The electrical properties of magnetic sensing devices fabricated from anisotropic materials are not easily extracted. Here we present a method for determining the resistance matrix for an anisotropic device with multiple electrical contacts placed in a perpendicular magnetic field. By using the methods developed by Van der Pauw and Wasscher, the analysis for the anisotropic system is reduced to the equivalent problem for an isotropic sample, which can then be solved using methods developed previously. As a result, the method works in the case of structures with an arbitrary number of asymmetric extended contacts at large magnetic field strength. In addition to the extraction of nonisotropic resistivities, the resistance matrix can be used to analyze the Hall effect for anisotropic plates.
References
[1]
Van der Pauw, L.J. (1958) A Method of Measuring Specific Resistivity and Hall Effect of Discs of Arbitrary Shape. Philips Research Reports, 13, 1-9.
[2]
Oliveira, F.S., Cipriano, R.B., da Silva, F.T., Romão, E.C. and dos Santos, C.A.M. (2020) Simple Analytical Method for Determining Electrical Resistivity and Sheet Resistance Using the Van der Pauw Procedure. Scientific Reports, 10, Article No. 16379. https://doi.org/10.1038/s41598-020-72097-1
[3]
de Lima, O.F, Ribeiro, R.A., Avila, M.A., Cardoso, C.A. and Coelho, A.A. (2001) Anisotropic Superconducting Properties of Aligned MgB2 Crystallites. Physical Review Letters, 86, 5974-5977. https://doi.org/10.1103/PhysRevLett.86.5974
[4]
Mihaly, G., Kezsmarki, I., Zamborszky, F. and Forro, L. (2000) Hall Effect and Conduction Anisotropy in Organic Conductors. Physical Review Letters, 84, 2670-2673. https://doi.org/10.1103/PhysRevLett.84.2670
[5]
Kazani, I., De Mey, G., Hertleer, C., Banaszczyk, J., Schwarz, A., Guxho, G. and Van Langenhove, L. (2011) Van der Pauw Method for Measuring Resistivities of Anisotropic Layers on Textile Substrates. Journal of Textile Research, 81, 2117-2124. https://doi.org/10.1177%2F0040517511416280
[6]
Wasscher, J.D. (1961) Note on Four Point Resistivity Measurements on Anisotropic Conductors. Philips Research Reports, 16, 301-306.
[7]
Wasscher, J.D. (1969) Electrical Transport Phenomena in MnTe, an Antiferromagnetic Semiconductor. Ph.D. Thesis, Technische Hogeschool, Eindhoven.
[8]
Montgomery, H.C. (1971) Method for Measuring Electrical Resistivity of Anisotropic Materials. Journal of Applied Physics, 42, 2971-2975. https://doi.org/10.1063/1.1660656
[9]
dos Santos, C.A.M., de Campos, A., da Luz, M.S., White, B.D., Neumeier, J.J., de Lima, B.S. and Shigue, C.Y. (2011) Procedure for Measuring Electrical Resistivity of Anisotropic Materials: A Revision of the Montgomery Method. Journal of Applied Physics, 110, Article ID: 083703. https://doi.org/10.1063/1.3652905
[10]
Miccoli, I., Edler, F., Pfnur, E. and Tegencamp, C. (2015) The 100th Anniversary of the Four-Point Probe Technique: The Role of Probe Geometries in Isotropic and Anisotropic Systems. Journal of Physics: Condensed Matter, 27, Article ID: 223201. https://doi.org/10.1088/0953-8984/27/22/223201
[11]
Borup, K.A., Fisher, K.F., Brown, D.R., Snyder, G.J. and Iversen, B.B. (2015) Measuring Anisotropic Resistivity of Single Cristals Using the Van der Pauw Technique. Physical Review B, 92, Article ID: 045210. https://doi.org/10.1103/PhysRevB.92.045210
[12]
Van der Pauw, L.J. (1961) Determination of the Resistivity Tensor and Hall Tensor of Anisotropic Conductors. Philips Research Reports, 16, 187-195.
[13]
Kyriakos, D.S., Valassiades, O.E., Papadimitriou, K.G. and Economou, N.A. (1981) Weak-Field Magnetoresistance Skewness and Galvanomagnetic Measurements on the (001)-Plane of n-Type Ge Single Crystal. Journal of Applied Physics, 52, 6178-6184. https://doi.org/10.1063/1.328519
[14]
Versnel, W. (1983) Electrical Characteristics of an Anisotropic Semiconductor Sample of Circular Shape with Finite Contacts. Journal of Applied Physics, 54, 916-921. https://doi.org/10.1063/1.332054
[15]
Shibata, H. and Oide, J.-I. (2000) Analysis of the Hall Effect Device Using an Anisotropic Material. Journal of Applied Physics, 88, 4813-4817. https://doi.org/10.1063/1.1289784
[16]
Homentcovschi, D. and Bercia, R. (2018) Analytical Solution for the Electric Field in Hall Plates. Zeitschrift für angewandte Mathematik und Physik, 69, Article No. 97. https://doi.org/10.1007/s00033-018-0989-7
[17]
Homentcovschi, D. and Murray, B.T. (2019) Explicit Resistance Matrix for a Hall Disk with Multiple Peripheral Contacts: Application to a Van der Pauw Type Method for Extended Contacts. Sensors and Actuators A, 294, 1-7. https://doi.org/10.1016/j.sna.2019.04.027
[18]
Homentcovschi, D. and Murray, B.T. (2020) Basic Relationships for Hall Half-Plane Structures with Multiple Extended Contacts on the Boundary: Applications to the Extraction of Physical Parameters and Optimization of Graphene and Vertical Hall Devices. Solid-State Electronics, 171, Article ID: 107837. https://doi.org/10.1016/j.sse.2020.107837
[19]
Ausserlechner, U. (2019) Relations between Hall Plates with Complementary Contact Geometries. Journal of Applied Mathematics and Physics, 7, 2836-2867. https://doi.org/10.4236/jamp.2019.711195
[20]
Kleiza, J. and Kleiza V. (2011) On the Applying of the Van der Pauw Method to Anisotropic Media. Acta Physica Polonica A, 119, 148-150. https://doi.org/10.12693/APhysPolA.119.148
[21]
Takahasi, H. and Mori, M. (1974) Double Exponential Formulas for Numerical Integration. Publications of the Research Institute for Mathematical Sciences, 9, 721-741. https://doi.org/10.2977/prims/1195192451
[22]
Nader, A. and Kouba, O. (2014) Influence of Contact’s width on the Resistance Measurement and Resistivity Anisotropic Determination Method for an Anisotropic Circular Disk. Journal of Applied Physics, 115, Article ID: 234909. https://doi.org/10.1063/1.4883764
[23]
Ausserlechner, U. (2018) An Analytical Theory of Piezoresistive Effects in Hall Plates with Large Contacts. Advances in Condensed Matter Physics, 2018, Article ID: 7812743. https://doi.org/10.1155/2018/7812743
[24]
Batista, M. (2019) Elfun18—A Collection of MATLAB Functions for the Computation of Elliptic Integrals and Jacobian Elliptic Functions of Real Arguments. SoftwareX, 10, Article ID: 100245. https://doi.org/10.1016/j.softx.2019.100245
[25]
Ausserlechner, U. (2017) Van-der-Pauw Measurement on Devices with Four Contacts and Two Orthogonal Mirror Symmetries. Solid-State Electronics, 133, 53-63. https://doi.org/10.1016/j.sse.2017.04.006
[26]
Nehari, Z. (1952) Conformal Mapping. McGraw-Hill, New York.
[27]
Cornils, M. and Paul, O. (2008) Reverse-Magnetic-Field Reciprocity in Conductive Samples with Extended Contacts. Journal of Applied Physics, 104, Article ID: 024505. https://doi.org/10.1063/1.2951895
[28]
Cornils, M., Rottmann, A. and Paul, O. (2010) How to Extract the Sheet Resistance and Hall Mobility from Arbitrary Shaped Planar Four-Terminal Devices with Extended Contacts. IEEE Transactions on Electron Devices, 57, 2087-2097. https://doi.org/10.1109/TED.2010.2053493
