arX

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0142

v1 [

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201

3

Theory of supercurrent transport in SIsFS Josephson junctions

S. V. Bakurskiy,1,2 N. V. Klenov,2 I. I. Soloviev,1 M. Yu. Kupriyanov,1 and A. A. Golubov3

1Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Leninskie gory, Moscow 119991, Russian Federation2Faculty of Physics, Lomonosov Moscow State University, Leninskie gory, Moscow 119992, Russian Federation

3Faculty of Science and Technology and MESA+ Institute for Nanotechnology,University of Twente, 7500 AE Enschede, The Netherlands

(Dated: June 20, 2018)

We present the results of theoretical study of Current-Phase Relations (CPR)JS(ϕ) in Josephson junctions ofSIsFS type, where ’S’ is a bulk superconductor and ’IsF’ is a complex weak link consisting of a superconductingfilm ’s’, a metallic ferromagnet ’F’ and an insulating barrier ’I’. At temperatures close to critical,T . TC,calculations are performed analytically in the frame of theGinsburg-Landau equations. At low temperaturesnumerical method is developed to solve selfconsistently the Usadel equations in the structure. We demonstratethat SIsFS junctions have several distinct regimes of supercurrent transport and we examine spatial distributionsof the pair potential across the structure in different regimes. We study the crossover between these regimeswhich is caused by shifting the location of a weak link from the tunnel barrier ’I’ to the F-layer. We show thatstrong deviations of the CPR from sinusoidal shape occur even in a vicinity of TC , and these deviations arestrongest in the crossover regime. We demonstrate the existence of temperature-induced crossover between 0andπ states in the contact and show that smoothness of this transition strongly depends on the CPR shape.

PACS numbers: 74.45.+c, 74.50.+r, 74.78.Fk, 85.25.Cp

I. INTRODUCTION

Josephson structures with a ferromagnetic layer becamevery active field of research because of the interplay betweensuperconducting and magnetic order in a ferromagnet leadingto variety of new effects including the realization of aπ-statewith phase differenceπ in the ground state of a junction, aswell as long-range Josephson coupling due generation of odd-frequency triplet order parameter1–3.

Further interest to Josephson junctions with magnetic bar-rier is due to emerging possibilities of their practical useas elements of a superconducting memory4−12, on-chipπ-phase shifters for self-biasing various electronic quantum andclassical circuits13−16, as well asϕ- batteries, the structureshaving in the ground state phase differenceϕg = ϕ , (0 <

|ϕ | < π) between superconducting electrodes17–25. In stan-dard experimental implementations SFS Josephson contactsare sandwich-type structures26−27. The characteristic voltageVC = JCRN (JC is critical current of the junction,RN is resis-tance in the normal state) of these SFS devices is typicallyquite low, which limits their practical applications. In SIFSstructures28−32 containing an additional tunnel barrier I, theJCRN product in a 0-state is increased9, however in aπ-stateVC is still too small33,34 due to strong suppression of the su-perconducting correlations in the ferromagnetic layer.

Recently, new SIsFS type of magnetic Josepshon junctionwas realized experimentally9–12. This structure represents aconnection of an SIs tunnel junction and an sFS contact in se-ries. Properties of SIsFS structures are controlled by the thick-ness of s layerds and by relation between critical currentsJCSIsandJCsFS of their SIs and sFS parts, respectively. If the thick-ness of s-layerds is much larger than its coherence lengthξSandJCSIs ≪ JCsFS, then characteristic voltage of an SIsFS de-vice is determined by its SIs part and may reach its maximumcorresponding to a standard SIS junction. At the same time,the phase differenceϕ in a ground state of an SIsFS junction

is controlled by its sFS part. As a result, both 0- andπ-statescan be achieved depending on a thickness of the F layer. Thisopens the possibility to realize controllableπ junctions hav-ing largeJCRN product. At the same time, being placed inexternal magnetic fieldHext SIsFS structure behaves as a sin-gle junction, sinceds is typically too thin to screenHext . Thisprovides the possibility to switchJC by an external field.

However, theoretical analysis of SIsFS junctions was notperformed up to now. The purpose of this paper is to developa microscopic theory providing the dependence of the char-acteristic voltage on temperatureT , exchange energyH in aferromagnet, transport properties of FS and sF interfaces andthicknesses of s and F layers. Special attention will be given todetermining the current-phase relation (CPR) between the su-percurrentJS and the phase differenceϕ across the structure.

II. MODEL OF SISFS JOSEPHSON DEVICE

We consider multilayered structure presented in Fig.1a. Itconsists of two superconducting electrodes separated by com-plex interlayer including tunnel barrier I, intermediate super-conducting s and ferromagnetic F films. We assume that theconditions of a dirty limit are fulfilled for all materials inthe structure. In order to simplify the problem, we also as-sume that all superconducting films are identical and can bedescribed by a single critical temperatureTC and coherencelengthξS. Transport properties of both sF and FS interfacesare also assumed identical and are characterized by the inter-face parameters

γ =ρSξS

ρFξF, γB =

RBFAB

ρFξF. (1)

HereRBF andAB are the resistance and area of the sF and FSinterfacesξS andξF are the decay lengths of S and F materials

2

dSL ds dSR

s( ) S(dF)s( ds)

(x)

x

0

0dF0-ds

dF

c)

b)

0

-ds dF0

(x)

x0

S s SI Fa)

FIG. 1: a) Schematic design of SIsFS Josephson junction. b),c)Typical distribution of amplitude|∆(x)| and phase differenceχ(x) ofpair potential along the structure.

while ρS andρF are their resistivities.Under the above conditions the problem of calculation of

the critical current in the SIsFS structure reduces to solutionof the set of the Usadel equations35. For the S layers theseequations have the form1–3

ξ 2S

ΩGm

ddx

(G2

mddx

Φm

)−Φm =−∆m, Gm =

Ω√Ω2+ΦmΦ∗

m

,

(2)

∆m lnTTC

+TTC

∞

∑ω=−∞

(∆m

|Ω| −ΦmGm

Ω

)= 0, (3)

wherem = S for x ≤ −ds andx ≥ dF ; m = s in the interval−ds ≤ x ≤ 0. In the F film(0≤ x ≤ dF) they are

ξ 2F

ddx

(G2

Fddx

ΦF

)− ΩΦF GF = 0. (4)

Here Ω = T (2n + 1)/TC are Matsubara frequencies nor-malized to πTC, Ω = Ω + iH/πTC, GF = Ω/(Ω2 +

ΦF,ωΦ∗F,−ω)

1/2, H is exchange energy,ξ 2S,F = (DS,F/2πTC)

andDS,F , are diffusion coefficients in S and F metals, respec-tively. Pair potential∆m and the Usadel functionsΦm andΦFin (2) - (4) are also normalized toπTC. To write equations (2)- (4), we have chosen thex axis in the directions perpendic-ular to the SI, FS and sF interfaces and put the origin at sFinterface. Equations (2) - (4) must be supplemented by theboundary conditions36. At x =−ds they can be written as

G2S

ddx

ΦS = G2s

ddx

Φs, (5)

γBIξSGsddx

Φs =−GS (ΦS −Φs) ,

whereγBI = RBIAB/ρSξS, RBI andAB are resistance and areaof SI interface. Atx = 0 the boundary conditions are

ξS

ΩG2

sddx

Φs = γξF

ΩG2

Fddx

ΦF , (6)

γBξF GFddx

ΦF =−Gs

(ΩΩ

Φs −ΦF

)

and atx = dF they have the form

ξS

ΩG2

Sddx

ΦS = γξF

ΩG2

Sddx

ΦF , (7)

γBξF GFddx

ΦF = GS

(ΩΩ

ΦS −ΦF

),

Far from the interfaces the solution should cross over to a uni-form current-carrying superconducting state37−39

ΦS(∓∞) = Φ∞ expi(χ(∓∞)− ux/ξS) , (8)

∆S(∓∞) = ∆0expi(χ(∓∞)− ux/ξS) , (9)

Φ∞ =∆0

1+ u2/√

Ω2+ |ΦS|2, (10)

resulting in order parameter phase difference across the struc-ture equal to

ϕ = ϕ(∞)−2ux/ξS, ϕ(∞) = χ(∞)− χ(−∞). (11)

Hereϕ(∞) is the asymptotic phase difference across the junc-tion, ∆0 is modulus of order parameters far from the bound-aries of the structure at a given temperature,u = 2mvsξS, mis the electron mass andvs is the superfluid velocity. Notethat since the boundary conditions (5) - (6) include the Mat-subara frequencyΩ, the phases ofΦS functions depend onΩand are different from the phase of the pair potential∆S at theFS interfacesχ(dF) andχ(0). Therefore it is the valueϕ(∞)rather thanϕ = χ(dF)− χ(0), that can be measured experi-mentally by using a scheme compensating the linear inx partin Eq. (11).

The boundary problem (2)-(11) can be solved numericallymaking use of (8), (10). Accuracy of calculations can be mon-itored by equality of currentsJS

2eJS(ϕ)πTAB

=∞

∑ω=−∞

iG2m,ω

ρmΩ2

[Φm,ω

∂Φ∗m,−ω

∂x−Φ∗

m,−ω∂Φm,ω

∂x

],

(12)

3

calculated at the SI and FS interfaces and in the electrodes.In the further analysis carried out below we limit our-

selves to the consideration of the most relevant case of low-transparent tunnel barrier at SI interface

γBI ≫ 1. (13)

In this approximation, the junction resistanceRN is fully deter-mined by the barrier resistanceRBI. Furthermore the currentflowing through the electrodes can lead to the suppression ofsuperconductivity only in the vicinity of sF and FS interfaces.That means, up to terms of the order ofγ−1

BI we can neglectthe effects of suppression of superconductivity in the regionx ≤−ds and write the solution in the form

ΦS(x) = ∆S(x) = ∆0. (14)

Here without any lost of generality we putχ(−∞) = χ(−ds−0) = 0 (see Fig. 1c).

Substitution of (14) into boundary conditions (5) gives

γBIξSGsddx

Φs =− Ω√Ω2+∆2

0

(∆0−Φs) . (15)

Further simplifications are possible in a several limiting cases.

III. THE HIGH TEMPERATURE LIMIT T ≈ TC

In a vicinity of critical temperature the Usadel equations inthe F layer can be linearized. Writing down their solution inthe analytical form and using the boundary conditions (6), (7)on sF and FS interfaces we can reduce the problem to the solu-tion of Ginzburg-Landau (GL) equations in the s and S layers.We limit our analysis by considering the most interesting casewhen the following condition is fulfilled:

ΓBI =γBIξS

ξS(T )≫ 1, (16)

and when there is strong suppression of superconductivity inthe vicinity of the sF and FS interfaces. The latter takes placeif the parameterΓ

Γ =γξS(T )

ξS, ξS(T ) =

πξS

2√

1−T/TC(17)

satisfies the conditions

Γp ≫ 1, Γq ≫ 1. (18)

Here

p−1 =8

π2 Re∞

∑ω=0

1

Ω2√

ΩcothdF

√Ω

2ξF

, (19)

q−1 =8

π2 Re∞

∑ω=0

1

Ω2√

Ω tanhdF

√Ω

2ξF

. (20)

Note that in the limith = H/πTC ≫ 1 anddF ≫√

2/hξFthe sums in (19), (20) can be evaluated analytically resultingin

β =p− qp+ q

=√

8sin

(dF

ξF

√h2+

3π4

)exp

(−dF

ξF

√h2

),

(21)

p+ q = 2√

2h(T/TC)2 , pq = 2h(T/TC)

4 . (22)

In general, the phases of the order parameters in s and Sfilms are functions of the coordinatex. In the considered ap-proximation the terms that take into account the coordinatedependence of the phases, are proportional to small parame-ters (Γq)−1 and (Γp)−1 and therefore provide small correc-tions to the current. For this reason, in the first approximationwe can assume that the phases in superconducting electrodesare constants independent of x. In the further analysis we de-note the phases at the s-film byχ and at the right S-electrodeby ϕ (see Fig.1c).

The details of calculations are summarized in the AppendixA. These calculations show that the considered SIsFS junctionhas two modes of operation depending on relation between slayer thicknessds and the critical thicknessdsc = (π/2)ξS(T ).For ds larger thandsc, the s-film keeps its intrinsic supercon-ducting properties (mode (1)), while fords ≤ dsc superconduc-tivity in the s-film exists only due to proximity effect with thebulk S electrodes (mode (2)).

A. Mode (1): SIs+ sFS junctionds ≥ dsc

We begin our analysis with the regime when the interme-diate s-layer is intrinsically superconducting. In this case itfollows from the solution of GL equations that supercurrentflowing across SIs, sF and FS interfaces (J(−ds), J(0) andJ(dF), respectively) can be represented in the form (see Ap-pendix A)

JS(−ds)

JG=

δs(−ds)

ΓBI∆0sin(χ) , JG =

π∆20AB

4eρSTCξS(T ), (23)

JS(0)JG

=JS(dF)

JG=

Γ(p− q)

2∆20

δs(0)δS(dF)sin(ϕ − χ) , (24)

where∆0 =√

8π2TC(TC −T )/7ζ (3) is bulk value of orderparameter in S electrodes,AB is cross sectional area of thestructure,ζ (z) is Riemann zeta function. Here

δs(0) =2b(p− q)cos(ϕ − χ)−2a(p+ q)

Γ[(p+ q)2− (p− q)2 cos2 (ϕ − χ)

] , (25)

δS(dF) =2b(p+ q)−2a(p− q)cos(ϕ − χ)

Γ((p+ q)2− (p− q)2cos2 (ϕ − χ)

) , (26)

4

are the order parameters at sF and FS interfaces, respectively(see Fig. 1b) and

a =−δs(−ds)

√1− δ 2

s (−ds)

2∆20

, b =∆0√

2, (27)

whereδs(−ds) is the solution of transcendental equation

K

(δs(−ds)

∆0η

)=

dsη√2ξs(T )

, η =

√2− δ 2

s (−ds)

∆20

. (28)

Here,K(z), is the complete elliptic integral of the first kind.Substitution ofδs(−ds) = 0 into Eq. (28) leads to the expres-sion for critical s layer thickness,dsc = (π/2)ξS(T ), whichwas used above.

For the calculation of the CPR we need to exclude phaseχof the intermediate s layer from the expressions for the cur-rents (23), (24). The value of this phase is determined fromthe condition that the currents flowing across Is and sF inter-faces should be equal to each other.

For large thickness of the middle s-electrode (ds ≫ dsc) themagnitude of order parameterδs(−ds) is close to that of a bulkmaterial∆0 and we may puta =−b in Eqs.(25) and (26)

δS(dF) = δs(0) =

√2∆0

Γ((p+ q)− (p− q)cos(ϕ − χ)), (29)

resulting in

JS(0) = JS(dF) =JGβ sin(ϕ − χ)

Γ(1−β cos(ϕ − χ))(30)

together with the equation to determineχ

ΓΓBI

sin(χ) =β sin(ϕ − χ)

1−β cos(ϕ − χ), β =

p− qp+ q

. (31)

From (29), (30) and (31) it follows that in this mode SIsFSstructure can be considered as a pair of SIs and sFS junctionsconnected in series. Therefore, the properties of the structureare almost independent on thicknessds and are determined bya junction with smallest critical current.

Indeed, we can conclude from (31) that the phaseχ of slayer order parameter depends on the ratio of the critical cur-rent, ICSIs ∝ Γ−1

BI , of its SIs part to that,ICsFS ∝ |β |Γ−1, ofthe sFS junction. The coefficientβ in (31) is a function ofF layer thickness, which becomes close to unity in the limitof smalldF and exhibits damped oscillations withdF increase(see analytical expression forβ (21)). That means that there isa range of thicknesses,dFn, determined by the equationβ = 0,at whichJS ≡ 0 and there is a transition from 0 toπ state insFS part of SIsFS junction. In other words, crossing the valuedFn with an increase ofdF provides aπ shift of χ relative tothe phase of the S electrode.

In Fig.2 we clarify the classification of operation modes anddemonstrate the phase diagram in the(ds,dF) plane, whichfollows from our analytical results (21)-(28). The calculationshave been done atT = 0.9TC for h = H/πTC = 10,ΓBI = 200and Γ = 5. The structures with s-layer smaller than critical

FIG. 2: The phase diagram of the operation modes of the SIsFSstructure in the(ds,dF ) plane. The bottom area corresponds to themode (2) with fully suppressed superconductivity in the s-layer. Thetop part of the diagram, separated from the bottom one by the solidhorizontal line, corresponds to the s-layer in superconductive state.It provides the value of s layer critical thickness,dsc. The upper-leftpart indicates themode (1a) with the weak place located at SIs tunnelbarrier. The upper-right area as well as thin valley around first 0−πtransition correspond to themode (1b) with the weak place locatedat sFS junction. Solid vertical lines provide loci of the boarders be-tween themodes (1a) and(1b). Vertical dashed lines show positionsof 0−π transitions. The calculations have been done forH = 10πTC ,ΓBI = 200 andΓ = 5 atT = 0.9TC.

thickness dsc = πξS(T )/2 correspond to themode (2) withfully suppressed superconductivity in the s layer. Conversely,the top part of diagram corresponds to s-layer in the super-conductive state (mode (1)). This area is divided into twoparts depending on whether the weak place located at thetunnel barrier I (mode(1a)) or at the ferromagnetic F-layer(mode(1b)). The separating black solid vertical lines in theupper part in Fig.2 represent the locus of points where thecritical currents of SIs and sFS parts of SIsFS junction areequal. The dashed lines give the locations of the points of 0to π transitions,dFn = π(n− 3/4)ξF

√2/h, n = 1,2,3..., at

whichJs = 0. In a vicinity of these points there are the valleysof mode (1b) with the width,∆dFn ≈ ξFΓΓ−1

BI h−1/2expπ(n−3/4), embedded into the areas occupied bymode (1a). Forthe set of parameters used for calculation of the phase dia-gram presented in Fig.2, there is only one valley with thewidth ∆dF1 ≈ ξFΓΓ−1

BI h−1/2expπ/4 located around thepointdF1 = (π/4)ξF

√2/h of the first 0 toπ transition.

1. Mode (1a): Switchable 0−π SIs junction

In the experimentally realized case8–11 Γ−1BI ≪ |β |Γ−1 the

condition is fulfilled and the weak place in SIsFS structure islocated at the SIs interface. In this approximation it followsfrom (31) that

χ ≈ ϕ − 2qΓ(p− q)ΓBI

sin(ϕ)

5

0 1 2 3 4 5

-1

0

1

0 1 2

0.5

1.0

(-d s)/

0

dF/ F

J CR

N

eTC/

dF/ F

FIG. 3: Critical currentJC of the SIsFS structure versus F-layer thick-nessdF calculated atT = 0.9TC , H = 10πTC , ΓBI = 200 andΓ = 5for s layer thicknessds = 2ξs(T ) slightly above the critical onedsc.Inset shows dependence of pair potentialδs(−ds) at the Is interfaceof the s-layer versus F-layer thicknessdF . Solid lines have been cal-culated fords ≫ dsc from Eqs. (32)-(33). The dashed line is theresult of calculations using analytical expressions (23)-(28) for thethickness of s-layerds = 2ξs(T ). Short-dashed line is the result ofnumerical calculations in the frame of Usadel equations (2)-(11).

in 0-state (dF < dF1) and

χ ≈ π +ϕ − 2qΓ(p− q)ΓBI

sin(ϕ)

in π-state (dF > dF1). Substitution of these expressions into(30) results in

JS(ϕ) =± JG

ΓBI

[sinϕ − Γ

ΓBI

1∓β2β

sin(2ϕ)]

(32)

for 0- andπ- states, respectively. It is seen that fordF < dF1the CPR (32) has typical for SIS tunnel junctions sinusoidalshape with small correction taking into account the suppres-sion of superconductivity in the s layer due to proximity withFS part of complex sFS electrode. Its negative sign is typi-cal for the tunnel Josephson structures with composite NS orFS electrodes39,40. FordF > dF1 the supercurrent changes itssign thus exhibiting the transition of SIsFS junction intoπstate. It’s important to note that in this mode the SIsFS struc-ture may have almost the same value of the critical currentboth in 0 andπ states. It is unique property, which can notbe realized in SFS devices studied before. For this reason wehave identified this mode as ”Switchable 0−π SIS junction”.

2. Mode (1b): sFS junction

Another limiting case is realized under the conditionΓ−1BI ≫

|β |Γ−1. It fulfills in the vicinity of the points of 0− to π−transitions,dFn, and for largedF values and high exchangefields H. In this mode (see Fig. 2) the weak place shifts to

sFS part of SIsFS device and the structure transforms into aconventional SFS-junction with complex SIs electrode.

In the first approximation onΓ/(β ΓBI)≫ 1 it follows from(30), (31) that

χ =ΓBI

Γβ sin(ϕ)

1−β cos(ϕ),

resulting in

JS(ϕ) =JGβ

Γ(1−β cosϕ)

(sinϕ − ΓBI

2Γβ sin(2ϕ)

(1−β cosϕ)

). (33)

The shape of CPR forχ → 0 coincides with that previouslyfound in SNS and SFS Josephson devices37. It transforms tothe sinusoidal form for sufficiently large thickness of F layer.For small thickness of the F-layer as well as in the vicinity of0−π transitions, significant deviations from sinusoidal formmay occurred.

Transition between themode (1a) and themode (1b) is alsodemonstrated in Fig.3. It shows dependence of critical cur-rentJC across the SIsFS structure versus F-layer thicknessdF .The inset in Fig.3 demonstrates the magnitude of an order pa-rameter at Is interface as a function ofdF . The solid lines inFig.3 give the shape ofJC(dF) andδ0(−ds) calculated from(32)-(33). These equations are valid in the limitds ≫ dsc anddo not take into account possible suppression of superconduc-tivity in a vicinity of tunnel barrier due to proximity with FSpart of the device. The dashed lines are the result of calcula-tions using analytical expressions (23)-(28) for the thicknessof the s-layerds = 2ξs(T ), which slightly exceeds the criti-cal one,dsc = (π/2)ξs(T ). These analytical dependencies arecalculated atT = 0.9 TC for H = 10πTC, ΓBI = 200, Γ = 5,γB = 0. The short-dashed curves are the results of numericalcalculations performed selfconsistently in the frame of the Us-adel equations (2)-(11) for corresponding set of the parame-tersT = 0.9 TC for H = 10πTC, γBI = 1000, γ = 1, γB = 0.3and the same thickness of the s layerdsc = (π/2)ξs(T ). In-terface parametersγBI = 1000, γ = 1 are chosen the same asfor the analytical case. The choice ofγB = 0.3 allows one totake into account the influence of mismatch which generallyoccurs at the sF and FS boundaries.

It can be seen that there is a qualitative agreement betweenthe shapes of the three curves. For smalldF the structure is inthe 0-statemode (1a) regime. The difference between dashedand short dashed lines in this area is due to the fact that theinequalities (18) are not fulfilled for very smalldF . The solidand short dashed curves start from the same value since fordF = 0 the sFS electrode becomes a single spatially homo-geneous superconductor. Fords = 2ξs(T ) the intrinsic super-conductivity in the s layer is weak and is partially suppressedwith dF increase (see the inset in Fig.3). This suppressionis accompanied by rapid drop of the critical current. It canbe seen that starting from the valuedF ≈ 0.4ξF our analyticalformulas (23)-(28) are accurate enough. The largerds, the bet-ter agreement between numerical and analytical results duetothe better applicability of the GL equations in the s layer. WithfurtherdF increase the structure passes through the valley ofmode (1b) state, located in the vicinity of the 0 toπ transi-tion, and comes into theπ−state of themode (1a). Finally for

6

a)

c)

b)

d)

FIG. 4: a) Magnitude of the critical currentJC in the SIsFS structure versus F-layer thicknessdF for two thickness of middle s-layer,ds =5ξS(T )> dsc (solid line) andds = 0.5ξS(T ) < dsc (dashed line) calculated atT = 0.9TC for H = 10πTC , ΓBI = 200 andΓ = 5. b)-d) CPR inthe vicinity of 0-π transitions. The corresponding insets show the enlarged parts ofJC(dF ) dependence enclosed in rectangles on the part a) ofthe Figure and marked by the letters b-d, respectively. The digits on the insets show the points at which theJS(ϕ) curves have been calculated.The dashed lines in the Figs.4b-d are the loci of critical points at which theJS(ϕ) dependence reaches its maximum valueJC(dF ).

dF & 1.6ξF there is a transition frommode (1a) to mode (1b),which is accompanied by damped oscillation ofJC(dF) withdF increase.

B. Mode (2): SInFS junction ds ≤ dsc

For ds ≤ dsc intrinsic superconductivity in thes layer iscompletely suppressed resulting in formation of the complex-InF- weak link area, where ’n’ marks the intermediate s filmin the normal state. In this parameter range the weak is alwayslocated in the tunnel barrier and the CPR has sinusoidal shape

JS(ϕ)=JG√

2

(p− q)sinϕ2pqΓΓBI cos ds

ξs(T )+[2pqΓ+(p+ q)ΓBI ]sin ds

ξs(T )

.

(34)

In a vicinity of the critical thickness,ds . dsc, the factorcos(ds/ξS(T )) in (34) is small and supercurrent is given bythe expression

JS(ϕ) =JG

2√

2

(p− q)sinϕ2pqΓ+(p+ q)ΓBI

. (35)

Further decrease ofds into the limitds ≪ dsc leads to

JS(ϕ) =JG√

2

(p− q)sinϕ2pqΓΓBI

. (36)

The magnitude of critical current in (36) is close to that in thewell-known case of SIFS junctions in appropriate regime.

7

C. Current-Phase Relation

In the previous section we have demonstrated that the vari-ation in the thickness of the ferromagnetic layer should leadto the transformation of CPR of the SIsFS structure. Fig.4a il-lustrates theJC(dF) dependencies calculated from expressions(23)-(28) atT = 0.9TC for H = 10πTC, γB = 0, ΓBI ≈ 200andΓ ≈ 5 for two thickness of the s layerds = 5ξS(T ) (solidline) andds = 0.5ξS(T ) (dashed line). In Figs.4b-d we en-large the parts ofJC(dF) dependence enclosed in rectangleslabeled by letters b, c and d in Fig.4a and mark by digits thepoints where theJS(ϕ) curves have been calculated. Thesecurves are marked by the same digits as the points in the en-large parts ofJC(dF) dependencies. The dashed lines in theFigs.4b-d are the loci of critical points at which theJS(ϕ) de-pendence reaches its maximum value,JC(dF).

Figure 4b presents themode (1b) valley, which divides themode (1a) domain into 0- andπ- states regions. In themode(1a) domain the SIsFS structure behaves as SIs and sFS junc-tions connected in series. Its critical current equals to the min-imal one among the critical currents of the SIs(JCSIs) and sFS(JCsFS) parts of the device. In the considered case the thick-ness of the s film is sufficiently large to prevent suppressionof superconductivity. Therefore,JCSIs does not change whenmoving from the point 1 to the point 2 alongJCdF depen-dence. At the point 2, whenJCSIs = JCsFS, we arrive at theborder between themode (1a) andmode (1b). It is seen thatat this point there is maximum deviation ofJS(ϕ) from the si-nusoidal shape. Further increase ofdF leads to 0-π transition,when parameterβ in (33) becomes small andJS(ϕ) practi-cally restores its sinusoidal shape. Beyond the area of 0 toπtransition, the critical current changes its sign and CPR startsto deform again. The deformation achieves its maximum atthe point 7 located at the other border between themodes (1a)and (1b). The displacement from the point 7 to the point 8along theJC(dF) dependence leads to recovery of sinusoidalCPR.

Figure 4c presents the transition from theπ-state ofmode(1a) to mode (1b) with dF increase. It is seen that the off-set from the point 1 to the points 2−5 alongJC(dF) results intransformation of the CPR similar to that shown in Fig.4b dur-ing displacement in the direction from the point 1 to the points2−6. The only difference is the starting negative sign of thecritical current. However this behavior of CPR as well as closetransition between modes lead to formation of the well pro-nounced kink at theJC(dF) dependence. Furthermore, con-trary to Fig.4b at the point 6, the junction is still in themode(1b) and remains in this mode with further increase indF .At the point 6 the critical current achieves its maximum valueand it decreases along the dashed line for largerdF .

Figure 4d shows the transformation of the CPR in the vicin-ity of the next 0 toπ transition inmode (1b). There is smalldeviation from sinusoidal shape at the point 1, which vanishesexponentially with an increase ofdF .

In the mode (2) (the dashed curve in Fig.4a) an intrinsicsuperconductivity in the s layer is completely suppressed re-sulting in the formation of a complex -InF- weak link regionand the CPR becomes sinusoidal (34).

IV. ARBITRARY TEMPERATURE

At arbitrary temperatures the boundary problem (2)-(11)goes beyond the assumptions of GL formalism and requiresself-consistent solution. We have performed it numericallyin terms of the nonlinear Usadel equations in iterative man-ner. All calculations were performed forT = 0.5TC, ξS = ξF ,γBI = 1000, γBFS = 0.3 andγ = 1.

Calculations show that at the selected transparency of tun-nel barrier(γBI = 1000) the suppression of superconductivityin the left electrode is negligibly small. This allows one toselect the thickness of the left S electrodedSL = 2ξS with-out any loss of generality. On the contrary, proximity of theright S electrode to the F layer results in strong suppressionof superconductivity at the FS interface. Therefore the pairpotential of the right S electrode reaches its bulk value only atthicknessdSR & 10ξS. It is for these reasons we have chosendSR = 10ξS for the calculations.

Furthermore, the presence of a low-transparent tunnel bar-rier in the considered SIsFS structures limits the magnitude ofcritical currentJC by a value much smaller compared to a de-pairing current of the superconducting electrodes. This allowsone to neglect nonlinear corrections to coordinate dependenceof the phase in the S banks.

The results of calculations are summarized in Fig.5. Fig-ure 5a shows the dependence ofJC of the SIsFS structure onthe F-layer thicknessdF for relatively largeds = 5ξS (solid)and smallds = 0.5ξS (dashed) s-film thickness. The letters onthe curves indicate the points at which the coordinate depen-dencies of the magnitude of the order parameter,|∆(x)|, andphase difference across the structure,χ , have been calculatedfor the phase differenceϕ = π/2. These curves are shown inthe panels b)-f) of the Fig.5 as the upper and bottom plots, re-spectively. There is direct correspondence between the letters,b, c, d, e, f, onJC(dF) curves and the labels, b), c), d), e), f),of the panels.

It is seen that qualitative behavior of theJC(dF) dependenceat T = 0.5TC remains similar to that obtained in the frame ofthe GL equations forT = 0.9TC (see Fig.4a). Furthermore, themodes of operation discussed above remain relevant too. Thepanels b)-f) in Fig.5 make this statement more clear.

At the point marked by letter ’b’, the s-film is sufficientlythick, ds = 5ξS, while F film is rather thin,dF = 0.3ξF , andtherefore the structure is in 0- state of themode (1a). In thisregime the phase mainly drops across the tunnel barrier, whilethe phase shifts at the s-film and in the S electrodes are negli-gibly small(see the bottom plot in Fig.5b).

At the point marked by the letter ’c’(ds = 5ξS, dF = ξF),the structure is in theπ- state of themode (1a). It is seen fromFig.5c that there is a phase jump at the tunnel barrier and anadditionalπ-shift occurs between the phases of S and s layers.

For dF = 3ξF (Fig.5d) the position of the weak place shiftsfrom SIs to sFS part of the SIsFS junction. Then the structurestarts to operate in themode (1b). It is seen that the phase dropacross SIs part is small, whileϕ −χ ≈ π/2 across the F layer,as it should be in SFS junctions with SIs and S electrodes.

At the points marked by the letters ’e’ and ’f’, thickness ofthe s-layerds = 0.5ξS is less than its critical value. Then su-

8

FIG. 5: a) Magnitude of critical currentJC of the SIsFS structure versus F-layer thicknessdF calculated atT = 0.5TC for H = 10πTC ,γBI = 1000, γ = 1 and two thickness of the s filmds = 5ξS (solid line) andds = 0.5ξS (dashed line). The letters onJC(dF ) give the points atwhich the coordinate dependencies of the magnitude of the order parameter,|∆(x)|, and phase difference across the structure,χ, have beencalculated. These curves are shown in the panels b)-f) of theFigure as the upper and bottom panels, respectively.

9

perconductivity in the s-spacer is suppressed due to the prox-imity with the F film and SIsFS device operates in themode(2). At dF = ξF (the dot ’e’ in Fig.5a and the panel Fig.5e)the position of the weak place is located at the SIs part of thestructure and there is additionalπ-shift of phase across the Ffilm. As a result, the SIsFS structure behaves like an SInFStunnelπ-junction. Unsuppressed residual value of the pair po-tential is due to the proximity with the right S-electrode andit disappears with the growth of the F-layer thickness, whichweakens this proximity effect. AtdF = 3ξF (Fig.5f) weakplace is located at the F part of IsF trilayer. Despite strongsuppression of the pair potential in the s-layer, the distribu-tion of the phase inside the IsF weak place has rather complexstructure, which depends on thicknesses of the s and F layers.

A. Temperature crossover from 0 toπ states

The temperature-induced crossover from 0 toπ states inSFS junctions has been discovered in26 in structures with si-nusoidal CPR. It was found that the transition takes place inarelatively broad temperature range.

Our analysis of SIsFS structure (see Fig.6a) shows thatsmoothness of 0 toπ transition strongly depends on the CPRshape. This phenomenon was not analyzed before since al-most all previous theoretical results were obtained withinalinear approximation leading in a sinusoidal CPR. To provethe statement, we have calculated numerically the set ofJC(T )curves for a number of F layer film thicknessesdF . We havechosen the thickness of intermediate superconductordS = 5ξSin order to have SIsFS device in themode(1a) and we haveexamined the parameter range 0.3ξF ≤ dF ≤ ξF , in which thestructure exhibits the first 0 toπ transition. The borders of thedF range are chosen in such a way that SIsFS contact is eitherin 0- (dF = 0.3ξF ) or π- (dF = ξF ) state in the whole temper-ature range. The correspondingJC(T ) dependencies (dashedlines in Fig. 6a) provide the envelope of a set ofJC(T ) curvescalculated for the considered range ofdF . It is clearly seenthat in the vicinity ofTC the decrease ofdF results in creationof the temperature range where 0-state exists. The point of0 to π transition shifts to lower temperatures with decreasingdF . For dF & 0.5ξF the transition is rather smooth since forT ≥ 0.8TC the junction keeps themode (2) (with suppressedsuperconductivity) and deviations of the CPR from sin(ϕ) aresmall. Thus the behavior ofJC(T ) dependencies in this casecan be easily described by analytic results from Sec.III C.

The situation drastically changes atdF = 0.46ξF (short-dashed line in Fig.6a). For this thickness the point of 0 toπ transition shifts toT ≈ 0.25TC. This shift is accompaniedby an increase of amplitudes of higher harmonics of CPR (seeFig.6b). As a result, the shape of CPR is strongly modified,so that in the interval 0≤ ϕ ≤ π the CPR curves are char-acterized by two values,JC1 andJC2, as is known from thecase of SFcFS constrictions41. In general,JC1 and JC2 dif-fer both in sign and magnitude andJC = max(|JC1| , |JC2|).For T > 0.25TC the junction in the 0-state andJC grows withdecrease ofT up to T ≈ 0.5TC. Further decrease ofT is ac-companied by suppression of critical current. In a vicinityof

T ≈ 0.25TC the difference between|JC1| , and|JC2| becomesnegligible and the system starts to develop the instabilitythateventually shows up as a sharp jump from 0 toπ state. Afterthe jump,|JC| continuously increases whenT goes to zero.

It is important to note that this behavior should always beobserved in the vicinity of 0− π transition, i.e. in the rangeof parameters, in which the amplitude of the first harmonic issmall compared to higher harmonics. However, the closer istemperature toTc, the less pronounced are higher CPR har-monics and the smaller is the magnitude of the jump. Thisfact is illustrated by dash-dotted line showingJC(T ) calcu-lated fordF = 0.48ξF . The jump in the curves calculated fordF ≥ 0.5ξF also exists, but it is small and can not be resolvedon the scale used in the Fig. 6a.

At dF = 0.45ξF (dash-dot-dotted line in Fig. 6) the junctionis always in the 0-state and there is only small suppression ofcritical current at low temperatures despite the realization ofnon-sinusoidal CPR.

Thus the calculations clearly show that it’s possible to re-alize a set of parameters of SIsFS junctions where thermally-induced 0-π crossover can be observed and controlled by tem-perature variation.

B. 0 to π crossover by changing the effective exchange energyin external magnetic field

Exchange field is an intrinsic microscopic parameter of aferromagnetic material which cannot be controlled directly byapplication of an external field. However, the spin splitting inF-layers can be provided by both the internal exchange fieldand external magnetic field42,43, resulting in generation of ef-fective exchange field, which equals to their sum. However,practical realization of this effect is a challenge since itis diffi-cult to fulfill special requirements42,43on thickness of S elec-trodes and SFS junction geometry.

Another opportunity can be realized in soft diluted fer-romagnetic alloys like Fe0.01Pd0.99. Investigations of mag-netic properties44 of these materials have shown that below14 K they exhibit ferromagnetic order due to the formation ofweakly coupled ferromagnetic nanoclusters. In the clusters,the effective spin polarization of Fe ions is about 4µB, corre-sponding to that in the bulk Pd3Fe alloy. It was demonstratedthat the hysteresis loops of Fe0.01Pd0.99 films have the formtypical to nanostructured ferromagnets with weakly coupledgrains (the absence of domains; a small coercive force; a smallinterval of the magnetization reversal, where the magnetiza-tion changes its direction following the changes in the appliedmagnetic field; and a prolonged part, where the component ofthe magnetization vector along the applied field grows gradu-ally).

Smallness of concentration of Pd3Fe clusters and their abil-ity to follow variation in the applied magnetic field may resultin generation ofHe f f , which is of the order of

He f f ≈ Hn↑V↑− n↓V↓

n↓V↓+ n↑V↑+(n− n↑− n↓)(V −V↓−V↑). (37)

Heren is concentration of electrons within a physically small

10

0.0 0.2 0.4 0.6 0.8 1.0

-1.0

-0.5

0.0

0.5

1.0

a)

5 J CR

N

eTC/

(0)

dF=0.3 dF=0.45 dF=0.46 dF=0.48 dF=0.5 dF=1.0

T/TC

0-state

-state

123

4

-1.0

-0.5

0.0

0.5

1.0

54

21

J SR

N

eTC/

(0)

0 2

3

b)

FIG. 6: a) Critical currentJC of the SIsFS structure versus temperatureT for various F-layer thicknessesdF in the vicinity of 0 toπ transition.Dashed envelopes show temperature dependence in the 0- (top) andπ- (bottom) states. b) CPR of the structure withdF = 0.46ξF for a set ofthe temperatures in the vicinity of 0-π transition. Each curve corresponds to the point marked in the panel (a). Note that the curves (3) and(4) almost coincide but correspond to different ground states of the junction, 0- andπ, respectively. The calculations have been performednumerically fords = 5ξS, H = 10πTC , γBI = 1000, γB = 0.3, γ = 1.

0 5 10 1510-4

10-3

10-2

10-1

1mode (1b)

J C

RN

eT

C/

H/ TC

mode (1a)

mode (2)

FIG. 7: The magnitude of critical currentJC of the SIsFS structureversus exchange fieldH for thick ds = 5ξS (solid), thinds = 0.5ξS(dashed) and intermediateds = 3ξS (dash-dotted) s-layer thickness.The plot demonstrates the possibility of 0-π transition by varying theeffective exchange field. The calculations have been performed forT = 0.5TC , dF = 2ξF , γBI = 1000, γB = 0.3, γ = 1.

volume V , in which one performs an averaging of Greensfunctions in the transformation to a quasiclassical descriptionof superconductivity,n↑,↓ andV↑,↓ are the values describingspin polarized parts ofn and parts of volumeV, which theyoccupy, respectively. Similar kind ofHe f f nucleates in NFor SF proximity structures, which are composed from thinlayers45–48. There is an interval of applied magnetic fieldsHext where the alloy magnetization changes its direction andthe concentrationsn↑,↓ depend on a pre-history of applicationof the field10,12, providing the possibility to controlHe f f byan external magnetic field.

Derivation of possible relationships betweenHe f f andHext

is outside of the scope of this paper. Below we will concen-trate only on an assessment of the intervals in whichHe f fshould be changed to ensure the transition of SIsFS devicefrom 0 to π state. To do this, we calculate theJC(H) depen-dencies presented in Fig.7. The calculations have been donefor the set of structures withdF = 2ξF and s-films thicknessranging from thick one,dS = 5ξS (solid line) up to an interme-diate valuedS = 2ξS (dashed-dotted line) and finishing withthin film havingdS = 0.5ξS (dashed line). It is clearly seenthat these curves have the same shape asJC(dF) dependenciespresented in the Sec.III. FordS = 5ξS andH . 7πTC the mag-nitude ofJC is practically independent onH, but it changesthe sign atH ≈ 1.25πTC due to 0 -π transition. It is seenthat for the transition, while maintaining the normalized cur-rent value at a level close to unity, changes ofH are requiredapproximately of the order of 0.1πTC or 10%. FordS = 2ξSandH . 3πTC, it is necessary to changeH on 20% to realizethe such a transition. In this case the value of normalized cur-rent is at the level 0.4. In mode 2 the transition requires 100%change ofH, which is not practical.

V. DISCUSSION

We have performed a theoretical study of magnetic SIsFSJosephson junctions. AtT ≤ TC calculations have been per-formed analytically in the frame of the GL equations. Forarbitrary temperatures we have developed numerical code forselfconsistent solution of the Usadel equations. We have out-lined several modes of operation of these junctions. For s-layer in superconducting state they are S-I-sfS or SIs-F-S de-vices with weak place located at insulator (mode (1a)) and atthe F-layer (mode (1b)), respectively. For small s-layer thick-ness, intrinsic superconductivity in it is completely suppressedresulting in formation of InF weak place (mode (2)). We haveexamined the shape ofJS(ϕ) and spatial distribution of the

11

module of the pair potential and its phase difference acrossthe SIsFS structure in these modes.

For mode (1) the shape of the CPR can substantially differfrom the sinusoidal one even in a vicinity ofTC. The devia-tions are largest when the structure is close to the crossoverbetween themodes (1a) and (1b). This effect results in thekinks in the dependencies ofJC on temperature and on param-eters of the structure (thickness of the layersdF , ds and ex-change energyH) as illustrated in Fig.4 onJC(dF) curves. Thetransformation of CPR is even more important at low temper-atures. ForT . 0.25TC a sharp 0−π transition can be realizedinduced by small temperature variation (Fig.6). This instabil-ity must be taken into account when using the structures asmemory elements. On the other hand, this effect can be usedin detectors of electromagnetic radiation, where absorption ofa photon in the F layer will provide local heating leading todevelopment of the instability and subsequent phonon regis-tration.

We have shown that suppression of the order parameter inthe thin s-film due to the proximity effect leads to decrease ofJCRN product in both 0− andπ−states. On the other hand, theproximity effect may also support s-layer superconductivitydue to the impact of S electrodes. Inmode (1a) JCRN productin 0- andπ-states can achieve values typical for SIS tunneljunctions.

In mode (2) sinusoidal CPR is realized. Despite that, thedistribution of the phase differenceχ(x) in the IsF weak placemay have a complex structure, which depends on thickness ofthe s and F layers. These effects should influence the dynam-ics of a junction in itsac-state and deserve further study.

Further, we have also shown that inmode (1a) nearly 10%change in the exchange energy can cause a 0−π transition,i.e. changing the sign ofJCRN product, while maintaining itsabsolute value. This unique feature can be implemented inmode (1a), since it is in it changes of the exchange energyonly determine the presence or absence of aπ shift betweens and S electrodes and does not affect the magnitude of thecritical current of SIs part of SIsFS junction.

In mode (1b), the F layer becomes a part of weak link area.In this case theπ shift, initiated by the change inH must beaccompanied by changes ofJC magnitude due to the oscil-latory nature of superconducting correlations in the F film.The latter may lead to very complex and irregular dependenceof JC(Hext), which have been observed in Nb-PdFe-Nb SFSjunctions(see Fig.3 in8). Contrary to that theJC(Hext ) curvesof SIsFS structure with the same PdFe metal does not demon-strate these irregularities10,11.

To characterize a junction stability with respect toHvariations it is convenient to introduce the parameterη =(dJC/JC)/(dH/H) which relates the relative change in thecritical current to the relative change in the exchange energy.The larger the magnitude ofη the more intensive irregulari-ties in an SFS junction are expected with variation ofH. Inthe Fig.8 we compare the SIsFS devices with conventionalSFS, SIFS and SIFIS junctions making use of two the mostimportant parameters: the instability parameterη andJCRNproduct, the value, which characterizes high frequency prop-erties of the structures. The calculations have been done inthe

-1 0

10-4

10-3

10-2

10-1

1

J CR

N

eTC/ SIFS

SIFIS

SFS

SIsFSmode (2)

SIsFSmode (1a)

instability

Magnetic structuresdF= 1 F

H= 10 TC

BI= 1000

B = 0.3, = 1

Exchange Field StabilitydJC/dH * H/JC

stable

SIS

FIG. 8: Comparison of different types of Josephson structures,marked by points on the phase plane, in terms ofJCRN and exchangefield stabilityη. All calculation have been performed forT = 0.5TC,dF = ξF , γBI = 1000, γB = 0.3, γ = 1. For SIsFS structuresds = 5ξSandds = 0.5ξS are taken inmode (1a) andmode (2), respectively.

frame of Usadel equation for the same set of junctions param-eters, namelyT = 0.5TC, H = 10πTC, dF = ξF , γBI = 1000,γB = 0.3, γ = 1.

It can be seen that the presence of two tunnel barriers inSIFIS junction results in the smallestJCRN and strong insta-bility. The SIFS and SIsFS structures in themode (2) demon-strate better results with almost the same parameters. Con-ventional SFS structures have two times smallerJCRN prod-uct, having higher critical current but lower resistivity.At thesame time, SFS junctions are more stable due to the lack oflow-transparent tunnel barrier. The latter is the main sourceof instability due to sharp phase discontinuities at the barrier’I’.

Contrary to the standard SFS, SIFS and SIFIS junctions,SIsFS structures achieveJCRN and stability characteristicscomparable to those of SIS tunnel junctions. This uniqueproperty is favorable for application of SIsFS structures in su-perconducting electronic circuits.

Acknowledgments

We thank V.V. Ryazanov, V.V. Bol’ginov, I.V. Vernik andO.A. Mukhanov for useful discussions. This work was sup-ported by the Russian Foundation for Basic Research, Rus-sian Ministry of Education and Science, Dynasty Founda-tion, Scholarship of the President of the Russian Federation,IARPA and Dutch FOM.

Appendix A: Boundary problem at T . TC

In the limit of high temperature

GS = Gs = GF = sgn(Ω) (A1)

12

and the boundary problem reduces to to the system of liner-ized equations. Their solution in the F layer,(0 ≤ x ≤ dF),has the form

ΦF =Csinh

√Θ(x− dF/2)

ξF+Dcosh

√Θ(x− dF/2)

ξF, (A2)

whereΘ = Ωsgn(Ω). For transparent FS and sF interfaces(γB = 0) from the boundary conditions (6), (7) and (A2) it iseasy to get that

ξs

γ√

Θddx

Φs(0) =−Φs(0)cothdF

√Θ

ξF+

ΦS(dF)

sinhdF√

ΘξF

, (A3)

ξS

γ√

Θddx

ΦS(dF) = ΦS(dF)cothdF

√Θ

ξF− Φs(0)

sinhdF√

ΘξF

. (A4)

and thus reduce the problem to the solution of Ginzburg-Landau (GL) equations in s and S films.

ξ 2S (T )

d2

dx2 ∆k −∆k(∆20−|∆k|2) = 0, ∆2

0 =8π2TC(TC −T )

7ζ (3),

(A5)

J =JG

∆20

Im

(∆

∗kξS(T )

ddx

∆k

), JG =

π∆20

4eρSTCξS(T ), (A6)

whereξS(T ) = πξS/2√

1−T/TC is GL coherence length andk equals tos or S for −ds ≤ x ≤ 0 andx ≥ dF , respectively. AtIs, sF and FS interfaces GL equations should be supplementedby the boundary conditions in the form37

ξS(T )ddx

∆k(z) = b(z)∆k(z), b(z) =Σ1(z)Σ2(z)

, (A7)

Σ1(z) =∞

∑ω=−∞

ξS(T )ddx

Φk(z)Ω2 , Σ2(z) =

∞

∑ω=−∞

Φk(z)Ω2 , (A8)

wherez =−ds, 0, dF . In typical experimental situationγBI ≫1, γ

√H ≫ 1 anddF

√H & ξF . In this case in the first approx-

imation

ΦS(dF) = 0, Φs(0) = 0,ddx

Φs(−ds) = 0

and in the vicinity of interfaces

ΦS(x) = ∆S(x) = BS(x− dF)

ξS(T ), dF . x ≪ ξS(T ), (A9)

Φs(x) = ∆s(x) =−Bsx

ξs(T ), − ξS(T )≪ x . 0, (A10)

Φs(x) = ∆s(x) = ∆s(−ds), −ds . x ≪−ds+ξS(T ), (A11)

whereBS, Bs, and∆s(−ds) are independent onx constants.Substitution of the solutions (A9) - (A11) into (15), (A3), (A4)gives

ΓBIξS(T )ddx

Φs(−ds) = ∆s(−ds)−∆0, (A12)

ΦS(dF) =Bs

Γ√

ΘsinhdF

√Ω

ξF

+BS coshdF

√Θ

ξF

Γ√

ΘsinhdF√

ΘξF

, (A13)

Φs(0) =Bs coshdF

√Θ

ξF

Γ√

ΩsinhdF√

ΘξF

+BS

Γ√

ΘsinhdF√

ΘξF

, (A14)

ΓBI =γBIξS

ξs(T ), Γ =

γBIξs(T )ξS

. (A15)

From definition (A7), (A8) of coefficientsb(z) and expres-sions (A12) - (A14) it follows that

ΓBIξs(T )ddx

∆s(−ds) =−(∆0−∆s(−ds)) , (A16)

ξs(T )ddx

∆s(0) =−q+ p2

Γ∆s(0)−q− p

2Γ∆S(dF), (A17)

ξS(T )ddx

∆S(dF) =q+ p

2Γ∆S(dF)+

q− p2

Γ∆s(0), (A18)

where

p−1 =8

π2 Re∞

∑ω=0

1

Ω2√

ΩcothdF

√Ω

2ξF

, (A19)

q−1 =8

π2 Re∞

∑ω=0

1

Ω2√

Ω tanhdF

√Ω

2ξF

. (A20)

In considered limit both suppression parametersΓBI ≫ 1andΓ ≫ 1 are large and from relations (15), (A3), (A4) inthe first approximation on these parameters we get that theboundary conditions (A16) - (A18) can be simplified to

ξS(T )ddx

∆s(−ds) = 0, ∆s(0) = 0, ∆S(dF) = 0. (A21)

Taking into account that in this approximation supercurrentj = 0 and∆S(∞) = ∆0 from (A5), (A21) it follows that

∆S(x) = δS(x)expiϕ , δS(x) = ∆0 tanhx− dF√2ξS(T )

, (A22)

while

∆s(x) = δs(x)expiχ , (A23)

13

whereδs(x) is the solution of transcendental equation

F

(δs(x)

δs(−ds),

δs(−ds)

∆0η

)=− xη√

2ξs(T ), η =

√2− δ 2

s (−ds)

∆20

(A24)and δs(−ds) is a solution of the same equation at the SIsboundaryx =−ds

K

(δs(−ds)

∆0η

)=

dsη√2ξs(T )

. (A25)

HereF(y,z) andK(z) are the incomplete and complete ellipticintegral of the first kind respectively.

Substitution of (A22), (A23) into (A16) - (A18) gives thatin the next approximation onΓ−1

BI andΓ−1

J(−ds) = JGδs(−ds)

ΓBI∆0sin(χ) (A26)

J(0) = J(dF) = JGΓ(p− q)

2∆20

δs(0)δS(dF)sin(ϕ − χ) , (A27)

where

δs(0) =− 2b(q− p)cos(ϕ − χ)+2a(q+ p)

Γ[(q+ p)2− (q− p)2cos2 (ϕ − χ)

] , (A28)

δS(dF) =2b(q+ p)+2a(q− p)cos(ϕ − χ)

Γ((q+ p)2− (q− p)2cos2 (ϕ − χ)

) , (A29)

are magnitudes of the order parameters at the FS interfacesand

a =−δs(−ds)

√1− δ 2

s (−ds)

2∆20

, b =∆0√

2(A30)

Phase,χ , of the order parameters of the s layer is determinedfrom equality of currents (A26), (A27).

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