Topological magnetic materials of the (MnSb_2Te_4)\cdot(Sb_2Te_3)_n van der Waals compounds family
S.V. Eremeev, I.P. Rusinov, Yu.M. Koroteev, A.Yu. Vyazovskaya, M. Hoffmann, P.M. Echenique, A. Ernst, M.M. Otrokov, E.V. Chulkov
TTopological magnetic materials of the (MnSb Te ) · (Sb Te ) n van der Waalscompounds family S. V. Eremeev,
1, 2, ∗ I. P. Rusinov, Yu. M. Koroteev,
1, 2
A. Yu. Vyazovskaya,
2, 3
M. Hoffmann, P. M. Echenique,
5, 6, 7
A. Ernst,
4, 8
M. M. Otrokov,
7, 9, † and E. V. Chulkov
5, 6, 7, 3, ‡ Institute of Strength Physics and Materials Science,Russian Academy of Sciences, 634021 Tomsk, Russia Tomsk State University, 634050 Tomsk, Russia Saint Petersburg State University, 198504 Saint Petersburg, Russia Institut f¨ur Theoretische Physik, Johannes Kepler Universit¨at, A 4040 Linz, Austria Departamento de F´ısica de Materiales UPV/EHU,20080 Donostia-San Sebasti´an, Basque Country, Spain Donostia International Physics Center (DIPC),20018 Donostia-San Sebasti´an, Basque Country, Spain Centro de F´ısica de Materiales (CFM-MPC), Centro Mixto CSIC-UPV/EHU,20018 Donostia-San Sebasti´an, Basque Country, Spain Max-Planck-Institut f¨ur Mikrostrukturphysik, Weinberg 2, D-06120 Halle, Germany IKERBASQUE, Basque Foundation for Science, 48011, Bilbao, Spain (Dated: February 5, 2021)Combining robust magnetism, strong spin-orbit coupling and unique thickness-dependentproperties of van der Waals crystals could enable new spintronics applications. Here, using densityfunctional theory, we propose the (MnSb Te ) · (Sb Te ) n family of stoichiometric van der Waalscompounds that harbour multiple topologically-nontrivial magnetic phases. In the groundstate, thefirst three members of the family, i.e. MnSb Te ( n = 0), MnSb Te ( n = 1), and MnSb Te ( n = 2), are 3D antiferromagnetic topological insulators (AFMTIs), while for n ≥ n = 0) or FM axion insulator states ( n ≥ Te is nottopologically trivial as was previously believed that opens possibilities of realization of a wealth oftopologically-nontrivial states in the (MnSb Te ) · (Sb Te ) n family. INTRODUCTION
The recently discovered first intrinsicantiferromagnetic topological insulator (AFMTI)MnBi Te has attracted a great deal of attention .The AFMTI state of matter is expected to give riseto a number of exotic phenomena such as quantizedmagnetoelectric effect , axion electrodynamics ,and Majorana hinge modes . Due to its layeredvan der Waals structure and interlayer AFM order,in the 2D limit MnBi Te has been predicted toshow an unusual set of thickness-dependent magneticand topological transitions, which drive it throughFM and (un)compensated AFM phases, as well asquantum anomalous Hall (QAH) and zero plateau QAHstates . This makes MnBi Te the first stoichiometricmaterial predicted to realize the zero plateau QAH stateintrinsically, which has been theoretically shown to host ∗ [email protected] † [email protected] ‡ [email protected] the exotic axion insulator (AXI) phase . Soon afterthe density functional theory (DFT) prediction , theevidence of both the AXI and QAHI states has indeedbeen reported in thin MnBi Te flakes in Refs. and ,respectively. Very recently, the QAH regime has alsobeen reported in the MnBi Te /Bi Te superlattices .Moreover, the quantized Hall effect under externalmagnetic field has been achieved in MnBi Te flakes,displaying the Chern numbers C = 1 and 2 .A recent study also predicts that the C = 3 state isachievable in the twisted MnBi Te bilayer . Finally,MnBi Te -derived family of compounds has beenrecently proposed as a magnetically tunable platformfor realizing various symmetry-protected higher-ordertypologies , hinged quantum spin Hall effect ,helical Chern insulator , as well as spin-polarized flatbands .The abovementioned results establish a new directionin the field of magnetic TIs that focuses on intrinsicallymagnetic stoichiometric compounds. For furthersuccessful development of this direction more layeredmagnetic materials need to be found. Currently, thequest for new MnBi Te -like systems follows the strategyof either codoping Bi sublattice with Sb or a r X i v : . [ c ond - m a t . m t r l - s c i ] F e b growing the homologous series (MnBi Te ) · (Bi Te ) n whose typical representatives are MnBi Te ( n = 1) andMnBi Te ( n = 2) , although MTIs withhigher n have also been reported very recently .In this paper, using the state-of-the-art DFTand tight-binding calculations we propose the(MnSb Te ) · (Sb Te ) n homologous series, whichhas not been considered previously. By means of thehighly-accurate total-energy calculations we show that n = 0 , , and 2 systems, i.e. MnSb Te , MnSb Te ,and MnSb Te , are interlayer antiferromagnets inwhich FM Mn layers are coupled antiparallel to eachother and the easy axis of staggered magnetizationpoints perpendicular to the layers. Such magneticordering makes these compounds invariant withrespect to the combination of the time-reversal (Θ)and primitive-lattice translation ( T / ) symmetries, S = Θ T / , which gives rise to the Z topologicalclassification of AFM insulators , Z being equal to1 for all these materials. Consistently, their S -breaking(0001) surfaces exhibit Dirac point gap, which is aconsequence of the out-of-plane direction of the easyaxis of staggered magnetization. At the same time,the S -preserving surfaces are gapless as expected forAFMTIs. For the MnSb Te compound ( n = 3), whoseinterlayer exchange coupling is so weak that it canhardly be reliably established within DFT, we predictthe AFMTI state assuming the interlayer AFM phase. Inthe FM phases, that can be achieved by the applicationof the external magnetic field, MnSb Te appears tobe a Weyl semimetal, while MnSb Te , MnSb Te ,MnSb Te , etc., are 3D FM AXIs according to the Z classification. Finally, we study the thin films ofthe (MnSb Te ) · (Sb Te ) n compounds and find AFMAXIs in the zero plateau QAH state, Chern insulatorsin the constrained FM state, as well as intrinsicQAHI in the uncompensated AFM state. Our studyprovides a compelling first-principles evidence thatMnSb Te , previously predicted to be trivial at normalconditions , is, in fact, a topological insulator orWeyl semimetal in the AFM or FM states, respectively.These findings significantly extend the emergent familyof the MTIs by predicting a number of solid materialscandidates. RESULTS
The growth of the trigonal R ¯3 m -group bulk MnSb Te has been reported very recently . This structure,comprised of the septuple layer blocks (SLs; Fig. 1a)turns out to be energetically more favorable than themonoclinic structure , previously observed for someother chalcogenides with the same formula . Magneticproperties of MnSb Te have been found to depend onthe degree of Mn-Sb intermixing: the larger the amountof the Mn atoms incorporated in the Sb layers, thestronger the tendency to the interlayer FM coupling is . At lowest intermixing levels as well as someMn deficiency, MnSb Te , similarly to MnBi Te isexperimentally found to be an interlayer antiferromagnet,in which FM Mn layers of SLs are coupled antiparallelto each other below the N´eel temperature of 19 K .This is in agreement with our total energy calculations(Table I), performed for the ideal crystal structuremodel. Importantly, we find that the ideal structureis energetically more favorable than those with Mn-Sbintermixing and therefore in what follows we considerthe ordered structure of MnSb Te . We further find apositive magnetic anisotropy energy E a of 0.115 meVper formula unit, indicating the easy axis with anout-of-plane orientation of the local magnetic moment(4.59 µ B ), which is, again, in agreement with theexperimental report .The combination of the trigonal structure and theinterlayer AFM order makes MnSb Te S -symmetric, S being the combination of the time-reversal Θ andprimitive-lattice translational symmetries T / , S =Θ T / . The presence of this symmetry allows introducing Z classification of AFM insulators . To determinewhether the system is insulating or not, bulk electronicstructure has been studied next for the above discussedmagnetic ground state. Our calculations reveal aninsulating character of the spectrum, the fundamentalband gap value being equal to 124 meV taking spin-orbitcoupling (SOC) into account (Fig. 1b). The Z invariant can be calculated based on the occupied bandsparities , in which way we find Z = 1 for MnSb Te ,meaning that it is an AFMTI. The latter means thatthe bulk band gap of the material should be inverted,which is indeed confirmed by performing the density ofstates (DOS) calculations lowering the SOC constant λ stepwise from its natural value λ to λ = 0 . λ (seeFig. 1c). As can be seen, the system demonstrates thetopological phase transition, i.e. passes through thezero gap state, when the SOC strength is decreasedto about 0.7 λ . Recent DFT studies report,however, that MnSb Te is topologically trivial, whichis in contrast to our results. As we discuss in detailin the Supplementary Information, the crucial factorcausing this difference is the MnSb Te crystal structure,which in our case is fully optimized taking the van derWaals interactions into account (see Methods section).To provide further support to our claim of the AFMTIstate in MnSb Te , apart from the projector augmentedwave method (PAW) calculations using VASP, we haveperformed DFT calculations using the full-potentiallinearized augmented plane-wave method (FLEUR code)as well as the Green’s functions method (HUTSEPOTcode). These calculations predict the band-invertedAFMTI phase in MnSb Te as well (SupplementaryInformation Figures S1 and S2).The bulk edge correspondence dictates that theinverted bulk band gap gives rise to a 2D spin-polarizedstate at the material’s surface, i.e. the topologicalsurface state (TSS). In an AFMTI case, this state -1.0-0.50.00.51.0 E - E F ( e V ) L Z F Z 0.00.020.040.060.080.10.120.14 E g ( e V )
30 40 50 60 70 80 90 100
SOC (%) -0.3-0.2-0.10.00.10.20.3 E - E F ( e V ) -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 k || (A -1 ) -0.3-0.2-0.10.00.10.20.3 E - E F ( e V ) -0.4 -0.2 0.0 0.2 0.4 k || (A -1 ) a b cd e f Γ Γ K ← → M − k y ← → + k y FIG. 1:
Bulk and surface band structures of AFM MnSb Te . (a) Atomic structure of bulk MnSb Te withred, gray and green balls showing Mn, Te and Sb atoms. (b) Bulk electronic band structure calculated for theinterlayer AFM state modelled in the rhombohedral unit cell. (c) The evolution of the bulk band gap E g with thechange of the SOC constant λ . (d) Electronic structure of the MnSb Te (0001) surface. Gray areas correspond tothe bulk band structure projected onto the surface Brillouin zone (BZ). (e) Spin texture of the gapped Dirac state inthe vicinity of the BZ center. (f) The tight-binding calculated electronic bandstructure of the S -preserving (10¯11)surface. The regions with a continuous spectrum correspond to the 3D bulk states projected onto a 2D BZ.may either be gapless or not, depending on whetherthe corresponding surface preserves the S symmetry.Namely, the breakdown of this symmetry lifts theDirac point (DP) protection and even an infinitesimalmagnetization component perpendicular to the surfacewill cause the DP gap opening . To checkwhether MnSb Te shows such a behavior, we havecalculated the electronic structures of the S -breaking [(0001)] and S -preserving [(10¯11)] surfaces. In the lattercase, since the computational cell is very large, thecalculations have been done using the ab-initio -basedtight-binding method, which is more cost-effectivecomputationally. In Fig. 1d one can see that thespectrum of the S -breaking (0001) surface is indeedgapped, with the DP gap being equal to 14.8 meV. Thespin texture of the gapped Dirac state (Fig. 1e) changes -0.5-0.4-0.3-0.2-0.10.00.10.20.30.40.5 E - E F ( e V ) M K A L H A 0.00.020.040.060.080.10.12 E g ( e V ) SOC (%) -0.3-0.2-0.10.00.10.20.3 E - E F ( e V ) -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 k || (A -1 ) SL-QL-QL-SL- -0.3-0.2-0.10.00.10.20.3 E - E F ( e V ) -0.2 0.0 0.2 k || (A -1 ) a b cd e Γ Γ K ← → M − k y ← → + k y FIG. 2:
Bulk and surface band structures of AFM MnSb Te . (a) Atomic structure of bulk MnSb Te , withyellow, blue and green balls showing Mn, Te and Sb atoms, respectively. The interlayer AFM ground state is alsoreflected. (b) Bulk band spectrum calculated for the hexagonal unit cell. (c) The evolution of the bulk gap with thechange of the SOC strength. (d) Surface band structure of MnSb Te (0001) for the SL (blue) and QL (red) blockterminations. (e) The tight-binding calculated electronic band structure of the S -preserving side surface.from a helical far from the ¯Γ point to the hedgehogone in the vicinity of the gap with a purely out-of-planespin alignment at ¯Γ, positive and negative for the lowerand upper branches, respectively. The S -preserving(10¯11) surface features a gapless Dirac cone (Fig. 1f), asexpected for an AFMTI. These results constitute anotherproof of the AFMTI phase in MnSb Te .The prediction of the AFMTI state in MnSb Te implies that there exists a number of related intrinsicallymagnetic insulators that should be topologicallynontrivial. Indeed, as a rule, compounds with a formula XY Z ( X =Ge, Sn, Pb; Y = Sb, Bi and Z = Se, Te), giverise to a series of the ( XY Z ) · ( Y Z ) n form that retaina nontrivial topology . Let us therefore considera hypothetical MnSb Te compound, whose crystalstructure is formed by alternating quintuple layers (QLs)of Sb Te and SLs of MnSb Te (space group P ¯3 m Te is ordered ferromagnetically, whilethe interlayer exchange coupling is antiferromagnetic (seeTable I and Fig. 2a). Further calculations show thatthe system has an easy axis pointing out of the MnTABLE I: Total energy differences of the AFM and FM configurations inside the Mn layer(∆ E || = E NCAF M − E F M ) and between the neighboring layers (∆ E ⊥ = E AF M − E F M ). In the former case, theAFM state is represented by the coplanar non-collinear configuration of local moments, in which the magnetizationsof the three sublattices form angles of 120 ◦ with respect to each other . E diff = E in − plane − E out − of − plane , is thetotal energy difference of the in-plane and out-of-plane (oop) directions of the magnetic moment, E d − d is the energyof the dipole-dipole interaction, E a = E diff + E d − d is the magnetic anisotropy energy, and E g is the bulk band gap. a c m ∆ E || ∆ E ⊥ E diff E d − d E a E g (˚A) (˚A) ( µ B ) (meV/Mn pair) (meV/Mn pair) (meV/Mn at) (meV/Mn at) (meV/ Mn at) (meV)MnSb Te Te Te plane, but the magnetic anisotropy energy, E a , appearsto be significantly lower than that in MnSb Te : 0.05meV. The reduction of E a arises due to the reduced E diff value, while the dipolar contribution stays thesame because both the local magnetic moment andin-plane lattice parameter are practically equal to thosein MnSb Te , see Table I. It is reasonable to suggest that,similarly to how it happens in the (MnBi Te ) · (Bi Te ) n series , the critical temperature of the magneticordering will drop roughly twice when going fromMnSb Te to MnSb Te . This is due to the weakeningof both the interlayer exchange coupling and magneticanisotropy energy of the system (Table I).Like in the MnSb Te case, the combination of the P ¯3 m Te meets the S -symmetry. According to the bulkelectronic structure calculations, MnSb Te features aninverted bulk band gap of 0.1 eV (Fig. 2b,c), while thecalculated Z value of 1 confirms nontrivial topology ofthe system, which similarly to MnSb Te appears to bean AFMTI.An important difference between the MnSb Te andMnSb Te systems is a possibility of having twoterminations at the MnSb Te (0001) surface: the QL andSL ones. Consistently with the bulk edge correspondenceprinciple, there is a TSS at both these terminations(Fig. 2d), but the dispersions of the two TSSs are stronglydifferent. The TSS of the SL termination is locatedwithin the fundamental band gap and shows the DP gapof 32.5 meV, which is due to the S -symmetry breakingat the MnSb Te (0001) surface and the out-of-planemagnetization of the Mn layer. The dispersion seenat the QL termination appears to be much morecomplex, owing to more complex localization of theQL-terminated TSS in TIs with alternating QL/SLstructure which shows relocation of the state fromsurface QL to subsurface SL block in the vicinity of the¯Γ point. Such a behavior usually results in deviation ofthe TSS dispersion from linear near ¯Γ and even Diracstate can demonstrate strongly pronounced kink-likedispersion . In MnSb Te , when the magnetic momentson Mn atoms are artificially constrained to zero (Fig. S3 in Supplementary Information), the TSS at the QLtermination also demonstrates strong deviation fromlinear dispersion at k (cid:107) ≈ .
035 ˚A − . At the QLtermination of AFM MnSb Te (0001) the Dirac state hasmagnetic gap of 46.3 meV at ¯Γ, however, the lower branchof the state appears at -0.08 eV, between the magneticallysplit subbands of the upper bulk valence band (Fig. 2(d)).In contrast, the S -preserving side surface hosts a singlegapless TSS (Fig. 2e).Another potential member of the(MnSb Te ) · (Sb Te ) n family is MnSb Te . Inthis compound, adopting a trigonal structure with the R ¯3 m space group, each MnSb Te SL is separted by twoSb Te QLs (Fig. 3a), leading to even weaker exchangeinteraction between SLs, which, nevertheless, staysantiferromangetic (Table I). The bulk spectrum has adirect Γ-point gap of 91 meV (Fig. 3b), which is invertedas confirmed by its dependence on the SOC strength(Fig. 3c). Our Z invariant calculations (MnSb Te is S -symmetric as well) yield a value of 1, meaning that itis an AFMTI, too.Sharing the same crystal structure with MnBi Te and non-magnetic PbBi Te , MnSb Te shouldhave three possible surface terminations, QL-QL-SL-,QL-SL-QL- and SL-QL-QL-, showing differentdispersions of the TSS (Fig. 3d). The spectra ofthe SL-QL-QL- and QL-SL-QL-terminated surface ofMnSb Te in general are similar to those for SL-QL-and QL-SL- terminations in MnSb Te , c.f. Fig. 2dand Fig. 3d. The TSS dispersion at the double-QLtermination, QL-QL-SL-, is similar to that of theQL-SL-QL- termination as well.A common feature of MnSb Te and MnSb Te is aweak interlayer exchange coupling. As it can be seen fromTable I, the total-energy difference between the interlayerAFM and FM structures, ∆ E ⊥ = E AF M − E F M ,in MnSb Te (MnSb Te ) is more than 6 (10) timessmaller than that in MnSb Te , indicating a dramaticdrop in the interlayer exchange coupling strength due toinsertion of one (two) Sb Te QL block(s) between eachpair of MnSb Te SLs. With further increase in numberof the Sb Te QL blocks to three and formation of the -0.5-0.4-0.3-0.2-0.10.00.10.20.30.40.5 E - E F ( e V ) L Z F Z
MnSb Te E g ( e V ) SOC (%) -0.3-0.2-0.10.00.10.20.3 E - E F ( e V ) -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 k || (A -1 ) SL-QL-QL-QL-SL-QL-QL-QL-SL- -0.5-0.4-0.3-0.2-0.10.00.10.20.30.40.5 E - E F ( e V ) L Z F Z
MnSb Te a b cd e Γ Γ K ← → M Γ Γ
FIG. 3:
Bulk and surface band structures of AFM MnSb Te . (a) Atomic structure of bulk MnSb Te ,with yellow, blue and green balls showing Mn, Te and Sb atoms, respectively. The interlayer AFM ground state isalso reflected. (b) Bulk electronic structure calculated for the rhombohedral unit cell. (c) The evolution of the bulkband gap E g with the change of the SOC strength. (d) Surface band structure of MnSb Te (0001) for theSL-QL-QL- (blue), QL-SL-QL- (red), and QL-QL-SL- (green) terminations. Gray areas correspond to the bulk bandstructure projected onto the surface Brillouin zone. (e) Bulk spectrum of MnSb Te in the AFM state.MnSb Te compound the interlayer exchange couplingbetween ferromagnetic SLs practically vanishes. Thisleads to a situation where below the Curie temperatureof the MnSb Te SLs their magnetizations are randomlyoriented either parallel or antiparallel to the [0001]direction. If calculated in the imposed interlayer AFMstate, MnSb Te shows an inverted bulk band gap ofabout 100 meV (Fig. 3e).We envision, that similarly to(MnBi Te ) · (Bi Te ) n , the (MnSb Te ) · (Sb Te ) n homologous series can contain members with n >
3, i.e.MnSb Te , MnSb Te , etc., that are going to be topologically-nontrivial as well. This is because addingmore Sb Te QLs is expected to make the band gapinversion stronger, since the material becomes moreSb Te -like. Thus, if an n ≥ Te , MnSb Te , and MnSb Te or its absencefor n ≥ -1.0-0.50.00.51.0 E - E F ( e V ) L Z F Z
MnSb Te : PBE -0.050.00.05 E - E F ( e V ) Z -1.0-0.50.00.51.0 E - E F ( e V ) L Z F Z
MnSb Te : mBJ -0.050.00.05 E - E F ( e V ) Z-1.0-0.50.00.51.0 E - E F ( e V ) M K A L H A
MnSb Te -1.0-0.50.00.51.0 E - E F ( e V ) L Z F Z
MnSb Te -1.0-0.50.00.51.0 E - E F ( e V ) L Z F Z
MnSb Te (a) (b) (c) (d)(e) (f) (g) Γ Γ Γ
Γ Γ Γ Γ Γ Γ Γ Γ Γ
FIG. 4:
Bulk band structure of ferromagnetic phase of MnSb Te family compounds. (a) Spectrum ofMnSb Te as calculated within GGA-PBE+ U ; (b) Magnified view of the MnSb Te spectrum near the Fermi levelin the vicinity of the Γ point; (c) and (d) the same as in panels (a) and (b) but calculated with mBJ functional withHubbard U ; Bulk spectra of ferromagnetic MnSb Te (e), MnSb Te (f), and MnSb Te (g) calculated along highsymmetry directions of corresponding Brillouin zones (hexagonal or rhombohedral) within GGA-PBE+ U .phase by applying an external magnetic field along the c axis. Such a change in a magnetic state may cause changein the topology. In order to identify the topologicalphases of the (MnSb Te ) · (Sb Te ) n family compoundsin FM state we calculated the strong topological invariant Z (see Methods section). Z = 1 or 3 correspond toWeyl semimetal phases, where an odd number of Weylpoints exist in half of the BZ. Z = 2 corresponds to theAXI, having the quantized topological magnetoelectricresponse in the bulk and chiral hinge modes.Similarly to MnBi Te , where in FM state aWeyl semimetal phase was predicted , ferromagneticMnSb Te , according to Z calculations, is a Weylsemimetal, too ( Z = 1), in agreement with Refs. .Namely, the GGA+ U bulk spectrum (Fig. 4a,b) isgapless and shows a band crossing near the Fermi level,forming the Weyl points ( k W ) at ± − alongthe ΓZ direction just above the Fermi level. However,since the inversion of the bands at Γ is only 2 meV,the topological phase can be sensitive to the choice ofthe exchange-correlation functional. As can be seenin Figs. 4c,d, applying the modified Becke-Johnson(mBJ) semilocal exchange potential retains the Weylphase, providing larger Γ gap (20 meV) and a larger k W = ± − as well as shifting the Weyl pointsslightly below the Fermi level.The bulk spectra of MnSb Te (Fig. 4e) andMnSb Te (Fig. 4f) in the FM phase demonstrate directband gaps at the BZ boundaries with magnitudes of84 and 58 meV, respectively, whereas the MnSb Te spectrum shows indirect Γ-Z gap of 96.5 meV (Fig. 4g). At the same time, all these spectra demonstrate bandinversion of the Sb and Te p z orbitals at the Γ point.Calculations of the topological indices for ferromagneticphase of the (MnSb Te ) · (Sb Te ) n compounds with n =1, 2, and 3 gives Z = 2 and hence all of them arein the FM AXI state, which can be expected for n > Te , are stronglythickness dependent. The spectrum of a single MnSb Te SL demonstrates the indirect gap of 419.7 meV, whichis significantly larger than that in the MnBi Te SL( ∼ .
32 eV) . The direct ¯Γ-gap amounts to 811.4 meV.The Chern number calculations reveal a C = 0 state,the system being a topologically trivial ferromagnet(Table II). Upon increasing thickness up to 2, 3and 4 SLs the interlayer AFM order (compensatedor uncompensated for even and odd SL thicknesses,respectively) sets in, but C = 0 again for all of them.For these systems, we find the ¯Γ-point band gaps of210.2, 69.5, and 65.8 meV, respectively (see Table II).However, if calculated in the artificial FM phase (thatcould be achieved by applying external magnetic field),the Chern number appears to be equal to − C is stillzero). This means that 4 SLs of MnSb Te is a Cherninsulator that is expected to show the quantum Halleffect in the external magnetic field but without Landaulevels, as it has been recently reported for MnBi Te flakes . Reversing the magnetization of the FMTABLE II: Thickness dependence of the MnSb Te films topology and the ¯Γ band gap. Thickness (SL) Topology ¯Γ-gap (meV)1 Trivial 811.42 Trivial 210.23 Trivial 69.54 AFM AXI 65.85 Trivial 18.66 AFM AXI 30.47 QAHI 4.88 AFM AXI 14.8 ∝ (bulk) 3D AFMTI 124.0 Te film yields the C = +1 Cherninsulator state, thus proving a so-called zero plateauquantum Hall behavior in the compensated AFM stateof this film, characteristic of the AFM AXI .At a thickness of 5 SLs MnSb Te appears to be inthe trivial insulator phase again, the band gap beingequal to 18.6 meV. Similarly to the 4-SL-thick film case,we also predict the intrinsic AFM AXI states for the 6-and 8-SL-thick films (meaning that in the FM state theyare expected to show the quantum Hall effect withoutLandau levels), while the 7-SL-thick MnSb Te film isin the QAH phase with a gap of 4.8 meV (Table II).Thus, like MnBi Te in the thin film limit theMnSb Te films also demonstrate alternating AXI/QAHItopological phases, however, owing to a weaker SOC inMnSb Te , they emerge at larger thicknesses than in theMnBi Te case.We have further constructed variousMnSb Te /(Sb Te ) n /MnSb Te sandwiches withdifferent number of the Sb Te QLs, n = 1 − n = 1 and 2 we find topologically trivialsituation for both parallel and antiparallel alignmentsof the SLs magnetizations, the n = 3 system appearsto be in the AFM AXI and Chern insulator phasesfor the AFM and FM states, respectively. Similarbehavior is also expected for n = 4 and 5. Note thatMnSb Te /(Sb Te ) n /MnSb Te can in principle beobtained by careful exfoliation of the thin flakes fromthe (MnSb Te ) · (Sb Te ) n single crystals. CONCLUSIONS
Using state-of-the-art ab initio calculations wehave predicted the existence of a new family ofmagnetic topological insulators of the van der Waalslayered compounds (MnSb Te ) · (Sb Te ) n . In theirgroundstate, the MnSb Te ( n = 0), MnSb Te ( n =1), and MnSb Te ( n = 2) materials appear to beantiferromagnetic topological insulators. For n ≥ Te and on), a special magnetictopological insulator phase is formed in which, below critical temperature, the magnetizations of the 2Dferromagnetically ordered Mn layers of the MnSb Te building blocks are disordered along the [0001] direction.We have further shown that imposing the overallferromagnetic state (which can be achieved by applyingexternal magnetic field) drives MnSb Te into a Weylsemimetal phase, while the compounds with higher n become 3D ferromagnetic axion insulators. Finally, inthe 2D limit, (MnSb Te ) · (Sb Te ) n films show a varietyof the topologically-nontrivial states, being among themintrinsic axion and quantum anomalous Hall insulators,as well as quantum Hall state, that could be achievedunder external magnetic field, but without Landau levels.Our results convincingly show that MnSb Te is nottrivial as it has been concluded by the previous abinitio works and we actually demonstrate the reasonwhy those studies found it to be trivial. Obviously, the(MnSb Te ) · (Sb Te ) n family represents a fertile groundfor realizing multiple and tunable exotic states and theexperimental realization of this family would represent agreat advance in the research of the magnetic topologicalmatter. Note.
Very recently, the first member of the(MnSb Te ) · (Sb Te ) n series, i.e. MnSb Te , has beenproved in the experiment as a magnetic topologicalinsulator . METHODSElectronic structure and total-energy calculations
Electronic structure calculations were carried outwithin the density functional theory using the projectoraugmented-wave (PAW) method as implemented in theVASP code . The exchange-correlation energy wastreated using the generalized gradient approximation .The Hamiltonian contained scalar relativistic correctionsand the spin-orbit coupling was taken into accountby the second variation method . The energy cutofffor the plane-wave expansion was set to 270 eV. Thecrystal structure of each compound was fully optimizedto find the equilibrium lattice parameters, cell volumes, c/a ratios as well as atomic positions. At that, aconjugate-gradient algorithm was used. In order todescribe the van der Waals interactions we made use ofthe DFT-D2 and the DFT-D3 approaches, whichgave similar results. The atomic coordinates were relaxedusing a force tolerance criterion for convergence of 0.01eV/˚A. Spin-orbit coupling was always included whenperforming relaxations.The Mn 3 d -states were treated employing the GGA+ U approach within the Dudarev scheme . The U eff = U − J value for the Mn 3 d -states was chosen to beequal to 5.34 eV, as in previous works .Further extensive testing was performed for MnSb Te ,MnSb Te and MnSb Te systems to check thestability of the results against U eff variation. Namely, thebulk crystal structure was fully optimized for U eff = 3 eVand then the magnetic ordering and electronic structurewere studied. It was found that neither the magneticground state nor the topological class change upon sucha variation of U eff and crystal structure. The magneticanisotropies of (MnSb Te ) · (Sb Te ) n were found to bestable against this variation as well.The bulk magnetic ordering was studied usingtotal-energy calculations, performed for the FM andtwo different AFM states. Namely, we considered aninterlayer AFM state and a noncollinear intralayer AFMstate, in which three spin sublattices form angles of 120 ◦ with respect to each other . To model the FM andinterlayer AFM structures in (MnSb Te ) · (Sb Te ) n , weused rhombohedral cells for n = 0 ,
2, and 3, while ahexagonal cell was chosen for n = 1 (all cells containedthe number of atoms corresponding to two formulaunits). For the n ≤ √ × √ R ◦ in-plane periodicity]. The magnetic anisotropy energywas calculated as explained in Ref. .The (MnSb Te ) · (Sb Te ) n semi-infinite surfaces weresimulated within a model of repeating films separatedby a vacuum gap of a minimum of 10 ˚A. The interlayerdistances were optimized for the utmost SL or QL blockof each surface.Additional bulk electronic structure calculations forthe AFM MnSb Te (Supplementary Figure S1) wereperformed within the full-potential linearized augmentedplane waves (FLAPW) formalism as implemented inFLEUR . We took the GGA+ U approach under thefully localised limit with the same U eff =5.34 eV aswe use in VASP calculations. The core states weretreated fully relativistically, while the valence stateswere computed in a scalar relativistic approximationwith taking into account the spin-orbit coupling. Themuffin-tin radii for Mn, and Te were chosen to be 2.70a.u., while for Sb we used 2.87 a.u. Inside each muffin-tinsphere, the basis functions were expanded in sphericalharmonics with angular momentum up to l max =10, andthe electron density and potential were expanded up to l max =8. The crystal wave functions were expanded intoaugmented plane waves with a cutoff of k max =3.7 a.u. − ,corresponding approximately to the 180 basis LAPWfunctions per atom.The bulk electronic structure of the AFM MnSb Te was also calculated using the HUTSEPOT code basedon the Green’s function method within the multiplescattering theory (Supplementary Figure S2). Todescribe both localization and interaction of the Mn 3 d orbitals, the Hubbard U eff values between 3 and 5 eVwere employed using the GGA+ U approach .Ab-initio-based tight-binding calculations wereperformed using the VASP package with the Wannier90interface . The Wannier basis chosen consisted ofsix spinor p -type orbitals of Sb and Te. The surfaceelectronic band structure was calculated within thesemi-infinite medium Green’s function approach . Topological invariants calculations
The Chern numbers were calculated using Z2Pack .The Z invariant for the 3D AFMTIs was calculatedusing the expression derived in Ref. [52]. When spatialinversion symmetry is present in the system, as it is in theMnSb Te -family compounds, the following Z invariant λ can be defined:( − λ = (cid:89) k inv ∈ B − TRIM ,n ∈ occ/ ζ n ( k inv ) , where ζ n is the parity of the n th occupied band andthe multiplication over occ /2 means only one state in aKramers’ pair is chosen. See Ref. [52] for further details.Calculations of Z topological invariant were doneaccording to Refs. as Z = (cid:88) α =1 n occ (cid:88) n =1 ζ n (Λ α )2 mod 4 , where Λ α are the eight inversion-invariant momenta, n is the band index, n occ is the total number of electrons,and ζ n (Λ α ) is the parity of the n -th band at Λ α . M. M. Otrokov, I. I. Klimovskikh, H. Bentmann,D. Estyunin, A. Zeugner, Z. S. Aliev, S. Gaß, A. U. B.Wolter, A. V. Koroleva, A. M. Shikin, M. Blanco-Rey,M. Hoffmann, I. P. Rusinov, A. Y. Vyazovskaya, S. V.Eremeev, Y. M. Koroteev, V. M. Kuznetsov, F. Freyse,J. S´anchez-Barriga, I. R. Amiraslanov, M. B. Babanly,N. T. Mamedov, N. A. Abdullayev, V. N. Zverev,A. Alfonsov, V. Kataev, B. B¨uchner, E. F. Schwier,S. Kumar, A. Kimura, L. Petaccia, G. Di Santo, R. C.Vidal, S. Schatz, K. Kißner, M. ¨Unzelmann, C. H. Min,S. Moser, T. R. F. Peixoto, F. Reinert, A. Ernst, P. M.Echenique, A. Isaeva, E. V. Chulkov, Prediction and observation of an antiferromagnetic topological insulator.
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