JAMP  Vol.7 No.9 , September 2019
The Classical Hall Effect in Multiply-Connected Plane Regions Part I: Topologies with Stream Function
Abstract: Typical Hall plates for practical magnetic field sensing purposes are plane, simply-connected regions with peripheral contacts. Their output voltage is the sum of even and odd functions of the applied magnetic field. They are commonly called offset and Hall voltage. Contemporary smart Hall sensor circuits extract the Hall voltage via spinning current Hall probe schemes, thereby cancelling out the offset very efficiently. The magnetic field response of such Hall plates can be computed via the electric potential or via the stream function. Conversely, Hall plates with holes show new phenomena: 1) the stream function exists only for a limited class of multiply-connected domains, and 2) a sub-class of 1) behaves like a Hall/Anti-Hall bar configuration, i.e., no Hall voltage appears between any two points on the hole boundary if current contacts are on their outer boundary. The paper studies the requirements under which these effects occur. Canonical cases of simply and doubly connected domains are computed analytically. The focus is on 2D multiply-connected Hall plates where all boundaries are insulating and where all current contacts are point sized.
Cite this paper: Ausserlechner, U. (2019) The Classical Hall Effect in Multiply-Connected Plane Regions Part I: Topologies with Stream Function. Journal of Applied Mathematics and Physics, 7, 1968-1996. doi: 10.4236/jamp.2019.79136.

[1]   Wick, R.F. (1954) Solution of the Field Problem of the Germanium Gyrator. Journal of Applied Physics, 25, 741-756.

[2]   Versnel, W. (1981) Analysis of Symmetrical Hall Plates with Finite Contacts. Journal of Applied Physics, 52, 4659-4666.

[3]   Ausserlechner, U. (2017) The Signal-to-Noise Ratio and a Hidden Symmetry of Hall Plates. Solid-State Electronics, 135, 14-23.

[4]   Homentcovschi, D. and Bercia, R. (2018) Analytical Solution for the Electric Field in Hall Plates. Zeitschrift für Angewandte Mathematik und Physik, 69, 97.

[5]   Isenberg, I., Russell, B.R. and Greene, R.F. (1948) Improved Method for Measuring Hall Coefficients. Review of Scientific Instruments, 19, 685-688.

[6]   Rudan, M., et al. (2006) Theory and Experimental Validation of a New Analytical Model for the Position-Dependent Hall Voltage in Devices with Arbitrary Aspect Ratio. IEEE Transactions on Electron Devices, 53, 314-322.

[7]   Raman, J., Rombouts, P. and Weyten, L. (2013) Method for Electric Field and Potential Calculations in Hall Plates. Electronics Letters, 49, 33-34.

[8]   Ausserlechner, U. (2016) A Method to Compute the Hall-Geometry Factor at Weak Magnetic Field in Closed Analytical Form. Electrical Engineering, 98, 189-206.

[9]   Haeusler, J. (1967) Die Untersuchung von Potentialproblemen bei tensorieller Leitfähigkeit der Halbleiter und Plasmen im transversalen Magnetfeld. Dissertation TH Stuttgart. (In German)

[10]   Mani, R.G. and Von Klitzing, K. (1994) Temperature-Insensitive Offset Reduction in a Hall Effect Device. Applied Physics Letters, 64, 3121-3123.

[11]   Antipov, Y. and Silvestrov, V. (2010) Hilbert Problem for a Multiply Connected Circular Domain and the Analysis of the Hall Effect in a Plate. Quarterly of Applied Mathematics, 68, 563-590.

[12]   Briane, M. and Milton, G.W. (2009) Homogenization of the Three-Dimensional Hall Effect and Change of Sign of the Hall Coefficient. Archive for Rational Mechanics and Analysis, 193, 715-736.

[13]   Kern, C., Kadic, M. and Wegener, M. (2017) Experimental Evidence for Sign Reversal of the Hall Coefficient in Three-Dimensional Metamaterials. Physical Review Letters, 118, Article ID: 016601.

[14]   Bierwagen, O., Ive, T., Van de Walle, C.G. and Speck, J.S. (2008) Causes of Incorrect Carrier-Type Identification in van der Pauw-Hall Measurements. Applied Physics Letters, 93, Article ID: 242108.

[15]   van Haaren, J.A.M.M. (1988) Macroscopic Current Diversion in Magnetic Fields in Metals and Semiconductors. Doctoral Dissertation, University of Nijmegen, Nijmegen.

[16]   Betz, A. (1964) Konforme Abbildung. 2nd Edition, Chapter IV, Springer Verlag, Berlin. (In German)

[17]   Oszwaldowski, M., Pieranski, P., Berus, T. and Jankowski, J. (2010) Reinterpretation of Hall Effect in Medium with Hole.

[18]   Szymański, K., Cieśliński, J.L. and Łapiński, K. (2013) Van der Pauw Method on a Sample with an Isolated Hole. Physics Letters A, 377, 651-654.

[19]   Oh, D., Ahn, C., Kim, M., Park, E.K. and Kim, Y.S. (2016) Application of the van der Pauw Method for Samples with Holes. Measurement Science and Technology, 27, Article ID: 125001.

[20]   Oswald, M., Oswald, J. and Mani, R.G. (2005) Voltage and Current Distribution in a Doubly Connected Two-Dimensional Quantum Hall System. Physical Review B, 72, Article ID: 035334.

[21]   Kriisa, A., Mani, R.G. and Wegscheider, W. (2010) Hall Effects in Doubly Connected Specimens. IEEE Transactions on Nanotechnology, 10, 179-182.

[22]   Popovic, R.S. (2004) Hall Effect Devices, Appendix E. IoP Publishing, Bristol.

[23]   Spal, R. (1980) A New DC Method of Measuring the Magnetoconductivity Tensor of Anisotropic Crystals. Journal of Applied Physics, 51, 4221-4225.

[24]   Sample, H.H., Bruno, W.J., Sample, S.B. and Sichel, E.K. (1987) Reverse-Field Reciprocity for Conducting Specimens in Magnetic Fields. Journal of Applied Physics, 61, 1079-1084.

[25]   Cornils, M. and Paul, O. (2008) Reverse-Magnetic-Field Reciprocity in Conductive Samples with Extended Contacts. Journal of Applied Physics, 104, Article ID: 024505.

[26]   Munter, P.J.A. (1990) A Low-Offset Spinning Current Hall Plate. Sensors Actuat A, 21-23, 743-746.

[27]   Ausserlechner, U. (2004) Limits of Offset Cancellation by the Principle of Spinning Current Hall Probe. Proceedings of IEEE SENSORS, Vienna, 24-27 October 2004, 1117-1120.

[28]   Madec, M., Kammerer, J.B., Hébrard, L. and Lallement, C. (2011) Analysis of the Efficiency of Spinning-Current Techniques Thru Compact Modeling. Sensors, Limerick, 28-31 October 2011, 542-545.

[29]   Motz, M., Ausserlechner, U., Scherr, W. and Katzmaier, E. (2006) An Integrated Hall Sensor Platform Design for Position, Angle and Current Sensing. Sensors, Daegu, 22-25 October 2006, 1008-1011.

[30]   Buehler, M.G. and Pearson, G.L. (1966) Magnetoconductive Correction Factors for an Isotropic Hall Plate with Point Sources. Solid-State Electronics, 9, 395-407.

[31]   Schott, C., Randjelovic, Z., Waser, J.M. and Popovic, R.S. (1998) 2D Nonlinearity Simulation of the Vertical Hall Sensor Using SESES. 1st International Conference on Modeling and Simulation of Microsystems, Semiconductors, Sensors and Actuators, Santa Clara, 6-8 April 1998, 643-648.

[32]   Weissman, M.B. (1980) Relations between Geometrical Factors for Noise Magnitudes in Resistors. Journal of Applied Physics, 51, 5872-5875.

[33]   Kern, C., Milton, G.W., Kadic, M. and Wegener, M. (2018) Theory of the Hall Effect in Three-Dimensional Metamaterials. New Journal of Physics, 20, Article ID: 083034.

[34]   Schinzinger, R. and Laura, P.A.A. (2003) Conformal Mapping. Dover Publications Inc., Mineola.

[35]   Komatu, Y. (1949) Existence Theorem of Conformal Mapping of Doubly-Connected Domains. Kodai Mathematical Seminar Reports, 1, 3-4.

[36]   Stafl, M. (1967) Electrodynamics of Electrical Machines. Academia, Publishing House of the Czechoslovak Academy of Sciences, Praha, Chapter 2.6.

[37]   Schuh, F. (1919) Theorem on the Term by Term Differentiability of a Series. KNAW, Proceedings, 22 I, 1919-1920, Amsterdam, 376-378.