Strong suppression of the resistivity near the transition to superconductivity in narrow micro-bridges in external magnetic fields

Xiaofu Zhang, Adriana E. Lita, Konstantin Smirnov, HuanLong Liu, Dong Zhu, Varun B. Verma, Sae Woo Nam, and Andreas Schilling Department of Physics, University of Zürich, Winterthurerstrasse 190, 8057 Zürich, Switzerland National Institute of Standards and Technology, 325 Broadway, Boulder CO 80305, USA Moscow State Pedagogical University, Malaya Pirogovskaya str. 22, Moscow 127055, Russia National Research University Higher School of Economics, Myasnitskaya str. 20, Moscow 101000, Russia

It is generally accepted that the superconductivity in superconducting bridges can be suppressed by gradually reducing their dimensions. While sufficiently thick and wide bridges reflect the properties of the bulk material, wide strips with a reduced thickness ≲ (where is the Ginzburg-Landau coherence length) can be viewed as quasi two-dimensional [1]. Their properties are then strongly influenced by the thickness d, with a certain reduction of the transition temperature to a zero-resistance state [2][3][4][5]. Upon further narrowing a bridge down towards to the one-dimensional (1D) limit < , the critical temperature decreases exponentially with the inverse of the cross section of the bridges [6], leading to a transition to insulating state in the 1D limit [6][7][8][9][10].
Placing a type-II superconducting strip into an external magnetic field, magnetic-field-induced vortices can exist as long as > 4.4 [11]. In very thin films, vortices can interact in a different way than in their bulk peers, namely via their stray fields in the surrounding space. The characteristic length scale for this interaction is given by the Pearl length = 2 2 ⁄ , which can be substantially larger than the London penetration depth [12]. In wide bridges, where the bridge width w is larger than all length scales that are relevant for superconductivity, the vortex-vortex interactions are long-range logarithmic as a function of distance r for r < , and they determine the superconducting properties in clean enough samples. It has been suggested that in narrow bridges w < , the vortex-vortex interaction becomes short-range exponential for vortex-vortex separations r > w/π [13], thereby excluding a Berezinskii-Kosterlitz-Thouless (BKT) transition [14,15]. While in this low-field limit, surface barriers also play an important role [16], they are negligible in the high-field limit B ~ Bc2. In this letter we study the transition to superconductivity for amorphous superconducting films in this high-field limit, as a function of the bridge width w for w < and w > .
We have fabricated micro-bridges based on four amorphous WSi and MoSi films of various thickness, with ten different bridge widths w ranging from 2 μm to 1000 μm (fabrication details are provided in the Supplemental Material [17]), and performed detailed transport measurements on them. Figure 1 shows the respective resistive transitions to the superconducting states in magnetic fields up to B = 5 T perpendicular to the films. To facilitate a comparison, the original resistance data have been converted to the respective sheet resistances . In order to eliminate any minor remaining variations in in the normal state due to uncertainties in the geometric dimensions, we normalized the data to the normal-state sheet-resistance values ( ) of the 100-µm-wide bridges at T = 7 K (T = 10 K for the 6.2-nm-MoSi film) and B = 0 T.
The bridges prepared from the 100-nm-thick WSi film have a zero-field critical temperature (0) ≈ 4.95 K, which is close to the maximum for amorphous WSi [18][19][20][21], thereby guaranteeing the high quality of our films. The corresponding critical temperature of the 4-nm-thick WSi film is reduced to (0) ≈ 3.42 K, in agreement with Refs. [18][19][20][21]. The 6.2-nm-and 4.5-nm-thin MoSi films show (0) ≈ 6.85 K and 5.15 K, respectively, which are the highest reported values for MoSi films in this thickness range to the best of our knowledge [22]. The material parameters relevant for superconductivity for these four films are tabulated in section S2 in the Supplemental Material [17]. In zero magnetic field (B = 0 T), all bridges made from a particular film show the same critical temperature and temperature dependence of ( ) because the respective coherence lengths are more than three orders of magnitude smaller than the width of the narrowest 2-μm-wide bridge [17,23].
With increasing magnetic field, the sheet resistance curves ( , ) are significantly broadened with a resistive "tail", and the transitions are shifted towards lower temperatures along with the reduction of the respective critical temperatures ( ). The ( , ) data of the bridges made from the 100-nm-thick WSi film show a shoulder-like drop with decreasing temperature before zero resistance is reached, with a sharp peak in the corresponding derivatives d /d (see Fig. 2). This feature is particularly pronounced in the wide bridges, but vanishes for → 2 and when the bridge width is far smaller than the Pearl length. It is reminiscent of corresponding drops in ( , ) that have been attributed to a first-order solidification of a vortex fluid to a vortex lattice in high-temperature superconductors [23]. While this observation is not the central subject of this letter, its occurrence is a strong indication for the high quality of our films and supports the notion that bulk vortex pinning in amorphous superconducting films is weak enough to allow for the occurrence of this transition [24].
All our data show a further unexpected striking phenomenon: upon lowering the temperature, the ( , ) data of the bridges for a particular film and magnetic field separate in such a way that the resistivity in narrow bridges is significantly suppressed near the transition, thereby leading to a narrowing of the transition to the zero-resistance state ( Fig. 1). For each of the films, this separation occurs at a well-defined, field-dependent temperature * ( ) [see Fig. 3(a) as an example]. It coincides with the temperature where the derivatives / for a given film and magnetic field for the different bridge widths w show a sharp maximum [ Fig. 3(b)], indicating that there is a change in the dissipative mechanism for electric-current flow. We shall see below that a nonlinear current-voltage characteristic also sets in around * ( ). As this temperature is independent of the bridge width, we may interpret it as an intrinsic temperature * ( ) of the film where the electric currents are immune to geometric effects or any vortex-pinning mechanisms.
For an interpretation of the observed change in the characteristics of ( , ) we first state that the reduction in ( , ) cannot be explained by the presence of conventional surface barriers [16]. It is known that such barriers inhibit dissipative vortex flow transverse to the current and therefore result in a reduction of the resistance. The maximum magnetic field below which surface barriers can play a role is given by the superheating field , which is in the limit k >> 1 given by ≈ 0 (4 ⁄ ) [25]. It amounts to ≈ 73 mT at most in our case [17] and is probably much smaller close to the critical temperature, where both and diverge. As ≫ in our investigations, with no signs of any weakening of the effect in the limit → 2 , we definitely conclude that conventional surface barriers are not responsible for the reduction of in ( , ).
Most interestingly, a qualitatively similar sharpening of the resistivity in ~ 300-µm-wide strips of  [26].
To quantify the observed reductions of The corresponding excess conductivity − 0 scales as 1⁄ for a given film and magnetic field B, and vanishes for an infinite film, with = 0 . As long as the spatial variation of ( ) from = to = 0 occurs over a short enough length scale ≪ , this result is insensitive to how exactly ( ) varies from = 0 to > . Any deviations from a 1⁄ -scaling for small bridge widths would indicate that w becomes comparable to s, and the measured conductivity would eventually saturate at > 0 for even smaller values of w. As we do not see any deviation from this scaling even for the narrowest bridges, the length scale s within this model (which may depend on magnetic field) must be smaller than 2 µm, both in the thick and in the thin films.
The fact that there exists a separation of the ( , ) data even in large magnetic fields calls for a discussion of possible vortex-pinning effects that are responsible for the reduction of the resistance.

The observation of a non-linear I-V in the mixed state of superconductors at low currents is in most
cases associated with vortex pinning or, more generally, with the presence of potential barriers inhibiting vortex motion [28]. information about the extension s of such channels and the microscopic details of the respective current-transport mechanism are yet to be clarified, however.
We note that both the vortex structure and the vortex interactions for w < may be entirely different from those in infinite films [13,29]. At present, it has, to our knowledge, not yet been considered how such arguments can be transferred to the case of a dense lattice of Pearl vortices with < ≪ and → 2 , nor what the possible consequences on measurable quantities might be.
However, it is conceivable that a reduced vortex interaction among Pearl vortices in narrow enough bridges [13] makes them much more susceptible to pinning, or that peculiarities near the edges of a bridge [29] can play an important role.
Finally we argue that the features reported here can only be observed in films of weakly-pinning superconductors. In Fig. 5, we show corresponding ( , ) data taken on a thin film (d ≈ 5 nm) of NbN, which is known to be a strongly-pinning superconductor [30]. It is obvious that the field-induced large reduction in resistivity as we measured it in the WSi and MoSi films is absent. We believe that if bulk pinning is strong enough, any additional, comparably weak increase of the electrical conductivity at the bridge edges is dominated by bulk pinning and therefore becomes unobservable. In

S1 fabrication of superconducting micro-bridges
The superconducting thin films adopted in our research were prepared by magnetron sputtering We firstly patterned the 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, as it is shown in Fig. S1 for the 4.5-nm-MoSi film. The bridges are electrically connected in a four-wire configuration by wire bonding (white spots on top of the gold contacts, see Fig. S1) for the transport measurement (Physical properties measurement system, Quantum Design Inc.).

S2 Material parameters of the investigated microbridges
Zero-field critical temperatures (0) of the investigated films. The Ginzburg-Landau coherence lengths (0), London penetration depths (0), Pearl lengths (0), superheating fields , and upper-critical fields 2 (0) are all extrapolated to T = 0. While was derived from an estimate of 2 based on resistance measurements, was obtained by the procedure described in Ref. [19].

S3 The current-voltage characteristics near the transition to superconductivity
The I-V curves of two 4.5-thick MoSi films (w = 1000 µm and 2 µm,