PHYSICAL REVIEW B 98, 085417 (2018) Towards Joule heating optimization in Hall devices Robert Benda, 1, 2 , * J. M. Rubi, 2 E. Olive, 3 and J.-E. Wegrowe 1 1 LSI, Ecole Polytechnique, CEA, CNRS, Palaiseau F-91128, France 2 Department of Fisica Fonamental, Universitat de Barcelona, Barcelona, Spain 3 GREMAN UMR-CNRS 7347, Université de Tours, INSA Centre Val de Loire, Parc de Grandmont, 37200 Tours, France (Received 8 February 2018; revised manuscript received 18 July 2018; published 10 August 2018) We show that the Joule heating in Hall devices can be optimized by engineering the shape of the device. By using a geometrical method based on conformal transformations, we show the existence of two equivalence classes for Hall devices, exemplified by the Hall bar and the Corbino disk. The elements inside each class are related by means of a conformal mapping. Whereas the power dissipated by the systems of the Hall bar class formed by simply connected systems is conformally invariant and equal for each element of the class, that for systems of the Corbino disk class whose elements are doubly connected depends on their geometry and can thus be controlled by modifying the shape of the device. This general property could be used for a better performance of Hall devices. DOI: 10.1103/PhysRevB.98.085417 The possibility of optimizing Joule heating losses by modi- fying the resistance of a Hall device is an issue of great practical importance. Controlling the resistance of Hall sensors may increase their sensibility thus making them more precise for sensing applications. To control the resistance of the device by modifying its geometry and thus to increase its sensibility was addressed in Refs. [1–18]. Magnetoresistive sensors having diverse geometries have been proposed [7,8]. Baker et al. found upper and lower bounds for the magnetoresistance R(B ) of planar devices subject to a perpendicular magnetic field B showing that the upper bound of the geometrical factor is attained for devices with only electrically conducting leads [7]. This is the case of the Corbino geometry [1] or of a topological equivalent structure which was reported in [8]. Wick [19] and more recently Zhang et al. [10] focused on the improvement of the voltage sensitivity of planar Hall devices in the light of the shape of the boundaries and the length of the electrodes. They used Schwarz-Christoffel transformations to relate polygonal devices to simpler geome- tries. As all simply connected Hall devices are conformally related, it is possible to choose a convenient boundary shape to obtain the properties of the system. Wick investigated the dependence of the Hall voltage on the aspect ratio and Zhang et al. proposed a method to optimize the geometrical factor in the Hall sensitivity for square and cross devices, relating them to a simple parameter in the image circular Hall device. Popovic presented a thorough analysis of the geometrical factors playing a role in the resistance and the sensitivity of Hall devices. He alluded to the possibility of suiting the shape of a Hall device to lower the Joule heating without changing the area of the active region [6]. This point is often put aside in the study of Hall devices. * robert.benda@polytechnique.edu In this article we show how Joule heat losses can be opti- mized by engineering the shape of the Hall device. We show that planar Hall devices can be divided into two equivalence classes which differ in their geometry. The first class (A) is formed by all simply connected devices obtained from the Hall bar by performing a conformal transformation. The elements of the second class (B) are the doubly connected devices obtained from a Corbino disk by means of that transformation. We show that the stationary states for the elements of both classes are very different in nature. Whereas for systems of class A electric charges can be accumulated at the boundaries (the current lines are then radial), for those of class B the deflection of the current lines impedes the charge accumulation. This fact has consequences in the power dissipated by the device. We show that, whereas the dissipated power of an element of A is conformally invariant, that for the elements of the systems of class B depends on the geometry. The dissipated power reaches a minimum for a particular realization of the device. The article is organized in the following way. In Sec. I we classify Hall devices into two equivalence classes and discuss their main properties. Section II is devoted to the analysis of the dissipated power for the devices belonging to both classes. In the Conclusion we summarize our main results. I. EQUIVALENCE CLASSES FOR HALL DEVICES The device is a planar electric conductor contacted to an electric generator, on which a magnetic field is applied perpendicular to the plane. Ohm’s law relates the electric current  J to the electric field  E eff =−  ∇ μ and reads  ˆ J =−  σ σ H −σ H σ   ∇ μ, (1) where σ is the longitudinal conductivity, the off-diagonal coefficient σ H is the Hall conductivity (which is proportional to the magnetic field), and μ is the electrochemical potential that accounts for both drift and diffusion effects. The expression of 2469-9950/2018/98(8)/085417(9) 085417-1 ©2018 American Physical Society