Anomalous current in diffusive ferromagnetic Josephson junctions

We demonstrate that in diffusive superconductor/ferromagnet/superconductor (S/F/S) junctions a finite, {\it anomalous}, Josephson current can flow even at zero phase difference between the S electrodes. The conditions for the observation of this effect are non-coplanar magnetization distribution and a broken magnetization inversion symmetry of the superconducting current. The latter symmetry is intrinsic for the widely used quasiclassical approximation and prevent previous works, based on this approximation, from obtaining the Josephson anomalous current. We show that this symmetry can be removed by introducing spin-dependent boundary conditions for the quasiclassical equations at the superconducting/ferromagnet interfaces in diffusive systems. Using this recipe we considered generic multilayer magnetic systems and determine the ideal experimental conditions in order to maximize the anomalous current.

In its minimal form the current-phase relation (CPR) characterizing the dc Josephson effect reads I(ϕ) = I c sin ϕ, where ϕ is the phase difference between superconducting electrodes and |I c | is the critical current that is the maximum supercurrent that can flow through the junction 1,2 .Ordinary Josephson junctions are characterized by I c > 0 yielding the zero phase difference ground state ϕ = 0.In certain cases, however, I c < 0 and the ground state corresponds to ϕ = π.Such π-junctions can be realized for example in superconductor/ferromagnet/superconductor (SFS) structures [3][4][5][6] , Josephson junctions with non-equilibrium normal metal interlayer 7 , d-wave superconductors 8 , semiconductor nanowires 9 , gated carbon nanotubes 10 or multi-terminal Josephson systems 11 .π-junctions has being suggested for building scalable superconducting digital and quantum logic [12][13][14] .
As for ϕ = 0, π junctions, no physical argument speaks against a CPR of the form 15 I(ϕ) = I c sin(ϕ + ϕ 0 ) . ( with an arbitrary phase shift ϕ 0 = πn and Josephson energy E J = −I c cos(ϕ + ϕ 0 ).In such a case the ground state corresponds to ϕ = −ϕ 0 and a finite supercurrent at zero phase difference I an = I c sin ϕ 0 , termed the anomalous current.This effect, referred as the anomalous Josephson effect (AJE), takes place only in systems with a broken time reversal symmetry.
The AJE has been predicted in junctions which combine conventional superconductors with magnetism and spin-orbital interaction [15][16][17][18][19][20][21][22][23] , between unconventional superconductors 24 , and topologically non-trivial superconducting leads 25 .In the presence of magnetic flux piercing the normal interlayer the superconducting proximity currents are generated which naturally leads to the phase shift of CPR 26,27 .Experimentally, a ϕ 0 -junction has been reported in nano-wire quantum dot 28 controlled by an external magnetic field and an electrostatic gate.
Another type of systems predicted to exhibit the AJE are conventional SFS junctions with a non-homogeneous magnetization texture [29][30][31][32][33][34][35][36] .In such systems the current is a functional of the magnetization distribution M , I = I(ϕ, M ).Time-inversion symmetry dictates that I(ϕ, M ) = −I(−ϕ, −M ).If the system has an additional magnetization inversion symmetry such that then I(ϕ, M ) = −I(−ϕ, M ) and obviously the system does not exhibit the AJE.In other words, it is necessary to break the symmetry (2) in order to obtain the ϕ 0 state.For example, for any coplanar magnetization distribution exists a global SU(2) spin rotation such that flips the direction of M , and the condition (2) is fulfilled.For this reason, the AJE requires a non-coplanar magnetization texture.This explains the AJE predicted for ballistic S/F/F/F/S systems with non-collinear magnetizations [32][33][34] .The anomalous current obtained in those works shows rapid oscillations as a function of the ferromagnetic thickness.These oscillations result from the Fabry-Perot interference of electronic waves reflected at the S/F and F/F interfaces.
In diffusive SFS structures, as those used in experiments 4,5,13,[37][38][39] the impurity scattering randomizes directions of electron propagation and hence one expects the suppression of the rapidly oscillating anomalous current.Studies, based on quasiclassics, of the diffusive Josephson junctions through various non-coplanar structures including helix 40 , magnetic vortex 41 and skyrmion 42 have shown no AJE.In contrast, in diffusive systems with half-metallic elements 29,31 and in junctions between magnetic superconductors with spin filters 35,36 a finite anomalous current has been predicted.From this apparent contradiction, the general condition for the AJE in diffusive systems still remains elusive.
In this letter we show for the first time that the AJE is a robust effect that can exists in any diffusive SFS systems with non-coplanar magnetization textures under quite general conditions.We demonstrate that the reason why anomalous currents have not been found in previous studies on diffusive SFS systems is due to the additional magnetization inversion symmetry (2) that the quasiclassical approximation 43,44 has with respect to the original Hamiltonian and that prevents the description of the AJE in ferromagnetic junctions.In a second part of the letter we consider a spin-filter at the S/F interfaces and demonstrate the existence of anomalous currents in diffusive SF structures.This allow us to study the AJE without having to renounce the widely used quasiclassical approximation 6,45 .
We start by analyzing the inherent symmetries of the Usadel equation, which is a diffusion-like equation for the quasiclassical Green functions (GF).In the Matsubara representation it has the form 6,44,45 where [a, b] = (ab − ba)/2, ω is the Matsubara frequency, h(r) is the exchange field which is parallel to the local magnetization M (r), σ = (σ 1 , σ2 , σ3 ) is the vector of Pauli matrices in spin space σ1,2,3 and τ1,2,3 are the Pauli matrices in Nambu space.The gap matrix is defined as ∆ = τ1 ∆e −iτ3ϕ , where ∆ and ϕ are the magnitude and phase of the order parameter.The 4×4 matrix GF in spin-Nambu space can be written in the form, which takes into account the general particle-hole symmetry of Eq.(3 with 2 × 2 components ĝ and f in the spin space and the time-reversed operation defined as X = σ2 X * σ2 .Eq. ( 3) is complemented by the normalization condition ǧ2 = 1.We introduce the following transformation which is a combination of two transformations g new = T ΘgΘ † T † : the time reversal transformation, T = iσ 2 K, with K being the complex conjugate operation, and the transposition of the electron and hole blocks of g, Θ = τ1 .Applying the transformation (5) to the Usadel Eq.(3) one obtains that On the other hand, the current is expressed as: where σ n = e 2 N F D is the normal metal conductivity and N F is the density of states at the Fermi level.The summation is done over Matsubara frequencies ω = πT (2n + 1), where n is the integer number and T is the temperature.It follows from Eqs. (5-6) that the current is invariant with respect to the magnetization inversion, j(h) = j(−h), as anticipated in Eq. (2).By combining this extra symmetry with the general time-reversal symmetry, j(ϕ, h) = −j(−ϕ, −h) one obtains that j(ϕ) = −j(−ϕ) and hence within, the quasiclassical approach, the AJE cannot take place for any spatial dependence of the exchange field h(r).On the other hand, we know from previous works that anomalous current may be generated at least in ballistic SFS junctions with noncoplanar configuration of the magnetization [32][33][34] .What is the origin of the apparent contradiction between the explicit ballistic calculations in those references and the magnetization reversal symmetry of the Usadel equation?Is the absence of AJE a specific feature of diffusive systems or is there a deeper reason for the above symmetry?
To answer these questions let us first recall the Bogoliubov-De Gennes (BdG) Hamiltonian: where ξ = p 2 /2m − E F is the quasiparticle energy relative to the Fermi energy E F .The general symmetries of the BdG Hamiltonian are well known 46 .In the quasiclassical limit, which is equivalent to the Andreev approximation 47 , transport properties are determined by particles living exactly at the Fermi surface.
In the BdG Hamiltonian this corresponds to the ξ = 0 case.In this, and only in this case, the BdG Hamiltonian acquires an additional symmetry with respect to the transposition of the electron and hole blocks, namely τ1 H BdG (ξ = 0, ϕ, h)τ 1 = H BdG (ξ = 0, −ϕ, h).According to Eq. ( 5) this symmetry together with the timereversal operation leads to the invariance of the current under magnetization inversion.Obviously, this invariance is a general feature of the quasiclassical theory, which holds true not only in the diffusive (Usadel) limit, but also for the full Eilenberger equation.In particular it explains why no AJE is obtained at the leading quasiclassical order in ballistic junctions with generic spin fields 48 .
Clearly in real materials quantum effects always break this symmetry to a degree determined by the accuracy of quasiclassical approximation, which is the ratio h/E F .Once this symmetry is broken the AJE may occur in any SFS system with arbitrary degree of non-magnetic disorder and non-coplanar magnetization distribution.The magnitude of the anomalous current will then be in leading order of the parameter h/E F .Typical experiments on SFS junctions showing the π-junction behavior, used weak ferromagnets 4,5,49 , for which h/E F 1. Therefore, at first glance, the AJE is hardly expected to be observed in these structures.This conclusion is however not fully correct, and there is indeed a way to enhance the anomalous Josephson currents in systems with weak ferromagnets if one introduces spin-filtering tunnel barriers at the S/F interfaces,i.e.barriers with spin-dependent transmission for up and down spins.As we show below such barriers breaks the quasiclassical symmetry, Eq. ( 2) and can lead to a measurable AJE in realistic SFS junctions.Spin-filtering barriers are described by the generalized Kuprianov-Lukichev boundary conditions 50 , that include spin-polarized tunnelling at the SF interfaces 51,52 γǧ∂ Here ∂ n = (n • ∇) is the normal derivative at the surface, γ = σ n R is the parameter describing the barrier strength, R is the normal state tunneling resistance per unit area, and ǧS is the Green function of the superconducting electrode.We assume that the magnetization of the barriers points in m direction.The spin-polarized tunneling matrix has the form Γ = tσ 0 τ0 +u(mσ)τ 3 , with )/2 and P being the spin-filter efficiency of the barrier that ranges from 0 (no polarization) to 1 (100% filtering efficiency).
By applying the transformation (5) to Eq. ( 8) one can easily check the sign of the barrier polarization does not change and hence where P = P m.On the other hand, the time-reversal transformation flips all the magnetic moments including the exchange field and the barrier polarizations Combining Eqs.(9,10) we see, that in principle, I(ϕ, h, P ) = −I(−ϕ, h, P ) and the zero-phase difference current at ϕ = 0 is not prohibited by symmetry.
From this analysis is clear that the general features of the CPR can be deduced from the symmetry relations (9,10).First we consider the S/FI/F/FI/S structure of Fig. 1a.Here FI stands for the spin-filtering barriers with magnetizations P r,l and F is the mono-domain weak ferromagnet with exchange field h.From previous works [32][33][34] one would expect that the anomalous current is proportional to the spin chirality χ = h • (P r × P l ).However, this term is prohibited by the symmetry (9) because of the change of sign of χ.Instead, one can construct the scalar I an ∝ (P r,l h)χ which is invariant to the sign change of h → −h and therefore is robust to the quasiclassical symmetry (9).In such a case the anomalous current is finite if all vectors are non-collinear and in addition h has a component parallel to at least one of the magnetizations P l,e .
To get an agreement with the results based on the Bogolubov -de Gennes calculations 33 , that yield I an = 0 for any non-coplanar spin texture and has to take into account the magnetic proximity effect [53][54][55][56]  In the next example we consider the same trilayer but we take into account the presence of an effective exchange field b r and b l in the superconducting electrodes, as a consequence of either the magnetic proximity effect, or an external field (Zeeman effect) 52,55 .Such an effective exchange field leads to a spin-split density of states in the superconductors 55 .If we assume a small exchange field, In this case we define the chiralities χ l,r = P l,r • (b r,l × h). which are invariant respect to the quasiclassical symmetry since they contain two exchange fields changing signs under the transformation (9).Thus, in this case the AJE is expected to be proportional to a linear combination of the chiralities χ l,r .
A similar behavior can be obtained for the structure shown in Fig. 1c.It is a S/FI/F/F /S junction with noncoplanar configuration of the one barrier polarization P l and two ferromagnetic layers h and h 1 .In this case the chirality (P l × h 1 )h = 0 is invariant under the symmetry (9) thus allowing for the existence of the AJE.
To quantify these effects we calculate the CPR analytically focusing on the weak proximity effect in the F layer that allows for a linearization of the Usadel equation with respect to the anomalous Green's function 45 .The latter can be written as a superposition of the scalar singlet amplitude f s and the vector of triplet states f t = (f x , f y , f z ), f = f s σ0 + f t σ.From Eq. (3) we get the following sys-tem of equations for ω > 0 supplemented by the linearized boundary condition obtained 57 from Eq. ( 8) in case the possible exchange field in the superconductor is parallel to the barrier polarization b P where {a, b} = (ab + ba)/2, Ĝs and Fs are the the normal and anomalous GF in the superconducting electrodes.The first term in the right hand side of Eq.( 13) is novel as compared to the boundary conditions for non-magnetic interfaces (P = 0).This term provides a π/2 phase rotation of the triplet superconducting components noncollinear with the barrier polarization P .It is precisely this phase rotation that may lead to an effective shift of the phase difference between the Cooper pairs across the junction resulting in the AJE.We calculate the amplitude I an for the structures shown in Fig. (1) in the practically relevant regime when the coherence length in the middle ferromagnetic layer ξ F = D/h is much shorter than that in a normal metal ξ N = D/T .The analytical result can be obtained by assuming that the length d of the junction is ξ F d ξ N .Under such conditions the Josephson current is mediated by long-range triplet superconducting correlations (LRTSC) 45 since short-range modes decay over ξ F .
For the S/FI/F/FI/S structure shown in Fig. 1a we neglect the magnetic proximity effect and assume the bulk GF in the S electrodes Ĝs = ω/ ω 2 + |∆| 2 ≡ G 0 , Fs = ∆G 0 /ω ≡ F 0 .Then the anomalous current is 57 ) where P = P r + P l and k ω = ω/D.As expected for this case, the anomalous current is proportional to (h P )χ, where χ = h • (P r × P l ) is the spin chirality.It is important to note that the usual contribution to the Josephson current I 0 = I(ϕ = π/2) determined by the LRTSC is proportional to I 0 ∝ γ −4 , and hence it dominates over the anomalous one, I 0 I an ∝ γ −5 .If we now take the inverse proximity effect into account, and assume effective exchange fields b r and b l in the superconductors [Fig.(1 )b], we obtain 57 where the chiralities χ r,l are defined above and F 0 = dF 0 /dω.The usual current carried by the LRTSC is, to the lowest order in transparency, given by 57 I 0 ∝ γ −2 (b r⊥ b l⊥ ), where b r,l⊥ = b r,l − h(b r,l h)/h 2 are the projections onto the plane perpendicular to the exchange field h.In contrast to the previous example, I 0 is given by the lower order in γ −1 since the LRTSC can tunnel directly from the superconducting electrodes modified by the exchange fields b r,l .Hence in general if (b r⊥ b l⊥ ) = 0 the anomalous current ( 15) is a factor ξ F /γ 1 smaller than I 0 .However if either b r or b l vanishes, then I 0 ∝ γ −4 and the anomalous current dominates.This leads to a large AJE with I = I an cos ϕ so that the Josephson current has its maximal value at zero phase difference.
In the practice the situation equivalent to e.g.b r = 0 and b l = 0 can be realized in a S/FI/F/F /S junction [Fig.1c].If the middle F layer satisfies the above condition, ξ F d ξ N , but the right layer, F is short enough such that d 1 ξ F , the zero-phase difference current is given by 57 where χ = (P l × h 1 )h = 0.As in our first example, the usual component of the current is proportional to 57 I 0 ∝ γ −4 and therefore I an I 0 .This type of S/FI/F/F /S structure provides the maximal AJE since the anomalous current is of the same order of the critical one I an ∼ I c .
All previous results are strictly speaking valid in the quasiclassical limit in which h/E F 1.However, in the case of strong ferromagnets, h/E F 1, the difference between Fermi velocities for spin up and down electrons can be described by an effective spin-filtering effect at the S/F interfaces, and therefore they also apply for for ballistic systems and strong ferromagnets.
To summarize, the proposed mechanism for the AJE and ϕ 0 ground states in SFS structures is rather generic and exists in any system with non-coplanar magnetization configuration.This conclusion is in contrast to a number of previous studies which did not obtain anomalous currents in diffusive and ballistic systems in the framework of quasiclassical approximation.We clarify this apparent controversy by demonstrating that the absence of AJE within quasiclassics is due to an additional symmetry which is only exact at the Fermi level.In order to restore the symmetries of the original Hamiltonian we have considered spin-filtering boundary conditions to the Usadel equations and found analytical expressions for the anomalous current in different geometries.Our results show that in structures as those shown in Figs.1b,c, the amplitude of the anomalous current comparable to critical one I an ∼ I c , and therefore the AJE may be observed in such junctions.6 Phys.Rev. Lett.61, 637 (1988) . 57See the Supplementary material file for the detailed derivation of Eqs.(14,15,16 ) for the anomalous current and the expressions for the usual component of the Josephson current for the systems shown in Fig. (1).

I. SUPPLEMENTARY MATERIAL:
DERIVATION OF CURRENT-PHASE RELATIONS.
Here we derive analytical expressions for the anomalous and usual Josephson current components in generic trilayer SFS structures shown in Figs.(1).We use Usadel Eqs.(11,12) with boundary conditions obtained from the linearization of the Eq.( 8).For the general spin structure of GF in the superconducting electrode the linearized boundary condition can be written as follows where Ĝs = t 2 Ĝs + u 2 (mσ Ĝs mσ) + 2ut{ Ĝs , mσ} In the presence of exchange field b in the superconducting electrode the GF are Ĝs = G 0 − i(σb)dG 0 /dω (20) Fs = F 0 − i(σb)dF 0 /dω.
If the exchange field is collinear with the barrier polarization b m the boundary condition (17) acquires the form of Eq.( 13).In the right hand side of Eq.( 13) the first and second terms are much smaller than the third one.
Both the first and second terms are proportional to the small tunnelling parameter γ −1 but have different symmetry.The third term can be safely neglected since it has the same symmetry as the left hand side and therefore does not provide any qualitative corrections.We keep the second term which is important to obtain anomalous Josephson effect.
To calculate the charge current in the ferromagnetic layer we use the expression which is obtained linarizing the general Eq.(7).
A. S/FI/F/FI/S structure First of all we consider the simplest possible tri-layer non-coplanar structure S/FI/F/FI/S, where FI stands for the spin-filtering barriers with magnetizations P r,l and F is the mono-domain weak ferromagnet with exchange field h.We calculate the CPR for the structure shown in Fig. (1)a assuming without loss of generality that the exchange filed is h = hz and P r,l can have arbitrary directions.Then we have the Usadel equations in components: In Eqs.(23,24) we neglected ω which is small compared to the exchange energy.
The boundary conditions at the left electrode and at the right electrode Using the above boundary conditions and the general expression for current (22) we get that To simplify the derivation we assume that the length is ξ F d ξ ω where ξ F = D/h and ξ ω = D/ω are the coherence lengths in normal and ferromagnetic regions.
The to the first order in tunnelling γ −1 we can calculate f s and f z near each interface independently without overlapping.For example at x = d/2 we have where k 2 1,2 = ±ih/D.Then we get where f * s = f * s (d/2).To find the current we need to determine corrections f s with the help of boundary conditions (83) due to the triplet components generated at the x = −d/2 boundary.In this way we search the corrections to the short-range solution fs , fz in the form (36, 37) with the amplitudes determined by the boundary condition (83,84).Thus we obtain fs = −i The components f t to be substituted in Eq.(90) can be found using the equation ( 67) These expressions can be rewritten in the coordinateindependent form eRI 0 2π = 1 − P 2 l (P l h 1⊥ ) where the chirality is given by χ = (P l × h 1 )h and h 1⊥ = h 1 − h(hh 1 )/h 2 is the perpendicular component of the exchange field h 1 .The usual current is given by the higher order corrections in the tunnel barrier transparency I 0 ∝ γ −4 than the anomalous one I an ∝ γ −3 .Therefore in the tunnelling limit I an I 0 .

FIG. 2 :
FIG. 2: (Color online) Generic non-coplanar tri-layer SFS systems: (a) Non-collinear spin-filtering barriers (FI) with polarizations Pr,l and metallic ferromagnetic layer (F) with exchange field h.(b) The same configuration as in (a) and Zeeman fields br,l in superconducting electrodes.(c) Spinfiltering barrier with polarization P and two layers of metallic ferromagnet with non-collinear magnetizations h1 (F) and h2 (F').

FIG. 1 .
FIG. 1. (Color online) Generic non-coplanar tri-layer SFS systems: (a) Non-collinear spin-filtering barriers (FI) with polarizations P r,l and metallic ferromagnetic layer (F) with exchange field h.(b) The same configuration as in (a) and Zeeman fields b r,l in superconducting electrodes.(c) Spinfiltering barrier with polarization P and two layers of metallic ferromagnet with non-collinear magnetizations h1 (F) and h2 (F').