Surface waves versus negative refractive index in layered superconductors
V.A. Golick, D.V. Kadygrob, V.A. Yampol'skii, A.L. Rakhmanov, B.A. Ivanov, Franco Nori
aa r X i v : . [ c ond - m a t . s up r- c on ] J a n Surface waves versus negative refractive index in layeredsuperconductors
V.A. Golick, D.V. Kadygrob, V.A. Yampol’skii,
1, 2, 3
A.L. Rakhmanov,
3, 4
B.A. Ivanov,
3, 5, 6 and Franco Nori
3, 7 V.N. Karazin Kharkov National University, 61077 Kharkov, Ukraine A.Ya. Usikov Institute for Radiophysics and Electronics,Ukrainian Academy of Sciences, 61085 Kharkov, Ukraine 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 Institute of Magnetism, Ukrainian Academy of Sciences, 03142 Kiev, Ukraine National T. Shevchenko University of Kiev, 03127 Kiev, Ukraine Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA
Abstract
We predict a new branch of surface Josephson plasma waves (SJPWs) in layered superconductorsfor frequencies higher than the Josephson plasma frequency. In this frequency range, the permit-tivity tensor components along and transverse to the layers have different signs, which is usuallyassociated with negative refraction. However, for these frequencies, the bulk Josephson plasmawaves cannot be matched with the incident and reflected waves in the vacuum, and, instead ofthe negative-refractive properties, abnormal surface modes appear within the frequency band ex-pected for bulk modes. We also discuss the excitation of high-frequency SJPWs by means of theattenuated-total-reflection method.
PACS numbers: 74.25.N-, 42.25.Bs T c layered cuprate superconductors are important candidates for negative-refractive-index (NRI) metamaterials (see, e.g., [1, 2]). Indeed, being uniaxial strongly anisotropicmaterials, they provide different signs of the permittivity tensor components along, ε ab , andtransverse, ε c , to the layers in a wide frequency range (see, e.g., [3]), providing a possibilityof NRI. These metamaterials are attracting considerable attention because they have the po-tential for subwavelength resolution and aberration-free imaging. Layered superconductorsare very promising metamaterials because they are relatively straightforward to fabricate(compared to double negative metamaterials) and do not require negative permeability.Experiments for the c -axis conductivity in layered superconductors prove the use of amodel in which the superconducting CuO layers are coupled by the intrinsic Josephsoneffect through the layers [4]. Thus, a Josephson plasma with anisotropic current capabilityis produced in layered superconductors. Moreover, the physical mechanisms of the cur-rents along and across the layers are fundamentally different. The current along the layersis similar to the current in bulk superconductors, while the current across the layers isJosephson-type.The Josephson current along the c -axis couples with the electromagnetic field inside theinsulating dielectric layers, forming Josephson plasma waves (JPWs) (see the review [5] andreferences therein). Thus, the propagation of electromagnetic waves through the layers isfavored by the layered structure. The study of these waves is very important because oftheir Terahertz frequency range, which is still hardly reachable for electronic and opticaldevices.Like in common plasma waves, JPWs propagate with frequencies above some thresholdvalue (the Josephson plasma frequency ω J ). However, in the frequency range below ω J , thepresence of the sample boundary can produce surface Josephson plasma waves (SJPWs) [6],which are analog to the surface plasmon polaritons in metals [7, 8]. Such waves can propagatealong the vacuum-layered superconductor interface and damp away from it.At frequencies ω higher than the Josephson plasma frequency ω J , the normal-to-the-layerscomponents of both the group velocity and the Poynting vector of propagating JPWs, havesigns opposite to the sign of the normal component of the wave-vector k s . This correspondsto a NRI. However, the condition ω > ω J is not sufficient for NRI. This NRI effect canbe observed at the vacuum-layered superconductor boundary only in a relatively narrowfrequency range , ω J < ω < ω vac1 = ω J [ ε/ ( ε − / , where ε is the interlayer dielectric2onstant. A similar limitation exists for any insulator-layered superconductor boundary, ifthe dielectric constant ε ext of the insulator is less than ε . The above conditions follow fromthe dispersion relation for the JPWs and the natural limitation for the tangential component q ( q = k sin θ < k, k = ω/c ) of the wave-vector for a wave incident at an angle θ from thevacuum onto the layered superconductor. A simple analysis shows that the above inequalityis compatible with the dispersion relation for JPWs only for frequencies ω J < ω < ω vac1 . Inother words, any wave with frequency ω > ω vac1 incident from the vacuum cannot propagatefurther in a layered superconductor. JPWs with ω > ω vac1 can only match evanescent wavesin the vacuum. As for frequencies ω J < ω < ω vac1 , the NRI can be observed, but only forincident angles θ higher than some critical value θ crit .For frequencies in the interval ω vac1 < ω < ω , ω = ω J γ, (1)we predict the existence of a new branch of surface waves (here γ = λ c /λ ab ≫ λ c = c/ω J ε / and λ ab are the magnetic-field penetrationdepths along and across the layers, respectively). Despite numerous works on this issue,SJPWs with frequencies higher than ω J were not discussed before. Here we study them andprove that the SJPWs spectrum consists of two branches. The low-frequency branch wasdescribed in detail in [6]. Its spectrum ω ( q ) follows the “vacuum light line”, q = ω/c , anddeviates from it at frequencies close to ω J . The new branch of SJPWs predicted here startsat the frequency ω = ω vac1 and follows the vacuum light line for ω ≪ ω . For frequencies ofthe order of ω , the dispersion curve ω ( q ) strongly deviates from the vacuum light line andstops at the frequency ω = ω when q ≈ /λ ab .Thus, SJPWs do not exist within the frequency gap ω J < ω < ω vac1 . On the otherhand, as shown here, this is actually the range where the NRI can be observed (for wavesincident from the vacuum onto the layered-superconductor boundary). Hence, some kind ofcomplementarity between the NRI and surface waves is established in this paper. Conditions for the observation of NRI .— Consider a layered structure consisting of su-perconducting and dielectric layers with thicknesses s and d , respectively (see Fig. 1). Westudy the transverse-magnetic JPWs propagating with wave-vector k s = ( q, , κ s ) and hav-ing the electric, E s = { E sx , , E sz } , and magnetic, H s = { , H s , } , components proportionalto exp[ i ( qx + κ s z − ωt )]. The coordinate system is shown in Fig. 1.3 IG. 1: (Color online) Geometry for studying waves in layered superconductors. The interface z = 0 divides the layered superconductor from an insulator with dielectric constant ε ext . The electromagnetic field inside the layered superconductor is determined by the dis-tribution of the gauge invariant phase difference ϕ ( x, z, t ) of the order parameter betweenneighboring layers. This phase difference can be described by a set of coupled sine-Gordonequations (see review [5]). In the continuum and linear approximation, ϕ ( x, z, t ) can beexcluded from the set of equations for electromagnetic fields, and the electrodynamics oflayered superconductors can be described in terms of an anisotropic frequency-dependentdielectric permittivity with components ε c (Ω) and ε ab (Ω) across and along the layers, re-spectively [2]. In Ref. [9], the effect of spatial dispersion in ε c , related to the capacitiveinterlayer coupling, was taken into account. Here we do not consider this effect because itis only important for a very narrow frequency range near ω J [9].In the limit s/d ≪
1, the equations for ε c (Ω) and ε ab (Ω) can be written as ε c (Ω) = ε (cid:18) − + iν c (cid:19) ,ε ab (Ω) = ε (cid:18) − γ + iν ab γ (cid:19) . (2)Here we introduce the dimensionless parameters Ω = ω/ω J , ν ab = 4 πσ ab /εω J γ , and ν c =4 πσ c /εω J . The relaxation frequencies ν ab and ν c are proportional to the averaged quasi-4article conductivities σ ab (along the layers) and σ c (across the layers), respectively; ω J =(8 πeDj c / ~ ε ) / is the Josephson plasma frequency. The latter is determined by the criticalJosephson current density j c , the interlayer dielectric constant ε , and the spatial period ofthe layered structure D = s + d ≈ d .Analyzing the relations for ε c (Ω) and ε ab (Ω), we conclude that their real parts havedifferent signs in a wide frequency range: ω J < ω < ω = ω J γ (or 1 < Ω < Ω = γ ≫ z -components of the group velocity and the Poynting vector of the bulk JPWs are directed opposite to the z -component of the wave-vector k s , and this, at firstsight, corresponds to having a NRI. However, a more careful analysis shows that this canonly be observed for a much narrower frequency interval. To verify this, one can considerthe well-known dispersion relation for the normal component κ s of the JPW wave-vector, κ s = ε ab (Ω) (cid:20) k − q ε c (Ω) (cid:21) , (3)that follows directly from Maxwell’s equations. Obviously, the JPWs can propagate onlywhen Re( κ s ) >
0, where “Re” stands for the real part. Neglecting dissipation, the permit-tivity ε ab is negative for the frequency region ω J < ω < ω considered here. Consequently,JPWs can propagate if the factor [ k − q /ε c (Ω)] in Eq. (3) is negative. Using Eq. (2) for ε c (Ω), one can conclude that this factor is negative only when 1 < Ω < q λ c . For awave incident, at an angle θ , from the insulator with dielectric constant ε ext onto the layeredsuperconductor, we have q = ( ωε / /c ) sin θ . Thus, a NRI can be observed for waves withincident angles higher than the critical value θ crit defined by the equation,sin( θ crit ) = p ε c (Ω) /ε ext . (4)It is important to note that, due to the negative sign of the permittivity ε ab in Eq. (3), theincident wave penetrates the superconductor for θ > θ crit , and totally reflects from it for θ < θ crit , contrary to the standard case of waves incident onto the interface dividing twousual right-handed media.For ε ext < ε , Eqs. (2), (4), and the inequality sin( θ crit ) ≤
1, provide the conditions, ω J < ω < ω = ω J (cid:18) εε − ε ext (cid:19) / . (5)Thus, the NRI for a layered superconductor bounded by an insulator with ε ext < ε canonly be observed in the frequency range ω J < ω < ω = ω J Ω . This frequency window5an be expanded if one uses an insulator with large enough permittivity ε ext . Only forinsulators with very high permittivity ε ext > ε , the negative refraction can occur in the wholefrequency interval ω J < ω < ω . The frequency range for the existence of the bulk JPWsin superconductors with capacitive interlayer coupling was derived in [10]. If the constantof this coupling tends to zero, the frequency range obtained in [10] coincides with Eq. (5).Below we consider waves in the insulator-layered superconductor system with ε ext < ε forfrequencies ω > ω . Surface Josephson plasma waves above ω .— When ω J < ω < ω and the factor in Eq. (3)is positive, the z -component κ s of the wave-vector k s becomes imaginary. This means thatthe wave damps into the layered superconductor. On the other hand, for q > ω √ ε ext /c the wave damps also into the insulator above the layered superconductor. These are thecharacteristic features of the surface waves discussed in this section.Consider an interface (the xy -plane) separating an insulator ( z > z ≤ x -axis (i.e., proportional to exp[ i ( qx − ωt )]) and decaying into both, the insulator and layered superconductor, away from the interface z = 0.Performing the standard procedure for searching surface waves (i.e., solving the Maxwellequations for the insulator and layered superconductor with proper boundary conditionsat the interface between them) we obtain the dispersion relation for the surface Josephsonplasma waves: κ (Ω) = Ω (cid:18) ε c (Ω) ε ext − ε ab (Ω) ε − ε c (Ω) ε ab (Ω) (cid:19) / , (6)or, neglecting dissipation, κ (Ω) = Ω (cid:18) γ − Ω + Ω ε ext /εγ − Ω + Ω ε / (Ω − ε (cid:19) / . (7)Here the dimensionless wave-vector is defined as κ = cq/ω J ε / . Equation (7) describes twobranches of the dispersion curve for the SJPWs (see Fig. 2). The first branch exists in thelow frequency range, 0 < ω < ω J , and it was studied before in [6]. The second (predictedhere) branch starts from the light line ω = cq/ε / (or Ω = κ ) at ω = ω (point A in Fig. 2),then follows this line, deviates from it at ω ∼ ω = γω J , and stops at the point where q = γω J ε / /c, ω = ω (point B in Fig. 2).Thus, there exists a frequency gap, ω J < ω < ω , in the spectrum of the SJPWs. Weemphasize that the NRI should only exist within this gap. When the permittivity ε ext of the6 IG. 2: (Color online) The dispersion curve (Ω = ω/ω J versus κ = cq/ω J ε / ) for SJPWs at thevacuum-layered superconductor interface. The values of the parameters are: γ = 200 , ε = 16.Inset: zoom-in of the spectrum near the point ( κ = 1 , Ω = 1). Points A and B correspond to thebeginning and end of the high-frequency branch. The green dashed line is the vacuum light lineΩ = κ . insulator increases, the point A in Fig. 2 moves towards point B, and the gap in the SJPWspectrum increases. When ε ext = ε (1 − /γ ) ≈ ε , the points A and B coincide, and thehigh-frequency branch in the SJPWs spectrum disappears.Note that the Josephson current is small with respect to the displacement current athigh frequencies, Ω ≫
1. In this case, we can omit 1 in the denominator in Eq. (7).This corresponds to the dispersion relation for a periodic layered structure without couplingbetween superconducting layers . Specifically, the interlayer Josephson coupling is responsiblefor the appearance of a frequency gap, ω J < ω < ω , in the spectrum of SJPWs. Excitation of the SJPWs above ω .— It is known that the excitation of surface waves isaccompanied by the so-called Wood anomalies of the reflectivity and transmissivity coeffi-7ients (see, e.g., [8]). These resonance anomalies can result in the complete suppression ofthe reflectivity by a proper choice of parameters. Here we consider the excitation of high-frequency SJPWs (Ω ≫
1) by a wave incident from a dielectric prism with permittivity ε p onto a layered superconductor separated from the prism by a vacuum gap of thickness δ (theso-called “attenuated-total-reflection” method for producing surface waves, see Fig. 3).The suppression of the specular reflectivity | R | = | H r /H i | due to the resonant excitationof the surface waves can be observed by changing the incident angle at a given frequency orby changing the frequency at a given incident angle, as demonstrated in Fig. 4 (a, b). Here H i and H r are the magnetic field amplitudes of the incident and reflected waves, respectively.Figure 4 (c) shows the sharp decrease of the reflectivity in the ( θ, Ω) plane.
Conclusions .— Here we predict the existence of a new branch of SJPWs in layered su-perconductors for the frequency range higher than the Josephson plasma frequency, whichis a very unusual phenomenon for plasma-like media. It is important that a NRI can onlybe observed for frequencies within the gap in the spectrum of the SJPWs. Thus, some kindof complementarity between NRI and surface waves is established in this paper. We havealso described the excitation of these SJPWs by means of the attenuated-total-reflectionmethod.We gratefully acknowledge partial support from the NSA, LPS, ARO, and NSF grantNo. 0726909. [1] M. Ricci et al. , Appl. Phys. Lett. , 034102 (2005); C. Du et al. , Phys. Rev. B , 113105(2006); B. Wood et al. , J. Phys.: Condens. Matter , 076208 (2007); F. Magnus et al. , NatureMaterials , 295 (2008); E. Narimanov, ibid. , 273 (2008); M.B. Romanowsky et al. , Phys.Rev. A , 041110 (2008); K.Y. Bliokh et al. , Rev. Mod. Phys. , 1201 (2008).[2] A.L. Rakhmanov et al. , arXiv:0907.3564 (2009).[3] V.A. Podolskiy and E. E. Narimanov, Phys. Rev. B , 201101 (2005); 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).[4] R. Kleiner et al. , Phys. Rev. Lett. , 2394 (1992); R. Kleiner and P. Muller, Phys. Rev. B , 1327 (1994).[5] S. Savel’ev et al. , arXiv:0903.2969; Rep. Prog. Phys. (to be published). IG. 3: (Color online) A dielectric prism with permittivity ε p is separated from a layered supercon-ductor by a vacuum gap of thickness δ . An electromagnetic wave with incident angle θ , exceedingthe limit angle θ t = arcsin( ε − / p ) for total internal reflection, can excite surface waves that sat-isfy the following resonance condition: ωε / p sin θ = cq . Here k i and k r are the wave-vectors ofthe incident and reflected waves associated with the magnetic field amplitudes H i and H r . Theresonance excitation of surface waves by the incident wave produces a strong suppression of thereflected wave.[6] S. Savel’ev et al. , Phys. Rev. Lett. , 187002 (2005); V. A. Yampol’skii, et al. , Phys. Rev. B , 224504 (2007); V. A. Yampol’skii et al. , ibid , 214501 (2009).[7] P. M. Platzman and P. A. Wolff, Waves and Interactions in Solid State Plasmas (Academic,London, 1973). IG. 4: (Color online) (a) The reflectivity coefficient | R | versus the incident angle θ calculatednumerically for the parameters ν ab = 10 − , γ = 200, ε = 16, ε p = 4, and Ω = 135. Theseparameters correspond to the solid square on the dispersion curve in Fig. 2. The thickness δ ofthe vacuum gap (see Fig. 3) is one wavelength, kδ = 2 π . The vertical dashed line at θ = 30 ◦ corresponds to the limit angle of the total internal reflection. (b) The reflectivity coefficient | R | versus frequency, Ω = ω/ω J , for θ = 30 . ◦ . (c) Color contour plot of the reflectivity coefficient | R | in the plane ( θ, Ω). The dispersion relation for the waves in the dielectric-vacuum-layeredsuperconductor system in Fig. 3 is presented by the solid white curve.
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