Link between Intrinsic Spin Hall Effect and Geometrical Curvature of the Fermi Surface

Spin and Anamalous Hall effects (SHE and AHE) are when orthogonal spin and charge currents are generated from an applied E-field in the absence of external B-fields. Both are of great interest from both a fundamental physics perspective as well as for technological applications. Currently, few good SHE materials are known and a method for searching for new SHE materials is lacking. Here we present a new way of finding SHE materials based on using the geometrical interpretation of the Berry curvature, in analogy to previous work in the field of the AHE. First, we compute the electronic structure of a material with and without spin orbit coupling (SOC) and then combine that with analysis of the non-SOC Fermi surface. We show that areas of high fermi surface curvature w/o SOC and corresponding spin Berry curvature w/SOC came from areas in the electronic structure where anti-crossing points exist w/o SOC and are gapped by the inclusion of SOC. We explain, mathematically, why this must be the case and propose a route to a quantitative SHC calculation based on the Fermi surface curvature. Also, we show that even this qualitative understanding can be used to rapidly screen through prospective SHE materials and quickly find strong candidates for calculation using the Kubo formalism.