If an external field is applied, the distribution functions will change from
their equilibirium values, and their correct form must be determined by
solving of the **Boltzmann transport equation** (see, e.g. [1],pp. 17-29).

The term on the left hand side is the total time derivative of *f* and
all its variables. It was first given by Liouville and is a formulation
of the continuity equation. The term on the right hand side was added by
Boltzmann and is called collision term.

A general solution of this equation is a hard task, but as a first
approximation we may assume steady state conditions and spatial
uniformity throughout the bulk of the investigated material. Thus,
the first and the third term are zero. Note that that the remaining term
on the left hand side contains a derivative for the velocity coordinates.
If we describe the electrons in the solid in terms of their wavefunctions,
we can recognize the semiclassical relationship *v = ħk/m* and apply
the chain rule to convert this term into a derivative for *k*. We find:

In the following, we will go through a series of simplifications and approximations. Let's start on the right hand side:

Let us briefly assume that the system is in thermal equilibrium without external field. We
denote this by a function *f*_{0} which is given by the Fermi-Dirac distribution.
This allows us to derive the following relation:

This statement is called **principle of detailed balance**.

Elastic scattering means that we do not change the energy of the particles,
but only their direction. Then, *E*(*k*) is equal to *E*(*k'*) and the detailed
balance tells us that also the scattering functions *S*(*k,k'*) and *S*(*k',k*) must
be equal.

We obtain a massive simplification of the collision integral:

The quantity *τ*(*k*) is called transport relaxation time. We could also
write *τ _{k,k'}*, but an exchange of variables shows that this
is equal to

Next, we look at the left hand side of the Boltzmann equation. The remaining term contains a velocity derivative. We can relate this acceleration to the force that the external electric field exerts on the charge:

As further approximation, we apply the gradient operation only to the undisturbed
distribution function *f*_{0}. The result can be solved
for the nonequilibrium correction *f*_{1}:

In the parabolic band approximation, the gradient of the energy yields a term
that is proportional to the wavevector. Then, the vectorial product with the
field will yield the cosine of the angle between these two directions.
Nevertheless, this looks still like an integral equation because the
term *τ*(*k*) depends *f*_{1}. Let's spell it out:

Inserting the relation for *f*_{1} into the definition of *τ*(*k*)
above, we have to evaluate the ratio that appears in the square brackets. We can use the
assumption of elastic scattering once more; since the scattering process does not change
the energy but only the direction of the particles, the derivative of *f*_{0} with
respect to energy is the same for *E*(*k*) and *E*(*k'*); consequently the
derivative cancels out of the ratio. Moreover, we asserted further up that the relaxation times
are also the same for *k* and *k'*. They drop out of the ratio as well!

All things together, the unwieldy fraction reduces to the following:

Remember that all the simplifications so far work towards the calculation of the integral
that appears in the calculation of the transport relaxation time. The problem is now
reduced to resolving the angles between the the applied field and the wavevectors
of the initial and of the final state. We choose a coordinate system where the initial
velocity is along the *z* direction and the electric field is contained in
the *yz* plane.

We can read off the spherical coordinates from the illustration above and resolve the vectorial products (see e.g. [2], pp. 47-48).

Carrying out the integration over *k* space in spherical coordinates, the dependence on
the polar angle will be averaged out. We obtain the following:

Thus, the big term in the square brackets reduces to [1-cos *θ*],
where *θ* is the angle between the directions before and after
the scattering event. We can think of this cosine-term as a weighting
function which tells us that scattering events with large angles
contribute most to the scattering integral (not really a big surprise!).

The probability for scattering after a time *τ* is
virtually one, but with respect to unit time it is just 1/*τ*.
However, this is exactly the definition of *S*(*k',k*):

If we know the differential cross section of our scattering process, we can
determine the transport relaxation time. Here, we replaced the dependence on
*k'* and *k* by the scattered angle *θ*. We explicitely write
the *δ*-function to express that we are dealing with elastic scattering processes
that conserve the energy. As such, it comes out of Fermi's Golden Rule when the
differential cross secion is determined from the scattering potential. Sometimes
it is dropped in the literature, but it is then a bit surprising why the integral for the
transport relaxation time is extended only over the angular part of *d*^{3}*k*
without further explanation.

We are through! After solving the integral for the transport relaxation time, we can obtain the first-order correction to the distribution function in the presence of an electric field with the expression that was given above:

Carrying out the derivative for the Fermi-Dirac distribution is a bit messy. Below is
an approximation using the Maxwell-Boltzmann distribution which applies generally for
non-degenerate situations. We revert to the expression for particles with kinetic
energy and *A* is a suitable normalization constant. Note that *f*_{1}
changes sign with the direction of the electric field.

Note that the integral weighted with the equilibrium function *f _{0}* will give zero because
there is no current without field; the only contribution will come from the

Sometimes the explicit route via the integral of the the current density
is not spelled out, but the authors calculate the mobility relaxation
time by averaging the transport relaxation time with a weighting
factor of *E*^{3/2} (e.g. eq. (5.4) in [3] or eq. (35) in [4]).
The form of this particular weighting factor comes from the fact that we
pick up a wavevector from the gradient term and another one from
the current density integral, resulting in *k*^{2} which
is proportional to *E*. An additional factor of *E*^{1/2}
comes from the usual conversion of *d ^{3}k* into

[1] P. L. Rossiter, The electrical resisitvity of metals and alloys, Cambridge University Press, 1991

[2] T. M. Tritt, Thermal Conductivity, Springer, 2005

[3] H. Brooks, Adv. Electr. Elecron Phys. 7, 85 (1955)

[4] R. B. Dingle, Phil. Mag. 46, p831 (1955)

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last changed: 6/7/2015