Nonlinearity in the effect of an inhomogeneous Hall angle Daniel W. Koon St. Lawrence University Canton, 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,  H, and the perturbation,  H, exceed the linear, small angle limit. In general, g(x,y) depends on position and on both  H, and  H.

The problem: ► Process of charge transport measurement averages local values of  and  H. ► 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.

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).

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).

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, (2006).

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

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 entire specimen ► What about Hall weighting function?  Simple formula?  Nonlinearity depend on position?  Nonlinearity depend on unperturbed Hall angle?

Solving for potential near non-uniform Hall angle:  H <<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.

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 (|  H | < 45º )  location of perturbation  unperturbed Hall angle (|  H | < 45º )

Small-angle limit: ► |  H |, |  H |  2°. (B=¼T for pure RT) ► Results were fit to the quadratic expression: ► Linear terms,  1 and  0 are plotted vs position of perturbation. (Nonlinearity depends on  H if and only if  1 ≠0.)

Small-angle results: ► Nonlinearity varies across the specimen, depends on unperturbed Hall angle,  H.

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

Empirical fit for center of square

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

Conclusions ► No simple expression for Hall nonlinearity.  Depends on position, (x,y)  Depends on both unperturbed Hall angle,  H, and perturbation,  H ► Weighting function blows up as |  tan  H |  ► For center of square, empirical fit found for |tan  H |<45°

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