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.
We assume small fields wich change the distribution function f only little from its equilibirium value f0.
We consider a relatively low carrier concentration where the probability of free states (1-f) can be approximated by one. In the stationary state the collision term must be zero which yields the following:
This statement is called principle of detailed balance, and we will stick to it in the following.
The quantity τ(k) is often called transport relaxation time; take care, this is NOT the same as the relaxation time we used in the definition of the mobility. See [1] for more on the different definitions. We need a few more steps to go. We try to find the non-equilibrium distribution function by introducin the external electric field into the left hand side of the Boltzmann equation by replacing the velocity derivative with the accelerating force:
Then, we approximate the argument of that gradient by using only the undisturbed distribution function f0 which finally enables us to solve for the nonequilibrium correction f1:
Before we can continue, we have to go through a loop of iteration because the τ(k) in the equation for f1 actually requirs the knowledge of f1.
Now we can take this newly found relation for f1 and plug it into the ratio that appears in the square brackets of the definition of τ(k). At this point the assumption of elastic scattering is useful. If the scattering process does not change the energy but only the direction of the particles, the derivative of f0 with respect to energy is the same for v and v'; consequently it cancels out of the ratio. At first sight this appears like a trade for just for another difficulty because the gradient is still there. However, it turns out that we can do something elegant now because this gradient operator works on velocity coordinates (thus its index v). In semiconductors it is very often a good approximation to use a parabolic relationship between energy and velocity. Then, this gradient yields a velocity vector multiplied with the mass. 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. However, if the terms in the integral do no longer depend on directions, but only on the energy, the same will be true for the outcome:
Thus, somewhere here, the relaxation times will also drop out of the ratio. Next, we deal with the vectorial products.
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 carry out the vectorial products.
Carrying out the integration over k space in spherical coordinates, the dependence on the polar angle will be averaged out. This rather central conclusion is only occasionally elaborated further in the literature (see e.g. [2], pp. 47-48). We obtain the following:
Finally, the square brackets reduce to [1-cos θ], where θ is the anlge between the velocity 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 scattering angles contribute most to the scattering integral (indeed, a big surprise!). The other remaining term in the collision integral is the scattering probability S(k,k').
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'):
Assuming that we know the differential cross section of the scattering process that our charge carriers suffer, we can determine an approximate non-equilibrium distribution function f0+f1 in the presence of an electric field. More on cross section in the next sections.
Note that the integral weighted with the equilibrium function f0 will give zero because there is no current without field; the only contribution will come from the f1 term. We have now finally a term for the mobility and only now we could also calculate the mobility relaxation time if we should feel inclined to do so.
Sometimes the different steps that lead to the calculation of the current density are treated rather shortly. Instead of explicitely going through the equation for the current density as we do here, some authors calculate the mobility relaxation by averaging with a weighting factor of E3/2 (e.g. eq. (5.4) in [3]). The form of this particular weighting factor comes from the fact that we pick up a velocity v from the gradient term and another from the current density integral, resulting in v2 which is proportional to E. An additional factor of E1/2 comes from the usual conversion of d3k into dE.