Electron—Phonon Interaction

• Gerald D. Mahan
Chapter
Part of the Physics of Solids and Liquids book series (PSLI)

Abstract

The Fröhlich Hamiltonian describes the interaction between a single electron in a solid and LO (longitudinal optical) phonons:
$$H = \sum\limits_{p\sigma } {{\varepsilon _p}} C_{p\sigma }^\dag {C_{p\sigma }} + {\omega _0}\sum\limits_q {a_q^\dag } {a_q} + \sum\limits_{qp\sigma } {\frac{{{M_0}}}{{\sqrt V }}} \;\frac{1}{{\left| q \right|}}C_{p + q,\sigma }^\dag {C_{p\sigma }}({a_q} + a_{ - q}^\dag )$$
$${M_0} = \frac{{4\pi \alpha h{{(h{\omega _0})}^{3/2}}}}{{\sqrt {2m} }},\;{\varepsilon _p} = \frac{{{p^2}}}{{2m}}$$
(7.1)
$$\alpha = \frac{{{e^2}}}{h}{(\frac{m}{{2h{\omega _0}}})^{1/2}}(\frac{1}{{{\varepsilon _\infty }}} - \frac{1}{{{\varepsilon _0}}})$$
(7.2)
This Hamiltonian was derived in Chapter 1, with the form of the interaction given in Sec. 1.3.5. The LO phonons are usually represented by an Einstein model, i.e., the phonon frequency ω0 = ωLO is taken to be a constant. Since there is a single electron, the Hamiltonian may also be written as
$$H = \frac{{{p^2}}}{{2m}} + {\omega _0}\sum\limits_q {a_q^\dag } {a_q} + \sum\limits_q {\frac{{{M_0}}}{{\sqrt v }}} \;\frac{{{e^{iq \cdot r}}}}{{\left| q \right|}}({a_q} + a_{ - q}^\dag )$$
(7.3)

where r and p are the conjugate coordinates of the electron. The unperturbed electron is taken to have free-particle motion with an effective mass m.

Keywords

Fermi Surface Dielectric Function Longitudinal Optical Phonon Diagonal Transition Strong Coupling Theory
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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