Layered superconductors as negative-refractive-index metamaterials
A.L. Rakhmanov, V.A. Yampol'skii, J.A. Fan, Federico Capasso, Franco Nori
aa r X i v : . [ c ond - m a t . s up r- c on ] J u l Layered superconductors as negative-refractive-indexmetamaterials
A.L. Rakhmanov,
1, 2
V.A. Yampol’skii,
1, 3
J.A. Fan, Federico Capasso, and Franco Nori
1, 5 Advanced Science Institute, The Institute of Physical andChemical Research (RIKEN), Saitama, 351-0198, Japan Institute for Theoretical and Applied ElectrodynamicsRussian Acad. Sci., 125412 Moscow, Russia A.Ya. Usikov Inst. for Radiophysics and Electronics,Ukrainian Acad. Sci., 61085 Kharkov, Ukraine School of Engineering and Applied Sciences,Harvard University, Cambridge, Massachusetts 02138, USA Department of Physics, University of Michigan, Ann Arbor, MI 48109-1040, USA
Abstract
We analyze the use of layered superconductors as anisotropic metamaterials. Layered super-conductors can have a negative refraction index in a wide frequency range for arbitrary incidentangles. Indeed, low- T c ( s -wave) superconductors allow to produce artificial heterostructures with low losses for T ≪ T c . However, the real part of their in-plane effective permittivity is very large.Moreover, even at low temperatures, layered high- T c superconductors have a large in-plane normalconductivity, producing large losses (due to d -wave symmetry). Therefore, it is difficult to enhancethe evanescent modes in either low- T c or high- T c superconductors. PACS numbers: 74.25.Nf, 42.25.Bs ε and permeability µ (see, e.g., [3]). However, these “double negative” structures require intricate design anddemanding fabrication techniques, are not “very subwavelength”, and suffer from spatialdispersion effects. Moreover, the implicit overlapping electric and magnetic resonances (see,e.g., [4]) often leads to resonant losses that, together with material losses, lead to significantdegradation in metamaterial functionality. This manifestation of loss can be quantified byexamining the figure of merit (FOM) in such materials, which is defined as | n ′ | /n ′′ where n ′ and n ′′ are the real and imaginary parts of the refractive index n , respectively. The FOMsof negative index materials in the visible and near-IR have experimentally ranged from 0.1up to 3.5 [3, 5].Another promising route to creating negative index metamaterials is to use stronglyanisotropic materials, in particular, uniaxial anisotropic materials with different signs of thepermittivity tensor components along, ε k , and transverse, ε ⊥ , to the surface (see, e.g., [6, 7]).These materials have been theoretically [8] and experimentally [9] demonstrated to supportsub-wavelength imaging, and they have also been proposed as a model system for scattering-free plasmonic optics [10] and subwavelength-scale waveguiding [11]. These materials areparticularly attractive because: they are relatively straightforward to fabricate, comparedto double negative metamaterials; they do not require negative permeability; and do notsuffer from magnetic resonance losses. The FOMs for such materials have been calculatedto be significantly greater than those measured in double negative materials [6, 12].Experimental schemes for creating strongly anisotropic uniaxial materials have typicallyinvolved the fabrication of subwavelength stacks of materials whose layers comprise alternat-ing signs of permittivity. For example, alternating stacks of Ag and Al O [9] and of dopedand undoped semiconductors [12] have been demonstrated to support strong anisotropyin the visible and infrared frequency ranges respectively. However, spatial dispersion canstrongly modify the optical response of the system relative to the ideal effective medium limitresponse [13]; strong local field variations exist due to the structure and length scales of plas-monic modes supported by negative-permittivity films, even in the limit of l ≪ a , where l
2s the length scale of the thin films in the material and a is the free-space electromagneticwavelength. This imposes limitations to subwavelength imaging and waveguiding in suchmaterials. Spatial dispersion may be reduced by making composite structures with thinnerlayers. However, there exist practical material deposition limitations to thin-film stacks in-volving film roughness and continuity. In addition, damping due to electron scattering atthe thin film interface becomes significant starting at length scales of a v F /c ∼ a /
100 where v F is the Fermi velocity in the material [14], limiting the minimum film thickness. It is clearthat composite structures are limited in practice as “ideal” strongly anisotropic materials.We analyze here the idea of using superconductors as metamaterials (see, e.g., [15,16, 17]). In particular, we consider layered cuprate superconductors [17] and artificialsuperconducting-insulator systems [18] as candidates for strongly anisotropic metamate-rials. Unlike the composite structures discussed earlier, layered superconductors are notlimited in performance by the spatial dispersion effects discussed in [13]. We will analyzethese materials in the specific context of subwavelength resolution, which can be achievedby the amplification of evanescent waves [2]. This amplification is high when n is close tounity and its imaginary part is small [2, 19]. For the incident p -polarized waves consideredhere, subwavelength resolution requires Im( ε ) ≪ exp( − k ⊥ L ), where k ⊥ is the wavevectorcomponent across the surface, and L is the plane lens-thickness [19]. For evanescent modeswith k ⊥ = 2 ω/c = 2 k and L/a = 0 .
1, we have Im( ε ) ≪ . T c cuprates the losses are high at any reasonablefrequency. In the case of artificial layered structures prepared from low- T c superconductors,the losses can be reduced significantly at low temperatures, T ≪ T c , where T c is the criticaltemperature. The frequency range for such a metamaterial is ¯ hω < T c superconductors. We prove that the in-plane permittivity for low- T c multi-layersis large, preventing the effective enhancement of evanescent waves. This is problematicbecause subwavelength resolution [2] requires the amplification of evanescent waves. Notethat Refs. 16 only focus on the zero-frequency DC case. Effective permittivity .— We study a medium consisting of a periodic stack of supercon-ducting layers of thickness s and insulating layers of thickness d with Josephson coupling be-tween successive superconducting planes. The number of layers is large, L/ ( s + d ) = N ≫ s is smaller than: the in-plane magnetic field penetration depth λ k , transverse skin3epth δ ⊥ ( ω ), and wavelength a ( ω ) ∼ πc/ω p | ε ( ω ) | . We calculate the effective permittivity, b ε = ( ε k , ε ⊥ ), of the layered system in the case of p -wave refraction.Layered superconductors with Josephson couplings can be described by the Lawrence-Doniach model, where the averaged current components can be expressed as [20] J ⊥ = J c sin ϕ n + σ ⊥ Φ ˙ ϕ n πc ( s + d ) , J k = c Φ p n π λ k + σ k E k , (1)where ϕ n is the gauge-invariant phase difference between the ( n + 1)th and n th supercon-ducting layers, p n is the in-plane superconducting momentum, J c = c Φ / (8 π dλ ⊥ ) is thetransverse supercurrent density, Φ is the magnetic flux quantum, and λ ⊥ is the transversemagnetic field penetration depths. Also σ ⊥ and σ k are the averaged transverse and in-planequasiparticle conductivities. The transverse E ⊥ and in-plane E k components of the electricfield are related to the gauge-invariant phase difference and superconducting momentumby [20, 21] (cid:0) − α ∇ n (cid:1) E ⊥ = Φ πc ( s + d ) ˙ ϕ n , E k = Φ πc ˙ p n , (2)where ∇ n f ( n ) = f ( n + 1) + f ( n − − f ( n ), α = εR D / ( sd ) is the capacitive couplingbetween layers, and R D is the Debye length. We linearize the first of Eqs. (1) and considera linear electromagnetic wave E k , ⊥ ( x, n, t ) = P q R dk dω (2 π ) E k , ⊥ ( k, q, ω ) exp( − iωt + ikx + iqn ),where q = πl/ ( N + 1), l = 0 , ± , ±
2, and the x -axis is in the plane of the layers. UsingEqs. (1) and (2), we obtain: J ⊥ E ⊥ = (cid:16) α e q (cid:17)(cid:20) σ ⊥ − εω p ( s + d )4 πidω (cid:21) , J k E k = σ k − εγ ω p πiω , (3)where ω p = c/ ( λ ⊥ √ ε ) is the Josephson plasma frequency, ε is the interlayer permit-tivity, γ = λ ⊥ /λ k , and e q = 2(1 − cos q ). Averaged over the sample volume, theMaxwell equation has the form c ∇ × H = 4 π J + ∂ D /∂t , where D k = ε k E k and D ⊥ = ε ⊥ E ⊥ . In the effective medium approximation, the components of the permittiv-ity tensor can be expressed as [22] ε k = ( dε + s ) / ( s + d ), and ε ⊥ = ε ( s + d ) / ( sε + d ),where we assume that ε superconductor = 1. Fourier transforming the above Maxwell equa-tion, we derive c [ ∇ × H ] ⊥ ( k, q, ω ) = − ε ⊥ E ⊥ and c [ ∇ × H ] k ( k, q, ω ) = − ε k E k , where ε k = ε k − (4 π/iω )( J k /E k ) and ε ⊥ = ε ⊥ − (4 π/iω )( J ⊥ /E ⊥ ). Therefore, we finally obtain ε ⊥ = ε ⊥ − π (cid:16) α e q (cid:17) σ ⊥ iω − ε (cid:16) α e q (cid:17) ω p ( s + d ) ω d ,ε k = ε k − πσ k iω − εγ ω p ω . (4)4hus, ε k < ε ⊥ > r(cid:16) α e q (cid:17) sε + dd < ωω p < γ r ε ( s + d ) dε + s . (5)If the incident angle is close to normal and anisotropy is large, γ ≫
1, we can find an estimateFOM ≈ (cid:12)(cid:12) Re( ε k ) / Im( ε k ) (cid:12)(cid:12) ≈ εγ ω / πσ k ω p . Electromagnetic waves propagate in the layeredsuperconductors if ω > ω p . Thus, the results obtained are valid if ω p < ω < ω c = 2∆ / ¯ h .Below we analyze separately the different cases of a typical high- T c layered superconductor,Bi Sr CaCu O δ (Bi2212), and also an artificial low- T c layered structure made from Nb. Layered high- T c superconductors .— In the case of Bi2212, it is known that s ≪ d = 1–2 nm, ε = 12, α ≈ .
1, and at low temperatures ( T ≪ T c = 90 K) ω p ≈ s − , γ = 500, σ k ≈ · Ω − cm − , and σ ⊥ ≈ · − Ω − cm − [20, 23]. In this case, Eqs. (4) canbe rewritten as ε ⊥ ≈ ε (cid:0) − ω /ω p (cid:1) +4 πiσ ⊥ /ω , ε k ≈ ε (cid:0) − γ ω /ω p (cid:1) +4 πiσ k /ω . The calculatedfrequency dependence of the permittivity for Bi2212 is shown in Fig. 1. The superconductinggap for Bi2212 is estimated as ∆ ≈ k B T c , with ω c ≈ × s − ≪ γω p . Thus, forany incident angle, Bi2212 has negative n in the frequency range from about 0.15 THz to7.5 THz, or in the wavelength domain 40 µ m < ∼ a < ∼ σ k is large,even at helium temperatures. As it is seen from the inset in Fig. 1(b), σ k = 0 when T → d -type symmetry of the order parameter. Inaddition, the usual dimensions of high-quality Bi2212 single crystals are less than 1 mm inthe in-plane direction and about 30–100 µ m in the transverse direction. Thus, it might bedifficult to use Bi2212 single crystals as metamaterials, or elements of a superlens. Low- T c artificial layered structures .— The thickness of the insulator in Josephson junc-tions is about a few nm. To attain a low-loss regime and reach the bulk critical temper-ature, the thickness of the superconducting layers should be larger or about the super-conductor coherence length ξ . For clean superconductors, ξ is about tens of nm. Thus,for low- T c artificial-layered structures, it is reasonable to analyze the case d ≪ s . In thislimit, λ k = λ p ( s + d ) /s ≈ λ , where λ is the bulk magnetic field penetration depth and α = εR D / ( sd ) ≪ σ i to satisfy the condition σ i ≪ σ s d/s at any reasonable temperature,where σ s is the quasiparticle conductivity of the superconductor. In this case we have ε ⊥ = 1 , ε k = 1 , σ ⊥ = σ i s/d and σ k = σ s . Equations (4) for the effective permittivity can5
20 40 I m (10 s ) I m
20 30 40 50-600-400-2000 R e / R e (10 s ) a (10 mm) R e (c)(b) (a) T (K) O h m c m FIG. 1: Dependence of the real and imaginary parts of the permittivity ˆ ε in Bi2212 on frequency ω (or wavelength a ), calculated from Eqs. (4): (a) real part of the in-plane permittivity ε k ( ω ).Inset: ratio of the real parts of the in-plane and transverse permittivities; (b) imaginary part of thein-plane permittivity. Inset: temperature dependence of the in-plane quasiparticle conductivity σ k ( T ) in Bi2212; solid triangles are low frequency data from Ref. 23, open squares correspond to14.4 GHz data from Ref. 24; (c) imaginary part of the transverse permittivity ε ⊥ ( ω ). now be rewritten as ε ⊥ ≈ (cid:18) − ε sω p dω (cid:19) + 4 πiσ i sωd , ε k ≈ ε (cid:18) − γ ω p ω (cid:19) + 4 πiσ s ω . (6)Therefore, the refraction index n is negative if p εs/d < ω/ω p < γ. (7)For artificial structures, γ can be easily made of the order of, or even much larger than, innatural layered superconductors. In contrast to d -wave high- T c superconductors, for bulk s -wave superconductors, the quasiparticle conductivity σ s tends to zero for decreasing T .6 I m t s n Nb t
012 0 0.5 1
FIG. 2: Calculated, from Eq. (8), temperature dependence of the imaginary part of ε k ( t ) = ε k ( T /T c ), for a Nb-based layered structure, with ω = 0 . ω c , ε = 10, s/d = 5, and γ = 500; here: ω p /ω c = 0 .
1, Re ( ε ⊥ ) = 0 . ε k ) ≈ − · . The inset shows the dependence [25] of σ s /σ n on t ≡ T /T c ; points: experimental data for Nb at about 60 GHz; solid line: Mattis-Bardeentheory in the weak-coupling BCS limit; dashed line: strong-coupling Eliashberg prediction [25]. Thus, in principle, the imaginary part of ε k could be made as small as necessary by coolingthe system.Consider now Nb superconducting layers. For estimates we can take [25]: T c = 9 . λ ( T = 0) = 44 nm, ξ = 38 nm, electron mean free path l e = 20 nm, and normal stateconductivity σ n = 0 . × Ω − cm − . Thus, a reasonable thickness for the superconductinglayers can be chosen as s = 30–40 nm ≪ a ( ω c ) > ∼ σ s ( ω, T ) can be calculated using the Mattis-Bardeen theory [26] (see inset inFig. 2). At low temperatures, T ≪ T c , in the weak-coupling BCS limit, we have ∆ =1 . k B T c . When ω < ω c and T ≪ T c , we can rewrite the Mattis-Bardeen formula forconductivity [25, 26] in the form σ s /σ n = ω c [1 − exp ( − . ω/ ( ω c t ))] /ω × Z ∞ ( u + 1 + 2 uω/ω c ) exp (cid:0) − . ut (cid:1)q ( u − (cid:2) ( u + 2 ω/ω c ) − (cid:3) du, (8)where t = T /T c . The results of our calculations are shown in Fig. 2. These calculationsdemonstrate that the losses in artificial structures made from low- T c superconductors can e extremely low . The maximum frequency ω c = 3 . k B T c / ¯ h for Nb corresponds to approx-imately 0.7 THz. From the results presented in Fig. 2, we can estimate that at ω ∼ ω c the imaginary part of ε k is lower than 10 − if T < ω > ω c ,the conductivity of the superconductor is about the conductivity of the normal metal andit cannot be easily used as a metamaterial with low losses. Note also that by an ap-propriate choice of insulator, s , and d , we can vary the parameters γ and ω p in a widerange. If we assume that ε ∼
10, then to fulfill conditions (7) for ω p < ω c we shouldprepare highly-anisotropic heterostructures with γ > . If the anisotropy is large, wecan find from Eq. (6) that Re ( ε k ) ≈ − c /λ ω . The absolute value of Re ( ε k ) is verylarge, | Re ( ε k ) | ≥ c /λ ω c ≈ × . These estimates suggest that low- T c superconductingmulti-layers might not work as practical metamaterials.The metamaterial properties of layered superconductors, either natural or artificial, canbe tuned varying the temperature or an in-plane magnetic field, which strongly affects thetransverse critical current density and, consequently, the plasma frequency. But applyinga magnetic field increases dissipation, which is undesirable. Note also that the estimatesmade above show that the value of FOM may be very large for the systems considered here,however, this does not mean necessarily that these media can be easily used as practicalmetamaterials. Cuprates in the normal state .— There is experimental evidence that cuprate supercon-ductors have strongly anisotropic optical characteristics in the normal state [27, 28]. Forexample, it was observed that La − x Sr x CuO supports negative permittivity along the CuOplanes at frequencies up to the mid- and near-IR range [27]. Moreover, these optical prop-erties could be finely tuned by varying the stoichiometry. Such natural materials are thuscandidates for practical anisotropic metamaterials. The use of cuprates in the normal statehave evident advantages, such as operating above ω c and to work at room temperature.However, the normal conductivity of cuprates is of the same order as their quasi-particleconductivity in the superconducting state (see, e.g., the inset in Fig. 1b and Ref. 23). Themetamaterial properties of cuprates in the normal state require a separate analysis and willbe performed elsewhere. Conclusions .— Here we analyze the properties of anisotropic metamaterials made fromlayered superconductors. We show that these materials can have a negative refraction indexin a wide frequency range for arbitrary incident angles. However, superconducting metama-8erials made from natural layered high- T c cuprates have a large in-plane normal conductivity,even at very low temperatures, due to d -wave symmetry of their superconducting order pa-rameter. Therefore, these are very lossy. Nevertheless, low- T c s -wave superconductors allowto produce metamaterials with low losses at low temperatures, T ≪ T c . But the real part oftheir in-plane permittivity is very large, reducing the enhancement of the evanescent modesand potentially limiting the use of superconducting structures as practical metamaterials.We gratefully acknowledge partial support from the NSA, LPS, ARO, NSF grant No.EIA-0130383, and JSPS-RFBR 09-02-92114, FC gratefully acknowledges useful discussionswith E. Narimanov. [1] V.G. Veselago, Sov. Phys. Usp. , 509 (1968); J.B. Pendry et al. , Physics Today , No 6,37 (2004); K.Y. Bliokh et al. , Rev. Mod. Phys. , 1201 (2008).[2] J.B. Pendry, Phys. Rev. Lett. , 3966 (2000).[3] V.M. Shalaev, Nature Photonics , 41 (2007);[4] R.A. Shelby et al. , Science , 77 (2001); H.O. Moser, et al. , Phys. Rev. Lett. , 063901(2005); T. Koschny et al. , ibid. , 107402 (2004); S. Zhang et al. ibid. , 137404 (2005).[5] J. Valentine et al. , Nature , 376 (2008);[6] V.A. Podolskiy et al. , Phys. Rev. B , 201101 (2005).[7] J.B. Pendry, Science , 1353 (2004); A. Alu et al. , J. Opt. Soc. Am. B , 571 (2006);O.V. Ivanov et al. , Crystallogr. Rep. , 487 (2000);[8] Z. Jacob et al. , Opt. Express , 8247 (2006).[9] Z. Liu, et al ., Science , 1686 (2007).[10] J. Elser et al. , Phys. Rev. Lett. , 066402 (2008).[11] A.A. Govyadinov et al. , Phys. Rev. B , 155108 (2006).[12] A.J. Hoffman et al. , Nature Materials , 946 (2007);[13] J. Elser et al. , Appl. Phys. Lett. , 191109 (2007).[14] A.A. Govyadinov et al. , J. Mod. Opt. , 2315 (2006).[15] M. Ricci et al. , Appl. Phys. Lett. , 034102 (2005); C. Du, et al. , Phys. Rev. B , 113105(2006).[16] B. Wood et al. , J. Phys.: Condens. Matter , 076208 (2007); F. Magnus et al. , Nature aterials , 295 (2008); E. Narimanov, ibid. , 273 (2008).[17] M.B. Romanowsky et al. , Phys. Rev. A , 041110 (2008).[18] A. Pimenov et al. , Phys. Rev. Lett. , 247009 (2005).[19] N. Garcia et al. , Phys. Rev. Lett. , 207403 (2002).[20] A.E. Koshelev et al. , Phys. Rev. B , 174508 (2001).[21] S.E. Savel’ev et al., arXiv:0903.2969 (2009).[22] L.D. Landau et al. , Electrodynamics of Continuous Media (Butterworth-Heinemann, Oxford,1995).[23] Yu.I. Latyshev et al ., Phys. Rev. B , 134504 (2003).[24] S.-F. Lee et al ., Phys. Rev. Lett. , 735 (1996); H. Kitano et al ., J. Low Temp. Phys. ,1241 (1999).[25] O. Klein et al ., Phys. Rev. B , 6307 (1994).[26] D.C. Mattis et al. , Phys. Rev. , 412 (1958).[27] S. Uchida et al ., Phys. Rev. B , 7942 (1991).[28] S. Tajima et al ., Phys. Rev. B , 16164 (1993)., 16164 (1993).