Induced Triplet Pairing in clean s-wave Superconductor/Ferromagnet layered structures
We study induced triplet pairing correlations in clean ferromagnet/superconductor/ferromagnet heterostructures. The pairing state in the superconductor is the conventional singlet s-wave, and the angle between the magnetizations of the two ferromagnetic layers is arbitrary. We use a numerical fully self-consistent solution of the microscopic equations and obtain the time-dependent triplet correlations via the Heisenberg equations of motion. We find that in addition to the usual singlet correlations, triplet correlations, odd in time as required by the Pauli principle, are induced in both the ferromagnets and the superconductor. These time-dependent correlations are largest at times of order of the inverse of the Debye cutoff frequency, , and we find that within that time scale they are often spatially very long ranged. We discuss the behavior of the characteristic penetration lengths that describe these triplet correlations. We also find that the ferromagnets can locally magnetize the superconductor near the interface, and that the local magnetization then undergoes strongly damped oscillations. The local density of states exhibits a variety of energy signatures, which we discuss, as a function of ferromagnetic strength and .
pacs:74.45.+c, 74.25.Bt, 74.78.Fk
Triplet Cooper pairing is no new phenomenon: it has long been recognized to be responsible for superfluiditysfhe () in as well as for superconductivity in some electronic materials. This occurs when the pairing interaction is in a partial wave with odd . However, recent observations have raised the possibility of induced triplet pairing correlations in s-wave superconductors. It is a matter of elementary physics that the Cooper pair wavefunction must be antisymmetric under exchange of the two electrons to satisfy the Pauli principle. For spatially symmetric s-wave superconductors with decoupled spatial and spin degrees of freedom, the spin singlet pair is the only possible antisymmetric state of an electron pair. Triplet pairing states on the other hand, where the spin state is symmetric, are obviously allowed when the pairing is spatially antisymmetric, such as in p-wave superconductors. Triplet states in systems with s-wave pairing, even in momentum or coordinate space, would naively appear to violate the Pauli principle. However, many years ago Berezinskii proposedberez () a triplet state in superfluid , which involved spatially symmetric correlations. Berezinskii’s triplet pairing correlations, involving different-time pairing, did not violate the Pauli principle by virtue of being odd in time, thus allowing a triplet state in a system with s-wave interactions. While such a state did not turn out to be appropriate to describe superfluidity in , its consideration has led the way to the study of cases where some sort of time-reversal symmetry breaking mechanism may allow an odd time triplet state to be induced in systems with spatially symmetric interactions.
Interest in exotic triplet pairing arises from many quarters. In the case of two component cold atomic gasesbulgac () with short-range s-wave interactions, in which the two species have the same mass but different chemical potentials, it may be possible to induce a triplet pairing that breaks time reversal symmetry. In electronic materials, an important issue is the possible existence of a long range proximity effect in Superconductor/Ferromagnet (SF) heterostructures. Interest in these heterostructures arises in turn from their possiblezutic () applications. Of particular interest is that the thermodynamic and transport properties of FSF trilayers are found to depend strongly on the relative orientation of the magnetization in the two F layers.gu (); moraru (); bell (); visani () This rather well-understoodhv72 (); dg66 (); tagirov (); buzdin99 () fact makes these structures candidates as spin valves.
There have been no unambiguous observations of induced triplet correlations in SF structures involving s-wave superconductors. However, there have been some enticing experimental hints in the form of long ranged proximity induced superconducting behavior in SF multilayers with strong exchange fields. The observed effects are over length scales much larger than those of the usual SF proximity effect and more like the much longer length scales associated with the standard proximity effect between a superconductor and a normal non-magnetic metal. These observations include measurements in superlatticespena () with ferromagnetic spacers and SQUIDsnelson () with ferromagnetic interlayers. Superconducting characteristics, such as a critical temperature and enhanced sub-gap conductance, have been observed in point contact conductance measurements on s-wave superconductor/half metallic systems.kriv () Perhaps most compelling is the observation of a Josephson current through a strong almost half-metallic, ferromagnet.klapwijk06 () All of these experiments indicate long range superconducting correlations that are not destroyed by a strong exchange field: a triplet state would obviously be consistent with these experiments. To fully understand the behavior of these systems, further studies involving, for example, quantities sensitive to the gap such as the local density of states (local DOS), are needed. Until both better theoretical models and more varied experimental observations are made available, one cannot conclusively say that there is indeed an induced triplet state in these systems, but at this point the facts fit this explanation and no better one has been proposed.berg05 ()
Many theoretical studies agree that it is possible to induce this exotic state in certain SF systems berg05 (); hbv (); berg01 (); berg03 (); eschrig05a (); fominov07 (); braude (); houzet07 (); fominov (); eschrig05b (); buzdin05 (); asano (); yokoyama () (perhaps even in the nonmagnetictanaka () case) with ordinary singlet pairing in S. Some studies use an arrangement in which the F and layers have different magnetization orientations. Others assume that a domain structure in a single F layer is responsible for the symmetry breaking. Yet others assume an FSF system with different in-plane magnetization orientations in the F layers. Whatever the mechanism for the symmetry breaking, such arrangements can induce, via proximity effects, triplet correlations of different kinds. Recently it was shown that a Josephson supercurrent can exist in a half metal by virtue of equal spin triplet pairs and spin flip scattering events at the interfaces. asano () To understand and probe the underlying triplet state, investigations have been done on conductance spectra in simpler FS structureslinder () with arbitrary magnetization alignment and with spin active interfaces, as well as in diffusiveyokoyama () SF junctions, through characterizing the possible superconductor symmetry classes consistent with Pauli’s principle. With the exception of our early workhbv () on SFS trilayers and some recent work on SFS Josephson junctions,asano () the studies above are done in the dirty limit through linearized Usadel-type or other quasiclassical equations. The disadvantage of a quasiclassical approach is that it is unsuitable for magnets with an exchange field on the order of the Fermi energy. Thus, it cannot properly model a strong ferromagnet, and it does not allow for atomic scale oscillations in the pair amplitude. A good quantitative explanation requires a self-consistent treatment of a fully microscopic model. Thus, as pointed out in a recent review,buzdin05 () the very existence of triplet correlations in clean FS structures was until very recently generally doubted, and these doubts have only very recentlyhbv () been dispelled.
In this paper we explore the phenomenon of induced triplet correlations, odd in time, of clean FSF structures, where S is an ordinary s-wave superconductor and the magnetizations in the two F layers are rotated by an arbitrary angle . We assume strong ferromagnets (up to the half-metallic limit), and smooth, sharp interfaces. In this geometry, triplet correlations with total spin projection on the axis of quantization of the Cooper pairs are in general possible and, when the relative magnetizations (which we assume as usual are both parallel to the interfaces) are not aligned, triplet components with are allowed also. To satisfy the Pauli principle, these spatially symmetric triplet pairing correlations must be odd in frequency or time.berez () That such correlations are allowed, does not mean that they must exist, nor that they must exist over an extended spatial range. We find, however, via a fully self-consistent solution to the microscopic Bogoliubov de-Gennesbdg () (BdG) equations that such correlations do indeed exist, and that the penetration depth associated with them can be very long. Our use of the BdG equations allows us to study strong ferromagnets. Self-consistency is fundamental: non-self-consistent solutions are found to violate the Pauli principle. Thus, the time consuming step of calculating fully self-consistent solutions is necessary to properly model the proximity effects which allow for the mixing of superconducting and ferromagnetic orderings that causes these induced correlations.
In Sec. II of this paper, we discuss the basic equations and our method for numerical self-consistent solution. There, the extraction of the all-important time dependence via solution of the Heisenberg equations of motion for the relevant operators is explained in detail. Expressions for all of the time-dependent triplet correlations are also derived. The equations for the local density of states (local DOS) and the local magnetic moment (which we use to discuss the reverse proximity effect, that is, the penetration of the magnetism into the superconductor) are also presented. In the next section (Sec. III) we begin by presenting an extensive discussion of the triplet correlations as a function of position, , and magnet strength. The appropriate penetration depths are extracted and discussed. Results for the DOS, the magnetic moment and the temperature dependence of both triplet and ordinary singlet correlations are also given. Finally, in Sec. IV, a brief conclusion and summary is given.
The geometry we consider consists of a planar FSF junction as depicted in Fig. 1. The thickness of the superconducting layer is and the F layers have thicknesses and . The system is assumed to be infinite in the plane perpendicular to the layers, which we label as our plane. The magnetizations of the F layers, which are in this plane, form an angle with the axis, which is that of the direction of quantization of the spins.
Our starting point is the Bogoliubov de-Gennes (BdG) equationsbdg () for the system under consideration. The derivation of the BdG equations for the case of interest requires some care with the conventions for all operator phase factors, which are not universally agreed upon in the literature, and which may give rise to different signs in some of the equations below. We write the effective BCS Hamiltonian, , as
where , are the set of Pauli matrices, spin is denoted by Greek indices, and as usual, we represent the magnetism of the F layers by an effective exchange Stoner energy which will in general have components in both the transverse ( directions. The spin independent scattering potential is denoted , and is the usual pair potential.
To diagonalize the effective Hamiltonian, the field operators and are expanded by means of a Bogoliubov transformation, which, for our phase convention, we writesong () as:
where and are the quasiparticle and quasihole amplitudes, and and are the Bogoliubov quasiparticle annihilation and creation operators, respectively.
We require that the transformations in Eqs. (2) diagonalize ,
One can also take the commutator . With the magnetizations in the plane as explained above, this gives the following,
where the single particle Hamiltonian is defined as,
A plane wave factor has been canceled in both sides of Eq. (5). The longitudinal momentum operator, , is given by, , is the kinetic energy of the transverse modes, is the self-consistent pair potential, and is a scalar potential representing interface scattering characterized by a delta function of strength . The ferromagnetic exchange field , vanishes in the S layers. We have and in the layer, where is the magnitude of the exchange field, while in , , and . We refer to Fig. 1 for details. The dimensionless parameter conveniently characterizes the strength of the magnetism. One thus has in the half metallic limit. If we take or , i.e. the magnetizations of both layers lie along the same direction, or if there is only one F layer, then and we recover the simpler form of the BdG equations usedhv69 (); hv72 () in other contexts. We can find the quasiparticle amplitudes on a different quantization axis in the plane forming an angle with , by performing a spin rotation with,
where we have introduced as a set of Pauli-like matrices in particle-hole space. It is convenient to use the matrices in conjunction with the ordinary Pauli matrices in spin space to rewrite Eqs. (5) in the more compact but perhaps less transparent way:
where , with the superindex denoting transposition.
The usual self consistency condition relates the spectrum obtained from Eq. (5) to the inhomogeneous pair potential by an appropriate sum over states:
where the prime on the sum indicates that only those positive energy states with energy less than the pairing interaction energy cutoff, , are included, and is the temperature. The function vanishes in the F layers while in the S layers it takes the value of the usual BCS singlet coupling constant in the S material.
where , and , the transpose of
contains the expansion coefficients associated with the set of orthonormal basis functions. We write , and for . The necessary matrix elements analogous to Eqn. (5) for different junction geometries and for strictly collinear magnetization orientations have been calculated in previous work.hv69 (); hv70 () The situation is more complicated for the case of a FSF trilayer considered here, with the magnetization angles of the two F layers forming an angle . The matrix elements are then written as,
where is a convenient dimensionless measure of interfacial scattering. We have also defined:
We now consider the appropriate quantities that characterize the induced triplet correlations. To do this, we define the following triplet pair amplitude functions in terms of the field operators,
We will later demonstrate that these amplitudes vanish at , as required by the Pauli principle.
To make use of these expressions, it is most convenient to use the Heisenberg picture. Thus we write in the Heisenberg representation:
where . The spatial dependence of the complex quantities and is, in our geometry, on the coordinate only. They vanish identically at .
We will focus on in our study of the induced triplet correlations on the time dependent quantities and . Their existence at is allowed by the Pauli principle. It is also important to sort out when it is allowed by the spin symmetries: when the axis of quantization of the Cooper pairs is the only axis of quantization in the system (i.e., when ) then it is not hard to see that the total spin operator of the Cooper pairs does not commute with the Hamiltonian. This is best seen directly from the matrix expression on the left side of Eqn. (5). On the other hand, and the Hamiltonian do commute in this case. However, when (and therefore ) is nonzero, then no component of commutes with the effective Hamiltonian. From this spin symmetry argument it follows that the induced amplitude may exist (at finite times) only at nonzero , while is allowed ant any . For , when the magnetizations are antiparallel and along the axis, no triplet amplitudes with nonzero component along that axis can exist. The matrix in Eq. (7) can be used to verify this by performing the corresponding spin rotations. That the existence of certain quantities is consistent with all symmetry properties does not mean that these quantities will indeed be nonvanishing, and it certainly tells us nothing about the possible range and behavior in space and time of these amplitudes. To determine this requires detailed calculations.
Also of considerable interest in F/S structures is the reverse proximity effect: the leakage of magnetism out of the magnets and into the superconductor. This can be characterized by the local magnetization . It is defined as,
where is the Bohr magneton. The vector has two components in the FSF geometry discussed. Both components depend on . They are:
It is convenient to normalize these components to , where , and .
The proximity effects can also be examined through the local DOS, , given by,
where . We will be concerned mainly with the DOS normalized to the DOS of a bulk (unpolarized) normal metal, .
In this section we present our results, obtained self consistently as explained above and in previoushv69 (); hv70 (); hbv () work. We have assumed a coherence length . We will choose a geometry in which the layers are relatively thick: (that is, two coherence lengths) and . These values ensure that the sample will be overall superconducting at temperatures up to about of the transition temperature , of a pure S bulk sample. This was not the case for the smaller values used in Ref. hbv, where the condensation energy was quite small (see e.g. figure 6 in Ref. hv70, ). This allows us to study the temperature dependence of the quantities involved over a broad range. The most important parameters are the angle and the magnet strength . We will vary in its full range between 0 and and give results for the values of of 0.25, 0.5, and unity. No triplet amplitudes arise at (when magnetism is absent). In the results presented we have , when proximity effects are in general maximized.
In Fig. 2 we present comprehensive results for the real parts of and , which we denote simply as and respectively. These are plotted in terms of the dimensionless variable . The amplitudes are normalized to the value of the usual singlet amplitude in a pure bulk S sample. The temperature is set to zero in this figure. In the main plots, half of the S region and a portion (three fifths) of the left () region are included. The corresponding portion on the side can be inferred from the geometry and symmetry considerations. Results are plotted at three values of and at a number of finite times between 0.4 and 8 as indicated in the legends. We have verified that at the computed triplet amplitudes vanish identically, in agreement with the Pauli principle. This is true, however, only when the calculation is performed to self-consistency: non-self consistent results invariably violate the Pauli principle near the interface. The results for are given at an angle while those for are at . At , vanishes identically since the component of the total spin is then a good quantum number. At , and short time scales, the spatial dependences of the two triplet components coincide, albeit with different signs in the two magnet regions, due to the magnetization vectors having equal projections on the and axes. At longer times, when the triplet amplitudes extend throughout the S layer and couple the two magnets, and deviate from one another. The insets in each panel amplify and clarify the region near the interfaces.
On the F side, both amplitudes peak very near the interface and then decay in an oscillatory manner, reminiscent of the behavior of the usual pair amplitude. Although the height of the first peak does not depend strongly on , the subsequent decay in the F material is faster for larger values of . This can be attributed to a decreased overall proximity effect: here we have assumed that at there would be no mismatch between the Fermi surface wavevectors of the two materials, implying that as increases the mismatch between either the up or the down Fermi wavevectors and , on the F side, and that in the S side increases. The location of this first peak depends very clearly on , its distance to the interface decreasing as approximately consistent with the general rule that the oscillatory spatial dependences on the F side are determined by the inverse of . The height of the first peak depends strongly on time and is maximum at times of about . It is quite obvious that at intermediate values of the penetration of the triplet correlation into the F material is rather long ranged.
On the superconducting side the behavior is quite different: the triplet correlations penetrate into the S material over a distance that rather quickly reaches two correlation lengths and then of course saturates at the sample size, without signs of decaying in time at these length scales. Furthermore this effect now increases sharply with and is maximal in the half metallic case. Thus, the magnets act as sources, so to speak, of triplet correlations that enter the S material and this effect is stronger when is larger.
It is instructive to extract characteristic penetration lengths from the above data using the definition,
where the integration is either over the S or the F region. In Fig. 3, the top two panels show the penetration lengths for the F material, at three values of and the same values of as for Fig. 2. The results are very similar whether they are calculated from the results for or from those for . The penetration length at constant time decreases with as already noted and shows signs of saturating with time at a value which for approaches that of the superconducting coherence length. On the S side (bottom panels) the situation is very different: the results for and are now clearly dissimilar with the penetration length for the former quantity being (for cases shown here) the larger one. This arises from the geometry and magnetization projections of each F layer on the axis, which are in opposite directions, forcing the triplet to possess a node at the center of the trilayer. No such requirement exists for , as it is spatially symmetric. Except for the case of at , we see no sign of saturation. In fact, the maximum value of displayed here corresponds to the case in which the entire intrinsically singlet superconductor layer, two coherence lengths thick and sandwiched between two magnets, is wholly pervaded by induced triplet correlations.
It is also of interest to consider the variation of the spatial maximum values of and with time. In Fig. 4 we show, for each time, the largest value of these quantities, which typically is attained near the interface, in either the F or S regions. By “maximum” value we mean the maximum of and , not to be confused with the absolute value of the complex quantities or . In this figure, the magnets are half metallic, . In the top panel, we plot the results for at both and . We see that at earlier times, the value at its peak just inside the F region (see Fig. 2, bottom left panel) exceeds the maximum value of this quantity in S. At longer times, however, there is a crossover as the size of the peaks in F decreases rather sharply, as explained above, while the size of the amplitude in S decreases only slowly, as the triplet correlations fill the S layer. It is apparent from careful examination of the panels in Fig. 2, that this crossover does not occur for smaller values of except possibly on a time scale much longer than that considered here. In the bottom panel, we present a similar study of , this time of course only at since this quantity vanishes identically for collinear magnetizations. The results are clearly very similar except that the results in the F region appear to saturate and do not decrease at long times. This is consistent with the earlier discussion where we saw that and overlap at , except at sufficiently long times. In all cases the maximum value of the quantity plotted crests near in agreement with previous remarks.
All of the above results have been given in terms of the real parts, and of the complex amplitudes and . The behavior of the corresponding imaginary parts is qualitatively very similar and thus we will present only two examples, in Fig. 5. We denote these imaginary parts by and respectively. In Fig. 5 we consider only the case (compare with Fig. 2) and . As in Fig. 2 the main plots include an extended region near one of the interfaces and the insets are close views of the interface itself, in this case one of the insets shows a more detailed view of the S side. On the F side, the behavior is reminiscent to that of the real parts, except that the very prominent peak seen in the real parts right at the interface is absent for the imaginary parts. On the S side, the sign is now initially negative and it changes to positive at of order unity. No such change was observed for the real parts. At longer times, the imaginary part of the triplet correlations also eventually penetrates several correlation lengths into the S sample, just as the real part does.
In the next two figures, we explore the reverse proximity effect (the spreading of the magnetism into the S layer) as a function of . This is best done by considering separately the two components of the local magnetic moment vector. First, in Fig. 6 we consider the component (see Eq. (21)) normalized to the absolute value of its bulk value in a pure F material. The results in this figure are for half metallic magnets, (). In the main plot of the figure we display the value of in the region very near an interface for several values of . Of course, vanishes at . At other values of it is large in the F material and it penetrates into S in an oscillatory way that is quite reminiscent of the corresponding penetration of the superconducting correlations into S. We see that the period of the spatial oscillations of the magnetization is independent of the angle between the two F layer magnetizations. Another discernible feature is that dampens out over relatively short length scales, consistent with past work.hv69 () The inset shows the overall behavior of in the entire sample, demonstrating also the opposite signs between in the two magnets in accordance with Fig. 1.
Similarly, in Fig. 7 we display the -component of (see Eq. (20)), normalized in the same way as , and for the same values of but including now three different values of . Again, the main plots display the behavior near the interface while the insets are for the entire sample. One can see here that the reverse proximity effect is very weak at small and largest in the half metallic case. The magnetic moment oscillates in the superconductor with a period that is independent of the direction and magnitude of the mutual magnetization in the F layers. The observed trends in hold also for the component.
We next study the energy dependence of the single particle quasiparticle spectrum by considering the local density of states (local DOS), , as defined in Eqn. (22). In Fig. 8 we consider the local DOS, integrated either over either the entire S or the entire F region, and normalized to its bulk value on a sample of the S material in its normal (non-superconducting) state. The results are displayed for three values of and several values of . The results reflect and confirm what we already have found out from analyzing the triplet amplitudes. On the F side, the proximity effect increases the correlations somewhat with , as a reduction in quasiparticle states emerges for low energies due to increased correlations as the relative magnetizations become increasingly antiparallel. As increases, the proximity effects weaken. Indeed, at the DOS is nearly flat except at the larger values of , suggesting that in the absence of a down spin band there may be a contribution from a possible triplet presence. On the superconducting side, the situation is somewhat different: the results never approach a limit where the local DOS would look similar to that of a bulk superconductor. Even at , there is never a gap and even when that situation is approached at larger , when the magnetizations in the two F layers are antiparallel, the shape of the DOS curve does not resemble the signature bulk superconductor result. The profound difference between the parallel () and antiparallel () cases is consistent in every respect with that previouslyhv72 () found.
The above results were all obtained in the low temperature limit. In the remaining figures we consider the temperature dependence. In Fig. 9 we plot directly the spatial behavior of and over a broad range of temperatures. As in previous figures for the triplet amplitudes, the main plot shows the behavior over a relatively extended region of the sample and the insets magnify the region near interface. In both cases we have taken and while for and for . It can be inferred that at low T the temperature dependence of the triplet amplitudes is weak, while as increases, the closer spacing in temperatures shown suggest that the correlations become destroyed at a much more rapid rate. In comparing the two panels, it is seen that for much of the temperature range, , but at higher temperatures (), becomes the larger of the two. The temperature dependence of the characteristic penetration depths is weak. These observations are further exemplified in Fig. 10, where the peak values of and (also at ) are shown as a function of . These quantities are determined by calculating and throughout the structure for a given temperature. The inset depicts the corresponding peak values of the ordinary self- consistent equal-time singlet amplitude , (see Eq. (9)). For both the triplet and singlet behavior, there is a strong dependence on as the temperature approaches the system’s transition temperature, which is about for our system. Technically, the determination of the self consistent amplitudes is more difficult at higher , when the number of iterations is in principle much higher. This increase in computational time can be reduced by up to an order of magnitude by taking as the initial spatial pair potential at a given the result for the previously obtained next lower temperature times a dependent factor derived from the linearized Ginzburg-Landau theorybdg ().
In accordance with what we have just seen, it is a natural extension to study how thermal effects might destroy the spatial characteristics of singlet correlations throughout the structure. In Fig. 11, we therefore display as a function of , for several temperatures, and in which the relative magnetizations are collinear () and at right angles (). Remarkably, one can see that the temperature dependence of the triplet components is somewhat weaker that that of the standard singlet amplitude. One can also clearly see that, as indicated above, the penetration of the triplet amplitudes into the F material over a length scale that is clearly much longer than that of the singlet amplitude. This is again a strong indication that experimental tunneling results indicating long ranged penetration effects in F/S structures are indeed evidence for induced triplet correlations.
In this paper, we have presented a detailed study of induced time dependent (odd in time) triplet pairing correlations in clean planar FSF junctions consisting of an ordinary -wave superconductor sandwiched between two relatively thick ferromagnets whose magnetizations are misoriented with respect to each other by an angle . Our microscopic formalism allowed us to investigate cases involving strong magnets, as well as atomic scale phenomenon, two things not possible in the widely used quasiclassical approaches. We have obtained results as a function of , time, the strength of the ferromagnets, and the temperature. We have presented results for the spatial behavior of the time-dependent triplet pair amplitudes, and , and for the corresponding penetration lengths extracted from them. We have found that these triplet correlations are indeed induced via the proximity effect, that they completely pervade even a superconductor several coherence lengths thick, and also substantially penetrate the ferromagnetic layers. These results have clear implications for the experimental work in which long range proximity effects in SF nanostructures have been reported, effects that have been speculated to be due to the existence of some kind of triplet pairing. Our calculations, in which the time dependence is studied from the Heisenberg picture, emphasize the need for full self-consistency of the solutions, without which we find that the Pauli principle is violated.
We have also considered the reverse proximity effect, which is of particular interest in this case due to the presence of two components of the magnetization, and we also have given results for the experimentally measurable local density of states, which revealed clear subgap energy signatures as a function of and . We have studied the temperature dependence of the triplet amplitudes (as well as the ordinary singlet amplitude) and found that the temperature dependence of the penetration depths associated with these triplet amplitudes is weak: these lengths remain large all the way up to the vicinity of the transition temperature of the system. This bodes well for further experimental observations and verification in these clean systems.
Acknowledgements.This project is funded in part by the Office of Naval Research (ONR) In-House Laboratory Independent Research (ILIR) Program and by a grant of HPC resources from the Arctic Region Supercomputing Center at the University of Alaska Fairbanks as part of the Department of Defense High Performance Computing Modernization Program.
- (1) D.D. Osheroff, R.C. Richardson and D.M. Lee, Phys. Rev. Lett. bf 28, 885 (1972).
- (2) V.L. Berezinskii, JETP Lett. 20, 287, (1974).
- (3) Aurel Bulgac, Michael McNeil Forbes, and Achim Schwenk, Phys. Rev. Lett. 97, 020402 (2006).
- (4) Igor Z̆utić, Jaroslav Fabian, S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004).
- (5) J. Y. Gu, C.-Y. You, J. S. Jiang, J. Pearson, Ya. B. Bazaliy, and S. D. Bader, Phys. Rev. Lett. 89, 267001 (2002).
- (6) I. C. Moraru, W. P. Pratt, N. O. Birge, Phys. Rev. Lett. 96, 037004 (2006).
- (7) C. Bell, S. Turşucu, and J. Aarts, Phys. Rev. B74, 214520 (2006).
- (8) C. Visani, V. Peña, J. Garcia-Barriocanal, D. Arias, Z. Sefrioui, C. Leon, J. Santamaria, N.M. Nemes, M. Garcia-Hernandez, J.L. Martinez, S.G.E. te Velthuis and A. Hoffmann, Phys. Rev. B75, 054501 (2007).
- (9) K. Halterman and O.T. Valls, Phys. Rev. B72, 060514(R) (2005).
- (10) P. G. de Gennes, Phys. Lett. 23, 10 (1966).
- (11) L.R. Tagirov, Phys. Rev. Lett. 83, 2058 (1999).
- (12) A.I. Buzdin, A.V. Vdyayev, and N.V. Ryzhanova, Europhys. Lett. 48, 686 (1999).
- (13) V. Peña, Z. Sefrioui, D. Arias, C. Leon, J. Santamaria, M. Varela, S. J. Pennycook, J. L. Martinez, Phys. Rev. B69, 224502 (2004).
- (14) K. D. Nelson, Z. Q. Mao, Y. Maeno, Y. Liu, Science 306, 1151 (2004).
- (15) V.N. Krivoruchko and V. Yu. Taernkov Phys. Rev. B75, 214508 (2007).
- (16) R. S. Keizer, S. T. B. Goennenwein, T. M. Klapwijk, G. Miao, G. Xiao, A. Gupta, Nature 439, 825, (2006).
- (17) F.S. Bergeret, A.F Volkov, and K.B. Efetov, Rev. Mod. Phys. 77, 1321 (2005).
- (18) K. Halterman, P.H. Barsic, and O.T. Valls Phys. Rev. Lett. 99, 127002 (2007).
- (19) F.S. Bergeret, A.F Volkov, and K.B. Efetov, Phys. Rev. Lett. 86, 3140 (2001).
- (20) F. S. Bergeret, A. F. Volkov, and K. B. Efetov, Phys. Rev. B68, 064513 (2003).
- (21) T. Champel and M. Eschrig, Phys. Rev. B72, 054523 (2005).
- (22) Ya. V. Fominov, A. F. Volkov, and K. B. Efetov Phys. Rev. B75, 104509 (2007).
- (23) V. Braude and Yu. V. Nazarov Phys. Rev. Lett. 98, 077003 (2007).
- (24) M. Houzet and A.I. Buzdin Phys. Rev. B76, 060504(R) (2007).
- (25) Ya. V. Fominov, A. A. Golubov, and M. Yu. Kupriyanov, JETP Lett. 77, 510 (2003).
- (26) T. Löfwander, T. Champel, J. Durst, and M. Eschrig, Phys. Rev. Lett. 95, 187003 (2005).
- (27) A.I. Buzdin, Rev. Mod. Phys. 77, 935 (2005).
- (28) Y. Asano, Y. Sawa, Y. Tanaka, and A. Golubov, Phys. Rev. B76, 224525 (2007).
- (29) T. Yokoyama, Y. Tanaka, and A.A. Golubov, Phys. Rev. B75, 134510 (2007).
- (30) Y. Tanaka, Y. Tanuma, and A. A. Golubov, Phys. Rev. B76, 054522 (2007).
- (31) J. Linder and A. Sudbø, Phys. Rev. B75, 134509 (2007).
- (32) P.G. de Gennes, Superconductivity of Metals and Alloys, (Addison-Wesley, Reading, MA, 1989).
- (33) J. B. Ketterson and S. N. Song, Superconductivity, (Cambridge University Press, Cambridge, UK 1999), p. 286.
- (34) K. Halterman and O.T. Valls, Phys. Rev. B69, 014517 (2004).
- (35) K. Halterman and O.T. Valls, Phys. Rev. B70, 104516 (2004).