Theoretical modeling of critical temperature increase in metamaterial superconductors

Recent experiments have demonstrated that the metamaterial approach is capable of a drastic increase of the critical temperature $T_{\mathrm{c}}$ of epsilon near zero (ENZ) metamaterial superconductors. For example, tripling of the critical temperature has been observed in $\mathrm{Al}-\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3$ ENZ core-shell metamaterials. Here, we perform theoretical modeling of $T_{\mathrm{c}}$ increase in metamaterial superconductors based on the Maxwell-Garnett approximation of their dielectric response function. Good agreement is demonstrated between theoretical modeling and experimental results in both aluminum- and tin-based metamaterials. Taking advantage of the demonstrated success of this model, the critical temperature of hypothetic niobium-, $\mathrm{Mg}{\mathrm{B}}_{2^{-}}$, and ${\mathrm{H}}_2\mathrm{S}$-based metamaterial superconductors is evaluated. The $\mathrm{Mg}{\mathrm{B}}_2$-based metamaterial superconductors are projected to reach the liquid nitrogen temperature range. In the case of a ${\mathrm{H}}_2\mathrm{S}$-based metamaterial $T_{\mathrm{c}}$ appears to reach ∼250 K.

Figure 1

(a) Predicted behavior of ${\lambda }_{\mathrm{eff}}/{\lambda }_{\mathrm{m}}$ as a function of metal volume fraction $n$ in a metal-dielectric metamaterial. (b) Plots of the hypothetic values of $T_{\mathrm{c}}$ as a function of metal volume fraction $n$, which would originate from either Eq. (10) or Eq. (12) in the absence of each other. The experimentally measured data points from [4] are shown for comparison on the same plot. The vertical dashed line corresponds to the assumed value of $n_{\mathrm{cr}}$. Note that the theoretical curves contain no free parameters.

Figure 2

(a) Temperature dependence of zero-field-cooled magnetization per unit mass for several $\mathrm{Al}-\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3$ core-shell metamaterial samples with various degrees of oxidation measured in magnetic field of 10 G. The highest onset of superconductivity at ∼3.9 K is 3.25 times larger than $T_{\mathrm{c}}=1.2\mathrm{K}$ of bulk aluminum [4]. (b) Corresponding FTIR reflectivity spectra of the core-shell metamaterial samples in the region of phonon-polariton resonance in $\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3$. All curves are normalized to 100% reflectivity at 12 μm wavelength. The normalized reflectivity of the metamaterial samples at 10.3 μm (where $\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3$ behaves as an almost perfect absorber) indicates the volume fraction of aluminum in the metamaterial sample.

Figure 3

(a) Theoretical plot of $T_{\mathrm{c}}$ versus volume fraction $n$ of aluminum in the $\mathrm{Al}-\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3$ core-shell metamaterial calculated using Eq. (18). The best fit to experimental data is obtained at $\alpha =0.89$. (b) Theoretical plot of $T_{\mathrm{c}}$ versus volume fraction $n$ of tin in the tin-$\mathrm{BaTi}{\mathrm{O}}_3$ ENZ metamaterial [3] calculated using Eq. (18). The best fit to experimental data is obtained at $\alpha =0.83$. The experimental data points are taken from Ref. [3].

Figure 4

(a) Theoretical plots of $T_{\mathrm{c}}$ versus volume fraction $n$ of niobium in a hypothetic niobium-based ENZ metamaterial calculated using Eq. (18) for different values of the α ratio. The experimental data points for bulk Nb and $\mathrm{N}{\mathrm{b}}_3\mathrm{Sn}$ are shown in the same plot for comparison. (b) Theoretical plots of $T_{\mathrm{c}}$ versus volume fraction $n$ of $\mathrm{Mg}{\mathrm{B}}_2$ in a hypothetic $\mathrm{Mg}{\mathrm{B}}_2$-based ENZ metamaterial calculated using Eq. (18) for different values of the α ratio. In both cases the vertical dashed lines correspond to the same value of $n_{\mathrm{cr}}$ as in the $\mathrm{Al}-\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3$ core-shell metamaterial superconductor.