SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

4.4. REAL DIODES:DEFECTS AND CARRIER GENERATION 165

EV

EFp

EC

Eip EFn

Ein

ψ(x)

ψ’(x)

EV

EF

EC

EF

Eip

Ein

(a)

(b)

Figure 4.11: Band diagram of ap-njunction (a) in equilibrium and (b) under forward bias

p(x)=pp 0 exp

(

−eψ(x)

kBT

)

(4.4.3)

Note thatψ(x)is the voltage measured downwards fromEipandψ′(x)is measured upwards
fromEinand that

n(x)·p(x)=nn 0 pp 0 exp

[

−e(ψ′(x)+ψ(x))

kBT

]

ψ(x)+ψ′(x)=Vbi+V (4.4.4)

Therefore,

n(x)·p(x)=nn 0 pn 0 exp

(

eV

kBT

)

or

n(x)·p(x)=n^2 iexp

(

eV

kBT

)

(4.4.5)