Effect of spin-orbit coupling in noncentrosymmetric half-Heusler alloys

The spin-orbit coupled electronic structure of two representative nonpolar half-Heusler alloys, namely 18-electron compound CoZrBi and 8-electron compound SiLiIn, has been studied in detail. An excursion through the Brillouin zone of these alloys from one high-symmetry point to the other revealed rich local symmetry of the associated wave vectors resulting in nontrivial spin splitting of the bands and consequent diverse spin textures in the presence of spin-orbit coupling. Our first-principles calculations supplemented with the low-energy $\mathbit{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbit{p}$ model Hamiltonian revealed the presence of the linear Dresselhaus effect at the $X$ point having ${D}_{2d}$ symmetry, and the Rashba effect with both linear and nonlinear terms at the $L$ point with ${C}_{3v}$ point group symmetry. Interestingly we have also identified nontrivial Zeeman spin splitting at the non-time-reversal invariant $W$ point and a pair of nondegenerate bands along the path $\mathrm{\ensuremath{\Gamma}}$ to $L$ displaying vanishing spin polarization due to the non-pseudo-polar point group symmetry of the wave vectors. Further, a comparative study of CoZrBi and SiLiIn suggests, in addition to the local symmetry of the wave vectors, the important role of the participating orbitals in deciding the magnitude of the spin splitting of the bands. Our calculations suggest that half-Heusler compounds with heavy elements displaying diverse spin textures may be ideal candidates for spin valleytronics, where spin textures can be controlled by accessing different valleys around the high-symmetry $k$-points.