Analytic solutions of the linear Boltzmann transport equation for multilayered systems

Instead of hard solving the linear response Boltzmann transport equation in the relaxation time approximation to study electron transport, an equivalent solution for the electron distribution can be easily determined by use of the effective mean-free-path model proposed in this paper. In the effective mean-free-path model, the electron distribution is related to the effective mean free path, and the difference between the effective mean free path and the mean free path is attributed to the effective diffusive scattering in the system. Given boundary conditions, the effective diffusive scattering can be specified. The process to determine the effective diffusive scattering in the effective mean-free-path model can be transformed into a diagrammatical method with a clear physical insight. This determination can be easily written down and is always explicitly expressed in analytic forms. Therefore, the analytic solutions for the electron distributions can be obtained in multilayered systems with various shapes and with realistic boundary conditions. All solutions derived from the effective mean-freepath model can be proven equivalent to those obtained by use of solving the complicated coupled equations from the linear response Boltzmann transport equation in the relaxation time approximation.