Inverse proximity effect and influence of disorder on triplet supercurrents in strongly spinpolarized ferromagnets
Abstract
We discuss the Josephson effect in strongly spinpolarized ferromagnets where triplet correlations are induced by means of spinactive interface scattering, extending our earlier work [R. Grein et al., Phys. Rev. Lett. 102, 227005 (2009)] by including impurity scattering in the ferromagnetic bulk and the inverse proximity effect in a fully selfconsistent way. Our quasiclassical approach accounts for the differences of Fermi momenta and Fermi velocities between the two spin bands of the ferromagnet, and thereby overcomes an important shortcoming of previous work within the framework of Usadel theory. We show that nonmagnetic disorder in conjunction with spindependent Fermi velocities may induce a reversal of the spincurrent as a function of temperature.
pacs:
72.25.Mk,74.50.+r,73.63.b,85.25.CpI Introduction
Ferromagnetsuperconductor hybrid systems are currently subject to intense research activity, as they were conjectured to host triplet superconductivity induced by the proximity effect(1); A. I. Buzdin (2005); (3); (4); (5). While the first successful experiments on these structures found evidence for socalled triplet correlations(6); (7); (8), whose hallmark are oscillations A. I. Buzdin (2005) of the critical current in Josephson junction devices, the existence of equal spin triplet pairing is currently in the focus of attention(1); (5). So far, the main experimental prove of such triplet correlations is based on the Josephson effect and on phase coherent electron transport in proximity structures (9); (10); (11); (12); (13); (16); (14); (15). After two first confirmations of longrange triplet amplitudes in 2006, (9); (10) an impressive series of affirmative experimental results was published in 2010.(11); (12); (13); (14); (15) The systems under investigation varied largely in terms of materials and fabrication, the common idea being that a breaking of spinrotation symmetry around the bulk magnetization axis must somehow be enforced at the superconductor (SC)ferromagnet (FM) contacts  which is in line with the general theoretical concepts behind this effect(17); (1); (18); (19); (20); (21). In this article, we elaborate on our earlier work concerning longrange triplet supercurrents(22), where we used a recent extension of the quasiclassical Green function technique(23), that allows us to consider ferromagnets whose exchange splitting is of the same order of magnitude as the Fermi energy consistently within quasiclassical theory(22); (24). In the following we include additional relevant effects like the suppression of the superconducting energy gap in proximity to the interfaces, the induced magnetization in the superconductor close to the ferromagnet, and disorder in the FM bulk. These calculations are performed numerically and cover all ranges of junction length, interface transparency, and impurity scattering. In the appendix, we provide detailed discussions of the intricacies involved in solving the selfconsistency equation in quasiclassical theory with spinactive boundary conditions. In particular, we point out a previously not discussed technical problem with the self consistent solution of the order parameter that arises for quasiclassical transport equations with boundary conditions where the quasiclassical propagator undergoes a jump discontinuity during reflection from the interface.
With regard to the inverse proximity effect, we show that while the suppression of the gap function in the SCelectrodes is indeed substantial for highly transparent junctions, this does not imply that higher harmonic contributions to the currentphase relation (CPR) are necessarily suppressed. As expected, impurity scattering reduces the Josephson current and in particular higher harmonic contributions to the CPR. We also find that there may be a reversal of the Josephson spincurrent as a function of temperature if impurity scattering is sufficiently strong.
Ii Quasiclassical Theory of Superconductivity
Let us first summarize the theory for a system where the band structure is not spinpolarized. The quasiclassical Green function is obtained from Gor’kov’s Green function(26) by an integration over momentum(27); (28); (29); (30):
(1) 
where is a quasiparticle renormalization parameter, and . The “hat” denotes a 44 matrix in combined spin and NambuGor’kov space. The propagator depends on the spatial coordinate , time , Matsubarafrequency(31) and the momentum , which lies on the Fermisurface. The Fermi velocity corresponding to Fermi momentum is denoted by . The resulting equation of motion, which can be obtained from the Dysonequation of Gor’kov’s Green function, is the Eilenbergerequation(27):
(2) 
which must be supplemented by a normalization condition: . denotes offdiagonal selfenergies in particlehole space and the diagonal ones.
The equilibrium current density in a system without spin polarization reads:
(3) 
where Tr denotes a trace over spin and particlehole (NambuGor’kov) degrees of freedom, and and are defined by:
(4)  
(5) 
is the density of states per spin at the Fermi level. In appendix A we detail our numerical Fermisurface averaging procedure.
For a ferromagnet these expressions need to be modified in order to take into account the different Fermi surfaces for the two spin channels. As discussed in our earlier work(22); (24), we model the ferromagnet by two spinscalar quasiclassical Green functions for each band. This is consistent with the quasiclassical approximation in the limit and naturally results in a threechannel matching problem at the SC/FM interface. We underline that an equivalent approach in the diffusive limit of quasiclassical theory (Usadelequation (25)) is currently not available for lack of multichannel boundary conditions. All definitions given above remain the same apart from the reduced matrix structure of the Green function and the fact that , , and now depend on the spin band in question.
We will make extensively use of a fundamental symmetry relating particlelike and holelike quantities in quasiclassical theory, for which we introduce the operation defined by
(6) 
ii.1 Riccati parameterization
We use a particular Riccati parameterization of the quasiclassical (QC) Green function(32); (33); (34); (35); (23) which has proven very useful in the past. For Matsubara’s Green functions,(31) it consists of 2 parameters (“coherence functions”) , , describing particlehole coherence in the superconducting state. The Green function reads in terms of these parameters(23):
(7) 
where , , and correspond to and , respectively. The transport equations are:
(8)  
For a superconductor or normal metal the , , , , , and are 2x2 spin matrices for each Fermi surface point , as all Fermi surfaces can be regarded as doubly spin degenerate on the scale of the Fermi energy. A weak band splitting due to an external field or as in ferromagnetic alloys can be incorporated as source field in these equations, leading to spindependent self energies. In contrast, for a ferromagnet with strong spin splitting of the energy bands the transport equations hold separately for each spin band with corresponding Fermi surface point , and the abovementioned quantities are all scalars. The important difference lies in the integration over in equation (1), which destroys all coherence between quasiparticle excitations living on different spin bands for the latter case of strong spin polarization, however allows for quasiparticle coherence between spin bands when the band splitting is on the lowenergy scale that constitutes the phase space for quasiparticles.
The selfenergies introduced here are related to those we used in Eq. (2) by
(9) 
There is a further symmetry relating the Matsubara coherence functions with positive and negative frequencies:
(10) 
As a result, we only need to consider in the following. All other quantities can then be obtained from the symmetry relations. The current density in the FM spinbands () is obtained from
(11) 
with the component of and denotes a Fermisurface average which is taken over momenta with positive projection on the axis only. To obtain this expression, we did not only exploit the symmetry relations but also the fact that for the spin channels an additional symmetry between the diagonal components of holds: .
ii.2 Boundary conditions
The boundary conditions for Riccati amplitudes at interfaces and surfaces are formulated in terms of the normal state scattering matrix . (35); (23) The transport equations for Riccati amplitudes can be solved by integrating them along their characteristics, which are straight lines in real space(35). Every amplitude has a unique stable direction of integration. One may hence group them into “incoming” and “outgoing” amplitudes with respect to a scattering region, which will be the interfaces here. The convention is that outgoing amplitudes are denoted by capital case letters and incoming ones by small case letters. The transport channels that participate in scattering are labeled by . In our case, these will be band indices, as the conservation of parallel momentum that we assume excludes the scattering between different momenta in the same band. The boundary conditions then relate outgoing to incoming amplitudes. For numerical calculations, the most convenient way of solving these boundary conditions is to calculate(23):
(12) 
where denotes a matrix product in channel space and:
(13) 
is the normal state scattering matrix, which – thanks to the conservation of parallel momentum – is blockdiagonal here, and the blocks can be labeled by . The outgoing Riccati amplitudes are then obtained from:
(14) 
As explained at length in our earlier publications,(23); (24) the matrices will be either or , depending on whether is larger or smaller than the Fermi wave vector of the minority band in the FM. Thus, all matrices defined above will have these dimensions, as channels with different do not mix.
Iii Solution of the Eilenberger equation
iii.1 Impurities
The simplest way to consider impurity scattering is to look at elastic, nonmagnetic scattering in the selfconsistent Bornapproximation. This gives the following selfenergies in the ferromagnet(21):
(15) 
Note that here , denote all offdiagonal selfenergies in Nambuspace, rather than a superconducting gap. The Riccati transport equations then need to be solved numerically, and in each step the impurity selfenergy is updated.
The scattering time is related to the scattering length by . Since corresponds to the mean free path of the quasiparticles, it is directly related to the impurity concentration and should be the same for both spinbands (we consider here spininactive scattering). The Fermivelocities, however, are different and hence . This argument can also be made quantitative by noting that the impurity self energy in Born approximation is proportional to , with impurity concentration , impurity scattering potential , and density of states at the Fermi level . As this defines , one obtains for the two spin directions . As shown in appendix A, , and thus (assuming equal effective masses for the spin bands) , i.e. the mean free path for both spin bands is equal.
iii.2 Gapequation
The meanfield gapequation in the SC is
(16) 
where the coupling constant is eliminated in favor of . In appendix B we discuss in detail some sophisticated problems when iterating this equation at an interface. We show that whenever the quasiclassical Green function undergoes a jumpdiscontinuity under reflection from the interface, the only selfconsistent solution of the above equation at the interface is . The discontinuity of the quasiclassical propagator in turn results from the fact that the microscopic reflection process at the interface falls outside the range of applicability of QC theory. To remedy this issue, we introduce a length cutoff and calculate the gapequation only for distances larger than from the interface. We show that if the cutoff length is chosen small enough, neither its precise value nor the profile of the SCgap for distances smaller than influences the results. We choose for our numerical calculations .
iii.3 Numerical solution
We found that a very stable way of solving the Riccati differential equations for spinpolarized systems is to resort to analytical solutions for constant selfenergies. I.e., instead of using a standard solver for ordinary differential equations along each trajectory, we approximate the selfenergies by a constant in each interval that corresponds to the gridspacing of our discretization in the direction, and use the analytical solution for homogeneous selfenergies (23):
(17) 
where is the homogeneous bulk solution, is the initial value and parameterizes the respective trajectory as . Moreover, , and
(18) 
where is given by the solution of:
(19) 
The value of at grid point is then calculated from the analytical solution with the value at grid point as initial value. These general formulas simplify inside the FM, because all quantities are scalar and commute. In the SC they simplify as well, since we only consider the order parameter selfenergy there and thus always have and . The homogeneous bulk solution is for these cases
(20) 
where .
As we include the selfconsistent equations for the impurity potential and the order parameters, we have to deal with a nonlinear problem, in which the solutions for different trajectories and Matsubara frequencies are coupled. We solve this problem iteratively as as illustrated in Fig. 2 (see also appendix B.2).
At each step, we assume a given set of initial values for the coherence functions at the outer boundaries of the system and at the interfaces and given values for the impurity potential and order parameters. We then integrate the transport equations and solve the boundary conditions at the interfaces to update all coherence functions and calculate the initial values for the next step of the iteration. Subsequently, all selfenergies are updated by evaluating the respective selfconsistent equation with the new coherence functions. The initial values at the outer boundaries are always assumed to be the respective SC bulksolution. The orderparameter phase difference enters via these boundary values. At the initial step, we chose the coherence functions and the impurity potential inside the FM to be zero and the SC gap to be constant up to the interface.
Iv Scattering Matrix
The normalstate scattering matrices entering the quasiclassical boundary conditions cannot be obtained within QC theory itself. We calculate them from plane wave matching in the normalstate using a simple model for the interface scattering potential that captures all relevant effects.
The interface Hamiltonian reads:
(21) 
where is the interface exchange field, and:
(22) 
plays the role of a spinindependent interface potential. Accordingly, the normalstate Hamiltonians of the ferromagnet and the superconductor read:
(23)  
(24) 
Obviously, is invariant under rotations in spinspace, while both and break spinrotation symmetry. For we assume for simplicity a parabolic dispersion with the same effective mass in all cases. The dispersions and and are shifted by a constant energy and , respectively, compared to . The unitary matrix , which maps the spineigenbasis of the interface to that of the ferromagnet, reads:
(25) 
The scattering matrix is obtained by diagonalizing the Hamiltonians and inferring the respective values of at the Fermilevel for a given . The corresponding wavefunctions in the three layers are then matched at the SC/I and the I/FM interface respectively, choosing a spinquantization axis in the SC aligned with that of the interface, while at the I/FM interface, the rotation must be taken into account.
The requirement that must be conserved across the interface implies that even if the ferromagnet is not fully spinpolarized, some trajectories will have an evanescent solution in the minority band of the FM, corresponding to the halfmetallic case. In the case were this minority band solution is propagating we obtain a scattering matrix, if it is evanescent, the matrix is as discussed in appendix C.
V Results
Our scattering geometry is shown in in Fig. 3. In what follows, the Fermimomentum of the majority (spinup) band is assumed to be identical to the Fermimomentum of the SC. This is simply a matter of convenience, as we must only distinguish two and not three types of trajectories in this case. The third type of trajectories would describe total reflection from the interface, which does not contribute to transport. Our methods can easily applied to the general case as well. The value of the minority band Fermimomentum is determined by the exchange field , the magnitude of which is assumed to be the same in the interface and in the FM bulk (however not the direction).
In what follows we consider two parameter sets for the interface. A “high”transparency interface with thickness and and a “low”transparency interface with , and . If and are the submatrices of related to transmission from the SC to the spinup band of the FM and vice versa, then the relevant transmission quantity for inducing triplet correlations in the spinup band is:
(26) 
and analogously we have for the spindown band. These quantities are plotted in Fig. 4 for the two interfaces discussed here. Their functional dependence on was investigated extensively in Ref. (24).
In order to discuss various symmetry components of the induced triplet pair correlations, We use the following decomposition of the quasiclassical Green functions:
(27) 
and obtain the induced magnetization from
(28) 
As measures of oddfrequency and oddmomentum triplet amplitudes we define
(29)  
(30) 
Here, is the swave component of the oddfrequency correlations. does not correspond to a particular spherical harmonic. Our model system only breaks translational invariance in the direction and is rotationally invariant in the plane. We therefore project out the correlations which are odd under .
We first investigate the inverse proximity effect, as shown in Fig. 5. Spinactive scattering at the SC/FM interface induces tripletpairing correlations and a magnetization inside the SCelectrodes close to the interface. Moreover, the order parameter is suppressed at the interface. While this is also caused by spinactive scattering, here the predominant cause for this suppression is the transparency of the interface. This can be seen in the topmost plots of Fig. 5 (left), where we compare a highly transparent interface to an interface in the tunneling limit. The reason why the spinmixing effect plays a minor role here, is that under the assumption of a box shaped scattering potential the magnitude of the interface spinmixing parameters that are responsible for creating tripletpair correlations are typically small (see our extensive discussion of this effect in Ref. (24)). This can be seen in the top right panel, where the absolute value of the induced magnetization is shown. The transparency is largely irrelevant here, as the inverse proximity effect is due to spinactive backscattering from the superconducting side, i.e. a property of the spindependent scattering phases in the reflection parts of the scattering matrix. The induced magnetization decays on the scale of the superconducting coherence length, . Note that this induced magnetization is a result of the superconducting inverse proximity effect (production of triplet pair correlations by the magnetic interface) and absent in the normal state.
In the lower panels of Fig. 5 we project our various symmetry components of the triplet pair correlations. Since the SC itself is spinrotation invariant, it makes sense to plot the invariant quantity rather than the individual components of , which are not invariant. Since we consider in Fig. 5 a rather clean superconductor (), oddmomentum and oddfrequency correlations are comparable in magnitude and decay on the same length scale . In the case of the high transparency interface, the induced tripletcorrelations and the magnetization show a slight dependence on the length of the junction (not shown), i.e. superconducting correlations that propagated through the FMlayer also contribute to the inverse proximity effect in that case.
We now study the influence of impurity scattering on the triplet pair correlations. In Fig. 6, we plot both the spatial profiles of the absolute value of the total momentum averaged equalspin triplet correlations, (swave, odd frequency) and of the correlations which are odd under , (odd momentum), for each spin direction inside the FMslab. As the impurity concentration increases, these correlations decay faster into the FM and the relative magnitude of decreases. However, the magnitude of the latter only becomes negligible for very high impurity concentrations. This shows that the Usadel approximation does require very short scattering lengths in order to be able to neglect oddparity correlations in the ferromagnet. Right next to the interfaces, to suppress to about 1/10 of requires a scattering length as short as . One needs to carefully estimate if this is then already on the interatomic length scale, a length scale outside the scope of quasiclassical theory (both EilenbergerLarkinOvchinnikov and Usadel theory), in which case the boundary layer must be treated microscopically. One option is to incorporate this region as an ’isotropization’ zone in effective boundary conditions for the Usadel equation.
We next discuss the Josephson effect in the structure shown in Fig. 1. With regard to the CPR, we first observe that higher harmonics, i.e. a nonsinusoidal CPR, can still be present in the hightransparency case, even if the suppression of pairingcorrelations in the SC is properly taken into account. To show this, we consider a junction with high transparency interfaces in Fig. 7. The CPR is calculated as a function of the impurity concentration. Decreasing the scattering length leads to a suppression of the critical current, but also of higher harmonics contributions, as can be seen in the normalized plots on the righthand side.
In Fig. 8, we consider a junction in the tunneling limit (, ) and calculate the critical current as a function of temperature. For a clean junction, , we find a strong spinpolarization of the current. For higher impurity concentrations, the current is not only suppressed globally, but the spin current, , also decreases relative to the total current. At high impurity concentrations, there may even be a crossover to a negative spincurrent at small temperatures. This can be understood as follows. From the cleanlimit solution, we see that the loss of particlehole coherence due to temperature penalizes the spindown current more than the spinup current. The mean free path due to impurity scattering is, however, the same for both bands, and the cumulative effect of the scattering only depends on the length of the trajectory. This can easily be understood by rewriting the transport equation as:
(31) 
Since all selfenergies are proportional to (the superconducting gap is zero in the FM), the factor cancels on the right hand side of this equation, which describes impurity scattering, and only remains in – which is the term that can be interpreted as the suppression due to finite temperatures, and which clearly is seen to be spindependent.
Vi Conclusions
We carried out a comprehensive study of the tripletJosephson effect and inverse proximity effect in SC/FM/SC junctions with highly spinpolarized FMlayers taking into account impurity scattering inside the FM in the selfconsistent Bornapproximation. We showed that oddmomentum triplet correlations are generically present in these junctions and are only suppressed at very high impurity concentrations. We found an induced magnetization in the SCelectrodes resulting from the inverse proximity effect and a suppression of the SCgap which is mainly related to the transparency of the interface for the interface model we considered here. Larger spinmixing phases would, however, also result in a substantial suppression of the gap close to the interface (21). These can be realized by considering a smooth scattering potential at the interface. Even if this suppression is taken into account, the CPR of hightransparency junctions may have higherharmonic contributions. Regarding the critical current, we see that impurity scattering suppresses the Josephson spincurrent and may even lead to an inversion of it at small temperatures and short scattering lengths.
Acknowledgements.
RG acknowledges financial support from the Karlsruhe House of Young Scientists. TL acknowledges support from the Swedish Research Council (VR). ME acknowledges support from EPSRC under grant reference EP/J010618/1.Appendix A Expressions for the Fermisurface average
For the spherical bands that we assume, the density of states at the Fermilevel is:
(32) 
The FSaverage for the selfenergies is calculated from:
(33) 
with and . And the FSaverage for the current:
(34) 
The number of points we use in our calculation (one hundred) is to small to make these expressions converge numerically. Thus, we use a numerical normalization factor which is calculated from Eq. (33) as . This ensures that all our FSaverages are properly normalized. To be precise, the numerical integration is implemented as a trapezoidal rule over one hundred points which are equidistant in .
Appendix B Numerical iteration of gap and boundary conditions
b.1 Discretization of the gapprofile
The numerical solver we use is not a standard RungeKutta scheme but relies on the analytical solution for constant selfenergies. The only numerical approximation made is assuming a stepwise variation of the selfenergies between gridpoints. Naturally, the question arises how this stepwise profile is to be interpolated when knowing the selfenergies at a given number of gridpoints. A subtlety here is that the coupled system of transport and selfconsistency equations becomes fundamentally inconsistent if the condition , where denotes the th gridpoint, is violated. This is because the asymptotic solution of the transport equation approaches the local homogeneous bulksolution as the Matsubara frequency goes to infinity. As we will see in appendix B.3, this is crucial for formulating the gapequation in a cutoff independent way. The interpolated profile we use is illustrated in Fig. 9 (left).
b.2 Iterative scheme
We assume that at the beginning of the iteration step, the incoming solutions at all interfaces and the selfenergies are known from the previous step. The procedure works as follows:

Solve the boundary conditions at all interfaces.

Integrate all outgoing coherence functions up to the next interface or outer boundary.

Update all selfenergies. We use the secant method to accelerate convergence.

Update all incoming coherence functions.
We use two abortion criteria, since the iteration in the SCs may converge a lot faster than in the central FM layer if the transparency of the interfaces is small. If the gapprofile is wellconverged, the iteration in the SC is stopped and continues only inside the FM.
b.3 Elimination of the highenergy cutoff
Generally, technical energy cutoffs that appear in expressions for selfenergies or observables in quasiclassical theory must be eliminated in favor of phenomenological parameters. This is usually achieved by adding and subtracting terms in Matsubara sums or energy integrals that cancel the divergent behavior of these expressions and allow to formally extend the sum or integral to infinity (29). The other half of such an “added zero” can typically be included in a phenomenological (measurable) parameter. While this procedure is standard for the weakcoupling gap equation in quasiclassical theory, it turns out to be problematic at interfaces. We are not aware of any discussion of this subtle issue in the literature before. We here discuss a method for dealing with this issue. We start from the selfconsistency equation:
(35) 
In what follows, we limit the discussion to singlet, swave superconductivity, i.e. , but the argumentation can easily be generalized. The interaction constant can be eliminated by linearizing the above equation close to (). Since the interaction constant is a bulkproperty, it is justified to consider the homogeneous bulk case, i.e.:
(36) 
This equation also shows why a technical cutoff must be included in (35), as the Matsubara sum diverges. This problem does not arise in the microscopic theory, where the asymptotic behavior of is given by .(36) The linearized equation reads:
(37) 
Given that , we then obtain the wellknown standard result in terms of the digammafunction ,
(38) 
At this point, one can see that keeping a technical cutoff is, at least in the Matsubara case, not tenable. should of course be an integer number. Obviously one can choose so that is integer here, but must be the same for all temperatures and since the spacing of Matsubara frequencies is temperature dependent, it will generally not be commensurate with . One can verify numerically that if such a cutoff is kept in , the gapprofile one finds is the BCSrelation with a jigsawpattern superimposed to it, which vanishes as . The standard way of dealing with this problem is to regularize the Matsubara sum. This is achieved by subtracting the asymptotic form of on both sides of (35):
(39)  
Both sides of this equation can now be regularized, if the expression (38) for is taken into account. The cutoff can be taken to infinity, which yields
(40) 
For the boundary value problem one may object that the counter term was adapted to the homogeneous bulkcase, but expanding the solution of the Riccatiequations in orders of , one finds (33):
(41)  
and therefore:
(42)  
Thus, even if the gapprofile is inhomogeneous, the asymptotic behavior remains the same as long as the gap is a smoothly varying function.
Up to this point, the procedure is wellknown and established. A problem does, however, arise at interfaces where the boundary conditions leads to a discontinuous behavior of the coherence functions under reflection. To see this, we consider the coherence function to leading order in . The incoming coherence function at the interface is then given by . To illustrate the point we here assume that the solution of the boundary conditions at the interface is:
(43) 
where denotes the reflection matrix on the SCside of the interface. This is not fully generic, since we assume that the transmission from the other side of the interface can be neglected. However, for the case we consider here this is reasonable as the coherence functions on the FM side are exponentially suppressed by propagation through the FM layer and can thus be neglected at high energies. Now it is easy to see that the asymptotic behavior of at the interface is given by:
(44) 
Here denotes an FSaverage over trajectories pointing towards/away from the interface. It is obvious that this function will only have the required asymptotic behavior if or . This implies that the righthand side of equation (40) will diverge if that is not the case. Adapting the cancellation term will not help either, since the coupling constant is fixed by the bulkbehavior, which implies that the lefthand side of (40) would diverge in that case. So eventually, one arrives at the conclusion that the only consistent solution for is . The problem here is the discontinuous jump of the function upon reflection at the interface, which is directly related to the fact that the interface region cannot be described within quasiclassical theory. While the use of effective boundary conditions is sufficient to circumvent this problem if the selfconsistency relation is neglected, one must go a step further if it is retained. Our solution to this problem is to assume that the selfconsistency equation must not be calculated if the distance to the interface is smaller than a length cutoff . To see how this resolves our issue, we have to calculate and . Here, is the incoming coherence function and obtained by integrating the Riccati equation from some initial value inside the bulk. Since the gapprofile is smooth for , this implies that it does have the correct asymptotic behavior at high energies. originates from the interface. If we assume that the gap has a constant value in some environment of , where can be arbitrarily small, the solution for will be given by (23)
(45) 
where and . This solution has the correct asymptotic behavior, irrespective of how the gapprofile varies for , as it is given by the homogeneous solution at sufficiently high energies. We found that the sum in (40) must be, however, calculated up to very high energies, since should be very short. Thus, we need to discuss highenergy contributions next.
b.4 Highenergy contribution
To numerically compute (40), we consider two numerical energy cutoffs on the righthand side:
(46)  
Up to we calculate the full coherence functions in the whole structure without any approximations. The highenergy contribution is only calculated in the SCelectrodes. To efficiently calculate this contribution, we consider the linearized Riccatiequation:
(47) 
which is easily solved exactly by a variation of constants:
(48) 
Again assuming that the gapprofile is constant, this yields:
(49) 
We also assume that the boundary conditions reduce to (43). It is easy to convince oneself that the linearized Riccatiequation is exact up to second order in . Up to second order, we also have:
(50) 
from which we calculate the contribution to the gap function.
As discussed above, the need for taking into account this highenergy correction factor arises from interface scattering. It is therefore not surprising that it vanishes rapidly inside the SCbulk. This fortunately implies that we must only calculate it very close to the interface.
b.5 Extension to zero temperature
For , we have
(51)  
(52) 
The zerotemperature Green function is thus obtained by replacing and all Matsubara sums must be replaced with the above integral. Care must be taken in regard to the normalization factor in the selfconsistency equation:
(53) 
with , since both terms in this sum become infinite. For , the second term approaches , and we have:
(54) 
Appendix C Calculation of the matrix
c.1 HMcase
First, we need the wavevectors which are obtained from solving the onedimensional Schrödingerequation for given , . These are and for , for , and and for , where is associated to the evanescent mode. These wavevectors may be real or imaginary, corresponding to either a conducting or an insulating interface. Moreover, we have the Fermivelocities and in the SC and the majority band of the FM, respectively, which are proportional to and respectively. After defining the auxiliary quantities
(55)  
(56)  
(57)  
(58) 
where is the thickness of the interface layer, and the matrices:
(59) 
where is the spinrotation from the spinbasis of the interface to that of the FMbulk, we obtain:
(60) 
with
(61)  
(62) 
c.2 FM case
In the FM case, the minority band of the FM has a propagating mode. We write and for the wavevectors of the planewaves in the FMbands and , for the corresponding group velocities. We use the same definitions for , and as above, but redefine:
(63)  