Plasmonic scaling of superconducting metamaterials

Superconducting metamaterials are utilized to study the approach to the plasmonic limit simply by tuning temperature to modify the superfluid density, and thus the superfluid plasma frequency. We examine the persistence of artificial magnetism in a metamaterial made with superconductors in the plasmonic limit, and compare to the electromagnetic behavior of normal metals as a function of frequency as the plasma frequency is approached from below. Spiral-shaped Nb thin film meta-atoms of scaled dimensions are employed to explore the plasmonic behavior in these superconducting metamaterials, and the scaling condition allows extraction of the temperature dependent superfluid density, which is found to be in good agreement with expectations.

Figure 1

Plot of the calculated dimensionless RHS (right-hand side) and LHS (left-hand side) of Eq. (2) against normalized temperature. The conductivity is calculated at 75 MHz using the weak-coupled Mattis-Bardeen theory (Ref. 26) in the Pippard limit. Other parameters for the calculation are given in the figure. The condition for artificial magnetism in a superconducting metamaterial is that the RHS exceeds ${\sigma }_1/{\epsilon }_0\omega$.

Figure 2

(a) Transmission $|S_{21}\left(f\right)|$ through one of the scaled Nb spirals with the largest scale factor ($s=2.5$) showing its resonance spectrum below and above the $T_c$ of Nb. The data above $T_c$ is characteristic of two loops coupled together by a mutual inductance. Below $T_c$ an additional transmission channel through the spiral appears, leading to the observed interference patterns (Refs. 24 and 25). The inset illustrates the scaled spiral samples; the $D_0=6$ mm spiral is outlined with the yellow box which all others are scaled from. (b) Gradual decay of the fundamental resonant transmission peak response with increasing temperature for the same sample. The inset shows a schematic of the measurement, two coax antennas sandwiching the meta-atom.

Figure 3

(a) Plot of $f_0\left(T\right)/f_0(T_{\min }$) vs $T/T_c$. Inset shows the same quantity over a larger temperature range. (b) Plot of $K$ for two scaled Nb spirals as a function of reduced temperature $T/T_c$. The analysis to estimate $n_s(T/T_c)$ is shown with red stairs. Extracted values of $n_s$ at specific $T/T_c$ are labeled in blue. Inset is the plot of extracted scaled superfluid density $n_s(T/T_c)$ versus scaled temperature $T/T_c$ from a set of scaled Nb spiral resonators. The solid lines are the best fit to the extracted $n_s(T/T_c)$ for each set, demonstrating the linear trend according to $n_s(T/T_c)\sim 1-T/T_c$.