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

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.