Introduction

One of the most important goals of condensed matter physics is reliably designing new materials with enhanced superconducting properties. Recently, a metamaterial strategy, consisting of deliberately engineering the dielectric properties of a nanostructured “metamaterial superconductor” that results in an enhanced electron pairing interaction that increases the value of the superconducting energy gap and the critical temperature, Tc, was suggested to achieve this goal1,2. Our recent experimental work3,4 has conclusively demonstrated that this approach can indeed be used to increase the critical temperature of a composite superconductor-dielectric metamaterial. For example, we have demonstrated the use of Al2O3-coated aluminum nanoparticles to form epsilon near zero (ENZ) core-shell metamaterial superconductors with a Tc that is three times that of pure aluminum4. However, this core-shell metamaterial superconductor geometry exhibits poor transport properties compared to its parent (aluminum) superconductor. A natural way to overcome this issue is the implementation of the hyperbolic metamaterial geometry (Fig. 1a), which has been suggested in refs 1,2. Hyperbolic metamaterials are extremely anisotropic uniaxial materials, which behave like a metal (Reεxx = Reεyy < 0) in one direction and like a dielectric (Reεzz > 0) in the orthogonal direction. Originally introduced to overcome the diffraction limit of optical imaging5, hyperbolic metamaterials demonstrate a number of novel phenomena resulting from the broadband singular behavior of their density of photonic states. The “layered” hyperbolic metamaterial geometry shown in Fig. 1a is based on parallel periodic layers of metal separated by layers of dielectric. This geometry ensures excellent transport properties in the plane of the layers. As noted in ref. 6, typical high Tc superconductors (such as BSCCO) exhibit hyperbolic behavior in a substantial portion of the far infrared and THz frequency ranges. In this report, we demonstrate that the artificial hyperbolic metamaterial geometry may also lead to a considerable enhancement of superconducting properties.

Figure 1
figure 1

Geometry and basic properties of hyperbolic metamaterial superconductors: (a) Schematic geometry of a “layered” hyperbolic metamaterial. (b) Electron-electron pairing interaction in a hyperbolic metamaterial is strongly enhanced near the cone in momentum space defined as .

Full size image

Electromagnetic properties are known to play a very important role in the pairing mechanism of superconductors7. According to the BCS theory, a Cooper pair is formed from two electrons with opposite spins and momenta that are loosely bound. This mechanism may be described as an attractive interaction of electrons that results from the polarization of the ionic lattice which these electrons create as they move through the lattice. Based on this interpretation, Kirzhnits et al. formulated their description of superconductivity in terms of the dielectric response function of the superconductor7. They demonstrated that the electron-electron interaction in a superconductor may be expressed in the form of an effective Coulomb potential

where VC is the Fourier-transformed Coulomb potential in vacuum and εeff(q, ω) is the linear dielectric response function of the superconductor treated as an effective medium. Following this “macroscopic electrodynamics” formalism, it appears natural to use the recently developed plasmonics8 and electromagnetic metamaterial9 tools to engineer and maximize the electron pairing interaction (Eq. 1) in an artificial “metamaterial superconductor” via deliberate engineering of its dielectric response function εeff(q, ω). For example, it was predicted in refs 1,2 that considerable enhancement of the attractive electron-electron interaction may be expected in such actively studied metamaterial scenarios as ENZ10 and hyperbolic metamaterials5. In both cases εeff(q, ω) may become small and negative in substantial portions of the relevant four-momentum (q, ω) space leading to an enhancement of the electron pairing interaction. Indeed, it was demonstrated in refs 1,2 that in the case of a hyperbolic metamaterial the effective Coulomb potential from Eq. (1) assumes the form

where εxx = εyy = ε1 and εzz = ε2 have opposite signs. As a result, the effective Coulomb interaction of two electrons may become attractive and very strong along spatial directions where

Demonstration of the resulting superconductivity enhancement in hyperbolic metamaterials would open up numerous new possibilities for metamaterial enhancement of Tc in such practically important simple superconductors as niobium and MgB2.

Here, we report the first successful realization of such an artificial hyperbolic metamaterial superconductor, which is made of aluminum films separated by thin layers of Al2O3. This combination of materials is ideal for the proof of principle experiments because it is easy to controllably grow Al2O3 on the surface of Al and because the critical temperature of aluminum is quite low (Tc Al = 1.2 K11), leading to a very large superconducting coherence length ξ = 1600 nm11. Such a large value of ξ facilitates the metamaterial fabrication requirements since the validity of macroscopic electrodynamic description of the metamaterial superconductor requires that its spatial structural parameters must be much smaller than ξ. It appears that the Al/Al2O3 hyperbolic metamaterial geometry is capable of superconductivity enhancement, which is similar to that observed for a core-shell metamaterial geometry4, while having much better transport and magnetic properties compared to the core-shell superconductors. The multilayer Al/Al2O3 hyperbolic metamaterial samples for our experiments (Fig. 2) were prepared using sequential thermal evaporation of thin aluminum films followed by oxidation of the top layer for 1 hour in air at room temperature. The first layer of aluminum was evaporated onto a glass slide surface. Upon exposure to ambient conditions a ∼2 nm thick Al2O3 layer is known to form on the aluminum film surface12. Further aluminum oxidation may also be achieved by heating the sample in air. The oxidized aluminum film surface was used as a substrate for the next aluminum layer. This iterative process was used to fabricate thick multilayer (up to 16 metamaterial layers) Al/Al2O3 hyperbolic metamaterial samples (throughout our paper a single metamaterial layer is understood as a layer of Al with a layer of Al2O3 on its top surface). A transmission electron microscope (TEM) image of the multilayer metamaterial sample is shown in Fig. 2. During TEM experiments samples were coated with gold and platinum to ensure conductivity of the surface during sample preparation. A focused ion beam (FIB) microscope was used to prepare a sample for transmission electron microscopy (TEM). Samples were analyzed using a JEOL 2200 FS TEM, acquiring bright field and high resolution images. Figure 2 shows an image from the hyperbolic stack, showing polycrystalline Al grains, with 1–2 nm thick amorphous Al2O3 spacing layers, corresponding with the designed structure. Some Al2O3 layers are difficult to discern due to slight sample warping from the preparation process. The inset shows that the interfacial layers are indeed amorphous, between polycrystalline grains of Al, that, in cross-section, exhibit Moire cross-hatching.

Figure 2
figure 2

Transmission electron microscope (TEM) image of a 16 layer metamaterial sample.

During the imaging experiments samples were coated with gold and platinum to ensure conductivity of the surface during sample preparation. The inset shows that the interfacial layers are amorphous, between polycrystalline grains of Al, that, in cross-section, exhibit Moire cross-hatching.

Full size image

To demonstrate that our multilayer samples exhibit hyperbolic behavior, we studied their transport and optical properties (Figs 3,4 and 5). The temperature dependences of the sheet resistance of a 16-layer 10 nm/layer Al/Al2O3 hyperbolic metamaterial and a 100 nm thick Al film are shown in Fig. 3a. As illustrated in the logarithmic plot in the inset, the electronic conductivity of the metamaterial approaches conductivity values of bulk aluminum (indicated by the arrow) and is far removed from the parameter space characteristic for granular Al films13, which is indicated by the gray area in the inset. These results were corroborated by measurements of IR reflectivity of these samples, shown in Fig. 3b. The IR reflectivity of the hyperbolic 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 Al2O3. While the reflectivity spectrum of bulk Al is almost flat, the spectrum of Al2O3 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 optical phonon of Al2O314. The step in reflectivity is due to the negative sign of εAl2O3 near the resonance. The absence of this step in our multilayers indicates that the aluminum layers in our samples are continuous and not intermixed with aluminum oxide. Such a step is clearly observed in reflectivity data obtained from a core-shell Al/Al2O3 sample shown in Fig. 3b where the aluminum grains are separated from each other by Al2O3. On the other hand, this step is completely missing in reflectivity spectra of the hyperbolic metamaterial samples (the step at 9 μm observed in the spectrum of a three-layer sample is due to phonon-polariton resonance of the SiO2 substrate). Thus, our transport and optical measurements confirm excellent DC and AC (optical) conductivity of the aluminum layers of the fabricated hyperbolic metamaterials.

Figure 3
figure 3

Measurements of DC and AC (optical) conductivity of the aluminum layers of the fabricated hyperbolic metamaterials: (a) Temperature dependences of the sheet resistance of a 16-layer 10 nm/layer Al/Al2O3 hyperbolic metamaterial and a 100 nm thick Al film. As illustrated in the logarithmic plot in the inset, the conductivity of the metamaterial approaches conductivity values of bulk aluminum and is far removed from the parameter space characteristic for granular Al films which is indicated by the gray area. (b) IR reflectivity of bulk aluminium, 3 and 8 layer hyperbolic metamaterial and the core shell metamaterial samples measured in the long wavelength IR (LWIR) (2.5–22.5 μm) range using an FTIR spectrometer. The step in reflectivity around 11 μm is related to the phonon-polariton resonance (PPR) and may be used to characterize the spatial distribution of Al2O3 in the metamaterial samples.

Full size image
Figure 4
figure 4

The calculated plots of the real parts of εx,y (a) and εz (b) of the multilayer Al/Al2O3 metamaterial. The metamaterial consists of 13 nm thick Al layers separated by 2 nm of Al2O3 in the LWIR spectral range. The calculations were performed using Eqs (4 and 5) based on the Kramers-Kronig analysis of the FTIR reflectivity of Al and Al2O3 in ref. 4. The metamaterial appears to be hyperbolic except for a narrow LWIR spectral band between 11 and 18 μm.

Full size image
Figure 5
figure 5

Ellipsometry and polarization reflectometry of Al/Al2O3 hyperbolic metamaterials.

(a) Comparison of measured pseudo-dielectric function using ellipsometry and theoretically calculated Reε1 and Imε1. Theoretical data points are based on real and imaginary parts of εAl tabulated in ref. 17. (b) Data points are measured p- and s-polarized reflectivities of the metamaterial sample at 2.07 eV (600 nm) and 2.88 eV (430 nm). Dashed lines are fits using Eqs (6, 7, 8, 9). ε1 and ε2 obtained from the fits confirm hyperbolic character of the metamaterial.

Full size image

The Kramers-Kronig analysis of the FTIR reflectivity spectra of Al and Al2O3 measured in ref. 4 also allowed us to calculate the ε1 and ε2 components of the Al/Al2O3 layered films in the LWIR spectral range using the Maxwell-Garnett approximation as follows:

where n is the volume fraction of metal and εm and εd are the dielectric permittivities of the metal and dielectric, respectively15. Results of these calculations for a multilayer metamaterial consisting of 13 nm thick Al layers separated by 2 nm of Al2O3 are shown in Fig. 4. The metamaterial appears to be hyperbolic except for a narrow LWIR spectral band between 11 and 18 μm. A good match between the Maxwell-Garnett approximation (Equations (4 and 5)) and the measured optical properties of the metamaterial is demonstrated by ellipsometry (Fig. 5a) and polarization reflectometry (Fig. 5b) of the samples.

Variable angle spectroscopic ellipsometry with photon energies between 0.6 eV and 6.5 eV on the Al/Al2O3 metamaterial have been performed using a Woollam Variable Angle Spectroscopic Ellipsometer (W-VASE). For a uniaxial material with optic axis perpendicular to the sample surface and in plane of incidence, ellipsometry provides the pseudo-dielectric function which, in general, depends both on ε1 and ε2. However, as demonstrated by Jellison and Baba16, the pseudo-dielectric function in this measurement geometry is dominated by the in-plane dielectric function ε1 and is independent of the angle of incidence. We find that the pseudo-dielectric function of the Al/Al2O3 metamaterial is indeed similar (but not the same) as that of aluminum i.e. metallic as expected from effective medium theory. We also find that the pseudo-dielectric function is rather insensitive to the angle of incidence. The measured results for the real and imaginary parts of the pseudo-dielectric function in Fig. 5a show good agreement with the model for the in-plane dielectric function (Equation (4)). The calculated data points are based on the real and imaginary parts of εAl tabulated in ref. 17. The measured sign of the real part of the pseudo-dielectric function is negative, which suggests metallic in-plane transport. The sign of the real part of ε2 (and therefore, the hyperbolic character of our samples) was determined by polarization reflectometry, since ellipsometry data are less sensitive to ε216. Polarization reflectometry also confirmed the negative sign of the real part of ε1 consistent with ellipsometry data. The metamaterial parameters were extracted from the polarization reflectometry data as described in detail in ref. 18. Reflectivity for s-polarization is given in terms of the incident angle θ by

where

Reflectivity for p-polarization is given as

where

We measured p- and s- polarized absolute reflectance on the metamaterial sample using the reflectance mode of the ellipsometer. The reflectance was measured at two photon energies, 2.07 eV (600 nm) and 2.88 eV (430 nm), as shown in Fig. 5b and was normalized to the measured reflectance of a 150 nm gold film. The absolute reflectance of the gold film was obtained from ellipsometry measurements. The estimated uncertainty in the absolute reflectance of the Al/Al2O3 metamaterial is one percent. In order to obtain the dielectric permittivity ε1 and ε2 values, we fit the s- polarized reflectance first and get the in-plane dielectric function ε1. We then use the in-plane dielectric function to fit the p- polarized reflectance to obtain the out-of-plane dielectric function, ε2. The data analysis was done using W-VASE software. At 2.07 eV (600 nm), ε1 = −7.17 + 1.86i and ε2 = 1.56 + 0.21i and at 2.88 eV (430 nm), ε1 = −2.15 + 0.50i and ε2 = 1.30 + 0.09i. It is clear that the real part of the out-of-plane dielectric function is positive while the real part of the in-plane dielectric function is negative, which confirms the dielectric nature along z-axis and metallic nature in the xy-plane i.e. a hyperbolic metamaterial.

The Tc and critical magnetic field, Hc, of various samples (Figs 6 and 7) were determined via four-point resistivity measurements as a function of temperature and magnetic field, H, using a Physical Property Measurement System (PPMS) by Quantum Design. Even though the lowest achievable temperature with our PPMS system was 1.75 K, which is higher than the critical temperature TcAl = 1.2 K of bulk aluminum, we were able to observe a pronounced effect of the number of layers on Tc of the hyperbolic metamaterial samples. Figure 6a shows measured resistivity as a function of temperature for the 1-layer, 3-layer and 8-layer samples each having the same 8.5 nm layer thickness. While the superconducting transition in the 1-layer sample was below 1.75 K and could not be observed, the 3-layer and 8-layer metamaterial samples exhibited progressively higher critical temperature, which strongly indicates the role of hyperbolic geometry in Tc enhancement. A similar set of measurements performed for several samples having 13 nm layer thickness is shown in Fig. 6b.

Figure 6
figure 6

Effect of the number of layers on Tc of the Al/Al2O3 hyperbolic metamaterial samples: (a) Measured resistivity as a function of temperature is shown for the 1-layer, 3-layer and 8-layer samples each having the same 8.5 nm layer thickness. (b) Measured resistivity as a function of temperature for the 1-layer, 3-layer, 8-layer and 16-layer samples each having the same 13 nm layer thickness.

Full size image
Figure 7
figure 7

Evaluation of the Pippard coherence length of the Al/Al2O3 hyperbolic metamaterial: (a) Measured resistivity as a function of temperature for a 16-layer 13.2 nm layer thickness metamaterial sample. The critical temperature appears to be Tc = 2.3 K. The inset shows resistivity of this sample as a function of parallel magnetic field at T = 1.75 K. (b) Resistivity of the same sample as a function of perpendicular magnetic field at T = 1.75 K. Assuming Hc2perp = 100G (based on the measurements shown in the inset) the corresponding coherence length appears to be ξ = 181 nm, which is much larger than the layer periodicity.

Full size image

The measurements of Hc in parallel and perpendicular fields are shown in Fig. 7. Figure 7a shows measured resistivity as a function of temperature for a 16-layer 13.2 nm layer thickness hyperbolic metamaterial sample. The critical temperature of this sample appears to be Tc = 2.3 K, which is about two times higher than the Tc of bulk aluminum (another transition at Tc = 2.0 K probably arise from one or two decoupled layers or edge shadowing effects where the thickness of the films is not uniform). The inset in Fig. 7a illustrates the measurements of Hcparallel for this sample. The critical field appears to be quite large (∼3T), which is similar to the values of Hcparallel observed previously in granular aluminium films19. However, it is remarkable that such high critical parameters are observed for the films, which are much thicker than granular Al films.

Measurements of the perpendicular critical field Hcperp for the same metamaterial sample, which are shown in Fig. 7b allowed us to evaluate the Pippard coherence length

Assuming Hc2perp = 100 G (based on the inset in Fig. 7b) the corresponding coherence length appears to be ξ = 181 nm, which is much larger than the layer periodicity. Other measured samples also exhibit the coherence length around 200 nm. Therefore, our use of effective medium approach is validated and our multilayer samples should obey the metamaterial theory.

We have also studied changes in Tc as a function of Al layer thickness in a set of several 8-layer Al/Al2O3 metamaterial samples, as shown in Fig. 8a. The quantitative behaviour of Tc as a function of n may be predicted based on the hyperbolic enhancement of the electron-electron interaction (Eq. (2)) and the density of electronic states, ν on the Fermi surface which experience this hyperbolic enhancement. Using Eqs (4 and 5), the effective Coulomb potential from Eq. (2) may be re-written as

Figure 8
figure 8

Effect of the aluminum volume fraction n on Tc of the Al/Al2O3 hyperbolic metamaterial samples: (a) Resistivity as a function of temperature for the 8-layer samples having different aluminum layer thicknesses. (b) Experimentally measured behavior of Tc as a function of n (which is defined by the Al layer thickness) correlates well with the theoretical fit (red curve) based on the hyperbolic mechanism of Tc enhancement. Experimental data points shown in black correspond to 8-layer samples, while blue ones correspond to 16-layer samples.

Full size image

Let us assume that the dielectric response function of the metal used to fabricate the hyperbolic metamaterial shown in Fig. 1 may be written as

where ωp is the plasma frequency, k is the inverse Thomas-Fermi radius and Ωn(q) are dispersion laws of various phonon modes20. Zeroes of the dielectric response function εm(q, ω) of the bulk metal (which correspond to its various bosonic modes) maximize the electron-electron pairing interaction given by Eq. (1). As summarized in ref. 21, the critical temperature of a superconductor is typically calculated as

where θ is the characteristic temperature for a bosonic mode mediating the electron pairing interaction (such as the Debye temperature θDin the standard BCS theory) and λeff is the dimensionless coupling constant defined by V(q, ω) = VC(q)ε−1(q, ω) and the density of states ν (see for example ref. 22):

where VC is the unscreened Coulomb repulsion. The integral in Eq. (14) is typically simplified to take into account only the contributions from the poles of the inverse dielectric response function ε−1(q, ω), while averaging is performed over all the spatial directions.

Let us consider the region of four-momentum (q, ω) space, where ω > Ω1(q). While εm = 0 at ω = Ω1(q), εm(q, ω) is large in a good metal and negative just above Ω1(q). Compared to the bulk metal, the poles of the angular-dependent νV product of the hyperbolic metamaterial are observed at shifted positions compared to the zeroes of εm and additional poles may also appear2. Based on Eq. (11), the differential of the product νV may be written as

where x = cos θ and θ varies from 0 to π. The latter expression has two poles at

As the volume fraction, n, of metal is varied, one of these poles remains close to εm = 0, while the other is observed at larger negative values of εm:

This situation is similar to calculations of Tc for ENZ metamaterials23. Since the absolute value of εm is limited (see Eq. (12)), the second pole disappears near n = 0 and near n = 1. Due to the complicated angular dependences in Eq. (16), it is convenient to reverse the order of integration in Eq. (14) and perform the integration over dω first, followed by angular averaging. Following the commonly accepted approach, while integrating over dω we take into account only the contributions from the poles given by

Eqution (16) and assume the value of Imεm = εm” to be approximately the same at both poles. The respective contributions of the poles to d(νV)/dx may be written as

Near n = 0 and n = 1 these expression may be approximated as

and

respectively. Note that at the ω = Ω1(q) zero of the dielectric response function of the bulk metal the effective Coulomb potential inside the metal may be approximated as

so that the coupling constant λeff of the hyperbolic metamaterial obtained by angular integration of the sum of Eqs (20) and (21) may be expressed via the coupling constant λm of the bulk metal:

where α is a constant of the order of 1 and x0 is defined by the maximum negative value of εm, which determines if the second pole (Equation (18)) exists at a given n. Based on Eq. (18),

If the second pole does not exist then x0 = 0 may be assumed. Based on Eq. (13), the theoretically predicted value of Tc for the hyperbolic metamaterial is calculated as

assuming the known values Tcbulk = 1.2 K and λm = 0.17 for bulk aluminum23. The predicted behaviour of Tc as a function of n is plotted in Fig. 8b. This figure demonstrates that the experimentally measured behaviour of Tc as a function of n (which is defined by the Al layer thickness) correlates well with the theoretical fit, which was obtained using Eq. (25) based on the hyperbolic mechanism of Tc enhancement.

The observed combination of transport and critical properties of the Al/Al2O3 hyperbolic metamaterials is very far removed from the parameter space typical of the granular aluminum films13,19. Together with the number of layer and layer thickness dependences of Tc and Hc shown in Figs 6,7 and 8, these observations strongly support the hyperbolic metamaterial mechanism of superconductivity enhancement described by Eqs (11. The developed technology enables efficient nanofabrication of thick film aluminum-based hyperbolic metamaterial superconductors with a Tc that is two times that of pure aluminum and with excellent transport and magnetic properties. While the observed Tc increase is slightly smaller than the one observed in ENZ metamaterials4, the hyperbolic metamaterial geometry exhibits superior transport and magnetic properties compared to the ENZ core-shell metamaterial superconductors. In addition, our theoretical model is applicable to previous experiments performed in NbN/AlN24 and Al/Si25 multilayer geometries. We should also note that unlike recent pioneering work on quantum metamaterials26, which are based on superconducting split-ring resonators and quantum circuits, our work aims at engineering of metamaterials with enhanced superconducting properties.

Our results open up numerous new possibilities for considerable Tc enhancement in other practically important simple superconductors, such as niobium and MgB2. However, due to their much smaller coherence length11,20 metamaterial structuring of these superconductors must be performed on a much more refined scale. The two-fold increase of Tc in an artificial hyperbolic metamaterial superconductor that we have observed suggests that the recently discovered hyperbolic properties of high Tc superconductors (such as BSCCO)6 may play a considerable role in the high values of Tc observed in cuprates.

Additional Information

How to cite this article: Smolyaninova, V. N. et al. Enhanced superconductivity in aluminum-based hyperbolic metamaterials. Sci. Rep. 6, 34140; doi: 10.1038/srep34140 (2016).