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 iS 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(Ϫit), 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 2c/ 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 4P and H ϭ B Ϫ boundary. For optically isotropic substances, these tensors each ϭ 4M 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. The Note that, unless the filling factor, that is, the ratio between the first one involves the order-of-magnitude estimate for .As volumes of the particle and the unit cell, is very small, higher-order M ᐉ ϽϽ ϭ discussed here, the effective magnetic susceptibility of metama- multipoles matter even if (22). For S 1, the following ᐉ terials composed of particles with large permittivity is signifi- expressions, including terms up to order / in the induced far field, cantly larger than that of natural diamagnets. The second, more contain all of the terms relevant to our problem (28) M subtle point concerns the uniqueness and significance of . 1 ␣ ᑟ ͑ ͒ ϩ Ѩᑟ Ѩ ͉ ϩ ᑜ ͑ ͒ ϩ Given that the magnetic-dipole moment depends on the point of ϭ 0 A / x ϭ G 0 ... pi ij j 3 ijk j k r 0 ij j reference chosen if the object possesses a time-varying electric jk dipole (see Eq. 1), it is apparent that the magnetic-dipole density ␥ ᑜ ͑ ͒ ϩ ᑟ ͑ ͒ ϩ [0. In metamaterials, it ϭ ij j 0 G*ij j 0 ... [3 is ill defined even if ͉cٌϫM͉ ϾϾ ͉ѨP/Ѩt͉ mi is better to define the magnetization as m/VC where VC is the j m volume and is the magnetic-dipole moment of a unit cell ᑟ ͑ ͒ ϩ ϭ A*ijk k 0 .... calculated using a point inside the cell as the origin of coordi- qij nates (20).† Because m 3 m Ϫ i⌬r ϫ p/2c under the trans- k formation r 3 r ϩ⌬r (p is the electric dipole of the unit cell), ϭ ϭ ͉ ͉ ϾϾ ͉ ͉ Here, p P/N is the electric dipole, m M/N is the magnetic the origin ambiguity is removed if m (d/ ) p . As shown later, dipole, q is the electric quadrupole tensor, and N is the number of this applies to large-permittivity systems. We finally note that, ij particles per unit volume; ␣ij and ␥ij are, respectively, the polariz- although M and other multipoles depend on the choice of origin, ability and the magnetizability (or magnetic polarizability) tensor of the charge and current densities (and, therefore, the reflection the particle. They, as well as Gij, have the dimensions of a volume. and transmission coefficients as well as the effective permittivity This multipole expansion forms the basis of the so-called Casimir and permeability) are, as they should, invariant at any order (21). formulation of electrodynamics of continuous media (15, 29, 30) where optical activity (as shown by, e.g., an isotropic ensemble of First Homogenization Step: Scattering by Small Particles chiral molecules) is associated with the tensors Gij and Aijk (28). Consider a periodic array of identical particles of arbitrary shape Instead, Landau and Lifshitz describe optical activity in terms of a ᐉ ϽϽ and dimension . The lattice constants are also small wavevector-dependent permittivity (17). We emphasize that m as compared with . As before, the complex permittivity of the well as q and the coefficients in the multipole expansion usually ij particles is S and their permeability is S. The particles are depend on the origin of coordinates, a fact that needs to be immersed in a host medium of permittivity (permeability) H addressed because the bulk effective parameters must certainly be ( H). There is a vast literature describing the many approaches invariant under a change of origin (21). to calculate effective-medium electromagnetic parameters (22– The coefficients in Eq. 3 can be expressed in a series involving 24), and many of the existing theories are closely related to powers of ᐉ/. This leads to a natural classification of the models developed in the late 1800s and early 1900s. We note in multipolar coefficients and, hence, of the associated electromag- particular the expressions for the effective permittivity obtained netic parameters. For molecular systems, the multipolar order- by Maxwell-Garnett (25) and by Bruggeman (26) that, in turn, ing can be deduced by using the following expressions from are closely related to the much older Lorentz–Lorenz formula time-dependent quantum perturbation theory: for time-dependent and the Clausius–Mosotti equation for static ͗0͉pˆ ͉s͗͘s͉pˆ ͉0͘ fields (27). ␣ ϭ i j ij ប͑ Ϫ ͒ [4a] s s †This is the approach used for molecular systems which, like metamaterials, consist of ͗0͉mˆ ͉s͗͘s͉mˆ ͉0͘ weakly interacting objects, with well-defined boundaries, for which the multipolar ex- ␥ ϭ i j pansion of the induced fields is well defined (except for the origin dependence). This is not ij ប͑ Ϫ ͒ [4b] s the case of metals or covalent solids where the multipoles depend also on the choice of cell. s
1694 ͉ www.pnas.org͞cgi͞doi͞10.1073͞pnas.0808478106 Merlin Downloaded by guest on September 29, 2021 ͗0͉pˆ ͉s͗͘s͉⌰ˆ ͉0͘ ϭ i jk Aijk ប͑ Ϫ ͒ [4c] s s
͗0͉pˆ ͉s͗͘s͉mˆ ͉0͘ ϭ i j Gij ប͑ Ϫ ͒ . [4d] s s
Here, the hat symbol represents a quantum operator, ͉0͘ and ͉s͘ denote, respectively, the ground and an excited state of the mole- cule, of frequency S, and ប is Planck constant. Using the estimates ͉͗ ͉ ͉ ͉͘ ϳ ᐉ ͉͗ ͉⌰͉ ͉͘ ϳ ᐉ2 0 pˆ s e mol and 0 ˆ s e mol, where e is the electron charge and ᐉmol is a characteristic size of the molecule, we get A/␣ϳᐉmol (it is understood that the origin of coordinates is close to the molecule). Moreover, the identity ͗ j͉vˆ͉s͘ϭi(jϪs)͗ j͉ pˆ͉s͘/e gives ͉͗ ͉ ͉ ͉͘ ϳ ᐉ ϳ ᐉ2 ͉␥ ␣͉ ϳ ͉ ␣͉2 ϳ ᐉ 2 0 mˆ s e molv/c e mol/ 0 and, thus, / G/ ( mol/ 0) , 2 where 0 is the resonant wavelength. Because e /ᐉmol ϳប0,it ␣ ϳ ᐉ3 follows that mol and, accordingly, ᐉ ᐉ2 ␣ ϳ ϾϾ ͭ Gmol ͮ ϳ mol ϾϾ␥ ϳ mol mol Vmol Vmol mol Vmol 2 Fig. 1. Split-ring parameters and geometry. Amol/ 0 0 0 [5] ordinary magnetic resonances must be extremely long-lived, ϭ ᐉ3 ⌫ Շ ⌬␥ ϳ ᐉ 2 where Vmol mol. Hence, the natural order places the polar- / 0 N ( / 0) , to have an appreciable effect on the izability as the single member of the top group, followed by G permeability. and A for the electric quadrupole and magnetic dipole. In
Electromagnetic Scattering by a Thin Split Ring SCIENCES agreement with the Landau–Lifshitz estimate for M, Eq. 2, the ᐉ2 2 magnetizability is of order mol/ 0 and belongs to the third group Small particles can show resonances analogous to those of APPLIED PHYSICAL that includes also the leading contributions to the electric electrical circuits consisting of an inductor (L) and a capacitor octopole and magnetic quadrupole (31). (C), cavity-like and plasmon resonances (32) as well as strongly Let us turn back our attention to metamaterials and consider overdamped resonances that occur when ␦ Ϸ ᐉ. The split ring a nonmagnetic particle in a uniform quasistatic magnetic field exhibits all four resonant forms. Consistent with our contention Ϫit ᑜ0e eˆz, as in the Landau–Lifshitz argument. Provided that magnetism requires large values of the permittivity, the 2 2 ͉S͉ ϽϽ /ᐉ , the induced magnetic field can be ignored so that magnetically active LC resonance (9) is underdamped only for տ ᐉ in cylindrical coordinates, P Ϸ (iEᑜ0/2c)reˆ (because S ϭ 1, S / . M ϭ 0). Thus, from Eq. 1 the magnetic moment is 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 ϭ 2 ᑜ 2ͯ ϫ ͑ ͒ ͯ ϳ ͑ᐉ2 2͒ ᑜ ͵ r m E 0/ reˆ dV / E 0V. thickness is g. The time dependence is exp (Ϫit), and we take the permeability of the substance making the ring to be S ϭ 1. [6] If the gap is momentarily ignored, a uniform, time-varying magnetic field oriented along the zˆ axis couples only to the Under arbitrary conditions, but ignoring resonant effects, symmetric mode, which is the only mode for which ѨEw/Ѩw ϭ 0 3 p ϳ Eᑟ0V and A ϳ Eᑟ0Vᐉ (V ϳ ᐉ ). Therefore, ͉␥/␣͉ϳ͉A/ (33). To calculate the fields across a cross-section of the wire, we ␣͉2 ϳ ᐉ2/2. Using arguments similar to those leading to Eq. 6,we regard it as straight. Ignoring radiation losses, of order (ᐉ/)3 ͉ ␣͉ ϳ ᐉ ͉ ͉ ϽϽ 2 ᐉ2 get G/ / . Consequently, for S / metamaterials fol- (34), the solution, except near the gap, is E ϭ E ϭ H ϭ Hw ϭ low the same natural order as molecules. The situation is, however, 0 and entirely different, and the natural order breaks down in the oppo- site limit. If the permittivity is unusually large and such that the skin ͑ ͱ ͒ ͑͒ ϭ 1 2I J0 k S ␦ ϭ ϽϽ ᐉ Ew D ͑͒ ϭ i ϫ depth /2 S is , where S is the extinction coefficient, w ͱ ͑͒ S ca S J1 the particle behaves like a superconductor (perfect conductor) with [7] ͑ ͱ ͒ respect to the magnetic (electric) field. The magnetic moment 2I J1 k S ϳᑜ ͉␥͉ϳ͉␣͉ H͑͒ ϭ ϫ becomes 0V, and thus (the electric moment is always ͑͒ ca J1 ϳᑟ0V). Alternatively, if the refractive index nS is sufficiently large so that n ϳ /ᐉ ϾϾ 1 and ϽϽ n , the particle exhibits ͌ S S S where I is the total current, k ϭ 2/, and ϭ ka S ϭ paramagnetic- or paraelectric-like behavior in that it can sustain ka(nSϩiS). The field at the gap is nearly uniform as for a cavity-like resonances that enhance the fields inside the particle. parallel-plate capacitor. Its value can be gained from Eq. 7 by using The determination of the coefficients in Eq. 3 is but the first ϪϪ1 Ϫ ϭ conservation of charge, (1 S )I iQ 0, where Q is the total step in a calculation of the optical constants (22). If local-field charge at one of the surfaces defining the gap. The result is effects can be ignored, that is, if E Ϸ ᑟ and B Ϸ ᑜ, we get for ϭ ␦ ϭ ␦ ϭ ͑ ͒ ϭ 2 ϭ Ϫ 2͑ ϪϪ1͒ optically isotropic particles ( ij ij and ij ij) Ew gap 4iI/ a 4Q/a 1 S . [8] H(1 ϩ 4N␣) and ϭ H(1 ϩ 4N␥). In cases where the ϭ ϪϪ1 2 resonant behavior of the polarizability or the magnetizability can The current I and, thus, the magnetic moment mz (1 S )I r0/c, Ᏹ ϷϪ 2ϩ ϩ Ᏹ ϭ be described by a simple pole at 0, i.e., ␣,␥ ϭ f␣,␥(0 Ϫ Ϫ can be obtained from i LI/c Ew(a)2 r0 iI/ C. Here, Ϫ1 i⌫) , the effect of the resonance on and can be quantified Ϫ(1/c)d⌽e/dt is the electromotive force, L ϭ 4r0 log(2r0/a) and by the dimensionless parameters N⌬␣ and N⌬␥, where ⌬␣, ⌬␥ ϭ C ϭ a2/4g are the inductance and the gap capacitance for static f␣,␥/0. Based on the previous discussion, it is apparent that fields, and ⌽e is the flux of the external magnetic field. We then find
Merlin PNAS ͉ February 10, 2009 ͉ vol. 106 ͉ no. 6 ͉ 1695 Downloaded by guest on September 29, 2021 2 4 ϳ ␦ i r of the form r0/(a 0), which represents the resistance associ- ␥ ϭ ͑ ϪϪ1͒ 0 ␦ zz 1 S 2 [9] ated with the skin depth . c Zspr When s ϭ 0, the multipolar coefficients exhibit cavity-like ͌ 2 resonances at Zspr() ϭ 0. For ϽϽ c/ LC,wegetJ1()Ϸ0or where Zspr ϭ Ϫi(L/c Ϫ1/C) is the split-ring impedance and 2 nSka Ϸ 5/4, 9/4, 13/4.. . These resonances are similar to ϭ i(4r0/c )J0()/J1(). From Eq. 8, we see that the magnetic dipole is linked through the current to an electric moment those of spherical inclusions, as discussed in ref. 37 and dem- oriented along yˆ and proportional to the magnetic field so that onstrated experimentally in refs 38 and 39 and small cubes (40); however, note that, unlike split rings, the highly symmetric r2g spheres and cubes do not allow for resonances of combined ϭϪ͑ ϪϪ1͒ 0 Gyz 1 . [10] electric- and magnetic-dipole character. Cavity resonances are S cZ spr paramagnetic in nature because the magnetic field is drawn into Reciprocally, and according to Eq. 3, an electric field along yˆ the particle. induces a magnetic moment parallel to zˆ. The associated current ͉͉ ϽϽ Ϸ I ϭϪ(g/Z* ) ᑟ generates, in turn, an electric-dipole moment Small Permittivity Limit. For 1, J0( )/J1( ) 2/ and, there- spr Ϫ Ϸ 2 ϾϾ ϾϾ ͌ along yˆ of magnitude (1Ϫ 1)Ig/ so that fore, i8 r0/a S. For S nS, and at frequencies c/ LC, S the transition between the limits ␦ ϾϾ a and ␦ ϽϽ a is accom- 2 Ϫ ig panied by a large change in the magnetizability from nearly zero ␣ ϭ Ϫ͑1 Ϫ 1͒ ␥ ϷϪ2 3 yy S Z* . [11] to zz r0. The cross-over manifests itself as a strongly spr overdamped diamagnetic resonance, as those observed in the
Note that this expression describes only the contribution to ␣yy simulations reported in ref. 41. ͉͉ Ϸ 2 2 ϾϾ 2 ϳ 2 due to induced charges at the surfaces defining the capacitor gap. Because 8 r0 / c 4 r0 /c L/c , the LC reso- There is an additional component, not considered here, associ- nance washes out in this limit. This effect has been reported in ated with induced charges that are distributed along the length recent experiments and simulations of metallic rings (42). For ϽϽ Ϫ1 Ϸ ϭ 2 of the ring. With the origin at the center of the ring, the electric metals at , R 2r0/a 0 turns into the dc resistance dipole gives rise to a quadrupole so that ⌰xx ϭ⌰yy ϭ 0 and ⌰xy ϭ of the ring. The LC resonance becomes overdamped because of the increase in ohmic losses for ␦ ϾϾ a. Also, notice that in the range ⌰yx ϭ (3/4)r0py. Hence, ϾϾ ϾϾ Ϫ1 ϷϪ2 2 ϷϪ 22 P , S P/ and 8 i r0/a P behaves like ϭ ͑ ͒ ␣ Ϸ 2 Axyy 3/4 r0 yy. [12] an inductance (for dielectrics, ir0/ a S is capacitive-like). For ϽϽ c/͌LC, we get from Eq. 9 We emphasize that all of these coefficients, with the exception of the polarizability, depend on the origin of coordinates (28). 2r2a 2 ͑ Ϫ 1͒ ␥ Ϸ ͑ 2 2 ͒2 ␣ Ϸ ͩ 0 ͪ S From Eqs. 9, 10, and 11, it follows that yy 2 r0/g yy ϩ . [14] 2 r0 g S ͉␥ ␣ ͉ϳ͉ ␣ ͉2 ϳ 4 22 [13] spr/ spr Gspr/ spr r0/g . This expression has a plasmon-like resonance at a frequency 0 so that 2r0 ϩ gS(0) ϭ 0, a condition that corresponds to This equation seemingly indicates that split rings also follow the ͛E.dl ϭ 0 (this resonance should not be confused with the molecular ordering, Eq. 5. However, we show below that this only 2 2 conventional plasmon resonances involving charges distributed applies to ͉S͉ ϽϽ /r0 . For large values of the permittivity, the 2 2 over the outer surfaces of the ring). Because a ͉S͉ ϽϽ and, resonant wavelength depends on the parameters of the particle 2 ϽϽ 2 ⌬␥ ϳ 4 2 2 ϽϽ 3 thus, a r0 g , we get for a Drude metal r0a / g r0. in such a way that the natural order is not obeyed. ⌬␣ Ϸ 2 ϾϾ⌬␥ ͉ ⑀ ͉ ϽϽ2 2 Note that a g for s /r0. These consider- ϾϾ ations also apply to results reported for horseshoe-shaped an- Large Permittivity Limit. Consider first the case when S /a tennas (43, 44). (small skin-depth limit). Then J0()/J1() ϷϪi and, thus, Ϸ 4r0k/c. This is the limit considered in ref. 9 and, more 2 2 Split Rings vs. Molecules. For split rings with ͉S͉ ϽϽ /ᐉ , the recently, in ref. 35. In this case, there is a resonance of the LC coefficients in Eq. 3 follow the same natural order found in ϭ ͌ ͉͉ ϽϽ ͌ type at 0 c/ LC. Because c L/C, the resonance is molecular systems. Indeed, the expressions for molecules, Eq. 5, ⌬␥ ϳ 3 long-lived. The split ring behaves as a diamagnet, with r0, become identical to those of thin split rings if one makes the reflecting the fact that the induced eddy or displacement cur- ᐉ 3 2 substitution mol r0/g. Nevertheless, split rings follow a very rents oppose the external field. For the electric dipole, we have different order if the permittivity is very large. If one compares ⌬␣ ϳ 2 ga . Hence, the magnetic-dipole strength is significantly the molecular and the split-ring equations for the magnetizabil- larger and, thus, the split ring does not follow the natural ity, Eq. 4b and Eq. 9, one obtains for large ͉S͉ ordering of Eq. 5. To understand this point further, it is ␥ ᐉϪ3 2 ᐉ convenient to relate the electric and magnetic moment of the mol mol e mol Ϫ ϳ ϫ ϽϽ 1 [15] particle to the change in the total energy of the corresponding ␥ r 3 បc field when the particle is introduced, as for static fields (17). spr 0 Within this context, ⌬␥ ϾϾ⌬␣ because the excluded or screened which is ϳ10Ϫ5 to 10Ϫ6 at optical frequencies. The reason why volume for the magnetic field is much larger than for the electric the magnetic activity is so much larger for high-permittivity split field. rings is, first, the fact that molecular magnetism is a weak ϭ Ϫ2 ϩ For Drude metals, S 1 P/ ( i/ ), where P is the relativistic effect and, second, the superdiamagnetic (superpara- plasma frequency and is the relaxation time. At frequencies magnetic) response of the ring for ␦ ϽϽ ᐉ (nS ϳ /ᐉ) that behaves ϾϾ ϾϾ Ϫ1 Ϸ Ϸ P , nS 0 and S p/ . Hence, behaves as an effectively as an object with S ϭ 0(S ϭϱ). Note that the effect 2 inductance, i.e., ϷϪiL/c , where L ϭ 4r0c/aP. As dis- of the permittivity on the electric moment is considerably less cussed in ref. 35, this expression accounts for the saturation of important because, to lowest order, the electric moment does not the split-ring magnetic response at high frequencies (36). Note depend on the wavelength. 2 that L LK ϭ 8r0(c/aP) , where LK is the kinetic inductance that was introduced ad hoc in ref. 36. At low frequencies, S Ϸ Optical Constants of Metamaterials: Lorentz–Lorenz and Lewin’s For- 4i0/ (0 is the dc conductivity). The real component of is mula. The Lorentz–Lorenz formula (27)
1696 ͉ www.pnas.org͞cgi͞doi͞10.1073͞pnas.0808478106 Merlin Downloaded by guest on September 29, 2021 1 ϩ 8N␣˜ /3 Table 1. Calculated effective-medium permeability of a system ϭ ϩ i ϭ ͩ ͪ [16] of spherical particles of radius R in a simple-cubic arrangement 1 2 H 1 Ϫ 4N␣˜ /3 of lattice constant d ͌ is a widely used approximation to the effective complex permit- Material , m s ϭ nS ϩ iS 1 ϩ i2 tivity of an optically isotropic system, such as spheres in a cubic Cu 3,000 975 ϩ i975 0.382 ϩ i0.005 lattice and a randomly oriented set of split rings.‡ Its magnetic KTa Nb O 500 17.3 ϩ i0.58 3.322 ϩ i1.226 counterpart is 0.982 0.018 3 PbTe 312.5 43.4 ϩ i43.0 0.487 ϩ i0.102 ϩ ϩ 1 ϩ 8N␥˜/3 SrTiO3 111.0 25 i25 0.571 i0.165 ϭ ϩ ϭ ͩ ͪ ϩ ϩ 1 i 2 H Ϫ ␥ [17] SiC 12.5 17 i17 0.678 i0.221 1 4 N˜/3 Sb 4.0 9.73 ϩ i13.77 0.811 ϩ i0.163 ϩ ϩ where ␣˜ and ␥˜ are the averages over all orientations. We include Ag 1.93 0.24 i14.09 0.834 i0.004 Ge 0.590 5.75 ϩ i1.63 1.041 ϩ i0.029 these expressions here to illustrate the effect a single-particle Si 0.288 4.09 ϩ i5.39 0.978 ϩ i0.053 resonance has on the collective behavior of the metamaterial. It
is apparent that, other than for the frequency shifts caused by The refractive index nS and the extinction coefficient S of the correspond- coupling between particles, poles in the polarizability and mag- ing materials are room temperature values at the wavelengths shown. Results ϭ ϭ ϭ netizability lead to resonances in and , with concomitant are for d 2R /20 and H 1.96. Note the paramagnetic response of the regions of anomalous dispersion (45). It is worthwhile noticing substances for which nS dominates over S. that Eq. 16 and Eq. 17 follow from Eq. 3 if one takes, respec- ᑟ ϭ ϩ ᑜ ϭ ϩ tively, (0) E 4 P/3 and (0) H 4 M/3. Eq. 17 appears dipole transitions (37). Recent experiments have corroborated to have been first derived by Lewin for a lattice of spheres (46). the validity of these results (38–40). These expressions have been rediscovered by many authors; see, for example, ref. 41. An extensive list of early references can be Concluding Discussion
found in ref. 37 and at www.wave-scattering.com/negative.html. The double constraint S ϾϾ /ᐉ ϾϾ 1 (or, nS ϳ /ᐉ ϾϾ 1if S ϽϽ nS) poses severe limitations for attaining magnetism at Scattering by Spheres arbitrarily high frequencies. Because they have a large extinction SCIENCES The behavior of spheres is very similar to that of split rings, coefficient, metals are to be favored at optical frequencies.
ϷϪ2 2 ϾϾ ϾϾ APPLIED PHYSICAL particularly in regard to the magnitude of the multipolar coef- Because S P/ for 1, the constraint becomes ficients in the low- and high-permittivity limits. The main ᐉ ϾϾ P (note that the skin depth is nearly constant in this range: ␦ Ϸ difference is that the higher spherical symmetry leads to a c/ P). The measured values of the permittivity for noble separation between magnetic and electric multipole resonances metals (48) indicate that magnetism can coexist with the effective- ϳ ϫ 14 ϳ and hence, that spheres do not exhibit resonances of the LC type. medium condition for frequencies up to 1.5 10 Hz ( 2.5 m). This estimate is in very good agreement with simulations The polarizability ␣sph and magnetizability ␥sph of a sphere of radius R are known exactly from the work of Mie (47). The of plasmonic metamaterials (18). A similar order-of-magnitude Bruggerman (26) and Maxwell-Garnet (25) expressions are estimate follows from the realization that the skin depth must be no less than a few lattice sites, say, ␦ տ ␦ ϳ 100 Å, or ᐉ ϾϾ based on Mie scattering and, as such, they also pertain to a min 2␦ . Although substances with large, but not exceedingly collection of spheres. The later model (25) corresponds to the min ϭ ͌ ϽϽ large, permittivity are not expected to lead to negative values of so-called static limit where kR S S 1. In this case, we get the permeability because of losses, metamaterials involving such ϽϽ ␥ Ϸ 32 ϳ for R and a nonmagnetic medium sph R /30 Vsph(R/ substances may nevertheless show a magnetic response that is 2͉ ͉ ϭ 3 Ϸ ) S (Vsph 4 R /3) and, from Eq. 17, 1, whereas the incompatible with the Landau–Lifshitz argument. This is dem- polarizability is given by the well-known expression onstrated in Table 1, which lists calculated values of using 3 ␣sph Ϸ R (S Ϫ H)/(S ϩ 2H). In contrast, for large and imag- Lewin’s formula (46) for a simple-cubic lattice of spheres. The ␣ ϳ 3 ␥ ϳϪ 3 Ϸ Ϫ ϩ inary, sph R and sph R /2. Thus, (1 NVS)/(1 optical parameters of Ag, Cu, KTa0.982Nb0.018O3, PbTe, SrTiO3, NVS/2) Ͻ 1 and /H Ϸ (1 ϩ 2NVS)/(1 Ϫ NVS) Ͼ 1 (46). Similar SiC, and Sb are, respectively, from refs. 48, 49, 50, 51, 52, 53, and to the cavity-like resonances of split rings, the magnetizability 54. Those for Ge and Si are from ref. 55. diverges for ϭ ,2,3, (we assume S ϭ H ϭ 1). Note that the positions of the resonances are slightly different for electric- ACKNOWLEDGMENTS. I thank V. Agranovich, G. Bouchitte´, N. Engheta, D. Felbacq, A. Grbic, G. Milton, J. Pendry, V. Podolskiy, G. Shvets, M. Stockman, and I. Tsukerman for useful discussions. This work was supported by a fellow- ship from the John Simon Guggenheim Memorial Foundation and by the Air ‡Although an oriented set of split rings exhibits bianisotropy, split rings are not chiral Force Office of Scientific Research Contract FA 9550-06-01-0279 through the objects, and thus, a random set does not exhibit optical activity. Multidisciplinary University Research Initiative Program.
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1698 ͉ www.pnas.org͞cgi͞doi͞10.1073͞pnas.0808478106 Merlin Downloaded by guest on September 29, 2021