Kinetic theory of semiconductor cascade laser based on quantum wells and wires

The paper presents a numerical solution of a system of nonlinear equations for the electron distribution functions in the upper and lower subbands between which lasing transitions occur and the number of nonequilibrium optical phonons in semiconducting cascade lasers based on quantum wells and wires. For the case of quantum wells, we propose an analytical solution of this system of equations, which is a generalization of the previously found solution [V. F. Elesin and Yu. V. Kopaev, Zh. Éksp. Teor. Fiz. 108, 2186 (1995) [JETP 81, 1192 (1995)]; V. F. Elesin and Yu. V. Kopaev, Sol. St. Commun. 96, 897 (1995)] in a wider range of injection rates. The threshold injection rate can be significantly reduced owing to reabsorption and accumulation of nonequilibrium optical phonons, nonparabolicity of the subbands and different effective masses of electrons in different subbands. In the case of quantum wires, the threshold injection rate is considerably lower, and its decrease is even larger than in quantum wells. It is remarkable that, owing to the lower electron-electron relaxation rate in the one-dimensional case, the decrease in the threshold injection rate may be two or three orders of magnitude. The relation between the density of states and threshold current has also been studied.