Nonlinearity in the effect of an Nonlinearity in the effect of an inhomogeneous Hall angle inhomogeneous Hall angle

Daniel W. KoonDaniel W. KoonSt. Lawrence UniversitySt. Lawrence University

Canton, NYCanton, NY

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

The problem:The problem:

► Process of charge transport measurement Process of charge transport measurement averages averages local values of local values of and and HH..

► They are They are weighted weighted averages.averages.

► Weighting functions Weighting functions have been studied, have been studied, quantified for variety of geometries.quantified for variety of geometries.

► All physical specimens are inhomogeneous. All physical specimens are inhomogeneous. Knowledge of Knowledge of weighting functionweighting function helps us helps us choose best measurement geometry.choose best measurement geometry.

Weighting functions for square vdP Weighting functions for square vdP geometry: resistivity and Hall anglegeometry: resistivity and Hall angle

Single-measurement resistive weighting function is negative in places.Single-measurement resistive weighting function is negative in places. Hall weighting function is broader than resistive weighting function.Hall weighting function is broader than resistive weighting function.

(a)(a) Resistivity: D. W. Koon & C. J. Knickerbocker, Rev. Sci. Instrum. 63 (1), 207 (1992); Resistivity: D. W. Koon & C. J. Knickerbocker, Rev. Sci. Instrum. 63 (1), 207 (1992); (b)(b) Hall effect: D. W. Koon & C. J. Knickerbocker, Rev. Sci. Instrum. 64 (2), 510 (1993). Hall effect: D. W. Koon & C. J. Knickerbocker, Rev. Sci. Instrum. 64 (2), 510 (1993).

Hall weighting function Hall weighting function for other van der Pauw geometries:for other van der Pauw geometries:

Hall weighting function, Hall weighting function, gg((x,yx,y), for (a) cross, (b) ), for (a) cross, (b) cloverleaf. cloverleaf.

Both geometries focus measurement onto a smaller Both geometries focus measurement onto a smaller central region.central region.D. W. Koon & C. J. Knickerbocker, Rev. Sci. Instrum. 67 (12), 4282 (1996).D. W. Koon & C. J. Knickerbocker, Rev. Sci. Instrum. 67 (12), 4282 (1996).

The problem (continued):The problem (continued):

► These results based on These results based on linearlinear assumption, assumption, i.e. that the perturbation does not alter the i.e. that the perturbation does not alter the E-field lines.E-field lines.

► NonlinearNonlinear results (and empirical fit) have results (and empirical fit) have been obtained for resistivity measurement been obtained for resistivity measurement on square van der Pauw geometry.on square van der Pauw geometry. D. W. Koon, “The nonlinearity of resistive impurity effects D. W. Koon, “The nonlinearity of resistive impurity effects

on van der Pauw measurements", Rev. Sci. Instrum., on van der Pauw measurements", Rev. Sci. Instrum., 7777, , 094703 (2006).094703 (2006).

Nonlinearity of the weighting Nonlinearity of the weighting functionfunction

Increasing Increasing Decreasing Decreasing

Fit curve (in white):Fit curve (in white):where where ≈0.66 for entire specimen.≈0.66 for entire specimen.

1

),(),( 0 yxfyxf

The problem (continued):The problem (continued):

► NonlinearNonlinear results have been obtained for results have been obtained for resistivity measurement on square van der Pauw resistivity measurement on square van der Pauw geometry. geometry.

► Nonlinearity can be modeled by Nonlinearity can be modeled by simplesimple, one-, one-parameter function for parameter function for entireentire specimenspecimen

► What about Hall weighting function?What about Hall weighting function? Simple formula?Simple formula? Nonlinearity depend on position?Nonlinearity depend on position? Nonlinearity depend on unperturbed Hall Nonlinearity depend on unperturbed Hall

angle?angle?

Solving for potential Solving for potential near non-uniform Hall angle:near non-uniform Hall angle:

HH <<1: <<1:

General case:General case:

Small perturbation is equivalent to point Small perturbation is equivalent to point dipole perpendicular to and proportional to dipole perpendicular to and proportional to local E-field. Linear.local E-field. Linear.But the perturbation changes the local E-field. But the perturbation changes the local E-field. Therefore there is a nonlinear effect.Therefore there is a nonlinear effect.

ProcedureProcedure

► Solve difference-equation form of modified Solve difference-equation form of modified Laplace’s Equation on 21x21 matrix in Excel Laplace’s Equation on 21x21 matrix in Excel by successive overrelaxation [SOR].by successive overrelaxation [SOR].

Verify selected results on 101x101 grids.Verify selected results on 101x101 grids.

► Apply pointlike perturbation of local Hall Apply pointlike perturbation of local Hall angle as function of…angle as function of…

size of perturbation (|size of perturbation (|HH| | << 4545ºº) )

location of perturbationlocation of perturbation

unperturbed Hall angle (|unperturbed Hall angle (|HH| | << 4545ºº))

Small-angle limit:Small-angle limit:

► ||HH|, ||, |HH| | 2°. (B=¼T for pure Si @ RT) 2°. (B=¼T for pure Si @ RT)

► Results were fit to the quadratic expression:Results were fit to the quadratic expression:

► Linear terms, Linear terms, 11 and and 00 are plotted vs are plotted vs position of perturbation. (Nonlinearity position of perturbation. (Nonlinearity depends on depends on HH if and only if if and only if 11≠0.)≠0.)

HHHHH

HH yxgyxg

12

201

0

1

),(),,,(

Small-angle results:Small-angle results:

► Nonlinearity varies across the specimen, Nonlinearity varies across the specimen, depends on unperturbed Hall angle, depends on unperturbed Hall angle, HH..

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

Empirical fit for center of Empirical fit for center of squaresquare

,)tan(tan1

),,,(2

0 HH

HH

BAg

yxg

and where

0066.0

063.02cos21

383.12sin 0

H

H

B

gA

-45-30-150153045

-45

-15

15

45

-0.5

0

0.5

1

1.5

2

g(3,3)

H

H

-45045

-45

0

450

0.5

1

1.5

2

2.5

g(3,11)

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

(3,3)(3,3)

-45-30-150153045

-45-30

-150

1530

45

1

1.5

2

2.5

g(11,11)

H

H

ConclusionsConclusions

► No simple expression for Hall nonlinearity.No simple expression for Hall nonlinearity.

Depends on position, (x,y)Depends on position, (x,y)

Depends on both unperturbed Hall Depends on both unperturbed Hall angle, angle, H,H, and perturbation, and perturbation, HH

► Weighting function blows up as |Weighting function blows up as |tantanHH||

► For center of square, empirical fit found for For center of square, empirical fit found for |tan|tanHH|<45°|<45°

Inconclusions (What’s next?)Inconclusions (What’s next?)

► Is there a general expression for how the Hall Is there a general expression for how the Hall weighting function varies with respect toweighting function varies with respect to

Unperturbed Hall angle, tanUnperturbed Hall angle, tanHH

Perturbation, Perturbation, tantanHH

Location, (x,y), of perturbationLocation, (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 |Can results be extended to |HH|, ||, |HH|>45°?|>45°?

► How do two simultaneous point perturbations How do two simultaneous point perturbations interact?interact?