Metamaterials and the Landau–Lifshitz permeability argument: Large permittivity begets high-frequency magnetism

Abstract

Homogeneous composites, or metamaterials, made of dielectric or metallic particles are known to show magnetic properties that contradict arguments by Landau and Lifshitz [Landau LD, Lifshitz EM (1960) Electrodynamics of Continuous Media (Pergamon, Oxford, UK), p 251], indicating that the magnetization and, thus, the permeability, loses its meaning at relatively low frequencies. Here, we show that these arguments do not apply to composites made of substances with Im ͌ S Ͼ Ͼ /ഞ or Re ͌ S ϳ /ഞ ( S and ഞ are the complex permittivity and the characteristic length of the particles, and Ͼ Ͼ ഞ is the vacuum wavelength). Our general analysis is supported by studies of split rings, one of the most common constituents of electromagnetic metamaterials, and spherical inclusions. An analytical solution is given to the problem of scattering by a small and thin split ring of arbitrary permittivity. Results reveal a close relationship between S and the dynamic magnetic properties of metamaterials. For ͦ ͌ Sͦ Ͻ Ͻ /a (a is the ring cross-sectional radius), the composites exhibit very weak magnetic activity, consistent with the Landau-Lifshitz argument and similar to that of molecular crystals. In contrast, large values of the permittivity lead to strong diamagnetic or paramagnetic behavior characterized by susceptibilities whose magnitude is significantly larger than that of natural substances. We compiled from the literature a list of materials that show high permittivity at wavelengths in the range 0.3-3000 m. Calculations for a system of spherical inclusions made of these materials, using the magnetic counterpart to Lorentz-Lorenz formula, uncover large magnetic effects the strength of which diminishes with decreasing wavelength.effective medium theory ͉ electromagnetic scattering ͉ negative refraction ͉ split rings M etamaterials are homogeneous artificial mixtures; that is, composites become metamaterials when probed at wavelengths that are significantly larger than the average distance between its constituent particles. The electromagnetic properties of metamaterials have received considerable attention in the past decade motivated, to a large extent, by proposals of negative-index superlensing (1-3) as well as by their promise for a variety of microwave and optical applications such as novel antennas, beam 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). This can only occur in the vicinity of a resonance or, for the permittivity of metals, below the plasma frequency. Because magnetic resonances are very weak and, thus, negative values of are extremely rare in nature, it should not come as a surprise that, with the possible exception of La 2/3 Ca 1/3 MnO 3 (8), there is no natural substance known to posses a negative index. Because of this, considerable efforts have gone into the search for this elusive phenomenon in artificial systems. Unlike natural substances, various structures have bee...

Low-Frequency Scattering Analysis and Homogenisation of Split-Ring Elements

“…from which it is seen that kR decreases if N (2a) 3 increases, but we must take the upper limit N (2a) 3 < 1 into account. In Figure 7, the quantity (71) is plotted as a function of N , using the dimensions of the rings given above.…”

“…The two preferred directions of the particle are the z-direction due to the plane of the split-ring, and the x-direction due to the direction of the gap. Further we note that the (dominating part of the) (a x , a y ) parameters scale as a 3 , the (a z , b x , b y , b z ) parameters scale as ad 2 , c y = d z scale as αad 2 , and c z = d y scale as a 3 . Also, everywhere the angle α appears, it is multiplied by ad 2 .…”

“…These conditions require that the distances between the particles must be larger than the particles themselves and, at the same time, much smaller than a wavelength, in order for the Lorentz field to be applicable. For a cubic unit cell this condition can be formulated as k 3 < N < 1/(2a) 3 .…”

“…In total, the Lorentz field approach leads to the classical Clausius-Mossotti/Lorentz-Lorenz relation [3,13] for the effective susceptibility matrix χ…”

“…Other approaches to this problem are scarce, e.g., [3,4], but there are some analytical results for two-dimensional versions of the geometry [5,6]. We note that high-dielectric resonators [7] may provide an efficient alternative to split-ring resonators as constituent "molecules" of a medium displaying extraordinary magnetic properties.…”