Unraveling Four-Phonon Contributions to Thermoelectricity in Ba₄Sb₂Se and Ba₄Sb₂Te

We conduct a systematic study that couples first-principles lattice dynamics, including three- and four-phonon processes, with parameter-free electronic transport simulations to quantify the thermoelectric response of Ba₄Sb₂Se and Ba₄Sb₂Te. Four-phonon processes expand the anharmonic phase space and suppress the lattice thermal conductivity. Directional and doping trends mirror the band structure. Multiple valence and conduction band valleys increase the Seebeck coefficient while keeping the carrier mobility competitive. With three- and four-phonon processes, the peaks of the thermoelectric figure of merit at 700 K are 0.9/1.0 and 0.7/1.0 for Ba₄Sb₂Se and 0.9/1.2 and 0.9/1.3 for Ba₄Sb₂Te in the a- and c-directions, respectively, with each pair of values ordered as hole/electron. Altogether, the high thermoelectric figures of merit obtained show that the studied materials offer a promising mid-temperature thermoelectric platform. Explicitly treating four-phonon scattering and selecting the optimal carrier and transport direction are key.