The nonlinear effect of resistive inhomogeneities on van der Pauw measurements Daniel W. Koon St. Lawrence University Canton, NY The “resistive weighting function”, f(x,y) [1], predicts the impact of local macroscopic inhomogeneities on van der Pauw resistivity measurements in the small-perturbation limit, assuming such effects to be linear. This talk will describe deviations from linearity for a square van der Pauw geometry, using an 11x11 grid network model of discrete resistors, and covering both positive and negative perturbations spanning two orders of magnitude in . The empirical expression provides a good fit over the entire range of both positive and negative changes in local resistivity. [1] D. W. Koon & C. J. Knickerbocker, Rev. Sci. Instrum. 63, 207 (1992).

Outline ► Background ► Model  A table-top, “analog computational” model of the square van der Pauw geometry. ► Experimental results ► Conclusions

The van der Pauw technique for resistivity measurement ► ► Allows for resistivity measurement from arbitrary geometry with arbitrary placement of current and voltage leads. ► ► Requires two independent resistance measurements (see figures above), but is often conducted using a single measurement. ► ► Sample thickness is the only relevant geometrical factor (not shape or lateral dimensions). ► ► But assumes uniform sample thickness & resistivity.

The resistive weighting function In general, the resistivity is not uniform, and so the measured resistivity,  m, is a weighted average of the local values of resistivity,  (x,y) : where f (x,y) is the “resistive weighting function”, a measure of the impact that a local inhomogeneity has on the measured resistivity.

Results for the square vdP geometry The resistive weighting function, f (x,y), for (a) single resistance measurement, (b) resistance measurement averaged over two independent readings. D. W. Koon & C. J. Knickerbocker, Rev. Sci. Instrum. 63 (1), 207 (1992).

Results for other vdP geometries: The resistive weighting function, f (x,y), averaged over two independent readings, for (a) cross, (b) cloverleaf. D. W. Koon & C. J. Knickerbocker, Rev. Sci. Instrum. 67 (12), 4282 (1996).

Systems studied ► Resistive weighting function  Square with electrodes at either edges or sides  Circular disc  Cross  Cloverleaf  Bar  4-point probe arrays (both linear and square) ► Hall weighting function (same geometries) ► Finite-thickness effects ► Experimental verification of both resistivity & Hall effect But, until now, no study of effect of large inhomogeneity.

Why should the effect of inhomogeneities be nonlinear? For a material of non-uniform resistivity, Locally tweaking the resistivity is equivalent to placing a dipole at that point which is parallel to and proportional to the local E-field. Since the perturbation alters the local E- field, - , we would expect this to be a nonlinear effect. D. W. Koon & C. J. Knickerbocker, Rev. Sci. Instrum. 63 (1), 207 (1992).

“Analog computer” model: ► Replace continuous 2D film with discrete individual resistors: ► Edge resistors: R ► Interior resistors: R/2 ► Experimental implementation:  11x11 grid array, with R/2=10k . (308 resistors, total)

Perturbation I: Decrease local resistivity (“short-like” impurities) ► Range:   = -1 to 0   = R/r = 0 to    = 0 to 1, where

Perturbation II: Increase local resistivity (“open-like” impurities) ► Range:   /  = 0 to    = - 1 to 0   = -1 to 0, where

Experimental results I: Weighting function in 11  11 grid The resistive weighting function f (x,y) (a) for a single measurement (b) for an average over two measurements. (Experimental results: 11x11 grid)

Experimental results II: Nonlinearity of f(x,y) at center ► Impact of “shorting out” the center of 11x11 grid on the measured resistance, R m, of the entire grid.

Experimental results III: Nonlinearity of the weighting function  Increasing  Decreasing   Fit curve (in white): Fit curve (in white):

Conclusions ► Large resistive inhomogeneities produce nonlinear effects.  Decreasing the local resistivity produces up to 3x the expected effect (compared to linear approximation).  Increasing the local resistivity produces less than the expected effect. ► These results are independent of location and well predicted by the expression:

Inconclusions (what’s next) ► Why this expression? ► What about additive effects? (simultaneous perturbation at 2+ locations)