Symmetries of certain double integrals related to Hall effect devices
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Symmetries of certain double integrals related to Hall effect devices Udo Ausserlechner1 · M. Lawrence Glasser2,3
· Yajun Zhou4,5
Received: 2 May 2018 / Accepted: 25 September 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2020
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. Keywords Incomplete elliptic integrals · Complete elliptic integrals · Hall effect Mathematics Subject Classification 33E05 (Primary) · 78A35 (Secondary)
Partial financial support is acknowledged to the Spanish Junta de Castilla y León (VA057U16) and MINECO (Project MTM2014-57129-C2-1-P).
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M. Lawrence Glasser [email protected] Udo Ausserlechner [email protected] Yajun Zhou [email protected] ; [email protected]
1
Infineon Technologies Austria AG, Siemensstrasse 2, 9500 Villach, Austria
2
Dpto. de Física Teórica, Facultad de Ciencias, Universidad de Valladolid, Paseo Belén 9, 47011 Valladolid, Spain
3
Department of Physics, Clarkson University, Potsdam, NY 13699, USA
4
Program in Applied and Computational Mathematics (PACM), Princeton University, Princeton, NJ 08544, USA
5
Academy of Advanced Interdisciplinary Studies (AAIS), Peking University, Beijing 100871, People’s Republic of China
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U. Ausserlechner et al.
1 Introduction As one can easily demonstrate oneself, if you spin a coin, oriented perpendicular to an inclined plane, due to the balance of gravity and the gyroscopic force, the coin will move across the plane rather than down it as it does when it is not spinning. The speed at which it moves is determined by various factors such as the tilt of the plane, the rate of spin and the surface conditions. The electrical analogue is the Hall effect: if an electron current is produced, by electrical contacts, across a conducting plate in a perpendicular magnetic field a current IH , and equivalently, a voltage VH , resulting from the balance between the strength of the current and the Lorentz force on the electrons, will be detectable between electrodes placed perpendicular to the current. The magnitude of this voltage will depend on the magnetic field strength, the electrical characteristics of the plate material and its geometry. For such a standard four-contact commercial semiconductor Hall device, having two perpendicular reflection lines, one of us [1–3] determined the analytic form of its geometrical factor G H , in the expression for VH , in terms of a double elliptic integral whose two moduli depended on adjustable characteristics of the system. From numerical evaluations of conformal transformations, it was found that G H exhibited an invariance which could be expressed as
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