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

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### 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.

• 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.

• 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 Hall anglefor 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): Hall angle

• 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).

Nonlinearity of the weighting function Hall angle

 Increasing r Decreasing r

Fit curve (in white):

where a≈0.66 for entire specimen.

The problem (continued): Hall angle

• 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?

Solving for potential Hall anglenear 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.

Procedure Hall angle

• 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)

Small-angle limit: Hall angle

• |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.)

Small-angle results: Hall angle

• Nonlinearity varies across the specimen, depends on unperturbed Hall angle, QH.

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

Conclusions square

• 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°

• 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?