### Mean Free Path Spectroscopy

Figure 1. Transition from diffusive
to ballistic heat transfer.

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**Figure 2. Example measurement from MFP spectroscopy: thermal conductivity accumulation
distribution
measurements and calculations
versus MFP for silicon ^{1}. **

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**Figure 3. Example reconstructed MFP distribution
obtained numerically ^{3}.**

Knowledge of phonon mean free paths (MFPs) is essential to understanding and engineering size effects in nanoscale heat transfer. Because they largely determine the thermal conductivity of materials and the figure of merit in thermoelectric devices, means of measuring this quantity is especially important for scientific and practical purposes.

Our lab has developed the first experimental method, mean free path spectroscopy, that is able to measure MFPs over a wide range of
length scales and materials^{1}. The technique uses
observations of quasi-ballistic heat transfer, in which thermal
lengths are comparable to MFPs, to determine how much different
phonons contribute to the thermal conductivity as a function of
their MFP. Fig. 1 illustrates the transition from diffusive to
ballistic heat transfer when the size of a heater is decreased.

To implement this technique, an experiment with spatial resolution comparable to MFPs (nanometers to microns) is required. Optical techniques, which offer micron spatial resolution and sub-picosecond time resolution, are well suited for this purpose. We employ several ultrafast optical techniques, including traditional TDTR and TG techniques (see optical techniques).

The experiments yield several thermal conductivity values
that vary with the length scale used to perform the
measurements^{1}. For example, Fig. 2 shows measurements
of silicon MFPs at cryogenic temperatures (symbols) compared to
the predictions of first-principles calculations.

Until
recently, how these measurements are related to the actual MFP
distribution was unclear. However, we have showed that the MFP
distribution can be recovered by solving a common inverse
problem using experimental results as input^{2}. A suppression function that is needed for this inversion is obtained computationally by solving the Boltzmann Transport Equation (BTE) using a variety of methods such as finite differences and Monte Carlo. Recently, we have introduced a theoretical framework based on the BTE that allows the MFP accumulation distribution to be quantitatively reconstructed from thermal measurements from TG experiments^{3}. Fig. 3 shows a numerical example of the reconstruction procedure to obtain the MFP distribution. Thus by using experiment and computation, we are able to measure MFPs in semiconductors, thermoelectrics, and other technologically important materials.

**References**