固体物理+晶格振动.ppt

Lecture11,Chapter2Thelatticedynamics:Thermalandopticalpropertiesofcrystals,Latticedynamics-a1dhomogeneoussolid,Considertheonedimensionalsolidabove,inanarbitrarystateofstrain,uisthedisplacementoftheindicatedelementfromitsequilibriumposition,theassociatedstrainise=du/dx,(iethefractionalchangeoflength),Atthelefthandendoftheelementwehaveastraine(x),Attherighthandendwehaveastrain,Theforcesatthetwoendsoftheelementopposeeachother,andtheresultantforceis,wherecisanelasticmodulus,Lecture11,Dynamicsina1dhomogeneoussolid,UsingF=ma,andassumingisthelinearmassdensityofthelinearsolid,=dm/dx,or,hence,with,Thisisthewell-knownwaveequationwithsolutionsintheformofatravellingwave,where,Thesewavesaresoundwaves,andvoisthevelocityofsoundinanhomogeneousmedium,Substitutingthesolutionintothewaveequationweobtainthedispersionrelation,mustbeanintegralmultipleof/LwhereListhelengthofthebody,sothedispersionrelationisnotstrictlycontinuousbutaseriesofcloselyspacedpointsormodesofvibration,Lecture11,Alineofatoms,Weknowthatmatterisdiscontinuous,sowehavetotakeintoaccounttheatomicstructure,Thisisbestdonebyconsideringthe1dsolidasalinearchainofatomsofmassMspacedatadistanceaconnectedbysprings,ofspringconstantC,whichobeyanidealHookeslaw,equilibrium,strained,Lecture11,Dynamicsofachainofatoms,Asbeforewecansetupaforceequation,Takingthenthatomwehaveforcesof,C(un-un-1)totheleft,C(un+1-un)totheright,So,usingF=mawehave,Thisequationofmotionisthesameforallatoms,onlythesubscriptsvary,Asbeforewecanassumewave-likesolutionsoftheform,Replacingthexcoordinatebyna,whereaisthelatticeparameterwehave,Lecture11,Dynamicsofachainofatoms,Itisreasonabletoassumethatall(identical)atomsvibratewiththesameamplitude,so,Sosubstitutingthethreecoupledequations,intotheequationofmotion,wehave,So,oncancellingfromeachsideoftheequationweobtain,so,Hence,or,Lecture11,Dynamicsofachainofatoms,Notethatforsmallthedispersionislinear,(asforthe1dsolid),Lecture11,DynamicsintheBrillouinzones,1stBrillouinzone,FromwhatweknowaboutBrillouinzonesthepointsAandB(relatedbyareciprocallatticevector)mustbeidentical,Thisimpliesthatthewaveformofthevibratingatomsmustalsobeidentical.,Lecture11,Equivalenceofpointsin1stand2ndzones,But:notethatpointBrepresentsawavetravellingright,andpointAonetravellingleft,Lecture11,Lecture11,Azoneboundarymode,ConsiderpointCatthezoneboundary,When=/a,=2a,andmotionbecomesthatofastandingwave(theatomsarebouncingbackwardandforwardagainsteachother,Lecture11,Lecture11,Zoneboundarymode,Athigherfrequencies(egneartheboundaryofthe1stBrillouinzone)thegroupvelocityandphasevelocityarenolongerequal,If0,vgvs,butif/a,vg0,Similarly,andif0,vpvs,butif/a,vp2vs/,Vgisthequantitythatrepresentsthetransferofsignalorenergy,soat=/anosignalorenergyispropagated:thewaveisastandingwave,ThissituationisequivalenttoaBraggreflectioninadiffractionexperiment,RememberthatthezoneboundaryisthelocusofpointsforwhichaBraggreflectionispossible,Lecture11,Simpledispersionrelations:,Nearestneighbourforces,Lecture12,Thediatomiclinearlattice,Ats,Misdisplacedasmallamountus,Ats+1/2,misdisplacedasmallamountvs+1/2,Theequationsofmotionforthetwoatomtypesare:,toright,toleft,toleft,toright,and,AsbeforeweassumeonlynearneighbourinteractionswithaspringconstantC,Lecture12,Thediatomiclattice,Therearesolutionstotheseequationsofmotionofthetype:,wheresisaninteger,Substitutingthesesolutionsbackintotheequationsofmotionwehave,so,whichwemustnowsolve,Lecture12,Thediatomiclattice,Solvingtheseequationsforuandvwehave,Solvingthisquadraticequationfor2gives,Twosolutions!,Lecture12,Thediatomiclattice,Wehavethereforefoundtwopossiblesolutionsgivingtwopossibledispersionrelationsforthediatomiclattice,When=0,ieatthezonecentre,When=/a,ieatthezoneboundary,+sign,-sign,Lecture12,Dispersionrelationforadiatomiclattice,Lecture12,Dispersionrelationforadiatomiclattice,Whatdothetwobranchesrepresent?,wefind,Returningtoeitherofthetwosolutionstothewaveequation,eg,wefind,EvidentlytheamplitudesofthemassesMandmareinphaseinbranchAofthedispersionrelationandoutofphaseinbranchB,Lecture12,Atthezoneboundary,Theequationsthatdefinetheamplitudeoftheoscillationsare,Fromwhich,andonlythelightatomsvibrate,andonlytheheavyatomsvibrate,Lecture12,Thelowenergybranch,InbranchAwherewecancalculatethewavemotionfor,say=/5a,andm/Mof0.6,.andthetwotypesofatomvibratewiththesameamplitude,directionandphase,ThisbranchiscalledtheAcousticBranch,Lecture12,Acousticmodeat=0.2/a,Lecture12,Theacousticmodeatthezoneboundary,Lecture12,Thehighenergybranch,InbranchBwherewecancalculatethewavemotionfor,say=/5aandm/Mof0.6,.andthetwotypesofatomvibrateagainstoneanother.,Thecentreofmassremainsstationary,andtheamplitudeoftheheavyatomsism/Mthatofthelightatoms,ThisbranchiscalledtheOpticBranch,Lecture12,Theopticmodeatthezonecentre,Lecture12,Theopticmodeat=0.2/a,Lecture12,Theopticmodeatthezoneboundary,Lecture12,Intheextendedzonescheme,Wehavedrawnthedispersionrelationinthereducedzonescheme-everythingmappedintothefirstBZ,Wecouldalsodrawitintheextendedzonescheme,WhenwedoitisapparentwhathappensasmM,(a)thegapstartstocloseandwhenm=Mitlookslikethemonatomiclattice,(b)also,whenm=MthefirstBZextendsto=2/a-thisistobeexpectedasweknowhaveaneffectivelatticeparameterofonlya/2,Lecture12,Discreteexcitations,Wehavealreadysuggestedthatthedispersionrelationisnotstrictlycontinuousbutaseriesofcloselyspacedpointsormodesofvibration-aconsequenceofthefinitelengthofthesample.Thesameisalsotruehere,S+N-1,S,S+1,S+2,Intermsoftheofphaseoftheoscillatorymotion,Thisrequires,Thesmallestseparationoftwovaluesofthatareallowedistherefore,andeachvalueisa“mode”or”state”,Notethatisindependentof,andisdeterminedbyN,hencesizeofsample,Thisisanindicationofthequantisationoflatticevibrations,Howeveralinearchain1cmlongwitha=1,has=1cm-1.ButtheBZis108cm-1wideandwillthushave108modesdistributedevenlythroughtheBZ,Lecture12,Phonons,Althoughwehavetreatedlatticevibrationsclassicallywecanseethattheyareinfactquantised,Thenormalmodesareharmonicandindependentandthereforelendthemselvestoquantummechanicaltreatment,Alatticevibrationmodeoffrequencywbehaveslikeasimpleharmonicoscillatorwithanenergy,iethereareuniformlyspacedenergylevelswithazeropointmotionof,Encanbeconsideredasthesumofnexcitationquantaaddedtothegroundstate,ThequantumofthermallyexcitedlatticevibrationenergyiscalledaPHONONinanalogytothethermallyexcitedphotonsofblackbodyelectromagneticradiation,Thetwosystemsarebothequivalenttoaquantumharmonicoscillator,Bothphotonsandphononsarebosons-theycanbecreatedanddestroyedincollisions,Lecture12,Spatiallocalisationofaphonon,Normalmodesareplanewavesextendingthroughoutthecrystalhencephononsarenotlocalisedparticles,Theirpositioncannotbedeterminedbecauseisknownexactly,Howeverareasonablylocalisedwavepacketcanbeconstructedwithinthelimitsoftheuncertaintyprinciplebycombiningmodesofdifferingwandl,Thusifwetakewaveswithaspreadofofsayp/10awewillconstructawavepacketlocalisedtowithin10unitcellsmovingwithagroupvelocityofdw/dk,Lecture12,Phononmomentum,Thephononmomentumisgivenbyp,Neutrons,photons,electrons,etccaninteractwiththephononasifitwereaparticlewiththismomentum,However,thisisnotatruemomentum,forexamplewehaveseenthatwithinreciprocalspace,Thephononthereforedoesnotcarryphysicalmomentum.,.aphononcoordinateinvolvestherelativecoordinatesoftheparticipatingatoms.,.astheatomsareonlyvibratingabouttheircentreofmassnolinearmomentumistransferred.,issometimescalledthecrystalmomentum,Lecture13,Phononinteractions,Wehaveseenthatforx-rayorneutrondiffractionwecanwrite,wherekisthewavevectoroftheincidentradiation,kthatofthescatteredradiationandGhklisareciprocallatticevector,Inthescatteringprocessthemomentumofthesystemasawholeisrigorouslyconserved,Ifthescatteringofthephotonorneutronisinelasticthenaphononcanbecreatedwithamomentumpsotheselectionrulebecomes,Alternatively,aphononmightbeabsorbedbythephotonorneutronduringthescatteringprocess,Lecture13,Studyingphonons,Lightscattering,Lightscattering,ieRamanscattering,isproducedbythepolarisationofatomsandsubsequentemissionofdipoleradiation,Themaximumwavevectortransferis2k=2x2p/l=2x10-3-1,whichisonly1/1000ofareciprocallatticevector.,Ramanscatteringcanthereforeonlybeusedtostudymodesclosetothezonecentre,X-rayinelasticscattering,Thewavelengthofx-rays(1)overcomesthisproblem,buttheassociatedenergyis10keV.,Latticevibrationenergiesaretypically1meV,sowewouldneedtomonochromatewithDl/l10-7,which,assumingBraggsLaw,impliesacrystalmonochromatorforwhichDd/d10-7bothinaccuracyandstability,Difficulttothepointofimpossible.,Butneutronsarejustright1neutronshave80meVenergy,implyingDl/l10-3,Lecture13,Neutronsandphonons,Conservationofmomentumgivesus,whiletheconservationofenergygives,whereistheenergyofthephonon(createdorabsorbedinthescattering),andthelatticeperiodicityalsoallowsustowrite,Notethatwecanwritetheneutronmomentumtransferas,Fromthediagram,whichcanbesolvedbymeasuringk,thefinalneutronmomentum,Lecture13,.andontheEwalddiagram,Thisisphononwouldbedifficulttomeasureasthetotalscatteringangle,f,isverysmall,andthescatteredneutronbeamisclosetotheincidentneutronbeam,Lecture13,.andontheEwalddiagram,Exactlythesamephononcanbemeasuredclosetothereciprocallatticepoint,wherethegeometryismuchmorefavourable,Lecture13,Thetripleaxisspectrometer,Lecture13,Thetripleaxisspectrometer,Lecture13,Thetripleaxisspectrometer,Lecture13,Thetripleaxisspectrometer,Lecture13,Dispersionrelationships,MonatomicFCCNeon,Lecture13,Dispersionrelationships,MonatomicBCCNa,Lecture13,Dispersionrelationships,“diatomic”Si(diamondstructure),Lecture13,PhononsinMgB2,Phononmodesintheanomalous40KsuperconductorMgB2,Lecture13,Anharmonicity,Themeasuredphonondispersionrelationssuggestthatthisisnotunreasonable,howevertheconsequencesforotherpropertiesaresevere:,Sofarwehaveconsideredlatticevibrationsonlywithinthe“harmonicapproximation”,ietheinteratomicpotentialistreatedasbeingparabolic,(ii)Thermalconductivityisinfinite,(i)Therecanbenothermalexpansionintheharmonicapproximation,Intheharmonicapproximationphononsdonotinteractwithoneanother,so,intheabsenceofanydefectstherewillbenohindrancetothetransferofheatenergybyphonons,Lecture13,Phonon-PhononCollisions,Intheharmonicapproximationtwoormorephononsmaypassthrougheachotherwithoutinteraction,iethephononmodesareuncoupled.,Whenanatomvibratesitexcitesaphonon.,Thisisbecausetheelasticconstantsareeverywhereth