∆D = 0, which corresponds to isolated layers, a phase transition from a trivial insulator to a Chern insulator state occurs when the exchange interaction exceeds ∆S ; the Weyl states emerge at smaller JS for interior magnetic layers because both exchange interactions JS and JD contribute. For exam- ple, for M = 1, N = 0, illustrated in Fig. 5A, the band energy at the Γ point is EΓ = ±(JS + JD ) ± ∆S , and the phase transi- tion to the Chern insulator occurs when JS + JD > ∆S , that is, at JS = ∆S /(1 + δ) with δ ≡ JD /JS . For the case of M = 1, N = 1 shown in Fig. 5B, the magnetic layers are isolated so that EΓ = ±JS ± ∆S and ±JD ± ∆S , and there are therefore two phase transition points at ∆D = 0, one at JS = ∆S and one at JD = ∆S . When the latter phase transition occurs, the total Chern number per period changes from C = 1 to C = 2. At finite ∆D quan- tum Hall states with different Chern numbers per period are separated by Weyl semimetal states. Fig. 5 C and D shows the cases of M = 1, N = 2 and M = 2, N = 1 for which the eigen- values in the ∆D to 0 limit are discussed in SI Appendix and Fig. 4. Topological phase diagram of ferromagnetic-configuration imply critical values of JS = ∆S for the M = 1, N = 2 case and (MnBi2X4)M (Bi2X3)N superlattices vs. M/N. This diagram was constructed √ by using the Dirac cone model to calculate energy bands and 2D Chern JS = ∆S / 1 + δ, ∆S /δ for the M = 2, N = 1 case. Fig. 5D shows numbers as a function of kz for all indicated superlattices. Weyl semimetal that ferromagnetic Mn2Bi6Te11 has a large chance to be a Weyl superlattices are placed in the orange region of the phase diagram and semimetal. insulators in the green region. MBS energy gaps (in millielectronvolts) at the Γ point are listed at the top of the corresponding superlattice Discussion icons for insulating superlattices and the bottom for the Weyl semimetal The coupled Dirac cone model developed here provides an superlattices in which the gap has been inverted. The solid red (purple) MTI TI excellent qualitative description of MnSbx Bi2−x X4 (MPX) lay- line plots Γ-point gaps for MBS (MBT) calculated using ∆S = ∆S , while TI ered van der Waals materials in both thin-film and bulk crystal the dashed red lines illustrate the cases in which ∆S deviates from MTI limits. The simplified Dirac cone model has the advantage that ∆S by ±10%. it can readily be used as a platform to address the influence of gating, disorder, and external magnetic fields, all relevant perturbations that are not easily accounted for using ab initio layers favors Weyl semimetals. Among (M,N) = (1,1) super- approaches. It can also help explain trends across the mate- MnBi Te Bi Se lattices, candidates include 2 4 combined with 2 3, rial family and simplify theories of systems with less trans- Bi Sb Sb Te 2 3, or 2 3, which are estimated to have stronger lational symmetry (films that are many layers thick or bulk same-layer hybridization than Bi2Te3 according to the DFT calculations summarized in SI Appendix, Fig. S8. The properties of (MnBi2X4)M (Bi2X3)N superlattices and thin films can be varied in a variety of different ways, for example AB by varying the pnictide fraction, applying pressure, or changing temperature. We study the efficacy of these tuning knobs by con- structing phase diagrams vs. JS and ∆D , as illustrated in Fig. 5, for several different stacking sequences keeping the ratio of JS to JD and ∆S fixed. To limit the number of parameters we used the same values for ∆S and ∆D in magnetic and nonmagnetic lay- ers. The motivation for illustrating the dependence of phase on this particular subset of model parameters is that 1) we expect both exchange interactions to decline with temperature as the alignment of the local moment spins decreases with increasing temperature and 2) we expect the van der Waals gap to narrow CD as pressure is applied, increasing ∆D . The dependence of ∆D on pressure is addressed explicitly in SI Appendix, Fig. S7. For M = 1 and N = 0 (that is, for bulk MnBi2X4), the phase diagram shown in Fig. 5A maps to that of the MTI/normal insulator superlattice model studied by Burkov and Balents (42). The cases of M/N = 0.5, 1, and 2 are shown in Fig. 5 B–D, where the positions of MBS and MBT systems in the phase diagram are marked by red and blue dots. Although antiferromagnetic MnSb2Te4 is a trivial insulator, unlike MBT, according to DFT calculations (47) and in agreement with our results, ferromagnetic MnSb2Te4 is a Weyl semimetal according to DFT calculations shown in SI Appendix, Fig. S2. The estimated model parameters are mF = JS + JD ≈ Fig. 5. Phase diagram of ferromagnetic (MnBi2X4)M/(Bi2X3)N superlattices with exchange interactions J (J ) and Dirac cone coupling energies ∆ 45 meV, ∆S ≈ 124 meV, and ∆D ≈ −166 meV for MnSb2Te4 S D D and thus it should lie between that of MBT and MBS in the phase in adjacent layers and ∆S in the same layer, where A–D are for (M,N) = (1,0)(MnBi X )(A), (M,N) = (1,1)(MnBi X )(B), (M,N) = (1,2)(MnBi X )(C), diagram of Fig. 5. 2 4 4 7 6 10 and (M,N) = (2,1)(Mn2Bi6X11)(D). Increasing temperature moves points In all cases the transition between normal insulator and Weyl down in this phase diagram whereas applying pressure moves points to the semimetal states occurs at weakest exchange coupling near ∆D = right. The zero-temperature MBS and MBT parameters are marked by red ∆S , which marks the boundary between normal insulator and and blue dots. D shows that ferromagnetic Mn2Bi6Te11 has large chance to topological insulator states in the nonmagnetic JS/D = 0 limit. be a Weyl semimetal.

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B Rev. insulators. superlattice. insulator. topological magnetic a ermgei oooia nuao hssi Bi2MnSe4. in phases (2019). insulator topological ferromagnetic (2017). study. ab-initio An inMnPn2Ch4compounds: structures effects. magnetoelectric and Hall anomalous (2015). 1316–1321 Crx(Bi0.1Sb0.9)2-xTe3. insulator topological magnetic Science (2019). 126–143 rdecrease or 2 nrni antctplgclisltr nvndrWaslayered Waals der van in insulators topological magnetic Intrinsic al., et mgn ia-asdsre rmmgei oataosi h ferro- the in atoms dopant magnetic from disorder Dirac-mass Imaging al., et Te unie nmlu alefc nmgei oooia insulators. topological magnetic in effect Hall anomalous Quantized al., et 429(2010). 245209 81, 16 (2010). 61–64 329, 4 rsa tutr,poete n aotutrn fanwlayered new a of nanostructuring and properties structure, Crystal al., et oooia xo ttsi h antcisltrMnBi insulator magnetic the in states axion Topological al., et fml materials. -family Nature a.Phys. Nat. xeietlosraino h unu nmlu alefc in effect Hall anomalous quantum the of observation Experimental al., et hs e.B Rev. Phys. rdcinadosraino natfroantctopological antiferromagnetic an of observation and Prediction al., et ihyodrdwd ada aeil o quantized for materials bandgap wide Highly-ordered al., et 1–2 (2019). 416–422 576, |∆ 8–8 (2010). 284–288 6, D | 415(2016). 045115 93, ota hyaenal qa nmagnitude, in equal nearly are they that so c.Adv. Sci. Science hs e.Lett. Rev. Phys. aw65(2019). eaaw5685 5, CrystEngComm 6–7 (2013). 167–170 340, DMater. 2D 041(2019). 206401 122, rc al cd c.U.S.A. Sci. Acad. Natl. Proc. .Aly Compd. Alloys J. 202(2017). 025082 4, 5253 (2013). 5532–5538 15, p opt Mater. Comput. Npj ) h field The T). a.Rv Phys. Rev. Nat. 2 172–178 709, ) tfol- It T). Te 4 ihthe with Phys. 112, 1, 5 8 .J Chen J. Y. 28. Li H. 26. Zhang S. 25. Gong Y. 24. Deng Y. 23. Chen Y. K. 22. Liu C. 21. Otrokov M. M. 20. Li B. 19. Lee H. S. 18. Yan Q. J. 17. Zeugner A. 16. Rienks L. D. E. 15. 7 .J Hao J. Y. 27. iecntnswr bandb eaigteui elutltefre for forces the until cell unit 1 than the smaller relaxing were by atom each obtained were constants tice 9 and calculations bulk for ec -43 eakoldegnru optrtm loain from allocations Center. time Computing computer Advanced under generous Texas Foundation the acknowledge Welch We the F-1473. by and Welch W911NF-16-1-0472 Grant under Office ACKNOWLEDGMENTS. Availability. Data eV, to set 600 was be 3d loop to self-consistency its set electronic for 10 the was included for basis was condition plane-wave break eV global the 5.25 the for = energy U cutoff using with The U) electron– (50), electrons. + on-site Hubbard (LDA (VASP) a (43) present, interaction package were approximation electron atoms simulation Mn gradient When initio (51). generalized (PBE-GGA) ab Perdew–Burke–Ernzerhof Vienna semilocal the in mented Calculations. DFT Methods and Materials conductivity. Hall by in required the change are quantized states changes the protected topologically antiferromagnet case to MTI either In zero a zero. from in to either illustrated layer conductivity As a Hall adding changes. layers S6, of system Fig. number the the antiferro- where in the edges present step on at that expected surface state states is the are magnetic to edge which explanation sensitive protected more 32), possible are topologically 18, experiments One ARPES 16, sensitive. lower-energy (10, larger the surface experiments (usually less ARPES photon-energy likely eV) higher Dirac 20 in Gapped model. than observed cone Dirac were coupled the cones and DFT films both antiferromagnetic by thick for predicted gaps surface-state the generally near energies 33), photon small (26–28, using those experiments (ARPES) photoemis- angle-resolved spectroscopy Some sion present. at confusing experiments, are photoemission which of interpretation the for model cone double fields include that weaker spacers. arrangements even nonmagnetic stacking in mag- chosen complex reachable suitably more be with Still field. should net the configurations the with netic aligned with unit each antiferromagnetically, of arranged spin are moments the −7 ti neetn ocnie h mlctoso h Dirac the of implications the consider to interesting is It niermgei oooia nuao MnBi insulator topological antiferromagnetic MnBi insulator Lett. topological Nano magnetic intrinsic the in effect Hall anomalous Lett. Phys. Chin. MnBi insulator MnBi pressurized insulator. topological antiferromagnetic MnBi (2019). antiferromagnet interlayer Waals MnBi insulator MnBi insulator topological antiferromagnetic Res. in Rev. effect Hall anomalous (2019). 064202 3, MnBi insulator heterostructures. MnBi mnbi and V n h ehsz o raigtekpitgi a e o9 to set was grid k-point the creating for size mesh the and eV, 2 ia ufc ttsi nrni antctplgclisltr eusn insulators topological magnetic intrinsic in states surface Dirac al., et optn antcitrcin nteatfroantctopological antiferromagnetic the in interactions magnetic Competing al., et outainisltradCenisltrpae natwo-dimensional a in phases insulator Chern and insulator axion Robust al., et Te xeietlraiaino nitiscmgei oooia insulator. topological magnetic intrinsic an of realization Experimental al., et 4 als ufc ia oei niermgei oooia insulator topological antiferromagnetic in cone Dirac surface Gapless al., et rsa rwhadmgei tutr fMnBi of structure magnetic and growth Crystal al., et unu nmlu alefc nitiscmgei topological magnetic intrinsic in effect Hall anomalous Quantum al., et 101(2019). 012011 1, 2n . pnsatrn n ocliersi tutr-nue intrinsic structure-induced spin noncollinear and scattering Spin al., et xeietlosraino h aecnrle eeslo the of reversal gate-controlled the of observation Experimental al., et oooia lcrncsrcueadistmeaueeouinin evolution temperature its and structure electronic Topological al., et hs e.X Rev. Phys. 0–1 (2019). 709–714 20, te hmclapcso h addt niermgei topological antiferromagnetic candidate the of aspects Chemical al., et upeso fteatfroantcmtli tt nthe in state metallic antiferromagnetic the of Suppression al., et 3n+1 2 2 2 ag antcgpa h ia on nBi in point Dirac the at gap magnetic Large al., et Te Te Te 781(2019). 076801 36, nqetikesdpnetpoete ftevnder van the of properties thickness-dependent Unique al., et 2 hr r odt neligti work. this underlying data no are There Nature PNAS h F acltossmaie eo eeimple- were below summarized calculations DFT The 4 4 4 Te . . . . hs e.X Rev. Phys. Science Lett. Rev. Phys. Mater. Chem. 4 igecrystal. single 408(2019). 041038 9, hswr a pnoe yteAm Research Army the by sponsored was work This 2–2 (2019). 423–428 576, | × 9–0 (2020). 895–900 367, oebr3 2020 3, November 9 409(2019). 041039 9, × × 7520 (2019). 2795–2806 31, o hnfim.FrMSadMTtelat- the MBT and MBS For films. thin for 1 10 624(2020). 167204 124, hs e.Mater. Rev. Phys. −3 a.Mater. Nat. 2 eV/ Te 4 2 A. Te ˚ films. 4 ±e ∼7 . | hs e.X Rev. Phys. 2–2 (2020). 522–527 19, o.117 vol. 2 921(2019). 094201 3, hs e.Lett. Rev. Phys. V aentobserved not have eV, /h rfrom or 2 Te | 400(2019). 041040 9, 4 o 44 no. . IAppendix, SI hs e.Mater. Rev. Phys. 2 Te ±e 107202 122, 2 3 2 Te /MnBi Te | 2 × 4 4 /h device. . 27229 9 Phys. 2 2 × Te as to 3 4 2

PHYSICS 29. J. Ge et al., High-Chern-number and high-temperature quantum Hall effect without 40. H. Zhang et al., Topological insulators in Bi2Se3, Bi2Te3 and Sb2Te3 with a single Dirac Landau levels. Nat. Sci. Rev. 7, 1280–1287 (2020). cone on the surface. Nat. Phys. 5, 438–442 (2009). 30. C. Hu et al., A van der Waals antiferromagnetic topological insulator with weak 41. G. Rosenberg, M. Franz, Surface magnetic ordering in topological insulators with bulk interlayer magnetic coupling. Nat. Commun. 11, 97 (2020). magnetic dopants. Phys. Rev. B 85, 195119 (2012). 31. L. Ding et al., Crystal and magnetic structures of magnetic topological insulators 42. A. A. Burkov, L. Balents, Weyl semimetal in a topological insulator multilayer. Phys. MnBi2Te4 and MnBi2Te7. Phys. Rev. B 101, 020412 (2020). Rev. Lett. 107, 127205 (2011). 32. R. C. Vidal et al., Surface states and rashba-type spin polarization in antiferromag- 43. B. Himmetoglu, A. Floris, S. de Gironcoli, M. Cococcioni, Hubbard-corrected DFT netic MnBi2Te4 (0001). Phys. Rev. B 100, 121104 (2019). energy functionals: The LDA+U description of correlated systems. Int. J. Quant. Chem. 33. P. Swatek et al., Gapless Dirac surface states in the antiferromagnetic topological 114, 14–49 (2013). insulator MnBi2Te4. Phys. Rev. B 101, 161109 (2020). 44. X. Wan, A. M. Turner, A. Vishwanath, S. Y. Savrasov, Topological semimetal and fermi- 34. T. Hirahara et al., Large-gap magnetic topological heterostructure formed by arc surface states in the electronic structure of pyrochlore iridates. Phys. Rev. B 83, subsurface incorporation of a ferromagnetic layer. Nano Lett. 17, 3493–3500 205101 (2011). (2017). 45. C. Nowka et al., Chemical vapor transport and characterization of MnBi2Se4. J. Cryst. 35. J. A. Hagmann et al., Molecular beam epitaxy growth and structure of self-assembled Growth 459, 81–86 (2017). Bi2Se3/Bi2MnSe4 multilayer heterostructures. New J. Phys. 19, 085002 (2017). 46. A. Burkov, Weyl metals. Annu. Rev. Condens. Matter Phys. 9, 359–378 (2018). 36. S. V. Eremeev, M. M. Otrokov, E. V. Chulkov, New universal type of interface in the 47. B. Chen et al., Intrinsic magnetic topological insulator phases in the Sb doped magnetic insulator/topological insulator heterostructures. Nano Lett. 18, 6521–6529 MnBi2Te4 bulks and thin flakes. Nat. Commun. 10, 4469 (2019). (2018). 48. D. T. Son, B. Z. Spivak, Chiral anomaly and classical negative magnetoresistance of 37. J. Wu et al., Natural van der Waals heterostructural single crystals with both magnetic Weyl metals. Phys. Rev. B 88, 104412 (2013). and topological properties. Sci. Adv. 5, eaax9989 (2019). 49. A. A. Burkov, Chiral anomaly and diffusive magnetotransport in Weyl metals. Phys. 38. R. C. Vidal et al., Topological electronic structure and intrinsic magnetization in Rev. Lett. 113, 247203 (2014). MnBi2Te7: A Bi2Te3 derivative with a periodic Mn sublattice. Phys. Rev. X 9, 041065 50. G. Kresse, J. Hafner, Ab initio molecular dynamics for liquid metals. Phys. Rev. B 47, (2019). 558–561 (1993). 39. I. I. Klimovskikh et al., Variety of magnetic topological phases in the 51. J. P. Perdew, K. Burke, M. Ernzerhof, Generalized gradient approximation made (MnBi2Te4)(Bi2Te3)m family. arXiv:1910.11653 (25 October 2019). simple. Phys. Rev. Lett. 77, 3865–3868 (1997).

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