Magnetic interference pattern in planar SNS Josephson junctions

Magnetic interference pattern in planar SNS Josephson junctions

G. Mohammadkhani,1M. Zareyan,1and Ya. M. Blanter2

1_{Institute for Advanced Studies in Basic Sciences (IASBS), P.O. Box 45195-1159, Zanjan 45195, Iran} 2_{Kavli Institute of NanoScience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands}

共Received 11 September 2007; published 25 January 2008兲

We study the Josephson current through a ballistic normal metal layer of thickness D, on which two superconducting electrodes are deposited within a distance L of each other. In the presence of an 共in-layer兲 magnetic field, we find that the oscillations of the critical current I_c共⌽兲 with the magnetic flux ⌽ are signifi-cantly different from an ordinary magnetic interference pattern. Depending on the ratio L/D and temperature, Ic共⌽兲 oscillations can have a period smaller than flux quantum ⌽0, nonzero minima, and damping rate much smaller than 1/⌽. A similar anomalous magnetic interference pattern was recently observed experimentally.

DOI:10.1103/PhysRevB.77.014520 PACS number共s兲: 74.78.Fk, 74.50.⫹r

I. INTRODUCTION

Existence of a supercurrent in a Josephson junction is the manifestation of the interference between the macroscopic wave functions 共superconducting order parameters兲 of the two contacted superconductors. The quantum interference can be modulated by an external magnetic field applied to the junction. As a result, the critical共maximum兲 supercurrent Ic共⌽兲 shows a well-known

Fraunhofer-diffraction-pattern-like dependence on the magnetic flux⌽ penetrating the junc-tion area. In a superconductor-insulator-superconductor共SIS兲 junction, the critical current,

Ic共⌽兲 = Ic共0兲

sin共␲⌽/⌽0兲

␲⌽/⌽0

, 共1.1兲

oscillates with the period of flux quantum⌽₀=␲បc/e and an amplitude decreasing as 1/⌽.1,2 _{The main features of the}

effect, i.e., damped oscillations of Icwith the magnetic flux,

take place in other types of Josephson weak links; however, the detailed behavior, including the period of the oscillations and the rate of damping, depends on the geometry as well as the nature of the weak link.

In a wide SNS共N being a normal metal layer兲 junction, Ic

has a similar magnetic interference pattern as SIS systems.3

On the other hand, Heida et al.,4 _{investigating}

S–two-dimensional-electron-gas–S 共S2DEGS兲 junctions of compa-rable width W and length L, have measured a 2⌽0periodicity

of the critical current instead of the standard⌽0periodicity.

The first explanation of this finding was due to Barzykin and Zagoskin,5_{who considered a S2DEGS junction with perfect}

Andreev reflections at NS interfaces and both absorbing and reflecting lateral boundaries, and obtained a 2⌽0 periodicity

in the limit L/W→⬁ 共i.e., in the limit of the point-contact geometry兲. Later, Ledermann et al.6 _{considered more}

realis-tic reflecting boundaries at the edges and found that, in the limit of strip geometry共L/W⯝1兲, the periodicity of the criti-cal current changes from⌽0to 2⌽0 as the flux through the

junction increases. In general, increase of the periodicity was attributed to the nonlocality of the supercurrent density in hybrid NS structures.

In this paper, we report on a different type of magnetic interference pattern in a planar SNS junction. The SNS junc-tion studied below consists of a thin normal metal layer of

thickness D on which two superconducting electrodes are deposited in a distance L of each other共Fig.1兲. An external

magnetic field is applied in the plane of the N layer, perpen-dicular to the direction of the Josephson current flow. Such a Josephson setup was recently used in the experiment by Keizer et al.7 _{to investigate the Josephson supercurrent}

through a S–half-metallic-ferromagnet–S共SHMFS兲 junction. They used NbTiN superconducting electrodes on top of a thin layer of CrO₂, which is a fully polarized共half-metallic兲 ferromagnet, i.e., it supports only one spin direction of elec-trons. Surprisingly, Josephson supercurrent was detected for the junction length L⬃300 nm in spite of strong pair break-ing in CrO₂, which is expected to suppress all singlet super-conducting correlations. Reference 7 also reports measure-ment of the magnetic interference pattern with magnetic field applied in the plane of the HMF layer. It is found that Ic共⌽兲

oscillates with the in-plane flux⌽ with a period of order ⌽0.

In contrast to a standard magnetic interference pattern共1.1兲,

Ichas nonzero values at the minima and the amplitude of the

oscillations decreases rather slowly compared to 1/⌽. The long range superconducting proximity in HMF can be explained in terms of the triplet superconducting correlations generated at spin-active HMFS interfaces as a result of the interplay between the singlet superconducting correlations and a noncollinear magnetization inhomogeneity.8,9 _While

1 0 1=∆eiϕ ∆

x

L

D

2 0 2=∆ eiϕ ∆

z

y

θ

φ

0

y

1

τ

2

τ

0

N

S

2

S

1 0 = τ Z=D Z=0 W

FIG. 1. 共Color online兲 Schematic of the SNS junction. The cur-rent flow between two superconductors 共S_1,2兲 through a normal layer共N兲. The NS interfaces are perfectly transparent.

the singlet superconducting correlations are destroyed over a short distance of the Fermi wavelength, the triplet compo-nents can survive over a distance of the order of the normal coherent length␰N=vF/2␲T, withvFbeing the Fermi

veloc-ity. In Ref. 7, by investigating the critical supercurrent for different distances between the electrodes, it was concluded that the length dependence is the same as in nonmagnetic SNS junctions. This strongly suggests, indeed, that triplet correlations are responsible for the observed Josephson cur-rent.

The aim of the present paper is to investigate Ic共⌽兲 in

such a planar SNS junction. Whereas this problem is inter-esting by itself, we also believe that it is relevant for the understanding of the experimental results of Ref.7. Indeed, penetration of triplet correlations in SHMFS junction is simi-lar to an ordinary 共singlet兲 superconducting proximity in SNS systems, i.e., both decay exponentially within the length scale␰N. By noting this fact, we will use the quasiclassical

Green’s function formalism to investigate the magnetic flux dependence of the supercurrent in the corresponding planar SNS junction 共see Fig.1兲. We find that the magnetic

inter-ference pattern is significantly different from that of the stan-dard one. The period of the oscillations can be smaller than ⌽0depending on the length-to-thickness ratio L/D. The

pe-riod tends to⌽0at higher magnetic fluxes and also for very

large L/D. We also obtain the two anomalous features ob-served in the experiment: the amplitude of the oscillations has a rather slow decrease with⌽ compared to the standard SIS case 共1.1兲, and the critical current as a function of the

flux, Ic共⌽兲, at low temperatures can have finite minima when

the total flux⌽ is an integer or noninteger multiple of the ⌽0

depending on the period of the oscillations.

In Sec. II, we introduce our model of a ballistic planar SNS contact and present solutions of the Eilenberger equa-tion for the quasiclassical Green’s funcequa-tions of a given elec-tronic trajectory. Introducing the effect of the in-plane mag-netic field through a gauge invariant phase, we obtain the expression of the critical supercurrent as a function of the magnetic flux. Section III is devoted to the analysis of the Ic共⌽兲 in terms of L/D for different temperatures. In Sec. IV,

we present the conclusion.

II. JOSEPHSON CURRENT IN SNS JUNCTION WITH AN IN-PLANE MAGNETIC FIELD

In this section, we calculate the Josephson current for a clean SNS junction in the presence of an external magnetic field H. The setup is schematically shown in Fig.1. It con-sists of a normal metallic共N兲 layer of thickness D and width W, on which two superconducting electrodes are deposited at a distance of L. This planar Josephson structure was studied experimentally in Ref. 7 with a half-metallic N layer. A phase difference␸ between order parameters of the super-conductors drives a Josephson supercurrent through the parts of N layer underneath the superconductors and the junction N part共−L/2艋y艋L/2兲. The magnetic field H=−Hxˆ is ap-plied in the plane of the N layer, perpendicular to the direc-tion of supercurrent flow. We consider a clean structure with all dimensions L, W, and D being smaller than the electronic

impurity mean free path ᐉimp, and ideally transparent NS

interfaces. At the same time, the Fermi wavelength is small compared to L, W, and D, and the superconducting coher-ence length ␰0=vF/2⌬0共T=0兲 共we use the system of units

withប=kB= 1兲. Under these conditions, the electronic

prop-erties of the system can be derived from the Eilenberger equations for the semiclassical matrix Green’s function gˆ,

−vFn ·ⵜgˆ =␻n关␶ˆ3,gˆ兴 + 关⌬ˆ共r兲,gˆ兴. 共2.1兲

The matrix Green’s function

gˆ =

冉

g_␻ n f␻n f_␻ n † − g_␻ n

冊

,

where the normal g and anomalous f Green’s functions de-pend on the Matsubara frequency␻n=␲T共2n+1兲, on the

co-ordinate r, and on the direction of motion n; ␶ˆi 共i=1,2,3兲

denotes the Pauli matrices in the Nambu space, and the ma-trix

⌬ˆ共r兲 =

冉

0 ⌬共r兲 ⌬*_{共r兲 0}

冊

represents the superconducting order parameter ⌬共r兲. The matrix Green’s function gˆ_␻

n satisfies the normalization

con-dition, gˆ2= 1, g_␻ n 2 + f_␻ nf␻n † = 1, 共2.2兲 where f_␻ n † _{共r,n兲= f} ␻n

*_{共r,−n兲. In components, the Eilenberger} equations have the form

−vFn ·ⵜg_␻n=⌬共r兲f_␻

n

† _−⌬*_共r兲f

␻n, 共2.3兲

−vFn ·ⵜf␻_n= 2␻nf␻_n− 2⌬共r兲g␻_n. 共2.4兲

We solve these equations along an electronic quasiclassical trajectory 共shown in Fig. 1兲, which is parametrized by −⬁

艋␶艋⬁.10,11 _{Assuming a weak external magnetic field, we}

neglect its effect on the orbital motion of the quasiparticles. The magnetic field then will have only phase effect, which we will include by introducing the gauge invariant phase as

␸=␸₀−2␲ ⌽0

冕

0

ᐉ

A · nd␶, 共2.5兲 where␸0is the phase difference between two

superconduct-ors in the absence of the magnetic field, the second part is the phase accumulated by the quasiparticle on the trajectory due to the magnetic field H =ⵜ⫻A, with ᐉ being the length of the trajectory inside the N layer. The vector potential is taken to be A = −Hzyˆ.

A typical trajectory consists of three parts: the part ex-tended from bulk of S₁ 共␶= −⬁兲 to a point at NS₁ interface ␶= 0; the part inside the N layer共0⬍␶⬍ᐉ兲, which extends

from a point at NS1interface to a point at NS2interface; and

the last part, which extends from the point␶=ᐉ to the bulk of S2 共␶=⬁兲. Our equations are supplemented by the boundary

conditions which determine the values of Green’s functions in the bulk S1and S2,

f_␻ n共␶= ⫿ ⬁兲 = ⌬0共T兲exp共i␸1,2兲 ⍀n , 共2.6兲 g_␻ n共␶= ⫿ ⬁兲 = ␻n ⍀n , 共2.7兲 where⍀n=

冑

⌬0 2_+␻ n 2_{, and⌬} 0共T兲 is the temperature-dependent

superconducting gap. We neglect the variation of the order parameter close to NS interfaces inside the superconductors and approximate the order parameter by the step function, ⌬共r兲=⌬0ei␸1␪共␶兲+⌬0ei␸2␪共␶− L兲. We can obtain the Green’s

function g_␻

n on a trajectory inside N that is constant. It

de-pends only on the length of that trajectoryᐉ and the phase difference␸0, which is given by

g_␻

n= tanh关共␻nᐉ/vF兲 + 共i␸0/2兲 + arcsinh共␻n/⌬0兲兴. 共2.8兲

Note that in the presence of the magnetic field, the phase difference␸₀ is replaced by␸ 关see Eq. 共2.5兲兴.

The supercurrent density can then be obtained by averag-ing Eq.共2.8兲 over all different possible classical trajectories.

This corresponds to an averaging over Fermi velocity direc-tions. In the presence of the planar magnetic field, we find

j共r兲 = eN共0兲TvF

兺

␻n

冕

n Im g_␻

n共ᐉ,vF兲sin␪d␪d␾.

共2.9兲 Here,N共0兲 is the density of states at the Fermi surface and Im g_␻

n共ᐉ,vF兲 denotes the imaginary part of the normal

com-ponent of the matrix Green’s function, given by Im g_␻

n=

⌬02共T兲sin␸

共⍀n2+␻n2兲cosh␹+ 2⍀n␻nsinh␹+⌬02cos␸

, 共2.10兲

␹= ␻nᐉ ␲Tc␰0

. 共2.11兲

To calculate the integral of the vector potential along the quasiclassical trajectories, we split it into the segments as shown in Fig.1,

冕

0 ᐉ A · nd␶=

冕

0 ␶1 A · nd␶+

冕

␶1 ␶2 A · nd␶+ ¯ +

冕

␶n−2 ␶n−1 A · nd␶+

冕

␶n−1 ᐉ A · nd␶. 共2.12兲 For the first term on the right side, we can write

冕

0 ␶1 A · nd␶= − H

冕

0 y₀/2 zdy .

Since the equation for this segment of the trajectory is z = D关1−y共D tan␪sin␾兲−1_{兴, we get for the integral}

冕

0 ␶1

A · nd␶= −HDy0

4 , y0= 2D tan␪sin␾, 共2.13兲 where y0 is shown in Fig. 1. Similarly, we find identical

results for the integrals of the vector potential over the other segments. Therefore, for a trajectory with lengthᐉ, the phase induced by the planar magnetic field is proportional to

⌽_ᐉ=

冕

0 ᐉ

A · nd␶=− HDNy0

2 , 共2.14兲

where N is the number of the triangles共each triangle consists of two segments兲 for the trajectory of length ᐉ passing through the point z, N =关L/共4D tan␪sin␾兲−z/共2D兲兴 +关L/共4D tan␪sin␾兲+z/共2D兲兴+2, with the square brackets denoting the integer part.

Thus, for the phase difference, we obtain ␸=␸0+␲

⌽ ⌽0

2ND tan␪sin␾

L . 共2.15兲

Here,⌽=HDL is the total flux through the junction. Substi-tuting this into Eq.共2.9兲 and taking the y component of the

current, we obtain the final expression

I共␸₀兲 I₀ = T Tc␰0␻

兺

n=−⬁ ␻n=⬁

冕

0 D

冕

0 ␲

冕

−1 1 ⌬0 2_共T兲sin

冋

_␸ 0+␲ ⌽ ⌽0 ᐉ L共1 − x 2_兲1/2_sin␾

册

_{共1 − x}2_兲1/2_{dx sin␾d␾dz}

共⍀n2+␻n2兲cosh␹+ 2⍀n␻nsinh␹+⌬02cos

冋

␸0+␲

⌽ ⌽0 ᐉ L共1 − x 2_兲1/2_sin␾

册

, Ic共⌽兲 = max0ⱕ␸0ⱕ2␲ I共␸0兲 I0 , 共2.16兲

with the notations cos␪= x and I0= 2evFN共0兲TcW␰0, and W

being the width of the normal layer in the x direction, which for a wide junction is taken to be much larger than D and L. The length ᐉ of the quasiclassical trajectory equals ᐉ = 2ND/x. For low temperatures, TⰆvF/L, the summation

over Matsubara frequencies in Eq.共2.16兲 can be replaced by

the integration

2␲T

兺

共...兲 →

冕

共...兲d␻→

冕

共...兲⌬₀共T兲cosh␮d␮, where␻=⌬0sinh␮. For a long SNS junction 共LⰇ␰0兲 and

low temperature, Eq.共2.16兲 is simplified as

I共␸₀兲 I₀ =

冕

₀ D

冕

0 ␲

冕

−1 1 E共␸,ᐉ兲共1 − x2_兲1/2_{dx sin␾d␾dz,} E共␸,ᐉ兲 = 2 ␰0

冉

␰0 ᐉ − 2␰₀2 ᐉ2

冊

兺

m=1 ⬁ 共− 1兲msin共m␸兲 m , 共2.17兲 where E共␸,ᐉ兲 is the Fourier series, given by 共see Ref.12兲

E共␸,ᐉ兲 = 1 ␰0

冉

␰0 ᐉ − 2␰₀2 ᐉ2

冊

冉

␸− 2␲

冋

␸ 2␲+ 1 2

册

冊

. 共2.18兲 III. DISCUSSION AND RESULTS

Equation共2.16兲 expresses the magnetic interference

pat-tern Ic共⌽兲 of a ballistic SNS junction in the presence of an

in-plane magnetic flux⌽. In this section, we analyze Ic共⌽兲 in

terms of the length-to-thickness ratio L/D and the tempera-ture T for D/␰0= 1/30.

Let us start with analyzing the case of very large L/D. Figures 2共a兲 and 2共b兲 show oscillations of Ic共⌽兲 for L/D

= 100 and at low共T=0.1Tc兲 and high 共T=0.95Tc兲

tempera-tures, respectively. At low temperatempera-tures, the critical current goes through nonzero minima at finite fluxes. The amplitude of supercurrent minima decreases with⌽ and drops to zero at ⌽Ⰷ⌽0. Compared to an ordinary magnetic interference

pattern, the oscillations are weakly damped since their am-plitude decreases with ⌽ much slower than 1/⌽. With in-creasing temperature, the amplitude of the oscillations de-creases. Also, the minimal values of the supercurrent decreases and vanishes as T→Tc, where⌬0共T兲ⰆT. Note that

at both low and high temperatures, the period of oscillations varies from 0.92⌽0共first minima兲 at low magnetic fluxes to

⌽0at high fluxes. The result that the period of oscillations is

temperature independent comes from the fact that the gauge invariant phase in the argument of the sine and cosine func-tions in Eq. 共2.16兲 does not contain any

temperature-dependent factors.

Figure3 presents the magnetic interference pattern for a lower L/D=10 and at the same temperatures as Fig.2. From these plots, we see that lowering L/D has two main effects. First, the rate at which the amplitude of Ic oscillation

de-creases with⌽ increases. Second, the period of oscillations at both low and high temperatures becomes smaller⯝0.72⌽0

at small fluxes. By increasing⌽, the period increases up to

⯝⌽0. Again as in Fig. 2, the value of supercurrent at the

minima vanishes as the temperature approaches Tc.

Still lower period of oscillations at small fluxes can be reached at low values of L/D. This is illustrated in Fig. 4, where the magnetic interference pattern is presented for L/D=5. Clearly, the decay is close to the ordinary pattern 共1.1兲, i.e., 1/⌽, and the period can be as small as half the

flux quantum.

The existence of the nonzero minima in the oscillations of Ic共⌽兲 is related to the nonsinusoidal phase dependence of the

Josephson current共2.16兲, which is more pronounced at low

temperatures TⰆTc. We have found that Ic共⌽兲 undergoes a

I

_c

/I

₀

Φ/Φ

0 ₀ -2 -4 2 4 0.0008 0.0016 0.0024 D/ =1/30 L/D=100ξ0 (a) -6 -8 6 8 T/Tc=0.1

I

_c

/I

₀

Φ/Φ

0 ₀ -2 -4 2 4 2 10-6 4 10-6 6 10-6 D/ =1/30_L/D=100ξ0 (b) × × × -6 -8 6 8 T/Tc=0.95

FIG. 2. The critical current dependence on the external magnetic field, applied in the plane of the normal metallic layer for different temperatures, L/D=100 and D/␰₀= 1/30. -5 -4 -3 -2 -1 0 1 2 3 4 5 0.007 0.014 0.021

I

_c

/I

₀

Φ/Φ

0 ξ0 D/ =1/30 L/D=10 T/Tc=0.1 (a)

I

c

/I

0

Φ/Φ

0 ₀ -2 -4 2 4 0.00025 0.0005 0.00075 ξ0 D/ =1/30 L/D=10 T/Tc=0.95 (b) -1 -3 1 3 5 -5

change of sign at a nonzero minimum. At such point, the amplitude of the first harmonic 共⬀sin␸0兲 of the Josephson

current vanishes, and Ic共⌽兲 is determined by the amplitude

of the higher共mainly second兲 harmonics which change sign upon crossing the minimum. A similar effect was found be-fore in ferromagnetic Josephson junctions共see Refs. 13–18, and references therein兲. As T approaches Tc, the ratio

⌬0共T兲/T goes to zero and the current-phase relation 关Eq.

共2.16兲兴 becomes sinusoidal and, consequently, the nonzero

minima disappear.

The dependence of the period of the oscillations on the magnetic flux and the geometry can be understood in terms of the difference between the magnetic flux⌽_ᐉ 关Eq. 共2.14兲兴

enclosed by a trajectory of lengthᐉ, and half of the flux ⌽ penetrating through the area DL. The difference comes from the fact that a trajectory which does not pass through the edges of S contacts has extra parts in the N layer which lies outside the area DL共see Fig.1兲. Writing ⌽_ᐉ=⌽/2+␦⌽_ᐉ, the difference␦⌽_ᐉvanishes only for the trajectories which pass through the edges of S contacts. A finite averaged具␦⌽_ᐉ典 over different trajectories means that the period of Ic共⌽兲

oscilla-tions, obtained from Eq.共2.16兲, differs from ⌽0. We note that

in the limit of thin N layer LⰇD or high magnetic fluxes ⌽Ⰷ⌽0, the contribution of the trajectories which are not

passing through the edges is negligibly small in the interfer-ence structure and the period of the oscillation approaches ⌽0共see Figs.2–4兲.

In contrast to the LⰇD case, for smaller L/D, the contri-bution of the trajectories not passing through the edges is important. Because of having larger length, a trajectory not passing through the edges has greater contribution in gaining the effect of the magnetic flux as compared to the corre-sponding trajectory 共having the same orientation ␪ and ␾兲 passing through the edges. Therefore, we expect that the ef-fect of magnetic flux is more pronounced for thicker N layers compared to the thinner ones, which explains why the de-crease of the amplitude of Ic共⌽兲 oscillations with ⌽ is faster

for smaller L/D.

IV. CONCLUSIONS

In conclusion, we have studied Josephson effect in a SNS structure made of a thin ballistic N layer of thickness D, on which two superconducting electrodes are deposited at the distance L between each other. A magnetic field is applied in the plane of the N layer, which modulates the superconduct-ing interference and leads to a decaysuperconduct-ing oscillatory variation of the critical supercurrent Ic共⌽兲 with the magnetic flux ⌽.

Using the quasiclassical Green’s functions approach, we have shown that such a magnetic interference pattern has three main differences with that of an ordinary pattern. First, at low temperatures, the oscillations of the critical current Ic共⌽兲 go through the minima at which the supercurrent has

nonzero values. Second, for a large L/D, the amplitude of the quasiperiodic oscillations of Ic共⌽兲 decays at a rate which

is much slower than 1/⌽. Third, at low magnetic fluxes, the oscillations can have a period smaller than the magnetic quantum flux ⌽₀ depending on L/D. These features have been experimentally observed recently.7

ACKNOWLEDGMENTS

We acknowledge useful discussions with D. Huertas-Hernando, A. G. Moghaddam, and Yu. V. Nazarov. M.Z. thanks G. E. W. Bauer for the hospitality and support during his visit to Kavli Institute of NanoScience at Delft, where this work was initiated. This work was supported in part by EC Grant No. NMP2-CT2003-505587共SFINX兲.

1_{B. D. Josephson, Rev. Mod. Phys. 36, 216共1964兲.} 2_{J. M. Rowell, Phys. Rev. Lett. 11, 200共1963兲.}

3_{T. N. Antsygina, E. N. Bratus, and A. V. Svidzinskii, Fiz. Nizk.} Temp. 1, 49共1975兲 关Sov. J. Low Temp. Phys. 1, 23 共1975兲兴. 4_{J. P. Heida, B. J. van Wees, T. M. Klapwijk, and G. Borghs, Phys.}

Rev. B 57, R5618共1998兲.

5_{V. Barzykin and A. M. Zagoskin, Superlattices Microstruct. 25,} 797共1999兲.

6_{U. Ledermann, A. L. Fauchère, and G. Blatter, Phys. Rev. B 59,} R9027共1999兲.

7_{R. S. Keizer, S. T. B. Goennenwein, T. M. Klapwijk, G. Miao, G.}

Xiao, and A. Gupta, Nature共London兲 439, 825 共2006兲. 8_{F. S. Bergeret, A. F. Volkov, and K. B. Efetov, Rev. Mod. Phys.}

77, 1321共2005兲.

9_{F. S. Bergeret, A. F. Volkov, and K. B. Efetov, Phys. Rev. Lett.} 86, 4096共2001兲.

10_{M. Zareyan, W. Belzig, and Yu. V. Nazarov, Phys. Rev. Lett. 86,} 308共2001兲.

11_{M. Zareyan, W. Belzig, and Yu. V. Nazarov, Phys. Rev. B 65,} 184505共2002兲.

12_{A. G. Golubov, M. Yu. Kupriyanov, and E. llíchev, Rev. Mod.} Phys. 76, 411 共2004兲; C. Ishii, Prog. Theor. Phys. 44, 1525 -4 -3 -2 -1 0 1 2 3 4 0.008 0.016 0.024

I

_c

/I

₀

Φ/Φ

₀ ξ0 D/ =1/30 L/D=5 T/Tc=0.1 (a) -4 -3 -2 -1 0 1 2 3 4 0.00035 0.0007 0.00105

I

_c

/I

₀

Φ/Φ

₀ D/ =1/30 L/D=5 T/Tc=0.95 ξ0 (b)

共1970兲.

13_{H. Sellier, C. Baraduc, F. Lefloch, and R. Calemczuk, Phys. Rev.} Lett. 92, 257005共2004兲.

14_{R. Mélin, Europhys. Lett. 69, 121共2005兲.}

15_{M. Houzet, V. Vinokur, and F. Pistolesi, Phys. Rev. B 72,} 220506共R兲 共2005兲.

16_{A. Buzdin, Phys. Rev. B 72, 100501共R兲 共2005兲.}

17_{G. Mohammadkhani and M. Zareyan, Phys. Rev. B 73, 134503} 共2006兲; V. Braude and Ya. M. Blanter, arXiv:0706.1746 共unpub-lished兲.

18_{S. M. Frolov, D. J. Van Harlingen, V. V. Bolginov, V. A.} Oboznov, and V. V. Ryazanov, Phys. Rev. B 74, 020503共R兲 共2006兲.