Temperature-dependent band renormalization in half-Heusler <i>M</i> NiSn and <i>M</i> CoSb ( <i>M</i>  = Ti, Zr, Hf)

Temperature-dependent band structures are crucial for optimizing the thermoelectric performance of half-Heusler materials operating at elevated temperatures. Here, we systematically investigate the band structure evolution of MCoSb and MNiSn (M = Ti, Zr, Hf) by explicitly incorporating electron–phonon renormalization, including lattice expansion and phonon vibrations. The results show that bandgaps decrease monotonically with increasing temperature (e.g., from 0 to 1000 K, the reductions in bandgap values are ∼0.27 eV for TiCoSb and ∼0.15 eV for TiNiSn, respectively), predominantly driven by phonon vibrations. Pronounced valence-band convergence is observed in p-type MCoSb at operating temperatures above 600 K, whereas no conduction-band convergence occurs in n-type MNiSn up to 1000 K. Intermediate-frequency phonons reduce bandgaps in both MCoSb and MNiSn, while high-frequency phonons have a stronger impact in MCoSb. This work provides clear insights into the temperature-dependent band structure evolution of half-Heusler thermoelectric materials.