Phonon Boltzmann Transport Equation based modeling of time domain thermo-reflectance experiments

Abstract

Time Domain Thermo-Reflectance (TDTR) experiments have been recently identified as a viable pathway toward extracting the phonon mean free path spectrum of semiconductor materials. However, this requires an intervening model. It is now widely believed that the frequency and polarization dependent phonon Boltzmann Transport Equation (BTE) is the most suitable model for this purpose. In this article, TDTR experiments are simulated using large-scale parallel computations of the phonon BTE in a two-dimensional computational domain. Silicon is used as the candidate substrate material. Simulations are performed for multiple pulse and modulation cycles of the TDTR pump laser. This requires resolution of a picosecond laser pulse within a computational timeframe that spans several hundreds of nanoseconds. The metallic transducer layer on top of the substrate is modeled using the Fourier law and coupled to the BTE within the silicon substrate. Studies are conducted for four different laser spot sizes and two different modulation frequencies. The BTE results are fitted to the Fourier law, and effective thermal conductivities are extracted. It is demonstrated that the time delay of the probe laser could have a significant effect on the fitted (extracted) thermal conductivity value. The modulation frequency is found to have negligible effect on the thermal conductivity, while the spot size variation exhibits significant impact. Both trends are found to be in agreement with experimental observations. The thermal conductivity accumulation function is also computed, and the effect of the mean free path spectrum on the thermal conductivity suppression is delineated.

Tutorial: Time-domain thermoreflectance (TDTR) for thermal property characterization of bulk and thin film materials

“…The thermal conductivity of some semiconductor materials extracted from TDTR experiments was found to depend on the modulation frequency or the laser spot size, suggesting that the heat diffusion equation cannot accurately describe the experiment. A more rigorous practice is to analyze the TDTR data based on phonon Boltzmann transport equation, which, despite some recent progress, 110,111,215,228 inevitably complicates the data reduction process. as two examples of the potential candidates) or attempting to perform TDTR measurements without metal transducers at high temperatures.…”

“…Besides the work by Wilson and Cahill,112 there are also a bunch of other theoretical and numerical studies attempting to elucidate the nature of frequency and spot size dependences in TDTR experiments. [209][210][211][212][213][214][215] Interested readers can refer to these references.…”