10. Appendix

624 Appendix B
F, the change in the distribution function gk is given by (5.19) as:
gk
f 0 k
Ek
qÙkvk · F
within the relaxation time approximation. The new distribution function fk is
given by (5.16): fk f 0 k gk. Thus
fk f 0 k gk f 0 k (Ek)
f 0 k
Ek
(qÙkvk · F) ≈ f 0 k (Ek qÙkvk · F)
The physical meaning of this result is that: under the influence of the field, the
functional form of the final distribution function fk is the same as the initial
distribution function f 0 k (Ek). However, the energy E ′ k of each electron is now
equal to its energy in the presence of the field, ie E ′ k Ek qÙkvk · F. Asa
result we can write
fk(E ′ k ) f 0 k (Ek) f 0 k (E′ k qÙkvk · F)
If we consider the distribution function of the electron as a function of its
velocity vk rather than its energy, then initially f 0 k is symmetric with respect
to vk 0 in the absence of F. In the presence of F the entire distribution
function would be displaced (without change in shape) along the vk axis by
an amount: qÙkF/m (where m is the electron mass). The entire distribution
appears to have acquired a drift velocity qÙkF/m. Hence f 0 k (Ek qÙkvk · F) is
known as a drifted distribution.
Solution to Problem 5.2(a)
(a) As an example of the application of the results in Problem 5.1 we will
consider the special case where f 0 k is the Boltzmann distribution function in
the absence of F: f 0 k A exp[Ek/kBT]. In addition, we will assume that
the electrons occupy a spherical band with the dispersion: Ek (1/2)m∗v2 k
where m∗ is the electron effective mass. The resultant distribution is known
as a Maxwell-Boltzmann distribution. Under the effect of the field F the perturbed
distribution function fk is given by:
fk ≈ f 0 k (Ek qÙkvk · F)
m∗v2 k
f 0 k
≈ f 0 k
2 qÙkvk · F
m ∗ (vk vd) 2
2
f 0 k
m ∗ (vk vd) 2
2
m∗ v 2 d
2
using the result of Prob. 5.1. This result can also be expressed as: fk
A exp[m ∗ (vk vd) 2 /2kBT] where vd qÙkF/m ∗ is the drift velocity. This
distribution function is known as a drifted Maxwell-Boltzmann distribu-