SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

166 CHAPTER 4. JUNCTIONS IN SEMICONDUCTORS:P-NDIODES

X = 0

x

ψ

ψ

x

Towards p Towards n

Maximum Recombination Plane

Figure 4.12: The maximum recombination plane is where the recombination is peaked within
the depletion region

Under steady state bias ofVthe termn+p+2niis minimized whenn=p.Thevalueofx
where this occurs is chosen to be zero. This is the maximum recombination plane (MRP).

n(0) =p(0) =niexp

(

eV

2 kBT

)

(4.4.6)

If we move away from the MRP, the electron and hole concentrations change proportionally to
the term

exp

(

±

eψ(x)

kBT

)

as shown in figure 4.12, Where

n(x)=n(0) exp

(

eψ(x)

kBT

)

and p(x)=p(0) exp

(

−

eψ(x)

kBT

)

Assuming the distancexis small so that we can assume the electric field is constant for purposes
of the analysis to beE=E(0).Thenψ=±Exand therefore,

U=

1

τ

·

pn−n^2 i

n+p+2ni

=

1

τ

·

n^2 iexp (eV /kBT)−n^2 i

n(0) exp (eψ/kBT)+p(0) exp (−eψ/kBT)+2ni

Neglectingn^2 iin the numerator and 2 niin the denominator we get

U=

1

τ

·

niexp (eV / 2 kBT)

2 cosh(eEx/kBT)

(4.4.7)