Page 181 - Electrical Properties of Materials

The p–n junction in equilibrium 163

electrons on the right-hand side is simply due to the fact that they need to do
some work against the electric field before they can reach the conduction band
on the left-hand side.
What can we say about the transition region? One would expect the electron
and hole densities to change gradually from high to low densities as shown in
Fig. 9.1(d). But what sort of relationship will determine the density at a given
point? And furthermore, what will be the profile of the conduction band in the
transition region? They can all be obtained from Poisson’s equation,

2
d U 1
= (net charge density), (9.1)
dx 2
∗
where U is the electric potential used in the usual sense. Since the density ∗ We are in a slight difficulty here be-
of mobile carriers depends on the actual variation of potential in the transition cause up to now potential meant the
potential energy of the electron, denoted
region, this is not an easy differential equation to solve. Fortunately, a simple
by V. The relationship between the two
approximation may be employed, which leads quickly to the desired result. quantities is eU = V, which means that
As may be seen in Fig. 9.1(d), the density of mobile carriers rapidly de- if you confuse the two things you’ll be
19
creases in the transition region. We are, therefore, nearly right if we maintain wrong by a factor of 10 .
that the transition region is completely depleted of mobile carriers. Hence we
may assume the net charge densities are approximately of the form shown in The transition region is often
Fig. 9.2(a). Charge conservation is expressed by the condition called the ‘depletion’ region.

N A x p = N D x n . (9.2)
–x p and x n are the widths of the de-
Poisson’s equation for the region –x p to 0 reduces now to the form
pletion regions in the p- and n-type
materials, respectively.
2
d U eN A
= . (9.3)
dx 2
Integrating once, we get
dU eN A
E =– =– (x + C), (9.4)
dx
C is an integration constant.
According to our model, the depletion region ends at –x p . There is no charge
imbalance to the left of –x p , hence the electric field must be equal to zero at
x =–x p . With this boundary condition eqn (9.4) modifies to

eN A
E = (x + x p ). (9.5)

Similar calculation for the n-type region yields

eN D
E = (x – x n ). (9.6)

The electric field varies linearly in both regions, as may be seen in Fig. 9.2(b).
It takes its maximum value at x = 0 where there is an abrupt change in its slope.
The variation of voltage may then be obtained from

x
U =– E dx, (9.7)
–x p