SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

B.1. BOLTZMANN TRANSPORT EQUATION 525

φ

θ'

θ

α

x

y

z

k

E

k'

Figure B.4: Coordinate system illustrating a scattering event.

in they-zplane. The wavevector after scattering is given byk

′
represented by the anglesαand
φ. Assuming that the energy bands of the material is isotropic,|vk|=|vk′|. We thus get

vk′·E

vk·E

=

cosθ

′

cosθ

(B.32)

We can easily see from figure B.4 that

cosθ

′

=sinθsinαsinφ+ cosθcosα

or
cosθ

′

cosθ

=tanθsinαsinφ+ cosα

When this term is integrated overφto evaluateτ, the term involvingsinφwill integrate to zero
for isotropic bands sinceW(k,k

′

)does not have aφdependence, only anαdependence. Thus

1

τ

=

∫

W(k,k

′

)(1−cosα)d^3 k

′

(B.33)