Driving magnetic skyrmions with microwave fields
Weiwei Wang, Marijan Beg, Bin Zhang, Wolfgang Kuch, Hans Fangohr
DDriving magnetic skyrmions with microwave fields
Weiwei Wang, Marijan Beg, Bin Zhang, Wolfgang Kuch, and Hans Fangohr ∗ Engineering and the Environment, University of Southampton, Southampton, SO17 1BJ, United Kingdom Institut f¨ur Experimentalphysik, Freie Universit¨at Berlin, Arnimallee 14, 14195 Berlin, Germany
We show theoretically by numerically solving the Landau-Lifshitz-Gilbert equation with a classicalspin model on a two-dimensional system that both magnetic skyrmions and skyrmion lattices can bemoved with microwave magnetic fields. The mechanism is enabled by breaking the axial symmetryof the skyrmion, for example through application of a static in-plane external field. The net velocityof the skyrmion depends on the frequency and amplitude of the microwave fields as well as thestrength of the in-plane field. The maximum velocity is found where the frequency of the microwavecoincides with the resonance frequency of the breathing mode of the skyrmions.
PACS numbers: 75.78.-n, 76.50.+g, 75.70.Ak, 75.10.Hk
Skyrmions, topologically stable magnetization textureswith particle-like properties, have recently attractedgreat attention [1–4] due to their potential use in futurespintronic devices [5]. The manipulation of skyrmionsis of great importance and interest: skyrmions canbe driven using spin-polarized current [6–9], magneticor electric field gradients [10, 11], temperature gradi-ents [12–14] and magnons [15–17]. Microwaves, on theother hand, have been broadly used in studying vari-ous magnetic phenomena, such as the ferromagnetic reso-nance (FMR) and spin wave excitations in skyrmion crys-tals [18–21]. However, the possibility of creating transla-tional motion of skyrmions has not been explored in theseexperiments [19–21]. In this Rapid Communication, weshow that both a single skyrmion and a skyrmion latticecan be moved by microwave fields if the axial symme-try of skyrmions is slightly broken by a static in-planeexternal field.We employ skyrmions stabilized by the Dzyaloshinskii-Moriya Interaction (DMI) [22, 23]. More precisely, thebulk DMI is considered so that a chiral skyrmion (vortex-like) rather than a hedgehog (radial) skyrmion configura-tion emerges [2, 24, 25]. We start with a classical Heisen-berg model on a two-dimensional regular square latticewith nearest-neighbor symmetric exchange interaction,the bulk-type DMI, and the Zeeman field [12, 18, 26]. Inaddition, a time-dependent magnetic field h ( t ) is appliedin the + z -direction. Accordingly, the system’s Hamilto-nian can be written as H = − J (cid:80) (cid:104) i,j (cid:105) m i · m j + (cid:80) (cid:104) i,j (cid:105) D ij · [ m i × m j ] − (cid:80) i | µ i | ( H + h ( t )) · m i , (1)where (cid:104) i, j (cid:105) represents a unique pair of lattice sites i and j , m i is the unit vector of the magnetic moment µ i = − (cid:126) γ S i with S i being the atomic spin and γ ( > J is the symmetric exchangeenergy constant. In the case of bulk DMI, the DMI vec-tor D ij can be written as D ij = D ˆ r ij , where D is the ∗ [email protected] DMI constant and ˆr ij is the unit vector between S i and S j . We use the DMI value with D/J = 0 .
18, which re-sults in the spiral period λ ∼ πJa/D ∼
25 nm for atypical lattice constant a = 0 . H : anin-plane component H y and a perpendicular component H z , i.e., H = (0 , H y , H z ). A nonzero H z is essential forstabilizing the skyrmion crystal [18]. xy z (a) (b) (c) (d) y x -0.0021-0.0015-0.0009-0.00030.0003 FIG. 1. (a) Skyrmion configuration in the presence of an in-plane field H y = 0 .
006 with D = 0 . J = 1 and H z = 0 . H y = 0. (c) The corresponding topologicalcharge density for the skyrmion shown in (a) when H y > N = 174 ×
150 sites, with an in-plane field H y = 0 . The spin dynamics at lattice site i is governed by theLandau-Lifshitz-Gilbert (LLG) equation, ∂ m i ∂t = − γ m i × H eff + α m i × ∂ m i ∂t (2)where α is the Gilbert damping and H eff is the effec-tive field that is computed as H eff = − (1 / | µ i | ) ∂ H /∂ m i .The Hamiltonian (1) associated with the LLG equa-tion (2) can be understood as a finite-difference-basedmicromagnetic model. Therefore, our simulation resultsare reproducible by setting the saturation magnetization M s = (cid:126) γS/a , exchange constant A = J/ a and DMIconstant for continuum form D a = − D/a (correspond-ing to the energy density ε dmi = D a m · ( ∇× m )) in micro-magnetic simulation packages such as OOMMF [27]. We a r X i v : . [ c ond - m a t . m t r l - s c i ] J u l TABLE I. Unit conversion table for J = 1 meV, S = 1 and a = 0 . x ˆ x = a = 0 . t ˆ t = (cid:126) S/J ≈ .
66 psVelocity v ˆ v = Ja/ ( hS ) ≈ . × m/sFrequency ω ˆ ω = J/ ( (cid:126) S ) ≈ . × GHzMagnetic field H ˆ H = J/ ( (cid:126) γS ) ≈ .
63 T have carried out simulations with and without dipolarinteractions, and the results are qualitatively the same.We report results without dipolar interactions for clarityof the model assumptions.A two-dimensional system of size N = 160 × N = 174 ×
150 sites for the skyrmion lattice, as shownin Fig. 1(d). We have chosen J = (cid:126) = γ = S = a = 1as simulation parameters [18, 28], therefore, the coeffi-cients to convert the external field H , time t , frequency ω and velocity v to SI units are ˆ H = J/ (cid:126) γS , ˆ t = (cid:126) S/J ,ˆ ω = J/ (cid:126) S and ˆ v = Ja/ (cid:126) S , respectively. Table I showsthe expressions, and particular values for the case of J = 1 meV, S = 1 and a = 0 . H z is fixed as H z = 0 .
02 which corresponds to 0 . S = 1 and J = 1 meV. We use Gilbert damp-ing α = 0 .
02 for all simulations except for the magneticspectra shown in Fig. 2 where α = 0 .
04 is chosen. Forthe single skyrmion dynamics, we apply the absorbingboundary conditions for damping [29] by setting α = 1 . H y = 0 .
006 is shown in Fig. 1(a). It is foundthat the radial symmetry is broken. Indeed, as shown inFig 1(c), the corresponding distribution of the topolog-ical charge density q ( x, y ) = (1 / π ) m · ( ∂ x m × ∂ y m ) isasymmetric. However, the total topological charge of asingle skyrmion remains constant Q = (cid:82) qdxdy = −
1. Asa comparison, Fig. 1(b) shows the topological charge den-sity q for a skyrmion with radial symmetry when H y = 0.Similar to the vortex [30], the distortion of the skyrmionis along the x -axis when an external field is applied inthe y -direction.The excitation of internal modes depends on the staticexternal field H z as well as the frequency and direc-tion of microwaves [18, 31]. The typical excited modesare the clockwise/counterclockwise rotation and breath-ing modes [18–20]. To study how the in-plane ap-plied field H y affects the excitation mode of a skyrmion,we calculate the magnetic absorption spectrum of theskyrmion. After applying a sinc-function field pulse h = h sinc( ω t ) = h sin( ω t ) / ( ω t ) to the stable skyrmionstate we record the spatially averaged magnetization evo-lution and from that we compute the dynamic suscepti-bility χ via a Fourier transformation [32, 33]. For in- w I m χ zz ( a r b . un i t ) H y =0 . H y =0 . H y =0 . H y =0 . (a) w H y =0 . H y =0 . H y =0 . H y =0 . (b) FIG. 2. Imaginary parts of the H y -dependent dynamical sus-ceptibility χ zz as a function of frequency for (a) a singleskyrmion, and (b) a skyrmion lattice. The spectra are ob-tained by applying a sinc-field pulse h = h sinc( ω t ) to thesystem with h = 1 × − and ω = 0 . π in the z -axis, themagnetization dynamics is recorded every dt = 5 for 8000steps. stance, the component χ zz is computed using m z whenthe pulse is parallel to the z -axis.Figure 2 shows the imaginary part of the dynami-cal susceptibility χ zz for a single skyrmion (a) and askyrmion lattice (b); each calculated for different in-planefields H y . We see in (b) for the skyrmion lattice and H y = 0 that the mode with frequency ω ≈ . ω ≈ . f = ω/ π ≈ .
95 GHz(using ˆ ω from Table I). The breathing mode frequencydecreases slightly as the in-plane field H y increases. Asecond peak emerges with increasing H y , and the fre-quency of the new mode is ω ≈ . H y = 0 . ω ≈ . .
02 emerges for the single skyrmion case as H y is increased. This second mode is the uniform mode withfrequency ω = γH where H (cid:39) ( H y + H z ) / is the am-plitude of the external field.In the presence of an in-plane applied field H y , theskyrmion is deformed, as shown in Figs. 1(a) and 1(c).Therefore, instead of the geometric center we measure theso-called guiding center [34] R = ( X, Y ) of a skyrmion: X = (cid:82) xqdxdy/ (cid:82) qdxdy and Y = (cid:82) yqdxdy/ (cid:82) qdxdy ,where q is the topological charge density. For a symmet-ric skyrmion, the guiding center is the same as its geo-metric center. In the rest of this work, we consider thescenario that a linearly polarized microwave is applied inthe z -direction, i.e., h ( t ) = h sin( ω t ) e z , where h and ω are the amplitude and frequency of the microwave,respectively.Figure 3(a) shows the displacement of the guiding t × − D i s p l a c e m e n t XY w V e l o c i t y v × v x v y h × V e l o c i t y v × v x v y H y × V e l o c i t y v × (b)(d)(a)(c) v x v y FIG. 3. (a) The displacements of the guiding center (
X, Y )for a single skyrmion. The simulation parameters are ω =0 . h = 2 × − and the in-plane field H y = 0 . v x and v y as functions of (b) the microwavefrequency ω , (c) the microwave amplitude h , and (d) thein-plane field H y . The fixed simulation parameters are thesame as in (a). center for a single skyrmion with ω = 0 .
017 and in-plane field H y = 0 . h = 2 × − , which corresponds to 1.73 mT for J = 1meV and S = 1. It can be seen that the x -component ofthe guiding center, X , changes significantly as a functionof time, while the displacement of Y is relatively small.Figure 3(b) plots the frequency-dependent skyrmion ve-locity and shows that a single skyrmion has maximumvelocity when ω = 0 . H y . Using the conversions presentedin Table I, the maximum velocity is v x ≈ . v x is positive for frequency ω = 0 . ω = 0 . v x on h is nonlinear: v x ∝ h which is proportional tothe power of the microwaves.Figure 3(d) describes the relation between theskyrmion velocity and the in-plane field H y . The velocityis zero if H y = 0, which is expected due to the symme-try of the skyrmion. The velocity of the skyrmion alsodepends on the direction of the in-plane field H y : the ve-locity is reversed when the direction of the in-plane fieldis reversed. Similarly, a change in the sign of the DMIconstant will also reverse the sign of the velocity, whichis different from the case of driving skyrmions with spin-polarized currents, where the sign of perpendicular veloc-ity (with respect to the current direction) of the skyrmionmotion is related to the sign of topological charge ratherthan the DMI constant sign.To understand why the skyrmion moves in the pres-ence of an in-plane field, we split the magnetization (a) (b) FIG. 4. The total spatial force density f i = ˜ m s · [ ∂ i ˜ m s ×(cid:104) m × H eff (cid:105) ] for (a) H y = 0, and (b) H y = 4 × − , where we haveused ˜ m s = (cid:104) m (cid:105) . The microwave frequency is ω = 0 . unit vector m into a slow part m s and a fast part n ,i.e., m = m s + n , where the slow part represents theequilibrium profile of the skyrmion while the fast partis responsible for the excited spin wave mode [12, 35].In the continuum approximation, the effective field is H eff = ˜ A ∇ m − ˜ D ∇ × m + H + h ( t ), where ˜ A = 2 A/M s and ˜ D = 2 D a /M s . In the presence of the microwaveswith specific frequency, bound spin-wave modes are ex-cited, and thus we expect (cid:104) ˙ n (cid:105) = 0 and (cid:104) n (cid:105) = 0 due tothe microwave synchronization, where T = 2 π/ω is theperiod of microwaves and the notation (cid:104) f (cid:105) = (cid:104) f (cid:105) ( t ) ≡ T − · (cid:82) t + Tt f ( t (cid:48) ) dt (cid:48) represents the time average of func-tion f ( t ) over a single period T . Furthermore, one ob-tains (cid:104) m s × ˙ n (cid:105) ≈ (cid:104) n × ˙ m s (cid:105) ≈ m s is theslow part. Therefore, by averaging the LLG equation (2)over a period T we arrive at (cid:104) ˙ m s (cid:105) = − γ (cid:104) m × H eff (cid:105) + α (cid:104) m s × ˙ m s (cid:105) , (3)where (cid:104) n × ˙ n (cid:105) = 0 is used since basically n is a sineor cosine function in time. We then consider the possibletranslational motion of the skyrmion such that m s ( r , t ) = m s ( r − v s t ), i.e., ˙ m s = − ( v s · ∇ ) m s , where the skyrmionvelocity v s = d R /dt is assumed to be a constant. If theskyrmion moves slowly, i.e., v s T (cid:28) L ( L is the typicalskyrmion size), we have (cid:104) ˙ m s (cid:105) ≈ − ( v s · ∇ ) ˜ m s (see Ap-pendix A) where ˜ m s = (cid:104) m s (cid:105) . Similarly, (cid:104) m s × ˙ m s (cid:105) ≈ ˜ m s × (cid:104) ˙ m s (cid:105) , and thus Eq. (3) can be rewritten as( v s · ∇ ) ˜ m s = γ (cid:104) m × H eff (cid:105) + α ˜ m s × ( v s · ∇ ) ˜ m s . (4)Following Thiele’s approach in describing the motion ofmagnetic textures [36], we replace the dots in (cid:82) ˜ m s · ( ∂ i ˜ m s × · · · ) dxdy by Eq. (4) to obtain [10, 12, 35] G × v s + (cid:98) D v s = F , (5)where i = x, y and G = 4 πQ e z . The tensor (cid:98) D ij = αη ij isthe damping tensor in which η ij = (cid:82) ( ∂ i ˜ m s · ∂ j ˜ m s ) dxdy = δ ij η is the shape factor of the skyrmion and η is close to4 π [35]. The force F is given by F i = − γ (cid:90) ˜ m s · (cid:2) ∂ i ˜ m s × (cid:104) m × H eff (cid:105) (cid:3) dxdy. (6)Figure 4(a) and (b) depict the total spatial force den-sity for H y = 0 and H y = 4 × − , respectively, wherewe have used ˜ m s = (cid:104) m (cid:105) . The force density is sym-metric if H y = 0 and thus the total force F is zero.However, when H y is nonzero the force distribution isasymmetric which results in the skyrmion motion due tothe nonzero net force. For small damping α (cid:28)
1, wehave v x ≈ F y / (4 πQ ). The total force calculated withparameters ω = 0 . H y = 0 .
004 and h = 2 × − is F y = − . × − , therefore, the established veloc-ity is v x = 3 . × − , which fits the simulation result( ∼ . × − ) well. Similarly, for ω = 0 .
023 using Eq. (6)we obtain F y = 5 . × − and find v x ≈ − . × − from Eq. (5); in agreement with the simulation results(the minimum of v x is − . × − ).It is of interest to circumstantiate the contributions ofthe total force F . In Appendix B we show that there arethree nontrivial terms (cid:104) m × H eff (cid:105) ≈ (cid:104) n × [ ˜ A ∇ n − ˜ D ∇ × n + h ( t )] (cid:105) . (7)The exchange term n × ˜ A ∇ n corresponds to magnon cur-rents [12, 35]. Compared to the skyrmion motion inducedby the temperature gradient, where the magnon cur-rent is generated by the temperature gradient, here themagnon current originates from the external microwavefields. Another difference is that in our case the contribu-tions from DMI and microwave fields are also significant. H y × V e l o c i t y v × (a) v x v y w V e l o c i t y v × (b) v x v y FIG. 5. (a) The velocities v x and v y as a function of anin-plane field H y for the skyrmion lattice at frequency ω =0 . ω with H y = 4 × − . The microwave amplitude is h = 2 × − . We repeat the velocity study for the skyrmion lattice.Fig. 5(a) plots the velocities v x and v y of the skyrmionlattice as functions of the in-plane external field H y . Thedependencies are similar to the single skyrmion case. Thefrequency-dependent velocities v x and v y are shown inFig. 5(b). As for the single skyrmion case, the velocitypeak coincides with the dominant dynamical susceptibil-ity peak in Fig. 2(b).In closing, we briefly comment on the importance ofsymmetry breaking in driving the skyrmions. The driv-ing force originates from the microwave field, which isperiodic in time and averages to zero. The symmetry-breaking field converts the periodic microwave field intoa net force and thus moves skyrmions effectively. A re-lated field with periodic driving forces is that of ratchet- like transport phenomena [37–39], where the net motionis obtained by breaking the spatial symmetry [37] or tem-poral symmetry [39]. We also note that preliminary sim-ulation results suggest that for magnon-driven skyrmions[15, 16] the introduction of a symmetry-breaking in-planefield affects the skyrmion’s motion and changes the Hallangle significantly.In summary, we have studied the skyrmion dynamicsdriven by microwaves in the presence of an in-plane ex-ternal field. We found that both a single skyrmion anda skyrmion lattice can be moved by a linearly polarizedmicrowave field if the axial symmetry of skyrmions isslightly broken. These results suggest a novel methodfor skyrmion manipulation using microwaves fields.We acknowledge financial support from EPSRC’s DTCgrant EP/G03690X/1. W.W. thanks the China Scholar-ship Council for financial support. The authors acknowl-edge the use of the IRIDIS High Performance ComputingFacility, and associated support services at the Universityof Southampton, in the completion of this work. Appendix A
Assume that a well-behaved function f ( x, t ) = f ( x − vt ) describes the dynamics of a soliton where v is a con-stant. As we can see, f satisfies ˙ f = − vf (cid:48) . For giventime T , if vT (cid:28) L where L is the typical size of the soli-ton (for example, L could be the domain wall width fora magnetic domain wall), we can find that (cid:104) ˙ f (cid:105) (0) = 1 T (cid:90) T ˙ f dt = 1 T [ f ( x − vT ) − f ( x )] ≈ − vf (cid:48) ( x − vT / , (A1)where we have used the Taylor series for f ( x ) and f ( x − vT ): f ( x ) ≈ f ( x − vT /
2) + f (cid:48) ( x − vT / vT / f ( x − vT ) ≈ f ( x − vT / − f (cid:48) ( x − vT / vT / . (A3)Similarly, we can see that ˜ f ≡ (cid:104) f (cid:105) (0) ≈ f ( x − vT /
2) andthus we have (cid:104) ˙ f (cid:105) (0) ≈ − v ˜ f . This relation actually holdsfor arbitrary t (cid:104) ˙ f (cid:105) ≈ − v ˜ f . (A4) Appendix B
By using the effective field explicitly and noticing that m = m s + n , the term (cid:104) m × H eff (cid:105) can be splited intofour parts (cid:104) m × H eff (cid:105) = (cid:104) T + T + T + T (cid:105) , (B1)where T = n × [ ˜ A ∇ n − ˜ D ∇ × n + h ( t )] is shown inEq (7), T = m s × ( ˜ A ∇ m s − ˜ D ∇ × m s + H ), T = n × ( ˜ A ∇ m s − ˜ D ∇ × m s + H ) and T = m s × [ ˜ A ∇ n − ˜ D ∇× n + h ( t )]. We expect T = 0 since m s represents theequilibrium state of the skyrmion. For the slow skyrmionmotion, replacing m s by ˜ m s and noticing that (cid:104) n (cid:105) = 0, we obtain (cid:104) T (cid:105) ≈ (cid:104) n (cid:105) × ( ˜ A ∇ ˜ m s − ˜ D ∇ × ˜ m s ) = 0 and (cid:104) T (cid:105) ≈ ˜ m s × (cid:104) ˜ A ∇ n − ˜ D ∇ × n + h ( t ) (cid:105) = 0. In this slowmotion approxmation, the fast part n can be computedas n ≈ m − (cid:104) m (cid:105) . [1] U. K. R¨ossler, A. N. Bogdanov, and C. Pfleiderer, Nature , 797 (2006).[2] S. M¨uhlbauer, B. Binz, F. Jonietz, C. Pfleiderer,A. Rosch, A. Neubauer, R. Georgii, and P. B¨oni, Sci-ence , 915 (2009).[3] N. Nagaosa and Y. Tokura, Nat. Nanotechnol. , 899(2013).[4] A. Fert, V. Cros, and J. Sampaio, Nat. Nanotechnol. ,152 (2013).[5] N. Romming, C. Hanneken, M. Menzel, J. E. Bickel,B. Wolter, K. von Bergmann, A. Kubetzka, andR. Wiesendanger, Science , 636 (2013).[6] J. Zang, M. Mostovoy, J. H. Han, and N. Nagaosa, Phys.Rev. Lett. , 136804 (2011).[7] S.-Z. Lin, C. Reichhardt, C. D. Batista, and A. Saxena,Phys. Rev. Lett. , 207202 (2013).[8] J. Iwasaki, M. Mochizuki, and N. Nagaosa, Nat. Com-mun. , 1463 (2013).[9] J. Iwasaki, M. Mochizuki, and N. Nagaosa, Nat. Nan-otechnol. , 742 (2013).[10] K. Everschor, M. Garst, B. Binz, F. Jonietz,S. M¨uhlbauer, C. Pfleiderer, and A. Rosch, Phys. Rev.B , 054432 (2012).[11] Y.-H. Liu, Y.-Q. Li, and J. H. Han, Phys. Rev. B ,100402 (2013).[12] L. Kong and J. Zang, Phys. Rev. Lett. , 067203(2013).[13] S.-Z. Lin, C. D. Batista, C. Reichhardt, and A. Saxena,Phys. Rev. Lett. , 187203 (2014).[14] M. Mochizuki, X. Z. Yu, S. Seki, N. Kanazawa,W. Koshibae, J. Zang, M. Mostovoy, Y. Tokura, andN. Nagaosa, Nat. Mater. , 241 (2014).[15] J. Iwasaki, A. J. Beekman, and N. Nagaosa, Phys. Rev.B , 064412 (2014).[16] C. Sch¨utte and M. Garst, Phys. Rev. B , 094423(2014).[17] Y.-T. Oh, H. Lee, J.-H. Park, and J. H. Han, Phys. Rev.B , 104435 (2015).[18] M. Mochizuki, Phys. Rev. Lett. , 017601 (2012).[19] Y. Onose, Y. Okamura, S. Seki, S. Ishiwata, andY. Tokura, Phys. Rev. Lett. , 037603 (2012). [20] Y. Okamura, F. Kagawa, M. Mochizuki, M. Kubota,S. Seki, S. Ishiwata, M. Kawasaki, Y. Onose, andY. Tokura, Nat. Commun. , 2391 (2013).[21] T. Schwarze, J. Waizner, M. Garst, A. Bauer,I. Stasinopoulos, H. Berger, C. Pfleiderer, andD. Grundler, Nat. Mater. , 478 (2015).[22] I. Dzyaloshinskii, J. Phys. Chem. Solids , 241 (1958).[23] T. Moriya, Phys. Rev. , 91 (1960).[24] A. Neubauer, C. Pfleiderer, B. Binz, A. Rosch, R. Ritz,P. G. Niklowitz, and P. B¨oni, Phys. Rev. Lett. ,186602 (2009).[25] Y. Zhou and M. Ezawa, Nat. Commun. , 4652 (2014).[26] S. D. Yi, S. Onoda, N. Nagaosa, and J. H. Han, Phys.Rev. B , 054416 (2009).[27] M. Donahue and D. Porter, “OOMMF User’s Guide, Ver-sion 1.0,” (1999), http://math.nist.gov/oommf/.[28] B. Zhang, W. Wang, M. Beg, H. Fangohr, and W. Kuch,Appl. Phys. Lett. , 102401 (2015).[29] W. Wang, M. Albert, M. Beg, M.-A. Bisotti,D. Chernyshenko, D. Cort´es-Ortu˜no, I. Hawke, andH. Fangohr, Phys. Rev. Lett. , 087203 (2015).[30] R. P. Cowburn, D. K. Koltsov, A. O. Adeyeye, M. E.Welland, and D. M. Tricker, Phys. Rev. Lett. , 1042(1999).[31] S.-Z. Lin, C. D. Batista, and A. Saxena, Phys. Rev. B , 024415 (2014).[32] R. Liu, J. Wang, Q. Liu, H. Wang, and C. Jiang, J.Appl. Phys. , 013910 (2008).[33] J.-V. Kim, F. Garcia-Sanchez, J. Sampaio, C. Moreau-Luchaire, V. Cros, and A. Fert, Phys. Rev. B , 064410(2014).[34] N. Papanicolaou and T. Tomaras, Nucl. Phys. B ,425 (1991).[35] A. A. Kovalev, Phys. Rev. B , 241101 (2014).[36] A. A. Thiele, Phys. Rev. Lett. , 230 (1973).[37] P. Reimann, Phys. Rep. , 57 (2002).[38] P. H¨anggi and F. Marchesoni, Rev. Mod. Phys. , 387(2009).[39] N. R. Quintero, J. A. Cuesta, and R. Alvarez-Nodarse,Phys. Rev. E81