520 APPENDIX B. BOLTZMANN TRANSPORT THEORY
Substituting for the partial time derivatives due to diffusion and external fields we get−vk·∇rfk−e
(
E+
vk×B
c)
·∇kfk=−∂fk
∂t)
scattering(B.10)
Substitutingfk=fk^0 +gk
−vk·∇rfk^0 −e(E+vk×B)∇kfk^0
=−∂f∂tk)
scattering+vk·∇rgk+e(E+vk×B)·∇kgk(B.11)
We note that the magnetic force term on the left-hand side of equation B.11 is proportional to
vk·e
(vk×B)and is thus zero. We remind ourselves that (the reader should be careful not to confuseEk,the
particle energy andE, the electric field)
vk=1
∂Ek
∂kand (in semiconductor physics, we often denoteμbyEF)
fk^0 =1
exp[
Ek−μ
kBT]
+1
Thus
∇rf^0 =−
[
exp(
Ek−μ
kBT)]
[
exp(
Ek−μ
kBT)
+1
] 2 ∇r(
Ek−μ(r)
kBT(r))
= kBT·∂f^0
∂Ek[
−
∇μ
kBT−
(Ek−μ)
kBT^2∇T
]
∇rf^0 =∂f^0
∂Ek[
−∇μ−(Ek−μ)
T∇T
]
(B.12)
Also
∇kf^0 =∂f^0
∂Ek·∇kEk= vk∂f^0
∂Ek