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Electron-Phonon Coupling: a Tutorial

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Electron-Phonon Coupling: a Tutorial Electron-phonon coupling: a tutorial W. Hübner, C. D. Dong, and G. Lefkidis University of Kaiserslautern and Research Center OPTIMAS, Box 3049, 67653 Kaiserslautern, Germany Targoviste, 29 August 2011 Outline 1. The harmonic oscillator real space energy basis 2. 1D lattice vibrations one atom per primitive cell two atoms per primitive cells 3. Electron‐phonon interactions localized electrons small‐polaron theory phonons in metals 4. Superconductivity 5. A numerical example: CO 6. Literature Outline 1. The harmonic oscillator real space energy basis 2. 1D lattice vibrations one atom per primitive cell two atoms per primitive cells 3. Electron‐phonon interactions localized electrons small‐polaron theory phonons in metals 4. Superconductivity 5. A numerical example: CO 6. Literature 1) The harmonic oscillator quantization of the oscillator in real space Eigenvalues of Projection of eigenvalue equation to X basis (Substitution by differential operators) leads to 22d 1 () mx 22 E 22mdx2 1) The harmonic oscillator quantization of the oscillator in real space 1) Dimensionless variables 22224 dmEbmb2 2 xb y leads to 22 2y 0 dy 12 2 mEb E b and 2 m 10 5 '' (2y 2 ) 0 -10 -5 5 10 -5 2 '' y 2 0 with solution Aymy e 2 since ''22my2 21mmm (1) my 2222 Aye1 24y Aye y yy 1) The harmonic oscillator quantization of the oscillator in real space '' 20 with solution Acos[ 2yB ] sin[ 2 y ] consistency requires 1.0 2 y0A cy O() y 0.5 thus (3) -10 -5 5 10 -0.5 2 ansatz: ()yuye () y 2 -1.0 leads to uyu''2(21)0 ' 1) The harmonic oscillator quantization of the oscillator in real space 4) Power‐series expansion: n uy() Cyn n0 nn2 n inserted into differential equation Cnnn[( 1) y 2 ny (2 1)] y 0 n0 n2 Cnnn (1) y n2 with index shift mn 2 mn Cmmn22( 2)( my 1) Cn ( 2)( ny 1) mn00 n we get yC[(2)(1)(212)]0nn2 n n C n n0 (2n 1 2 ) feeding back in the original leads to recursion: CC nn2 (2)(1)nn 1) The harmonic oscillator quantization of the oscillator in real space so we have (12)yyyy2435 (12) (41 2) (21 2) (21 2)(61 2) uy() C0 1 C1 y (0 2)(0 1) (0 2)(0 1) (2 2)(2 1) (1 2)(1 1) (1 2)(1 1) (3 2)(5 1) Problems => way out: termination of series required 24 n yy2 22CCyCy02 4 ... Cyn 2 ()yuye () 35 n e CCyCy13 5 ... Cyn 1) The harmonic oscillator quantization of the oscillator in real space consequence energy quantization of the harmonic oscillator by backwards substitution 1 Enn () 2 Examples: Hy0 () 1 Hy1() 2 y 2 Hy2 () 212 y 2 3 Hy3 () 12 y y 3 244 H 4 ()yyy 121 4 3 Outline 1. The harmonic oscillator real space energy basis 2. 1D lattice vibrations one atom per primitive cell two atoms per primitive cells 3. Electron‐phonon interactions localized electrons small‐polaron theory phonons in metals 4. Superconductivity 5. A numerical example: CO 6. Literature 1) The harmonic oscillator quantization of the oscillator in energy basis Oscillator in energy basis 2 P 1 22 mX E EE 22m Direct way: Fourier transform from real to momentum space No savings compared to direct solution of Schrödinger equation in real space 1) The harmonic oscillator quantization of the oscillator in energy basis commutator X , PiIi definition and adjoint 12 12 12 12 m 1 m 1 aXiP aXiP 22 m 22 m further aa,1 New operator (dimensionless) H Haaˆ (12) 1) The harmonic oscillator quantization of the oscillator in energy basis Commutator of creation and annhiliation operators with Hamiltonian ˆ aH,,12, aaa aaa a ˆ aH, a Raising and lowering properties ˆˆˆˆ Ha aH[, a H ] aH a ( 1) a aC 1 1 But eigenvalues non‐negative requirement a 0 0 1 no further lowering allowed aa 0 0 0 2 1) The harmonic oscillator quantization of the oscillator in energy basis 1 0 0 (12),0,1,2,...nn Enn (12), n 0,1,2,... 2 A possible second family must have the same ground state, thus it is not allowed an C n 1 * n and adjoint equation na n1 Cn form scalar product of both equation * naan n11 n CCnn ˆ * nH12 n CCnn 2 2 12 i nnn Cn Cnn Cnen 1) The harmonic oscillator quantization of the oscillator in energy basis an n12 n1 an (1)1 n12 n aanann121 nnnnn 12 12 with number operator Naa HNˆ 12 further 12 12 nan''1 n nn nnn', 1 12 12 na'(1)'1(1) n n nn n nn', 1 1) The harmonic oscillator quantization of the oscillator in energy basis position and momentum operators 12 12 X ()aa P ()aa 2m 2m nnn012... 12 000... 01 0 0... n 0 12 10012 002 0 12 n 1 12 a 02 0 a 00 03 n 2 12 . 003 . . . 1) The harmonic oscillator quantization of the oscillator in energy basis What do we learn? 0112 0 0... 01 12 0 0... 12 12 12 12 1020 1020 12 12 12 12 12 12 02 03 m 02 0 3 X Pi 2m 00312 0 2 00312 0 . . . . Analogously for derived operators 12 0 0 0 ... 03200 H 0052 . . Outline 1. The harmonic oscillator real space energy basis 2. 1D lattice vibrations one atom per primitive cell two atoms per primitive cells 3. Electron‐phonon interactions localized electrons small‐polaron theory phonons in metals 4. Superconductivity 5. A numerical example: CO 6. Literature 2) 1D lattice vibrations (phonons) 1 atom per primitive cell force on one atom FCuus psps() p du2 M s Cu() u equation of motion of atom 2 psps dt p solution in the form of traveling wave is() pKa i t uueesp EOM reduces to 2()isKa i t i s p Ka isKa i t 2MCe (1)ipKa Mue e Cp () e e ue p p p translational symmetry 2 2 ipKa ipKa 2 finally leads to MCee (2) CpKap (1 cos ) p M p0 p0 2) 1D lattice vibrations (phonons) 1 atom per primitive cell 2 w d 2 1.0 since paCp sin pKa 0 0.8 dK M p0 0.6 0.4 nearest‐neighbor interaction only 0.2 221 K (4CM )sin ( Ka ) 0.5 1.0 1.5 2.0 2.5 3.0 1 2 2 (2CM )(1 cos Ka ) 1 1 (4CM )12 sin( Ka ) dispersion relation 1 2 w K, dw dK 0.4 0.2 K 1 2 3 4 5 6 -0.2 -0.4 Outline 1. The harmonic oscillator real space energy basis 2. 1D lattice vibrations one atom per primitive cell two atoms per primitive cells 3. Electron‐phonon interactions localized electrons small‐polaron theory phonons in metals 4. Superconductivity 5. A numerical example: CO 6. Literature 2) 1D lattice vibrations (phonons) 2 atoms per primitive cell du2 dv2 2 EOMs M s Cv(2) v u and M s Cu(2) u v 11dt 2 s ss 21dt 2 s ss isKa i t isKa i t ansatz uuees and vvees 2 iKa 2 iKa substituting M1uCve(1 ) 2 Cu and M 2vCue(1)2 Cv 11 2 2C 2 iKa 2(1)CM C e MM12 leads to 1 0 Ce(1iKa ) 2 CM 2 C 2 2 222 Ka MM12 2) 1D lattice vibrations (phonons) 2 atoms per primitive cell Lattice with 1 atom per primitive cell gives only 1 acoustic branch Lattice with 2 atom per primitive cell gives 1 acoustic and 1 optical branch http://dept.kent.edu/projects/ksuviz/leeviz/phonon/phonon.html Outline 1. The harmonic oscillator real space energy basis 2. 1D lattice vibrations one atom per primitive cell two atoms per primitive cells 3. Electron‐phonon interactions localized electrons small‐polaron theory phonons in metals 4. Superconductivity 5. A numerical example: CO 6. Literature 3) Electron-phonon interaction: Hamiltonian The basic interaction Hamiltonian is HH peei H H 2 pi 112 Haapq 12 HVrVrR() ( ) qq Hee ei i ei i j q iij22mrij iij Taylor series expansion for the displacements the electron‐phonon interaction reads and the Fourier transform of the potential 1 iq r 1 iq r Vrei() Vqe ei () Vrei() i qVqe ei () N q N q 3) Electron-phonon interaction: Hamiltonian we need to calculate (0) i iq r iq Rj Vr() qVqeei () Qe j N qj by using (0) ii iq Rj 12 Qe Q () a a jqG12 qG qG qG NNjGG2 MNqG and MN we can write the Hamiltonian in the form ir() q G 12 Vr() e VqGqG ()()() a a ei q qq qG, 2q 3) Electron-phonon interaction: Hamiltonian by integrating the potential over the charge density of the solid 3 12 H drrVr()() ( qGVqGqG ) ( )( ) ( ) a a ep ei q qq qG, 2q or in an abbreviated form 1 H ()qGM a a ep 12 qG qq qG, with 12 MqG VqGqGei()()() q 2q Outline 1. The harmonic oscillator real space energy basis 2. 1D lattice vibrations one atom per primitive cell two atoms per primitive cells 3. Electron‐phonon interactions localized electrons small‐polaron theory phonons in metals 4. Superconductivity 5. A numerical example: CO 6. Literature 3) Electron-phonon interaction: localized electrons If the electrons are localized the Hamiltonian becomes iq r e i iG ri HH H aa 12 a a ( qGMe ) pepq qq12 q q 0 qG qiG here the electron density operator is the Fourier transform the localized charge density 2 ()qG dre3()ir q G () rr dre 3() ir q G () qG 00 0 i 2 ()qG dre3()ir q G () r 00 i rearranging terms 1 iq ri HaaaaeFr12 ( ) qqiqq12 q q qi with the periodic function iG r F ()rqGMe ( ) q 0 qG G 3) Electron-phonon interaction: localized electrons we now transform the creation and annhiliation operators * Fr () Fr () 1 q iq ri 1 q iq ri Aae Aa e qq12 and qq12 q i q i and rewrite the Hamiltonian 2 2 Fq 1 iq ri HAAe 12 q qq qqi q which has the eigenstates and eigenvalues n q 2 2 A Fq q 1 iq ri 0 and En12 e 12 qq qqi q nq ! 3) Electron-phonon interaction: deformation potential Traditionally in semiconductors one parametrizes electron‐phonon interactions (long wavelengths) •deformation‐potential coupling to acoustic phonons •piezoelectric coupling to acoustic phonons •polar coupling to optical phonons the deformation‐potential coupling takes the form HD()() 12 qqaa ep qq q 2q 3) Electron-phonon interaction: piezoelectric interaction The electric field is proportional the the stress Ek MS ijk ij ij Stress is the symmetric derivative
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