Weyl Semimetal Phase in Noncentrosymmetric Transition-Metal Monophosphides

PHYSICALREVIEWX5,011029(2015)

WeylSemimetalPhaseinNoncentrosymmetricTransition-MetalMonophosphides

HongmingWeng,1,2,*ChenFang,3ZhongFang,1,2B.AndreiBernevig,4andXiDai1,2

BeijingNationalLaboratoryforCondensedMatterPhysics,andInstituteofPhysics,

ChineseAcademyofSciences,Beijing100190,China

2

CollaborativeInnovationCenterofQuantumMatter,Beijing100084,China

3

DepartmentofPhysics,MassachusettsInstituteofTechnology,Cambridge,Massachusetts02139,USA

4

DepartmentofPhysics,PrincetonUniversity,Princeton,NewJersey08544,USA

(Received12January2015;published17March2015)Basedonfirst-principlecalculations,weshowthatafamilyofnonmagneticmaterialsincludingTaAs,TaP,NbAs,andNbPareWeylsemimetals(WSM)withoutinversioncenters.WefindtwelvepairsofWeylpointsinthewholeBrillouinzone(BZ)foreachofthem.Intheabsenceofspin-orbitcoupling(SOC),bandinversionsinmirror-invariantplanesleadtogaplessnodalringsintheenergy-momentumdispersion.ThestrongSOCinthesematerialsthenopensfullgapsinthemirrorplanes,generatingnonzeromirrorChernnumbersandWeylpointsoffthemirrorplanes.Theresultingsurface-stateFermiarcstructuresonboth(001)and(100)surfacesarealsoobtained,andtheyshowinterestingshapes,pointingtofascinatingplaygroundsforfutureexperimentalstudies.

DOI:10.1103/PhysRevX.5.011029

SubjectAreas:CondensedMatterPhysics,

MaterialsScience,TopologicalInsulators

1

I.INTRODUCTION

Mosttopologicalinvariantsincondensed-matternon-interactingphasesaredefinedonclosedmanifoldsinmomentumspace.Forgappedsystems,boththeCherninsulatorandZ2topologicalinsulatorphasescanbedefinedusingtheBerryphaseandcurvatureineithertheentireorhalfofthetwo-dimensional(2D)Brillouinzone(BZ),respectively[1,2].Asimilarideacanbegeneralizedtogaplessmetallicsystems.Inthree-dimensional(3D)sys-tems,besidestheBZ,animportantclosedmanifoldinmomentumspaceisa2DFermisurface(FS).TopologicalmetalscanbedefinedbyChernnumbersofthesingle-particlewavefunctionsattheFermisurfaceenergies[3–5].SuchnonzeroFSChernnumbersappearwhentheFSenclosesaband-crossingpoint—theWeylpoint—whichcanbeviewedasasingularpointofBerrycurvatureor“magneticmonopole”inmomentumspace[6–9].MaterialswithsuchWeylpointsneartheFermilevelarecalledWeylsemimetals(WSM)[7–10].

Weylpointscanonlyappearwhenthespin-doubletdegeneracyofthebandsisremovedbybreakingeithertimereversalTorspacialinversionsymmetryP(infact,WeylpointsexistifthesystemdoesnotrespectT·P).Inthesecases,thelow-energysingle-particleHamiltonianarounda

*

hmweng@http://wendang.chazidian.com

PublishedbytheAmericanPhysicalSocietyunderthetermsoftheCreativeCommonsAttribution3.0License.Furtherdistri-butionofthisworkmustmaintainattributiontotheauthor(s)andthepublishedarticle’stitle,journalcitation,andDOI.

Weylpointcanbewrittenasa2×2“Weylequation,”whichishalfoftheDiracequationinthreedimensions.Accordingtothe“no-gotheorem”[11,12],foranylatticemodel,theWeylpointsalwaysappearinpairsofoppositechiralityormonopolecharge.TheconservationofchiralityisoneofthemanywaystounderstandthetopologicalstabilityoftheWSMagainstanyperturbationthatpre-servestranslationalsymmetry:TheonlywaytoannihilateapairofWeylpointswithoppositechiralityistomovethemtothesamepointinBZ.SincegenericallytheWeylscansitfarawayfromeachotherintheBZ,thisrequireslargechangesofHamiltonianparameters,andtheWSMisstable.TheexistenceofWeylpointsneartheFermilevelwillleadtoseveraluniquephysicalproperties,includingtheappear-anceofdiscontinuousFermisurfaces(Fermiarcs)onthesurface[7–9],theAdler-Bell-Jackiwanomaly[10,13–15],andothers[16,17].

ThefirstproposaltorealizeWSMincondensed-mattermaterialswassuggestedinRef.[7]forRn2Ir2O7pyro-chlorewithall-in/all-outmagneticstructure,where24pairsofWeylpointsemergeasthesystemundergoesthemagneticorderingtransition.ArelativelysimplersystemHgCr2Se4[9]wasthenproposedbysomeofthepresentauthors,whereapairofdouble-Weylpointsduetoquadraticbandcrossingappearwhenthesystemisinaferromagneticphase.Anotherproposalinvolvesafine-tunedmultilayerstructureofnormalinsulatorsandmag-neticallydopedtopologicalinsulators[18].TheseproposedWSMsystemsinvolvemagneticmaterials,wherethespindegeneracyofthebandsisremovedbybreakingtime-reversalsymmetry.Asmentioned,theWSMcanalsobe

2160-3308=15=5(1)=011029(10)011029-1PublishedbytheAmericanPhysicalSociety

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generatedbybreakingthespatialinversionsymmetryonly,amethodwhichhasthefollowingadvantages.First,comparedwithmagneticmaterials,nonmagneticWSMaremuchmoreeasilystudiedexperimentallyusingangle-resolvedphotoemissionspectroscopy(ARPES)asalignmentofmagneticdomainsisnolongerrequired.Second,withoutthespinexchangefield,theuniquestructureofBerrycurvatureleadstoveryunusualtransportpropertiesunderastrongmagneticfield,unspoiledbythemagnetismofthesample.

Currently,thereareseveralrepresentativeproposalsforWSMgeneratedbyinversionsymmetrybreaking.Thefirstoneisasuperlatticesystemformedbyalternativelystack-ingnormalandtopologicalinsulators[19,20].Thesecondoneinvolvestelluriumorseleniumcrystalsunderpressure[21].ThethirdoneisthesolidsolutionsofABi1?xSbxTe3(A¼LaandLu)[22]andTlBiðS1?xRxÞ2(R¼SeorTe)[23]tunedaroundthetopologicaltransitionpoints[24].Thefourthoneisamodelbasedonzinc-blendestructure[25]withthefine-tuningoftherelativestrengthbetweenSOCandtheinversionsymmetry-breakingterm.Butnoneoftheaboveproposalshasbeenrealizedexperimentally.Inthepresentstudy,wepredictthatTaAs,TaP,NbAs,and

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NbPsinglecrystalsarenaturalWSM,http://wendang.chazidian.comparedwiththeexistingproposals,thisfamilyofmaterialsiscom-pletelystoichiometricand,therefore,areeasiertogrowandmeasure.UnlikeinthecaseofpyrochloreiridatesandHgCr2Se4,whereinversionisstillagoodsymmetryandtheappearanceofWeylpointscanbeimmediatelyinferredfromtheproductoftheparitiesatallthetime-reversalinvariantmomenta(TRIM)[26–28],intheTaAs,familyparityisnolongeragoodquantumnumber.However,theappearanceofWeylpointscanstillbeinferredbyanalyzingthemirrorChernnumbers(MCN)[29,30]andZ2indices[26,31]forthefourmirrorandtime-reversalinvariantplanesintheBZ.Similartomanyothertopologicalmaterials,theWSMphaseinthisfamilyisalsoinducedbyatypeofband-inversionphenomena,which,intheabsenceofspin-orbitcoupling(SOC),leadstonodalringsinthemirrorplane.OncetheSOCisturnedon,eachnodalringwillbegappedwiththeexceptionofthreepairsofWeylpointsleadingtofascinatingphysicalpropertieswhichincludecomplicatedFermiarcstructuresonthesurfaces.

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FIG.1.CrystalstructureandBrillouinzone(BZ).(a)ThecrystalsymmetryofTaAs.(b)ThebulkBZandtheprojectedsurfaceBZforboth(001)and(100)surfaces.(c)ThebandstructureofTaAscalculatedbyGGAwithoutincludingthespin-orbitcoupling.(d)ThebandstructureofTaAscalculatedbyGGAwiththespin-orbitcoupling.

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II.CRYSTALSTRUCTUREANDCALCULATIONMETHODS

Asallfourmentionedmaterialsshareverysimilarbandstructures,intherestofthepaper,wewillchooseTaAsastherepresentativematerialtointroducetheelectronicstructuresofthewholefamily.TheexperimentalcrystalstructureofTaAs[32]isshowninFig.1(a).Itcrystalizesinbody-centered-tetragonalstructurewithnonsymmorphicspacegroupI41md(No.109),whichlacksinversionsymmetry.Themeasuredlatticeconstantsarea¼b¼3.4348Åandc¼11.641Å.BothTaandAsareat4aWyckoffposition(0,0,u)withu¼0and0.417forTaandAs,respectively.WehaveemployedthesoftwarepackageOpenMX[33]forthefirst-principlescalculation.Itisbasedonnorm-conservingpseudopotentialandpseudo-atomiclocalizedbasisfunctions.Thechoiceofpseudopotentials,pseudo-atomicorbitalbasissets(Ta9.0-s2p2d2f1andAs9.0-s2p2d1),andthesamplingofBZwitha10×10×10gridhavebeencarefullychecked.Theexchange-correlationfunctionalwithinageneralizedgradientapproximation(GGA)parametrizedbyPerdew,Burke,andErnzerhofhasbeenused[34].Afterfullstructuralrelaxation,weobtainthelatticeconstantsa¼b¼3.4824Å,c¼11.8038Åandoptimizedu¼0.4176fortheAssite,inverygoodagreementwiththeexperimentalvalues.TocalculatethetopologicalinvariantsuchasMCNandsurfacestatesofTaAs,wehavegeneratedatomiclikeWannierfunctionsforTa5dandAs4porbitalsusingtheschemedescribedinRef.[35].

III.RESULTS

A.Bandstructureswithandwithoutspin-orbitcoupling

WefirstobtainthebandstructureofTaAswithoutSOCbyGGAandplotitalongthehigh-symmetrydirectionsinFig.1(c).WefindclearbandinversionandmultiplebandcrossingfeaturesneartheFermilevelalongtheZN,ZS,andΣSlines.ThespacegroupoftheTaAsfamilycontainstwomirrorplanes,namely,MxandMy[shadedplanesinFig.1(b)]andtwoglidemirrorplanes,namely,MxyandM?xy[illustratedbythedashedlinesinFig.1(b)].TheplanespannedbyZ,N,andΓpointsisinvariantundermirrorMy,andtheenergybandswithintheplanecanbelabeledbymirroreigenvaluesÆ1.FurthersymmetryanalysisshowsthatthetwobandsthatcrossalongtheZtoNlinebelongtooppositemirroreigenvalues,andhence,thecrossingbetweenthemisprotectedbymirrorsym-metry.Similarbandcrossingscanalsobefoundalongotherhigh-symmetrylinesintheZNΓplane,i.e.,theZSandNSlines.Altogether,thesebandcrossingpointsforma“nodalring”intheZNΓplaneasshowninFig.2(b).UnlikeforthesituationintheZNΓplane,inthetwoglidemirrorplanes

PHYS.REV.X5,011029(2015)

(MxyandM?xy),thebandstructureisfullygapped,withaminimumgapofroughly0.5eV.

TheanalysisoforbitalcharactershowsthatthebandsneartheFermienergyaremainlyformedbyTa5dorbitals,whichhavelargeSOC.IncludingSOCinthefirst-principlecalculationleadstoadramaticchangeofthebandstructureneartheFermilevel,asplottedinFig.1(d).Atfirstglance,itseemsthatthepreviousbandcrossingsintheZNΓplaneareallgapped,withtheexceptionofonepointalongtheZNline.Detailedsymmetryanalysisrevealsthatthebands“2”and“3”inFig.1(d)belongtooppositemirroreigenvalues,indicatingthealmost-touchingpointalongtheZNlineiscompletelyaccidental.Infact,thereisasmallgapofroughly3meVbetweenbands“2”and“3”asillustratedbytheinsetofFig.1(d).TheZNΓplanethenbecomesfullygappedonceSOCisturnedon.

B.Topologicalinvariantsformirrorplane

andWeylpointsSincethematerialhasnoinversioncenter,theusualparitycondition[26–28]cannotbeappliedtopredicttheexistenceofWSM.Wethenresorttoanotherstrategy.Aspreviouslymentioned,thespacegroupofthematerialprovidestwomirrorplanes(MxandMy),wheretheMCNcanbedefined.IfafullgapexistsfortheentireBZ,theMCNwoulddirectlyrevealwhetherthissystemisatopologicalcrystallineinsulatorornot.Interestingly,asshownbelow,ifthesystemisnotfullygapped,wecanstillusetheMCNtofindoutwhetherthematerialhostsWeylpointsintheBZornot.Besidesthetwomirrorplanes,wehavetwoadditionalglidemirrorplanes(MxyandM?xy).AlthoughtheMCNisnotwelldefinedfortheglidemirrorplanes,theZ2indexisstillwelldefinedhereastheseplanesaretime-reversalinvariant.WethenapplytheWilson-loopmethodtocalculatetheMCNsforthetwomirrorplanesandZ2indicesforthetwoglidemirrorplanes.Here,wejustbrieflydescribetheessenceofthismethod.Foramoredetailedexplanationofthemethod,pleaserefertoRefs.[5,37].AWilsonloopisanarbitraryclosedk-pointloopinBZ,evaluatedaroundwhich,theoccupiedBlochfunctionsacquireatotalBerryphaseθðwÞ,withwbeingtheloopindex.OnecandefineaseriesofparallelWilsonloopswtofullycoveraclosed2Dmanifoldin3Dmomentumspace,suchasacutplaneinBZoraclosedFSasstatedinthebeginningofthispaper.Then,theevolutionofθðwÞalongtheseparallelWilsonloops(itturnsouttobea1Dproblem)givesinformationontheband-structuretopologyontheclosed2Dmanifold.Forexample,todeterminetheMCNsforthemirrorplaneMx,wedefineWilsonloopsalongthekxdirectionwithfixedkz.Alltheoccupiedbandsatkpointsinthisplanecanbeclassifiedintotwogroupsaccordingtotheireigenvaluesundermirroroperation,ior?i.Takingthosehavingeigenvaluei,theevolutionofBerryphasesalongtheperiodickzdirectioncanbeobtained,andtheMCNissimplyitswindingnumber.

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TheresultsareplottedinFig.2(d),whichshowsthatMCNis1fortheZNΓplane(My)andtheZ2indexisevenortrivialfortheZXΓplane(Mxy).Then,ifweconsiderthe(001)surface,whichisinvariantundertheMymirror.ThenontrivialhelicalsurfacemodeswillappearbecauseofthenonzeroMCNintheZNΓplane,whichgeneratesasinglepairofFScutsalongtheprojectivelineoftheZNΓplane[thexaxisinFig.2(c)].WhethertheseFermicutswilleventuallyformasingleclosedFermicircleornotdependsontheZ2indexforthetwoglidemirrorplanes,whichareprojectedtothedashedbluelinesinFig.2(c).SincetheZ2indicesfortheglidemirrorplanesaretrivial,asconfirmedbyourWilson-loopcalculationplottedinFig.2(d),therearenoprotectedhelicaledgemodesalongtheprojectivelinesoftheglidemirrorplanes[dashedbluelinesinFig.2(c)],andtheFermicutsalongthexaxisinFig.2(c)mustendsomewherebetweenthexaxisandthediagonallines[dashedbluelinesinFig.2(c)].Inotherwords,theymustbeFermiarcs,indicatingtheexistenceofWeylpointsinthebulkbandstructureofTaAs.

PHYS.REV.X5,011029(2015)

FromtheaboveanalysisoftheMCNandZ2indexofseveralhigh-symmetryplanes,wecanconcludethatWeylpointsexistintheTaAsbandstructure.WenowdeterminethetotalnumberofWeylpointsandtheirexactpositions.Thisisahardtask,astheWeylpointsarelocatedatgenerickpointswithoutanylittle-groupsymmetry.Forthispurpose,wecalculatetheintegraloftheBerrycurvatureonaclosedsurfaceinkspace,whichequalsthetotalchiralityoftheWeylpointsenclosedbythegivensurface.BecauseofthefourfoldrotationalsymmetryandmirrorplanesthatcharacterizeTaAs,weonlyneedtosearchfortheWeylpointswithinthereducedBZ—one-eighthofthewholeBZ.WefirstcalculatethetotalchiralityormonopolechargeenclosedinthereducedBZ.Theresultis1,whichguaranteestheexistenceof,andoddnumberof,Weylpoints.TodeterminepreciselythelocationofeachWeylpoint,wedividethereducedBZintoaverydensek-pointmeshandcomputetheBerrycurvatureorthe“magneticfieldinmomentumspace”[35,38]onthatmesh,asshowninFig.3.Fromthis,wecaneasilyidentifytheprecise

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FIG.2.NodalringsandWeylpointsdistribution,aswellasZ2andMCNformirrorplanes.(a)3Dviewofthenodalrings(intheabsenceofSOC)andWeylpoints(withSOC)intheBZ.(b)Sideviewfrom[100]and(c)topviewfrom[001]directionsforthenodalringsandWeylpoints.OncetheSOCisturnedon,thenodalringsaregappedandgiverisetoWeylpointsoffthemirrorplanes(seemovieinSupplementalMaterial[36]).(d)Toppanel:FlowchartoftheaveragepositionoftheWanniercentersobtainedbyWilson-loopcalculationforbandswithmirroreigenvalueiinthemirrorplaneZNΓ.(d)Bottompanel:TheflowchartoftheWanniercentersobtainedbyWilson-loopcalculationforbandsintheglidemirrorplaneZXΓ.Thereisnocrossingalongthereferenceline(thedashedline),indicatingtheZ2indexiseven.

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PHYS.REV.X5,011029(2015)

TABLEI.ThetwononequivalentWeylpointsinthexyzcoordinatesshowninFig.1(b).ThepositionisgiveninunitsofthelengthofΓ-ΣforxandyandofthelengthofΓ-Zforz.

Weylnode1

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appearanceofWeylpointscanalsobederivedfromak·pmodelwithdifferenttypesofmasstermsinducedbySOC,whichwillbeintroducedindetailintheAppendix.Thebandstructuresfortheotherthreematerials—TaP,NbAs,andNbP—areverysimilar.TheprecisepositionsoftheWeylpointsforallthesematerialsaresummarizedinTable.I.

C.Fermiarcsandsurfacestates

UniquesurfacestateswithunconnectedFermiarcscanbefoundonthesurfaceofaWSM.Thesecanbeunder-stoodinthefollowingway:ForanysurfaceofaWSM,wecanconsidersmallcylindersinthemomentumspaceparalleltothesurfacenormal.Inthe3DBZ,thesecylinderswillbecutbythezoneboundary,andtheirtopologyisequivalenttothatofaclosedtorusratherthanthatofopencylinders.IfacylinderenclosesaWeylpoint,byStokestheorem,thetotalintegraloftheBerrycurvature(Chernnumber)ofthisclosedtorusmustequalthetotal“monopolecharge”carriedbytheWeylpoint(s)enclosedinside.Onthesurfaceofthematerial,suchacylinderwillbeprojectedtoacyclesurroundingtheprojectionpointoftheWeylpoint,andasingleFermisurfacecutstemmingfromthechiraledgemodelofthe2DmanifoldwithChernnumber1(or?1)mustbefoundonthatcircle.Byvaryingtheradiusofthecylinder,itiseasytoshowthatsuchFSsmuststartandendattheprojectionoftwo(ormore)Weylpointswithdifferent“monopolecharge”;i.e.,theymustbe“Fermiarcs”[7,9,16].IntheTaAsmaterialsfamily,onmostofthecommonsurfaces,multipleWeylpointswillbeprojectedontopofeachother,andwemustgeneralizetheaboveargumenttomultipleprojectionsofWeylpoints.ItiseasytoprovethatthetotalnumberofsurfacemodesattheFermilevelcrossingaclosedcircleinsurfaceBZmustequalthesumofthe“monopolecharge”oftheWeylpointsinsidethe3Dcylinderthatprojectstothegivencircle.AnotherfactcontrollingthebehaviorofthesurfacestatesistheMCNintroducedinthepreviousdiscussion,whichlimitsthenumberofFSscuttingcertainprojectionlinesofthemirrorplane(whenthecorrespondingmirrorsymmetriesarestillpreservedonthesurface).

ByusingtheGreen’sfunctionmethod[5]basedonthetight-binding(TB)Hamiltoniangeneratedbytheprevi-ouslyobtainedWannierfunctions,wehavecomputedthe

FIG.3.BerrycurvaturefrompairsofWeylpoints.(a)ThedistributionoftheBerrycurvatureforthekz¼0plane,wheretheblueandreddotsdenotetheWeylpointswithchiralityofþ1and?1,respectively;(b)sameas(a)butforthekz¼0.592πplane.Theinsetsshowthe3DviewofhedgehoglikeBerrycurvaturenearthetwoselectedWeylpoints.

positionoftheWeylpointsbysearchingforthe“source”and“drain”pointsofthe“magneticfield.”TheWeylpointsinTaAsareillustratedinFig.2(a),wherewefind12pairsofWeylpointsinthevicinityofwhatusedtobe,intheSOC-freecase,thenodalringsontwoofthemirror-invariantplanes.Foreachofthemirror-invariantplanes,afterturningonSOC,thenodalringswillbefullygappedwithintheplane,butisolatedgaplessnodesslightlyoffplaneappear,asillustratedinFig.2(b).TwopairsofWeylpointsarelocatedexactlyinthekz¼0plane,andanotherfourpairsofWeylpointsarelocatedoffthekz¼0plane.Consideringthefourfoldrotationalsymmetry,itistheneasytounderstandthatthereareatotalof12pairsofWeylpointsinthewholeBZ.TheWeylpointsinthekz¼0planeareabout2meVabovetheFermienergyandformeighttinyholepockets,whiletheothersareabout21meVbelowtheFermileveltoform16electronpockets.The

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