Spin supercurrent in Josephson contacts with noncollinear ferromagnets

We present a theoretical study of the Josephson coupling of two s-wave superconductors which are connected through a diffusive contact consisting of noncollinear ferromagnetic domains. First, we consider a contact with two domains with magnetization vectors misoriented by an angle $\theta$. Using the quantum circuit theory, we find that in addition to the charge supercurrent, a spin supercurrent, which is even in $\phi$ and odd in $\theta$, with a spin polarization normal to the magnetization vectors flows between the domains. Furthermore, with asymmetric insulating barriers at the interfaces of the junction, the system may experience an antiferromagnetic-ferromagnetic phase transition for $\phi=\pi$. Secondly, we discuss the spin supercurrent in an extended magnetic texture with multiple domainwalls. We find the position-dependent spin supercurrent. The magnitude of the spin supercurrent strongly depends on the phase difference between the superconductors and the number of domain walls. Our results demonstrate the possibility to couple the superconducting phase to the magnetization dynamics.


Abstract.
We present a theoretical study of the Josephson coupling of two superconductors which are connected through a diffusive contact consisting of noncollinear ferromagnetic domains. The leads are conventional s-wave superconductors with a phase difference of ϕ. First, we consider a contact with two domains with magnetization vectors misoriented by an angle θ. Using the quantum circuit theory, we find that in addition to the charge supercurrent, which shows a 0−π transition relative to the angle θ, a spin supercurrent with a spin polarization normal to the magnetization vectors flows between the domains. While the charge supercurrent, is odd in ϕ and even in θ, the spin supercurrent is even in ϕ and odd in θ. Furthermore, with asymmetric insulating barriers at the interfaces of the junction, the system may experience an antiferromagnetic-ferromagnetic phase transition for ϕ = π. Secondly, we discuss the spin supercurrent in an extended magnetic texture with multiple domainwalls. We find the position-dependent spin supercurrent. While the direction of the spin supercurrent is always perpendicular to the plane of the magnetization vectors, the magnitude of the spin supercurrent strongly depends on the phase difference between the superconductors and the number of domain walls. In particular, our results reveal a high sensitivity of the spin-and charge-transport in the junction to the number of domain walls in the ferromagnet. We show that superconductivity in coexistence with non-collinear magnetism, can be used in a Josephson nanodevice to create a controllable spin supercurrent acting as a spin transfer torque on a system. Our results demonstrate the possibility to couple the superconducting phase to the magnetization dynamics and, hence, constitutes a quantum interface e.g. between the magnetization and a superconducting qubit.

Introduction
The Josephson effect refers to the coherent transfer of Cooper pairs between two weakly coupled superconductors [1]. In a Josephson contact with a normal metal between the superconductors, the underlying microscopic mechanism is Andreev scattering [2] at the two normal-metal-superconductor interfaces which converts electron and hole excitations of opposite spin directions into each other by creating a Cooper pair.
The resulting dissipationless electrical current is driven by the difference between the phases of the superconducting order parameters across the contact. From the fact that superconductivity is a coherent state of spontaneously broken U(1)-symmetry, it follows that the Josephson effect is a response of this coherent state to an inhomogeneity over the junction which is produced by the variation of the phase. An analogous nonsuperconducting effect is predicted to exist in the magnetic tunnel barrier between two ferromagnets with the SO(3) symmetry breaking coherent states [3,4]. In this case the misorientation angle θ of the two magnetization vectors is the driving potential for a dissipationless spin supercurrent, similar to the exchange interaction.
In this Article we develop the circuit theory of the superconducting spin Josephson effect in an inhomogeneous ferromagnetic (F) contact between two conventional superconductors. We show that when the F contact consists of two domains whose magnetization vectors enclose an angle θ, in addition to the charge supercurrent, a spin supercurrent will also appear. This spin supercurrent is created by the simultaneous existence of the superconducting states and a noncollinear orientation θ of the magnetization vectors. Interestingly, we find that the spin supercurrent and the corresponding spin transfer torque is directed perpendicular to the plane of the two magnetization vectors, such that it would lead to a precession of the magnetizations around each other. In addition to extensive theoretical [5,6,7,8] and experimental [9,10,11,12,13] studies of the spin-transfer torque in F spin-valve and domain structures, there have been studies devoted to the spin-transfer torque in structures with superconducting parts [14,15,16,22,17,18,19,20,21,22,23,24,25,26]. The essential effect is the production of long-ranged spin-triplet superconducting correlations by the interplay between the induced spin-singlet correlations and the noncollinearity of the magnetization profile in F contact [27,28,29,30,31,32,33]. Compared to previous studies, we present a quantum circuit theory calculation which takes the spatial variation of Green's functions as well as the nonlinearity of the proximity effect fully into account. By this method we specifically are able to obtain the inhomogeneity of the spin supercurrent and to define a spin-transfer torque in non-collinear ferromagnetic Josephson contacts.
Several experimental works on this triplet proximity effect have been done [34,35]. Recently, Khaire et al. [36] reported the observation of the long-range supercurrent in Josephson junctions which is controllable by varying the thickness of one of the ferromagnetic domains. Also, Robinson et al. [37,38] detected the flow of a long-range supercurrent in the ferromagnetic Josephson junction with a magnetic Ho-Co-Ho trilayer and found an enhancement of the critical currents in the antiparallel configuration of the Junctions with a trilayer Fe/Cr/Fe barrier.
In analogy to the conventional charge Josephson effect, the spin Josephson effect has the tendency to remove the inhomogeneity of the order parameter vector of the spin-triplet superconducting state in the F-contact. This is analogous to the spin Josephson effect in contacts between two unconventional triplet superconductors, where the Cooper pair spin current appears in conjunction with the usual charge supercurrent [39,40,41,42].
The spin-dependent circuit theory has already been used in Ref. [32] to study the density of states, and the Josephson supercurrent in S/F/S heterostructures, which are shown to be dependent on the configuration of the magnetization in F. Here, we further study the spin supercurrent and spin transfer torque in such Josephson junctions. We demonstrate the dependence of the charge and spin supercurrent on the phase difference ϕ and the angle between the magnetizations θ: the spin supercurrent is an even function of ϕ and an odd function of θ, the charge supercurrent satisfies the inverse relations relative to the ϕ and θ. Further, we study the equilibrium configuration of the exchange field vectors as a function of the phase difference and the temperature. We obtain phase diagrams which show the antiferromagnetic-ferromagnetic phase transitions in the system which hasn't been announced in any other literatures.
We also discuss the generality of this effect for other ferromagnetic contacts with a more complex inhomogeneity of the direction of the magnetization vector. In particular, when the ferromagnetic contact consists of an in-plane rotating magnetization vector between two homogeneous domains with antiparallel magnetization, we find that the spin supercurrent is highly sensitive to the value of the wave vector. We show that one can tune the spin supercurrent acting as a spin transfer torque, by changing the phase difference between the superconductors or with variations of the wave vector. Also, we investigate the position dependence of the spin current in S1F1DWF2S2 which, to the best of our knowledge, hasn't been studied in any other texts. We find that the behavior of spin supercurrent relative to the position strongly depends on the phase difference between the superconductors. We extend the quantum circuit theory by the description, how spin-transfer torques can be calculated within the method.

Model and basic equation
We describe the basic theory and the model first for a two-domain ferromagnetic contact between two conventional superconductors, as is shown schematically in Fig. 1a. The exchange field of one domain F 1 makes an angle θ with that of the other domain F 2 . We restrict our study to the time-independent case and do not consider changes in the magnetic structure in this article. The generalization to the structure with a continuous magnetization texture in Fig. 1b is straightforward and described in the end of this section. To proceed our work, we make use of the quantum circuit theory which is a finite-element technique for calculating the quasiclassical Green's functions in diffusive nanostructures [43,44,45,46,47,48]. In this technique, we represent each F domain and S reservoirs by a single node, which is characterized by an energy-dependent 4 × 4matrix Green's functionǦ i , in Nambu and spin (=↑, ↓) spaces [47,48]. Furthermore, the two nodes in the F domains are assumed to be weakly coupled to each other by means of a tunneling contact. In terms of its spin-space matrix componentsĝ andf , the matrix Green's function is written aš We consider the equilibrium condition where a misorientation angle θ and phase difference ϕ may drive Josephson spin and charge currents between two adjacent nodes. These equilibrium Josephson currents can be extracted from the matrix current defined asǏ where g ij is the tunneling conductance of the contact between two nodes and i and j denote the connected nodes. This approach works also, if we divide the ferromagnetic region into n nodes, as is necessary in the case of Fig. 1. Then we have to take . Where, the conducting part of F domains is discretized into n nodes. For a S1F1F2S2 structure, n is two. The conductance of the tunnel barrier between S1(2) and F1 (2) is denoted by g S1(2)F 1 (2) and, g F 1F 2 is the conductance of the whole F1F2 contact. The matrix current obeys the following law of current conservation in matrix form HereǏ ωi = −G Q (ω/δ i ) τ 3 ,Ǧ i , with δ i being the electronic level spacing of the node, is the matrix of the leakage current which takes into account the dephasing of electrons and holes due to their finite dwell time in the node i, ,Ǧ i is the corresponding matrix current representing the leakage caused by the spin-splitting due to an exchange field h i (G Q ≡ e 2 /(2πh) is the quantum of the conductance). The third term represents the matrix currents from the neighboring nodes i-1,i+1. From Eq. (3) we can find the spin-torque, e.g. in z-direction as τ zi = I zi,i+1 + I zi,i−1 .
Equation (3) is given for all nodes and is supplemented by boundary conditions, which are the values ofǦ in the S reservoirs. We neglect the inverse proximity effect in the reservoirs and set the matrix Green's function in S1 and S2 to the bulk values: Here,∆ 1,2 = |∆| exp (±iϕ/2)σ 1 are, respectively, the superconducting order parameter matrix in S1 and S2 (σ i denote the Pauli matrices in spin space) and ω = πT (2m + 1), with m being an integer, is the Matsubara frequency. The temperature dependence of the amplitude of the order parameter is well approximated by |∆|= 1.76T c tanh(1.74 T c /T − 1). We note that the matrix Green's function satisfies the normalization conditionǦ 2 =1. We have solved these equations numerically by an iteration method. In our calculation we start by choosing a trial form of the matrix Green's functions of the nodes, for a given φ, T, and the Matsubara frequency m=0. Then, using Eq. (3) and the boundary conditions iteratively, we refine the initial values until the Green's functions are calculated in each of two nodes with the desired accuracy. Note that in general for any phase difference φ, the resulting Green's functions vary from one node to another, simulating the spatial variation along the F contact (see Ref. [49]). From the resulting Green's functions and Eq. (2) we find the matrix currents. Then, we calculate the charge supercurrent I and the components of the spin supercurrent vector I from the relations in which tr . . . = (iπT /2e) ω Tr . . . with Tr denoting the trace in Nambu-spin spaces. In the next steps we change to the next Matsubara frequency and use the results from the previous one ω m−1 as initial guess. We find the respective contribution to the spectral currents and continue to higher frequencies until the required precision of the summation over m is achieved. In the following we scale the length of the system, L, in units of the diffusive superconducting coherence length ξ, and use the dimensionless parameters of h/T c and t = T /T c , as measures of the amplitude of the exchange field h and the temperature T .To describe a continuous domain wall, we use as parameter the wave vector Q associated with one full winding of the magnetization by 2π. Hence the total number of windings of the magnetization is given by QL F /2π, where L F is the length of the inhomogeneous region (see Fig. 1). The currents are expressed in units of I 0 , the amplitude of the critical Josephson charge current at h = 0 and t = 0.

Results and discussions
From the numerical calculations we obtain the spin supercurrent with a polarization directed normal to the plane of the magnetization vectors. Further, we find a transition of the favorable configuration of the domain, from antiparallel to the parallel as the exchange field of the asymmetric domains increases. Also, we show that in a system with more complex configuration of the direction of the magnetization, the profile and penetration depth of the spin supercurrent are highly dependent on the number of the rotations that the magnetization vector has undergone across the domain wall.

S1F1F2S2 junction
Using the method described in Sec. 2 we have calculated the spin and charge supercurrents for the two domains F contact of Fig. 1a, when the exchange field vectors are taken to be in the x − y plane. We found that the spin supercurrent has a polarization which is aligned along the z axis, namely perpendicular to the plane of the exchange fields of F 1 and F 2 . Our results for the dependence of the spin I z and charge I supercurrents on the misorientation angle θ, the phase difference ϕ and the temperature t are shown in Fig. 2. We found that, in general, the spin supercurrent obeys the symmetry relations I z (ϕ) = I z (−ϕ) and I z (θ) = −I z (−θ), which are the analogs of the relations I(ϕ) = −I(−ϕ) and I(θ) = I(−θ) for the charge supercurrent (see Fig. 2 a,b). These behaviors suggest that one can change the direction of the spin supercurrent, which is proportional to the induced spin transfer torque [7], by changing the phase difference between two superconductors when θ is fixed. We note that a nonzero spin supercurrent is provided by a noncollinear orientation of the exchange field vectors and the existence of the superconductivity (|∆| = 0), even for ϕ = 0.
We define the critical spin supercurrent, I zcr (ϕ), as the maximum of the absolute value of the spin supercurrent as a function of θ for a given ϕ, in similarity to the definition of the charge critical supercurrent. We may also use a distinct definition, which we denote by I zmax (θ), as the absolute value of the spin supercurrent for a value of ϕ which maximizes the charge supercurrent as a function of ϕ, for a given θ. Figure. 2c shows the behavior of I zcr as a function of ϕ for different temperatures. At a given temperature t, I zcr (ϕ) shows a change of sign at a phase difference which depends on t. This change of sign may be recognized as the signature of a transition between 0 and π spin Josephson couplings, in analogy to the charge 0 − π transition in F Josephson junctions [50,51,52,53,54,55,56,57,58,59]. The corresponding critical charge current shows a 0−π transition with varying θ. Note that both I zcr (ϕ) and I cr (θ) have a nonzero value at the transition point at low temperatures, as the signature of nonzero second harmonic in the current-phase and the current-angle relations [60,61,62]. In Fig. 2d we have also plotted I zmax (θ), which shows a change of sign at θ = π for all temperatures [19]. We have also studied the dependence of the spin supercurrent on the absolute value of the exchange field. The results are shown in Fig. 3, in which I zcr is plotted as a function of h/T c for different ϕ and θ. These results show that the sign and the amplitude of the spin supercurrent can be also modulated by varying h/T c , which can be used for further tuning of the corresponding spin transfer torque. We note that for strong ferromagnets with h ≫ ∆, the spin supercurrent vanishes. This is due to the suppression of the amplitude of the Andreev reflection at S 1 F 1 and S 2 F 2 interfaces in this limit, which suppresses the proximity effect.
It is also interesting to study the equilibrium configuration of the exchange field vectors as a function of the phase difference and the temperature. The equilibrium angle can be obtained by minimizing the free energy, F , of the contact as a function of θ. We have calculated the θ-dependence of F by integrating the spin supercurrent over θ and  charge supercurrent over φ: Our calculation shows that the exchange field vectors favor either parallel (θ = 0) or antiparallel (θ = π) configurations, depending on ϕ, t and h/T c . The behavior of this superconductivity-induced exchange coupling differs for the two cases of a contact with symmetric barriers with g S1F 1 = g S2F 2 and an asymmetric contact with a very different g S1F 1 and g S2F 2 . For a symmetric system, we have found that the coupling is antiferromagnetic (θ = π) for ϕ = 0, but becomes ferromagnetic (θ = 0) for ϕ = π. This behavior is found to hold irrespective of the values of L/ξ, h/T c . However, for an asymmetric system it is possible to change the coupling from ferromagnetic to antiferromagnetic and vice versa by varying L/ξ, h/T c , or t, for the phase difference ϕ = π. In Fig. 4, we have shown the ferromagnetic-antiferromagnetic coupling phase diagram of the system in the plane of h/T c and t, when ϕ = π and for some different values of L/ξ. This phase diagram is similar to the 0 − π Josephson couplings phase diagram of a homogenous F contact between two superconductors, see Ref. [49]. As we show in the inset of Fig. 4, the minimum of F as a function of θ shifts from θ = π for low values of h/T c to θ = 0 at higher h/T c s, when L/ξ = 1 and t = 0.4. The temperature induced transition between the ferromagnetic and the antiferromagnetic phases is also possible but only over a finite interval △h of the amplitude of the exchange field of the F domains. This width of the temperature induced transition increases with decreasing L/ξ. For ϕ = 0, the coupling between the exchange fields of the two domains is found to be always antiferromagnetic in an asymmetric structure, which is very similar to the symmetric case.

S1F1DWF2S2 junction with Neel domain wall
The superconducting spin Josephson effect described above may take place in F contacts with a more complex profile of the exchange field vector. An interesting case is a finite width F domain wall between two domains, where the exchange field has a continuous spatial rotation between two homogeneous F domains. Here, we present the results of our calculation of the spin supercurrent and the spin transfer torque for a Neel domain wall junction, which is shown schematically in Fig. 1b. Note that the spin transfer torque acting on the local magnetization on node i is obtained from τ zi = I zi,i+1 + I zi,i−1 , as we have shown earlier. We model the local angle of the exchange field vector with respect to the y-axis to vary as α(x) = (QL F )(x/L F ). We have obtained the position-dependent spin supercurrent, which is perpendicular to the plane of the magnetization and flows in the homogeneously and inhomogeneously magnetized parts of the system. To the best of our knowledge, there have been no previous studies investigating the position dependence of the spin current and the spin transfer torque. Figures 5(a,b) show the behavior of the spin transfer torque versus position for a system with L F /L = 1/3. We can see that the system has an interesting behavior depending strongly on the values of QL F . As we see in Fig. 5a, the spin transfer torque penetrating into the homogeneous ferromagnets becomes of negligible constant value. This shows that the spin supercurrent in F 1 and F 2 has a linear position dependence when φ = 0. In particular, this is true while the value of the spin supercurrent is comparable to the one in the nonhomogeneous parts.
Also Fig. 5b shows that the penetrating spin transfer torque in the homogeneous parts is nearly zero for QL F = π, 3π and is much smaller than the one in the domain wall region for QL F = 2π, 4π when φ = π. In addition, we note that the the spin transfer torque has always a symmetric position dependence around x = L/2, whereas the spin supercurrent always shows an asymmetric behavior. Our calculations demonstrate further that the behavior of the spin current and spin transfer torque versus position for a system with QL F π and φ = π is similar to that of the system with QL F = (n + 1)π and φ = 0, see Fig. 5. This observation of a symmetry between the magnetic winding number and the superconducting phase is presently not fully understood, but will be the subject of future research.
Finally, we have also studied Josephson systems without the homogeneous ferromagnetic parts F 1 and F 2 , which corresponds to L F = L. In this way we would like to check how the spin currents in the homogenous part, which are appreciable in size, but give rise to a negligible spin-transfer torque, influence the spin-torque on the magnetization texture. We show the corresponding dependence of the spin supercurrent on the phase difference and the position in Fig. 6 for a wall with rotation angle π. The first thing to note, is that we observe a qualitatively similar phase dependence as in the case with two homogeneous ferromagnets. In particular, the I z (φ) relation is always a symmetric. Also, for a specific position, the sign of the spin transfer torque can be changed by varying the phase difference between the superconductors. In addition, for a simple Neel domain wall, when QL F = π, we see that the behavior of the spin transfer torque can be approximately described by the relation cos[(x/L)(QL F ) + π/2] for φ = 0, which turns into cos[3(x/L)(QL F ) + π/2] for φ = π. These findings have the same symmetry, as we mentioned in the previous discussion for a system with homogenous ferromagnet attached. Also, this result shows the strong dependence of the spin transfer torque on the phase difference between the superconductors, which can be interpreted as a direct interplay between the induced spin-singlet correlations of the superconductors and the inhomogeneity of the magnetization in the domain wall. In fact this interplay leads to the generation of the triplet correlations in the contact region, whose inhomogeneity drives the spin supercurrent and therefore the spin-transfer torque. Figures 7(a,b) show the spin transfer torque as function of the phase difference, φ, and the wave vector, QL F (which is more or less given by the number of 180 degree domain walls). In Figs. 7(a,b) the behavior for different positions x/L = 1/2 and x = L/6 are shown. For the position closer to the superconductor, x/L = 1/6, the spin transfer torque goes to zero much faster for larger wave vectors than in the middle of the ferromagnet at x = L/2. Hence, we see that the suppression of the spin transfer torque strongly depends on the position in the domain wall. Also, Fig. 7(a,b) shows that the spin transfer torque oscillations versus wave vector strongly depend on the phase difference and also on position. While they start for small QL F in exactly opposite fashion for both positions, the oscillations in the middle of the domain wall are well behaved also for large QL F , whereas the behavior is more complex in the ferromagnetic part close to the superconductor. In that limit no well defined oscillation period can be identified. Furthermore, while the direction of the spin supercurrent is fixed, we find that the sign and magnitude of the spin supercurrent and spin transfer torque can be modulated by changing the wave vector or the phase difference. We have obtained that superconductivity in coexistence with non-collinear magnetism can be used in a Josephson nanodevice to create the tunable spin supercurrent which acts as a spin transfer torque on the junctions magnetization.

Conclusion
We have studied the Josephson effect in a diffusive contact consisting of two ferromagnetic domains with noncollinear magnetizations which connects two conventional superconductors. Using quantum circuit theory, we have shown that the spin supercurrent will flow through the contact, due to the generation of an inhomogeneous spin-triplet superconducting correlations. The polarization of the spin supercurrent is directed normal to the plane of two magnetization vectors in a way that the resulting spin transfer torque intends to align the magnetization of the two domains. The spin and charge current-phase-angle relations obey the specific symmetry relations versus the phase difference ϕ and the misorientation angle θ. Whereas, the charge supercurrent satisfies the odd-even relationship on the ϕ and θ, the spin supercurrent is an even-odd function of φ and θ. From these relations we have predicted a transition between 0 and π Josephson coupling by varying the misorientation angle θ. We found a transition of the favorable configuration of the domain, from antiparallel to the parallel as the exchange field of the domains increases. This transition occurs for asymmetric systems with φ = π. Also, the domains in the symmetric systems settle in a parallel configuration when φ = π.
We have further discussed the generation of the spin supercurrent in magnetic contacts with more complex configuration of the direction of the magnetization vector. For a domain wall between two domains with antiparallel magnetizations, we have shown that the profile and penetration depth of the spin supercurrent are highly dependent on the number of the rotations that the magnetization vector has undergone across the domain wall. In particular, we show that, while the direction of the spin supercurrent is always perpendicular to the plane of the magnetization vectors, the sign and the magnitude of the spin supercurrent strongly depends on the phase difference between the superconductors and the value of the wave vector. We present a Josephson nanodevice