An Analytical Theory of Piezoresistive Effects in Hall Plates with Large Contacts

Figure 10

(a–d) Mapping a circular device with anisotropic conductivity by one affine and three conformal transformations onto an equivalent rectangular device with isotropic conductivity. (a) shows a circular disk shaped device with two pairs of contacts and two perpendicular mirror-symmetry axes and . Its resistivity is anisotropic with the principal axes x and y. This describes the action of biaxial load along - and -axes on chips in (100)-silicon aligned like in Figure 1. The elliptical device in (b) with isotropic resistivity is equivalent to the device in (a). The shape in (b) is obtained by the affine transformation (12a). The conformal transformation rotates the device clockwise by 90° and scales it isotropically: and with . A second-conformal transformation   (26a) and (26b) maps the interior of the ellipse in (c) onto the upper half of the t-plane in (d). A third-conformal transformation maps the upper half of the t-plane into the interior of the rectangle in (e). The contacts along the perimeter of the shapes relate to the contacts on the real t-axis. Corresponding points have equal indices. The sequence of points with indices 0, 1, 2, 3, and 4 is always counterclockwise while the conductive region is at the left hand side of this path. The asterisk means the conjugate complex.