### Definition of the dominant phonon wavelength

Many physical interpretations of heat transport depend on the phonons wavelengths compared to the geometrical dimensions of the system. While it is relatively easy to characterize the dimensions by microscopy imaging (SEM, AFM, ...), characterize all the spectrum of phonons wavelengths is very challenging.

We are theoritically revisite the concept of dominant phonon wavelength. The idea is to use the wavelength of the phonon spectrum which contributes the most to the heat transport to identify the main physical phenomenon controlling heat transport. The dominant phonon wavelength, even though simplistic, is widely used because it is easily available and gives a good approximation.

### System at temperature T

When the system is at a temperature T given by a thermal reservoir, the dominant phonon wavelength \(\lambda_d\) is given by the maximum of the Planck distribution for phonons. The Planck distribution for phonons is the heat energy spectrum, equivalent to the theory of the blackbody radiations. The expression is

\(\lambda_d =\frac{h c}{\alpha k_B T}\)

where \(h\), \(c\) and \(k_B\) are the Planck constant, the sound velocity and the Boltzmann constant, respectively. \(\alpha\) is also a constant which is the position of the maximum. For 3D a system \(\alpha=2.82\) and for 2D a system \(\alpha=1.59\).

### System between two thermal reservoirs

When the system is placed between two thermal reservoirs, the two phonon populations interact to get a net heat flux from the hot reservoir at \(T_h\) to the cold reservoir at \(T_c\) which modify the Planck distrubtion. Then, the dominant phonon wavelength increases when the temperature difference decreases; a phenomon we called phonon blueshif.

The generalized expression of \(\lambda_d\) can still be written as before but T corresponds to \(T_h\) and \(\alpha\) now depends on both \(T_h\) and \(T_c\) (see paper below).

Further information: Ramiere A., Volz S. & Amrit J. Heat flux induced blueshift of dominant phonon wavelength and its impact on thermal conductivity. AIP Advances **7**, 015017 (2017)