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Considering the case of photons in a semiconductor, Fig. 5 shows two representative series of processes that might be involved. One type involves interband excitation to create electron-hole pairs and the other involves intraband transitions, each with their associated spontaneous and stimulated inverse processes. Let PCi,Vj be the coefficient describing the rate of exciting an electron state Vj in the valence band when fully occupied to a completely empty state Ci in the conduction band. By the same type of argument as for the Einstein coefficients, it follows that PCi,Vj = PVj,Ci while the corresponding coefficient for stimulated emission is fpt PVj,Ci.

The ratio of stimulated emission to absorption for the band to band processes in Eq. 30) is given by: R= fCi ( 1 − fVi ) = e( − hf + µ CV ) / kT . 32). 41) Since bosons satisfy a similar relationship to Eq. 42) l where ε are the phonon energies and µ are chemical potentials. 6 General Cell Analysis 49 Substituting back into Eq. 44) e If phonons are assumed to have zero chemical potential and the loss of one electron from the conduction band to the valence band is being considered, this reduces to the final value of Eq.

Why don’t we see something unusual at sub-bandgap wavelengths for these devices? The first term on the denominator of Eq. 38) will be negative under such conditions. There would seem to be no fundamental reason why the critical condition mentioned previously could not occur first for such sub-bandgap photons. The energy emitted by the cell would then occur at far infrared wavelengths! The reason this does not seem to occur is that the achievable negative magnitude of αCV decreases very rapidly with decreasing hf.