# Nonlinearity%20in%20the%20effect%20of%20an%20inhomogeneous%20Hall%20angle%20Daniel%20W.%20Koon%20St.%20Lawrence%20University%20Canton,%20NY - PowerPoint PPT Presentation

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## Nonlinearity%20in%20the%20effect%20of%20an%20inhomogeneous%20Hall%20angle%20Daniel%20W.%20Koon%20St.%20Lawrence%20University%20Canton,%20NY

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### ... empirical fit) have been obtained for resistivity measurement on square van der Pauw geometry. ... Empirical fit for center of square ... – PowerPoint PPT presentation

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Title: Nonlinearity%20in%20the%20effect%20of%20an%20inhomogeneous%20Hall%20angle%20Daniel%20W.%20Koon%20St.%20Lawrence%20University%20Canton,%20NY

1
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, 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 a0.66 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 entire specimen
• 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 ltlt1
• 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
Laplaces 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 lt 45º)
• location of perturbation
• unperturbed Hall angle (QH lt 45º)

10
Small-angle limit
• QH, dQH ?? 2. (B¼T for pure Si _at_ 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
tanQHlt45

16
Inconclusions (Whats 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, dQHgt45?
• How do two simultaneous point perturbations
interact?