Physics and Engineering of Radiation Detection

288 Chapter 5. Solid State Detectors

Similarly for the n-side we get

E(x)=−

dΦ

dx

=

eND



(x−xn)for0≤x<xn (5.1.61)

Fig.5.1.26 shows these functions as well as the field profile in a realistic pn junction.

−xp xn

x

E

Figure 5.1.26: Electric field intensity profile of

the idealized charge density shown in Fig.5.1.25

(solid line) together with a more realistic profile

(dotted line).

To determine the profile of the electric potential and the depletion depth, we can
integrate the above two equations again to get

Φ(x)=

−eN D

[

x^2

2 −xxn

]

+A 1 :0≤x<xn n-side

eNA

[

x^2

2 +xxp

]

+A 2 : −xp<x≤0p-side.

(5.1.62)

The integration constantsA 1 andA 2 can be determined by noting that the applied
reverse bias appears as a potential difference across the junction, which can be taken
as 0 atx=−xpandV 0 atx=xn. In such a case the potential profile inside the
junction becomes

Φ(x)=

−eN 2 D(x−xn)^2 +V 0 :0≤x<xn n-side

eNA

2 (x+xp)

(^2) : −xp<x≤0p-side.

(5.1.63)

This potential has been plotted in Fig.5.1.27

−xp xn

x

φ

V 0

Figure 5.1.27: Variation of electric potential

with respect to distance from the center of a

pn junction.

An interesting result can be obtained if we use the condition that the potentials at
x= 0 must be equal. This gives

V 0 =

e

2 

[

NAx^2 p+NDx^2 n

]

(5.1.64)