Quantum-Mechanical Rate Equations for Semiconductor Lasers

Abstract

Quantum-mechanical rate equations are derived for semiconductor lasers (SL). Fluctuation operators with shot-noise character describe the quantum nature of the transitions. These equations are treated in the high-temperature limit for pure and highly doped III-V compound semiconductors. Numerical calculations are carried out for GaAs. From the mean rate equations we determine (a) the temperature dependence of the threshold pump rate for pure bulk SL and the threshold current for SL junctions and (b) the temperature and pump dependences of the mean light intensity and of the mean quasi-Fermi-level. By linearizing the fluctuations around the mean values, the noise spectrum for the light intensity is obtained. The general form of the noise spectrum is the same as that obtained by McCumber for a four-level laser system. Above threshold a sharp resonance is found in the GHz region. The temperature and pump dependences of the spectrum and especially of the resonance frequency are calculated in detail. The results from the mean equations and from the noise calculations which are obtained for highly doped GaAs are compared with experimental results for junction lasers, and good agreement is found. For pure SL the present numerical results are in good agreement with former analytical results of Haug and Haken for the mean intensity and for the low-frequency part of the noise spectrum, which have been found for the regions below and above threshold. The results for pure bulk SL are applicable to experiments with optical or electron-beam excitation.