Abstract
Singly connected Hall plates with N peripheral contacts can be mapped onto the upper half of the z-plane by a conformal transformation. Recently, Homentcovschi and Bercia derived the General Formula for the electric field in this region. We present an alternative intuitive derivation based on conformal mapping arguments. Then we apply the General Formula to complementary Hall plates, where contacts and insulating boundaries are swapped. The resistance matrix of the complementary device at reverse magnetic field is expressed in terms of the conductance matrix of the original device at non-reverse magnetic field. These findings are used to prove several symmetry properties of Hall plates and their complementary counterparts at arbitrary magnetic field.
Highlights
The purpose of this work is to derive relations between Hall plates and their complementary counter-parts
We gave a simple derivation of the electric field in the infinite upper half plane with N contacts on the real axis
The original formula has N real coefficients, but we could show that the N-th one vanishes
Summary
The purpose of this work is to derive relations between Hall plates and their complementary counter-parts. In [8] it was implicitly mentioned that the product of input resistances of original and complementary Hall plates of that particular symmetry (i.e., input resistance equals output resistance) at zero magnetic field equals twice the square of the sheet resistance (see the paragraph after (50) in [8]). —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 Figure 6 in this work). This new approach shows how the stagnation points are linked to the electric field in the Hall plate Appendix C gives a numerical example, where the theory is compared to results of finite element simulations
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