Electron-phonon-coupling電子聲子耦合

Phonons&

electron-phononcouplingClaudiaAmbrosch-Draxl

DepartmentfürMaterialphysik,Montanunversit?tLeoben,AustriaInstitutfürPhysik,Universit?tGraz,AustriaAspectsofe-phCouplingChargetransportHeattransportThermalexpansionElectron(hole)lifetimesSuperconductivitySomeimportantphenomenaeffectiveelectron-electroninteraction?kqk+qk'k'-qOutline

ThefrozenphononapproachLatticedynamicsAtomicforcesBasicsPhononsandelectron-phononcoupling

SymmetryVibrationalfrequenciesNormalvectorsRamanscatteringLinear-responsetheoryComparisonwithexperimentLAPW/WIEN2kspecificaspectsandexamplesTheFrozen-PhononApproach1Dcase:Calculateenergies

FitexpansioncoefficientsHarmonicapproximationTheHarmonicApproximationGeneralcase:Forceconstant:changeoftheforceactingonatomainunitcellnindirectioni,whendisplacingatombinunitcellmindirectionj.Displacementwave:TheHarmonicApproximationEquationofmotion:Vibrationalfrequenciesw:bydiagonalizationofthedynamicalmatrixD

Natomsperunitcell

3NdegreesoffreedomSetof3Ncoupledequations#displacementsTotalenergyForcesN=2103

YBa2Cu3O7:N=135Agmodes70321194ComputationalEffortNatomsperunitcellHarmoniccaseonly!Interpolationonly–nofit!TheHellmann-FeynmanTheorem

ManyparticleSchr?dingerequationelectroniccoordinatesioniccoordinatesgroundstatewavefunctionwithrespecttofixedionsTheHellmann-FeynmanForceHellmann-Feynmanforce:totalclassicalCoulombforceactingonthenucleusastemmingfromallotherchargesofthesystem=electrostaticforcestemmingfromallothernuclei+electrostaticforcestemmingfromtheelectronicchargescomponentoftheelectricfieldcausedbythenuclearchargeForcesinDFTAtomicforce:PulaycorrectionsTotalenergy:ForcesintheLAPWBasisHellmann-Feynmanforce:classicalelectrostaticforceexcertedonthenucleusbytheothernucleiandtheelectronicchargedistributionIBSforce:incompletebasissetcorrectionduetotheuseofafinitenumberofposition-dependentbasisfunctionsCorecorrection:contributionduetothefactthatforcoreelectronsonlythesphericalpartofthepotentialistakenintoaccountTheLAPWMethodAtomicspheres:atomic-likebasisfunctionsInterstitial:planewavebasissite-dependent!Example:YBa2Cu3O7O(1)Cu(1)BaYCu(2)O(4)O(3)O(2)Orthorhombiccell:PmmmSymmetryBilbaoCrystallographicServer::///Factorgroupanalysis:5Ag+8B1u+5B2g+8B2u+5B3g+8B3uAgB1uB2gB2uB3gB3uDynamicalmatrixRaman-active

Infrared-active

q=0ForcesinYBa2Cu3O7Forcecontributionsforamixeddistortion:-O(2),-O(4)YBa2Cu3O7:PhononFrequenciesAgmodesYBa2Cu3O7:NormalVectorsAgmodesYBa2Cu3O7:LatticeVibrationsoxygen

modesBa

/CumodesRamanActivePhononsTheoryRamanScatteringIntensitiesCAD,H.Auer,R.Kouba,E.Ya.Sherman,P.Knoll,M.Mayer,Phys.Rev.B65,064501(2002).ExperimentProbinge-phCouplingStrengthO(4)modeProbinge-phCouplingStrengthResonance:Peakat2.2eVAlloxygenmodesO(4)displacement!ProbingNormalVectorsBa-Cumodes:Experiment:site-selectiveisotopesubstitutionIsotopeSubstitutionRamanscatteringintensities:InfluenceofmassandeigenvectorsIsotopeSubstitutionRamanscatteringintensities:Changeofe-phcouplingstrengththroughnormalvectorsCAD,H.Auer,R.Kouba,E.Ya.Sherman,P.Knoll,M.Mayer,Phys.Rev.B65,064501(2002).RelevantforsuperconductivityE.Ya.ShermanandCAD,Eur.Phys.J.B26,323(2002).q-dependentPhononsEquilibriumq≠0q=0Supercellsvs.PerturbationTheoryUnitcellcommensuratewiththeq-vector(supercell)ComputationallyverydemandingSupercellmethod:Linearresponsetheory:Startingpoint:undisplacedstructureTreatq-dependentdisplacementasperturbationSelf-consistentlinear-responsetheoryKeepsinglecellComputationaleffortnearlyindependentofq-vectorAnharmoniceffects

neglectedN.E.Zein,Sov.Phys.Sol.State26,1825(1984).

S.Baroni,P.Gianozzi,andA.Testa,Phys.Rev.Lett.58,1861(1987).LinearResponseTheoryAtomicdisplacement:smallpolarizationvectorSuperpositionofforwardandbackwardtravellingwaveStaticfirst-orderperturbationwithindensity-functionalperturbation

theory(DFPT)Determinefirst-orderresponseontheelectroniccharge,effectivepotentialandKohn-ShamorbitalsLinearResponseTheoryIterativesolutionofthreeequations:Determineq-dependentatomicforcesanddynamicalmatrixAlternativelycomputesecondorderchanges(DM)directly

e-phMatrixElementsElectron-phononmatrixelement:Needtoevaluatematrixelementslike:Scatteringprocessofanelectronbyaphononwithwavevector

qMatrixelementsincludingPulay-liketerms:S.Y.SavrasovandD.Y.Savrasov,Phys.Rev.B54,16487(1996).R.Kouba,A.Taga,CAD,L.Nordstr?m,andB.Johansson,Phys.Rev.B64,184306(2002).e-phCouplingConstantsCouplingconstantforaphononbranchn:bccSR.Kouba,A.Taga,CAD,L.Nordstr?m,and

B.Johansson,Phys.Rev.B64,184306(2002).J.K.Dewhurst,S.Sharma,andCAD,68,020504(R)(2003);

H.Rosner,A.Kitaigorodotsky,andW.E.Pickett,Phys.Rev.Lett.88,127001(2002).

WhatCanWeLearn?Comparisonwithexperiment….P.Puschnig,C.