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SOLUTION OF THE BOLTZMANN EQUATION BY THE RELAXATION TIME APPROXIMATION

THE RELAXATION TIME APPROXIMATION_

As noted previously, equation 4.10 is an integro-differential equation and therefore some drastic approximations have to be made to obtain an analytical solution. However, the labour is worthwhile not only because of the greater numerical accuracy compared to empirical models but mainly because of the greater physical insight obtained by the analytical solutions.

We are mostly interested in solutions where the derivative df Idt is zero. Note that these are steady-state solutions, not equilibrium solutions, energy is being driven into the system by the fields. The most difficult term is obviously the integral term of the RHS of 4.10. The so-called relaxation time approximation simplifies this last term by putting

i.e. the rate of change due to collisions is proportional to the deviation from the equilibrium distribution f0—the Fermi-Dirac distribution

What is the meaning of equation 4.11 and when is it valid?

Let us write for a homogeneous system

where /' is the deviation from the equilibrium value /0 and assume that the system is perturbed by some field which is turned off at some time later. Then the first two terms of the RHS of 4.10 are zero and we have

i.e. the system returns back to its equilibrium distribution exponentially with time, with x being the approximate time constant. So the relaxation approximation looks like a reasonable assumption. But when is it valid?

For the relaxation approximation to be valid, the deviation from equilibrium must be small, which in turn means that the fields acting on the system must be small. Since the latter statement is not always quantifiable, it is a safer condition that the relaxation time x, which in principle can be a function of k, must be independent of the strength of the perturbation causing the departure from equilibrium. A rather long argument shows that if only elastic processes are involved then the use of the relaxation time x(k) is a valid approximation. The use of the relaxation time approximation for inelastic processes is not valid. We will not delve into this as we are going to abandon the Boltzmann equation in the next chapter. But given the limitations described above, we are now ready to obtain solutions of the

Boltzmann equation that shed more physical insight in what we have done so far and analyze situations where the previous semiempirical approach was insufficient.

Let us first rewrite for the collision term

and for the deviation from equilibrium f

where Ф can be a function of к and r. Using the definition 4.14, equation 4.10 becomes in the steady state df tdt = 0, with the velocity restored back in the equation.

Now a series of approximations are necessary to solve 4.16. We first assume that the perturbing fields do not induce any space dependent variation in f. There can be a space dependent variation in the total / but only if there is originally one due to a thermal gradient. In this case, both the temperature T(r) and the Fermi level EF(r) will be functions of r. Then /o(r) will be a function of r. The use of the Fermi-Dirac distribution, with T(r) and Ep(r) being space dependent, is only an approximation that is valid if the variations in T(r) and Ef (r) are so weak that the region around any r may be considered as being in pseudo equilibrium with its neighbouring regions. In mathematical terms

where К = Boltzmann’s constant.

The reason for writing equation 4.15 in this product form should now be clear. We expect dfo/dE to appear as a common term in all our manipulations. From equation 4.17 after differentiating the quotient we get

or

For the term V*/ we have,

Substituting equations 4.18a, and 4.19 into 4.16 we get

Note that in 4.20 we have dropped the term multiplying the electric field £ with V*/' since it is a term multiplying the perturbing field with the perturbation and hence it must be of second order to the first term involving the electric field. Note also that the second term in 4.20 is zero because it involves the inner product of two perpendicular vectors. Rearranging 4.20 we get

We now need to express the second term on the LHS of 4.21 in a product form with

oE

so that the latter can be canceled from both sides of 4.21. The following manipulations are elementary

where in the last step of 4.22 we have used the definition f = -ФЭ/0E. Combining now 4.21 and 4.22 we get

where in 4.23 we have reinstituted the к and r dependence where appropriate. Equation 4.23 is the linearized Boltzmann equation that will form the basis of our subsequent investigations.

 
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