Magnetoresistance and Hall coefficient of inhomogeneous metals

In strong magnetic fields, most metals have highly anisotropic transport coefficients, and these have long been known to be much influenced by sample inhomogeneities. This paper reports a detailed theoretical study of such effects. Various approximations for calculating the effective transport coefficients of inhomogeneous solids are rederived from a unified point of view; these are then used for a variety of model calculations appropriate to metals with open Fermi surfaces. A small concentration of crystallites with open orbits embedded in a free-electron metal is shown to give rise to a strictly linear transverse magnetoresistance (TMR) at strong magnetic fields. The linear coefficient is strongly dependent on the orientation of the open orbit in the plane perpendicular to the magnetic field $\overrightarrow{\mathrm{H}}$. Extended-orbit crystallites in a free-electron metal produce a TMR which is initially linear, but saturates at sufficiently strong field. The Hall coefficient $R_H$ is unchanged from its free-electron value to first order in the concentration of defects. A striking geometrical effect is predicted, the TMR from open-orbit crystallites saturating in geometries such that current distortions are unable to propagate parallel to $\overrightarrow{\mathrm{H}}$. The TMR and Hall coefficient of a free-electron metal containing a larger concentration of open-orbit crystallites is calculated in the effective-medium theory (EMT). The TMR is found to saturate at strong fields, in agreement with previous results of Stachowiak, while the Hall coefficient falls off as 1/${\mathrm{H}}^2$ at strong fields, except in flat-plate samples with $\overrightarrow{\mathrm{H}}$ perpendicular to the plate, in which case it is predicted to saturate at its free-electron value for a sufficiently large concentration of open-orbit crystallites, but to fall off quadratically for lower concentrations. In contrast, calculations within a non-self-consistent approximation give a strictly linear TMR and a Hall coefficient which saturates at a value below the free-electron coefficient. Possible explanations for the discrepancy are discussed. Calculations in the EMT for a model polycrystal with extended-orbit crystallites reveal a broad field region of quasi-linear magnetoresistance, as found previously by Stachowiak, and a reduced Hall coefficient, as well as a conspicuous geometrical effect. The possible relation of these model calculations to experiments of polycrystalline noble metals is examined, but no quantitative theory for these metals is given.