80 CHAPTER 3. OPTICAL TRANSMITTERS
Figure 3.2: Conduction and valence bands of a semiconductor. Electrons in the conduction band
and holes in the valence band can recombine and emit a photon through spontaneous emission
as well as through stimulated emission.
whereρcvis thejoint density of states, defined as the number of states per unit volume
per unit energy range, and is given by [18]
ρcv=
( 2 mr)^3 /^2
2 π^2 ̄h^3(h ̄ω−Eg)^1 /^2. (3.1.11)In this equation,Egis the bandgap andmris the reduced mass, defined asmr=
mcmv/(mc+mv), wheremcandmvare the effective masses of electrons and holes in
the conduction and valence bands, respectively. Sinceρcvis independent ofE 2 in Eq.
(3.1.10), it can be taken outside the integral. By contrast,A(E 1 ,E 2 )generally depends
onE 2 and is related to the momentum matrix element in a semiclassical perturbation
approach commonly used to calculate it [2].
The stimulated emission and absorption rates can be obtained in a similar manner
and are given by
Rstim(ω)=∫∞EcB(E 1 ,E 2 )fc(E 2 )[ 1 −fv(E 1 )]ρcvρemdE 2 , (3.1.12)Rabs(ω)=∫∞EcB(E 1 ,E 2 )fv(E 1 )[ 1 −fc(E 2 )]ρcvρemdE 2 , (3.1.13)whereρem(ω)is the spectral density of photons introduced in a manner similar to Eq.
(3.1.1). Thepopulation-inversion condition Rstim>Rabsis obtained by comparing Eqs.
(3.1.12) and (3.1.13), resulting infc(E 2 )>fv(E 1 ). If we use Eqs. (3.1.8) and (3.1.9),
this condition is satisfied when
Efc−Efv>E 2 −E 1 >Eg. (3.1.14)