Spin torques and magnetic texture dynamics driven by the supercurrent in superconductor/ferromagnet structures

We introduce the general formalism to describe spin torques induced by the supercurrents injected from the adjacent superconducting electrodes into the spin-textured ferromagnets. By considering the adiabatic limit for the equal-spin superconducting correlations in the ferromagnet we show that the supercurrent can generate both the field-like spin transfer torque and the spin-orbital torque. These dissipationless spin torques are expressed through the current-induced corrections to the effective field derived from the system energy. The general formalism is applied to show that the supercurrent can either shift or move the magnetic domain walls depending on their structure and the type of spin-orbital interaction in the system. These results can be used for the prediction and interpretation of the experiments studying magnetic texture dynamics in superconductor/ferromagnet/superconductor Josephson junctions and other hybrid structures.


I. INTRODUCTION
It has been commonly recognized that reducing Joule heating effects and power consumption are among the main priorities for the development of electrically controlled magnetic memory devices [1][2][3][4] . Since the first spin transfer torque (STT) experiments 5,6 much effort has been invested to optimize the switching currents, thermal stability and tunnel magnetoresistance of the magnetic tunnel junctions 3,7,8 . Thermal effects are also of the crucial importance for the operation of the other type of STT memory -magnetic racetrack memory 9-11 based on the electrical control over the domain wall (DW) motion. The progress in improving these spin memory devises depends crucially on the competition between the thermal stability of DWs and large current densities required to overcome the pinning forces 4, [12][13][14][15][16][17][18] . As an alternative route to the low-power manipulation of magnetic textures the current-driven magnetic skyrmion dynamics has attracted large interest [19][20][21] .
In applications which require very large currents, for example in powerful magnets, using superconducting materials have been proven to be an effective solution to eliminate Joule heating effects. In view of the energysaving spintronics it is quite appealing to employ the spin torques generated by the dissipationless spin-polarized superconducting currents (supercurrents). The existence of spin-polarized supercurrents is ubiquitous to the spin-textured superconductor/ferromagnet (SC/FM) hybrid structures resulting from long-range spin-triplet proximity [22][23][24][25] .
Recently there have been many works studying spinpolarized supercurrents in various SC/FM systems (for the review see Refs. 26,27 ). However the supercurrentinduced spin torques have been characterised theoretically only in several model systems: in Josephson junctions through the single-domain magnets [28][29][30] , two 31 and three 32 FM layers, in ferromagnetic spin-singlet 33 and spin-triplet superconductors 34 . The general understanding of the supercurrent-spin texture interaction has been lacking since there is no direct connection between the above examples and practically interesting systems -bulk non-homogeneous FMs. That is, the possibility of moving DWs and skyrmions by injecting the supercurrent in real ferromagnets has been an open question for a long time despite of the large attention to the subject.
This challenging question is addressed in the present paper. We employ the adiabatic approximation which is widely used for the description of kinetic processes in spin-textures metallic ferromagnets including the calculation of conductivity 19 and spin transfer torques 35 in the inhomogeneous FMs. We bring this approach to the realm of superconducting systems to describe their transport properties governed by equal-spin superconducting correlations. For that we go beyond the commonly used quasiclassical theory of hybrids 25,36 which has been designed to treat only the weak ferromagnets with the exchange splitting much less than the Fermi energy. Instead of that we employ the recently developed approach of generalized quasiclassical theory 37 which allows for the description of proximity effect in strong ferromagnets with the exchange splitting much larger than other energy scales and comparable to the Fermi energy.
We show that the spin-polarized superconducting current can induce magnetization dynamics, described in general by the Landau-Lifshitz-Gilbert (LLG) equatioṅ where γ = 2µ B is the electron gyromagnetic ratio. The second term in the r.h.s. is the Gilbert damping. The superconducting spin current J can induce two types of spin torques which can be written as the correction to effective field −γM ×H ef f = N st + N so . The first term here is the adiabatic spin-transfer torque 15,38,39 while the arXiv:1804.00718v1 [cond-mat.supr-con] 2 Apr 2018 second term N so is the spin-orbital (SO) torque 40 . The non-adiabatic (antidamping) STT 14 is not produced by the supercurrent since it breaks the time-reversal symmetry of LLG equation and can be considered as a correction to the dissipative Gilbert damping 41 . That is the antidamping STT should be connected with the quasiparticle contribution which is beyond the scope of our present study. The paper is organized as follows. In Sec.II the general equations for spin dynamics and spin torques generated by supercurrent are considered. In Sec.III we derive expressions for the spin-transfer and spin-orbital torques using the generalized quasiclassical theory. In Sec.IV we derive Josephson energy in SFS junctions and use it to provide an alternative derivation of supercurrent spin torques. Sec. V is devoted to the DW dynamics in SFS Josephson junctions induced by the supercurrent spin torques. Our conclusions are given in Sec.VI.

II. SPIN TORQUE GENERATED BY THE SUPERCURRENT
We use s-d model with the localized magnetization M and that of the itinerant electrons M s = −µ B s, where s is the electron spin and µ B is the Bohr magneton. The dynamics of localized spins is determined by the usual LLG equation with the contribution to the effective field resulting from exchange interaction with conductivity electrons 42 The last term here is the source of spin torque and should be found from the kinetic equation for conductivity electrons. The kinetic theory for the conduction electrons in metals can be formulated in terms of the matrix Green's functionǦ =Ǧ(r 1 , r 2 , t 1 , t 2 ) which has the following explicit structure in the Keldysh spaceǦ = Ǧ RǦK 0Ǧ A , whereǦ R/A/K are the retarder/advanced/Keldysh components. The general quantum kinetic equation reads: Here we define the •-product as (Â •B)(t 1 , t 2 ) = is given by the Eq.(5). The self-energy termΣ includes the effects related to disorder scattering as well as the off-diagonal superconducting self-energies. The conduction electron spin polarization s, charge j and spin J i currents are given by The strategy of studying magnetization dynamics consists of solving the coupled LLG (2) and kinetic equations (3)-(5) together with the expression for the magnetic moment (6). However, the general problem is too complicated for the analysis. In the next section (III) we discuss the simplification of the kinetic equation using the so-called generalized quasiclassical approximation 37 adopted to treat the non-stationary problems.
Besides that, significant simplification can be obtained in the linear response limit when the dynamics of magnetization is slow so that the characteristic frequency is small as compared to the energy gap in the quasiparticle spectrum. In this case we can make use of the quasistationary equation for the electron magnetization which is obtained from Eq.(3). Multiplying it byσ from the left and taking the trace we obtain Here we introduce the vector B j = (B xj , B yj , B zj ), which is determined by j-th coordinate component of the tensorB and J j = (J x j , J y j , J z j ) is the j-coordinate component of the spin current (8). Next, the driving term in the LLG equation (2) can be found neglecting the term with time derivative in Eq.(9) In this work we are interested in the quasi-equilibrium spin torques generated solely by the supercurrent without the contribution of non-equilibrium quasiparticles. That means the normal component of the current and the electric field are assumed to be absent. Generally, the last term with the collision integral in Eq.(10) has contributions both from the off-diagonal order parameter and the spin-orbital scattering self-energy. The present work is based on the following simplifying assumptions allowing to put Tr 4 [σǏ] K = 0. First, we are interested in the spin torques occurring in the normal metal ferromagnetic interlayer where the order parameter is absent.
Second, the exchange splitting between spin subbands is assumed to be large enough to suppress spin-flip transitions between them. Below we demonstrate that in this regime the first term in the r.h.s. of Eq. (10) produces the adiabatic STT 35,38,43 , while the second term yields the spin-orbit torque 40,44,45 . In order to find these contributions we calculate the spin supercurrent through the spatially-inhomogeneous ferromagnet. In the next section it is shown that in the adiabatic limit valid for the description of strong ferromagnet this calculation can be done analytically in the in the most general way using the technique developed in Ref. 37 .

III. GENERALIZED QUASICLASSICAL THEORY
Eilenberger equation for equal-spin correlations. To find the results in the adiabatic approximation it is convenient to work in the local reference frame, where the spin quantization axis is aligned with the local direction of the exchange field in the ferromagnet. Then we use the transformationǦ loc =Û †ǦÛ whereÛ =Û (r, t) is in general the time-and space-dependent unitary 2 × 2 matrix that rotates the spin quantization axis z to the local frame determined by the exchange field, so that h z. To implement the adiabatic approximation we introduce the equal-spin pairing components of the GF Here the projection operators to spin-up and spin-down states defined by the index σ = ±1 are given byγ σ0 = τ 0σ0 + στ 3σ3 ,γ σ1 =τ 1σ1 − στ 2σ2 ,γ σ2 =τ 2σ1 + στ 1σ2 , γ σ3 =τ 3σ0 +στ 0σ3 . The generalized quasiclassical theory is formulated in terms of the spin-less propagatorŝ whereĜ σ ES =Ĝ σ ES (p, r) is the GF in the mixed representation, ξ pσ = p 2 /2m F + σh − µ and the notation means that the integration takes into account the poles of GF near the corresponding Fermi surface. Then in the adiabatic approximation, that is neglecting the coupling between equal-spin and mixed-spin correlations 37 we obtain the generalized Keldysh-Eilenberger equation Here the spin-dependent Fermi velocities v ± = 2(µ ± h)/m F are determined on each of the spin-split Fermi surfaces . The spin-dependent gauge field is given by the superposition of two terms Z = Z m + Z so , where Z m i = −iTr σ zÛ † ∂ iÛ /2 is the texture-induced part and the term Z so i = m F (mB i ) (where m = M /M ) appears due to the SOC.
One can see that the Eilenberger-type equations for the spin-up/down correlations contain an additional U(1) gauge field Z which is added to the usual electromagnetic vector potential A with the opposite effective charges for spin-up and spin-down Cooper pairs. On a qualitative level it is equivalent to the adiabatic approximation in the single-particle problems that allows to describe the quantum system evolution in terms of the Berry gauge fields 46 .
Charge and spin currents. The Eilenberger equations (13) are supplemented by the expressions for the charge current j and the spin current J k , where k denoted the spin index. The former is given by where ν σ are the spin-resolved DOS and .. denotes the averaging over the spin-split Fermi surface. The spin current in rotated frame is given bỹ Diffusive limit. Let us consider the system with large nonmagnetic impurity scattering rate as compared to the superconducting energies determined by the bulk energy gap ∆. In this experimentally relevant diffusive limit it is possible to derive the generalized Usadel theory with the help of the normalization condition (ĝ σ •ĝ σ )(t 1 , t 2 ) = δ(t 1 − t 2 ) which holds due to the commutator structure of the quasiclassical equations (13).
The impurity self-energy in the Born approximation is given byΣ σ = ĝ σ /2iτ σ . In the dirty limit we have The solution of Eq. (17) can be found asĝ σ = ĝ σ + g a σ p σ /p σ , where the anisotropic part of the solutionĝ a σ is small with respect to ĝ σ . Making use of the relation { ĝ σ ,ĝ a σ } t = 0, which follows from the normalization condition, one obtainŝ Substituting to Eq.(13) and omitting the angle brackets we get the diffusion equation where D σ are the spin-dependent diffusion coefficients, in the isotropic case given by D σ = τ σ v 2 σ /3. The current and spin current are obtained substituting expansion (18) to Eqs.(15)-(16) Further simplification can be obtained as follows. First, due to the normalization condition we introduce the parametrization of Keldysh component in terms of the Then, switching to the mixed representation in time-energy domainĝ σ (t 1 , t 2 ) = ∞ −∞ĝ σ (ε, t)e −iε(t1−t2) dε/2π, where t = (t 1 + t 2 )/2 we keep only the lowest order terms in the time derivatives.
As an example of the above procedure one can obtain from (20) the charge current in the normal state j = e 2 (ν + D + − ν − D − )E e driven by the emergent electric field 19,47 E e = −∂ t Z. We however will neglect these effects and take into account only the quasi-equilibrium contributions to the currents given by σ is the spectral current and f 0 (ε) = tanh(ε/2T ) is equilibrium distribution function.
Supercurrent-induced torque. In the quasi-equilibrium regime when the time derivative of the GF in the mixed representation can be neglected Eqs. (13) or (19) yield the conservation of spin current in rotating frame ∇ ·J z = 0. The spin current in the laboratory frame is given by J k = R kzJ z which can be written in the form It is not conserved due to the spatially-dependent magnetization of d-electrons m = m(r). Substituting Eq. (24) into Eq.(10) we obtain the torque, induced by the supercurrent in the quasiequilibrium regime: Here N st is the supercurrent spin transfer torque, which takes only the form of the adiabatic torque in the considered approximation, and N so is the spin-orbit torque.
Its particular structure strongly depends on the type of the spin-orbit coupling, realized in the system. Below we show that due to the coherent nature of the spinpolarized superconducting current the same result can be obtained from the energy functional of the system yielding the correction to the effective field.

IV. SUPERCURRENT SPIN TORQUES AS CORRECTIONS TO THE EFFECTIVE FIELD
Above we have derived general expressions (26,27) for the superconducting spin torques starting from the kinetic equation treated in the adiabatic limit. For any particular system one can find the spin torques solving generalized Eilenberger/Usadel equations for the quasiclassical propagators and calculating the spin current according to Eq. (21).
An alternative approach to obtain superconducting spin torques is based on the description of magnetization dynamics in terms of the phenomenological expression for the effective field H ef f = −δF/δM , where F = F (M ) is the system energy as a functional of the magnetization distribution. The LLG equation without dissipation terms is given byṀ This approach cannot be applied to derive spin transfer torques in the normal state where the conduction electron magnetization is not coherent. In contrast to the normal system superconducting electrons are in the macroscopically coherent state. Therefore the total energy of the system written in terms of the macroscopic variables describes the interaction between the condensate spin and the ferromagnetic order parameter. Based on the above discussion one can conclude that the superconducting spin transfer torques (26) and (27) can be obtained from the energy arguments. To demonstrate this we consider a generic example of the Josephson system consisting of superconducting leads coupled through the ferromagnet with non-homogeneous magnetization texture. In general this task is rather complicated and requires extensive numerical calculations for each particular system considered. However, in strong ferromagnets the general expressions for Josephson spin and charge currents for different magnetic textures of the interlayer can be obtained using the machinery of the generalized quasiclassical theory 37 .
We consider the 1D magnetic texture M = M (x) in the interlayer of the thickness d between two superconducting interfaces, located at x = ±d/2. The superconducting order parameter phase difference between them is χ. The current-phase relation for this setup has been found 37 as the superposition of partial currents carried by the spin-up and spin-down Cooper pairs: where the amplitudes j σ are determined by the boundary conditions at FM/SC interfaces 37 . The rotated-frame spin currentJ z ≡J z x is given by the difference of the partial spin-up/down currents J z (χ) = 1 2e σ=± σj σ sin χ + 2σ The Josephson energy can be obtained according to the usual relation The current-phase relation (29) is given by j(χ) = 2|e|(dF J /dχ). Therefore calculating the correction to effective fieldH ef f = −δF J /δM (see details in Ap-pendixA) we obtain Substituting the result (32) to the LLG equation we get the spin transfer torques identical to the Eqs. (25,26,27). The above energy consideration demonstrates that the direct coupling between the magnetization and superconducting current exist even in the limit when the spontaneous charge current is absent. Indeed, the spontaneous phase shift of the Josephson current-phase relation (29) is given by tan χ 0 = tan ϕ(j Therefore χ 0 = 0 in the limit when the spin subbands are formally degenerate j + = j − . At the same time the Josephson spin current (30) and correspondingly the spin torque in the Eq.(32) are non-zero. This result generalizes the previously suggested mechanism of the supercurrent-induced spin-orbital torque stemming from the χ 0 = χ 0 (m) dependence 29,30 . In our case it is not only the phase shift, but in addition the overall critical current in Eq.(29) which depends on the magnetization j c = j 2 + + j 2 − + 2j + j − cos(2ϕ) through ϕ = ϕ(m). This provides the non-zero effective field even in the case of degenerate bands.

V. SUPERCURRENT DRIVEN MAGNETIC TEXTURE DYNAMICS
A. General case of the texture dynamics driven by the adiabatic STT The first striking consequence of the dissipationless supercurrent spin torques is the possibility to realize the quasiequilibrium magnetic texture dynamics driven solely by the adiabatic STT generated by the superconducting current. In the absence of dissipation the LLG equation (1) have the solution in the form of the travelling wave m = m(x − ut) with the constant velocity determined by the spin current u = 2µ BJ z /M . For the periodical magnetic structure , e.g. magnetic helix that yields locally rotating magnetization with the frequency defined by ω ∼ u/L, where L is the period. We can estimate u for the parameters of half-metallic CrO 2 nanostructures which are the promising elements for the dissipationless superconducting spintronics 48 . The maximal Josephson current density through the CrO 2 nanowire is j ∼ 10 9 A/m 2 , which determines the spin currentJ z = j/(2e). Taking into account the saturation magnetization M = 4.75 × 10 5 A/m we get the speed of the order of u = 1 m/s. However, these time-dependent quasiequilibrium solutions do not correspond to the ground state. It can be reached only in the presence of the Gilbert damping which transforms the magnetic texture in such a way to compensate the effective field generated by the spin-polarized supercurrent. Therefore, eventually the systems will stop at the stationary state when H ef f = 0. As we show below, in case if the initial state contains DW the ground state modified by the supercurrent can correspond either to the distorted DW or to the homogeneous state when the DW is eliminated from the sample.

B. Domain wall motion
Now we consider the magnetic texture of the ferromagnet in the form of the DW. We are interested in its dynamics induced by the supercurrent spin torques, discussed above. The two particular types of DW are considered: head-to head DW and Neel DW.
The particular shape of the DW is dictated by the combination of the anisotropy energy and the exchange energy. We start with the head-to-head DW. In this case the corresponding energy term can be written as follows: where K > 0 and K ⊥ > 0 are the anisotropy constants for the easy and hard axes, respectively. A ex is the constant describing the inhomogeneous part of the exchange energy. The effective magnetic field It is convenient to parametrize the magnetization as follows: m = (cos θ, sin θ cos δ, sin θ sin δ), (34) where in general the both angles depend on (x, τ ). At zero applied supercurrent the equilibrium shape of the DW is given by δ = π/2 and where d w = A ex /K is the DW width. The above ansatz corresponds to the head-to-head DW, lying in the xz-plane. The tail-to-tail DW can be obtained by θ → θ + π.
Let us consider the behavior of the head-to head DW under the applied supercurrent and the presence of SOC given by the superposition of the Rashba-type term 2m F µ B B x /M = (0, −β R , 0) and the Dresselhaus-type term 2m F µ B B x /M = (β D , 0, 0). First, we follow the Walker's procedure 49 by assuming that δ = δ(t) and the DW is moving according to the time-dependent shift x 0 (t) = t 0 v(t )dt in the Eq. (35).
Substituting this ansatz to the LLG equation we obtain that this type of the solution exists only in the absence of Rashba SOC β R = 0. We assume that the distortion of the wall is small during the wall motion, that is δ = π/2+δ 1 , where |δ 1 | << 1. In this case taking into account that d w ∇ x θ = sin θ for the DW we obtain In this case Eqs. (36) and (37) yield the following equation for v(t): Taking into account the initial condition determined by the Eq. (37) (1 + α 2 )v(t = 0) = u − 2d w αβ DJ z , which follows from δ 1 (t = 0) = 0 we determine the solution of Eq. (38) in the form: where t d = (1 + α 2 )M/(αγK ⊥ ) is the characteristic time scale. The solution for the moving DW expressed by Eqs. (39,40) exactly coincides with the solution found for the DW motion in normal ferromagnets under the influence of the adiabatic and nonadiabatic torques 50 . But, nevertheless, there is an important physical difference between the spin-orbit torque, considered here, and the nonadiabatic spin torque. As it can be seen from Eq. (27), the SO torque is equivalent to the torque, generated by an external applied field γH = −4µ B m FJ z j B j /M . Consequently, it moves DWs of opposite types (+/-and -/+) to opposite directions as opposed to the action of the nonadiabatic torque, which moves all the DWs in one and the same direction. At the same time, it is seen from Eqs. (36) and (37) that the Rashba SO torque is equivalent to the field perpendicular to the wall plane, therefore it does not move the DW and only distorts it. The solution (39,40) is only valid for small enough electric and, correspondingly, spin currents, applied to the system. If the current is large enough, the condition |δ 1 | 1 is violated and Eqs. (36) and (37) are not valid. It was shown 15 that in this regime forJ z >J crit the DW can be moved even by the adiabatic torque only.
We consider the regime of arbitrary values of the applied current numerically by solving Eq. (2) together with the expressions for the torque Eqs. (25)- (27) and the effective field H ef f , found from Eq. (33). The results for the case of small enough applied currents, when our analytical solutions are valid, are represented in Fig. 1. The figure demonstrates the displacement of the DW center as a function of time. The black curve corresponds to the case of no spin-orbit torque. The blue and pink curves are for the Rashba case β R = 0, β D = 0. They demonstrate that the Rashba spin-orbit torque does not move the DW in this case, as it was mentioned above. The green and red curves demonstrate the influence of the Dresselhaus SO torque on the DW motion. In agreement with our analytical calculations, the numerics gives that at t t d the DW moves with the constant velocity. The direction of the motion is determined by the sign of β D or, in other words, by the sign of the effective magnetic field. In this case it is possible that the DW reverses the direction of its motion if the adiabatic spin torque tends to displace it in the direction opposite to the one dictated by the effective field. This case is illustrated by the red curve in Fig.(1). The regime of large applied currentsJ z >J crit , when the DW can be moved by the adiabatic torque only, is shown in Fig. 2. We have obtained that the value of J crit is rather close to Kd w . Therefore the critical electric current density is of the order eKd w / ∼ 10 10 A/m 2 which is an order of magnitude larger the Josephson critical current obtained in experiment 48 . Again, the black curve in Fig. 2 shows the displacement of the DW in the absence of the SO torques. The initial dynamics of the DW at small t coincides with Fig. 1, but at larger values of t the situation changes, so that in this regime the DW moves, but its velocity is not constant. The Rashba SO torque does not cause any essential influence on the DW dynamics, as in the case of the small applied currents. But the effect of Dresselhaus SOC is significant and at the first glance unexpected. Indeed, as shown in Fig. 1 the torque generated by this type of SOC e.g. for β D < 0 moves the DW to the direction x > 0. But in the above-threshold regime it can also reduce the averaged DW velocity (green curve in Fig. 2), that is the combined action of the adiabatic ST torque and SO torque cannot be viewed just as a simple sum of independent motions due to the both reasons. Vice versa, the SO torque generated at β D > 0, which by itself tends to move the DW to the direction x < 0, can slightly enhance the average DW velocity, as it is demonstrated by the red curve.
The influence of the Dresselhaus SOC on the DW average velocity is represented in Fig. 3 in more detail. Fig. 3(a) demonstrates the displacement of the DW as a function of time t for several values of β D > 0. It is seen that there is a weak increase of the average velocity at β D > 0, but the more important and pronounced effect is that increasing β D leads to the decrease of the velocity oscillation period. The case β D < 0 is shown in Fig. 3(b), where once can see that the dependence of the average DW velocity on β D is nonmonotonous. While at smaller values of |β D | the average velocity is indeed reduced with respect to the case β D = 0, at larger values of |β D | the velocity starts to increase and exceed its value at β D = 0 considerably. This behavior can be understood in the framework of the analogy between the SO torque and the magneticfield induced torque. For the situation when the DW moves under the combined action of the spin-induced torque and field-induced torque it is known that the steady motion of the DW withδ = 0 is only possible for a range of fields and currents 51 . The lines in the (J z , H) -plane, separating the regions of steady motion and precession motionδ = 0, are called by the Walkerlike stability lines 51 . This limit condition for the steady motion is strictly equivalent to the Walker breakdown 49 condition in the case where only an external magnetic field is applied. For the problem under consideration the increase of β D absolute value at fixed current is equivalent to the increase of the applied field (at fixed current). When at zero β D the system is in the precession regime, as in Fig. 2, the increase of |β D | at β D < 0 moves the system towards the steady motion region, where the wall velocity is higher. Therefore, the transition from the precession regime to the steady regime in Figs.1 and 2 is analogous to crossing the Walker-like stability lines for problem of DWs motion under the combined action of the spin-induced torque and field-induced torque.  Let us now consider the Neel DW. In this case the combination of the anisotropy energy and the exchange energy takes the form: It is convenient to parametrize the magnetization as: m = (sin θ sin δ, cos θ, sin θ cos δ).
At zero applied supercurrent the equilibrium magnetization profile is described by Eqs. (35) and δ = π/2. It can be shown that the problem of the Neel DW motion in the presence of the Rashba SO coupling is mathematically equivalent to the considered above motion of the head-to-head DW in the presence of the Dresselhaus SO coupling with the substitution β D → −β R . Therefore, in this case the Rashba SO torque plays the part of the field-induced torque moving DWs. The above analysis demonstrates that the dynamics of a DW under an applied supercurrent depends strongly (i) on the particular type of the DW and (ii) on the particular type of the SO coupling, which induces the spin-orbit torque. The stationary motion of the DWs induced by small supercurrents is possible even in the absence of the nonadiabatic torque if the spin-orbit torque is present in the system. Due to the presence of the Gilbert damping the motion of a DW by a supercurrent is not a disspationless process. Interestingly the DW motion generates voltage across the junction in the regime when the charge current is fixed but its magnitude is smaller than the Josephson critical current of the system. The voltage amplitude can be roughly estimated from the balance of the energy dissipation rate in the magnetic subsystem due to the Gilbert damping and the power put in by the current source. The characteristic energy dissipation rate can be estimated asḞ ∼ ∆F/t d , where ∆F is the difference between the free energies of the equilibrium state of the DW at zero current and the nonequilibrium state of the distorted wall in the presence of the current. For small distortions of the DW ∆F can be obtained as follows: Substituting the equilibrium profile of the DW θ(x) given by the Eq.(35) into Eq. (43), we obtain: where S p is the cross-section area of the ferromagnet. The voltage, generated at the Josephson junction can be estimated as V ∼Ḟ /S p I c which yields where I c is the Josephson current density and we have assumed that α 1. For estimations we use the material parameters of the CrO 2 nanostructures 52 which are the promising systems for the dissipationless spintronics 48 . Taking the maximal Josephson current density through the CrO 2 nanowire to be j ∼ 10 9 A/m 2 , the saturation magnetization M = 4.75 × 10 5 A/m, d w = 10 −6 cm, K = 1.43 × 10 5 erg/cm 3 and K ⊥ = 3K, we obtain V ∼ 0.1δ 2 1 [mV ] , where we took into account the typical values of the Gilbert damping α ∼ 0.01. The amplitude of DW distortion angle can be varied in wide limits, e.g. δ 2 1 ∼ 10 −4 − 10 −3 for the red curve in Fig.1, δ 2 1 ∼ 10 −3 for the green curve in Fig.1 and δ 2 1 ∼ 10 −1 for the dashed green curve in Fig.3b.

VI. CONCLUSION
To conclude, we have calculated the spin transfer torques acting on the magnetic textures from the spinpolarized superconducting current flowing through the ferromagnetic material. For this we take the advantage of the widely used adiabatic approximation, bringing it from the realm of single-electron dynamics into the field of superconductivity governed by the propagation of the spin-triplet Cooper pairs generated at the SC/FM interface. This approximation enables us to find the analytical expression for the spin torques in the most general case of the spin texture and develop the efficient formalism of the generalized quasiclassical theory for calculating the charge and spin supercurrents through the inhomogeneous magnetic systems. We show that the supercurrentdriven dynamics of DWs crucially depends on the type and magnitude of the spin-orbital coupling. The obtained results demonstrate that the DW motion by the supercurrent is a phenomenon realistic for the recently developed Josephson junctions through CrO2 nanowires.