Multipole Representations of Spin-splitting Terms in Non-magnetic Materials

The band structures of non-centrosymmetric non-magnetic materials display relativistic spin-splitting, rendering them promising candidates for electrically controllable spintronic devices due to spin-momentum locking. While common lowest order terms have been studied in detail, higher-order terms remain little explored. In this study, starting from a centrosymmetric cubic point group, we classify all possible spin-splitting terms using irreducible representations (irreps), identifying terms that go beyond the commonly studied linear Rashba and Dresselhaus effects. We elucidate the connection between spin-orbit coupling terms and electric multipoles or toroidal moments, establishing the different lattice distortion patterns that give rise to the same multipoles. We also perform first-principles density functional theory (DFT) calculations on select materials to quantify the magnitude of the spin-splitting for each lattice distortion pattern.