To develop new component concepts, like thermal sensors based on 2D effects [1], which can be realized on a basis of nanowires or superlattice structures, an appropriate understanding of the heat transport is essential. Thus, that the material parameters of those multi-layer components are very temperature-dependent, the heat transport has a strong influence onto the electronic and photonic functions of the device. In order to achieve an efficient device in terms of the temperature and energy management, the component needs to be designed in such a way, that the heat dissipation is low and fast [2].

Typically, a Boltzmann methodology is used to model such transport mechanisms [3]. But since this method often approximate the phonon dispersion through a power series focusing on a linear approximation, it results in an inaccurate description of the dispersion, especially at the edges of the first Brillouin zone. An inclusion of higher orders of the power series with regard to the dispersion relation could lead to a more precise approximation of the phonon transport. In particular regarding the diffusive heat transport, in this case the phonon-phonon scattering with occasionally occurring Umklapp-processes at the edges of the Brillouin zone can be described adequately.

In order to set up a heat transport model, firstly, a phonon Hamiltonian is introduced considering inversion symmetry with which a von Neumann type equation can be constructed in real space . The orders of the power series used to approximate the phonon dispersion is represented by the derivation order of the real space coordinates of the spatially dependent Hamiltonian. Utilizing a transformation onto center of mass coordinates  and after the application of the Wigner-transform a Wigner type formalism  in the phase space results. Through the inclusion of phonon-phonon scattering, a model for the diffusive heat transport can be realized.

This Wigner formalism leads to an intuitive approach and description of the ballistic and diffusive heat transport. Different approximation orders with respect to phonon dispersion are investigated and evaluated. A comparison is made with conventional Boltzmann approaches.

References

M.S. Dresselhaus, et al., New Directions for Low-Dimensional Thermoelectric Materials, Adv. Mater. 19, 1043-1053 (2007).
W. Kim, R. Wang, A. Majumdar, “Nanostructuring expands thermal limits”, Nano Today 2, 40-47 (2007).
Baoyi Hu, Wenlong Bao, Guofu Chen, Zhaoliang Wang, Dawei Tang, “Boltzmann transport equation simulation of phonon transport across GaN/AlN interface”, Computational Materials Science 230, 112485 (2023).