Spin currents in crystals with spin-orbit coupling: multi-band effects in an effective Hamiltonian formalism

Abstract: When focusing on a few essential bands in an effective description of a material to calculate observable quantities, the respective operators have to be adjusted accordingly. Ignoring contributions arising from integrating out remote bands can lead to qualitatively wrong results. We present a detailed analysis of the interband mixing effects on spin currents. Specifically, we calculate the intrinsic spin current in a time-reversal invariant noncentrosymmetric crystal in the presence of electron-lattice spin-orbit coupling. Starting from formally exact microscopic expressions, we derive the spin current operator restricted to one or more essential bands by iterative elimination of the contributions from distant bands. We show that the standard definition of the spin current operator in terms of the group velocity obtained from an effective band Hamiltonian cannot be justified using a microscopic theory. The modified expression for the spin current operator contains additional terms, which dominate the equilibrium spin current in a uniform crystal. We show that the magnitude of these additional terms can considerably exceed the spin current obtained using the standard definition.