SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

140 CHAPTER 3. CHARGE TRANSPORT IN MATERIALS

δn(0)

δn(x=L)

δn(x)

L

x

0

Carrier injection

Figure 3.27: Electrons are injected atx=0into a sample. Atx=0, a fixed carrier concentration
is maintained. The figure shows how the excess carriers decay into the semiconductor.

whereLn(Lp)defined asDnτn(Dpτp)are called the diffusion lengths We will see below that
the diffusion length represents the distance an electron (hole) will travel before it recombines
with a hole (electron). Let us examine the schematic of the equation derived above. Consider
the case where an excess electron densityδn(0)is maintained at the semiconductor atx=0,as
shown in figure 3.27. At some pointLin the semiconductor the excess carrier density is fixed at
δ(L). We are interested in finding out how the excess density varies with position. The general
solution of the second-order differential equation 3.9.11 is

δn(x)=A 1 ex/Ln+A 2 e−x/Ln

Using the boundary conditions atx=0andx=L, we find that the coefficientsA 1 andA 2 are

A 1 =

δn(L)−δn(0)e−L/Ln

eL/Ln−e−L/Ln

A 2 =

δn(0)eL/Ln−δn(L)

eL/Ln−e−L/Ln

(3.9.12)