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The Physics and Applications of Superconducting Metamaterials

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The Physics and Applications of Superconducting Metamaterials The Physics 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 superconductivity, 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 magnetic field. 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 1 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 2 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 2 2 4 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 Josephson effect. 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 magnetic flux 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
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