The Impact of Spin–Orbit Coupling on the Structural, Mechanical, Electronic, and Optical Properties of MCoBi (M = Ti, Zr, Hf) Half-Heusler Compounds: A FP-LAPW Study

In this work, the structural, mechanical, electronic, and optical properties of the 18-valence-electron half-Heusler compounds MCoBi (M = Ti, Zr, Hf) are investigated using the all-electron full-potential linearized augmented plane wave (FP-LAPW) method. The relativistic effect of spin–orbit coupling (SOC) is systematically incorporated and analyzed. Structural optimization confirms the stability of the Type-III phase, with lattice parameters in good agreement with available experimental data. Mechanically, the inclusion of SOC reduces both stiffness and brittleness, indicating a clear softening effect on the material behavior. SOC significantly lifts the degeneracy of the electronic bands, resulting in splitting (ΔSO) at the valence band maximum of approximately 0.10 eV and 0.12 eV for TiCoBi and ZrCoBi, respectively, which subsequently reduces the band gap; however, HfCoBi exhibits a remarkably weak splitting. The impact of SOC is further evidenced in the optical response across all three compounds. The absorption coefficient reaches high values (>105 cm−1) in the visible spectrum. Furthermore, a dramatic reduction in the intensity of plasmon resonance frequencies is observed, with values dropping to approximately 0.10–0.11 for the studied compounds. These findings highlight the potential of these materials for future electronic and optoelectronic device applications.