Semiconductor optical gain
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.
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.
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.
In recent years, the microscopic many-body model based on the semiconductor Bloch equations (SBE) has been very successful.
If only stationary gain spectra in the linear regime are of interest, one can neglect the time dependence of the distribution functions
In contrast to the free-carrier description, this model contains contributions due to many-body Coulomb interactions such as
The easiest approach is to replace the collision term by a phenomenological relaxation rate (
[1] However, though this approximation is often used, it leads to somewhat unphysical results like absorption below the semiconductor band gap.
A more correct but also much more complex approach considers the collision term kinetically and thus contains in- and out-scattering rates for the microscopic polarizations.
[2] In this quantum kinetic approach, the calculations require only the basic input parameters (material band structure, geometrical structure, and temperature) and provide the semiconductor gain and refractive index spectra without further free parameters.
In detail, the above-mentioned equation of motion of the polarization is solved numerically by calculating the first two terms on the right hand side from the input parameters and by computing the collision contributions.
to obtain the complex macroscopic polarization which then provides the gain and the refractive index spectra in semiconductor laser theory.
It should be mentioned that present-day modeling assumes a perfect semiconductor structure in order to reduce the numerical effort.
Disorder effects like composition variations or thickness fluctuations of the material are not microscopically considered but such imperfections often occur in real structures.
The predictive quality of microscopic modeling can be verified or disproved by optical-gain measurements.
If experiments exhibit unexpected gain features, one can refine the modeling by including systematically new effects.
[7] This method uses a strong laser source for optical excitation of the sample under investigation.
of the amplified spontaneous emission (ASE) of the sample out of this edge is measured as a function of the stripe length
The stripe-length method provides reasonable qualitative results for semiconductor samples which have not yet been processed towards electrically pumped laser structures.
Then, the spectrum of the emitted ASE is strongly governed by the Fabry–Pérot modes of the diode laser resonator.
If the length of the device and the reflectivities of the facets are known, the gain can be evaluated from the maxima and the minima of the Fabry–Pérot peaks in the ASE spectrum.
This requires, however, that the ASE data are recorded with a spectrometer of sufficient spectral resolution.
The transmission method[3] requires a weak broadband light source that spectrally covers the region of interest for the gain spectra.
[4] For the experimental spectra, the injection current was varied while for the theoretical curves different carrier densities were considered.
The theoretical spectra were convoluted with a Gaussian function with an inhomogeneous broadening of 19.7 meV.
While for the data shown in the figure, the inhomogeneous broadening was adapted for optimum agreement with experiment, it may also unambiguously determined from low-density luminescence spectra of the material under study.
[5] Almost perfect quantitative agreement of theoretical and experimental gain spectra can be obtained considering that the device heats up slightly in the experiment at higher injection currents.
