Spin-orbit coupling and electron-phonon scattering - Fachbereich ...

Chapter 5
Introduction to electron-phonon
interaction
In a solid, vibrational excitations of the crystal lattice are describes as the creation
of phonons. These quasi-particles have, like electrons, a definite energy dispersion
E s (q) = ω s (q), where q is the wave vector of the phonon, ω is its frequency, and
s denotes the different phonon branches (acoustic or optical), and polarisations
(longitudinal or transversal). Electron scattering by crystal lattice vibrations is
then described by emission and absorption of phonons.
In the following, we are interested in the low-energy and low-temperature
properties of electron-phonon scattering. Therefore, we restrict ourselves to singlephonon
scattering events with long wavelength acoustic phonons. Excitation energies
of optical phonons are typically on a much higher energy scale, e.g. in GaAs
the lowest optical phonon excitation has an energy of more than 30meV [142].
In this introduction we follow the presentation of Ref. [139]. A discussion of the
scattering by optical phonons and the effect of multi-phonon processes, which we
omit in this context, can also be found there.
In general, any deformation of the crystal lattice may induce a perturbation
δV (r,t) of the potential in which the electron moves. Independent of the actual
microscopic mechanisms, this perturbation can be written in linear order in the
deformation as
δV (r,t) = ∑V a (r) · u a (t), (5.1)
a
where u a (t) is the displacement of the atom at lattice site a in the crystal. The
vector components V i a(r) describe the change in lattice potential at site r for a
unit displacement of atom a along the i-direction. If we denote the eigenmodes of
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