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

86 Introduction to electron-phonon interaction
vibrations of atom a of the crystal with (u a ) sq , we can write the displacement as
[
u a (t) = ∑ asq (u a ) sq e −iωsqt + c.c. ] . (5.2)
s,q
After quantisation of the acoustical problem which determines (u a ) sq , the coefficients
in expansion (5.2) correspond to creation and annihilation operators, a sq
and a † sq, of a phonon in mode (sq). Combining Eqs. (5.2) and (5.1) yields
δV (r,t) = ∑
s,q
[
a sq∑
a
]
V a (r) · (u a ) sq e −iωsqt + h.c.
(5.3)
[
]
=: ∑ V sq (r)a sq e −iωsqt + h.c. . (5.4)
s,q
We can regard V sq (r) as the perturbation which an electron experiences when a
phonon in mode (sq) is present. With this understanding, phonon induced transition
probabilities between different electron states (lk) can be calculated by
Fermi’s golden rule
W ±sq
lk→l ′ k ′ = 2π
with matrix elements
M −sq
lk→l ′ k ′ =
∣
∣
( ∣M ±sq ∣∣
2
lk→l ′ k ′ N sq + 1 2 ± 1 )
δ ( ε lk − ε l
2
′ k ′ ∓ ω sq)
, (5.5)
Z
V
d 3 rψ ∗ l ′ k ′(r)V sq(r)ψ lk (r) =
(
M +sq
l ′ k ′ →lk) ∗.
(5.6)
The Bose factors N sq = 1/[exp(ω sq /K B T ) − 1] mark the different probabilities
for phonon absorption (W −sq
lk→l ′ k ′ ∝ N sq ) and emission (W +sq
lk→l ′ k ′ ∝ N sq +1) [induced
and spontaneous emission]. From Eq. (5.5) follows that spontaneous emission of
phonons leads to electron scattering even at T = 0, provided that the electrons are
not in equilibrium.
To evaluate Eqs. (5.5) and (5.6) we need to know the electron wavefunctions
ψ lk (r) and the potential V sq (r). The latter depends on the eigenmodes of vibration
and their induced interaction potential, see Eqs. (5.3) and (5.4). So far we
explicitly paid attention to the lattice structure of the solid. When we average δV
over an area of the crystal which is larger than the lattice constant a 0 but smaller
than the phonon wavelength λ∼1/q, we can decompose δV into
δV = δ ¯V + δṼ . (5.7)
Here, δ ¯V is the averaged potential and δṼ describes the fluctuations which average
to zero. The physical effect of both terms on the electron is different: δṼ is a