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.
D. W. Koon and C. J. Knickerbocker, "What do you measure when you measure resistivity?", Rev. Sci. Instrum. 63 (1), 207-210 (1992).
A formalism for calculating the sensitivity of Hall measurements to local inhomogeneities of the sample material or the magnetic field is developed. This Hall weighting function g(x,y) is calculated for various placements of current and voltage probes on square and circular laminar samples. Unlike the resistivity weighting function, it is nonnegative throughout the entrie sample, provided all probes lie at the edge of the sample. Singularities arise in the Hall weighting function near the current and voltage probes except in the case where these probes are located at the corners of a square. Implications of the results for cross, clover, and bridge samples, and the implications of our results for metal-insulator transition and quantum Hall studies are discussed.
D. W. Koon and C. J. Knickerbocker, "What do you measure when you measure the Hall effect?", Rev. Sci. Instrum. 64 (2), 510-513 (1993).
The effect of macroscopic inhomogeneities on resistivity and Hall angle measurements is studied by calculating weighting functions (the relative effect of perturbations in a local transport property on the measured global average for the object) for cross, cloverleaf, and bar-shaped geometries. The "sweet spot", the region in the center of the object that the measurement effectively samples, is smaller for crosses and cloverleafs than for the circles and ssquares already studied, and smaller for the cloverleaf than for the correcponding cross. Resistivity measurements for crosses and cloverleafs suffer from singularities and negative weighting, which can be eliminated by averaging two independent resistance measurments, as done in the van der Pauw technique. Resistivity and Hall measurements made on suficiently narrow bars are shown to effectively sample only the region directly between the voltage probes.
D. W. Koon and C. J. Knickerbocker, "Effects of macroscopic inhomogeneities on resistive and Hall measurements on crosses, cloverleafs, and bars", Rev. Sci. Instrum. 67 (12), 4282-4285 (1996).
The authors extend their study of the effect of macroscopic impurities on resistive and Hall measurements to include objects of finite thickness. The effect of such impurities is calculated for a series of rectangular parallelepipeds with two current and two voltage contacts on the corners of one square face. The weighting functions display singularities near these contacts, but these are shown to vanish in the two-dimensional limit, in agreement with previous results. Finally, it is shown that while Hall measurements principally sample the plane of the electrodes, resistivity measurements sample more of the interior of an object of finite thickness.
D. W. Koon and C. J. Knickerbocker, "Resistive and Hall weighting functions in three dimensions", Rev. Sci. Instrum. 69 (10), 3625-7 (1998).
We have measured the resistivity weighting function -- the sensitivity of a four-wire resistance measurement to local variations in resistivity -- for a square specimen of photoconducting material. This was achieved by optically perturbing the local resistivity of the specimen while measuring the effect of this perturbation on its four-wire resistance. The weighting function we measure for a square geometry with electrical leads at its corners agrees well with calculated results, displayed two symmetric regions of negative weighting which disappear when van der Pauw averaging is performed.
D. W. Koon and Winston K. Chan, "Direct measurement of the resistivity weighting function", Rev. Sci. Instrum., 69 (12), 4218-0 (1998).
We have directly measured the Hall weighting function -- the sensitivity of a four-wire Hall measurement to the position of macroscopic inhomogeneities in Hall angle -- for both a square-shaped and a cloverleaf specimen. Comparison with the measured resistivity weighting function for a square geometry [D. W. Koon and Winston K. Chan, Rev. Sci. Instrum. 69, 12 (1998).] proves that the two measurements sample the same specimen differently. For Hall measurements on both a square and a cloverleaf, the function is nonegative with its maximum in the center and its minimum of zero at the edges of the square. Coverting a square into a cloverleaf is shown to dramatically focus the measurement process onto a much smaller portion of the specimen. While our results agree qualitatively with theory, details are washed out, owing to the finite size of the magnetic probe used.
Julia K. Scherschligt and D. W. Koon, " Measuring the Hall weighting function for square and cloverleaf geometries ", Rev. Sci. Instrum. 71, 587 (2000).
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