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$\sqrt{{\epsilon }_{\text{S}}}$ ≫ λ/ℓ or Re$\sqrt{{\epsilon }_{\text{S}}}$ ∼ λ/ℓ (ε_S and ℓ are the complex permittivity and the characteristic length of the particles, and λ ≫ ℓ 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 ε_S and the dynamic magnetic properties of metamaterials. For |$\sqrt{{\epsilon }_{\text{S}}}$ | ≪ λ/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 μm. 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.

Metamaterials 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 ε and permeability μ 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 μ are extremely rare in nature, it should not come as a surprise that, with the possible exception of La_2/3Ca_1/3MnO₃ (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 been identified that exhibit significant bianisotropy (9, 10), associated with resonances of mixed electric–magnetic character, or unusually strong magnetic resonances that can be tuned to regions where ε is negative (11). These studies, a large fraction of which centers on split-ring resonators, have led to a large body of literature devoted to metamaterials magnetism covering the range from microwave to optical frequencies (12–16).

Although the magnetic behavior of metamaterials undoubtedly conforms to Maxwell's equations, the reason why artificial systems do better than nature is not well understood. Claims of strong magnetic activity are seemingly at odds with the fact that, other than magnetically ordered substances, magnetism in nature is a rather weak phenomenon at ambient temperature.* Moreover, high-frequency magnetism ostensibly contradicts well-known arguments by Landau and Lifshitz that the magnetization loses its physical meaning at rather low frequencies (17).

Here, we discuss the relevance of the Landau–Lifshitz argument for metamaterials and present a comparison between composites and their natural counterparts, molecular systems, which accounts for the profound differences between their magnetic properties. We show that a necessary condition for artificial magnetism is that the metamaterials be made of substances with κ_S ≫ λ/ℓ or n_S ∼ λ/ℓ where n_S + iκ_S = $\sqrt{{\epsilon }_{\text{S}}}$; ε_S and ℓ are the complex permittivity and the characteristic length of the particles in the composite, and λ ≫ ℓ is the vacuum wavelength. For inclusions with a large κ_S (n_S), the metamaterials may exhibit diamagnetic- (paramagnetic-)like resonances and, at non-zero frequencies, values of the permeability that are negative or comparable to that of superconductors (superparamagnets) in static fields. We note that the large-permittivity condition is consistent with recently reported simulations of plasmonic systems (18) and with the existence of a lower bound for the lattice size of negative-index systems (19), whose proof involves arguments very different from those of ours.

Landau–Lifshitz Permeability Argument

The total magnetic moment of an object can be obtained from the expression for the current density j = c∇×M+∂P/∂t; M is the magnetization, P is the polarization, t is the time, and c is the speed of light. Assuming a time dependence of the form exp(−iωt), the magnetic moment can be written as the volume integral

where ω = 2πc/λ is the angular frequency. Because the gradient of an arbitrary function can be added to M without affecting j, Landau and Lifshitz argue that the physical meaning of M, as being the magnetic moment per unit volume, requires that the magnetization-induced current be significantly larger than that due to the time-varying polarization. To determine the range for which this condition applies, they consider a situation that minimizes the P-contribution to the current, namely, a small object of dimension l ≪ λ placed in a quasistatic magnetic field so that |E| ∼ ωl |H|/c ≪ |H|. Here, E = D − 4πP and H = B − 4πM are, respectively, the electric field and the auxiliary magnetic field, whereas D = εE and B = μH are the electric-displacement field and the magnetic field appearing in Maxwell's equations of continuous media. Thus,

where χ_M is the magnetic and χ_E ∼1 is the dielectric susceptibility. For diamagnets at optical frequencies, Landau and Lifshitz use the estimate χ_M ∼ v²/c² ∼ d²/λ², where v is a characteristic speed of the electrons and d is the lattice parameter. This gives |c∇ × M|/|∂P/∂t| ∼ (d/l)² ≪ 1, which provides a compelling reason for ignoring M and setting μ = 1 at high frequencies (17).

There are two pieces to the Landau–Lifshitz argument. The first one involves the order-of-magnitude estimate for χ_M. As discussed here, the effective magnetic susceptibility of metamaterials composed of particles with large permittivity is significantly larger than that of natural diamagnets. The second, more subtle point concerns the uniqueness and significance of M. Given that the magnetic-dipole moment depends on the point of reference chosen if the object possesses a time-varying electric dipole (see Eq. 1), it is apparent that the magnetic-dipole density is ill defined even if |c∇ × M| ≫ |∂P/∂t|≠0. In metamaterials, it is better to define the magnetization as m/V_C where V_C is the volume and m is the magnetic-dipole moment of a unit cell calculated using a point inside the cell as the origin of coordinates (20).^† Because m → m − iωΔr × p/2c under the transformation r → r + Δr (p is the electric dipole of the unit cell), the origin ambiguity is removed if |m| ≫ (d/λ)|p|. As shown later, this applies to large-permittivity systems. We finally note that, although M and other multipoles depend on the choice of origin, the charge and current densities (and, therefore, the reflection and transmission coefficients as well as the effective permittivity and permeability) are, as they should, invariant at any order (21).

First Homogenization Step: Scattering by Small Particles

Consider a periodic array of identical particles of arbitrary shape and dimension ℓ ≪ λ. The lattice constants are also small compared with λ. As before, the complex permittivity of the particles is ε_S and their permeability is μ_S. The particles are immersed in a host medium of permittivity (permeability) ε_H (μ_H). There is a vast literature describing the many approaches to calculate effective-medium electromagnetic parameters (22–24), and many of the existing theories are closely related to models developed in the late 1800s and early 1900s. We note in particular the expressions for the effective permittivity obtained by Maxwell-Garnett (25) and by Bruggeman (26) that, in turn, are closely related to the much older Lorentz–Lorenz formula for time-dependent and the Clausius–Mosotti equation for static fields (27).

The solutions to Maxwell's equation in periodic arrangements (photonic crystals) are of the form e^iK.r F_K(r) where K is the Bloch–Floquet wavevector and F is a periodic function that possesses the same periodicity as the lattice. At low frequencies, ω = c_KK, where c_K is a parameter that depends on the direction of K, and the system can be described as a continuous medium in terms of the refractive-index tensor. The effective permittivity and the permeability tensor, ε_ij(ω) and μ_ij(ω), are introduced in the computation of the reflected and transmitted fields at a boundary. For optically isotropic substances, these tensors each have a single independent component, ε and μ, so that c_K = c/$\sqrt{\epsilon \mu }$ (for arbitrary K). Hence, the refractive index is n= $\sqrt{\epsilon \mu }$ whereas the wave impedance, which defines the reflectivity of a semi-infinite slab, is Z = $\sqrt{\mu /\epsilon }$. The low-frequency requirement reads K ≪ K_BZ, or λ ≫ 2d$\sqrt{\epsilon \mu }$, where K_BZ is the magnitude of a wavevector at the edge of the Brillouin zone and d is a lattice constant. This is a necessary condition for a periodic composite to be considered homogeneous. An independent and usually weaker condition is k ≪ K_BZ/$\sqrt{{\epsilon }_{\text{H}}{\mu }_{\text{H}}}$.

The (local) electric field (r)e^−iωt and magnetic field (r)e^−iωt in the immediate vicinity of a particle result from contributions caused by external sources and scattering from other particles. In self-consistent methods (22), the first step to compute bulk parameters is the calculation of the induced electric and magnetic multipoles. Note that, unless the filling factor, that is, the ratio between the volumes of the particle and the unit cell, is very small, higher-order multipoles matter even if ℓ ≪ λ (22). For μ_S = 1, the following expressions, including terms up to order ℓ/λ in the induced far field, contain all of the terms relevant to our problem (28)

Here, p = P/N is the electric dipole, m = M/N is the magnetic dipole, q_ij is the electric quadrupole tensor, and N is the number of particles per unit volume; α_ij and γ_ij are, respectively, the polarizability and the magnetizability (or magnetic polarizability) tensor of the particle. They, as well as G_ij, have the dimensions of a volume. This multipole expansion forms the basis of the so-called Casimir formulation of electrodynamics of continuous media (15, 29, 30) where optical activity (as shown by, e.g., an isotropic ensemble of chiral molecules) is associated with the tensors G_ij and A_ijk (28). Instead, Landau and Lifshitz describe optical activity in terms of a wavevector-dependent permittivity (17). We emphasize that m as well as q_ij and the coefficients in the multipole expansion usually depend on the origin of coordinates, a fact that needs to be addressed because the bulk effective parameters must certainly be invariant under a change of origin (21).

The coefficients in Eq. 3 can be expressed in a series involving powers of ℓ/λ. This leads to a natural classification of the multipolar coefficients and, hence, of the associated electromagnetic parameters. For molecular systems, the multipolar ordering can be deduced by using the following expressions from time-dependent quantum perturbation theory:

Here, the hat symbol represents a quantum operator, |0〉 and |s〉 denote, respectively, the ground and an excited state of the molecule, of frequency ω_S, and ℏ is Planck constant. Using the estimates |〈0|p̂|s〉| ∼ eℓ_mol and |〈0|Θ̂|s〉| ∼ eℓ_mol², where e is the electron charge and ℓ_mol is a characteristic size of the molecule, we get A/α∼ℓ_mol (it is understood that the origin of coordinates is close to the molecule). Moreover, the identity 〈j|v̂|s〉 = i(ω_j−ω_s)〈j|p̂|s〉/e gives |〈0|m̂|s〉| ∼ eℓ_molv/c ∼ eℓ_mol²/λ₀ and, thus, |γ/α| ∼ |G/α|² ∼ (ℓ_mol/λ₀)², where λ₀ is the resonant wavelength. Because e²/ℓ_mol ∼ ℏω₀, it follows that α ∼ ℓ_mol³ and, accordingly,

where V_mol = ℓ_mol³. Hence, the natural order places the polarizability as the single member of the top group, followed by G and A for the electric quadrupole and magnetic dipole. In agreement with the Landau–Lifshitz estimate for χ_M, Eq. 2, the magnetizability is of order ℓ_mol²/λ₀² and belongs to the third group that includes also the leading contributions to the electric octopole and magnetic quadrupole (31).

Let us turn back our attention to metamaterials and consider a nonmagnetic particle in a uniform quasistatic magnetic field ₀e^−iωtê_z, as in the Landau–Lifshitz argument. Provided |ε_S| ≪ λ²/ℓ², the induced magnetic field can be ignored so that in cylindrical coordinates, P ≈ (iωχ_E0/2c)rê_φ (because μ_S = 1, M = 0). Thus, from Eq. 1 the magnetic moment is

Under arbitrary conditions, but ignoring resonant effects, p ∼ χ_E0V and A ∼ χ_E0Vℓ (V ∼ ℓ³). Therefore, |γ/α|∼|A/αλ|² ∼ ℓ²/λ². Using arguments similar to those leading to Eq. 6, we get |G/α| ∼ ℓ/λ. Consequently, for |ε_S| ≪ λ²/ℓ² metamaterials follow the same natural order as molecules. The situation is, however, entirely different, and the natural order breaks down in the opposite limit. If the permittivity is unusually large and such that the skin depth δ = λ/2πκ_S is ≪ ℓ, where κ_S is the extinction coefficient, the particle behaves like a superconductor (perfect conductor) with respect to the magnetic (electric) field. The magnetic moment becomes ∼₀V, and thus |γ|∼|α| (the electric moment is always ∼₀V). Alternatively, if the refractive index n_S is sufficiently large so that n_S ∼ λ/ℓ ≫ 1 and κ_S ≪ n_S, the particle exhibits paramagnetic- or paraelectric-like behavior in that it can sustain cavity-like resonances that enhance the fields inside the particle.

The determination of the coefficients in Eq. 3 is but the first step in a calculation of the optical constants (22). If local-field effects can be ignored, that is, if E ≈ and B ≈ , we get for optically isotropic particles (ε_ij = εδ_ij and μ_ij = μδ_ij) ε = ε_H(1 + 4πNα) and μ = μ_H(1 + 4πNγ). In cases where the resonant behavior of the polarizability or the magnetizability can be described by a simple pole at ω₀, i.e., α,γ = f_α,γ(ω₀ − ω − iΓ)⁻¹, the effect of the resonance on ε and μ can be quantified by the dimensionless parameters NΔα and NΔγ, where Δα, Δγ = f_α,γ/ω₀. Based on the previous discussion, it is apparent that ordinary magnetic resonances must be extremely long-lived, Γ/ω₀ ≲ NΔγ ∼ (ℓ/λ₀)², to have an appreciable effect on the permeability.

Electromagnetic Scattering by a Thin Split Ring

Small particles can show resonances analogous to those of electrical circuits consisting of an inductor (L) and a capacitor (C), cavity-like and plasmon resonances (32) as well as strongly overdamped resonances that occur when δ ≈ ℓ. The split ring exhibits all four resonant forms. Consistent with our contention that magnetism requires large values of the permittivity, the magnetically active LC resonance (9) is underdamped only for κ_S ≳ λ/ℓ.

Consider a thin split ring of circular cross-section, placed in vacuum, with the parameters and coordinate axes shown in Fig. 1. The ring radius is r₀, the cross-sectional radius is a, and the gap thickness is g. The time dependence is exp (−iωt), and we take the permeability of the substance making the ring to be μ_S = 1. If the gap is momentarily ignored, a uniform, time-varying magnetic field oriented along the ẑ axis couples only to the symmetric mode, which is the only mode for which ∂E_w/∂w = 0 (33). To calculate the fields across a cross-section of the wire, we regard it as straight. Ignoring radiation losses, of order (ℓ/λ)³ (34), the solution, except near the gap, is E_ξ = E_θ = H_ξ = H_w = 0 and

where I is the total current, k = 2π/λ, and η = ka$\sqrt{{\epsilon }_{\text{S}}}$ = ka(n_S+iκ_S). The field at the gap is nearly uniform as for a parallel-plate capacitor. Its value can be gained from Eq. 7 by using conservation of charge, (1−ε_S⁻¹)I−iQω = 0, where Q is the total charge at one of the surfaces defining the gap. The result is

The current I and, thus, the magnetic moment m_z = (1−ε_S⁻¹)Iπr₀²/c, can be obtained from ℰ ≈ −iωLI/c²+E_w(a)2πr₀+iI/ωC. Here, ℰ = −(1/c)dΦ_e/dt is the electromotive force, L = 4πr₀ log(2πr₀/a) and C = a²/4g are the inductance and the gap capacitance for static fields, and Φ_e is the flux of the external magnetic field. We then find

where Z_spr = ζ−i(ωL/c²−1/ωC) is the split-ring impedance and ζ = i(4πr₀ω/ηc²)J₀(η)/J₁(η). From Eq. 8, we see that the magnetic dipole is linked through the current to an electric moment oriented along ŷ and proportional to the magnetic field so that

Reciprocally, and according to Eq. 3, an electric field along ŷ induces a magnetic moment parallel to ẑ. The associated current I = −(g/Z_spr*) generates, in turn, an electric-dipole moment along ŷ of magnitude (1−ε_S⁻¹)Ig/ω so that

Note that this expression describes only the contribution to α_yy due to induced charges at the surfaces defining the capacitor gap. There is an additional component, not considered here, associated with induced charges that are distributed along the length of the ring. With the origin at the center of the ring, the electric dipole gives rise to a quadrupole so that Θ_xx = Θ_yy = 0 and Θ_xy = Θ_yx = (3/4)r₀p_y. Hence,

We emphasize that all of these coefficients, with the exception of the polarizability, depend on the origin of coordinates (28). From Eqs. 9, 10, and 11, it follows that

This equation seemingly indicates that split rings also follow the molecular ordering, Eq. 5. However, we show below that this only applies to |ε_S| ≪ λ²/r₀². For large values of the permittivity, the resonant wavelength depends on the parameters of the particle in such a way that the natural order is not obeyed.

Fig. 1.

Split-ring parameters and geometry.

Large Permittivity Limit.

Consider first the case when κ_S ≫ λ/a (small skin-depth limit). Then J₀(η)/J₁(η) ≈ −i and, thus, ζ ≈ 4πr₀k/cη. This is the limit considered in ref. 9 and, more recently, in ref. 35. In this case, there is a resonance of the LC type at ω₀ = c/$\sqrt{LC}$. Because c|ζ| ≪ $\sqrt{L/C}$, the resonance is long-lived. The split ring behaves as a diamagnet, with Δγ ∼ πr₀³, reflecting the fact that the induced eddy or displacement currents oppose the external field. For the electric dipole, we have Δα ∼ πga². Hence, the magnetic-dipole strength is significantly larger and, thus, the split ring does not follow the natural ordering of Eq. 5. To understand this point further, it is convenient to relate the electric and magnetic moment of the particle to the change in the total energy of the corresponding field when the particle is introduced, as for static fields (17). Within this context, Δγ ≫ Δα because the excluded or screened volume for the magnetic field is much larger than for the electric field.

For Drude metals, ε_S = 1−ω_P²/ω(ω+i/τ), where ω_P is the plasma frequency and τ is the relaxation time. At frequencies ω_P ≫ ω ≫ τ⁻¹, n_S ≈ 0 and κ_S ≈ ω_p/ω. Hence, ζ behaves as an inductance, i.e., ζ≈−iωL_ζ/c², where L_ζ = 4πr₀c/aω_P. As discussed in ref. 35, this expression accounts for the saturation of the split-ring magnetic response at high frequencies (36). Note that L_ζ≠L_K = 8πr₀(c/aω_P)², where L_K is the kinetic inductance that was introduced ad hoc in ref. 36. At low frequencies, ε_S ≈ 4πiσ₀/ω (σ₀ is the dc conductivity). The real component of ζ is of the form ∼r₀/(aδσ₀), which represents the resistance associated with the skin depth δ.

When κ_s = 0, the multipolar coefficients exhibit cavity-like resonances at Z_spr(ω) = 0. For ω ≪ c/$\sqrt{LC}$, we get J₁(η)≈0 or n_Ska ≈ 5π/4, 9π/4, 13π/4.. . These resonances are similar to those of spherical inclusions, as discussed in ref. 37 and demonstrated experimentally in refs 38 and 39 and small cubes (40); however, note that, unlike split rings, the highly symmetric spheres and cubes do not allow for resonances of combined electric- and magnetic-dipole character. Cavity resonances are paramagnetic in nature because the magnetic field is drawn into the particle.

Small Permittivity Limit.

For |η| ≪ 1, J₀(η)/J₁(η)≈2/η and, therefore, ζ≈i8πr₀/a²ωε_S. For κ_S ≫ n_S, and at frequencies ω ≫ c/$\sqrt{LC}$, the transition between the limits δ ≫ a and δ ≪ a is accompanied by a large change in the magnetizability from nearly zero to γ_zz≈−π²r₀³. The cross-over manifests itself as a strongly overdamped diamagnetic resonance, as those observed in the simulations reported in ref. 41.

Because |ζ| ≈ 8πr₀ω/η²c² ≫ 4πr₀ω/c² ∼ ωL/c², the LC resonance washes out in this limit. This effect has been reported in recent experiments and simulations of metallic rings (42). For metals at ω ≪ τ⁻¹, ζ ≈ R = 2r₀/a²σ₀ turns into the dc resistance of the ring. The LC resonance becomes overdamped because of the increase in ohmic losses for δ ≫ a. Also, notice that in the range ω_P ≫ ω ≫ τ⁻¹, ε_S ≈ −ω_P²/ω² and ζ ≈ −8πiωr₀/a²ω_P² behaves like an inductance (for dielectrics, ζ≈ir₀/ωa²ε_S is capacitive-like).

For ω ≪ c/$\sqrt{LC}$, we get from Eq. 9

This expression has a plasmon-like resonance at a frequency ω₀ so that 2πr₀ + gε_S(ω₀) = 0, a condition that corresponds to ∮E.dl = 0 (this resonance should not be confused with the conventional plasmon resonances involving charges distributed over the outer surfaces of the ring). Because a²|ε_S| ≪ λ² and, thus, a²r₀ ≪ gλ², we get for a Drude metal Δγ ∼ r₀⁴a²/ λ²g ≪ r₀³. Note that Δα ≈ a²g ≫Δγ for -ϵ_s-≪λ²/r₀². These considerations also apply to results reported for horseshoe-shaped antennas (43, 44).

Split Rings vs. Molecules.

For split rings with |ε_S| ≪ λ²/ℓ², the coefficients in Eq. 3 follow the same natural order found in molecular systems. Indeed, the expressions for molecules, Eq. 5, become identical to those of thin split rings if one makes the substitution ℓ_mol → r₀²/g. Nevertheless, split rings follow a very different order if the permittivity is very large. If one compares the molecular and the split-ring equations for the magnetizability, Eq. 4b and Eq. 9, one obtains for large |ε_S|

which is ∼10⁻⁵ to 10⁻⁶ at optical frequencies. The reason why the magnetic activity is so much larger for high-permittivity split rings is, first, the fact that molecular magnetism is a weak relativistic effect and, second, the superdiamagnetic (superparamagnetic) response of the ring for δ ≪ ℓ (n_S ∼ λ/ℓ) that behaves effectively as an object with μ_S = 0 (μ_S = ∞). Note that the effect of the permittivity on the electric moment is considerably less important because, to lowest order, the electric moment does not depend on the wavelength.

Optical Constants of Metamaterials: Lorentz–Lorenz and Lewin's Formula.

The Lorentz–Lorenz formula (27)

is a widely used approximation to the effective complex permittivity of an optically isotropic system, such as spheres in a cubic lattice and a randomly oriented set of split rings.^‡ Its magnetic counterpart is

where α̃ and γ̃ are the averages over all orientations. We include these expressions here to illustrate the effect a single-particle resonance has on the collective behavior of the metamaterial. It is apparent that, other than for the frequency shifts caused by coupling between particles, poles in the polarizability and magnetizability lead to resonances in ε and μ, with concomitant regions of anomalous dispersion (45). It is worthwhile noticing that Eq. 16 and Eq. 17 follow from Eq. 3 if one takes, respectively, (0) = E + 4πP/3 and (0) = H + 4πM/3. Eq. 17 appears to have been first derived by Lewin for a lattice of spheres (46). These expressions have been rediscovered by many authors; see, for example, ref. 41. An extensive list of early references can be found in ref. 37 and at www.wave-scattering.com/negative.html.

Scattering by Spheres

The behavior of spheres is very similar to that of split rings, particularly in regard to the magnitude of the multipolar coefficients in the low- and high-permittivity limits. The main difference is that the higher spherical symmetry leads to a separation between magnetic and electric multipole resonances and hence, that spheres do not exhibit resonances of the LC type. The polarizability α_sph and magnetizability γ_sph of a sphere of radius R are known exactly from the work of Mie (47). The Bruggerman (26) and Maxwell-Garnet (25) expressions are based on Mie scattering and, as such, they also pertain to a collection of spheres. The later model (25) corresponds to the so-called static limit where θ = kR$\sqrt{{\mu }_{\text{S}}{\epsilon }_{\text{S}}}$ ≪ 1. In this case, we get for R ≪ λ and a nonmagnetic medium γ_sph ≈ R³θ²/30 ∼ V_sph(R/λ)²|ε_S| (V_sph = 4πR³/3) and, from Eq. 17, μ ≈ 1, whereas the polarizability is given by the well-known expression α_sph ≈ R³(ε_S − ε_H)/(ε_S + 2ε_H). In contrast, for θ large and imaginary, α_sph ∼ R³ and γ_sph ∼ −R³/2. Thus, μ ≈ (1 − NV_S)/(1 + NV_S/2) < 1 and ε/ε_H ≈ (1 + 2NV_S)/(1 − NV_S) > 1 (46). Similar to the cavity-like resonances of split rings, the magnetizability diverges for θ = π, 2π, 3π, (we assume μ_S = μ_H = 1). Note that the positions of the resonances are slightly different for electric-dipole transitions (37). Recent experiments have corroborated the validity of these results (38–40).

Concluding Discussion

The double constraint κ_S ≫ λ/ℓ ≫ 1 (or, n_S ∼ λ/ℓ ≫ 1 if κ_S ≪ n_S) poses severe limitations for attaining magnetism at arbitrarily high frequencies. Because they have a large extinction coefficient, metals are to be favored at optical frequencies. Because ε_S ≈ −ω_P²/ω² for ωτ ≫ 1, the constraint becomes λ ≫ ℓ ≫ λ_P (note that the skin depth is nearly constant in this range: δ ≈ c/ω_P). The measured values of the permittivity for noble metals (48) indicate that magnetism can coexist with the effective-medium condition for frequencies up to ∼1.5 × 10¹⁴ Hz (λ ∼2.5 μm). This estimate is in very good agreement with simulations of plasmonic metamaterials (18). A similar order-of-magnitude estimate follows from the realization that the skin depth must be no less than a few lattice sites, say, δ ≳ δ_min ∼ 100 Å, or ℓ ≫ 2πδ_min. Although substances with large, but not exceedingly large, permittivity are not expected to lead to negative values of the permeability because of losses, metamaterials involving such substances may nevertheless show a magnetic response that is incompatible with the Landau–Lifshitz argument. This is demonstrated in Table 1, which lists calculated values of μ using Lewin's formula (46) for a simple-cubic lattice of spheres. The optical parameters of Ag, Cu, KTa_0.982Nb_0.018O₃, PbTe, SrTiO₃, SiC, and Sb are, respectively, from refs. 48, 49, 50, 51, 52, 53, and 54. Those for Ge and Si are from ref. 55.

Table 1.

Calculated effective-medium permeability of a system of spherical particles of radius R in a simple-cubic arrangement of lattice constant d

Material	λ, μm	$\sqrt{{\epsilon }_{\text{S}}}$ = nS + iκS	μ1 + iμ2
Cu	3,000	975 + i975	0.382 + i0.005
KTa0.982Nb0.018O3	500	17.3 + i0.58	3.322 + i1.226
PbTe	312.5	43.4 + i43.0	0.487 + i0.102
SrTiO3	111.0	25 + i25	0.571 + i0.165
SiC	12.5	17 + i17	0.678 + i0.221
Sb	4.0	9.73 + i13.77	0.811 + i0.163
Ag	1.93	0.24 + i14.09	0.834 + i0.004
Ge	0.590	5.75 + i1.63	1.041 + i0.029
Si	0.288	4.09 + i5.39	0.978 + i0.053

The refractive index n_S and the extinction coefficient κ_S of the corresponding materials are room temperature values at the wavelengths shown. Results are for d = 2R = λ /20 and ε_H = 1.96. Note the paramagnetic response of the substances for which n_S dominates over κ_S.