Mesoscopic conductance fluctuations in InAs nanowire-based SNS junctions

We report a systematic experimental study of mesoscopic conductance fluctuations in superconductor/normal/superconductor (SNS) devices Nb/InAs-nanowire/Nb. These fluctuations far exceed their value in the normal state and strongly depend on temperature even in the low-temperature regime. This dependence is attributed to high sensitivity of perfectly conducting channels to dephasing and the SNS fluctuations thus provide a sensitive probe of dephasing in a regime where normal transport fails to detect it. Further, the conductance fluctuations are strongly non-linear in bias voltage and reveal sub-gap structure. The experimental findings are qualitatively explained in terms of multiple Andreev reflections in chaotic quantum dots with imperfect contacts.

As a consequence of the quantum mechanical interference of electron wavefunctions the low-temperature conductance G of mesoscopic samples fluctuates when varying the chemical potential or an applied magnetic field. These conductance fluctuations were demonstrated more than 20 years ago as one of the first examples of mesoscopic quantum phenomena in sub-micron samples [1,2]. Through the Landauer formula G = (2e 2 /h) i T i the conductance can be expressed in terms of a samplespecific set of transmission eigenvalues {T i }, and the variance of the conductance fluctuations, Var G, provides important information about the statistical properties of the transmissions, such as the distribution ρ(T ) and correlations. An important energy scale for electron interference in random systems is the so-called Thouless energy E Th being the shift in chemical potential µ c ∼ E Th sufficient to uncorrelate the transport properties. At high temperatures T , strong dephasing due to inelastic scattering with rate γ φ ≫ E Th subdivides the sample into many uncorrelated parts and the conductance fluctuations are suppressed by self-averaging. As the temperature is lowered Var G increases and it is a remarkable result that when T, γ φ ≪ E Th (usually γ φ ≪ T at low T ), Var G saturates to a value on the order of (e 2 /h) 2 , independent of the sample size and degree of disorder. For this reason the phenomenon is denoted universal conductance fluctuations (UCF) [1,2] and in this regime transport remains practically insensitive to dephasing.
A fundamentally different situation occurs if the leads to the normal (N) sample turn superconducting (S). In this case, a gap ∆ opens at the Fermi level, and a sub-gap energy electron incident on the S interface cannot penetrate into the lead, but is instead coherently Andreev reflected (AR) as a hole upon injection of a Cooper pair. Instead of being a sum of {T i }, the transport properties now depend on Andreev states modified by finite-voltage V in way highly dependent on the transmissions {T i } [3][4][5][6][7] and Landau-Zener transitions between the states lead to quasi-particle current [4]. These are most probable when levels come close for T ≈ 1 and φ ≈ π as schematically illustrated in Fig. 1(a). We will show that this has important consequences for the statistical properties of the differential conductance, because its fluctuations develop extreme sensitivity to the statistics of the almost perfect channels, T ≈ 1.
This Letter presents the first study focused on this intriguing interplay of interference and Andreev processes and its consequences for the statistical properties of mesoscopic junctions. Enabled by recent progress in nanoscale device fabrication [8][9][10][11][12] we measure the lowtemperature fluctuations of differential conductance in short mesoscopic SNS devices based on semiconducting nanowires contacted by Niobium (Nb) leads. We systematically study the temperature and bias dependence of the fluctuation amplitude, the correlation potential µ c , Vg at various temperatures (for clarity the 6.5K, 17K, 180K, and 290K traces have been off-set by 1, . . . , 4e 2 /h, respectively). Inset: Scanning electron micrograph of a typical device. (c) Device schematic. and the average differential conductance, and find that the normal-lead universal limit for the fluctuations is broken in SNS devices as was also recently pointed out by Doh et al. [12]. In addition we here show that unexpectedly, the fluctuations maintain a strong dependence on T even at low temperatures (T ≪ E Th ) where the normalstate fluctuations are saturated. To explain the data we theoretically analyze how dephasing modifies the statistics of the almost perfect channels T ≈ 1 and find that transmissions T ≈ 1 are suppressed. This mechanism explains the strong temperature dependence of the SNS fluctuations and shows that they provide a much more sensitive probe of dephasing than normal UCF. Furthermore, varying the bias, we find that the fluctuation amplitude diverges as a power-law as V → 0 and we observe, for the first time, that multiple Andreev reflections (MAR) lead to sub-gap structure (SGS) in the fluctuation amplitude and in µ c . The finite bias results are compared with computations based on MAR-theory in chaotic quantum dots with imperfect contacts with good qualitative agreement between theory and experiment. From this we conclude that the results are generic for mesoscopic SNS fluctuations.
The nanowires are grown by molecular beam epitaxy and transferred to a doped Si substrate capped with 200 nm SiO 2 . Contacts to individual wires are defined by e-beam lithography, DC sputtering of 70 nm Nb following a brief etch in BHF (see Refs. [10,11] for details). The leads have a critical temperature T c ≈ 1.7 K resulting in a gap ∆ = 1.76T c ≈ 0.25 meV at low temperature. The inset to Fig. 1(a) shows a scanning electron micrograph of a typical device; the wires have diameters d ∼ 80 − 100 nm and the distance between the contacts is L ∼ 100 nm. The nanowires are n-type and the device discussed here has mobility µ ∼ 10 3 cm 2 /Vs, carrier density n ∼ 4 × 10 17 cm −3 , mean free path l e ∼ 18 nm, diffusion constant D ∼ 60 cm 2 /s, and Thouless energy E Th ∼ 0.4 meV estimated from the transfer characteristics G(V g ). Due to the design of the outer circuit, the measurable supercurrent is strongly suppressed allowing a study of the quasi-particle current alone (within the RCSJ/"tilted washboard" model of Josephson junctions the device constitute a strongly underdamped junction). We measure the two-terminal differential conductance G ≡ dI/dV using standard lock-in techniques (V ac = 12 µV, 77 Hz) while varying the bias V , back-gate potential V g (applied to the doped substrate), and temperatures from 300 K to 300 mK. In the following, data from one device is presented, but similar results have been obtained on two additional Nb-based and one Albased device demonstrating the generality of the phenomena. These results and details of the device parameters, the properties of the Nb contacts, and the device design can be found in the supplement [13].
Disorder in the InAs crystal together with a multifaceted wire surface [14] presumably make the system chaotic and the barriers formed in the NS interface dominate the resistance. Therefore we compare data with predictions from theory of MAR [4,7] and energy independent scattering Random Matrix Theory (RMT) for multichannel chaotic dots with imperfect contacts, see Fig. 1 (c) [15]. This RMT is valid for both diffusive and ballistic dots if E Th of dot+contacts is large, eV, ∆, T ≪ E Th . Thus we ignore the energy dependence of T i (relaxing this assumption makes our numerics impractical and discussion more involved [6]). For the instructive case of perfect contacts we analytically find the effect of weak dephasing (T ≪ E Th ) on the distribution ρ(T ) and of small bias eV ≪ ∆ on Var G at T = 0. For the general case of imperfect contacts (N = 16 channels, transparencies Γ L = Γ R chosen to match the experiment) the biasdependence of transport statistics is computed at T = 0. For details of the theory and a discussion of the role of contact asymmetry see Ref. [13]. Figure 1(b) shows examples of the measured G(V g ) for V = 0 V for various temperatures. For T 100 K they exhibit a large number of reproducible, aperiodic fluctuations allowing a statistical analysis of the data. To characterize the fluctuations, we extract for each trace the average G ≡ G Vg , the variance Var G = G 2 − G 2 , and from the correlation function 17]. The normal state behavior at temperatures below T c is measured by applying a magnetic field B = 0.5 T to suppress the superconductivity of the leads.
Let us first consider the role of temperature T . Figure  2 shows the temperature dependence of the extracted parameters at zero bias. For T > T c = 1.7 K, G is almost constant 3e 2 /h showing that the current is not carried by thermally excited carriers. At T = 1.7 K when the leads turn superconducting G increases as a consequence of Andreev reflections. The increase occurs over a range 1 K T ≤ T c corresponding to the T -dependence of the superconducting gap ∆(T ) (included in the figure) which, below 1 K, is very weak and G is effectively saturated.
The fluctuation amplitude displays a different dependence on T : Upon lowering T from room temperature, Var G increases as T −1.7 (solid line). This reflects the self averaging discussed above and the saturation at T ∼ 5 K agrees with E Th ∼ 5 K estimated from the transfer characteristics. The transition to superconducting leads at T = T c is accompanied by a sudden increase of Var G, but unexpectedly it keeps increasing all the way to the lowest T ; the upper inset to Fig. 2 emphasizes the low-T behavior of Var G. Thus, the T -dependence of Var G is not governed by ∆(T ). Interestingly for T 1 K Var G seems to rejoin the T −1.7 relationship that was followed above 5 K. We note that the normal-state saturation value 0.09(e 2 /h) 2 (measured with B = 0.5 T) is of the order of the theoretical normal-state universal value [18]. In the superconducting state, however, Var G reaches 2.5(e 2 /h) 2 at 300 mK, ∼ 30 times larger than the normal state value [19].
To describe this behavior we consider the Andreev states which are formed when the leads turn superconducting. These appear at energies sensitive to the phase difference of the leads φ and the transparency of the channels, ǫ i,± = ±∆(1 − T i sin 2 φ/2) 1/2 [3] as illustrated in Fig. 1(a). A finite bias V ≪ ∆/e leads to a quasiparticle current ∝ exp(−π∆(1 − T i )/eV ) since the resulting time evolution of the phase difference, φ = 2eV t/ , induces Landau-Zener transitions between low energy pairs ǫ i,± ≈ 0 [4]. Such transitions are most probable for T i → 1 and φ ≈ π, and in contrast to the normal case, transport is therefore dominated exponentially by the almost perfect channels. We therefore study the role of dephasing on the statistics of T → 1. Using the dephasing-probe model [20], Ref. [21] numerically demonstrated that in a single-channel dot dephasing suppresses ρ(T ) for T → 1. Using this approach, we consider the limit γ φ ≪ δ, (δ is the level spacing) and for 1−T ≪ γ φ /δ we have derived an exponential suppression of the transmission density, ρ(T ) ∝ exp[−β(γ φ /2δ)/(1 − T )] (β is the Dyson parameter). Extending to the multi-channel limit N ≫ 1 we find that dephasing leads to the appearance of a temperature-dependent upper bound T + = 1 − γ φ /2πE Th such that ρ(T ) = 0 for T > T + (see lower inset to Fig. 2). For normal transport this is accompanied by a practically undetectable correction −γ φ /πE Th to the UCF [21]. However, for SNS transport due to the exponential sensitivity of the current to T 's near 1 the appearance of T + makes transport strongly temperature dependent and unlike normal UCF theoretically Var G diverges, Var G → ∞, T, V → 0 (the V -dependence is discussed below). In conclusion, lowering T decreases γ φ (T ) and increases T + and thus allows the exponential contributions from progressively more transparent channels to play a role in the transport and its fluctuations thus increasing Var G. We are, at present, not able to predict the functional form of the increase and the power-law relationship Var G ∝ T −1.7 suggested by the experiment remains unexplained. Also, the combined inclusion of dephasing and imperfect contacts remains a challenging theoretical problem.
Consider now the T -dependence of µ c presented in Fig.  2. Generally, µ c reflects the dependence of the T 's on V g and provides information about the statistics of {T i } complimentary to Var G. For T > E Th the correlations depend on dephasing and thermal averaging and µ c decreases with T and saturates for T ≪ E Th to a value ∼ 0.35 meV in agreement with the previous estimates of E Th . As T is lowered from T c we observe a further dramatic increase in the sensitivity to V g (see Fig. 1) and µ c decreases significantly below its normal-state saturation value (µ c ∝ T 0.7 ). In the S-state µ c probes correlations of the nearly perfect channels as discussed above. However, the functional form of the T -dependence (and, in particular, the T = 0 value of µ c ) is a complicated and fully open theoretical problem, which needs further work.
We now discuss the bias-dependence of the transport and its fluctuations. The nature of the important ARprocesses depends strongly on V , because a sequence of n AR-processes that transfer a quasiparticle across the junction, is energetically possible only when eV n ≥ 2∆. At the same time, a large number of AR requires a high transparency. Hence, as the bias voltage is decreased an enhanced sensitivity to the tail ρ(T → 1) is indeed expected, and it is interesting to study the characteristics of the fluctuation pattern as a function of bias voltage. Figure 3(a) shows measurements of G(V ) for two different V g illustrating the strong dependence on V g . Upon lowering |V | the differential conductance shows an increase at V ≈ ±0.5 meV corresponding to enhanced quasi-particle transport when the peaks in the DOS of the leads line-up at V = ±2∆/e. The SGS at lower bias is the consequence of the bias thresholds for MAR as described above. Figure 3(b) shows a grey-scale representation of ∼ 5000 such traces covering 0 ≤ V g ≤ 9.5 V. The enhanced quasi-particle transport for |V | ≤ 2∆/e is seen throughout the plot and the pattern of tilted bands of high differential conductance observed for |V | > 2∆/e is characteristic of conventional gate and bias dependent fluctuations in disordered mesoscopic samples [22]. Extracting again G , Var G, and µ c for each bias value leads to the result of Fig. 3(c,left) which also includes normal-state data measured at 2 K. The average G increases at |V | < 2∆/e and the SGS clearly survives the averaging. The correlation potential µ c decreases slowly as V is lowered from ∆/e, whereas Var G displays a pronounced peak for V → 0. Both contain structure resembling MAR. To understand these results we theoretically consider multichannel samples at T = 0 first for eV ≪ ∆, when the generic distribution tail ρ(T → 1) ∝ 1/ √ 1 − T gives a nonlinear dependence, I ∝ eV /∆ [4]. To find the fluctuations around this value in fully coherent wires or quantum dots with ideal contacts we use the known correlators of T , T ′ [23] leading to Var G ∝ 1/V 2 . The appearance of a divergence agrees with the experiment, however, as seen in Fig. 3(d) the experiment finds V −0.8 . Therefore a more realistic model of the experiment should take into account the barriers formed in the NS interfaces. Figure  3(c,right) shows G , Var G and µ c computed for T = 0. Importantly we now find Var G close to V −1 qualitatively similar to the experimental trend. Interestingly, we see that the SGS for V = 2∆/ne n = 1, 2, . . . appears not only in the average current, but also in the fluctuations Var G and µ c . The SGS peaks in µ c do not merge to form a divergence as for Var G, but rather decrease slowly for V → 0 in qualitative agreement with the experiment. No prior theoretical results exists for µ c (V ) and therefore the found agreement with the experiment is quite satisfactory. We note, that at lowest bias the computed values of Var G are considerably larger than the measured values (factor 20), and that computed MAR peaks appear considerably sharper and higher than observed in the experiment. We attribute this to our simplifying assumptions of symmetric contacts and energy-independent elastic scattering. Most importantly, however, the strong suppression of Var G induced by dephasing, is absent in the T = 0 model. A quantitative agreement with the experiment is therefore not expected. Future studies could investigate this by repeating the measurement of Fig. 3 at various temperatures in devices with individually tunable barriers.
In conclusion, we present the first systematic study of the fluctuations of SNS-transport. We find a very large enhancement of the fluctuation amplitude compared to normal-state UCF and an extreme temperaturesensitivity Var G ∝ T −1.7 even for temperatures where the normal-state fluctuations are saturated. We argue theoretically that this can be understood as the combined effect of the almost perfectly transmitting channels dominating the transport and the cut-off of transmissions close to one with increasing dephasing. Thus, SNS fluctuations provide a sensitive probe of quantum interference which might be used for measuring weak dephasing, unavailable from normal UCF. Moreover, we reveal that the statistical properties of SNS fluctuations exhibit subgap structure as a function of bias. Good qualitative agreement is found with numerical calculations based on scattering RMT and MAR theory.
We thank J.B. Hansen for experimental support and P. Samuelsson for discussions. This work was supported by the Carlsberg Foundation, Lundbeck Foundation and the Danish Science Research Councils (TSJ).

SUPPORTING INFORMATION
Additional information relevant for the main manuscript is provided.
Presenting experimental results from three additional devices, including one with a different superconducting metal as contact material, we establish our findings as general to disordered SNS junctions. Furthermore the details of the device characteristics are discussed and we present further details on the Random Matrix Theory used to analyze the finite-bias results. We discuss the construction of the scattering matrix of the sample and the role of contact asymmetry and dephasing for the statistics of the transmission eigenvalues.

DATA FROM ADDITIONAL DEVICES
In the main manuscript (MM) results were presented from measurements of one device (S0); an InAs nanowire contacted by superconducting Niobium leads. Figure  4(a-f) shows data from one device (S1) with superconducting leads based on a Ti/Al/Ti trilayer [11] with ∆ ≈ 115 µeV and two additional Niobium-based devices (S2,S3). The qualitative behavior analyzed in relation to Fig. M3 (in the following references to figures in the main manuscript is written as Fig. Mx) is again observed for all three devices with the key features being: 1) The enhancement of G at the quasiparticle onset |V | = 2∆/e, and sub-gap structure at lower bias. 2) The strongly peaked fluctuation amplitude as |V | → 0 V (see below) and structure in Var G at lower bias, and 3) The slow decrease of the correlation potential V c for |V | → 0 V and sub-gap structure also in V c . Furthermore, for comparison, Fig. 4(g-h) shows the corresponding measurements for a similar nanowire device contacted by normal-metal Titanium/Gold leads [10]. The low-bias behavior of G , Var G, and V c remains featureless, thus confirming the significant role of the superconductors. The increased noise of the results in Fig. 4 in comparison with the results of the main manuscript is due to poorer statistics for S1-S3. Nevertheless the results supports the generic nature of the discussed phenomena.
In Fig. 5 the bias dependence of the fluctuation amplitude for samples S0-S4 is displayed on logarithmic scales where each data point represents an average of positive and negative bias values. For the samples with S-leads (S0-S3) the Var G ∝ V p dependence discussed in the main manuscript is again observed with best-fit powers −1.13 ≤ p ≤ −0.79 (inset) in good agreement with our numerical results for the fluctuations of multichannel quantum dots with imperfects contacts (details below). We ascribe the spread in p to sample-specific transparency and asymmetry of the contacts which have great influence on the fluctuation amplitude and its biasdependence (see the discussion in the main manuscript and details below). We note, that the gap ∆ S1 ≈ 115 µeV of the leads for sample S1 is significantly smaller than the Nb-based devices. Thus, at low bias the results will be more sensitive to thermal effects and noise as well as influence from by the ac-bias (V ac ∼ 12 µV ∼ 0.1∆ Al /e) required by the lock-in technique. Therefore Var G saturates at a larger bias ∼ 0.2∆ for S1 than for the other devices.
The figure also shows the corresponding data for sample S4 having normal leads. As compared to S0-S3, the fluctuation amplitude for S4 is effectively independent of bias and has a magnitude similar to that of the superconducting samples when these are biased outside the gap (|V | > 2∆/e).

DEVICE PARAMETERS
The characteristic parameters of the devices (carrier density n, Fermi wave vector k F , Fermi velocity v F , mean free path l e , number of channels N , diffusion constant D, and Thouless energy E Th ) were estimated from the measurements of the transfer characteristics and bias spectroscopy in the Coulomb blockade regime. In the following we describe the analysis for sample S0 but the parameters of S1 − S4 are obtained similarly and the values are collected in Table I. Figure 6(a) shows the linear conductance G vs. V g at a temperature of 17K where conductance fluctuations are not yet dominant (The analysis is insensitive to the temperature as the G vs. V g traces for T 150 K are very similar. Below ∼ 10 K, however, UCF makes an accurate determination of the transconductance problematic). As seen in Fig. 6(a) the wire is depleted from carriers at low gatevoltages. The threshold appears at V G,T ∼ −8 V from which on G increases linearly with V g until V g ∼ −5 V. Within the charge control model [24], which is widely used for analysis of nanowire FET's [25], the transconductance g m = ∂G/∂V g ≈ 0.6 e 2 /hV is given by g m = µC g /L 2 , where µ is the mobility, C g the capacitance to the backgate, and L ∼ 100 nm the length of the device. The capacitance is found from the V g -separation δV g ∼ e/C g of Coulomb blockade conductance peaks which appear close to pinch-off [10] (this slightly underestimates C g due to a finite level-spacing, however, this correction is not significant for the analysis). Figure 6(b) shows a measurement of dI/dV vs. bias and gate for V g ∼ −7 V exhibiting the characteristic Coulomb-blockade diamonds with δV g ∼ 65 meV yielding C g ≈ 2.5 aF in good agreement with similar studies in other nanowire devices [10]. This C g -value is somewhat lower than the result of an electrostatic cylinder-over-plane model where C g = 2πǫ0ǫrL ln(2h/r) ≈ 9 aF (ǫ 0 ,ǫ r ,L,r,h are the free-space permittivity, relative dielectric constant of SiO 2 , length of wire-segment between the leads, the wire radius and center-to-plane dis- Var G ∝ V p is observed for devices S0-S3 which have superconducting leads and the best-fit exponents are collected in the table. In contrast, the results of S4, which has normalstate (Ti/Au) leads display a strong saturation (for S4, ∆ was set to 0.25 meV to allow easy comparison with the Nb-based devices S0,S2, and S3). All results are measured at 300 mK. tance, respectively). However, this is expected since the electric field from the back-gate in our device is effectively screened by the large metal leads (see SEM-image on Fig. M1). As C g depends mainly on the geometry of the sample we do not expect it to depend significantly on V g . Using C g = 2.5 aF we get a mobility µ ≈ 960 cm 2 /Vs. The carrier density in the wire at a gate potential V g can be estimated from the charge induced by the gate-voltage with respect to the pinch-off, n = C g (V g − V G,T )/πr 2 Le (here r is the wire radius and L the length of the wire segment between the leads). This gives n ≈ 2.5 − 5.7 × 10 17 cm −3 for V g = 0 − 10 V. We note that these estimates of mobility and density are similar to other studies of InAs nanowire devices [8,26]. The Fermi wave vector k F is calculated using the 3D expression for the Fermi energy k F = (6nπ 2 ) 1/3 and using the bulk value for the effective electron mass in InAs m * = 0.026m e (m e being the electron mass) we find also the Fermi velocity v F = k F /m * . Finally, since µ = el e /v F m * we find the mean free path l e , the diffusion constant D = 1 3 v F l e , and the Thouless energy E Th = D/L 2 . In order to transform changes in V g into the corresponding change in chemical potential the parameter α = C g /C total is needed. In analogy with the standard procedure for finding α in the Coulomb blockade regime it is determined as the typical slope of the high-conductance ridges observed in Fig. M3(b) giving   Table I which includes also the results of a similar analysis for devices S1-S4.

PROPERTIES OF THE NB FILM
The results reported in the main manuscript are from a device with an InAs nanowire contacted by Niobium leads deposited by DC sputtering. In Fig. 7(c) shows traces of dI/dV vs. V at constant gate V g = 2.5 V as the temperature is lowered from 1.8 K to 0.3 K. Below T c ∼ 1.7 K the low-bias conductance increases and at lower temperatures discernable sub-gap structure develops. The dashed curve shows the temperature dependence of (twice) the gap 2∆ BCS (T ) saturating to 2 × 1.75T C = 0.5 meV at low temperature. This agrees with the observed increase of the conductance for |V | ∼ 0.5 mV as the peaks in the density of states of the leads line up at 2∆/e. The critical temperature measured in the device is considerably lower than that of bulk Nb which has a critical temperature of 9.2 K. Such differences between bulk properties of the lead material and the actual properties of the nanoscale devices is often observed. For example, in Refs. [11,27] aluminum was used for contacting carbon nanotubes and InAs nanowires, respectively, with an observed transition temperature of 750 mK considerably lower than T c of bulk Al (1.2 K). In the case of the present device the dramatic decrease of T c may be due an impure Nb film in the interface between contact and nanowire due to a reaction of the sputtered Nb with outgassing from the electron-beam resist (PMMA) on the substrate or traces of oxygen during sputtering [28,29]. The critical magnetic field of Niobium can exceed several Tesla depending on the quality and geometry of the film, making Niobium a good candidate for nano-structure based SQUID's [30] where robustness to an external magnetic field is desired. In our case, however, the critical field turns out to be relatively small which is beneficial as we can then measure the normal-state behavior for temperatures below T c . Figure 7(a) and (b) shows dI/dV vs. V and V g with B = 0 T and B = 0.5 T, respectively. For B = 0 the conductance exhibits pronounced sub-gap peaks due to Andreev reflections (similar to Fig. M3(b)). The sub-gap structure disappears upon the application of ∼ 350 mT, however, at B = 0.5 T, a small conductance decrease remains for |V | ≤ 0.1 mV as also seen in Fig. 7(d). This feature repeats for all gate-voltages and indicates that some reminiscence of superconductivity may still exist. From the measurements for T > T c in Fig. M3 and of sample S4 in Fig. 4 it is known that in the normal state the statistical properties (Var G, G and µ c ) are bias-independent for small V . Therefore, to ensure that the normal-state data reported for T < T c in Fig. M2 (solid points) are free from superconducting correlations they are measured with B = 0.5 T and a small bias V = 0.2 mV.

CONSIDERING THE SUPPRESSED SUPERCURRENT
The theoretical supercurrent through the wire is given by [31] I c = π∆ BCS /2eR n where R n is the normal state resistance of the device. In our case R n ∼ (3e 2 /h) −1 which yields a theoretical supercurrent of I c ∼ 30 nA. The measurable supercurrent, however, depends on the external circuit and in the extended  RCSJ/"tilted-washboard" model [32] it depends on the quality factor Q, where Q −1 = ω p (RC + 2e 1 IC Rn ) and ω p = 2eI C / (C(1 + R/R n ) + C j ). Here C j is the junction capacitance and I J = I c sin(φ) is the phase-dependent super-current through the junction. C is the relatively large area bonding-pad capacitance and R is the resistance of the on-chip wiring connecting the bonding pads to the device. The circuit is schematically shown in Fig. 8, where R W ∼ 50 Ω is the resistance of the wiring of the cryostat. In Ref. [27] efforts were made to optimize these parameters for a large measurable supercurrent in carbon nanotube based junctions (essentially by making R large). In the present case of the Nb-NW-Nb junctions we have C j ∼ 2 fF, R n ∼ 10kΩ, C ∼ 1 − 2 pf, and R 10Ω, and with these parameters we get Q ∼ 10 showing that the measurable supercurrent is strongly suppressed. This is consistent with the experimental results: At all gate voltages we observe the reminiscence of the highly suppressed supercurrent as a narrow weak peak in dI/dV at zero bias (see Fig. M3(a)). The peak is, however, negligible compared to the quasiparticle conductance which allows us to analyze the results in terms of the quasiparticle transport and Andreev reflections alone.

POSSIBLE EFFECTS OF THERMAL NOISE AND Vac
In principle the temperature dependence presented in Fig. M2 could be affected by thermal noise and the finite excitation voltage of the lock-in amplifier V ac = 12µeV. Due to down-mixing by the nonlinear SNS device, thermal noise may result in a dc voltage V N (T ) ∝ √ T . Since a bias-dependence of Var G is observed in Fig.  M3 at T = 0.3 K down to V ∼ V ac , we conclude that Only at lowest temperature do our data reach 2.6, and at higher T values significantly lower than the Nyquist upper bound are observed. Thus, we conclude that the power-law extracted from our data cannot be governed by a thermal noise. Further, the values of Var G are affected by V ac only at the lowest temperature.

CONSTRUCTION OF S-MATRIX
Since we expect that the fluctuational phenomena we consider are generic, particular geometry of the sample is not expected to make qualitative difference. On one hand, the barriers on the NS boundaries are the re-gions where the main voltage drop occurs. The mesoscopic sample itself can then be considered point-like. On the other, the samples are disordered due to impurities. These two features allow us to consider a chaotic quantum dot as a realistic geometry for the experiment. Importantly, the quantum dot model allows us to study how the fluctuations depend on the voltage V g on a nearby electrostatic gate and on the imperfect contacts. Such information, unavailable from other models, is important for the comparison to the experiment: An open dot with N ballistic channels is fully characterized by its N × N scattering matrix U. Usually U is assumed to be uniformly distributed over the ensemble of unitary matrices of relevant symmetry (Dyson Circular Ensembles with β = 1, 2, 4). However, a gate voltage V g coupled to the dot via capacitor C, see Fig. 9, shifts the chemical potential (the bottom of the band) and thus affects the matrix U. Variations in V g lead to a shift of the dot's Hamiltonian and therefore The unit matrix of N × N size is denoted by 1 1 N and the coupling M × N matrix W consists of the matrix ( √ M δ/π)1 1 N and zeros in the lower M − N rows [33]. For N ≫ 1 universal results (independent of M ) are reached only when M → ∞. In the limit N ≪ M the Hamiltonian Gaussian Ensemble was shown to give the same uniform distribution for U as the Circular Ensembles [34]. Equation (1) is essential for finding the role of V g in numerical mesoscopic averaging.
2) Effect of imperfect contacts. For ballistic contacts, the scattering matrix U is taken from the Circular Ensemble directly, or by using Eq. (1). For a dot with imperfect channels with transparency {G i }, 0 ≤ G i ≤ 1, i = 1, .., N the scattering matrix S is distributed according to the Poissonian ensemble [15]. A representative of this ensemble can be obtained from the matrix U of an open dot after including possible multiple reflections from the contacts, see Fig. 9: Indeed, expanding the last term to n-th order in U accounts for n − 1 internal reflections before the electron exits through one of the contacts. We use quantum dots to model the SNS samples and find the fluctuations of conductance G = dI/dV . Each sample is specified by its S-matrix, and we find its set of transmission eigenvalues {T i }. Using the scattering theory which includes multiple Andreev reflections (MAR) theory in SNS structures, developed by Averin and Bardas [35], we compute the current I as a sum of currents in each channel for the slightly shifted voltages to find the sample-specific G. Repeating this calculation for many different S-matrices allows to find statistical properties of G.
In general, the mesoscopic averaging can be performed after measurements in many samples, or by using a single sample and varying the gate voltage V g . Justification for this widely used procedure comes from the hypothesis that energy averaging equals ensemble averaging. Therefore, the mesoscopic averaging denoted by ... should be understood as ... Vg .
Experimentally, correlations for traces taken at the same bias voltage V and gate voltages shifted by δV g are quantified by the correlator F [17], The variance of the conductance is then given by Var G = F (V, 0). For large δV g the conductances become completely uncorrelated, F (V, δV g ) → 0, and the correlation . 9: Imperfect contacts for a dot with N1,2 channels are modeled by transmissions G1,2 for the contacts. The S matrix in Eq. (2) accounts for multiple reflections due to G1,2 = 1.
The gate voltage Vg varies the potential in the dot via a capacitor Cg. The coupling of the dot to a probe with N φ channels and transmission G φ corresponds to the dephasing rate γ φ = N φ G φ δ. Uniform dephasing for a given γ φ is reached at potential µ c is defined as the shift in chemical potential µ c which diminishes the correlator twice, In normal transport through a dot with ballistic contacts µ c is naturally measured in units of its escape rate, or the Thouless energy of an open dot, E Th = N δ/2π = /τ d . For example, for normal linear conductance µ c = E Th , β = 1, 2 and it reaches its maximum [(2 1/2 +3 1/2 )(2 1/2 −1)] 1/2 E Th ≈ 1.14E Th in the crossover between β = 1, 2. If the contacts are imperfect with equal transmission G, the Thouless energy (and thus µ c ) diminishes according to the time an electron typically spends inside the dot, E Th = N Gδ/2π. In SNS transport, to find µ c (V ) we have to numerically solve Eq. (4) (iteratively by the Newtonian method). We substitute the gate voltage averaging in Eq. (3) by averaging over several hundred (100-400) samples at each iteration, their S matrices being found combining the procedure in 1)-2). While the correlator F is mathematically well-defined in Eq. The Random Matrix Theory (RMT) described here uses energy-independent S matrices for electrons with (generally) different kinetic energies. This assumption is valid if the electrons are close to the Fermi level. A typical energy of an electron is limited by the largest energy among the temperature scale k B T , the superconducting gap in the contacts, ∆(T ), and the bias eV . The scattering matrix S can be taken energy-independent if T, ∆, eV ≪ E Th = N Gδ/2π due to large level spacing δ and good conduction of the sample, N G ≫ 1.

EFFECT OF IMPERFECT CONTACTS
Analytical results for Var G in SNS transport can be obtained after combining transmission correlators K(T , T ′ ) [23] with the MAR theory of Averin and Bardas [35], who showed that for eV ≪ ∆ only channels with high transmissions, T ≈ 1, are important. Landau-Zener transitions between Andreev bound states lead to nonlinear current I ∝ (eV /∆) 1/2 in diffusive wires [36]. We point out that ρ(T ) ∝ 1/ √ 1 − T is generic for mesoscopic samples with perfect connection to the leads, such as diffusive wires or quantum dots with ballistic contacts, and obtain Var G is obtained from Eq. (5) after differentiating both sides of equation ∂ V ∂ V ′ and setting V ′ = V . The final result is Var G ∝ 1/V 2 , eV ≪ ∆. Numerical results for multi-channel dots with ballistic contacts, N = 16, indeed show this behavior. However, this instructive example does not take the contacts into account, which are important for the current experiment. The opposite limit is a random mode mixer, or the Fabry-Perrot interferometer, where electrons gain random phases traversing between the contacts. The internal reflections are absent and the main resistance comes from the contacts ('opaque mirrors') [37] [38]. This model is thus relevant for almost perfect conductors and its results are also universal. The distribution ρ(T ) of such a random Fabry-Perrot interferometer depends on the transparency G 1,2 of the contacts, and is bound by The transmission can be parameterized by a uniformly distributed phase ϕ ∈ [0, 2π] as T = T + T − /(T + cos 2 ϕ/2 + T − sin 2 ϕ/2). Even though the correlator K(T , T ′ ) ∝ 1/(β sin 2 (ϕ − ϕ ′ )/2) is formally independent of the cut-offs T ± , they do affect fluctuations for eV ≪ ∆: Asymmetry in transmissions of the contacts G 1 = G 2 modulates T ± in Eq. (7) and suppresses currents and their fluctuations. Importantly, the appearance of a fixed cutoff for perfect channels affects the current fluctuations exponentially. One reason for the appearance of T + < 1, the contact asymmetry, is obvious from the last example, and below we consider it more quantitatively for chaotic quantum dots. The universality of the results (5), (8) does not hold in a general situation: the shape of ρ(T ) and the cut-off values T ± , found by the methods of Refs. [39], [18] depend on conductance g N and the contact asymmetry, N 1 = N 2 or G 1 = G 2 . Symmetric contacts with N 1 = N 2 still yield ρ(T → 1) ∝ 1/ √ 1 − T , even for G 1 = G 2 ≪ 1, but for asymmetric contacts we can have ρ(T → T + − 0) ∝ T + − T . For transmission distribution ρ(T ) we can take into account the contact asymmetry following the method of Ref. [18]. We find that the perfect channels do not vanish, ρ(1) = 0, if the contacts are not very asymmetric, and ρ(T ) near T = 1 is suppressed compared to the universal distribution as whereG ≡ G 1 +G 2 −G 1 G 2 . Indeed, for N 1,2 = N/2 perfect channels exist when the contact transparencies are close, In the limit G 1,2 ≪ 1 this condition can be easily violated by a relatively small difference in G. We can not analytically predict the behavior of the current fluctuations, since K(T , T ′ ) is unknown and most probably non-universal. However, we assume that our contacts are not very asymmetric and ρ(1) = 0. Even if asymmetry in contacts does affect Var G in our experiment, it can not account for a strong temperature dependence of our data. To explain the strong T -dependence of our results, we consider the effect of dephasing on the transmission statistics..

EFFECT OF DEPHASING
As discusses above, in the low-temperature limit, γ φ , T ≪ E Th , and small bias voltages, eV ≪ ∆, the statistics of T close to perfect transmission become extremely important for SNS transport. As seen in Eq. (8), the appearance of a cut-off T + < 1 strongly suppresses Var G. Our samples are expected to combine effects of imperfect contacts and dephasing. For simplicity we now consider the role of dephasing alone, assuming ballistic contacts to the reservoirs. Temperature is assumed to be sufficiently low to make the dephasing rate small, γ φ = h/τ φ ≪ E Th . The dephasing probe model, proposed by Büttiker [20], has been extensively used in the literature. In this model the quantum coherence in the dot is destroyed by attaching a probe, where the voltage can be either externally controlled or remain floating. This probe exchanges electrons with the dot, and one distinguishes non-uniform (or localized) vs. uniform dephasing by such a probe depending on its coupling to the dot, see Fig. 9.
Uniform dephasing denotes a probe having large number of poorly coupled channels, G φ → 0, N φ → ∞. However, the dephasing rate γ φ = N φ G φ δ remains fixed. Known results for uniform dephasing correspond to the results of the imaginary potential model in the Hamiltonian approach [21]. Non-uniform dephasing, on the other hand, usually takes perfect coupling to the dot, G φ = 1, and finite N φ . The advantage of this approach is the technical simplicity due to lack of back-reflection from the probe, but on the other hand γ φ should be strictly quantized. For dots which are not artificially dephased by a local probe we assume the uniform dephasing model to be more realistic.
For small dephasing in a dot with arbitrary N only the uniform dephasing model can be used. To illustrate the role of γ φ in the tails of ρ(T ) we first take a single-channel quantum dot, N = 2, and later consider the limit N ≫ 1. Coherent quantum dots with N = 2 have only one transmission eigenvalue T and ρ(T ) = (β/2)T β/2−1 , β = 1, 2 [23]. For weakly dephased dots we use the intermediate results of Ref. [21] and express the dimensionless conductance g = hG/2e 2 close to g = 1 as a sum where u, u ′ are 2 × 2 unitary random matrices and u ′ = u T , β = 1. If the particles were absorbed by the probe, only the first term in the sum (10) would have been present. The second term is due to reinjection of particles by the probe and results from the requirement of particle conservation. The parameters N φ G φ /x 1,2 characterize the coupling strength of the probe to the dot, x → ∞ corresponds to weak coupling. The distribution ρ(T ) for N φ G φ ≪ 1 is found after integration over the uniform distribution of u, u ′ in the unitary group and the 0. distribution P (x 1 , x 2 ) (see Ref. [21] for its general form), For 1 − T ≪ N φ G φ we find an exponential suppression, can be understood as the density of classical particles at a point T ∈ [0, 1] and temperature 1/β in an external potential Even though this potential is weak, it repels the transmission T from the perfect T = 1. Only at N φ G φ ≪ 1 − T ≪ 1 does the density return to ∼ 1, see Fig. 10.
If W (λ) + W 0 (λ) and/or asymmetry in the contacts, N 1 = N 2 , lead to cut-off values λ = a, b such that ρ(λ) = 0, λ = a, b, we find [40] where z = N min −N/2 ≤ 0 is defined by the contact asymmetry. The distribution ρ(λ) depends on the exact shape of W (λ). Integration of (13) over λ satisfies the normalization condition, dλρ(λ) = N min , since the W (λ ′ ) contribution is canceled after integration over λ. When W = 0, the boundary is given by √ b 0 + 1 = −N/2z, and the minimal possible value of λ, a 0 = 0. For symmetric dots, z = 0, one reproduces bimodal distribution ρ(T ) = N/(2 T (1 − T )). However, if the potential W (λ) is unknown, the resulting distribution ρ(λ) must reproduce results obtained in the uniform dephasing model, Var G in particular [21]. The transmission repulsion of coherent dots allows us to add another equation relating Var G and a, b: Together with boundary conditions on ρ(λ) the total system of equations reads If the potential only repels λ from 0, the second and third equations in (15) for symmetric dots, z = 0, give However, a different assumption about functional dependence of W (λ) on N φ G φ leads to different dependence on λ. Therefore, additional information about W (λ) is needed.
On the other hand, we could assume that the potential W (λ) maintains the functional form given by (12), W (λ) = (N φ G φ /2)(1 + 1/λ). This potential only repels λ from the perfect transmission, λ = 0, and for a symmetric dot, z = 0, the second equation in (15) gives b → ∞. First equation in (15) allows us to express a as a function of dephasing strength and find the total distribution: Instead of generic ρ(T ) ∝ 1/ √ 1 − T , T → 1 the distribution behaves as ρ(T ) ∝ (T + − T ) 1/2 , T ≈ T + , see Fig. 10. However, the cut-off value a in Eq. (17) does not satisfy the third equation in (15). In fact, Eq. (12) is only an asymptote λ → 0 and the part which repels λ from ∞ is absent. This part might become important for strong dephasing, see numerical results of Ref. [21]. The potential (12) leads to formation of the gap in transmissions, ρ(T ) = 0, T + < T < 1, but it probably overestimates the size of this gap.
The system of equations (15) defines a, b through the unknown potential W (λ). The function W (λ) cannot be universal, since in the strong dephasing limit N φ G φ ≫ N the last equation gives a ≈ b, but then the first two equations can not be simultaneously satisfied. This shows that, in general, the phenomenological treatment of dephasing in form of a wisely chosen potential W (λ) is not complete. The repulsion between transmissions should itself change from its universal form − ln |λ − λ ′ | as a result of dephasing and a more accurate treatment should be taken.
However, from the analytical results for N = 2 we expect that the potential W (λ) is the main effect of weak dephasing, N φ G φ ≪ N and modification of transmission repulsion would affect the size of the gap only perturbatively with small parameter N φ G φ /N ≪ 1. Without finding the exact form of W (λ), we find from the third equation in (15) perturbations to the coherent values of the boundaries a 0 , b 0 . If the contact asymmetry affects only T − , we find Comparing Eqs. (17,18) we conclude that the potential Eq. (12) overestimates the dephasing effect and gives too large a value of a for N φ G φ ≪ N . Close to the cut-off ρ(λ) ∝ √ λ − a, λ ≈ a independently of the exact form of W (λ) and we conclude that for ρ(T ) ≈ T + − T , T ≈ T + effects of dephasing and asymmetric contacts are similar.
The important result (18) shows that the dephasinginduced gap for transmissions close to T = 1 survives the limit N ≫ 1, 1 − T + ≈ γ φ /2πE Th = τ d /τ φ . The shape of ρ(T ) ∝ (T + − T ) 1/2 , T ≈ T + for N ≫ 1 should be contrasted with ρ(T ) ∝ exp[−βN φ G φ /2(1 − T )], T ≈ 1 for N = 2. We conclude that lowering temperature T , γ φ diminishes and thus opens conducting channels. In a normal system the widening of ρ(T ) and the growth in Var G usually come together. In some models, like the Fabry-Perrot interferometer with T ± given in Eq. (7), or a weakly dephased quantum dot with T ± presented in Eq. (18), Var G is defined by (T + − T − ) 2 only, see Eq. (14). Even though the connection of Var G to T ± is generally much more complicated [18], we suggest that a wider distribution is a signature of enhanced Var G. Since the normal current is only proportional to T , I(T ) ∝ V T , this enhancement is hardly noticeable in a dephased dot at small γ φ /E Th ≪ 1, see Eq. (14). However, the SNS current is exponentially dominated by perfect channels, I(T ) ∝ exp(−π∆(1 − T )/eV ), and a weak dephasing γ φ /E Th ≪ 1 remains important for SNS transport fluctuations. In a realistic model of our SNS experiment both dephasing and imperfect contacts should be taken into account.