Atomistic origin of low thermal conductivity in quaternary chalcogenides Cu(Cd, Zn)$_2$InTe$_4$

Crystalline semiconductors with intrinsically low lattice thermal conductivity ($\mathcal{K}$) are vital for device applications such as barrier coatings and thermoelectrics. Quaternary chalcogenide semiconductors such as CuCd$_2$InTe$_4$ and CuZn$_2$InTe$_4$ are experimentally shown to exhibit low $\mathcal{K}$, yet its microscopic origin remains poorly understood. Here, we analyse their thermal transport mechanisms using a unified first-principles framework that captures both the Peierls (particle-like propagation, $\mathcal{K}_P$) and coherence (wave-like tunneling, $\mathcal{K}_C$) mechanisms of phonon transport. We show that extended antibonding states below the Fermi level lead to enhanced phonon anharmonicity and strong scattering of heat-carrying phonon modes, suppressing $\mathcal{K}$ in these chalcogenides. We show that $\mathcal{K}_P$ dominates the total thermal conductivity, while $\mathcal{K}_C$ remains negligible even under strong anharmonicity of the phonon modes. The heavier Cd ions in CuCd$_2$InTe$_4$ induce greater acoustic-optical phonon overlap and scattering compared to CuZn$_2$InTe$_4$, further lowering thermal conductivity of the former. Additionally, grain boundary scattering in realistic samples contributes to further suppression of thermal transport. Our findings establish the atomistic origins of low $\mathcal{K}$ in quaternary chalcogenides and offer guiding principles for designing low-thermal-conductivity semiconductors.