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Spin-orbit coupled electronic structure of two representative non-polarhalf-Heusler alloys, namely 18 electron compound CoZrBi and 8 electron compoundSiLiIn have been studied in details. An excursion through the Brillouin zone ofthese alloys from one high symmetry point to the other revealed rich localsymmetry of the associated wave vectors resulting in non-trivial spin splittingof the bands and consequent diverse spin textures in the presence of spin-orbitcoupling. Our first principles calculations supplemented with low energy$\boldsymbol{k.p}$ model Hamiltonian revealed the presence of linearDresselhaus effect at the X point having $D_{2d}$ symmetry and Rashba effectwith both linear and non-linear terms at the L point with $C_{3v}$ point groupsymmetry. Interestingly we have also identified non-trivial Zeeman spinsplitting at the non-time reversal invariant W point and a pair ofnon-degenerate bands along the path $\Gamma$ to L displaying vanishing spinpolarization due to the non-pseudo polar point group symmetry of the wavevectors. Further a comparative study of CoZrBi and SiLiIn suggest, in addition,to the local symmetry of the wave vectors, important role of the participatingorbitals in deciding the nature and strength of spin splitting. Ourcalculations identify half-Heusler compounds with heavy elements displayingdiverse spin textures may be ideal candidate for spin valleytronics where spintextures can be controlled by accessing different valleys around the highsymmetry k-points.

Effect of Spin Orbit Coupling in non-centrosymmetric half-Heusler alloys