Superdiffusive heat conduction in semiconductor alloys -- I. Theoretical foundations

Semiconductor alloys exhibit a strong dependence of effective thermal conductivity on measurement frequency. So far this quasi-ballistic behaviour has only been interpreted phenomenologically, providing limited insight into the underlying thermal transport dynamics. Here, we show that quasi-ballistic heat conduction in semiconductor alloys is governed by L\'evy superdiffusion. By solving the Boltzmann transport equation (BTE) with ab initio phonon dispersions and scattering rates, we reveal a transport regime with fractal space dimension $1 < \alpha < 2$ and superlinear time evolution of mean square energy displacement $\sigma^2(t) \sim t^{\beta} (1 < \beta < 2)$. The characteristic exponents are directly interconnected with the order $n$ of the dominant phonon scattering mechanism $\tau \sim \omega^{-n} (n>3)$ and cumulative conductivity spectra $\kappa_{\Sigma}(\tau;\Lambda)\sim (\tau;\Lambda)^{\gamma}$ resolved for relaxation times or mean free paths through simple relations $\alpha = 3-\beta = 1 + 3/n = 2 - \gamma$. The quasi-ballistic transport inside alloys is no longer governed by Brownian motion, but instead dominated by L\'evy dynamics. This has important implications for the interpretation of thermoreflectance (TR) measurements with modified Fourier theory. Experimental $\alpha$ values for InGaAs and SiGe, determined through TR analysis with a novel L\'evy heat formalism, match ab initio BTE predictions within a few percent. Our findings lead to a deeper and more accurate quantitative understanding of the physics of nanoscale heat flow experiments.