Electron transport in a ferromagnet-superconductor junction on graphene

Electron transport in a ferromagnet-superconductor junction on graphene

Yasuhiro Asano,1 _{Toshihiro Yoshida,}1_{Yukio Tanaka,}2_{and Alexander A. Golubov}3

1_{Department of Applied Physics, Hokkaido University, Sapporo 060-8628, Japan} 2_{CREST-JST and Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan} 3_{Faculty of Science and Technology, University of Twente, 7500 AE, Enschede, The Netherlands}

共Received 13 March 2008; revised manuscript received 16 June 2008; published 18 July 2008兲

In a usual ferromagnet connected with a superconductor, the exchange potential suppresses the supercon-ducting pairing correlation. We show that this common knowledge does not hold in a ferromagnet-superconductor junction on a graphene. When the chemical potential of a graphene is close to the conical point of energy dispersion, the exchange potential rather assists the charge transport through a junction interface. The loose-bottomed electric structure causes this unusual effect.

DOI:10.1103/PhysRevB.78.014514 PACS number共s兲: 74.50.⫹r, 74.25.Fy, 74.70.Tx

I. INTRODUCTION

A long time ago Fulde-Ferrell1_{and Larkin-Ovchinnikov}2 have showed that superconducting order parameter becomes inhomogeneous under the action of an exchange potential. Such a superconducting state is called Fulde-Ferrell-Larkin-Ovchinnikov 共FFLO兲 state and is currently a hot topic in condensed matter physics. A spin-singlet Cooper pair con-sists of two electrons with momenta k and −k on the Fermi surface. The exchange potential shifts such momenta to k + q/2 and −k+q/2 due to Zeeman effect. As a consequence, a Cooper pair acquires a center-of-mass momentum q and the superconducting order parameter oscillates in real space such as cos共q·r兲. Essentially similar effect has been pre-dicted and observed in ferromagnet-superconductor共FS兲 and superconductor-ferromagnet-superconductor 共SFS兲 proxim-ity structures.3–7_{In metallic SFS junctions, the proximity} ef-fect in a ferromagnet causes unusual Josephson coupling. An SFS junction undergoes a transition from 0 state to␲state as temperatures or length of a ferromagnet change because the pairing function in a ferromagnet changes its sign almost periodically as a function of spatial coordinate. In metallic FS junctions, the exchange potential Vexsuppresses the An-dreev reflection.8 _{Therefore the conductance of an FS} junc-tion decreases with the increase in V_ex and vanishes at V_ex =␮, where␮is the Fermi energy in the absence of exchange potential. For Vex⬎␮, a ferromagnet becomes half metallic and the usual proximity effect is impossible in a ferromagnet.9–13_{This picture is valid when the minority spin} conduction band has its bottom. Thus the above picture would be changed for the loose-bottomed band structure in graphene.

Graphene is a novel material attracting much attention recently. Carbon atoms on graphene are separated into two sublattices共A and B兲 due to its honeycomb lattice structure. In undoped graphene, the Fermi level lies on the conical point of the linear dispersion relation. In slightly doped graphene, two independent valleys appear on the Fermi sur-face. A considerable number of theoretical studies are made on transport properties of graphene because electronic struc-tures can be described by the massless Dirac Hamiltonian.14 The superconducting graphene junction was first discussed by Beenakker.15_{When the Fermi level in normal graphene is}

smaller than the pair potential of superconducting graphene, the Andreev reflection has the specular reflection property instead of the usual retroreflection. As a result, an electron-hole pair in the normal segment loses its retroreflection prop-erty even in the presence of time-reversal symmetry. Thus an unusual electronic transport is expected in superconducting graphene junctions.16

In this paper, we discuss electric current in FS and SFS junctions on graphene. When the exchange potential is much larger than the Fermi energy共i.e., VexⰇ␮兲 in a ferromagnetic segment, the Andreev reflection always has the specular re-flection property. As a consequence, the conductance spectra of FS junction become insensitive to Vex. In addition, very small shot noise is expected because the Andreev reflection probability is unity at the zero bias irrespective of Vex. In SFS junctions, the amplitude of Josephson current increases with increasing Vex. In all cases, the exchange potential as-sists electron transport in graphene, in contrast to usual FS and SFS junctions.

II. CONDUCTANCE IN FS JUNCTIONS

Let us consider a graphene sheet on an insulator-superconductor junction as shown in Fig. 1共a兲, where the width of graphene sheet is W. There are several ways to

Superconductor Insulator Graphene B // y Superconductor x=L x=0 (a) FS (b) SFS Insulator Graphene B // y Superconductor x=0

FIG. 1. 共Color online兲 Schematic figures of FS and SFS junc-tions on graphene. The width of graphene is W.

introduce the exchange potential onto a ferromagnetic seg-ment of graphene. Several theories suggested intrinsic ferro-magnetic correlations.17,18_{It is also possible to substitute the} insulator by a ferromagnetic insulator such as EuO.19 _A rough estimation20 _{indicates the exchange splitting on} graphene would be 5 meV. Although the Zeeman splitting due to external magnetic field is not large 共1 meV per 20 T兲,20 _{magnetic field in the y direction can tune the} ex-change splitting. In addition to these, a recent paper sug-gested the spin field-effect transistor using graphene21 _and the spin injection into graphene.22Thus introducing spin im-balance on graphene would be possible by combination of conventional technique or novel ideas of spin control. A junction on graphene is described by the Dirac-Bogoliubov–de Gennes equation15

冋

hˆ₀− sV_ex␴ˆ₀ s⌬共x兲␴ˆ₀ s⌬共x兲ⴱ␴ˆ_{0 − hˆ0− sVex␴}ˆ₀

册

⌽s共r兲 = E⌽s共r兲, 共1兲 hˆ0= − iបvFⵜ ·␴ˆ −关␮+ V共x兲兴␴ˆ0, 共2兲 V共x兲 =

再

0 :x⬍ 0 − U₀ :x⬎ 0

冎

, 共3兲 ⌬共x兲 = ⌬ei_{␸⌰共x兲, 共4兲} ⌽s=

冋

us vs

册

, us共r兲 =

冋

uA s uB s

册

, vs共r兲 =

冋

vA s vB s

册

, 共5兲 whereⵜ=共⳵x,⳵y兲, vFis a constant velocity,⌰共x兲 is the step function, ␴ˆ₀ is the 2⫻2 unit matrix, ␴j for j = 1 – 3 is the Pauli matrix representing sublattice space, and␮and Vexare the chemical potential and the exchange potential of a ferro-magnetic segment, respectively. Here and in what follows, we take the unit of ប=kB= 1, where kB is the Boltzmann constant. The spin degree of freedom of a quasiparticle is represented by s =⫾1; s=1 indicates a subspace of spin-up electron and spin-down hole, and s = −1 indicates a subspace of spin-down electron and spin-up hole. In this paper, ¯ˆ indicates a 2⫻2 matrix in sublattice space and ¯

ˇ

means a 4⫻4 matrix in Nambu⫻sublattice space. The amplitude of the pair potential in a superconducting segment is⌬ and we denote its zero temperature value by⌬₀. We assume that the superconducting segment is heavily doped so that the prox-imity effect induces superconductivity on graphene15 _共i.e.,

U0Ⰷ⌬0兲. In superconducting segment, the wave functions are given by ⌿S,e,␯= ␹ˇ

冑

2

冤

␯u0 u0 ␯v0 v0

冥

ei␯p0x_f q共y兲, 共6兲 ⌿S,h,␯= ␹ˇ

冑

2

冤

−␯u0 u0 −␯v0 v0

冥

e−i␯p0x_f q共y兲, 共7兲 u0=

冑

1 2

冉

1 + ⍀ E

冊

, v0=

冑

1 2

冉

1 − ⍀ E

冊

, 共8兲 p0= U0+␮ vF cos cs+ i␬s, sin cs= vFq U0+␮ , 共9兲 ␬s=⌬0共U0+␮兲sin␤ p₀vF2 , cos␤=

冏

E ⌬0

冏

, 共10兲 sin␤=

再

冑

1 −共E/⌬0兲 2 _{:兩E兩 ⬍ ⌬} 0 − i

冑

共E/⌬₀兲2− 1 :兩E兩 ⬎ ⌬0

冎

共11兲 ␹ˇ =

冢

ei␸/2 _{0 0 0} 0 ei␸/2 0 0 0 0 e−i␸/2 0 0 0 0 e−i␸/2

冣

, 共12兲 where⍀=

冑

E2_−⌬ 0

2_{,␸is the phase of superconductor, f}

q共y兲 is the wave function in the y direction with a momentum q,

e共h兲 in the subscript of ⌿ represents electron 共hole兲 branch,

and␯= 1 or −1 indicates moving direction of a quasiparticle in the x direction. In a ferromagnetic segment, the wave function is described by ⌿F,s,e,␯= 1

冑

2

冤

e−i␯a/2 ␯ei␯a/2 0 0

冥

ei␯pexfq共y兲, 共13兲 ⌿F,s,h,␯= 1

冑

2

冤

0 0 −␯e−i␯b/2 ei␯b/2

冥

ei␯phxfq共y兲, 共14兲 sin a = vFq E +␮+ sVex , pe= E +␮+ sVex vF cos a, 共15兲 sin b = vFq E −␮+ sVex , ph= E −␮+ sVex vF cos b. 共16兲 The wave functions in the two segments are connected at the junction interface,

⌿F,s,e,++ ree⌿F,s,e,−+ rhe⌿F,s,h,−= tee⌿S,e,++ the⌿S,h,+. 共17兲 Here ree and rhe are, respectively, the normal and Andreev reflection coefficients at the interface with a ferromagnet. They have a form

rhe= cos ae−i␸/X, 共18兲 ree= i

冋

cos␤sin

冉

a + b 2

冊

+ i sin␤sin

冉

a − b 2

冊

册

冒

X, 共19兲 X = cos␤cos

冉

a − b 2

冊

+ i sin␤cos

冉

a + b 2

冊

. 共20兲 The differential conductance of junction at zero temperature follows from the Blonder-Tinkham-Klapwijk23_formula,

GFS= 2e2 h _s=

兺

_⫾1

兺

q 关1 − 兩ree兩2+兩rhe兩2兴s,E=eV, 共21兲

兺

_q →Ns 2

冕

−␲/2 ␲/2 da cos a, Ns= 兩␮+ eV + sV_ex兩W ␲vF , 共22兲 where Ns is the number of propagating channels of an inci-dent electron. The number of propagating channels of outgo-ing hole is given by

Ms=

兩−␮+ eV + sVex兩W

␲vF

. 共23兲

As is seen from the above expression, the conductance is expected to have some characteristic structure at such values of eV, which result in Ns= 0 or Ms= 0.

In Fig. 2, we show the conductance of FS junction as a function of bias voltages. The vertical axis is normalized by a constant value independent of␮and Vex

G0= 2e2 h W ␲␰0 , ␰0= vF ⌬0 . 共24兲

We first summarize the conductance of NS junction15_{in Fig.}

2共a兲, where Vex= 0. The conductance for␮/⌬0= 0.5 and that for ␮/⌬0= 1, respectively, have bending structure at eV = 0.5⌬₀and eV =⌬₀because Ms= 0 holds at these points. The amplitude of conductance increases with the increase in ␮ for ␮⬎⌬0 because the number of propagating channels ba-sically increases with␮. The conductance spectra are close to those in usual NS junctions with highly transparent inter-face. In graphene NS junctions, however, the Andreev reflec-tion probability in the subgap region is always smaller than

unity. Second, we focus on the conductance in the limit of ␮→0 as shown in Fig.2共b兲. In usual metallic ferromagnets, ␮ⰆVexdescribes a half metal within the single-band picture. In a FS junction on a graphene, electronic structures are me-tallic for both spin directions because the conduction band is not limited from the bottom. When the Fermi energy of a spin-up electron lies on the upper cone, that of a spin-down electron is in the lower cone as shown in Fig. 3共a兲. The similar argument is also given by Ref.24. The line shape of conductance spectra in Fig. 2共b兲 can be understood by the following argument. As shown in Fig.3共b兲, a in Eq.共15兲 and

b in Eq. 共16兲 represent, respectively, an incident angle of an

electron and a reflection angle of a hole. At␮= 0, the relation

a = b holds for all E and Vex. As a result, the integral with respect to a in Eqs. 共21兲 and 共22兲 becomes independent of

Vex. Only Nsand Ms关Eqs. 共22兲 and 共23兲兴 depend on Vex. The differences in line shapes of conductance in Fig.2共b兲mainly come from the dependence of Ns and Ms on eV. At Vex = 0.5⌬₀, N_s=1 increases with increasing eV. On the other hand, N_s=−1first decreases to zero at eV = V_exthen increases with increasing eV. Thus the conductance spectra show bending structure at eV = Vex. For Vex⬎⌬0, the line shape of conductance is independent of Vexand the amplitude is sim-ply proportional to V_ex. It should be noted that the Andreev reflection probability is unity at eV = 0 for Vex⬎⌬0. There-fore, according to Ref. 8, very small shot noise is expected near the zero bias. This feature remains unchanged as far as a relation ␮ⰆVexbeing satisfied.

Here we briefly discuss the specular Andreev reflection in FS junction. The velocity of right moving incident electron is

vF共cos a,sin a兲. On the other hand, the velocity of left mov-ing hole isvF共−cos b,sin b兲. Since a=b at␮= 0, the velocity component in the x direction changes its sign in the Andreev

FIG. 2. Differential conductance of versus bias voltage. We choose V_ex= 0 in共a兲 and␮=0 in 共b兲.

hole-down Ferromagnetic segmentµ=0 electron-up electron-down hole-up (p ,q)_e (_{−p ,q)}h (a) Superconductor a b (b) electron hole

FIG. 3. 共Color online兲 共a兲 The dispersion relation of a quasipar-ticle at ␮=0 in ferromagnetic segment. 共b兲 Incident angle of an electron a and reflection angle of a hole b.

reflection, whereas that in the y direction remains unchanged. Thus the Andreev reflection is always specular in a FS junc-tion on graphene for␮ⰆVex.

At␮= 0.5⌬0, the conductance spectra show more variety as seen in Fig.4共a兲. At V_ex/⌬₀= 0.5, the conductance at zero bias has very small value because the conditions M_s=1= 0 and

Ns=−1= 0 are satisfied at eV = 0. The conductance increases with increasing eV near zero bias because the bias voltage increases the number of propagating channels. For Vex/⌬0 = 1, the conductance has a bending structure at eV =␮ be-cause Ns=−1becomes zero there. For Vex= 2⌬0, the line shape of conductance is close to that of Vex⬎⌬0 in Fig. 2共b兲. In Fig. 4共b兲, we also show the conductance for ␮= 2⌬0. At

Vex= 0.5⌬0, the conductance spectra are similar to those in NS junction in Fig.2共a兲. The subgap conductance decreases with increase in Vexas shown in Fig.4共b兲 for the cases Vex =⌬0and Vex= 2⌬0. This is because the number of propagat-ing channels decreases with increaspropagat-ing Vex. For Vex⬎␮, the conductance spectra are close to those in Fig.2共b兲.

In realistic junctions on graphene, the amplitude of pair potential ⌬0 would be expected to be on the order of 1 K. The constant slope of the dispersion relationបvFis estimated to be 7⫻10−6 _{m K.}14_{Thus the coherence length␰}

0becomes on the order of 1 ␮m. The typical amplitude of the conduc-tance is共e2/h兲Nswith Ns= W兩␮+ eV +⫾Vex兩/共បvF兲 being the number of propagating channels. When we choose W is about 1 ␮m, the conductance becomes the quantum conduc-tance e2_{/h for 兩␮+ eV +⫾V}

ex兩⯝1 meV.

III. JOSEPHSON EFFECT IN SFS JUNCTIONS Let us consider SFS junction as shown in Fig.1共b兲, where we add a superconductor on the left-hand side of FS junc-tion. We assume that the length of an insulator L is large enough so that the electric current through the insulator com-pletely vanishes. Then the superconductors are electrically coupled only through the graphene sheet. The potentials of junctions in Eqs.共1兲 and 共2兲 are replaced by

V共x兲 =

冦

− U₀ :xⱕ 0 0 :0⬍ x ⬍ L − U0 :xⱖ L

冧

, 共25兲 ⌬共x兲 =

冦

⌬ei␸L :xⱕ 0 0 :0⬍ x ⬍ L ⌬ei␸R :xⱖ L

冧

. 共26兲

The continuity equation of electron density implies

⳵tn共r兲 + ⵜ · j共r兲 = 0, 共27兲 j共r兲 =− evF 2

兺

s T

兺

␻n ⫻ Tr

冋

冉

␴ˆ 0ˆ 0ˆ ␴ˆ

冊

G ˇ 共s兲_{共r,r,␻n兲}

册

_, 共28兲 where G ˇ

共s兲 _{is the Matsubara-Green function of SFS junction} at r =共x,y兲 with 0ⱕxⱕL and␻n=共2n+1兲␲T is the

Matsub-ara frequency with n and T being an integer number and a temperature, respectively. The trace is carried out over Nambu and sublattice spaces. The Josephson current in SFS junction is obtained by integrating Eq.共28兲 with respect to y.

In direct-current Josephson effect, the Josephson current does not depend on x. Thus we choose x = 0 in what follows. Without losing generality, we obtain the total Josephson cur-rent in SFS junctions, J = e 2បs=

兺

⫾1

兺

q T

兺

␻n ⌬ ⍀n 关rhe− reh兴s, 共29兲 where⍀n=

冑

␻n2+⌬2. For propagating channels, the summa-tion is replaced by

兺

_q →Ns 2

冕

_−␲/2 ␲/2 da cos a, Ns= 兩␮+ sVex兩W ␲vF . 共30兲 The current expression coincides with Furusaki-Tsukada formula.25,26In Eq.共29兲, reh and rheare the Andreev reflec-tion coefficients, which should be calculated in an SFS junc-tion. From appropriate boundary condition at x = 0 and L, they become 关rhe− reh兴s=1=⌬⍀nD⌶, 共31兲 ⌶ = A␻n 2 +⌬2R P␻n4+␻n2⌬2Q +⌬4R2, 共32兲 P = A2+ C2, Q =共2RA + C2兲, 共33兲 R =共A + B + D cos␸兲/2, 共34兲

with␸being the phase difference between the two supercon-ductors, and

A = cos共peL兲cos a cos共phL兲cos b − sin共peL兲sin共phL兲, 共35兲

B = sin共peL兲sin a sin共peL兲sin b, 共36兲

C = cos共peL兲cos a sin共phL兲 + cos共phL兲cos b sin共peL兲, 共37兲

FIG. 4. Conductance spectra in FS junctions for␮=0.5⌬0in共a兲 and␮=2⌬0in共b兲.

D = cos a cos b. 共38兲

Here a, b, pe, and ph are given by Eqs.共15兲 and 共16兲 with

E→0. The results for s=−1 are obtained by Vex→−Vex in above results. For short junctions LⰆ␰0=vF/⌬0, we obtain

J = e

兺

s=⫾1

兺

q sin␸D

冑

Q2− 4PR2⫻

冋

␻+2A − R⌬2 ␻+ tanh

冉

␻+ 2T

冊

−␻− 2 A − R⌬2 ␻− tanh

冉

␻− 2T

冊

册

, 共39兲 where ␻_⫾2=⌬2_共Q⫾

冑

_Q2_{− 4PR}2_{兲/2P. At V} ex= 0, we recover the previous results for SNS junction.27

In Fig. 5, we show current-phase relation共CPR兲 of SFS junction at T = 0 for ␮= 0 in Fig.5共a兲and ␮= 10⌬0 in Fig.

5共b兲, where we choose L = 0.2␰0. The vertical axis is normal-ized by a constant value of J0= e⌬0W/␲␰0, which is indepen-dent of␮and V_ex, and is about 0.01 ␮A for W = 1 ␮m and ⌬0= 1 K. We first pay attention to the CPR in SNS junction,27_{which is represented by a broken line in Fig.5共b兲.} In SNS junctions, the Josephson current has a CPR, which slightly deviates from the sinusoidal relation for␮Ⰷ⌬₀. On the other hand in SFS junction, the CPR shows complicated structures depending on Vex, as shown in Fig. 5共a兲. At Vex =⌬0, the Josephson current mainly flows through evanescent channels and CPR is similar to that in the short diffusive SNS junction.27_{At V}

ex= 5⌬0, the contribution of propagating channel dominates the Josephson current, and the CPR sud-denly deviates from a linear relation around the critical phase difference␸= 0.35␲. In the limit of␮= 0, the Josephson cur-rent at T = 0 becomes J = e⌬0cos共␸/2兲

兺

q T

兺

␻n 兩␻+兩 − 兩␻−兩 2⌬0bscscos共peL兲 ⫻关共1 − 2cs2兲 + sgn兵cos2共peL兲 − sin2共␸/2兲其兴, 共40兲 with bs=

冑

1 − Tnsin2共␸/2兲, 共41兲 cs=

冑

1 − Tncos2共peL兲, 共42兲 Tn= cos2a

cos2a cos2共peL兲 + sin2共peL兲

, 共43兲

␻⫾=⌬0

冑

Tn关bscos共peL兲 ⫾ cssin共␸/2兲兴. 共44兲 The energies of the bound states are given by ⫾␻_⫾. For ␸ ⬎0 and peLⱗ1, ␻− changes its sign at ␸c, satisfying cos2共peL兲−sin2共␸c/2兲=0. Namely, the two Andreev bound states ⫾␻−go across the Fermi level at ␸c. As a result, the Josephson current changes its sign. The transition is not so sharp because␸cdepends on an incident angle of a quasipar-ticle. This situation is similar to the 0 −␲transition in diffu-sive superconductor/ferromagnet/constriction/ferromagnet/ superconductor junction.7 _{Although SFS junctions on} graphene is free from impurity scatterings, it shows a char-acteristic property of diffusive junctions. The Josephson cur-rent are always negative and positive for all ␸⬎0 at Vex = 10⌬0 and 20⌬0, respectively. This tendency can be also observed even in the presence of ␮ as shown in Fig. 5共b兲 with␮= 10⌬₀.

The Josephson current oscillates as a function of V_ex as shown in Fig.6. At␮= 0 in Fig.6共a兲, the amplitude of criti-cal current totally increases with the increase in Vexbecause the number of propagating channels increases with increas-ing V_ex. At the same time, the critical current shows oscilla-tions, which indicate the 0 −␲ transition. Strictly speaking, the oscillations are not identical to the 0 −␲ transition in usual SFS junctions because the CPR deviates far from the sinusoidal relation. In Fig. 6, we first calculate the CPR for fixed V_exin 0⬍␸⬍␲ then determine the critical current. If the critical current flows to the +x direction, we assign such junction as 0 junction in Fig. 6. In the same way, if the critical current flows to the −x direction, we label such junc-tion as␲ junction. At the 0 −␲ transition points, the critical current still have a large value because of the higher harmonics.28_{At V}

ex= 0, the critical current does not become zero because the contribution of the evanescent channels makes the Josephson current a finite value.

For␮= 20⌬₀, as shown in Fig.6共b兲, the amplitude of the critical current first decreases with increasing Vex then in-creases. At Vex=␮, the critical current becomes a small value because the number of propagating channels becomes zero. The Josephson coupling through the evanescent channels gives a finite critical current there. We analytically confirm that the junction is always in the 0 state in the limit of Vex = 0.

In Fig. 7共a兲, we show the critical current at ␮= 0 and T = 0, where we choose L = 0.1␰0and 0.5␰0. With Fig.6共a兲, we

FIG. 5. Current-phase relation of SFS junction at T = 0 for ␮ = 0 in共a兲 and␮=10⌬0in共b兲.

FIG. 6. Critical Josephson current versus exchange potential at

find that the period of oscillations becomes shorter for larger

L. At the same time, the amplitude of the critical current

becomes smaller for larger L. Figure7共b兲 shows the critical current for L = 0.2␰0 for several choices of temperatures:

T/Tc= 0.1, 0.5, and 0.8 from top to bottom. The 0 −␲ transi-tion points remain almost unchanged at finite temperatures. Thus 0 −␲ transition by changing temperature is absent in SFS junctions on graphene, as well as in usual SFS junction in the clean limit.7

IV. CONCLUSION

We have studied electron transport in ferromagnet-superconductor 共FS兲 junctions and superconductor-ferromagnet-superconductor 共SFS兲 on graphene. Cooper pairs can carry electric current even when exchange potential 共Vex兲 is much larger than the Fermi energy 共␮兲 in a ferro-magnet because electric structures of graphene lose a bottom. The exchange potential rather enhances the electron transport on graphene. The Andreev reflection is always specular re-flective for ␮= 0. As a consequence, the line shape of con-ductance spectra in FS junctions is independent of Vex for sufficiently large Vex. In addition, the Andreev reflection probability at the zero bias is unity, which leads to small shot noise near the zero bias. In SFS junction, the Josephson cur-rent shows complicated curcur-rent-phase relationships. The SFS junction exhibits the 0 −␲ transition behavior as changing

Vex.

ACKNOWLEDGMENTS

This work was partially supported by the Dutch FOM, the NanoNed program under Grant No. TCS7029 and Grant-in-Aid for Scientific Research from The Ministry of Education, Culture, Sports, Science and Technology of Japan 共Grants No. 17071007, No. 19540352, and No. 20029001兲.

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FIG. 7. Critical Josephson current versus exchange potential at

T = 0 for L = 0.2␰0 and 0.5␰0 in 共a兲. Critical Josephson current is shown for several choices of temperature for L = 0.2␰0in共b兲.