Superconducting proximity effects in magnetic metals

We explain the basic physics behind oscillatory effects in superconductor/metallic ferromagnet ( S / F ) sand-wiches, and describe the important effects of the spin orbit scattering in these systems. We ﬁ nd that spin-orbit scattering plays a major role in the physics of the superconducting proximity effect with a conducting ferro-magnet. As examples, we present calculations of the T c of an S / F bilayer and the Josephson current ! near T c " of an S / F / S trilayer. # S0163-1829 ! 97 " 06521-1 $


I. INTRODUCTION
What happens when a ferromagnetic layer (F) is placed in contact with a superconductor (S)?The purpose of this work is to consider the answer to this question for the specific case in which the ferromagnet is a good conductor and the superconductor is an s-wave superconductor.We also restrict our consideration to the most relevant case when the Curie temperature of the ferromagnet is much greater than the superconducting T c .
This situation has been treated by several authors, most recently by Buzdin et al., [1][2][3][4][5] who have observed that the exponentially decaying Cooper pair density in the ferromagnet also has an oscillatory character, indicating that the Cooper pair acquires a spatially dependent phase in the ferromagnetic layer.This causes an exchange field dependent oscillation in the critical current of SFS sandwiches, and in the T c of SF bilayers and multilayers.When the Josephson coupling energy is negative, one has a so-called junction, for which the minimum energy configuration corresponds to a phase shift of in the macroscopic phase difference across the junction.Despite several experimental studies, 6 there is no definitive experimental evidence for these predictions.
The purpose of this paper is to review these earlier calculations, so as to reveal more clearly the underlying physics, and to extend them so as to include a more general treatment of the important effects of the spin-orbit scattering.As we shall see, spin-orbit scattering plays a major role in the physics of the proximity effect with a ferromagnet. 7,12We do not consider here a specific example of the experimental situation, which is unclear and controversial.We simply note that spin-orbit scattering is relevant in conductors containing large Z elements.An assessment of the current experimental situation in light of our results will be presented in a subsequent paper.
Before turning to the detailed microscopic theory of these effects, it is well to review the basic physics behind them, and to present simple physical arguments which justify them.The fundamental feature to be justified is the oscillating pair density.
For simplicity, we first consider the situation in which spin is a good quantum number ͑i.e., there is no spin-orbit interaction͒.Imagine a Cooper pair being adiabatically transported across an SF interface with its electron momenta aligned with the interface normal.Upon entering the F region, where the pair is not an eigenstate, it becomes an evanescent state, decaying exponentially on the length scale 0 , the normal metal coherence length.In addition, the up spin electron in the pair lowers its potential energy by h, the exchange field energy in the ferromagnet, while the down spin electron raises its potential energy by the same amount.In order for each electron to conserve its total energy, the up spin electron must increase its kinetic energy, while the down spin electron must decrease its kinetic energy, to make up for these additional potential energies in F.
So for a pair, shown on top of Fig. 1, entering into a ferromagnetic region results in acquiring a center of mass momentum Qϭ2h/v F .The fermionic antisymmetry requires us to consider the pair described above together with the pair which has the down spin and up spin electrons in-terchanged in momentum space ͑shown on the bottom of Fig. 1͒.The latter gains a center of mass momentum ϪQ upon crossing the SF boundary.Combining the two pairs into a singlet combination we see that the overall effect of the exchange field in the F region on a singlet Cooper pair is to give it a spatial modulation.Hence if the wave function of the pair in a superconductor is ⌽(x 1 Ϫx 2 ), where x 1 and x 2 are the coordinates of the two electrons, in a ferromagnet the wave function becomes cos͓Q(x 1 ϩx 2 )͔⌽(x 1 Ϫx 2 ).
In the more general case when the electrons in a pair have their momenta at an angle with respect to the interface normal ͑see Fig. 2͒, the additional momentum that each electron gains after crossing the SF boundary is ⌬p x ϭh/v F cos and ⌬p y ϭ⌬p z ϭ0.Here we used the fact that momentum is conserved in the direction parallel to the interface to reason that the electrons may change their momenta in the x direction only.The modulation factor of the pair shown in Fig. 2 in the F region is cos͓h(x 1 ϩx 2 )/v F cos͔.
The overall Cooper pair distribution is then obtained by accounting for all possible angles of incidence for the pair, so it is proportional to

͑1͒
͓We assume, for simplicity, that v F /(2h)ϭ m0 Ӷ 0 , so that the overall exponential decay of the Cooper pair in F over 0 may be neglected.͔Thus the Cooper pair distribution oscillates on the scale set by the length m0 .This establishes simply the physical origin of the oscillations.
The physical picture of the proximity effect in a clean ferromagnetic conductor is therefore very similar to the Fulde-Ferrel-Larkin-Ovchinikov ͑FFLO͒ effect. 8,9In the FFLO state a superconducting order parameter is generated in the presence of an exchange field, and it turns out that energetically a spherically symmetric distribution of the electrons is less favorable than the distribution extended along one of the directions perpendicular to the exchange field.So Cooper pairs with shifted center of mass momenta appear, and an inhomogeneous distribution function, similar to the one we described earlier, develops.
We now want to understand the effect of elastic potential ͑nonmagnetic and non-spin-orbit͒ scattering.As usual we only need to consider the processes in which the two electrons in a Cooper pair are scattered by the same impurity into states with opposite momenta, since all the other scattering events are incoherent in the pairing process.An interesting question now is what happens to the center of mass momentum that the Cooper pairs acquire upon entering the ferromagnet ͓and ultimately the modulation factor cos͓h/ (v F cos )(x 1 ϩx 2 )] for singlet pairs͒.In a clean limit we found that this center of mass momentum was a function of the relative momentum of the two electrons and remained constant throughout the whole trajectory of the pair in the F region.Now, as a result of the multiple scattering processes, the center of mass momentum will vary along the trajectory of the pair together with the relative momentum.It is important to realize, however, that the center of mass momenta that we need to consider are always in the x direction, because the scattering events leading to the other directions are incoherent.And since scattering on impurities does not change the energy of the electrons, we can again use the energy argument introduced above to deduce that in the dirty limit we have the same local relationship between the direction of the relative momentum of the electrons and the magnitude of the center of mass momentum Qϭ2h/v F cos as in the clean limit.This observation allows us to treat the case of isotropic and strong potential scattering ͑l, the mean free path due to scattering, is much smaller than any other length in the problem͒ similarly to the clean case, only instead of integrating over the angles of incidence of the pairs we integrate over all possible intermediate orientations.The only thing that we will need to add for the case of impurities, present in a ferromagnet, is an extra decay of the Cooper pairs due to scattering.This decay has the important feature that it depends on the direction of the momenta of the electrons in the Cooper pair and, in the dirty limit, the net scattering rate for the pairs is proportional to cos/, yielding an effective mean free path for pairs of l/cos. 10The pair distribution function, accounting for this scattering effect, is thus proportional to which integrates approximately, yielding a Cooper pair density proportional to Re ͩ e Ϫ͑1Ϫi ͒x/ m ϩ3i/8 ͱx m /l 2 ͪ , where m ϭͱl m0 /2.In the integral ͑1͒ the rapid oscillations of the integrand for small values of cos ensure that the most important pair trajectories contributing to this effect are those which are nearly perpendicular to the SF interface.Formula ͑2͒ shows FIG. 2. Cooper pair in superconductor and ferromagnet for a general direction of the relative momentum of two electrons.⌬ p x ϭh/v F cos.
that if there is strong elastic scattering in F, however, the values of cos less than one can contribute as well, because of scattering into the low direction.The average cos value contributing dominantly then decreases.This lowers the effective period of the oscillations, as well as introducing decay of the oscillations due to the fact that momentum parallel to the interface is no longer conserved within the F layer.The oscillations are now damped on the same length scale at which they oscillate.The effect of the scattering is to average over the effective magnitude of the exchange field from a minimum value equal to h(cosϭ1), to a maximum which approaches infinity (cosϭ0), resulting in a shorter oscillation period.
In the presence of spin-orbit scattering, the spin singlet Cooper pair wave function decays ͑to a spin triplet͒ by spinflip scattering, and the ϩQ momentum pairs mix with the ϪQ momentum ͑spin-exchanged͒ pairs.The decay to a triplet is a pair breaking effect, giving the pairs a finite lifetime because the intermediate state in spin-orbit scattering is a triplet whose energy depends upon the exchange field.Another effect of the spin orbit scattering is to mix the Copper pair with its spin-exchanged counterpartner.This causes a pair to ''see'' an exchange field which changes sign at a rate proportional to 1/ so , decreasing the average h field experienced by the pair, hence increasing the period of the oscillations.
For very strong spin-orbit scattering, the Cooper pair can no longer be regarded as a spin singlet for any reasonable length of time, in particular for the time ⌬tϭ1/h, when so Ͻ1/h, so that the oscillation effect disappears.Put another way, the electrons which make up the pair can no longer be regarded as eigenstates of spin in this case.
For pedagogical purposes, we ignore potential scattering for the moment, and consider the effects of spin-orbit scattering alone.In actuality, spin-orbit scattering is inevitably accompanied by potential scattering.The additional influence of potential scattering in the dirty limit will be accounted for subsequently.
The decay and mixing processes change the energy of the pairs, and introduce a lifetime effect through equations for the forward ( f Q ) and backward, spin-exchanged ( f ϪQ ) portions of the Cooper pair wave function ͑see Sec.III of this paper͒: ͑In the absence of the exchange field h, one sees immediately that f Q ϭ f ϪQ and the spin orbit scattering has no effect on the pairs.͒Solving for E yields a ͑complex͒ pair energy Eϭ2i/ so Ϫͱ͑2h ͒ 2 Ϫ͑2/ so ͒ 2 ϭv F / m0 and hence a complex momentum QϭE/͑v F cos ͒ϭ1/͑ m0 cos ͒.
Accounting for both directions of momentum, we get a Cooper pair density proportional to ͵ 0 1 d͑cos ͒Re͑ e ix/͑ m0 cos ͒ ͒.
One can see that the existence of oscillations requires h Ͼ1/ so .If we include strong elastic scattering as before, then this becomes, approximately where m ϭͱl m0 /2.

II. EILENBERGER EQUATIONS IN THE PRESENCE OF THE EXCHANGE FIELD AND THE SPIN ORBIT SCATTERING
In this section we briefly review the derivation of the Eilenberger equations and show how they can be generalized to account for the presence of an exchange field and spin orbit scattering.
We perform all the calculations in the Matsubara imaginary time formalism and our four-coordinate x stands for ͑,r͒.Following Maki 11 we introduce a spinor representation of the fermion operators:

͑3͒
In the absence of the impurities the Gor'kov equations for a 4ϫ4 matrix Green's function G ˆ(x 1 ,x 2 )ϭ Ϫ͗T ⌿(x 1 )⌿ † (x 2 )͘ can be written in two forms, corresponding to writing the equations of motion of the particles at ˆand ˆare the Pauli matrices in the particle-hole and spin spaces correspondingly and is the BCS coupling constant.The Gor'kov equations carry the information about both the macroscopic fields and the excitation spectrum.In particular, it is the the center of mass spatial dependence that gives the macroscopic fields and the relative coordinate dependence that gives the excitation spectrum.Since we are not interested in the excitation spectrum but only in the macroscopic pair wave function, we integrate over the relative coordinates of two particles sacrificing our knowledge of the excitation spectrum for the sake of getting simpler equations.
We want to separate the center of mass and relative motions inside the Green's functions.So, from r 1 and r 2 we go to Rϭ(r 1 ϩr 2 )/2, the position of the center of mass, and rϭr 2 Ϫr 1 , the relative coordinate of the two particles.We also make a Fourier transform in the imaginary time domain using the fact that the time homogeneity is not broken and everything depends on 2 Ϫ 1 only, We assume that the macroscopic fields h and ⌬ vary on the length scales bigger than the coherence length of material ͑S or F͒, and so we can replace the actual argument of the two functions RϮr/2 by just R. Later we will be using the resulting equations when this condition is not rigorously satisfied.However, one can convince oneself that this procedure is a quasiclassical approximation and only results in averaging over the fast oscillations on the length scale of 1/k F .
We subtract the first equation of ͑6͒ from the second and get We can integrate over the energies of the relative motion of the two particles using two simple transformations where n is a unit vector that carries information about the direction of the relative motion of two particles and vϭp/m Ӎv F n.
At this point it is straightforward to introduce the effect of the impurities.It is simply a matter of insertion of the self-energy in Born approximation into ͑9͒,
Components of the matrix G ˆare given by .

͑12͒
When writing ͑12͒ we assumed purely singlet pairing, and we used the integral of the equation ͑10͒ together with the asymptotic form of the Green's functions in the bulk of the superconductor to get the equality of the spin reversed Green's functions ͑so, that we have only two g's instead of four͒.And we also introduced numerical factors in order to have a simple assymptotic form of the f and g functions in the bulk of the superconductor: f Ϯ ϭ⌬/ͱ 2 ϩ⌬ 2 and g Ϯ ϭ/ͱ 2 ϩ⌬ 2 ͑for the case when h ϭ0͒.Using the functions introduced in ͑12͒, Eqs.͑10͒ and ͑11͒ become Equations ͑14͒ are the generalization of the Eilenberger equations for the case when an exchange field and spin-orbit scattering are present.A heuristic way to obtain these equations would be to generalize the results from Maki 11 and Likharev. 13ne can see from ͑14͒ that in the absence of the spin-orbit scattering the plus and minus components do not mix with each other.It is only so that mixes the time reversed states.

III. USADEL EQUATIONS CLOSE TO T c
The Eilenberger equations can be considerably simplified when the mean free path for potential scattering lϭv F 1 is much shorter than the superconducting coherence length.This simplification appears because all the Green's functions corresponding to the different directions of the relative motion of the electron ͑different n's͒ get smeared out on the distances of the order of l, and since the characteristic scale of the Green's functions variations is we obtain that the spherical harmonics expansion is rapidly converging so that we can restrict ourselves to only the first two harmonics.
We will not derive the general case of the Usadel equations in the presence of the spin-orbit scattering but will only restrict ourselves to the temperatures very close to T c .This case is much simpler for investigation and contains all the new physics of introduction of the spin orbit scattering.For temperatures sufficiently close to T c we can take g Ϯ ϵ1 and ͑10͒ becomes We solve Eqs.͑15͒-͑17͒ for the case when the parameters vary only as a function of x.This corresponds to the ''sandwich'' geometry of interest here.
Let be the angle between n and x axis.From the full Legendre polynomial expansion of f Ϯ (,,v) ϭ ͚ lϭ0 ϱ f l (,x) P l (cos) by the reasons described earlier we take only the first two terms We integrate Eq. ͑15͒ over all 's directly and after being multiplied by cos to arrive at the following equations:

͑20͒
We are in the limit of a very dirty superconductor, when 1/ 1 ӷT c , h, 1/ so , so Eq.͑20͒ simplifies to Inserting into ͑19͒ we get where Dϭ 2 3 1 v F 2 .In the absence of the spin orbit scattering we recover the equations of Ref. 3.
We now want to check the self-consistency of our approximation, namely that the higher harmonics in the Legendre polynomial expansion are small.From Eq. ͑21͒ we see that f 1 / f 0 ϰl/L where L is the characteristic scale on which f Ϯ changes.Analogously, we can get that f (nϩ1)/ f (n) ϰl/L by multiplying Eq. ͑15͒ by cos n and integrating over all angles.In the superconductor Lϰ 0 and in the ferromagnet LϰͱD/h and in both cases l/LӶ1, so that our approximation is valid.

IV. BOUNDARY CONDITIONS FOR THE EILENBERGER AND USADEL EQUATIONS
The boundary conditions for the Eilenberger equations follow from the continuity conditions for the normal and anomalous Greens functions.Ivanov et al. showed 14 that at the sharp planar interface the Eilenberger functions are continuous along the flight trajectories on which the electrons can pass from one metal to the other and are equal on trajectories corresponding to the incident and reflected waves In Eq. ͑23͒ we assumed that xϭ0 is the boundary between two metals and in Eq. ͑24͒ xϭ0 is a perfectly reflecting boundary.
The Usadel functions are the isotropic part of the Eilenberger functions and obviously they should also satisfy the continuity condition on any boundary.Another condition on these functions comes from the requirement of the continuity of the current and reduces to the conservation of the quantity DNٌF where N is the density of states and D is the diffusion coefficient.
Interfacial scattering breaks the validity of the semiclassical approximation.Several authors have found effective boundary conditions that include the effect of ␦-function scattering. 15,16However, those do not appear to be applicable for our particular case.In Sec.VII we will describe qualitatively the effect of the interfacial scattering on the Josephson current in the SFS system.
In the sections to follow we assume for simplicity that the Fermi velocities in the two materials are the same.Whatever difference there may be in these velocities will lead to reflections at the interface, and can also be modeled by a ␦-function potential, as we will do in the subsequent article.
For now we will justify our assumption by noticing that there is little difference between the Fermi velocities of most conducting ferromagnets and superconductors, leading to a negligible reflection at the interface.

V. PROXIMITY EFFECT IN A FERROMAGNET
As described in the introduction an exchange field in the ferromagnet leads to the oscillations of the induced superconducting order parameter ͑see Fig. 3͒.In this section we show that the presence of spin-orbit scattering not only modifies the oscillation length but also leads to an extra de-FIG.3. Superconducting order parameter on the SF interface.cay of the order parameter and, that a critical strength of the spin orbit scattering completely suppresses the oscillations.Clearly the role of the spin-orbit scattering needs to be addressed in interpreting the experimental results on the S-F-S Josephson junctions and S-F bilayers.
We restrict ourselves to the case of dirty superconductors at temperatures close to T c , when Eqs. ͑22͒ are valid.In our model the superconductor is characterized by ⌬ which we take constant everywhere in the superconductor, which has no exchange field and no spin orbit scattering.In the ferromagnet, the BCS coupling is identically zero, so that ⌬ F ϭ0.However, the induced order parameter ͗ ↓ (x) ↑ (x)͘ is finite.
The Usadel equations in the superconductor are given by and in the ferromagnet

͑26͒
We need to solve these equations with the boundary conditions which correspond to continuity of the order parameter and the current.Another obvious condition is that at xϭϪϱ, in the bulk of the superconductor, the f Ϯ functions approach their equilibrium values of ⌬/.
Since hӷT c in most cases of interest, in Eq. ͑26͒ we can omit the term.As it turns out this way we are only losing the usual decay of the induced order parameter in the normal metal at the distances n ϭͱD/⌬, because we are interested in the effects of the exchange field, not in the conventional proximity decays.We look for the solution of Eq. ͑26͒ in the form and after substitution into ͑26͒, we get the eigenvalue equation for k ͩ The imaginary part of k M defines the oscillations.We can see from ͑30͒ that spin orbit scattering modifies the charac-teristic length of the oscillations and completely destroys them for 1/ so Ͼh.The analogous expression for the case of strong spin orbit scattering has been obtained by Ref. 17.
For future reference we find a complete solution of the Usadel equations in the ferromagnet and the superconductor, which means that we need to find the eigenvectors corresponding to each eigenvalue.
For the ferromagnet So, the general form of the solution in the ferromagnet is where ␣ϭ 1/ so ͱh 2 Ϫ1/ so 2 Ϫh .͑34͒ And for the superconductor and the solution is

VI. T c FOR AN S-F BILAYER
In this section we apply the general equations developed above to determine the superconducting transition temperature of the superconductor-ferromagnet bilayer ͑see Fig. 4 for geometry͒.The difference between our consideration and that by Ref. 2 is the inclusion of the spin-orbit scattering.
For this problem it is more convenient to work with rather than with f Ϯ .We have the following equations for F's.
In the superconductor, xϽ0: and we have a self-consistency equation ⌬(x) ϭ ͚ F ϩ (,x).
In the ferromagnet, xϾ0: And in the last two equations we also neglected in comparison with h which means that the thickness of the ferromagnetic layer should be smaller than ͱD/⌬.
Let us take solutions of F ϩ in the superconductor of the form This will satisfy Eq. ͑38͒ if we take For F Ϫ we take

͑44͒
Introducing ϭC 2F /C 1F , the boundary condition at the S-F interface xϭ0 becomes a set of two equations on and k s ͑since the equations are linear it is sufficient to require the continuity of FЈ-F at the interface͒ .

͑45͒
In order for our assumption of the separation of and x dependences in F ϩ to be valid we must have k s and correspondingly independent of .As can be easily seen from ͑45͒ this is only true when s k s ӷ F k M or s k s Ӷ F k M .We will consider the first case, and the second one can be done in a similar fashion.

͑49͒
The transition temperature for the bilayer may be found from

͑50͒
In Fig. 5 we show how the oscillations of T c are modified by the presence of the spin orbit scattering.
We can see that the oscillations of T c as a function of the ferromagnetic layer thickness found by Ref. 2 are considerably modified by the presence of the spin orbit scattering.

VII. JOSEPHSON EFFECT IN THE SFS SANDWICH
We assume that the phase of the left superconductor is Ϫ/2 and the phase of the right one is /2.Fermi velocity, electron masses, and all other parameters are taken equal for both superconductors and the ferromagnet ͑see Fig. 6 for geometry͒.
As we have shown earlier, in the presence of the spinorbit scattering f ϩ and f Ϫ are not eigenfunctions of the Usadel equations.It is more convenient therefore to use the basis e ˆ1ϭͩ i␣ 1 ͪ, e ˆ2ϭͩ 1 Ϫi␣ ͪ ͑51͒ that we have derived in Sec.V.In the bulk of the superconductor and now we can solve for e ˆ1 and e ˆ2 components separately.
Matching the values of the functions and the derivatives on the boundaries we get after simple calculations that the Josephson current near T c is given by We can again see that the spin orbit scattering considerably modifies the answer.Interfacial scattering at the two interfaces modifies this result.Using Bogolyubov-de Gennes equations as in Ref. 18  one can show that in the lowest order in transmission coefficients the right-hand side of Eq. ͑53͒ is multiplied by the product of the transmission coefficients, which is an intuitively clear result.We plan to present the details of these calculations elsewhere.
x ͒ϭC Ϫ ͑ ͒cos͓ͱ͉͉/D͑ xϩd s ͔͒.͑42͒Both ͑40͒ and ͑42͒ satisfy the condition F Ϯ Ј (xϭϪd s )ϭ0 So, for s k s ӷ F k M , ͑48͒we can write k M ϭ͉k M ͉e i .And if now we assume ͉k M ͉d M cosу1 and take into account s k s ӷ F k M expression ͑47͒ simplifies to Ϫ2͉k M ͉cosd M sin͑2͉k M ͉sin ͔͒.