Cooper-pair insulator phase in superconducting amorphous Bi films induced by nanometer-scale thickness variations

Ultrathin films near the quantum insulator-superconductor transition (IST) can exhibit Cooper-pair transport in their insulating state. This Cooper-pair insulator (CPI) state is achieved in amorphous Bi films evaporated onto substrates with a topography varying on lengths slightly greater than the superconducting coherence length. We present evidence that this topography induces film thickness and corresponding superconducting coupling constant variations that promote Cooper-pair island formation. Analyses of many thickness-tuned ISTs show that weak links between superconducting islands dominate the transport. In particular, the IST occurs when the link resistance approaches the resistance quantum for pairs. These results support conjectures that the CPI is an inhomogeneous state of matter.

Figure 1

Characteristics of thin films on nanohoneycomb substrates. (a) and (b) Insulator superconductor transitions tuned by thickness of $a$-Bi on more and less ordered hole arrays. Film thicknesses in (a) from the top down are 0.89, 0.94, 1.05, 1.08, 1.13, and $1.21\pm 0.1$ nm and in (b) are 1.08, 1.13, 1.21, 1.25, and $1.47\pm 0.1$ nm. Insets: Postexperiment SEM images. Center-to-center hole spacing: $100\pm 5$ nm. Radii: (a) 27 and (b) $28\pm 3$ nm. The scale bar spans 200 nm. (c) Conductance normalized to the conductance scale for weak localization phenomena [$G_{00}=e^2/\left(2{\pi }^2\hslash \right)\approx 1/81\text{k}\Omega$] and (d) activation energies for films on four different NHC substrates. Gray triangles in (c) show data for a reference film. Identical shapes are used when measurements were performed during the same experimental run (on side-by-side NHC substrates). Unfilled and filled symbols represent data from more and less ordered hole arrays, respectively. Arrows on the $x$ axis indicate critical thicknesses for conductivity and SC, and open arrows indicate critical thicknesses in reference films for conductivity, 0.5 nm, and superconductivity, 0.7 nm. The gray arrow in (d) indicates the measured thickness of the first SC film in the filled circle series. This value is a more accurate measure of $d_{\mathrm{IST}}$ than the linear extrapolation. In this case only, we use this value in place of the intercept. Data from the experiment in (a) and (b) are shown as circles. For squares, $r_{\text{hole}}=16$ (unfilled) and 15 (filled) $\pm$3 nm. The dashed lines are least-squares linear fits to the data. (e) Normal-state ($T=8$ K) critical sheet resistance for the last insulating and first SC films of the IST of nine different NHC substrates. The horizontal dashed line at $3R_{\mathrm{Q}}$ is a guide to the eye.

Figure 2

NHC film topography. (a) AFM image showing the topography of the substrate in Fig. 1a. The scale bar (white line) spans 100 nm. (b) Height profile along the trace (black line) in (a). (c)–(e) Film thicknesses along the trace in (a) for $d^{\mathrm{dep}}$ at (c) the onset of conduction, (d) a representative insulating film [bold in Fig. 1a], and (e) the IST. Pink and blue lines indicate the critical thicknesses for conductivity and SC, respectively, for a reference film. In (c), the SC coherence length $\xi$ in a reference film of comparable thickness is shown for feature size comparison. (f) Average link thickness for NHC films at their transitions to conductivity ($d_{\mathrm{Gc}}$, pink) and SC ($d_{\mathrm{IST}}$, blue) for six different substrates. Critical thicknesses for SC are shown by plotting the link thicknesses of the last insulating and first SC films spanning the IST.

Figure 3

Film thickness variations in NHC films. Calculated thickness variations in films deposited on the NHC substrate of Figs. 1a and 2a. Film thickness coloring represents conductivity on a reference film: White is nonconductive ($d<d_{\mathrm{Gc}}^{\mathrm{ref}}$), pink is conductive ($d>d_{\mathrm{Gc}}^{\mathrm{ref}}$), and blue is superconductive ($d>d_{\mathrm{IST}}^{\mathrm{ref}}$). (a)–(c) Film thickness variation plots (a) at the onset of conduction, (b) for a representative insulating film [bold in Fig. 1a], and (c) at the IST. The critical temperatures for SC in reference films ($T_{\mathrm{c}}$) corresponding to these film thicknesses are shown by the color bar on the right-hand side. $T_{\mathrm{c}}=0$ marks the IST and the maximum $T_{\mathrm{c}}$ seen in each of the panels is (a) 0.4 K, (b) 1.3 K, and (c) 2.2 K. (d) $T_{\mathrm{c}}$ (in K) contours shown for a section of the film at $d^{\mathrm{dep}}=d_{\mathrm{IST}}$. (e) Illustration of a wire array of resistors $R^{\mathrm{link}}$ connecting SC islands. The basic element of the array is shown in black.

Figure 4

Coalescence of superconducting islands. (a) SC island distributions showing individual island size, the number of islands of that size, and the fraction of the film area occupied by islands of that size for the substrate of Figs. 1a, 2a, 2b, 2c, 2d, 2e, and 3. SC film regions are identified for $d(x,y)>d_{\mathrm{IST}}^{\mathrm{ref}}=0.7$ nm on a $1\times 1$ $\mu$m image. Five different $d^{\mathrm{dep}}$ are shown, three of which match those in Figs. 2 and 3. $d^{\mathrm{dep}}=$ 1.07 nm is the IST. Vertical dashed lines are plotted to indicate a coherence area $\pi {(\xi /2)}^2$ with $\xi =15$ nm, and the hole-array unit-cell area $A_{\mathrm{cell}}=9900$ nm${}^2$. (b) Film area coverage of the largest SC island for each deposition in (a) vs $d^{\mathrm{dep}}$. The IST is shown as a vertical dashed line at $d^{\mathrm{dep}}=1.07$ nm. The shaded region indicates a 10$\%$ uncertainty in the thickness measurement. (c) The analysis of (a) and (b) is repeated for films on all six different NHC substrates and here we plot the deposition thickness at which 50% of the film area is covered by a single SC island $d_{50}\%$, compared to the experimental value $d_{\mathrm{IST}}$ for each substrate. The open circle plots the data in (a) and (b).