Evidence for enhanced phase fluctuations in nanostructured niobium thin films

In a superconducting nanostructure, phase fluctuations are prominent and give rise to finite resistance below superconducting transition temperature $T_c$. By using a monolayer polymer/nanosphere hybrid we developed previously, we fabricated a large array of interconnected niobium (Nb) honeycomb lattices with the thinnest interconnected linewidth $d$ ranging from 36 nm to 89 nm. The honeycomb cells form a highly ordered triangular lattice with more than ${10}^8$ unit cells extending over few ${\mathrm{mm}}^2$ area, which enables the detailed transport study at nanometer scales. We found $T_c$ gradually drops with decreasing $d$ due to the phase-slip effect, while the critical field at lower temperature tends to follow that of a continuous Nb thin film. One likely scenario is to consider a model system of numerous superconducting islands interconnected by short phase-slip junctions, where the phase coherence is dictated by the phase slippage in the nanoconstriction. This was strongly supported by the excellent fitting to the thermally activated phase-slip model and also the unusual phenomena of transition width narrowing in high fields.

Figure 1

(a) SEM image of a Nb honeycomb lattice with isolated Nb islands. The red dotted hexagon indicates a single cell of the lattice. (b) An AFM image of a interconnected Nb honeycomb lattice (INHL). The bottom panel shows the cross-sectional profile along the red dotted line showing a Nb thin film thickness $t\approx 34$ nm and interconnected linewidth $d\approx 74$ nm. (c) The normalized resistance $R/R_n$ versus T for four INHL samples with different $d$'s ranging from 36 nm to 89 nm. Low-$T$ blowup view is shown in (d) that shares the same color code as (c). (e) The superconducting transition temperature $T_c$ as a function of $d$. The upper and lower inset figures are SEM images for $d=89$ and 36 nm, respectively, where the white scale bars equal 500 nm. The red dotted lines are the $T_c$ value for a continuous Nb thin film. As $d$ decreases, both the $T_c$ and the residual resistivity ratio drops. The superconducting transition width is also getting wider for smaller $d$.

Figure 2

(a) Sheet resistance $R_{\square }$ as a function of $H/H_m$ for $d=36$ nm INHL close to $T_c$. The rapid and macroscopic increase of $R_{\square }$ at fractional values of $H/H_m$ is a direct consequence of kinetic energy loss due to the screening supercurrents. (b) A phase diagram of INHL for continuous Nb film and $d=36$ nm, 61 nm, and 89 nm. The color code represents the normalized resistance $R/R_n$, where red and blue regions refer to the normal state and the superconducting state, respectively. The solid black lines are the curves for $R/R_n=0.5$. Apparent oscillations at the phase boundary were observed for all the INHLs.

Figure 3

(a) LAMH fittings (red lines) to the $R/R_n-T$ curves in four INHL samples with difference $d$'s under investigation. (b) The $d$ dependence of the two fitting parameters: $T_c$ and zero-temperature free energy barrier $\mathrm{\Delta }{F}_0$, obtained from LAMH model. The solid squares are the $T_c$ values obtained from the resistance data with $R/R_n=0.001$. $\mathrm{\Delta }{F}_0$ is practically linear with $d$, but it appears to exponentially decay with $d$ as $d\le 60$ nm. (c) $R/R_n-T$ curves under perpendicular magnetic fields $H$. The solid and dotted lines are the data for $d=61$ nm and continuous Nb thin film, which shares the same color codes representing different $H$ values ranging from zero to 6 kOe. For $d=61$ nm, the superconducting transition width apparently decreases with increasing $H$, which is in big contrast to the continuous Nb thin film that shows little variation of transition width with $H$. (d) The normalized transition width $\mathrm{\Delta }{T}_c/T_c$ versus $H$. A rapid decrease of $\mathrm{\Delta }{T}_c/T_c$ with increasing $H$ appears in all INHL samples, where it reaches a local minimum at the critical field $H_{\mathrm{x}}$. $H_{\mathrm{x}}$ values, indicated by the black arrows, are strongly $d$-dependent and grow larger for smaller $d$.

Figure 4

Critical field $H_{\mathrm{x}}$ versus $d$ that separates two distinct phases of flux lines. In liquid phase, the flux lines can move around within the INHL and cross each other without costing much energy as illustrated in the lower-left image. Large phase fluctuations can appear in the liquid phase. On the contrary, as $H$ increases, the collective pinning effect drives the flux lines into a solid phase, where flux lines are mostly pinned either inside the cell or within the INHL, and thus the phase fluctuations are strongly suppressed.