Phonon transport in disordered alloys: A Multiple-scattering approach

Abstract

I present a formalism to calculate the configuration-averaged lattice thermal conductivity for substitutional random alloys. The method is based on multiple-scattering approach which can capture the effect of disorder-induced configuration fluctuations on single-particle and two-particle phonon Green functions in random alloys. The randomness of the system is dealt within the augmented space theorem. This is combined with a generalized Feynman diagrammatic technique to extract various useful results in the form of mathematical expressions. I show the structure of all possible scattering diagrams up to the fourth order and subsequently illustrate how to obtain Dyson's equation from a resummation of the diagrammatic series. I also study how disorder scattering affects two-particle Green functions associated with thermal response. It was shown explicitly how the disorder scattering renormalizes both the phonon propagators as well as the heat currents. I derive the relation between these renormalized heat currents and the self-energy of the propagators. I have also studied a different class of scattering diagrams which are not related to the self-energy but rather to the vertex corrections. The configuration-averaging scheme is straightforward to apply to other relevant quantities such as joint density of states, thermal diffusivity, etc., for random alloys. The developed formalism is applied to a realistic ${\mathrm{Au}}_{1\ensuremath{-}x}{\mathrm{Fe}}_{x}$ binary alloy. The effect of disorder-induced corrections (as compared to the simple virtual crystal approximation) turns out to be appreciable in this alloy. The configuration-averaged lattice thermal conductivity for ${\mathrm{Au}}_{50}{\mathrm{Fe}}_{50}$ shows a quadratic behavior in low-temperature regime ($T\ensuremath{\le}30$ K), which increases smoothly to a $T$-independent saturated value at high $T$. Simulated thermal diffusivity $D(\ensuremath{\nu})$ helps to numerically estimate the mobility $\mathrm{edge}\phantom{\rule{0.28em}{0ex}}({\ensuremath{\nu}}_{c})$ which in turn evaluates the fraction of localized states. $D(\ensuremath{\nu})$ is found to decrease smoothly (almost linearly) in the high-$\ensuremath{\nu}$ range, which when fitted to ${({\ensuremath{\nu}}_{c}\ensuremath{-}\ensuremath{\nu})}^{\ensuremath{\alpha}}$ gives the critical $\mathrm{exponent}\phantom{\rule{0.28em}{0ex}}(\ensuremath{\alpha})$ to be 1.018 for ${\mathrm{Au}}_{50}{\mathrm{Fe}}_{50}$ alloy. This agrees fairly well with the scaling and other theories of Andersen localization.