SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

3.9. CURRENT CONTINUITY 141

This gives for the excess carrier concentration

δn(x)=

δn(0) sinh

(

L−x

Ln

)

+δn(L)sinh

(

x

Ln

)

sinh

(

L

Ln

) (3.9.13)

There are two important cases that occur in bipolar devices, we will examine them here:
(i)LLnandδn(L)=0: In this case the semiconductor sample is much longer thanLn.This
happens in the case of thelongp-ndiode , which will be discussed in chapter 4 For this case we
have
δnp(x)=δnp(0)e−x/Ln (3.9.14)

Thus the carrier density simply decays exponentially into the semiconductor.
(ii)LLn: This case is very important in discussing the operation of bipolar transistors and
narrowp-ndiodes. Using the smallxexpansion forsinh(x)

sinh(x)=x+

x^3

3!

+

x^5

5!

+...

and retaining only the first-order terms we get

δnp(x)=δnp(0)−

x[δnp(0)−δnp(L)]

L

(3.9.15)

i.e., in this case the carrier density goeslinearly from one boundary value to the other.
Note that once the carrier density is known the diffusion current can be simply obtained by
taking its derivative.
Let us examine the case where excess carriers are injected into a thick semiconductor sample.
As the excess carriers diffuse away into the semiconductor they recombine. The diffusion length
Lnrepresents the distance over which the injected carrier density falls to 1/eof its original value.
It also represents the average distance an electron will diffuse before it recombines with a hole.
This can be seen as follows.
The probability that an electron survives up to pointxwithout recombination is, from equation
3.9.15,
δnp(x)
δnp(0)

=e−x/Ln (3.9.16)

The probability that it recombines in a distanceΔxis

δnp(x)−δnp(x+Δx)

δnp(x)

=−

Δx

δnp(x)

dδnp(x)

dx

=

1

Ln

Δx (3.9.17)

where we have expandedδnp(x+Δx)in terms ofδnp(x)and the first derivative ofδnp. Thus
the probability that the electron survives up to a pointxand then recombines is

P(x)Δx=

1

Ln

e−x/LnΔx (3.9.18)