Apr 26, 2012 - In SNS junctions the supercurrent depends not only on the phase across .... metal island, RK = h/e2 is the resistance quantum,. ÏD = L...

2 downloads 16 Views 512KB Size

arXiv:1202.3358v3 [cond-mat.supr-con] 26 Apr 2012

1

Low Temperature Laboratory, Aalto University, P.O. Box 15100, FI-00076 AALTO, Finland 2 Institute for Theoretical Physics and Astrophysics, University of W¨ urzburg, D-97074 W¨ urzburg, Germany (Dated: November 8, 2018)

We study a diffusive superconductor-normal metal-superconductor junction in an environment with intrinsic incoherent fluctuations which couple to the junction through an electromagnetic field. When the temperature of the junction differs from that of the environment, this coupling leads to an energy transfer between the two systems, taking the junction out of equilibrium. We describe this effect in the linear response regime and show that the change in the supercurrent induced by this coupling leads to qualitative changes in the current-phase relation and for a certain range of parameters, an increase in the critical current of the junction. Besides normal metals, similar effects can be expected also in other conducting weak links. PACS numbers: 74.45.+c, 74.25.N-, 74.50.+r

Superconducting Josephson junctions are non-linear circuit elements and therefore their state and the supercurrent carried through them depends sensitively on the properties of the field driving them. Besides the average current or voltage across the junction, fluctuations in the field also modify the junction response. In traditional superconductor-insulator-superconductor (SIS) junctions, this can be described via the fluctuating phase difference φ(t) inserted in the dc Josephson relation IJ = IC sin(φ(t)) within the resistively and capacitively shunted junction (RCSJ) model [1]. As a result of these fluctuations, the junctions may switch to the dissipative state at bias currents lower than the critical current IC , and therefore the measured critical current is often lower than its theoretical value that does not include the fluctuations. The difference between the two is proportional to the ratio of the temperature Tenv describing the fluctuations and the Josephson energy EJ = ~IC /(2e) of the junction. In contrast to such a simplified picture, fluctuations can nevertheless have a significant effect on the critical current even if the condition EJ kB T is satisfied. Such an effect arises in superconductor–normal metal– superconductor (SNS) junctions, where the insulator is replaced by a normal metal (N) layer. These junctions also support finite supercurrents, but the effect of the electromagnetic field on this system is more complicated. In SNS junctions the supercurrent depends not only on the phase across the junction and its fluctuations, but also on the state of the electron system inside the normal metal [2]. The latter is determined by a balance of energy currents between the electrons on the normal metal island and the other degrees of freedom in the system: phonons in the metal film, electrons inside the superconducting leads and — via the fluctuations — the electromagnetic environment of the junction (electron-photon coupling) [3]. At temperatures low compared to the superconduct-

ing energy gap, Andreev reflection [4] suppresses the energy transfer to the electrons in the leads. Therefore, the state of the electron system depends on the balance between electron-phonon and electron-photon coupling. In this Letter, we derive a linear-response collision integral describing electron-photon scattering in an SNS junction, and show that the effect of fluctuations on the SNS supercurrent is controlled by a parameter different from kB Tenv /EJ , and that increasing the temperature of the environment can lead either to a decrease or an increase in the SNS supercurrent. The latter effect is directly related to the Eliashberg stimulated superconductivity [5, 6] in SNS junctions. Besides changing the critical current, we show that this effect modifies the current-phase relation as the field absorption is greatly phase dependent.

FIG. 1. Circuit model for an electrical environment with admittance Yenv coupling to a superconductor-normal metalsuperconductor junction with admittance YSNS .

To specify our analysis, we consider the system depicted in Fig. 1. There, the SNS junction, described by its normal-state resistance RSNS , length L, and diffusion constant D, is coupled to an environment with admittance Yenv . At first we consider this impedance as

2 generic, but specify it more when discussing examples. Moreover, we assume that the SNS junction can be dc phase biased (e.g., in a SQUID setup) with phase φ and consider only the limit of a long junction, L ξ0 = p ~D/(2∆), where ∆ is the superconducting energy gap. The fluctuations related to the dissipative part of Yenv and those of the junction itself give rise to a fluctuating voltage ∆V (ω) and a total fluctuating current ∆I(ω) over the junction. The electron-photon coupling then results into a dissipated power Peγ = h∆V (ω)∆I(ω)i into (or out of) the junction. The details of ∆V and ∆I are sensitive to the superconducting correlations in the SNS junction, which we take into account in the following. First we note that the coupling of the electrons on the SNS junction to the electromagnetic environment can be envisaged as a photon exchange between two separate electron systems, described by energy distribution functions fSNS and fenv . Therefore, the collision integral for this process can be written in the form (below, ~ = kB = 1) Z SNS env Ieγ () = dωd0 K(ω, , 0 )[f+ω f0 −ω (1 − fSNS )(1 − fenv ) 0 SNS − fSNS fenv (1 − f+ω )(1 − fenv 0 0 −ω )].

Here the kernel K(ω, , 0 ) describes the coupling strength, and includes the effects of the superconducting correlations. We consider a macroscopic linear normalmetal noise source, for which the radiation absorption is energy independent, and which is in internal equilibrium, described by temperature Tenv . In that case the kernel does not depend on 0 and we can carry out the integral over 0 to get Z SNS Ieγ () = dωωK(ω, )[f+ω (1 − fSNS )(nenv ω + 1) (1) SNS env − fSNS (1 − f+ω )nω ], which describes electron-boson (photon) coupling. Here nenv is the Bose distribution function of the photons at ω temperature Tenv and the two parts of the collision integral describe photon emission and absorption, respectively. We consider the effect of electron-photon interaction on the supercurrent flowing through the SNS junction at a certain phase difference φ across it, Z ∞ 1 d jS (, φ)(1 − 2f ()), (2) IS (φ) = eRSNS 0 where jS (, φ) = Im[jE (, φ)] is the spectral supercurrent [2] and f () is the electron distribution function. In what follows, we consider linear response changes δf of the distribution function due to the electron-photon coupling, and solve the kinetic equation Ieγ () = Ieph () = −Γeph ν()δf ().

(3)

Here the collision integral Ieph () describing electronphonon scattering is assumed to be the dominant source of energy relaxation. The latter form is valid in the linear response regime; ν() is the spatially averaged density of states inside the normal metal normalized to the normal state density of states at the Fermi level and Γeph is the electron-phonon scattering rate. Energy diffusion into the superconductors can be disregarded due to Andreev reflection [4] when we consider energies much below the superconducting energy gap ∆. Equation (3) is therefore a valid approximation for long junctions L ξ0 , where the relevant physics takes place around the Thouless energy ET = ~D/L2 . In the linear response regime, the form of the kernel K(ω, ) in Eq. (1) can be argued by considering the ac response of the junction [7, 8] to a fluctuating potential in the environment. We get K(ω, ) = Kqp (ω, ) + Ksc (ω, )+Kdy (ω, ), containing three parts due to quasiparticle, supercurrent, and dynamic responses on the ac potential. This yields

Re(Yenv ) 1 4 h[1 + g()g( + ω)∗ RK τD ω 2 |Yenv + YSNS |2 2 1 1 + F ()F ( + ω)∗ + F˜ ()F˜ ( + ω)∗ ]−1 i−1 2 2 1 (4) − ∂φ Re[jE () + jE ( + ω)] 2 [j() − j( + ω)∗ ]2 ET −Im 2(ω − 2iΓ) hg() + g( + ω)∗ i 4 Re(Yenv ) ≡ k(, ω, φ), RK τD ω 2 |Yenv + YSNS |2

K(ω, ) =

where h·i denotes a spatial average over the normal metal island, RK = h/e2 is the resistance quantum, τD = L2 /D is the diffusion time, YSNS (ω) is the admittance of the SNS junction [7], g() and F () are the normal and anomalous Green’s functions inside the normalmetal island, jE () is the spectral supercurrent [2], and ET = D/L2 is the Thouless energy of the junction. These quantities can be calculated from the equilibrium Usadel [9, 10] equation. Note that this approach disregards the equilibrium effect of phase fluctuations on the supercurrent [1]. It is typically relevant when Tenv is of the order EJ , or when R|| ≡ Re[Yenv + YSNS ]−1 is of the order of RK . In what follows, we thus assume R|| RK and a large enough critical current to satisfy Tenv EJ . Below, we describe the external noise source by assuming it to consist of a resistance Renv in parallel with a capacitance C (as in Fig. 1). The change in the supercurrent due to electron-photon coupling at linear response

3 is then given by Z ∞ Z ∞ dω jS (, φ) Renv 1 d × δIS (φ) = eRSNS RK Γe−ph τD −∞ −∞ ω ν(, φ) k(, ω, φ) +ω ω sech sech sinh ϕ(˜ ω , r, ωC ) 2TSNS 2TSNS 2TSNS ω ω × coth − coth , (5) 2Tenv 2TSNS where the circuit parameters constitute a frequency2 dependent term ϕ = |1 − iω/ωC + r˜ ySNS (ω)| describing the matching between the SNS junction and the environment, containing the parameters r = Renv /RSNS , ωC = 1/(Renv C) and y˜SNS ≡ RSNS YSNS . The presence of a finite ωC cuts the contribution from high frequencies — if ωC > max{TSNS , Tenv }, the cutoff is provided by the temperature. In the opposite limit, we can expand the coth(·) functions at low frequencies, and find that the effect is proportional simply to Tenv − TSNS . From Eq. (5) we find that the overall magnitude of the change induced in the supercurrent by electron-photon coupling is described by the parameter α ≡ Renv /(RK Γe−ph τD )(Tenv − TSNS )/ET = (Renv /RK )(Tenv − TSNS )/(Γe−ph ). The characteristics of the effect depend mostly on the following four parameters: temperature TSNS of the SNS junction, phase φ across the junction, the charge relaxation rate ωC ≡ (Renv C)−1 and the matching factor r ≡ Renv /RSNS . In the following, we analyze their effect in more detail. The effect of electron-photon coupling on the supercurrent at phase φ = π/2 (close to the phase giving the maximum supercurrent) as a function of the temperature TSNS of the phonons in the SNS junction is depicted in Fig. 2. The inset shows the overall supercurrent as a function of temperature in the presence and absence [2, 11] of the electron-photon coupling (corresponding, hence, to the cases Tenv > TSNS and Tenv = TSNS , respectively). We find out that at low TSNS , the supercurrent decreases as the SNS junction heats up due to the absorption of power from the electromagnetic environment. However, at higher temperatures, kB TSNS & 5ET , the electron-photon coupling to a high-temperature noise source leads to an increase in the supercurrent. This is a true nonequilibrium effect and resembles the stimulation of superconductivity encountered also in the presence of monochromatic driving of the junction [5, 12]. Note that this happens at the linear response of the junction to the electron-photon coupling: increasing Tenv further eventually leads to a decrease of the overall supercurrent. The strongest enhancement of the supercurrent can be found for phases around φ ≈ π/2. This effect can be traced to the existence of a minigap of size ∼ ET in the excitation spectrum (and the kernel k(E, φ)). On the other hand, for phases φ ≈ π, the minigap closes and the electron-photon coupling only suppresses the supercurrent. This characteristics is shown in Fig. 3, which shows

FIG. 2. (Color online): Electron-photon coupling induced change in the supercurrent vs. temperature of the SNS junction for φ = π/2, r = 1 and ~ωC = 5ET . The blue solid line shows the result calculated with the coherent kernel k(E, φ) from Eq. (4) and the red dashed line the result that would be obtained in the incoherent limit where k(E, φ) = 1. Inset shows the total supercurrent in the absence of electronphoton scattering (blue solid line) and a sketch of the effect of electron-photon scattering with Tenv > TSNS (red dashed line). The arrows point the direction of the change in the supercurrent as the noise temperature of the environment is increased. Strictly speaking, the dashed line is for kB T . 5ET outside of the linear response regime, but it captures the qualitative effect correctly. Note that in practice when considering δIS (T ), one should take into account the temperature depen3 . dence of the electron-phonon scattering Γeph ∝ TSNS

the supercurrent change δIS as a function of the phase. A similar shape of the current-phase relation has been found for monochromatic driving, both theoretically [5] and experimentally [13]. The effect of electron-photon coupling is naturally strongest when the resistance describing the electromagnetic environment equals the SNS normal-state resistance, i.e., r = Renv /RSNS ≈ 1, and as much noise as possible is coupled to the junction, and therefore ωC is as large as possible [14]. For completeness, we show the effect of varying these parameters in Fig. 4. We find out that the major effect on the current increase comes from frequencies ω ≈ ET , so that a further increase of the cutoff frequency beyond a few ET does not affect the increase much. On the other hand, the incoherent reduction of the supercurrent (at low temperatures, for example) increases in strength as the noise bandwidth is increased. We also point out that the “optimal” matching of noise takes place at Renv somewhat larger than the normal-state resistance RSNS of the SNS junction, but the order of magnitude of the effect depends quite weakly on their ratio. Let us estimate the typical parameters for the electron-

4 1

0

0.5 0

3

−0 .5 −1 0

2

−1 −2

0 −1

0.5

−0.5 −1

−3

−1

−1.5

−3

−2

−4 0

−2

−2.5 10

−2

0

0

1

−2

10

−1

2

10

0

10

4

1

10

2

6

8

10

0

FIG. 3. (Color online): Electron-photon coupling induced change in the supercurrent vs. phase φ with r = 1 and ~ωC = 5ET and three temperatures kB TSNS : 3ET (blue solid line), 10ET (red dashed line) and 15ET (black dashdotted line). Inset shows the normalized current-phase relation IS (φ)/ maxφ IS (φ) in the absence of electron-photon scattering (solid lines) and a sketch of the effect of electronphoton scattering with Tenv > TSNS (dashed lines) at the same three temperatures. There, the arrows point the direction of the supercurrent change as Tenv is increased.

photon coupling in SNS junctions [11], where the authors report at low temperatures a 7 % difference between their experimental results and the theory that does not take into account phase fluctuations. A Cu wire of length 1 µm, diffusion constant D = 0.02 m2 /s and normalstate resistance RSNS = 0.2 Ω has a Thouless energy ET ≈ 13 µeV and a zero-temperature critical current of 650 µA. This corresponds to the Josephson energy EJ = ~IC /(2e) = 1.3 eV, allowing to increase the (noise) temperature of the electromagnetic environment to very large values before any phase diffusion could be observed. Increasing TSNS decreases the critical current and thereby EJ , but the observation of phase diffusion would require quite high TSNS . On the other hand, the change in the supercurrent due to electron-photon coupling 2 is δIS /IS ∼ aRSNS kB (Tenv − TSNS )/(cRK Renv ~Γeph ), where we assume RSNS < Renv , a is the dimensionless number plotted in Figs. 2-4 at perfect matching, and c = eIS RSNS /ET ≈ 10 at TSNS . ET /kB . For Cu, a typical electron-phonon scattering rate at T = 100 mK is 20 kHz [15], corresponding to the temperature scale of ~Γeph /kB ≈ 0.15 µK. Therefore, for a typical Renv = 50 Ω, we get δIS /IS = a/c ∼ 0.07 for Tenv − TSNS = 8 K. As the SNS junctions are typically connected to a measurement equipment residing at higher temperatures, the noise coupling from such equip-

FIG. 4. (Color online): Electron-photon coupling induced change in the supercurrent at φ = π/2 vs. the parameters of the circuit at the temperatures indicated in the figure. Main figure: effect of the changing charge relaxation rate ωC = 1/(Renv C) acting as an effective high-frequency cutoff on the electron-photon coupling. The curves have been calculated with Renv = RSNS . Inset shows the effect of changing the ratio Renv /RSNS while keeping ~ωC = 5ET (note the logarithmic scale on the horizontal axis). The two figures show the same quantity (with the same scaling).

ment may well result in noise temperatures of this order of magnitude. Moreover, many experiments are conducted on higher-resistance samples than those considered above, in which case the required temperature difference decreases. Therefore, our results may explain the typically encountered difference between the experimental results and the standard theoretical predictions [16] as being caused by electron-photon coupling. However, to really probe the effect we are predicting, the environmental noise should be systematically varied while measuring the supercurrent. The previous can be done for example by passing a large heating current through a macroscopic shunt resistor of the SNS junction. Conclusions. We have shown that whereas the typical and well-known mechanism of the effect of phase fluctuations on the supercurrent through superconductornormal-metal-superconductor junctions, dependent on the parameter kB T /EJ , can often be disregarded, the heat current due to the temperature difference between the electromagnetic environment and the SNS junction leads to much more pronounced effects. At low temperatures kB TSNS . 5ET , this results into a suppression of the observed supercurrent, but what is more remarkable, for kB TSNS & 5ET , we predict an increased supercurrent, competing with the exponentially suppressed bare supercurrent. Our predictions should be tested by simply varying the temperature of the electromagnetic environment while keeping that of the SNS junction constant. Be-

5 sides weak links fabricated of normal metals, similar effects can be expected for other types of conducting weak links, such as those made of graphene, carbon nanotubes or semiconductor nanowires. We thank M.A. Laakso, J.C. Cuevas and F.S. Bergeret for discussions. This work was supported by the Finnish Foundation for Technology Promotion, the Academy of Finland and the European Research Council (Grant No. 240362), and the Emmy-Noether program of the Deutsche Forschungsgemeinschaft.

[1] M. Tinkham, Introduction to superconductivity, 2nd Ed., Dover, New York (2004) [2] T. T. Heikkil¨ a, J. S¨ arkk¨ a, and F.K. Wilhelm, Phys. Rev. B 66, 184513 (2002). [3] D. R. Schmidt, R. J. Schoelkopf, and A. N. Cleland, Phys. Rev. Lett. 93, 045901, (2004); M. Meschke, W. Guichard, and J. P. Pekola, Nature 444, 187 (2006); T. Ojanen and T.T. Heikkil¨ a, Phys. Rev. B 76, 073414 (2007); L.M.A. Pascal, H. Courtois, and F. W. J. Hekking, ibid. 83, 125113 (2011). [4] A.F. Andreev, Zh. Eksp. Teor. Fiz. 46, 1823 (1964) [Sov. Phys. JETP 19, 1228 (1964)]. [5] P. Virtanen, T. T. Heikkil¨ a, F. S. Bergeret, and

J. C. Cuevas, Phys. Rev. Lett. 104, 247003 (2010). [6] G.M. Eliashberg, JETP Lett. 11, 114 (1970). [7] P. Virtanen, F. S. Bergeret, J. C. Cuevas, and T. T. Heikkil¨ a Phys. Rev. B 83, 144514 (2011) [8] See the supplementary material. [9] K. Usadel, Phys. Rev. Lett. 25, 507 (1970). [10] These quantities may be calculated for example with the Usadel solver publicly available at [http://ltl.tkk.fi/∼theory/usadel1/]. [11] P. Dubos, et al., Phys. Rev. B 63, 064502 (2001). [12] J.M. Warlaumont, J.C. Brown, T. Foxe, and R.A. Buhrman, Phys. Rev. Lett. 43, 169 (1979). [13] M. Fuechsle, et al., Phys. Rev. Lett. 102, 127001 (2009). [14] The theory detailed in [7] leading to Eq. (4) assumes a position independent vector potential inside the junction. This assumption breaks down at high frequencies, ~ω & 10ET , and therefore we limit ourselves to cutoff frequencies below this scale. We do not expect significant qualitative changes to the collision integral even beyond this point, but quantitative details are altered. [15] F. Giazotto, et al., Rev. Mod. Phys. 78, 217 (2006). [16] Besides Ref. [11], see for example H. Courtois, M. Meschke, J.T. Peltonen, and J.P. Pekola, Phys. Rev. Lett. 101, 067002 (2008) and C. Pascual Garcia and F. Giazotto, Appl. Phys. Lett. 94, 132508 (2009). Note that some of the deviations can probably be explained also by extra scattering at the NS interface.

Appendix: Environment-controlled change in the current

In this supplementary material, we give details on how the effect of fluctuations on a superconductor–normal metal–superconductor junction can be derived from microscopic theory. We describe the effect of fluctuations by considering the Keldysh path integral action [1] of the circuit of Fig. 5: S[Φ, χ] = SSNS [Φ + χ] + Senv [Φ] ,

(6)

Rt

where Φ(t) = ~e dt V (t) is the electromagnetic phase drop across the SNS junction, with the quantum and classical cl/q components Φ (t) = (Φ+ (t) ± Φ− (t))/2 related to its values Φ± on the two Keldysh branches. We also add a generating field χ, so that the current in the SNS can be written as Z δSSNS [Φ + χ] I(t) = D[Φ]eiSenv [Φ]+iSSNS [Φ] |χ=0 . (7) δχq (t) We assume the environment is characterized by an admittance Yenv describing a circuit element at equilibrium. We also assume that the saddle point of the action corresponds to a constant superconducting phase difference ϕ0 over the junctions, corresponding to a dc supercurrent I0 through the SNS. In terms of fluctuations φ = Φ − Φ0 around the saddle point Φ0 , the environment action can be written as: Z

∞

Senv [φ] = −I0

dt φq (t) +

−∞

Z

∞

−∞

dω 2π

† cl 0 [iωYenv (ω)]∗ φcl (ω) φ (ω) , iωYenv (ω) 2iω coth( 2Tωenv ) Re Yenv (ω) φq (ω) φq (ω)

(8)

which produces the correlators hφφi expected of a classical circuit element. Here and below, we use natural units in which e = ~ = kB = 1. Consider now the action of the SNS junction similarly expanded in fluctuations: † cl Z ∞ Z ∞ dω φcl (ω) 0 [iωYSNS (ω)]∗ φ (ω) + A[φ] (9) SSNS [φ, 0] = I0 dt φq (t) + ω q iωY (ω) 2iω coth( ) Re Y (ω) φ (ω) φq (ω) 2π SNS SNS 2TSN S −∞ −∞ = SSN S,0 [φ] + A[φ] ,

(10)

6

FIG. 5. SNS junction and its electromagnetic environment. Φ is the electromagnetic phase across both elements.

where I0 = Ieq (ϕ0 ) is the equilibrium supercurrent. The form of the second-order term is fixed by the fact that it describes the linear response of the SNS junction around equilibrium. It is similar to Eq. (8), but with the admittance (for which approximations are known[2]) and temperature replaced by those of the SNS junction. The term A describes higher-order corrections to the behavior of the SNS due to the fluctuations. When Tenv 6= TSN S , part of these corrections comes from nonequilibrium associated with the energy transfer from one subsystem to the other by phase fluctuations. The next step would be to compute SSNS [φ] based on a microscopic model. This problem is however equivalent to finding the full counting statistics [3] of the SNS junction under a general time-dependent drive, which for long junctions is a difficult problem. Below, we argue that nevertheless, in the limit of small phase fluctuations, the physics we are interested in here is described by the response of the junction to classical fluctuations. We first expand the higher-order SNS part A in Eq. (7) in φ, and obtain: (

Z I(t) =

D[φ]e

=

iSenv [φ]+iSSN S,0 [φ]

δSSNS [φ + χ] |χ=0 δχq (t)

∞

˜ + χ] δ A[φ |χ=0 + . . . I0 + 2 dt V (t − t )φ (t ) + δχq (t) −∞ Z

0

R

0

cl

0

+ O(φ3 ) = hISN S [φ]iφ + O(φ3 ) ,

) (11) (12)

φ

where A˜ contains only the third-order terms, V R is the Fourier transform of iωYSNS (ω), and the averages are computed with the quadratic part of the action. The second term on the first line vanishes, hφiφ = 0, but the third is finite. Note the structure of this approach: one first computes the current ISN S [φ] through the junction using a fixed time dependence of the phase fluctuation φ, and finally averages the result over Gaussian fluctuations as determined by the admittances. We now observe the following: Eq.

(8) implies that the temperature of the environment Tenv appears in Eq. (12) only in correlation functions φcl φcl φ . Therefore, if we consider only the effect of Tenv on the current, we find that in the leading order in the phase fluctuations, the change in the current due to Tenv 6= TSN S is δI(t) ≡ I(t) − I(t)|Tenv =TSN S = hISN S [φ]iφcl − hISN S [φ]iφcl |Tenv =TSN S ,

(13)

where the field averages are taken considering φ as a classical field, φq = 0. This observation considerably simplifies the approach: we can first compute the current for a given time dependence of a classical phase difference over the junction, and then average the result over Gaussian fluctuations. The effect of such classical fluctuations on the supercurrent can be obtained as an extension of our earlier results [2, 4] for the effect of a monochromatic classical drive. This is outlined in the next section. We now comment on how small the phase fluctuations must be for the validity of our model. The criterion is that truncating the expansion Eq. (12) must remain accurate. The first requirement is that the average phase fluctuations should be small, hφ(t)φ(0)i 1. Assuming total parallel admittance Y = R−1 + 1/(iωL) of the SNS junction and the environment, this is equivalent to the restrictions R RK and LkB T /(~RK ) 1. The former is satisfied for typical SNS junctions. If the inductance comes from the Josephson inductance of the SNS junction, the latter is equivalent to EJ kB T , which is the typical condition for fluctuations to have a small effect. There is also a requirement that the nonequilibrium corrections to the SNS current are small enough to remain in the linear regime. As noted in the main text, this condition can be written as R/RK × kB (Tenv − TSN S ) Γe−ph , where Γe−ph is the electron-phonon relaxation rate, which should dominate energy relaxation inside the SNS junction.

7 Effect of classical phase fluctuations

The effect of small classical phase fluctuations on the dc current in a SNS junction can be studied by expanding the time-dependent Usadel equation [5, 6] in the fluctuating electric field associated with the time-dependent phase difference φ(t). Such a calculation was done in Ref. 4 for a monochromatic excitation φ(t) = φ0 cos(ω0 t). A kinetic equation for an arbitrary small perturbation φ(t) can however also be derived following the same steps. Our starting point here is the kinetic equation for the dc component of the electron distribution obtained in Ref. 4, which does not make assumptions about the time-dependence of the small perturbation: 8Γe−ph ν()[h(, + ω) − h0 (, + ω)] = tr[−iAτ3 , ˆj K ]◦ (, + ω) + O(A3 ) ,

(14)

where ω → 0, and the commutator involves a convolution over energy arguments. Here, h is the energy mode ˇ g )K is the current related to the Keldysh Green’s function (longitudinal) electron distribution function, ˆj K ≡ (ˇ g ◦ ∇ˇ ˇ gˇ, and ∇ is the gauge-invariant gradient. In particular, the charge current is proportional to tr τˆ3 ˆj K . Moreover, A(ω, ω 0 ) = φ(ω −ω 0 )/L is the Fourier-transformed vector potential corresponding to a constant electric field associated density of states in the absence of with the fluctuation φ in a junction of length L, ν() the position-averaged . fluctuations, and h0 (, 0 ) = 2πδ( − 0 )h0 (), h0 () = tanh 2TSN S Averaging Eq. (14) over the fluctuating fields, we find: Z ∞ E d1 D 0 0 8Γe−ph ν()[hh(, )iφ − h0 (, )] = −i A( − 1 ) tr τˆ3 ˆj K (1 , 0 ) − A(1 − 0 ) tr τˆ3 ˆj K (, 1 ) . (15) φ −∞ 2π In Ref. 2 we showed that in linear order in the field, the quantity tr τˆ3 ˆj K can be approximated by K tr τˆ3 ˆj K (, 0 ) ' tr τˆ3 ˆjeq (, 0 ) + A( − 0 )M (, 0 ) ,

(16)

with a known linear response coefficient M (, 0 ) = M (0 , )∗ . Combining this result with the kinetic equation, we find Z ∞ dω Sφ (ω, Tenv , TSN S )L−2 Im M (, + ω) (17) 8Γe−ph ν()[h() − h0 ()] = 2 2π −∞ Z ∞ −1 = π −1 dω Sφ (ω, Tenv , TSN S )τD 2k(ω, )[h0 () − h0 ( + ω)] , (18) −∞

0

0

where hh(, )iφ = 2πδ( − )h(), the factor k(ω, ) is defined in Eq. (4) in the main text, and Sφ (ω, Tenv , TSN S ) = 4π

Re[Yenv (ω)] coth

ω 2Tenv

+ Re[YSNS (ω)] coth

ωRK |Yenv (ω) + YSNS (ω)|2

ω 2TSN S

,

(19)

is the symmetrized phase fluctuation spectrum from the field correlators, hA(ω)A(ω 0 )iφ = L−2 φcl (ω)φcl (ω 0 ) = L−2 2πδ(ω + ω 0 )S(ω, Tenv , TSN S ). Here, RK = h/e2 = 2π in natural units. As we argued in Ref. 4, when the electron-phonon relaxation is small compared to the inverse dwell time in the junction, Γe−ph ET = ~D/L2 , the change in the supercurrent through the junction is mainly determined by the change in the distribution function. Applying now the result in Eq. (13) gives Z ∞ 1 d jS ()δh() , (20) δI = eRSN S −∞ Γe−ph ν()δh() = Γe−ph ν()[h() − h()|Tenv =TSN S ] Z ∞ 1 −1 = dω [Sφ (ω, Tenv , TSN S ) − Sφ (ω, TSN S , TSN S )]τD k(, ω)[h0 () − h0 ( + ω)] 4π −∞ Z h ω ω i 1 ∞ = dω ωK(ω, ) coth − coth [h0 () − h0 ( + ω)] 4 −∞ 2Tenv 2TSN S Z ∞ env = dω ωK(ω, )[f+ω (1 − f )(nenv ω + 1) − f (1 − f+ω )nω ] ,

(21) (22) (23) (24)

−∞

where f = 1−h20 () is the equilibrium Fermi function, and nenv = [eω/Tenv − 1]−1 the Bose function. We therefore ω find that the change in the current is determined by an electron-boson collision integral, and we obtain the kernel

8 K(ω, ) given in Eq. (4) of the main text. The approach we used to derive this result here, however, is restricted to the leading order in the field amplitude and small nonequilibrium effects.

[1] G. Sch¨ on and A. D. Zaikin, Phys. Rep. 198, 237 (1990); M. Kindermann, Y.V. Nazarov, and C.W.J. Beenakker, Phys. Rev. Lett. 90, 246805 (2003). [2] P. Virtanen, F.S. Bergeret, J.C. Cuevas, and T.T. Heikkil¨ a, Phys. Rev. B 83, 144514 (2011). [3] Observe that SSNS [φ] is essentially the generating function of the full counting statistics. [4] P. Virtanen, T.T. Heikkil¨ a, F.S. Bergeret, and J.C. Cuevas, Phys. Rev. Lett. 104, 247003 (2010). [5] K.D. Usadel, Phys. Rev. Lett. 25, 507 (1970). [6] A.I. Larkin and Y.N. Ovchinnikov, in Nonequilibrium superconductivity, edited by D. Langenberg and A. Larkin (Elsevier, Amsterdam, 1986), p. 493.