Direct calculation of modal contributions to thermal conductivity via Green-Kubo modal analysis

We derived a new method for direct calculation of the modal contributions to thermal conductivity, which is termed Green-Kubo modal analysis (GKMA). The GKMA method combines the lattice dynamics formalism with the Green-Kubo formula for thermal conductivity, such that the thermal conductivity becomes a direct summation of modal contributions, where one need not define the phonon velocity. As a result, the GKMA method can be applied to any material/group of atoms, where the atoms vibrate around stable equilibrium positions, which includes non-stoichiometric compounds, random alloys, amorphous materials and even rigid molecules. By using molecular dynamics simulations to obtain the time history of each mode's contribution to the heat current, one naturally includes anharmonicity to full order and can obtain insight into the interactions between different modes through the cross-correlations. As an example, we applied the GMKA method to crystalline and amorphous silicon. The modal contributions at each frequency result from the analysis and thereby allow one to apply a quantum correction to the mode heat capacity to determine the temperature dependence of thermal conductivity. The predicted temperature dependent thermal conductivity for amorphous silicon shows the best agreement with experiments to date. The GKMA method provides new insight into the nature of phonon transport, as it casts the problem in terms of mode-mode correlation instead of scattering, and provides a general unified formalism that can be used to understand phonon-phonon interactions in essentially any class of materials or structures where the atoms vibrate around stable equilibrium sites.