Downloaded by guest on September 30, 2021 a Lei Chao multilayers topological Magnetized www.pnas.org/cgi/doi/10.1073/pnas.2014004117 MnBi Weyl of design the on superlattices. comment to configuration, and magnetic quantum arrangement, thickness, stacking the and film of on effect, absence Hall or thin-film cone anomalous of presence Dirac dependence the the coupled particularly for the properties, explanation employ an provide then to We model calculations. the- (DFT) density-functional cone initio ory ab estimate with Dirac and comparing quintu- by applicability only or parameters model’s its septuple the retains each demonstrate that analysis of We layer. surfaces model Our ple both on order. simplified freedom a of arbitrary degrees of in use layers makes quintuple insulator ical example, (for (Sb by insulator Mn formed topological films magnetized crys- Wang) intrinsic thin Jian bulk stacking (quasi-2D) and of McQueeney quasi–two-dimensional properties Rob by and topological reviewed and tals 2020; magnetic 6, July the review discuss for We (sent 2020 18, September MacDonald, H. 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X-Mn-X an by replaced is layer x Bi 4 a | x 1−x Bi h Chen Shu , ) ) antzdtplgclinsulator topological magnetized Bi 2 2 2 X X 1−x Te ) 2 3 4 MnBi X aiyo aee a e al aeil,which materials, Waals der van layered of family 3 4 Iti lswt antceeet.How- elements. magnetic with films thin TI ) vntog h ui eprtr is temperature Curie the though even , ∼100 2 ihX=S,e etpelyr n topolog- and layers septuple Se,Te) = X with X 2 4 Te a,b 2 salyrdmtra opsdo seven- of composed material layered a is X e htaegnrlyepce 1,15) (10, expected generally are that meV 4 3 ls(3,ala eprtrsexceed- temperatures at all (23), films n la .MacDonald H. Allan and , | unul aesi hc h mid- the which in layers quintuple elsemimetal Weyl | ∼20 oooia superlattices topological ,i uhhigher much is K, ∼5 o thick- for T ∼30 a,1 ∼15 Kin mK K | doi:10.1073/pnas.2014004117/-/DCSupplemental at online information supporting contains article This 1 the under Published interest.y competing no declare University.y authors the Peking The J.W., wrote and A.H.M. University; per- State and Iowa C.L. A.H.M. R.M., and Reviewers: and data; S.C., analyzed C.L., A.H.M. and research; paper.y S.C., designed C.L., A.H.M. research; and formed C.L. contributions: Author sdt drs h rpriso hnfim n ukcrys- bulk and films thin of (Sb Mn properties stacking by the formed tals address to used exchange interlayer 29). fer- weak 23, chang- thin to (21, the interactions by antiferromagnetic overcoming in but from through effects configuration quantization, romagnetic magnetic Landau Hall 34– the establishing quantum 31, ing by 30, establish (8, not crystals fields films bulk fields Magnetic from magnetic exfoliation 39). by mechanical aligned Mn by be the or can weak, films are thin ∼5 layers in antiferromag- moments septuple the layer between Because antiferromagnetically interactions layers. and netic septuple layer adjacent septuple between each ferromag- within ordered ions netically Mn with order antiferromagnetic A-type ncoe ie;sbtttoso ,S,T ntediffer- the on Te Se, S, concentrations. = defect growth-controlled X and sites; chalcogen of ent fraction substitutions Sb sites; the in Important pnictogen changes understood. include and the recognized variations across be materials it to properties that materials topological importantly of and family more magnetic and in layers trends allows many and containing sequences stacking films complex facilitates thin with it crystals that of are descriptions model simplified the the parame- of material-dependent advantages model’s The the ters. of values fix accurate to to usually ini- also is able and description ab are simplified with highly we this surface comparing calculations that establish By each (DFT) theory layer. on density-functional quintuple freedom tio or of septuple degrees each cone of Dirac only contains (Sb bulk mea- that scattering show neutron 30) 12), 26, (9, (17, surements predictions theoretical with owo orsodnemyb drse.Eal [email protected] Email: addressed. be may correspondence whom To b a e al ope uligbok,adbl electronic bulk and blocks, building depen- the thick- coupled structure. the of film Waals sequences on on der stacking effect light van the Hall configuration, shed anomalous magnetic density- to quantum ness, initio it the ab of use dence by We validated theory. model superlat- functional simplified of a study theoretical employs MnBi a from estab- formed on has tices report We work insulator. 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PHYSICS Our model can also be used as a starting point for theories that cones result from exchange interactions with Mn local moments account for gating electric fields, external magnetic fields, disor- and break time-reversal symmetry der, and other perturbations that are difficult to describe using X ab initio approaches. mi = JiαMα, [2] In this paper we focus on Bi2X3 and MnBi2X4 with X = α Se, Te, which have received greatest attention to date. We find that bulk ferromagnetic MBX is a nearly ideal Weyl , where α is a layer label. We limit our attention here to the that thin-film ferromagnets are two-dimensional (2D) Chern case of magnetization perpendicular to the van der Waals layers insulators with Chern numbers that grow and gap sizes that so that Mα = ±1 specifies the sense of magnetization on layer decline with film thickness, and that antiferromagnetic thin films α if it is magnetic, and Mα = 0 on nonmagnetic layers. Each with sufficiently large odd layer numbers are Chern insula- can have two nearby Mn layers, one from the same tors with Chern number |C | = 1. We further find that ferro- layer with exchange splitting JS and one from the adjacent layer magnetic [MnBi2X4]M [Bi2X3]N superlattices formed by insert- with exchange splitting JD , shown as in Fig. 1, if these layers ing nonmagnetic quintuple layers in the stack are ordinary are magnetic. In our surveys of MTI superlattice properties we insulators for M /N smaller than about 3, but become Weyl retain only ∆S , ∆D , JS , and JD , which are normally dominant, for larger M /N . For superlattices with (M,N) = as model parameters; the model allows different values of ∆S (1,1), there is a large chance to achieve a in TI and MTI layers and allows ∆D at magnetic/nonmagnetic phase with MnBi2Te4 as magnetic topological insulator lay- to differ its value at magnetic/magnetic or ers and Bi2Se3, Bi2Sb3, or Sb2Te3 as topological insulator nonmagnetic/nonmagnetic heterojunctions. layers. The model simplifies when only one type of layer is present. When that layer is magnetic and the magnetic configuration is Coupled Dirac Cone Model ferromagnetic, our model reduced to the topological insulator We construct a model for a magnetized topological insulator multilayer model proposed by Burkov and Balents (42). Instead multilayer by including Dirac cone degrees of freedom not only of inserting a normal insulating layer between magnetic topolog- on the surface layers (40, 41), but also on the top and bottom of ical insulator layers to form a flexible three-dimensional system, each magnetic septuple layer and nonmagnetic quintuple layer as imagined in ref. 42, in this model the coupling between Dirac as illustrated in Fig. 1. We allow for arbitrary stacking of mag- cones on different layers is across the van der Waals gap. As netic and nonmagnetic layers either to form a thin film or, if we discuss below, some of the flexibility that would be associ- repeated, to form a bulk crystal. We allow for exchange interac- ated with nonmagnetic spacer layers of variable thickness can be tions with the Mn local moments and arbitrary spin-independent recovered by inserting nonmagnetic TI layers between magnetic hybridization between different surfaces, denoting the hopping TI layers. parameter between the ith surface and the j th surface by ∆ij . The Hamiltonian is therefore Comparison to DFT Calculations The relevance of the model that retains only Dirac cone degrees X h i  of freedom to realistic systems can be tested by comparing with H = (−) v (ˆz × σ) · k + m σ δ ~ D ⊥ i z ij ab initio DFT electronic structure calculations, which we per- k⊥,ij [1] formed using the LDA + U approximation (43) with U = i † + ∆ij (1 − δij ) c ck j , 5.25 eV on Mn sites. k⊥i ⊥ Nonmagnetic. As a first test of the model, we examine the bands where spin labels have been left implicit, i and j are Dirac cone of bulk Bi2Se3 as estimated by DFT. We find that we can obtain labels with even integers reserved for layer bottoms and odd ones a reasonable approximation to the DFT bands retaining only the hybridization parameters ∆S for hopping within the same layer for layer tops, ~ is the reduced Planck constant, and vD is the velocity of the Dirac cones. The most important hybridization and ∆D for hopping to the adjacent surface of the adjacent layer. 1 parameters, hopping within the same layer (∆S ) and hopping The fit can be improved by adding ∆D , for hopping to the same across the van der Waals gap between adjacent layers (∆D ), are surface of the adjacent layer. Retaining only these three param- highlighted in Fig. 1. The mass gaps mi of the individual Dirac eters the model predicts two spin-degenerate bands along the Γ to Z line in the Brillouin zone with dispersions q 2 2 1 E(kz ) = ± ∆S + ∆D + 2∆S ∆D cos kz d + 2∆D cos kz d, [3]

with d the distance between layer centers and kz the momen- tum along the Γ to Z line. The corresponding DFT bands are illustrated in Fig. 2. Because of details of the stacking arrange- ments that are not captured by the simplified model, the Bi2Se3 lattice used in the DFT calculations repeats only after three lay- ers, instead of after a single layer. Nevertheless, we find excellent agreement between the two sets of bulk bands once the DFT bands are unfolded to the larger Brillouin zone. The property that the gap between the low-energy bands is larger at Z than at Γ implies that ∆S and ∆D have opposite signs. The param- eter ∆1 is added to account for the small difference between Fig. 1. Coupled Dirac cone model. (Left) Magnetized topological insulator D multilayers composed of M magnetic septuple layers and N nonmagnetic the average of the DFT conduction and valence band energies at Γ and Z . The model band energies at the Γ point (kz d = 0) quintuple layers. (Right) Dirac cone model in which Dirac cones localized on 1 the top and bottom of each layer are hybridized with remote Dirac cones and the Z point (kx d = π) are, respectively, ±|∆S + ∆D | + 2∆D 1 and altered by exchange interactions with the local moments present in the and ±|∆S − ∆D | − 2∆D . Fitting to these four energies we obtain 1 magnetic layers. ∆S = 143 meV, ∆D = −280 meV, and ∆D = −11 meV. The

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Table in-plane and on S1 bands Fig. DFT Appendix, the parameter (SI of velocity momenta dependence Dirac the from model’s estimated The is analysis. following the in i yrdzd n hrfr apd ia oe nteinterior. the in cones Because Dirac cones gapped, no Dirac therefore layer and has surface hybridized, between layer via hopping to (bottom) 2 due top is (Fig. gap layers the because thin-film adjacent small of from is cone partner gap Dirac hybridization thin-film (bottom) The hybridization top the network. to the cones due Dirac is gap the bulk of a the supports which agreement in good picture This physical bandgap. the in of instead bands Bi Brillouin larger the to from dispersion unfolded band ∆ be DFT must calcula- bulk bands the DFT DFT Fitting from zone. the result cell, the unit is per line layers Bi blue bulk The the (A) Since model. tions. cone Dirac the and ihrdadbu os,ecp ntesnl-unul ae case layer single-quintuple the in except for dots), blue (shown results and the DFT red from with films with agreement thin good of in gaps are the model 2B, simplified Fig. in illustrated As layers. ple than larger is ∆ for parameters fitted corresponding hopping represent links layers. gray between and and represented purple within layers the individual and of ellipses magenta/green surfaces by top/bottom bonding cone the Dirac label the of t/b illustration network, schematic the In dispersion. band bulk corresponding The state. band valence the than layer the Bi of at middle state the band in conduction single-layer the that noting 2. Fig. e tal. et Lei Bi and seen model. As simple this layer. with quintuple wider 2A a Waals Fig. across der in is van latter narrow the a whereas across gap is since parameter surprising hopping not former is topology, the band nontrivial the for hence and Bi for meV 488 r u ftedslydrne ti rud87mVfrBi for meV 847 around is it range; calculations displayed DFT 2D from the the gaps of at the out scale, gaps displayed are thin-film the for In number. models layer cone vs. point Dirac and DFT between parison AB 2 D D sn hs aaeesetmtdfo h ukbnsat bands bulk the from estimated parameters these Using 2 2 Te Bi Te Te = = 3 Z 8 e,and meV, −280 tyields fit 2 2 3 3 ecluae h a tte2D the at gap the calculated we , 0 e,and meV, −200 adsrcueo Bi of structure Band Se Se o h atrmtra h ia on isi h valence the in lies point Dirac the material latter the For . hnfim ihtikessrnigfo n osxquintu- six to one from ranging thicknesses with films thin ∆ 3 4 enwtr otemgei ae trigwt bulk with starting case, magnetic the to turn now We D 1 n ntesnl-addul-unul ae ae for cases layer double-quintuple and single- the in and h noddDTbnsaei xeln agreement excellent in are bands DFT unfolded the MS and (MBS) ssalcmae to compared small is 2 ∆ Te |∆ S 3 4meV, 74 = h ia oemdl eeotie yfitn othe to fitting by obtained were models cone Dirac The . S | hc srsosbefrbn neso at inversion band for responsible is which , 2 ∆ Se D 1 3 MnBi ∆ = ae tcigarneethstrequintuple three has arrangement stacking layer 2 Se ∆ D 1 1mV h inof sign The meV. −11 3 D = ln the along = 2 0 e,and meV, −200 0mV h rprythat property The meV. −10 Te 4 ∆ MT.Blww ee othe to refer we Below (MBT). D eicueonly include we , Γ Bi to 2 Te Z Γ Γ iefo F calculations DFT from line to 3 on for point ∆ are D 1 Z ∆ Γ = yields S a rae weight greater has Com- (B) meV. −10 a enfie by fixed been has ∆ 2 B, S Se ∆ Bi ∆ 3 4meV, 74 = .The Inset). S n around and S 4 meV, 143 = 2 Se and 3 |∆ and ∆ D D Γ Γ Γ | ofiuain(oe)(F/x. mdl (DFT/exp.) E (model) (DFT/exp.) (model) configuration E r h aei vr ae n h oe adeeg dispersion energy band model the the magnetic and along layer external every in energies same modest exchange the the are relatively configuration ferromagnetic a the For applying can field. configuration by ferromagnetic realized the be weak, extremely are layers ewe odcinadvlnebnsa the at bands valence and conduction between Magnetic MnBi for parameters millielectronvolts Model 1. Table at points Weyl the between nonzero over are curvature Berry the 45). 30, 16, vary (10, that meV meV experimentally 86 200 and to estimated MBS 50 gaps from for to meV compared 85 MBT, around for are model this calcu- from gaps antiferromagnetic lated (k bulk (SI The gap points trends. thin-film S3 ) Weyl Fig. 2D-Γ-point Appendix, or of (AHEs), effects positions Hall the anomalous have on and influence millielectronvolts several little below approximately discussions are phase-diagram they qualitative since Weyl the in the dropped at but in bands addressed crossing is the point, between difference longer-range of magnitude role The MBT. parameters for yield meV which hopping 12 calculations, and good MBS DFT small for in our the meV in are 23 although smaller bands even results, are model DFT gaps The point the points. with semimetal Weyl overall Weyl exchange two agreement simple the only a with by generating (44) 13), inverted (12, are previously MBT, noted for meV splitting 44 around illustrated the and Because are 3A. configuration Fig. ferromagnetic in the for parameters 1. Table in dispersions summarized band DFT the bulk with of expressions these comparing By splitting exchange J the configuration antiferromagnetic (J at on-layer energies The the is interactions. of exchange splitting sum the exchange is and ferromagnetic the where au fteatfroantcsaegpadDTcluainrslsfor results antiferromagnetic calculation AF, states. experiment. DFT ferromagnetic from FM, and available states; not gap were state that antiferromagnetic quantities the of value ∆ ∆ fit to configurations magnetic Because J J D S gap gap Γ Γ S D S nFg 3B Fig. In The o h ounlbldDTep eueteeprmnal available experimentally the use we DFT/exp. labeled column the For − (FM) (AF) J D MnBi Γ E m m lentsi info ae olyradtegap the and layer to layer from sign in alternates Γ (k to F F to z h pnsltigcoe tafiievleof value finite a at closes splitting spin the , epo DCennmesotie yintegrating by obtained numbers Chern 2D plot we = ) eed nyon only depends Z 2 X Z 4 addsesoscluae ihteemodel these with calculated dispersions band E ± ieis line X=S,T) eetattemdlparameters model the extract we Te), Se, = (X gap 3 — −232 AF q B B B MBT MBT MBS MBS E E 9 4— 84 — 190 5— — 25 32 30 85 Γ Z ∆ ∆ 2( = i.S2 Fig. Appendix, SI = = S 2 D n pitgp,aon 0mVfrMBS for meV 30 around gaps, Γ-point ∆ + ±(∆ ±(∆ hc r epnil o velocity a for responsible are which , q k x ∆ D 2 and S S J D 2 J 2∆ + S '100 ∆ + − '23 S + and S + ∆ k n egbrn-ae (J neighboring-layer and ) m y D D J 2 S AF 2 D ) X safnto of function a as ) J ∆ ± 4 k ± D ems osdrother consider must we , D z − X=S,T)i nt of units in Te) Se, = (X NSLts Articles Latest PNAS m d m neednl.Frthe For independently. cos ∆ 0 = F k F yDTcalculations DFT by z 2 — −127 S . k = Γ ). 412 44 650–200 86 9— — 29 36 z and d ±k on is point ± m w k w F m z h Weyl The . ,thin-film ), d = F k = , z J which , m S | π k + AF z f7 of 3 are as , [4] [6] [5] J D Γ- = D )

PHYSICS AC

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Fig. 3. (A) Bulk band dispersion of ferromagnetic MBX calculated from the Dirac cone model with the DFT-extracted parameters summarized in Table 1. Ferromagnetic MnBi2X4 (X = Se,Te) is a Weyl semimetal, with the Weyl point located at kz = ±kw , where kw ≈ 3π/50d for MBS and kw ≈ 3π/20d for MBT. (B) Dirac cone model kz-dependent 2D Chern numbers for bulk MBX calculated from the model, showing jumps from 1 to 0 at the Weyl points. (C) The 2D Chern numbers of thin-film MBX calculated from the Dirac cone model which show that the Chern numbers of ferromagnetic MBX jump to no-zero values beyond four septuple layers thickness for MBS and two septuple layers for MBT. The antiferromagnetic configurations have nonzero Chern numbers for odd-septuple-layer films when the thickness is larger than eight layers for MBS and two layers for MBT. (D) Gap of thin-film MBX at the 2D-Γ point as a function of the number of septuple layers.

momentum kw is nonzero for mF > |∆D | − |∆S | and moves from jumps from 1 to 2 when the thickness reaches nine (29) septuple Γ to Z with increasing mF , reaching π/d when mF = |∆D | + layers. 2 |∆S |. The bulk Hall conductivity per layer is (46) kw /(π/d)e /h. Superlattice Phase Diagrams Since the exchange interactions mF in MBX are larger than |∆D | − |∆S |, but quite small compared to |∆S | + |∆D |, the Hall The van der Waals character of MBX materials allows for conductivity per septuple layer in bulk ferromagnetic MBX is property variation by changing the stacking sequence of mag- small compared to the quantum value e2/h. netic septuple and nonmagnetic quintuple layers (30, 35, 37– In Fig. 3C we plot the total Chern numbers of thin films 39). Importantly, the magnetic field needed to convert between with ferromagnetic and antiferromagnetic configurations as a antiferromagnetic and ferromagnetic stacking arrangements can function of the number of septuple layers in the film. For ferro- be reduced simply by inserting nonmagnetic quintuple spac- magnetic configurations, the thin films have zero Chern number ers between the magnetic septuple layers. When magnetic and for thicknesses up to the critical number of septuple layers, which nonmagnetic layers are simply alternated, for example, it has is four for MBS and two for MBT. As the thickness increases been shown experimentally that the magnetic field needed the Chern numbers increase indefinitely, increasing by one for to convert to a ferromagnetic configuration is reduced from every ∼ π/d/kw added septuple layers so that the quantized Hall ∼5 to ∼0.22 T (30). conductivity of very thick films approaches the bulk value when We consider the family of bulk crystals in which a template normalized per layer (SI Appendix, Fig. S4). As the Chern num- with M MnBi2X4 septuple layers and N Bi2X3 quintuple layers ber increases the energy gaps tend to decrease, as illustrated is repeated. Several of these superlattices have already been real- in Fig. 3D, where we have assigned a negative sign to the gap ized experimentally, including (M,N) = (1,1)(MnBi4Te7) (30, for odd integer Chern numbers. The oscillating gap size simply 37–39), (M,N) = (1,2)(MnBi6Te10) (37, 39), and (M,N) = (1,3– reflects the size quantization of the bulk Weyl semimetal bands, 6) (MnBi8Te13, MnBi10Te16, and MnBi12Te19) (39). In Fig. 4 we as detailed in SI Appendix, Fig. S5. The behavior of antiferromag- summarize the energy gaps and topological phases of ferromag- netic configurations is quite distinct. Here the Chern numbers netic configuration (MnBi2X4)M /(Bi2X3)N superlattices. We are nonzero only for odd numbers of septuple layers and, in that find that the energy gap is mainly determined by the ratio of mag- case, only when a critical film thickness equal to two septuple netic to nonmagnetic layers M /N and is less dependent on the MTI TI layers for MBT and eight septuple layers for MBS is exceeded. order of the layers. For example, when we set ∆S = ∆S , we The antiferromagnetic configuration gaps are identical for even find that M /N larger than around 3 leads to bulk Weyl semimet- MTI TI and odd numbers of layers in the thick-film limit (SI Appendix, als. Decreasing ∆S /∆S favors Weyl semimetal phases with Fig. S5). These semianalytic predictions of the Dirac cone model a decrease of around ±10% reducing the critical M /N ratio are consistent with recent experiments (21, 23, 29) which have, to 2. We therefore expect that stronger hybridization between in particular, shown that for MBT films the total Chern number within the nonmagnetic topological insulator

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Fig. eiea codn oDTcluain hw in S2 shown calculations Fig. DFT to according ferromagnetic semimetal results, in our and with (47) calculations agreement DFT to red according MBT, by unlike marked insulator, are 5 antiferromagnetic diagram Although Fig. dots. phase blue the in and in shown = systems M/N MBT are of and 2 MBS cases The and (42). 1, Balents 0.5, and Burkov by studied model 5A bulk Fig. for in is, (that 0 = N and narrow in to explicitly gap addressed is Waals pressure der increasing van applied, increasing is the with pressure expect as decreases we the spins 2) as moment and expect temperature temperature local we with the 1) decline of that to alignment is interactions parameters on exchange model phase both of of dependence subset the particular illustrating this for motivation The ers. in summarized calculations than hybridization same-layer ∆ aevle for values same of ratio the J keeping sequences stacking different several for eprtr.W td h fcc fteetnn nb ycon- by vs. knobs diagrams tuning changing phase these or of structing efficacy pressure, the applying study example We fraction, for temperature. ways, pnictide different the of variety varying a by in varied be can films thin D S MTI S nalcsstetasto ewe omlisltradWeyl and insulator normal between transition the cases all In h rpriso (MnBi of properties The 2 meV, hc ak h onaybtennra nuao and insulator normal between boundary the marks which , and Sb by 2 X 3 Γ 4 h siae oe aaeesare parameters model estimated The . ±10%. or , ∆ ) oooia hs iga fferromagnetic-configuration of diagram phase Topological M pitgp o B MT acltdusing calculated (MBT) MBS for gaps Γ-point on r itda h o ftecrepnigsuperlattice corresponding the of top the at listed are point ∆ S ast hto h T/omlisltrsuperlattice insulator MTI/normal the of that to maps (Bi S xd olmttenme fprmtr eue the used we parameters of number the limit To fixed. 2 Sb ≈ X 3 124 2 ) N ∆ Te ueltie vs. superlattices S 3 e,and meV, hc r siae ohv stronger have to estimated are which , and k z o l niae ueltie.Wy semimetal Weyl superlattices. indicated all for ∆ MnBi D i.S8. Fig. Appendix, SI 2 MnBi J X nmgei n omgei lay- nonmagnetic and magnetic in S ∆ 4 Bi and ) M ∆ D 2 M o 1 = M For . S7 Fig. Appendix, SI X /N 2 2 D − ≈ hr h oiin of positions the where B–D, Te Te 4 (Bi h eedneof dependence The . ∆ hsdarmwsconstructed was diagram This . ,tepaedarmshown diagram phase the ), 3 4 D 166 2 silsrtdi i.5, Fig. in illustrated as , obndwith combined X codn oteDFT the to according 3 MnSb MnSb ) e for meV N ueltie and superlattices m ∆ ∆ J 2 S TI 2 F S Te Te S MTI = /D IAppendix, SI eitsfrom deviates 4 4 MnSb ∆ = J satrivial a is 0 = saWeyl a is S Bi + S TI ∆ ∆ while , limit. J 2 J 2 D Se D D S Te on to = ≈ 3 4 , e eidcagsfrom number Chern changes total period the occurs, per transition phase latter the When that so at isolated points are transition layers magnetic the ±J 5B, Fig. in shown when J occurs insulator Chern the to tion the at for ple, ml rtclvle of values critical imply the are in period values 5 per Fig. of numbers states. cases semimetal Chern Weyl different by with separated states Hall tum interactions insulator exchange Chern both a smaller at to emerge exceeds insulator states interaction exchange trivial the when a occurs state from transition phase limit the For ih.Tezr-eprtr B n B aaeesaemre yred semimetal. by Weyl marked a be are parameters dots. MBT the blue to and and points MBS moves zero-temperature pressure applying The whereas right. diagram phase this (2,1)(Mn in down = (M,N) and and (1,0)(MnBi layers adjacent in interactions exchange with bulk or trans- thick less layers with many systems are mate- of that the theories across (films trends simplify symmetry initio explain lational and ab help family using relevant influence also for all rial the can accounted fields, address It easily to magnetic not approaches. platform external are a that and perturbations as disorder, that used gating, advantage be the of crystal readily has bulk model can and cone thin-film it Dirac both an simplified in The provides materials limits. Waals here der developed van ered model of description cone qualitative excellent Dirac coupled The Discussion semimetal. ferromagnetic that J i.5. Fig. AB S S CD ∆ = ∆ = S ± Γ hs iga ffroantc(MnBi ferromagnetic of diagram Phase ∆ S S / 1+ /(1 M 2 on is point S M X √ 4 and 1, = ,(,)=(1,1)(MnBi = (M,N) (A), ) + 1 1, = D ∆ ∆ δ hw htfroantcMn ferromagnetic that shows δ ) ±J N D D N ∆ , with E 0 = 0 = o0lmtaedsusdin discussed are limit 0 to Γ ∆ 2 = D Mn 2 S Bi ∆ = D ± /δ 6 S hc orsod oioae aes a layers, isolated to corresponds which , h adenergy band the 5A, Fig. in illustrated , δ X 2 ±(J 0 = ntesm ae,where layer, same the in ∆ and J 11 Bi ≡ o the for J J S .Icesn eprtr oe points moves temperature Increasing (D). ) S S S 6 (J J C n at one , ∆ = n hr r hrfr w phase two therefore are there and , S Te D D o neirmgei aesbecause layers magnetic interior for M n ia oeculn energies coupling cone Dirac and ) J + 1 = /J 11 S S M J 2, = S D and 4 a ag hnet eaWeyl a be to chance large a has o h aeof case the For . o the for X to 2, = ) J 7 ± MnSb ,(,)=(1,2)(MnBi = (M,N) (B), ) S N C J ∆ = ∆ 1 = D N 2 = J S 2 NSLts Articles Latest PNAS 2 S Bi M n h hs transi- phase the and , 1 = otiue o exam- For contribute. X x S tfinite At . 6 4 + Bi o hc h eigen- the which for Te ) n n at one and 1, = M C J ae i.5D Fig. case. 2−x 11 /(Bi A–D D and IAppendix SI a ag hneto chance large has 2 > N X X 3 M r o MN = (M,N) for are ∆ 4 ∆ 2 = ) D N MX lay- (MPX) S S superlattices 1, = ∆ hti,at is, that , hw the shows h Weyl the ; J aeand case D D 6 X | ∆ = N shows quan- 10 E f7 of 5 (C ) and Γ 1 = ∆ S = ), D .

PHYSICS crystals with complex stacking arrangements) and theories of the moments are arranged antiferromagnetically, with the net interaction effects. We anticipate that in both ferromagnetic spin of each unit aligned with the field. Still more complex mag- and antiferromagnetic thin films, gating can be used to drive netic configurations should be reachable in even weaker fields transitions between insulators with different total Chern num- with suitably chosen stacking arrangements that include double bers, including between quantum anomalous Hall insulators with nonmagnetic spacers. nonzero total Chern numbers and ordinary insulators with total It is interesting to consider the implications of the Dirac Chern numbers equal to zero. In bulk crystals we anticipate cone model for the interpretation of photoemission experiments, the simplified model will enable quantitative descriptions of which are confusing at present. Some angle-resolved photoemis- the chiral magneto-transport anomaly (48, 49) of ferromagnetic sion spectroscopy (ARPES) experiments (26–28, 33), generally states and the magneto-electric response of even-layer those using small photon energies near ∼7 eV, have not observed antiferromagnetic states. the surface-state gaps predicted for thick antiferromagnetic films We have used the Dirac cone model here to explain the main by both DFT and the coupled Dirac cone model. Gapped Dirac trends in topological properties, namely that 1) bulk antiferro- cones were observed in higher photon-energy (usually larger magnetic MPX compounds are topological insulators, 2) bulk than 20 eV) ARPES experiments (10, 16, 18, 32), which are ferromagnetic MPX compounds are Weyl semimetals, 3) thin- likely less surface sensitive. One possible explanation is that film antiferromagnets are 2D Chern insulators when the number the lower-energy ARPES experiments are more sensitive to the of layers is odd and exceeds a critical value and ordinary insula- topologically protected edge states expected on the antiferro- tors otherwise, and 4) thin-film ferromagnets that exceed a critical magnetic state surface at step edges where the number of layers thickness are 2D Chern insulators whose Chern numbers grow present in the system changes. As illustrated in SI Appendix, and gaps decline with film thickness. We have reached these Fig. S6, adding a layer to a MTI antiferromagnet changes the conclusions based mainly on the properties of van der Walls het- Hall conductivity either from zero to ±e2/h or from ±e2/h to erojunctions with MBT and MBS building blocks, but based on zero. In either case topologically protected states are required by the phase diagrams in Fig. 5 they appear to be fairly robust. We the quantized change in Hall conductivity. have used the model to verify that it is not possible to realize Weyl semimetals by a repeated stacking of odd layer number antifer- Materials and Methods romagnets separated by nonmagnetic spacers, even though they DFT Calculations. The DFT calculations summarized below were imple- are 2D Chern insulators when isolated. To move more deeply into mented in the Vienna ab initio simulation package (VASP) (50), using the topologically nontrivial part of the phase diagram, the best semilocal Perdew–Burke–Ernzerhof generalized gradient approximation prospect seems to be in engineering constituent layers to increase (PBE-GGA) (51). When Mn atoms were present, a Hubbard on-site electron– electron interaction (43) (LDA + U) with U = 5.25 eV was included for its 3d ∆S or decrease |∆D | so that they are nearly equal in magnitude, for example by tuning the pnictide fraction. electrons. The cutoff energy for the plane-wave basis was set to be 600 eV, the global break condition for the electronic self-consistency loop was set to Since the MPX compounds are antiferromagnetic in their 10−7 eV, and the mesh size for creating the k-point grid was set to 9 × 9 × 3 ground states, it is necessary to apply an external magnetic field for bulk calculations and 9 × 9 × 1 for thin films. For MBS and MBT the lat- to align the moments and reach the ferromagnetic configura- tice constants were obtained by relaxing the unit cell until the forces for tion. Because the antiferromagnetic exchange interaction across each atom were smaller than 1 × 10−3 eV/A.˚ the van der Waals gap is weak, the magnetic field needed to align the magnetic moments is relatively small (∼5 T). The field Data Availability. There are no data underlying this work. strength necessary to align moments in layers separated by a ACKNOWLEDGMENTS. This work was sponsored by the Army Research single nonmagnetic spacer layer is much smaller (∼0.2 T). It fol- Office under Grant W911NF-16-1-0472 and by the Welch Foundation under lows that there is a broad field range in which the ground-state Welch F-1473. We acknowledge generous computer time allocations from configuration will consist of units with no spacers within which the Texas Advanced Computing Center.

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PHYSICS