rf SQUID metamaterials

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A rf superconducting quantum interference device SQUID array in an alternating magnetic field is ..... of Education and the European Union and Grant No.
APPLIED PHYSICS LETTERS 90, 163501 共2007兲

rf superconducting quantum interference device metamaterials N. Lazaridesa兲 Department of Physics, University of Crete, P.O. Box 2208, 71003 Heraklion, Greece; Institute of Electronic Structure and Laser, Foundation for Research and Technology-Hellas, P.O. Box 1527, 71110 Heraklion, Greece; and Department of Electrical Engineering, Technological Educational Institute of Crete, P.O. Box 140, Stavromenos, 71500, Heraklion, Crete, Greece

G. P. Tsironis Facultat de Fisica, Department d’Estructura i Constituents de la Materia, Universitat de Barcelona, Av. Diagonal 647, E-08028 Barcelona, Spain; Department of Physics, University of Crete, P.O. Box 2208, 71003 Heraklion, Greece; and Institute of Electronic Structure and Laser, Foundation for Research and Technology-Hellas, P.O. Box 1527, 71110 Heraklion, Greece

共Received 19 January 2007; accepted 13 March 2007; published online 16 April 2007兲 A rf superconducting quantum interference device 共SQUID兲 array in an alternating magnetic field is investigated with respect to its effective magnetic permeability, within the effective medium approximation. This system acts as an inherently nonlinear magnetic metamaterial, leading to negative magnetic response, and thus negative permeability above the resonance frequency of the individual SQUIDs. Moreover, the permeability exhibits oscillatory behavior at low field intensities, allowing its tuning by a slight change of the intensity of the applied field. © 2007 American Institute of Physics. 关DOI: 10.1063/1.2722682兴 A rf superconducting quantum interference device 共SQUID兲 consists of a superconducting ring interrupted by a Josephson junction 共JJ兲.1 When driven by an alternating magnetic field, the induced supercurrents around the ring are determined by the JJ through the celebrated Josephson relations. This system exhibits rich nonlinear behavior, including chaotic effects.2 Recently, quantum rf SQUIDs have attracted great attention, since they constitute essential elements for quantum computing.3 In this direction, rf SQUIDs with one or more zero and/or ␲ ferromagnetic JJs have been constructed.4 In this letter we show that rf SQUIDs may serve as constitual elements for nonlinear magnetic metamaterials 共MMs兲, i.e., artificial, composite, and inherently nonmagnetic media with 共positive or negative兲 magnetic response at microwave frequencies. Classical MMs are routinely fabricated with regular arrays of split-ring resonators 共SRRs兲, with operating frequencies up to the optical range.5 Moreover, MMs with negative magnetic response can be combined with plasmonic wires that exhibit negative permittivity, producing thus left-handed 共LH兲 metamaterials characterized by negative refraction index. Superconducting SRRs promise severe reduction of losses, which constrain the evanescent wave amplification in these materials.6 Metamaterials involving superconducting SRRs and/or wires have been recently demonstrated experimentally.7 The effect of incorporating superconductors in LH transmission lines has been also studied.8 Naturally, the theory of metamaterials has been extended to account for nonlinear effects.9–13 Nonlinear MMs support several types of interesting excitations, e.g., magnetic domain walls,11 discrete breathers,12 and envelope solitons.13 Regular arrays of rf SQUIDs offer an alternative for the construction of nonlinear MMs due to the nonlinearity of the JJ. Very much like the SRR, the rf SQUID 关Fig. 1共b兲兴 is a resonant nonlinear oscillator, and similarly it responds in a manner analogous to a magnetic “atom” in a time-varying a兲

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magnetic field with appropriate polarization, exhibiting a resonant magnetic response at a particular frequency. The SRRs are equivalently RLC circuits in series, featuring a resistance R, a capacitance C, and an inductance L, working as small dipoles. In turn, adopting the resistively and capacitively shunted junction model for the JJ,1 the rf SQUIDs are not dipoles but, instead, they feature an inductance L in series with an ideal Josephson element 共i.e., for which I = Ic sin ␾, with ␾ the Josephson phase兲, shunted by a capacitor C and a resistor R 关Fig. 1共c兲兴. However, the fields they produce are approximately those of small dipoles, although quantitatively they are affected by flux quantization in superconducting loops. Consider a rf SQUID with loop area S = ␲a2 共radius a兲, in a magnetic field of amplitude He0, frequency ␻, and intensity Hext = He0 cos共␻t兲 perpendicular to its plane 共t is the time variable兲. The field generates a flux ⌽ext = ⌽e0 cos共␻t兲 threading the SQUID loop, with ⌽e0 = ␮0SHe0 and ␮0 the permeability of the vacuum. The flux ⌽ trapped in the SQUID ring is given 共in normalized variables兲 by f = f ext + ␤i,

共1兲

where f = ⌽ / ⌽0, f ext = ⌽ext / ⌽0, i = I / Ic, ␤ = ␤L / 2␲ ⬅ LIc / ⌽0, I is the current circulating in the ring, Ic is the critical current of the JJ, L is the inductance of the SQUID ring, and ⌽0 is

FIG. 1. Schematic drawing of the SQUID array, along with the equivalent circuit for a rf SQUID in external flux ⌽ext.

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Appl. Phys. Lett. 90, 163501 共2007兲

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FIG. 2. 共Color online兲 共a兲 Coefficient D vs the applied flux amplitude f e0, for ␤ = 0.15 共red-solid curve兲; ␤ = 0.10 共green-dashed curve兲; ␤ = 0.05 共bluedotted curve兲. 共b兲 The amplitude of the flux f 0 vs f e0, for ⍀ = 0.9 共red-solid curve兲, ⍀ = 1.1 共green-dashed curve兲, and ␥ = 0.001, ␤ = 0.15. The black circles and blue diamonds correspond to the numerically obtained f 0 for ⍀ = 0.9 and 1.1, respectively.

the flux quantum. The dynamics of the normalized flux f is governed by the equation df d2 f + ␥ + ␤ sin共2␲ f兲 + f = f ext , d␶ d␶2

共2兲

where C and R are the capacitance and resistance, respectively, of the JJ, ␥ = L␻0 / R, ␶ = ␻0t, ␻20 = 1 / LC, and f ext = f e0 cos共⍀␶兲,

共3兲

with f e0 = ⌽e0 / ⌽0 and ⍀ = ␻ / ␻0. The small parameter ␥ actually represents all of the dissipation coupled to the rf SQUID. An approximate solution for Eq. 共2兲 may be obtained for ⍀ close to the SQUID resonance frequency 共⍀ ⬃ 1兲 in the nonhysteretic regime ␤L ⬍ 1. Following Ref. 14 we expand the nonlinear term in Eq. 共2兲 in a Fourier-Bessel series of the form ⬁

␤ sin共2␲ f兲 = − 兺

n=1

共− 1兲n Jn共n␤L兲sin共2␲nf ext兲, n␲

共4兲

where Jn is the Bessel function of the first kind, of order n. By substiting Eq. 共3兲 in Eq. 共4兲 and carrying out the FourierBessel expansion of the sine term, one needs to retain only the fundamental ⍀ component in the expansion.15 This leads to the simplified expression

␤ sin共2␲ f兲 ⯝ D共f e0兲cos共⍀␶兲,

共5兲

⬁ where D共f e0兲 = −2兺n=1 关共−1兲n / n␲兴Jn共n␤L兲J1共2␲nf e0兲. By substitution of Eq. 共5兲 in Eq. 共2兲, the latter can be solved for the flux f = f 0 cos共⍀␶ + ␪兲 in the loop, with

f0 =

冑␥ ⍀ 2

f e0 − D 2

+ 共1 − ⍀ 兲

2 2

,

␪ = tan−1





− ␥⍀ , 1 − ⍀2

兩f 0兩 = 共f e0 − D兲/兩1 − ⍀2兩.

where we plot separately the three terms of Eq. 共1兲 in time. The quantities f, f ext, and ␤i are shown for two periods T = 2␲ / ⍀ in each case, after they have reached a steady state. For ⍀ ⬍ 1 关Figs. 3共a兲 and 3共b兲兴, the flux f 共green-dashed curves兲 is in phase with f ext 共blue-dotted curves兲, while for ⍀ ⬎ 1 关Figs. 3共c兲 and 3共d兲兴 the flux f is in antiphase with f ext. The other curves 共red-solid curves兲 correspond to ␤i, the response of the SQUID to the applied flux. Away from the resonance, the response is 共in absolute value兲 less than 关Fig. 3共a兲, for ⍀ = 0.63兴 or nearly equal 关Fig. 3共d兲, for ⍀ = 8.98兴 to the magnitude of f ext. However, close to resonance, the response ␤i is much larger than f ext, leading to a much higher flux f 关Figs. 3共b兲 and 3共c兲 for ⍀ = 0.9 and ⍀ = 1.1, respectively兴. Moreover, in Fig. 3共c兲, f is in antiphase with f ext, showing thus extreme diamagnetic 共negative兲 response. The numerically obtained amplitudes f 0 关depicted as black circles for ⍀ = 0.9 and blue diamonds for ⍀ = 1.1 in Fig. 2共b兲兴 are in fair agreement with the analytical expression, Eq. 共6兲. The agreement becomes better for larger f e0. We now consider a planar rf SQUID array consisting of identical units 关Fig. 1共a兲兴, and forming a lattice of unit-cell side d; the system is placed in a magnetic field Hext ⬅ H perpendicular to SQUID plane. If the wavelength of H is much larger than d, the array can be treated as an effectively continuous and homogeneous medium. Then, the magnetic induction B in the array plane is B = ␮0共H + M兲 ⬅ ␮0␮rH,

共8兲

3

共6兲

where ␪ is the phase difference between f and f ext. The dependence of D and f 0 on f e0 for low field intensity is illustrated in Figs. 2共a兲 and 2共b兲, respectively. For larger f e0 the coefficient D approaches zero still oscillating, while f 0 approaches a straight line with slope depending on ⍀ and ␥. For ␥ Ⰶ 1 and not very close to the resonance, ␪ ⯝ 0. It is instructive to express the ␥ = 0 solution as f = ± 兩f 0兩cos共⍀␶兲,

FIG. 3. 共Color online兲 Time dependence of the flux f 共green-dashed curves兲, the applied flux f ext 共blue-dotted curves兲, and the response ␤i 共red-solid curves兲, for ␤ = 0.15, ␥ = 0.001, f e0 = 1.14, and 共a兲 ⍀ = 0.63; 共b兲 ⍀ = 0.9; 共c兲 ⍀ = 1.1; 共d兲 ⍀ = 9.0.

共7兲

The plus 共minus兲 sign, corresponding to a phase shift of 0 共␲兲 of f with respect to f ext, is obtained for ⍀ ⬍ 1 共⍀ ⬎ 1兲. Thus, the flux f may be either in-phase 共⫹ sign兲 or in antiphase 共⫺ sign兲 with f ext, depending on ⍀. This is confirmed by numerical integration of Eq. 共2兲, as shown in Fig. 3,

where M = SI / d is the magnetization induced by the current I circulating a SQUID loop and ␮r the relative permeability of the array. Introducing M into Eq. 共8兲 and using Eqs. 共1兲, 共2兲, and 共7兲, we get

␮r = 1 + ˜F共±兩f 0兩/f e0 − 1兲,

共9兲

where ˜F = ␲2共␮0a / L兲共a / d兲3. The coefficient ˜F has to be very ˜ Ⰶ 1兲, so that magnetic interactions between indismall 共F vidual SQUIDs can be neglected in a first approximation. Recall that the plus sign in front of 兩f 0兩 / f e0 should be taken for ⍀ ⬍ 1, while the minus sign should be taken for ⍀ ⬎ 1. In Fig. 4 we plot ␮r both for ⍀ ⬍ 1 关Fig. 4共a兲兴 and ⍀ ⬎ 1 关Fig. 4共b兲兴, for three different values of ˜F. In real arrays, that coefficient could be engineered to attain the desired value. In both Figs. 4共a兲 and 4共b兲, the relative permeability ␮r oscillates for low intensity fields 共low f ext兲, while it tends to a

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Appl. Phys. Lett. 90, 163501 共2007兲

N. Lazarides and G. P. Tsironis

FIG. 4. 共Color online兲 Relative permeability ␮r vs f e0, for ˜F = 0.01 共red-solid curves兲, ˜F = 0.02 共green-dashed curves兲, ˜F = 0.03 共blue-dotted curves兲, and 共a兲 ⍀ = 0.99; 共b兲 1.01.

constant at larger f ext. In Fig. 4共a兲 共⍀ ⬍ 1兲, the relative permeability ␮r is always positive, while it increases with increasing ˜F. In Fig. 3共b兲, however, ␮r may assume both positive and negative values, depending on the value of ˜F. With appropriate choice of ˜F, it becomes oscillatory around zero 关green-dashed curve in Fig. 4共b兲兴, allowing tuning from positive to negative ␮r with a slight change of f ext. In conclusion, we have shown that a planar rf SQUID array exhibits large magnetic response close to resonance, which may be negative above the resonance frequency, leading to effectively negative ␮r. For low field intensities 共low f ext兲, ␮r exhibits oscillatory behavior which gradually dissappears for higher f ext. This behavior may be exploited to construct a flux-controlled metamaterial 共as opposed to voltagecontrolled metamaterial demonstrated in Ref. 16兲. The physical parameters required for the rf SQUIDs giving the dimensionless parameters used above are not especially formidable. A rf SQUID with L ⯝ 105 pH, C ⯝ 80 fF, and Ic ⯝ 3 ␮A, would give ␤ ⯝ 0.15 共␤L ⯝ 0.94兲. For these parameters, a value of the resistance R ⯝ 3.6 k⍀ is required in order to have ␥ ⯝ 10−3, used in the numerical integration of Eq. 共2兲. However, our results are qualitatively valid for ␥ even an order of magnitude larger, in which case R ⯝ 360 ⍀. We note that ␻0 = ␻ p / 冑␤L, where ␻ p is the plasma frequency of the JJ. For the parameters considered above, where ␤L is slightly less than unity, the frequencies ␻0 and ␻ p are of the same order. However, ␻ p does not seem to have any special role in the microwave response of the rf

SQUID. Du et al. have studied the quantum version of a SQUID array as a LH metamaterial, concluded that negative refractive index with low loss may be obtained in the quantum regime.17 Consequently, ␮r can be negative at some specific frequency range. However, their corresponding expression for ␮r is linear, i.e., it does not depend on the amplitude of the applied flux, and thus it does not allow flux tuning. Moreover, experiments with SQUID arrays in the quantum regime, where individual SQUIDs can be described as twolevel systems, are much more difficult to realize. The authors acknowledge support from the grant “Pythagoras II” 共KA. 2102/TDY 25兲 of the Greek Ministry of Education and the European Union and Grant No. 2006PIV10007 of the Generalitat de Catalunia, Spain. K. K. Likharev, Dynamics of Josephson Junctions and Circuits 共Gordon and Breach, Philadelphia, 1986兲. 2 K. Fesser, A. R. Bishop, and P. Kumar, Appl. Phys. Lett. 43, 123 共1983兲. 3 M. F. Bocko, A. M. Herr, and M. J. Feldman, IEEE Trans. Appl. Supercond. 7, 3638 共1997兲. 4 T. Yamashita, S. Takahashi, and S. Maekawa, Appl. Phys. Lett. 88, 132501 共2006兲. 5 T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, Science 303, 1494 共2004兲. 6 M. C. Ricci, H. Xu, S. M. Anlage, R. Prozorov, A. P. Zhuravel, and A. V. Ustinov, IEEE Trans. Appl. Supercond. 共to be published兲, e-print condmat/0608737. 7 M. C. Ricci, N. Orloff, and S. M. Anlage, Appl. Phys. Lett. 87, 034102 共2005兲. 8 H. Salehi, A. H. Majedi, and R. R. Mansour, IEEE Trans. Appl. Supercond. 15, 996 共2005兲. 9 A. A. Zharov, I. V. Shadrivov, and Y. Kivshar, Phys. Rev. Lett. 91, 037401 共2003兲; M. Lapine, M. Gorkunov, and K. H. Ringhofer, Phys. Rev. E 67, 065601 共2003兲. 10 N. Lazarides and G. P. Tsironis, Phys. Rev. E 71, 036614 共2005兲; I. Kourakis and P. K. Shukla, ibid. 72, 016626 共2005兲. 11 I. V. Shadrivov, A. A. Zharov, N. A. Zharova, and Y. S. Kivshar, Photonics Nanostruct. Fundam. Appl. 4, 69 共2006兲. 12 N. Lazarides, M. Eleftheriou, and G. P. Tsironis, Phys. Rev. Lett. 97, 157406 共2006兲. 13 I. Kourakis, N. Lazarides, and G. P. Tsironis, e-print cond-mat/0612615. 14 S. N. Erné, H.-D. Hahlbohm, and H. Lübbig, J. Appl. Phys. 47, 5440 共1976兲. 15 A. R. Bulsara, J. Appl. Phys. 60, 2462 共1986兲. 16 O. Reynet and O. Acher, Appl. Phys. Lett. 84, 1198 共2004兲. 17 C. Du, H. Chen, and S. Li, Phys. Rev. B 74, 113105 共2006兲. 1

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