Electronicstructures oftopologicalquantummaterials studiedbyARPES
LexianYanga,HaifengYangb,andYulinChena,b,c,*
aStateKeyLaboratoryofLowDimensionalQuantumPhysicsandDepartmentofPhysics,TsinghuaUniversity, Beijing,China
bSchoolofPhysicalScienceandTechnology,ShanghaiTechUniversity,Shanghai,China
cDepartmentofPhysics,UniversityofOxford,Oxford,UnitedKingdom
∗Correspondingauthor:e-mailaddress:yulin.chen@physics.ox.ac.uk Contents
Abbreviations
2D two-dimension(al)
3D three-dimension(al)
ARPES angle-resolvedphotoemissionspectroscopy
MBS Majoranaboundstates
QH quantumHall
QSH quantumspinHall
SOC spin-orbitalcoupling
STM scanningtunnelingmicroscope
TCI topologicalcrystallineinsulator
TCS topologicalchiralsemimetal
TDS topologicalDiracsemimetal
TI topologicalinsulator
TMD transitionmetaldichalcogenide
TNLS topologicalnodallinesemimetal
SemiconductorsandSemimetals,Volume108Copyright # 2021ElsevierInc.
ISSN0080-8784Allrightsreserved.
https://doi.org/10.1016/bs.semsem.2021.07.004
CHAPTERONE
1. IntroductiontoARPES2 1.1 Basicconcept 2 1.2 Generalprinciple 5 1.3 Experimentalinstrument 10 1.4 ARPESspectrum 15 2. ARPESstudiesontopologicalquantummaterials 16 2.1 Topologicalinsulatingphases 17 2.2 Topologicalsemimetals 22 2.3 Topologicalsuperconductors 32 3. Summaryandperspective 34 References 35
1
TQM(s) topologicalquantummaterial(s)
TRS time-reversalsymmetry
TSC topologicalsuperconductor
TSS topologicalsurfacestates
TWS topologicalWeylsemimetal
UHV ultrahighvacuum
UV ultraviolet
1.IntroductiontoARPES
1.1Basicconcept
ARPESisbasedonthephotoelectriceffectHeinrichHertzdiscoveredin 1887whenstudyingthesparkdischargeeffecttoconfirmMaxwell’selectromagnetictheory(Hertz,1887).Hefoundthatthemaximumkinetic energy ofphotoelectronsisindependentonthelightintensitybutproportionaltothefrequencyoftheincidentlight.Moreover,thelightfrequency mustbehigherthanamaterial-dependentthresholdvaluetoliberateelectronsfromsolids.Lateron,AlbertEinsteinsuccessfullyresolvedthesemysteriesbythesimpleconceptofphotonandwasawardedtheNobelPrizein 1905(Einstein,1905).Inhistheory,themaximumkineticenergyofphotoelectrons reads:
,(1) where hυ isthephotonenergy, ϕ iscalledtheworkfunctionofthesolid material(H € ufner, 2003).
From theperspectiveofelectronicstructure,electronsinsolidsarebound atthebindingenergy EB withrespecttotheFermienergy EF (thephotoelectronsatmaximumkineticenergyareexcitedfrom EF ofthesolids).Thegeneralrelationshipbetweenthekineticenergyofphotoelectronsandthe bindingenergycanthusbewrittenas(Fig.1A): E
Therefore,ifwecanaccuratelymeasurethekineticenergyofphotoelectrons,wecancalculatethebindingenergyofelectronsinsolids.
Likewise,themomentumofelectronsinsolidscanbededucedfromthe momentumofphotoelectrons( K jj¼ 2mE kin p )with m theelectronmass,if thephotoemissionprocessrespectsthemomentumconservationlaw.
υ ϕ
E max kin ¼ h
kin ¼ hυ ϕ E B jj: (2)
2 LexianYangetal.
Fig.1 BasicworkingprincipleofARPES.(A)Theenergeticsofphotoemissionprocess.(B)Schematicoftheemissionandcollectionof photoelectrons.
However,thisisnotcompletelytruesincethetranslationalsymmetryperpendiculartothesamplesurfaceisbroken.Fortunately,thetranslational symmetryparalleltothesamplesurfaceisstillrespectedthustheparallelelectronmomentumofphotoelectronsisconservedduringphotoemission.
Usinganelectronanalyzer,wecandirectlyrecordthekineticenergyand emissionangles(θ , φ)ofphotoelectronsasschematicallyshownin Fig.1B.Consequently,theparallelelectronmomentum kk insolidscan becalculatedaccordingto
For theelectronmomentumintheverticaldirection, kz,althoughitisnot conservedduringthephotoemissionprocess,wecanapproximatelydetermineitunderreasonableassumptions.Themostcommonlyusedisthe free-electronfinalstateassumption,basedonwhichthe kz isdeducedas:
where E0 istheenergyofthevalencebandbottom.Notethat Ef and E0 are withrespectto EF,while Ekin iswithrespecttothevacuumenergylevel. Thus Ef ¼ Ekin + ϕ (see Fig.1A).UsingEqs.(3, 4),weobtain:
where V0 ¼j E0 j + ϕ iscalledtheinnerpotential.Underthisassumption,itis easytonotethat kz dispersioncanbemeasuredbyphoton-energydependentARPESmeasurements,fromwhichtheinnerpotentialcanbedeterminedbymatchingtheperiodicityofthe kz dispersionwiththatofBrillouin zone(Damascellietal.,2003; Hufner, 2003; Luetal.,2012).Thefactthat kz dependson θ suggeststhattheelectronsphotoemittedbyadeterminant photonenergyarefromacurved kz sphere.Ithasbeendemonstratedthatthe assumptionoffree-electronfinalstateisnotonlysuitableforsimplemetals butalsoapplicableforcomplexcompoundssuchascorrelatedmaterials. Photon-energydependentmeasurementisparticularlyusefulfortheidentificationoftopologicalsurfacestates(TSS)ofTQMs(Chen,2012; Chen et al.,2020; Lvetal.,2019a,2021; Yangetal.,2018; Zhangetal., 2020),sincethesurfacestatesshowno kz dispersion,incontrasttothebulk statesthatusuallyshowobvious kz dispersion.
kk ¼ K k ¼ 1 ħ 2mE kin p sinθ cosφ b x + sinφ b y ðÞ: (3)
E f k ðÞ ¼ ħk2 2m E 0 jj¼ ħ2 k2 k + k2 z 2m E 0 jj,(4)
kz jj ¼ 1 ħ 2mE kin cos 2 θ + V 0 ðÞ p ,(5)
4 LexianYangetal.
1.2Generalprinciple
AfterphenomenologicallydescribingtheenergyandmomentumconversionprocessinARPESexperiment,wenowdiscussthemicroscopicquantumprocessofphotoemission.Thephotoemissioncanbetreatedasan opticaltransitionfroman N-electroninitialstatetoafinalstateconsisting of N-1electronsandaphotoelectron.Theinitial N-electronstateis describedbyamany-bodywavefunctionthatsatisfiesthesurfaceboundary condition.Itisoneoftheeigenstatesofthe N-electronsystem.Thefinal stateisdefinedbyoneoftheeigenstatesoftheionized(N-1)-electronsystem andthecomponentofthewavefunctionofthephotoelectron(thatisusually approximatedbyaplane-wavepropagatinginvacuumwithanamplitude componentinthesolid).Tocalculatetheintensityofthephotoelectrons, weneedtoknowthetransitionprobabilityfromtheinitialstatetothefinal stateafterphoto-excitation,whichcanbeapproximatelygivenbyFermi’s goldenrule:
where Ψf N and Ψi N (EfN and EiN)arewavefunctions(energies)oftheinitial andfinal N-electronsystems. Hint describestheopticalperturbationofthe system:
A and p are thevectorpotentialoftheexcitationlightandthemomentumof theelectron,respectively.Withdipoleapproximation,thevectorpotential oftheultra-violetlightcanberegardedasconstantattheatomicscale,therefore r A 5 0andEq. (7) reads:
To accuratelydescribeandcalculatethephotoemissionprocess,oneneeds totreatitasaone-stepquantumprocess.Thebulk,surface,andvacuum informationhavetobeincludedintheHamiltoniandescribingthesystem, whichinvolvesnotonlythebulkandsurfacestatesbutalsotheevanescent statesandsurfaceresonancestates.Suchaone-stepmodelistoocomplicated tobequantitativelysolved.Instead,aphenomenologicalthree-stepmodelis
ωfi ¼ 2π ħ ΨN f jH int jΨN i DE δ E N f E N i hυ ,(6)
H int ¼ e 2mc A p + p A ðÞ: (7)
H int ¼ e 2mc A p + p, A ½ + p A ðÞ ¼ e mc A p iħr A5 e mc A p: (8)
5 ElectronicstructuresofTQMsstudiedbyARPES
usuallyusedforsimplification.Inthismodel,theexcitationofphotoelectronsisdecomposedintothreesubsequentprocessesasshownin Fig.2:
(i) Theincidentphotonstransferenergyintotheelectrons,andtheelectronsareexcitedabovethevacuumlevel;
(ii) Theexcitedelectronstransporttowardthesurfaceofthesolids throughelasticorinelasticscattering;
(iii) Theexcitedelectronsnearthesurfaceovercomethesurfacebarrier andescapefromthesamplesurfaceintothevacuum,becomingthe photoelectrons.
Theoverallphotoemissionintensity I isproportionaltotheproductofthe probabilityofthethreesubsequentprocesses:
Thefirststep,photo-excitationofelectrons,canberegardedastheverticaltransitionfromtheinitialstatetothefinalstateinthefirstBrillouinzone, thatis, kf ¼ ki (themomentumofthephotonsisneglected).Thisstep containsalltheinformationaboutthei ntrinsicelectronicstructureof thesample.Tofurthersimplifytheproblem,weusuallyprocessthisstep insingle-electronpicturebyneglectingtheCoulombinteractionbetween electrons.Tocalculatethetransitionprobability,wecanwritetheinitial
I ∝ ω i ðÞ ω ii ðÞ ω iii ðÞ : (9)
Fig.2
6 LexianYangetal.
Three-stepmodelofphotoemissionprocess.
stateasthecombinationoftheorbitalfromwhichthephotoelectronis excited(ϕi k)andthewavefunctionoftheremaining N-1electrons Ψi N 1 :
, (10)
wheretheoperator A anti-symmetrizesthewavefunction.Inthesame manner,thewavefunctionoffinalstatecanbewrittenastheproductof thewavefunctionsofthephotoelectron ϕf Ekin ¼ ϕf k andtheremaining N-1 electrons Ψf N 1 :
: (11)
Hereanimportantassumptionhasbeenapplied:thesuddenapproximation. Inactualphotoemissionprocess,boththephotoelectronandtheremaining systemcanrelax,makingthecalculationextremelydifficult.Weassumethat thephotoemissiontakesplacesuddenlyandthephotoelectrondoesnot interactwiththeremainingsystem.Inotherwords,theelectronisremoved suddenlyandthepotentialenergyofthesystemdoesnotchangediscontinuously.Althoughthesuddenapproximationiswidelyusedinmany-body problems,itisonlyapplicableforhigh-energyelectronsinprinciple.
Thewavefunctionoftheremaining(N-1)-electronsystemafterphotoexcitation Ψf N 1 canbechosenastheexcitedstateofthesystemwith eigenstateof Ψm N 1 andeigenvalueof EmN 1.Thetotaltransitionprobability willbegivenbythesummationoveralltheexcitedstates.
Theenergiesoftheinitialandfinalstatescanalsobecorrespondingly writtenas:
The matrixelementinFermi’sgoldenrulecannowbewrittenas:
Definingtheoverlappingintegralbetweentheinitialandfinal(N-1)electronsystem:
ΨN i ¼Aϕk i ΨN 1 i
ΨN f ¼AϕEkin f ΨN 1 f ¼Aϕk f ΨN 1 f
E N i ¼ E N 1 i E k B (12)
E N f ¼ E N 1 f + E kin : (13)
and
ΨN f jH int jΨN i DE ¼ ϕk f jH int jϕk i DE ΨN 1 m jΨN 1 i ≡ M k f ,i ΨN 1 m jΨN 1 i (14) Mf, i k
isdefinedasthedipolematrixelement.
c m,i ≡ ΨN 1 m jΨN 1 i ,(15) 7 ElectronicstructuresofTQMsstudiedbyARPES
thetotalphotoemissionintensityisthesummationoverallthestates:
j cm,i j2 istheprobabilitythattheremaining(N-1)-electronsystemisinthe intrinsicexcitedstate Ψm N 1 aftertheelectronatinitialstateof ϕi k isremoved. Inanon-interactingsystem, j cm,i j2 isunityonlyforoneparticular m stateand zeroforothers.Inotherwords,thefinalandinitialremainingsystemwith (N-1)electronsarethesame: Ψf N 1 ¼ Ψi N 1 and hΨf N 1 j Ψi N 1i¼ 1.Inthis case,ARPESspectrumiscomposedofaseriesofdeltafunction(Fig.3B):
Thematrixelement Mf,i k doesnotaffectthedispersioninARPESmeasurements,butitwillinducetheredistributionofARPESintensity.
Forsystemswithinteractions,theoverlappingintegral (15) isnotzerofor several m states.Therefore,thephotoemissionspectrumisnotcomposedof deltafunctionsanymore.Instead,thereexistsatellitepeaksaccompanying
Fig.3 (A)Banddispersioninthesystem.(B)Innon-interactingsystems,ARPESspectrumconsistsofaseriesofdeltafunctions.(C)Insystemswithinteraction,thereexist satellitepeaksaccompanyingthemainpeaks.
I k, E ðÞ¼ X f , i ωfi ¼ X f , i M k f ,i 2 X m c m,i jj2 δ E N 1 m + E kin E N 1 i + E k B hυ : (16)
I k, E ðÞ¼ X f , i M k f ,i 2 δ E kin + E k B hυ : (17)
8 LexianYangetal.
themainpeaksasshownin Fig.3C.Theoretically,itisusefultodescribe ARPES spectralfunctionwithGreen’sfunction.Thepropagationofasingle electroninaninteractingmany-bodysystemcanbedescribedbythe time-orderedGreen’sfunction G t t 0 ðÞ.ItcanberegardedastheprobabilityamplitudethatafterinjectingaBlochelectronintothesystemattime zero,theelectronremainsinthesamestateat j t t0 j.ByFouriertransformation,wecanuseGreen’sfunctioninmomentum-energyspace: G k, ω ðÞ¼ G + k, ω ðÞ + G k, ω ðÞ,wherethe+and superscripts represent theaddingandremovingelectrons.At T ¼ 0K,wehave:
G k, ω ðÞ¼ X m
ΨN 1 m jc k jΨN i 2
ω E N 1 m + E N i iη , (18)
where ck istheelectroncreator/annihilatorand η isaninfinitesimal. Single-particlespectralfunction A(k, ω)describestheprobabilityofremovingoraddinganelectronwiththeenergy E andmomentum k fromortothe interacting N-electronsystem,whichcanbedefinedas:
A k, ω ðÞ¼ A+ k, ω ðÞ + A k, ω ðÞ¼ 1 π ImG k, ω ðÞ, (19) where G(k, ω) ¼ G+(k, ω)+[G (k, ω)]∗ istheretardedGreen’sfunction. Theplusandminussuperscriptscorrespondtotheelectronaddingor removingspectralfunctions,whichcanbemeasuredusinginversephotoemissionandphotoemissionrespectively.Inthelimit η ! 0+,usingthe formula x iη ðÞ 1 ¼P 1=x ðÞ iπδ x ðÞP ð meanstheCauchyprinciple integral),Eq. (19) canberewrittenas: A k, ω ðÞ ¼
ComparingEq. (20) with (16),weimmediatelynotethat:
I k, E ðÞ ¼ X f , i M k f , i 2 A k, E ðÞ (21)
ByintroducingtheFermi-Diracdistributionfunction f(ω, T ), A (k, ω) ¼ A(k, ω)f(ω, T ).Therefore,at T 6¼ 0K,
I k, E ðÞ¼ I 0 k,hυ, A ðÞ A k, E ðÞ fE , T ðÞ: (22)
I0(k,hυ, A) ∝ j Mf,i k j2 representstheinfluenceofthematrixelementon ARPESspectrum.Itdependsonthephotonenergy,photonpolarization, andexperimentalgeometry,etc.ItisnoteworthythatARPESdirectly
N
m jc k jΨN
2
E N
m + E N
X m Ψ
1
i
δω
1
i : (20)
9 ElectronicstructuresofTQMsstudiedbyARPES
measurestheone-particlespectralfunctionofthesystem. f(E, T )inEq. (22) clearlysuggeststhatregularARPEScanonlydetecttheoccupiedstatesin the system.Inpractice,ARPESspectrumisfurthercomplicatedbytheresolutionoftheexperimentalinstrument, R(k, E),whichisconvolutedinthe spectrum:
Next,weconsiderthesecondstep,thetransportofexcitedelectronstoward thesamplesurface,whoseinfluenceonARPESspectrumcanbedescribed bythemeanfreepathofelectrons.Itreflectstheprobabilitythattheexcited electronremainstheenergyandmomentuminthetransportprocess.Itis inevitablethatsomeelectronswillbeinelasticallyscattered,whichinduces backgroundsignalinARPESspectrum.Asdemonstratedbyexperiments, themeanfreepathofelectronsisimmunetothematerialsbutstrongly dependsontheelectronkineticenergy.Forthecommonlyusedphoton energyinARPESmeasurements,20–200eV,themeanfreepathofphotoelectronsisabout2–50A ˚ ,suggestingthatARPESisasurfacesensitiveprobe, whichmakesitpowerfulinthestudyofTQMs.
Inthethirdstep,excitedelectronsescapethesamplesurface.Theescape probabilityismainlydeterminedbytheelectronkineticenergyandthesurfacebarrierortheworkfunctionofthesolidmaterials.Duringthetransmission,electronslosemomentumperpendiculartothesamplesurface,which requires ħ2 k2 z 2m E 0 + ϕ toguaranteenon-zerotransmissionprobability.
1.3Experimentalinstrument
ARPESinstrumentusuallyconsistsoffourpivotalparts:lightsource,electronanalyzer,vacuumsystem,andsamplemanipulator,asshownin Fig.4 In thissection,wewillbrieflyintroducetheseparts.
1.3.1Lightsource
Thelightsourcedeliversmonochromaticphotonstoexcitetheelectronsin solids.Thephotonenergymustexceedtheworkfunctionofthematerials, andfurtherneedstobelargerthan6eVtominimizethephotoemissionfinal stateeffect(thatis,thefree-electronassumptionofthephotoelectronis applicable).Therearemanydifferentkindsoflightsourcesvaryinginforms andoperatingconditions,includinggas-dischargelamp,synchrotronradiation,andlaser.Amongthem,synchrotronradiationisthemostcommonly usedlightsourceforARPESexperiment.Asynchrotronconsistsofastorage
I k, E ðÞ¼ I 0 k,hυ, A ðÞ A k, E ðÞ fEðÞ R k, E ðÞ: (23)
10 LexianYangetal.
Fig.4 AtypicalARPESsetup.Left:sideview.Right:topview.
ringinwhichseveralbunchesofelectronbeamstravelcircularly.Theaccelerationoftheelectronbeamsemitscollimatedradiationrangingfrom microwavetohard-X-ray.ThetypicalenergyrangeusedforARPESexperimentis20–200eV.TheadvantageofthesynchrotronradiationforARPES measurementisthecollimation,coherence,monochromaticity,andhigh intensity.Moreover,thephotonenergyandpolarizationofsynchrotron radiationiseasytochange,whichisdesirablenotonlyforthecompletemeasurementofthewhole3Dsetoftheelectronicstructurebutalsofor distinguishingthebulkandsurfacestatesofTQMs.
Thesecondwidelyusedlightsourceisthegas-dischargelamp,usually thehelium-dischargelamp.Theheliumatomsareexcitedbymicrowave radiationandemitultraviolet(UV)lightssubsequently.Itcandeliverphotonsatdifferentenergies.Thestr ongestandmostcommonlyusedarethe photonsat21.2eV(HeI α)and40.8eV(HeII α)thatareselectedbya monochromatorandguidedtothesamplethroughacapillary.Thehelium lampiseasytooperateandlab-based(thusnotlimitedbythebeamtime). However,thebeamintensityandbeams ize(typically0.5mmindiameter) limitsitsapplicationinthepursuitofextremelyhigh-resolution measurements.
Recently,lasersourcehasbeenusedforARPESmeasurementswith superbenergyandmomentumresolutions.Thankstotheswiftdevelopment ofnon-linearoptics,theUVlasersuitableforARPESmeasurementcanbe easilygeneratedbyfrequencydoublingmethodusingnonlinearopticalcrystalssuchas β-BaB2O4 (BBO)andKBe2BO3F2 (KBBF).Thetypicalenergy oftheUVlaserisabout6–7eV.Thenarrowbandwidthofthelasersource haspushedtheenergyresolutionofARPEStothelimitoflessthan1meV. Thehighphotonfluxalsogreatlyenhancestheexperimentefficiency. Moreover,thegoodcoherenceoflasermakesitpossibletofocusthebeam downto1 μm,whichgreatlyenhancethemomentumresolutionofARPES. Nevertheless,thekineticenergyofphotoelectronsexcitedbyUVlaseris usuallylessthan3eV,whichmaybringuptheproblemoftheabovementionedfinalstateeffect;andspatialchargingeffect(Zhouetal.,2005) needs tobeavoidedduringmeasurements.
1.3.2Vacuumsystem
ThesurfacesensitivityofARPESrequiresacleansurfacetomeasurethe intrinsicelectronicstructureofthesample.Toavoidthesampleaging inducedbysurfaceabsorptionofresidualgasmolecules,thesamplehasto beprocessedandmeasuredinultrahighvacuum(UHV)environment.
12 LexianYangetal.
Typically,acompleterunofARPESmeasurementtakesabout24h,which requiresapressurelessthan1 10 10 mbar(preferablylessthan3 10 11mbar)toavoidseveresamplecontamination.Ontheotherhand,asthephotoelectronsneedtotravelalongdistancebeforecollectedbythedetectorof theanalyzer,theUHVconditionisnecessarytoguaranteethatthephotoelectronsarenotscatteredbythegasmolecules.
TherequirementoftheUHVconditiongreatlycomplicatedARPES instrument.AsetofARPESsystemconsistsofaUHVchambermadeby stainlesssteelandseveraldifferentkindsofvacuumpumps.Thevacuum chamberalsocontainsadouble-layer μ-metallinertoreducethemagnetic fieldinsidethechambersincetheresidualmagneticfieldmayaffectthetrajectoryofphotoelectrons.Furthermore,toobtainUHVlessthan1 10 10mbar,onehastobaketheUHVchambertohigherthan100 °Cforseveral days,whichistime-consumingandlaborious.Therefore,asampleloading, preparingandtransfersystemisnecessary(Fig.4),whichmakesARPES instrument morecomplicated.
1.3.3Samplemanipulator
Asimpliedbyitsname,thesamplemanipulatorisusedtoholdthesample withthecapabilityofmulti-axisrotationsandtranslations.Rotatingthe sampleisnecessarytoletphotoelectronsemittedfromdifferentanglesfly intotheanalyzerasshownin Fig.1 (someARPESapparatusalsoestablishes the rotationoftheanalyzertoachievethisgoal).Themanipulatoralsocontainsacryostatandaheaterfortemperature-dependentARPESmeasurements.Thetemperaturecanrangefromaslowas1Ktoashighasseveral hundredK.Nowadaystheimprovementofthemanipulatorincludinglower temperature,higherstability,andadvancedautomationhasgreatlybroadenedtheexperimentalcapabilityofARPES.
1.3.4Electronanalyzer
Theelectronanalyzeristhecore(alsothemostexpensive)partofARPES instrument.Itcollectsphotoelectronsandmeasurestheirenergyandemissionangles.Atpresent,two-dimensionalelectrostatichemisphericalanalyzeristhemostcommonlyused. Fig.5Aschematicallyshowsatypical hemispherical analyzerwiththetrajectoriesofphotoelectronsemittedfrom differentanglesindicatedbyred,blackandbluecolors(Sobotaetal.,2020). The photoelectronsfirstflythroughacylindricalelectroniclensthatrectifies theelectronenergytoapropervaluesothattheycanpassthroughtheelectricfieldappliedbetweentwoconcentrichemispheres.Electronswith
13 ElectronicstructuresofTQMsstudiedbyARPES
differentkineticenergy(momentum)willdispersealongthelongitudinal (lateral)directionofthedetector.Therefore,asingleshotofthehemisphericalanalyzercollectsatwo-dimensionaldispersionofthebandstructure.
Fig.5Bshowsarealimageofahemisphericalanalyzerwithadiameterof about618mm.Recently,newfunctionshasbeendevelopedforthehemisphericalanalyzer.Forexample,byapplyingadeflectionvoltage,theanalyzercancollectelectronsemittedfromalargeanglerange( 15°)without rotatingthesample.
Anothercommonanalyzerisangle-resolvedtime-of-flight(ARTOF) analyzer.Itmeasurestheflyingtimeofphotoelectronstodeducetheir energyandusesatwo-dimensionaldetectortomeasurethetwoemission angles(θx, θy)ofphotoelectrons,asschematicallyshownin Fig.5C (Sobotaetal.,2020).Therefore,itcanmeasureacompletetwo-dimensional banddispersionwithoutrotatingthesample(oranalyzer).Nevertheless,it requirespulsedlightsourcewitharelativelylowrepetitionrate,whichlimits itsefficiency.
ARPEShasplayedanimportantroleinthestudyofquantummaterials includingmetals,semiconductors,high-temperaturesuperconductors, stronglycorrelatedmaterials,andtopo logicalquantummaterials.Inturn, italsobenefitsfromthedevelopmentofcondensedmatterphysics.To obtainmoreanddeeperinsightsintothematerials,newcapabilityof ARPEShasbeeninvented.Forexample,thespinpolarizationofthe
14 LexianYangetal.
Fig.5 Schematic(A)andrealimage(B)ofahemisphericalelectronanalyzer.Schematic (C)andrealimage(D)ofanangle-resolvedtime-of-fightelectronanalyzer. Panels(B–D): AdaptedfromthewebsiteofScientaOmicron(scientaomicron.com).
electronicstructurecanbemeasured byattachingaspindetectorafterthe electronanalyzer;theelectronicdynamicscanbemeasuredintimedomainbytime-resolvedARPES;thereal-spacedistributionoftheelectronicstructurecanbevisualiz edbythemicro-ornano-ARPES.
1.4ARPESspectrum
Inthissection,weintroducehowtheARPESdataiscollectedinapractical experiment,supposingweareusingthehemisphericalanalyzer.Inasingle shot,weobtainaphotoelectrondistributiononthedetector(Fig.6A), whichcanbeaccuratelyconvertedtotheenergy-anglespace(Fig.6B). Withhigh-qualitysamplesurface,sharpdispersionalongthetiltanglecan beobtained.Thecolormapofthedatacanbechangedforbettervisualizationofthedispersion(Fig.6C).Sucha3Ddataset(I, Ekin,tilt)isusually calleda“cut.”Byscanningthepolarangle(rotatingthesamplealongpolar angle, Fig.1B),wecanobtainaseriesofcutsfromdifferentpolarangles. Combiningthesecutstogether,weobtainafour-dimensionaldataset (I, Ekin,tilt,polar)(Fig.6D).FollowingEq. (3),thedatasetcanbeconverted
Fig.6 ARPESdatacollection.(A)Arawdataimageonthedetector.(B)Therawdata imageconvertedintotheangle-energyspace.(C)Thesimulateddispersionwitha Diraccone.(D)Mapofthetwo-dimensionalelectronicstructurebyscanningthepolar angle.(E)Three-dimensionalillustrationoftheARPESdatainmomentum-energyspace. (F)ArealisticARPESdatacollectedonWTe2.
15 ElectronicstructuresofTQMsstudiedbyARPES
intothemomentum-energyspace,whichcontainsboththebanddispersion alongdifferentmomentumdirectionsandtheconstantenergycontours includingtheFermisurface,asshownin Fig.6E.Arealisticdatasetcollected fromatype-IIWeylsemimetalcandidateWTe2 isshownin Fig.6F(Qihang etal.,2017).
2.ARPESstudiesontopologicalquantummaterials
ThestudyofTQMscanbetracedbacktothediscoveryofthequantumHall(QH)effectbyKlausvonKlitzingin1980(Dordaetal.,1980). SymmetrybreakingfailedtodescribetheQHstate;instead,topologywas adoptedandatopologicalinvariantcalledtheChernnumberstoodout (Bellissardetal.,1994; Thoulessetal.,1982).Duetothetopologicalproperty,electronsattheedgeshowquantizedHallconductance,whichare insensitivetothesmoothchangeofmaterialparameterssuchastheimpurity concentrationandsize.TheQHfamilyofstates(includingthefractional quantumHall,orFQHstate)weretheonlyclassofTQMformorethan 20years,untilthequantumspinHall(QSH)effectwaspredicted (Bernevigetal.,2006)andexperimentallyidentified(K€ onigetal.,2007). Afterthat,tremendousresearcheffortshavebeendevotedtosearching fornewTQMsthatarecharacterizedbythetopologyoftheirbandstructure,andvariousTQMshavebeendiscoveredasshownbythetimelinefor thediscoveryofTQMsin Fig.7 (Chenetal.,2020).
Ingeneral,TQMscanbeclassifiedintomainlythreetypesaccordingto thebulkbandgap.Thefirsttypehasabulkbandgap,suchasthetopological insulators(TIs)andtopologicalcrystallineinsulators(TCI).Thesecondtype hasnobulkbandgap;instead,thebulkconductionandvalencebandstouch atdiscretemomentumpointsorcontinuousmomentumlines.Thistype
16 LexianYangetal.
Fig.7 TimelineforrecentdevelopmentofARPESstudiesonTQMs.TheTQMsthatarein thedashedrectangulararenotinvestigatedbyARPESyetduetothedimensionlimit.
includesthetopologicalDiracandWeylsemimetals(TDSsandTWSs),the topologicalnodallinesemimetals(TNLSs),andthetopologicalchiralsemimetals(TCSs),etc.Thethirdtypeistopologicalsuperconductors(TSCs)has anemergentsuperconductinggapinthebulkandTSSonthesurface.
DifferentTQMsarecharacterizedbytheirownuniquebulkandsurfaceband structures,whichcanbedirectlymeasuredanddistinguishedbyARPES.
2.1Topologicalinsulatingphases
2.1.1Topologicalinsulator
Soonafterthetheoreticalprediction,theQSHeffect,orthetwodimensional(2D)TI,wasdiscoveredinHgTe/CdTequantumwell (Bernevigetal.,2006; K€ onig etal.,2007).TheQSHstatecanbedescribed by a Z2 topologicalinvariantwhosevalueiseither0or1.IntheQSHstate, thestrongspin-orbitcoupling(SOC)servesasananalogtotheexternal magneticfieldintheQHstate.WhilethebulkoftheQSHstateisinsulating, helicalconductingmodesemergeintheedge.TheconceptoftheTIissoon extendedto3D,makingitpossibletodirectlyvisualizeitstopologicalelectronicstructureusingARPES.TheTIphasecanbeeasilyrecognizedbyits uniquebandsequencewithoutstrenuouscalculationofthe Z2 topological invariant.Incontrasttothenormalinsulator,thebulkstatesina3DTIshow aninvertedbandgap,inwhichtherearerobustmetallicsurfacestatesonall thesurfacesofthecrystal,asshownin Fig.8A.Notably,sincethe time-reversal symmetry(TRS)isrespectedinthesystem,thesurfacestates formgaplessDiracfermionwithlineardispersions.ToestablishastrongTI, itisrequiredthatthereareanoddnumberofsurfaceDiracfermionslocated atthetime-reversalinvariantpointsofthe3DBrillouinzone.
Thefirstexperimentallyidentified3DTIistheBi1 xSbx alloy.Thesubstitutionofbismuthbyantimonyresultsinachangeofbandsequenceand inducesabandinversionat0.07 x 0.22.ARPESobservesanoddnumberofsurfacestates,verifyingitsTInature(Hsiehetal.,2008).Nevertheless, this materialisanalloywithcomplicatedbandstructureandsmallbulkgap, makingitundesirableforthetransportmeasurementandfutureapplication. Soon,theV2-VI3 binarycompounds(Bi2Se3,Bi2Te3,andSb2Te3)with largebulkgapandsimplesurfacebandstructurewereputforward (Zhangetal.,2009).Bi2Se3 andBi2Te3 werepredictedtobethesimplest 3DTIs.ThestrongSOCinthesematerialsinducestheinversionofbismuth/antimonyandselenium/tellurium pz orbitalswithalargeinverted bulkgapandsinglesurfaceDiracconeatthe Γ point. Fig.8BandCshow the banddispersionsofBi2Se3 andBi2Te3 measuredwithARPES.Bulk
17 ElectronicstructuresofTQMsstudiedbyARPES
bandgapaslargeas200meV(165meV)wasobservedinBi2Se3 (Bi2Te3), whilethesurfacestatesformgaplesslineardispersioninthebulkgap (Chenetal.,2009,2010a; Hsiehetal.,2009; Xiaetal.,2009).SimilartopologicalelectronicstructureswerealsoobservedintheternaryIII-V-VI2 familyofcompounds(TlBiTe2 andTlBiSe2)(Binghaietal.,2010; Chen etal.,2010b).Asshownin Fig.8D,TlBiSe2 showsalargebulkgap ( 200meV)thatissimilartoBi2Se3 withasinglesurfaceDiracconenear Γ.
TheTRSnotonlyprotectsthegaplesssurfacestatesbutalsorequiresa spin-momentumlockingofthesurfaceDiracfermions,asschematically shownbythehelicalspintexturein Fig.9A.Inthatcase,electronsof TSScannotbeback-scatteredoreliminatedbynonmagneticimpurities. Asaresult,theTSSarerobustandthecounter-propagatingelectronsmust havetheoppositespinpolarization,promisingthetransportofnetspincurrentwithoutdissipation,whichisofgreatapplicationpotentialinlowenergy-consumingelectronicsandspintronics.
18 LexianYangetal.
Fig.8 ARPESstudiesontopologicalinsulators.(A)Schematicoftheelectronicstructure oftopologicalinsulators.(B–D)ARPESmeasuredbandstructureof(B)Bi2Se3,(C)Bi2Te3, and(D)TlBiSe2.
Fig.9 Spin-polarizationoftheTSSofTIs.(A)Schematicofthesurfacestateswith spin-momentumlocking.(B)Schematicofthespindetectorinspin-ARPESmeasurements.(C)Measuredycomponentofthespin-polarizationofthesurfacestatesof Bi2Te3 at20meVbelowEF.(D)Fittedvaluesofthespin-polarizationvectorfor +kx and kx.(EandF)SpinpolarizationofTSSofBi2Se3 measuredwithdifferentphoton polarizations.
ThespinstructureofTIscanbedirectlymeasuredbyspin-resolved ARPES,whichcanbeestablishedbyattachingaspindetectortotheelectronanalyzer,asschematicallyshownin Fig.9B.Thespinpolarizationof photoelectronscanberesolvedbyMottorexchangescatteringonatarget (usuallyheavymetalsormagneticfilms).Electronswithoppositespinpolarizationwillbescatteredintodifferentchannelsofthespindetectors.Thanks tothedevelopmentofhighbrilliancelightsourcesandhigh-efficiencyspin detectors,theprobeefficiencyofspin-ARPESisgreatlyenhancedrecently, makingitcapabletoresolvesubtlespinstructureofenergybandsofTQMs.
Fig.9Cshowsthemeasuredy-componentofthespinpolarizationofthe TSSofBi2Te3 (Hsiehetal.,2009).Theelectronswithoppositemomentum showoppositespinpolarizationalongthe y direction.Bycontrast,noclear signalofspinpolarizationwasobservedalongthe x and z direction(not shownhere).TheoverallspinstructureoftheTSSfittedfromthe
19 ElectronicstructuresofTQMsstudiedbyARPES
experimentaldataisshownin Fig.9D.ThespinsofTSSpointtothe(k b z ) direction,inperfectconsistencewiththeoreticalpredictionofspinmomentumlocking.Itisnoteworthythatthesubsequentspin-ARPES measurementrevealsasurprisingphoton-polarizationdependentspinpolarizationoftheTSSinBi2Se3 ( Jozwiaketal.,2013).Asshownin Fig.9Eand F,the y componentofthespinpolarizationmeasuredwith s-and p-polarized photonsarereversed,whichisbeyondtheexpectationofthetopological SOCoriginoftheTSSandisstillunderdebate.Itwasattributedtoeither aspin-dependentinteractionofthehelicalTSSwithphotonsduetothe strongSOCoranorbital-dependentspintextureinBi2Se3 ( Jozwiak etal.,2013; Xieetal.,2014).
2.1.2Weaktopologicalinsulator
In3D,theTIsareactuallydescribedbyfour Z2 topologicalinvariants (ν0;ν1ν2ν3).Accordingly,theTIscanbefurtherclassifiedasstrongTIand weakTI.Whilethe ν0 indexisnontrivialforstrongTIs,itisvanishing forweakTis;instead,(ν1ν2ν3)indexesarenontrivialforweakTIs. Consequently,theweakTIsareanalogoustostackedQSHinsulatorsand haveTSSonlyonsomespecificsurfaces,indrasticcontrasttothestrong TIs,asschematicallyshownin Fig.10A.SuchaweakTIphaseisrecently identifiedinquasi-one-dimensional β-Bi4I4 (Noguchietal.,2019).Abinitiocalculationsshowsabandinversion,suggestingitstopologicallynontrivialnature.ARPESmeasurementsdirectlyrevealaDirac-cone-like featureonlyonthe(100)surfacebutnotonthe(001)surface(Fig.10B andC),confirmingtheweakTIphaseof β-Bi4I4
Fig.10 (A)SchematicsofweakTIthatisanalogoustostackedQSHlayers.(BandC) ARPESmeasurementon(100)and(001)surfacesof β-Bi4I4,respectively.
20 LexianYangetal.
2.1.3Topologicalcrystallineinsulator
Notlongafterthediscoveryof3DTIs,anewexampleofsymmetryprotectedtopologicalphasewasintroduced:thetopologicalcrystallineinsulator(TCI)(Fu,2011),inwhichthecrystalsymmetry,suchasthemirror symmetryandthenon-symmorphiccrystalsymmetry,protectsthetopologicalelectronicstructure,similartoTRSinTIs.Themultifariouscrystalsymmetriespromisearichplatformtosearchandengineerthetopologicalphases inthisclassofTQMs.Moreover,SOCisnotcrucialintheTCIs,thusheavy elementwithstrongSOCsuchasbismuthisnotnecessaryinTCIs,which greatlybroadensthematerialsfamilyofTQMs.
TheTCIphaseistopologicallynontrivialinthesensethatthebulkstate ofTCIscannotbeadiabaticallytransformedtoanatomicinsulatorwithout breakingtheprotectingcrystalsymmetry;andthereexistgaplessboundary statesprotectedbytherespectivesymmetry.IncontrasttotheTRSprotectedTIs,however,theboundarystatesdependonthesurfaceorientationandmaybegappediftheprotectingcrystallinesymmetryisreducedon certaincrystallinesurfaces.
Fig.11AschematicallyshowsamirrorTCIinwhichthemirrorsymmetryprotectstheTSS.MirrorTCIswerefirstpredictedanddiscoveredin Pb1 xSnxX (X ¼ Se,Te)compounds(Hsiehetal.,2012; Jinetal.,2017; Tanakaetal.,2012; Xuetal.,2012).Themirrorsymmetryprotectsintotal fourDiracconeswithhelicalspintextureintheBrillouinzone. Fig.11B showsonepairofthesurfaceDiracconesnearthesurfaceBrillouinzone boundaryofPb0.6Sn0.4TemeasuredbyARPES,identifyingitsTCIphase.
Besidesmirrorsymmetry,othercrystalsymmetriescanalsoprotectthe topologicalelectronicstructureofTCIs.Forexample,rotationalsymmetry canalsoprotecttheTCIphasein α-Bi4Br4;theglidemirrorsymmetrycan
21 ElectronicstructuresofTQMsstudiedbyARPES
Fig.11 (A)SchematicofthesurfacestatesofaMirrorTCI.(B)ARPESmeasuredband structureofPb0.6Sn0.4Te.
protectthehourglass-likeTSSinthenon-symmorphiccrystalKHgSb; thewallpapergroupsonthe2Dsurfacesof3Dmaterialscanprotectthe wallpaperfermionsinSr2Pb3 (Wiederetal.,2018);thespatiotemporalsymmetries canprotectthehigh-orderTIsinmanycrystalssuchasBi(Schindler et al.,2018a),SnTe,surfacemodifiedBiSe(BiTe)andBi2TeI(Schindler et al.,2018b),andBi4Br4 (Noguchietal.,2021);andmanyotherinteresting TCI phases,whichawaitstheoreticalandexperimentaldiscovery.
2.2Topologicalsemimetals
2.2.1TopologicalDiracsemimetal
Whilethebulkconductionandvalencebandsformaninvertedbandgapina 3DTI,theycanalsotouchatsomeisolatedpointsenforcedbycrystalline symmetries,formingnoveltypeofTQMs—3DtopologicalDiracsemimetal (TDS)(Armitageetal.,2018).ThesetouchingpointsarecalledDiracpoints, aroundwhichelectronenergybandsdisperselinearlyalongallthreemomentumdirections,whichcanbedescribedbythemasslessDiracequation.A3D Diracpointmusthavefourfolddegeneracy.Thisiscommonlyguaranteed bythecombinationofTRSandinversionsymmetry(theformerandlatter dictatetheeigenvaluesofonebandfollow En, "(k) ¼ En, #( k)and En, #(k) ¼ En, #( k),respectively,thuscrossingpointsoftwobandsmustexhibitfourfolddegeneracy).3DDiracpointsnaturallymanifestthemselvesasthe non-trivialbulktopologyofa3DTDS,aseach3DDiracpointcanbeviewed asapairofWeylpoints(monopole/anti-monopoleofBerrycurvature)overlappinginmomentumspace.SuchbulktopologyfurthergivesrisetoTSSon thesurfacesimilarto3DTIs,whichtogetherwiththebulkDiracconeslead tomanyexoticphysicalphenomena,e.g.,ultrahighcarriermobility(Liang et al.,2015),chiralanomaly(Xiongetal.,2015),quantumoscillationsfrom surfaceFermiarcs(Molletal.,2016; Potteretal.,2014),3DquantumHall effect(Zhangetal.,2019),etc.
As a3DDiracpointisdescribedinthe4Dspaceasafunctionofvariables (E,kx, ky, kz),itsvisualizationneedsprojectionsofbandstructuresontotwo linearlyindependent3Dsubspaces(like E kx ky and E ky kz)that mustbothexhibit2DDiraccones(Fig.12A,C,andE).
Na3Biisarepresentative3DTDS(alsothefirstexperimentallyconfirmedTDS)inwhich C3 crystallinerotationsymmetrystabilizesonepair of3DDiracpointsalong Γ -A direction(Wangetal.,2012).The3D Fermi surfaceobtainedbyARPESmeasurementsclearlyresolvesapairof Diracpointslocatedsymmetricallyaboutthe Γ pointalongthe kz axis (Fig.12B)(Liuetal.,2014a). Fig.12Cschematicallyshowstheprojection
22 LexianYangetal.
Fig.12 3DTDSphaseinNa3Bi.(A)SchematicofaDiracfermionwithlineardispersionin TDSs.(B)3DFermisurfacemapofNa3BidirectlyrevealingapairofDiracpointsalong thekz direction.(C)Theprojectionofthe3DDiracconeonto(E,kx,ky)subspace.(D)3D volumeplotofthebanddispersionofNa3Biinthekx ky plane.(EandF)Thesameas (CandD)butin(E,ky,kz)subspace.
ofthe3DDiracconeinthe E kx ky subspace.ARPESmeasurement clearlyrevealedthelineardispersionaroundtheDiracpointalongthe kx and ky directions(Fig.12D).Similarly,theprojectionofthe3DDiraccone inthe E ky kz subspaceisshownin Fig.12EandF.Interestingly,despite nearlyisotropic2DDiracconeinthe kx ky plane,thereisasubstantial reductionoftheFermivelocityalongthe kz direction,resultinginan unusual“squeezed”Diracconeinthe3Dmomentumspace.Theupperside oftheDiracconecanbeapproachedbyraisingtheFermilevelusingsurface dopingofalkalimetalatoms,furtherconfirmingthe3DTDSnatureof Na3Bi(Liuetal.,2014a).Lateron,Cd3As2—stableinambientconditions incontrasttoNa3Bi—wasidentifiedtoexhibit3DDiracfermions (Borisenkoetal.,2014; Liuetal.,2014b; Neupaneetal.,2014; Wang etal.,2013),basedonwhichultrahighmobility( 9 106 cm 2 V 1 S 1 at 5K)(Liangetal.,2015),Weyl-orbit-basedchiralitytransfer(Molletal., 2016)and3DquantumHalleffect(Zhangetal.,2019)werereported.
Besides,theTSSformingFermiarcthatareprotectedbythebulktopologywerealsoobservedontheside-cleaved(100)surfaceofNa3Bi(Xuetal., 2015a).Sincethe3DDiracpointiscomposedoftwoWeylpointswith
23 ElectronicstructuresofTQMsstudiedbyARPES
oppositechirality,therearetwosurfaceFermiarcsconnectingtheprojectionsof3DDiracpointsandformingacircle-likestructure.Itisworthto emphasizethatthecircle-likeFermiarcsareintrinsicallydifferentfrom theTSSonthesurfaceofTIssinceDiracpointsarethesingularpointsin themomentumspace.Thus,thecircleisactuallybrokenandthespinpolarizationvanishesneartheDiracpoint.
TheTDSsdiscussedabovearenon-magneticsystemspreservingboth TRSandinversionsymmetry,whicharenecessaryforthefourfolddegeneracyof3DDiracpoints.Breakingtime-reversalsymmetrymaynaturally splitone3DDiracpointintoapairoftwofoldWeylpoints(tobediscussed below).However,3DDiracpointscanstillsurviveinsomemagneticsystemsundersubtlesymmetryprotections.OneproposedcandidateisantiferromagnetCuMnAs(Tangetal.,2016).WhereasbothTRSandinversion symmetry arebrokeninitsantiferromagneticstate,theircombinationisstill preserved,guaranteeingthedoubledegeneracyofbands.Thetwofoldscrew rotationfurthersymmetrystabilizesthebandcrossingpointswithfourfold degeneracyalongthe X-U direction.AnothermagneticTDScandidateis EuCd2As2 withinterlayerantiferromagnetism,inwhichthe3DDiracpoints alongthe Γ -A directionisstabilizedbythenonsymmorphictime-reversal (thecombinationofafractiontranslationandtime-reversalsymmetry) androtationsymmetries(Huaetal.,2018).ARPESmeasurementshave been carriedoutonthesecandidates,however,decisiveevidencesof3D DiracpointsandTSSarestilllacking,partlysufferingfromthep-typenature ofcrystals.Thesemagnetictopologicalmaterialsareunderintenseinvestigationsrecently.
2.2.2TopologicalWeylsemimetal
TheDiracequationdescribingtheDiracfermionswithlineardispersions canbesimplifiedbysettingthemasstermtozero.Bydoingthis, HermannWeylpredictedthemasslessWeylfermionsharboringdefinite chirality.Uptodate,theWeylfermionstillawaitsexperimentaldiscovery inhigh-energyphysics.Nevertheless,itisrecentlyproposedthattheWeyl fermionscanexistinthecondensedmattersystemsaslow-energyexcitations,givingrisetotheintriguingtopologicalWeylsemimetal(TWS)phase (Armitageetal.,2018; Lvetal.,2021).
In a3DTWS,thebulkmasslesschiralWeylfermionsdisperselinearly alongallthemomentumdirections,whichcanberegardedasmagnetic monopolesinmomentumspace.Theuniquebulktopologyguarantees theexistenceofTSSonthesurfaceformingunclosedFermiarconthe
24 LexianYangetal.
Fig.13 (A)SeparationofaDiracfermionintoapairofWeylfermionswithoppositechiralitybybreakingTRSorinversionsymmetryofthecrystal.AFermiarcterminatesatthe Weylpoints.(B)FermisurfacestructureofTaAsshowingtheFermiarc.(C)ThedispersionofthebulkWeylfermions.(D)TheevolutionoftheFermisurfaceinTaAsfamily. (E)FermisurfaceofmagneticWeylsemimetalCo3Sn2S2.(F)Thedispersionofthebulk WeylfermionsinCo3Sn2S2.
FermisurfacethatterminatesatapairofWeylpoints.TWSsexhibitmany intriguingproperties,suchasthechiralanomaly,negativemagnetoresistance,andWeylorbitsinquantumoscillations,promisinggreatapplication potentials.
Asshownin Fig.13A,aDiracpointcanseparateintoapairofWeyl pointsofoppositechiralityifTRSorinversionsymmetryisbroken. Therefore,theTWSphasecanberealizedbybreakingTRSorinversion symmetryinTDSs.Followingthisstrategy,differentschemestorealize theTWSphasehavebeenproposed,includingbutnotlimitedto: (1)IntensiveefforthasbeenmadetosearchforintrinsicTWScrystalswith TRSorinversionsymmetrybroken.On2011,Wanetal.proposedthatthe magneticpyrochloresY2Ir2O7 withall-in-all-outmagneticorderisan intrinsicmagneticTWSwith12pairsofWeylpointsinthe3DBrillouin zone(Wanetal.,2011).Xuetal.predictedtheTWSphasewithquadratic bandcrossingsinhalf-metallicHgCr2Se4 (Xuetal.,2011).However,itis challengingtoexperimentallyverifytheTWSphaseinthesematerials
25 ElectronicstructuresofTQMsstudiedbyARPES
partiallyduetotheircomplexcrystalandmagneticstructure.(2)TheTWS phasecanberealizedbyapplyingstrongmechanicalstrainorhighmagnetic fieldtosplittheDiracfermionsinTDSs.However,thesemethodsrequire complicatedexperimentalinstrument,whichlimitstheapplicationof theTWSs.(3)ByalternativelystackingtheTIsandnormalinsulators (Hala ´ sz andBalents,2012),whichrequiresprecisesuperlatticefabrication. (4) ATWSphaseisexpectedinthephaseboundarybetweenatrivialinsulatorandaTIinnon-centrosymmetriccrystals(Singhetal.,2012). Unfortunately, alltheseproposedschemesarenotyetexperimentallyrealizeduptodate.
On2015,non-centrosymmetricTaAsfamilyofcompounds(including TaAs,NbAs,NbP,andTaP)weretheoreticallypredictedtobeTWSs (Huangetal.,2015; Wengetal.,2015).Thereare12pairsofWeylpoints in the3DBrillouinzoneofTaAs.ARPESsoonrevealedthecharacteristic FermiarcconnectingapairofWeylpointsofoppositechirality(Lvetal., 2015a; Xuetal.,2015b; Yangetal.,2015).Asshownin Fig.13B,thereare three Fermisurfacesheetsintheuppersideofthedashedline,whileonly twoFermisurfacesheetswereobservedinthelowerside,suggestingthat oneoftheFermisurfacesheetsterminatesattheWeylpoints(Yang et al.,2015).Lateron,thespintextureoftheFermiarcwasresolvedby spin-resolved ARPES,compellinglyconfirmingtheTWSnatureofTaAs (Lvetal.,2015b; Xuetal.,2016).Inadditiontotheobservationofthe Fermi arc,thelineardispersionsofthebulkWeylfermionsinTaAswere alsovisualizedusingsuitablephotonenergy,asperfectlyreproducedby thecalculationin Fig.13C(Lvetal.,2015c; Xuetal.,2015b; Yang et al.,2015).
Other membersoftheTaAsfamilywerealsoexperimentallyconfirmed tobeTWSs(Liuetal.,2016; Xuetal.,2015c).Afamilypicturetakenby ARPES isshownin Fig.13D.WithstrongerSOCinthecompounds(from NbP, TaPtoTaAs),thelengthoftheFermiarcandtheseparationbetween Weylpointssequentiallyincreases.Therefore,SOCplaysasaneffectivetuningparameterforthetopologicalelectronicstructureoftheTWSphase.
ItisworthtoemphasizethattheconnectionpatternoftheFermiarc (howtheFermiarcchoosesandconnectsapairofWeylpoints)isnotconstrainedbythetopologicalpropertyoftheTWS,butdependentonthesurfaceconditionsuchasterminationandsurfacedoping.Indeed,experiment hasshownthatthepatternoftheFermiarconthesurfaceofNaAscanbe effectivelymanipulatedbysurfacedopingofalkalimetalatoms(Yang et al.,2019).
26 LexianYangetal.
Comparedwithnon-centrosymmetricTWSs,magneticTWSsprovide anattractiveplatformtoinvestigatetheinterplaybetweenmagnetismand topology,whichcaninducemanyexotictopologicalquantumsates,such asquantumanomalousHallstate,axioninsulatorstate,idealTWSphase (withaminimumnumberofWeylpoints,twoWeylpointsinthe Brillouinzone).AlthoughtheintrinsicTWSwasfirstproposedinmagnetic materials,topologicalelectronicstructureofTRS-breakingTWSswasnot experimentallyconfirmeduntil2018,whenaseriesofnewmagneticTWS candidatesweretheoreticallyproposedandthenconfirmedbyARPES (Belopolskietal.,2019; Liuetal.,2018,2019; Moralietal.,2019; Wang et al.,2018; Xuetal.,2018).Asarepresentativeexample,Kagomecrystal Co3Sn2S2 hasthreepairsofWeylpointsinthe3DBrillouinzone(Liuetal., 2018, 2019; Moralietal.,2019; Wangetal.,2018; Xuetal.,2018).The Fermi surfacemeasuredbyARPESin Fig.13Eshowsline-segment-shape surface FermiarcsconnectingtheprojectionofWeylpoints,ingoodagreementwiththecalculations(Liuetal.,2019).Bydopingpotassiumatomson the samplesurface,theFermilevelcanberaisedandtheWeylpointat 50meVabove EF canbeapproached.Thelineardispersionacrossthe Weylpointisclearlyshownin Fig.13F,confirmingtheTWSnatureof Co3Sn2S2
2.2.3Type-IItopologicalWeyl/Diracsemimetals
Thelow-energyexcitationsinTDSsandTWSsarethelow-energycounterpartsoftheDiracandWeylfermionsinhigh-energyphysics,inwhichthe Lorentzsymmetryisrespected.TheLorentzsymmetryrequiresthatthe physicallawsarethesameforalltheobservers,nomatterofyourobserving directionandmovingspeed.OneoftheconsequencesofLorentzsymmetry isthewell-knownprincipleofconstancyofthelightvelocity.
Interestingly,inthecondensedmattersystems,theLorentzsymmetryis notrespectedduetothelow-energynatureofthequasiparticleincrystals. Thesituationisnotsosevereinthetype-ITWSsincetheweakviolationof theLorentzsymmetrycanbeadiabaticallyremoved.Inotherwords,the Lorentzsymmetryisapproximatelyrespected.Insomeparticularcrystals, however,theLorentzsymmetrymaybestronglyviolatedthatthelow energyexcitationcannotbesmoothlytransformedtoaLorentzinvariant Weylfermion,realizingtheso-calledtype-IIWeylfermionswithstrongly tiltedWeylcone(Fig.14A)(Soluyanovetal.,2015).Thematerialsharboring thesetype-IIWeylfermionsisthuscalledtype-IITWSs.Incontrastto thetype-ITWSthatshowspoint-likeFermisurface,thetiltedWeylcone
27 ElectronicstructuresofTQMsstudiedbyARPES
Fig.14 Schematicsof(A)type-IITWSand(B)type-IITDS.(C)SurfaceFermiarconthe FermisurfaceofMoTe2.(D)Banddispersionoftype-IIDiracfermioninPtSe2 alongkz.
inducesintricatebulkFermisurfacewithbothelectronandhole-like pockets.Similarly,atype-IITDSfeaturestiltedDiracconeasshownin Fig.14B.
Transitionmetaldichalcogenide(TMD)MoTe2 wasthefirstexperimentallyconfirmedtype-IITWS(Dengetal.,2016; Huangetal.,2016a; Jiangetal.,2017).Itfeaturesmuchfewerandsimplerdistributionof Weylpointsandcrystallizesintoalayeredstructure,makingitmorefeasible forfutureapplicationinelectronicdevicesbasedonTWSs.InMoTe2,the electronandholebandstouchattheWeylpoints,forming4pairsoftilted WeylconesintheBrillouinzone.EachWeylpointcarriesatopological chargeof+1or 1dependingonitschirality,whichprotectsthesurface Fermiarcstatesonthe(001)surface.However,theWeylpointsconnected byFermiarclocatesatdifferentenergies(6and59meV)above EF andthe coexistingtrivialsurfacestatesleaveonlyasmallmomentumandenergy windowfortheFermiarcstates,whichchallengestheexperimentalconfirmationofthetype-IITWSphase.Fortunately,theFermiarcstatedisperses below EF,makingitvisiblebyARPES,asshownin Fig.14C,ingoodagreementwiththeoreticalcalculations.Lateron,thesurfaceFermiarcabove EF
28 LexianYangetal.
wasdirectlyobservedbypump-probetechniqueinMo025W0.75Te2 (Belopolskietal.,2016).Duetothecomplicatedsurfaceelectronicstructure anditsterminationdependence,itisdifficulttoobservethebulkWeylfermionsinMoTe2.However,fromexperimentalperspective,thetype-II natureoftheTWScannotbeconfirmedbythesoleobservationofFermi arcsurfacestates.Inanothertype-IITWS,LaAlGe,boththeFermiarc andthetype-IIWeylpointsaredirectlyobservedbyARPES,makingit anotherprototypicaltype-IITWS(Xuetal.,2017).
Soonafterthediscoveryoftype-IITWS,thetype-IITDSphaseislikewisepredictedinmanymaterialsincludingtransitionmetaldichalcogenides (PtSe2 classandNiTe2)(Ghoshetal.,2019; Huangetal.,2016b), AMgBi (A ¼ K,Rb,Cs)familyofcompounds(Leetal.,2017),andHeuslercompounds(Mondaletal.,2019),etc.InPtSe2,atiltedDiracconeisrevealed alongthe Γ-AdirectionbyARPES,asshownin Fig.14D(Lietal.,2017).
2.2.4Topologicalnodallinesemimetals
InTDSsandTWSs,thevalencebandandconductionbandtouchatdiscrete momentumpoints,formingthefourfolddegenerateDiracfermionsortwofolddegenerateWeylfermions.Suchbandtouchingcanfurthercontinuouslyextendalongalineorloopinthemomentumspace,formingtheso calledtopologicalnodal-linesemimetals(Fangetal.,2015,2016),asschematicallyshownin Fig.15A.TheDiracnodallinecanbeprotectedbydifferentsymmetriesinthesystem,suchasmirrorreflectionornonsymmorphic crystalsymmetry.Forexample,inZrSiS,themirror-glidesymmetryprotects thenodallinesattheBrillouinzoneboundary,asvisualizedbyARPES in Fig.15B(Chenetal.,2017; Fuetal.,2019; Schoopetal.,2016). Correspondingly,iftherespectivesymmetryisbroken,thenodallinemay
29 ElectronicstructuresofTQMsstudiedbyARPES
Fig.15 (A)Schematicofanodalline.NotethatthebanddispersionformsaDiraccone ink1-k2 plane(left),whichextendsinthethirdmomentumdirection.(B)Nodalline inZrSiS.
