Superconducting Properties in Arrays of Nanostructured β-Gallium

Samples of nanostructured β-Ga wires were synthesized by a novel method of metallic-flux nanonucleation. Several superconducting properties were observed, revealing the stabilization of a weak-coupling type-II-like superconductor (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${T}_{c}$$\end{document}Tc \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\approx $$\end{document}≈ 6.2 K) with a Ginzburg-Landau parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\kappa }_{GL}$$\end{document}κGL = 1.18. This contrasts the type-I superconductivity observed for the majority of Ga phases, including small spheres of β-Ga with diameters near 15 μm. Remarkably, our magnetization curves reveal a crossover field \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H}_{D}$$\end{document}HD, where we propose that the Abrikosov vortices are exactly touching their neighbors inside the Ga nanowires. A phenomenological model is proposed to explain this result by assuming that only a single row of vortices is allowed inside a nanowire under perpendicular applied field, with an appreciable depletion of Cooper pair density at the nanowire edges. These results are expected to shed light on the growing area of superconductivity in nanostructured materials.

(χ = M/H) and normalized to π −1/4 at the saturated maximum shielding of zero field cooling (ZFC) measurements, for the lowest applied fields. The paramagnetic background in the normal state region was not subtracted since it does not interfere with the analysis.
The sharp transitions at low fields that define a T c of 6.2 K (Fig. 1) and the nanowire diameter of 140 nm (within the range of sizes 21,22 that favors the stabilization of β-Ga) are indications that we obtained a pure β-Ga phase. Perhaps most importantly, the graph shown in the inset of Fig. 1 displays the specific heat of the nGa sample, measured on warming. The peak at T m = 256.8 K represents the latent heat associated 22 with melting of the pure β-Ga phase. Figure 2 displays a set of MH curves for T = 2 K, 4 K and 5 K. These curves were obtained by subtracting the normal paramagnetic background coming from the alumina template and nanowires. First, a normal state reference curve, obtained for the nGa sample at 6.2 K, was subtracted from each curve measured in the superconducting state for the same sample. In this process the signal magnitude was properly corrected to account for the temperature dependence. Second, we performed an additional subtraction of the paramagnetic contribution coming from the unfilled alumina template, which was measured at each T of interest. Therefore, the final MH curves shown in Fig. 2 are attributed solely to the Ga nanowires.
The MH curves show a large hysteresis between the ascending (M ↑) and descending (M ↓) branches, as indicated by the arrows near the 2 K curve. Equilibrium magnetization curves can be evaluated by the average 23,24 M eq = (M ↑ + M ↓)/2. One calculated example (at 2 K) is plotted as a black dashed line in Fig. 2. A penetration field  Scientific REPORTS | 7: 15306 | DOI:10.1038/s41598-017-15738-2 H p is defined at the point where M eq departs from the Meissner state straight line, and an upper critical field H u is defined at the merging point with the normal state baseline.
A relevant fact is that only a very small hysteresis appears between the MH curves for the first increase from H = 0 (virgin state) and subsequent field increases. This could be due to a negligible bulk pinning of vortices as they enter the nanowires in a similar way 25 , independently of the field cycling. Under decreasing field, however, a practically zero magnetization is observed, as expected from the Bean-Livingston (BL) surface barrier mechanism 24,26 , until a crossover value H D is reached and diamagnetic shielding currents show up. This strong asymmetry, between the ascending and descending branches of MH curves, indicates 24 the dominance of the BL barrier over the negligible bulk pinning. It is important to mention that MH curves measured with H parallel to the nanowires (not presented here) do not show a crossover field like H D .
To further explore the magnetization behavior of the nGa samples a minor hysteresis loop 27,28 , was measured on top of the second MH curve at = T K 2 , represented by open stars in the first quadrant of Fig. 2, with an inverted vertical scale at the right axis. This curve starts at H = 0 going up to 550 Oe (arrow 1), then is reversed down to 370 Oe, then reversed up to 550 Oe (arrow 2) and then reversed down to H = 0. This completes the full loop, which almost overlaps with the first measured MH curve represented by open up-triangles. The relevant feature in the whole process is the minor hysteresis loop between 550 Oe and 370 Oe, showing that a substantial portion of the reversed branches (down and up) are almost parallel to the Meissner straight line. These portions are marked in the graph by two straight line segments that indicate the dominance of the surface barrier against the entrance of vortices 29 . Figure 3(a) presents plots for the fields H p , H u , and H D whose data (see Table 1) were extracted from Figs 1 and 2 within an experimental error of 5%. Notice the good agreement between H u lines extracted from MH curves (closed stars) and MT curves (closed squares).

Discussion
We present in this section different possible interpretations for our data. It is important to notice that our magnetization curves represent a global response of the total nanowire array. Due to the high uniformity of the nGa sample, however, we infer that all properties calculated in this section are the same for each individual nanowire, which are separated by the insulating matrix. Also, because the quantized flux lines cross each nanowire along its length and have comparable diameter sizes, there will be a depletion of Cooper pair density 30 at the wire edges. This produces effective values for the coherence length ξ ξ Here, ξ T ( ) and λ T ( ) are the usual 3D parameters from Ginzburg-Landau (GL) theory and ξ f T ( ), λ f T ( ) are depletion parameters to be determined from the experimental data. Assuming that H u is similar to the bulk nucleation field H c2 from GL theory, we estimate H u (0) = 923 Oe, at = T 0, from the WHH formula 31 , where Φ 0 = . × − 2 07 10 7 Gcm 2 is the flux quantum. Then, the effective coherence length at T = 0 becomes ξ (0) e ≈ 60 nm. This means that a vortex core at T = 0 has a diameter just slightly smaller than that of the Ga nanowire. Table 1 and Fig. 3 also show values for the thermodynamic critical field (H c ), calculated with an equation that balances the isothermal magnetic work and the condensation energy involved in the superconducting transition 32 : T 0 and the thermal energy at T c , which is an important parameter from BCS theory, can now be calculated by 36 : where the slope of H T ( ) c at T c was evaluated (Fig. 3) to be around −129 Oe/K. This value of 3.61 for β-Ga is close to the BCS prediction of 3.53 for weak-coupling superconductors 32,37 , and is similar to In (3.63), Sn (3.6) and Ta (3.6).
A Model for H D . -From the calculated properties above, we conclude that our nGa sample is consistently well described as a weak-coupling type-II-like superconductor. The estimated values of ξ ≈ (0) 60 e nm and λ ≈ (0) 88 e nm suggest that at very low T the diameter of the vortices nearly matches the nanowire diameter of 140 nm. When temperature increases ξ T ( ) e and λ T ( ) e become increasingly depleted 30 . This effect is especially pronounced along the nanowire diameter, because there is no severe size restrictions along the nanowire length. This leads to the conclusion that only one row of vortices is allowed inside the nanowire. This is similar to the reported scenario for Pb nanowires of diameters near 390 nm, under perpendicular H 13 . For thicker Pb nanowires 13 or millimeter-sized disks 17 a classical type-I intermediate state with multiquanta domains are observed.
We propose a simple phenomenological model assuming that the crossover field H D corresponds to the situation in which the vortices are exactly touching their neighbors as depicted in Fig. 4. Because this happens in the descending branch of MH curves, it is helpful to recall that overlapped vortices are nucleated at H u and become gradually separated as H decreases. This occurs because part of the vortices leaves the nanowire easily, with no surface barrier, as discussed before. When ≤ H H D , the superconducting regions are enhanced between the vortices, producing a fast increase of the diamagnetic response as observed (see Fig. 2).
In Fig. 4, when H D is reached, a geometrical relation between the nanowire length (L) and the number of enclosed vortices inside (N v ) will be λ = ν L N 2 . The density of vortices will be = ν n N Ld / , using the maximal longitudinal cross-section area of the nanowire. Since    Fig. 3(a). Combining this result with equation (3) f t t ( ) 0 551(1 ) 0 25 , which is plotted in Fig. 3  As shown in Table 1 and Fig. 3(b) h D decreases just slightly and almost linearly as T increases in the measured range. A fit to the experimental data gives = .
The weakly decreasing monotonic behavior of ξ f and λ f , as T increases in the measured range, is in fact inversely proportional to the depletion degree of Cooper pair density in the vortex volume. This is the expected trend. As T grows both λ T ( ) and ξ T ( ) must increase, eventually exceeding the nanowire diameter. This causes an enhanced depletion at the edges 30 .
We have also tried to interpret our data as type-I superconductivity, similar to the approach used in ref. 6 for β-Ga microspheres; by assuming the upper critical field (H u ) to be H cI and the crossover field (H D ) to be a supercooling field (H sc  , which is plotted as solid down triangles in Fig. 3(b). The solid line through these points is a fit of the expected two fluid model expression 14,15 Clearly this temperature dependence does not fit well our data. Also, one can obtain H (0) cI ≈ 820 Oe by fitting a parabolic expression to our H u data in Fig. 3(a). Using the GL expression 25 π λξ = Φ H /(2 2 ) c 0 and ξ λ κ = / sc one gets λ ≈ (0) 315 nm and ξ ≈ (0) 90 nm. These results sound unlikely and are very different from those found in ref. 6 for β-Ga microspheres. Particularly, in the type-I interpretation, the BCS energy ratio would be 2 ∆ ≈ kT (0)/ c 6, because the slope dH dT ( / ) cI Tc ≈ −215 Oe/K has to be used instead of dH dT ( / ) c T c ≈ −129 Oe/K in equation (2). However this energy ratio value is unrealistically high, even for a strong coupling superconductor.

Conclusion
Samples of nanostructured β-Ga wires were successfully prepared by a novel method of metallic-flux nanonucleation. Several superconducting properties were determined from magnetization measurements and are well described as a weak-coupling type-II-like superconductor with a Ginzburg-Landau parameter κ GL = 1.18.
Possibly the unexpected type-II-like behavior reported here is favored by the nanoscopic scale of the Ga nanowires, stabilized in very particular geometrical conditions. To our knowledge, no such effect has yet been verified for Ga. Particularly we have introduced a model to interpret a clearly defined crossover field (H D ), using simple ideas based on the GL theory and vortex behavior. Although the obtained results seems plausible, we feel that a more accurate and fundamental treatment is lacking, especially to explain the depletion parameters ξ f and λ f , introduced to take account of the partial suppression of the vortex volume (or Cooper pair density 30 ) at the nanowire edges.
We also tried to interpret the data as a classical type-I superconductor, but the results were not so convincing. We then conclude that possibly our β-Ga nanowires, under perpendicular applied field, favors a type-II-like behavior that calls for further investigation. We are planning to study new β-Ga samples with different nanowires diameters, as well as samples of Sn and In, prepared by the same method employed here. Finally, we hope this work will motivate new studies regarding nanostructured superconductors 38 .

Methods
The MFNN technique [18][19][20] has been successfully developed to nucleate crystalline nanowires inside the pores of an alumina template. The nanoporous template presents several advantages, such as an excellent pore-size control over large areas and large aspect-ratio pores that exhibit a highly regular spatial pattern. Our present samples consist of small pieces of the alumina template filled with pure Ga (nGa), having typically an area of 2 by 2 mm 2 and thickness of 80 µm. Figure 5(a),(b), show Scanning Electron Microscope (SEM) images of a small top view area and a longitudinal view of one Ga nanowire, respectively. The nanowires were exposed by gently crushing a filled template. Figure 5(b) shows a nanowire with uniform diameter of 140 nm and length around 3.8 µm. This is only a small portion from one of the original wires embedded in the template, which are typically 80 µm long. The distances between the centers of the neighboring nanowires are fixed at 250 nm, forming a nearly perfect triangular array as shown in Fig. 5(a).