The critical current of disordered superconductors near 0 K

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 Cooper-pairs 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 heat-balance 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 heat-balance equation we are able to accurately predict the critical currents in a variety of experimental conditions. The heat-balance 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.

The authors report low temperature critical current measurements in high resistance a:InO films. The main result of the experiments is that the critical current behavior and actual critical current values can be determined via a heat-balance model that has previously been used to explain breakdown in high resistance non-superconducting films. Overall, I find the authors analysis somewhat vague and essentially identical to what they have previously published. Moreover, in my view, there is a serious inconsistency in their interpretation of the measured critical current traces. I understand that once a sufficient amount of power is delivered to the electron gas, the gas temperature can suddenly jump to a very high value in a very high resistance non-superconducting system. This bistable behavior is predicted by Equation 1. But the authors fail to explain how power is delivered to the superconducting electron gas in the first place since P=IV and V must be zero in the zero resistance state. Indeed, it is more reasonable to assume that once the superconding state is compromised via pair-breaking, or vortex motion, or more likely heating at the contacts, then power begins to be delivered to the electron gas and the system quickly "runs away" to the normal state. So, in my view the heat-balance mechanism is not the source of the critical current behavior but instead a consequence of the breakdown of the zero resistance phase via some other mechanism. Some other comments are: 1. What role does the applied field play? Is the critical current mechanism the same in zero field for this heat-balance model?
2. More details need to be provided on how the measurements were done. If the critical currents where measured via dc currents then contact heating is almost certainly a concern. One usually uses a pulsed technique to circumvent Joule heating at the contacts. 3. The critical field behavior at low currents needs to be shown. I do not recommend publication.
Reviewer #3: In the manuscript, the authors presented the results of experimental studies of the critical current ($I_c$) of the a:InO superconducting film at low temperatures in magnetic fields close to the upper critical field. The current $I_c$ was defined as the current at which the current-voltage characteristic has a voltage jump. When interpreting the experiments, the authors show that the value of $I_c$ can be obtained from the heat balance equation (1). They believe that the voltage jump is of a thermal nature and occurs if the balance condition between heat generation and heat removal is violated.
Both the experiment results of the authors and the interpretation of the experiment results are of interest. It is especially important that the heat balance equation describes the experiment if the exponent $\beta$= 5.1, which is very close to the theoretical value $\beta$= 5 (see Refs.14, 15). Therefore, in my opinion, the article by A. Doron et al. entitled "The critical current of disordered superconductors near T=0" can be published in Nature Communications. At the same time, I have two comments, the discussion of which could clarify the results of the article. 1) Equation (1) describes not only two stable states of a current-carrying superconducting film, but also two current values at which a voltage jump occurs (with increasing and decreasing current). The authors report that they observe "limited hysteresis" of critical currents. I believe that it is important to say more specifically which part of the current-voltage characteristic is described in terms of electron self-heating and where the discrepancy between theory and experiment begins.
2) The authors suggest that limited hysteresis of critical currents can be caused by the occurrence of hot spots. Indeed, regions with weakened superconducting properties may exist in inhomogeneous superconductors. When sufficiently large currents flow in these areas, the formation of resistive domains is possible. The theory of resistive domains is well known (see Ref. 12 and the references therein). It seems to me that it would be interesting to compare the results of the authors with the theory of resistive domains.
We thank the Reviewers for their thoughtful review and insightful comments regarding our manuscript. We would like to point out that both reviewers 1 and 3 support the publication of our findings. Reviewer 1 states "I find the qualitative arguments of the authors persuasive and the method of extracting the electron temperature new and interesting. The jumps shown in Fig. 1 are hardly consistent with the conventional vortex depinning, but rather indicative of switching between weakly and highly dissipative resistive state which can result from an electron overheating bistability. In my opinion the results are novel and can be of interest to a broader cond.mat. community" and reviewer 3 wrote "Both the experiment results of the authors and the interpretation of the experiment results are of interest". Both reviewers recommended the publication of our manuscript after we address several points that they raised. We have addressed all of the points raised by the reviewers. One suggestion both reviewers 1 and 3 raised, regarding treating the lack of hysteresis using the theory of resistive domains, turned out to be consistent with our results, accounting for the lack of hysteresis. We thank both reviewers for this useful suggestion that improved our manuscript.
On the other hand, Reviewer 2 was not convinced by our results and raised several issues mostly regarding the source of dissipation in our system. We have addressed all of the issues the reviewer raised and made several clarifications in the manuscript.
In view of what we regard as strong support to our work by both reviewers 1 and 3, and our detailed response to every point all three reviewers made (listed below) we request that our manuscript will be published with minimum delay.
Below we begin by presenting the most significant change made following the suggestion that was common to both reviewers 1 and 3, then we reply to each comment made by each reviewer and we end with a summary of the changes made in the new version of the manuscript.
Response to a common comment of reviewers 1 and 3: Both reviewers 1 and 3 had a similar suggestion that appears to be very relevant. Reviewer 1 wrote: "The big resistive jumps and the lack of hysteresis may be more consistent with a switching wave between the cold and hot states propagating at I=Ip This is a very interesting point that is consistent with our lack of hysteresis. We have considered this possibility initially and wrongfully chose to dismiss it, because we naively expected to be more or less the average between the lower and upper theoretical bounds of the bi-stability interval (found from the graphical solution of the heat-balance equation), which in the new version of the manuscript we define as and . As can be seen in the The central part of our analysis is detailed in Figure 2a-e (displayed in the next page) and we also changed section D of the discussion section in the main-text and added to the supplemental material a new section, Sec. S8, titled "Lack of Hysteresis -Propagating switching waves".
In chapter 3 of Ref. 12 the authors discuss the propagation of a switching wave in bi-stable conductors. They write that "in most cases the transitions between these states (these phases)" [the cold and hot phases] "are initiated by local perturbations, which result in the nucleation of a new phase, which then propagates to cover the whole specimen". They proceed to assume that in the sample there are domains at a of 1 and of 3 > 1 and calculate the velocity of the domain wall as a function of . They show that for > the domain with 3 will expand and that it will collapse at < where is found from the  = ΓΩ ( − ℎ ) (red) and = 2 ( ) vs plotted on a log-log scale for = 8 A. We mark the two stable solutions as 1 and 3 and the unstable solution as 2 . The equal area rule is satisfied if the two shaded areas in the figure are equal. Changing will move the green curve as a whole without changing its shape therefore increasing will decrease the shaded area between 1 and 2 and increase the area between 2 and 3 . From the figure it appears that the area between 1 and 2 is greater than the area between 2 and 3 , this is actually an artifact of plotting this figure on a log-log scale. b. Here we plot the same data as in a on a linear scale. It can be seen that the shaded area between 2 and 3 is actually much greater than the area between 1 and 2 therefore that meets the equal area law should be smaller than 8 A. Note that here ≈ 6.2 A and ≈ 41 A therefore we see that ∼ which can account for the lack of hysteresis. c.
, which is the integrated area between the curves from 1 up to some vs. . plotted for different values of between 6.4 A and 6.7 A. It can be seen that ( 3 , 6.58 ) = 0 therefore = 6.58 A.
1. The reviewer writes: "The authors should avoid the confusing jargon of "trapping" and "escaping" currents but clearly explain instead the meaning of Ic and I2 using Fig. S4 in the main text."

Answer:
We have made the appropriate changes and defined the 's according the states direction of the transition between the low resistive (LR) and high resistive (HR) states. We define of the transition from the LR to the HR state as → and the transition from the HR to the LR state as → .
2. The reviewer writes: "The critical current Ic was not clearly defined: it was mentioned in a passing comment that Ic is a "trapping" current but with no explanation of what it is. Fig 3 shows  Answer: We agree with the reviewer's comment. In the revised version we added a subfigure to Fig 1 (Fig 1b) where we plot the same data that was displayed in Fig S4 of the previous SM (which is Fig S7 in the revised version) on both linear and logarithmic scales. In this subfigure we mark the low resistive (LR) and high resistive (HR) states, and we define the transitions from the HR to the LR state as → (in the previous version we referred to this transition as the "trapping" current) and from the LR to the HR state as → (previously referred to as "escaping" current). In the main text describing the figure we clarified that because in our measurements the hysteresis is not pronounced and → ≈ → (we further discuss the hysteresis in your next comment), throughout the manuscript, wherever there is no need to distinguish between → and → we refer to both of them as . In addition, in the text describing Fig 3 we added an explanation that the experimental used in the figure is → and that the plotted theoretical is , which is the lowest where the heat-balance equation has two stable solutions.
3. The reviewer writes: "The lack of hysteresis seems inconsistent with the definition of Ic as a current at which a highly resistive state first appears. Hysteresis can occur at Ic <I < I2 where I2 is the current at which the low resistance state disappears." …"The scenario of "stochastic switching" between Ic and I2 mentioned in sec D of the Discussion does not really explain the lack of hysteresis and why the resistive transition should occur at Ic but not at higher currents. " Answer: We agree that the theoretical approach we follow does not prohibit hysteresis. The exact values of the measured critical currents are not predicted by the heat-balance model, from which we can only extract an interval of currents, ∈ [ , ], where the heat balance equation has two stable solutions (if < min the system can only be in the lowresistive state and if > it can only be in the high-resistive state). The model sets the following inequality: ≤ → ≤ → ≤ . What we measure in practice is that ≈ → ≈ → which means that not only there is almost no hysteresis but both transitions occur "prematurely", namely near the low end of the bi-stable current interval [ , ]. In the added analysis suggested by the reviewer (see above) we see that an that satisfies a non-equilibrium variant of the equal area law is also very similar to .
In the previous submission we wrote that the transition can occur stochastically anywhere within this interval, by "stochastically" we meant that a priori the measured is not determined by the theory, only the boundaries on are determined. We believe that our terminology might have been misleading and we have avoided using the term stochastic in the new version. We also elaborated more on the relation between the theoretically predicted , and the measured → , → in order to help the reader understand.
4. The reviewer writes: "The authors mentioned that hysteresis is weak so the difference between Ic and I2 is not that significant, but Fig S4 suggests that Ic = 6.2 μA but I2 is certainly larger than 17. μA so the difference between Ic and I2 is significant."

Answer:
We interpret the reviewer's definition of I2 as → , which is a measured quantity. If this is indeed what the reviewer meant, we think that the statement is inaccurate and we would like to explain Fig  2 )) and is a parameter we vary while solving for . A solution of the heat balance equation at a given is where the curve corresponding to intersects with the red curve (i.e. the two sides of the equations are equal), we track these crossing points while varying the parameter .
Starting from the purple curve which corresponds to = −3 we see that the red and purple curves intersect once, at ∼ ℎ i.e. ( = −3 , ℎ = 60 ) ∼ 60 . As the figure is plotted on a log scale, while increasing | | further, 2 maintains the same shape and only moves vertically as a whole. At = −6.2 (blue line) the two sides of the equation intersect twice, once as before at ∼ ℎ and once at ∼ 150 where the two curves are tangent. This value of is the lowest where there are two stable solutions for the heatbalance equation, i.e. = 6.2 . In practice we see (for example in light blue in Fig S7a) that this theoretically predicted is very similar to the measured quantities → and → (which we now understand as ≈ ). We emphasize again, → ≈ → ≈ → is not a priori set by the theory, the fact that it holds suggests that for some reason (maybe due to the equal area rule) our system "prefers" to be in the HR state whenever such a state exists. The green curve marks at = −17 . It can be seen that there are three crossing points between the green and red curves corresponding to three solutions to the heat-balance equation (the middle solution is unstable). As there are three solutions it means that = −17 is still in the bi-stability interval ≤ | − 17| ≤ therefore theoretically the system could still "choose" to be in the LR state with ∼ ℎ or at the HR state with (−17 ) ∼ 350 . We emphasize that the crossing point at (−17 ) ∼ 350 does not mean that I2, which we interpret as the measured → , is 17 , it only means that 17 < . In practice I2≡ → is ∼ 6.2 . The black curve marks at = = −41 . It can be seen that there are only two crossing points between the black and red curves where for > the low solution does not exist. . It can be seen that at this value the low black and red curves are tangent at the low solution and if we increase | | beyond it the low solution will not exist anymore leaving only the high solution. Answer: We agree with the reviewer and in fact the possibility that in high superconductors due to electron heating might be much smaller than has been previously studied. One important manifestation of heating effects in high superconductors is "thermal runaway" which is crucial in application such as superconducting magnets [1] [2] where the substance is heated as a whole relative to the coolant. This problem is typically solved by cladding such materials with a metal like copper that has good thermal conductivity. Heating due to electron-phonon de-coupling was studied in YBCO [3] [4] [5] where Ref. [3] concludes that near is due to a flux-flow instability and at low 's it is due to what they refer to as the "hot-spots effect" which is Joule-heating (the term is taken from Ref. [6]).

The reviewer writes: "The field dependence of the depairing current density near Hc2 was calculated long ago by Boyd, Phys Rev. 145, 255 (1966). This work should be cited"
Answer: Thank you for the correction, we now cite this work appropriately.

The reviewer writes: "Are those a:InO films clean or dirty, that is, whether the m.f.p larger of smaller than xi? What is Tc and how broad the zero-field resistive transition is. A bit more info about the superconducting parameters would be useful"
Answer: Relative to typical highly disordered a:InO films the films studied here are considered to be of low disorder level. Having said that they are still highly disordered superconductors with a m.f.p of less than 1nm while ∼ 5nm (using the Drude formula: our samples have = 4 Ω ⋅ and assuming a charge density typical for our a:InO films of = 5 ⋅ 10 20 −3 we get a mean free path of ~0.5nm where calculated from 2 ∼ 12T is ~5 ). Unfortunately we have measured only for one of the samples (the 26nm thick sample) where =2.4K and the width of the transition (between 90% and 10% of the normal state resistance) is Δ ≈ 0.35K. The reason we have measured only for one sample is technical, as we are measuring in a dilution refrigerator it takes a special effort to reach ′ of above 1.5K. We assume that of the other films is similar. We have added all these values to Sec. S1 of the supplemental material.

The reviewer writes: "The wavelength of thermal phonons at 20 mK is of the order of a few μm is much larger than the thicknesses of the a:InO films, so both the Kapitza conductance and the parameters beta and gamma should strongly depend on the film thickness. Yet the analysis in SI
gives beta varying non systematically from 5 to 10 (see

in my view the heat-balance mechanism is not the source of the critical current behavior but instead a consequence of the breakdown of the zero resistance phase via some other mechanism"
Answer: This is indeed a good point and in the initial submission we did not explain how power is delivered to the superconducting electrons in the first place. Following this comment, we added a short explanation of this issue in the main-text. The point that was missing to the reviewer is that highly disordered, thin film type-II superconductors in the presence of a magnetic field above 1 ( 1 is practically zero for a:InO) are expected to have zero resistance only at = 0 [7] [8] therefore, as we are measuring at a finite , there is no true zero resistance state. For example, In Fig. 1b of the new submission we plot vs.
of our sample at = 12T and = 60mK where the main-frame is on a linear scale and it seems that the LR state is a zero resistance state while in the inset we plot the same data on a logarithmic scale showing that there is some small but finite residual resistance (of ~0.1Ω) even at low ′ which are orders of magnitude below . We emphasize that this measurements is a four-terminal measurement were the source and drain leads are 1/3 of a mm away from the voltage measuring leads therefore the measured residual resistance at low currents cannot be due to the contacts. This residual resistance transfers power to the superconducting electron gas.

The reviewer writes: "What role does the applied field play? Is the critical current mechanism the same in zero field for this heat-balance model?"
Answer: This is a good question, unfortunately we are not sure what role the field plays. At zero field we are unable to characterize due to technical limitations, in order to measure a response we needed to use 's that are of several mA's, the problem with that is that our refrigerator is wired with resistive constantan wires with a resistance of ~ 50Ω inside the mixing chamber therefore at 1 the dissipation from the constantan wires in our mixing chamber is comparable to the cooling power of the refrigerator and the mixing chamber heats up as a whole.
3. The reviewer writes: "More details need to be provided on how the measurements were done. If the critical currents where measured via dc currents then contact heating is almost certainly a concern. One usually uses a pulsed technique to circumvent Joule heating at the contacts" Answer: First, following the request for additional details on how the measurements were done we have added to the supplemental material an "Experimental setup" section (Sec. S2) where we describe our transport setup. Regarding the use of a pulsed technique, it is a good and interesting idea to compare our results, acquired by combining a constant dc-with a low frequency ac-(which is a standard method, for example see Refs. [9] [4] [10]) with results using a pulsed technique. At the time being our experimental setup is no suited for such measurements but we plan to make the appropriate adjustments. We would also like to note that pulsed methods have several shortcomings [11]. Regarding the heating occurring at the contacts, although we can never completely rule out the possibility that the nucleation of a hot domain occurs at the contacts, we find it rather unlikely for three reasons: i. If one would expect the measured 's to be purely a result of heating at the contacts, we would not expect to have a dependence. As plotted in Fig. 1c is strongly dependent. If the reviewer's claim is that the role of the contact resistance is to cause a local nucleation of a hot domain that eventually expands to the whole sample then although we cannot rule out this possibility, we think that this mechanism is not necessary here as our highly disordered superconductor has a small residual resistance in the LR state (as we wrote above) which already acts as a source for dissipation. ii. The source and drain contacts of our sample are rather large and they have an overlap area of 333 m on 50 m with the a:InO which is evaporated on top of the contacts (see Fig. S1 of the new version of the supplemental material which is a sketch of the sample and its contacts). The measurements we perform are 4-terminal measurements where we measure the voltage drop between contacts that are at a distance of 367 m away from the source and drain contacts. iii. The fact that we show that we can use the 4-terminal resistance measurement, which does not measure the contact resistance and was measured at low 's (much below ) where heating is not relevant, to predict makes it very unlikely that the effect we present is caused by heating at the contacts.

The reviewer writes: "The critical field behavior at low currents needs to be shown"
Answer: Good idea, thank you, we have added a Fig. S3 where we plot the critical field behavior as a function of at low currents for all samples.
Response to the comments of Reviewer 3: 1. The reviewer writes: "Equation (1) describes not only two stable states of a current-carrying superconducting film, but also two current values at which a voltage jump occurs (with increasing and decreasing current). The authors report that they observe "limited hysteresis" of critical currents. I believe that it is important to say more specifically which part of the current-voltage characteristic is described in terms of electron self-heating and where the discrepancy between theory and experiment begins" Answer: This is a very good point, in the supplemental material of our new submission we have added Fig. S8 where we compare between the measured and simulated results and focus on these deviations. We reference this figure from the discussion regarding the deviations due to the simplified Ohmic approximation in the main-text.
2. The reviewer writes: "The authors suggest that limited hysteresis of critical currents can be caused by the occurrence of hot spots. Indeed, regions with weakened superconducting properties may exist in inhomogeneous superconductors. When sufficiently large currents flow in these areas, the formation of resistive domains is possible. The theory of resistive domains is well known (see Ref. 12 and the references therein). It seems to me that it would be interesting to compare the results of the authors with the theory of resistive domains" Answer: See "Response to common comment of reviewer 1 and 3" above.
Summary of the main changes in the manuscript:  We changed our terminology from trapping and escape critical current to → , → (following a comment of reviewer 1).  We added Fig.1b where we define the → , → (following a comment of reviewer 1).  We defined and as the theoretical limits of the bi-stability interval (following a comment of reviewer 1).  We have added (see discussion section D of main-text and supplemental material Sec. S8) an analysis showing that the observation that → ≈ → ≈ is consistent with a propagation of a switching wave in our bi-stable conductor (as suggested by reviewers 1 and 3).  We have added to Fig.S7b the black curve which corresponds to (following a comment of reviewer 1).  We have corrected the citation to Eq.S1 (following a comment of reviewer 1).  We have added to Sec. S1 of the supplemental material the values of the mean-free-path, , width of and the coherence length (following a comment of reviewer 1).  We have emphasized that in highly disordered, thin film type-II superconductors in the presence of a magnetic field above 1 ( 1 is practically zero for a:InO) are expected to have zero resistance only at = 0. We also show that in the inset of Fig.1b (following a comment of reviewer 2).  We added to the supplemental material an "Experimental setup" section (Sec. S2) where we describe our transport setup (following a comment of reviewer 2).  We have added a Fig. S3 where we plot the critical field behavior as a function of at low currents for all samples (following a comment of reviewer 2).  We added Fig. S8 where we compare between the measured and simulated results and focus on the deviations at low 's (following a comment of reviewer 3).
In the revised version some points of my previous report have been addressed, although mostly in the supplemental material (SM). The main text still has presentation issues and does not really convey the main message to a general readership of Nature Communications. It sounds like the authors just postulate the electron overheating scenario and try to proselytize the reader by making general statements and talking about different critical currents without clear explanation what they are. It is misfortunate because plenty of convincing experimental evidences are given in the SM which appears to be more of a real research paper than the main text.
Most figures with experimental data in the main text show redundant (and sometimes duplicate) semilog plots of dV/dI from which the overheating bistability does not readily follows. Then all of the sudden the authors show the key Fig. 3 and claim that the overheating model is in excellent agreement with experiment. I do not know how many people will be convinced by this way of presentation, but in my opinion the key experimental evidences of the electron overheating bistability should be presented in the main text before Fig. 3. One way of doing so would be to show an example of graphic solution of the thermal balance eq based on the analysis of experimental data (similar to Fig.  S7b in the SM) and use it not only to clearly demonstrate the thermal bistability and the magnitude of electron overheating, but also to explain how the minimum and maximum currents I_c^{min} and I_c^{max} are defined, and what are the equal area theorem and the associated current Ip in one sentence, relegating all technical details of analysis of particular cases to the SM. This would also help the reader to understand the physics of switching waves and the lack of hysteresis mentioned in the discussion section without going back and forth between the main text and the SM on each principal issue. I think that this work can be of interest to the Nat Com readership, but the authors should make an effort to convince not just a few specialists but a broader condensed matter superconductivity/condmat community that the overheating scenario is indeed essential.
Some inaccurate/peculiar statements should be fixed: page 1: "Lorentz force acting on the vortices exceeds their binding energy" -this sounds like current somehow unbind pairs of vortices, not to mention that a force cannot exceed energy because they have different dimensionalities. Perhaps, something like "the Lorentz force acting on vortices exceeds their depinning force" would be more informative. Page 1. "near the high critical B" -why cannot the standard "near the upper critical field" be used? Also, the list of references was cut and pasted from the SI, but Refs. 44-59 were not mentioned in the main text.
Reviewer #2 (Remarks to the Author): Dear Editor-I have considered the authors' response to my criticisms of the orginal manuscript. Unfortunately, I am unpersuaded. Contact heating is a very serious concern in critical current measurements. This was well understood more than 50 years ago by one of the pioneers of experimental studies of thin film superconductors, Rolf Glover III, who stated "The second trouble is due to warming of the film by Joule heating which can occur if small normally conducting regions are present, for example where electrical contact is made to the film. The problem is acute since the current densities exceed 10^6 A/cm^2" (Rev. Mod. Phys. 36, 299 (1964).) Indeed, I have personally measured critical currents in thin film superconductors and found that low duty cycle pulse measurements can give critical current values that are more than an order of magnitude greater than those of a dc measurement. There are groups all over the world that have the capability to perform pulsed critical current measurements so not having the capability "in house" is not a legitimate excuse. The critical current data in the manuscript must be verified via pulsed measurements or, perhaps by incorporating superconducting contacts, before the authors rush to publication.
I do not recommend publication in Nature Communications nor in any other reputable journal for that matter.
Reviewer #3 (Remarks to the Author): The authors gave acceptable answers to all my comments and significantly clarified the manuscript.
After an additional analysis, the authors argue that it is the motion of the switching wave between the S and N states in the a:InO film that is consistent with their observations. In my opinion,the revised manuscript NCOMMS-19-30499A can be published in Nature Communications.

Response to referees
Summary of the main changes in the manuscript:  Following the suggestion of Reviewer 1 the graphical solution section of the supplemental material ( Fig.S7a and S7b and their description of the previous submission) was moved to the main text ( Fig.3a and 3b). In addition to the figures the graphical solution is also described in the text at the end of the 3 rd page and at the beginning of page 4.  Following the suggestion of Reviewer 1 we added to the main text Fig.4d where we show how to extract the minimum propagation current Ip.
 Following the suggestion of Reviewer 1 and the comments of Reviewer 2 we added to the main-text discussion section E where we discuss the effects of the contacts.
 We have corrected the three "inaccurate/peculiar statements should be fixed" pointed out by o References to the supplemental material were altered to match the guidelines.
Response to the comments of Reviewer 1: 1. The reviewer writes: "The main text still has presentation issues…. plenty of convincing experimental evidences are given in the SM which appears to be more of a real research paper than the main text… in my opinion the key experimental evidences of the electron overheating bistability should be presented in the main text before Fig. 3. One way of doing so would be to show an example of graphic solution of the thermal balance eq based on the analysis of experimental data (similar to Fig. S7b in the SM) and use it not only to clearly demonstrate the thermal bistability and the magnitude of electron overheating, but also to explain how the minimum and maximum currents I_c^{min} and I_c^{max} are defined, and what are the equal area theorem and the associated current Ip in one sentence... This would also help the reader to understand the physics of switching waves and the lack of hysteresis mentioned in the discussion section without going back and forth between the main text and the SM on each principal issue".
The referee reiterates "Instead of showing the redundant and pretty standard dV/dI semilog plots, they should've shown an example of the electron heat balance plot and analyzed three key currents: I_min at which the hot phase can exist, I_max at which the cold SC phase disappears and the minimum propagating current Ip, where I_min > Ip > I_max".