Quantum mechanical prediction of four-phonon scattering rates and reduced thermal conductivity of solids

Recently, first principle-based prediction of lattice thermal conductivity $\kappa$ from the perturbation theory has achieved significant success. However, it only includes three-phonon scattering due to the assumption that four-phonon and higher-order processes are generally unimportant. Also, directly evaluating the scattering rates of four-phonon and higher-order processes has been a long-standing challenge. In this work, however, we have developed a formalism to explicitly determine quantum mechanical scattering probability matrices for four-phonon scattering in the full Brillouin Zone, and by mitigating the computational challenge we have directly calculated four-phonon scattering rates. We find that four-phonon scattering rates are comparable to three-phonon scattering rates at medium and high temperatures, and they increase quadratically with temperature. As a consequence, $\kappa$ of Lennard-Jones argon is reduced by more than 60% at 80 K when four-phonon scattering is included. Also, in less anharmonic materials -- diamond, silicon, and germanium, $\kappa$ is still reduced considerably at high temperature by four-phonon scattering. Also, the thermal conductivity of optical phonons is dominated by the fourth and higher orders phonon scattering even at low temperature.