The ability to carry electric current with zero dissipation is the hallmark of superconductivity1. 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 fields2. 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 non-equilibrium state, where superconductivity coexists with dissipation, mediated by the so-called Andreev quasiparticles. Besides the promise for applications such as single-photon detectors3, the effect is a vivid example of a controllable non-equilibrium state of a quantum liquid. Thus our findings provide an accessible generic platform to investigate conceptual problems of out-of-equilibrium quantum systems.
With applications ranging from infrared detectors4 to prototypical qubits5,6, superconducting nanocircuitry has emerged in recent years as a fascinating area of research. Its fundamental significance lies in, for example, phase-sensitive studies of pairing mechanisms in novel superconductors7,8 and access to a wealth of non-equilibrium quantum phenomena9. 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 state10,11,12. Observing and studying such dissipative superconductivity has turned out to be a challenge. The culprits are non-equilibrium 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 wires15,16 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 non-equilibrium quasiparticles into the environment. By choosing Zn as the growth material17, we are able to fabricate quasi-1D wires whose length L is significantly shorter than the inelastic relaxation length Lin, yet much longer than the coherence length ξ ≈ 250 nm. These bring us to a situation in which quasiparticles form a peculiar non-equilibrium distribution, governed by Andreev reflections from the boundaries with the superconducting leads. As a result, we observe a non-hysteretic 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 well-filtered dilution refrigerator (Methods). Over a range of currents, flanked by the bottom and top threshold values Ib < I < It, the voltage across the nanowire exhibits a nearly flat plateau at V0 = 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 It and Ib are factors of 30–50 smaller than the estimated depairing critical current of the wire18. Collecting Ib and It together with Ic, 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 normal-state resistance, Rn, and were measured in a different weakly filtered refrigerator. Despite some smearing due to unfiltered noise, the plateau voltage V0 remains almost unchanged for all the Zn wires. The only exception is Al sample D, with the voltage plateau at V0 ≃ 93.2 ± 1.3 μV. Rescaling this voltage with the Bardeen–Cooper–Schrieffer superconducting gap 2Δ0 ≈ 3.52 Tc, we found that the plateaux in all samples fall close to the same universal line eV0/Δ0 = 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 time-domain 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 τsc and τvp 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 τvp and the reduction of the superconducting lifetime τsc, albeit at a smaller rate. It is worth mentioning that the two lifetimes are almost independent of temperature until T ∼ 400 mK, above which τvp increases and τsc decreases exponentially (see Supplementary Methods).
The observed dissipative state exhibits a counter-intuitive magnetic field dependence. One could expect that the magnetic field suppresses superconductivity, thus decreasing τsc and possibly increasing τvs. 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 Ib 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 counter-intuitive non-equilibrium 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 magnetic-field-induced superconductivity and anti-proximity effect19,20,21,22. It is now apparent that the magnetic-field-induced 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 resistance18,23, as opposed to a constant voltage. The plateau cannot be a giant Shapiro step24,25, caused by leaking high-frequency noise. Indeed, Zn and Al samples have different V0 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 non-equilibrium quasiparticles generated by phase slips18. In our samples (in contrast with previous studies13,14,15,16,18) the inelastic relaxation length Lin exceeds the wire length17, 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 energy27, resulting in a peculiar non-equilibrium distribution. Because of the self-consistency relation, such a non-equilibrium distribution suppresses the order parameter Δ inside the wire (relative to its equilibrium value Δ0). Although the quasiparticles are far from equilibrium, the order parameter is fixed to its local self-consistent value everywhere apart from a distance ∼ξ around the phase slip. The condensate chemical potential μ, given by the Josephson relation μ = ℏ〈∂tϕ〉/2, thus exhibits a discontinuity eV at the phase slip location18. Because the leads absorb high-energy 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, eV0 ≈ 2Δ. In this case the train of phase slips permanently suppresses the order parameter in the narrow middle region, self-propelling the dissipative state (that is, in such a self-consistent 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, FL, and transverse, FT, components, which are its odd and even parts with respect to the two chemical potentials μ1,2 = ±eV/2. At the boundaries with the leads x = x1,2(ε) and the energy window Δ < |ε − μ1,2| < Δ0 they obey Andreev boundary conditions:
In the absence of inelastic relaxation the continuity relation (known also as the Usadel equation28,29) reads ∂xFL/T(ε, x) = JL/T(ε), with x-independent energy, JL, and charge, JT, currents. Together with equation (1) this leads to an x-independent FL(ε) (for |ε| < Δ0), satisfying the energy diffusion equation27∂ε2FL = 0. As the self-consistency relation
(ωD is Debye frequency) involves only FL, 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 FL(|ε| < Δ) = 0 for the energy diffusion27, resulting in the distribution function depicted in Fig. 3. On substitution into the self-consistency relation (2), it results in a transcendental equation for δ = Δ/Δ0, which at T = 0 has two solutions (bistability): δsc = 1 and δvp = 0.17.
The second of these solutions implies a dissipative state with eV0 = 2Δ ≈ 0.34Δ0. It is sustained if a normal current Ib ≈ 1.64(V0/Rn) is applied to the wire (see Supplementary Methods). Note that for ξ ≪ L the current Ib is much smaller than the depairing critical current, allowing the wire to support a supercurrent. (This is different from long whiskers18, where phase slips extend over a few inelastic lengths, resulting in a negligible coherent supercurrent.) An excess current I − Ib is thus carried as the supercurrent, without an additional voltage increase—hence the observed voltage plateau. The current exceeding It ≈ 0.72Δ0/eRn stabilizes another solution of the self-consistency 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 normal-state 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 Lin ≈ 12 μm (ref. 17) brings the currents Ib, t as well as the voltage plateau V0 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 Δlead, leaving that of the wire (almost) intact. This narrows the interval for the energy diffusion30, bringing the distribution closer to the equilibrium one. This in turn increases the bottom threshold current Ib. For Δlead ≲ 0.78Δ0 the self-consistent 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 It. In fact, suppression of Δlead is also the primary mechanism of the voltage plateau termination at T ≳ 650 mK (Fig. 1b).
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 1-μm-wide Zn electrodes (inset to Fig. 1a). The Zn electrodes were 10 μm long and in turn were connected to pre-patterned Au contacts. Both the nanowire and the electrodes were fabricated in a single step of the quench-deposition at substrate temperatures of 77 K, depositing through a resist mask patterned using electron-beam lithography. The normal-state 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 Kelvinox-400) 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 He-3 insert (Quantum Design), with no filtering system other than having the electrical lines twisted in pairs.
To bypass effects of noise, the time-domain 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 (τsc) and the voltage plateau (τvp 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 set-up: ∼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 τsc and τvp reported in the main text are mean values of the stochastic residence time sequence, collected over 1,800 s.
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Discussions with X. Wang are warmly acknowledged. Experimental work at Minnesota was supported by the DOE Office of Basic Energy Sciences under Grant No. DE-FG02-02ER4600. 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 NSF-funded Materials Research Facilities Network (http://www.mrfn.org) via the MRSEC program. AK was supported by DOE Contract No. DE-FG02-08ER46482.
The authors declare no competing financial interests.
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Chen, Y., Lin, YH., Snyder, S. et al. Dissipative superconducting state of non-equilibrium nanowires. Nature Phys 10, 567–571 (2014). https://doi.org/10.1038/nphys3008
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