凝聚态光物理学3

Interbandabsorption3.1Interbandtransitions3.2Thetransitionratefordirectabsorption3.3Bandedgeabsorptionindirectgapsemiconductors3.4Bandedgeabsorptioninindirectgapsemiconductors3.5Interbandabsorptionabovethebandedge3.6Measurementofabsorptionspectra3.7Semiconductorphotodectors313.1InterbandtransitionIsolatedatomdiscreteenergylevelSolidband(delocalizedstate)Forsemiconductororinsulator,photonexciteselectronfromfilledvalencetotheemptyconductionband,thetranstionenergyis

Thereisacontinuousrangeoffrequency;Thereisathresholdh>Eg=(Ef

—Ei)min;

Creationofanelectron-holepair;DirectandIndirectbandgap.23.1InterbandtransitionInadirectbandmaterial,boththeconductionbandminimumandthevalencebandminimumoccuratthezonecentrewherek=0;Inaindirectbandgapmaterial,theconductionbandminimumdoesnotoccuratk=0,butisusuallyatthezoneedgeorclosetoit.PhotonabsorptionAbsorptionandemissionofphonon3.2Thetransitionratefordirectabsorption

TheopticalabsorptioncoefficientWi->ftransitionrate,3.1InterbandtransitionWherethematrixelementM,the

densityofstatesg(h).(Fermi’sgoldenrule)3

thematrixelementM,(semiclassicalapproach)

3.2ThetransitionratefordirectabsorptionTheelectronstatewavefunction:Perturbation:Dipolemoment:Lightwave:Initial:Final:

thejointdensityofstatesThisgives:43.3BandedgeabsorptionindirectgapsemiconductorsElectronlevelinacovalentcrystalmadefromfour-valentatomsuchasGeorbi-narycompoundssuchasGaAs.ThesAndpstatesoftheatomshybridizetoformbondingandantibondingmolecularorbitals,whichthenevolveintothecon-ductionandvalencebandsofthesemi-conductor4s24p2Selectionrules:Theparityoftheinitialandfinalstatesmustbedifferent.j=-1,0or+1.(Totalangularmomentummustchangebyoneunit)l=1.ms=0.(Spinquantumnumbersneverchange).Electric-dipoletransition:hightransitionrate,shortradiativelifetime(10-9–10-8s)–fluorescence.Magneticdipoleorelectricquadrpole:smallertransitionratesandlongerradiativelifetime(10-6supwards)–theslowemissionbyelectricdipole-forbiddeniscalled–phosphorescence.

3.3.1Theatomicphysicsoftheinterbandtransitions53.3.2ThebandstructureofadirectgapIII-Vsemiconductor3.3BandedgeabsorptionindirectgapsemiconductorssantibondingpbondingS-likeconductionandthreep-likevalenceband(heavyholeband)(lightholeband)(split-offholeband)BandstructureofGaAs.ThedispersionofthebandsisshownfortwodirectionsoftheBrillouinzonecentre:XandL.Thepointco-respondstothezonecentrewithawavevectorof(0,0,0),whiletheXandLpointscorrespondrespectivelytothezoneedgesalongthe(100)and(111)directions.ThevalencebandsareBelowtheFermilevelandarefullofelectrons.63.3Bandedgeabsorptionindirectgapsemiconductors3.3.3ThejointdensityofstatesThedispersionofband(E—krelationship)Theenergyconservationofaheavyholeoralightholetransition:Generally,等于能量时的状态对密度)(Fig.173.3.4ThefrequencydependenceofthebandedgeabsorptionSquareoftheopticalabsorptioncoefficientversusphotonenergyforthedirectgapIII-VsemiconductorInAsatroomtemperature.Thebandgapcanbededucedtobe0.35eVbyextrapolatingtheabsorptiontozero.FrequencydependenceapproximatelyobeyedThecoulombattractionofexcitonsneglected;Impurityordefectstateswithingapneglected;Theparabolicbandapproximationonlyvalidneark=TheFranz-KeldysheffectTwomaineffectsonbandedgeabsorptionbyapplicationofanexternalelectricfieldE:Electro-opticeffect:Theelectricfieldmodulatedopticalconstants(n,).(K-Krelationship).Electroreflectance:Thereflectivitycanbechangedduetomodulatedopticalconstants(n,).83.3.6BandedgeabsorptioninamagneticfieldTheelectronsinmagneticfield:

Thequantizedenergy(Landaulevels):

(cyclotronfrequency)Theenergieselectronsandholeswithinthebandsaregivenby:Quantizedmotioninthe(x,y)plane,freemotioninthezdirection.E=0atthetopofthevalenceband:TheinterbandtransitioncreatesanelectronintheconductionbandandaholeinvalencebandSelectionrule:n=n’,

kZ=k’Z.(themomentumisnegligible)TheabsorptionspectrumwithkZ=0givenby:Thetransitionenergy:Twoconsequences:TheabsorptionedgeshiftsbyheBZ/2;Equallyspacedpeaksinthespectrum.

TransmissionspectrumofgermaniumofgermaniumforB=0andB=3.6Tat300K.Theelectroneffectivemasscanbedeterminedfromtheenergiesofminima.Fig.293.4BandedgeabsorptioninindirectgapsemiconductorTheindirecttransitioninvolvebothphotonsandphonons(h,hq):Thisisasecond-orderprocess,thetransitionrateismuchsmallerthanfordirectabsorption.ComparisonoftheabsorptioncoefficientofGaAsandSiliconneartheirbandedges.GaAshasadirectbandgapat1.42eV,whilesiliconhasanindirectgapat1.12eV.Theabsorptionrisesmuchfasterwithfrequencyinadirectgapmaterial,andexceedstheindirectmaterial.Absorptioncoefficientofindirectbandgap:Thedifferences:Threshold;Frequencydependence.Thedifferencesprovideawaytodeterminewhetherthebandgapisdirectornot.103.4BandedgeabsorptioninindirectgapsemiconductorBandstructureofgermanium.1.ThelowestconductionbandminimumoccursattheLpoint(k=/a(1,1,1),notat(k=0);2.Indirectgap=0.66eV,directgap()=0.8eV;TheabsorptioncoefficientofgermaniumTable3.1111.vshclosetothebandgapat0.66eV;2.Thestraightlineextrapolatesbackto0.65eV,whichindicatesthataphonon(TA)ofenergy~0.01eVhasbeenabsorbedandq(phonon)=k(electron)at

L-pointoftheBrillouinzone;3.Ataildownto0.6eV,thisiscausedbyabsorptionofthehigherfrequencyandalsomultiphononsabsorption;4.Thetemperaturedependenceoftheabsorptionedge:(Bose-EinsteinFormula)AsTdecrease,phononsdecreasegradually.AtverylowT,nophononsexcitedwithenoughenergy.ThusatthelowestT,theindirectabsorptionedgeisdeterminedbyphononemissionratherthanphononabsorption;5.Thedirectabsorptiondominatesovertheindirectprocessesonceh>0,8eV,isthebandgapforthetransitionatthe12TheinterbandabsorptionspectrumofsiliconThebandstructureofsiliconEgisindirectandoccursat1.1eV;E1andE2aretheseparationofthebandsattheLandXpoints,wheretheconductionandvalenceareapproximatelyparallelalongthe(111)and(100).E1=3.5eVistheminimumdirectseparation,andcorrespondstothesharpincreaseinabsorptionatE1,andE2correspondtotheabsorptionmaximumat4.3eV.AbsorptionattheseenergiesisveryhighduetotheVanHovesingularitiesinthejointdensityofstates(bandareparallel.Efordirecttransitiondoesnotdependonk,dE/dk=0,g(E)diverges(criticalpoint)3.5Interbandabsorptionabovethebandedge1.Theopticalpropertiesatthebandedgedeterminetheemissionspectra;2.ThespectrumcanbeworkedoutbydE/dkfrom

thefullbandstructure.133.6MeasurementofabsorptionspectraThemeasurementofabsorptioncoefficient:Measuretransmissioncoefficient;

2.MeasurereflectivityspectraR(h)Self-consistentfittingofthereflectivityspectrausingtheKramers-Kronigformula.143.7SemiconductorphotdectorsTheoperatingprinciples:Lightwithphotonenergygreaterthanthebandgapisabsorbedinthesemiconductor,andthiscreatefreeelectronsintheconductionbandandfreeholeinthevalenceband.Thepresenceofthelightcanthereforebedetectedeitherbymeasuringachangeinresistanceofthesampleorbymeasuringanelectricalcurrentinanexternalcircuit.Thep-i-nphotodiodeisoperatedinreversebiaswithapositivevoltageVoappliedtothen-region.ThisgeneratesofastrongDCelectricfieldEacrossthei-region.Absorptionofphotonsinthei-regioncreatesfreeelectron–andholethatareattractedtothen-regionandp-regionsrespectivelythenflowintothecircuitbythefield,generatingthephotocurrentIpc3.7.1PhotodiodesThefractionoflightabsorbedinalengthl:PhotocurrentIpc:quantumefficiency,thefluxofphotonsperunittime(A/W)If153.7Semiconductorphotdectors3.7.2PhotoconductivedevicesThedevicereliesonthechangeoftheconduc-tivityofmaterialwhenilluminatedbylight.Theconductivityincreaseduetothegenerationoffreecarriersafterabsorptionofphotonsbyinterbandtransitions.Comparedwithphotodiodes,thedetectorsaresimpler,buttendtohaveslowresponsetimes.3.7.3Photovoltaicdevices(solarcell)Thedevicegeneratesaphotovoltagewhenirradiatedbylight.Thisinturncanbeusedtogenerateelectricalpowerinanexternalpower.16Exercises(3):IndiumphosphideisadirectgapIII-Vsemiconductorwithabandgapof1.35eVatroomtemperature.Theabsorptioncoefficientat775nmis3.5*106m-1.Aplateletsample1mthickismadewithantireflectioncoatedsurfaces.Estimatethetransmissionofthesampleat620nm.(0.37%)Thebandparametersofthefour-bandmodelshowninFig.1aregivenforGaAsinTableC.2.i)Calculatethekvectoroftheelectronexcitedfromtheheavyholebandtotheco