SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

518 APPENDIX B. BOLTZMANN TRANSPORT THEORY

n(r)

Position

Time t = 0

r =δtvk

n(r)

Position

Time t =δt

δtvk

r

Figure B.1: At timet=0particles at positionr−δtvkreach the positionrat a later timeδt.
This simple concept is important in establishing the Boltzmann transport equation.

1. Due to the motion of the electrons (diffusion), carriers will be moving into and out of any
   volume element aroundr.


2. Due to the influence of external forces, electrons will be changing their momentum (or
   k-value) according todk/dt=Fext.


3. Due to scattering processes, electrons will move from onek-state to another.

We will now calculate these three individual changes by evaluating the partial time derivative
of the functionfk(r)due to each source.

B.1.1 Diffusion-Induced Evolution offk(r) ...................

Ifvkis the velocity of a carrier in the statek, in a time intervalt, the electron moves a distance
tvk. Thus the number of electrons in the neighborhood ofrat timeδtis equal to the number of
carriers in the neighborhood ofr−δtvkat time 0, as shown in figure B.1
We can thus define the following equality due to the diffusion

fk(r,δt)=fk(r−δtvk,0) (B.2)

or

fk(r,0) +

∂fk

∂t

·δt = fk(r,0)−

∂fk

∂r

·δtvk

∂fk

∂t

∣∣

∣∣

diff

= −

∂fk

∂r

·vk (B.3)