What do you measure when you measure the Hall effect?

Abstract

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 entire 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.

A Theoretical Study on Van Der Pauw Measurement Values of Inhomogeneous Compound Semiconductor Thin Films

“…If tendencies and degrees of deviations in measurements obtained by the van der Pauw method for inhomogenous samples from those of homogeneous samples are made clear, then the utility of the measurement method and reliability of the measurement values will be increased. There have been several reports on the influence of inhomogeneity on van der Pauw measurements of such unintentional inhomogeneous samples [6][7][8][9][10][11]. However, concrete values on such deviations have not been sufficiently reported.…”

Sensitivity of resistive and Hall measurements to local inhomogeneities

“…29 While we have derived Eqs. (4) and (5) for the case of a conducting plane of infinite area, the equations appear to be of general validity for all simply connected specimens, as calculating the functions on the right-hand side and comparing them to previously calculated (and experimentally measured 21,22 ) weighting functions for a both circular and square vdP [18][19][20] and both linear and square 4PP 18,24,25 geometries confirms. We have included two of these cases, a square vdP arrangement and both a linear and square 4PP array on an infinite conducting plane, in Figs.…”

“…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.…”

“…There are a few points, however, that we can easily map. The center of the circle, for example, maps to the center of the unit square, and so, 18,19 We can also map the diagonals (both y ¼ 0 and x ¼ 0) of the circular disc for the Hall weighting function …”

“…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.…”