Solid State Theory
Exercise 11
FS 11
Prof. M. Sigrist
Transport in metals
Exercise 11.1
Relaxation time approximation
In this exercise we will show that the socalled singlerelaxationtime approximation,
∂f
(
k
)
∂t
coll
=

Z
d
d
k
0
(2
π
)
d
W
(
k
,
k
0
)[
f
(
k
)

f
(
k
0
)]
→

f
(
k
)

f
0
(
k
)
τ
,
(1)
is a true solution to the Boltzmann equation under certain conditions.
We consider a homogeneous twodimensional metal with an isotropic Fermi surface (
ε
k
=
~
2
k
2
/
2
m
) at zero temperature. The impurity scattering responsible for a finite resistivity
is described by a delta potential in real space,
V
imp
(
r
) =
V
0
δ
(
r
)
.
(2)
The system is subject to a homogeneous and timeindependent electric field along the
x
axis.
a) Show that the transition rates
W
(
k
,
k
0
) for the impurity potential (2) are constant
in
k
space.
b) Write down the static Boltzmann transport equation for this setup in the form
“driftterm” = “collisionintegral”
(3)
and take advantage of the zerotemperature limit and the symmetries of the system
to eliminate all but angular variables.
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 Spring '11
 Sigrist
 Electron, Fundamental physics concepts, Dielectric, Boltzmann transport equation, penetration depth, Boltzmann equation

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