Phonon-transmission rate, fluctuations, and localization in random semiconductor superlattices: Green’s-function approach

We analytically study phonon transmission and localization in random superlattices by using a Green's-function approach. We derive expressions for the average transmission rate and localization length, or Lyapunov exponent, in terms of the superlattice-structure factor. This is done by considering the backscattering of phonons, due to the complex mass-density fluctuations, which incorporates all of the forward-scattering processes. These analytical results are applied to two types of random superlattices and compared with numerical simulations based on the transfer-matrix method. Our analytical results show excellent agreement with the numerical data. A universal relation for the transmission fluctuations versus the average transmission is derived explicitly, and independently confirmed by numerical simulations. The transient of the distribution of transmission to the log-normal distribution for the localized phonons is also studied.