The and Applications of Superconducting Metamaterials

Steven M. Anlage1,2

1Center for Nanophysics and Advanced Materials, Physics Department, University of Maryland, College Park, MD 20742-4111 2Department of Electrical and Computer Engineering, University of Maryland, College Park, MD 20742

We summarize progress in the development and application of metamaterial structures utilizing superconducting elements. After a brief review of the salient features of , the advantages of superconducting metamaterials over their normal metal counterparts are discussed. We then present the unique electromagnetic properties of superconductors and discuss their use in both proposed and demonstrated metamaterial structures. Finally we discuss novel applications enabled by superconducting metamaterials, and then mention a few possible directions for future research.

speculates about future directions for these Our objective is to give a basic metamaterials. introduction to the emerging field of superconducting metamaterials. The I. Superconductivity discussion will focus on the RF, microwave, Superconductivity is characterized by and low-THz frequency range, because only three hallmark properties, these being zero DC there can the unique properties of resistance, a fully diamagnetic Meissner effect, superconductors be utilized. Superconductors and macroscopic quantum phenomena.1,2,3 have a number of electromagnetic properties The zero DC resistance hallmark was first not shared by normal metals, and these discovered by Kamerlingh Onnes in 1911, and properties can be exploited to make nearly has since led to many important applications ideal and novel metamaterial structures. In of superconductors in power transmission and section I. we begin with a brief overview of energy storage. The second hallmark is a the properties of superconductors that are of spontaneous and essentially complete relevance to this discussion. In section II we diamagnetic response developed by consider some of the shortcomings of normal superconductors in the presence of a static metal based metamaterials, and discuss how . As the material enters the superconducting versions can have superconducting state it will develop currents dramatically superior properties. This section to exclude magnetic field from its interior. also covers some of the limitations and This phenomenon is known as the Meissner disadvantages of superconducting effect, and distinguishes superconductors from metamaterials. Section III reviews theoretical perfect conductors (which would not show a and experimental results on a number of spontaneous Meissner effect). Finally, unique metamaterials, and discusses their macroscopic quantum effects arise from the properties. Section IV reviews novel quantum mechanical nature of the applications of superconducting metamaterials, superconducting correlated electron state. while section V includes a summary and

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In some circumstances, the the electron scattering rate 1/τ , well outside superconducting electrons can be described in the range of frequencies considered here. The terms of a macroscopic quantum imaginary part of the normal state 4 wavefunction. This wavefunction has a conductivity ( σ 2 ( f ) ) is much smaller than magnitude whose square is interpreted as the the real part (by a factor of ωτ <<1), starts at local density of superconducting electrons (ns), zero at zero frequency, and shows a peak near and whose phase is coherent over macroscopic ω ~ 1/τ . The small value of σ translates dimensions. This phase coherent 2 into a near-zero value of the kinetic wavefunction gives rise to quantum inductance of a normal metal, at least for interference and tunneling effects which are frequencies ω << 1/τ . Hence the inductance quite extraordinary and unique to the of a normal metal wire is dominated by its superconducting state. Examples of geometrical inductance, which arises from macroscopic quantum phenomena include energy stored in the magnetic field created by fluxoid quantization, and the DC and AC a normal current. Josephson effects at tunnel barriers and weak Figure 1(b) shows the conductivity of links. the same metal in the superconducting state in Superconductors have interesting and the local limit at zero temperature. The real unique electromagnetic properties. Consider part of the conductivity is now zero for Fig. 1, which shows an idealized sketch of the frequencies between DC and the energy gap complex conductivity of (a) a normal metal ∆ and (b) a superconductor. One can think of 2 , which is a measure of the binding energy the electromagnetic response of a of the Cooper pairs making up the superfluid. superconductor in terms of a crude but At zero frequency there is now a delta- σ somewhat effective model, known as the two function in 1 , whose strength is proportional 4 fluid model. The superconductor consists of to the superfluid density in the material, ns. a super-fluid of Cooper-paired electrons This feature is responsible for the infinite coexisting with a second fluid of ‘normal’ or conductivity at DC. Surprisingly, an ideal non-superconducting electrons. superconductor has zero AC (real) Electromagnetic fields interact with these two conductivity at zero temperature, between DC fluids in different ways. The superfluid has a and the gap frequency. Losses arise from purely reactive (inductive) response to absorption of photons with energy greater than external fields, and is responsible for 2∆, or from absorption by quasi-particles establishing and maintaining the diamagnetic (broken Cooper pairs) that exist at finite Meissner state below the critical temperature temperature above the energy gap, or due to Tc. The normal fluid acts like a collection of pair-breaking impurities in the material. The ordinary electrons and is responsible for the AC electrodynamics of a superconductor is bulk of the dissipative properties of the dominated by the imaginary part of the superconducting state, through absorption of conductivity, which at finite temperature is photons. much greater than the real part in magnitude Figure 1(a) shows a sketch of the and is strongly frequency dependent

normal-state conductivity, based on the Drude ( σ 2 ~ 1/ω ). The response of a model in the Hagen-Rubens limit (ωτ << 1). superconductor is thus primarily diamagnetic

The real part of the conductivity ( σ 1 ( f ) ) and inductive. The inductance arises from starts at its DC value at zero frequency and screening currents that flow to maintain the monotonically decreases with increasing Meissner state in the bulk of the material. frequency, particularly at frequencies beyond These currents flow within a ‘penetration

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depth’ λ of the surface. The penetration depth all electrons in the metal. The absolute square varies inversely with the superfluid density in of the macroscopic quantum wavefunction is 4 2 2 interpreted as the local superfluid density the two-fluid model as λ = m /(µ0 ns e ) , 2 where m and e are the electronic mass and ns ~ Ψ . When two superconductors are charge respectively, and λ has values ranging brought close together and separated by a thin from roughly 10’s to 100’s of nano-meters at insulating barrier, there can be quantum zero temperature, depending on the material. mechanical tunneling of Cooper pairs between 1 The superfluid density ns (T) is a the two materials. This tunneling results in monotonically decreasing function of two types of . The first temperature up to the critical temperature, produces a DC current between the two where it is equal to zero, as such it acts as an superconductors which depends on the gauge- order parameter of the superconducting state. invariant phase difference between their Thus the penetration depth diverges as Tc is macroscopic quantum wavefunctions as approached from below, reflecting the I = Ic sin(ϕ(t)) , where weakening of the superconducting state and 2π 2    the penetration of deeper into ϕ(t) = θ1(t) −θ2 (t) − A(r,t) • dl is the Φ ∫ the superconductor. The inductance of a 0 1 gauge-invariant phase difference between superconductor has two contributions, one   from the energy stored in magnetic fields superconductors 1 and 2, A(r,t) is the vector outside and inside the superconductor, and the potential in the region between the other from kinetic energy stored in the superconductors, arising from magnetic fields, dissipation-less supercurrent flow. The latter Ic is the critical current of the junction, and contribution is known as kinetic inductance.4 Φ = h / 2e is the flux quantum ( h is Planck’s In the case of a thin and narrow wire with 0 cross-sectional dimensions each less than λ , constant and e is the electronic charge). The the kinetic inductance can be written DC Josephson effect states that a DC current 5 2 will flow between two superconductors approximately as: L = µ λ /t , where t is Kin 0 connected by a tunnel barrier, in the absence the film thickness. As the superfluid density of a DC voltage, and the magnitude and is diminished to zero (by means of increased direction of that current depend on a nonlinear temperature to Tc, or applied current to the function of the phase differences of their critical current, or applied magnetic field to macroscopic quantum wavefunctions, as well the critical field), the kinetic inductance will as applied magnetic field. diverge. In this limit, when the The AC Josephson effect relates a DC superconductor is part of an electrical circuit it voltage drop on the junction to a time-varying can make a dramatic change to the operation gauge-invariant phase difference, and of that circuit as the kinetic inductance varies. therefore to an AC current in the junction:

dϕ / dt = (2π / Φ )V , where V is the DC I.1 Macroscopic Quantum Effects 0 As mentioned above, under certain voltage across the tunnel barrier. The AC circumstances a superconductor can be impedance of a Josephson junction contains described by a macroscopic phase-coherent both resistive and reactive components in general.6 At low frequencies and currents, the complex quantum wavefunction Ψ = n eiθ . s impedance of a Josephson junction can be This wavefunction inherits its phase coherence treated with the resistively and capacitively from the underlying microscopic BCS shunted junction (RCSJ) model.4 In the wavefunction describing the Cooper pairing of presence of both DC and AC applied currents 3

in the junction, the nonlinear inductance of the These quantized tubes of magnetic flux are junction can be described approximately as4 known as magnetic vortices. They have a  2  dynamics and interactions in superconductors L = Φ /2πI 1− (I / I )  , where I is 1,2 JJ 0  c DC c  DC that is quite interesting and non-trivial. The the DC current through the junction. This electrodynamics of magnetic vortices expression holds in the limit of small AC stimulated by microwave currents has been studied extensively.1,4 ,8 Vortices, when they currents I AC << I DC . are present, contribute both resistive and By creating two Josephson junctions in reactive contributions to the impedance of a parallel, forming a superconducting loop, it is superconductor. possible to create quantum interference effects Superconductivity can be found for electrical transport through the two throughout the periodic table and in many junctions. Such a device is called a compounds. Traditional (pre-1986) Superconducting Quantum Interference 4 superconductors are generally referred to as Device, or SQUID for short. The critical “low-T ” superconductors because their current of a SQUID is a nonlinear function of c transition temperatures were in the range applied magnetic flux in the SQUID loop, and between a few mK to about 25 K. The “high- its AC inductance is also a nonlinear function 7 T ” superconductors are mainly copper-oxide of applied field and current. c based, and have transition temperatures up to Superconductors have two important 150 K. More recently a number of interesting length scales, the coherence length ξ , and the novel superconductors have been discovered magnetic penetration depth λ , mentioned in intermediate temperature ranges from 5 K above. The coherence length is a measure of 9 to 50 K, including MgB2, and iron-pnictide the characteristic scale for variations in the superconductors.10 Technologically important magnitude of the macroscopic quantum wave superconductors include Nb (Tc = 9.2 K) and function. Like the penetration depth, it is a various alloys (e.g. Nb-Ti) that are used to microscopic quantity at low temperatures and create superconducting magnets for magnetic diverges as the critical temperature is resonance imaging and particle accelerators. approached from below. The ratio of these Superconductivity exists within a two length scales, the Ginzburg-Landau dome in a three-dimensional parameter space parameter κ ≡ λ /ξ , is approximately spanned by temperature, electrical current temperature independent, and distinguishes density and magnetic field. The limit in type-I superconductors (κ <1/ 2 ) from type- temperature, Tc, is the temperature beyond II superconductors ( κ >1/ 2 ).1 The which the superconducting electronic difference between these two types of correlations are destroyed by thermal agitation. superconductors lies in the free energy The limit in current is called the critical required to create an interface between normal current density, Jc. This is a measure of when and superconducting phases inside the the kinetic energy of a super-current flow material. Type-II superconductors have a equals the free energy difference between the negative free energy for creating such normal and superconducting states. The limit interfaces, and thus the interfaces proliferate in magnetic field, the upper critical field (Hc2 spontaneously when the material is exposed to for type-II superconductors) is a measure of a sufficiently strong magnetic field. The when the magnetic vortices crowd close proliferation ends at the quantum limit, where enough together such that their cores (of each bounded normal phase unit supports dimension ξ ) begin to overlap and superconductivity is fully destroyed. In exactly one quantum of magnetic flux, Φ0 .

4 addition, there is a limit in the frequency and SRRs are decreased, losses increase as domain beyond which superconducting ρ / r 2 and ρ /t , respectively, where ρ is the electrodynamics alone cannot describe the resistivity of the metal, r is the wire radius,  is surface impedance or complex conductivity of the size of the SRR, and t is the thickness of a superconductor. The gap frequency the material making up the SRR.14 = ∆ f gap 2 / h , is set by the ‘energy gap’ Decreasing the normal metal wire radius of an ∆ 2 required to directly destroy a Cooper pair ε eff ( f ) < 0 artificial dielectric to increase its and create two quasi-particles. For most plasma frequency will result in significant superconductors the gap frequency lies increase of losses. As the normal metal SRR between about 10 GHz (low-Tc) and a few dimensions ( t ,  ) decrease, losses will THz (high-Tc) at zero temperature. Note that increase, and the frequency bandwidth of the energy gap monotonically decreases with µeff ( f ) < 0 will eventually vanish. These increasing temperature, going to zero at Tc. deleterious effects do not happen with II. The Advantages of superconducting wires and SRRs because the Superconducting Metamaterials resistivity is orders of magnitude smaller, and Metamaterials are typically the electromagnetic response is dominated by constructed of “atoms” that have engineered the reactive impedance. Superconductors will electromagnetic response. The properties of only break down when the dimensions the artificial atoms are often engineered to become comparable to the coherence length, or when the induced currents approach the produce non-trivial values for the effective 6 9 2 permittivity and effective permeability of a critical current density (Jc ~ 10 – 10 A/cm ). lattice of identical atoms. Such values include relative permittivities and permeabilities that II.1 Low Loss Metamaterials are less than 1, close to zero, or negative. For Novel applications of negative index concreteness, we shall consider below the of refraction (NIR) metamaterials require that scaling properties of metamaterials made of they be made very small compared to the traditional “atomic” structures, like those used wavelength and with very low losses. For example, the proposal of Ziolkowski and in the early metamaterials literature. 15 Traditional metamaterials utilize wires to Kipple to improve the radiation efficiency of influence the dielectric properties by ultra-small dipole antennas requires manipulating the effective plasma frequency metamaterials with overall dimensions on the of the medium.11 The magnetic properties of order of 1 mm operating at 10 GHz. Such Split-Ring Resonators (SRRs) are utilized to metamaterials must be made with elements create a frequency band of sub-unity, negative (wires and split-ring resonators) on the scale or near-zero magnetic permeability.12 of 10’s of micro-meters. Large losses will Substantial losses are one the key destroy the desired NIR behavior so care must limitations of conventional metamaterials. As be taken to see if existing metamaterials discussed in detail below, Ohmic losses place designs can be scaled down to the required a strict limit on the performance of dimensions. Here we examine such a scaling metamaterials in the RF – THz frequency of a conventional normal metal metamaterial range. In contrast to normal metals, made up of split-ring resonators (to provide superconducting wires and SRRs can be negative permeability) and wires (to provide substantially miniaturized while still negative permittivity). maintaining their low-loss properties.13 For comparison, as the size of normal metal wires II.1.a Scaling of SRR Properties

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Consider what happens when normal µeff Eq. (1) is proportional to metal components are reduced in size to ρ / r ~ ρ /(crt) . Since the radius of the achieve the metamaterial dimensions 1 mentioned above. Currently split-ring- SRR (r) will scale like the lattice parameter ℓ resonators (SRRs) for operation at 10 GHz are as the SRR array is shrunk, the loss will scale fabricated using very thick plated metal roughly as ρ / ct . As the size of the SRR (thickness t ~ 25-50 µm) on lossy dielectric decreases, the width of the SRR c, and substrates (such as FR4 and G10). These thickness of the film t must also diminish. SRRs (Fig. 2) have outer diameters of about w The thickness must decrease because one will = 2.5 mm, and gap widths on the scale of g = not be able to make SRRs with dimensions c, 300-500 µm.16 d, g ~ 5 µm using existing films with Let us examine what happens if the thickness t ~ 50 µm. As the film thickness size of the SRR is reduced by a factor of 10, and SRR dimension shrink, the loss term in or more. The effective relative magnetic µeff will grow dramatically. For example, an 12 permeability of an SRR array is given by order of magnitude decrease in SRR πr 2 /  2 dimension (from w = 2.5 mm to 250 µm) will µ = 1− , (1) eff 2 produce roughly a factor of 50-100 increase in 2ρ1 3c0 1+ i − loss. The situation is worse because the losses ωrµ 2 3 2c 0 πω r ln are dominated by the skin effect in the GHz d range and beyond. In this limit the losses are where r is the inner radius of the inner SRR, ℓ proportional to the surface resistance is the lattice spacing of the cubic SRR array, ( Rs = ρ /δ in the local limit, where δ is the c0 is the , and c and d are defined in Fig. 2. skin depth), and the skin depth is typically First, the resonant frequency of the smaller than the film width and thickness. To achieve the objectives of the proposed antenna 3 SRR is given by; ω = c . The application, even further shrinkage of the SRR 0 0 2c πr 3 ln would be required. Normal metal losses are d already significant in conventional designs, so region of Re[ µeff ] < 0 is just above this clearly normal metal SRRs will not be resonant frequency. This suggests that if the practical for ultra-small metamaterials. lattice parameter scales with the radius  ~ r and the ratio c/d is kept fixed, the II.1.b Scaling of Wire Array resonant frequency of the SRR will scale Properties A wire array of lattice spacing a and roughly as ω ~ 1/ r as r decreases. However, 0 wire radius r has a plasma frequency given this expression lacks the capacitive effect of c 2π the gap g, which will modify the scaling 11 0 approximately by, ω p = . The dependence. One can make g quite small to a ln(a / r) increase the capacitance and keep the SRR real part of the effective permittivity of the resonant frequency at 10 GHz as the SRR wire array is negative for frequencies below shrinks. the plasma frequency. Again, as the wire Equation (1) shows the losses in the array lattice parameter shrinks, the plasma metal ring are parameterized by the sheet edge will increase roughly as ω p ~ 1/ a . This resistance per unit width ρ = ρ/(ct), where ρ 1 scaling will maintain an ε < 0 range at 10 is the resistivity of the metal and t is the film eff thickness. The loss term (imaginary part) in GHz, as desired. The complex effective

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relative electric permittivity is given penetration depth λ, the kinetic inductance will be enhanced. As discussed above and below, this is the inductance associated with by;11 . The the kinetic energy of the dissipation-less 2 2 2 loss term is given by ε 0 a ω p ρ /πr , where ρ supercurrent flow. If the cross-sectional area is reduced so that the width or thickness is on = 1/σ is the resistivity of the normal metal the scale of the coherence length ξ, or if a wire. Given the fact that ω ~ 1/ a , this leads p narrow tunnel barrier is created in the to a dielectric loss that scales as ρ / r 2 as the superconductor, then the Josephson effect also wire radius shrinks. Again one runs in to comes in to play.1 Thus a new contribution to trouble as the metamaterial is shrunk, with a the inductance can arise from currents flowing

strong inverse-square increase of losses as the through a Josepshon junction, LJJ . These two wire radius is diminished. forms of inductance allow superconducting It is clear that the properties of metamaterials to be even further miniaturized, conventional normal metal SRR and wire compared to normal metal-based arrays suffer from a dramatic increase in loss metamaterials. as their dimensions are uniformly shrunk to achieve sizes required for non-trivial II.3 Tunability and Nonlinearity applications. This makes normal metals Because of the unique quantum- impractical for any application involving an mechanical properties of superconductors, order of magnitude or more decrease in the their inductance is highly tunable and can be size of a metamaterial that begins with made quite nonlinear. First, the superfluid minimally acceptable losses. Alternatives to density can be a strong function of applied this dilemma must be sought, and magnetic field (both DC and RF) and current, superconducting metamaterials are one creating the opportunity to vary the kinetic obvious choice. and Josephson inductances. The use of temperature-dependent tuning of a negative II.2 Compact Superconducting permeability region of superconducting split Metamaterials ring resonators has been demonstrated.13,14,19 Superconductors have small values of The use of a DC current to tune the kinetic the surface resistance at microwave 2 inductance of superconductors has also been frequencies. Typical surface resistance considered.5,20 values for Nb at 2 K and 1 GHz are in the 17 Superconductors also support nano-Ohm range, while high temperature quantized magnetic vortex excitations, and superconductors have a surface resistance on these bring with them inductance and loss the scale of 100 micro-Ohms at 77 K and 10 8 18 when agitated by RF currents. Subsequent GHz. These small losses (and associated work on superconducting metamaterials σ 2 -dominated electrodynamics) allow showed that the negative permeability region superconductors to overcome the scaling can also be tuned by changing the RF current limitations discussed above. In addition, circulating in the SRRs.21 This was attributed superconductors have other unique advantages, to RF magnetic vortices entering into the and these advantages are especially evident SRRs at sharp inside corners of the patterned when they are scaled down in size. When the structure. This hypothesis was confirmed cross-sectional area of a current-carrying through imaging of the RF current flow in a superconductor is reduced such that one or high-temperature superconducting SRR.21,22 both are on the scale of the magnetic DC magnetic vortices have also been added to

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superconducting SRRs, and frequency shifts Superconductors can also be very in the negative permeability region have been sensitive to environmental influences. observed.21 It was hypothesized that the Variations in temperature, stray magnetic field, frequency shifts came about from an or strong RF power can alter the inhomogeneous distribution of magnetic flux superconducting properties and change the entering and leaving the SRR as the field was behavior of the metamaterial. Careful ramped. Microscopic imaging of the flux temperature control and high quality magnetic penetration by means of magneto-optical shielding are often required for reliable imaging23 verified the strongly performance of superconducting devices. inhomogeneous nature of flux penetration.21 Magnetic vortex lattices in superconductors III. Novel Superconducting have also been proposed as a way to generate Metamaterial Implementations photonic crystals, where the dielectric contrast A number of novel implementations of is created between the superconducting bulk superconducting metamaterials have been and non-superconducting cores of the achieved in addition to superconducting split vortices.24,25 Such materials are considered in rings and wires,13,14, 21. Here we present more detail below. results on several classes of superconducting The Josephson inductance is a strong metamaterials. function of applied currents and fields. Tuning of Josephson inductance in a III.1 Superconductor/Ferromagnet transmission line geometry was considered Composites theoretically by Salehi.20,26 The tunability of Ferromagnetic resonance offers a SQUID inductance by external fields was natural opportunity to create a negative real considered by Du, et al.27 The Josephson part of the effective permeability of a inductance is also known to be very sensitive gyromagnetic material for frequencies above to vortices propagating in an extended resonance.32 However the imaginary part of Josephson junction.28,29,30 the permeability is quite large near the resonance and will limit the utility of such II.4 The Limitations of Re[µeff] < 0 materials. Combining such a Superconducting Metamaterials material with a superconductor can help to The greatest disadvantage of minimize losses,33 and to introduce a superconductors is the need to create and conducting network with Re[εeff] < 0 at the maintain a cryogenic environment, and to same time. A superlattice film of high bring signals to and from the surrounding temperature superconductor and manganite room temperature environment. The use of ferromagnetic layers was created and shown liquid cryogens (such as Nitrogen and to produce a band of negative index in the Helium) is inconvenient and increasingly vicinity of 90 GHz.34 The material displayed expensive. Closed-cycle cryocooler systems Re[n] < 0 near an applied field of 3 T at 90 have become remarkably small, efficient and GHz, although the imaginary part of the index inexpensive since the discovery of high-Tc (Im[n]) was of comparable magnitude to the superconductors. Such systems are now able real part. to operate for 5 years un-attended, and can accommodate the heat load associated with III.2 DC Magnetic Superconducting microwave input and output transmission lines Metamaterials 31 to room temperature. The concept of a DC magnetic cloak has been proposed and examined by Wood and

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Pendry.35 The general idea is to take a solid and negative permeability will disappear for diamagnetic superconducting object and de-phasing rates larger than a critical value. divide it into smaller units, arranging them in A treatment of RF SQUIDs interacting such a way as to tailor the magnetic response. with classical electromagnetic waves was The cloak would shield a region of space from presented by Lazarides and Tsironis.41 They external DC magnetic fields, and not disturb considered a two-dimensional array of RF the magnetic field distribution outside of the SQUIDs in which the Josephson junction was cloaking structure. The cloak involves the use treated as a parallel combination of resistance, of superconducting plates to provide capacitance and Josephson inductance. Near diamagnetic response, causing the radial resonance, a single RF SQUID can have a component of effective permeability (µr) to lie large diamagnetic response. In an array, there between 0 and 1. An additional component is is a frequency and RF-magnetic field region in required to enhance the tangential components which the system displays a negative real part of magnetic permeability so that µθ and µφ are of effective permeability. The permeability is both greater than 1, and paramagnetic in fact oscillatory as a function of applied substances were suggested in the original magnetic flux, and will be suppressed with proposal. An experimental demonstration of applied fields that induce currents in the the first step (µr < 1) in creating such a cloak SQUID that exceed the critical current of the was made using an arrangement of Pb thin Josephson junction. Related work on a one- film plates.36 Subsequent theoretical work has dimensional array of superconducting islands refined the DC magnetic cloak design and that can act as quantum bits (qubits) was 42 suggested that it be implemented with high- considered by Rakhmanov, et al. When temperature superconducting thin films.37 interacting with classical electromagnetic radiation, the array can create a quantum III.3 SQUID Metamaterials photonic crystal that can support a variety of A SQUID is a natural quantum analog of the nonlinear wave excitations. A similar idea split ring resonator. In fact the Radio based on a SQUID transmission line was Frequency (RF) SQUID, developed in the implemented to perform parametric 43 44 1960’s, is essentially a quantum SRR in which amplification of microwave signals. , the classical capacitor is replaced with a Josephson junction. It’s original purpose was III.4 Radio Frequency to measure small RF magnetic fields and Superconducting Metamaterials operate as a flux to frequency transducer.38,39 Superconducting thin film wires can create The first proposal to use an array of RF large inductance values without the associated SQUIDs as a metamaterial was made by Du, losses found in normal metals. This offers the Chen and Li.40,27 Their calculation assumes opportunity to extend metamaterial atom the SQUID has quantized energy levels and structures to lower frequencies where larger considers the interaction of individual inductance values are required to build microwave photons with the lowest lying resonant structures. In the past, three- states of the SQUID potential. For small dimensional structures have been employed to 45 detuning of the microwave photon frequency create artificial magnetism below 100 MHz. above the transition from the ground state to The use of two-dimensional spiral resonators the first excited state, the medium will present to create negative effective permeability atoms a negative effective permeability. However, has been developed in the sub-GHz domain 46 the frequency region of negative permeability using thick normal metal wires. However, is diminished by a non-zero de-phasing rate, such spirals are too lossy to resonate below

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100 MHz, for reasons similar to those As such, they can be used for both PC discussed above in section II.1. structures24,54 and metamaterials.63 It is Superconducting thin film spirals have low known that the c-axis plasma frequency can be losses, but also have enhanced inductance tuned by a DC magnetic field,64,65 hence the from the kinetic inductance of the superfluid entire PC band structure can also be tuned by flow. Superconducting spiral metamaterials a magnetic field.54 operating near 75 MHz have been developed Feng, et al.,66 examined a PC made up with Nb thin films, and show strong tunability of superconducting cylinders embedded in a as the transition temperature is approached.47 dielectric medium. They found that Such metamaterials may be useful for temperature tuning of the superfluid density, magnetic resonance imaging,45,48 near-field and therefore of the effective superfluid imaging,49 and compact RF resonator plasma frequency, can produce a tunable band applications.50 of all-angle negative refraction, as well as tunable refracting beams. The angle of III.5 Superconducting Photonic refraction can be tuned from positive to Crystals negative values, and achieve up to 45o of Photonic crystals (PCs) are generally sweep. This work did not consider the losses constructed from a modulated dielectric or plasmonic effects due to quasiparticles, and function contrast with a spatial scale on the did not take into account the frequency order of the wavelength.51 As such, they fall bandwidth limitations imposed by the outside the domain of what are usually called superconducting gap 2∆. Subsequent work metamaterials, but we shall consider them showed that neglect of the quasiparticles here nonetheless. PCs show intricate band introduces significant errors, although a structure arising from multiple scattering of strongly temperature dependent quasiparticle light from the dielectric contrast in the scattering rate (as seen in certain cuprate material. Superconducting photonic crystals superconductors67) can restore some of the have been proposed and discussed by a temperature tunability.58 number of Development of dielectric contrast researchers24,25,52,53,54,55,56,57,58,59,60,61 The from an Abrikosov vortex lattice to create a original suggestions for superconducting PCs photonic crystal is a novel idea first proposed pre-dated most of the work on metamaterials. by Takeda, et al.68, and studied by Kokabi, et The earliest ideas proposed propagating al.,69 and Zandi, et al.70 Magnetic vortices modes in anisotropic cuprate superconducting form a long-range ordered lattice in an ideal PCs,24 and a new spectroscopic feature (a superconductor, creating a regular dielectric phonon-polariton-like gap) in a contrast between normal metal core tubes and superconducting PC.25,52 In the latter case, it the superconducting background. Takeda, et was found that the gap is temperature al. assumed the cores were described by a dependent, due to the temperature dependence constant permittivity, while outside it is a of the superfluid density, and therefore dissipation-less Drude metal. They found that tunable.52,53,54,62,57 the PC band structure, and gaps, can be tuned High temperature cuprate by variation of the magnetic field (which superconducting materials have strongly changes the PC lattice parameter) and the anisotropic dielectric properties. They show Ginzburg-Landau parameter, κ.68 Kokabi, et metallic behavior for currents flowing in the al.,69 and Zandi, et al.70 calculated the Cu-O planes, and have dielectric-like superconducting electron density using the properties for light polarized in the c-direction. self-consistent Ginzburg-Landau formalism,

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and calculated the resulting PC band structure superconducting films and showed resonances for both isotropic69 and anisotropic70 with indices between +2 and -6, including superconductors. zero, over the broad frequency range from about 5 to 24 GHz. Superconductors are IV. Novel Applications Enabled by particularly attractive for use in ultra-compact Superconducting Metamaterials resonators because otherwise the Ohmic losses Superconducting metamaterials have of a deep sub-wavelength structure would been utilized to realize a number of unique suppress the Q and render the device applications. In addition, theorists have made practically useless. Quality factors of the predictions for other exciting applications of negative order resonances were on the order of superconducting metamaterials which have 3000 at 30 K, while those for positive order not yet been realized. resonances were below 400.19 The combination of left-handed and A related theoretical proposal was to right-handed propagation media creates add Josephson inductance to the opportunities for new types of resonant superconducting dual transmission line to structures. Engheta predicted that a resonant enhance nonlinearity. This can result in a structure created by laminating two materials tunable dispersion relation for propagating with opposite senses of phase winding can waves in the structure.26 Considering a create a new class of resonant structures.50 He metamaterial made up of a Josephson proposed a resonator consisting of two flat quantum bit (qubit) array leads to a number of conducting plates separated by a sandwich of interesting predictions, including the left-handed and right-handed metamaterials. development of a quantum photonic crystal A wave propagating in the direction normal to that derives its properties from the quantum the plates will suffer a combination of forward states accessible to the qubits.42 and reverse phase windings before reaching A one-dimensional SQUID array the other reflecting boundary. Under these nonlinear transmission line has been used as a conditions the wave could undergo a net phase parametric amplifier, tunable for microwave shift of 0 radians and still create a resonance signals between 4 and 8 GHz, and providing condition. The net phase shift could also be a up to 28 dB of gain.43,44 This amplifier is positive or negative multiple of 2π radians as suitable for use in detecting signals from low well, each creating a resonant condition. The temperature qubits operating at microwave result is an ultra-compact resonator whose frequencies, and can squeeze quantum noise.44 overall dimension is no longer constrained by the wavelength of the resonant wave. The V. Future Directions and first realization of this ultra-compact resonator Conclusions with superconductors was accomplished by There are many exciting future Wang and Lancaster.19,71 They utilized a directions of research in superconducting lumped-element dual transmission line metamaterials. Their low loss, compact structure implemented in a co-planar structure, and nonlinear properties make them waveguide geometry. Such structures have ideal candidates for realization of the proven very useful for creating practical landmark predictions of metamaterial theory, backward-wave microwave devices,72 and including the near-perfect lens, evanescent superconducting versions were also studied by wave amplification, hyper-lensing, Salehi, Majedi and Mansour.20 The transformation optics and illusion optics. superconducting sub-wavelength resonator There are many exciting possibilities was implemented with cuprate to extend the frequency coverage of

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superconducting metamaterials from the low- example, the anisotropic dielectric function in 63 RF range (below 10 MHz) to the upper limits the layered high-Tc cuprate superconductors of superconductivity in the multi-THz domain. offers an opportunity to realize a hyperlens in Low frequencies offer the opportunity to the THz and far-infrared domain.74 The create deep sub-wavelength structures to act cuprates also have a built-in plasmon as energy storage devices, imaging devices, or excitation for electric fields polarized along as filters. The THz domain brings many their c-axis (the nominally insulating exciting possibilities for spectroscopy and direction),64,65 which can be exploited for imaging as well. plasmonic applications. Conventional Josephson-based metamaterials offer superconductors have also shown many interesting opportunities for novel meta conventional propagating plasmon material structures. Their nonlinear response excitations,75 suggesting that low loss can be used to introduce parametric microwave plasmonics, analogous to optical amplification73 of negative-index photons, plasmonics, can be developed using further reducing the deleterious effects of loss. superconducting thin films. They also have extreme tunability with both DC and RF magnetic fields due to changes in Acknowledgements: This project has the Josephson inductance. Their properties been supported by the US Office of Naval can also be made broad-band using a plurality Research through grant # N000140811058, of junction critical currents in the design. and the Center for Nanophysics and Advanced Many superconductors have intrinsic Materials (CNAM) at the University of properties that make them quite suitable for Maryland. use as metamaterials or photonic crystals. For

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Figure Captions

Fig. 1 Schematic plot of the complex conductivity of (a) normal metal and (b) superconductor versus frequency ω in the Hagen-Rubens limit (ωτ << 1). We assume ideal materials at zero temperature in this sketch.

Fig. 2 (a) A “single” planar split-ring resonator (SRR) with dimensions of the film (black) shown. Based on the sketch in Ref. (16). (b) A stack of SRRs illustrating the lattice spacing ℓ.

3.0 (a) σ = σ1− iσ22.5

2.0

1.5 σ1(ω) 1.0

0.5 σ2(ω) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 ω

3.0 σ = σ − σ (b) 1 i 22.5 σ2(ω) ~ 1/ω 2.0

1.5

1.0

(π n e2/m)δ(ω)0.5 s σ1(ω)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 ω 2∆ / 

Figure 1

13

(b) g (a) ℓ w

c d

Figure 2

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