Superconducting triplet pairing in Al/Al_\mathbf{2} O_\mathbf{3} /Ni/Ga junctions
Andreas Costa, Madison Sutula, Valeria Lauter, Jia Song, Jaroslav Fabian, Jagadeesh S. Moodera
SSuperconducting triplet pairing in Al/Al O /Ni/Ga junctions Andreas Costa, ∗ Madison Sutula,
2, 3
Valeria Lauter, Jia Song, Jaroslav Fabian, † and Jagadeesh S. Moodera
2, 5, ‡ Institute for Theoretical Physics, University of Regensburg, 93040 Regensburg, Germany Francis Bitter Magnet Laboratory and Plasma Science and Fusion Center,Massachusetts Institute of Technology, MA 02139, USA Department of Materials Science and Engineering, Massachusetts Institute of Technology, MA 02139, USA Neutron Scattering Division, Neutron Sciences Directorate,Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA Department of Physics, Massachusetts Institute of Technology, MA 02139, USA (Dated: February 8, 2021)Ni/Ga bilayers are a versatile playground for exploring the competition of the strongly antagonistic fer-romagnetic and superconducting phases. Systematically characterizing this competition’s impact on highlyballistic Al/Al O /Ni/Ga junctions’ transport properties from both the experimental and theoretical viewpoints,we identify novel conductance peak structures, which are caused by superconducting triplet pairings at theNi/Ga interface, and which are widely adjustable through the Ni–Ga thickness ratio. We demonstrate that theseconductance anomalies persist even in the presence of an in-plane magnetic field, which provides—togetherwith the detection of the paramagnetic Meissner effect in Ga—the clear experimental evidence that the observedconductance features serve indeed as the triplet pairings’ unique transport fingerprints. Our work demon-strates that Ni/Ga bilayers have a strong potential for superconducting spintronics applications, in particular fortriplet-pairing engineering. Introduction.
Magnetic superconducting junctions formelementary building blocks for superconducting spintron-ics [1–5], with potential applications in quantum comput-ing [6–13]. Early conductance measurements on ferromag-net/superconductor point contacts [14, 15] demonstrated thatAndreev reflection can be used to quantify the ferromag-net’s spin polarization [16]. Nowadays, more complex struc-tures, such as magnetic Josephson-junction geometries [17],in which Yu–Shiba–Rusinov states [18–20] can strongly in-fluence the supercurrent [21, 22] and even induce current-reversing 0- 𝜋 transitions [23, 24], are being exploited. Awealth of unique physical phenomena and transport anomalieshas been predicted to emerge in such junctions, covering thepotential formation of Majorana states [25–33], significantlymagnified current magnetoanisotropies [34–37], as well asthe efficient generation and detection of spin-polarized tripletCooper pair currents [3, 38].Particularly appealing materials for superconducting spin-tronics are Ni/Ga (Bi) bilayers [39, 40], as strong proxim-ity effects turn the intrinsically weakly ferromagnetic Ni filmsuperconducting. The coexistence of the two nominally antag-onistic ferromagnetic and superconducting phases within theNi region might strongly modify transport properties such asthe differential conductance. Most remarkable is the possibil-ity to generate spin-triplet states, as previous studies [38, 41–58] concluded that ferromagnetic exchange can induce odd-frequency superconductivity as a signature of triplet pairing.The two main factors that generate triplet pairing in proxim-itized 𝑠 -wave superconductors are noncollinearly magnetizeddomains and spin-orbit coupling effects. While triplet currentsoriginating from noncollinear magnetizations were success-fully implemented in various systems [43, 47, 48, 59–67]—e.g., in Nb/Py/Co/Py/Nb junctions through tilting the thinpermalloy (Py) spin-mixers’ magnetizations [66]—generatingthem through spin-orbit coupling [52, 53] is more challenging and often requires specific magnetization configurations (withrespect to the spin-orbit field) to induce detectable triplet pair-ings [68–71].In this Letter, we systematically investigate theconductance of high-quality superconducting magneticAl/Al O /Ni/Ga junctions and demonstrate, in combinationwith realistic theoretical simulations, the generation of tripletpairings due to the presence of Ni/Ga bilayers. Specifically,we observe unexpected conductance peak structures, whichstrongly depend on the Ni–Ga film-thickness ratio. This ra-tio controls the effective spin-orbit coupling’s strength at theNi/Ga interface, and provides an extremely efficient knob toadjust the triplet pairing. An applied in-plane magnetic fielddoes not significantly influence the conductance spectra, whichis a clear evidence that the superconducting triplet pairingsdominate these peculiar features. Furthermore, we detect theparamagnetic Meissner effect in Ga, which is yet another keyevidence for the presence of an odd-frequency superconduct-ing triplet state [50, 72–74]. We believe that our findings pavethe way for employing Ni/Ga-based magnetic superconductingtunnel junctions as platforms for investigating outstanding su-perconducting phenomena, such as Majorana states [25–33],magnetoanisotropies [34–37], or triplet supercurrents [3, 38],in a controllable way. Experimental and theoretical methods.
All investigatedjunctions, with cross-section areas of 150 μ m × μ m, wereprepared by means of thermal evaporation inside an UHV sys-tem with a base pressure of 2 × − mb, using in-situ shadow-masking techniques. During the growth process, thin filmsof Al, Ni, and Ga were evaporated on clean glass slides atroughly 80 K. The junctions’ ultrathin Al O barriers, sep-arating adjacent Al and Ni/Ga films, were likewise createdin situ through exposing Al to controlled oxygen plasma.Before taking the junctions out of the UHV chamber, theywere protected by 10 nm thick Al O layers. In one run, a r X i v : . [ c ond - m a t . s up r- c on ] F e b we could thereby prepare 72 junctions with different Ni andGa film thicknesses (keeping all other growth parameters thesame), allowing us to study the Ni and Ga films’ individualimpact on the junctions’ transport characteristics in a well-controllable manner. To perform the tunneling conductancemeasurements, we attached 24 distinct junctions to a probewith electrical leads and immersed the system into a pumpedliquid-helium bath to achieve temperatures below 1 K. Boththe Al and Ni/Ga thin films turned superconducting, withcritical temperatures (and superconducting gaps) strongly de-pending on their thicknesses [75]. For instance, we deduceda critical temperature of about 2 . vs. junc-tion bias data was obtained using standard lock-in technique.To unravel the conductance anomalies’ physical origin, wecompare the experimental outcomes against theoretical simu-lations. For the latter, we discretize the Al/Ni/Ga junctions on atwo-dimensional tight-binding grid and compute the tunnelingdensity of states (DOS) [75], which can be directly mapped tothe taken conductance data, within the python transport pack-age Kwant [79]. The aforementioned Al O barriers, andalso EuS spacers present in some samples [80, 81]—that cansignificantly enhance triplet-pair currents further [82, 83]—are not expected to be relevant to characterize the experi-mental results and not part of our theoretical modeling. Asreference parameters, we take the constant number of lat-tice sites 𝐿 Al =
200 inside the Al region (mimicking 4 nmthick Al), as well as Ga’s zero-temperature superconduct-ing gap | Δ S , Ga ( )| = .
358 meV that we experimentally es-timated from one specific sample; all remaining material pa-rameters [i.e., “thicknesses” 𝐿 Ni and 𝐿 Ga , and gaps | Δ S , Al ( )| and | Δ S , Ni ( )| ] are then rescaled following their experimentalvalues [75]. Ferromagnetic-exchange effects.
Summarizing our richtunneling conductance–bias voltage data, tuning the individ-ual films’ thicknesses allows for a universal switching betweendifferent transport regimes.As long as the outer Ga region is thick enough (semi-infinite in theory), all the junctions’ conductance charac-teristics are fully describable within the earlier establishedOctavio–Tinkham–Blonder–Klapwijk modeling [84–86], es-sentially resulting in the formation of a well-distinct subhar-monic conductance gap structure with less pronounced peaksat voltages 𝑉 ≈ ± | Δ S , Al ( )|/( 𝑛𝑒 ) , and more marked onesat 𝑉 ≈ ±| Δ S , Ga ( )|/ 𝑒 and 𝑉 ≈ ±[| Δ S , Al ( )| + | Δ S , Ga ( )|]/ 𝑒 ; 𝑒 denotes the positive elementary charge and 𝑛 a positive in-teger. The intermediate Ni film’s superconducting gap doesnot have a great impact owing to the largely extended Ga elec-trode (whose gap is moreover nearly five times larger thanNi’s). As we show in the Supplementary [75], we can ex- Experiment -6 -4 -2 0 2 4 6
Theory conductanceshoulders FIG. 1.
Measured tunneling conductance–bias voltage characteristicsof Al (4 nm)/Al O (0 . . . .
07 K. Inset:
Calculated zero-temperature tunnel-ing DOS as a function of particle energy 𝐸 and for different in-dicated numbers of lattice sites (“thicknesses”) 𝐿 Ni of weakly fer-romagnetic Ni; 𝐿 Al / 𝐿 Ga = / = ( )/(
20 nm ) is keptconstant. Colored arrows highlight qualitatively reproducible con-ductance (DOS) peaks (shoulders). perimentally recover this regime in junctions containing about60 nm Ga.Decreasing Ga’s thickness leads to an intricate interplaybetween all materials’ superconducting gaps and Ni’s fer-romagnetic exchange splitting, notably altering the junc-tions’ conductance features. As an example, Fig. 1 dis-plays Al (4 nm)/Al O (0 . . . weaklocal conductance maxima (called “ conductance shoulders ”)in addition to the still present (sharp) conductance peak(s).Since such conductance shoulders have not yet been addressedin earlier works, we support the experiment with numeri-cal Kwant simulations of the junctions’ zero-temperaturetunneling DOS—noting that the transport anomalies aremicroscopically caused by the formation of resonant An-dreev bound states [87, 88] that we can likewise identify asDOS peaks.Already thin Ni films raise a pronounced DOS peak at 𝑒𝑉 ≈| Δ S , Ga ( )| , followed by a much weaker “DOS shoulder”, whoseposition is now determined by all superconducting gaps (asopposed to the case in which Ga extends to infinity). Al-though these DOS peaks’ positions do quantitatively not ex-actly coincide with the experiment (due to simplifications in themodel), the related conductance peaks are qualitatively clearlyresolvable in the experimental measurements. The secondconductance shoulder experimentally observed in between thetwo computed DOS peaks must be ascribed to an additionalpeak splitting triggered by Ni’s exchange coupling. Since wesubstituted a rather small Ni spin polarization of less than 0 . Experiment -6 -4 -2 0 2 4 6
Theory FIG. 2.
Measured tunneling conductance–bias voltage char-acteristics of Al (4 nm)/Al O (0 . . . Calculated zero-temperature tunnel-ing DOS—assuming strong superconducting triplet pairings atthe Ni/Ga interface—and for different indicated numbers of lat-tice sites (“thicknesses”) 𝐿 Ni of weakly ferromagnetic Ni; 𝐿 Al / 𝐿 Ga = / = ( )/(
40 nm ) is kept constant. Colored arrows high-light qualitatively reproducible conductance (DOS) peaks (shoul-ders). magnetism in Ni suffices to explain the measurements—recovering the related peak splitting in the tunneling DOSrequires rather large Ni thicknesses (recall that we estimatedabout 1 % spin polarization from the samples). As it com-pletely disappears when turning Ni non ferromagnetic (evenat Ni thicknesses far above Ga’s), the additionally appearingconductance shoulder must indeed originate from the inter-play between Ni’s ferromagnetism and superconductivity, andserves therefore as a clear transport fingerprint of (weak) fer-romagnetism inside Ni. From the microscopic viewpoint, theformation of the respective DOS shoulder might signify theintriguing competition between Andreev bound states [87, 88]and Yu–Shiba–Rusinov states [18–20] in magnetic Joseph-son junctions [21, 22]. The conductance peak close to zero biasis not clearly resolvable in the DOS simulations, which is prob-ably a consequence of our simplified model (neglecting, i.e.,the influence of the EuS layer present in this sample). Signatures of triplet pairing.
Further increas-ing the Ni–Ga thickness ratio from 𝑑 Ni / 𝑑 Ga = .
04 in the previous to 𝑑 Ni / 𝑑 Ga ≈ . O (0 . . Experiment H = 0 H = 1 A/m FIG. 3.
Measured tunneling conductance–bias voltage charac-teristics of Al (4 nm)/Al O (0 . . . ) /Al O (12 nm) junctions at 1 K. Inset: Measured tun-neling conductance–bias voltage characteristics of the same sample,measured in an in-plane magnetic field of strength 𝐻 = − . Rusinov) bound states.Comparing the conductance modulations against our the-oretical tunneling-DOS simulations, we deduce that the ex-perimental findings are qualitatively only reproducible ifwe assume that not just singlet superconducting pairing ispresent (as up to now), but moreover strong triplet pairingsat the Ni/Ga interface arise. To see clear effects, the triplet-pairing potentials substituted into our DOS simulations areexaggeratedly large (i.e., nearly four times larger than thesinglet-pairing potentials), explaining the quantitative devi-ations between the DOS and conductance peaks’ (shoulders’)positions. Nickel’s spin polarization remains as tiny as be-fore (below 0 . interfacial triplet pairings and ferromagneticproximity effects . Analogously, introducing superconductingtriplet correlations in the bulk of Ni and Ga (instead of at theirinterface) does also not fully cover the experimental outcomes,and would turn the zero-bias peaks into dips. As all the crucialphysics must consequently happen right at the Ni/Ga interface,growing the same sample once again interchanging the Ni andGa films does not visibly alter the conductance spectrum.Let us now verify that those interface effects result in the the-oretically proposed (strong) superconducting triplet pairings,exploiting that the ferromagnetic and singlet superconduct-ing phases are strongly antagonistic. Applying sufficientlylarge magnetic fields to a singlet superconductor typicallytends to break up its Cooper pairs and aligns all unpairedelectrons’ spins parallel to generate net magnetic moments.As a consequence, the superconducting order gets slowly de-stroyed and the associated order parameter, i.e., the singletsuperconducting gap, is suppressed. Since the observed con-ductance peaks’ positions in our study are essentially deter-mined by the individual films’ superconducting gaps, applyinga magnetic field would not only reduce the gaps themselves, butconcurrently visibly displace the apparent conductance peaks.Measuring the conductance–voltage characteristics ofAl (4 nm)/Al O (0 . . . ) /Al O (12 nm) junctions once in the absenceand once in the presence of an external in-plane mag-netic field, however, does not reveal any substantial impactof the magnetic field on the conductance peaks, as illustratedin Fig. 3. The striking physics, leading to the described con-ductance anomalies, must therefore indeed originate from in-terfacial triplet pairings, whose gaps are much less sensitiveto external magnetic fields. The overall conductance modula-tions are now even richer than before (additional peak split-tings), mostly due to the EuS layer whose strong ferromag-netic exchange coupling [80, 81] further amplifies the peak-splitting effects. These splittings turn the previously discussedzero-bias conductance peak into another series of closely ad-jacent conductance peaks. Polarized-neutron reflectometry.
To investigate the Meiss-ner effect in Ni/Ga films and directly obtain their struc-ture [by low-angle X-ray reflectometry (XRR) studies]and magnetization-depth profile, we employ the polarized-neutron reflectometry (PNR) method. PNR was previouslysuccessfully applied to observe the diamagnetic Meissner ef-fect, as well as vortex-line distributions, in niobium and YBCObilayer films [89, 90]. Here, we report the detection of theparamagnetic Meissner effect in Ga—another key propertyof the odd-frequency superconducting triplet state generatedthrough the proximity coupling of the Ni/Ga bilayer. For thePNR-data analysis, we distinguish between two scenarios: thefirst is consistent with the conventional dia magnetic Meiss-ner screening and the second corresponds to the para magneticMeissner response in Ga. The results for both cases are shownin Fig. 4(a) and 4(b). Fitting the PNR (obtained at 5 K, whichis below that bilayer’s superconducting critical temperature)and XRR data, we observe that the Ni (5 . . = ( 𝑅 + − 𝑅 − )/( 𝑅 + + 𝑅 − ) reveals that the best fit to thedata requires 26 emu / cc induced magnetization over roughly6 nm in Ga, while the magnetization in the Ni film is about136 emu / cc and uniform. Thus, we are able to directly see theinduced ferromagnetic order’s influence inside the Ga layerright at the interface with the Ni film, which must be at-tributed to the paramagnetic Meissner response. PNR provideshence the expected evidence that superconducting Ni/Ga bi-layers exhibit the paramagnetic Meissner response in Ga as a (a)(b) FIG. 4. Polarized-neutron reflectometry (PNR) results forNi (5 . = ( 𝑅 + − 𝑅 − )/( 𝑅 + + 𝑅 − ) ; the best fit to the data correspondsto the magnetization profile shown as the solid black line in (b)—paramagnetic Meissner state—, revealing 26 emu / cc induced mag-netization over about 6 nm in Ga, while the Ni film’s magnetizationis about 136 emu / cc and uniform. Contrary, the diamagnetic Meiss-ner state in the spin-asymmetry (SA) plot and its magnetic scatteringlength density (MSLD) profile, referring to the dashed black lines,strongly deviate from the experimental data. clear signature of the theoretically predicted superconductingtriplet state [50, 72–74]. Conclusions.
We demonstrated that the coexistence ofthe nominally opposing ferromagnetic and superconductingphases in Ni/Ga bilayers gives rise to intriguing modula-tions of Al/Al O /Ni/Ga junctions’ tunneling conductance–bias voltage relations, which are essentially controlled by theNi–Ga film-thickness ratio and can be reproduced in a well-defined manner in several distinct junctions. Mapping the-oretical tunneling-DOS simulations to the experimental con-ductance data, we predicted that large enough Ni–Ga thick-ness ratios support sizable superconducting triplet pairingsat the Ni/Ga interface, most likely stemming from interfa-cial spin-orbit couplings. To further characterize the specifictriplet-pairing mechanism, we suggest to subsequently focusmore on the junctions’ Josephson-transport characteristics. Acknowledgments.
The experimental work performed inthe U.S. was supported by the NSF Grant DMR 1700137,ONR Grants N00014-16-1-2657 and N00014-20-1-2306,and John Templeton Foundation Grants 39944 and 60148.The theoretical work at Regensburg (A.C. and J.F.) re-ceived funding from the Elite Network of Bavaria throughthe International Doctorate Program Topological Insulatorsand Deutsche Forschungsgemeinschaft (DFG, German Re-search Foundation)—Project-ID 314695032—SFB 1277,Subproject B07. The undergraduate M.S. was supported bythe UROP program funds at Massachusetts Institute of Tech-nology. This research used resources at the Spallation Neu-tron Source, a DOE Office of Science User Facility operatedby the Oak Ridge National Laboratory. ∗ E-Mail: [email protected] † E-Mail: [email protected] ‡ E-Mail: [email protected][1] I. Žutić, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. , 323(2004).[2] J. Fabian, A. Matos-Abiague, C. Ertler, P. Stano, and I. Žutić,Acta Phys. Slovaca , 565 (2007).[3] M. 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Exp.Theor. Phys. , 455 (1966).[89] V. Lauter-Pasyuk, H. J. Lauter, V. L. Aksenov, E. I. Kornilov,A. V. Petrenko, and P. Leiderer, Physica B , 166 (1998).[90] V. Lauter-Pasyuk, H. J. Lauter, M. Lorenz, V. L. Aksenov, andP. Leiderer, Physica B , 149 (1999).[91] G. B. Arnold, J. Low Temp. Phys. , 143 (1985).[92] G. B. Arnold, J. Low Temp. Phys. , 1 (1987).[93] Y. M. Blanter and F. W. J. Hekking, Phys. Rev. B , 024525(2004).[94] M. Kuhlmann, U. Zimmermann, D. Dikin, S. Abens, K. Keck,and V. M. Dmitriev, Z. Phys. B , 13 (1994).[95] V. Lauter-Pasyuk, Collect. Soc. Fr. Neutron , s221 (2007). UPPLEMENTAL MATERIALSuperconducting triplet pairing in Al/Al O /Ni/Ga junctions Andreas Costa, ∗ Madison Sutula,
2, 3
Valeria Lauter, Jia Song, Jaroslav Fabian, † and Jagadeesh S. Moodera
2, 5, ‡ Institute for Theoretical Physics, University of Regensburg, 93040 Regensburg, Germany Francis Bitter Magnet Laboratory and Plasma Science and Fusion Center,Massachusetts Institute of Technology, MA 02139, USA Department of Materials Science and Engineering, Massachusetts Institute of Technology, MA 02139, USA Neutron Scattering Division, Neutron Sciences Directorate,Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA Department of Physics, Massachusetts Institute of Technology, MA 02139, USA
In this Supplemental Material, we introduce two distincttheoretical approaches applied to understand the experimen-tally investigated Al/Al O /Ni/Ga junctions’ puzzling tunnel-ing conductance–voltage characteristics. Section I summa-rizes the individual layers’ most essential material parame-ters that were extracted from (some of) the prepared samplesand enter our theoretical simulations as a fundamental input.In Sec. II, we adapt the well-established analytical Octavio–Tinkham–Blonder–Klapwijk (OTBK) model [S1, S2] to coverthese junctions and extract some valuable general conduc-tance features, which are able to partially explain the experi-mental results. Afterwards, we outline the greatest drawbacksof the OTBK description in view of our systems and elaborateon more sophisticated numerical simulations, employing thepython tight-binding transport code Kwant [S3], in Sec. III.Section IV clarifies the influence of likewise present ferromag-netic proximity effects on the discussed outcomes, before weeventually analyze additional polarized-neutron reflectometrydata in Sec. V to provide a deeper insight into the magnetiza-tion profile at the Ni/Ga interface. CONTENTS
I. Junction parameters iII. Analytical OTBK calculations vs. experimentalmeasurements iiIII. Numerical Kwant simulations—generic recipe ivIV. Role of purely ferromagnetic proximity effects viV. Polarized-neutron reflectometry viiReferences viii ∗ E-Mail: [email protected] † E-Mail: [email protected] ‡ E-Mail: [email protected] Al insulator Ni Ga current
FIG. S1. Schematical sketch of the investigated Al/insulator (i.e.,Al O )/Ni/Ga junctions. The Al and Ga electrodes are intrinsi-cally superconducting, while proximity effects additionally turn theweakly ferromagnetic Ni film likewise superconducting. Applyingthe bias voltage 𝑉 between the electrodes and measuring the relatedtunneling current (indicated by the dark green arrow), we probe thesamples’ tunneling conductance–bias voltage characteristics. I. JUNCTION PARAMETERS
As we stated in the main text, Al/Al O /Ni/Ga junc-tions (see Fig. S1 for a schematic sketch) offer unprecedentedpossibilities to investigate the intriguing interplay of the nomi-nally antagonistic ferromagnetic and superconducting states ofmatter. Apart from the already intrinsically superconductingAl and Ga components, proximity effects turn also the weaklyferromagnetic Ni film additionally superconducting. The indi-vidual layers’ (zero-temperature) superconducting energy gapsand critical temperatures vary usually a lot with the distinctfilms’ thicknesses. We extracted their values from (some of)the experimental samples. For junctions built on 4 nm Al, thelatter’s critical temperature was estimated to be 𝑇 C , Al = . 𝑇 C , Ni = 𝑇 C , Ga = . . TABLE S1. Material parameters estimated from (some of) the experimental samples and used for our theoretical simulations.
Al electrode zero-temperature superconducting gap | Δ S , Al ( )| = .
18 meVcritical temperature 𝑇 C , Al = . Ni film zero-temperature superconducting gap | Δ S , Ni ( )| = .
277 meVcritical temperature 𝑇 C , Ni = . NiGa interfacial alloy zero-temperature superconducting gap | Δ S , NiGa ( )| = .
251 meVcritical temperature 𝑇 C , NiGa = . Ga electrode zero-temperature superconducting gap | Δ S , Ga ( )| = .
358 meVcritical temperature 𝑇 C , Ga = . will discuss in the following, and are summarized in a compactway in Tab. S1.The Ni region truly counts to the most crucial components ofthe junctions, as it does not only turn superconducting throughproximity effects, but shows additionally weak intrinsic ferro-magnetism, and might hence raise clear spin-polarized trans-port features. Its spin polarization, which we classified bymeans of state-of-the-art Meservey–Tedrow spectroscopy [S4–S6], rises with increasing Ni film thickness. For instance,samples with about 4 nm thick Ni revealed spin polarizationsslightly below one percent. II. ANALYTICAL OTBK CALCULATIONS
VS.
EXPERIMENTAL MEASUREMENTS
To theoretically access Al/Al O /Ni/Ga junctions’ generictunneling conductance–bias voltage characteristics and com-pare them against the experimental data, we apply twodistinct methods. The first approach essentially gener-alizes the earlier established Octavio–Tinkham–Blonder–Klapwijk (OTBK) modeling [S1, S2] to a simple toy systemthat captures all essential parts of the experimental samples.More specifically, we idealize the junctions in terms of effec-tive one-dimensional Al/N/Ni/NiGa/Ga point contacts, whosethin nonsuperconducting normal-metal link (N) of thickness 𝑑 gets separated by two strong ultrathin (deltalike) barriers fromthe neighboring Al and Ni regions to resemble the samples’Al O tunneling barriers. Following earlier works, we cal-culate the overall tunneling-current flow (and thereby thetunneling conductance) in the same manner as explained inRefs. [S1] and [S2], taking the material parameters sum-marized in Tab. S1. The perhaps greatest strength of theOTBK approach is that we can reliably control all single junc-tion parameters and deduce their resolved impact on the wholejunctions’ conductance characteristics. Since experimentalobservations suggested that the Ni/Ga bilayers may addition-ally form thin Ni–Ga alloys at their interfaces, our model alsoaccounts for the latter (though they are extremely thin and theirgap is thus small that they do not visibly impact the calculatedconductances). Figures S2(A)–(C) illustrate the numerically evaluatedtunneling conductance–bias voltage relations of symmetric Ga/N/Ni/NiGa/Ga junctions (meaning that we are concernedwith two similar massive Ga electrodes), held at a tempera-ture of 0 . 𝑑 Ni = . 𝑑 Ni = . 𝑑 Ni = . 𝑃 Ni = O tunneling barriers,we assume realistic barrier heights of 0 .
75 eV together withthicknesses of about 0 .
80 nm, coinciding with experiment.As long as the Ni region remains thin enough, we recoverS/N/S junctions’ unique subharmonic conductance gap struc-ture [S1, S2], manifesting itself in pronounced conduc-tance maxima at all bias voltages 𝑉 ≈ | Δ S , Ga ( )|/( 𝑛𝑒 ) withinteger 𝑛 ( 𝑒 denotes the positive elementary charge). Thephysical origin of that subharmonic gap structure is analyzedin Refs. [S7] and [S8]. Multiple Andreev reflections [S9] ofincident electrons (holes) at the N/Ga and N/Ni interfaces,respectively, open additional transmission channels throughwhich unpaired electrons (strictly speaking, electronlike quasi-particles) can tunnel from one into the second Ga electrode.Only at bias voltages fulfilling 𝑉 ≈ | Δ S , Ga ( )|/( 𝑛𝑒 ) , thesetunnelings involve large-density of states (DOS) states to be-come likely enough to raise sizable tunneling current contri-butions; the latter become eventually visible in terms of localmaxima in the junctions’ conductance–voltage characteristics.Nevertheless, we need to account for another peculiarity of An-dreev reflections in magnetic junctions. Each electron (hole)approaching the N/Ni boundary and being Andreev reflectedpicks up the extra phase 𝛿𝜙 = ( Δ XC , Ni 𝑑 Ni )/( ℏ 𝑣 F , Ni ) [S10],where Δ XC , Ni ( 𝑑 Ni ) indicates Ni’s exchange splitting (thick-ness) and 𝑣 F , Ni its Fermi velocity. Since Andreev reflec-tions are strongly phase sensitive, additional phases of in-cident electrons (holes) may notably change the probabili-ties to really undergo Andreev reflections. For certain val-ues of 𝛿𝜙 , it might even become possible that Andreev re-flections become suddenly forbidden at their initially favoredii (a)(b)(c)(A)(B)(C) (d) Experiment
OTBK theory
FIG. S2. (a)–(c)
Calculated dependence of Al/N/Ni/NiGa/Ga junctions’ tunneling conductance 𝐺 , normalized to Sharvin’s conductance 𝐺 S ,on the applied bias voltage 𝑉 [the associated electron energy 𝑒𝑉 is given in multiples of Ga’s zero-temperature superconducting gap | Δ S , Ga ( )| ]for two indicated Ni spin polarizations 𝑃 Ni and Ni thicknesses (a) 𝑑 Ni = . 𝑑 Ni = . 𝑑 Ni = . . Measured tunneling conductance–voltage characteristics of one specific Al (4 nm)/Al O (0 . .
58 K. (A)–(C) Same as in (a)–(c), but for symmetric
Ga/N/Ni/NiGa/Ga junctions. Black arrows highlight anomalous conductance variations , i.e.,increasing Ni thicknesses may turn conductance maxima into minima. At small voltages, the tunneling current becomes tiny and calculatingits values accurately enough is numerically demanding; we shaded this region and do not present any data there. bias voltages 𝑉 ≈ | Δ S , Ga ( )|/( 𝑛𝑒 ) , and happen then primar-ily at voltages well below or above. As a consequence, theconductance-enhancing electron tunnelings are no longer pos-sible at 𝑉 ≈ | Δ S , Ga ( )|/( 𝑛𝑒 ) , turning prior conductance max-ima into conductance dips and simultaneously raising newmaxima at other voltages. The full effect is predominantlycontrollable by means of 𝛿𝜙 and thus by altering either Δ XC , Ni or 𝑑 Ni . In our simulations, we observe such clear modificationswhen increasing 𝑑 Ni from 𝑑 Ni = . 𝑑 Ni = . Δ XC , Ni (essentially its spin polarization 𝑃 Ni ) constant. Theinitially arising conductance maximum at 𝑒𝑉 ≈ | Δ S , Ga ( )| really turns into a conductance minimum (dip), as the ad-ditionally accumulated 𝛿𝜙 heavily suppresses conductance- enhancing multiple Andreev reflections there.Coming back to the experimentally studied junctions, we re-place the left Ga electrode by Al and repeat the same theoreticalcalculations, now distinguishing between the two Ni spin po-larizations 𝑃 Ni = .
25 % and 𝑃 Ni = dissimilar electrodes’ superconductinggaps strongly alter the subharmonic gap structure and splitthe series of conductance maxima into well-distinct subseries.Transferring those results to our case, we expect local con-ductance peaks forming at 𝑒𝑉 ≈ ± | Δ S , Al ( )|/( 𝑛 ) with pos-vitive integers 𝑛 , 𝑒𝑉 ≈ ±| Δ S , Ga ( )| , and 𝑒𝑉 ≈ ±[| Δ S , Al ( )| +| Δ S , Ga ( )|] —at least as long as the ferromagnetic exchangein Ni can be neglected (i.e., at sufficiently small Δ XC , Ni and/or 𝑑 Ni ). The latter two peaks can be clearly identifiedin Fig. S2(a)—though they are both located at slightly smallervoltages than expected due to the superconducting gaps’ rescal-ing at finite temperatures. The peak subseries originatingfrom Al, however, cannot be properly resolved since the tun-neling current becomes thus tiny at small bias voltages thatcalculating its values within the required accuracy would bedemanding. For that reason, we shaded this area in the plotsand do not present any numerical data there. The interfa-cial Ni–Ga alloy has no visible influence on the outcomes, asit is assumed to be ultrathin (0 . O (0 . .
58 K [see Fig. S2(d)] confirms that the Ni spin polar-ization extracted from Meservey–Tedrow spectroscopy (lessthan one percent) is a reasonable value. The Ni thicknessof the experimental sample, 𝑑 Ni = 𝑑 Ni = 𝑃 Ni = 𝑒𝑉 ≈ ±| Δ S , Ga ( )| could not beexperimentally observed, suggesting that Ni’s spin polariza-tion in the sample must indeed be smaller than one percent.Nevertheless, we cannot quantitatively fit our model to the ex-perimental data since the junction parameters given in Tab. S1were only determined for some samples and are not preciselyknown for all of them. The experimental results in Fig. S2(d)indeed reveal (slightly) different superconducting gaps thanthose we substituted into our calculations. The reduction toone-dimensional junctions is another issue our theory mightface. Earlier studies [S12] demonstrated that the tunneling-current flow through three-dimensional junctions is not ho-mogeneous (e.g., due to inhomogeneities in the oxide barriers)and rather carried by different transverse channels. Each chan-nel could be subject to distinct interfacial scattering (barrier)parameters, additionally shifting the conductance peaks andexplaining deviations between theory and experiment. Extend-ing our model to three dimensions and accounting for differenttransverse momentum-dependent barrier parameters is feasi-ble, but goes beyond the scope of this work. At the moment, itis sufficient to understand that Al/Al O /Ni/Ga junctions’ pe-culiar tunneling conductance–voltage characteristics are pre-dominantly governed by the interplay between the strongly differing superconducting gaps of the Al and Ga electrodes onthe one, and Ni’s ferromagnetic exchange interaction on theother hand. III. NUMERICAL KWANT SIMULATIONS—GENERIC RECIPE
Most surprisingly, the bias voltages at which conduc-tance maxima arise in Figs. S2(a)–(c) are not visibly influ-enced by Ni’s superconducting gap, although we claimed thatproximity effects turn Ni superconducting and an unprece-dented competition with its intrinsic ferromagnetic propertiesmight occur. The main reason for that illustrates another greatdeficit of the OTBK formulation. Since this approach re-gards the junctions’ Al and Ga regions as massive, infinitelyextended, electrodes, the superconducting gap of the muchthinner Ni layer—simultaneously being several times tinierthan Ga’s gap—will not have a substantial impact on the cal-culated conductance peak positions. Even enhancing Ni’s su-perconducting gap within a reasonable range would not leadto significant changes of the obtained conductance data.The experimental results depicted in Fig. S2(d) wererecorded for a sample whose 60 nm thick Ga region indeed re-markably overcomes the Ni layer’s thickness of just about 3 nm,and can hence be treated to some extent like a semi-infiniteGa electrode. This observation justifies the good agreementbetween the OTBK modeling’s theoretical predictions and theexperimental outcomes for that particular junction configura-tion. Nevertheless, as soon as the Ga electrode becomes muchthinner and the Ni layer gains more importance in determiningthe samples’ overall transport characteristics, the OTBK de-scription might face serious trouble and we can no longercompare those calculations with the experimental outcomes.As this is essentially the case with all junctions analyzed in themain text, we urgently need to elaborate on a second and morepowerful method that can efficiently model electrical transportthrough more realistic physical systems and notably extendsour analytical possibilities.For that purpose, we implement the experimentally stud-ied Al/Ni/Ga stackings within the python transport pack-age Kwant [S3], basically discretizing their continuousBogoljubov–de Gennes Hamiltonian on a discrete tight-binding lattice. For simplicity, we consider a two-dimensionalsquare lattice with spacing 𝑎 = [ a . u . ] between two ad-jacent lattice sites; each site with the real-space coordi-nates ( 𝑧, 𝑦 ) = ( 𝑎𝑖, 𝑎 𝑗 ) is then uniquely identified by its integerlattice “coordinates” ( 𝑖, 𝑗 ) . Figure S3(a) shows a graphicalrepresentation of the chosen tight-binding lattice. We denotethe numbers of lattice sites along the longitudinal ˆ 𝑧 -directioninside the Al, Ni, and Ga junction regions by 𝐿 Al , 𝐿 Ni , and 𝐿 Ga ,respectively, whereas we assume in total 𝑊 lattice sites alongthe transverse ˆ 𝑦 -direction. Contrary to the aforementionedOTBK modeling, we do not include the Al O tunneling barri-ers and the interfacial Ni–Ga alloy into our Kwant descriptionsince we are predominantly interested in deducing qualitativetrends, which would not be dramatically affected by those twocomponents. Al Ni Ga
W L Al L Ni L Ga –t–t (a) KWANT: pure singlet pairing (b)
KWANT: interfacial triplet pairing
Al Ni Ga
W L Al L Ni L Ga –t–t FIG. S3. (a) Schematical illustration of Al/Ni/Ga junctions’ tight-binding modeling within the Kwant python transport package, starting froma square lattice with constant spacing 𝑎 = [ a . u . ] . The numbers of horizontal lattice sites inside the junctions’ different regions are denotedby 𝐿 Al , 𝐿 Ni , and 𝐿 Ga , respectively, and that along the transverse direction by 𝑊 . Colored dots represent the on-site energies , determined by theBogoljubov–de Gennes Hamiltonian stated in Eq. (S1), while 𝑡 measures the strength of the nearest-neighbor hoppings . All superconductingpairing potentials capture only pure singlet pairings. (b) Same as in (a), but assuming that the superconducting pairing potentials in the Ni andGa layers combine singlet with interfacial triplet pairings [within the lattice sites ( 𝐿 Ni ; 𝐿 Ga ) ]; the latter are emphasized by colored arrows. The on-site energies at lattice site ( 𝑖, 𝑗 ) are then given by the discretized Nambu-space Bogoljubov–de Gennes Hamiltonianˆ H BdG ( 𝑖, 𝑗 ) = " ( 𝑡 − 𝜇 ) ˆ 𝜏 + (cid:12)(cid:12) Δ singletS , Al (cid:12)(cid:12) ˆ 𝜏 Θ ( 𝑖 ) Θ ( 𝐿 Al − 𝑖 )+ Δ XC , Ni ˆ 𝜏 Θ ( 𝑖 − 𝐿 Al − ) Θ ( 𝐿 Al + 𝐿 Ni + 𝐿 Ni − 𝑖 )+ (cid:12)(cid:12) Δ singletS , Ni (cid:12)(cid:12) ˆ 𝜏 Θ ( 𝑖 − 𝐿 Al − ) Θ ( 𝐿 Al + 𝐿 Ni + 𝐿 Ni − 𝑖 )+ (cid:12)(cid:12) Δ tripletS , Ni (cid:12)(cid:12) ˆ 𝜏 Θ ( 𝑖 − 𝐿 Al − 𝐿 Ni − ) Θ ( 𝐿 Al + 𝐿 Ni + 𝐿 Ni − 𝑖 )+ (cid:12)(cid:12) Δ singletS , Ga (cid:12)(cid:12) ˆ 𝜏 Θ ( 𝑖 − 𝐿 Al − 𝐿 Ni − 𝐿 Ni − ) Θ ( 𝐿 Al + 𝐿 Ni + 𝐿 Ni + 𝐿 Ga + 𝐿 Ga − 𝑖 )+ (cid:12)(cid:12) Δ tripletS , Ga (cid:12)(cid:12) ˆ 𝜏 Θ ( 𝑖 − 𝐿 Al − 𝐿 Ni − 𝐿 Ni − ) Θ (cid:0) 𝐿 Al + 𝐿 Ni + 𝐿 Ni + 𝐿 Ga − 𝑖 (cid:1) × Θ ( 𝑗 ) Θ ( 𝑊 − − 𝑗 ) (S1)and the nearest-neighbor hoppings ( h 𝑖, 𝑗 i indicates nearest-neighbor lattice sites) byˆ H hop (h 𝑖, 𝑗 i) = − 𝑡 ˆ 𝜏 , (S2)where ˆ 𝜏 = − − , ˆ 𝜏 = − − , ˆ 𝜏 = , and ˆ 𝜏 = . (S3)Thereby, the hopping parameter represents 𝑡 =ℏ /( 𝑚𝑎 ) — 𝑚 refers to the effective quasiparticle masses—,noting that it is most convenient to use such units that 𝑎 = 𝑡 = [ a . u . ] , at least as long as one focuses rather on qualitativethan on quantitative analyses.Apart from the discrete single-particle energies 𝜀 ( 𝑖, 𝑗 ) = i ( 𝑡 − 𝜇 ) ˆ 𝜏 , measured with respect to the chemical poten-tial 𝜇 (taken to be the same throughout the junction), and Ni’sferromagnetic exchange gap Δ XC , Ni (the magnetization vec-tor points along the ˆ 𝑧 -direction), we need to account forthe junction regions’ distinct superconducting pairing poten-tials (gaps). Within the OTBK approach, all superconductingpairing potentials are simply approximated by steplike singlet ( 𝑠 -wave) functions. However, analyzing our numerical simu-lations and comparing them against the experimental resultsconvinces that pure singlet pairings alone are not yet suffi-cient to reproduce all observed conductance features. Someof the samples instead reveal clear indications of additionallypresent triplet pairings at the Ni/Ga interface , which mightresult as a consequence of strong spin-orbit interactions due tothe structure-inversion asymmetry. Covering both singlet andtriplet correlations, the Bogoljubov–de Gennes Hamiltoniancontains not only singlet superconducting pairing potentials,coupling spin-up and spin-down electrons to form spin-singletCooper pairs , but also triplet-pairing potentials that facilitate spin-triplet Cooper pairs consisting of two equal-spin elec-trons. The corresponding singlet superconducting gaps areabbreviated by (cid:12)(cid:12) Δ singletS , Al (cid:12)(cid:12) , (cid:12)(cid:12) Δ singletS , Ni (cid:12)(cid:12) , and (cid:12)(cid:12) Δ singletS , Ga (cid:12)(cid:12) , while the in-terfacial triplet gaps are (cid:12)(cid:12) Δ tripletS , Ni (cid:12)(cid:12) and (cid:12)(cid:12) Δ tripletS , Ga (cid:12)(cid:12) . To ensure thattriplet correlations arise indeed solely in the vicinity of theNi/Ga interface, their respective pairing potential terms are nonzero only in 𝐿 Ni = 𝐿 Ni / 𝐿 Ga = 𝐿 Ga / FIG. S4.
Calculated zero-temperature tunneling DOS for differentindicated proximity-induced exchange splittings Δ XC , Ga in 1/5 ofGa’s lattice sites around the Ni/Ga interface; Ni is weakly ferromag-netic (i.e., Δ XC , Ni = . | Δ S , Ga ( )| corresponds to a spin polarizationof less than 0 .
03 %) and contains 𝐿 Ni =
400 lattice sites (resemblinga thickness of 8 nm). All other parameters are the same as in Fig. 1 ofthe main text, i.e., also no triplet superconducting gaps are assumedto be present. Ferromagnetic proximity effects can partially causemore clearly pronounced zero-bias conductance peaks (emphasizedby the violet arrow), but are not yet sufficient to reproduce all theexperimentally observed conductance peak splittings at larger volt-ages (black arrow) that we attributed to interfacial triplet pairings inthe main text. FIG. S5. Polarized-neutron spin-asymmetry ratio SA = ( 𝑅 + − 𝑅 − )/( 𝑅 + + 𝑅 − ) for Ni/Ga bilayers with various Ni thicknesses. this particular range is not motivated by the experiment, whichcould not satisfactorily characterize all the physics happeningat the Ni/Ga boundary so far, but rather an ad-hoc assumptionto recover the generic trends visible in the recorded conduc-tance data. Moreover, we could not yet specify the orbital symmetry of the triplet-pairing potential (e.g., 𝑝 -wave) fromthe current measurements.To obtain the theoretical results discussed in the main text,we implement the aforementioned tight-binding Bogoljubov–de Gennes Hamiltonian in Kwant and use Kwant’s internalKernel Polynomial Method (KPM) to extract the junctions’spatially integrated zero-temperature tunneling DOS (normal-ized to its normal-state counterpart), which gets essentiallyprobed by conductance measurements and is therefore ex-pected to reveal closely related features. For the singlet super-conducting gaps, we use Ga’s value of | Δ S , Ga ( )| = .
358 meVas a reference and rescale all others according to Tab. S1,whereas we put exaggeratedly large triplet gaps—nearly fourtimes as large—into our simulations in order to resolve clearramifications of triplet pairings. Along the transverse direc-tion, we include 𝑊 =
500 lattice sites. Although changing 𝑊 does not significantly impact the outcomes, using rather largenumbers is reasonable to minimize unphysical numerical fluc-tuations in the generated DOS data. IV. ROLE OF PURELY FERROMAGNETICPROXIMITY EFFECTS
We argued in the main text that the rich conductance vari-ations, appearing as the Ni–Ga thickness ratio continues toincrease, signify superconducting triplet pairings around theNi/Ga interface. To exclude that similar conductance anoma-lies might simply be caused by the pure ferromagnetic prox-imity effects that are additionally expected following ourpolarized-neutron reflectometry analyses (see Sec. V), we as-sume that the Ni interlayer furthermore induces the nonzeroferromagnetic exchange gap Δ XC , Ga in Ga’s proximity re-gion (i.e., at all lattice sites indicated by 𝐿 Ga ), while the afore-ii a b (a) (b) FIG. S6. Polarized-neutron reflectometry (PNR) and X-ray reflectometry (XRR) results for Ni (5 . 𝑄 for spin-up ( 𝑅 + ) and spin-down ( 𝑅 − ) neutron-spin states measured in 1 kOe after a zero-field cooling to 5 K. (b) Complementary X-ray reflectometry (XRR) data has been used to verify the films’ depth morphology. From the fit tothe PNR and XRR, we deduce that the Ni (5 , . mentioned triplet pairings are simultaneously no longer there.The tunneling DOS, numerically evaluated from Kwant forthat case and shown in Fig. S4, suggests that even unreal-istically great ferromagnetic proximity effects alone—raisingproximity-induced gaps that are larger than Ni’s actual ex-change gap—are not enough to recover the experimentallyobserved features. The latter succeeds only if we allow for in-terfacial triplet pairings at the Ni/Ga boundary, as claimed inthe main text. Anyhow, ferromagnetic proximity effects maystrengthen the conductance peak at zero bias voltage, whichbecomes also evident in the experimental data and remainsonly barely visible when considering interfacial triplet pair-ings alone. Therefore, we conclude that we are probably con-cerned with a superposition of both effects in the experiment—triplet correlations around the Ni/Ga interface and (at leastweak) ferromagnetic proximity inside the Ga electrode—,which could also provide another reasonable explanation forthe deviations between our theoretical toy model and the ex-perimental results. V. POLARIZED-NEUTRON REFLECTOMETRY
To better understand the Ni/Ga interfacial interaction, andto directly explore the depth profile of the magnetism at theinterface, we employ a depth-sensitive polarized-neutron re-flectometry (PNR) technique. Being electrically neutral, spin-polarized neutrons penetrate the entire multilayer structures,and probe magnetic and structural composition of the filmsthrough the buried interfaces down to the substrate [S13]. ThePNR experiments were performed on the Magnetism Reflec-tometer at the Spallation Neutron Source at Oak Ridge Na-tional Laboratory [S14]. A neutron beam with a wavelength band of 2 . . . . Δ 𝜆 impingeson the film at a grazing incidence angle 𝜃 , where it interactswith atomic nuclei and the spins of unpaired electrons. The re-flected intensity is measured as a function of wave vector trans-fer, 𝑄 = 𝜋 sin 𝜃 / 𝜆 , for the two neutron polarizations 𝑅 + and 𝑅 − , indicating a neutron spin parallel ( + ) or antiparallel ( − ) tothe direction of the external field 𝐻 ext . To separate the nuclearfrom the magnetic scattering, the data is presented in termsof the spin-asymmetry ratio SA = ( 𝑅 + − 𝑅 − )/( 𝑅 + + 𝑅 − ) , asdepicted in Fig. S5, as well as in Figs. 4(a) and 4(b) in themain text. A value of SA = nuclear and magnetic scattering length densities ( NSLD and
MSLD ) cor-respond to the depth profiles of the chemical and in-planemagnetization vector distributions, respectively. The magne-tization can then be calculated from the MSLD data using therelation 𝑀 ( emu / cc ) = MSLD ( Å )/( . × − ) .Complementary X-ray reflectometry (XRR) data has beenused to verify the depth morphology of the films. The relatedexperiments were carried out on the bilayers with Ga thick-ness fixed at 25 nm, while the Ni thickness varied (i.e., theNi thickness was 0 .
8, 2 .
4, 4 .
0, and 5 . . . 𝑅 + )and spin-down ( 𝑅 − )]. For samples containing 2 . . . . ± / cc induced magnetizationover roughly 6 nm in Ga, while the magnetization inside theNi film was about 136 ± / cc and uniform.Spin-polarized tunneling studies investigatingAl/Al O /Ni/Ga tunnel junctions through Meservey–Tedrow spectroscopy and with a Zeeman-split supercon-ducting Al spin detector revealed a very similar magneticbehavior as a function of the Ni/Ga bilayers’ Ni thickness—forNi films thinner than 4 nm, it was hard to detect any spin po-larization 𝑃 (could be as small as 0 . 𝑃 couldbe clearly measured for 4 nm and larger Ni thicknesses—,being fully consistent with our PNR observations. Thetunneling measurements were performed at 0 . . [S1] M. Octavio, M. Tinkham, G. E. Blonder, and T. M. Klapwijk,Phys. Rev. B , 6739 (1983).[S2] K. Flensberg, J. Bindslev Hansen, and M. Octavio, Phys. Rev.B , 8707 (1988).[S3] C. W. Groth, M. Wimmer, A. R. Akhmerov, and X. Waintal,New J. Phys. , 063065 (2014).[S4] P. M. Tedrow and R. Meservey, Phys. Rev. Lett. , 192 (1971).[S5] P. M. Tedrow and R. Meservey, Phys. Rev. B , 318 (1973).[S6] R. Meservey and P. M. Tedrow, Phys. Rep. , 173 (1994).[S7] G. B. Arnold, J. Low Temp. Phys. , 143 (1985).[S8] G. B. Arnold, J. Low Temp. Phys. , 1 (1987). [S9] A. F. Andreev, Zh. Eksp. Teor. Fiz. , 1823 (1964); J. Exp.Theor. Phys. , 1228 (1964).[S10] Y. M. Blanter and F. W. J. Hekking, Phys. Rev. B , 024525(2004).[S11] J. M. Rowell and W. L. Feldmann, Phys. Rev. , 393 (1968).[S12] M. Kuhlmann, U. Zimmermann, D. Dikin, S. Abens, K. Keck,and V. M. Dmitriev, Z. Phys. B , 13 (1994).[S13] V. Lauter-Pasyuk, Collect. Soc. Fr. Neutron , s221 (2007).[S14] V. Lauter-Pasyuk, H. J. Lauter, V. L. Aksenov, E. I. Kornilov,A. V. Petrenko, and P. Leiderer, Physica B248