where and are the effective densities

where and are the effective densities of states of electrons and holes in a two-dimensional subband, respectively;
Let us assume that the equality is satisfied under optical excitation, and find the dependence of the position of the quasi Fermi levels of electrons and holes on the temperature and the concentration of charge carriers. The number of photons emitted per unit volume per unit time in the frequency interval from ν to ν+ dν for charge carrier transitions from the akt inhibitor level i to the hole level j can be written as follows [8]: where is the quantum well width, n is the refractive index, is the reduced mass, are the overlap integrals (calculated in Ref. [9]), P is the Kane matrix element of the momentum operator. This matrix element is expressed through the bulk band gap and the electron mass m0 in the following way:
Thus, the total number of emitted photons for all possible transitions (see Fig. 5) is written as follows:
Substituting expression (7) into formula (9) and taking into account the dependence of the distribution function on the Fermi level (which, in turn, depends on the temperature and concentration of charge carriers), we have determined the theoretical dependence of the peak luminescence intensity on the charge carrier concentration for each structure at a temperature of 77K (see Fig. 4). By scaling the abscissa, we have found the values of charge carrier concentrations corresponding to a good agreement between the experimental points and the theoretical curve. It is evident from Fig. 4 that the concentration of the charge carriers involved in radiative recombination is lower in the structure with 4-nm-wide quantum wells because, according to calculations of the band diagram, under optical pumping, charge carriers are excited only in the ground quantum-confinement levels. Charge carrier concentrations in structures with 7- and 9-nm-wide quantum wells were not significantly different due to slight variations in the band diagrams. Fig. 4 shows that the dependence of peak photoluminescence intensity on pump intensity is approximately linear for the structure with 5nm wide quantum wells. This corresponds to average concentrations of charge carriers participating in photoluminescence. Evidently, the structure with 5nm wide quantum wells exhibited the lowest charge carrier concentration of all samples. As proved above, the injected charge carriers in this structure are involved in non-radiative resonant Auger recombination, which reduces their contribution to radiative recombination. Thus, resonant non-radiative Auger recombination can decrease the concentration of charge carriers involved in radiative recombination by almost an order of magnitude. This phenomenon reduces the quantum yield and the effectiveness of lasers. In order to eliminate non-radiative Auger recombination, the design of semiconductor injection lasers emitting at a wavelength of about 3µm must involve carefully calculating the band diagram and checking whether the condition that the effective band gap be equal to the energy spacing between the ground state of heavy holes and the first level of the band split by spin–orbit interaction is not fulfilled.
Conclusion The dependence of photoluminescence intensity in the spectral peak on the optical pumping intensity for nanostructures with different InGaAsSb/AlGaAsSb quantum well widths is investigated. This dependence had a linear behavior for the structure with 5nm wide quantum wells where resonant Auger recombination was expected to be observed. In order to analyze the obtained experimental results, we calculated non-equilibrium charge carrier concentrations as a function of optical pumping level. The study was conducted with the financial support of the Government of St. Petersburg, the Ministry of Education and Science of the Russian Federation (government task), an RFBR grant no. 16-02-00863, and a grant of the President of the Russian Federation for young candidates of science MK-4616.2016.2.