## A one-dimensional tunable magnetic metamaterial |

Optics Express, Vol. 21, Issue 19, pp. 22540-22548 (2013)

http://dx.doi.org/10.1364/OE.21.022540

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### Abstract

We present experimental data on a one-dimensional super-conducting metamaterial that is tunable over a broad frequency band. The basic building block of this magnetic thin-film medium is a single-junction (rf-) superconducting quantum interference device (SQUID). Due to the nonlinear inductance of such an element, its resonance frequency is tunable in situ by applying a dc magnetic field. We demonstrate that this results in tunable effective parameters of our metamaterial consisting of 54 rf-SQUIDs. In order to obtain the effective magnetic permeability *μ*_{r,eff} from the measured data, we employ a technique that uses only the complex transmission coefficient *S*_{21}.

© 2013 OSA

## 1. Introduction

1. M. C. Ricci, N. Orloff, and S. M. Anlage, “Superconducting metamaterials,” Appl. Phys. Lett. **87**, 034102 (2005). [CrossRef]

2. M. C. Ricci, H. Xu, R. Prozorov, A. P. Zhuravel, A. V. Ustinov, and S. M. Anlage, “Tunability of Superconducting Metamaterials,” IEEE Trans. Appl. Supercond. **17**, 918–921 (2007). [CrossRef]

4. J. Wu, B. Jin, Y. Xue, C. Zhang, H. Dai, L. Zhang, C. Cao, L. Kang, W. Xu, J. Chen, and P. Wu, “Tuning of superconducting niobium nitride terahertz metamaterials,” Opt. Express **19**, 12021–12026 (2011). [CrossRef] [PubMed]

5. N. Lazarides and G. P. Tsironis, “rf superconducting quantum interference device metamaterials,” Appl. Phys. Lett. **90**, 163501 (2007). [CrossRef]

7. A. I. Maimistov and I. R. Gabitov, “Nonlinear response of a thin metamaterial fim containing Josephson junction,” Optics Commun. **283**, 1633–1639 (2010). [CrossRef]

8. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. **84**, 4184–4187 (2000). [CrossRef] [PubMed]

9. P. Jung, S. Butz, S. V. Shitov, and A. V. Ustinov, “Low-loss tunable metamaterials using superconducting circuits with Josephson junctions,” Appl. Phys. Lett. **102**, 062601 (2013). [CrossRef]

## 2. The rf-SQUID

*L*

_{j}is tunable by a magnetic field. In addition to

*L*

_{j}, the geometric inductance of the loop

*L*

_{geo}contributes to the total inductance

*L*

_{tot}of the rf-SQUID. The full equivalent electric circuit is depicted in Fig. 1(b). The red circle marks the electric circuit analogue of the Josephson junction for which the resistively capacitively shunted junction model is used [11]. Like the SRR, the rf-SQUID can be interpreted as an LC-oscillator. Unlike the SRR however, the total inductance and thus the resonance frequency of the rf-SQUID is tunable, assuming that the ac magnetic field component is small.

_{x}/Nb trilayer process. The Josephson junction is circular with a diameter of 1.6

*μ*m, its critical current

*I*= 1.8

_{c}*μ*A. From this value, the zero field Josephson inductance is calculated to be

*L*

_{j}= 183 pH. This value is approximately twice as large as the geometric inductance of the loop

*L*

_{geo}= 82.5 pH. Thus, the rf-SQUID considered in this work is nonhysteretic [10, 11]. The junction is shunted with an additional parallel plate capacitor with a capacitance

*C*

_{shunt}= 2.0 pF which is two orders of magnitude larger than the intrinsic capacitance of the Josephson junction. Due to this shunt capacitor, the resonance frequency of the rf-SQUID is reduced and tunable between approximately 9 GHz and 15 GHz.

## 3. The SQUID Metamaterial

*μ*m, which is twice the width of the single SQUID and more than ten times the distance between each SQUID and the central conductor of the waveguide. Therefore, the inductive coupling between adjacent SQUIDs is approximately one order of magnitude smaller than the coupling to the CPW and can be neglected.

12. S. Butz, P. Jung, L. V. Filippenko, V. P. Koshelets, and A. V. Ustinov, “Protecting SQUID metamaterials against stray magnetic field”, Supercond. Sci. Technol. **26**, 094003 (2013). [CrossRef]

*T*= 4.2 K.

## 4. Experimental Results

*S*

_{21}) as a function of frequency

*ν*and magnetic flux Φ

_{e0}. The microwave power at the sample is approximately

*P*≈ −90 dBm, including losses in the coaxial cables. The calibration of the measurement is done by applying a flux of Φ

_{e0}= Φ

_{cal}= Φ

_{0}/2. At this flux value the resonance frequency is shifted to its lowest possible value, which lies between 9 and 10 GHz. The built-in “thru” calibration function of the VNA is used to subtract the corresponding reference data from the rest of the measurement (see also Appendix B). The resulting transmission magnitude for such a measurement is presented in Fig. 3. For clarity, only data above 10 GHz are shown.

_{e0}is used instead of magnetic field. For our SQUIDs, a flux of one flux quantum Φ

_{0}=

*h*/(2

*e*) = 2.07 × 10

^{−15}Vs corresponds to a field of 1.2

*μ*T.

9. P. Jung, S. Butz, S. V. Shitov, and A. V. Ustinov, “Low-loss tunable metamaterials using superconducting circuits with Josephson junctions,” Appl. Phys. Lett. **102**, 062601 (2013). [CrossRef]

_{e0}= −0.185Φ

_{0}. It allows a more detailed look at transmission magnitude and phase around the resonance frequency of

*ν*

_{0}= 13.88 GHz. We observe that the resonance dip is indeed one collective resonance dip, in which no single SQUID lines are distinguishable. Its width and shape result from the superposition of the resonances of the individual SQUIDs. The steep rise of the phase dependence underlines the collective behaviour. However, as expected from the weak coupling between adjacent SQUIDs, there is no mutual synchronization. The quality factor of this resonance is

*Q*

_{collective}= 100. When we compare this value to the quality factor of a single SQUID line (we used one of the weak stray lines mentioned earlier), the quality factor of the single SQUID is more than twice as large

*Q*

_{single}= 215. A detailed investigation of the quality factors, however, is hindered by the very shallow resonance of the single SQUID which is less than 1dB.

*μ*

_{r,eff}from the measured transmission data depicted in Fig. 3. In Fig. 4(b), real and imaginary part of the effective magnetic permeability

*μ*

_{r,eff}are plotted for the same frequency interval and flux value as the transmission data in Fig. 4(a). The magnetic permeability shows a frequency dependence typical for a metamaterial consisting of resonant elements [13

13. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. **41**, 2075–2084 (1999). [CrossRef]

*μ*

_{r,eff}increases from unity to almost 2. When reaching

*ν*

_{0}it drops drastically to values below zero, which is due to the change of phase between incoming signal and magnetic response of the SQUIDs. From there it increases again to unity. The comparatively slow increase of Re(

*μ*

_{r,eff}) between 12.5 GHz and 13.5 GHz as well as the small dip slightly above 13.5 GHz is due to a sample holder resonance at 13.6 GHz which couples to the SQUIDs and affects the resonance. The deviation from zero of the imaginary part of

*μ*

_{r,eff}reflects the increased losses at resonance.

*μ*

_{r,eff}) described above, extents qualitatively over the full frequency and flux range of the resonance curve displayed in Fig. 3. The data show clearly how the SQUID metamaterial enables us to reach any desired value of

*μ*

_{r,eff}for a given frequency. The minimum and maximum achieved values are Re(

*μ*

_{r,eff})

_{min}= −2 and Re(

*μ*

_{r,eff})

_{max}= 3, respectively. Unfortunately, due to sample holder resonances, especially around 13.6 GHz and close to the maximum of the curve at approximately 14.5 GHz, these values cannot be reached for every measured frequency. Figure 5(b) shows the real part of the magnetic permeability for a fixed frequency of

*ν*= 13.83 GHz, highlighted by the black dashed line in Fig. 5(a). For this frequency, Re(

*μ*

_{r,eff}) can be tuned to any value between −1.5 and +2.

## 5. Effective Parameter Retrieval

*S*

_{21}for data presented in Fig. 4(b) and Fig. 5, we treat the system as a transmission line problem; the circuit diagram is shown in Fig. 6(a).

*L′*and

*C′*are the respective characteristic inductance and capacitance per unit length of transmission line. Then we simplify the unit cell by projecting the influence of the inductively coupled SQUIDs onto a change of the effective magnetic permeability seen by the transmission line and thus the line inductance (cf. Fig. 6(b)).

*Z*and scattering matrix

*S*for the chain by cascading the unit cell’s ABCD matrix. Here,

*A*=

*A*(

*μ*

_{r,eff}) and

*Ã*are the ABCD matrices of the unit cell and the chain of N unit cells, respectively.

*Z*

_{0}is the port impedance on both sides. Since the transmission line properties

*L′*and

*C′*of the CPW are known, the only free parameter is the effective, relative permeability

*μ*

_{r,eff}. The reverse process, however, (calculating

*μ*

_{r,eff}from the scattering matrix) is more complicated because it involves choosing the correct root when solving a system of coupled nonlinear equations (see Appendix A and B). Similar to other well known methods [14

14. J. Baker-Jarvis, M. D. Janezic, B. F. Riddle, R. T. Johnk, P. Kabos, C. L. Holloway, R. G. Geyer, and C. A. Grosvenor, “Measuring the permittivity and permeability of lossy materials: solids, liquids, metals, building materials, and negative-index materials,” *NIST Technical Note*1536 (Boulder, CO, USA), (2005).

*μ*

_{r,eff}using only one element of the scattering matrix which simplifies the experimental process significantly.

## 6. Conclusion

*S*

_{21}, we were able to extract the effective magnetic permeability from the measured data. These results show that we have created a material with tunable effective magnetic permeability in a frequency range between 10 GHz and 14.5 GHz. Further experiments will include an improved sample layout and environment in order to obtain results with higher quality factors and less parasitic resonances.

## Appendix A: Retrieval of *μ*_{r}_{,}_{eff}

*μ*

_{r,eff}mentioned in Section 5, we describe the loaded waveguide as a lossless transmission line (see Fig. 6). The ABCD matrix of one unit cell [15] is Here,

*Z*

_{L}=

*iωμ*

_{r,eff}

*L′l*is the impedance of the inductance and

*Z*

_{C}= 1/(

*iωC′l*) is the impedance of the capacitance for a unit cell of length

*l*.

*L′*and

*C′*are the inductance and capacitance per unit length of transmission line, respectively. The eigenvalues and eigenvectors of the ABCD matrix can be used to rewrite it in the form This simplifies the total ABCD matrix

*Ã*when cascading N unit cells: Using this equation and the well-known relation [15] of

*Ã*to the scattering matrix

*S*, we construct a system of four coupled, nonlinear equations

*S*

_{11},

*S*

_{12},

*S*

_{22}, and

*μ*

_{r,eff}since

*S*

_{21}is the measured quantity and all other parameters (like

*L′*and

*C′*) are known from either design considerations or simulations.

## Appendix B: Data calibration & reference planes

*S*

_{21}using the method described above, special care has to be taken to choose the correct calibration technique. In this approach, the reference planes of the calibration have to be just before and after the array of unit cells. In a real experiment, the best set of reference planes achievable for a full calibration of such a measurement is located at the microwave connectors closest to the sample at cryogenic temperatures. This, however, does not only require a more complex experimental setup [16

16. J.-H. Yeh and S. M. Anlage, “In situ broadband cryogenic calibration for two-port superconducting microwave resonators”, Rev. Sci. Instrum. **84**, 034706 (2013). [CrossRef] [PubMed]

S^{in} | describes the input part of the setup, from port 1 of the network analyzer to the beginning of the SQUID loaded section of the CPW. |

S^{stl} | describes the SQUID loaded section of the CPW. |

S^{out} | describes the output part of the setup from the end of the SQUID loaded section of the CPW to port 2 of the network analyzer. |

| is the transmission through the loaded waveguide section without the effect of the SQUID resonance. Consequently, this factor does not depend on magnetic field. |

α (ω, Φ_{e0}) | describes the change in the transmission through the loaded waveguide due to the SQUID resonance. Therefore, this factor is frequency and field dependent. |

*μ*

_{r,eff}retrieval. If we restrict our investigation to a limited frequency range bounded by

*ω*

_{min}and

*ω*

_{max}, we can usually find a value of the flux Φ

_{e0}= Φ

_{cal}for which

*α*(

*ω*, Φ

_{cal}) ≈ 1. Thus, by dividing all the measured data in the specified frequency range by the corresponding value at the calibration flux, we can extract

*α*: (Choosing this “calibration” flux effectively limits the validity of the measured data to the mentioned frequency range.) As a last step, we have to reconstruct

## Acknowledgments

## References and links

1. | M. C. Ricci, N. Orloff, and S. M. Anlage, “Superconducting metamaterials,” Appl. Phys. Lett. |

2. | M. C. Ricci, H. Xu, R. Prozorov, A. P. Zhuravel, A. V. Ustinov, and S. M. Anlage, “Tunability of Superconducting Metamaterials,” IEEE Trans. Appl. Supercond. |

3. | J. Gu, R. Singh, Z. Tian, W. Cao, Q. Xing, M. He, J. W. Zhang, J. Han, H.-T. Chen, and W. Zhang, “Terahertz superconductor metamaterial,” Appl. Phys. Lett. |

4. | J. Wu, B. Jin, Y. Xue, C. Zhang, H. Dai, L. Zhang, C. Cao, L. Kang, W. Xu, J. Chen, and P. Wu, “Tuning of superconducting niobium nitride terahertz metamaterials,” Opt. Express |

5. | N. Lazarides and G. P. Tsironis, “rf superconducting quantum interference device metamaterials,” Appl. Phys. Lett. |

6. | C. Du, H. Chen, and S. Li, “Stable and bistable SQUID metamaterials,” J. Phys.: Condens. Matter |

7. | A. I. Maimistov and I. R. Gabitov, “Nonlinear response of a thin metamaterial fim containing Josephson junction,” Optics Commun. |

8. | D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. |

9. | P. Jung, S. Butz, S. V. Shitov, and A. V. Ustinov, “Low-loss tunable metamaterials using superconducting circuits with Josephson junctions,” Appl. Phys. Lett. |

10. | K. K. Likharev, |

11. | M. Tinkham, |

12. | S. Butz, P. Jung, L. V. Filippenko, V. P. Koshelets, and A. V. Ustinov, “Protecting SQUID metamaterials against stray magnetic field”, Supercond. Sci. Technol. |

13. | J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. |

14. | J. Baker-Jarvis, M. D. Janezic, B. F. Riddle, R. T. Johnk, P. Kabos, C. L. Holloway, R. G. Geyer, and C. A. Grosvenor, “Measuring the permittivity and permeability of lossy materials: solids, liquids, metals, building materials, and negative-index materials,” |

15. | D. M. Pozar, |

16. | J.-H. Yeh and S. M. Anlage, “In situ broadband cryogenic calibration for two-port superconducting microwave resonators”, Rev. Sci. Instrum. |

**OCIS Codes**

(310.2790) Thin films : Guided waves

(350.4010) Other areas of optics : Microwaves

(160.3918) Materials : Metamaterials

**ToC Category:**

Metamaterials

**Citation**

S. Butz, P. Jung, L. V. Filippenko, V. P. Koshelets, and A. V. Ustinov, "A one-dimensional tunable magnetic metamaterial," Opt. Express **21**, 22540-22548 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-19-22540

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### References

- M. C. Ricci, N. Orloff, and S. M. Anlage, “Superconducting metamaterials,” Appl. Phys. Lett.87, 034102 (2005). [CrossRef]
- M. C. Ricci, H. Xu, R. Prozorov, A. P. Zhuravel, A. V. Ustinov, and S. M. Anlage, “Tunability of Superconducting Metamaterials,” IEEE Trans. Appl. Supercond.17, 918–921 (2007). [CrossRef]
- J. Gu, R. Singh, Z. Tian, W. Cao, Q. Xing, M. He, J. W. Zhang, J. Han, H.-T. Chen, and W. Zhang, “Terahertz superconductor metamaterial,” Appl. Phys. Lett.97, 071102 (2010). [CrossRef]
- J. Wu, B. Jin, Y. Xue, C. Zhang, H. Dai, L. Zhang, C. Cao, L. Kang, W. Xu, J. Chen, and P. Wu, “Tuning of superconducting niobium nitride terahertz metamaterials,” Opt. Express19, 12021–12026 (2011). [CrossRef] [PubMed]
- N. Lazarides and G. P. Tsironis, “rf superconducting quantum interference device metamaterials,” Appl. Phys. Lett.90, 163501 (2007). [CrossRef]
- C. Du, H. Chen, and S. Li, “Stable and bistable SQUID metamaterials,” J. Phys.: Condens. Matter20, 345220 (2008). [CrossRef]
- A. I. Maimistov and I. R. Gabitov, “Nonlinear response of a thin metamaterial fim containing Josephson junction,” Optics Commun.283, 1633–1639 (2010). [CrossRef]
- D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett.84, 4184–4187 (2000). [CrossRef] [PubMed]
- P. Jung, S. Butz, S. V. Shitov, and A. V. Ustinov, “Low-loss tunable metamaterials using superconducting circuits with Josephson junctions,” Appl. Phys. Lett.102, 062601 (2013). [CrossRef]
- K. K. Likharev, Dynamics of Josephson Junctions(Gordon and Breach Science, 1991).
- M. Tinkham, Introduction to Superconductivity (2nd Edition) (Dover Publications Inc., 2004).
- S. Butz, P. Jung, L. V. Filippenko, V. P. Koshelets, and A. V. Ustinov, “Protecting SQUID metamaterials against stray magnetic field”, Supercond. Sci. Technol.26, 094003 (2013). [CrossRef]
- J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech.41, 2075–2084 (1999). [CrossRef]
- J. Baker-Jarvis, M. D. Janezic, B. F. Riddle, R. T. Johnk, P. Kabos, C. L. Holloway, R. G. Geyer, and C. A. Grosvenor, “Measuring the permittivity and permeability of lossy materials: solids, liquids, metals, building materials, and negative-index materials,” NIST Technical Note1536 (Boulder, CO, USA), (2005).
- D. M. Pozar, Microwave Engineering (2nd Edition) (John Wiley & Sons Inc., 1998) pp. 208–211
- J.-H. Yeh and S. M. Anlage, “In situ broadband cryogenic calibration for two-port superconducting microwave resonators”, Rev. Sci. Instrum.84, 034706 (2013). [CrossRef] [PubMed]

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