3.8. CARRIER GENERATION AND RECOMBINATION 131
electron-hole system and is
Ncv(E)=√
2(m∗r)^3 /^2 (E−Eg)^1 /^2
π^2 ^3(3.8.9)
If we express the energy in eV, and the absorption coefficient incm−^1 for most direct gap semi-
conductors the absorption coefficient is approximately
α(ω)∼ 5. 6 × 104(ω−Eg)^1 /^2
ωcm−^1 (3.8.10)For indirect gap materials the absorption coefficient is an order of magnitude smaller than the
result given above since in first order transitions momentum is not conserved. Thus for materials
like Si and Ge near bandedge absorption is weak.If there are electrons in the conduction band and
holes in the valence band they can recombine to emit photons. If the occupation of an electron
state is unity and the occupation of the corresponding hole state is also unity the recombination
rate is given by
Wem=1
τ 0=
e^2 nr
6 πom 0 c^3 ^2(
2 p^2 cv
m 0)
ω (3.8.11)Using typical values of the momentum matrix elementpcvfor direct gap materials the result is
Wem=1
τ 0=10^9 Egs−^1 (3.8.12)When electrons and holes are injected into the conduction and valence bands of a semiconductor,
they recombine with each other. In general the occupation of electrons and holes is given by
the quasi-Fermi levels. Theemission rateor the electron-hole recombination rate is (units are
cm−^3 s−^1 )
Rspon=1
τo∫
d(ω)Ncv{fe(Ee)}{fh(Eh)} (3.8.13)The spontaneous recombination rate is quite important for both electronic and optoelectronic
devices. It is important to examine the rate for several important cases. We will give results for
the electron hole recombination for the following cases: i)Minority carrier injection:Ifnp
and the sample is heavily doped, we can assume thatfe(Ee)is close to unity. We then have for
the rate at which holes will recombine with electrons,
Rspon ∼=1
τo∫
d(ω)Ncvfh(Eh)∼=1
τo∫
d(ω)Nhfh(Eh)(
m∗r
m∗h) 3 / 2
∼=^1
τo(
m∗r
m∗h) 3 / 2
p (3.8.14)Thus the recombination rate is proportional to the minority carrier density (holes in this case).
ii)Strong injection:This case is important when a high density of both electrons and holes is
injected and we can assume that bothfeandfhare step functions with values 1 or zero. We get
for this case
Rspon=
n
τo=
p
τo