Let us demonstrate that tuning electron-electron interaction is indeed possible using the epsilon-near-zero (ENZ) metamaterial approach, which is based on intermixing metal and dielectric components in the right proportions. Following [18], a simplified dielectric response function of a metal may be written as

$$\begin{array}{l}{\epsilon}_{\text{m}}\mathrm{(}q,\text{\hspace{0.17em}}\omega \mathrm{)}=\mathrm{(}1-\frac{{\omega}_{\text{p}}^{2}}{{\omega}^{2}+i\omega \Gamma -{\omega}_{\text{p}}^{2}{q}^{2}/{k}^{2}}\mathrm{)}\mathrm{(}1-\frac{{\Omega}_{1}^{2}\mathrm{(}q\mathrm{)}}{{\omega}^{2}+i\omega {\Gamma}_{1}}\mathrm{)}\\ \text{}\mathrm{...}\mathrm{(}1-\frac{{\Omega}_{\text{n}}^{2}\mathrm{(}q\mathrm{)}}{{\omega}^{2}+i\omega {\Gamma}_{\text{n}}}\mathrm{)}\end{array}$$(5)

where *ω*_{p} is the plasma frequency, *k* is the inverse Thomas-Fermi radius, Ω_{n}(*q*) are dispersion laws of various phonon modes, and Γ_{n} are the corresponding damping rates. Zeroes of the dielectric response function of the bulk metal (which correspond to various bosonic modes) maximize electron-electron pairing interaction given by Eq. (1). In the discussion below we will limit our consideration to the behavior of *ε*_{m}(*q*, *ω*) near *ω*=Ω_{1}(*q*) in the vicinity of the Fermi surface, so that a more complicated representation of *ε*_{m}(*q*, *ω*) by the Lindhard function, which accurately represents *ε*_{m}(*q*, *ω*) in the *q*→0 and *ω*→0 limit [18], does not need to be used. We are going to use only the fact that *ε*_{m} changes sign near *ω*=Ω_{1}(*q*) by passing through zero. Expression for *ε*_{m}(*q*, *ω*) given by Eq. (5) does exhibit this behavior.

As summarized in [19], the critical temperature of a superconductor in the weak coupling limit is typically calculated as

$${T}_{\text{c}}=\theta \mathrm{exp}\mathrm{(}-\frac{1}{{\lambda}_{\text{eff}}}\mathrm{)},$$(6)

where *θ* is the characteristic temperature for a bosonic mode mediating electron pairing (such as the Debye temperature *θ*_{D} in the standard BCS theory, or *θ*_{ex}=*ħω*_{ex}/*k* or *θ*_{pl}=*ħω*_{pl}/*k* in the theoretical proposals which suggest excitonic or plasmonic mediation of the electron pairing interaction), and *λ*_{eff} is the dimensionless coupling constant defined by *V*(*q*, *ω*)=*V*_{C}(*q*) *ε*^{−1}(*q*, *ω*) and the density of states *ν* (see for example [20]):

$${\lambda}_{\text{eff}}=-\frac{2}{\pi}\nu {\displaystyle \underset{0}{\overset{\infty}{\int}}\frac{d\omega}{\omega}}\u3008{V}_{\text{C}}\mathrm{(}q\mathrm{)}\mathrm{Im}{\epsilon}^{-1}\mathrm{(}\overrightarrow{q},\text{\hspace{0.17em}}\omega \mathrm{)}\u3009,$$(7)

where *V*_{C} is the unscreened Coulomb repulsion and the angle brackets denote average over the Fermi surface.

Compared to the bulk metal, zeroes of the effective dielectric response function *ε*_{eff}(*q*, *ω*) of the metal-dielectric metamaterial are observed at shifted positions compared to the zeroes of *ε*_{m}(*q*, *ω*) [2], and additional zeros may also appear. According to the Maxwell-Garnett approximation [21], mixing of nanoparticles of a superconducting “matrix” with dielectric “inclusions” (described by the dielectric constants *ε*_{m} and *ε*_{d}, respectively) results in the effective medium with a dielectric constant *ε*_{eff}, which may be obtained as

$$\mathrm{(}\frac{{\epsilon}_{\text{eff}}-{\epsilon}_{\text{m}}}{{\epsilon}_{\text{eff}}+2{\epsilon}_{\text{m}}}\mathrm{)}=\mathrm{(}1-n\mathrm{)}\mathrm{(}\frac{{\epsilon}_{\text{d}}-{\epsilon}_{\text{m}}}{{\epsilon}_{\text{d}}+2{\epsilon}_{\text{m}}}\mathrm{)},$$(8)

where *n* is the volume fraction of metal (0≤*n*≤1). The explicit expression for *ε*_{eff} may be written as

$${\epsilon}_{\text{eff}}=\frac{{\epsilon}_{\text{m}}\mathrm{(}\mathrm{(}3-2n\mathrm{)}{\epsilon}_{\text{d}}+2n{\epsilon}_{\text{m}}\mathrm{)}}{\mathrm{(}n{\epsilon}_{\text{d}}+\mathrm{(}3-n\mathrm{)}{\epsilon}_{\text{m}}\mathrm{)}},$$(9)

and it is easy to verify that

$${\epsilon}_{\text{e}\mathrm{ff}}^{-1}=\frac{n}{\mathrm{(}3-2n\mathrm{)}}\frac{1}{{\epsilon}_{\text{m}}}+\frac{9\mathrm{(}1-n\mathrm{)}}{2n\mathrm{(}3-2n\mathrm{)}}\frac{1}{\mathrm{(}{\epsilon}_{\text{m}}+\mathrm{(}3-2n\mathrm{)}{\epsilon}_{\text{d}}/2n\mathrm{)}}$$(10)

For a given value of the metal volume fraction *n*, the ENZ conditions (*ε*_{eff} ≈ 0) may be obtained around *ε*_{m} ≈ 0 and around

$${\epsilon}_{\text{m}}\approx -\frac{3-2n}{2n}{\epsilon}_{\text{d}}$$(11)

Since the absolute value of *ε*_{m} is limited by some value *ε*_{m,max} [see Eq. (5)], the latter zero of *ε*_{eff} disappears at small *n* as *n*→0 at some critical value of the volume fraction *n*=*n*_{cr}=1.5*ε*_{d}/*ε*_{m,max}. Let us evaluate Im(*ε*^{−1}_{eff}(*q*, *ω*)) near this zero of *ε*_{eff} at *n*>*n*_{cr}. Based on Eq. (10),

$$\mathrm{Im}\mathrm{(}{\epsilon}^{-1}{}_{\text{eff}}\mathrm{)}\approx \frac{9\mathrm{(}1-n\mathrm{)}}{2n\mathrm{(}3-2n\mathrm{)}{{\epsilon}^{\u2033}}_{\text{m}}}$$(12)

where ${{\epsilon}^{\prime}}_{\text{m}}=\text{Re}{\epsilon}_{\text{m}}$ and ${{\epsilon}^{\u2033}}_{\text{m}}=\text{Im}{\epsilon}_{\text{m}}.$ Assuming that *ν*~*n* (which is justified by the fact that there are no free charges in the dielectric phase of the metamaterial) and using Eq. (7), we may obtain the resulting expression for *λ*_{eff} as a function of *λ*_{m} and *n*:

$${\lambda}_{\text{eff}}\approx \frac{9\mathrm{(}1-n\mathrm{)}}{2\mathrm{(}3-2n\mathrm{)}}{\lambda}_{\text{m}},$$(13)

where *λ*_{m} is also determined by Eq. (7) in the limit *ε*→*ε*_{m}, and we have assumed the same magnitudes of ${{\epsilon}^{\u2033}}_{\text{m}}$ for the metamaterial zero described by Eq. (11) and for the *ε*_{m} ≈ 0 conditions. This assumption will be re-evaluated in Section 5 and confronted with experimental data for aluminum-based ENZ metamaterial, which indicate that ${{\epsilon}^{\u2033}}_{\text{m}}$ changes by ~11%. Figure 1A shows the predicted behavior of *λ*_{eff}/*λ*_{m} as a function of *n*. Based on this prediction, we may expect enhancement of superconducting properties of the metal-dielectric metamaterial in the *n*_{cr}<*n*<0.6 range. For comparison, Figure 1A also shows the behavior of *λ*_{eff} near the *ε*_{m} ≈ 0 pole of the inverse dielectric response function of the metamaterial. According to Eq. (9), near *ε*_{m} ≈ 0 the effective dielectric response function of the metamaterial behaves as

Figure 1: Theoretical modeling of ENZ metamaterial superconductors.

(A) Predicted behavior of *λ*_{eff}/*λ*_{m} as a function of metal volume fraction *n* in a metal-dielectric metamaterial. (B) Plots of the hypothetical values of *T*_{c} as a function of metal volume fraction *n*, which would originate from either Eq. (13) or Eq. (15) 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*_{cr}. Note that the theoretical curves contain no free parameters.

$${\epsilon}_{\text{eff}}\approx \frac{3-2n}{n}{\epsilon}_{\text{m}}$$(14)

Therefore, near this pole

$${\lambda}_{\text{eff}}\approx \frac{{n}^{2}}{\mathrm{(}3-2n\mathrm{)}}{\lambda}_{\text{m}}$$(15)

Let us apply the obtained simple estimates to the case of Al-Al_{2}O_{3} core-shell metamaterial studied in [4]. Assuming the known values *T*_{cbulk}=1.2 K and *θ*_{D}=428 K of bulk aluminum [8], Eq. (6) results in *λ*_{m}=0.17, which corresponds to the weak coupling limit. In order to make the mechanism behind the enhancement of *T*_{c} in the Al-Al_{2}O_{3} core-shell metamaterial superconductor abundantly clear, we have plotted the hypothetical values of *T*_{c} as a function of metal volume fraction *n*, which would originate from either Eq. (13) or Eq. (15) in the absence of each other. The corresponding values calculated as

$${T}_{\text{c}}={T}_{\text{cbulk}}\mathrm{exp}\mathrm{(}\frac{1}{{\lambda}_{\text{m}}}-\frac{1}{{\lambda}_{\text{eff}}}\mathrm{)}$$(16)

are shown in Figure 1B. The vertical dashed line corresponds to the assumed value of *n*_{cr}.

The experimentally measured data points from Ref. [4] are shown for comparison on the same plot. The match between the experimentally measured values of enhanced *T*_{c} and the theoretical curve obtained based on Eq. (13) is quite impressive, given the fact that Eqs. (13) and (16) do not contain any free parameters. Such a good match unambiguously identifies metamaterial enhancement as the physical mechanism of critical temperature tripling in the Al-Al_{2}O_{3} core-shell metamaterial superconductor. It is impressive that such simple and well-known nanophotonics tools as the effective medium theory may be used to considerably enhance the critical temperature of simple superconductors.

The proof of principle experiments conducted with both random [3] and core-shell [4] ENZ metal-dielectric metamaterials indeed validates the effective medium-based theory outlined above. Our initial superconducting metamaterial samples [3] were prepared using commercially available tin and barium titanate nanoparticles obtained from the US Research Nanomaterials, Inc. The nominal diameter of the BaTiO_{3} nanoparticles was 50 nm, while tin nanoparticle size was specified as 60–80 nm. Both nanoparticle sizes fall substantially below the superconducting coherence length in pure tin *ξ*_{Sn}~230 nm [17]. Our choice of materials was based on results of numerical calculations of the real part of the dielectric constant of the Sn/BaTiO_{3} mixture shown in Figure 2. These calculations were based on the measured dielectric properties of Sn [22] and BaTiO_{3} [23], respectively, and on the Maxwell-Garnett expression, Eq. (9), for the dielectric permittivity of the mixture. These calculations indicated that it was possible to achieve broadband ENZ conditions in the 30–50% range of the volume fraction of BaTiO_{3} in the frequency range of relevance around *hν*~*kT*_{c} of pure tin.

Figure 2: Numerical calculations of the real part of the dielectric constant of the Sn/BaTiO_{3} mixture as a function of volume fraction of BaTiO_{3} performed with 10% steps.

The Sn/BaTiO_{3} superconducting metamaterials were fabricated by combining the given amounts of Sn and BaTiO_{3} nanoparticle powders by volume into a single test tube filled with de-ionized water. The resulting suspensions were sonicated and magnetically stirred for 30 min, followed by water evaporation. The remaining mixtures of Sn and BaTiO_{3} nanoparticles were compressed into test pellets using a hydraulic press. A typical scanning electron microscopy (SEM) image of the resulting Sn/BaTiO_{3} composite metamaterial is shown in Figure 3. The original compressed nanoparticles are clearly visible in the image. The resulting average metamaterial composition has been verified after fabrication using an SEM with energy-dispersive X-ray spectroscopy (EDS). The compositional analysis of an area of the samples of about 1 μm in diameter is consistent with this nominal composition. Such EDX spectra were used to establish the homogeneous character of the fabricated composite metamaterials.

Figure 3: Random mixture metamaterials.

(A) SEM image of the composite Sn/BaTiO_{3} metamaterial with 30% volume fraction of BaTiO_{3} nanoparticles. Individual compressed nanoparticles are clearly visible in the image. (B) Schematic diagram of the metamaterial sample geometry.

The superconducting critical temperature *T*_{c} of various Sn/BaTiO_{3} metamaterials was measured via the onset of diamagnetism for samples with different volume fractions of BaTiO_{3} using a MPMS SQUID magnetometer. The zero-field-cooled (ZFC) magnetization per unit mass for several samples with varying concentrations of BaTiO_{3} is plotted in Figure 4A. The temperatures of the onset of the superconducting transition and the temperatures of the midpoint of the transition are plotted in Figure 5. The *T*_{c} increased from the pure Sn value of 3.68 K with increasing BaTiO_{3} concentration to a maximum Δ*T*_{c}~0.15 K or 4% compared to the pure tin sample for the 40% sample followed by *T*_{c} decreasing at higher volume fractions. A pure tin sample was prepared from pressed tin nanoparticles of the same diameter using the same method of preparation. The value of *T*_{c} agreed with expected value of pure Sn. The magnetization of the composite samples in the superconducting state is comparable in magnitude with the pure tin sample. The increase of *T*_{c} and its dependence on effective dielectric constant determined by the concentration of BaTiO_{3} agrees qualitatively with the Maxwell-Garnett theory-based calculations described above (see Figure 2). Moreover, the decrease in *T*_{c} at higher volume fraction of BaTiO_{3}, which is apparent from Figure 5 also agrees well with the Maxwell-Garnett theory, since *ε*_{eff}(*q*, *ω*) changes sign to positive for higher BaTiO_{3} concentrations.

Figure 4: Temperature dependence of zero-field-cooled magnetization per unit mass for several samples with varying concentration of (A) BaTiO_{3} and (B) SrTiO_{3} measured in magnetic field of 10 G.

Figure 5: Temperatures of the onset of the superconducting transition (blue squares) and of the midpoint of the transition (red circles) plotted as a function of volume fraction of BaTiO_{3}.

The line is a guide to the eye.

To verify the reproducibility of these results, our measurements were repeated for several sets of nanocomposite samples fabricated using strontium titanate nanoparticles (instead of barium titanate) using the same fabrication technique described above. The superconducting critical temperature *T*_{c} of the Sn/SrTiO_{3} metamaterial samples was measured for samples with different volume fractions of SrTiO_{3} nanoparticles as shown in Figure 4B producing volume fraction dependencies, which are similar to those obtained for Sn/BaTiO_{3} samples. These results established that the metamaterial approach to dielectric response engineering can indeed increase the critical temperature of a composite superconductor-dielectric metamaterial. However, despite this initial success, the observed critical temperature increase was modest. It was argued in [2] that the random nanoparticle mixture geometry may not be ideal because simple mixing of superconductor and dielectric nanoparticles results in substantial spatial variations of *ε*_{eff}(*q*, *ω*) throughout a metamaterial sample. Such variations lead to considerable broadening and suppression of the superconducting transition.

To overcome this issue, it was suggested that an ENZ plasmonic core-shell metamaterial geometry, which has been developed to achieve partial cloaking of macroscopic objects [24], should be implemented [2]. The cloaking effect relies on mutual cancellation of scattering by the dielectric core (having *ε*_{d}>0) and plasmonic shell (with *ε*_{m}<0) of the nanoparticle, so that the effective dielectric constant of the nanoparticle becomes very small and close to that of vacuum (a plasmonic core with a dielectric shell may also be used). This approach may be naturally extended to the core-shell nanoparticles having negative ENZ behavior, as required in the superconducting application. Synthesis of such individual ENZ core-shell nanostructures followed by nanoparticle self-assembly into a bulk ENZ metamaterial (as shown in Figure 6) appears to be a viable way to fabricate an extremely homogeneous metamaterial superconductor.

Figure 6: Schematic geometry of the ENZ metamaterial superconductor based on the core-shell nanoparticle geometry.

The nanoparticle diameter is d=18 nm. The inset shows typical dimensions of the fabricated bulk aluminum-based core-shell metamaterial.

The design of an individual core-shell nanoparticle is based on the fact that scattering of an electromagnetic field by a sub-wavelength object is dominated by its electric dipolar contribution, which is defined by the integral sum of its volume polarization [24]. A material with *ε*>1 has a positive electric polarizability, while a material with *ε*<1 has a negative electric polarizability (since the local electric polarization vector, *P*=(*ε*−1)*E*/4*π*, is opposite to *E*). As a result, the presence of a plasmonic shell (core) cancels the scattering produced by the dielectric core (shell), thus providing a cloaking effect. Similar consideration for the negative ENZ case leads to the following condition for the core-shell geometry:

$${r}_{\text{c}}^{3}{\epsilon}_{\text{c}}\approx -\mathrm{(}{r}_{\text{s}}^{3}-{r}_{\text{c}}^{3}\mathrm{)}{\epsilon}_{\text{s}},$$(17)

where *r*_{c} and *r*_{s} are the radii, and *ε*_{c} and *ε*_{s} are the dielectric permittivities of the core and shell, respectively. Eq. (17) corresponds to the average dielectric permittivity of the core-shell nanoparticle being approximately equal to zero. Working on the negative side of this equality will ensure negative ENZ character of each core-shell nanoparticle. A dense assembly of such core-shell nanoparticles will form a medium that will have small negative dielectric permittivity. Moreover, in addition to obvious advantage in homogeneity, a core-shell based metamaterial superconductor design enables tuning of the spatial dispersion of the effective dielectric permittivity *ε*_{eff}(*q*, *ω*) of the metamaterial, which would further enhance its *T*_{c} [2]. Spatial dispersion of a metamaterial is indeed well known to originate from plasmonic effects in its metallic constituents. In a periodic core-shell nanoparticle-based ENZ metamaterial spatial dispersion is defined by the coupling of plasmonic modes of its individual nanoparticles. This coupling enables propagating plasmonic Bloch modes and hence nonlocal effects.

The successful realization of such an ENZ core-shell metamaterial superconductor using compressed Al_{2}O_{3}-coated aluminum nanoparticles has been reported in [4], leading to tripling of the metamaterial critical temperature compared to the bulk aluminum. This material is ideal for the proof of principle experiments because the critical temperature of aluminum is quite low (*T*_{c Al}=1.2 K [17]), leading to a very large superconducting coherence length *ξ*=1600 nm [17]. Such a large value of *ξ* facilitates the metamaterial fabrication requirements while Al_{2}O_{3} exhibits very large positive values of dielectric permittivity up to ${\epsilon}_{{\text{Al}}_{2}{\text{O}}_{3}}\sim 200$ in the terahertz frequency range [25]. These results provide an explanation for the long known, but not understood, enhancement of the *T*_{c} of granular aluminum films [26], [27].

The 18-nm-diameter Al nanoparticles for these experiments were acquired from the US Research Nanomaterials, Inc. Upon exposure to the ambient conditions, a ~2-nm-thick Al_{2}O_{3} shell is known to form on the aluminum nanoparticle surface [28], which is comparable to the 9-nm radius of the original Al nanoparticle. Further aluminum oxidation may also be achieved by heating the nanoparticles in air. The resulting core-shell Al_{2}O_{3}-Al nanoparticles were compressed into macroscopic, ~1-cm diameter, ~0.5-mm-thick test pellets using a hydraulic press, as illustrated in the inset in Figure 6.

The IR reflectivity of such core-shell metamaterial samples was measured in the long-wavelength IR (LWIR) (2.5–22.5 μm) range using an FTIR spectrometer, and compared with reflectivity spectra of Al and Al_{2}O_{3}, as shown in Figure 7. While the reflectivity spectrum of Al is almost flat, the spectrum of Al_{2}O_{3} exhibits a very sharp step-like behavior around 11 μm that is related to the phonon-polariton resonance, which results from coupling of an infrared photon with an optic phonon of Al_{2}O_{3} [29]. The step in reflectivity is due to the negative sign of ${\epsilon}_{{\text{Al}}_{2}{\text{O}}_{3}}$ near resonance. This step-like behavior may be used to characterize the volume fraction of Al_{2}O_{3} in the core-shell metamaterial. In the particular case shown in Figure 7, the volume fraction of Al_{2}O_{3} in the core-shell metamaterial may be estimated, based on the Maxwell-Garnett approximation, as ~39%, which corresponds to (*r*_{s}–*r*_{c})~0.18*r*_{c}. At *r*_{c}~9 nm, the corresponding thickness of Al_{2}O_{3} appears to be (*r*_{s}–*r*_{c})~1.6 nm, which matches expectations based on [28].

Figure 7: Comparison of the FTIR reflectivity spectrum of a typical core-shell Al_{2}O_{3}-Al metamaterial sample with reflectivity spectra of bulk Al and Al_{2}O_{3} samples.

The step in reflectivity around 11 μm may be used to characterize the volume fraction of Al_{2}O_{3} in the core-shell metamaterial. The increased noise near 22 μm is related to the IR source cutoff.

The Kramers-Kronig analysis of the FTIR reflectivity spectra of the Al-Al_{2}O_{3} sample also allows us to evaluate *ε*_{eff}(0, *ω*) for the metamaterial in the LWIR spectral range. Plots of the real part of *ε* for pure Al and for the Al-Al_{2}O_{3} core-shell metamaterial based on the Kramers-Kronig analysis of the data in Figure 7 are plotted in Figure 8. The plot in Figure 8A clearly demonstrates that ${\epsilon}_{{\text{Al-Al}}_{2}{\text{O}}_{3}}<<{\epsilon}_{\text{Al}}$ so that the ENZ condition was achieved in the sense that the initial dielectric constant of aluminum was reduced by a factor ~1000. On the other hand, Figure 8B demonstrates that the dielectric constant of the Al-Al_{2}O_{3} core-shell metamaterials remains negative and relatively small above 11 μm. In particular, the large negative contribution to *ε* from the aluminum cores is compensated by the large positive contribution from the Al_{2}O_{3} shells leading to the upturn of ${\epsilon}_{{\text{Al-Al}}_{2}{\text{O}}_{3}}$ that is observed near 20 μm in Figure 8B, which is caused by the large positive value of ${\epsilon}_{{\text{Al}}_{2}{\text{O}}_{3}}$ in this spectral range. Note that while both metamaterials shown in Figure 8B exhibit much smaller *ε* compared to the bulk aluminum, the metamaterial prepared using less oxidized aluminum nanoparticles exhibits considerably larger negative *ε*. The relatively large noise observed in the calculated plot of *ε*_{Al} in Figure 8A is due to the fact that the aluminum reflectivity is close to 100% above 7 μm so that the Kramers-Kronig-based numerical analysis of the reflectivity data does not work reliably for pure aluminum samples in this spectral range. Another limitation on the accuracy of the analysis is the use of the finite spectral range (2.5–22.5 μm) of the FTIR spectrometer rather than the infinite one assumed by the rigorous Kramers-Kronig analysis. These limitations notwithstanding, we note that our result for pure aluminum is in good agreement with the tabulated data for *ε*_{Al} reported in [30]. Therefore, these results reliably confirm the ENZ character of the core-shell Al-Al_{2}O_{3} metamaterial. It is also interesting to note that the same FTIR technique applied to the tin-BaTiO_{3} nanocomposite metamaterials studied in [3] also confirms their expected ENZ character, as illustrated in Figure 9. In both cases, the goal of metamaterial engineering was to create an effective superconducting medium with negative ENZ response. While both Figures 8 and 9 confirm that this goal has been achieved, the core-shell geometry of the developed Al-Al_{2}O_{3} metamaterial has a clear advantage. The core-shell geometry guarantees a homogeneous spatial distribution of the effective dielectric response function, leading to tripling of *T*_{c} for the Al-Al_{2}O_{3} core-shell metamaterial, compared to ~5% increase of *T*_{c} of the tin-BaTiO_{3} random nanocomposite metamaterial developed in [3].

Figure 8: Plots of the real part of *ε* for pure Al and for the Al-Al_{2}O_{3} core-shell metamaterial based on the Kramers-Kronig analysis of the FTIR reflectivity data from Figure 2.

(A) Comparison of *ε*′ for pure Al and for the Al-Al_{2}O_{3} metamaterial clearly indicates that ${\epsilon}_{{\text{Al-Al}}_{2}{\text{O}}_{3}}<<{\epsilon}_{\text{Al}}.$ (B) Real part of *ε* for two different Al-Al_{2}O_{3} core-shell metamaterials based on the Kramers-Kronig analysis. While both metamaterials shown in (B) exhibit much smaller *ε* compared to the bulk aluminum, the metamaterial prepared using less oxidized aluminum nanoparticles exhibits considerably larger negative *ε*.

Figure 9: Plots of the real part of *ε* for pure tin and for the ENZ tin-BaTiO_{3} nanocomposite metamaterial studied in [3].

(A) Comparison of *ε*′ for compressed tin nanoparticles and for the tin-BaTiO_{3} nanocomposite metamaterial. (B) Real part of *ε* for the tin-BaTiO_{3} nanocomposite metamaterial.

The *T*_{c} of various Al-Al_{2}O_{3} core-shell metamaterials was determined via the onset of diamagnetism for samples with different degrees of oxidation using a MPMS SQUID magnetometer. The ZFC magnetization per unit mass versus temperature for several samples with various volume fractions of Al_{2}O_{3} is plotted in Figure 10A, whereas the corresponding reflectivity data are shown in Figure 10B. Even though the lowest achievable temperature with our MPMS SQUID magnetometer was 1.7 K, we were able to observe a gradual increase of *T*_{c} that correlated with an increase of the Al_{2}O_{3} volume fraction as determined by the drop in reflectivity shown in Figure 10B. The reliability of the MPMS SQUID magnetometer at *T*>1.7 K was checked by measurements of the *T*_{c} of bulk tin at *T*_{c}=3.7 K, as described in Ref. [3], in excellent agreement with the textbook data [17].

Figure 10: Measured parameters of the core-shell metamaterial superconductors.

(A) Temperature dependence of ZFC magnetization per unit mass for several Al-Al_{2}O_{3} core-shell metamaterial samples with increasing degree of oxidation measured in magnetic field of 10 G. The highest onset of superconductivity at ~3.9 K is marked by an arrow. This temperature is 3.25 times larger than *T*_{c}=1.2 K of bulk aluminum. (B) Corresponding FTIR reflectivity spectra of the core-shell metamaterial samples. Decrease in reflectivity corresponds to decrease of the volume fraction of aluminum.

The observed increase in *T*_{c} also showed good correlation with the results of the Kramers-Kronig analysis shown in Figure 8B: samples exhibiting smaller negative *ε* demonstrated higher *T*_{c} increase. The highest onset temperature of the superconducting transition reached 3.9 K, which is more than three times as high as the critical temperature of bulk aluminum, *T*_{c Al}=1.2 K [17]. All of the samples exhibited a small positive susceptibility that increased with decreasing temperature, consistent with the presence of small amounts of paramagnetic impurities. The discussed *T*_{c} values were determined by the beginning of the downturn of *M*(*T*), where the diamagnetic superconducting contribution starts to overcome paramagnetic contribution, making this temperature the lower limit of the onset of superconductivity. Further oxidation of aluminum nanoparticles by annealing for 2 h at 600°C resulted in a *T*_{c} less than 1.7 K, our lowest achievable temperature. Based on the reflectivity step near 11 μm (see Figure 10B), the volume fraction of Al_{2}O_{3} in this sample may be estimated as ~50%, which corresponds to (*r*_{s}–*r*_{c})~0.26*r*_{c}. For *r*_{c}~9 nm, the corresponding thickness of Al_{2}O_{3} was (*r*_{s}–*r*_{c})~2.4 nm.

Thus, the theoretical prediction of a large increase of *T*_{c} in ENZ core-shell metamaterials shown in Figure 1 has been confirmed by direct measurements of *ε*_{eff}(0, *ω*) of the fabricated metamaterials and the corresponding measurements of the increase of *T*_{c}. These results strongly suggest that increased aluminum *T*_{c}’s that were previously observed in very thin (<50-nm thickness) granular aluminum films [26], [27] and disappeared at larger film thicknesses were due to changes in the dielectric response function rather than quantum size effects and soft surface phonon modes [27]. As clearly demonstrated by the experimental data and the discussion above, the individual Al nanoparticle size is practically unaffected by oxidation, thus excluding the size effects as an explanation of giant *T*_{c} increase in our bulk core-shell metamaterial samples. The developed technology enables efficient nanofabrication of bulk aluminum-based metamaterial superconductors with a *T*_{c} that is three times that of pure aluminum and with virtually unlimited shapes and dimensions. These results open up numerous new possibilities of considerable *T*_{c} increase in other simple superconductors.

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