Prediction of Exotic Electron Transport Properties using the Shape Effect of the Fermi Surface

The intrinsic anomalous Hall and spin Hall effects (AHE/SHE) provided evidence of exotic electronic transport phenomena relating to a topological connection between bands around the Fermi energy. Currently, the calculation of Berry curvature and the Kubo formalism based on Green functions is used to quantify bands’ topological connection and predict the AHE, SHE and other effects. Here we show that the topological connection of the bands also leads to particular shapes of the corresponding Fermi surfaces (FS) and that geometric analysis of these shapes can be used to predict transport properties. For the AHE/SHE we developed a qualitative indicator, Η_F, based on Gaussian curvature of the FS and found that Η_F is linearly correlated with the experimentally measured AHE (R-square 0.97) in a variety of famous AHE compounds as well correlated with the Kubo calculated SHE. We show that consideration of the geodesic flow on the FS gives rise to FS based equations in a semiclassical style, which contain information about the electron transport properties aside from longitudinal conduction. Going beyond just AHE/SHE, a full understanding of the Shape Effect of the FS (SEFS) opens the door to simple prediction of exotic electron transport property in materials in a novel and facile way.