New exact results for the Hall coefficient and magnetoresistance of inhomogeneous two-dimensional metals

Abstract

Several new theorems are proven for the magnetotransport coefficients of inhomogeneous normal-superconducting ($\frac{N}{S}$) and normal-insulating ($\frac{N}{I}$) films in a perpendicular magnetic field, under the assumption that macroscopic quantum effects such as Josephson coupling can be neglected. For $\frac{N}{S}$ composites, at any field $H$, the Hall coefficient $R_H\mathrm{}(p,H)\mathrm{}\propto {(p_c-p)}^{2s}$ and the resistivity $\rho \mathrm{}(p,H)\mathrm{}\propto {(p_c-p)}^s$ where $p$ is the volume fraction of superconductor and $p_c$ is the percolation threshold. For $\frac{N}{I}$ composites, as long as there is a connected metallic path across the sample, we show that both $R_H\mathrm{}(p,H)\mathrm{}$ and the ratio $\frac{\rho \mathrm{}(p,H)\mathrm{}}{\rho \mathrm{}(p,0\mathrm{})\mathrm{}}$ are independent of $p$ and equal to the corresponding quantities for the normal metal.