15 Electron-Phonon Coupling - cond-mat.de

1 15 Electron-Phonon CouplingRolf HeidInstitute for Solid State PhysicsKarlsruhe Institute of TechnologyContents1 Introduction22 Electron-Phonon vertex .. ohlich Hamiltonian ..33 Normal-state functions and perturbation .. self-energy .. s theorem .. self-energy and linewidth ..114 Phonon-mediated electron-electron interaction .. formalism .. theory .. gap equations ..185 Density functional theory approach226 Summary25A Phonon quantization26E. Pavarini, E. Koch, R. Scalettar, and R. Martin (eds.)The Physics of Correlated Insulators, Metals, and SuperconductorsModeling and Simulation Vol. 7 Forschungszentrum J ulich, 2017, ISBN 978-3-95806-224-5 Heid1 IntroductionThe Electron-Phonon interaction is, besides the Coulomb interaction, one of the fundamentalinteractions of quasiparticles in solids.

2 It plays an important role for a variety of physicalphenomena. In particular in metals, low-energy electronic excitations are strongly modifiedby the Coupling to lattice vibrations, which influences, , their transport and thermodynamicproperties. Electron-Phonon Coupling (EPC) also provides in a fundamental way an attractiveelectron-electron interaction, which is always present and, in many metals, is the origin of theelectron pairing underlying the macroscopic quantum phenomenon of lecture addresses the consequences of Electron-Phonon Coupling in both the normal andthe superconducting state of metals. In Section 2, the basic Hamiltonian describing the coupledelectron-phonon system is introduced.

3 In Section 3, a closer look onto normal state effects ina metal is taken, focusing on the renormalization of quasiparticles, which allows to experimen-tally quantify the strength of the interaction. Section 4 is devoted to phonon-mediated super-conductivity. First a derivation of the effective attractive interaction among electrons mediatedby phonon exchange is given. Then we analyze the role of Electron-Phonon Coupling for su-perconductivity in the context of the strong- Coupling Migdal-Eliashberg theory in some Section 5, we discuss the density-functional based technique to calculate electron-phononcoupling quantities and present two examples to illustrate its predictive power.

4 Throughout thisChapter, only nonmagnetic states are considered and atomic units~= 2me=e2/2 = 1as wellaskB= 1are Electron-Phonon Electron-Phonon vertexThe lowest-order process involving the Electron-Phonon interaction is the scattering of a singleelectron by a simultaneous creation or annihilation of a single phonon, as diagrammaticallyshown in Fig. 1. The probability for the scattering process is called the Electron-Phonon vertexg. We will briefly sketch its derivation starting from rather general grounds. For more detailsone can refer to the book of Grimvall [1].Due to the large ratio of the ionic and electronic mass, the dynamics of the ions and the electronscan be systematically expanded in terms of the small parameter = (m/M)1/4, which resultsin a partial decoupling [2, 3].

5 To lowest order in , called the adiabatic or Born-Oppenheimerapproximation, the total wavefunction of the coupled electron-ion system can be written as aproduct (r,R) = (R) (r;R), whererandRdenote the sets of electron and ion coordinates,respectively. The electronic wavefunction obeys the equation[Te+Vee+He i(R)] n(r;R) =En(R) n(r;R),(1)whereTeandVeedenote the kinetic energy and Coulomb interaction of the electron system,respectively. Eq. (1) depends parametrically on the ionic positionsRvia the electron-ion inter-actionHe 1:Diagrammatic representation of the basic Electron-Phonon scattering process. Blacklines represent electrons, the blue zigzag line a phonon, and the red circle the screened Electron-Phonon vertex appears in first order beyond the adiabatic approximation.

6 One canshow that it induces off-diagonal matrix elements among the electronic eigenstates nand hasthe form n| RV|n .(2)The operator RVstands for the linear change of the potential felt by the electrons under adisplacement of an atom from its rest position:R=R0+u. If the potentialVis the bareelectron-ion potentialV0, then RV=u V0|R0. Eq. (2) represents the bare vertex. However,in solids, and in particular in metals, the bare electron-ion potential is screened by the otherelectrons. Screening also alters the vertex significantly. Within linear response theory thisoperator takes the form RV=u 1 V0|R0,(3)where 1is the inverse dielectric matrix [4], which is a measure of the screening.

7 Note that inEq. (3), the screening operator does not commute with the gradient operation, and thus can notbe written in terms of the gradient of a screened Fr ohlich HamiltonianWe now aim to develop a systematic perturbative treatment of the mutual influence of the elec-tronic and phononic subsystems in a solid. Thereby the question arises, what are the propernoninteracting quasiparticles to start with. The correct answer requires to know the solution tosome extent. As we will see, electronic states are significantly influenced by lattice vibrationsmostly in close vicinity of the Fermi energy. It is therefore appropriate to start with electronsmoving in the static potential of a rigid ion lattice , without any renormalization by the latticevibrations.

8 On contrast, the bare vibrations of the ion lattice would be a bad starting point,because they are strongly altered by the screening of the electrons. This screening must be builtinto the description of the harmonic lattice vibrations which defines the noninteracting the discussion of Electron-Phonon Coupling effects in periodic solids, a good starting pointis theFr ohlich Hamiltonian, which reads in second quantizationH=He+Hph+He ph.(4) HeidHere the electron system is described by noninteracting quasi-particles with dispersion quasiparticles are considered to be the stationary solutions of band electrons in a per-fect periodic lattice , and include already the renormalization from Coulomb interactionHe= k k c k ck.

9 (5)Hereck (c k ) are the annihilation (creation) operators for an electronic state with momentumk, band index , spin , and band energy k .The lattice Hamiltonian is expressed in terms of quantized harmonic vibrations, and representsnoninteracting phononsHph= qj qj(b qjbqj+12),(6)wherebqj(b qj) are the annihilation (creation) operators for a phonon with momentumq, branchindexj, and energy qj. Phonons are the quanta of the normal mode vibrations (for more detailssee Appendix A). The operator of atom displacements is expressed in terms of the phononoperators byuls =eiqR0ls1 Nq qjAqjs (bqj+b qj)withAqjs = s (qj) 2Ms qj.(7)Atoms are characterized by two indices denoting the unit cell (l) and the atoms inside a unitcell (s), respectively, withMsthe corresponding atom mass.

10 Denotes Cartesian indices, and s (qj)is the eigenvector of the normal modeqj. The number of points in the summation third term describes the lowest-order Coupling between electrons and phonons derived fromEq. (3). Using the relationship Eq. (7) it has the formHe ph= k qjgqjk+q ,k c k+q ck (bqj+b qj).(8)gqjk+q ,k is the Electron-Phonon matrix element and describes the probability amplitude forscattering an electron with momentumkfrom band to a state with momentumk+qinband under the simultaneous absorption (emission) of a phonon with momentumq( q) andbranch indexjgqjk+q ,k = s Aqjs k+q | qs V|k .(9)Here again the screened first-order variation enters the matrix elements.