Intraband Auger processes and simple models of the ionization balance in semiconductor quantum-dot lasers

The importance of intraband Auger processes in determining the ionization balance in quantum dots is reported. The numerically inexpensive binary-encounter model for a Coulomb collision between identical particles is found to be a good estimator of the intraband Auger rates out of a quantum dot. Intraband and the conventional interband Auger processes differ in that the former involve only intraband transitions whereas the latter always involve a radiationless interband transition. As such, intraband Auger rates do not involve the evaluation of the very small overlap integral of a conduction band with a valence band Bloch wave function and are thus much larger than interband Auger rates, especially for large-band-gap semiconductors like GaAs. Though intraband Auger processes are not strong enough to establish a quasiequilibrium within the entire conduction band at the room-temperature free-carrier concentrations (${10}^{16}$ ${\mathrm{cm}}^{\mathrm{-}3}$) and bound energy separations (greater than an LO phonon energy) commonly assumed in the quantum-dot literature, they are capable of placing almost as many bound carriers in states near the band edge as would be predicted erroneously by a quasiequilibrium Fermi-Dirac distribution.

Such large bound state occupations are important for quantum-dot laser design. A sufficient condition for a quasiequilibrium to exist within all of an energy (conduction or valence) band is found to be the existence of many inverse Auger processes faster than interband spontaneous emission, which occurs for total (bound plus free) electron concentrations greater than 5×${10}^{17}$ ${\mathrm{cm}}^{\mathrm{-}3}$ at room temperature in 100 Å radius GaAs/${\mathrm{Al}}_{0.3}$${\mathrm{Ga}}_{0.7}$As quantum dots whose centers are separated by 400 Å. The nonlocal thermodynamic equilibrium populations in quantum dots can be understood from a simple model in which states connected by fast Auger or phonon processes are in Saha-Boltzmann equilibrium. All other states have occupation factors which are determined by the ratio of intraband collisional to interband radiative lifetimes, as described by a Fokker-Planck equation modeling diffusion in energy of bound particles.