Title: Hierarchical Characterization of Thermoelectric Performance in Copper-Based Chalcogenide CsCu$_3$S$_2$: Unveiling the role of Anharmonic Lattice Dynamics

Abstract: We explicitly consider both phonon energy shifts and broadening arising from both cubic and quartic anharmonicities, as well as diagonal/non-diagonal terms of heat flux operators in thermal conductivity. Our findings show that the strong anharmonicity of CsCu$_3$S$_2$ primarily arises from the presence of $p$-$d$ anti-bonding hybridization between Cu and S atoms, coupled with the random oscillations of Cs atoms. Notably, the competition between phonon hardening described by the loop diagram and softening induced by the bubble diagram significantly influences particle-like propagation, predominantly reflected in group velocity and energy-conservation rule. Additionally, the electrical transport properties are determined by employing the precise momentum relaxation-time approximation (MRTA). At high temperatures, the thermoelectric performance of $p$-type CsCu$_3$S$_2$ reaches its optimum theoretical value of 0.94 along the in-plane direction based on advanced phonon renormalization theory. In striking contrast, the harmonic approximation theory significantly overestimates the thermoelectric efficiency at the same temperatures, rendering it an impractical expectation. Conversely, the first-order renormalization approach leads to a serious underestimation of the thermoelectric properties due to the over-correction of phonon energy. Our study not only reveals the pivotal role of anharmonic lattice dynamics in accurately assessing thermoelectric properties but also underscores the potential thermoelectric applications for novel copper-based chalcogenides.