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We study the stationary state of Hall-bar devices composed of a load circuit connected to the lateral edges of a Hall-bar. We follow the approach developed in a previous work (Creff et al. J. Appl. Phys 2020) in which the stationary state of a ideal Hall bar is defined by the minimum power dissipation principle. The presence of both the lateral circuit and the magnetic field induces the injection of a current: the so-called Hall current. Analytical expressions for the longitudinal and the transverse currents are derived. The same analysis is performed on the spin-Hall effect, in the framework of the two spin-channel model.
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Jean-Eric Wegrowe, Félix Faisant, Melvin Creff, "Hall effect : do not forget the surface currents," Proc. SPIE 11805, Spintronics XIV, 1180529 (1 August 2021); https://doi.org/10.1117/12.2596454