Compact formula for the density of states in a quantum well

In this paper we derive a formula for the density of states in the presence of inelastic scattering in the quantum well of a double-barrier structure as a function of a characteristic time of the motion of electrons (namely, the round-trip time in the well) and of transmission probabilities for the whole structure and for each barrier. In the model we use, the scattering processes due to phonons, impurities, and interface roughness are taken into account by a unique phenomenological parameter, the mean-free path, which plays the role of a relaxation length. We also show that, for lower rates of incoherent processes, the derived formula reduces to the one obtained by means of the Breit-Wigner formalism.