Electron-phonon coupling: a tutorial - …

1 Electron-phonon coupling : a tutorialW. H bner, C. D. Dong, and G. LefkidisUniversity of Kaiserslautern and Research Center OPTIMAS, Box 3049, 67653 Kaiserslautern, GermanyTargoviste, 29 August 2011 Outline1. The harmonic oscillator real space energy basis2. 1D lattice vibrations one atom per primitive cell two atoms per primitive cells3. electron phonon interactions localized electrons small polaron theory phonons in metals4. Superconductivity5. A numerical example: CO6. LiteratureOutline1. The harmonic oscillator real space energy basis2. 1D lattice vibrations one atom per primitive cell two atoms per primitive cells3. electron phonon interactions localized electrons small polaron theory phonons in metals4. Superconductivity5. A numerical example: CO6.

2 Literature1) The harmonic oscillatorquantization of the oscillator in real spaceEigenvalues ofProjection of eigenvalue equation to X basis(Substitution by differential operators) leads to222221()22dmxEmdx 1) The harmonic oscillatorquantization of the oscillator in real space1) Dimensionless variablesxby 12bm 22mEbE ''2(2) 0y ''20y 22224222220dmEbmbydy leads toand22myAy e with solutionsince22''222222421 (1)1mymyymmmAyeAyeyyy -10-5510-55101) The harmonic oscillatorquantization of the oscillator in real spacewith solution''20 cos[ 2 ] sin[ 2 ]AyBy consistency requires20()yAcy O y thus (3)22() ()yyuye ansatz:leads to'''2(21)0uyu ) The harmonic oscillatorquantization of the oscillator in real space4) Power series expansion: 0()nnnuyCy 20[( 1)2 (2 1) ] 0nnnnnCnn ynyy 22(1)nnnCnny 2mn 2200( 2)( 1)( 2)( 1)mnmnmnCm m y Cn n y inserted into differential equationwith index shiftwe get2(2 1 2 )(2)(1)nnnCCnn feeding back in the original leads to recursion:20[(2)(1) (212)]0nnnnyC nnCn 1) The harmonic oscillatorquantization of the oscillator in real spaceso we haveProblems=> way out: termination of series required22242202 43513 () ().

3 NyynnnCCyCyCyyuyeeCCy CyCy 243501(1 2)(1 2) (4 1 2)(2 1 2) (2 1 2)(6 1 2)() 1(0 2)(0 1) (0 2)(0 1) (2 2)(2 1)(1 2)(1 1) (1 2)(1 1) (3 2)(5 1)yyyyuy CC y 1) The harmonic oscillatorquantization of the oscillator in real spaceconsequence energy quantization of the harmonic oscillator by backwards substitution1()2nEn Examples:0() 1Hy 1() 2 Hyy 22() 21 2 Hyy 332() 123 Hyy y 2444() 121 43 Hyyy Outline1. The harmonic oscillator real space energy basis2. 1D lattice vibrations one atom per primitive cell two atoms per primitive cells3. electron phonon interactions localized electrons small polaron theory phonons in metals4. Superconductivity5. A numerical example: CO6. Literature1) The harmonic oscillatorquantization of the oscillator in energy basisOscillator in energy basis222122 PmX E EEm Direct way.

4 Fourier transform from real to momentum spaceNo savings compared to direct solution of Schr dinger equation in real space1) The harmonic oscillatorquantization of the oscillator in energy basiscommutator ,XPiIi 1212122maXi Pm 1212122maXiPm ,1aa definitionand adjointfurtherNew operator (dimensionless) (12)HHaa 1) The harmonic oscillatorquantization of the oscillator in energy basisCommutator of creation and annhiliation operators with HamiltonianRaising and lowering properties ,,12,aHaaaaaa a ,aHa [, ]( 1)HaaHa HaH aa 11aC But eigenvalues non negative requirementno further lowering allowed00a 00aa 012 1) The harmonic oscillatorquantization of the oscillator in energy basis012 0(12), 0,1,2.

5 Nn (12), 0,1,2,..nEnn A possible second family must have the same ground state, thus it is not allowed1nan C n *1nnanC *11nnnaannnCC * 12nnnHn CC 2nnnn C 2nCn 12inCne and adjoint equationform scalar product of both equation1) The harmonic oscillatorquantization of the oscillator in energy basis121an n n 12(1) 1an nn 1212 121aanannnnnnn Naa 12HN with number operatorfurther1212', 1''1nnnan n nnn 1212', 1'(1)'1(1)nnna n nnnn 1) The harmonic oscillatorquantization of the oscillator in energy basisposition and momentum operators12()2 Xaam 12()2 Paam 12121201 0 000 What do we learn?Analogously for derived operators1) The harmonic oscillatorquantization of the oscillator in energy basis1212121212121201 0 032003 1212121212121201 0 0 32003 12 0 0 0.

6 0320 Outline1. The harmonic oscillator real space energy basis2. 1D lattice vibrations one atom per primitive cell two atoms per primitive cells3. electron phonon interactions localized electrons small polaron theory phonons in metals4. Superconductivity5. A numerical example: CO6. Literature2) 1D lattice vibrations (phonons)1 atom per primitive cell()spsp spFCuu 22()spsp spduMCuudt ()is pKa i tspuue e force on one atomequation of motion of atomsolution in the form of traveling waveEOM reduces to2()()isKai ti s p KaisKai tppMueeC eeue 2(1)ipKappMCe translational symmetry20(2)ipKaipKappMCee 202(1 cos)ppCpKaM finally leads to2) 1D lattice vibrations (phonons)1 atom per primitive cellsince202sin0ppdpaCpKadKM 21(2)(1 cos )CMKa 2211211(4)sin ()21(4) sin()2 CMKaCMKa nearest neighbor interaction onlydispersion K,dw dKOutline1.

7 The harmonic oscillator real space energy basis2. 1D lattice vibrations one atom per primitive cell two atoms per primitive cells3. electron phonon interactions localized electrons small polaron theory phonons in metals4. Superconductivity5. A numerical example: CO6. Literature2) 1D lattice vibrations (phonons)2 atoms per primitive cell2112(2)ssssduMCv vudt 2212(2)ssssdvMCuuvdt isKai tsuuee isKai tsvvee 2 EOMsansatzandandsubstituting21(1) 2iKaMuCv eCu 22(1)2iKaMvCueCv andleads to 21222(1)0(1) 2iKaiKaCMC eCeCM 212112 CMM 222122 CKaMM 2) 1D lattice vibrations (phonons)2 atoms per primitive cellLattice with 1 atom per primitive cell gives only 1 acoustic branchLattice with 2 atom per primitive cell gives 1 acoustic and 1 optical The harmonic oscillator real space energy basis2.

8 1D lattice vibrations one atom per primitive cell two atoms per primitive cells3. electron phonon interactions localized electrons small polaron theory phonons in metals4. Superconductivity5. A numerical example: CO6. Literature3) Electron-phonon interaction: HamiltonianThe basic interaction Hamiltonian ispeeiHH H H 12qqpqqHaa 221122ieiijijpHemr ()( )eiieiijiijHVr VrR Taylor series expansion for the displacementsthe electron phonon interaction readsand the Fourier transform of the potential1()()iq reieiqVrVqeN 1()()iq reieiqVr iqVqeN 3) Electron-phonon interaction: Hamiltonianwe need to calculateby using(0)()()jiq Riq reijqjiVrqV qeQeN (0)1212()2jqGqGiq RjqGqGjGGqGiiQeQaaNNMN andwe can write the Hamiltonian in the form ()12,()()()( )2qqir q GeiqqGqVreVqGqGa a MN 3) Electron-phonon interaction.

9 Hamiltonianby integrating the potential over the charge density of the solid 312,()()( ) ( )( ) ()2qqepeiqqGqHdr rVrqGVqGqGa a or in an abbreviated form12()()( )2eiqqGqMVqGqG 12,1()qqepqGqGHqGM a a withOutline1. The harmonic oscillator real space energy basis2. 1D lattice vibrations one atom per primitive cell two atoms per primitive cells3. electron phonon interactions localized electrons small polaron theory phonons in metals4. Superconductivity5. A numerical example: CO6. Literature3) Electron-phonon interaction: localized electronsIf the electrons are localized the Hamiltonian becomes 01212( )iiqqqqiq riG rpepqqGqiGeHH Haaa aqGM e here the electron density operator is the Fourier transform the localized charge density23()3()000()()()ir q Gir q GiqG drer rdreqG 23()00()()ir q GiqG drer rearranging terms 12112( )iqqqqiq rqqiqiHaaaaeFr with the periodic function0()( )iG rqqGGFrqGMe 3) Electron-phonon interaction: localized electronswe now transform the creation and annhiliation operatorsand rewrite the Hamiltonian 22112iqqqiq rqqqiqFHAAe which has the eigenstates and eigenvalues 120!

10 QqnqAn 22112iqiq rqqqqiqFEne andand12()1iqiq rqqiqFrAae *12()1iqiq rqqiqFrAae 3) Electron-phonon interaction: deformation potentialTraditionally in semiconductors one parametrizes electron phonon interactions(long wavelengths) deformation potential coupling to acoustic phonons piezoelectric coupling to acoustic phonons polar coupling to optical phononsthe deformation potential coupling takes the form 12()()2qqepqqHDqqaa 3) Electron-phonon interaction: piezoelectric interactionThe electric field is proportional the the stresskijk ijijEMS Stress is the symmetric derivative of the displacement field 1211()222iqqjiq riijijj iqjiqQQSqqaaexx () ()rQr The field is longitudinal and can hence be written as the gradient of a potentialThis potential is proportional to the displacement 12() () ()2qqiq rqqriMqe a a 12() ()()2qqepqqHiMqqaa leading to121()iq rkkqqkEr iqex 3) Electron-phonon interaction.