Sensitivity of resistive and Hall measurements to local inhomogeneities

We derive exact, analytic expressions for the sensitivity of resistive and Hall measurements to local inhomogeneities in a specimen's material properties in the combined linear limit of a weak perturbation over an infinitesimal area in a small magnetic field. We apply these expressions both to four-point probe measurements on an infinite plane and to symmetric, circular van der Pauw discs, obtaining functions consistent with published results. These new expressions speed up calculation of the sensitivity for a specimen of arbitrary shape to little more than the solution of two Laplace equation boundary-value problems of the order of N³ calculations, rather than N² problems of total order N⁵, and in a few cases produces an analytic expression for the sensitivity. These functions provide an intuitive, visual explanation of how, for example, measurements can predict the wrong carrier type in n-type ZnO.