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

88 Introduction to electron-phonon interaction
with the transversal and longitudinal velocities of sound c t and c l , respectively.
For bulk media follows that the eigenmodes are given by plane waves (sq),
u(r,t) = ∑û s (q)e i(q·r−ω sq t) , (5.13)
s,q
where s = {l,t} denotes the different longitudinal and transversal polarisations,
and û s (q) is the corresponding polarisation vector. With proper normalisation one
can show that the bulk matrix elements (5.6) for the absorption of a phonon (sq)
via deformation and piezo-electric potentials are given by [139]
M def
lk→l ′ k ′ = δ k ′ ,k+q
M pz
lk→l ′ k ′ = δ k ′ ,k+q
[ ] 1
1 2
√
√ iΞ q, (5.14)
V 2ρ M c s
[ ] 1
1 2
√
V 2ρ M c s
eβ
√ q
. (5.15)
We mention two important points. (i) The phase difference of π/2 between the
matrix elements shows that the piezo-electric and deformation potentials are indeed
independent scattering mechanisms, hence they do not interfere. (ii) Comparison
of the amplitudes of the matrix elements yields,
|M pz | 2 ( ) eβ 2
|M def | 2 = . (5.16)
Ξq
Thus, in piezo-electric bulk materials, the scattering of electrons by long wavelength
(q → 0) acoustic phonons is typically dominated by the piezo-electric coupling
potential.