Abstract
The ability to carry electric current with zero dissipation is the hallmark of superconductivity^{1}. This very property makes possible such applications ranging from magnetic resonance imaging machines to Large Hadron Collider magnets. But is it indeed the case that superconducting order is incompatible with dissipation? One notable exception, known as vortex flow, takes place in high magnetic fields^{2}. Here we report the observation of dissipative superconductivity in far more basic configurations: superconducting nanowires with superconducting leads. We provide evidence that in such systems, normal current may flow in the presence of superconducting order throughout the wire. The phenomenon is attributed to the formation of a nonequilibrium state, where superconductivity coexists with dissipation, mediated by the socalled Andreev quasiparticles. Besides the promise for applications such as singlephoton detectors^{3}, the effect is a vivid example of a controllable nonequilibrium state of a quantum liquid. Thus our findings provide an accessible generic platform to investigate conceptual problems of outofequilibrium quantum systems.
Main
With applications ranging from infrared detectors^{4} to prototypical qubits^{5,6}, superconducting nanocircuitry has emerged in recent years as a fascinating area of research. Its fundamental significance lies in, for example, phasesensitive studies of pairing mechanisms in novel superconductors^{7,8} and access to a wealth of nonequilibrium quantum phenomena^{9}. Key elements of such circuitry—superconducting nanowires—are known to be susceptible to strong fluctuations. Their most spectacular manifestations are phase slips of the superconducting order parameter, which lead to dissipation within a nominally dissipationless superconducting state^{10,11,12}. Observing and studying such dissipative superconductivity has turned out to be a challenge. The culprits are nonequilibrium quasiparticles massively generated by phase slips. If not removed efficiently, they tend to overheat the nanowire, driving it into the normal state. For example, thin MoGe (refs 13, 14) and Al wires^{15,16} seem to be switched into the normal state by a single phase slip. The return to the superconducting state requires a significant decrease of the drive current, leading to hysteretic I–V characteristics.
In the experiments reported here we overcome the excess heating by an improved fabrication process (Methods). Electrically transparent interfaces between the wire and the leads allow a fast escape of nonequilibrium quasiparticles into the environment. By choosing Zn as the growth material^{17}, we are able to fabricate quasi1D wires whose length L is significantly shorter than the inelastic relaxation length L_{in}, yet much longer than the coherence length ξ ≈ 250 nm. These bring us to a situation in which quasiparticles form a peculiar nonequilibrium distribution, governed by Andreev reflections from the boundaries with the superconducting leads. As a result, we observe a nonhysteretic dissipative state, which still exhibits distinct superconducting features such as a supercurrent and a sensitivity to weak magnetic fields.
In Fig. 1a, we show I–V characteristics of sample A, measured at temperatures from 50 to 750 mK in a wellfiltered dilution refrigerator (Methods). Over a range of currents, flanked by the bottom and top threshold values I_{b} < I < I_{t}, the voltage across the nanowire exhibits a nearly flat plateau at V_{0} = 52.5 ± 1.2 μV (for T ≲ 450 mK). This indicates a peculiar dissipative state that is distinctly different from the normal state. It is important to note that both I_{t} and I_{b} are factors of 30–50 smaller than the estimated depairing critical current of the wire^{18}. Collecting I_{b} and I_{t} together with I_{c}, where the system becomes normal, we construct a temperature–current phase diagram, shown in Fig. 1b. It shows that as the temperature increases, the voltage plateau is compressed and eventually disappears at approximately 650 mK. This temperature dependence resembles that of the superconducting order parameter, suggesting that the dissipative voltage plateau state is associated with the superconducting order.
Performing measurements on different devices, we found a remarkable universality associated with the voltage plateau. In Fig. 1c–f, we compare the I–V characteristics of sample A with two other Zn samples, B and C, as well as an Al sample D, all measured at 450 mK. These samples differ from sample A both in their geometry and normalstate resistance, R_{n}, and were measured in a different weakly filtered refrigerator. Despite some smearing due to unfiltered noise, the plateau voltage V_{0} remains almost unchanged for all the Zn wires. The only exception is Al sample D, with the voltage plateau at V_{0} ≃ 93.2 ± 1.3 μV. Rescaling this voltage with the Bardeen–Cooper–Schrieffer superconducting gap 2Δ_{0} ≈ 3.52 T_{c}, we found that the plateaux in all samples fall close to the same universal line eV_{0}/Δ_{0} = 0.43 ± 0.05 (the ratio for all samples is listed in the Supplementary Methods).
Another remarkable feature of the voltage plateau state is its onset through a region of stochastic bistability. It is revealed by timedomain measurements, with the voltage measured using a repetition rate of 3 Hz under a sustained constant current. In Fig. 2a we show a time trace of the measured voltage at I = 1.95 μA and T = 50 mK. The system exhibits random switching between the superconducting and voltage plateau states with a characteristic timescale of a few seconds, indicating an intrinsic bistability. To quantify the stability of the two competing states we define lifetimes τ_{sc} and τ_{vp} as the averaged residence times in the superconducting and the voltage plateau states respectively. Figure 2b shows the dependence of the lifetimes on the applied current throughout the transition range. Increasing the current leads to an exponential growth of the voltage plateau lifetime τ_{vp} and the reduction of the superconducting lifetime τ_{sc}, albeit at a smaller rate. It is worth mentioning that the two lifetimes are almost independent of temperature until T ∼ 400 mK, above which τ_{vp} increases and τ_{sc} decreases exponentially (see Supplementary Methods).
The observed dissipative state exhibits a counterintuitive magnetic field dependence. One could expect that the magnetic field suppresses superconductivity, thus decreasing τ_{sc} and possibly increasing τ_{vs}. In fact, the exact opposite happens. As shown in Fig. 2c, a magnetic field of merely a few gauss stabilizes the superconducting state, increasing its lifetime by more than an order of magnitude and simultaneously decreasing the voltage plateau lifetime by two orders of magnitude. This behaviour is consistently observed through the entire transition range of currents.
The enhancement of superconductivity by magnetic field is even more apparent by inspecting the I–V characteristics at different fields (Fig. 2d). It is evident that the field shifts the bottom critical current I_{b} to higher values, stabilizing the superconducting state. Such a stabilization is in fact a result of the suppression of the voltage plateau state. This is best seen in the critical current versus magnetic field phase diagram (Fig. 2e). The range of currents supporting the voltage plateau decreases rapidly from below until the plateau collapses at 19 G. Correspondingly the phase space of the superconducting state expands. We thus observe a rather counterintuitive nonequilibrium phenomenon: keeping the current within the voltage plateau regime and increasing the magnetic field brings the system from the dissipative into the superconducting state. This is consistent with the reported magneticfieldinduced superconductivity and antiproximity effect^{19,20,21,22}. It is now apparent that the magneticfieldinduced superconductivity originates with the collapse of the voltage plateau state, providing an intriguing connection between the two effects.
It is crucial to distinguish the observed voltage plateau state from other phenomena. It is different from phase slip centres, seen in long superconducting whiskers and characterized by a constant differential resistance^{18,23}, as opposed to a constant voltage. The plateau cannot be a giant Shapiro step^{24,25}, caused by leaking highfrequency noise. Indeed, Zn and Al samples have different V_{0} values, requiring noise of very different frequencies within the same measurement apparatus. It also cannot be attributed to a running state of an underdamped Josephson junction ^{26}. The underdamped regime would require a capacitance orders of magnitude larger than that of our system (Supplementary Methods). External capacitance is also excluded by the fact that the same results were obtained in two refrigerators with very different circuitry. Moreover, contrary to the observed plateau, voltage across an underdamped junction is expected to grow with the increased current bias.
Key to understanding these observations are nonequilibrium quasiparticles generated by phase slips^{18}. In our samples (in contrast with previous studies^{13,14,15,16,18}) the inelastic relaxation length L_{in} exceeds the wire length^{17}, allowing the quasiparticles to spread over the entire wire. At the interfaces with the leads the quasiparticles experience Andreev reflections, which mix particles and holes, as shown in Fig. 3. This leads to quasiparticle diffusion over energy^{27}, resulting in a peculiar nonequilibrium distribution. Because of the selfconsistency relation, such a nonequilibrium distribution suppresses the order parameter Δ inside the wire (relative to its equilibrium value Δ_{0}). Although the quasiparticles are far from equilibrium, the order parameter is fixed to its local selfconsistent value everywhere apart from a distance ∼ξ around the phase slip. The condensate chemical potential μ, given by the Josephson relation μ = ℏ〈∂_{t}ϕ〉/2, thus exhibits a discontinuity eV at the phase slip location^{18}. Because the leads absorb highenergy quasiparticles, the concentration of the latter is largest in the centre, pinning the phase slip to the midpoint of the wire. For phase slips to occur, the voltage must exceed the energy gap (Fig. 3), namely, eV_{0} ≈ 2Δ. In this case the train of phase slips permanently suppresses the order parameter in the narrow middle region, selfpropelling the dissipative state (that is, in such a selfconsistent dynamical regime the rate of phase slips is not exponentially small).
For a quantitative description (see Supplementary Methods for the full calculation), it is convenient to parametrize the quasiparticle distribution function F(ε, x) = 1 − 2n(ε, x) by its longitudinal, F_{L}, and transverse, F_{T}, components, which are its odd and even parts with respect to the two chemical potentials μ_{1,2} = ±eV/2. At the boundaries with the leads x = x_{1,2}(ε) and the energy window Δ < ε − μ_{1,2} < Δ_{0} they obey Andreev boundary conditions:
In the absence of inelastic relaxation the continuity relation (known also as the Usadel equation^{28,29}) reads ∂_{x}F_{L/T}(ε, x) = J_{L/T}(ε), with xindependent energy, J_{L}, and charge, J_{T}, currents. Together with equation (1) this leads to an xindependent F_{L}(ε) (for ε < Δ_{0}), satisfying the energy diffusion equation^{27}∂_{ε}^{2}F_{L} = 0. As the selfconsistency relation
(ω_{D} is Debye frequency) involves only F_{L}, it results in an almost constant Δ(x) ≈ Δ. The phase slips in the middle of the wire excite quasiparticles and holes, equilibrating their populations. This provides the boundary condition F_{L}(ε < Δ) = 0 for the energy diffusion^{27}, resulting in the distribution function depicted in Fig. 3. On substitution into the selfconsistency relation (2), it results in a transcendental equation for δ = Δ/Δ_{0}, which at T = 0 has two solutions (bistability): δ_{sc} = 1 and δ_{vp} = 0.17.
The second of these solutions implies a dissipative state with eV_{0} = 2Δ ≈ 0.34Δ_{0}. It is sustained if a normal current I_{b} ≈ 1.64(V_{0}/R_{n}) is applied to the wire (see Supplementary Methods). Note that for ξ ≪ L the current I_{b} is much smaller than the depairing critical current, allowing the wire to support a supercurrent. (This is different from long whiskers^{18}, where phase slips extend over a few inelastic lengths, resulting in a negligible coherent supercurrent.) An excess current I − I_{b} is thus carried as the supercurrent, without an additional voltage increase—hence the observed voltage plateau. The current exceeding I_{t} ≈ 0.72Δ_{0}/eR_{n} stabilizes another solution of the selfconsistency and energy diffusion equations: the one with a vanishing order parameter in the middle of the wire. It essentially terminates the supercurrent, resulting in the resistance being close to the normalstate resistance. The fact that these threshold values are approximately 30% less than those observed is attributed to residual inelastic processes, neglected above. Indeed, the latter lead to particle–hole recombination, driving the distribution towards the equilibrium one. A reasonable estimate L_{in} ≈ 12 μm (ref. 17) brings the currents I_{b, t} as well as the voltage plateau V_{0} within 10% of the observed values.
This picture also naturally accounts for the observed effects of the magnetic field and temperature. The field mostly suppresses the order parameter of the leads Δ_{lead}, leaving that of the wire (almost) intact. This narrows the interval for the energy diffusion^{30}, bringing the distribution closer to the equilibrium one. This in turn increases the bottom threshold current I_{b}. For Δ_{lead} ≲ 0.78Δ_{0} the selfconsistent solution of Fig. 3 with δ≠1 is no longer possible. The superconducting state of the wire is thus stabilized all the way up to I_{t}. In fact, suppression of Δ_{lead} is also the primary mechanism of the voltage plateau termination at T ≳ 650 mK (Fig. 1b).
Methods
Investigated Zn nanowires were 100–120 nm wide, 65–110 nm high, with the length 1.5 μm < L < 6 μm (Supplementary Table 1) connected to four 1μmwide Zn electrodes (inset to Fig. 1a). The Zn electrodes were 10 μm long and in turn were connected to prepatterned Au contacts. Both the nanowire and the electrodes were fabricated in a single step of the quenchdeposition at substrate temperatures of 77 K, depositing through a resist mask patterned using electronbeam lithography. The normalstate resistivity varies in the range ρ = (6.2–8.4) × 10^{−8} Ω m (in comparison the bulk value for Zn is ρ = 5.9 × 10^{−8} Ω m).
The electrical measurements were carried out in two different refrigerators. The first was a dilution refrigerator (Oxford Kelvinox400) with a minimum temperature of 50 mK. The associated electrical lines were heavily filtered at room temperature using RC filters with a cutoff frequency of approximately 100 Hz and at low temperature using thermo coaxial cable filters with a cutoff frequency of approximately 1 GHz. The second was a Physical Properties Measurement System equipped with a He3 insert (Quantum Design), with no filtering system other than having the electrical lines twisted in pairs.
To bypass effects of noise, the timedomain data was taken only in refrigerator 1. The electrical measurements were performed with a bandwidth of 12 Hz and a repetition rate of 3 Hz. At a fixed current in the transition regime, the voltage was continuously measured for 1,800 s, exhibiting random switching in real time. To extract characteristic lifetimes of the superconducting (τ_{sc}) and the voltage plateau (τ_{vp} states the first 100 s of data was discarded to bypass possible transients. After that, whenever the voltage crossed above or below a threshold value (defined as three times the noise floor of the measurement setup: ∼80 nV), the switching into or out of the voltage plateau state was recorded. The time interval between two consecutive switching events is defined as the residence time in one state. Finally, the values of τ_{sc} and τ_{vp} reported in the main text are mean values of the stochastic residence time sequence, collected over 1,800 s.
References
De Gennes, P. Superconductivity of Metals and Alloys 2nd edn (Westview Press, 1999).
Blatter, G. et al. Vortices in hightemperature superconductors. Rev. Mod. Phys. 66, 1125–1388 (1994).
Gol’tsman, G. N. et al. Picosecond superconducting singlephoton optical detector. Appl. Phys. Lett. 79, 705–707 (2001).
Eisaman, M. D., Fan, J., Migdall, A. & Polyakov, S. V. Invited Review Article: Singlephoton sources and detectors. Rev. Sci. Instrum. 82, 071101 (2011).
Clarke, J. & Wilhelm, F. K. Superconducting quantum bits. Nature 453, 1031–1042 (2008).
Devoret, M. H. & Schoelkopf, R. J. Superconducting circuits for quantum information: An outlook. Science 339, 1169–1174 (2013).
Wollman, D. A., Van Harlingen, D. J., Lee, W. C., Ginsberg, D. M. & Leggett, A. J. Experimental determination of the superconducting pairing state in YBCO from the phase coherence of YBCOPb dc SQUIDs. Phys. Rev. Lett. 71, 2134–2137 (1993).
Tsuei, C. C. et al. Pairing symmetry and flux quantization in a tricrystal superconducting ring of Y Ba2Cu3O7−δ . Phys. Rev. Lett. 73, 593–596 (1994).
Mooij, J. E. & Nazarov, Y. V. Superconducting nanowires as quantum phaseslip junctions. Nature Phys. 2, 169–172 (2006).
Bezryadin, A. Superconductivity in Nanowires: Fabrication and Quantum Transport 1st edn (Wiley, 2012).
Altomare, F. & Chang, A. M. OneDimensional Superconductivity in Nanowires 1st edn (Wiley, 2013).
Arutyunov, K. Yu, Golubev, D. S. & Zaikin, A. Superconductivity in one dimension. Phys. Rep. 464, 1–70 (2008).
Sahu, M. et al. Individual topological tunnelling events of a quantum field probed through their macroscopic consequences. Nature Phys. 5, 503–508 (2009).
Shah, N., Pekker, D. & Goldbart, P. M. Inherent stochasticity of superconductor–resistor switching behavior in nanowires. Phys. Rev. Lett. 101, 207001 (2008).
Li, P. et al. Switching currents limited by single phase slips in onedimensional superconducting Al nanowires. Phys. Rev. Lett. 107, 137004 (2011).
Singh, M. & Chan, M. H. W. Observation of individual macroscopic quantum tunneling events in superconducting nanowires. Phys. Rev. B 88, 064511 (2013).
Stuivinga, M., Mooij, J. E. & Klapwijk, T. M. Currentinduced relaxation of charge imbalance in superconducting phaseslip centers. J. Low Temp. Phys. 46, 555–563 (1982).
Skocpol, W., Beasley, M. R. & Tinkham, M. Phaseslip centers and nonequilibrium processes in superconducting tin microbridges. J. Low Temp. Phys. 16, 145–167 (1974).
Chen, Y., Snyder, S. D. & Goldman, A. M. Magneticfieldinduced superconducting state in Zn nanowires driven in the normal state by an electric current. Phys. Rev. Lett. 103, 127002 (2009).
Chen, Y., Lin, YH., Snyder, S. D. & Goldman, A. M. Stabilization of superconductivity by magnetic field in outofequilibrium nanowires. Phys. Rev. B 83, 054505 (2011).
Tian, M. et al. Suppression of superconductivity in zinc nanowires by bulk superconductors. Phys. Rev. Lett. 95, 076802 (2005).
Tian, M., Kumar, N., Wang, J., Xu, S. & Chan, M. H. W. Influence of a bulk superconducting environment on the superconductivity of onedimensional zinc nanowires. Phys. Rev. B 74, 014515 (2006).
Tidecks, R. CurrentInduced Nonequilibrium Phenomena in QuasiOneDimensional Superconductors 1st edn (Springer, 1990).
Dinsmore, R. C. III, Bae, MH. & Bezryadin, A. Fractional order Shapiro steps in superconducting nanowires. Appl. Phys. Lett. 93, 192505 (2008).
Bae, MH., Dinsmore, R. C. III, Sahu, M. & Bezryadin, A. Stochastic and deterministic phase slippage in quasionedimensional superconducting nanowires exposed to microwaves. New J. Phys. 14, 043014 (2012).
Barone, A. & Paterno, G. Physics and Applications of the Josephson Effect 1st edn (Wiley, 1982).
Nagaev, K. E. Frequencydependent shot noise in long disordered superconductor∖normalmetal∖superconductor contacts. Phys. Rev. Lett. 86, 3112 (2001).
Usadel, K. D. Generalized diffusion equation for superconducting alloys. Phys. Rev. Lett. 25, 507–510 (1970).
Kamenev, A. Field Theory of NonEquilibrium Systems (Cambridge Univ. Press, 2011).
Vodolazov, D. Y. & Peeters, F. M. Enhancement of the retrapping current of superconducting microbridges of finite length. Phys. Rev. B 85, 024508 (2012).
Acknowledgements
Discussions with X. Wang are warmly acknowledged. Experimental work at Minnesota was supported by the DOE Office of Basic Energy Sciences under Grant No. DEFG0202ER4600. Samples were fabricated in the Nano Fabrication Center, which receives funding from the NSF as a part of the National Nanotechnology Infrastructure Network, and were characterized in the Characterization Facility, University of Minnesota, a member of the NSFfunded Materials Research Facilities Network (http://www.mrfn.org) via the MRSEC program. AK was supported by DOE Contract No. DEFG0208ER46482.
Author information
Authors and Affiliations
Contributions
Y.C. and A.M.G. conceived and designed the experiment. Y.C. and S.D.S. fabricated the devices. Y.C and YH.L. performed the measurements. A.K. provided theoretical analysis and cowrote the manuscript with Y.C. and A.M.G. All authors contributed to the discussion and presentation of the results.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Supplementary information
Supplementary Information
Supplementary Information (PDF 808 kb)
Rights and permissions
About this article
Cite this article
Chen, Y., Lin, YH., Snyder, S. et al. Dissipative superconducting state of nonequilibrium nanowires. Nature Phys 10, 567–571 (2014). https://doi.org/10.1038/nphys3008
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1038/nphys3008
This article is cited by

Magnetic field enhanced critical current and subharmonic structures in dissipative superconducting gold nanowires
Quantum Frontiers (2022)

CurrentInduced Metastable States Close to T$$_{c}$$ in NbTi Superconducting Bridges
Journal of Superconductivity and Novel Magnetism (2021)

Measurement of the gap relaxation time of superconducting NbTi strips on a sapphire substrate
Applied Physics A (2019)

Novel voltage signal at proximityinduced superconducting transition temperature in gold nanowires
Science China Physics, Mechanics & Astronomy (2018)

Thermal and quantum depletion of superconductivity in narrow junctions created by controlled electromigration
Nature Communications (2016)