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Materials based on cubic tetrahedrites (Cu$_{12}$Sb$_{4}$S$_{13}$) are usefulthermoelectrics with unusual thermal and electrical transport properties, suchas very low and nearly temperature-independent lattice thermal conductivity($\kappa_{L}$). We explain the microscopic origin of the glass-like$\kappa_{L}$ in Cu$_{12}$Sb$_{4}$S$_{13}$ by explicitly treating anharmonicityup to quartic terms for both phonon energies and phonon scattering rates. Weshow that the strongly unstable phonon modes associated with trigonallycoordinated Cu atoms are anharmonically stabilized above approximately $100$ Kand continue hardening with increasing temperature, in accord with experimentaldata. This temperature induced hardening effect reduces scattering of heatcarrying acoustic modes by reducing the available phase space for three-phononprocesses, thereby balancing the conventional $\propto T$ increase inscattering due to phonon population and yielding nearly temperature-independent$\kappa_{L}$. Furthermore, we find that very strong phonon broadening lead to aqualitative breakdown of the conventional phonon-gas model and modify thedominant heat transport mechanism from the particle-like phonon wave packetpropagation to incoherent tunneling described by off-diagonal terms in theheat-flux operator, which are typically prevailing in glasses and disorderedcrystals. Our work paves the way to a deeper understanding of glass-likethermal conductivity in complex crystals with strong anharmonicity.

Microscopic Mechanisms of Glass-Like Lattice Thermal Transport in Cubic Cu$_{12}$Sb$_{4}$S$_{13}$ Tetrahedrites