One ofthe most powerful tools for determining the band structure of amaterial is the photoemission process,bywhichanincomingphotonkicksanelectronoutofthesolid.
Invacuum,theelectronwilltravelballisticallywiththemomentumandenergyithadwhen itleftthematerial.Acurrentofelectronsejectedinthiswayfromthematerialcanthenbe analyzedfortheirdirectionofmotionandkineticenergy.Thismeasurementisknownas angle-resolved photoemission spectroscopy(ARPES).
Typically,themomentumofthephotonisnegligiblecomparedtothemomentumofthe electron.Theabsorptionofthephotoncanthereforebeviewedasa“vertical”process,in whichtheelectronmovestohigherenergywhilestayingatnearlythesame k-vector.The high-energyelectroncanthenhaveenoughenergytoovercometheworkfunctionofthe materialandleavethecrystal.
Inthinkingoftheprocessbywhichtheelectronleavesthesolid,thequestionimmedi- atelyarises ofwhatconservationrules toapply.We havealreadyseenthat ±k isnotthe truemomentumofanelectron;thisisgivenby(1.6.10),
³ψ´k|´p|ψ´k²= ±´k − i±
à
d3ru∗n´k∇un´k. (1.10.1) Whentheelectroncrossestheboundaryofthesolid,doweconservemomentum,ordowe conserve ±k?Theansweristhatweconserve ±k inthedirectionparalleltothesurface,not thetotalelectronmomentum.Thiscanbeunderstoodasaconsequenceofthewavenature oftheelectrons,inanalogywithSnell’slaw,whichisdiscussedindetailinChapter3.We write ´k = ´kº+ k⊥ˆz,where ´kºis thewavevectorcomponentparallelto thesurfaceand k⊥isthecomponentperpendiculartothesurface.Thespacingofthewavefrontsalonga direction ´x onthesurfaceisgivenbythecondition ´kã´x = kºx = 2πn,where n isaninteger.
Thedistancebetweenpointsofphase2π istherefore ±x = 2π/kº.Thisspacingmustbe thesameforthewave bothinsideand outsideofthesolid,aconditiongenerallyknown as phase matching.Although ´kº isconserved,thetotalmomentumoftheelectronisin generalnotconserved.Therefore,thecrystalmustrecoilslightly,takingupthedifference oftherealmomentumwhentheelectronleaves.
We thereforehavetwo rules,conservationof energyand conservationofthe k-vector paralleltothesurface.Thedirectionofanejectedelectroncanbemeasured,yieldingtwo angles,namelytheangle θ relativetothenormaltothesurface,andanazimuthalangle φ whichgivesthedirectionparalleltotheplaneofthesurface.Thetwocomponentsof ´kº
arethengivenby
´kº= (k sin θ cos φ, k sin θ sin φ). (1.10.2) Themagnitudeof k oftheelectroninthevacuumoutsidethematerialcanbeknownby measuringthekineticenergy Ekinoftheejectedelectrons,whichyields
k =º
2mEkin/±2. (1.10.3)
We thereforecanknow ´kº fully.For atwo-dimensional material,the band energy En is onlyafunctionof ´kº.Wecanthenobtainthebandstructureofatwo-dimensionalmaterial directly,knowingthephotonenergy ±ω:
En(´kº) = Ekin− ±ω. (1.10.4) ARPESiscapableofabsolutebandmappingoftwo-dimensionalelectronicstatesforthis reason.
63 1.10 Angle-ResolvedPhotoemissionSpectroscopy
Equation(1.10.4)givesthebandenergyrelativetothevacuumenergy,where Ekin = 0.
Inpractice,itisofteneasiertodeterminetheenergyrelativetotheFermileveloftheelec- tronsinthematerial,whichcanbedefinedbyaccuratelymeasuringtheelectricpotential differencebetweenthe sample and theanalyzer. Wewill discuss theFermi statisticsof electronsindetailinChapter2.
For three-dimensional,bulk materials,wemust makesomeadditional assumptionsin ordertouseARPEStostudybandenergies.Ingeneral,onemustassumesomemodelfor thebandstructureandfitthedatatothismodel.Forexample,wecanmakeasimplemodel thatabandnearzonecenterhasenergygivenby
En(´k) = ±2|´k|2
2meff − E0, (1.10.5)
where meffisaneffectivemass(seeSection1.9.4)and E0givestheenergyatzonecenter, whichisnegativerelativetotheenergyofanelectronatrestinvacuum.Forafixedvalue of kº,therewillbearangeofkineticenergiescorrespondingtodifferentvaluesof k⊥.The numberofelectronsemittedwithagivenenergy Ekin isproportionalto
N(Ekin) ∝ à ∞
−∞dk⊥ δ(Ekin− En− ±ω). (1.10.6) Toresolvethe δ-functionthatenforcesenergyconservation,wemustchangethevariable ofintegrationtoanenergy.Wewrite
En= ±2(k2º+ k⊥2)
2meff − E0, (1.10.7)
whichhastheJacobian
∂En
∂k⊥ =±2k⊥
meff = ±2 meff
º2meff(En+ E0)/±2− k2º. (1.10.8)
Wethushave N(Ekin) ∝
à ∞
±2kº2/2meff−E0dEn meff
±2º
2meff(En+ E0)/±2− k2ºδ(Ekin− En− ±ω)
= meff
±2º
2meff(Ekin− ±ω + E0)/±2− k2ºà(Ekin− ±ω + E0− ±2k2º/2meff), (1.10.9) where à(E)istheHeavisidefunction(seeAppendixC).
Recallingthat k2º= k2sin2θ,wecanwrite
N(Ekin) ∝ 1
ºEkin(1 − (m/meff)sin2θ) − ±ω + E0
ì à(Ekin(1 − (m/meff)sin2θ) − ±ω + E0).
Thisisapeakedfunctionwithamaximumat Ekin= (±ω − E0)/(1 − (m/meff)sin2θ).The effectivemass meffcanbefoundbyfittingthisfunctiontotheangle-resolvedARPESdata,
–0.2 0.0
0.1 Ekin 0.2
0.3
θ 0.0
±Fig.1.33 ARPESemissionspectraasafunctionofangle,forthesimpleeffective-mass0.2 modelgiveninthetext,for
m/meff= 10and ±ω − E0= 0.1.Thedashedlinefollowsthecurve Ekin = (±ω − E0)/(1 − (m/meff)sin2θ).
asshowninFigure1.33.SimilartothevanHovesingularitiesdiscussedinSection1.8,the infinityinthisequationdoesnotcauseaproblembecauseitisintegrable.
Complicating effects.TheabovediscussiongivesthegeneralwaythatARPEScanbe usedtodeducebandstructure.However,thereareseveraladditionaleffectsthatmustbe takenintoaccountinanalyzingrealdata.
• Internal scattering and surface reflection. Photoemission can be approximately described as a three-step process. In the first step, electrons in bands are excited by photonabsorptionintonearlyfreeelectronstateswiththesamemassaselectronsout- sidethematerial.Inthesecondstep,theseelectronspropagatetothesurface.Inthethird step,theelectronsaretransmittedthroughthesurfacetothevacuumoutside.
Breakingphotoemissiondownintothesestepsallowsustoaccountforwaysinwhich electronscanbelostbeforetheyreachtheoutside.First,electronspropagatingtoward the surface may scatter with imperfections in the lattice, changing their energy and momentum. Thescattering rate for theseprocessesmaydepend ontheenergy ofthe electrons(asdiscussedinChapter5.)Thescatteringratesforelectronsasafunctionof energyhavebeenmeasuredandtabulatedformanymaterials,andareoftenpresented asa“universalcurve.”Second,theelectronsmaybereflectedfromthesurface,andthe probabilityofreflectionmayalsodependontheelectronenergy.
• Line broadening. The energy spectra of the optical transitions can be broadened by many-bodyeffects,as discussedinSection8.4. Asimple waytoseewhythis broad- eningoccursistorecallthattheuncertaintyprincipleofquantummechanicsdoes not allowtheenergyofastatetobedefinedmoreaccuratelythan ±/τ,where τ isthetime spent byan electron in thestate. Interactions withother electrons (or withphonons, discussedinChapter4)canreducethetimespentinastate.
65 1.11 BandsandMolecularBonds
• Spacechargeeffects.Ifthecurrentofphotoemittedelectronsislarge,theywillrepeleach other,leadingtoblurringoftheangle-resolvedandenergy-resolveddata.Thiseffectcan bemitigatedexperimentallybyreducingthefluenceofphotons,sothatthephotoemis- sioncurrent is sufficientlylowthat the Coulombinteraction among photoelectrons is negligible.
• Fermilevel.Clearly,onlystatesinbandsthatareoccupiedbyelectronscancontribute tothephotoemission.ARPESmeasurementstherefore onlyworkforstatesbelowthe Fermileveloftheelectrons.AsdiscussedinChapter2,whenthetemperatureincreases, theFermilevelissmearedoutoverarangeoforder EF± kBT.
Exercise1.10.1 Supposethatabandenergyisgivenby(seeSection1.9.2)
En(´k) =−E0+ U12(cos kxa + coskya + cos kza). (1.10.10) Determinethespectral line shape N(Ekin) for angle-resolvedphotoemission from thisband,followingtheapproachfortheparabolicbanddiscussedinthissection.