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?
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