SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

136 CHAPTER 3. CHARGE TRANSPORT IN MATERIALS

GL

ra rb

rc rd

Figure 3.25: Possible Recombination processes

or the net capture rate of electrons = net capture rate of holes. This leads us to:

vthσnnNt[1−f(Et)]−vthniexp (Et−i/kBT)Ntf(Et)=

vthσppNtf−vthσpniexp [(Ei−Et)/kBT]Nt[1−f(Et)]

Since we are in non-equilibrium,f(Et), the distribution function for the traps has to be cal-
culated from the above equation, where we have substituted forrathroughrd,

f(Et)=

σnn+σpniexp (Ei−t/kBT)

σn[n+niexp (Et−i/kBT)] +σp[p+n·exp (Ei−t/kBT)]

(3.8.37)

where for compactness we have used the notation: Ei−t=Ei−Etand vice versa. Re-
substituting to find a net rate of recombination:

U=ra−rb=rc−rd (3.8.38)

leads to:

U=

σpσnvthNt

(

pn−n^2 i

)

σn[n+niexp (Et−i/kBT)] +σp[p+niexp (Ei−t/kBT)]

(3.8.39)

Let us now consider some special cases:

1. forσn=σp=σ

U=σvthNt

pn−n^2 i

n+p+2nicosh (Et−i/kBT)

(3.8.40)

2. forσn=σp=σpand whenEt=Ei

U=

1

τ

pn−n^2 i

n+p+2ni

(3.8.41)

We see clearly thatpn−n^2 iis the driving force for recombination. We can also see thatn+p+2ni
is a resistance to recombination term, which is minimized whenn+pis minimized. For low
level injection, we assume thatnnpnand

nnniexp ((Et−Ei)/kBT) (3.8.42)