Theory of the Josephson effect in unconventional superconducting junctionswith diffusive barriers

T. Yokoyama,1 Y. Tanaka,1 and A. A. Golubov2

1Department of Applied Physics, Nagoya University, Nagoya, 464-8603, Japanand CREST, Japan Science and Technology Corporation (JST) Nagoya, 464-8603, Japan

2Faculty of Science and Technology, University of Twente, 7500 AE, Enschede, The Netherlands�Received 6 November 2006; revised manuscript received 9 January 2007; published 30 March 2007�

We study theoretically the Josephson effect in junctions based on unconventional superconductors withdiffusive barriers, using the quasiclassical Green’s function formalism. Generalized boundary conditions atjunction interfaces applicable to unconventional superconductors are derived by calculating a matrix currentwithin the circuit transport theory. Applying these boundary conditions, we have calculated the Josephsoncurrent in structures with various pairing symmetries. A number of predictions are made: �a� nonmonotonictemperature dependence in d-wave superconductor/diffusive normal metal/d-wave superconductor �D/DN/D�junctions, �b� anomalous current-phase relations in p-wave superconductor/diffusive normal metal/p-wavesuperconductor �P/DN/P� junctions, �c� second harmonics in D/DN/D and P/DN/P junctions, �d� a double-peakstructure of the critical current in D/DF/D junctions, and �e� enhanced Josephson current by the exchange fieldin S/DF/P junctions. We have also investigated peculiarities of the Josephson coupling in D/DF/D, P/DF/P, andS/DF/P junctions. An oscillatory behavior of the supercurrent and the second harmonics in the current-phaserelation is studied as a function of the length of the diffusive ferromagnet.

DOI: 10.1103/PhysRevB.75.094514 PACS number�s�: 74.20.Rp, 74.50.�r, 74.70.Kn

I. INTRODUCTION

The Josephson effect1 has been studied in various types ofjunctions2–4 motivated by fundamental interest and potentialapplications for future technology. In superconductor/diffusive normal metal/superconductor �S/DN/S� junctionsthe critical current increases monotonically with decreasingtemperature3–6 because the proximity effect is enhanced atlow temperatures. Superconducting junctions with ferromag-netic interlayers have shown rich physics due to the interplayof the proximity effect and the exchange field.7,8 When theDN is replaced by a diffusive ferromagnet �DF�, it was pre-dicted that � junctions can be realized.9–16 The physical rea-son for a � state is nonzero momentum of induced Cooperpairs in the ferromagnet,12 similar to the so-called Fulde-Ferrel-Larkin-Ovchinnikov state17,18 in a magnetic supercon-ductor. SFS � junctions have been realized experimentallyby several groups.19–29

In d-wave superconductor junctions, one of the most re-markable phenomena is the formation of midgap Andreevresonant states �MARSs� at interfaces.30 The MARSs stemfrom a sign change of pair potentials of d-wavesuperconductors.31 In d-wave superconductor/insulator/d-wave superconductor junctions, � junctions emerge due tothe formation of the MARSs.32,33 In order to clarify the roleof the proximity effect and MARSs, Tanaka et al. have ex-tended circuit theory34 to junctions with unconventionalsuperconductors.35–38 The conservation of matrix current en-ables one to apply the generalized Kirchhoff rules to uncon-ventional superconducting junctions and to derive the bound-ary conditions for the Usadel equation39 widely used indiffusive superconducting junctions. Application of thistheory to DN/d-wave superconductor �DN/D� junctions hasshown that the formation of MARSs strongly competes withthe proximity effect in DNs.35,36 It was also demonstrated

that the formation of MARSs coexists with the proximityeffect in DN/p-wave superconductor �DN/P� junctions,which produces a giant zero-bias conductance peak.37

Recently this theory has been extended to diffusive Jo-sephson junctions with unconventional superconductors.40 Itis clarified that a nonmonotonic temperature dependence ofthe critical current appears in D/DN/D junctions due to thecompetition between the proximity effect and the formationof MARSs. Concerning unconventional superconductingjunctions with clean ferromagnets, the Josephson effect isstudied in Ref. 41. However, the Josephson effect in uncon-ventional superconducting junctions with diffusive ferromag-nets has not yet been studied, although the proximity effectin these junctions was studied recently.42 Moreover, detailedderivation of the boundary conditions is not given and onlyD/DN/D junctions are considered in Ref. 40. Since the prox-imity effect and MARSs strongly influence the density ofstates,35–37 they should also crucially influence the Josephsoneffect in diffusive unconventional superconducting junctions.

The purpose of the present paper is to study the Josephsoneffect in various types of conventional and unconventionalsuperconducting junctions with DN or DF interlayers. Wefirst provide the technical details of the derivation in Ref. 40.Then, solving the Usadel equations, we apply the generalapproach to study the influence of the exchange field, theproximity effect, and the formation of MARSs on the Jo-sephson current simultaneously. A number of peculiarities inthe Josephson current are found depending on the pairingsymmetry: a nonmonotonic temperature dependence inD/DN/D junctions, anomalous current-phase relations inP/DN/P junctions, second harmonics in the current-phase re-lation and their half-periodic oscillations as a function of thelength of the DF in D/DN/D and P/DN/P junctions, transi-tions to a � state in D/DF/D, P/DF/P, and S/DF/P junctions,a double-peak structure in the temperature dependence of the

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critical current in D/DF/D junctions, and enhancement ofJosephson current by the exchange field in S/DF/P junctions.

II. FORMULATION

We consider a junction consisting of unconventional su-perconductors �USCs� connected by a quasi-one-dimensionalDN �or DF� with a resistance Rd and a length L much largerthan the mean free path. The DN/USC interface located atx=0 has the resistance Rb�, while the DN/USC interface lo-cated at x=L has the resistance Rb. We model infinitely nar-row insulating barriers by the delta function U�x�=H��x−L�+H���x�. The resulting transparencies of the junctionsTm and Tm� are given by Tm=4 cos2 � / �4 cos2 �+Z2� andTm� =4 cos2 � / �4 cos2 �+Z�2�, where m is a channel index,Z=2H /vF and Z�=2H� /vF are dimensionless constants, � isthe injection angle measured from the interface normal to thejunction, and vF is the Fermi velocity.

In order to study the Josephson effect in diffusive USCjunctions, we first concentrate on the quasiclassical Keldysh-

Nambu Green’s function in a DN defined by GN�x�. Its re-

tarded part RN�x� can be expressed as

RN�x� = cos � sin � �1 + sin � sin � �2 + cos � �3, �1�

with Pauli matrices in the electron-hole space �1, �2, and �3.

Since RN�x� obeys the Usadel equation, the following equa-tions are satisfied:

D� �2

�x2� − � ��

�x�2

cos � sin �� + 2i�� + �− �h�sin � = 0,

�2�

�

�x�sin2 �� ��

�x�� = 0 �3�

for majority �minority� spin with the diffusion constant D

and exchange field h. The boundary condition of GN�x� at theDN/USC interface is given by37,40

L

Rd�GN�x�

�GN�x��x

�x=L−

= −I����

Rb, �4�

I��� = 2�G1,B� , �5�

B = �− T1�G1,H−−1� + H−

−1H+ − T12G1H−

−1H+G1�−1�T1�1 − H−−1�

+ T12G1H−

−1H+� �6�

with G1= GN�x=L−�, H±= �G2+± G2−� /2, and T1=T / �2−T

+2�1−T�, where G2± is the asymptotic Green’s function inUSCs as defined in our previous papers.37 � means the av-erage over the various angles of injected particles at the in-

terface. The retarded components of G1 and G2± are given by

R1,2± where R2± is expressed by R2±=g±�3+ f±�2 with g±

=� /��2−±2 and f±=± /�±

2 −�2, where � denotes the qua-

siparticle energy measured from the Fermi energy. + �−� isthe effective pair potential experienced by quasiparticles

with an injection angle � ��−��. We also denote by Rp, Rm,

and IR the retarded parts of H+, H−, and I.Now we discuss the boundary condition of the retarded

part of the Keldysh-Nambu Green’s function at the DN/USCinterface. The left side of the boundary condition of Eq. �4�can be expressed as

L

RdRN�x�

�

�xRN �x� x=L

=Li

Rd��−

��

�xsin � −

��

�xsin � cos � cos ���1

+ � ��

�xcos � −

��

�xsin � cos � sin ���2

+��

�xsin2 � �3� . �7�

In the right side of Eq. �4�, IR can be expressed by usingseveral spectral vectors:

IR = 4iT1�dR · dR�−1�−1

2�1 + T1

2��s2+ − s2−�2�s1 �s2+

+ s2−�� · � + 2T1s1 · �s2+ s2−��s1 �s2+ s2−�� · �

+ 2T1s1 · �s2+ − s2−��s1 �s2+ − s2−�� · � − i�1 + T12��1

− s2+ · s2−��s1 �s2+ s2−�� · � + 2iT1�1 − s2+ · s2−�

�s1 · �s2+ − s2−�s1 − �s2+ − s2−�� · �� , �8�

dR = �1 + T12��s2+ s2−� − 2T1s1 �s2+ − s2−�

− 2T12s1 · �s2+ s2−�s1 �9�

with R1=s1 · � and R2±=s2± · �.37

The spectral vectors s1 and s2± are given by

s1 = �sin � cos �

sin � sin �

cos ��, s2± = � f± cos �

f± sin �

g±� �10�

where � denotes the phase of the USC. Then IR is reduced to

IR = 2iT��2 − T� + T� f S sin � cos�� − �� + gS cos ��

− TfS sin � sin�� − ���−1��− gS sin � sin �

+ f S cos � sin � − fS cos � cos ���1 + �− f S cos � cos �

+ gS sin � cos � − fS cos � sin ���2 + � f S sin � sin��

− �� + fS sin � cos�� − ����3� , �11�

and hence we find the following form of the matrix current:

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IR�

= i�− I1 sin � sin � + I2 cos � sin � − I3 cos � cos �

− I2 cos � cos � + I1 sin � cos � − I3 cos � sin �

I2 sin � sin�� − �� + I3 sin � cos�� − ��� · � ,

I1 = � 2TmgS

A�, I2 =� 2TmfS

A�, I3 = � 2TmfS

A� ,

A = �2 − Tm� + Tm� f S sin � cos�� − �� + gS cos ��

− TmfS sin � sin�� − �� ,

gS =g+ + g−

1 + f+f− + g+g−, f S =

f+ + f−

1 + f+f− + g+g−,

fS =i�f+g− − f−g+�

1 + f+f− + g+g−. �12�

Finally the boundary conditions are given by

LRb

Rd

�

�x� = − I1 sin � + I2 cos � cos�� − ��

− I3 cos � sin�� − �� , �13�

LRb

Rdsin �

�

�x� = − I2 sin�� − �� − I3 cos�� − �� . �14�

For the calculation of the thermodynamical quantities, weuse the Matsubara representation: �→ i�. We parametrizethe quasiclassical Green’s functions G and F using the func-tion :

G� =�

��2 + � −�*

= cos � , �15�

F� = �

��2 + � −�*

= �

�G� = sin �e−i�, �16�

F−�* =

−�*

��2 + � −�*

= −�

*

�G� = sin �ei� �17�

with Matsubara frequency �. Then the Usadel equationreads39

�2�TC

G�

�

�x�G�

2 �

�x �� − �� − �+ �ih� � = 0 �18�

for majority �minority� spin with �=�D /2�TC and criticaltemperature TC. The following relations are satisfied:

sin � cos � =G�

2�� � + −�

* � , �19�

sin � sin � =iG�

2�� � − −�

* � . �20�

Then the boundary condition is expressed as

G�

�

�

�x � =

Rd

RbL�−

�

�I1 + e−i��I2 + iI3�� ,

I1 = � 2TmgS

A�, I2 =� 2TmfS

A�, I3 = � 2TmfS

A� ,

A = 2 − Tm + Tm�gSG� + f S�B cos � + C sin ��

− fS�C cos � − B sin ��� ,

B =G�

2�� � + −�

* �, C =iG�

2�� � − −�

* � , �21�

at x=L.This boundary condition is quite general since with a

proper choice of ±, it is applicable to any unconventionalsuperconductor with Sz=0 in a time reversal symmetry con-serving state. Here, Sz denotes the z component of thetotal spin of a Cooper pair. For s-, d-, and p-wave supercon-ductors we choose ±=�T�, �T�cos�2��2��, and�T�cos�����, respectively.

In the following we will calculate the Josephson currentusing this boundary condition at x=0 and L, where � is theexternal phase difference across the junctions, and � and �denote the angles between the normal to the interface and thecrystal axes of the USCs for x�0 and x�L, respectively. Itis important to note that the solution of the Usadel equationis invariant under the transformation �→−� or �→−�. Thisis made clear by replacing � with −� in the angular averag-ing.

The Josephson current can be expressed using � and −�

* .4 Below R, T, and IC denote Rd+Rb+Rb�, the tempera-ture, and the critical current, respectively, and we considersymmetric barriers with Rb=Rb� and Z=Z� for simplicity.

III. RESULTS

A. Junctions with DNs

Let us first focus on the junctions with DNs. Figure 1shows the current-phase relation for T /TC=0.1, Rd /Rb=0.1,and ETh /�0�=0.2 in �a� s-wave, �b� d-wave, and �c� p-wavesuperconducting junctions with �� ,��= �0,0�. In s-wavejunctions, the IR product is suppressed with the increase of Zbecause the proximity effect is suppressed. In d-wave junc-tions, the proximity effect and hence IR are enhanced withthe increase of Z because of the cancellation of the positiveand negative parts of the pair potential in the angular aver-aging. As Z increases, the contribution from the positive partexceeds that from the negative part and hence the cancella-tion becomes weak.40 In p-wave junctions, the IR product isstrongly enhanced with the increase of Z because of the for-mation of resonant states. It is known that the proximityeffect and MARSs can coexist.37 At �� ,��= �0,0�, the prox-imity effect is mostly enhanced. As Z increases, the contri-bution of the MARSs becomes remarkable and hence theproximity effect gets strongly enhanced.37 Consequently itsmagnitude is an order of magnitude larger than that in

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s-wave junctions. The results at lower temperatures areshown in Fig. 2. The IR product is enhanced with decreasingtemperature because the proximity effect is enhanced. In thiscase the current-phase relation in p-wave junctions has aform close to sin�� /2�, in contrast to the standard sinusoidalrelation. This is a peculiar property of the formation of theresonant states in p-wave junctions where constructive inter-ference occurs near �=�.43

Next we study the Josephson effect for other misorienta-tional angles. As � or � increase, the IR product is mono-tonically suppressed due to the suppression of the proximityeffect as shown in Figs. 3�a� and 3�b�. In d- �p�-wave junc-tions, the first harmonics disappear and hence the IR productis proportional to −sin 2� at �� ,��= �� /4 ,0� ��� /2 ,0�� asshown in Figs. 3�c� and 3�d�. This can be explained in thelimiting case as follows. Near TC, the IR product, whichstems from the first harmonics, is proportional tocos 2� cos 2� in d-wave junctions because angular averag-ing gives cos�2�−2����cos 2�.44 Thus the first harmonicsdisappear at �=� /4. A similar argument is also applicable top-wave junctions.

Let us discuss the results for the critical current. In Fig. 4,temperature dependence of the critical current is plotted forvarious Rd /Rb and ETh /�0� in �a� s-wave, �b� d-wave, and�c� p-wave superconducting junctions with Z=10 and�� ,��= �0,0�. As Rd /Rb and ETh /�0� increase, ICR in-creases for all the junctions because the proximity effect isenhanced. In p-wave junctions, the critical current is stronglyenhanced at low temperatures compared to the s-wave andd-wave junctions. When the misorientational angles in

d-wave junctions are changed, nonmonotonic temperaturedependence appears as shown in Fig. 5�a�. This nonmono-tonic behavior can be explained in terms of the competitionbetween the proximity effect and the formation of MARSs. Itis known from the previous studies that for �=�=0 theproximity effect exists but the MARS is absent at the inter-faces. On the other hand, for �=�=� /4, only the MARSexists and the proximity effect is absent.35,36 In other cases,both the proximity effect and MARSs are present. With thedecrease of temperature, the formation of MARSs stronglysuppresses the proximity effect. This results in the suppres-sion of the Josephson current at low temperatures. Therefore,a nonmonotonic temperature dependence appears when boththe proximity effect and MARSs coexist.

The above statement can be confirmed by calculation ofthe dependence of the anomalous Green’s function F on theMatsubara frequency � as shown in Figs. 5�b� and 5�c� atx=L /2 and �=� /2 for �� ,��= �� /8 ,0�. At low temperature�T /TC=0.01� the magnitude of Im F is suppressed at lowenergy in contrast to the case of high temperature �T /TC=0.2 and 0.3�. This result illustrates strong suppression of theproximity effect by the formation of MARSs at low T, whichleads to the nonmonotonic temperature dependence. Notethat this nonmonotonic dependence can appear only for largeZ when the role of MARSs is essential.40

It is interesting to study the junctions composed of super-conductors with different symmetries. Here we studyS/DN/D junctions with Z=10, Rd /Rb=1, and ETh /�0�=0.1.We choose TCD /TCS=5 in Fig. 6�a� and TCD /TCS=10 in Fig.6�b� where TCS �TCD� denotes the critical temperature of thes-wave �d-wave� superconductors. In this case the nonmono-

FIG. 1. �Color online� Current-phase relation for T /TC=0.1,Rd /Rb=0.1, and ETh /�0�=0.2. �a� s-wave junctions. �b� d-wavejunctions. �c� p-wave junctions.

FIG. 2. �Color online� Current-phase relation for T /TC=0.02,Rd /Rb=0.1, and ETh /�0�=0.2. �a� s-wave junctions. �b� d-wavejunctions. �c� p-wave junctions.

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tonic temperature dependence also occurs due to thecompetition as shown in Fig. 6. In S/insulator/D junctions,the nonmonotonic temperature dependence was observed ex-perimentally in Ref. 45. We can qualitatively explain thesedata by regarding the barrier as a diffusive normally conduct-ing material.

We study the dependence of the critical current on barrierthickness L at various temperatures in �a� s-wave, �b�d-wave, and �c� p-wave superconducting junctions with Z=10, Rd /Rb=0.1, and �� ,��= �0,0� in Fig. 7. The ICR prod-

FIG. 3. �Color online� Current-phase relationfor T /TC=0.01, Z=10, Rd /Rb=0.1, andETh /�0�=0.2. �a� and �c� d-wave junctions. �b�and �d� p-wave junctions.

FIG. 4. �Color online� Temperature dependence of the criticalcurrent. �a� s-wave junctions. �b� d-wave junctions. �c� p-wavejunctions. Curves a. Rd /Rb=2 and ETh /�0�=1. Curves b. Rd /Rb

=0.5 and ETh /�0�=1. Curves c. Rd /Rb=2 and ETh /�0�=0.1.Curves d. Rd /Rb=0.5 and ETh /�0�=0.1.

FIG. 5. �Color online� �a� Temperature dependence of the criti-cal current for �� ,��= �� /8 ,0�. Curve a. Rd /Rb=2 and ETh /�0�=1. Curve b. Rd /Rb=0.5 and ETh /�0�=1. Curve c. Rd /Rb=2 andETh /�0�=0.1. Curve d. Rd /Rb=0.5 and ETh /�0�=0.1. �b� realand �c� imaginary parts of anomalous Green’s functions F withRd /Rb=2 and ETh /�0�=1.

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uct is proportional to exp�−CL /�� for large L /� for all the

junctions as shown in Fig. 7. Here C is a constant indepen-

dent of L. As temperature is lowered, the magnitude of C is

reduced. From our results, we also find the relation C�T−1/2. The results for junctions with other misorientationalangles for T /TC=0.01, Z=10, and Rd /Rb=0.1 are shown in

Fig. 8. As � and � increase, ICR is suppressed. However, Cis independent of these values, which indicates that MARSs

do not influence the effective coherence length � / C. This isbecause the effective coherence length reflecting the penetra-tion of Cooper pairs is determined by the Usadel equationand therefore is independent of MARSs.

B. Junctions with DFs

Here we consider junctions with DFs. We will study threetypes of junctions: D/DF/D, P/DF/P and S/DF/P junctions.Figure 9 shows the current-phase relation in D/DF/D junc-

FIG. 6. �Color online� Temperature dependence of the criticalcurrent in S/DN/D junctions with Z=10, Rd /Rb=1, and ETh /�0�=0.1. TCD /TCS= �a� 5 and �b� 10.

FIG. 7. �Color online� Length dependence of the critical currentwith Z=10 and Rd /Rb=0.1. �a� s-wave junctions. �b� d-wave junc-tions. �c� p-wave junctions.

FIG. 8. �Color online� Length dependence of the critical currentwith T /TC=0.01, Z=10, and Rd /Rb=0.1. �a� d-wave junctions. �b�p-wave junctions.

FIG. 9. �Color online� Current-phase relation in D/DF/D junc-tions for T /TC=0.01, Z=10, Rd /Rb=1, and ETh /�0�=0.1.

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tions for T /TC=0.01, Z=10, Rd /Rb=1, and ETh /�0�=0.1.At �� ,��= �0,0� where the MARSs are absent, the exchangefield causes a 0-� transition as predicted for s-wave junc-tions �see Fig. 9�a��. Similarly, the second harmonic changesits sign at �� ,��= �� /4 ,0�, where the proximity effect isabsent at x=0, as shown in Fig. 9�b�. Figure 10 displays thetemperature dependence of the critical current in D/DF/Djunctions with Rd /Rb=1 and ETh /�0�=0.1. At �� ,��= �0,0�, the exchange field causes a 0-� transition as shownin Fig. 10�a�. At �� ,��= �� /8 ,0�, the exchange field alsocauses a 0-� transition, and, as a result, a double-peak struc-ture appears for h /�0�=0.4 as shown in Fig. 10�b�. Thepeak at lower temperature stems from the competition be-tween proximity effect and MARSs. The peak at higher tem-perature stems from the 0-� transition. With the decrease ofZ, the magnitude of ICR is suppressed while the 0-� transi-tion temperature is almost independent of Z �see Fig. 10�c��.At �� ,��= �� /8 ,0�, the peak at lower temperature disap-pears for small Z as shown in Fig. 10�d� because the exis-tence of the insulating barrier is essential for the formation ofMARSs.

The barrier thickness dependence of the critical current inD/DF/D junctions is plotted in Fig. 11 with T /TC=0.1, Z=10, Rd /Rb=1, and ETh /�0�=0.1. For h=0, the ICR producthas an exponential dependence on L. As h /�0� increases, IC

oscillates as a function of L /�. The period of the oscillationbecomes shorter with increasing h as shown in Fig. 11�a�. As� and � increase, ICR is suppressed while the period of theoscillations remains constant, that is, the period is indepen-dent of the MARSs. The second harmonics have a shorter�almost half� oscillation period than that of the first harmon-ics, similar to the predictions for S/DF/S junctions14,15 �seethe result for �� ,��= �� /4 ,0� in Fig. 11�b��.

Next we consider the P/DF/P junctions. The current-phaserelation in P/DF/P junctions for T /TC=0.01, Z=10, Rd /Rb=1, and ETh /�0�=0.1 is plotted in Fig. 12. With increasingh, the dependence of IR changes from sin�� /2� to sin 2� andfinally to −sin � at �� ,��= �0,0� as shown in Fig. 12�a�. The

phase dependences originate from the formation of the reso-nant states, the disappearance of the first harmonics at the0-� transition and the emergence of the � junctions, respec-tively. At �� ,��= �� /2 ,0� where the MARSs are absent atx=0, the second harmonics change the sign with the increaseof h as shown in Fig. 12�b�. The critical current as a functionof T is shown in Fig. 13�a�. The 0-� transition occurs due tothe exchange field. Similarly, as h /�0� increases, ICR oscil-lates as a function of L /�. The period of the oscillation be-comes short with increasing h as shown in Fig. 13�b�.

Finally we study S/DF/P junctions for Z=10, Rd /Rb=1,ETh /�0�=0.1, and �=0. The current-phase relation atT /TC=0.01 has the form of −sin 2� for h=0 due to the dif-ference of the parities of two superconductors41 as shown inFig. 14�a�. As h increases, the shape of IR transforms from

FIG. 10. �Color online� Temperature depen-dence of the critical current in D/DF/D junctions.Z=10 in �a� and �b�. h /�0�=0.4 in �c� and �d�.

FIG. 11. �Color online� Length dependence of the critical cur-rent in D/DF/D junctions with T /TC=0.1, Z=10, Rd /Rb=1, andETh /�0�=0.1. We choose h /�0�=4 in �b�.

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−sin 2� to cos � since the first harmonics recover by break-ing the symmetry between up and down spins. The tempera-ture dependence of the critical current is plotted in Fig.14�b�. The magnitude of ICR is enhanced by the increase of hdue to the recovery of the first harmonics, in contrast tojunctions between superconductors with equal parities.

IV. CONCLUSIONS

In this paper, we studied the Josephson effect in junctionsbetween unconventional superconductors with diffusive bar-

riers. The Usadel equations in the barrier region were solvedwith the generalized boundary conditions applicable to un-conventional superconductors at the interfaces. Applyingthese boundary conditions, we calculated the Josephson cur-rent in various types of junctions: S/DN/S, D/DN/D, P/DN/P,S/DN/D, D/DF/D, P/DF/P, and S/DF/P junctions. Our mainconclusions can be summarized as follows.

�1� The dependences of Josephson current on the interfacebarrier strength Z are different for S/DN/S, D/DN/D, andP/DN/P junctions. The Josephson current is suppressed bythe increase of Z in S/DN/S junctions while it is enhanced bythe increase of Z in D/DN/D and P/DN/P junctions. InD/DN/D and P/DN/P junctions, the proximity effect is en-hanced by the increase of Z due to the cancellation of thepositive and negative parts of the pair potential in the angularaveraging and the coexistence of MARSs and proximity ef-fect, respectively. The coexistence also induces anomalouscurrent-phase relation in P/DN/P junctions. When the prox-imity effect is absent at one interface, the second harmonicsdominate in D/DN/D and P/DN/P junctions. The competitionbetween MARSs and proximity effect causes a nonmono-tonic temperature dependence of the critical current inD/DN/D junctions. Similar dependence can be seen inS/DN/D junctions.

�2� In S/DN/S, D/DN/D, and P/DN/P junctions, the criti-cal current has an exponential dependence on the length ofthe DN. The prefactor of the length is independent of theMARSs.

�3� In D/DF/D, P/DF/P, and S/DF/P junctions, the � statecan be realized. A double-peak structure in temperature de-pendence of the critical current occurs in D/DF/D junctionsdue to 0-� transition and the competition between MARSsand proximity effect. In S/DF/P junctions, the Josephson cur-rent can be enhanced by the exchange field, in contrast to

FIG. 12. �Color online� Current-phase relation in P/DF/P junc-tions for T /TC=0.01, Z=10, Rd /Rb=1, and ETh /�0�=0.1.

FIG. 13. �Color online� �a� Temperature and �b� length depen-dence of the critical current in P/DF/P junctions. Z=10, Rd /Rb=1,and �� ,��= �0,0�. We choose ETh /�0�=0.1 in �a� and T /TC=0.1in �b�.

FIG. 14. �Color online� �a� Current-phase relation at T /TC

=0.01 and �b� temperature dependence of the critical current inS/DF/P junctions for Z=10, Rd /Rb=1, ETh /�0�=0.1, and �=0.

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other types of junctions, due to the recovery of the first har-monics.

�4� In D/DF/D and P/DF/P junctions, the critical currenthas an oscillatory behavior as a function of the length of theDF. The period of the oscillation becomes short for largeexchange fields while it is independent of the MARSs. Thesecond harmonics show almost half periodicity compared tothe first harmonics.

ACKNOWLEDGMENTS

The authors thank T. Akazaki of NTT Basic ResearchLaboratories for useful discussions. T.Y. acknowledges sup-port by JSPS. This work was supported by the NTT basicresearch laboratory, NAREGI Nanoscience Project, the Min-

istry of Education, Culture, Sports, Science and Technology,Japan, the Core Research for Evolutional Science and Tech-nology �CREST� of the Japan Science and Technology Cor-poration �JST�, a Grant-in-Aid for Scientific Research on Pri-ority Area “Novel Quantum Phenomena Specific toAnisotropic Superconductivity” �Grant No. 17071007� fromthe Ministry of Education, Culture, Sports, Science andTechnology of Japan, a Grant-in-Aid for Scientific Researchon B �Grant No. 17340106� from the Ministry of Education,Culture, Sports, Science and Technology of Japan, and the21st Century COE “Frontiers of Computational Science.”The computational aspect of this work was performed at theResearch Center for Computational Science, Okazaki Na-tional Research Institutes, and the facilities of the Supercom-puter Center, Institute for Solid State Physics, University ofTokyo and the Computer Center.

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