Tunable Negative Permeability in a Three-Dimensional Superconducting Metamaterial

We report on highly tunable radio-frequency (rf) characteristics of a low-loss and compact three-dimensional (3D) metamaterial made of superconducting thin-film spiral resonators. The rf transmission spectrum of a single element of the metamaterial shows a fundamental resonance peak at $\sim 24.95\mathrm{MHz}$ that shifts to a 25% smaller frequency and becomes degenerate when a 3D array of such elements is created. The metamaterial shows an in situ tunable narrow frequency band in which the real part of the effective permeability is negative over a wide range of temperature. This narrow frequency band gradually possesses near-zero and positive values for the real part of permeability as the superconducting critical temperature is approached. The studied 3D metamaterial can be used for increasing power-transfer efficiency and tunability of electrically small rf antennas.

Figure 1

Comparison of transmission $|S_{12}|$ through a single spiral (green curve), 2D array of $3\times 6$ spirals (red curve), and 3D array of the spirals formed by stacking four substrates including 2D arrays of spirals (blue curve). The lowest resonant frequency band decreases and develops fine splitting by adding more elements to the metamaterial. The maroon curve shows the data taken at the $T_c$ of Nb, 9.25 K, for a single spiral, where the resonant peak disappears due to enhanced ohmic losses. 2D and 3D arrays also show the same feature when the sample is heated above 9.25 K. The inset shows a schematic of the 3D metamaterial sandwiched between two rf loop antennas which are 27.2 mm apart along the $z$ axis.

Figure 2

Real ${\mu }_{\text{eff}}^{\prime }(\omega )$ and imaginary ${\mu }_{\text{eff}}^{\prime \prime }(\omega )$ parts of the effective complex permeability retrieved from the experimental scattering matrix at 4.3 K for (a) a single spiral and (b) a 3D array of such spirals. The dashed maroon and cyan curves are Lorentzian fits to the retrieved data according to Eq. (1), demonstrating a significant agreement. The insets are shorter frequency-range plots of the data showing the details in the vicinity of the resonant frequency. A single smooth resonance peak of the individual spiral splits into multiple resonances for the 3D metamaterial.

Figure 3

Temperature evolution of (a) real ${\mu }_{\text{eff}}^{\prime }(\omega )$ and (b) imaginary ${\mu }_{\text{eff}}^{\prime \prime }(\omega )$ parts of the retrieved permeability for the 3D metamaterial. Both the frequency band where the ${\mu }_{\text{eff}}^{\prime }(\omega )$ is negative and $f_r$ shift to smaller values of frequency with increasing temperature due to the enhanced inductance and losses.

Figure 4

Temperature evolution of the extracted resonance frequency $f_r$ (blue squares) and damping factor $\mathrm{\Gamma }$ (red diamonds) by fitting the retrieved permeability of the 3D metamaterial according to Eq. (1). In the vicinity of $T_c$, $f_r$ shows a steep drop as $\mathrm{\Gamma }$ dramatically increases. The inset is a plot showing the quantities expected to be proportional to the normal fluid density $n_N(T)$.