Symmetries of certain double integrals related to Hall effect devices

Abstract

One encounters iterated elliptic integrals in the study of Hall effect devices, as a result of conformal mappings of Schwarz-Christoffel type. Some of these double elliptic integrals possess amazing symmetries with regard to the physical parameters of the underlying Hall effect devices. We give a unified mathematical treatment of such symmetric double integrals, in the context of Hall effect devices with three and four contacts.

Relations between Hall Plates with Complementary Contact Geometries

“…If such Hall plates have a single mirror symmetry, and if we supply the original and the complementary Hall plate with the same supply voltage, their output voltages are identical. This was proven for weak magnetic field in [10] [11] with considerable mathematical effort, and with the above theory we will prove it also for strong magnetic field. For the original Hall plate in Figure 6 we write 11 12 supply supply…”

“…This was proven for weak magnetic field in [10] [11] with considerable mathematical effort, and with the above theory we will prove it also for strong magnetic field. For the original Hall plate in Figure 6 we write 11 12 supply supply…”

“…Then-analogous to above-the change of the potentials on the output contacts due to reversal of magnetic field polarity are identical in both original and complementary devices, if both devices are supplied with the same supply voltage on the other two contacts, and if the magnetic field is weak (see also Figure 6 in this work). This property also means that the ratio of Hall output signal over thermal noise under the constraint of fixed supply voltage and fixed input resistance is the same in the original Hall plate and in the complementary Hall plate [10] [11].In Section 2 we reconsider the General Formula of [12] for the electric field in the upper half of the z-plane with N contacts on the real axis. Thereby, we present a different derivation than the one given in [12].…”

An Analytical Theory of Hall-Effect Devices with Three Contacts

“…In (30) we only have to determine the transformation of current density from q-plane in Fig. (8d) to -plane in Fig.…”

“…(9d), which is detailed in Appendix A. The subtraction in (30) means that the current flowing along the boundary between the supply contacts C 1 and C 3 reduces the Hall output signal. Finally, the weak magnetic field limit of the geometry factor of a 3C-Hall-effect device with single mirror symmetry is given by…”