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

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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

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.

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.

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
  • (b) Hall effect D. W. Koon C. J.
    Knickerbocker, Rev. Sci. Instrum. 64 (2), 510

Hall weighting function for other van der Pauw
  • 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
  • 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
  • ? Increasing r
    Decreasing r ??
  • Fit curve (in white)
  • where a0.66 for entire specimen.

The problem (continued)
  • Nonlinear results have been obtained for
    resistivity measurement on square van der Pauw
  • 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
  • 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.

  • 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)

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.)

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

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)
  • 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

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