Effect of Random Inhomogeneities on Electrical and Galvanomagnetic Measurements

Abstract

This paper is designed to supplement the existing extensive literature on the conductivity of a randomly inhomogeneous medium, by treating the effects of inhomogeneities on piezoelectric, galvanomagnetic, and thermoelectric measurements. The scale of the inhomogeneities is supposed small compared with the dimensions of the specimen being measured, but large compared with mean free path, Debye length, etc. Formulas for all the effects are derived which are asymptotically exact in the limit of small fractional fluctuations in the local conductivity, etc. Comparison with other approximations and application to various exactly soluble cases show that these formulas are often roughly valid for quite sizable fluctuations. For material which, if uniform, would show a high field saturation of transverse magnetoresistance, the presence of appreciable inhomogeneities in the Hall constant will cause the magnetoresistance to increase indefinitely with field. This effect is due to the current distortions arising from the large and fluctuating Hall fields. For the special case of an isolated inclusion, these distortions are shown to extend, at high fields, to distances in the direction of the magnetic field which are many times the diameter of the inclusion. Under some conditions it appears that even the random distribution of impurities in a semiconductor can give rise to perceptible fluctuations on a scale large enough for concepts of macroscopic conduction to be applicable. Since fluctuations of even smaller scale are still larger, the total effect of fluctuations cannot be properly treated by the present methods; however, when the macroscopic part of the fluctuations is large, conventional impurity-scattering theories must be inadequate. Applications to polycrystalline metals and semiconductors are discussed briefly.