Annals of Physics 351 (2014) 850–856

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Brownian motion of massive skyrmions inmagnetic thin films

Roberto E. Troncoso a,∗, Álvaro S. Núñez b

a Centro para el Desarrollo de la Nanociencia y la Nanotecnología, CEDENNA, Avda. Ecuador 3493,Santiago 9170124, Chileb Departamento de Física, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Casilla 487-3,Santiago, Chile

a r t i c l e i n f o

Article history:Received 27 August 2014Accepted 12 October 2014Available online 18 October 2014

Keywords:SkyrmionsBrownian motionSpin-transportMagnetization dynamics

a b s t r a c t

We report on the thermal effects on the motion of current-drivenmassive magnetic skyrmions. The reduced equation for themotionof skyrmion has the form of a stochastic generalized Thiele’sequation. We propose an ansatz for the magnetization texture of anon-rigid single skyrmion that depends linearly with the velocity.By using this ansatz it is found that the skyrmion mass tensor isclosely related to intrinsic skyrmion parameters, such as Gilbertdamping, skyrmion-charge and dissipative force.Wehave found anexact expression for the average drift velocity as well as the mean-square velocity of the skyrmion. The longitudinal and transversemobility of skyrmions for small spin-velocity of electrons is alsodetermined and found to be independent of the skyrmion mass.

© 2014 Elsevier Inc. All rights reserved.

1. Introduction

Skyrmions are vortex-like spin structures that are topologically protected and have recently beenthe focus of intense research in condensed matter [1,2]. Skyrmions were originally conceived innuclear physics to describe interacting pions [1], however have found versatile application in avariety of different areas in physics [3–5]. A series of works reports their recent observation inchiral magnets [6–11]. There is a great interest in their dynamics due to the potential applications

∗ Corresponding author.E-mail addresses: r.troncoso.c@gmail.com (R.E. Troncoso), alnunez@dfi.uchile.cl (Á.S. Núñez).

http://dx.doi.org/10.1016/j.aop.2014.10.0070003-4916/© 2014 Elsevier Inc. All rights reserved.

R.E. Troncoso, Á.S. Núñez / Annals of Physics 351 (2014) 850–856 851

in spintronics that arise from the rather low current densities that are necessary to manipulate theirlocation [12]. Among other systems that have been reported hosting skyrmion textures they wereobserved in bulk magnets MnSi [6,7], Fe1−xCoxSi [8,9,13], Mn1−xFexGe [14] and FeGe [15] by meansof neutron scattering and Lorentz transmission electron microscopy. Regarding their dimensions, bythe proper tuning of external magnetic fields, sizes of the order of a few tens of nanometers havebeen reported. Spin transfer torques can be used to manipulate and even create isolated skyrmionsin thin films as shown by numerical simulations [16–18]. In thin films skyrmions have been observedat low temperatures, however energy estimates predict the stability of isolated skyrmions even atroom temperatures [19]. Under that regime themotion of skyrmions is affected by fluctuating thermaltorques that will render their trajectories into stochastic paths verymuch like the Brownian dynamicsof a particle. Proper understanding of such Brownian motion is a very important aspect of skyrmiondynamics. Numerical simulations [20,21] and experimental results [22], suggest that the skyrmionposition can be manipulated by exposure to a thermal gradient and that the skyrmions also display athermal creepmotion in a pinning potential [23]. The thermally activatedmotion of pinned skyrmionshas been studied in Ref. [24]where it has been reported that thermal torques can increase themobilityof skyrmions by several orders of magnitude. In this work we present a study of the random motionof magnetic skyrmions arising from thermal fluctuations. In our analysis we include an assessment ofthe deformation of the skyrmion that arises from its motion. This deformation induces an inertia-liketerm into the effective stochastic dynamics of the skyrmion. We present a theory that allows us toestablish a relation between the fluctuating trajectory of the skyrmion and its effective mass.

2. Brownian skyrmion dynamics

We begin our analysis from the stochastic Landau–Lifschitz–Gilbert (LLG) equation [25,26] thatrules the dynamics of the magnetization direction . Into this equation we need to include theadiabatic, given by −vs · ∇, and non-adiabatic, given by βvs · ∇, spin-transfer torques [27,28]where the strength of the non-adiabatic spin-transfer torque is characterized by the parameter β . Inthose expressions vs = −

pa3/2eM

j stands for the spin-velocity of the conduction electrons, p is the

spin polarization of the electric current density j, e(> 0) the elementary charge, a the lattice constant,andM the magnetization saturation. With those contributions the stochastic Landau–Lifshitz–Gilbertequation becomes:

∂

∂t+ vs · ∇

= × (Heff + h) + α ×

∂

∂t+

β

αvs · ∇

, (1)

where Heff =1h

δEδ

is the effective field, with E representing the energy of the system, and α theGilbert damping constant. An important aspect of this equation is the inclusion of the white Gaussianfluctuating magnetic field h, describing the thermal agitation of the magnetization and obeying thefluctuation–dissipation theorem [25,29]. The strength of the noise, σ = 2αkBTa2/h, is proportional tothe thermal energy kBT , theGilbert damping parameterα, and the volume of the finite element grid a2.

Solutions of the Landau–Lifshitz–Gilbert equations exhibit analogies with a very large vari-ety of physical systems such as particle-like or spin waves. For example, it has been shown thatspin-waves display the Bose–Einstein condensation transition phenomena in magnetic thin films[30–32]. Additionally, particle-like solutions, representing compact magnetic textures moving as co-herent entities with a well defined velocity, have known since long ago. Among other examples wecan found the dynamics of domain walls [33,34] and of Bloch points [35,36]. The account of the dy-namics of skyrmion textures is best handled by the use of the collective coordinates approach. Underthis framework the complex dynamics of the magnetization texture, (r, t) is reduced to the evolu-tion of a small number of degrees of freedom given by the skyrmion position and its velocity. In thisway the magnetization field associated to a single-skyrmion moving along the trajectory x(t) is rep-resented by a magnetization profile (r, t) = (r − x(t), v(t)). The explicit time-dependence of themagnetization, coming from the dependence on velocity v(t), includes the effects of deformations ofthe skyrmion [37–39]. The calculation for the static skyrmion profile, 0(r), has been addressed else-where [40], by means of a minimization of the magnetic energy. In this energy the contributions from

852 R.E. Troncoso, Á.S. Núñez / Annals of Physics 351 (2014) 850–856

the exchange, perpendicular anisotropy, and Dzyaloshinskii–Moriya energies must be included. Re-placement of the collective coordinates ansatz and integration over space reduce the LLG equation toan equation of motion for the collective variables. This equation has the form of a stochastic massiveThiele’s equation

Mv(t) − gz × v(t) + αDv(t) = F + η(t). (2)

Neglecting the contribution of the noise term (η(t)) Eq. (2) turns into the generalized Thiele’sequation [41].Wehighlight the termproportional to the acceleration, quantified by the effectivemass,that corresponds to a matrix Mij = Mij + Mij, comprised by the elementsMij =

dr ·

∂

∂xi×

∂

∂vj

,

arising from the conservative dynamics, and Mij = αdr

∂

∂xi·

∂

∂vj

arising from the dissipative

contribution to the Landau–Lifshitz–Gilbert equation. The second term in Eq. (2) describes theMagnusforce [7] exerted by themagnetic texture on themoving skyrmion. In the case of an isolated skyrmiong = 4πW whereW = −1 stands for thewinding number, or skyrmion charge. The third contributionrepresents the dissipative forcewhich is defined through the relationDij =

dr ∂

∂xi·

∂

∂xj, that becomes

Dij = Dδij in the case of the highly symmetrical case of an isolated skyrmion.The dynamics of the skyrmion in Eq. (2) is forced by a deterministic term F = −gz × vs + βDvs −

∇V , that contains a contribution from the flowing electrons and a force arising from a potential V [x]that reflects the inhomogeneities in the skyrmion’s path, e.g., magnetic impurities, local anisotropiesor geometric defects. We conclude with the last term of right-hand side of Eq. (2), that describes thefluctuating force on the skyrmion. The strength of the Gaussianwhite noise turns out to be σD , there-fore the effective diffusion constant of the skyrmion depends not only on the Gilbert damping andthe temperature but also on the dissipative parameter D . By solving the stochastic Thiele equation(2), for the homogeneous case (V [x] = 0), we determine the time evolution of both longitudinaland transverse components of the velocity of the fluctuating skyrmion (as shown in Fig. 1(a) and (b)respectively) at temperature T = 100 K. Typical skyrmion speeds of ∼0–1 m/s are reached for spin-velocities on the order of 1m/s. Themassive skyrmion dynamicswas calculated for a Gilbert-dampingparameter α = 0.1, the β parameter β = 0.5α, the dissipative force D = 5.577π (from Ref. [16]),and where the values used for the mass are taken from Ref. [42]. Moreover, it is numerically solvedthe mean drift velocity, i.e., the steady-state current-induced skyrmion motion, which is displayedin Fig. 1(c) as a function of the spin-velocity vs = vsx of electrons. In addition, we are interested inthe probability distribution P[x, v; t] associated to the skyrmion dynamics, which is defined as theprobability density that a skyrmion at time t , is in the position x with a velocity v. The equation ofmotion satisfied by such distribution is known as Fokker–Planck equation and its derivation, as wellas its solution in simple cases, constitutes a standard issue in the analysis of stochastic processes [43].

3. Origin of the skyrmion mass

In comparison with the Langevin-like dynamics predicted in Ref. [24], Eq. (2) contains as a mainingredient a term proportional to the acceleration which is quantified by the effective mass matrixM. Generally speaking, it is linked to the explicit time-dependence of the magnetization direction.The mass of skyrmions can be determined perturbatively in linear response theory as follows. In theskyrmion center of mass reference frame we see that the magnetization in Eq. (1) is affected by anadditional magnetic field δH(r) = (r) × (v · ∇)(r). We note that the strength of the effective fieldis confined within the perimeter of the skyrmion. However, the effective torque takes a maximumvalue in the center of skyrmion and thus, it suffers a distortion of its shape due to the current-inducedmotion. This motivates us to propose an ansatz for the magnetization texture of a non-rigid singleskyrmion and its dependence on the velocity as

(r, v) = 0(r) + λ ξ 0(r) × (v · ∇)0(r), (3)

with 0 corresponds to the static and rigid skyrmion texture. The deformation of skyrmion size isparameterized by the second term, where λ is the dimensionless parameter that determines thestrength of the velocity induced deformations. In this expression ξ = hl2sk/Ja

2 with J the exchange

R.E. Troncoso, Á.S. Núñez / Annals of Physics 351 (2014) 850–856 853

a

b

c

Fig. 1. Fluctuating skyrmion velocities in the longitudinal and transverse directions, (a) and (b) respectively. The Browniandynamics Eq. (2) is solved for a spin-velocity of electrons vs = 1 m/s along x direction and for a characteristic time scalet0 = Mxx ≈ 6 ns (from Ref. [42]). In (c) the average velocities are presented as function of spin-velocity vs = vs x, boththe longitudinal (black line) and transverse (dashed line) components. The results are shown for a Gilbert damping α = 0.1,β = 0.5α, and at temperature 100 K.

constant, a the lattice constant and lsk the characteristic skyrmion size. It is worth noting that thiscontribution is linear in velocity and conserves the norm of themagnetization to leading order in λ. InFig. 2we present schematically the distortion of skyrmions exerted by the effective field δH . However,without loss of generality, we assume a motion of the skyrmion along x direction. The deformation inthe skyrmion texture consists of an in plane distortion, that resembles a dipolar field, and an out ofplane contribution that is antisymmetric. The nature of the mass of skyrmions can be traced back byusing the ansatz given by Eq. (3). By replacing it on the expressions for M up to linear order in λ, wefind that the mass tensor has a symmetric and an antisymmetric contribution, that are respectivelyrelated to the dissipative force D and gyrotropic constant g . The symmetric part represents the usualscalar mass, given by Mxx = Myy = λξD . In addition, the antisymmetric part Mxy = −Myx = λξαgrepresents a dissipation-like contribution proportional to the acceleration. This contribution can berelated with the notion of gyro-damping [44] that increases dissipation for clockwise motion andreduces it for counterclockwise motion. By comparing our results for Mxx with earlier theoretical[37,38] and experimental [42] estimates of the skyrmion mass we obtain λ ∼ 0.01. In this case weobtain, using a typical skyrmion velocity of 1m/s in Eq. (3), a characteristic strength of the deformationof the skyrmion in the range of |δ| ∼ 10−3.

4. Skyrmion mobility

The skyrmion dynamics is induced by an electric current density via spin-transfer torquemechanism. As mentioned previously, the flow of electrons exerts a force F that drives the skyrmionand its response is described by the stochastic generalized Thiele’s equation [24]. Next, we discuss therole of mass in the dynamics of skyrmions induced by currents at finite temperature. The probabilitydistribution for the velocity of the skyrmion can be readily found for V (x) = 0 by solving theassociated Fokker–Planck equation, P[v] = N exp

−

12γ (v − v)2

, where γ = λξα(g2

+ D2)/σD

and v = µ∥vs + µ⊥z × vs. In this equation µ∥ = (g2+ αβD2)/(g2

+ α2D2) and µ⊥ =

−(α − β)gD/(g2+ α2D2) are the longitudinal and transverse skyrmion mobilities respectively.

We are interested in the average drift velocity, which is determined directly from the probabilitydistribution as ⟨vi⟩ = vi. Another relevant quantity as themean-squared velocity is evaluated directly,and is found to obey ⟨vivj⟩−⟨vi⟩⟨vj⟩ = σD

λξα

g2

+ D2−1

δij that relates the fluctuations on thevelocity in terms of temperature and the skyrmion mass. The mean drift velocity scales linearly withthe spin-velocity of electrons and, unlike the current-induced domain wall dynamics [34], skyrmionsdo not exhibit an intrinsic pinning [16,17]. Note that these values for the spin-velocities correspond to

854 R.E. Troncoso, Á.S. Núñez / Annals of Physics 351 (2014) 850–856

Fig. 2. Top: Pictorial representation of the skyrmion magnetic texture. The color encodes the out of plane component ofthe magnetization. It changes from being fully align with the +z direction in the center to a complete alignment with theopposite direction in the outer rim. The arrows represent the behavior of the in plane component of the magnetization. Forthe case displayed those components swirl like a vortex. Bottom: Schematic plot for the skyrmion distortion generated by themotion of the skyrmion. In color we have represented the out of plane component of the deformation δΩz . This contribution isconcentrated in the direction transverse to the motion nearby the perimeter of the skyrmion (indicated by a dashed line). Thearrows correspond to the in plane (δΩx, δΩy). (For interpretation of the references to color in this figure legend, the reader isreferred to the web version of this article.)

electric current densities on the order of 1010 A/m2. As we analytically show, the average velocity ofthe free skyrmion is independent of the mass. However, it is related only on intrinsic parameters,such as Gilbert damping, skyrmion-charge, and dissipative force. Following our results, a detailedcharacterization of the strength of velocity fluctuations can be used to determine the values of theskyrmion mass.

5. Conclusions

We have investigated themechanisms bywhich thermal fluctuations influence the current-drivenmassive skyrmion dynamics. Based on the stochastic Landau–Lifschitz–Gilbert equation we derivedthe Langevin equation for the skyrmion. The equation has the form of a stochastic generalized

R.E. Troncoso, Á.S. Núñez / Annals of Physics 351 (2014) 850–856 855

Thiele’s equation that describes the massive dynamics of a single-skyrmion at finite temperature.We proposed an ansatz for the magnetization texture of a non-rigid single skyrmion that dependslinearly with the velocity. This ansatz has been derived based upon an effective field that distortsthe skyrmion texture. In particular, it implies that the deformation of the skyrmion shape consistsof an in plane distortion and an out of plane contribution that is antisymmetric. Furthermore, byutilizing this ansatz it is found that the skyrmion mass tensor is related with intrinsic parameters,such as Gilbert damping, skyrmion-charge, and dissipative force. This simple result provides a pathfor a theoretical calculation of the skyrmionmass. We have found an exact expression for the averagedrift velocity as well as the mean-square velocity of the skyrmion. The longitudinal and transversemobilities of skyrmions for small spin-velocity of electrons were also determined. We showed thatthe average velocity of skyrmions, unlike the mean-square velocity, is independent of the mass and itvaries linearly with the spin-velocity. In future work, we plan to use the formalism developed in thiswork to the study of the transport of massive skyrmions in disordered media.

Acknowledgments

The authors acknowledge funding from Proyecto Basal FB0807-CEDENNA, Anillo de Ciencia yTecnonología ACT 1117, and by Núcleo Científico Milenio P06022-F.

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