Enhanced superconductivity in aluminum-based hyperbolic metamaterials

One of the most important goals of condensed matter physics is materials by design, i.e. the ability to reliably predict and design materials with a set of desired properties. A striking example is the deterministic enhancement of the superconducting properties of materials. Recent experiments have demonstrated that the metamaterial approach is capable of achieving this goal, such as tripling the critical temperature TC in Al-Al2O3 epsilon near zero (ENZ) core-shell metamaterial superconductors. Here, we demonstrate that an Al/Al2O3 hyperbolic metamaterial geometry is capable of a similar TC enhancement, while having superior transport and magnetic properties compared to the core-shell metamaterial superconductors.

Scientific RepoRts | 6:34140 | DOI: 10.1038/srep34140 where V C is the Fourier-transformed Coulomb potential in vacuum, and ε eff (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 plasmonics 8 and electromagnetic metamaterial 9 tools to engineer and maximize the electron pairing interaction (Eq. 1) in an artificial "metamaterial superconductor" via deliberate engineering of its dielectric response function ε eff (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 ENZ 10 and hyperbolic metamaterials 5 . In both cases ε eff (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 ε xx = ε yy = ε 1 and ε zz = ε 2 have opposite signs. As a result, the effective Coulomb interaction of two electrons may become attractive and very strong along spatial directions where 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 Al 2 O 3 . This combination of materials is ideal for the proof of principle experiments because it is easy to controllably grow Al 2 O 3 on the surface of Al, and because the critical temperature of aluminum is quite low (T c Al = 1.2 K 11 ), leading to a very large superconducting coherence length ξ = 1600 nm 11 . 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/Al 2 O 3 hyperbolic metamaterial geometry is capable of superconductivity enhancement, which is similar to that observed for a core-shell metamaterial geometry 4 , while having much better transport and magnetic properties compared to the core-shell superconductors. The multilayer Al/Al 2 O 3 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 Al 2 O 3 layer is known to form on the aluminum film surface 12 . 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/Al 2 O 3 hyperbolic metamaterial samples (throughout our paper a single metamaterial layer is understood as a layer of Al with a layer of Al 2 O 3 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 Al 2 O 3 spacing layers, corresponding with the designed structure. Some Al 2 O 3 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.
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/ Al 2 O 3 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 films 13 , 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 μ m) range using an FTIR spectrometer, and compared with reflectivity spectra of Al and Al 2 O 3 . While the reflectivity spectrum of bulk Al is almost flat, the spectrum of Al 2 O 3 exhibits a very sharp step-like behavior around 11 μ m that is related to the phonon-polariton resonance, which results from coupling of an infrared photon with an optical phonon of Al 2 O 3 14 . The step in reflectivity is due to the negative sign of ε Al2O3 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/Al 2 O 3 sample shown in Fig. 3b where the aluminum grains are separated from each other by Al 2 O 3 . On the other hand, this step is completely missing in reflectivity spectra of the hyperbolic metamaterial samples (the step at 9 μ m observed in the spectrum of a three-layer sample is due to phonon-polariton resonance of the SiO 2 substrate). Thus, our transport and optical measurements confirm excellent DC and AC (optical) conductivity of the aluminum layers of the fabricated hyperbolic metamaterials.
The Kramers-Kronig analysis of the FTIR reflectivity spectra of Al and Al 2 O 3 measured in ref. 4 also allowed us to calculate the ε 1 and ε 2 components of the Al/Al 2 O 3 layered films in the LWIR spectral range using the Maxwell-Garnett approximation as follows: where n is the volume fraction of metal, and ε m and ε d are the dielectric permittivities of the metal and dielectric, respectively 15 . Results of these calculations for a multilayer metamaterial consisting of 13 nm thick Al layers separated by 2 nm of Al 2 O 3 are shown in Fig. 4. The metamaterial appears to be hyperbolic except for a narrow LWIR spectral band between 11 and 18 μ m. 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/Al 2 O 3 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 ε 1 and ε 2 . However, as demonstrated by Jellison and Baba 16 , the pseudo-dielectric function in this measurement geometry is dominated by the in-plane dielectric function ε 1 and is independent of the angle of incidence. We find that the pseudo-dielectric function of the Al/Al 2 O 3 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 ε Al 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 ε 2 (and therefore, the hyperbolic character of our samples) was determined by polarization reflectometry, since ellipsometry data are less sensitive to ε 2 16 . Polarization reflectometry also confirmed the negative sign of the real part of ε 1 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 ts 1 Reflectivity for p-polarization is given as 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/Al 2 O 3 metamaterial is one percent. In order to obtain the dielectric permittivity ε 1 and ε 2 values, we fit the s-polarized reflectance first, and get the in-plane dielectric function ε 1 . We then use the in-plane dielectric function to fit the p-polarized reflectance to obtain the out-of-plane dielectric function, ε 2 . The data analysis was done using W-VASE software. At 2.07 eV (600 nm), ε 1 = − 7.17 + 1.86i and ε 2 = 1.56 + 0.21i, and at 2.88 eV (430 nm), ε 1 = − 2.15 + 0.50i and ε 2 = 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 T c and critical magnetic field, H c , 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 T cAl = 1.2 K of bulk aluminum, we were able to observe a pronounced effect of the number of layers on T c 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 T c enhancement. A similar set of measurements performed for several samples having 13 nm layer thickness is shown in Fig. 6b. The measurements of H c 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 T c = 2.3 K, which is about two times higher than the T c of bulk aluminum (another transition at T c = 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 H c parallel for this sample. The critical field appears to be quite large (∼ 3T), which is similar to the values of H c parallel observed previously in granular aluminium films 19 . 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 H c perp for the same metamaterial sample, which are shown in Fig. 7b allowed us to evaluate the Pippard coherence length Assuming H c2 perp = 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 T c as a function of Al layer thickness in a set of several 8-layer Al/Al 2 O 3 metamaterial samples, as shown in Fig. 8a. The quantitative behaviour of T c 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 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 Assuming H c2 perp = 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. where ω p is the plasma frequency, k is the inverse Thomas-Fermi radius, and Ω n (q) are dispersion laws of various phonon modes 20 . Zeroes of the dielectric response function ε m (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 θ D in the standard BCS theory), 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 ref. 22): where V C 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 > Ω 1 (q). While ε m = 0 at ω = Ω 1 (q), ε m (q, ω) is large in a good metal and negative just above Ω 1 (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 ε m , and additional poles may also appear 2 . 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 ε m = 0, while the other is observed at larger negative values of ε m : This situation is similar to calculations of T c for ENZ metamaterials 23 . Since the absolute value of ε m 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 Imε m = ε m " 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 respectively. Note that at the ω = Ω 1 (q) zero of the dielectric response function of the bulk metal the effective Coulomb potential inside the metal may be approximated as  If the second pole does not exist then x 0 = 0 may be assumed. Based on Eq. (13), the theoretically predicted value of T c for the hyperbolic metamaterial is calculated as assuming the known values T cbulk = 1.2 K and λ m = 0.17 for bulk aluminum 23 . The predicted behaviour of T c as a function of n is plotted in Fig. 8b. This figure demonstrates that the experimentally measured behaviour of T c 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 T c enhancement. The observed combination of transport and critical properties of the Al/Al 2 O 3 hyperbolic metamaterials is very far removed from the parameter space typical of the granular aluminum films 13,19 . Together with the number of layer and layer thickness dependences of T c and H c shown in Figs 6,7 and 8, these observations strongly support the hyperbolic metamaterial mechanism of superconductivity enhancement described by Eqs (11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25). The developed technology enables efficient nanofabrication of thick film aluminum-based hyperbolic metamaterial superconductors with a T c that is two times that of pure aluminum and with excellent transport and magnetic properties. While the observed T c increase is slightly smaller than the one observed in ENZ metamaterials 4 , 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/AlN 24 and Al/Si 25 multilayer geometries. We should also note that unlike recent pioneering work on quantum metamaterials 26 , 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 T c enhancement in other practically important simple superconductors, such as niobium and MgB 2 . However, due to their much smaller coherence length 11,20 metamaterial structuring of these superconductors must be performed on a much more refined scale. The two-fold increase of T c in an artificial hyperbolic metamaterial superconductor that we have observed suggests that the recently discovered hyperbolic properties of high T c superconductors (such as BSCCO) 6 may play a considerable role in the high values of T c observed in cuprates.