Phonon mean free path spectroscopy by Raman thermometry

In this work, we exemplify on a bulk silicon sample that Raman thermometry is capable of phonon mean free path (PMFP) spectroscopy. Our experimental approach is similar to the variation of different characteristic length scales $l_{\text{c}}$ during thermal reflectance measurements in the time or frequency domain (TDTR and FDTR) and transient thermal grating (TTG) spectroscopy. In place of $l_{\text{c}}$, we vary the laser focus spot size ($w_{\text{e}}$) and the light penetration depth ($h_{\alpha }$) during one-laser Raman thermometry (1LRT) measurements, enabling control over the size of the temperature probe volume $V$. For our largest $w_{\text{e}}$ values, the derived effective thermal conductivities ${\kappa }_{\text{eff}}$ converge towards the bulk thermal conductivity ${\kappa }_{\text{bulk}}$ for silicon, which we confirm by two-laser Raman thermometry and ab initio theory. However, towards smaller $w_{\text{e}}$ values, we observe a pronounced increase for the ${\kappa }_{\text{eff}}$ values, which amounts up to a factor of 5.3 at 293 K and even 8.3 at 200 K. We mainly assign this phenomenon to quasi-ballistic phonon transport and discuss any prominent impact of other factors. As a result, we can compare our measured ${\kappa }_{\text{eff}}\left(w_{\text{e}}\right)$ trends with the thermal accumulation function ${\kappa }_{\text{cum}}$ and its dependence on the phonon mean free path $l_{\text{ph}}$, which we derive from ab initio solutions of the linearized phonon Boltzmann transport equation (BTE). Since the variation of $w_{\text{e}}$ can be experimentally cumbersome, we also suggest varying $h_{\alpha }\left(\lambda \right)$ via the applied Raman laser wavelength $\lambda$ during 1LRT. In this regard, we present proof-of-principle 1LRT measurements, yielding a step-like ${\kappa }_{\text{eff}}\left(\lambda \right)$ trend for four different $\lambda$ values, which we also interpret in terms of quasi-ballistic phonon transport. Interestingly, we find that our ${\kappa }_{\text{eff}}\left(w_{\text{e}}\right)$ scaling is opposite to previous TTG results, which can be explained by the actual physical quantity probed. For small $w_{\text{e}}$ or $h_{\alpha }$ values, 1LRT mimics the situation of a local and/or surfacic heat source in a large matrix, which enables probing of real local $\kappa$ values that exceed ${\kappa }_{\text{bulk}}$. From a theoretical point of view, this was first predicted by Chiloyan et al., (2020), who calculated ${\kappa }_{\text{eff}}\left(w_{\text{e}}\right)$ for different initial phonon distributions in good agreement with our data. To show the generality of our findings, we probed ${\kappa }_{\text{eff}}\left(w_{\text{e}}\right)$ not only for bulk silicon, but also germanium, which is in-line with our previous findings on GaN, see Elhajhasan et al., (2023). Our results shall seed future PMFP spectroscopy based on 1LRT, which can directly be benchmarked against state-of-art theory to probe the effect of, e.g., any nano-structuring by comparison of ${\kappa }_{\text{cum}}$ trends and not only $\kappa$ values, aiming to test our understanding of the intricate phonon transport physics.