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Metamaterials and the Landau–Lifshitz Permeability Argument: Large Permittivity Begets High-Frequency Magnetism

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Metamaterials and the Landau–Lifshitz Permeability Argument: Large Permittivity Begets High-Frequency Magnetism Metamaterials and the Landau–Lifshitz permeability argument: Large permittivity begets high-frequency magnetism Roberto Merlin1 Department of Physics, University of Michigan, Ann Arbor, MI 48109-1040 Edited by Federico Capasso, Harvard University, Cambridge, MA, and approved December 4, 2008 (received for review August 26, 2008) Homogeneous composites, or metamaterials, made of dielectric or resonators, have led to a large body of literature devoted to metallic particles are known to show magnetic properties that con- metamaterials magnetism covering the range from microwave to tradict arguments by Landau and Lifshitz [Landau LD, Lifshitz EM optical frequencies (12–16). (1960) Electrodynamics of Continuous Media (Pergamon, Oxford, UK), Although the magnetic behavior of metamaterials undoubt- p 251], indicating that the magnetization and, thus, the permeability, edly conforms to Maxwell’s equations, the reason why artificial loses its meaning at relatively low frequencies. Here, we show that systems do better than nature is not well understood. Claims of these arguments do not apply to composites made of substances with ͌ ͌ strong magnetic activity are seemingly at odds with the fact that, Im ␧S ϾϾ ␭/ഞ or Re ␧S ϳ ␭/ഞ (␧S and ഞ are the complex permittivity ␭ ϾϾ ഞ other than magnetically ordered substances, magnetism in na- and the characteristic length of the particles, and is the ture is a rather weak phenomenon at ambient temperature.* vacuum wavelength). Our general analysis is supported by studies Moreover, high-frequency magnetism ostensibly contradicts of split rings, one of the most common constituents of electro- well-known arguments by Landau and Lifshitz that the magne- magnetic metamaterials, and spherical inclusions. An analytical solution is given to the problem of scattering by a small and thin tization loses its physical meaning at rather low frequencies (17). split ring of arbitrary permittivity. Results reveal a close relation- Here, we discuss the relevance of the Landau–Lifshitz argu- ment for metamaterials and present a comparison between SCIENCES ship between ␧S and the dynamic magnetic properties of meta- ͌ APPLIED PHYSICAL materials. For ͦ ␧Sͦ ϽϽ ␭/a (a is the ring cross-sectional radius), the composites and their natural counterparts, molecular systems, composites exhibit very weak magnetic activity, consistent with which accounts for the profound differences between their the Landau–Lifshitz argument and similar to that of molecular magnetic properties. We show that a necessary condition for crystals. In contrast, large values of the permittivity lead to strong artificial magnetism is that the metamaterials be made of ͌ diamagnetic or paramagnetic behavior characterized by suscepti- substances with ␬S ϾϾ ␭/ᐉ or nS ϳ ␭/ᐉ where nS ϩ i␬S ϭ ␧S; ␧S bilities whose magnitude is significantly larger than that of natural and ᐉ are the complex permittivity and the characteristic length substances. We compiled from the literature a list of materials that of the particles in the composite, and ␭ ϾϾ ᐉ is the vacuum ␮ show high permittivity at wavelengths in the range 0.3–3000 m. wavelength. For inclusions with a large ␬S (nS), the metamate- Calculations for a system of spherical inclusions made of these rials may exhibit diamagnetic- (paramagnetic-)like resonances materials, using the magnetic counterpart to Lorentz–Lorenz for- and, at non-zero frequencies, values of the permeability that are mula, uncover large magnetic effects the strength of which dimin- negative or comparable to that of superconductors (superpara- ishes with decreasing wavelength. magnets) in static fields. We note that the large-permittivity condition is consistent with recently reported simulations of ͉ ͉ effective medium theory electromagnetic scattering plasmonic systems (18) and with the existence of a lower bound ͉ negative refraction split rings for the lattice size of negative-index systems (19), whose proof involves arguments very different from those of ours. etamaterials are homogeneous artificial mixtures; that is, Mcomposites become metamaterials when probed at wave- Landau–Lifshitz Permeability Argument lengths that are significantly larger than the average distance The total magnetic moment of an object can be obtained from between its constituent particles. The electromagnetic properties the expression for the current density j ϭ cٌϫMϩѨP/Ѩt; M is the of metamaterials have received considerable attention in the past magnetization, P is the polarization, t is the time, and c is the decade motivated, to a large extent, by proposals of negative-index speed of light. Assuming a time dependence of the form superlensing (1–3) as well as by their promise for a variety of exp(Ϫi␻t), the magnetic moment can be written as the volume microwave and optical applications such as novel antennas, beam integral steerers, sensors, and cloaking devices (4, 5). The refractive index of a material is negative if both the effective-medium ␧ ␮ ␻ permittivity and permeability are themselves negative (6, 7). ͵ͩM Ϫ i r ϫ Pͪ dV [1] This can only occur in the vicinity of a resonance or, for the 2c permittivity of metals, below the plasma frequency. Because magnetic resonances are very weak and, thus, negative values of Author contributions: R.M. designed research, performed research, and wrote the paper. ␮ are extremely rare in nature, it should not come as a surprise The author declares no conflict of interest. that, with the possible exception of La Ca MnO (8), there is 2/3 1/3 3 This article is a PNAS Direct Submission. no natural substance known to posses a negative index. Because of this, considerable efforts have gone into the search for this Freely available online through the PNAS open access option. 1 elusive phenomenon in artificial systems. Unlike natural sub- E-mail: [email protected]. Ϫ6 Ϫ7 *The magnetic susceptibility of diamagnets is typically in the range ␹M ϭϪ10 to 10 stances, various structures have been identified that exhibit Ϫ5 with record values for bismuth (␹M ϭϪ1.3 ϫ 10 ) and pyrolytic graphite (␹M ϭϪ3.2 ϫ significant bianisotropy (9, 10), associated with resonances of 10Ϫ5). Paramagnetic behavior is associated with spin degrees of freedom, and, as a result, Ϫ4 Ϫ5 mixed electric–magnetic character, or unusually strong magnetic paramagnets exhibit a somewhat larger susceptibility, ␹M Ϸ10 to 10 at room tem- resonances that can be tuned to regions where ␧ is negative (11). perature, that is well described by the Curie–Weiss law. These studies, a large fraction of which centers on split-ring © 2009 by The National Academy of Sciences of the USA www.pnas.org͞cgi͞doi͞10.1073͞pnas.0808478106 PNAS ͉ February 10, 2009 ͉ vol. 106 ͉ no. 6 ͉ 1693–1698 Downloaded by guest on September 29, 2021 where ␻ ϭ 2␲c/␭ is the angular frequency. Because the gradient The solutions to Maxwell’s equation in periodic arrangements iK.r of an arbitrary function can be added to M without affecting j, (photonic crystals) are of the form e FK(r) where K is the Landau and Lifshitz argue that the physical meaning of M,as Bloch–Floquet wavevector and F is a periodic function that being the magnetic moment per unit volume, requires that the possesses the same periodicity as the lattice. At low frequencies, magnetization-induced current be significantly larger than that ␻ ϭ cKK, where cK is a parameter that depends on the direction due to the time-varying polarization. To determine the range for of K, and the system can be described as a continuous medium which this condition applies, they consider a situation that in terms of the refractive-index tensor. The effective permittivity ␧ ␻ ␮ ␻ minimizes the P-contribution to the current, namely, a small and the permeability tensor, ij( ) and ij( ), are introduced in object of dimension l ϽϽ ␭ placed in a quasistatic magnetic field the computation of the reflected and transmitted fields at a so that ͉E͉ ϳ ␻l ͉H͉/c ϽϽ ͉H͉. Here, E ϭ D Ϫ 4␲P and H ϭ B Ϫ boundary. For optically isotropic substances, these tensors each ␧ ␮ ϭ 4␲M are, respectively, the electric field and the auxiliary mag- have a single independent component, and , so that cK ͌ ͌ netic field, whereas D ϭ ␧E and B ϭ ␮H are the electric- c/ ␧␮ (for arbitrary K). Hence, the refractive index is nϭ ␧␮, displacement field and the magnetic field appearing in Maxwell’s whereas the wave impedance, which defines the reflectivity of a semi-infinite slab, is Z ϭ ͌␮/␧. The low-frequency requirement equations of continuous media. Thus, ͌ reads K ϽϽ KBZ,or␭ ϾϾ 2d ␧␮, where KBZ is the magnitude of ϫ ͉ ␹ ͑ ͒ ␭ 2 ␹ ٌ ͉ c M c M H/l M a wavevector at the edge of the Brillouin zone and d is a lattice ϳ ϳ ͩ ͪ [2] ͉ѨP/Ѩt͉ ␻␹ E l ␹ constant. This is a necessary condition for a periodic composite E E to be considered homogeneous. An independent and usually ϽϽ ͌␧ ␮ where ␹M is the magnetic and ␹E ϳ1 is the dielectric suscepti- weaker condition is k KBZ/ H H. Ϫ ␻ Ϫ ␻ bility. For diamagnets at optical frequencies, Landau and Lif- The (local) electric field ᑟ(r)e i t and magnetic field ᑜ(r)e i t in 2 2 2 2 shitz use the estimate ␹M ϳ v /c ϳ d /␭ , where v is a charac- the immediate vicinity of a particle result from contributions caused teristic speed of the electrons and d is the lattice parameter. This by external sources and scattering from other particles. In self- gives ͉cٌϫM͉/͉ѨP/Ѩt͉ ϳ (d/l)2 ϽϽ 1, which provides a compelling consistent methods (22), the first step to compute bulk parameters reason for ignoring M and setting ␮ ϭ 1 at high frequencies (17). is the calculation of the induced electric and magnetic multipoles. There are two pieces to the Landau–Lifshitz argument.
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