What do you measure when you measure resistivity?

Abstract

Resistivity measurement is a weighted averaging of local resistivities. We develop a formalism to calculate the weighting function, applying it to square van der Pauw samples and to linear and square four-point probe arrays. In each case, some regions of the sample are negatively weighted, but these regions can be reduced or eliminated by van der Pauw averaging. We discuss negative weighting, which we feel is responsible for spurious reports of superconductivity above room temperature. We show how a square four-point array can be more effective at measuring local resistivity than a linear one. Finally, we show how to apply our formalism to anisotropic materials.

Sensitivity of resistive and Hall measurements to local inhomogeneities

“…This group's work showed in a rigorous fashion much of what had already been largely assumed by researchers: the advantages of using square specimens rather than circular ones, 18 of using cloverleafs and crosses rather than circular and square discs, 20 and of placing electrodes at the corners of a square specimen rather than along its edges. 18 Another group of researchers [DTU group], studying sheet resistance and Hall (micro-Hall) effect measurements with linear 4PP arrays, developed a complementary notation [24][25][26] and has numerically calculated sensitivities of measured configuration resistances, R i;m , to local variations of not only R S and R HS but also of the specimen's microscopic materials properties, such as sheet carrier density, N S , and mobility, l.…”

“…In earlier works [18][19][20][21][22][23] the SLU group has referred to the sensitivity functions calculated from Eq. (3) as the "resistivity weighting function" and the "Hall weighting function," stating that the latter was due to inhomogeneities in the Hall angle.…”

“…7,8 This problem vanishes for true vdP dual measurements. 18 For the remainder of this document we will focus mostly on singleconfiguration measurements since this method is very commonly used in research, due to simplicity.…”

“…5,9 One would expect similar issues with another four-wire measurement technique, the four-point-probe [4PP] approach, 3,4 which allows researchers to move an array of four point electrodes throughout the interior of a film, making it ideal for testing the uniformity of semiconductor wafers during micro-or nanofabrication. 10 The study of the effects of macroscopic inhomogeneity on charge transport properties dates back over sixty years to the study of Hall measurement sensitivity to inhomogeneous magnetic fields, [11][12][13][14][15][16][17] but more recently one group of researchers [SLU group], studying this problem for vdP geometries, defined, numerically calculated, and then directly measured what they have called resistive and Hall weighting functions, f and g, [18][19][20][21][22][23] for a variety of specimen shapes, quantifying the sensitivity of charge transport measurement to local inhomogeneities in R S and R HS . This group's work showed in a rigorous fashion much of what had already been largely assumed by researchers: the advantages of using square specimens rather than circular ones, 18 of using cloverleafs and crosses rather than circular and square discs, 20 and of placing electrodes at the corners of a square specimen rather than along its edges.…”

“…10 The study of the effects of macroscopic inhomogeneity on charge transport properties dates back over sixty years to the study of Hall measurement sensitivity to inhomogeneous magnetic fields, [11][12][13][14][15][16][17] but more recently one group of researchers [SLU group], studying this problem for vdP geometries, defined, numerically calculated, and then directly measured what they have called resistive and Hall weighting functions, f and g, [18][19][20][21][22][23] for a variety of specimen shapes, quantifying the sensitivity of charge transport measurement to local inhomogeneities in R S and R HS . This group's work showed in a rigorous fashion much of what had already been largely assumed by researchers: the advantages of using square specimens rather than circular ones, 18 of using cloverleafs and crosses rather than circular and square discs, 20 and of placing electrodes at the corners of a square specimen rather than along its edges. 18 Another group of researchers [DTU group], studying sheet resistance and Hall (micro-Hall) effect measurements with linear 4PP arrays, developed a complementary notation [24][25][26] and has numerically calculated sensitivities of measured configuration resistances, R i;m , to local variations of not only R S and R HS but also of the specimen's microscopic materials properties, such as sheet carrier density, N S , and mobility, l.…”