234 CHAPTER 5. SEMICONDUCTOR JUNCTIONS
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(b)
(a)
E
E
E
Figure 5.10: (a) Space-charge density and (b) electric field profiles in ann-pheterostructure.To findE 1 ,m,E 2 ,m,Wn,andWp, the following equations, along with equation 5.6.3, can be
used:
NdWn=NaWp (5.6.4)
E 1 ,m
Wn
=
qNd
1andE 2 ,m
Wp=
qNa
2(5.6.5)
Vbi= area underE=1
2
[WnE 1 ,m+WpE 2 ,m]=Vd 1 +Vd 2 (5.6.6)These are essentially the same set of equations used to deriveWn,Wp,andEfor ap-nhomo-
junction. Equation 5.6.4 is our charge neutrality condition, and equation 5.6.5 and equation
5.6.6 result from solving Poisson’s equation.
From these equations, we obtain the following set of relationships:
Wp(Vbi)={
2 1 2 Vbi
e[
Nd
Na(Na 2 +Nd 1 )]} 1 / 2
(5.6.7)
Wn(Vbi)={
2 1 2 Vbi
e[
Na
Nd(Na 2 +Nd 1 )]} 1 / 2
(5.6.8)
W(Vbi)=Wp(Vbi)+Wn(Vbi)=[
Wn^2 (Vbi)+Wp^2 (Vbi)+2Wn(Vbi)Wp(Vbi)