Geometric interpretation of the weak-field Hall conductivity in two-dimensional metals with arbitrary Fermi surface

The Hall conductivity xy of a two-dimensional metal in the weak-field, semiclassical, limit has a simple geometric representation. xy (normalized to e2/h, where e is the electron charge and h is Plancks constant), is equal to twice the number of flux quanta 0 threading the area Al, where Al is the total Stokes area swept out by the scattering path length l(k) as k circumscribes the Fermi surface (FS). From this perspective, many properties of xy become self-evident. The representation provides a powerful way to disentangle the distinct contributions of the three factors, FS area-to-circumference ratio, anisotropy in lk, and negative FS curvature. The analysis is applied to the Hall data on 2H-NbSe2 and the cuprate perovskites. Previous model calculations of xy are critically reexamined using the new representation.