Spin-Hall effect in topological materials: Evaluating the proper spin current in systems with arbitrary degeneracies

TL;DR

This paper introduces a new density functional theory method to accurately evaluate the conserved spin current in topological materials, addressing the challenge of spin non-conservation and arbitrary degeneracies, with implications for spintronics.

Contribution

The paper develops a novel method to compute the conserved spin current in systems with complex band structures, including all matrix elements of the position operator, improving upon previous approaches.

Findings

Proper spin current is generally smaller and can have a different sign than the conventional spin current.

The energy dependence of the proper spin current is flat and featureless across studied materials.

The magnitude of the proper spin current is comparable across topological insulators, 2D quantum spin-Hall insulators, antiferromagnets, and metals.

Abstract

The spin-Hall effect underpins some of the most active topics in modern physics, including spin torques and the inverse spin-Hall effect, yet it lacks a proper theoretical description. This makes it difficult to differentiate the SHE from other mechanisms, as well as differentiate band structure and disorder contributions. Here, by exploiting recent analytical breakthroughs in the understanding of the intrinsic spin-Hall effect, we devise a density functional theory method for evaluating the conserved (proper) spin current in a generic system. Spin non-conservation makes the conventional spin current physically meaningless, while the conserved spin current has been challenging to evaluate since it involves the position operator between Bloch bands. The novel method we introduce here can handle band structures with arbitrary degeneracies and incorporates all matrix elements of the position operator, including the notoriously challenging diagonal elements, which are associated with Fermi surface, group velocity, and dipolar effects but often diverge if not treated correctly. We apply this method to the most important classes of spin-Hall materials: topological insulators, 2D quantum spin-Hall insulators, non-collinear antiferromagnets, and strongly spin-orbit coupled metals. We demonstrate that the torque dipole systematically suppresses contributions to the conventional spin current such that, the proper spin current is generally smaller in magnitude and often has a different sign. Remarkably, its energy-dependence is relatively flat and featureless, and its magnitude is comparable in all classes of materials studied. These findings will guide the experiment in characterizing charge-to-spin interconversion in spintronic and orbitronic devices. We also discuss briefly a potential generalisation of the method to calculate extrinsic spin currents generated by disorder scattering.