Measurement of the phonon mean free path spectra and the universality in the high temperature limit

Here, we use broadband frequency domain thermoreflectance (BB-FDTR) to measure thermal conductivity accumulation functions ($k_{\mathrm{accum}})$ of Si, GaAs, GaN, AlN, and SiC at temperatures of 80 K, 150 K, 300 K, and 400 K and show that they collapse to a universal accumulation function ($k_{\mathrm{univ}})$ in the high temperature limit. BB-FDTR is a novel technique developed to measure the spectral contributions of phonons to bulk thermal conductivity as a function of phonon MFP i.e., $k_{\mathrm{accum}}$. BB-FDTR uses a heterodyne approach allowing for continuous resolution of the phonon MFP spectrum spanning two orders of magnitude (0.3 - 8 $\mu $m in Si at $T \quad =$ 300 K). Results in Si and GaAs compare favorably to numerical predictions (Esfarjani, et al., PRB, 2011) (Luo et al., arXiv, 2012) and show that phonons with long MFPs (\textgreater 1 $\mu $m) contribute significantly to the bulk thermal conductivity at $T \quad =$ 300 K. Next, we present a method to predict $k_{\mathrm{accum}}$ as the temperature of the material approaches its Debye temperature. Using the measured spectra at $T \quad =$ 400 K and assuming Umklapp scattering as the dominant scattering mechanism, $k_{\mathrm{univ}}$ was found to exist in GaAs, GaN, and Si after normalizing the phonon MFP. The existence of $k_{\mathrm{univ}}$ suggests that the phonon MFP spectrum is a universal feature of matter in the high temperature limit, and can be used to predict $k_{\mathrm{accum}}$ for any crystalline semiconductor near its Debye temperature.