Dissipation and supercurrent fluctuations in a diffusive NS ring
A mesoscopic hybrid Normal/Superconducting (NS) ring is characterized by a dense Andreev spectrum with a flux dependent minigap. To probe the dynamics of such a ring we measure its linear response to a high frequency flux, in a wide frequency range, with a multimode superconducting resonator. We find that the current response contains, beside the well known dissipationless Josephson contribution, a large dissipative component. At high frequency compared to the minigap and low temperature we find that the dissipation is due to transitions across the minigap. In contrast, at lower frequency there is a range of temperature for which dissipation is caused predominantly by the relaxation of the Andreev states’ population. This dissipative response, related via the fluctuation dissipation theorem to a non intuitive zero frequency thermal noise of supercurrent, is characterized by a phase dependence dominated by its second harmonic, as predicted long ago limpitsky ; martin96 but never observed so far.
A phase coherent non superconducting conductor (N) connected to two superconductors (an SNS junction) gives rise to the formation of Andreev states (AS) which are coherent superpositions of electron and hole states confined in the N metal and carry the Josephson-supercurrent kulik . They strongly depend on the phase difference between the two superconductors. The quasi-continuous Andreev spectrum of a diffusive metallic wire exhibits a phase modulated induced gap, the minigap, which closes at odd multiples of (for perfectly transmitting contacts), and can be approximated by heikkila ; spivak ; blatter . In the case of a wire longer than the superconducting coherence length, , with the Thouless energy , and the diffusion time across the N wire. Whereas most investigations of diffusive SNS junctions rely on non linear transport measurements of current voltage curves, switching current and ac Josephson effect dubos ; lehnert , only few experiments probe the Andreev spectrum at equilibrium in a phase biased configuration with NS rings threaded by an Aharonov Bohm flux. These include the tunnel spectroscopy of the minigap lesueur and the measurement of the flux dependent Josephson supercurrent using SQUIDillichev or Hall probe magnetometry strunk . Beyond probing the equilibrium AS spectrum in a static magnetic flux, the investigation of the dynamics associated to this spectrum, a far more complex question, has been addressed experimentally only recentlychiodi2011 . Theoretically, it was predicted that, in contrast to tunnel Josephson junctions noiseJJ and because of the smallness of the induced gap, SNS junctions should exhibit low frequency supercurrent fluctuations at equilibrium martin96 . According to the fluctuation dissipation theorem, in the linear response regime, such equilibrium fluctuations lead to a dissipative current under an ac flux excitation kulikintau ; virtanen . In this Letter, we present the linear current response of a phase biased NS ring and account for both the non dissipative component and the more surprising dissipative one, over a wide frequency range. We identify two fundamental processes leading to dissipation, the microwave-induced transitions across the minigap, and the energy relaxation of Andreev level populations.
To this end, we couple a NS ring to a superconducting resonator, and phase bias it with a dc Aharonov Bohm flux and a small ac flux at the resonator’s eigenfrequencies . The linear current response is characterized by the complex susceptibility , where is the NS ring’s admittance. This susceptibility is extracted from the variations of the resonator’s eigenmodes (frequency and quality factor). The ring’s dissipationless response is deduced from the periodic flux variations of , whereas the dissipation corresponds to .
A first experiment chiodi2011 found a large dissipative response as well as a non dissipative one that differed notably from the adiabatic susceptibility, the simple flux derivative of the ring’s Josephson current (the inverse kinetic inductance). These results were partially explained by the theory of the proximity effect virtanen . However, the shape of the flux dependences of did not vary with frequency (in the range explored), so that the different components of the ring’s dynamical response could not be accessed. In particular, with the inelastic scattering rate much smaller than the lowest eigen-frequency, the dissipative response associated to the relaxation of Andreev states could not be detected.
In the present experiment, we report on a NS ring with enhanced temperature dependent inelastic scattering rate , thanks to a thin Pd layer at the NS interface, between the normal gold mesoscopic wire and the superconducting niobium loop. The higher inelastic scattering rate, combined to a broader frequency and temperature range, lead to the identification of the two fundamental contributions to the supercurrent relaxation. At frequencies above the inelastic scattering rate, dissipation is due to microwave-induced excitations across the minigap. In the opposite regime of lower frequency, which could not be reached previously, dissipation is due to the relaxation of Andreev level populations.
Here we reveal this second contribution, proportional to the sum of the squared Andreev level currents. Accordingly we measure a response whose period in flux is nearly periodic. The extra cusps we find at odd multiples of reflects the closing of the minigap. This characteristic phase dependence, which is precisely that of the low frequency, thermal supercurrent noise, is in complete agreement with theoretical predictions formulated long ago limpitsky ; martin96 ; virtanen .
The experimental set-up is shown in Fig.1a, the resonator consists in a double meander line etched out of a 1 micron thick niobium film sputtered onto a sapphire substrate. The NS ring connects the two lines at one end of the resonator, turning it into a line with a fundamental frequency of 190 MHz, and harmonics 380 MHz apart. A weak capacitive coupling to the microwave generator preserves the high quality factor of the resonances, which can reach 5 up to 10 GHz. The NS ring is fabricated by electron beam lithography. The Au wire (4 micron long, 0.3 micron wide and 50 nm thick) is first deposited by e-beam deposition of high purity gold. The S part is deposited in a second alignment step by sputtering of a Pd/Nb bilayer (6 nm Pd, 100 nm Nb). The resulting uncovered length of the Au wire is 1. The ring is connected to the Nb resonator in a subsequent step, using ion-beam assisted deposition of a tungsten wire in a focused ion beam (FIB) microscope. This process creates a good superconducting contact between the resonator and the Pd/Nb part of the ring. The 6 nm-thick Pd buffer layer ensures a good transparency at the NS interface, as demonstrated by the amplitude of the critical current measured with dc transport measurements on control SNS junctions fabricated simultaneously (Fig.1c). It also enhances the inelastic scattering rate because of Pd’s spin-wave like excitations (paramagnons) raffy ; dumoulin ; elke . Considering that the phase coherence time extracted from weak localization measurements on a 6nm thick Pd thin film raffy , was of the order of ns at 1 K, which is longer than the estimated diffusion time = through the Au wire between the S contacts, we do not expect a reduction of the critical current as confirmed by measurements in the control samples.
The quantities we measure are the variations with dc flux of the resonator’s quality factor and eigen-frequencies and . They are simply related to the oscillating phase dependent part of the complex susceptibility, characterized by and , where the superconducting phase is related to the flux threading the ring by where is the superconducting flux quantum. The relation reads chiodi2011 :
The coupling inductance is due to the S part of the N/S ring; 0.3 is the inductance of the resonator. These expressions are valid at temperatures such that the kinetic inductance of the SNS junction is larger than the ring’s geometrical inductance (outside this range screening of the applied flux, both dc and ac, needs to be considered chiodi2011 ). This sets the lower limit to the temperature, so that experiments were conducted between 0.4 and 1.5 K. The frequencies probed ranged between 190 MHz and 3 GHz.
We find spectacular variations of both the amplitude and shape of and as frequency and temperature are changed. At the lowest frequencies and highest temperatures investigated (see Fig.1 and Fig.2a) the dissipationless is well described by a pure -periodic cosine, as expected for the adiabatic susceptibility of the Josephson current which is purely sinusoidal at these moderately high temperatures, much larger than heikkila ; strunk . As shown in Fig.1b,1c, The amplitude perfectly reflects the expected, roughly exponential, decay of the Josephson critical current dubos , that was also measured in the control wire. We find for both samples which corresponds to . In contrast, the dissipation, characterized by , is nearly periodic (see Fig.2b) at the largest temperatures investigated and acquires a strong periodic component at lower temperature. When increasing the frequency, (Fig.1b and Fig.2a and 2c ) contains additional harmonics, with peaks at odd multiples of and moreover a local maximum at , mod for the highest temperatures. On the other hand, at 2 GHz (Fig2.c and 2.d ) and low temperature, and have identical shapes, with peaks at , mod , reflecting the underlying minigap that varies like .
In the following we exploit this complex evolution of with frequency and temperature to extract the different mechanisms at work in the dynamics of Andreev states. To this end we make use of theoretical predictions virtanen based on Usadel equations, and recent numerical simulations dassonnevilletheo inspired by the analysis of the ac response of normal mesocopic rings buttiker ; trivedi ; reulet . The response function of a NS ring has been shown to contain three contributions: . The adiabatic, zero frequency, Josephson contribution is purely real and is the derivative of the Josephson current . The second contribution, the diagonal susceptibility , is the first non adiabatic, frequency dependent contribution. It describes the Debye-like relaxation of the (phase dependent) thermal populations of the Andreev states, with a typical inelastic relaxation time according to the simple model proposed for the dynamics of persistent currents in normal ringsbuttiker ; trivedi ; reulet :
where the square of , the current carried by the n-th Andreev level of energy , appears. Finally, the non-diagonal contribution describes quasi-resonant microwave-induced transitions between two Andreev levels, involving (in contrast with ) non diagonal matrix elements of the current operator dassonnevilletheo . The contribution to the phase dependent susceptibility dominates when , as in Fig.2d. and then have similar shapes which follow approximately the minigap with peaks at and a dependencedassonnevilletheo . This high frequency regime was the only one accessed in the previous experiments on Au wires directly connected to W superconducting wires. In those experiments the energy relaxation time, limited by electron electron interactions, of the order of , was very long due to the superconducing contactschiodi2011 ; blanter96 . Therefore those measurements were always in the regime where is negligible, (Eq.2). In contrast, the Pd layer beneath the Nb contacts in the present samples considerably reduces the inelastic scattering time, leading to a substantial contribution of for the resonator’s first five eigenfrequencies. We now focus on this contribution analyzed in Fig.3 and Fig.4.
We first present the predicted flux dependence of , given by the function F
which reads in the continuous spectrum limit . Here and are respectively the spectral current and the density of states of the SNS junction. This function was introduced by Lempitsky limpitsky to describe the I(V) characteristics of SNS junctions, and was calculated numerically using Usadel equations by Virtanen et al.virtanen . At large temperature compared to , can be approximated by the following analytical form: . It is dominated by its second harmonics with in addition a sharp linear singularity at odd multiples of (see Fig.4). This is due to the dominant contribution of Andreev levels close to the minigap whose flux dependence is singular like in a highly transmitting superconducting single channel point contact martin96 .
We now show that the particular flux dependence of the function can explain the experimental data of Fig.1 and 2. We first follow the frequency dependence of the amplitude of at fixed temperature, and check that the shape of does not change with frequency and is the same as that of , as predicted for the temperature and frequency regime where the contribution of can be neglected. As shown on Fig.3b it is then possible to fit the frequency dependence of the amplitude of by the expected and determine the characteristic time for several temperatures according to Eq.2 . We find values of varying between and ns, quite similar to what was deduced from weak localisation measurements in Pd thin films raffy . Moreover the power law decrease of in , is in reasonable agreement with what is expected for paramagnons which constitute the dominant inelastic scattering at low temperature in Pd which is close to a ferromagnetic transition. It is also interesting to note that our results can be described by a single inelastic time, independent of , whereas a phase dependent is expected for electron phonon collisions in SNS junctions giazotto . This is probably due to the fact that temperature is larger than in our case.
A similar analysis can be done on , the quality of the calibration is however not as good as on . Moreover we still lack a good analytical prediction for , which gives a large contribution to at low temperature and high frequency. We have overcome this difficulty by subtracting for frequencies larger than 1.7 GHz the flux dependence of estimated from the high frequency data (2.8 GHz). The resulting amplitude agrees with the expected frequency dependence in as shown in the inset of Fig.3c. One can also compare the independently measured flux dependences of and with theoretical predictions from the Usadel equations, . This is done in Fig.4 for several frequencies and a good agreement is found.
With this set of experiments, we have thus shown that the frequency and temperature dependences of the response function of NS rings in a time dependent flux are consistent with a simple Debye relaxation model of the population of the Andreev levels. Using fluctuation dissipation theorem one can estimate the related thermodynamic current noise as:
The measurement of the ac current linear response of an NS ring to an ac flux thus reveals two fundamental mechanisms contributing to dissipation at finite frequency. One of them, predominant at high frequency and low temperature, describes the physics of microwave induced transitions above the minigap. We have clearly identified and characterized the second cause of dissipation, the thermal relaxation of the populations of the Andreev states. It is described by an inelastic rate which is extremely sensitive to the nature of the NS interface. This dissipative response is directly related to the low frequency thermal noise of the Josephson current, with a flux dependence proportional to the average square of the spectral (or single level) current, and can be precisely described by theoretical predictions. These results show that linear ac measurements in a wide range of frequency, close to equilibrium, reveal physical properties of SNS junctions that are not accessible by standard transport measurements dominated by non linear effects. The type of experiments presented here is uniquely suited to investigate more exotic systems, for instance with the normal diffusive wire replaced by a ballistic wire, leading to a discrete Andreev spectrum known to be extremely sensitive to spin orbit interactions.
We acknowledge A. Kasumov and F. Fortuna for help with the FIB and M. Aprili, F. Chiodi, R. Deblock, M. Feigelman, T.T. Heikkil, K. Tikhonov and P. Virtanen for fruitful discussions.
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