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Anomalous temperature dependence of the supercurrent through a chaotic Josephson junction P. W. Brouwer and C. W. J. Beenakker Instituut-Lorentz, University of Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands

Abstract We calculate the supercurrent through a Josephson junction consisting of a phase-coherent metal particle (quantum dot), weakly coupled to two superconductors. The classical motion in the quantum dot is assumed to be chaotic on time scales greater than the ergodic time τerg , which itself is much smaller than the mean dwell time τdwell . The excitation spectrum of the Josephson junction has a gap Egap , which can be less than the gap ∆ in the bulk superconductors. The average supercurrent is computed in the ergodic regime τerg ≪ ¯h/∆, using random-matrix theory, and in the non-ergodic regime τerg ≫ ¯ h/∆, using a semiclassical relation between the supercurrent and dwell-time distribution. In contrast to conventional Josephson junctions, raising the temperature above the excitation gap does not necessarily lead to an exponential suppression of the supercurrent. Instead, we find a temperature regime between Egap and ∆ where the supercurrent decreases logarithmically with temperature. This anomalously weak temperature dependence is caused by long-range correlations in the excitation spectrum, which extend over an energy range ¯h/τerg greater than Egap ≃ ¯ h/τdwell . A similar logarithmic temperature dependence of the supercurrent was discovered by Aslamazov, Larkin, and Ovchinnikov, in a Josephson junction consisting of a disordered metal between two tunnel barriers. Published in: Chaos, Solitons & Fractals 8, 1249 (1997). (Pergamon/Elsevier; special issue on Chaos and Quantum Transport in Mesoscopic Cosmos) PACS numbers: PACS numbers: 74.50.+r, 74.80.Fp, 05.45.+b

1

I.

INTRODUCTION

The dissipationless flow of a current through a superconductor–normal-metal– superconductor (SNS) junction is a fundamental demonstration of the “proximity effect”: a normal metal borrows characteristic properties from a nearby superconductor. The energy gap ∆ in the bulk induces a suppression of the density of states inside the normal metal near the Fermi level, depending on the phase difference φ between the superconductors. The resulting φ-dependence of the free energy F implies the flow of a current I = (2e/¯h)dF/dφ in equilibrium. In contrast to the original Josephson effect in tunnel junctions, the separation of the superconductors in an SNS junction can be much greater than the superconducting coherence length. Recent experiments on mesoscopic Josephson junctions [1, 2, 3, 4, 5, 6] have revived theoretical interest in this subject[7, 8, 9], which goes back to work by Kulik [10] and Aslamasov, Larkin, and Ovchinnikov [11]. (For more references, see the review 12.) In this paper, we consider the case that the normal region consists of a chaotic quantum dot. A quantum dot is a small metal particle, within which the motion is phase coherent, weakly coupled to the superconductors by means of point contacts. We assume that the classical dynamics in the quantum dot is chaotic on time scales longer than the time τerg needed for ergodic exploration of the phase space of the quantum dot. (In order of magnitude, τerg ≃ L/vF for a quantum dot of size L without impurities, vF being the Fermi velocity.) On energy scales smaller than h ¯ /τerg , the spectral statistics of a chaotic quantum dot is described by random-matrix theory [13, 14]. On larger energy scales, the non-ergodic dynamics on time scales below τerg becomes dominant [15]. The condition of weak coupling means that the mean dwell time τdwell in the quantum dot is much greater than τerg . (The ratio τdwell /τerg is of the order of the ratio of the total surface area to the area of the point contacts.) Although Josephson junctions are commonly known as “weak links” [16], we will refer to junctions where τdwell ≃ τerg as “strongly coupled” junctions, to distinguish them from the weakly coupled junctions (τdwell ≫ τerg ) considered here. A weakly coupled SNS junction consisting of a dirty normal metal separated from the two superconductors by high tunnel barriers was studied in the original paper by Aslamasov, Larkin, and Ovchinnikov [11]. Their theory was restricted to the high-temperature regime kT ≫ h ¯ /τdwell . In contrast to strongly coupled Josephson junctions, where the supercurrent > h is suppressed exponentially for kT ∼ ¯ /τdwell [9, 17], it was found that the supercurrent < h depends logarithmically on temperature for kT ∼ ¯ /τerg , while exponential suppression > only sets in when kT ∼ h ¯ /τerg . In the present paper we find a qualitatively similar hightemperature behavior of the supercurrent in the case that the weak coupling is ensured by point contacts rather than tunnel barriers. In addition, we are able to go down to zero temperature, where we find that the supercurrent acquires a logarithmic dependence on the minimum of τdwell /τerg and τdwell ∆/¯h, over and above the conventional dependence on min(¯h/τdwell , ∆) known from strongly-coupled Josephson junctions. Our paper builds on earlier work with Melsen and Frahm [18], where we computed the density of states of a chaotic quantum dot which is weakly coupled to a superconductor, and found a gap at the Fermi level of width Egap ≃ h ¯ /τdwell (provided h ¯ /τdwell ≪ ∆). Although the supercurrent can be expressed as an integral over the density of states, direct application of the results of Ref. 18 is not possible, as they were only derived under the

that the temperature scale for the exponential suppression of the supercurrent is set by τerg and not by τdwell . This work is organized as follows: In Sec. II we review the scattering theory of the Josephson effect [19, 20], on which our calculation is based. We distinguish two regimes, the ergodic and non-ergodic regime, depending on the relative magnitude of ∆ and h ¯ /τerg . In Secs. III and IV, we consider the ergodic regime τerg ≪ h ¯ /∆, where we can use random>h matrix theory. The non-ergodic regime τerg ∼ ¯ /∆ is treated in Sec. V, using a semiclassical relation between the supercurrent and the dwell-time distribution. We conclude with an overview of our results in Sec. VI. II.

SCATTERING MATRIX FORMULA FOR THE SUPERCURRENT

We consider the SNS junction sketched in Fig. 1. The two superconductors S1 and S2 have order parameters ∆eφ1 , ∆eiφ2 , with phase difference φ = φ1 − φ2 . The contacts to the normal metal N have N1 , N2 propagating modes at the Fermi energy EF . We denote N = N1 + N2 . Elastic scattering by the normal metal at energy E = EF + ε is characterized by an N × N unitary matrix S(ε). Excitations in N with energy ε > 0 consist of electrons (occupied states lying ε above the Fermi level) and holes (empty states lying ε below the Fermi level). Their scattering matrix SN (ε) has dimension 2N ×2N, with the block structure ee eh S(ε) 0 S S SN (ε) = ≡ . (1) ∗ 0 S (−ε) S he S hh

The off-diagonal blocks are zero, because the normal metal does not scatter electrons into holes. The supercurrent couples electron and hole excitations through the mechanism of Andreev reflection [21]: An electron approaching the NS interface from the normal side at ε < ∆ is reflected as a hole, and vice versa. The scattering matrix SA (ε) for Andreev reflection is given by (assuming ∆ ≪ EF ) 0 eiΦ SA (ε) = α(ε) , (2a) e−iΦ 0 p α(ε) = e−i arccos(ε/∆) = ε/∆ − i 1 − ε2 /∆2 . (2b)

Here Φ is a diagonal matrix with elements Φjj = φ1 = φ/2 for 1 ≤ j ≤ N1 and Φjj = φ2 = −φ/2 for N1 + 1 ≤ j ≤ N1 + N2 . The matrix SA has √ been defined for ε < ∆. Its definition may be extended to ε > ∆, when α(ε) = ε/∆ − ε2 − ∆2 . Notice that SA is no longer unitary for ε > ∆. The matrices SN and SA determine the excitation spectrum of the Josephson junction for all ε, inPthe following way [19]. For ε < ∆, the spectrum is discrete. The density of states ρ(ε) = n δ(ε − εn ) consists of delta functions at the solutions of the equation det [1 − SA (εn )SN (εn )] = 0.

For ε > ∆, the spectrum is continuous, with density of states 1 1 d ln det [1 − SA (ε)SN (ε)] − ln det SA (ε)SN (ε) . ρ(ε) = − Im π dε 2 3

(3)

(4)

S

1

N

S

2

FIG. 1: Schematic drawing of a chaotic Josephson junction.

If ε is replaced by ε + i0+ , Eq. (4) describes both the continuous and the discrete spectrum [22]. The excitation spectrum determines the free energy F of the Josephson junction, Z ∞ F = −2kT dε ρ(ε) ln [2 cosh(ε/2kT )] + [φ-independent terms], (5) 0

where T is the temperature. The φ-independent terms include [23] a spatial integral over |∆(~r)|2 , which does not depend on φ in the step-function model for the pair potential ∆(~r). Only the φ-dependent terms contribute to the supercurrent I=

2e dF . h ¯ dφ

(6)

We use the analyticity of SA and SN in the upper half of the complex ε-plane to rewrite the expression for the supercurrent in a more convenient form. Under a change ε → −ε, the determinant det[1 − SA (ε + i0+ )SN (ε + i0+ )] goes over into its complex conjugate. Combination of Eqs. (4)–(6), and extension of the ε-integration from −∞ to ∞, results in e d I= 2kT iπ¯h dφ

Z

∞+i0+

dε ln [2 cosh(ε/2kT )]

−∞+i0+

d ln det[1 − SA (ε)SN (ε)]. dε

(7)

We now perform a partial integration and close the integration contour in the upper half of the complex plane. The integrand has poles at the Matsubara frequencies iωn = (2n + 1)iπkT . Summing over the residues one finds ∞

I=−

d X 2e ln det[1 − SA (iωn )SN (iωn )]. 2kT h ¯ dφ n=0

(8)

Eq. (8) is the starting point for our evaluation of the average supercurrent through a chaotic Josephson junction. III.

SUPERCURRENT THROUGH A CHAOTIC JOSEPHSON JUNCTION

We consider the case that the normal region has a chaotic classical dynamics on time scales greater than the ergodic time τerg . In this section, we assume that τerg ≪ h ¯ /∆, so that we may use random-matrix theory to evaluate the ensemble average of the supercurrent. > h We postpone to Sec. V a discussion of the regime τerg ∼ ¯ /∆, in which the non-ergodic dynamics on time scales shorter than τerg starts to play a role. We assume that the normal metal is weakly coupled to the superconductors, so that the mean dwell time τdwell ≫ τerg . No assumption is made regarding the relative magnitudes of τdwell and h ¯ /∆. 4

We use a relationship between the scattering matrix S of the normal metal and its Hamiltonian H [24, 25], S(ε) = 1 − 2πiW † (ε − H + iπW W † )−1 W.

(9)

The Hamiltonian H (representing the isolated normal metal region) is taken from the Gaussian ensemble of random-matrix theory [26], 1 −2 2 P (H) ∝ exp − Mλ tr H , (10) 4 where M is the dimension of H (taken to infinity at the end) and λ is a parameter that determines the average level spacing δ = λπ/2M of the excitation spectrum in the normal region. (This spacing δ is half the level spacing of H, because it combines electron and hole levels together.) The matrix H is real and symmetric. The coupling matrix W is an M × N matrix with elements[27, 28] 1/2 p 1 −1 1 − Γ . (11) Wmn = δmn (2Mδ)1/2 2Γ−1 − 1 − 2Γ n n n π Here Γn is the transmission probability of mode n in the contacts to the superconductor. For ballistic contacts, Γn = 1, while Γn ≪ 1 for tunneling contacts. We now substitute Eq. (9) for S into Eq. (1) for SN and then substitute SN into Eq. (8) for the supercurrent. Using also Eq. (2) for SA , we find after some straightforward matrix algebra that ∞

2e d X ln det[iωn − H + W(iωn )], I = − 2kT h ¯ dφ n=0

where we have introduced the 2M × 2M matrices H 0 H = , 0 −H π∆ (ε/∆)W W T W eiΦ W T W(ε) = √ . W e−iΦ W T (ε/∆)W W T ∆2 − ε2

(12)

(13a) (13b)

The matrix H − W(ε) is the effective Hamiltonian of Refs. 18, 29 (where the regime ε ≪ ∆ was considered, in which the ε-dependence of W(ε) can be neglected). We define the 2M × 2M Green function G(ε) = [ε − H + W(ε)]−1 ,

(14)

which determines the density of states according to ∂W . ρ(ε) = −π Im tr G(ε + i0) 1 + ∂ε −1

(15)

Eq. (15) is equivalent to Eq. (4). The expression for the supercurrent in terms of G(ε) is ∞

d X 2e ln det G(iωn ) 2kT I = h ¯ dφ n=0

∞ X 2e d = − 2kT tr G(iωn ) W(iωn ). h ¯ dφ n=0

5

(16)

The average supercurrent follows from the average Green function hG(ε)i, since W is a fixed matrix. The average over the random Hamiltonian H (determining G) is done with the help of the diagrammatic technique of Refs. 30, 31. We consider the regime M, N, |ε|/δ ≫ 1 in which only planar diagrams need to be considered. Resummation of these diagrams leads to a self-consistency equation which is similar to Pastur’s equation [32], −1 hG(ε)i = ε + W(ε) − (λ2 /M)P(ε) ⊗ 11M .

(17)

The symbol ⊗ indicates the direct product between the M × M unit matrix 11M and the 2 × 2 matrix htr G ee i −htr G eh i . (18) P= −htr G he i htr G hh i We seek the solution of Eqs. (17)–(18) which satisfies εG(ε) → 112M if |ε| ≫ λ. It is convenient to define a self energy ee eh λ Σ Σ tr G ee tr G eh Σ= = . Σhe Σhh M tr G he tr G hh

(19)

(20)

Eqs. (17)–(20) contain a closed set of equations from which hΣi can be determined. We are interested in the limit M → ∞, λ → ∞, keeping N and ε/δ = 2εM/λπ fixed. In this limit, the equations for hΣi become hΣee i = hΣhh i, hΣeh ihΣhe i − hΣee i2 = 1, N X πε eh hΣ i + Kj εhΣeh i + ∆eiΦjj hΣee i = 0, 2δ j=1

N X πε he hΣ i + Kj εhΣhe i + ∆e−iΦjj hΣee i = 0. 2δ j=1

(21a) (21b)

(21c)

The function Kj (ε) is defined through √ Γj /Kj = (4 − 2Γj ) ∆2 − ε2 + Γj ∆e−iΦjj hΣeh i + ∆eiΦjj hΣhe i + 2εhΣee i . (21d)

(We substituted Eq. (11) for the matrix W .) The boundary condition (19) becomes ineffective in the limit λ → ∞. Instead, we seek the solution of Eq. (21) with hΣee i = hΣhh i → −i for ε → i∞, corresponding to a constant density of states ρ(ε) = 1/δ for |ε| ≫ ∆. From hΣi, we find hGi and hence the ensemble averaged supercurrent hIi, ∞ X N X 2e kT ∆ sign(Φjj ) Kj eiΦjj hΣhe (iωn )i − e−iΦjj hΣeh (iωn )i . hIi = i¯h n=0 j=1

(22)

Eqs. (21) and (22) contain all the information needed to determine the average supercurrent through a chaotic Josephson junction.

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An analytic solution of Eq. (21) is possible in certain limiting cases. Here we discuss the case of high tunnel barriers, Γj ≪ 1 for all j. Then we may approximate Kj = (1/4)Γj (∆2 − ε2 )−1/2 and find ee

hh

hΣ i = hΣ i = −ε(

√

∆2

−

ε2

√ 2 − 12 2 2 2 2 2 , + ET ) |Ω| ∆ − ε ∆ − ε + ET

√ 2 − 12 2 2 2 hΣ i = Ω ∆ |Ω| ∆ − ε , ∆2 − ε2 + ET eh

√ 2 − 12 2 2 2 , hΣ i = Ω ∆ |Ω| ∆ − ε ∆2 − ε2 + ET he

∗

N

(23a) (23b) (23c)

N

δ X δ X Ω(φ) = Γj eiΦjj , ET = Γj = Ω(0). 2π j=1 2π j=1

(23d)

The energy ET is related to the mean dwell time through ET = h ¯ /2τdwell [33]. The excitation gap in the spectrum of the Josephson junction is of order |Ω(φ)| when τdwell ≫ h ¯ /∆ [18]. Substitution of Eq. (23) into Eq. (22) yields the supercurrent

where

∞ 2 − 12 p X 2π ∆2 sin φ 2 2 2 p , kT GET hIi = |Ω| ∆ + ωn ∆2 + ωn2 + ET 2 + ω2 e ∆ n n=0 2e2 G= h

PN1 PN i=1

j=N1 +1

PN

k=1 Γk

Γi Γj

(24)

(25)

is the conductance of the Josephson junction when the superconductors are in the normal state. For arbitrary transmission probabilities Γj , it is necessary to solve Eq. (21) numerically. We have studied the case that both point contacts have an equal number of modes (N1 = N2 = N/2), and that all transmission probabilities are equal (Γj = Γ for all j). The average supercurrent at zero temperature for Γ = 0.1 and Γ = 1 is shown in Fig. 2. IV.

ERGODIC REGIME

The general result (21)–(22) describes the supercurrent in the ergodic regime τerg ≪ h ¯ /∆. Within this regime, we can distinguish two further regimes, depending on whether the dwell time τdwell = h ¯ /2ET is short or long compared to h ¯ /∆. We discuss these two regimes in two separate subsections. A.

Short dwell-time regime

In the short dwell-time regime (when τdwell ≪ h ¯ /∆, or equivalently ET ≫ ∆), the magnitude of the critical current Ic = maxφ I(φ) is set by the energy gap ∆ in the bulk superconductor: Ic ≃ G∆/e at zero temperature. The temperature dependence of Ic can be neglected as long as kT ≪ ∆, i.e. for temperatures T much less than the critical temperature 7

FIG. 2: Average supercurrent at zero temperature, computed from Eqs. (21) and (22) for the case N1 = N2 = N/2, Γj = Γ for all j. Left panels: Γ = 1; right panels: Γ = 0.1. The upper panels show hIi in the short dwell-time regime for ET /∆ = 1 (bottom curve), 10, and 100 (top curve). The bottom panels show hIi in the long dwell-time regime for ET /∆ = 0.01 (top curve), 0.1, and 1 (bottom curve). The conductance G = (2e2 /h)N Γ/4 and the Thouless energy ET = N Γδ/2π. Notice that hIi is in units of G∆/e in the top panels, and in units of GET /e in the bottom panels.

Tc of the bulk superconductor. In the case of tunneling contacts, evaluation of Eq. (24) with ET ≫ ∆ ≫ kT yields G∆ K γ sin(φ/2) sin φ p hIi = . (26) e 1 − γ 2 sin2 (φ/2)

The conductance G was defined in Eq. (25), the function K is the complete elliptic integral of the first kind, and we abbreviated !1/2 N !−1 N1 N X X X γ=2 Γi Γj Γk . (27) i=1 j=N1 +1

k=1

The parameter γ equals 1 for two identical point contacts with mode-independent tunnel probabilities. The result (26) could also have been obtained directly from the general formula for the zero-temperature supercurrent in the short dwell-time regime [19], Z t sin φ e∆ 1 dt ρ(t) p , (28) hIi = 2¯h 0 1 − t sin2 (φ/2) which relates hIi to an integral over the transmission eigenvalues t of the junction in the normal state, with density ρ(t). The transmission eigenvalue density for a chaotic cavity 8

with two identical tunneling contacts (N1 = N2 = N/2, Γj = Γj+N/2 for j = 1, 2, . . . , N/2) is given by [34] N/2 X Γj (2 − Γj ) p ρ(t) = . (29) 2 π(Γ − 4Γ t + 4t) t(1 − t) j j j=1

One can check that the integral (28) equals Eq. (26) with γ = 1 if Γj ≪ 1 for all j. For two identical ballistic point contacts (N1 = N2 = N/2, Γj = 1 for all j), the density is ρ(t) = N(2π)−1 [t(1 − t)]−1/2 [35, 36], which yields 2e∆ GE i arsinh[tan(φ/2)], i cotan(φ/2) . hIi = iπ¯h sin(φ/2)

(30)

Here G = N/4 and E is the elliptic integral of the second kind. B.

Long dwell-time regime

In the long dwell-time regime (when τdwell ≫ h ¯ /∆, or equivalently ET ≪ ∆), the magnitude of the critical current is set by the Thouless energy, but retains a logarithmic dependence on ∆: Ic ≃ (GET /e) ln(∆/ET ). The temperature dependence of Ic can be neglected as long as kT ≪ ET . If kT ≫ ET (but still T ≪ Tc ) the critical current decreases, though only logarithmically: Ic ≃ (GET /e) ln(∆/kT ). For the case of tunneling contacts, we find from Eq. (24) the expressions ! GET 2∆/ET hIi = kT ≪ ET , (31a) sin φ ln p e 1 − γ 2 sin2 (φ/2) 2∆ GET + cEuler kT ≫ ET , T ≪ Tc , (31b) sin φ ln hIi = e πkT where cEuler ≈ 0.58 is Euler’s constant. For ballistic contacts, we do not have a simple expression as Eq. (31), but the parametric dependence of I on ∆, ET , and kT is the same as for tunneling contacts (cf. Fig. 2). The logarithmic dependence on ∆ of the supercurrent when ET ≪ ∆ arises because the Thouless energy ET is not an effective cutoff for the Matsubara sum (8) or, equivalently, for the energy integration (7). Spectral correlations exist up to energies of order h ¯ /τerg ≫ ET . eh These long-range spectral correlations are responsible for the weak decay Σ ∝ 1/ω of the self-energy and ρ − δ −1 ∝ 1/ε2 of the density of states. The superconducting energy gap ∆ has to serve as a cutoff energy for the otherwise logarithmically divergent Eqs. (7) and (8), which explains the logarithm ln ∆ in Eq. (31). V.

NON-ERGODIC REGIME

>h When τerg ∼ ¯ /∆, a random-matrix theory of the Josephson effect is no longer possible, because the non-ergodic dynamics on time scales shorter than τerg starts to play a role. To study the average supercurrent in this non-ergodic regime, we return to Eq. (8). On

9

substitution of Eqs. (1) and (2) we obtain an expression for I in terms of the scattering matrix S of the normal region, ∞ X 2π I = kT F (ωn ), e n=0

F (ω) = −

4e2 d tr ln[1 − α(iω)2 S(iω)eiΦ S ∗ (−iω)e−iΦ ]. h dφ

(32a) (32b)

The evaluation of the scattering matrix at the imaginary energy iωn is equivalent to the evaluation of the scattering matrix at the Fermi level in the presence of absorption, with rate 1/τabs = 2ωn /¯h = (2n + 1)2πkT /¯h. We first consider temperatures kT ≫ ET . Since ωn ≫ ET = h ¯ /2τdwell for all n in this high temperature regime, absorption is strong, τabs ≪ τdwell . The formal correspondence between Matsubara frequency and absorption rate helps to understand that, to lowest order in τabs /τdwell = ET /ω, the diagonal elements of S(iω) are given by the reflection amplitudes of the tunnel barriers in the contacts, Sjj = (1 − Γj )1/2 , while the off-diagonal elements satisfy Z ∞ Γi Γj 2 h|Sij (iω)| i = PN dτ Pij (τ ) exp(−2ωτ /¯h), i 6= j. (33) k=1 Γk 0 The function Pij is the classical distribution of dwell times for particles that enter the quantum dot through mode j and exit through mode i. Because of the smallness of h|Sij (iω)|2 i = O(ET /ω), it is sufficient to keep only the lowest order term in an expansion of hF (ω)i in the off-diagonal scattering matrix elements, N1 N X 4e2 X 2α(iω)2h|S(iω)ij |2 i sin φ hF (ω)i = − . h i=1 j=N +1 [1 − α(iω)2 (1 − Γi )][1 − α(iω)2(1 − Γj )]

(34)

1

Eqs. (32)–(34) permit a semiclassical calculation of the average supercurrent in the nonergodic regime for temperatures kT ≫ ET , where random-matrix theory fails. The only input required is the classical distribution of dwell times. On time scales greater than τerg , the distribution Pij is exponential with the same mean dwell time τdwell = h ¯ /2ET for all i, j: Pij (τ ) =

2ET exp(−2ET τ /¯h). h ¯

(35)

The non-chaotic dynamics on time scales shorter than τerg enters through a non-universal < τerg . We consider the case of a ballistic dynamics (size L of the normal form of Pij for τ ∼ region much less than the mean free path ℓ). The ergodic time τerg ≃ L/vF is then a lower cutoff on Pij , since the minimum dwell time L/vF is the time needed to cross the system ballistically. A qualitative estimate of hIi is obtained if we set Pij (τ ) = 0 for τ < L/vF and approximate it by Eq. (35) for larger times. Substitution of this dwell-time distribution into Eq. (33) gives Γi Γj ET exp(−2ωL/¯hvF ), ω ≫ ET , i 6= j. h|Sij (iω)|2 i = PN ω Γ k k=1 10

(36)

We next compute hF (ω)i from Eq. (34), replacing α(iω) by its value −i for ω ≪ ∆. The result is ˜ T GE hF (ω)i = exp(−2ωL/¯hvF ) sin φ, (37a) ω PN1 PN −1 −1 2 2e j=N1 +1 4Γi Γj (2 − Γi ) (2 − Γj ) i=1 ˜ = G . (37b) PN h Γ k k=1 ˜ = G for the case of high tunnel barriers (Γj ≪ 1 for all j). We can now Notice that G calculate the average supercurrent from Eq. (32). Eq. (37) is valid for ET ≪ ω ≪ ∆, ET ≪ h ¯ vF /L ≪ ∆, and is sufficient to determine the supercurrent in the temperature range ET ≪ kT ≪ ∆. Substitution of Eq. (37) into Eq. (32) gives

˜ T GE sin φ ln (¯hvF /πkT L) , ET ≪ kT ≪ h ¯ vF /L ≪ ∆, (38a) e ˜ T GE 2 sin φ exp(−2πkT L/¯hvF ), ET ≪ h ¯ vF /L ≪ kT ≪ ∆. (38b) hIi = e Eq. (38) has the same temperature dependence as the result of Ref. 11 for the double-barrier SNS junction. < ET . In this temperature regime, the Matsubara We now turn to low temperatures kT ∼ < ET , for which the off-diagonal scattering matrix elements sum (32) contains terms with ωn ∼ Sij (iωn ) are not small and the approximation (34) is no longer valid. However, since ET ≪ h ¯ /τerg , these Matsubara frequencies are well within the validity range of random-matrix < ET and the theory. Therefore, we can use the results of Sec. III to compute hF (ω)i for ω ∼ > ET . These two results match at ω ≃ ET , because the semiclassical formula (34) for ω ∼ validity range ω ≪ h ¯ /τerg of random-matrix theory and the validity range ω ≫ ET of the semiclassical theory overlap (assuming τerg ≪ τdwell = h ¯ /2ET ). For the case of high tunnel barriers, random matrix theory gives [cf. Eq. (24)] hIi =

hF (ω)i = p

GET sin φ , ω≪h ¯ vF /L ≪ ∆, |Ω(φ)|2 + ω 2

(39)

while the semiclassical formula (34) gives

GET sin φ exp(−2ωL/¯hvF ), ET ≪ ω ≪ ∆. (40) ω [The function Ω(φ) was defined in Eq. (23d).] The two results (39) and (40) have a common range of validity ET ≪ ω ≪ h ¯ vF /L, within which they can be matched. The result is a formula valid for all ω ≪ ∆, for a ballistic quantum dot with high tunnel barriers: hF (ω)i =

hF (ω)i = p

GET sin φ

|Ω(φ)|2 + ω 2

exp(−2ωL/¯hvF ), ω ≪ ∆.

(41)

After substitution of Eq. (41) into Eq. (32) we obtain the average supercurrent in the lowtemperature regime, " ! # h ¯ vF GET p sin φ ln − cEuler , kT ≪ ET ≪ h ¯ vF /L ≪ ∆. (42) hIi = e LET 1 − γ 2 sin2 (φ/2)

[The parameter γ was defined in Eq. (27).] The results (38) and (42) cover the entire temperature range below Tc . 11

Ic × e/G

weak coupling (τdwell ≫ τerg ) strong coupling ergodic (τerg ≪ ¯h/∆) non-ergodic (τerg ≫ ¯h/∆) (τdwell ≃ τerg )

short dwell time (τdwell ≪ ¯ h/∆)

∆

—

∆

long dwell time

∆ ET ln E T

¯h ET ln E τ T erg

ET

(τdwell ≫ ¯ h/∆)

TABLE I: Parametric dependence of the zero-temperature critical current Ic on the three time scales τdwell , τerg , and h ¯ /∆. The Thouless energy ET = h ¯ /2τdwell . VI.

CONCLUSION

In Table I we summarize the parametric dependence of the critical current at zero temperature on the three time scales τdwell , τerg , and h ¯ /∆. We show the three new regimes for a weakly coupled normal region (τdwell ≫ τerg ), and have included for comparison also the two old regimes for a strongly coupled normal region (τdwell ≃ τerg ). Apart from a logarithmic factor, the critical current is given by Ic ≃ (G/e) min(¯h/τdwell , ∆) in each of the five regimes. There is an additional logarithmic dependence on min(τdwell /τerg , τdwell ∆/¯h) in two of the three new regimes. Upon raising the temperature, the critical current is suppressed at a characteristic temperature given by min(¯h/τerg , ∆). At lower temperatures, Ic has a >h logarithmic T -dependence as long as T ∼ ¯ /τdwell and becomes T -independent at still lower T. In this work, we did not address the sample-to-sample fluctuations of the supercurrent, but calculated only the ensemble average. For strongly coupled diffusive Josephson junctions (τdwell ≃ τerg , L ≫ ℓ), the root-mean-squared of the fluctuations is a factor e2 /hG smaller than the average critical current [19, 37]. Preliminary calculations in the ergodic regime indicate that the same is true for weakly coupled Josephson junctions (τdwell ≫ τerg ), i.e. the r.m.s. fluctuations of Ic are given by the entries in Table I, times e/h. We close with a remark on quantum dots with an integrable classical dynamics, such as < ET , the excitation spectrum of rectangular or circular ballistic cavities. For energies ε ∼ an integrable Josephson junction is quite different from its chaotic counterpart [18]: The density of states ρ(ε) of a chaotic cavity in contact with a superconductor shows a gap of size ET around the Fermi level ε = 0, while ρ(ε) vanishes linearly when ε → 0 for a rectangular or circular cavity. It is an interesting open problem, to compute the supercurrent through an integrable cavity and compare with the results for the chaotic case obtained in this paper. Acknowledgments

We thank A. I. Larkin and Yu. N. Ovchinnikov for alerting us to the logarithmic temperature dependence of the supercurrent found in Ref. 11. This work was supported by the “Stichting voor Fundamenteel Onderzoek der Materie” (FOM) and by the “Nederlandse

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organisatie voor Wetenschappelijk Onderzoek” (NWO).

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