3.9. CURRENT CONTINUITY 137
asET∼Eifor efficient recombination. Then the recombination rate becomes:
U=
σpσnvthNt
σnnn
[
nnpn−n^2 i
]
(3.8.43)
=σpvthNt
[
pn−n^2 i/nn
]
(3.8.44)
=σpvthNt[pn−pn 0 ] (3.8.45)
U=
Δpn
τp
(3.8.46)
where the minority carrier lifetime,τpis defined as
1
τp
=σpvthNt (3.8.47)
Here the rate limiting step is the capture of the minority carrier. This is also achieved by recog-
nizing the hole capture rate,rcis the dominant step. In ann−type semiconductor, sinceEFis
close to the conduction band andf(ET)→ 1 which makesraandrdboth negligible. Typical
values ofσare 10 −^15 − 10 −^16 cm−^2.
Generation occurs whenn^2 ipn. From equation 3.8.39
U=−
σpσnvthNtn^2 i
σn[n+niexp (Et−i/kBT)] +σp[p+niexp (Ei−t/kBT)]
=−
σpσnvthNtni
σnexp (Et−i/kBT)+σpniexp (Ei−t/kBT)
For the caseσn=σp=σ
U=−
σvthNtni
2 cosh (Et−i/kBT)
(3.8.48)
Thus, generation rate peaks when the trap energy is at mid-gap:
U=−
ni
2 τ
(3.8.49)
whenEi→Etthe lifetime
τ=
1
σvthNT
3.9 CURRENT CONTINUITY(The law of conservation
of electrons and holes separately)
In the previous sections we have considered several elements of non-equilibrium phenomena
in semiconductors. These include drift and diffusion, carrier generation and recombination.