Introduction

A dissipationless current is known to flow through a thin (subnanometer) insulating barrier between two superconductors S with a superconducting phase difference, the well-known Josephson effect.1 This supercurrent may also flow through a long (micrometer) non-superconducting metal wire at low temperatures, a spectacular consequence of the quantum phase coherence throughout the normal metal. The way the supercurrent responds to a dc phase difference, the current-phase relation,2 was only recently measured with a Hall probe.3 It reflects the phase dependence of the ABS,4 entangled electron-hole states which form in the normal metal as a consequence of the superconducting mirror-like boundary conditions. Discrete phase dependent ABS were recently detected in a carbon nanotube between S electrodes5. In a diffusive SNS junction, with N longer than the superconducting coherence length, the ABS spectrum is a quasi-continuum of levels with a small energy gap Eg2,7 independent of Δ. This minigap depends solely on the wire length L and the diffusion constant D, via the Thouless energy , where τD = L2/D is the diffusion time through the N wire. Eg is fully modulated by the phase difference ϕ between the superconducting order parameters on both sides: it is maximal at ϕ = 0 with and goes linearly to zero at ϕ = π, as recently measured by scanning tunneling spectroscopy.8 The phase dependent Josephson current IJ (ϕ) at equilibrium sums the contributions of each ABS of energy ɛ n, via , the current carried by level n with occupation factor pn :

Its amplitude (the critical current Ic ) is 11GNETh /e at T = 0 where GN is the normal state conductance of the normal wire and is thus independent of Δ, the superconducting gap of the electrodes. Ic decreases roughly exponentially with an increasing temperature on the scale of Eg . The phase dependence of IJ is non-sinusoidal at low temperatures, but turns sinusoidal at T > Eg .2,3 This behavior presents striking analogies with persistent current of purely normal rings.6

Whereas the dc properties of the proximity effect (PE) at equilibrium in long SNS junctions are well understood theoretically and experimentally, its dynamics is a more complicated and yet unresolved issue. In particular the ac current response to a time dependent phase which affects both ABS energy levels and their populations still needs to be determined. At least two relaxation times should play a role, one related to the relaxation of the populations following a change of the energy levels, another related to the dynamics of the Andreev levels and possibly also a contribution due to the generation of quasiparticles. Which of the electron-electron, electron-phonon, diffusion or dephasing times are relevant in these processes? Experimentally, there are many ways in which to impose an out of equilibrium situation. Because of the Josephson relation linking voltage to the time derivative of the phase, a voltage bias across the junction causes a time dependent phase and the I(V) curve is a probe of the dynamics of the PE. Irradiating the junction with an RF field, which produces Shapiro steps in the I(V), is another. Both the critical current in current biased experiments9,11 and the complete dc Josephson current vs phase relation3 have been shown to be very sensitive to RF excitation at frequencies of the order of or larger than 1/τD . In addition, the electron-phonon scattering rate has also been shown to set the frequency scale for the hysteresis in current biased experiments11.

In contrast to these previous experiments on SNS junctions3,9,10,11 probing the dynamics in strongly non-linear regimes, in this paper we present a linear response experiment at high frequencies. We probe the dynamics of ABS by inductively coupling an NS ring to a multimode superconducting resonator, thereby implementing both a phase bias and current detection at high frequency. Whereas at low temperature the response is essentially non dissipative and given by the flux derivative of the Josephson current a quite different behavior is observed at large temperature compared to the minigap. The frequency dependence of the response reveals, in addition to the dissipation-less current, a strong dissipative component, with a Debye-type frequency dependence and relaxation time of the order of τD . The dc phase dependence of the high frequency current response is very different from the dc supercurrent and strongly non-sinusoidal. We show that this high harmonics content is well explained by phase dependent Andreev levels with a time-independent population, suggesting that the energy relaxation time is by far longer than the nanosecond time-scale probed by the experiment.

Results

The experiment consists in inductively coupling an NS ring to a multimode superconducting stripline resonator operating between 300 MHz and 6 GHz, see Figs. 1 and 2. Two different NS rings were probed, differing only by the length of the S loop.

Figure 1
figure 1

Left: Principle of the experiment, the ac current in the resonator (double meander line) induces a flux modulation through the NS ring.

The complex susceptibility of the ring modifies the resonator impedance. Right: Electron micrograph of the NS rings inserted in the niobium resonator, (meander line structure below). The S part large ring is attached to the resonator whereas the small one is not. This difference only affects the calculation of the inductive coupling between the ring and the resonator.

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Figure 2
figure 2

Raw data χ′m(Φext) and χ″m(Φext), see Eq. (2), (thick lines) measured on the large ring, as a function of the applied external flux and same data corrected for the geometrical inductance as a function of internal flux: χ′(Φint) (triangles) and χ″(Φint) (circles), at 670 mK and 1K and f1 = 365 MHz. At 1 K χ and χm are barely distinguishable.

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The N part of the NS ring is a mesoscopic high purity Au wire 1.5 × 0.3 µm2, 50 nm thick. The estimated normal state resistance is RN = 1/GN = 0.5 Ω corresponding to a conductance g = 2 × 104 in units of 2e2/h. The normal wire is patterned between the resonator meander lines by e-beam lithography and thermally evaporated onto the substrate. The S part is constructed with a W nanowire deposited using a focused ion beam. The sample is thus an ac-squid embedded in the resonator. The dc superconducting phase difference ϕ at the boundaries of the N wire is imposed by a magnetic flux Φ dc created by a magnetic field perpendicular to the ring plane:

with Φ0 = h/2e the superconducting flux quantum. In addition, an ac flux δΦ w cos(wt) is generated by the ac current in the resonator. At low enough frequency the current in the NS ring should follow adiabatically the oscillating flux and the ac response should be entirely in phase, given by the flux derivative of the Josephson current ∂IJ (Φ dc )/∂Φ, also called the inverse kinetic inductance. But at high frequencies the ac current δIw in the ring can have both an in-phase and out-of-phase response to the ac flux:

where χ = χ′ + iχ″, the complex susceptibility of the ring, is related to its complex admittance Y = 1/Z by χ = iwY. The aim of the experiment is to determine χ(ϕ) of the ring at the successive eigenfrequencies of the resonator. It is deduced from the periodic flux dependences of the resonance frequencies and inverse quality factors measured with high accuracy (see methods). It is important to distinguish the internal flux Φint from the applied flux Φ ext related by Φint = Φ ext + LgIJ (Φint) where Lg ∼ 15 pH is the geometrical inductance of the ring. This screening effect concerns both the ac and dc components of the flux leading to the following relation between the intrinsic susceptibility of the ring χ and the measured one χ m :

These differences between χ m (Φext) and χ(Φint) are undetectable when Lg is small compared to the kinetic inductance (see Fig. 2 data on the large ring at 1 K). This is not the case at lower temperatures when the parameter β = 2πLgIc (T)/Φ0 is of the order of unity or larger. This is why we have investigated different size of rings with normal junctions of identical sizes (see Fig. 1), the small ring (Lg = 2.5 pH) is such that β ≤ 0.5 down to 50 mK. The large loop (Lg ∼ 15 pH) is more adapted to the investigation of higher temperature data since hysteresis, corresponding to β ≥ 1, occurs below 600 mK. We nevertheless exploited the low temperature data on this large loop to determine the Thouless energy of the N wire which is one of the key parameters. Ic (T) is first deduced from the temperature dependence of the hysteresis amplitude δΦ ext (T) = 2LgIc (T). ETh is then obtained from the expression of , calculated from Ref.14. We find .

Below we discuss the phase dependence of χ(ϕ) with ϕ deduced from Φint through Eq. (2) and replacing Φ dc by Φint. Fig. 3 and Fig.4 present χ′(ϕ) and χ″(ϕ) for different temperatures and resonator eigenmodes. Each curve is the average of 30 to 100 magnetic field scans. At a low temperature compared to the minigap the response is essentially non-dissipative with no oscillations detectable in χ″(Φ). χ′(Φ) is practically the flux derivative of the Josephson current, see Fig. 3. A very different behavior is found at temperatures larger than T = 0.5 K as shown in Fig. 4. The first surprise is the observation, beside the non-dissipative response χ′, of a larger flux dependent dissipative response χ″, over the entire frequency range investigated (365 MHz to 3 GHz). In addition, χ′(ϕ) is not a pure cosine even though the dc Josephson current is sinusoidal at these temperatures, implying that χ′(ϕ) is not simply χ J (ϕ) = (2π/Φ0)∂IJ (ϕ)/∂ϕ. We define the quantities δχ′ and δχ″ as the flux dependent parts of χ′ and χ″. The frequency dependence of these quantities are plotted in Fig. 5 for different temperatures above the minigap. We find that δχ(w) follows a simple Debye-type dependence, δχ(w) = δχ(0)/(1 + iwτ), with: δχ(0), of the order of χ J as expected for the w = 0 limit and the relaxation time τ equal to 0.6 ± 0.2 ns, roughly 7 times τD . When expressed in terms of admittance like in the theoretical work in21 δY(w) = δχ(w)/iw, our findings suggest that the flux dependent part of exhibits an effective Drude like frequency dependence which is independent of frequency at w≪1/τ. We find no significant temperature dependence of τ between 0.5 and 1 K. However our low temperature results indicate that this Debye-type relaxation mechanism does not describe the dynamics below the Thouless energy.

Figure 3
figure 3

(a)and (b) External and internal flux dependence of χ′ measured on the small ring at 50 mK and 600 MHz in comparison with theoretical predictions assuming an adiabatic response of the ring.

A quantitative agreement between theory is obtained taking Ic = 110 µA which is comparable to the expected value Ic = 150 µA = 10.8ETh /Φ0. There is no detectable flux dependent signal on χ″. (c) Comparison between the integrated signal and the predicted current phase relation.

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Figure 4
figure 4

Phase dependence of χ′ and χ″ for different temperatures 0.55, 0.67 and 1 K all greater than the minigap and 2 different frequencies f=365MHz and 1500MHz.

The amplitude of the Josephson currents calculated are IJ (0.67K) = 18 µA and IJ (1 K) = 6 µA. Circles: fit with , see expressions (5) and (6) calculated from Usadel equation21 at temperatures corresponding to kB T = 5, 6 and 8ETh giving the best agreement with experimental data. The amplitude of the theoretical curves has been rescaled by a factor 0.3. All the curves have been arbitrarily shifted along the vertical axis.

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Figure 5
figure 5

Frequency dependence of δχ′ and δχ″ for two different temperatures above the minigap.

Continuous lines: fits according to Debye relaxation laws with τ = 0.6 ns. Inset: frequency dependence of the ratio δχ″/δχ′ for several temperatures. The linear dependence in 2πf confirms the validity of the Debye relaxation fits. Circles: T = 1 K, crosses: T=0.67 K.

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Discussion

We now discuss possible explanations of this linear-response data at temperatures larger than the minigap. One cause of dissipation is the relaxation of the populations of Andreev levels which was invoked to explain non-equilibrium effects in voltage biased configurations such as fractional Shapiro steps.15 This relaxation is characterized by the inelastic time τ in which is the shortest of the electron-electron and electron-phonon scattering times. At temperatures larger than the minigap, it yields a frequency dependent contribution proportional to the sum over the spectrum of the square of the single level current. Note the similarity between this relaxation mechanism and the finite temperature persistent current relaxation mechanism in Aharonov Bohm mesoscopic N rings discussed in Ref.16,17,18. The susceptibility is hence

where the non-adiabatic contribution can be written in the continuous spectrum limit in terms of the spectral current J(ɛ) and the density ρ(ɛ) of the the Andreev levels as: . This function exhibits a sharp anomaly at ϕ = π related to the closing of the minigap and when wτin≫1 this anomaly contributes substantially to χ(ϕ), reproducing our experimental findings at temperatures larger than the minigap (see Figs. 4 and 6). In the temperature range of the experiment, T ≤ 1 K, τ in should be the electron-electron scattering time24,25. Given the S boundaries τ in is probably of the order or even longer than its value in a finite length quasi 1D wire, itself much longer than the phase coherence time of an infinite wire or of the same wire connected to normal reservoirs. Since in the experiment , at 1 K, much longer that the nanosecond time scales investigated. Thus already at MHz frequencies Andreev level populations do not relax and are effectively frozen at their dc values. Equation 5 leads to and in this limit.

Figure 6
figure 6

Top: schematic representation of the phase dependent minigap closing linearly at ϕ = π.

Bottom: dotted line, adiabatic low frequency responseχ0(Φ) of an NS ring ∂IJ /∂Φ at kB T = 8ETh . Continuous line: non-adiabatic contribution obtained when the frequency is large compared to the inverse relaxation time of the populations. F(ϕ,T)9,21 was calculated using Usadel equations20 in the limit , . Bold line: total response calculated, χin(ϕ) = χ0(ϕ) − F(ϕ,T), for kBT = 8ETh and wτin = 100, reproducing experimental results as shown on Fig. 4.

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Whereas the above model describes the measured phase dependence χ′(ϕ) quite well (Fig. 4) for the large temperature data, it predicts a frequency independent amplitude in the investigated frequency range, which is not what is observed experimentally. Moreover the predicted χ″ is more than two orders of magnitude smaller than the experimental values and has a predominant π periodic component (due to the squared spectral current in the non-adiabatic contribution) which contrasts with the 2π measured periodicity. The experimental Debye relaxation of Fig. 5 must therefore be due to a second, different relaxation time τ much shorter than τ in. It is indeed expected that as the driving frequencies become of the order of the inverse diffusion time, the Andreev levels cannot follow the phase oscillations. The relevance of the diffusion time for superconductivity in a disordered superconductor in the vicinity of Tc was already discussed long ago9,22. The authors found an effective Ginzburg-Landau time in a finite wire τGL = (π2/4)τD . This justifies to fit the experimental results at w≫1/τ in with

We show in Fig. 4 that the phase dependence of χ′ and χ″ are well reproduced by Eq. (6) with τ = 0.6 ns and , the infinite frequency limit of (5). We use a single rescaling factor 0.3 for all data, attributed to errors in the estimation of the mutual inductance between the ring and the resonator. Recent work26 on the finite frequency response of a SINIS junction where the N mesoscopic wire is isolated from the electrodes by an insulating barrier lead to a frequency dependent response with a characteristic time proportional to τD and similar flux dependences for χ′ and χ″, as observed here. These results refer to the situation of temperature above the minigap, since its value is strongly suppressed by tunnel barriers. On the other hand, the computation of the high frequency impedance of a SNS junction with highly transmitting N/S interfaces recently performed in21 emphasizes the excitation of quasiparticles above the minigap as another dissipation mechanism. These results agree with our low temperature data and high temperature data at moderate frequency but the Debye-type frequency dependence relaxation is not reproduced. Thus it seems plausible that the unexpected dissipation mechanism we found is related with diffusion of thermally excited (above the minigap) quasiparticles across the N part of the structure.

In conclusion we have measured the high frequency linear response of NS rings. At kBT≪Eg the response is essentially non-dissipative and given by the flux derivative of the Josephson current. At kBT≫Eg our results show the existence of a strong dissipative component in addition to the expected dissipationless current. Moreover we also observe a strong frequency variation of the flux dependent kinetic inductance beyond adiabaticity, revealing the existence of two different relaxation times: (i) The inelastic relaxation time of the populations of the Andreev levels, much longer than the experimental investigated time scales. It explains the harmonics content of the dc flux dependence of χ′(Φ) resulting from the freezing of those populations which do not follow the time dependent flux, (ii) Another, much faster time scale, responsible for the observed dissipative response χ″(Φ) and the high frequency decrease of χ′(Φ) related through a Debye-type frequency dependence. This second temperature independent time scale in the ns range may be related to the diffusion time through the N segment. On the theory side, this calls for finding a relaxation time scale of the order of the diffusion time, which does not naturally show up in the frequency dependent Usadel equations in21. We also show that the phase dependent dynamics of a N wire between S reservoirs is fundamentally different from the dynamics of the same N wire between normal reservoirs. These measurements revealing the dynamics of the Andreev states could be extended to a wide range of SNS systems where the N part may be a semiconducting nanowire, a carbon nanotube, or a graphene ribbon.

Methods

Resonator and measurement of the susceptibility of the ring

The resonator is made of two parallel superconducting Nb meander lines (2 µm wide, LR = 20 cm long and 4 µm apart, see Fig. 1) on a sapphire substrate [12]. One line is weakly coupled to a RF generator via a small on chip capacitance, adjusted to preserve the high Q of the resonator (80 000 at 20 mK), the other one is grounded. The resonance conditions are , where n is an integer and λn the electromagnetic wavelength. The fundamental resonance frequency f1 is 365 MHz. We were able to detect more than 30 harmonics. The resonator is enclosed in a gold plated stainless-steel box, shielding it from electromagnetic noise and cooled to mK temperature. The high Q ensures an extreme sensitivity: , enabling very accurate ac impedance measurements of mesoscopic objects. This technique inspired by experiments performed at lower frequency on short Josephson junctions27,28 was previously employed in contactless measurements of purely normal (N) Aharonov Bohm rings.12 A single NS ring gives more signal than 104 N rings, for two reasons: First, the supercurrent in an NS ring is g times larger than the persistent current in a N ring of the same size as the N wire. Second, the length of the S wire can be adjusted to optimize the ring/resonator coupling. The dc flux dependent susceptibility is deduced from the flux induced variations in the resonance frequencies and inverse quality factors δfn (Φ dc ) and of the resonator:

Here is the resonator inductance, the mutual inductance between the ring and the two resonator lines was M ∼ 5 pH and 0.2 pH for the large and the small rings, respectively and kn accounts for the spatial dependence of the ac field amplitude at frequency fn .

SNS junctions using a focused ion beam

The superconducting electrodes on both sides of the Au wire are made from W nanowires deposited using a focused Ga ion beam (FIB) to decompose a tungstene carbonyle vapor. The superconducting transition temperature of these W wires is 4 K, their critical current is 1 mA and critical field is greater than 7 T.29 The superconducting gap measured by scanning tunneling spectroscopy30 is Δ = 0.8 meV. This technique provides a good NS interface, thanks to the light etching concomitant to the W deposition. We have checked that SNS junctions fabricated using this technique exhibit supercurrents and Shapiro steps comparable to long SNS junctions made with more conventional techniques.13