Advanced Solid State Physics - Student Project
Manuel Zingl - SS 2012
Boltzmann equation
To calculate currents, a function f is defined which gives the probability that a state ~k
is occupied at position ~r and time t.
Give an expression for the electrical current density in terms of f and the
density of states D(~k).
The electrical current density is defined as
~jelec = −e
Z
e
~v (~k)D(~k)f (~k)d~k = − 3
4π
Z
~v (~k)f (~k)d~k
with
D(~k) =
2
.
(2π)3
(1)
D(~k) is the density of states in ~k space per unit volume where the factor of two takes
the spin degeneracy into account. This general expression for the current density is
basically the sum over all velocities times the occupation probability for each ~k state.
The velocity ~v (~k) can be obtained from the dispersion relation E(~k) and in particular
for the free electron model ~v (~k) is given by
h̄2~k 2
E(~k) =
2m
~v (~k) =
1
h̄~k
∇k E(~k) =
.
h̄
m
(2)
Write down the Boltzmann equation that must be solved to find f .
In a steady state there is no change in f (~k, ~r, t) and thus the total derivative of the
probability distribution function f (~k, ~r, t)
df (~k, ~r, t)
∂f
∂f dkx
∂f dky
∂f dkz
∂f dx ∂f dy ∂f dz
=
+
+
+
+
+
+
dt
∂t
∂kx dt
∂ky dt
∂kz dt
∂x dt
∂y dt
∂z dt
∂f
d~k
d~r
!
=
+
· ∇k f +
· ∇f = 0
∂t
dt
dt
(3)
has to be zero. Additionally using
~v =
d~r
dt
and
h̄
d~k
= F~ext
dt
(4)
and introducing a collision term (for its meaning see below) leads to the Boltzmann
equation
∂f
F~ext · ∇k f
∂f =−
− ~v · ∇f +
.
∂t
h̄
∂t coll
(5)
The first term on the right hand side describes the effect on f due to external forces and
fields (e.g. electromagnetic fields). The process of diffusion is described by the second
term and the last term is the heuristically introduced collision term.
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Advanced Solid State Physics - Student Project
Manuel Zingl - SS 2012
What is the meaning of the collision term in the Boltzmann equation?
This term is introduced to take the transition between different ~k states into account.
The simplest model is the relaxation time approximation
f0 (~k) − f (~k)
∂f =
∂t coll
τ (~k)
(6)
where f0 is the equilibrium distribution function (Fermi-Dirac distribution) and τ denotes the relaxation time. The physical interpretation of this relaxation time is the time
associated with the rate of return to the equilibrium distribution when all external fields
or thermal gradients are turned off. A more sophisticated model for the collision term
might invoke Fermi’s golden rule to describe the transition rates between different states.
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