Abstract
The long-range spin-triplet supercurrent transport is an interesting phenomenon in the superconductor/ferromagnet (
) heterostructure containing noncollinear magnetic domains. Here we study the long-range superharmonic Josephson current in asymmetric 
junctions. It is demonstrated that this current is induced by spin-triplet pairs 
− 
or 
+ 
in the thick 
layer. The magnetic rotation of the particularly thin 
layer will not only modulate the amplitude of the superharmonic current but also realise the conversion between 
− 
and 
+ 
. Moreover, the critical current shows an oscillatory dependence on thickness and exchange field in the 
layer. These effect can be used for engineering cryoelectronic devices manipulating the superharmonic current. In contrast, the critical current declines monotonically with increasing exchange field of the 
layer and if the 
layer is converted into half-metal, the long-range supercurrent is prohibited but 
still exists within the entire 
region. This phenomenon contradicts the conventional wisdom and indicates the occurrence of spin and charge separation in present junction, which could lead to useful spintronics devices.
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Introduction
Superconductor/ferromagnet (
) hybrid structure has recently attracted considerable attention because of the potential applications in spintronics and quantum information1,2,3 as well as the display of a variety of unusual physical phenomena4,5,6,7. In general, if a weak F is adjacent to an s-wave S and there is no interfacial spin-flip scattering, the normal Andreev reflection will generate at 
interfaces. The process involves an electron incident on the 
interface from the F at energies less than the superconducting energy gap. The incident electron forms a Cooper pair in the S with the retroreflection of a hole of opposite spin to the incident electron. Consequently, the conventional spin-singlet Cooper pair decays at a short range in ferromagnetic region. In 
Josephson junctions with homogeneous magnetization, through the normal Andreev reflection occurring at two 
interfaces, a Cooper pair is transferred from one S to another, creating a supercurrent flow across the junction8. As a consequence of the exchange splitting of the Fermi level of the F, the Cooper pair shows an oscillatory manner superimposed on an exponential decay in the F. Correspondingly, the Josephson current displays a damped oscillation with increasing the thickness or the exchange field of the F, leading to the appearance of the so-called “0-π transition”1,2. In general, the normal Andreev reflection will be suppressed by the exchange field of the F, so the Josephson current just can transport a short distance.
In contrast, if one insert a thin spin-active F layer with noncollinear magnetization into the 
interface, it is found that the noncollinear magnetization can lead to a spin-flip scattering, then the reflected hole has the same spin as the incident electron, which is identified as anomalous Andreev reflection. When this reflection takes place at two 
interfaces, the parallel spin-triplet Cooper pairs 
are generated in the central F layer and can penetrate into F layer over a long distance unsuppressed by the exchange interaction, so that the proximity effect is enhanced. The induced long-range current manifests itself as a large first harmmonic (
) in the spectral decomposition of the Josephson current-phase relation 
8.
It is worth to point that, if the central F layer is converted into fully spin-polarized half-metal, in which electronic bands exhibit insulating behavior for one spin direction and metallic behavior for the other, the normal Andreev reflection will be inhibited completely due to inability to form a pair in the S and impossibility of single-particle transmission. However, the strength of the anomalous Andreev reflection can not be strongly influenced by the spin-polarization of the F and the transport processes of 
(or 
) in the F region will continue to take place. In response, several different inhomogeneous configurations have been proposed for studying such enhanced proximity effect9,10,11,12,13,14,15. The corresponding experiments have proved these physical process and observed the strong enhancement of the long-range spin-triplet supercurrents16,17,18,19,20,21.
Different from the configurations mentioned above, it has proposed a long-range proximity effect develops in highly asymmetric 
junction composed of thick 
layer and particularly thin 
layer with noncollinear magnetizations at low temperatures22,23,24. This effect arises from two normal Andreev reflections occurred at normal 
interface and two anomalous Andreev reflections at spin-active 
interface. The long-range spin-triplet correlations in this junction give the dominant second harmonic (
) in current-phase relation 23, which is known as superharmonic Josephson current22. Recently, Iovan et al.25 experimentally observed the long-range supercurrent through above junction. This second harmonic can be manifested as half-integer Shapiro steps that can be experimentally observed26 and the two times smaller flux quantum will be obtained, leading to more sensitive quantum interferometers (SQUIDs)27. It should be stressed that refs 22, 23, 24 did not discuss the difference of long-range triplet pairing fashion between asymmetric 
junction and symmetric 
. Moreover, it is high desirable to clarify the effect of the misorientation angle on the triplet pairing correlations in the 
junction, as well as the influence of the thickness and the exchange field in two ferromagnetic layers on the Josephson current and the long-range spin-triplet correlations.
In this work, we study the relation between the long-range superharmonic Josephson current and the spin-triplet pairing correlations in 
junction. It is proposed that the superharmonic Josephson current is induced by the spin-triplet pairs 
− 
or 
+ 
in the long 
layer. The variation of the misorientation angle between two magnetizations will not only turn the amplitude of the superharmonic current but also realize the conversion between 
− 
and 
+ 
. This can be used to control the superharmic current and the pairing fashion in the 
layer through modulating the magnetic structure of the 
layer. Besides, the critical current shows an oscillatory dependence on the thickness and exchange field of the highly thin 
layer. These effect can be used for engineering cryoelectronic devices manipulating spin-polarized supercurrent. In contrast, the critical current decreases monotonically with increasing exchange field of the 
layer. Specifically, if the 
layer is converted into half-metal, the long-range Josephson current will be completely prohibited, but 
still exist in 
region. This phenomenon indicates the occurrence of spin and charge separation in present 
junction which could lead to useful spintronics devices. These results also contradict the traditional view: the long-range Josephson current is determined by the parallel spin-triplet pairs in the multilayer junction with noncollinear magnetization alignment between ferromagnetic layers. At last, it is also found that the magnetization of the 
layer will bring about a same direction magnetization in the 
layer on condition that the magnetic moment of the 
layer is weak.
To be more precise, we consider the Josephson junction consists of two s-wave superconducting electrodes and ferromagnetic bilayer with noncollinear magnetizations. The schematic picture of the 
device is presented in Fig. 1. One assume that the transport direction is along the y axis and the system satisfies translational invariance in the x-z plane. The thicknesses of 
layer and 
layer are 
and 
, respectively. The exchange field 
due to the ferromagnetic magnetizations in the 
layer is described by 
. Here 
is the tilt angle from the z axis and 
is the horizontal angle respect to x axis.
Schematic illustration of the S/F1/F2/S Josephson junction containing a bilayer ferromagnet.
Thick arrows in F1 layer and F2 layer indicate the directions of the magnetic moments. The phase difference between the two s-wave Ss is 
.
Results
Based on the extended the Blonder-Tinkham-Klapwijk (BTK) approach28,29,30,31, the dc Josephson current in the 
junction can be expressed as follows
where 
are the Matsubara frequencies with 
and 
. 
are the perpendicular components of the wave vectors for electron-like (hole-like) quasiparticles in superconducting regions and 
with 
are the scattering coefficients of the normal Andreev reflection under the condition of four different incoming quasiparticles, electron-like quasiparticles (ELQs) and hole-like quasiparticles (HLQs) with spin up and spin down. Then the critical current is derived from 
.
By applying the Bogoliubov’s self-consistent field method32,33, the triplet pair amplitudes are defined as follows34:
where 
and equal-spin pair amplitude will be denoted by 
. The singlet pair amplitude writes as 
. In this paper, the singlet and triplet pair amplitudes are all normalized to the value of the singlet pairing amplitude in a bulk superconducting material. The LDOS is given by34
where 
is the derivative of the Fermi function. The LDOS is normalized to unity in the normal state of the S material. In addition, the local magnetic moment in the 
geometry has three components34.
where 
and 
are the Bohr magneton and the Fermi function, respectively. It is convenient to normalize these components to 
.
Unless otherwise stated, in BTK approach we use the superconducting gap 
as the unit of energy. The Fermi energy is 
, the interface transparency is 
and 
. We measure all lengths and the exchange field strengths in units of the inverse of the Fermi wave vector 
and the Fermi energy 
, respectively. The magnetization in the 
layer is fixed along the z direction (
, 
), while the 
is a free layer in which the magnetization points any direction. In Bogoliubov’s self-consistent field method, we consider the low-temperature limit and take 
, 
. The other parameters are the same as the ones mentioned before.
Discussion
Superharmonic currents versus misalignment angle
From Fig. 2 one can clearly see that the critical current reaches maximum for perpendicular magnetizations (
) and decreases to minimum as the magnetizations are parallel (
) or antiparallel (
) to each other. However, the variation of the angle 
can not lead to the change of critical current while keeping 
constant. It is known that characteristic variations of the critical current 
with the misaligned angles (
, 
) are related to the nature of pairing correlations. Figure 3 shows the spatial distribution of the spin-triplet pair amplitudes for different misalignment angle 
at fixed 
. It is found that the real part of 
and 
can not penetrate entire 
layer, but their image parts can be distributed throughout this region. With increasing 
, the left parts of 
are almost unchanged, however, their right parts gradually decrease. Correspondingly, the amplitudes of 
increase and turn to maximum at 
. The main reason is because the x-projection of misaligned magnetic moment in the 
layer can generate two separate effects: spin-mixing and spin-flip scattering process9. The former will result a mixture of singlet pairs and triplet pairs with zero spin projection 
− 
+ 
, where 
, 
is the Fermi velocity and R is the distance from the 
interface. The latter can convert 
into the parallel spin-triplet pairs 
3. These parallel spin pairs will penetrate coherently over a long distance into the 
layer. So the transport of 
can make a significant contribution to superharmonic Josephson current. Meanwhile, the period of this current becomes π and satisfies the second harmonic current-phase relation 
22,24. By contrast, in the Josephson junction with ferromagnetic trilayer only spin-triplet pairs 
(or 
can transmit in central ferromagnetic layer, which provide the main contribution to the long-range first harmonic current35.
Critical current as a function of the orientation angle (θ2, φ2) of the F2 layer.
Here we set 
, 
, 
and 
.
The spin-triplet pair amplitudes f0 and f1 plotted as a function of the coordinate kFy for several values of θ2 in the case of φ2 = 0.
The left panels show the real parts while the right ones show the imaginary parts. The dotted vertical lines represent the location of the 
, 
and 
interfaces. Here 
, 
, 
, 
, 
and 
. All panels utilize the same legend.
As plotted in Fig. 4, in the case of collinear orientation of magnetizations (
), the current 
is weak enough and present a first harmonic feature. At this time, the long-range spin-triplet pairs 
are absent, so the LDOS in the 
layer is almost equal to its normal metal value. With increasing 
, the magnitude of the second harmonic current is enhanced by the increased number of 
. Specifically, for orthogonal magnetizations (
), the second harmonic current grows big enough. Correspondingly, the LDOS is significantly enhanced with two distinguishable peaks. Moreover, the spatial profile of the local magnetic moments are plotted for several values of 
in Fig. 5. What’s most interesting is that the component 
grows very quickly in the 
region with increasing 
and also displays the penetration of the same component into the 
region. The induced 
in the 
region does not only change magnitude as a function of position, but it also rotates direction. However, the component 
in the 
region will gradually decrease with 
and remains almost unchanged in the 
region.
(a) the Josephson current-phase relation Ie(ϕ) for four values of the relative angle θ2 between magnetizations. (b) The normalized LDOS in the F1 layer (
) plotted versus the dimensionless energy 
for different θ2 and the results are calculated at 
. Other parameters are the same as in Fig. 3.
The x (top panels) and z components (bottom panels) of the local magnetic moment plotted as a function of the coordinate kFy for different θ2.
The left panels show the behaviours over the extended F1 regions while the right ones show the detailed behaviours in the F2 layer. Other parameters are the same as in Fig. 3.
As stated above, the variation of the horizontal angle 
can not influence the Josephson current as the tilt angle 
has a fixed value. However, the change of 
will induced a conversion of pairing fashion in the 
region. As shown in Fig. 6, on the condition of 
, 
decrease gradually from a finite value to zero with increasing 
, but 
exhibit the opposite characteristics. These phenomena can be explained as follows: since the magnetic direction of the 
layer is oriented along the x axis (
, 
), 
+ 
in the 
layer can be converted into 
in the 
layer. In contrast, if the magnetic moment of the 
layer is along y axis (
, 
), 
+ 
will be transformed into 
+ 
, which can also penetrate into the 
region a long distance and make a major contribution to the second harmonic current. At the same time, when the magnetization direction of the 
layer rotates from the x axis to the y axis, the induced magnetic moment in the 
layer would correspondingly turn from 
to 
, as seen in Fig. 7. In what follows, we focus on the dependence of the critical current on the thickness and exchange fields of two ferromagnetic layers under the condition of 
.
The spin-triplet pair amplitudes f1 [(a,b)] and f2 [(c,d)] plotted as a function of the coordinate 
for several values of 
in the case of 
. The left panels [(a,c)] show the real parts while the right ones [(b,d)] show the imaginary parts. Other parameters are the same as in Fig. 3.
The x (top panels) and y components (bottom panels) of the local magnetic moment plotted as a function of the coordinate kFy for different φ2.
The left panels show the behaviours over the extended F1 region while the right ones show the detailed behaviours in the F2 region. Other parameters are the same as in Fig. 3.
Superharmonic currents versus thickness and exchange field of the spin-active 
layer
layerFigure 8 shows the dependence of the critical current Ic on the length 
and exchange field 
for different misalignment angle 
when the 
layer has fixed values 
and 
. One can see that Ic is sufficiently weak and decays in an oscillatory manner in parallel (
) and antiparallel (
) alignments of the magnetizations. This is because the exchange field in the F2 layer induces a splitting of the energy bands for spin up and spin down. This effect can make 
oscillate with a period 
and simultaneously decay exponentially on the length scale of 
1. Here, 
is the magnetic coherence length. In this case, only the spin-singlet pairs 
− 
and spin-triplet pairs 
+ 
exist in the ferromagnetic layer. These two types of pairs can be suppressed by the exchange field of ferromagnetic layer and mainly provide the contribution to the first harmonic current.
Critical current (a) as a function of 
and θ2 for 
and (b) as a function of 
and θ2 for 
. We set 
, 
and 
.
On the other hand, if the orientations of the magnetic moments are perpendicular to each other (
), Ic also displays the oscillated behaviour with increasing 
, but its order of magnitude is larger than for collinear magnetizations. This characteristic behaviour can be attributed to the spatial oscillations of 
+ 
in the F2 region with period Q · R. It is well known that the Cooper pair in the F2 layer will acquire a total momentum Q because of the spin splitting of the energy bands. As described in ref. 36, for a fixed Q the amplitude of 
+ 
will vary with the length R (=kFL2) of the F2 layer. As a result, the oscillated 
+ 
can be converted into 
− 
in the F1 layer by the spin-flip scattering and then 
− 
can propagate over long distance in the F1 layer and lead to the enhanced superharmonic current. Similarly, if one fixes 
and changes 
, the same features about the critical current can be obtained (see Fig. 8(b)). It is worth mentioning that this oscillatory behaviour could be different from the oscillation of the critical current with the thickness of F2 layer in 
junction36, because the supercurrent in the central F1 layer derives from the contribution of 
and manifests itself as a dominant first harmonic in the Josephson current-phase relation.
Superharmonic currents versus length and exchange field of the long F1 layers
In Fig. 9 the dependence of the critical current Ic on exchange field 
and length 
are plotted for 
. Compared with the Josephson junctions with homogeneous magnetization, Ic in this asymmetric junctions decreases slowly with increasing 
on the weak or moderate exchange fields. This feature illustrates that 
− 
will propagate coherently over long distances in the F1 layer. Furthermore, Ic are almost monotonically decreasing with 
for various 
and will be prohibited completely at 
. It indicates that the superharmonic current will be suppressed by the exchange field of the F1 layer. This phenomenon is clearly different from the first harmonic current in the half-metal Josephson junction with interface spin-flip scattering9,16, because the first harmonic current induced by 
can not be suppressed by the exchange splitting.
Critical current as a function of 
and 
We set 
, 
, 
and 
.
In order to clearly explain the contribution of the spin-triplet pairs to the superharmonic current, we choose a fixed length 
for discussion, as illustrated by the red line in Fig. 9. Under such conditions, we plot the distribution of the spin-triplet pairing functions f0, f1, 
and 
for three exchange fields 
, 0.5 and 1.0 in Fig. 10. With increasing 
, the magnitude of f0 and f1 in the F1 region are all reduced and f0 drops to zero at 
. The reason can be summarized as follows: for weak exchange field 
the triplet correlations 
and 
will generate in the F2 region and then combine into f1 in the F1 region. f1 decay spatially with approaching the 
interface due to the fact that the pairs 
and 
are recombined into the pairs 
and 
by the normal Andreev reflections. For 
, 
and 
near the 
interface are both restrained. By contrast, 
adjacent to the 
interface increases instead. Moreover, because 
on the left side of F1 layer is suppressed, the recombination effect at the 
interface becomes weakened, in which case the superharmonic current will decrease. For a fully spin-polarized half-metal (
), Fig. 10(d) shows that 
will be completely suppressed, but 
does not vanish and it’s magnitude seems to be a slight increase in the vicinity of the 
interface (see Fig. 10(c)). These characters can be attributed to the contributions from two important phenomena taking place at the 
interface: normal Andreev reflections and normal reflections, as shown in Fig. 11 (a,b), respectively.
The imaginary parts of f0 (a), f1 (b), 
(c) and 
(d) plotted as a function of the coordinate 
for several 
. We set 
, 
, 
, 
, 
, 
and 
.
Two types of transference about the pairs of correlated electrons and holes.
(a) The first one consists of two normal Andreev reflections occurred at 
interface and two anomalous Andreev reflections at 
interface in the case of weak exchange field in the F1 layer. (b) The second one consists of two normal reflections at 
interface and two anomalous Andreev reflections at 
interface while the F1 layer is converted into half-metal.
If the exchange field 
is weak enough, the normal Andreev reflections will mainly occur at the 
interface, which provide the main contribution to Ic. In this case, the number of the pairs 
approximately equal to 
and then 
and 
can combine into 
. Subsequently, 
− 
can be converted into 
− 
in the left S. With increasing 
, the normal Andreev reflections are gradually being replaced by the normal reflections and the difference in the number of 
and 
will enlarge simultaneously. As a result, the transition from 
− 
to 
− 
occurred at the 
interface will be weakened. In the fully spin-polarized case (
) the absence of the spin down electrons makes it impossible to generate the normal Andreev reflections at 
interface and therefore the Josephson current is completely suppressed but 
still exist. As depicted in Fig. 11(b), the electron transfer process is analogous to the unconventional equal-spin Andreev-reflection process reported in Ref. 37. Look at the whole picture, it is easy to understand the above process: 
injecting from the right S is converted into 
in the F1 layer and 
will be consequently reflected normally back as 
at the 
interface. Then 
is transformed into 
by the spin-flip scattering of the F2 layer. At last, 
transports to the right S. In the whole process, none of Coopers can penetrate into the left S, so the Josephson current would be suppressed completely.
In order to facilitate the experimental observations for the future, we plot the current-phase relation and the LDOS in the F1 layer at three points 
, 0.5 and 1.0 in Fig. 12. With increasing 
, the superharmonic current 
decreases and two distinguishable peaks in the LDOS will become weak correspondingly. It’s particularly noteworthy that if 
Josephson current was completely suppressed but the LDOS displays a sharp zero energy conductance peak which marks the presence of 
. It can be measured in principle by STM experiments. And this feature is different from the conventional views: (i) The long-range triplet Josephson current is proportional to the parallel spin-triplet pairs 
or 
. (ii) If the long-range triplet supercurrent passes through the Josephson junction, there will present the zero energy conductance peak in the LDOS of F. Finally, we discuss the influence of 
on the local magnetic moment. As can be seen from Fig. 13, in the F1 region Mz will grow with the increase of 
, but the induced Mx could be suppressed. For 
, Mz reaches maximum but Mx will disappear. By contrast, Mx in the F2 region hardly changes with 
and Mz will partly permeate into the F2 layer.
(a) the Josephson current-phase relation Ie(ϕ) for different 
. (b) The normalized LDOS in the F1 layer (
) plotted versus the dimensionless energy 
and the results are calculated at 
. Other parameters are the same as in Fig. 10.
The x (top panels) and z components (bottom panels) of the local magnetic moment plotted as a function of the coordinate kFy for different h1/EF.
The left panels show the behaviours over the extended F1 region while the right ones show the detailed behaviours in the F2 layer. Other parameters are the same as in Fig. 10.
To summarize, we have studied the long-range superharmonic Josephson current and the spin-triplet pairing correlations in the asymmetric 
junction. We have shown that the superharmonic current was induced by the spin-triplet pairs 
− 
or 
+ 
in the long F1 layer. The rotation of the magnetic moment in the thin spin-active F2 layer will not only modulate the amplitude of the superharmonic current through the junctions, but also realize the conversion from 
− 
to 
+ 
in the F1 layer. Besides, the critical current oscillates with the length and exchange field in the F2 layer. These features provide an efficient way to control the superharmonic current and the spin-triplet pairing fashion by changing the magnetic moment of the F2 layer. Specifically, the critical current almost decreases monotonically with the exchange field of the F1 layer and if the F1 layer is converted into half-metal, the Josephson current disappear completely but the spin-triplet pairs 
still exist within the entire F1 layer. This behavior is different from the conventional view about the relationship between the long-range current and the parallel spin-triplet pairs in the junctions with ferromagnetic trilayers. These results therefore indicated that the spin and charge degrees of the freedom can be separated in practice in the junction with ferromagnetic bilayers and suggested the promising potential of these junctions for spintronics applications.
Methods
The BCS mean-field effective Hamiltonian is given by1,32
where 
is the single-particle Hamiltonian, 
and 
are creation and annihilation operators with spin α. 
and EF denote Pauli matrix and the Fermi energy, respectively. 
describes the superconducting pair potential with 
. Here 
accounts for the temperature-dependent energy gap. It satisfies the BCS relation 
, where 
is the energy gap at zero temperature and Tc is the superconducting critical temperature. 
is the unit step function and 
is the phase of the left (right) S.
By making use of the Bogoliubov transformation 
and the anticommutation relations of the quasiparticle annihilation and creation operators 
and 
, we have the Bogoliubov-de Gennes (BdG) equation1,32
Blonder-Tinkham-Klapwijk approach
The BdG equation (10) can be solved for each superconducting electrode and each F layer, respectively. For an incident spin up electron in the left S, the wave functions in the S leads and the Fp layer are
Here 
, 
, 
, 
are basis wave functions. Quasiparticle amplitudes are defined as 
and 
with 
. The perpendicular components of the ELQs (HLQs) wave vector in S leads and Fp layer are given by 
and 
with 
, respectively. It is worthy to note that the parallel component 
is conserved in transport processes of the quasiparticles. The matrix can be defined as38
The coefficients 
, 
, 
and a1 describe normal reflection, the normal reflection with spin-flip, anomalous Andreev reflection and normal Andreev reflection, respectively. 
(r = 1–8) are quasiparticles wave function amplitudes in the Fp layer. Likewise, c1, d1, 
and 
are the quasiparticles transmission amplitudes in the right superconducting electrode. All scattering coefficients can be determined by solving the continuity conditions of the wave function and its derivative at the interface
Here 
are dimensionless parameters describing the magnitude of the interfacial resistances. 
are local coordinate values at the interfaces and 
is the Fermi wave vector. From the boundary conditions, we obtain a system of linear equations that yield the scattering coefficients.
Bogoliubov’s self-consistent field method
We put the 
junction in a one-dimensional square potential well with infinitely high walls, then the eigenvalues and eigenvectors of the BdG equation (10) have the following changes: 
and 
. Accordingly, the corresponding quasiparticle amplitudes can be expanded in terms of a set of basis vectors of the stationary states39, 
= 
and 
with 
. Here, q is a positive integer and 
. LS1 and LS2 are the thicknesses of the left and right superconducting electrodes, respectively. The superconducting pair potential in the BdG equation (10) is determined by the self-consistency condition32
where the primed sum of En is over eigenstates corresponding to positive energies smaller than or equal to the Debye cutoff energy ωD and the superconducting coupling parameter g(y) is a constant in the superconducting regions and zero elsewhere. Iterations are performed until self-consistency is reached, starting from the stepwise approximation for the pair potential.
Additional Information
How to cite this article: Meng, H. et al. Long-range superharmonic Josephson current and spin-triplet pairing correlations in a junction with ferromagnetic bilayers. Sci. Rep. 6, 21308; doi: 10.1038/srep21308 (2016).
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Acknowledgements
This work is supported by the State Key Program for Basic Research of China under Grants No. 2011CB922103 and No. 2010CB923400, the National Natural Science Foundation of China under Grants No. 11174125, No. 11074109, No. 51106093 and No. 11447112, the Scientific Research Program Funded by Shaanxi Provincial Education Department Grant No. 12JK0972 and No. 15JK1132, the Scientific Research Foundation of Shaanxi University of Technology Grant No. SLG-KYQD2-01. J. Wu would like to thank the Shenzhen Peacock Plan and Shenzhen Fundamental Research Foundation Grant No. JCYJ20150630145302225.
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H.M. and J.S.W. conceived the research and performed the calculations. X.Q.W. and M.Y.R. gave scientific advice. H.M. wrote the main manuscript text and J.S.W. made an improvement. Y.J.R. prepared all the figures. All authors discussed and reviewed the manuscript.
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Meng, H., Wu, J., Wu, X. et al. Long-range superharmonic Josephson current and spin-triplet pairing correlations in a junction with ferromagnetic bilayers. Sci Rep 6, 21308 (2016). https://doi.org/10.1038/srep21308
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DOI: https://doi.org/10.1038/srep21308
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