Double-slit Fraunhofer pattern as the signature of the Josephson effect between Berezinskii superconductors through the ferromagnetic vortex
M. A. Silaev


 



 
 Department of
Physics and Nanoscience Center, University of Jyväskylä, P.O.
Box 35 (YFL), FI-40014 University of Jyväskylä, Finland


 
 I apply the recently developed formalism of generalized quasiclassical theory  to show that using 
 hybrid superconducting systems with non-collinear strong ferromagnets one can realize the Josephson junction between Berezinskii-type superconductors.   The reported calculation  reproduces main features observed in the recent experiment, namely the the 
 slightly asymmetric double-slit Fraunhofer interference pattern of the Josephson current through the ferromagnetic vortex. 
 The double-slit structure results from the spatially inhomogeneous Berezinskii state with the amplitude controlled by the 
 local angle between magnetic moments in two ferromagnetic layers. The critical current asymmetry  by the sign of magnetic field 
 can signal the presence of spontaneous supercurrents generated by the non-coplanar magnetic texture near the 
 core of the ferromagnetic vortex core. I demonstrate that ferromagnetic vortex can induce spontaneous vorticity in  
 the odd-frequency order parameter manifesting the possibility  of the emergent magnetic field to create topological defects.
   


  Double-slit Fraunhofer pattern as the signature of the Josephson effect between Berezinskii superconductors through the ferromagnetic vortex. 
    M.A. Silaev
    December 30, 2023
===============================================================================================================================================



During the recent years large attention has been devoted to the studies of long-range proximity
effect and spin-polarized Josephson currents carried by the spin-triplet Cooper pairs in superconductor/ferromagnet/superconductor (S/F/S) heterostructures<cit.>. 
This interest is motivated the
possible applications of spin-polarized superconducting currents in spintronics. Much effort is invested to the studies of tunable spintronic elements where the Josephson current <cit.> or the
 critical temperature <cit.> are controllable by the magnetic degrees of freedom. One of the possible ways for the implementation of such a devise has been suggested in my work <cit.> 
 employing the well-controlled properties of the nanomagnets with vortex-like magnetization patterns. In that proposal I have demonstrated that it is possible to gain the effective control over the long-range proximity effect by tuning the position of ferromagnetic (FM) vortex 
with the help of external in-plane magnetic field. 


 
 

Recently the conceptually similar devise has been realized experimentally<cit.>. 
Schematically this system is shown in the Fig.(<ref>)a. It consists of two spin-textured ferromagnets 
F and F' with the magnetizations M and M^' and the superconducting layer S. The thin layer F' is in the
contact with superconductor and contains a gap. The Josephson current flows through the layer F which contains FM vortex.
It has been observed that the presence of FM vortex in the system results in the non-trivial modification of the 
critical current dependence I_c= I_c(H) as a function of the external magnetic  field H.
The striking features produced by the FM vortex is that this dependence 
becomes similar to the double-slit Fraunhofer interference 
pattern instead of the usual single-slit form produces by the homogeneous junction. 

 I demonstrate below that this behaviour can be considered as
 the direct experimental signatures of the Josephson effect between Berezinskii-type superconductors characterized by the spin-triplet 
 odd-frequency s-wave order parameters 
 <cit.>. Such superconducting state is 
 induced in the F layer in result of the combined effect of Zeeman splitting in the S electrode and filtering of equal-spin (ES) 
 Cooper pairs in F. Qualitatively the unusual Fraunhofer patterns are explained by the 
 intrinsic inhomogeneity of the induced Berezinskii state which depends on the relative orientation of the 
 local magnetic moments in F and F' layers,  M and M^' respectively.  

 The amplitude of induced order parameter is defined by the polar angle θ between M and M^', see Fig.<ref>b.
 The variation of θ=θ (x) resulting from the magnetic texture along the junction leads to the 
 the experimentally observed  double-slit pattern of the  critical current<cit.>.
 Besides that, the azimuthal angle α produces spin-dependent phases of the order parameter components.
 The variation of α arises in non-coplanar magnetic textures and in strong ferromagnets with lifted spin degeneracy it produces 
 spontaneous supercurrents <cit.>. 
 The spatial dependence of α= α (x) along the junction is therefore equivalent to the 
 emergent localized magnetic field and results in the critical current asymmetry I_c(H) ≠ I_c(-H) which 
 can signal the presence of spontaneous supercurrents trough the FM vortex. 
  
  
 The system shown in Fig. <ref>a can be described analytically using 
several assumptions. First, the layers are supposed to be thin enough to neglect the variation magnetization fields along the z
coordinate. Next, the layer F' is assumed to induce an effective exchange field
within the superconducting electrode h_s∥ M^' . This model is justified by the small thickness of the F'
layer which allows to neglect the variation of correlation functions along z. The Green's functions in F'/S
layer can be found considering just the  superconducting layer with the thickness-averaged exchange field<cit.>
h_s =  h_F^'d_F^'/(d_F^' +d_S) where 
h_F^'∥ M^' is the exchange field in F', d_F^' and d_S
 are the thickness of the F' and S layers correspondingly. For simplicity the density of states is taken equal in S and F'. 
This exchange field leads to the significant critical temperature suppression of the S/F'/F structure observed in the 
experiment<cit.>. 
The role of this exchange field is to produce the mixed-spin triplet Cooper pairs which can be converted into the
equal-spin correlations (ESC) in the ferromagnetic layer due to the spin-dependent tunnelling <cit.>.

 The F layer hosting FM vortex has rather large thickness as compared to the mean free path, so that only the the equal-spin correlations (ESC)  
 residing on separate spin-split Fermi surfaces can penetrate to the full thickness. The generalized Usadel equation describing ESC quasiclassical 
 propagators ĝ_σ derived in Ref.(<cit.>) reads
 
    D_σ∂̂(ĝ_σ∂̂ĝ_σ) - [Δ̂_σ+ωτ̂_3, ĝ_σ] =0

 where σ= ± 1 is the spin subband index, D_σ are the spin-dependent diffusion coefficients. 
 The covariant differential operator is
 
  
    ∂̂_ r = ∇_ r + i σ Z  [τ̂_3, ·] - 
      i e  A [τ̂_3, ·] ,

  
  where A is the electromagnetic vector potential,  
  Z is the adiabatic spin gauge field, and D_σ are the spin-dependent diffusion coefficients. 
  The generalized Usadel equation (<ref>) is supplemented by the expression for the current
  
  
    j = iπ T e ∑_σ =±∑_ων_σ D_σ Tr[τ_3 ĝ_σ∂̂_ rĝ_σ] ,

  where ν_± is the spin-up/down density of states (DOS).
  
  The effective spin-dependent order parameter Δ̂_σ in Eq.(<ref>) describing the ES components 
  of the proximity-induced order parameter can be obtained from the non-diagonal part of the general tunnelling self-energy  
  <cit.>
  
  
    Σ̂= γΓ̂F̂Γ̂^† ,

  
  where F̂ = (g_01σ_0 + g_31σ n_h)e^iχτ_3τ̂_1 is the anomalous Green's function in the  
  superconductor with exchange field, n_h =  h_s/h_s. In the absence of spin-orbital relaxation the spin-singlet and 
  spin-triplet parts are given by 
  g_01= [F_bcs(ω+ih_s)+F_bcs(ω-ih_s)]/2 and 
  g_31 = [F_bcs(ω+ih_s)-F_bcs(ω-ih_s)]/2, where 
  F_bcs(ω) = Δ/√(ω^2+ Δ^2).
  The spin-polarized tunnelling matrix has the form 
  Γ̌= t σ̂_0 τ̂_0 + u σ m τ̂_3 where m =  M/M is the direction 
  of magnetization in the ferromagnet and γ = σ_n R is the parameter describing the barrier strength, 
  R is the normal state tunnelling resistance per unit area. The normalized tunnelling coefficients are
  t=√(1+√(1-P^2))/2, u=√(1+√(1-P^2))/2 and P is the effective spin-filtering coefficient 
  that ranges from 0 (no polarization) to 1 (100 % filtering efficiency).
  
  Then the ES component of the tunnelling self-energy is given by
  
 
    Σ̂_ES = γ g_31 e^iχτ̂_3
     [ τ̂_1 σ h_s + P τ̂_2 σ ( h_s× m)] .

 
 Projecting Σ̂_ES to the spin-up and spin-down states with respect to the quantization axis set 
 by the magnetization m we get the ES components of the effective spin-triplet order parameter
 
 
    Δ̂_σ = h_⊥Δ_Fστ_1 e^iτ_3 (χ + σα) ,

 
 with the amplitude given by Δ_Fσ = -( 1+ σ P) g_31 and the prefactor
 h_⊥ = sinθ which is the projection of exchange field to the plane perpendicular to the magnetization direction,
  θ is the polar angle of h_s in the coordinate system 
 defined by m. The distribution of sinθ can be obtained by the micromagnetic simulations<cit.> 
 for the realistic geometry. The additional spin-dependent phase α in Eq.(<ref>) 
 is defined by the azimuthal angle of h_s as shown in Fig.(<ref>)b. 
 For the in-plane textures when both m and h_s lie in the xy plane the additional phase in 
 Eq.(<ref>)is absent α =0.  
 
However near FM vortex core the texture becomes non-coplanar which leads to 
the gradients of α, which are coupled to the spin gauge field<cit.> Z so that the combination 
 
    V_s = σ( ∇α - 2 Z)

 has the meaning of the texture-induced part of the superfluid velocity. 
In strong ferromagnets with broken spin degeneracy V_s produces spontaneous charge supercurrents<cit.> 
resulting in the shift of the Fraunhoffer pattern as shown below. 

 Due to the symmetry g_31 (ω) =  - g_31 (-ω) the pairing amplitude in Eq. (<ref>) represents 
 the odd-frequency spin-triplet s-wave superconducting  order parameter suggested by Berezinskii <cit.> and intensively studied 
 afterwards<cit.>. 
 The odd superconducting correlations has been studied in  several setups with proximity- induced superconductivity in ferromagnets  <cit.>, 
 normal metals<cit.>, topological insulators
 <cit.> and non-equilibrium systems<cit.>.  
 Usually due to the broken translational or/and spin-rotation symmetries the total Cooper wave 
 function in proximity systems is a superposition of odd- and even- frequency components which inevitably coexist at one and the same 
 point<cit.>, except the discrete set of points in the 
 cores of proximity-induced vortices<cit.>.
 In contrast, the order parameter in Eq.(<ref>) 
 represents the pure Berezinskii state without admixtures of spin-singlet and/or  even-frequency components. 
 Therefore the setup shown in Fig.(<ref>)a consisting of two non-collinear strong ferromagnets emulates 
 the Josephson effect between two Berezinskii superconductors through the FM vortex. 
 Note that the absence of even-frequency spin-singlet pairings in the Eq.(<ref>) is not exact, 
 since the singlet correlations can strictly speaking penetrate even to the strong ferromagnetic layers. 
 However in the dirty regime the singlet amplitude is exponentially suppressed at 
 distances larger than the mean free path from the F/S interface, so that the presence of such components does not affect  
 transport properties in the thick ferromagnetic layer.  
 
 Experimentally the signature of Josephson current between proximity-induced Berezinskii superconductors can be 
 obtained due to the non-trivial structure of the order parameter (<ref>), which amplitude depends on the 
 angle θ between magnetic moments in F and F' layers. Such a dependence is peculiar for the spin-triplet 
 odd-frequency pairings since  the spin-singlet component is not sensitive to the exchange field rotations. As shown below, 
 that results in the striking modification of the Fraunhoffer pattern in the critical current as a function of external magnetic field 
 I_c=I_c(H) which reflects the inhomogeneity of the magnetic texture in the S/F'/F system. 
 This behaviour coincides qualitatively with recent experimental observations<cit.>.
 
  
 
 
   




















  
The Josephson effect in the setup shown in Fig.(<ref>)a can be analysed using the 
formalism of Eqs.(<ref>,<ref>,<ref>). 
Due to the small distance between superconducting electrodes
 one can use the model 1D system shown in Fig.<ref>b which 
 is the long junction with gap
functions in the leads given by (<ref>). There are two key
points to understand the experimental results. First, the
effective gap amplitude is Δ_σ∝sinθ
that is determined by the angle θ between m and h_s. As shown by
the magnetization pattern data<cit.>, θ changes quite
strongly along the junction both due to the vortex-like pattern in F and the distortions of the mono-domain state 
in F' near the trench that cuts S and F' layers. The model distribution of θ along the Josephson junction 
that is used in calculations is shown in Fig.<ref>c and described by 
 θ (x) = (π/2) θ̃(x/L_J), where
 
    θ̃(x̃) =  tanh(x̃/w_c) 
     tanh( x̃ -0.5/w_e) tanh( x̃ + 0.5/w_e).

 The basic features of this profile is that θ=0 at the FM vortex center x=0 and at the edges x=± L_J/2, 
 while reaching ±π/2 at the middle points. The dimensionless widths w_e and w_c determine the shape of the Fraunhofer pattern.  
 
  Second, near the FM vortex core the magnetization pattern 
 is non-coplanar which generates both the gauge field Z and the spin-dependent
 phase α=α (x) distribution along the junction. Using the model magnetization distribution
 m= ( m_⊥cosφ,  m_⊥sinφ, m_z), where φ is the real-space polar angle
 one obtains Z = m_z ∇φ /2
 and α = arctan [m_z tan (φ - φ_0)], where φ_0 is the azimuthal angle of 
 h_s = h_s (cosφ_0, sinφ_0, 0). Note that for |m_z|=1 the superfluid velocity (<ref>)
 is zero V_s = 0. However this is not the case at |m_z |< 1. The effective phase difference can be found integrating 
 the superfluid velocity across the junction Δα = ∫_-d^d dy V_Sy, where 2d is the junction width. 
 For φ_0=0 and provided that m_z≈ const at the scale |y|<d  one obtains in this way
 
    Δα (x)= 2[ arctan(m_z tanφ) - m_z φ ]

 where φ = arctan(d/x). 
    
   

The Josephson current can be calculated analytically assuming that the
proximity effect is weak and one can use the linearized theory formulated for the component 
 f_σ =  Tr[(τ_x + iτ_y)ĝ_σ]/2. 
The linearized Usadel equation and expression for the current across the junction read
  
   
    D_σ∇̂^2 f_σ - 
       2 ω f_σ - Δ_σ =0 
       
        j = π T e ∑_σ=±∑_ων_σ D_σ Im (f_σ∇̂f_σ^*)

   
 Since the distance between superconducting electrodes is small,
 proximity-induced vortices<cit.> cannot form in the junction. 
 Then the order parameter distribution can be approximated by the step-wise function
 
    Δ_σ(y) =  h_⊥Δ_Fσ
     [cos(χ_σ/2) + i Step(y) sin(χ_σ/2)].

 It is convenient to choose the gauge so that A_y= 0 and neglect A_x component due to the small
 junction width. Then the total 
 phase difference is given by 
 χ_σ (x) =Δχ + σΔα (x) + ϕ x/L_J, where  Δχ= χ_R - χ_L and
 Δα (x)= α_R - α_L are the usual and spin-dependent phase differences, 
 ϕ = Φ/Φ_0 where Φ is the total magnetic flux through the junction area including 
 the leads, Φ_0 is the flux quantum. 
 
The field Z is treated as locally homogeneous and gauged it out by using the effective phase difference (<ref>). 
 

Within the above assumptions the linearized Usadel Eq.(<ref>) can be solved 
analytically and Eq.(<ref>) yields the following  expression for the Josephson current density
 
    j (x) = ∑_σ= ± j_σ (sinθ)^2 sinχ_σ ,
     
       j_σ = π T e ν_σ D_σ/8∑_ωΔ_Fσ^2/ξ_Nσω^2  ,

 where
 ξ_Nσ = √(2|ω|/D_σ). The total critical current is the given by I_c = √(I_1^2 + I_2^2), where 
 
    I_1 = ∑_σ j_σ∫_-L_J/2^L_J/2 (sinθ)^2 
     cos(ϕ x + σΔα )dx 
     
    
     I_2 = ∑_σ j_σ∫_-L_J/2^L_J/2 (sinθ)^2 
     sin(ϕ x + σΔα )dx .
  
 
  Fig.(<ref>) presents the dependencies I_c = I_c (ϕ) calculated according to the Eqs.(<ref>,<ref>) with the help model distributions 
  (<ref>,<ref>) for θ(x) and Δα(x), respectively. 
 
The upper panel shows the standard single-slit pattern from the homogeneous junction (dashed curve) and the double-slit pattern produced by the 
junction with FM vortex (solid curve). Here the spin-dependent phase is absent Δα=0.  
The distribution of θ (x) is taken in the form (<ref>) with free parameters w_e,c. 
The homogeneous case is obtained in the limit w_e, w_c → 0. Increasing the widths w_e,v leads to the gradual transform of the interference pattern which at w_e=w_c= 0.3 acquires the double-slit form shown in the upper panel of the  Fig.(<ref>).
 Note that it is necessary to take into account the suppression of 
θ(x) both at the FM vortex core and at the boundary. In case when w_e→ 0 the interference pattern tends to the deformed homogeneous picture. 


 The spin-dependent phase  Δα (x) can produce the   
 Josephson current (<ref>) even in the absence of the usual phase difference and 
 external magnetic field. Since the phase shift has opposite signs in spin-up/down subbands the net effect on the current shows up only due to the 
 certain amount of spin-filtering in the system<cit.>, that is
 j_+ ≠ j_-. This is possible only if the spin-up and spin-down diffusion coefficients and/or 
 DOS are different. The model distribution (<ref>) produced by the FM vortex leads to the non-trivial modification of
 the Franhofer pattern, shown by the solid line in Fig.(<ref>), lower panel. The parameters here are 
 d= 0.2 L_J, j_- = 0.5 j_+. Asymmetric curve I_c(H)≠ I_c(-H) is qualitatively similar to the one produced by the 
 localized magnetic field or the internal  phase shifts  in junctions between chiral p-wave superconductors<cit.>.
 Here the asymmetry is finite due to the spin filtering j_+ ≠ j_- and thus it signal the presence of spontaneous currents. 
 
  
   For the the S/F'/F setup with magnetic configurations similar to Fig.(<ref>)a the effect of spontaneous current is rather tiny
 since the induced superfluid velocity is zero V_s = 0 both at the center where  m_z = 0 and outside the FM vortex core 
 where m_z=1. 
 The situation becomes completely different for z · h_s ≠ 0. For example if 
 h_s = h_s  z one can see that the spin-dependent phase is constant ∇α =0 so that 
 V_s = σ m_z ∇φ /2 and has singularity at r=0.  The current is however 
 finite j ∼ h_⊥^2/r since h_⊥ = √(1-m_z^2)∼ r. This behaviour is similar to the 
 orbital supercurrents around Abrikosov vortices in bulk superconductors.  Note that this texture-induced spontaneous 
 current is not related to the anomalous Josephson effect <cit.>   since it exists without the weak link in the superconducting layer. 
 
 The singularity of Z can be removed by the gauge transform introducing the vorticity to
 the spin-dependent phase α = φ. Therefore in such system FM vortex 
 generates singly-quantized superconducting vortices in the proximity-induced Berezinskii superconductor.
 The physics of such vortices and superconducting kinks that are generated by magnetic domain walls in the 
 multilayer setups similar to the Fig.(<ref>)a  is potentially quite rich  
 but is beyond the scope of the present paper. 
   
 
 
  To summarize, in this letter I explain the recent observation of the unusual magnetic field dependence of
   critical current of the Josephson junction through ferromagnetic vortex. 
  The key theoretical finding is that the experimental results are consistent with the existence of 
  proximity-induced Berezinskii superconductors in the S/F'/F systems with non-collinear strong ferromagnets. 
  The effective order parameter amplitude is defined by the angle between magnetic moments in F and F' layers.
  The inhomogeneous distribution of this angle results in the double-slit interference  
  pattern of the critical current. 
  Besides that, the non-coplanar magnetic texture near the FM vortex core generates 
  spin-dependent phase gradients and the emergent gauge field which can be combined into the invariant 
  combination yielding the spin-dependent part of the superfluid velocity V_s.  
  The lifted degeneracy of DOS and diffusion coefficients in spin subbands in strong ferromagnets
   converts V_s to the spontaneous supercurrent  which is shown to result in the critical current asymmetry I(H) ≠ I(-H). 
  
 Finally, I show that FM vortex can induce spontaneous vorticity in the Berezinskii state, 
 provided that the magnetization in F' layer has an out-of-plane direction. That finding demonstrates that 
 besides affecting transport properties of textured magnets<cit.> the emergent gauge field can show up in the 
 orbital motion of spin-triplet Cooper pairs and even produce topological defects in the superconducting order parameter .
 The result obtained here for the FM vortex are in general valid for magnetic skyrmions as well, owning to the similarity in their
 magnetization distributions.   
    
 
 I thank T. T Heikkilä ,  A. Mel'nikov, I. Bobkova and A. Bobkov
 for stimulating discussions.
  The work was supported by the Academy of Finland.
  
  
 







68
fxundefined [1]
 ifx#1

fnum [1]
 #1firstoftwo
 secondoftwo
 
fx [1]
 #1firstoftwo
 secondoftwo
 





noop [0]secondoftwo

ref[1]@startlink#1@href
href[1]#1@endlink
anitize@url [0]`
12`$12`&12`#12`1̂2`_12`%12
startlink[1]
endlink[0]

rl [1]href #1












@bib@innerbibempty

[Buzdin(2005)]Buzdin2005
  author author A. I. Buzdin, https://link.aps.org/doi/10.1103/RevModPhys.77.935
  journal journal Rev. Mod. Phys. volume 77, pages 935 (year
  2005)NoStop
[Bergeret et al.(2005)Bergeret, Volkov, and Efetov]Bergeret2005
  author author F. S. Bergeret, author A. F. Volkov,  and author K. B. Efetov, http://link.aps.org/doi/10.1103/RevModPhys.77.1321
  journal journal Rev. Mod. Phys. volume 77, pages 1321 (year
  2005)NoStop
[Eschrig(2015)]Eschrig2015a
  author author M. Eschrig, http://stacks.iop.org/0034-4885/78/i=10/a=104501
  journal journal Reports on Progress in Physics  volume 78, pages 104501 (year 2015)NoStop
[Linder and Robinson(2015)]Linder2015
  author author J. Linder and author J. W. A. Robinson, http://dx.doi.org/10.1038/nphys3242 journal journal Nat Phys volume
  11, pages 307 (year 2015)NoStop
[Ryazanov et al.(2001)Ryazanov, Oboznov, Rusanov, Veretennikov, Golubov, and Aarts]Ryazanov2001
  author author V. V. Ryazanov, author V. A. Oboznov, author A. Y. Rusanov, author A. V. Veretennikov, author A. A. Golubov,  and author J. Aarts, https://link.aps.org/doi/10.1103/PhysRevLett.86.2427
  journal journal Phys. Rev. Lett. volume 86, pages 2427 (year
  2001)NoStop
[Frolov et al.(2008)Frolov,
  Stoutimore, Crane, Van Harlingen, Oboznov, Ryazanov,
  Ruosi, Granata, and Russo]Frolov2008
  author author S. M. Frolov, author M. J. A. Stoutimore, author T. A. Crane, author D. J. Van Harlingen, author V. A. Oboznov, author V. V. Ryazanov, author A. Ruosi,
  author C. Granata,  and  author M. Russo, http://dx.doi.org/10.1038/nphys780 journal journal Nat Phys volume 4, pages
  32 (year 2008)NoStop
[Feofanov et al.(2010)Feofanov, Oboznov, Bol/'ginov,
  Lisenfeld, Poletto, Ryazanov,
  Rossolenko, Khabipov, Balashov, Zorin, Dmitriev, Koshelets, and Ustinov]Feofanov2010
  author author A. K. Feofanov, author V. A. Oboznov, author V. V. Bol/'ginov, author J. Lisenfeld, author S. Poletto,
  author V. V. Ryazanov, author A. N. Rossolenko, author M. Khabipov, author
  D. Balashov, author
  A. B. Zorin, author
  P. N. Dmitriev, author
  V. P. Koshelets,  and author
  A. V. Ustinov, http://dx.doi.org/10.1038/nphys1700 journal journal Nat Phys volume 6, pages
  593 (year 2010)NoStop
[Iovan et al.(2014)Iovan,
  Golod, and Krasnov]Iovan2014
  author author A. Iovan, author T. Golod,  and  author V. M. Krasnov, https://link.aps.org/doi/10.1103/PhysRevB.90.134514 journal
  journal Phys. Rev. B volume 90,  pages 134514 (year 2014)NoStop
[Golubov et al.(2002)Golubov, Kupriyanov, and Fominov]Golubov2002
  author author A. A. Golubov, author M. Y. Kupriyanov,  and author Y. V. Fominov, https://doi.org/10.1134/1.1475721 journal
  journal Journal of Experimental and Theoretical Physics
  Letters volume 75, pages 190
  (year 2002)NoStop
[Robinson et al.(2010a)Robinson, HalГЎsz, Buzdin, and Blamire]Robinson2010a
  author author J. W. A.  Robinson, author G. B.  HalГЎsz, author A. I.  Buzdin,  and author M. G.  Blamire, https://link.aps.org/doi/10.1103/PhysRevLett.104.207001 journal journal Phys. Rev. Lett. volume 104, pages 207001 (year
  2010a)NoStop
[Robinson et al.(2010b)Robinson, Witt,  and Blamire]Robinson2010
  author author J. W. A.  Robinson, author J. D. S.  Witt,  and author M. G.  Blamire, http://science.sciencemag.org/content/329/5987/59.abstract journal journal Science volume
  329, pages 59 (year
  2010b)NoStop
[Anwar et al.(2010)Anwar,
  Czeschka, Hesselberth, Porcu, and Aarts]Anwar2010
  author author M. S. Anwar, author F. Czeschka,
  author M. Hesselberth, author M. Porcu,  and author
  J. Aarts, https://link.aps.org/doi/10.1103/PhysRevB.82.100501 journal
  journal Phys. Rev. B volume 82,  pages 100501 (year 2010)NoStop
[Keizer et al.(2006)Keizer,
  Goennenwein, Klapwijk, Miao,
  Xiao, and Gupta]Keizer2006
  author author R. S. Keizer, author S. T. B. Goennenwein, author T. M. Klapwijk, author G. Miao,
  author G. Xiao,  and author A. Gupta, http://dx.doi.org/10.1038/nature04499 journal journal Nature volume 439, pages
  825 (year 2006)NoStop
[Tagirov(1999)]Tagirov1999
  author author L. R. Tagirov, https://link.aps.org/doi/10.1103/PhysRevLett.83.2058
  journal journal Phys. Rev. Lett. volume 83, pages 2058 (year
  1999)NoStop
[Fominov et al.(2003)Fominov, Golubov, and Kupriyanov]Fominov2003
  author author Y. V. Fominov, author A. A. Golubov,  and author M. Y. Kupriyanov, https://doi.org/10.1134/1.1591981 journal journal Journal of Experimental and Theoretical
  Physics Letters volume 77, pages 510
  (year 2003)NoStop
[Zdravkov et al.(2013)Zdravkov, Kehrle, Obermeier, Lenk, Krug von Nidda, MГјller,
  Kupriyanov, Sidorenko, Horn,
  Tidecks, and Tagirov]Zdravkov2013
  author author V. I. Zdravkov, author J. Kehrle,
  author G. Obermeier, author D. Lenk, author
  H.-A. Krug von Nidda, author
  C. MГјller, author M. Y. Kupriyanov, author A. S. Sidorenko, author S. Horn, author R. Tidecks,  and author L. R. Tagirov, https://link.aps.org/doi/10.1103/PhysRevB.87.144507
  journal journal Phys. Rev. B volume 87, pages 144507 (year
  2013)NoStop
[Fominov et al.(2010)Fominov, Golubov, Karminskaya,
  Kupriyanov, Deminov, and Tagirov]Fominov2010
  author author Y. V. Fominov, author A. A. Golubov, author T. Y. Karminskaya, author M. Y. Kupriyanov, author R. G. Deminov,  and author L. R. Tagirov, https://doi.org/10.1134/S002136401006010X journal journal JETP Letters volume
  91, pages 308 (year 2010)NoStop
[Singh et al.(2015)Singh,
  Voltan, Lahabi, and Aarts]Singh2015
  author author A. Singh, author S. Voltan,
  author K. Lahabi,  and author J. Aarts, https://link.aps.org/doi/10.1103/PhysRevX.5.021019 journal
  journal Phys. Rev. X volume 5,  pages 021019 (year 2015)NoStop
[Flokstra et al.(2010)Flokstra, van der Knaap, and Aarts]Flokstra2010
  author author M. Flokstra, author J. M. van der
  Knaap,  and author J. Aarts,  https://link.aps.org/doi/10.1103/PhysRevB.82.184523 journal journal Phys. Rev. B volume
  82, pages 184523 (year 2010)NoStop
[Silaev(2009)]Silaev2009
  author author M. A. Silaev, http://link.aps.org/doi/10.1103/PhysRevB.79.184505
  journal journal Phys. Rev. B volume 79, pages 184505 (year
  2009)NoStop
[Lahabi et al.(2017)Lahabi,
  Amundsen, Ouassou, Beukers,
  Pleijster, Linder, Alkemade,  and Aarts]AartsExperiment
  author author K. Lahabi, author M. Amundsen,
  author J. Ouassou, author E. Beukers, author
  M. Pleijster, author
  J. Linder, author P. Alkemade,  and author J. Aarts, @noop  journal journal arXiv:1705.07020  (year 2017)NoStop
[Berezinskii(1974)]Berezinskii
  author author V. L. Berezinskii, @noop  journal journal
  JETP Letters volume 20, pages 287
  (year 1974)NoStop
[Kirkpatrick and Belitz(1991)]Kirkpatrick1991
  author author T. R. Kirkpatrick and author D. Belitz, https://link.aps.org/doi/10.1103/PhysRevLett.66.1533
  journal journal Phys. Rev. Lett. volume 66, pages 1533 (year
  1991)NoStop
[Belitz and Kirkpatrick(1992)]Belitz1992
  author author D. Belitz and author T. R. Kirkpatrick, https://link.aps.org/doi/10.1103/PhysRevB.46.8393
  journal journal Phys. Rev. B volume 46, pages 8393 (year
  1992)NoStop
[Belitz and Kirkpatrick(1999)]Belitz1999
  author author D. Belitz and author T. R. Kirkpatrick, https://link.aps.org/doi/10.1103/PhysRevB.60.3485
  journal journal Phys. Rev. B volume 60, pages 3485 (year
  1999)NoStop
[Balatsky and Abrahams(1992)]Balatsky1992
  author author A. Balatsky and author E. Abrahams, https://link.aps.org/doi/10.1103/PhysRevB.45.13125
  journal journal Phys. Rev. B volume 45, pages 13125 (year
  1992)NoStop
[Abrahams et al.(1995)Abrahams, Balatsky, Scalapino, and  Schrieffer]Abrahams1995
  author author E. Abrahams, author A. Balatsky,
  author D. J. Scalapino,  and  author J. R. Schrieffer, https://link.aps.org/doi/10.1103/PhysRevB.52.1271 journal
  journal Phys. Rev. B volume 52,  pages 1271 (year 1995)NoStop
[Coleman et al.(1994)Coleman, Miranda, and Tsvelik]Coleman1994
  author author P. Coleman, author E. Miranda,   and author A. Tsvelik, https://link.aps.org/doi/10.1103/PhysRevB.49.8955 journal
  journal Phys. Rev. B volume 49,  pages 8955 (year 1994)NoStop
[Fominov et al.(2015)Fominov, Tanaka, Asano, and Eschrig]Fominov2015
  author author Y. V. Fominov, author Y. Tanaka,
  author Y. Asano,  and author M. Eschrig, https://link.aps.org/doi/10.1103/PhysRevB.91.144514 journal
  journal Phys. Rev. B volume 91,  pages 144514 (year 2015)NoStop
[Bobkova and Barash(2004)]Bobkova2004
  author author I. V. Bobkova and author Y. S. Barash, https://doi.org/10.1134/1.1839298 journal
  journal Journal of Experimental and Theoretical Physics
  Letters volume 80, pages 494
  (year 2004)NoStop
[Braude and Nazarov(2007)]Braude2007
  author author V. Braude and author Y. V. Nazarov, http://link.aps.org/doi/10.1103/PhysRevLett.98.077003
  journal journal Phys. Rev. Lett. volume 98, pages 077003 (year
  2007)NoStop
[Grein et al.(2009)Grein,
  Eschrig, Metalidis, and SchГ¶n]Grein2009
  author author R. Grein, author M. Eschrig,
  author G. Metalidis,  and  author G. SchГ¶n, http://link.aps.org/doi/10.1103/PhysRevLett.102.227005 journal journal Phys. Rev. Lett. volume 102, pages 227005 (year
  2009)NoStop
[Mironov and Buzdin(2015)]Mironov2015
  author author S. Mironov and author A. Buzdin, http://link.aps.org/doi/10.1103/PhysRevB.92.184506
  journal journal Phys. Rev. B volume 92, pages 184506 (year
  2015)NoStop
[Silaev et al.(2017)Silaev,
  Tokatly, and Bergeret]Silaev2017
  author author M. A. Silaev, author I. V. Tokatly,
   and author F. S. Bergeret,  https://link.aps.org/doi/10.1103/PhysRevB.95.184508 journal journal Phys. Rev. B volume
  95, pages 184508 (year 2017)NoStop
[Bobkova et al.(2017)Bobkova, Bobkov, and Silaev]BobkovaSilaev2017
  author author I. V. Bobkova, author A. M. Bobkov,
   and author M. A. Silaev,  @noop  journal journal arXiv:1706.04239   (year 2017)NoStop
[Bergeret et al.(2001a)Bergeret, Volkov, and Efetov]Bergeret2001
  author author F. S. Bergeret, author A. F. Volkov,  and author K. B. Efetov, https://link.aps.org/doi/10.1103/PhysRevLett.86.3140
  journal journal Phys. Rev. Lett. volume 86, pages 3140 (year
  2001a)NoStop
[Bergeret et al.(2012a)Bergeret, Verso,  and Volkov]Bergeret2012
  author author F. S. Bergeret, author A. Verso,   and author A. F. Volkov,  http://link.aps.org/doi/10.1103/PhysRevB.86.214516 journal journal Phys. Rev. B volume
  86, pages 214516 (year
  2012a)NoStop
[Bergeret et al.(2012b)Bergeret, Verso,  and Volkov]Bergeret2012a
  author author F. S. Bergeret, author A. Verso,   and author A. F. Volkov,  https://link.aps.org/doi/10.1103/PhysRevB.86.060506 journal journal Phys. Rev. B volume
  86, pages 060506 (year
  2012b)NoStop
[Kopnin and Melnikov(2011)]Kopnin2011
  author author N. B. Kopnin and author A. S. Melnikov, https://link.aps.org/doi/10.1103/PhysRevB.84.064524
  journal journal Phys. Rev. B volume 84, pages 064524 (year
  2011)NoStop
[Kopnin et al.(2013)Kopnin,
  Khaymovich, and MelвЂ™nikov]Kopnin2013
  author author N. B. Kopnin, author I. M. Khaymovich,  and author A. S. MelвЂ™nikov, https://link.aps.org/doi/10.1103/PhysRevLett.110.027003 journal journal Phys. Rev. Lett. volume 110, pages 027003 (year
  2013)NoStop
[Volovik(1987)]Volovik1987
  author author G. E. Volovik, http://stacks.iop.org/0022-3719/20/i=7/a=003 journal journal Journal of Physics C: Solid State Physics  volume 20, pages L83 (year
  1987)NoStop
[Bergeret et al.(2001b)Bergeret, Volkov, and Efetov]Bergeret2001a
  author author F. S. Bergeret, author A. F. Volkov,  and author K. B. Efetov, https://link.aps.org/doi/10.1103/PhysRevLett.86.4096
  journal journal Phys. Rev. Lett. volume 86, pages 4096 (year
  2001b)NoStop
[Tanaka and Golubov(2007)]Tanaka2007a
  author author Y. Tanaka and author A. A. Golubov, https://link.aps.org/doi/10.1103/PhysRevLett.98.037003
  journal journal Phys. Rev. Lett. volume 98, pages 037003 (year
  2007)NoStop
[Tanaka et al.(2007a)Tanaka, Golubov,
  Kashiwaya, and Ueda]Tanaka2007
  author author Y. Tanaka, author A. A. Golubov,
  author S. Kashiwaya,  and  author M. Ueda, https://link.aps.org/doi/10.1103/PhysRevLett.99.037005 journal journal Phys. Rev. Lett. volume 99, pages 037005 (year
  2007a)NoStop
[Tanaka et al.(2007b)Tanaka, Tanuma,  and Golubov]Tanaka2007b
  author author Y. Tanaka, author Y. Tanuma,   and author A. A. Golubov,  https://link.aps.org/doi/10.1103/PhysRevB.76.054522 journal journal Phys. Rev. B volume
  76, pages 054522 (year
  2007b)NoStop
[Yokoyama(2012)]Yokoyama2012
  author author T. Yokoyama, https://link.aps.org/doi/10.1103/PhysRevB.86.075410
  journal journal Phys. Rev. B volume 86, pages 075410 (year
  2012)NoStop
[Black-Schaffer and Balatsky(2012)]Black-Schaffer2012
  author author A. M. Black-Schaffer and author A. V.  Balatsky, https://link.aps.org/doi/10.1103/PhysRevB.86.144506 journal
  journal Phys. Rev. B volume 86,  pages 144506 (year 2012)NoStop
[Triola and Balatsky(2016)]Triola2016
  author author C. Triola and author A. V. Balatsky, https://link.aps.org/doi/10.1103/PhysRevB.94.094518
  journal journal Phys. Rev. B volume 94, pages 094518 (year
  2016)NoStop
[Yokoyama et al.(2007)Yokoyama, Tanaka, and Golubov]Yokoyama2007
  author author T. Yokoyama, author Y. Tanaka,   and author A. A. Golubov,  https://link.aps.org/doi/10.1103/PhysRevB.75.134510 journal journal Phys. Rev. B volume
  75, pages 134510 (year 2007)NoStop
[Alidoust et al.(2017)Alidoust, Zyuzin, and Halterman]Alidoust2017
  author author M. Alidoust, author A. Zyuzin,   and author K. Halterman, https://link.aps.org/doi/10.1103/PhysRevB.95.045115 journal
  journal Phys. Rev. B volume 95,  pages 045115 (year 2017)NoStop
[Cuevas and Bergeret(2007)]Cuevas2007
  author author J. C. Cuevas and author F. S. Bergeret, https://link.aps.org/doi/10.1103/PhysRevLett.99.217002
  journal journal Phys. Rev. Lett. volume 99, pages 217002 (year
  2007)NoStop
[Alidoust and Linder(2013)]Alidoust2013
  author author M. Alidoust and author J. Linder, https://link.aps.org/doi/10.1103/PhysRevB.87.060503
  journal journal Phys. Rev. B volume 87, pages 060503 (year
  2013)NoStop
[Alidoust and Halterman(2015)]Alidoust2015
  author author M. Alidoust and author K. Halterman, 10.1063/1.4908287 journal
  journal Journal of Applied Physics volume 117, pages 123906 (year
  2015)NoStop
[Kidwingira et al.(2006)Kidwingira, Strand, Van Harlingen, and  Maeno]Kidwingira2006
  author author F. Kidwingira, author J. D. Strand, author D. J. Van Harlingen,  and author Y. Maeno, http://science.sciencemag.org/content/314/5803/1267.abstract journal journal Science volume
  314, pages 1267 (year 2006)NoStop
[Buzdin(2008)]Buzdin2008
  author author A. Buzdin, http://link.aps.org/doi/10.1103/PhysRevLett.101.107005
  journal journal Phys. Rev. Lett. volume 101, pages 107005 (year
  2008)NoStop
[Reynoso et al.(2008)Reynoso, Usaj, Balseiro, Feinberg, and Avignon]Reynoso2008
  author author A. A. Reynoso, author G. Usaj,
  author C. A. Balseiro, author D. Feinberg,  and author M. Avignon, http://link.aps.org/doi/10.1103/PhysRevLett.101.107001 journal journal Phys. Rev. Lett. volume 101, pages 107001 (year
  2008)NoStop
[Zazunov et al.(2009)Zazunov, Egger, Jonckheere, and  Martin]Zazunov2009
  author author A. Zazunov, author R. Egger,
  author T. Jonckheere,  and  author T. Martin, http://link.aps.org/doi/10.1103/PhysRevLett.103.147004 journal journal Phys. Rev. Lett. volume 103, pages 147004 (year
  2009)NoStop
[Liu and Chan(2010)]Liu2010
  author author J.-F. Liu and author K. S. Chan,  http://link.aps.org/doi/10.1103/PhysRevB.82.184533 journal journal Phys. Rev. B volume
  82, pages 184533 (year 2010)NoStop
[Brunetti et al.(2013)Brunetti, Zazunov, Kundu, and  Egger]Brunetti2013
  author author A. Brunetti, author A. Zazunov,
  author A. Kundu,  and author R. Egger, http://link.aps.org/doi/10.1103/PhysRevB.88.144515 journal
  journal Phys. Rev. B volume 88,  pages 144515 (year 2013)NoStop
[Yokoyama et al.(2014)Yokoyama, Eto, and Nazarov]Yokoyama2014
  author author T. Yokoyama, author M. Eto,   and author Y. V. Nazarov,  http://link.aps.org/doi/10.1103/PhysRevB.89.195407 journal journal Phys. Rev. B volume
  89, pages 195407 (year 2014)NoStop
[Kulagina and Linder(2014)]Kulagina2014
  author author I. Kulagina and author J. Linder, http://link.aps.org/doi/10.1103/PhysRevB.90.054504
  journal journal Phys. Rev. B volume 90, pages 054504 (year
  2014)NoStop
[Nesterov et al.(2016)Nesterov, Houzet, and Meyer]Nesterov2016
  author author K. N. Nesterov, author M. Houzet,   and author J. S. Meyer, http://link.aps.org/doi/10.1103/PhysRevB.93.174502 journal
  journal Phys. Rev. B volume 93,  pages 174502 (year 2016)NoStop
[Konschelle et al.(2015)Konschelle, Tokatly, and Bergeret]Konschelle2015
  author author F. Konschelle, author I. V. Tokatly,  and author F. S. Bergeret, http://link.aps.org/doi/10.1103/PhysRevB.92.125443
  journal journal Phys. Rev. B volume 92, pages 125443 (year
  2015)NoStop
[Moor et al.(2015a)Moor, Volkov,  and Efetov]Moor2015
  author author A. Moor, author A. F. Volkov,   and author K. B. Efetov,  http://link.aps.org/doi/10.1103/PhysRevB.92.214510 journal journal Phys. Rev. B volume
  92, pages 214510 (year
  2015a)NoStop
[Moor et al.(2015b)Moor, Volkov,  and Efetov]Moor2015a
  author author A. Moor, author A. F. Volkov,   and author K. B. Efetov,  http://link.aps.org/doi/10.1103/PhysRevB.92.180506 journal journal Phys. Rev. B volume
  92, pages 180506 (year
  2015b)NoStop
[Bobkova et al.(2016)Bobkova, Bobkov, Zyuzin, and  Alidoust]Bobkova2016
  author author I. V. Bobkova, author A. M. Bobkov,
  author A. A. Zyuzin,  and  author M. Alidoust, http://link.aps.org/doi/10.1103/PhysRevB.94.134506 journal
  journal Phys. Rev. B volume 94,  pages 134506 (year 2016)NoStop
[Szombati et al.(2016)Szombati, Nadj-Perge, Car, Plissard, Bakkers, and Kouwenhoven]Szombati2016
  author author D. B. Szombati, author S. Nadj-Perge, author D. Car,
  author S. R. Plissard, author E. P. A. M. Bakkers,  and  author L. P. Kouwenhoven,  http://dx.doi.org/10.1038/nphys3742 journal journal Nat Phys volume 12, pages
  568 (year 2016)NoStop
[Nagaosa and Tokura(2013)]Nagaosa2013
  author author N. Nagaosa and author Y. Tokura, http://dx.doi.org/10.1038/nnano.2013.243 journal journal Nat Nano volume 8,  pages 899 (year 2013)NoStop


 



