SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

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 

(4.2.20)