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

Abstract

Four-terminal transducers can be used to measure the magnetic field via the Hall effect or the mechanical stress via the piezoresistance effect. Both effects are described by an anisotropic conductivity tensor with small offdiagonal elements. This has led other authors to the conclusion that there is some kind of analogy. In both cases the output voltage depends on the geometry of the device and the size of the contacts. For Hall plates this influence is accounted for by the Hall-geometry factor. The alleged analogy proposes that the Hall-geometry factor also applies to four-terminal stress transducers. This paper shows that the analogy holds only for a limited class of devices. Moreover, it is shown that devices of different geometries may have identical magnetic field sensitivity but different mechanical stress sensitivities. Thus, shape optimization makes sense for mechanical stress sensors. In extreme cases the output voltages of vertical Hall-effect devices may have notable magnetic field sensitivity but zero mechanical stress sensitivity. As byproduct, exact analytical formulae for the equivalent resistor circuit of rectangular and circular devices with two perpendicular mirror symmetries are given. They allow for an accurate description of how mechanical stress and deformation affect the output offset voltage and the magnetic sensitivity of Hall-effect devices.

The Classical Hall Effect in Multiply-Connected Plane Regions Part II: Spiral Current Streamlines

“…Our new theory can be applied to Hall plates with anisotropic conductivity as it is caused by mechanical stress in a cubic crystal. Then, in a preceding isotropization step, one can replace the original geometry by a distorted geometry with isotropic conductivity (see [16] and references therein). This procedure does not change the number of holes in a multiply-connected Hall plate.…”

“…The meaning of (14a) is: Since odd J is a vector field free of sources, its flux through any closed contour C in the multiply-connected Hall plate vanishes, and therefore odd J can be expressed as the curl of a magnetic field In (17a), we were allowed to add odd J in the integrand because integration of odd ⋅ J n over a closed path gives zero according to (16). The circulation of a specific pattern of spiral current streamlines is computed in Appendix A.…”

Drift of the sensitive direction of Hall-effect devices in (1 0 0)-silicon caused by mechanical shear stress

“…After all shear stress diverts the current streamlines and this may well affect 0 H G . In [13] it was shown that shear stress (or in general any in-plane biaxial stress) changes 0 H G for Hall plates unless they have 90° symmetry. In the case of Vertical Hall devices inplane shear stress xy  tilts the current streamlines out of the (x,z) cross-section plane.…”

“…Yet a small part may also come from piezo-resistance effect, which diverts the current flow lines thereby changing the Hall geometry factor (cf. section 5 of [13]). Stress is also a major source of offset error (zero point error) of Hall plates and to a lesser extent of Vertical Hall devices [13,13b].…”

Resistance Matrix for an Anisotropic Hall Plate with Multiple Extended Asymmetric Contacts on the Boundary

“…We note that anisotropy generated by piezoresistive effects is not analyzed here. This subject was addressed in the excellent paper by Ausserlechner [23].…”