  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.
