Magnetic Moment vs Angular Momentum: Spin Hall Response in Bismuth

Spin currents can carry either spin angular momentum or its associated magnetic moment, which are no longer strictly proportional in multiband systems. Using a multiband $k \cdot p$ model, we compute the intrinsic spin Hall conductivity tensors of elemental Bi. The magnetic-moment tensor emerges about two orders of magnitude larger and far less anisotropic than the angular-momentum tensor, while quasiparticle damping activates otherwise longitudinal components. The magnetic-moment spin Hall angle exceeds unity, demonstrating that a clear distinction between the two currents is indispensable for multiband systems.