SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

B.2. AVERAGING PROCEDURES 527

The perturbation in the distribution function is

gk =

−∂f^0

∂Ek

τvk·eE

≈

f^0

kBT

vk·eE (B.40)

If we consider a field in thex-direction, the average current in thex-direction is from equation
B.39 and B.40

〈Jx〉=

e^2

kBT

∫

τvx^2 f^0

d^3 k

(2π)^3

|E|x (B.41)

The assumption made on the drift velocity ensures thatv^2 x=v^2 / 3 ,wherevis the total velocity
of the electron. Thus we get

〈Jx〉=

e^2

3 kBT

∫

τv^2 f^0 (k)

d^3 k

(2π)^3

|E|x (B.42)

Now we note that

1

2

m∗〈v^2 〉 =

3

2

kBT

⇒kBT = m∗〈v^2 〉/ 3

also

〈v^2 τ〉 =

∫

v∫^2 τf^0 (k)d^3 k/(2π)^3

f^0 (k)d^3 k/(2π)^3

=

∫

v^2 τf^0 (k)d^3 k/(2π)^3

n

(B.43)

Substituting in the right-hand side of equation B.42, we get (using 3 kBT=m

〈

v^2

〉

)

〈Jx〉 =

ne^2

m∗

〈v^2 τ〉

〈v^2 〉

|E|x

=

ne^2

m∗

〈Eτ〉

〈E〉

|E|x (B.44)

Thus, for the purpose of transport, the proper averaging for the relaxation time is

〈〈τ〉〉=

〈Eτ〉

〈E〉

(B.45)

Here the double brackets represent an averaging with respect to the perturbed distribution func-
tion while the single brackets represent averaging with the equilibrium distribution function.
For calculations of low-field transport where the conditionv^2 x=v^2 / 3 is valid, one has to
use the averaging procedure given by equation B.45 to calculate mobility or conductivity of the