B.1. BOLTZMANN TRANSPORT EQUATION 519
B.1.2 External Field-Induced Evolution offk(r) ................
The crystal momentumkof the electron evolves under the action of external forces according
to Newton’s equation of motion. For an electric and magnetic field (EandB), the rate of change
ofkis given by
k ̇=e
[E+vk×B] (B.4)In analogy to the diffusion-induced changes, we can argue that particles at timet=0with
momentumk−k ̇δtwill have momentumkat timeδtand
fk(r,δt)=fk−k ̇δt(r,0) (B.5)which leads to the equation
∂fk
∂t∣∣
∣∣
ext. forces= −k ̇∂fk
∂k=−e
[
E+
v×B
c]
·
∂fk
∂k(B.6)
B.1.3 Scattering-Induced Evolution offk(r)...................
We will assume that the scattering processes arelocal andinstantaneous and change the state
of the electron fromktok
′
.LetW(k,k′
)define the rate of scattering from the statektok′if the statekis occupied andk
′
is empty. The rate of change of the distribution functionfk(r)
due to scattering is
∂fk
∂t)
scattering=
∫ [
fk′(1−fk)W(k′
,k)−fk(1−fk′)W(k,k′
)]d (^3) k′
(2π)^3
(B.7)
The(2π)^3 in the denominator comes from the number of states allowed in ak-space volume
d^3 k
′. The first term in the integral represents the rate at which electrons are coming from an
occupiedk
′
state (hence the factorfk′) to an unoccupiedk- state (hence the factor ( 1 −fk)).
The second term represents the loss term.
Under steady-state conditions, there will be no net change in the distribution function and the
total sum of the partial derivative terms calculated above will be zero.
∂fk
∂t)
scattering+
∂fk
∂t)
fields+
∂fk
∂t)
diffusion=0 (B.8)
Let us define
gk=fk−fk^0 (B.9)
wherefk^0 is the equilibrium distribution.
We will attempt to calculategk, which represents the deviation of the distribution function
from the equilibrium case.