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On the Hall Effect in Three-Dimensional Metamaterials Christian Kern On the Hall effect in three-dimensional metamaterials 2019 Dissertation On the Hall effect in three-dimensional metamaterials Zur Erlangung des akademischen Grades eines doktors der naturwissenschaften von der KIT-Fakultät für Physik des Karlsruher Instituts für Technologie (KIT) genehmigte dissertation von MSc ETH Christian Frederik Kern geboren in Stuttgart Tag der mündlichen Prüfung: 8. Februar 2019 Referent: Prof. Dr. Martin Wegener Korreferent: Prof. Dr. Jörg Schmalian Contents Publications ........................ 1 1 Introduction ...................... 3 2 Fundamentals ..................... 7 2.1 The classical Hall effect . 8 2.2 Tensorial properties and symmetry . 22 2.3 The Hall effect in anisotropic materials . 29 2.4 Metamaterials . 32 2.5 The Hall effect in composites and metamaterials . 37 3 Theory of Hall metamaterials ............. 41 3.1 Fundamentals . 42 3.2 Sign-inversion of the effective Hall coefficient . 48 3.3 Anisotropic structures . 62 3.4 Bounds on the effective parameters . 75 3.5 Magnetic permeability distributions . 77 v contents 4 Methods ........................ 87 4.1 Three-dimensional laser lithography . 88 4.2 Atomic layer deposition . 96 4.3 Electron-beam physical vapor deposition . 107 5 Probe-station based experiments ............ 109 5.1 Fabrication . 110 5.2 Measurement setup . 113 5.3 Sign-inversion of the effective Hall coefficient . 116 5.4 Parallel Hall effect . 122 6 Hall-bar devices .................... 127 6.1 Fabrication . 128 6.2 Measurement setup . 133 6.3 Investigating the role of unit cell orientation . 135 7 Conclusions &Outlook ................ 141 Bibliography ....................... 145 Acknowledgments ..................... 161 vi Publications Parts of this thesis have already been published ... in scientific journals: M. Kadic, R. Schittny, T. Bückmann, C. Kern, and M. Wegener, “Hall- • Effect Sign Inversion in a Realizable 3D Metamaterial”, Phys. Rev. X 5, 021030 (2015) C. Kern, M. Kadic, and M. Wegener, “Parallel Hall effect from three- • dimensional single-component metamaterials”, Appl. Phys. Lett. 107, 132103 (2015) C. Kern, M. Kadic, and M. Wegener, “Experimental Evidence for Sign • Reversal of the Hall Coefficient in Three-Dimensional Metamaterials”, Phys. Rev. Lett. 118, 016601 (2017) C. Kern, V. Schuster, M. Kadic, and M. Wegener, “Experiments on the • Parallel Hall Effect in Three-Dimensional Metamaterials”, Phys. Rev. Applied 7, 044001 (2017) M. Wegener, M. Kadic, and C. Kern, “Hall-effect metamaterials and • ’anti-Hall bars’”, Phys. Today 70, Issue 10, 14 (2017) M. Briane, M. Kadic, C. Kern, G. W. Milton, M. Wegener, and D. Whyte, • “Surprises Regarding the Hall Effect: An Extraordinary Story Involving an Artist, Mathematicians, and Physicists”, SIAM Newsjournal 7, Issue 10, 1 (2017) C. Kern, M. Kadic, M. Wegener, “Kern, Kadic, and Wegener Reply”, • Reply to a comment by J. Oswald on “Experimental Evidence for Sign Reversal of the Hall Coefficient in Three-Dimensional Metamaterials”, Phys. Rev. Lett. 120, 149702 (2018) 1 publications C. Kern, G. W. Milton, M. Kadic, and M. Wegener, “Theory of the Hall • effect in three-dimensional metamaterials”, New J. Phys. 20, 083034 (2018) C. Kern and M. Wegener, “Three-dimensional metamaterial Hall-bar • devices”, Phys. Rev. Mater. 3, 015204 (2019) . at scientific conferences (only own presentations): C. Kern, M. Kadic, and M. Wegener, “Sign reversal of the Hall coeffi- • cient in three-dimensional metamaterials” (invited talk), Auxetics 2016, Szymbark, Poland, September 2016 C. Kern, M. Kadic, G. W. Milton, and M. Wegener, “The Hall effect in • composites” (key speaker), ETOPIM11, Krakow, Poland, July 2018 C. Kern, M. Kadic, and M. Wegener, G. W. Milton, “Hall Effect Meta- • materials” (invited talk), Metamaterials 2018, Espoo, Finland, August 2018 C. Kern, G. W. Milton, M. Kadic, and M. Wegener, “On the integration • and characterization of chainmail-inspired Hall effect metamaterials”, Metamaterials 2018, Espoo, Finland, August 2018 Additional related work has already been published ... in scientific journals: K. Edelmann, L. Gerhard, M. Winkler, L. Wilmes, V. Rai, M. Schumann, • C. Kern, M. Meyer, M. Wegener, and W. Wulfhekel, “Light collection from a low-temperature scanning tunneling microscope using inte- grated mirror tips fabricated by direct laser writing”, Rev. Sci. Instrum. 89, 123107 (2018) S. Mannherz, A. Niemeyer, F. Mayer, C. Kern, and M. Wegener, “On • the limits of laminates in diffusive optics”, Opt. Express 26, 34274 (2018) 2 Chapter 1 1 Introduction In his 1879 paper on what we today know as the Hall effect [1], or more precisely, the classical Hall effect, Edwin Hall quotes a passage of Maxwell’s famous treatise on electricity and magnetism [2]. There, Maxwell states that, "It must be carefully remembered, that the mechanical force which urges a conductor carrying a current across the lines of magnetic force, acts, not on the electric current, but on the conductor which carries it." Contrary to Maxwell’s statement, Hall was able to show that a magnetic field directly affects the current rather than the conductor carrying it [1, 3]. In his experiments, Hall studied samples made from rectangular metal platelets. He imposed an electric current using a battery and a magnetic field using an electromagnet. Using a galvanometer, he was able to measure a transversal voltage, which we today refer to as the Hall voltage. According to the modern textbook explanation, the effect is caused by the action of the Lorentz force on the charge carriers that leads to a charge accumulation and an opposing electric field, the so-called Hall electric field. The voltage corresponding to this electric field is the Hall voltage, which is proportional to the imposed current and the magnetic induction and inversely proportional to the thickness of the sample. The proportionality constant is the so-called Hall coefficient. Today, sensors based on the Hall effect are regularly used to measure magnetic fields. Such Hall sensors are employed in many devices that we use in our everyday lives. Most commonly, they are used in combination with a permanent magnet for a variety of sensing applications [4]. In smartphones, they are used to determine whether a flip cover is closed. In every modern car, Hall sensors are employed, for example, in order to measure rotational 3 1 introduction speeds [5] or to determine whether a seat belt buckle is latched [6]. A different application of the Hall effect is its use in the characterization of semiconductors. In bulk semiconductors, the sign of the Hall coefficient and, hence, the sign of the Hall voltage is given by the sign of the charge carriers. Therefore, by measuring the Hall voltage, one can identify the predominant type of charge carriers. It turn out that this statement is false if one considers a more general class of materials, so-called metamaterials. For a certain type of structures that were inspired by medieval chainmail, the sign of the effective Hall coefficient is not identical to the sign of the charge carriers. In general, metamaterials are finely structured materials that are composed of several constituent materials and that, on a larger length scale, effectively act like homogeneous materials with properties that go beyond those of the constituent materials. Their effective properties are determined by their microscopic structure and, therefore, can be tailored to specific needs. Usu- ally, the microscopic structure is a periodic arrangement of certain building blocks very much like a conventional crystal is a periodic arrangement of atoms. Metamaterials vastly extend our set of available material properties. Furthermore, using metamaterials, it is possible to realize a wide variety of effective material properties starting from a relatively small set of constituent materials. In this thesis, I am studying how one can use three-dimensional metama- terials to tailor the effective conductivity in the limit of weak magnetic fields. I am referring to such metamaterials as Hall metamaterials. As it turns out, very unusual effective properties can be realized. A prime example is the sign-inversion of the effective Hall coefficient mentioned above. Moreover, I will show that it is possible to realize any arbitrary effective Hall coefficient using a single-constituent porous metamaterial. Very interesting effects ap- pear in the anisotropic case. For example, certain anisotropic metamaterials exhibit the parallel Hall effect. In such metamaterials, the Hall electric field is parallel rather than perpendicular to the external magnetic field. In general, one can show that the influence of a weak magnetic field on the behavior of a conductive material is described by a rank-two tensor, the so-called Hall tensor. Throughout this thesis, I am exploring both theoret- ically and experimentally how the effective Hall tensor of a metamaterial can be tailored by structure. On the experimental side, I am fabricating 4 and characterizing Hall metamaterials on the microscale. A main goal is the experimental demonstration of the sign-inversion of the effective Hall coefficient and the parallel Hall effect. All of these efforts aim at approaching the ultimate goal of Hall metamate- rials, that is, to obtain full control over the effective electrical properties of a conductive material in a weak magnetic field by tailoring its microscopic structure. Outline of this thesis In chapter 2, I will discuss the fundamentals that serve as a basis for the subsequent discussion of Hall metamaterials. Theoretical
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