1 Introduction

Distinct from sharp superconducting transition in bulk materials, the current-induced breakdown of superconductivity in one-dimensional (1D) systems tends to show more complicated behavior due to stronger fluctuation effect and topological defects such as phase slips [1, 2]. Phase slips bring dissipation into the superconducting state and may lead to step structures in I–V curves near transition temperature \(T_{c}\) by forming phase slip centers [3–7]. In fact, a few or even single phase slip event might induce an avalanche effect and drive the wire into the normal state [8–10]. Due to the slow thermal exchange at low temperature, it is often difficult to remove the Joule heating generated by phase slip events, giving rise to hysteretic I–V curves in the nanowire [8, 9, 11]. On the contrary, if the heat can be efficiently drained away from phase slip centers, the wire can remain superconducting while undergoing phase slip events [10]. Besides, voltage switching has been observed in dissipative Zn and Al nanowires between the superconducting state and a voltage plateau state, attributed to non-equilibrium quasiparticles generated by phase slips [12].

When contacted by superconducting electrodes, normal materials can acquire superconductivity via the proximity effect due to the non-local property of the Cooper pairs [13–20]. Specifically, in diffusive superconductor-normal metal-superconductor (S-N-S) junctions, the interplay of proximity effect and multiple Andreev reflections could give rise to minigap states [17]. In our prior work, we have reported the transport properties of individual 70 nm diameter crystalline gold nanowire contacted by four superconducting tungsten electrodes [18]. Among nanowires with different length, the short nanowire exhibits a sharp superconducting transition, while the longer wires show nonzero saturated resistance at low temperature. Interestingly, for gold nanowire of intermediate length, the breakdown of proximity-induced superconductivity occurs in two distinct steps. A minigap feature is observed under a low magnetic field around 0.3 T, then the superconductivity of the whole nanowire is destroyed above 3 T. The minigap state in a S-N-S nanostructure is also detected by a tunneling spectroscopy study on a nanowire contacted by superconducting electrodes [21].

Considering the dramatic role of phase slips in quasi-1D superconductors, it is natural and important to explore the effect of phase slips in proximity-induced 1D superconducting systems. Here, we report a systematic transport study of current-induced breakdown of superconductivity in 1.0 μm and 1.2 μm-long gold nanowires contacted by superconducting tungsten electrodes. We detect resistance switching and tail-like metallic state in resistance curves as a function of current, indicating the existence of phase slips in proximity-induced superconducting nanowires. Enhanced upper critical current is observed under low magnetic field, which may be attributed to the formation of spin-triplet pairing in superconducting gold nanowire. More intriguingly, subharmonic structures with a small characteristic energy emerge on the tail-like structures, suggesting possible axions-triggered process in phase slip center Josephson junction.

2 Resistance switching and tail-like metallic state

Our crystalline gold nanowires were fabricated by electrodeposition [18] (upper inset of Fig. 1(a)). After dispersing the nanowires on Si3N4/Si substrates, tungsten electrodes were deposited by focused ion beam (FIB) on individual gold nanowire [18, 22] for standard four-probe measurements (lower inset of Fig. 1(a)). The FIB-deposited tungsten electrodes, which are amorphous and composed of ∼40% tungsten, ∼40% carbon and ∼20% gallium, exhibit a superconductivity transition at around 5 K [23, 24]. At 1.8 K, with decreasing the measurement current, the resistance versus current (\(R(I)\)) curve of the 1.0 μm long-Au nanowire shows a superconducting transition due to the proximity effect of the superconducting electrodes (Fig. 1(a)). Unexpectedly, the superconducting transition at 1.8 K exhibits a resistance switching region between the superconducting state and a dissipative state (\(\approx 45\ \Omega \)) ranging from 15 μA to 17 μA. The resistance switching is reproducible when sweeping current from opposite directions (Fig. 1(a)-(c)). Moreover, the temperature fluctuation during the measurement is negligible (Fig. 1(b) and (c)), excluding the possibility that resistance switching is induced by external temperature fluctuations.

Figure 1
figure 1

Resistance switching of proximity induced superconducting gold nanowire. (a) Resistance as a function of current in a 1.0 μm-long gold nanowire measured at 1.8 K. Resistance switching is observed between the superconducting state and a dissipative state. The current scanning rate is 0.01 μA/s. Upper inset, scanning electron microscopy (SEM) image of crystalline gold nanowires fabricated by electrodeposition. The scale bar is 100 nm. The diameters of the nanowires are around 70 nm. Lower inset, SEM image of an individual gold nanowire contacted by four tungsten electrodes for standard four-probe measurements. The length (L) of nanowire is defined as the distance between the inner edges of the two voltage electrodes. (b) and (c) Zoom-in of oscillating region and temperature fluctuation when (b) increasing or (c) decreasing the current

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At higher temperatures from 2.0 to 4.0 K, we have observed tail-like metallic state instead of a sharp superconducting transition from normal state to zero resistance state (Fig. 2(a)). Superconducting state with zero resistance arises in gold nanowire below the critical current \(I_{c1}\), while resistance rises rapidly and gold nanowire turns into the normal state above \(I_{c2}\). In the intermediate current region between \(I_{c1}\) and \(I_{c2}\), tail-like structures appear where the resistance increases slowly with increasing current and is well below the normal resistance.

Figure 2
figure 2

Temperature and magnetic field dependence of the current induced breakdown of superconductivity in 1.0 μm-long gold nanowire. (a) Current-dependent resistance at 1.8 K, 1.9 K, 2.0 K, 2.3 K, 2.8 K, 3.0 K, 3.3 K, 3.4 K, 3.5 K, 3.6 K, 3.7 K, 3.8 K, 4.0 K, 4.3 K and 4.8 K, respectively. At temperatures ranging from 2.0 K to 4.0 K, tail-like structures appear between the superconducting state (Ic1) and the sharp transition current (Ic2). Ic1 and Ic2 are labeled by \(R(I)\) curve at 2.8 K. (b) Current-dependent resistance at various magnetic fields measured at 1.8 K. (c) Current-dependent resistance in the low magnetic field region. The lower critical current (\(I_{\mathrm{c}}^{\mathrm{low}}\)) is determined as the critical current below which the nanowire is in superconducting state or tail-like metallic state. The upper critical current (\(I_{\mathrm{c}}^{\mathrm{up}}\)) is determined as the critical current where the nanowire turns to normal state. \(I_{\mathrm{c}}^{\mathrm{up}}\) and \(I_{\mathrm{c}}^{\mathrm{low}}\) are labeled by R(I) curve at 0 T. (d) The dependence of \(I_{\mathrm{c}}^{\mathrm{up}}\) and \(I_{\mathrm{c}}^{\mathrm{low}}\) on the magnetic field. Error bars are labeled to represent the current range of resistance switching regions

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The appearance of resistance switching region and tail-like state in \(R(I)\) curve is closely related to the dissipation of phase slip centers [9, 10, 12, 25, 26]. In proximity-induced superconducting gold nanowire, since the superconducting gap in the middle of the wire is much smaller than that at the S-N boundary, it is more fragile and sensitive to external disturbances. In the central region of the wire, the heat released by phase slip events could raise local temperature up to \(T_{c}\) and results in a local superconducting to normal state transition, giving rise to the detected resistive state. The middle segment of the nanowire turns to normal phase slip center at a lower critical current \(I_{c1}\), while a higher current \(I_{c2}\) is needed to destroy the superconductivity of the entire wire. A tail-like structure emerges between \(I_{c1}\) and \(I_{c2}\) where the resistance gradually increases as the phase slip center size expands. The resistance switching could be observed at specific circumstances when the heat generation and dissipation are nearly balanced. Since the phase slip center is thermal decoupled from superconducting regions of nanowire, the heat dissipation is mainly from the thermal phonon coupling with the substrate [10, 27, 28]. Noteworthily, in the proximity-induced superconducting nanowire the central normal part is a much better thermal conductor than the superconducting parts, thus the normal part is easier to be cooled down than the superconducting parts. When heat dissipation is higher than heat generation at phase slip center, the hot normal part would be cooled down and return to superconducting state. In the subsequent phase slip event, the middle of wire would switch to normal state again. Such cycles can explain the observed resistance switching in our superconducting gold nanowires as shown in Fig. 1.

3 Magnetic field enhanced upper critical current

Magnetic field dependence of the current induced breakdown of superconductivity is investigated in the 1.0 μm-long nanowire at 1.8 K (Fig. 2(b)-(d)). The resistance switching region disappears and tail-like structure develops at 0.3 T (Fig. 2(b)). Superconductivity is gradually suppressed with rising magnetic field. Figure 2(c) shows current-dependent resistance at low magnetic field region. The resistance switching region is very sensitive to external magnetic field. Current range of switching region is suppressed to 0.2 μA at 0.05 T from 2 μA at 0 T and totally disappears at 0.1 T. Meanwhile an intermediate plateau-like state (\(\approx 90\ \Omega \)) develops under magnetic field. This structure forms at 0.05 T and broadens at 0.1 T; then it gradually shrinks and disappears under larger field. The lower critical current (\(I_{\mathrm{c}}^{\mathrm{low}}\)), determined as the critical current below which the nanowire is in superconducting state or tail-like metallic state, has a monotonously decreasing relation with increasing field. Unexpectedly, the upper critical current (\(I_{\mathrm{c}}^{\mathrm{up}}\)), determined as the critical current where the nanowire turns to normal state, shows a counter-intuitive field dependence where \(I_{\mathrm{c}}^{\mathrm{up}}\) increases with rising field below 0.1 T, indicating field-enhanced upper critical current in proximity-induced superconducting gold nanowire. The evolution of \(I_{\mathrm{c}}^{\mathrm{up}}\) and \(I_{\mathrm{c}}^{\mathrm{low}}\) with respect to magnetic field is summarized in Fig. 2(d). An enhancement of superconductivity has been observed in Zn nanowire when the superconductivity in bulk Sn electrode is suppressed under the applied magnetic field, known as the anti-proximity effect [19]. Magnetic field enhanced superconductivity has also been reported in Zn nanowire with Zn leads [12, 29], where the applied field suppresses the voltage plateau and stabilizes the superconducting state, giving rise to field enhanced lower critical current. Noteworthily, in our proximity-induced superconducting gold nanowire, only the upper critical current is enhanced while the lower critical current decreases with increasing magnetic field, which is distinct from the previous report in Zn nanowires. Additionally, compared to the voltage plateau that disappears at 0.0019 T in Zn nanowires [12], the resistance plateau in our gold nanowire is more stable and survives up to 0.3 T.

In conventional superconductors, the magnetic field is expected to suppress the superconductivity via the Zeeman effect and the orbital effect. However, counter-intuitive field dependence of superconductivity might also arise experimentally in some scenarios [12, 19, 29–33]. Some theoretical pictures have ever been proposed for the field enhanced superconductivity [34–40]. First, the applied field may enhance the effect of environmental dissipation, thus suppressing the phase slips and stabilizing the superconductivity in a nanowire [34]. This theory was proposed for the anti-proximity effect in nanowires, but it could not specifically explain why only the upper critical current is enhanced under field in our gold nanowire. Second, onset of superconductivity could be enhanced due to a positive conductance correction of the superconducting fluctuation effect [35, 36]. Superconducting fluctuation has been shown to give rise to an enhanced onset \(T_{c}\) and anomalous field-temperature phase boundary in ultrathin crystalline lead films [30]. However, fluctuation enhanced onset superconductivity could occur in the absence of magnetic field, which is in contrast to the field enhanced upper critical current in gold nanowire. A current-dependent superconducting fluctuation theory in 1D system is still lacking, calling for further theoretical studies in such system. Third, spin-triplet pairing was proposed to arise in ferromagnetic or spin-orbit coupling systems [37–40]. Spin-triplet pairing has been revealed in ferromagnet-superconductor hybrid structure, featured as a minimum critical current at zero field [32], which is similar to our observation. However, our proximity-induced superconducting gold nanowire is absent from ferromagnetic element, ruling out ferromagnetic scenario as the possible origin. Instead, spin-orbit coupling in a superconducting system could also lead to a pairing state which is a mixture of spin-singlet and spin-triplet state [39, 40]. Given the large spin-orbit coupling of gold [41, 42], the observed minimum critical current at zero magnetic field may suggest the formation of spin-triplet pairing in our gold nanowire. Recently, the gold surface has been reported to host topological surface state [41], and signature of Majorana zero modes has been revealed in a platform utilizing the surface state of gold [42]. Our observation together with the previously reported topological state of gold promises new opportunity to explore topological superconductivity [43] in gold nanowires with superconducting electrodes.

4 Subharmonic structures

Temperature dependence of the current induced breakdown of superconductivity is also investigated in the 1.2 μm-long gold nanowire (Fig. 3). The 1.2 μm-long gold nanowire exhibits a relatively sharp superconducting transition at 1.8 K without showing a resistance switching region (Fig. 3(a)). Compared to the 1.0 μm-long nanowire, the 1.2 μm-long nanowire shows a smaller normal state resistance and a larger critical current, which could be due to the slight diameter variation of the nanowires. The individual differences in gold nanowires give rise to the difference in the dissipation with the surrounding environment, which might explain why the resistance switching region is present in the 1.0 μm-long wire while absent in the 1.2 μm-long wire. At higher temperatures between 2.0 K and 4.0 K, tail-like metallic state appears in both the 1.0 μm and 1.2 μm-long gold nanowires (Fig. 2(a) and 3(a)). Moreover, tiny resistance peaks and dips are found to superimpose on the tail structure in both nanowires. Resistance peaks and dips appear in a wide temperature range in the 1.2 μm-long gold nanowire (see Fig. 3(b) for a zoom-in of \(R(I)\) curves in the tail region at temperatures ranging from 2.4 K to 3.6 K in the 1.2 μm-long gold nanowire).

Figure 3
figure 3

Temperature dependence of the current induced breakdown of superconductivity in 1.2 μm-long gold nanowire. (a) Current-dependent resistance at 1.8 K, 1.9 K, 2.0 K, 2.2 K, 2.4 K, 2.6 K, 2.8 K, 3.0 K, 3.2 K, 3.4 K, 3.6 K, 3.8 K, 4.0 K, 4.2 K and 4.8 K, respectively. At temperatures ranging from 2.0 K to 4.0 K, tail-like structures appear between the superconducting state (Ic1) and the sharp transition current (Ic2). Ic1 and Ic2 are labeled by \(R(I)\) curve at 2.4 K. (b) Zoom-in of the tail-like structure (red dash frame in figure (a)) reveals resistance peaks and dips between 2.4 K and 3.6 K

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To better demonstrate the feature of resistance peaks and dips, Fig. 4(a) displays the I–V characteristic of tail structure in the 1.2 μm-long nanowire ranging from 2.4 K and 3.5 K where resistance peaks and dips are frequently detected. The differential conductance (\(dI/dV\)) is plotted as a function of voltage in Fig. 4(b) from 2.8 K to 3.2 K. Surprisingly, the locations of peaks and dips correspond well for curves at various temperatures. Their locations can be approximately expressed as \(V= ( m/n ) V_{0} = ( m/n ) [ 2 \Delta _{0} /e ]\) [44], where m, n are integers, and \(\Delta _{0}\) is a characteristic energy. Most peaks and dips lay on the “\(m/n\)” lines in Fig. 4(b) when \(V_{0}\) is set as 94 μV. The extracted \(\Delta _{0} =47\ \mu \mathrm{eV}\) is much smaller than \(\Delta _{\mathrm{sc}} \approx 1.76kT_{c} =682\ \mu \mathrm{eV}\), where the onset \(T_{c}\) of the proximity-induced superconducting gold nanowire is around 4.5 K. The small characteristic energy may be related to the minigap structure in the middle of the nanowire. In our prior work, the minigap structure has been observed at low magnetic field, which is significantly smaller than the critical field to destroy the superconductivity of whole nanowire (Fig. 4(c)) [18].

Figure 4
figure 4

Subharmonic structures on tail-like structures of the 1.2 μm-long gold nanowire. (a) Tiny structures superimposed on tail-like metallic state in I–V characteristic between 2.4 K and 3.5 K in intervals of 0.1 K. (b) Derived \(dI/dV\) curves as a function of voltage at 2.8 K, 2.9 K, 3.0 K, 3.1 K and 3.2 K. Curves are shifted for clarity by −0.5, −0.25, 0, 0.25, 0.5 \(\Omega ^{-1}\) for temperature of 2.8 K, 2.9 K, 3.0 K, 3.1 K and 3.2 K, respectively. The positions of peaks and dips can be approximately expressed as \(( m/n ) V_{0}\) when \(V_{0}\) is set as 94 μV. “\(m/n\)” lines with \(n\leq 3\) are labeled with black dash lines. (c) Magnetoresistance at 3.0 K exhibits minigap structure (red dash frame) with characteristic magnetic field of the minigap structure well below the critical field to destroy the superconductivity of whole nanowire. Inset: 3D map of the minigap structure of the 1.2 μm nanowire from 2.4 K to 3.5 K. Figure (c) is reproduced from [18]. (d) Schematic for axions-triggered process at the phase slip center of a S-N-S junction as the possible origin of the subharmonic structures

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The structure in the I–V characteristic is reminiscent of Shapiro steps. Shapiro steps originate from the resonant absorption of microwave with frequency v, which results in integer voltage steps (i.e., differential conductance peaks) at \(2eV=mhv\) (\(m=0, 1, 2,\dots \)) [45]. Subharmonic Shapiro steps at \(2eV=(m/n)hv\) (\(n\geq 2\)) have also been observed, where the \(1/n\) step is associated with the multiple Andreev reflection process that transfers n Cooper pairs across the junction [46]. However, our measurement system is absent from any microwave source and such high frequency noise, ruling out subharmonic Shapiro steps as possible origin of our observation. In the absence of external microwave radiation, multiple Andreev reflection process could give rise to subharmonic structures at \(V= ( 1/n ) [ 2\Delta /e ]\) in S-N-S weak links [47–50], where Δ is the superconducting gap. The observed small characteristic energy of \(\Delta _{0} =47\ \mu \mathrm{eV}\) may be related to the minigap structure in gold nanowire. However, multiple Andreev reflection alone usually could not account for the subharmonic structures at \(V= ( m/n ) [ 2 \Delta _{0} /e ]\) with \(m\geq 2\) [48–50], calling for new physical explanation.

It was predicted that dark-matter axions could produce detectable signals in a resonant Josephson junction [51]. Our observed subharmonic structures might be related to the axions-triggered process (see Fig. 4(d) for a schematic). To be specific, an axion entering the phase slip center Josephson junction could decay into photons, which transfer the momentum to the electron-hole pair. The electron and hole then perform multiple Andreev reflection for n times, where n is the minimum integer that satisfies \(neV\geq 2 \Delta _{0}\). At the end of the process, the electron and hole could leave the junction or annihilate back into the photon, which recombines with another photon into an axion. When the Josephson frequency \(\omega _{J} \equiv 2 eV/ \hslash \) coincides with the frequency of axion \(\omega _{\mathrm{a}} \equiv m_{a} c^{2} / \hslash \), this process could be resonantly enhanced and lead to detectable signals. As a comparison with Shapiro steps in the presence of microwave radiation, the energy in axion scenario (\(m_{a} c^{2}\)) is provided by axion mass (\(m_{a}\)) instead of microwave radiation (hv). The axions-triggered process in S-N-S resonant junction could give rise to harmonic signal at \(2eV=m m_{a} c^{2}\) [51, 52] or subharmonic signal at \(2eV=(m/n) m_{a} c^{2}\) (\(n=2, 3\)) as observed in our gold nanowire. Given \(V_{0} =94\ \mu \mathrm{V}\) in our results, an axion mass of \(188\ \mu \mathrm{eV}\) is obtained, which is in the typical parameter range of the theoretical predictions [51]. Axion, as a hypothesized particle in the strong CP problem and a leading candidate for dark matter, is a research highlight in particle physics, astrophysics and cosmology. Although tremendous efforts have been made in the search of axions [53], it is challenging to experimentally confirm the existence of the dark matter axions. Our results together with previous reports suggest the S/N/S junction may offer an alternative approach in axions detection. Still, further experimental works are highly desired to confirm or refute the possible connections between the axions and the signal detected in the S/N/S junction. For instance, angular dependent experiments are helpful to explore the possible evolution of axion-generated signals with the relative spatial orientation between the Josephson junction and the galactic axions flow. Moreover, if the signal originates from the dark-matter axions passing through Earth from the halo of galaxy, it should exhibit a seasonal modulation of about 10% in intensity since the velocity of Earth moving through galactic halo changes seasonally as Earth rotates around the Sun, with a maximum in June and a minimum in December.

5 Conclusion

In summary, we have systematically investigated the current-induced breakdown of superconductivity in gold nanowire contacted by superconducting electrodes. We detect resistance switching and tail-like metallic state in resistance curves as a function of current, demonstrating the prominent role of phase slips in quantum transport of the proximity-induced superconducting nanowire. Magnetic field enhanced upper critical current is observed, which may be attributed to the formation of spin-triplet pairing in gold nanowire. Spin-orbit coupling gold nanowire with superconducting electrodes could provide a new platform to study topological superconductivity. Moreover, subharmonic structures with a small characteristic energy are observed to superimpose on the tail-like structures. A potential link between the detected signal in gold nanowire and dark-matter axions may stimulate further studies in dissipative superconducting nanowires.