Hydrodynamic theory for spatially inhomogeneous semiconductor lasers. I. A microscopic approach

Abstract

Starting from the microscopic semiconductor Bloch equations including the Boltzmann transport terms in the distribution function equations for electrons and holes, we derived a closed set of diffusion equations for carrier densities and temperatures with self-consistent coupling to Maxwell's equation and to an effective optical polarization equation. The coherent many-body effects are included within the screened Hartree-Fock approximation, while scatterings are treated within the second Born approximation including both the in- and out-scatterings. Microscopic expressions for electron-hole $(e$-$h)$ and carrier--LO-phonon $(c$-LO) scatterings are directly used to derive the momentum and energy relaxation rates. These rates, expressed as functions of temperatures and densities, lead to microscopic expressions for self- and mutual-diffusion coefficients in the coupled density-temperature diffusion equations. Approximations for reducing the general two-component description of the electron-hole plasma to a single-component one are discussed. In particular, we show that a special single-component reduction is possible when e-$h$ scattering dominates over $c$--LO-phonon scattering. The ambipolar diffusion approximation is also discussed and we show that the ambipolar diffusion coefficients are independent of e-$h$ scattering, even though the diffusion coefficients of individual components depend sensitively on the e-$h$ scattering rates. Our discussions lead to deeper insights into the roles played in the single-component reduction by the electron-hole correlation in momentum space induced by scatterings and the electron-hole correlation in real space via internal static electrical field. Finally, the theory is completed by coupling the diffusion equations to the lattice temperature equation and to the effective optical polarization, which in turn couples to the laser field.