152 CHAPTER 4. JUNCTIONS IN SEMICONDUCTORS:P-NDIODES
The Poisson equation in the depletion approximation for various regions is
d^2 V(x)
dx^2
=0 −∞<x<−Wp (4.2.9)
d^2 V(x)
dx^2
=
eNa
−Wp<x< 0 (4.2.10)
d^2 V(x)
dx^2
= −
eNd
0 <x<Wn (4.2.11)
d^2 V(x)
dx^2
=0 Wn<x<∞ (4.2.12)
Solving these equations gives the electric field in thep-side of the depletion region
E(x)=−
dV
dx
=−
eNax
−
eNaWp
−Wp<x< 0 (4.2.13)
The electric field reaches a peak value atx= 0. The potential is given by integrating the field,
V(x)=
eNax^2
2
+
eNaWpx
+
eNaWp^2
2
+Vp −Wp<x< 0 (4.2.14)
For then-side of the depletion region andn-side of the neutral region, we use the conditions
V(x)=Vn Wn<x<∞
E(x)=0 (4.2.15)
whereVnis the potential at the neutraln-side. The electric field and potential on then-side is
found to be
E(x)=
eNdx
−
eNdWn
0 <x<Wn (4.2.16)
V(x)=−
eNdx^2
2
+
eNdWnx
−
eNdWn^2
2
+Vn 0 <x<Wn (4.2.17)
The potential difference between points−Wpand 0 is
V(0)−V(−Wp)=
eNaWp^2
2
(4.2.18)
Similarly,
V(Wn)−V(0) =
eNdWn^2
2
(4.2.19)
Thus the built-in potential is
V(Wn)−V(−Wp)=Vbi=
eNdWn^2
2
+
eNaWp^2
2