Non-orthogonal Spin-Momentum Locking

Spin-momentum locking is a unique intrinsic feature of strongly spin-orbit coupled materials and a key to their promise of applications in spintronics and quantum computation. Much of the existing work, in topological and non-topological pure materials, has been focused on the orthogonal locking in the vicinity of the $\Gamma$ point where the directions of spin and momentum vectors are locked perpendicularly. With the orthogonal case, enforced by the symmetry in pure systems, mechanisms responsible for non-orthogonal spin-momentum locking (NOSML) have drawn little attention, although it has been reported on the topological surface of $\alpha$-$Sn$. Here, we demonstrate that, the presence of the spin-orbit scattering from dilute spinless impurities can produce the NOSML state in the presence of a strong intrinsic spin-orbit coupling in the pristine material. We also observe an interesting coupling threshold for the NOSML state to occur. The relevant parameter in our analysis is the deflection angle from orthogonality which can be extracted experimentally from the spin-and-angle-resolved photoemission (S-ARPES) spectra. Our formalism is applicable to all strongly spin-orbit coupled systems with impurities and not limited to topological ones. The understanding of NOSML bears on spin-orbit dependent phenomena, including issues of spin-to-charge conversion and the interpretation of quasiparticle interference (QPI) patterns as well as scanning-tunneling spectra (STS) in general spin-orbit coupled materials.