The quantum mechanical extension of the Boltzmann transport equation and optical absorption in semiconductors

Abstract

Free carrier absorption at frequencies just below the fundamental absorption edge in the semiconducting compounds InAs, InP, GaAs, and CdTe, requires a quantum mechanical treatment for its correct description. This frequency region, which lies in the near infrared, is characterized by a theoretically predicted and experimentally observed wavelength dependence of the absorption coefficient, which is close to λ 3. At sufficiently low frequencies, however, the quantum treatment reduces to the quasiclassical Boltzmann one, giving a λ 2 dependence of the absorption coefficient, which is tentatively indicated by experimental measurements on the same materials. This paper examines the intermediate and low frequency region of free carrier absorption, and the connection of the quantum theory with the quasiclassical Boltzmann equation in the low frequency limit, by deriving the quantum result for free carrier absorption with inelastic scattering from the generalized Boltzmann equation obtained by a density matrix approach. Use of the density matrix results in a generalized Boltzmann equation, which has been shown to reduce to the quasiclassical Boltzmann equation in the limit of elastic scattering and photon energies small compared with characteristic energies of the system. It is shown here that the generalized Boltzmann equation gives the results obtained by a quantum mechanical treatment based on time-dependent perturbation theory, and hence, gives the absorption coefficient, or optical conductivity, over the entire frequency range below the fundamental absorption edge. The rate equation for the photon occupation number, for direct interband and for indirect free carrier absorption, is shown to follow from the equation of motion of the quantum density matrix and use of the generalized Boltzmann equation.