Size dependent nature of the magnetic-field driven superconductor-to-insulator quantum-phase transitions

The nature of the magnetic-field driven superconductor-to-insulator quantum-phase transition in two-dimensional systems at zero temperature has been under debate since the 1980s, and became even more controversial after the observation of a quantum-Griffiths singularity. Whether it is induced by quantum fluctuations of the superconducting phase and the localization of Cooper pairs, or is directly driven by depairing of these pairs, remains an open question. We herein experimentally demonstrate that in weakly-pinning systems and in the limit of infinitely wide films, a sequential superconductor-to-Bose insulator-to-Fermi insulator quantum-phase transition takes place. By limiting their size smaller than the effective penetration depth, however, the vortex interaction alters, and the superconducting state re-enters the Bose-insulating state. As a consequence, one observes a direct superconductor-to-Fermi insulator in the zero-temperature limit. In narrow films, the associated critical-exponent products diverge along the corresponding phase boundaries with increasing magnetic field, which is a hallmark of the quantum-Griffiths singularity.

complex order parameter with an amplitude (related to the energy gap ∆ or the Cooper-pair density " ! ), and a phase #. The destruction of the superconducting state in high magnetic fields can occur through the suppression of " ! to zero. However, phase fluctuations of the order parameter can also destroy the zero-resistance state 2 . When the external field exceeds the lower-critical field in type-II superconductors, the magnetic field enters the superconductors via quantized flux lines (vortices). Sufficiently mobile vortices, as a manifestation of phase fluctuations, will generate dissipation and can drive superconductors into normal conductors.
The magnetic-field driven superconductor-to-insulator transition (SIT) in thin films at zero temperature is a well-documented quantum-phase transition [3][4][5] and has been observed in a myriad of experiments [6][7][8][9][10][11][12][13][14][15][16] . A long-standing controversy is the question whether the SIT is due to the loss of long-range coherence of # (bosonic scenario), or due to the breakdown of Cooper pairs that suppresses " ! to zero (fermionic scenario) [3][4][5] . The bosonic scenario describes the SIT as a result of quantum-phase fluctuations, in which the superconducting and the insulating states with different symmetry are separated by a single quantum-critical point 17,18 . On the superconducting side of the SIT, the Cooper pairs are mobile and the vortices are localized into a vortex lattice or glass, while on the insulating side, the vortices are mobile but the Cooper pairs are localized into isolated superconducting islands, forming a Bose-insulating state 19,20 . In the fermionic scenario, the SIT is driven by the breaking of Cooper pairs and the localization of electrons in high enough fields, forming a Fermi insulator 21-23 . Despite the majority of experiments demonstrate a quantum-fluctuation induced superconductor-to-Bose insulator quantum-phase transition 7-9,14,24,25 , a direct superconductor-to-Fermi insulator quantum-phase transition that is induced by the breakdown of Cooper pairs has also been reported 10,11,26 .
Due to the weak pinning in amorphous superconductors, the manifestation of intrinsic vortex interactions can be directly probed, and they therefore represent an optimal platform to study vortexrelated phase transitions . Depending on the bridge width w, we observe all of the above mentioned magnetic-field induced phenomena in thin amorphous superconducting films, such as superconductorto-Bose insulator quantum-phase transitions, direct superconductor-to-Fermi insulator phase transitions, and a quantum-Griffiths singularity. Our results can therefore in part explain the seeming absence of universality of these transitions.

Results and discussion
Preparation and characterization of the WSi bridges. We fabricated a series of superconducting microbridges on a single 10 × 10 mm " amorphous WSi thin film with thickness d = 4 nm 36,37 , spanning the range of bridge widths w over almost three decades from 2 µm to 1000 µm. The amorphous nature of our films is illustrated in Fig. 1a. Standard four-electrode transport measurements taken in zero magnetic field on the as-grown films and on the fabricated long bridges are shown in Fig. 1b instead of a temperature independent value in the zero-temperature limit as we observed it in the wider bridges. By carefully checking all these downturns in 3 ! (+) shown in Fig. 1

and in the Supplementary
Figs. 1 and 2, we find that superconductivity recovers above 4 # $ in the narrow bridges, i.e., the initially Bose-insulating state turns into the superconducting state again. When cooling the these bridges down from the normal state (see Supplementary Fig. 3 for the 3 ! (+) of the 10-and 20-µm wide bridges in 6 T), superconducting fluctuations appear around 6 K with ∂3 ! ∂+ ⁄ > 0 at first; then the 3 ! (+) curves reach a minimum, similar to those in the wide bridges, corresponding to a finite-temperature superconductor-to-Bose insulator phase transition. The Bose insulating state with ∂3 ! ∂+ ⁄ < 0 , however, does not persist down to the zero-temperature limit, but is terminated by re-entering into the superconducting state with ∂3 ! ∂+ ⁄ > 0.
The figures 1d-1f show the 3 ! (+) of the 50-µm, 10-µm and the 2-µm-wide bridges in the same field range as in Fig. 1c for the 1000-µm-wide bridge, demonstrating that the SIT is completely different from that in the infinite 2D systems. The complete evolution of the SIT as a function of bridge width is shown in the Supplementary Fig. 1  We first performed the scaling analysis on the 1000-µm-wide bridge, representing the infinitely large case, where the SIT boundary from 4 # $ to 4 # " is constituted by a series of + %&' plateaus ( Fig. 4). Near 4 # $ , on both sides of the transition, the sheet resistance can be rescaled as 3 ! (4, +) = has been observed in many 2D superconducting systems, and confirms the universal behaviour of the superconductor-to-Bose insulator quantum-phase transitions in large 2D superconducting systems 7-9,13,14 . Performing a similar scaling analysis for the pair-breaking critical field 4 # " at temperatures between 2.9 K to 3.4 K (Fig. 4c), we obtain the best data collapse with $% = 0.67~2 3 ⁄ (Fig. 4d). These resulting critical-exponent products and the critical behaviour, in general, are consistent with our previous results for infinitely large 2D bridges 14 .
As for the narrow bridges, the 4 # * phase boundary corresponds to the + %() (4 # * ) plateaus (Figs. 2b-2d and Supplementary Fig. 10). In Fig. 5, we show the corresponding scaling analyses for the 2-µm-wide bridge. The first + %() (4 # * ) plateau appears at 4 # * = 6.1 T, as it is shown in Fig. 5a, at temperatures between 1.18 K and 1.34 K. By utilizing the scaling analysis, the 3 ! (4) data within the critical regime are perfectly rescaled onto a single curve, as it is shown in Fig. 5b, resulting in a product of the critical exponents $% = 0.74. The largest critical point that could be investigated with our equipment is 4 # * = 7.1 T (shown in Fig. 5c, at temperatures from 0.34 K to 0.45 K). The best data collapse yields a relatively large critical-exponent product of 2.34 (Fig. 5d). To reveal the effects of the bridge dimensions on the critical behaviour, we performed corresponding scaling analyses every 0.1 T for all the bridges (details about the scaling analysis procedure at the 4 # * (+ %() ) boundary are shown in the Supplementary Fig. 6), and the resulting field dependences of $% for these bridges are summarized in Fig. 6. Different from the superconductor-to-Bose insulator phase transitions in large 2D systems, in which $% lies between 2 3 ⁄ and 4 3 ⁄ 14 , the product $% at the 4 # * (+ %() ) boundary grows dramatically with increasing magnetic field, and it is expected to diverge at the zero-temperature superconductor-to-Fermi insulator quantum-critical point at 4 # * (0) = 4 # 0 , thereby revealing the signature of a quantum-Griffiths singularity 27-33 .
Inspired by the activated scaling law for the quantum-Griffiths singularity 27 , we fitted the extracted $% for all the bridges according to $% = L • (4 # 0 − 4 # * ) ,12 (solid lines in Fig. 6; the fitting results are summarized in Table 1  as the bridge width is narrower than the Pearl length Q(0), but sill much larger than the coherence length x(0) ≈ 8 nm 37 , superconductivity is recovered from the Bose-insulating state, forming a direct superconductor-to-Fermi insulator quantum-phase transitions in the zero-temperature limit. At finite temperatures + > 0, the Bose-insulating phase shrinks with decreasing bridge width, terminating at 4 # 0 = 7.25 T, but it exists up to + # (0) at 4 # " for all bridges (see Supplementary Fig. 11). By further reducing the bridge width towards the one dimensional limit, however, we expect that the Boseinsulating state will eventually entirely disappear (Fig. 2e), so that the phase diagram is divided into only two distinct regions (with X3 ! X+ ⁄ > 0 and X3 ! X+ ⁄ < 0, respectively), as it is observed for a quantum-Griffiths singularity 27-33 . A direct superconductor-to-Fermi insulator phase transitions occurs at all temperatures up to + # (0) and beyond as long as superconducting fluctuations are still present.

Conclusion
We have experimentally demonstrated that in infinite 2D superconducting systems with intrinsically weak vortex pinning, a superconductor-to-Bose insulator quantum-phase transition occurs, which can be attributed to the condensation of vortices. The recovery of superconductivity and the appearance of a quantum-Griffiths singularity in the zero-temperature limit in narrow bridges are most probably due to size-effect induced disorder and a resulting breaking of transitional and rotational symmetry of the vortex lattice. The quantum-Griffiths singularity should therefore be universally expected in sub-2D superconducting systems with characteristic dimensions smaller than Q(0). Our experimental findings may solve part of the controversy concerning the nature of magnetic-field driven superconductor-toinsulator quantum-phase transitions in 2D superconducting systems.

Methods
The superconducting thin films adopted in our research were prepared by magnetron sputtering We at first patterned Ti/Au contacts on the as-grown films by lift-off technique. Then the micro-bridges were defined by optical lithography, followed by reactive ion-etching. The bridge widths range from 2 µm to 1000 µm. In order to exclude to any possible formation of pinning centers by high-energy electron irradiation, we here only applied optical lithography to fabricate these bridges instead of electron-beam lithography. The bridge length was 4000 µm and 3000 µm long for the 1000-and the 500-µm-wide bridges, respectively. For all other bridges, the length was 800 µm. Images of similar micro-bridges can be found in ref. 36. The resistivity measurements were done in a Physical Property Measurement System (PPMS, Quantum Design Inc.) equipped with a 3 He option. In order to make the results for the different bridge widths comparable, all measurements were performed with same low current density j = 1.25 MA/m 2 , which is around four orders of magnitude below the depairing criticalcurrent density. The bias currents were therefore ranging from 10 nA (for the 2-µm-wide bridge) to 5 µA (for the 1000 µm-wide bridge).
Weakly-pinning amorphous superconductors, such as WSi, InOx, and MoGe, are the most optimal platforms for investigating the quantum nature of superconductor-to-insulator quantum phase transitions, the details of which depend on the intrinsic interactions among vortices. In the stronglypinning peers, such as NbN, the pinning centers inside the materials prevent vortices from freely moving, making the interactions between free vortices inaccessible. Although atomically ordered superconducting crystalline films such as exfoliated NbSe2 monolayers would also be good candidates.
The generally available thin flakes usually have limited size, however, and the effective penetration depth can be orders of magnitude larger than the flake size, so that the limit w >> Q(0) is hardly accessible.

Supplementary Table 1.
Fitting results from the activated scaling law for all bridges.
Online Video 1 for a clearer visualization of the evolution of 3 ! (4, +) with the bridge width.
Online Video 2 showing the evolution of the magnetic phase diagrams as a function of bridge width.     Inset: temperature dependence of the scaling parameter t. The strong suppression of the resistivity near the normal-to-superconductor transition below the corresponding critical temperature + # (4) is due to the narrowing of the width of the superconducting bridges well below Q(0) ≈ 350 µm. dependence for fields above 4 # 0 . The most significant difference between the Bose-insulating and the Fermi-insulating states (d and e) is that the ?3 ! ?+ ⁄ in the Bose-insulating state shows a qualitatively much more pronounced divergence towards zero temperature than that in the Fermi-insulating state.