SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

4.4. REAL DIODES:DEFECTS AND CARRIER GENERATION 167

U

X = 0

Figure 4.13: Schematic of the net recombination rate as a function of distance from the MRP

To calculate the total recombination current we need to integrate over the volume of the depletion
region. Since the recombination rate curve is highly peaked aboutx=0,theMRP,asshownin
figure 4.13, the following approximations remain valid.

1. Linearizing the potentialψ=±Exsince only small values ofxcontribute to the integral.

2.

∫Wn

−Wp→

∫

∞

Therefore,

ISNS=IR=eA

∫

U(x)dx (4.4.8)

Making these substitutions and solving equation 4.4.8, we find:

IR=

eniA

2 τ

exp

(

eV

2 kBT

)∫+∞

−∞

dx

cosh [eE(0)x/kBT]

(4.4.9)

=

eniA

2 τ

·

πkBT

eE(0)

exp

(

eV

2 kBT

)

(4.4.10)

=IGR◦ exp

(

eV

2 kBT

)

(4.4.11)

Atzeroappliedbias,agenerationcurrentofIGbalancesouttherecombinationcurrent.