Solid State Physics-4

2020/2/14SSPLectureNotesFall1ChapterIV:Phonons1.CrystalVibrations2020/2/14SSPLectureNotesFall2TheimportantelementaryexcitationsinsolidsaresymbolizedinFig.1.2020/2/14SSPLectureNotesFall3§1.VibrationsofCrystalswithMonatomicBasisStudy:Theelasticvibrationsofacrystalwithoneatomintheprimitivecell;Find:Thefrequencyofanelasticwaveintermsofthewavevectorthatdescribesthewaveandelasticconstants.Themathematicalsolutionissimplestinthe[100],[110],and[111]propagationdirectionsincubiccrystals.2020/2/14SSPLectureNotesFall4Whenawavepropagatesalongoneofthesedirections,entireplanesofatomsmoveinphasewithdisplacementseitherparallelorperpendiculartothedirectionofthewavevector.2020/2/14SSPLectureNotesFall5(1)One-dimensionaldispersionrelation2020/2/14SSPLectureNotesFall6Wecandescribewithasinglecoordinateusthedisplacementoftheplanesfromitsequilibriumposition.Thisbecomesanone-dimensionalproblem.Becauseforeachwavevectortherearethreemodes,oneoflongitudinalpolarization(seeFig.2)andtwooftransversepolarization(seeFig.3).2020/2/14SSPLectureNotesFall7psspsuu1p1s)()(11sssssuuuuFWeassumeaccordinglythattheforceontheplanesscausedbythedisplacementoftheplaneisproportionaltothedifferenceForbrevityweconsideronlynearest-neighborinteractions,sothatAccordingtoNewton’slaw,thetotalforceonscomesfromplanesistheforceconstantbetweennearest-neighborplanesandwilldifferforlongitudinalandtransversewaves.Itisconvenienthereaftertoregardβasdefinedforoneatomoftheplane,sothatFsistheforceononeatomintheplanes.oftheirdisplacements.2020/2/14SSPLectureNotesFall8)2(1122sssssuuuFdtudM)exp(]exp[),(tiiqxutxusssudtud222)2(112ssssuuuuMTheequationofmotionoftheplanesiswhereMisthemassofanatom.WelookforsolutionswithalldisplacementshavingthetimedependenceThenwehave,and(2)thenbecomes.(3),(2)2020/2/14SSPLectureNotesFall9])1(exp[1asiquus}2{)1()1(2isqaqasiqasiisqaeeeuueMThisisadifferenceequationinthedisplacementsuandhastravelingwavesolutionsoftheform:whereaisthespacingbetweenplanesandqisthewavevector.Thevaluetouseforawilldependonthedirectionofq.With(4),wehavefrom(3):,(5),(4)2020/2/14SSPLectureNotesFall10]1[cos2]2cos2[]2[2qaqaeeMiqaiqa]cos1[2]1[cos2)(2qaMqaMqaq0sin22qaMadkdAndfurtherlywehave;(6)TheboundaryofthefirstBrillouinzoneliesatWeshowfrom(7)thattheslopeofωversusq.qiszeroatthezoneboundary:,(8)(7)aq.0sinqaat,forhere.2020/2/14SSPLectureNotesFall11Thespecialsignificanceofphononwavevectorsthatlieonthezoneboundaryisdevelopedin(12)below.2020/2/14SSPLectureNotesFall12;21sin422qaMqaMq21sin4)(Byatrigonometricidentity三角恒等式(7)maybewrittenasandAplotofωversusqisgiveninFig.4.Thisisthe1Ddispersionrelation.(9)2020/2/14SSPLectureNotesFall13(2)FirstBrillouinZoneiqasseuu1Whatrangeofqisphysicallysignificantforelasticwaves?OnlythoseinthefirstBrilouinzone.Fromeq.(4)theratioofthedisplacementsoftwosuccessiveplanetsisgivenbyTherange-πto+πforthephasekacoversallindependentvaluesoftheexponential.Thereisabsolutelynopointinsayingthattwoadjacentatomsareoutofphasebymorethanπ:arelativephaseof1.2πisphysicallyidenticalwitharelativephaseof-0.8π,andarelativeof4.2πisidenticalwith0.2π.Weneedbothpositiveandnegativevaluesofkbecausewavescanpropagatetotherightortotheleft..(10)2020/2/14SSPLectureNotesFall14qaaqaaqmax0amaxqa/Therangeofindependentvaluesofkisspecifiedby,orThisrangeisthefirstBrillouinzoneofthelinearlattice,asdefinedinChapterII.TheextremevaluesareThereisarealdifferenceherefromanelasticcontinuum:inthecontinuumlimitandValuesofkoutsideofthefirstBrillouinzone(seeFig.5)merelyreproducelatticemotionsdescribedbyvalueswithinthelimits.(10)2020/2/14SSPLectureNotesFall15a/2'qanqq/2'Wemaytreatavalueofqoutsidetheselimitsbysubtractingtheintegralmultipleofthatwillgiveawavevectorinsidetheselimits.Supposeqliesoutsidethefirstzone,butarelatedwavevectorisdefinedbylieswithinthefirstzone,wherenisaninteger.2020/2/14SSPLectureNotesFall16aiqnqainiiqasseeeeuu')2(21an/2a/2Thenthedisplacementratio(10)becomes:Thusthedisplacementcanalwaysbedescribedbyawavevectorwithinthefirstzone.Wenotethatisareciprocallatticevectorbecauseisareciprocallatticevector.Thusbysubtractionofanappropriatereciprocallatticevectorfromqwealwaysobtainanequivalentwavevectorinthefirstzone.(11)2020/2/14SSPLectureNotesFall17aq/maxisqasueusasqmaxssuisuu)1()exp(1suAttheboundariesoftheBrillouinzonethesolutiondoesnotrepresentatravelingwave,butastandingwave.Atthezoneboundaries,whenceThisisastandingwave:alternateatomsoscillateinoppositephases,becauseaccordingtowhethersisanevenoranoddinteger.Thewavemovesneithertotherightnottotheleft..(12)2020/2/14SSPLectureNotesFall18aq/maxndsin22/11,/2,nqada2ThissituationisequivalenttoBraggreflectionofx-rays:whentheBraggconditionissatisfiedatravelingwavecannotpropagateinalattice,butthroughsuccessivereflectionsbackandforth,astandingwaveissetup.ThecriticalvaluefoundheresatisfiestheBraggcondition:wehave,sothatWithx-raysitispossibletohavenequaltootherintegersbesidesunitybecausetheamplitudeoftheelectromagne