Superconductor-insulator-ferromagnet-superconductor Josephson junction: From the dirty to the clean limit
N. G. Pugach, M. Yu. Kupriyanov, E. Goldobin, R. Kleiner, D. Koelle
aa r X i v : . [ c ond - m a t . s up r- c on ] S e p Superconductor-insulator-ferromagnet-superconductor Josephson junction: From thedirty to the clean limit
N. G. Pugach, ∗ M. Yu. Kupriyanov, E. Goldobin, R. Kleiner, and D. Koelle Faculty of Physics, M.V. Lomonosov Moscow State University, 119992 Leninskie Gory, Moscow, Russia Skobeltsyn Nuclear Physics Institute, M.V. Lomonosov Moscow State University, 119992 Leninskie Gory, Moscow, Russia Physikalisches Institut–Experimentalphysik II and Center for Collective Quantum Phenomena,Universit¨at T¨ubingen, Auf der Morgenstelle 14, D-72076 T¨ubingen, Germany (Dated: February 12, 2018)The proximity effect and the Josephson current in a superconductor-insulator-ferromagnet-superconductor (SIFS) junction are investigated within the framework of the quasiclassical Eilen-berger equations. This investigation allows us to compare the dirty and the clean limits, to in-vestigate an arbitrary impurity scattering, and to determine the applicability limits of the Usadelequations for such structures. The role of different types of the FS interface is analyzed. It is shownthat the decay length ξ and the spatial oscillation period 2 πξ of the Eilenberger function mayexhibit a nonmonotonic dependence on the properties of the ferromagnetic layer such as exchangefield or electron mean free path. The results of our calculations are applied to the interpretation ofexperimentally observed dependencies of the critical current density on the ferromagnet thicknessin Josephson junctions containing a Ni layer with an arbitrary scattering. PACS numbers: 74.45.+c, 74.50.+r, 74.78.Fk
I. INTRODUCTION
Superconductor-ferromagnet-superconductor (SFS)Josephson junctions are a subject of intensive theoreticaland experimental studies . In particular, the questionof the applicability range of predictions from dirty andclean limit theories and the treatment of the crossoverbetween these two limits has been recognized as animportant problem for the theoretical description of SFSstructures .For the majority of experimental realizations of SFSstructures the exchange energy H of the ferromag-netic materials is rather large . As a consequence,the characteristic magnetic length ξ H = ~ v f / H . ℓ f ,where ℓ f is the electron mean free path and v f is theFermi velocity in the F layer . Under this conditionthe numerous theoretical predictions based on the Us-adel equations have a rather restricted range of valid-ity; i.e. a more general approach based on the Eilenbergerequations has to be developed.A simple expression for the critical current density J C , which is valid in the clean limit, was derived in Ref. 18.Still the analysis of the Eilenberger equations for moregeneral cases, remained a difficult problem. A significantprogress along this direction has been achieved a decadeago , where the solution of the Eilenberger equations forarbitrary scattering has been expressed in an integralform . It was supposed that the SF interface trans-parency D is small enough, providing the opportunity touse the linearized equations. Thus, a general expressionfor the Josephson junction supercurrent has been derivedand used for numerical evaluations. The integral rep-resentation of the solution of the Eilenberger equationsalso permits to reproduce analytical expressions for J C obtained earlier within both clean and dirty limits .Recently, the same results have been achieved using the Ricatti parametrization of the one-dimensional Eilen-berger equations . However, the obtained expressions are so complicated that they are difficult analyze anduse in practice. Although the use of a one-dimensionalequation significantly simplifies the problem, it containsa non-controllable assumption about the insignificance ofthe angular distribution of the Eilenberger function.Despite of this progress, it is still not clear within whichrange of parameters and with what accuracy it is possibleto use simple expressions for the clean and dirty lim-its. The question about the influence of transport prop-erties of SF interfaces is also still open. Since the answerto these questions is rather important for experimental-ists, we formulate here a problem for a particular caseof superconductor-insulator-ferromagnet-superconductor(SIFS) junctions. In comparison with SFS, in SIFS struc-tures superconductivity is induced in the F layer onlyfrom one S electrode. This essentially simplifies the anal-ysis compared to that for SFS junctions.On the one hand, SIFS junctions are interesting bythemselves as potential elements for superconductinglogic circuits because it is possible to vary their crit-ical current density in the π state within a rather widerange, still keeping a high J C R N product , where R N is the junction normal resistance per square. Further-more, in comparison with SFS junctions, SIFS junctionshave very small damping which decreases exponentiallyat T →
0. This makes them useful for superconductingcircuits where low damping is required. On the otherhand, an SIFS junction represents a convenient modelsystem for a comparative study of 0- π transitions for anarbitrary ratio of characteristic lengths in the F layer: theF-layer thickness d f , the mean free path ℓ f , the charac-teristic magnetic length ξ H , and the nonmagnetic coher-ence length ξ = ~ v f / πk B T, where T is the tempera-ture.To calculate J C of an SIFS junction, it is sufficient tostudy the proximity effect in the FS bilayer and to cal-culate the magnitude of the Eilenberger functions at theIF interface. For simplicity we will restrict ourselves tothe study of the situation when the anomalous Green’sfunction induced in the F layer is small enough, permit-ting us to use the linearized Eilenberger equation for thedescription of the superconducting properties induced inthe ferromagnet. Such an approximation is valid if theFS interface has a small transparency or if T is close tothe critical temperature T c of the S electrodes.This article is organized as follows. In sec. IIwe describe our model based on the linearized Eilen-berger equation supplemented with Zaitsev boundaryconditions for an SIFS junction. Different types of FSboundaries are analyzed. Section III presents dependen-cies of anomalous Green’s functions and the critical cur-rent density on the F-layer parameters. The comparisonwith experimental results is presented in Section IV. Sec-tion V concludes this work. The calculation details canbe found in the Appendix. II. MODELA. Proximity effect in the FS electrode
To analyze the proximity effect in an FS bilayer forarbitrary values of the electron mean free paths ℓ s and ℓ f in the S and F layer, respectively, it is convenient tointroduce the following functionsΦ + = 12 (cid:2) f ( x, θ s , ω ) + f + ( x, θ s , ω ) (cid:3) , (1)Φ − = 12 (cid:2) f ( x, θ s , ω ) − f + ( x, θ s , ω ) (cid:3) , (2)where f ( x, θ s , ω ) and f + ( x, θ s , ω ) are the quasiclassicalEilenberger functions . Then we rewrite the Eilenbergerequations for the S layer, located at 0 ≤ x ≤ ∞ , in theform: k s ℓ s µ s ∂ ∂x Φ + − Φ + = − τ s + h Φ + i | ω | τ s + 1 , (3)where h Φ + i = Z Φ + dµ, (4)Φ − = − ℓ s sign( ω ) k s µ s ∂∂x Φ + . (5)The insulating layer of negligible thickness is located at x = − d f . In the F layer (located at − d f ≤ x ≤
0) theequations have the form k f ℓ f µ f ∂ ∂x Φ + − Φ + = − h Φ + i k f , (6) Φ − = − ℓ f sign( ω ) k f µ f ∂∂x Φ + . (7)Here ω = πT (2 n + 1) are Matsubara frequencies, θ s,f are the angles between the FS interface normal and thedirection of Fermi velocities v s,f in the S and F layers, re-spectively, µ f,s = cos θ f,s , ∆ is the superconducting orderparameter, which is assumed to be zero in the ferromag-net, τ s,f = ℓ s,f /v s,f are electron scattering times, andcorresponding wave vectors k s = 2 | ω | τ s + 1 , k f = 2[ | ω | + iH sign( ω )] τ f + 1 = 2 | ω | ℓ f /v f + i Hℓ f /v f sign( ω ) + 1 =1 + ℓ f /ξ ω + i sign( ω ) ℓ f /ξ H , ξ ω = v f / | ω | . We use theunits where ~ = 1 and k B = 1. Here we also take intoaccount that the normal Eilenberger function in the ferro-magnet g ≈ sign( ω ) . We also neglect multiple reflectionsfrom the FS and the IF interfaces, which is reasonable if d f > ℓ f , i.e. the ballistic regime is not considered. Thefully ballistic case with a double barrier SIFIS junctionwas examined in Ref. [25].Equations (3)–(7) must be supplemented by theboundary conditions. At x → ∞ the function Φ + shouldapproach its bulk valueΦ + ( θ s , ω ) = ∆ p ∆ + ω , (8)where ∆ is the magnitude of superconductor order pa-rameter far from the SF interface. At x = − d f theboundary condition ddx Φ + ( − d f , θ f , ω ) = 0 (9)guarantees the absence of a current across the insulatinglayer.The boundary conditions at the FS interface stronglydepend on its transport properties. Below we will assumefor the FS interface transparency D ( µ f ) ≪
1. Then tothe first approximation we may neglect the suppressionof superconductivity in the S electrode and rewrite theZaitsev boundary conditions in the form ℓ f k f µ f ∂∂x Φ + = D ( µ f ) ∆ p ∆ + ω . (10)From the structure of Eilenberger equations (3)–(7)and the boundary conditions (8)–(10) it follows thatΦ + ( θ s , ω ) = Φ + ( θ s , − ω ) , (11a)Φ + ( θ f , ω ) = Φ ∗ + ( θ f , − ω ) , (11b)Φ − ( θ s , ω ) = − Φ − ( θ s , − ω ) , (12a)Φ − ( θ f , ω ) = − Φ − ( θ f , − ω ) . (12b)These symmetry relations permit us to consider the so-lution of the boundary problem formulated above onlyfor ω > . B. Solution of the Eilenberger equations in the Flayer.
It is convenient to look for a solution of Eilenbergerequations in the F layer (7) in the formΦ + = ∞ X m = −∞ Q m cos (cid:18) πm ( x + d f ) d f (cid:19) + B cosh (cid:18) x + d f ℓ f | µ f | k f (cid:19) , (13)which automatically satisfies the boundary condition (9).The relation between the coefficients Q m , and B can be found by substitution of the ansatz (13) into the Eilen-berger equation (6). Multiplying the obtained equationsby cos[ πk ( x + d f ) /d f ] , and integrating them over x , onecan easily find the relation between the coefficients Q m , and B , see the Appendix. B = B ( µ f ) can be foundfrom the boundary condition (10). Thus we arrive to thefollowing expression for the Eilenberger function in theferromagnetΦ + = ∆ p ∆ + ω ∞ X m = −∞ D D ( µ ) µq ( − m M ( µ ) E µ M ( µ f ) (cid:2) k f − (cid:10) M (cid:11)(cid:3) cos (cid:18) πm ( x + d f ) d f (cid:19) + D ( µ f )sinh (cid:16) qµ f (cid:17) cosh (cid:18) q x + d f d f µ f (cid:19) , (14)where q = d f k f ℓ f = d f (cid:18) ℓ f + 1 ξ ω + i ξ H (cid:19) . Then the Eilenberger function averaged over the angle θ f has the form h Φ + i = ∆ p ∆ + ω ∞ X m = −∞ D D ( µ ) µq ( − m M ( µ ) E µ h k f − qπm arctan (cid:16) πmq (cid:17)i cos (cid:18) πm ( x + d f ) d f (cid:19) , (15)There are several interface models with a different D ( µ f ) dependence.First, the FS interface may be represented by a thick diffusive barrier. Then the incident electron scatters in anyarbitrary direction with equal probability, independent on the incident angle, i.e., D ( µ f ) = D . (16)The second model considers an FS interface with a potential barrier, which appears due to different Fermi velocitiesin the S and F layer. The transmission coefficient in this case has the form D ( µ f ) = 4 v s µ s v f µ f ( v s µ s + v f µ f ) ≈ v f v s µ f = D µ f , if v s ≫ v f . (17)It is necessary to note, that the incidence angle and the reflection angle are related by the expression v s sin θ s = v f sin θ f . (18)Therefore, only the electrons which are at almost normal incidence to the interface ( µ s ≈
1) may penetrate throughthe barrier. This model seems to be mostly reasonable for description of recent experiments with Nb electrodes( v s ∼ . . . · cm / s) and 3 d ferromagnets or its Cu alloys (for Ni v f ≈ . · cm / s).Third, one can model the FS interface as a high and narrow ( δ -function like) potential barrier between two metalswith close Fermi-velocities. In this case D ( µ f ) = 4 v s µ s v f µ f ( v s µ s + v f µ f + W ) ≈ v s v f W µ f = D µ f , if v s ≈ v f , (19)where W is the strength of δ -function like barrier.For the above mentioned three models of the FS interface we have three possible expressions for Φ + with differentaverage values h Φ + i , see (A7)-(A9) for details.First, for D ( µ f ) = D Φ + = ∆ D p ∆ + ω ∞ X m = −∞ q π m ( − m ln h π m q + 1 i cos (cid:16) πm ( x + d f ) d f (cid:17)(cid:16) m π µ f q + 1 (cid:17) h k f − qπm arctan (cid:16) πmq (cid:17)i + cosh (cid:16) q x + d f d f µ f (cid:17) sinh (cid:16) qµ f (cid:17) . (20)Second, for D ( µ f ) = µ f D Φ + = ∆ D p ∆ + ω ∞ X m = −∞ qπ m ( − m (cid:16) − qπm arctan πmq (cid:17) cos (cid:16) πm ( x + d f ) d f (cid:17)(cid:16) m π µ f q + 1 (cid:17) h k f − qπm arctan (cid:16) πmq (cid:17)i + µ f cosh (cid:16) q x + d f d f µ f (cid:17) sinh (cid:16) qµ f (cid:17) . (21)Third, for D ( µ f ) = µ f D Φ + = ∆ D p ∆ + ω ∞ X m = −∞ q π m ( − m (cid:16) − q π m ln (cid:16) π m q + 1 (cid:17)(cid:17) cos (cid:16) πm ( x + d f ) d f (cid:17)(cid:16) m π µ f q + 1 (cid:17) h k f − qπm arctan (cid:16) πmq (cid:17)i + µ f cosh (cid:16) q x + d f d f µ f (cid:17) sinh (cid:16) qµ f (cid:17) . (22)These are the solutions of the linearized Eilenberger equa-tion in the ferromagnet. The corresponding expressionsfor h Φ + i are given by Eqs. (A11)-(A13) in the Appendix.Let us consider how these expressions reproduce thelimiting cases of strong and weak scattering, for whichthe solutions are well-known . C. Dirty limit
If the electron mean free path is the smallest charac-teristic length i.e. ℓ f ≪ ξ H , ξ , d f , the frequent nonmag-netic scattering permits averaging over the trajectories .Then it is possible to write a closed system of equationsfor averaged functions. The linearization of the Usadelequations is allowed at the same conditions as for theEilenberger equation.The linearized Usadel equation in the ferromagnet hasthe form ξ f ∂ ∂x h Φ + i + ( ω + iH ) πT c h Φ + i = 0 (23)where ξ f = D f / πT c , and D f = ℓ f v f / h Φ + i = A cosh x − d f ξ f r ω + iHπT c ! (24) The boundary condition at x = 0 reads γ B ξ f ∂∂x h Φ + i = ∆ p ∆ + ω . (25)Substituting the expression (24) into the Usadel equation(23) one can find the coefficient A = ∆ γ B p ∆ + ω q ω + iHπT c (cid:16) d f ξ f q ω + iHπT c (cid:17) . (26)Let us consider the previously obtained averaged Eilen-berger function (15). For small ℓ f the parameter q islarge and1 + ℓ f /ξ ω + iℓ f /ξ H − qπm arctan (cid:18) πmq (cid:19) ≈≈ q " π m + d f ξ f ( ω + iH ) πT c It is seen that the sums in (14) and (15) converge at π m ≈ ω + iH ) d f /d f . At these values of mπ m µ f q ≈ ω + iH ) d f d f µ f d f ℓ f ≈ ω + iH ) v f µ f ℓ f ≪ + ≈ ∆ p ∆ + ω r πT c ω + iH cosh (cid:16)q πT c ω + iH x + d f ξ f (cid:17) γ B sinh (cid:16)q πT c ω + iH d f ξ f (cid:17) + D ( µ f )sinh (cid:16) qµ f (cid:17) cosh (cid:18) q x + d f d f µ f (cid:19) , (27)where the suppression parameter γ B describes the elec- tron transmission through the FS interface, see expres-sions (A14)–(A16). After averaging over θ f the expres-sion (27) coincides with (24) with A given by (26). Itturns out that in the dirty limit the first term of the ex-pression (14) for the Eilenberger function plays the mainrole, as the second term reduces due to the spatial aver-aging (15), and the Eilenberger function Φ + = h Φ + i . D. Clean limit
For a larger electron mean free path ℓ f the sum in (A4)converges at π m µ f q ≈ , πmq ≈ µ f . In the limit ξ H ≪ ξ ω we have k f − qπm arctan (cid:18) πmq (cid:19) ≈ ℓ f /ξ ω + iℓ f /ξ H andΦ + = ∆ p ∆ + ω (cid:18) ℓ f ξ ω + iℓ f ξ H (cid:19) − * µD ( µ ) µ f − µ µ f cosh (cid:16) qm ( x + d f ) µ f d f (cid:17) sinh qµ f − µ cosh (cid:16) qm ( x + d f ) µd f (cid:17) sinh qµ + µ ++ D ( µ f )sinh (cid:16) qµ f (cid:17) cosh (cid:18) q x + d f d f µ f (cid:19) . (28)In this limit, the first term in the square brackets inEq. (14) is small in comparison with the second one.Thus, one may conclude that the first term in Eq. (14)plays the main role in the dirty limit, while the secondterm of Eq. (14) describes mainly the clean limit whentrajectories of motion are essential. E. Josephson current
To calculate the Josephson current J of a SIFS ferro-magnetic tunnel junction we start from the expression J = 4 eN (0) vπT X ω> Z − (Im g ) cos θd (cos θ ) == 4 eN (0) vπT X ω> Z (Im g a ) µdµ. (29)Here N (0) is the density of state at the Fermi surface, v is the Fermi velocity, g is the matrix element of theantisymmetric part of the Eilenberger function that isdefine by the following expression:ˆ g a = 12 (cid:20) g ( θ ) − g ( − θ ) f ( θ ) − f ( − θ ) f + ( θ ) − f + ( − θ ) − g ( θ ) + g ( − θ ) (cid:21) . We apply the tunnel hamiltonian approach and use theboundary conditions on the dielectric interface. Theseboundary conditions are matching the quasiclassical elec-tron propagators g and f on both sides of the boundary. The boundary conditions may essentially depend on thequality of the interface. In the case of a nonmagneticspecularly reflecting boundary between two metals theseconditions read ˆ g a ( R ˆ g c + ˆ g c − ) = D I ˆ g c − ˆ g c + . (30)ˆ g a = ˆ g a = ˆ g a . (31)Here D I and R I = 1 − D I are the interface transparencyand reflectivity coefficients, the index 1(2) labels thefunctions on the right (left) side from the boundary plane,and the symmetric parts of the quasiclassical Eilenbergerfunctions are defined by the equalitiesˆ g c − = 12 (ˆ g c − ˆ g c ) , ˆ g c + = 12 (ˆ g c + ˆ g c ) , (32)ˆ g c , = 12 (cid:18) g , ( θ ) + g , ( − θ ) f , ( θ ) + f , ( − θ ) f +1 , ( θ ) + f +1 , ( − θ ) − g , ( θ ) − g , ( − θ ) (cid:19) . Assuming that the insulating layer transparency issmall, D I ≪ , R I ≈
1, and taking into account thatˆ g = ˆ g = ˆ1, we can expand the boundary conditions(30) in powers of D I .For SIFS tunnel structures with s -wave pairing in theelectrodes, in this limit it immediately follows that thefunctions g ( θ ) = g s ( θ ) and f ( θ ) = f s ( θ ) are indepen-dent on θ and coincide with the expressions for spatiallyhomogeneous superconducting electrodes f f = F f ( θ ) exp { + iϕ/ } , f s = F s exp {− iϕ/ } ,F s = ∆ p ω + ∆ , g s = ω p ω + ∆ . (33)Here ∆ and ϕ are the absolute value and the phasedifference of the order parameter in the electrodes. Theboundary conditions (30) reduce in this case toˆ g a = D I ˆ g c − ˆ g c + = D I g c ˆ g c − ˆ g c ˆ g c ) = (34)= D I (cid:18) f f +2 − f f +1 g f − f g )2( f +1 g − g f +2 ) f +1 f − f +2 f (cid:19) . Then the expression for the Josephson current (29) hasthe form J = 12 eN (0) vπT ∞ X ω = −∞ Z π/ − π/ [ f +2 ( θ ) f ( θ ) −− f ( θ ) f +1 ( θ )] cos θ d (cos θ ) . (35)Using the definition (1) and its symmetry properties (11)and (12), one can find the result for the Josephson current J of the tunnel junction J = J C sin ϕ , where J C = 8 πTeR N X ω> Φ + ( µ s ) Z D I ( µ f ) Re [Φ + ( µ f )] µ f dµ f . (36)The Eilenberger function of the left electrode Φ + ( µ s ) isalso defined by (8).The thin insulating layer is considered as a high poten-tial barrier for electrons. The transmission probabilityis inversely proportional to the exponent of a distancepassed by an electron, i.e. ∼ exp( − d I / cos θ ) , where d I isthe dielectric thickness. Namely, it can be found from thewell known expression for the transmission coefficientof a square potential barrier of a thickness a , that D I ( a ) = 4 k k k k + ( k + k ) sinh ( k a ) ≈≈ k k ( k + k ) exp( − k a ) , (37)if k a ≫
1. Here k and ik are wave vectors of parti-cles outside and inside the barrier. The expressions forthe transmission coefficient of the FS boundary (17),(19)were obtained in the same way. Taking into account that a = d I /µ, one can write the dependence D I ( µ ) in theform D I ( µ ) = f D exp (cid:18) − αµ (cid:19) . (38)Here α is a decaying coefficient, that depends on thethickness and material of the insulator. It is usefulfor a numerical calculation to redefine the value f D = D exp( − α ) , and finally D I ( µ ) = D exp (cid:18) − α − µµ (cid:19) . (39) III. DISCUSSIONA. Main cases
First, we start from the analysis of the Eilenbergerfunction Φ + ( x, µ f ), which describes the superconduct-ing properties of a ferromagnetic Josephson junction. Inthe simplest case Φ + ( x, µ f ) is an exponential functionΦ + ( x ) ∝ exp( x/ξ ) , x
0, or a combination of exponen-tial functions, with a complex coherence length ξ − = ξ − + iξ − . (40)Here ξ describes the decay of the superconducting cor-relations at some distance from the FS boundary, while ξ defines the period of LOFF oscillations . For thedescribed FS structure, Φ + ( x, µ f ) is given by the expres-sions (14) and (20)-(22). Now we try to describe themain features of the proximity effect in an FS structure.If the superconducting layer is thick enough to be consid-ered as semi-infinite, and if in-plane nonuniformities con-nected with sample preparation and the domain structureof the ferromagnetic film can be neglected, the structurehas one geometric parameter — the ferromagnet thick-ness d f . The properties of the F material give three morecharacteristic lengths: the electron mean free path inthe ferromagnet ℓ f , the nonmagnetic coherence length ξ , and the characteristic magnetic length ξ H . The mainquestions at this stage are: can the Eilenberger functionbe approximated by an exponential function, and what isthe relation between ξ , ξ and the characteristic lengthsmentioned above.The Fermi velocity for usual ferromagnetic metals is v f ∼ · cm / s, and if even the temperature T ∼ T c , ξ ∼
100 nm, i.e. ξ is much larger than other parametersof the problem in usual cases. The value of ξ is diffi-cult to decrease. The gradual increase of ℓ f leads to thefollowing four cases:1. ℓ f ≪ ξ H , d f — the well investigated dirty limit.This condition allows averaging of the Eilenberger func-tions over trajectories and using the Usadel equations.Φ + is given by the expression (24), and for a strong ex-change field ( H ≫ k B T c ), ξ = ξ = p ℓ f ξ H / ℓ f ∼ ξ H , d f — the intermediate case. The Us-adel equations cannot be used, and one needs to solvethe Eilenberger equations taking into account the Φ + ( θ )dependence. Up to now the linearized equations weresolved only for an SFS junction , and the obtained ex-pressions are so complicated that they are very difficultto analyze. This is the most interesting case for our anal-ysis, which allows to find out when this case reduces tothe dirty limit and under which conditions the Usadelequations are applicable.3. ξ H ≪ ℓ f < d f , ξ — the clean limit. Here, Φ + ( x, θ )is defined mainly by the second term of Eq. (14), and ξ ∼ ℓ f , ξ ≈ ξ H .4. ℓ f ≫ d f — the ballistic regime cannot be consideredwithin the framework of our approach, excluding the case ξ ≪ d f when multiple reflections from the interfacescan also be neglected. The ballistic SFS junction wasconsidered earlier , and the dependence J C ( d f )cannot be presented as an exponent in the general case . B. Analysis of cases 1-3
In all plots presented below we use the same normaliza-tion for all lengths: x , d f , l f , ξ , ξ H , ξ , ξ . All of themare normalized to some unit length Ξ, which, in fact, canbe chosen arbitrarily, e.g. Ξ = 1 nm. Note that only theratios between the lengths given above are important,i.e., if one divides the above list of lengths by any arbi-trary constant Ξ all the results remain unchanged. Forexample, in all plots we use ξ = 81. a. Influence of different types of FS boundary trans-parency. Figure 1 shows the spatial distribution of thefunctions h Φ + ( x ) i calculated from (4) for ℓ f = 7 , ξ H = 3, d f = 10 and J C ( d f ) calculated from (36) for ℓ f =7 , ξ H = 3, and the insulating layer decay parameter α = 5. It is clearly seen that the angular dependence ofSF interface transparency influences both, the amplitudeand period of oscillations of the presented curves. Forthe particular choice of parameters this influence can beeasily explained.For the case of a transparency which is independenton µ f , the Eilenberger functions are initiated from theSF interface in all directions, thus leading to the largestamplitude of h Φ + ( x ) i oscillations. Simultaneously, thecontributions to h Φ + ( x ) i coming from rapidly decayingEilenberger functions with a large argument θ f result inthe appearance of the smallest period of h Φ + ( x ) i oscilla-tions. The stronger the angular dependence of the SF in-terface transparency coefficient, the smaller are the con-tributions to average values from the rapidly decayingEilenberger functions, and as the result the larger is theperiod and the smaller is the amplitude of h Φ + ( x ) i os-cillations. It is necessary also to mention that h Φ + ( x ) i decays more rapidly with | x | than Φ + ( x, l f . From (36) it follows that in general the contributionto the junction critical current comes not only fromΦ + ( x, + ( x, µ f ) located in a narrowdomain of µ f nearby µ f = 1 . Due to that, the J C ( d f )curve appears to be sensitive against the angular depen-dence of the SF interface transparency. We see that thestronger this dependence the smaller is the amplitude ofoscillations and the larger are the distances of the 0 to π transition points in J C ( d f ) from the SF interface, whilethe period of J C ( d f ) oscillations is practically insensi-tive against the form of D ( µ f ) and coincides with thatof Φ + ( x, . If the ferromagnetic layer is thick enough,the sum over ω in (36) converges rapidly, and the maincontribution is given by the first term of the sum, whichis determined by the real part of the Eilenberger functionΦ + ( x, + ( x, D ( µ f ) = D µ f for the transmissioncoefficient of the FS interface, as it is most applicable tomaterials used in experiments. -10 -8 -6 -4 -2 08 10 12 14 16 18 20 22 24 D( f )=D D( f )=D D( f )=D | R e + ( x ) | / D i coordinate x(a)(b) J C R N / T c D i e - , i = , , F-layer thickness d f FIG. 1: (Color online) (a) Averaged Eilenberger function h Φ + ( x ) i and (b) J C ( d f ) R N product for different types ofthe FS interface (16)-(19). The F-layer parameters are ℓ f =7 , ξ H = 3 (a,b), d f = 10 (a), the insulating layer decay pa-rameter is α = 5 (b); T ≈ . T c . b. Spatial profile of the Eilenberger function. Thedirect comparison (see Fig. 2) of the results of numericalcalculations of the real part of the Eilenberger functionΦ + ( x, app + ( x, ≈ Φ + (0 , d f /ξ ) cosh x + d f ξ (41)confirms that it is possible to approximate Φ + ( x,
1) bythis simple formula. It satisfies the boundary conditionon the dielectric interface (9) and contains the factorΦ + (0 , d f > ℓ f or ξ . Thisapproximation yields the values of ξ , i.e. ξ and ξ . c. Dependence of ξ on the junction parameters. Theattempt to find ξ in the area of intermediate scatteringwas made earlier . It was supposed in Ref. [36] that ξ -10 -8 -6 -4 -2 00.00.20.40.60.8 -30 -25 -20 -15 -10 -5 00.00.20.40.60.8 H =10 H =2.1 (a) R e + ( x , ) | / D coordinate x H =3.8d f =30 d f =20 d f =10 (b) R e + ( x , ) | / D coordinate x FIG. 2: (Color online) The exact Φ + ( x,
1) (solid lines) andapproximated Φ + ( x, app (dashed lines) Eilenberger function(a) for different values of ξ H corresponding to a local maxi-mum, minimum, and smooth region of the dependence ξ ( ξ H );see Fig.3(b), at ℓ f = 7 , d f = 10; (b) for different thicknesses d f at ℓ f = 7 , ξ H = 3 . T ≈ . T c . may be found as a solution of the equation iℓ f /ξ = arctan( iℓ f /k f ξ ) , (42)which determines the poles of the sum in Eqs.(14), (15)for Φ + ( x ) and h Φ + ( x ) i , respectively. Unfortunately, theinverse tangent is a multivalent function. In the generalcase this fact prevents to represent Φ + ( x ) and h Φ + ( x ) i asonly the sum of residues of summable functions in (14),(15). An exception occurs in the range of parametersfor which the ferromagnet is close to the dirty limit. Inthis case the sum in (14) and (15) converts faster thanthe multivalent nature of the inverse tangent becomes es-sential. These difficulties had been first pointed out inRef. [37], where it was also demonstrated that the solu-tion of the Eilenberger equations used in Ref. [36] doesnot transfer correctly to the solution of the Eilenbergerequations in the clean limit.To avoid these mathematical difficulties we have de-cided to find out the dependence ξ ( ξ H , ℓ f ) from the directfitting of the real part of Φ + ( x ) calculated numericallyfrom (14) by approximating formula (41). The result ofthis procedure is demonstrated in Fig. 3. It shows thedependencies of ξ and ξ on the magnetic length ξ H atdifferent ℓ f . The solutions from Ref. [36] are also pre- () * * x (a) l f =1d f =10 l f =7d f =20 (c) () l f =7d f =50 (d) ( H ) H xxx l f =7d f =10 (b) () FIG. 3: (Color online) Decay length ξ and oscillation param-eter ξ vs magnetic length ξ H for different values of ℓ f and d f , that are denoted in the plots. The corresponding valuesof ξ from Ref. [36] are also shown and marked by ∗ . sented in Fig. 3 for comparison. They coincide with ourvalues when ξ H ≫ ℓ f i.e. in the dirty limit. We havetaken two cases for the numerical investigation: ℓ f = 1,that is close to the dirty limit, and ℓ f = 7, that is closeto the clean case.It is seen, that ξ oscillates with the exchange energy H (so as ξ H ∼ H − ), an interesting result, that was notnoted earlier. These oscillations appear in rather cleansamples ℓ f & d f /π and for ξ H < d f /π . They origi-nate mainly from the second term of the expression (14),which plays the main role in the clean case. The func-tion ξ ( ξ H ) changes with increasing d f (cf. Fig. 3(b–d)).At large enough d f , electrons have a time to scatter inthe ferromagnet, their trajectories shuffle, and the dom-inant role of the ”clean“ term vanishes, the oscillationsdisappear, see Fig. 3(d). We note that the smaller isthe probability of dephasing of the Eilenberger functionsdue to electron scattering in the F layer, the stronger arethe interference effects between the IF and FS interfaces.This interference demands, for given values of ξ and ξ ,the fulfillment of the boundary conditions at these in-terfaces, which fix the value of the phase derivatives at x = 0 and x = d f . As it follows from Eq. (20)-(22), theseconditions cannot be satisfied on the class of monotonic ξ and ξ functions. Therefore the transition from thedirty to the clean limit should be accompanied by a non-monotonic behavior of ξ and ξ as a function of ξ H forfixed ℓ , see Fig. 3, or as a function of ℓ for fixed ξ H , seeFig. 4.In the dirty limit the spatial oscillation length ξ ≈ p ℓ f ξ H /
3, while in the clean limit ξ ≈ ξ H . Near thepoint marked by X in Fig. 3 there is a crossover betweenthe dependencies corresponding to dirty and clean lim-its, respectively, i.e. this region can be considered as aboundary between the clean and the dirty cases. ( l f ) mean free path l f ( l f ), H =5 ( l f ), H =5 ( l f ), H =10 ( l f ), H =10 FIG. 4: (Color online) Decay length ξ (solid lines) and oscil-lation parameter ξ (dashed lines) vs electron mean free path ℓ f in the ferromagnet for different values of ξ H ; here d f = 10, T ≈ . T c . The dependence of ξ on the mean free path ℓ for differ-ent ξ H is presented in Fig. 4. Usually the decay length ξ as well as the oscillation period 2 πξ increase withthe mean free path, which corresponds to the results inRef. [5]. However, the dependence ξ ( ℓ f ) may be non-monotonic for some values of the ferromagnet thickness.In the presented figures one can see, that ξ may belarger or smaller than ξ and may behave nonmonotoni-cally. This depends not only on the material constants offerromagnetic material, but also on the thickness of theF layer.The found values ξ and ξ together with the simpledependence (41) could be widely used for various estima-tions, approximate calculations, and fitting experimentaldata, for example for a measurement of the density ofstates (DOS) in an FS bilayer or calculation of thick-ness dependence of the critical current of SIFS Josephsonjunctions. d. J C ( d f ) in the limit of large d f . As an example,we consider the critical current of SIFS Josephson junc-tions in the limit of large F layer thickness d f ≫ ξ . In this limit the main contribution to the critical current in(36) is given by the term at ω = πT. Suppose further thatonly electrons, that are incident in the direction perpen-dicular to the FI interface, provide the current across it,from (36), (41) we have J C = 8 πT D eR N Re ∆ Φ + (0 , p ( πT ) + ∆ cosh( d f /ξ ) (43)The comparison with the exact numerical calculationhas shown that this expression (43) well approximates J C ( d f ) starting from d f & ξ . The thickness depen-dence of J C calculated from (43) at the value ξ taken at d f = 50, see Fig.3(d), for l f = 7, ξ H = 2 . T ≈ . T c is shown in Fig. 5 by the solid line. The dashed anddashed dotted curves in Fig. 5 give the results obtainedby numerical calculation, which have been done for thesame parameters with the use of the exact expressionfor Φ + ( − d f , µ f ) in (36) and at different insulating layerthicknesses described by the parameter α. It is clearly seen that the larger is α the closer is thethe result to the approximation formula (43). We mayalso conclude that the difference between the asymptoticsolid curve and the dashed curves in Fig. 5 calculated forfinite values of α occurs only in amplitude and positionsof the 0 to π transition points, while the decay length andperiod of J C ( d f ) oscillations are nearly the same. Thismeans, that if in experiment we are mainly interested inthe estimation of the ferromagnet material constant, ξ, we may really use for the data interpretation the simpleexpression (43) and consider the coefficient Φ + (0 ,
1) init as a phenomenological parameter, which depends notonly on the properties of ferromagnetic material, but alsoon d f in a rather complicated way, see Fig.2(b), as well ason the form of the transparency coefficients at the FI andFS interfaces. Although our results for SIFS junctionsgive the same ξ and ξ as for SFS junctions (at leastfor a thick F layer), different boundary conditions resultin different order parameter amplitude and phase. Thisresults in a different J C ( d f ) dependence, similar to theresults obtained earlier in dirty limit .It is interesting to note that in the case of a ratherthin insulating barrier and rather clean ferromagnet, ξ and ξ as measured from the critical current of SIFSjunctions may differ from ξ as measured from the DOSon the free F surface of an analogous FS bilayer. This isbecause the DOS measured by low-temperature scanningtunneling spectroscopy (which may only yield contribu-tions from electrons with normal incidence) is directlygiven by Φ + ( − d f , µ f = 1), while J C includes contribu-tions from other angles. e. Applicability of the Usadel equation. The condi-tion that allows using the Usadel equation is well known— this is strong nonmagnetic scattering , namely, ℓ f ≪ ξ H , d f , ξ . What does the symbol ” ≪ “ mean exactly?In order to illustrate this, Fig. 6 shows the dependence J C ( ℓ f ) given by the expression (21). Two parts of (21),called for convenience ”dirty” and ”clean”, are given bythe first and the second terms of the expression (21). The0 J C R N / T c D e - F-layer thickness d f approx. FIG. 5: (Color online) J C R N product as a function of theferromagnet thickness d f at different insulating layer thick-nesses described by the parameter α , and the approximationby expressions (36),(41) at the value ξ taken at d f = 50, seeFig.3(d), here l f = 7, ξ H = 2 . T ≈ . T c ,. -600-400-2000200 0 1 2 3 4 5-5051015 J C R N e / T c D (a)(b) Usadel appr. whole J C R N "dirty" term "clean" term J C R N e / T c D mean free path l f FIG. 6: (Color online) J C R N product vs ferromagnet meanfree path l f ( ξ H = 10, T ≈ . T c , α = 5) for d f = 10 (a)and d f = 30 (b). Each plot shows J C R N obtained from theUsadel function (15), from the Eilenberger function Φ + (21),as well as from only the first ”dirty” term or only the second”clean” term of (21). critical current density as given by the Usadel function(27), describing the dirty limit, is also shown. J C ( ℓ f )coincides with the Usadel solution at ℓ f ≈ . ξ H , thenthe first term in square brackets in Eq. (21) dominatesand the µ f -dependence becomes nonsignificant due tothe spatial averaging as a result of multiple scattering.This means, that the Usadel equations become appropri-ate to use if the parameter Hτ f .
1. This result was intuitively clear, but demanded a proof due to plenty ofinvestigations, where the Usadel equations was used at Hτ f . IV. COMPARISON WITH EXPERIMENT
There are only few experiments on SIFS tunnel junc-tions with moderate scattering in the F-layer . InRef. [10] experimental data points for J C ( d f ) are rathersparse, while in Ref. [39] the density of data points perperiod is much larger. We thus decided to compare ourtheory to the data of Ref. [39].This article also contains attempts to fit the ex-perimental data by theoretical curves, however, withoutmuch success. Some of these attempts, see Fig.5(a) inRef. [39], use a theory developed for a very clean (bal-listic) SFS junction. The predicted oscillation period of J C ( d f ) was significantly smaller than the one in the ex-periment. Figure 5(b) in Ref. [39] shows the same ex-perimental data together with fits using dirty limit the-ory. The best fit was achieved with ξ = 0 .
66 nm and ξ = 0 .
53 nm. However, the theory for a dirty ferromag-netic junction cannot explain ξ > ξ , and all of thesetheories, both clean and dirty, do not take into accountthe insulating layer.In Ref. [39] and Ref. [40] it was found, that the mag-netic anisotropy of the F layer changes from perpendicu-lar to in-plane with an increase of the F-layer thickness.Perpendicular magnetic anisotropy in a polycrystallinethin film may occur due to several reasons: the mecha-nism described by Neel related to the absence of near-est neighbor F-layer atoms near interfaces, the magneto-striction mechanism, the surface roughness, and the oneassociated with microscopic-shape anisotropy . The lat-ter may also vary for samples sputtered under differentangles . Perpendicular magnetic anisotropy can changeto an in-plane anisotropy due to a competition with theshape anisotropy of a film. However, none of the mech-anisms listed above can be expected to give a significantjump of the exchange magnetic energy H at the transi-tion between the two types of anisotropy. There is someshift of the experimental curve, see Fig.2(a) in Ref. [39],at d f = 3 . H . Therefore,we try to describe the whole experimental curve J C ( d f )using the same value of H .The common problem for all explanations of this ex-periment is the location of the first minimum of J C ( d f )at a rather large d f ≈ ξ ≈ . attribute thisto the existence of a nonmagnetic dead layer with ratherlarge thickness d dead = 2 .
26 nm in the Ni layer.We note, that d dead , in general, depends on d f . Weassume that d dead = d f up to some value d max andthen either d dead stays constant at a saturation value d satdead = d max with further increase of d f , or it is reduced1to a constant value d satdead < d max . For the latter casethere are the following reasons. The dead layer may con-sist of a nonmagnetic Ni alloy with magnetic clusters orislands, which are not connected by the exchange inter-action. A very thin film of a ferromagnetic material maybe paramagnetic. When one covers it by extra magneticlayers, some part of its thickness may magnetize. First,magnetic clusters uncoupled by the exchange interactioncan couple if they are covered by extra magnetic layers.Second, a thin homogeneous film of a magnetic materialof the thickness of a few monolayers exhibits nonmagneticproperties alone, but becomes magnetic entirely when itsthickness increases, due to the transition from the two-dimensional to the three-dimensional case.The dead layer is described as a normal layer, thatyields J C ( d f ) ∼ exp( − d f ξ N ). The best fitting presentedin Fig. 7(a),(b) by straight short-dashed lines, gives thevalue ξ N = 0 .
68 nm. The normal metal coherencelength ξ N depends both on the nonmagnetic and para-magnetic scattering. The latter may be significant inthe dead layer in the presence of many magnetic inho-mogeneities (clusters and impurities). Following a the-ory developed for SFS junctions taking into accountmagnetic scattering, one can find the expression for thecoherence length of a normal metal (at H = 0) withmagnetic impurities ξ N = q D f πT +1 /τ m ) , where τ m is themagnetic scattering time. We assume that the mean freepath has the same value as in the rest of the magneticNi layer, ℓ f = 0 . τ m = 1 . · − s. For comparison, the nonmag-netic scattering time is found to be almost 8 times less.Fig. 7(a),(b) shows that the obtained value of ξ N is veryclose to the decay length of the magnetic SIFS junctionat d f > d max . In the absence of the exchange field itspair-breaking role is played by the magnetic scattering inthis structure.We consider two possibilities to explain the experimen-tal data . First, we assume that the detected minimumof J C ( d f ) is the first one, see Fig. 7(a). Treating thedead layer as a normal layer ( H = 0), our best fit yields d satdead = d max = 2 .
37 nm. The fitting yields the fol-lowing values of the F-layer parameters: ℓ f = 0 . ξ H = 0 . H = 1782 K or H = 154 meV.As a second scenario we assume that the detected min-imum of J C ( d f ) is the second one, see Fig. 7(b). Thefirst J C ( d f ) minimum may not show up if the effectivethickness of the dead layer decreases as soon as d f isabove the threshold value d max . Then, at the value of d f where J C ( d f ) is supposed to have the first minimum, theF layer may still be nonmagnetic, while at large valuesof d f it has some small value d satdead . For that scenario d satdead = 0 . ℓ f = 0 . ξ H = 0 .
956 nm, that corre-sponds to H = 1119 K or H = 96 . T = 4 . T c = 8 K were taken like in the experiment , that yields ξ = 81 nm. Fermivelocities v f = 2 . · cm / s and v s = 6 . · cm / swere taken. Since v s > v f , we use the FS boundary con-dition in the form (17). The thickness of the insulatingAl O barrier was not measured exactly; still the value α = 5 seems to be realistic, see Eq.(39). The values ofthe exchange field for both cases are within the rangeobtained for Ni in SFS experiments . The rela-tion ℓ f ∼ ξ H does not allow using the Usadel equations,however, this is not far from the dirty limit.Comparing both fits shown in Fig. 7(a),(b) we cannotgive a definite answer which minimum (the first or thesecond) was observed in the experiment , but we areable to explain the fact that ξ > ξ in any case. Suchrelation between ξ and ξ cannot be obtained from adirty limit theory.The Josephson junctions used in the experiment con-tain an extra normal layer of Cu between the I and Flayer. A normal layer cannot change ξ , which mainlydepends on properties of the magnetic layer. However,it may change the boundary conditions. Therefore, ourtheory cannot be directly applied for the determinationof the exact value of the dead layer, but may explain theobtained value of ξ .For the analysis of the first minimum position and thedead layer estimation we may also use the theory devel-oped earlier for the dirty limit, since the estimated valueof ℓ f ∼ ξ . From the analysis of the uniform part of thesolutions obtained in Ref. [43] for dirty SINFS and SIFNSjunctions, it follows that the first minimum of the J C ( d f )dependence must be at ( d f − d dead ) /ξ ≈ π/
4. If we as-sume that the detected minimum of J C ( d f ) is the firstone, then, the estimation yields d satdead = d max = 1 . J C ( d f ) is the second one, thenthe first one must be located at d f ≈ . d satdead = 0 . J C ( d f ) at the thickness d f ≈ . π transition was not detected,possibly, due to the fact that d satdead < d max . Generallyspeaking, if we assume that d dead ( d f ) jumps from d max to d satdead at d max then one should see a jump on I C ( d f )dependence. However in our case it not observed, seeFig.7(a) and (b). We suppose that this is related to thefact that ξ N ≈ ξ .Figure 7(c) contains also calculated curves, plotted forthe same model parameters as in Fig. 7(b) but for acleaner ferromagnet. Usually the purpose of the exper-imental investigations of SIFS structures is the designof a π -Josephson junction with a large J C R N . R N isdetermined mainly by the insulating barrier. Our calcu-lations show that by increasing l f one can increase J C by1-2 orders of magnitude, see Fig.7(c). It would also bereasonable to delete extra normal layers to achieve the π -phase at a smaller d f , and consequently, to have larger J C ; see also Refs. [38,44].2 J C ( A / c m ) Exper. data Theory Dead layer extrap. (a) l f =0.6 nm H =0.6 nm d satdead =d max =2.37 nm l f =0.6 nm H =0.956 nm d satdead =0.2 nm J C ( A / c m ) Exper. data Theory Dead layer extrap. (b)(c) l f =0.6 nm l f =1 nm l f =3 nm J C ( A / c m ) thickness, d f (nm) FIG. 7: (Color online) (a) and (b) show a comparison of J C ( d f ) from the presented theory with experiment . Dashedlines are fits to the experimental data in the range d f < .
37 nm, where the F layer can be interpreted as nonmagnetic.(c) Dependencies J C ( d f ) at lower scattering. Parameters arelike in (b). The regime for which multiple reflections cannotbe neglected are marked by dotted lines. V. CONCLUSION
The SIFS ferromagnetic tunnel Josephson junction hasbeen investigated within the framework of the quasiclas-sical Eilenberger equations, that allow a description ofboth, the clean and the dirty limits, as well as an ar-bitrary scattering. The Eilenberger function Φ + ( x, µ f )may be approximated by the simple formula (41) withinthe entire range of considered parameters. The decaylength ξ and the oscillation period 2 πξ depends notonly on the mean free path ℓ f and the exchange energy H in the ferromagnet, but also on the ferromagnet thick- ness d f , and can be nonmonotonic as a function of ξ H or ℓ f .The approximation of J C ( d f ) by an exponential func-tion or its combinations has some restrictions. The ap-plicability of the Usadel equation has been establishedfor ℓ f . . ξ H .The developed approach has been used to fit exper-imental data providing a satisfactory fitting of the J C ( d f ) dependence. It allows to explain the values ξ > ξ and to give some practical recommendations onhow to increase the J C R N product. Acknowledgments
We gratefully acknowledge M. Weides for providingexperimental data and helpful discussions, as well asA.B. Granovskiy and N.S. Perov. This work was sup-ported by the Russian Foundation for Basic Research(Grants 10-02-90014-Bel-a, 10-02-00569-a, 11-02-12065-ofi-m), by the Deutsche Forschungsgemeinschaft (DFG)via the SFB/TRR 21 and by project Go-1106/03, by theDeutscher Akademischer Austauschdienst.
Appendix A
Substituting the expression (13) into the Eilenbergerequation (6), multiplying the obtained equations bycos [ πk ( x + d f ) /d f ] , and integrating them over x one canfind the relation between coefficients Q m and B (10),namely Q m = h Q m i + D B ζ f d f ( − m M sinh d f ζ f E k f M , (A1)where M = π m µ q + 1 , (cid:28) M (cid:29) = qπm arctan πmq . Averaging both sides of (A1) over the angle θ we get h Q m i = (cid:28) B ζ f d f ( − m M sinh d f ζ f (cid:29) D k f M Eh − D k f M Ei . (A2)A substitution of (A2) into (A1) finally gives the relationbetween the coefficients in expression (13). Q m = D B ζ f d f ( − m M sinh d f ζ f E M (cid:2) k f − (cid:10) M (cid:11)(cid:3) . (A3)Expressions (A3) and (13) permit to rewrite the solutionof the Eilenberger equations in the closed form3Φ + = ∞ X m = −∞ D B ( µ ) ( − m M ( µ ) µq sinh qµ E µ M ( µ f ) (cid:2) k f − (cid:10) M (cid:11)(cid:3) cos πm ( x + d f ) d f + B ( µ f ) cosh (cid:18) q x + d f d f µ f (cid:19) , (A4)and for an isotropic component h Φ + i of this solution to get h Φ + i = k f ∞ X m = −∞ D B ( µ ) ( − m M ( µ ) µq sinh qµ E µ (cid:2) k f − (cid:10) M (cid:11)(cid:3) cos πm ( x + d f ) d f . (A5)The form of the solution of the Eilenberger equations in the S layer essentially depends on transport properties ofthe FS interface.If the transparency D ( µ f ) is small, then in the first approximation on D ( µ f ), we can neglect the suppressionof superconductivity in the S region and consider the order parameter and the Eilenberger functions as constantsindependent on space coordinates, which are equal to their bulk values, thus leading to the boundary condition fordetermination of integration constants B ( µ f ) in the form of Eq. (10). From the boundary condition (10) we get B ( µ f ) = D ( µ f ) ∆ p ∆ + ω (cid:16) qµ f (cid:17) , (A6)and obtain the analytical solution in the form (14) that, from our point of view, is more convenient for the furtheranalysis than solutions previously used in Refs. [4,5].For the described three forms (16)–(19) of D ( µ f ) dependencies we get (cid:28) D ( µ ) µM ( µ ) (cid:29) µ = D Z µdµπ m µ /q + 1 = D q π m ln (cid:18) π m q + 1 (cid:19) , (A7) (cid:28) D ( µ ) µM ( µ ) (cid:29) µ = D Z µ dµπ m µ /q + 1 = D q π m (cid:18) − qπm arctan πmq (cid:19) , (A8) (cid:28) D ( µ ) µM ( µ ) (cid:29) µ = D Z µ dµπ m µ /q + 1 = D q π m (cid:20) − q π m ln (cid:18) π m q + 1 (cid:19)(cid:21) . (A9)(A10)Substituting the averages (A7)–(A9) into the expression (14) we get the Eilenberger function for different FS interfacetransparencies (20)–(22). The functions (20)–(22) averaged over the angle have the following forms.First, for D ( µ f ) = D h Φ + i = ∆ D p ∆ + ω q ( k f −
1) + ∞ X m =1 q π m ln (cid:16) π m q + 1 (cid:17) k f − qπm arctan πmq cos πmxd f . (A11)Second, for D ( µ f ) = µD h Φ + i = ∆ D p ∆ + ω q ( k f −
1) + 2 ∞ X m =1 qπ m (cid:16) − qπm arctan πmq (cid:17) k f − qπm arctan πmq cos πmxd f . (A12)Third, for D ( µ f ) = µ D h Φ + i = ∆ D p ∆ + ω q ( k f −
1) + ∞ X m =1 qπ m h − q π m ln (cid:16) π m q + 1 (cid:17)i k f − qπm arctan πmq cos πmxd f . (A13)Our calculation yields the following value of the sup-pression parameter for the Usadel equation: γ B = ℓ f h µD ( µ ) i − / p d f / πT c . It differs from the one ob-4tained early by a factor 2. We suppose, that this isa consequence of an approximation we have made inboundary conditions (10) at the FS interface. We haveneglected all spatial variations in the S part as well as theback influence of the ferromagnet on the superconductor,and this factor 2 is the price for this approximation. Ifwe use the boundary condition (10) in its full form , weget for γ B : γ B = 23 ℓ f q d f πT c h µD ( µ ) i − . Our three models of the FS interface yield γ B = 23 ℓ f D q d f πT c h µ i − = 43 ℓ f D q d f πT c , (A14) γ B = 23 ℓ f D q d f πT c (cid:10) µ (cid:11) − = 2 ℓ f D q d f πT c , (A15) γ B = 23 ℓ f D q d f πT c (cid:10) µ (cid:11) − = 83 ℓ f D q d f πT c . (A16) ∗ Electronic address: [email protected] A. Golubov, M. Kupriyanov, and E. Il’ichev, Rev. Mod.Phys. , 411 (2004). A. I. Buzdin, Rev. Mod. Phys. , 935 (2005). F. Bergeret, A. Volkov, and K. Efetov, Rev. Mod. Phys. , 1321 (2005). F. S. Bergeret, A. F. Volkov, and K. B. Efetov, Phys. Rev.B , 134506 (2001). J. Linder, M. Zareyan, and A. Sudbo, Phys. Rev. B ,064514 (2009). V. V. Ryazanov, V. A. Oboznov, A. Y. Rusanov, A. V.Veretennikov, A. A. Golubov, and J. Aarts, Phys. Rev.Lett. , 2427 (2001). V. A. Oboznov, V. V. Bol’ginov, A. K. Feofanov, V. V.Ryazanov, and A. I. Buzdin, Phys. Rev. Lett. , 197003(2006). Y. Blum, A. Tsukernik, M. Karpovski, and A. Palevski,Phys. Rev. Lett. , 187004 (2002). H. Sellier, C. Baraduc, F. Lefloch, and R. Calemczuk,Phys. Rev. B , 054531 (2003). F. Born, M. Siegel, E. K. Hollmann, H. Braak, A. A. Gol-ubov, D. Y. Gusakova, and M. Y. Kupriyanov, Phys. Rev.B , 140501 (2006). M. Weides, M. Kemmler, E. Goldobin, D. Koelle,R. Kleiner, H. Kohlstedt, and A. Buzdin, Appl. Phys. Lett. , 122511 (2006). M. Weides, M. Kemmler, H. Kohlstedt, R. Waser,D. Koelle, R. Kleiner, and E. Goldobin, Phys. Rev. Lett. , 247001 (2006). J. Pfeiffer, M. Kemmler, D. Koelle, R. Kleiner,E. Goldobin, M. Weides, A. K. Feofanov, J. Lisenfeld, andA. V. Ustinov, Phys. Rev. B , 214506 (2008). J. W. A. Robinson, S. Piano, G. Burnell, C. Bell, andM. G. Blamire, Phys. Rev. Lett. , 177003 (2006). J. W. A. Robinson, S. Piano, G. Burnell, C. Bell, and M. G. Blamire, Phys. Rev. B , 094522 (2007). K. D. Usadel, Phys. Rev. Lett. , 507 (1970). G. Eilenberger, Z. Phys. , 195 (1968). A. I. Buzdin, L. Bulaevskii, and S. Panyukov, JETP Lett. , 178 (1982), Pis’ma v ZhETF , 147 (1982). A. Buzdin, B. Bujicic, and M. Kupriyanov, JETP , 124(1992), Zh. Eksp. Teor. Fiz. , 231 (1992). E. Terzioglu and M. R. Beasley, IEEE Trans. Appl. Super-cond. , 48 (1998). L. B. Ioffe, V. B. Geshkenbein, M. V. Feigel’man, A. L.Fauche`ere, and G. Blatter, Nature (London) , 679(1999). A. V. Ustinov and V. K. Kaplunenko, J. Appl. Phys. ,5405 (2003). G. P. Pepe, R. Latempa, L. Parlato, A. Ruotolo, G. Au-sanio, G. Peluso, A. Barone, A. A. Golubov, Y. V. Fomi-nov, and M. Y. Kupriyanov, Phys. Rev. B , 054506(2006). A. V. Zaitsev, JETP , 1015 (1984), Zh. Exp. Teor. Fiz. , 1742 (1984). Z. Radovic, N. Lazarides, and N. Flytzanis, Phys. Rev. B , 014501 (2003). L. F. Mattheiss, Phys. Rev. B , 373 (1970). V. Shelukhin, A. Tsukernik, M. Karpovski, Y. Blum,K. B. Efetov, A. F. Volkov, T. Champel, M. Eschrig,T. L¨ofwander, G. Sch¨on, et al., Phys. Rev. B , 174506(2006). M. Y. Kupriyanov and V. F. Lukichev, Sov. Phys. JETP , 1163 (1988). L. Landau and E. Lifshits, Quantum Mechanics (Perga-mon Press, Oxford, 1961). A. I. Larkin and Y. N. Ovchinnikov, JETP , 762 (1965),[Zh. Eksp. Teor. Fiz. P. Fulde and R. A. Ferrell, Phys. Rev. , A550 (1964). A. Vedyayev, C. Lacroix, N. Pugach, and N. Ryzhanova, Europhys. Lett. , 679 (2005). Z. Radovi´c, L. Dobrosavljevi´c-Gruji´c, and B. Vujiˇci´c, Phys.Rev. B , 214512 (2001). F. Konschelle, J. Cayssol, and A. I. Buzdin, Phys. Rev. B , 134505 (2008). M. Zareyan, W. Belzig, and Y. V. Nazarov, Phys. Rev.Lett. , 308 (2001). D. Yu.Gusakova, A. A. Golubov, and M. Y. Kupriyanov,JETP Lett. , 418 (2006), [Pis’ma v ZhETF , 487(2006)]. A. F. Volkov, F. S. Bergeret, and K. B. Efetov (2006),arXiv:cond-mat/0606528v1 [cond-mat.supr-con]. A. S. Vasenko, A. A. Golubov, M. Y. Kupriyanov, andM. Weides, Phys. Rev. B , 134507 (2008). A. A. Bannykh, J. Pfeiffer, V. S. Stolyarov, I. E. Batov,V. V. Ryazanov, and M. Weides, Phys. Rev. B , 054501(2009). M. Weides, Appl. Phys. Lett. , 052502 (2008). L. Neel, Compt. Rend. , 1613 (1953). L. Maissel and R. Glang, eds.,Handbook of thin film technology (McGraw Hill Hookcompany, 1970). N. G. Pugach, M. Y. Kupriyanov, A. V. Vedyayev,C. Lacroix, E. Goldobin, D. Koelle, R. Kleiner, and A. S.Sidorenko, Phys. Rev. B , 134516 (2009). A. Buzdin, JETP Lett.78