Optical gain is the most important requirement for the realization of a semiconductor laser because it describes the optical amplification in the semiconductor material. This optical gain is due to stimulated emission associated with light emission created by recombination of electrons and holes. While in other laser materials like in gas lasers or solid state lasers, the processes associated with optical gain are rather simple, in semiconductors this is a complex many-body problem of interacting photons, electrons, and holes. Accordingly, understanding these processes is a major objective as being a basic requirement for device optimization. This task can be solved by development of appropriate theoretical models to describe the semiconductor optical gain and by comparison of the predictions of these models with experimental results found.
Theory for optical gain in semiconductors Since defining semiconductor's optical gain is an ambitious undertaking, it is useful to build the understanding by steps. The basic requirements can be defined without the major complications induced by the Coulomb interaction among electrons and holes. To explain the actual operation of semiconductor lasers, one must refine this analysis by systematically including the Coulomb-interaction effects.
Free-carrier picture For a simple, qualitative understanding of optical gain and its spectral dependency, often so-called free-carrier models are used which is discussed considering the example of a bulk laser here. The term free carrier means that any interactions between the carriers are neglected. A free-carrier model provides the following expression for the spectral dependence g ( ε ) {\displaystyle g(\varepsilon )}
g ( ε ) = g 0 ε [ f e ( ε ) + f h ( ε ) − 1 ] , {\displaystyle g(\varepsilon )=g_{0}{\sqrt {\varepsilon }}\,[f^{\mathrm {e} }(\varepsilon )+f^{\mathrm {h} }(\varepsilon )-1]~,}
with the reduced-mass energy ε {\displaystyle \varepsilon } , the quasi-Fermi-distribution functions for the conduction-band f e {\displaystyle f^{\mathrm {e} }} and for the valence-band f h {\displaystyle f^{\mathrm {h} }} , respectively, and with g 0 {\displaystyle g_{0}} given by:
g 0 ( ε ) = ν | μ ( ε ) | 2 4 ε 0 π n ( 2 m r ℏ 2 ) 3 / 2 , {\displaystyle g_{0}(\varepsilon )={\frac {\nu |\mu (\varepsilon )|^{2}}{4\varepsilon _{0}\pi n}}\left({\frac {2m_{\mathrm {r} }}{\hbar ^{2}}}\right)^{3/2}~,}
with ν {\displaystyle \nu } being the frequency, | μ ( ε ) | 2 {\displaystyle |\mu (\varepsilon )|^{2}} the dipole-matrix element, m r {\displaystyle m_{\mathrm {r} }} the reduced mass, ε 0 {\displaystyle \varepsilon _{0}} the vacuum permittivity, and n {\displaystyle n} the refractive index. Thus, the shape of the gain spectrum g ( ε ) {\displaystyle g(\varepsilon )} is determined by the density of states, proportional to ε {\displaystyle {\sqrt {\varepsilon }}} , for bulk material and the quasi-Fermi-distribution functions. This expression gives a qualitative impression of the dependence of the gain spectra on the distribution functions. However, a comparison to experimental data shows immediately that this approach is not at all suited to give quantitative predictions on the exact gain values and the correct shape of the spectra. For that purpose, a microscopic model including many-body interactions is required. In recent years, the microscopic many-body model based on the semiconductor Bloch equations (SBE) has been very successful.
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