Abstract
An increasing current through a superconductor can result in a discontinuous increase in the differential resistance at the critical current. This critical current is typically associated either with breaking of Cooperpairs or with the onset of collective motion of vortices. Here we measure the current–voltage characteristics of superconducting films at low temperatures and high magnetic fields. Using heatbalance considerations we demonstrate that the current–voltage characteristics are well explained by electron overheating enhanced by the thermal decoupling of the electrons from the host phonons. By solving the heatbalance equation we are able to accurately predict the critical currents in a variety of experimental conditions. The heatbalance approach is universal and applies to diverse situations from critical currents to climate change. One disadvantage of the universality of this approach is its insensitivity to the details of the system, which limits our ability to draw conclusions regarding the initial departure from equilibrium.
Introduction
One of the central properties superconductors is their critical current (I_{c})^{1,2,3} the maximal current (I) they are able to maintain. The sudden onset of resistance (R) at I_{c} is usually associated with one of two mechanisms: depairing, which occurs when the kinetic energy of a Cooperpair exceeds its binding energy (the superconducting gap)^{4,5}, or depinning of vortices, when the Iinduced Lorentz force acting on vortices exceeds their depinning force, setting them in motion^{4,6,7}. Typically, in typeII superconductors under the application of magnetic field (B), depinning occurs at lower I rendering the depairing current a theoretical upperbound^{4,8}.
Due to the practical significance of I_{c} the bulk of the scientific effort was centered around increasing its value at finite temperatures (T’s) rather than on its fundamental, T = 0, value. In a recent publication, I_{c}’s of superconducting amorphous indium oxide films (a:InO) have been studied at low T’s and high B’s near the high critical field of superconductivity, B_{c2} (~13 T)^{9}. The authors found that I_{c} ~ ∣B − B_{c2}∣^{α}, with α ≈ 1.6 that is close to the meanfield value of 3/2 indicating, as they pointed out, that I_{c} is a result of the combined action of depairing and depinning where the increasing I initially suppresses the order parameter (by pairbreaking), helping the Lorentz force to overcome the pinning. While by using this approach they were able to suggest a resolution to the ubiquitous linear B_{c2}(T) as T → 0^{10,11}, their theory is not yet refined enough to offer a quantitative prediction to the value of I_{c} itself.
Our purpose in this article is to suggest that a different mechanism, which is Joule selfheating, is behind I_{c}, this mechanism inevitably becomes more dominant as T → 0. Selfheating occurs when the power dissipated by the measurement I exceeds the rate of heat removal from the electrons. To analyze this process we model our experiment as being comprised of four independent subsystems (Fig. 1a) that are thermally coupled via lumped thermal resistors (\(\tilde{R}\)’s): The electrons, the host a:InO phonons, the substrate phonons and the liquid helium mixture (in which our sample is immersed in our dilution refrigerator). While our system as a whole is driven out of thermal equilibrium by our measurement I, it maintains a steadystate where we assume that we can treat each subsystem as being at local equilibrium albeit at different T’s represented by T_{el}, T_{ph}, T_{sub} and T_{0} as indicated in Fig. 1a.
Our electronic subsystem is thermally linked to its phonons via \({\tilde{R}}_{{\rm{elph}}}\), mediated by electronphonon coupling^{12,13,14,15}. The a:InO phonons are, in turn, linked to the substrate’s phonons via acoustic transfer at the interface between the different solids^{16}, which transfer their heat to the helium mixture through a thermalboundary resistance at the interface known as Kapitza resistance, \({\tilde{R}}_{{\rm{K}}}\)^{17,18,19}.
Under steady state conditions the power (P) flowing across each boundary \(\tilde{R}\) is equal to the Joule heating P ≡ I ⋅ V delivered to the electronic subsystem. A finite P flowing through the \(\tilde{R}\)’s results in a Tdifference between each pair of subsystems. If one of the \(\tilde{R}\)’s is significantly larger than the others it will constitute a thermal bottleneck, impeding the cooling process, and the largest T difference will develop across it. A straightforward analysis, given in Supplementary Note 6, reveals that the thermal bottleneck is between the electrons and the phonons (\({\tilde{R}}_{{\rm{el}}{\rm{ph}}}\)) and henceforth we assume that all other subsystems are in equilibrium with each other (the functional form of Equation (1) is general and also applies to the other \(\tilde{R}\)’s illustrated in Fig. 1a, therefore we can write the terms of Equation (1) as if \({\tilde{R}}_{{\rm{elph}}}\) is the thermal bottleneck and not lose generality). The Tdifferences across \({\tilde{R}}_{{\rm{elph}}}\) is determined by a heatbalance equation:
where Ω is the sample’s volume and Γ and β are parameters characterizing \({\tilde{R}}_{{\rm{el}}{\rm{ph}}}\)^{12,14,15}.
It turns out that Equation (1) can lead to dramatic behavior. If a rise in T_{el} that results from an increase in I causes a sufficiently steep increase in R(T_{el}), which is certainly the case in our typeII superconductor (in our experiment \(R({T}_{{\rm{el}}})\approx {R}_{{\rm{0}}}{e}^{{T}_{{\rm{0}}}/{T}_{{\rm{el}}}}\)), then below a critical T_{ph} value the heatbalance equation acquires two stable solutions for T_{el}(I): a low T_{el} solution where T_{el} ≳ T_{ph} and a high T_{el} solution where T_{el} ≫ T_{ph}^{12,20}. The jump at I_{c} is simply a manifestation of the system switching discontinuously between the two stable T_{el} solutions, and the sudden increase in V at I_{c} results from V = I_{c}R(low T_{el} → high T_{el}).
We present the results of a systematic study of I_{c} in superconducting a:InO at 0.5 > T > 0.01 K and B’s 12 ≥ B ≥ 9 T (where B_{c2} ≃ 13T), for samples of various thicknesses in both perpendicular B (B_{⊥}) and inplane B (B_{∣∣}). We demonstrate that the Vjump at I_{c} results from the behavior expected from the heatbalance Equation (1). Furthermore, the value of I_{c} can be accurately determined using only measurements done at I → 0 and at I ≫ I_{c}. We also show that I_{c} is not consistent with either depairing or depinning mechanisms, nor with their combined action.
Results
I _{c} at low T and high B
In Fig. 1b, c we depict several lowT current–voltage characteristics (I–V’s) typical of our study obtained from the 280 nm sample. In Fig. 1b we plot dV/dI vs. I measured at B_{⊥} = 12 T and at T = 60mK. The measured dV/dI clearly separates to two regimes, a lowresistive (LR) state at low ∣I∣’s and a high resistive (HR) state at high ∣I∣’s. The transition between these states occurs discontinuously at two different I_{c} value; we define the measured I where the HR → LR transition occurs as \({I}_{{\rm{c}}}^{{\rm{H}}\to {\rm{L}}}\) and the measured I where the LR → HR transition occurs as \({I}_{{\rm{c}}}^{{\rm{L}}\to {\rm{H}}}\) (as marked in the Fig. 1b). Typically our measured I–V’s did not show a pronounced hysteresis as \({I}_{{\rm{c}}}^{{\rm{H}}\to {\rm{L}}}\approx {I}_{{\rm{c}}}^{{\rm{L}}\to {\rm{H}}}\) (see “Discussion” section) therefore throughout the manuscript we refer to both as I_{c}. In the inset of Fig. 1b we replot the data of the main graph on a semilogarithmic scale. Plotted this way it can be seen that in the LR state, even at low I’s, there is a small but finite R. This is because a thin film typeII superconductor in the presence of high disorder and B reaches R = 0 only at T = 0^{6,21}. In Fig. 1c we plot dV/dI vs. I on a semilogarithmic scale measured at 20mK and at 6 Bvalues between B = 9.5–12 T (B_{c2} for this sample, plotted in Supplementary Fig. 2, was ~13 T). All curves exhibit a jump of several orders of magnitudes in dV/dI at a welldefined, and Bdependent I_{c}: Increasing B towards B_{c2} results in a decrease in I_{c}. Similar B dependence of I_{c} is observed in all of our samples.
Heatbalance analysis
We next demonstrate, using the heatbalance approach (Equation (1)), that the I–V’s are well described in terms of electron selfheating^{12,13,14,15,22,23,24}. Inspecting Equation (1) we see that the only unknown variable is T_{el}, which we need to obtain independently. For that, we assume that all deviations from Ohm’s law are due to an increase in T_{el} and not from other nonlinear effects (we shall review the flaws of this assumption in the discussion). Under this assumption, we convert the raw I –V’s obtained from our 280 nm film at B_{⊥} = 12 T at several T’s plotted in Fig. 2a to effective T_{el} and plot the results in Fig. 2b^{20,23,25} (see Supplementary Note 3 for a detailed description of the heatbalance analysis).
Finally we plot, in Fig. 2c, \(P+\Gamma \Omega {T}_{{\rm{ph}}}^{\beta }\) vs. \(\Gamma \Omega {T}_{{\rm{el}}}^{\beta }\) alongside the fit to Equation (1) (dashed black line), which our data follow for more than 4 decades, and from which we extract the parameters β = 5.1 and ΓΩ = 1.48 × 10^{−5 }W ⋅ K^{−β} (the values of β for our samples at various B’s are given in Supplementary Table 1). The systematic deviations at low P’s are addressed in Supplementary Note 7.
Calculation of I _{c} from the heatbalance analysis
Encouraged by the excellent fit of our data to the heatbalance theory, we further provide a quantitative test for its validity by using it to predict the values of I_{c} for our superconductors. This is achieved by numerically solving Equation (1) to obtain the theoretical lower and upper bounds of the bistability Iinterval, \({I}_{{\rm{c}}}^{\min }\) and \({I}_{{\rm{c}}}^{\max }\), for our B and T range as demonstrated in Fig. 3.
In Fig. 3a we plot the measured dV/dI vs. I for the 280 nm thick sample at T_{ph} = 60 mK and B_{⊥} = 12 T. In Fig. 3b we plot both sides of Equation (1), the magenta curve is the righthandside (\(\Gamma \Omega ({T}_{{\rm{el}}}^{\beta }{T}_{{\rm{ph}}}^{\beta })\)) where β and ΓΩ were extracted from the heatbalance analysis (this curve is simulated). The other curves are I^{2}R(T_{el}), the lefthand side of Equation (1) (plotted for four different I values marked in Fig. 3a), where R is the measured R(T) at equilibrium (the dashed portions are a low T extrapolation of the R(T)). A valid T_{el} solution is where each of the four Jouleheating curves intersect with the magenta curve. In the inset of Fig. 3b we plot these possible T_{el} solutions as a function of the driving I.
At low I (−3 μA) Equation (1) has only one solution at T_{el} ≈ 0.06 K, close to T_{ph}. This T_{el} solution is plotted in the inset of Fig. 3b as a thick dashed line.
Increasing ∣I∣ to I = −6.2 μA brings about a dramatic development. At this I the blue curve intersects the magenta curve twice as they become tangent at T_{el} ≈ 150 mK. We identify this I value with \({I}_{{\rm{c}}}^{\min }\), the theoretical lowerbound of the Iinterval where a thermal bistability can exist. The main result of this work is that this theoretical \({I}_{{\rm{c}}}^{\min }\) matches the measured I_{c}. This is demonstrated in Fig. 3a where I = −6.2 μA is marked by a dashed blue line that coincides with the measured \({I}_{{\rm{c}}}^{{\rm{H}}\to {\rm{L}}}\).
Increasing ∣I∣ beyond \( {I}_{{\rm{c}}}^{\min }\) drives the sample into the bistable regime where there are three solutions to Equation (1). For example, at I = −17 μA there are three crossing points between the brown and magenta curves in Fig. 3b marking three different T_{el} solutions for Equation (1). The middle solution is an unstable fixedpoint and the low and high T_{el} solutions are stable. The unstable T_{el} solution is marked in the inset of Fig. 3b by a thin dashed line and the stable high T_{el} solution is marked by a continuous thick line. The black curve (I = −41 μA) corresponds to \({I}_{{\rm{c}}}^{\max }\) where the black and magenta curves intersect twice as they become tangent at T_{el} ~ 70 mK. Above \({I}_{{\rm{c}}}^{\max }\) the lowT_{el} stable solution and the unstable high T_{el} solution coincide and vanish leaving the system only with the highT_{el} stable solution.
In Fig. 3c we plot the theoretical \({J}_{{\rm{c}}}^{\min }\) which is the critical current density corresponding to \({I}_{{\rm{c}}}^{\min }\) (squares) for samples with various thicknesses together with our measured J_{c} (crosses and ∣∣’s representing B_{⊥} and B_{∣∣} respectively where we have used the measured \({I}_{{\rm{c}}}^{{\rm{H}}\to {\rm{L}}}\)). For all samples and B values there is a remarkable quantitative agreement between theory and experiment. We emphasize that the measured value of J_{c} was not used in the heatbalance analysis and so our accurate prediction of J_{c} is a good test for the validity of this theoretical framework. Note that this result shows that although the transition can theoretically occur anywhere between \({I}_{{\rm{c}}}^{\min }\) to \({I}_{{\rm{c}}}^{\max }\), in practice it occurs at \({I}_{{\rm{c}}}^{\min }\). In the discussion section and in Supplementary Note 8 we show that this is consistent with a switching wave that propagates through our superconductors^{22}.
Discussion
Similarities with insulators  The heatbalance approach we used throughout this article is a general concept that can account for thermal bistabilities in various systems such as superconductors^{12,22,26,27,28,29}, insulators^{20,25}, and even in earth’s T^{30,31}. Here its use was inspired by earlier studies of the Bdriven insulating phase of a:InO^{32}. There, the discontinuities in the I–V’s were attributed to bistable T_{el} assuming that \({\tilde{R}}_{{\rm{el}}{\rm{ph}}}\) dominates the electrons cooling rate at low T’s^{20,25,33}. In Fig. 4a and b we plot V vs. I of one of our superconducting samples alongside I vs. V obtained from the Bdriven insulating phase of a more disordered a:InO sample. The colorcoding indicates the measurement T. We draw attention to the qualitative similarity between both measurements, and to the fact that the values of the parameters β and Γ do not vary significantly between superconducting and insulating samples (see Supplementary Table 1).
The Ohmic assumption: In our analysis, we assumed that all deviations from Ohmic transport are due to heating and other mechanisms leading to nonlinearity, while present, are less effective and do not influence our main results. For example, we do not take into account intrinsic nonlinearities that are known to exist in typeII superconductors at finite T and B^{5,34,35,36}. Our analysis, therefore, fails to quantitatively account for the onset of nonlinearity at I < I_{c} limiting its range of applicability to I ≥ I_{c} and I ≪ I_{c}. In Supplementary Fig. 5, we demonstrate these deviations. A similar discrepancy was also reported in the heatbalance description of the insulating phase in a:InO^{20,25}. We note that selfheating also applies in the presence of intrinsically nonlinear effects and a complete description of our I–V’s awaits a theory that integrates both selfheating and intrinsic nonlinearities.
Other mechanisms for I_{c}: The main result of this work is that, at low T’s, I_{c} is a result of thermal bistability. While we clearly demonstrated this by accurately predicting I_{c} under various conditions, it is also important to show that the other mechanisms for I_{c} are not relevant in our experiments. We focus here on the role played by vortices at a finite B^{4,6,7} and refer the reader to Supplementary Note 2 where we show that the mechanism of depairing of Cooperpairs is unlikely.
To examine whether vortex depinning can be the mechanism causing our I_{c}’s we oriented B in the sample’s plane (B_{∣∣}) and conducted two measurements of I_{c}: one where B_{∣∣} is aligned parallel to the sourcedrain I (I_{SD}) and one where B_{∣∣} was at an angle of φ ≈ 45^{∘} from I_{SD} (φ is defined in the inset of Fig. 4c). Because the coherence length of our films ξ ~ 5 nm^{37} is smaller than the film thickness vortices penetrating the plane of the sample experience a Lorentz force \(\propto {I}_{{\rm{SD}}}\sin (\varphi )\). In Fig. 4c we plot dV/dI vs. I of the 26 nm thick sample at T = 13 mK and at B_{∣∣} = 11 T where the dashed black line and the continuous red line correspond to φ ≈ 45^{∘} and φ ≈ 0^{∘}, respectively. It is apparent that the entire dV/dI curves, and in particular I_{c}, are completely independent of φ demonstrating that I_{c} is not due to collective depinning of vortices (similar insensitivity of transport properties to φ was reported in high T_{c} superconductors^{38,39,40,41}. While these results were still interpreted in terms of vortex motion, the different theoretical models rely heavily on the large anisotropy in high T_{c}’s. The contrast between the φ dependence of high T_{c}’s and an amorphous MoGe alloy, which is a conventional type II superconductor, is demonstrated in ref. ^{41}).
Our I_{c} results are not different from those recently presented in^{9}. These authors offered an interpretation very different from ours. They claim that I_{c} is a result of a combination of depairing and depinning. Their main experimental evidence are that J_{c} ∝ ∣B − B_{c2}∣^{α} with α ~ 1.6 which is similar to the meanfield depairing value of 3/2 and that J_{c} is comparable to the depairing J_{c} (smaller by a factor of 4 according to their calculation). We do not intend to counter their claims. We do think, on the other hand, that our analysis better describes the data for three reasons: (1.) contrary to their results, the value of α is actually nonuniversal (see Supplementary Fig. 3). (2.) the depairing J_{c} is actually 10–15 times larger than their measured J_{c} and 10–400 times greater than in our measurements (see Supplementary Note 2). (3.) unlike their model the heatbalance analysis provides a good quantitative prediction to J_{c}. In the supplementary material of ref. ^{9} Sacépé et al. discuss the possibility of a thermal bistability and provide several arguments against this interpretation. In Supplementary Note 7, we respond to these arguments.
Lack of hysteresis due to a propagating switching current: The heatbalance analysis can only determine the bounds of the Iinterval, \({I}_{{\rm{c}}}^{\min }\) and \({I}_{{\rm{c}}}^{\max }\), where Equation (1) has two stable solutions^{20}. The actual transition can occur anywhere within the interval \({I}_{{\rm{c}}}^{\min }\le {I}_{{\rm{c}}}^{{\rm{H}}\to {\rm{L}}}\le {I}_{{\rm{c}}}^{{\rm{L}}\to {\rm{H}}}\le {I}_{{\rm{c}}}^{\max }\), depending on a dynamic interplay between the electrons, the phonons and the disorder where \({I}_{{\rm{c}}}^{\max }\) can be orders of magnitude greater than \({I}_{{\rm{c}}}^{\min }\) (see Supplementary Table 3). However, the limited hysteresis and the results displayed in Fig. 3c indicate that both \({I}_{{\rm{c}}}^{{\rm{H}}\to {\rm{L}}}\) and \({I}_{{\rm{c}}}^{{\rm{L}}\to {\rm{H}}}\) occur near \({I}_{{\rm{c}}}^{\min }\).
A common mechanism driving the transition between LR and HR states in bistable conductors (such as our samples at \({I}_{{\rm{c}}}^{\min }<I<{I}_{{\rm{c}}}^{\max }\)) is a propagating switching wave^{22}. It turns out that such a switching wave is consistent with our observation that \({I}_{{\rm{c}}}^{{\rm{H}}\to {\rm{L}}}\approx {I}_{{\rm{c}}}^{{\rm{L}}\to {\rm{H}}}\approx {I}_{{\rm{c}}}^{\min }\). In this scenario, we consider a bistable material in the LR state (similar arguments apply to the HR state) where, due to disorder, there is local nucleation of high T_{el} domains embedded in the low T_{el} medium. In ref. ^{22} it is shown that above some minimum propagating current (I_{p}) these hot T_{el} domains expand and that for I < I_{p} these domains shrink. I_{p} is extracted using the equal area condition \(S({T}_{{\rm{3}}},I)\equiv \mathop{\int}\nolimits_{{T}_{{\rm{1}}}}^{{T}_{{\rm{3}}}}({P}_{{\rm{HB}}}{P}_{{\rm{Joule}}})d{T}_{{\rm{el}}}=0\) where \({P}_{{\rm{HB}}}\equiv \Gamma \Omega ({T}_{{\rm{el}}}^{\beta }{T}_{{\rm{ph}}}^{\beta })\), P_{Joule} = I^{2}R(T_{el}) and T_{1} and T_{3} are the low and high T_{el} solutions of Equation (1) respectively. In Fig. 4d we plot S(T_{el}, I) for several I values and it can be seen that I_{p} = 6.58 μA satisfies S(T_{3}, I_{p}) = 0. Note hat \({I}_{{\rm{p}}} \sim {I}_{{\rm{c}}}^{\min }\ll {I}_{{\rm{c}}}^{\max }\), a result that accounts for our observation that \({I}_{{\rm{c}}}^{{\rm{H}}\to {\rm{L}}}\approx {I}_{{\rm{c}}}^{{\rm{L}}\to {\rm{H}}}\approx {I}_{{\rm{c}}}^{\min }\). In Supplementary Note 8, we provide a detailed analysis of how we extract I_{p}.
Effects of the contacts: In the heatbalance analysis presented above we assumed that heating is a result of the nonvanishing R of our typeII superconductor at finite B and T. A different possibles source of heating that can potentially lead to the destruction of superconductivity at high I is dissipation that originates at the contacts due to their finite R. To reduce contact R we prepared our samples with a large overlap area of 333μm on 50 μm between the source and drain Ti/au contacts and the a:InO (see “Methods”). We find this possibility unlikely for two reasons: First, in our heatbalance analysis we obtain I_{c} relying strictly on very low bias 4terminal resistance measurements that are independent of contact resistances and were measured at ∣I∣ ≪ I_{c} where heating is irrelevant. The accuracy of our analysis, as displayed in Fig. 3c, and its correspondence with the 4terminal R makes it unlikely that the effect we present is caused by heating at the contacts. We emphasize that because the electron thermal length in our samples at 10mK is L_{T} ≈ 0.2 μm (approximated using a free electron gas approximation and a typical charge density of n = 5 × 10^{20} cm^{−3}), while our samples are 1mm long, the onset of the finite 4terminal R cannot be due to electron heating from the contacts. Second, because the contact R is not typically strongly B dependent, one would equally not expect I_{c} to be strongly B dependent, which it clearly is, see Fig. 1c. Due to these arguments we find it more likely that the heating we report above is in the bulk of our finite R typeII superconductor.
In summary, we have showed that the I_{c}’s of superconducting a:InO films measured at low T’s and high B’s are well described by thermal bistabilities originating from a model of heatbalance between electrons and phonons (Equation (1)). Using this model we predicted quantitatively I_{c} for samples of different thicknesses for both B_{⊥} and B_{∣∣}.
Methods
Sample fabrication
a:InO was deposited in an Oxygen rich environment of 3 × 10^{−5} Torr by egun evaporation of high purity In_{2}O_{3} pellets onto a Si/SiO_{2} substrate (a boron doped silicon wafer with ρ < 5 mΩ ⋅ cm with a 580 nm thick oxide layer). The sample thickness was measured in situ during evaporation using a crystal monitor and verified later by atomic force microscopy. The contacts of the samples are Ti/Au, prepared via optical lithography prior to the In_{2}O_{3} evaporation.
In Fig. 5a we present an illustration of the measured samples. The samples are Hallbar shaped where the distance between source and drain contacts is 1 mm and the width of each sample is 1/3 mm. Adjacent V contacts are located 0.8 squares apart (267 μm). Our study was performed using four such a:InO films of thicknesses 26, 57, 100, and 280 nm. Each sample was thermally annealed post deposition to a roomT resistivity (ρ) of 4 ± 0.2 mΩ ⋅ cm, which places them in the relatively lowdisorder range of a:InO.
Measurement setup
The samples were measured in an Oxford instruments Kelvinox dilution refrigerator with a base T of 10 mK, equipped with a zaxis magnet. In order to apply B’s in both perpendicular and inplane orientations we mounted our samples on a probe with a rotating head. While measuring, all lines were filtered using roomT RC filters with a cutoff frequency of 200 KHz.
The transport method we use to measure the zero bias R (defined as \({{\rm{lim}}}_{I\to 0}\frac{V}{I}\)) is a 4terminal method. In this measurement, the sample is probed using an input ac I (low frequency of ~ 10Hz) and we measure the resulting ac V drop between a pair of contacts along the I path. The 4terminal configuration is described in Fig. 5b where during a zero bias R measurement we fix \({I}_{{\rm{dc}}}^{{\rm{in}}}=0\) and set a different I_{ac} for samples of different thicknesses while maintaining a current density of J ≈ 0.1 A ⋅ cm^{−2}. To measure a nonOhmic response we use the same configuration as in Fig. 5b and measure how the differential resistance \(\frac{dV}{dI}\equiv \frac{d{V}_{{\rm{ac}}}}{d{I}_{{\rm{ac}}}}\) varies as a function of \(I\equiv {I}_{{\rm{dc}}}^{{\rm{in}}}\).
Data availability
The data that support the findings of this study are available in Mendeley with the identifier https://doi.org/10.17632/24kvcdtkjn.1. The source data underlying Figs. 1b, c, 2a–c, 3a–c, 4a–d, and Supplementary Figs. 1a–e, 2, 3, 4a–g, 5a, b, 6b–f, 7a, b, 8a–e, and 9 a, b are provided as a Source data file.
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Acknowledgements
We are grateful to K. Michaeli, M.V Feigel’man, and B. Sacépé for fruitful discussions. This research was supported by The Israel Science Foundation (ISF Grant no. 556/17), the United States  Israel Binational Science Foundation (BSF Grant no. 2012210) and the Leona M. and Harry B. Helmsley Charitable Trust.
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A.D. and T.L. prepared the samples. A.D., T.L., I.T., and F.G. performed the experiments. A.D. and D.S. carried out the analysis, interpretation of the results, and wrote the paper with the input of all coauthors. D.S. supervised the project.
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Doron, A., Levinson, T., Gorniaczyk, F. et al. The critical current of disordered superconductors near 0 K. Nat Commun 11, 2667 (2020). https://doi.org/10.1038/s41467020164628
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DOI: https://doi.org/10.1038/s41467020164628
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