132 CHAPTER 3. CHARGE TRANSPORT IN MATERIALS
iii)Weak injection:In this case we can use the Boltzmann distribution to describe the Fermi
functions. We have
fe·fh∼= exp{
−
(Ec−EFn)
kBT}
exp{
−
(EFp−Ev)
kBT}
·exp{
−
(ω−Eg)
kBT}
(3.8.16)
The spontaneous emission rate now becomes
Rspon=1
2 τo(
2 π^2 m∗r
kBTm∗em∗h) 3 / 2
np (3.8.17)If we write the total charge as equilibrium charge plus excess charge,
n=no+Δn;p=po+Δn (3.8.18)we have for the excess carrier recombination (note that at equilibrium the rates ofrecombination
and generation are equal)
Rspon∼=1
2 τo(
2 π^2 m∗r
kBTm∗em∗h) 3 / 2
(Δnpo+Δpno) (3.8.19)IfΔn=Δp, we can define the rate of a single excess carrier recombination as
1
τr=
Rspon
Δn=
1
2 τo(
2 π^2 m∗r
kBTm∗em∗h)
(no+po) (3.8.20)At low injectionτris much larger thanτo, since at low injection, electrons have a low probability
to find a hole with which to recombine. iv)Inversion condition:Another useful approximation
occurs when the electron and hole densities are such thatfe+fh=1.Thisistheconditionfor
inversionwhentheemissionandabsorptioncoefficientsbecomeequal. If we assume in this case
fe∼fh=1/ 2 , we get the approximate relation
Rspon∼=n
4 τo∼= p
4 τo(3.8.21)
The recombination lifetime is approximately 4τoin this case. This is a useful result to estimate
the threshold current of semiconductor lasers.
Example 3.11Optical radiation with a power density of 1. 0 kW/cm^2 impinges on GaAs.
The photon energy is 1.5 eV and the absorption coefficient is 3 × 103 cm−^1. Calculate the
carrier generation rate at the surface of the sample. If thee−hrecombination time is 1 ns,
calculate the steady state excess carrier density.
At the surface the carrier generation rate isG(0) =(3× 103 cm−^1 )(10^3 Wcm−^2 )
(1. 5 × 1. 6 × 10 −^19 J)
=1. 25 × 1025 cm−^3 s−^1
The excess carrier density is
δn=δp=1. 25 × 1025 × 10 −^9 =1. 25 × 1016 cm−^3