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The problem:

Nonlinearity in the effect of an inhomogeneous Hall angle Daniel W. Koon St. Lawrence University Canton, NY.

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The problem:

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  1. Nonlinearity in the effect of an inhomogeneous Hall angle Daniel W. KoonSt. Lawrence UniversityCanton, NY The differential equation for the electric potential in a conducting material with an inhomogeneous Hall angle is extended outside the small-field limit. This equation is solved for a square specimen, using a successive over-relaxation [SOR] technique, and the Hall weighting function g(x,y) -- the effect of local pointlike perturbations on the measured Hall angle -- is calculated as both the unperturbed Hall angle, QH, and the perturbation, dQH, exceed the linear, small angle limit. In general, g(x,y) depends on position and on both QH, and dQH.

  2. The problem: • Process of charge transport measurement averages local values of r and QH. • They are weighted averages. • Weighting functions have been studied, quantified for variety of geometries. • All physical specimens are inhomogeneous. Knowledge of weighting function helps us choose best measurement geometry.

  3. Weighting functions for square vdP geometry: resistivity and Hall angle • Single-measurement resistive weighting function is negative in places. • Hall weighting function is broader than resistive weighting function. (a) Resistivity: D. W. Koon & C. J. Knickerbocker, Rev. Sci. Instrum. 63 (1), 207 (1992); (b) Hall effect: D. W. Koon & C. J. Knickerbocker, Rev. Sci. Instrum. 64 (2), 510 (1993).

  4. Hall weighting function for other van der Pauw geometries: • Hall weighting function, g(x,y), for (a) cross, (b) cloverleaf. • Both geometries focus measurement onto a smaller central region. D. W. Koon & C. J. Knickerbocker, Rev. Sci. Instrum. 67 (12), 4282 (1996).

  5. The problem (continued): • These results based on linear assumption, i.e. that the perturbation does not alter the E-field lines. • Nonlinear results (and empirical fit) have been obtained for resistivity measurement on square van der Pauw geometry. • D. W. Koon, “The nonlinearity of resistive impurity effects on van der Pauw measurements", Rev. Sci. Instrum., 77, 094703 (2006).

  6. Nonlinearity of the weighting function  Increasing r Decreasing r Fit curve (in white): where a≈0.66 for entire specimen.

  7. The problem (continued): • Nonlinear results have been obtained for resistivity measurement on square van der Pauw geometry. • Nonlinearity can be modeled by simple, one-parameter function for entirespecimen • What about Hall weighting function? • Simple formula? • Nonlinearity depend on position? • Nonlinearity depend on unperturbed Hall angle?

  8. Solving for potential near non-uniform Hall angle: QH <<1: General case: Small perturbation is equivalent to point dipole perpendicular to and proportional to local E-field. Linear. But the perturbation changes the local E-field. Therefore there is a nonlinear effect.

  9. Procedure • Solve difference-equation form of modified Laplace’s Equation on 21x21 matrix in Excel by successive overrelaxation [SOR]. • Verify selected results on 101x101 grids. • Apply pointlike perturbation of local Hall angle as function of… • size of perturbation (|dQH| <45º) • location of perturbation • unperturbed Hall angle (|QH| <45º)

  10. Small-angle limit: • |QH|, |dQH|  2°. (B=¼T for pure Si @ RT) • Results were fit to the quadratic expression: • Linear terms, a1 and b0 are plotted vs position of perturbation. (Nonlinearity depends on QH if and only if a1≠0.)

  11. Small-angle results: • Nonlinearity varies across the specimen, depends on unperturbed Hall angle, QH.

  12. Larger-angle results: Hall weighting function at center of square

  13. Empirical fit for center of square

  14. Results: Hall weighting function:center (11,11), edge (3,11), corner (3,3)

  15. Conclusions • No simple expression for Hall nonlinearity. • Depends on position, (x,y) • Depends on both unperturbed Hall angle, QH, and perturbation, dQH • Weighting function blows up as |dtanQH| • For center of square, empirical fit found for |tanQH|<45°

  16. Inconclusions (What’s next?) • Is there a general expression for how the Hall weighting function varies with respect to • Unperturbed Hall angle, tanQH • Perturbation, dtanQH • Location, (x,y), of perturbation either in the small-angle limit or in general? • Can results be extended to |QH|, |dQH|>45°? • How do two simultaneous point perturbations interact?

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