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The nonlinear effect of resistive inhomogeneities on van der Pauw measurements Daniel W. Koon St. Lawrence University Ca

The nonlinear effect of resistive inhomogeneities on van der Pauw measurements Daniel W. Koon St. Lawrence University Canton, NY.

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The nonlinear effect of resistive inhomogeneities on van der Pauw measurements Daniel W. Koon St. Lawrence University Ca

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  1. The nonlinear effect of resistive inhomogeneitieson van der Pauw measurementsDaniel W. KoonSt. Lawrence UniversityCanton, 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 dr/r. 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).

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

  3. 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.

  4. The resistive weighting function In general, the resistivity is not uniform, and so the measured resistivity, rm , is a weighted average of the local values of resistivity, r(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.

  5. 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).

  6. 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).

  7. 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.

  8. 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, -F, we would expect this to be a nonlinear effect. D. W. Koon & C. J. Knickerbocker, Rev. Sci. Instrum. 63 (1), 207 (1992).

  9. “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=10kW. (308 resistors, total)

  10. Perturbation I:Decrease local resistivity(“short-like” impurities) • Range: • dr/r = -1 to 0 • ds/s = R/r = 0 to  • = 0 to 1, where

  11. Perturbation II:Increase local resistivity(“open-like” impurities) • Range: • dr/r = 0 to  • ds/s = - 1 to 0 • = -1 to 0, where

  12. 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)

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

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

  15. 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:

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

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