SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

522 APPENDIX B. BOLTZMANN TRANSPORT THEORY

t =τ

f(k)

t = 0.5 τ

t = 0

k= 0

k

Figure B.2: This figure shows that at timet=0, the distribution function is distorted by some
external means. If the external force is removed, the electrons recover to the equilibrium distri-
bution by collisions.

Note that we have not defined howτis to be calculated. We have merely introduced a simpler
unknown that still needs to be determined. Thek-space distribution function may be written as

fk = fk^0 −

(

∂fk^0

∂Ek

)

eτvk·E (B.19)

= fk^0 −

(

∇kfk^0

)

·

∂k

∂Ek

·eτvk·E (B.20)

Using the relation



∂k

∂Ek

·vk=1

We h ave

fk = fk^0 −

(

∇kfk^0

)

·

eτE



= fk^0

(

k−

eτE



)

(B.21)

This is a very useful result which allows us to calculate the non-equilibrium functionfkin
terms of the equilibrium functionf^0. The recipe is very simple—shift the original distribution
function forkvalues parallel to the electric field byeτE/. If the field is along thez-direction,
only the distribution forkzwill shift. This is shown schematically in figure B.3. Note that for
the equilibrium distribution function, there is an exact cancellation between positive velocities