# Surprises Regarding the Hall Effect: An Extraordinary Story Involving an Artist, Mathematicians, and Physicists

In February 2017, the front cover of Physics Today featured a beautiful microstructure of interlocking rings that, for the first time in history, reversed the sign of the Hall coefficient in a material with electrons (rather than holes) as the predominant charge carrier. The Hall effect, initially observed by Edwin Hall in 1879, predates the discovery of electrons. If one applies a magnetic field perpendicular to a conducting material—such as a copper plate—and a voltage across the plate’s length, current flows. But an additional voltage develops across the width of the plate, transverse to the current flow; this is called the Hall voltage. It develops because the electrons sense a force (the Lorentz force) perpendicular to both the magnetic field and the electron velocity that needs to be balanced by something — in this case, by the force due to the electric field generated by the Hall voltage (electrons move to one side of the plate until the required voltage develops). For small fields, the electric field induced by the Hall voltage is then linearly related to the product of the applied current and the applied magnetic field through a proportionality constant known as the Hall coefficient. Assuming that the electrons travel at constant velocity in parallel straight lines, this observation leads to the textbook statement claiming that the sign of the Hall coefficient reveals the charge carrier’s sign in classical physics. However, the standard argument’s weakness is the assumption that electrons travel in (approximately) parallel straight lines, which is certainly not the case in materials with microstructure.

Development of the counterexample to the textbook claim—that the sign of the Hall voltage determines the sign of the Hall coefficient—is an interesting story in its own right. The combined efforts of many, including mathematicians, physicists, and even an artist, yielded the example that overturned classical beliefs. The story begins with a seemingly unrelated question. In a two-dimensional periodic conducting composite (where the electrical potential $$V$$ satisfies $$\nabla\cdot\sigma\nabla V=0$$,  $$\sigma({\bf x})$$ is a smooth scalar-valued periodic electrical conductivity that is positive and finite everywhere, and $$\nabla V$$ is periodic with a nonzero average), one does not expect the vector-valued electric field $$-\nabla V$$ to vanish inside the medium. Going further, one can think of a $$2\times 2$$ matrix-valued electric field $${\bf E}({\bf x})$$ with elements $$E_{ij}=-\partial V_j/\partial x_i$$—where each $$V_j$$ solves the conductivity equations—but with an average value over $$-\nabla V_j$$'s unit cell of periodicity that is a unit vector directed along the $$x_j$$ direction, $$j=1,2$$. Thus, the average value of $${\bf E}({\bf x})$$ is the identity matrix. Applying an average electric field in a direction $$\bf v$$ generates a physical vector-valued electric field $${\bf E}({\bf x})\cdot{\bf v}$$; if the field does not vanish for any $$\bf x$$ and $${\bf v}\ne 0$$, the determinant of $${\bf E}({\bf x})$$ cannot vanish and thus must maintain its sign throughout the material. Giovanni Alessandrini and Vincenzo Nesi [2] proved this result, which then led to improved bounds on the effective conductivity of composites. This naturally inspired the question of whether a similar result holds true for the determinant of the $$3\times 3$$ matrix-valued electric field $${\bf E}({\bf x})$$ for three-dimensional composites.

Graeme Milton first thought about interlocking rings while on a bus in Rome, and Marc Briane and Vincenzo Nesi [8] developed the idea into a rigorous counterexample. A clue to the significance of highly-conducting interlocking rings is that when they are at different electrical potentials, the potential does not monotonically decrease (or increase) along the line joining their centers. Houman Owhadi pointed out that Alano Ancona [3] used a similar interlocking geometry to prove the loss of injectivity of the vector-valued potential associated with the matrix electric field $${\bf E}({\bf x})$$.

Thanks to David Bergman’s work [4] on the Hall effect, perturbation theory proves that if one has a zero Hall coefficient outside a small test region in the unit cell and a nonzero constant Hall coefficient in the test region, the effective Hall coefficient is determined by the integral of a principal cofactor of the matrix $${\bf E}({\bf x})$$ over the test region. The appropriate cofactor is determined by the direction of the magnetic field, assumed aligned with one of the coordinate axes. For conduction in a plane with the magnetic field perpendicular to that plane, Alessandrini and Nesi’s result [2] about the positivity of the determinant of $${\bf E}({\bf x})$$ implies that if the local Hall coefficient is nonnegative everywhere, then the effective Hall coefficient is also nonnegative; no sign reversal can occur [5]. In the quest to find a three-dimensional counterexample, one would ideally want a geometry with cubic symmetry so that its response is isotropic. It is easy to visualize a plane filled by interlocking rings, which looks a bit like chain mail (medieval armor). When we contacted chain mail artist Dylon Whyte for permission to reproduce a beautiful picture he had created of this geometry, he asked—to our amazement—if we had considered the three-dimensional geometry of interlocking rings. That turned out to be exactly what was needed (see Figure 1).

Figure 1. Semiconductor metamaterials: from art to physics via mathematics. 1a. Photograph of two-layer chain mail, courtesy of Dylon Whyte. 1b. Original blueprint of the three-constituent metamaterial suitable for reversing the sign of the Hall coefficient according to [6]. The black spheres correspond to a semiconductor constituent, the gray tori are made of ideal metal, and the homogeneous background is a weakly conducting material. Image courtesy of Christian Kern. 1c. Experimental test series composed of different single-constituent, microstructured metamaterial samples. 1d. Photograph of a fully functional Hall-effect metamaterial sample in the measurement probe station. 1c and 1d are courtesy of [11].

The geometry that we proved would reverse the sign of the Hall coefficient [6] with three phases—highly-conducting interlocking rings, a background conducting region of much lower conductivity, and small spheres of high Hall-coefficient material—in the gaps between the rings where a cofactor of $${\bf E}({\bf x})$$ changes sign (see Figure 1b). It was too complex to easily construct, even with state-of-the-art manufacturing. But upon numerically analyzing the conducting ring structure, with void outside the rings and the rings themselves having a nonzero Hall coefficient, we found that we could achieve a sign reversal of the Hall coefficient—even in this simplified microstructure—in the presence of conducting bridges between the rings, on their inside sides [9].

Although this geometry still involved the cubic array of interlocking rings, the physical intuition for the Hall coefficient reversal was quite different, and based on the Hall voltage sign on the inside of the rings. Ramesh G. Mani and Klaus von Klitzing [13] noted that connecting leads to the inside surface of holes gives a reversed Hall voltage. However, we emphasize that their observation had nothing to do with homogenization and effective moduli. The Hall coefficient reversal using this single-phase interlocking ring structure, numerically predicted in [9], was then experimentally confirmed by Christian Kern [11] (see Figures 1c and 1d). It is quite incredible to see the actual microstructures produced by three-dimensional laser printing.

This is not the end of the story on Hall-effect materials. Another model predicted the parallel Hall effect [7], where the induced electric field is parallel to the magnetic field. A simplified microstructure exhibiting this behavior was designed and experimentally tested in [10] and [12], which noted that a split cylindrical shell of such a material might be useful for measuring the magnetic field’s curl. Other interesting developments arise on the mathematical side as well. The vanishing of electric fields at certain points has turned out to be important to hybrid inverse problems, and conductivities $$\sigma({\bf x})$$—involving geometries similar to interlocking rings, for which the stability of the reconstructions will inevitably degrade—exist for any finite number of prescribed boundary conditions [1].

Acknowledgments: Muamer Kadic was supported by the Labex ACTION program (contact ANR-11-LABX-0001-01), and Graeme Milton was supported by the NSF through grant DMS-0411035.

References
[1] Alberti, G.S., Bal, G., & Di Cristo, M. (2017). Critical Points for Elliptic Equations with Prescribed Boundary Conditions. Arch. Ration. Mech. Anal., 226(1), 117-141.
[2] Alessandrini, G., & Nesi, V. (2001). Univalent σ-harmonic mappings. Arch. Ration. Mech. Anal., 158(2), 155-171.
[3] Ancona, A. (2002). Some results and examples about the behavior of harmonic functions and Green’s functions with respect to second order elliptic operators. Nagoya Math. J., 165, 123-158.
[4] Bergman, D.J. (2010). Self-duality and the low field Hall effect in 2D and 3D metal-insulator composites. In G. Deutscher, R. Zallen, & J. Adler (Eds.), Percolation Structures and Processes (pp. 297-321). Jerusalem: Israel Physical Society.
[5] Briane, M., Manceau, D., & Milton, G.W. (2008). Homogenization of the two-dimensional Hall effect. J. Math. Ana. App., 339(2), 1468-1484.
[6] Briane, M., & Milton, G.W. (2009). Homogenization of the Three-dimensional Hall Effect and Change of Sign of the Hall Coefficient. Arch. Ration. Mech. Anal., 193(3), 715-736.
[7] Briane, M., & Milton, G.W. (2010). An Antisymmetric Effective Hall Matrix. SIAM J. Appl.  Math., 70(6), 1810-1820.
[8] Briane, M., Milton, G.W., & Nesi, V. (2004). Change of sign of the corrector’s determinant for homogenization in three-dimensional conductivity. Arch. Ration. Mech. Anal., 173(1), 133-150.
[9] Kadic, M., Schittny, R., Bückmann, T., Kern, C., & Wegener, M. (2015). Hall-Effect Sign Inversion in a Realizable 3D Metamaterial. Phys. Rev. X, 5, 021030.
[10] Kern, C., Kadic, M., & Wegener, M. (2015). Parallel Hall effect from three-dimensional single-component metamaterials. Appl. Phys. Lett., 107, 132103.
[11] Kern, C., Kadic, M., & Wegener, M. (2017). Experimental Evidence for Sign Reversal of the Hall Coefficient in Three-Dimensional Metamaterials. Phys. Rev. Lett., 118, 016601.
[12] Kern, C., Schuster, V., Kadic, M., & Wegener, M. (2017).  Experiments on the Parallel Hall Effect in Three-Dimensional Metamaterials. Phys. Rev. Applied, 7, 044001.
[13] Mani, R.G., & von Klitzing, K. (1993). Realization of dual, tunable, ordinary- and quantized-Hall resistances in doubly connected GaAs/AIGaAs heterostructures. Z. Phys. B, 92(3), 335-339.

Marc Briane is a mathematician at the Institut de Recherche Mathématique de Rennes, France. Muamer Kadic is a physicist at the Institut FEMTO-ST, CNRS at the Université de Bourgogne Franche-Comté, Besançon, France. Christian Kern and Martin Wegener are physicists at the Institute of Applied Physics and Institute of Nanotechnology at Karlsruhe Institute of Technology, Germany. Graeme Milton is a distinguished professor of mathematics at the University of Utah. Dylon Whyte is a chainmail artist in Ontario, Canada.