Validity of the Relation Between Spontaneous and Stimulated Emissions in Semiconductors

The Einstein relation between spontaneous emission and absorption was originally derived for a system consists of a two-state subsystem representing matter and harmonic fields representing radiation. The derivation is based on the detailed balance between these two subsystems under thermal equilibrium. The relationship was later investigated in connection with the interactions between radiation field and solids or semiconductors. The simple derivation dose not hold for semiconductors in general. In certain limiting cases, simple relation was obtained. The validity of this relation is important not only because of its fundamental role connecting two of the most fundamental optical processes in semiconductors, but mostly also because of its wide use as a practical method to measure the optical gain of a semiconductor. The validity of this relation for semiconductors has been an issue of controversial for some time. In this paper we numerically examine the validity of this relationship for several different lineshapes including Lorentzian, Gaussian, Sech, and a convoluted double Lorentzians (CDL). We find out that at relatively low density above transparency level, all first three lineshapes violate the Einstein relation. The relation is approximately valid at high density. At very high density, the validity of the Einstein relation holds well for all three lineshapes. The reason behind this observation is explained. The CDL lineshape has been shown analytically to obey the Einstein relationship previously. We show that for a 2D semiconductor with parabolic bands, the CDL lineshape can be integrated analytically. This analytic lineshape is compared with a simple Lorentzian lineshape.