SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

120 CHAPTER 3. CHARGE TRANSPORT IN MATERIALS

nR

xo– xo

x

xo+

nL

LR

mean

free

path

C

ARRIER CONCENTRATION

Figure 3.16: The concentration profile of electrons as a function of space. The termsnL,nR,L,
andRare defined in the text. The distance is the mean free path for electrons; i.e., the distance
they travel between collisions.

In figure 3.16 is shown a concentration profilen(x, t)of electrons at timet,. We are going
to calculate the electron fluxφ(x, t)across a planex=xoat any instant of time. Consider a
region of space a mean free path to each side ofxo, from which electrons can come across the
x=xoboundary in timeτsc. Electrons from regions further away will suffer collisions that will
randomly change their direction. Since in the two regions labeledLandRin figure 3.16, the
electrons move randomly, half of the electrons in regionLwill go acrossx=xoto the right and
half in the regionRwill go acrossx=xoto the left in timeτsc. The flux to the right is

φn(x, t)=

(nL−nR)

2 τsc

(3.6.1)

wherenLandnRare the average carrier densities in the two regions. Since the two regionsL
andRare separated by the distance , we can write

nL−nR∼=−

dn

dx

· (3.6.2)

The total flux is

φn(x, t)=−

2

2 τsc

dn(x, t)

dx

=−Dn

dn(x, t)

dx

(3.6.3)