Thermally Driven Ratchet Motion of Skyrmion Microcrystal and Topological Magnon Hall Effect
M. Mochizuki, X. Z. Yu, S. Seki, N. Kanazawa, W. Koshibae, J. Zang, M. Mostovoy, Y. Tokura, N. Nagaosa
aa r X i v : . [ c ond - m a t . s t r- e l ] A p r Thermally Driven Ratchet Motion of Skyrmion Microcrystal andTopological Magnon Hall Effect
M. Mochizuki , , ∗ X. Z. Yu , S. Seki , , , N. Kanazawa , W.Koshibae , J. Zang , M. Mostovoy , Y. Tokura , , , and N. Nagaosa , , Department of Physics and Mathematics,Aoyama Gakuin University, Sagamihara, Kanagawa 229-8558, Japan PRESTO, Japan Science and Technology Agency,Kawaguchi, Saitama 332-0012, Japan RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan Department of Applied Physics, Quantum-Phase Electronics Center,The University of Tokyo, Bunkyo-ku Tokyo 113-8656, Japan Department of Applied Physics, The University of Tokyo,Bunkyo-ku, Tokyo 113-8656, Japan Department of Physics and Astronomy,Johns Hopkins University, Baltimore, Maryland 21218, USA and Zernike Institute for Advanced Materials,University of Groningen, Nijenborgh 4,9747 AG, Groningen, The Netherlands ∗ Electronic address: [email protected] pontaneously emergent chirality is an issue of fundamental importance acrossthe natural sciences [1]. It has been argued that a unidirectional (chiral) rota-tion of a mechanical ratchet is forbidden in thermal equilibrium, but becomespossible in systems out of equilibrium [2]. Here we report our finding that atopologically nontrivial spin texture known as a skyrmion - a particle-like ob-ject in which spins point in all directions to wrap a sphere [3] - constitutessuch a ratchet. By means of Lorentz transmission electron microscopy (LorentzTEM) we show that micron-sized crystals of skyrmions in thin films of Cu OSeO and MnSi display a unidirectional rotation motion. Our numerical simulationsbased on a stochastic Landau-Lifshitz-Gilbert (LLG) equation suggest that thisrotation is driven solely by thermal fluctuations in the presence of a tempera-ture gradient, whereas in thermal equilibrium it is forbidden by the Bohr-vanLeeuwen theorem [4, 5]. We show that the rotational flow of magnons drivenby the effective magnetic field of skyrmions gives rise to the skyrmion rotation,therefore suggesting that magnons can be used to control the motion of thesespin textures. The formation of triangular arrays of skyrmions in chiral-lattice magnets under an appliedmagnetic field was theoretically predicted [6–8] and experimentally observed in MnSi [9, 10],Fe − x Co x Si [11–13], FeGe [14], and Cu OSeO [15–17]. The magnetization in such crystalsis antiparallel to the magnetic field B at the center of each skyrmion and is parallel to B at its periphery (see Figs.1a-c). Recently, real-space images of nanosized skyrmions havebeen successfully obtained by the Lorentz TEM. While scanning temperatures and magneticfields in the experiments, we have encountered a peculiar dynamical phenomenon, that is,in a wide temperature interval excluding only the lowest temperatures, micro-scale regionsof the skyrmion crystal (SkX) show, in addition to Brownian motion, a clearly discernibleunidirectional rotation.In Supplementary Movies 1 and 2, we show examples of this phenomenon observed in thin-plate ( ∼ OSeO , respectively. These compounds arechiral-lattice magnets with a common space group P2
3. In zero field they undergo a phasetransition from paramagnetic to helimagnetic phase at 29.5 K and 60 K, respectively. Thehelix period λ m is, respectively, 18 nm and 50 nm. Thin-plate ( <
100 nm-thick) specimensof these compounds host stable SkX phases with the skyrmion lattice spacing ∼ √ λ m . The2eld strength for the Movies 1 and 2 is 175 mT and 65 mT, respectively, and B is appliedin the negative z -direction perpendicular to the plate plane: B k − z . The observed (static)Lorentz-TEM image of SkX in MnSi specimen is displayed in Figs. 1c and d. In Figs. 1e-l, weexemplify time evolution (snapshots) of Fourier components of the magnetic configurationswhere a hexagon composed of six Bragg peaks rotates clockwise, indicating the clockwiserotation of the SkX domains.The rotation rate depends on the irradiation density of the electron beam of the micro-scope (Supplementary Information). The skyrmion rotation is observed only above a criticalirradiation density and the rotation rate grows as the density increases. This indicates thatthis rotation is a non-equilibirum phenomenon induced by the electron beam. The similaritybetween the chiral rotations of SkX micro-domains and their hexagon-shaped Fourier com-ponents observed in MnSi (Supplementary Movie 1) and Cu OSeO (Supplementary Movie2) is remarkable in view of different origins of magnetism in these two compounds (MnSi is ametal, while Cu OSeO is an insulator) and the difference in skyrmion parameters, e.g. thetransition temperature and the SkX lattice spacing. Hence, such a ratchet motion shouldbe viewed as a generic feature of skyrmion systems. Then the question arises why and howit occurs.A circular magnetic field induce by the electron beam of the Lorentz TEM is estimatedto be five orders of magnitude smaller than the geomagnetic field, and thus cannot causethe rotation. Recently it was demonstrated that a spin-polarized electric current parallelto the sample plane can drive similar rotations of SkX domains in the presence of thermalgradient [20, 21]. In our case, however, a possible effect of electric currents can be excludedsince the electron beam of Lorentz TEM is three orders of magnitude smaller than thethreshold current of 10 -10 A/m for the current-driven skyrmion motions [20–23] and, inaddition, its direction is perpendicular rather than parallel to the film, which further reducesspin-torque effects. Furthermore, the energy of the electrons is very high and the interactionwith the spins in the specimen is very weak except through the magnetic field induced bythe magnetization. This conclusion is corroborated by the fact that the skyrmion rotationis also observed in the insulating Cu OSeO . Thus, the unidirectional rotation is apparentlyinduced by thermal effects.In order to clarify the nature of this phenomenon, we performed numerical simula-tions. The SkX phase in chiral-lattice magnets is described by a classical Heisenberg3odel on the two-dimensional square lattice [18], which contains ferromagnetic-exchangeand Dzyaloshinskii-Moriya (DM) interactions as well as the Zeeman coupling to B =(0, 0, B z ) normal to the plane [19]. The Hamiltonian is given by, H = − J X m i · m j − D X i ( m i × m i +ˆ x · ˆ x + m i × m i +ˆ y · ˆ y ) − B · X i m i , (1)where the magnetization vector m i is defined as m i = − S i / ¯ h with the norm m = | m i | . Thespin turn angle θ in the helical phase is determined by the ratio D/J and for
D/J =0.27used here, θ =11 ◦ corresponding to the period of ∼
33 sites. With a typical lattice constantof 5 ˚A, this gives λ m ∼
17 nm, which is comparable to the helical period ∼
18 nm in MnSi(our conclusions, however, are not affected by the choice of the value of
D/J ). Our Monte-Carlo analysis shows that the SkX phase emerges in the range of 1 . × − < | B z | /J m < . × − between the helical and ferromagnetic phases.In this study, we treat SkX confined in a micro-scale circular disk with the diameter2 R = 137 sites, as shown in Fig. 2a. We impose the open boundary condition and sim-ulate thermally-induced dynamics of this skyrmion microcrystal by numerically solving astochastic LLG equation [24, 25] using the Heun scheme [26]. The equation is given by d m i dt = −
11 + α h m i × (cid:0) B eff i + ξ fl i ( t ) (cid:1) + α G m m i × (cid:2) m i × (cid:0) B eff i + ξ fl i ( t ) (cid:1)(cid:3)i , (2)where α G is the Gilbert-damping coefficient and B eff i = − h ∂ H ∂ m i is the deterministic field.The Gaussian stochastic field ξ fl i ( t ) describes effects of thermally fluctuating environmentinteracting with m i , which satisfies (cid:10) ξ fl i,β ( t ) (cid:11) = 0 and (cid:10) ξ fl i,β ( t ) ξ fl j,λ ( s ) (cid:11) = 2 κδ ij δ βλ δ ( t − s ),where β and λ are Cartesian indices. The fluctuation-dissipation theorem gives a relationbetween κ and temperature T : κ = α G k B T /m [26]. The initial spin configuration (Fig. 2a)is prepared by the Monte-Carlo thermalization at low temperature and by further relaxingit in the LLG simulation at T =0. Starting from this initial configuration, we generate therandom numbers corresponding to the stochastic force ξ fl i ( t ) and solve equation (2). In whatfollows we use units in which the lattice constant a = 1, the exchange energy J = 1, theBoltzmann constant k B = 1 and ¯ h = 1.In thermal equilibrium we only find Brownian motion of skyrmions and no unidirectionalrotation. We then include a radial temperature gradient to examine whether it can give rise4o chiral rotation. In the Lorentz-TEM experiment, the electron beam is irradiated onto athin-plate specimen, which inevitably raises temperature of the beam spot with respect tothe outer region, resulting in the temperature gradient as shown in Fig. 2 b . We consider aconstant temperature gradient with T = T at the edge to T = T + ∆ T at the center in thecircular-disk system, i.e., − dT /dr =∆ T /R .We display snapshots of the calculated real-space magnetization dynamics at selectedtimes in Figs. 3a-d, and the trajectory of a selected skyrmion indicated by solid rectanglesin Fig. 3d. We find persistent rotation of SkX (Supplementary Movies 3 and 4). Figures 3e-lshow the time evolution of Fourier transform of the spin structure - the rotating hexagoncomposed of six Bragg peaks (Supplementary Movie 5). In the simulation, we apply B k − z in accord with a setup of the Lorentz-TEM experiment, and find clockwise rotations inagreement with the experimental observations. Remarkably, this unidirectional rotation isdriven purely by the thermal gradient because no other motive forces are considered in oursimulation.This nonreciprocal dynamics of SkX can be traced back to the algebra of spin operators,which determines the direction of spin precession in an applied B . Equations of motion forthe center-of-mass coordinates of a skyrmion have the form (Supplementary Information) M ¨ Y + α G Γ ˙ Y + 4 πQS ˙ X = − ∂U∂Y + 4 πQJ magnon x , M ¨ X + α G Γ ˙ X − πQS ˙ Y = − ∂U∂X − πQJ magnon y , (3)where M is the skyrmion mass, Q is the topological charge ( Q = − B z < α G is the Gilbert-damping constant, Γ ≈ . πS , U is the external potential, and J magnon = ( J magnon x , J magnon y ) is the magnon current density defined in Supplementary In-formation, which drives skyrmion motion through the spin transfer torque. The extensionto a many skyrmion problem is straightforward.First we note that, if one replaces the magnon current by the stochastic Langevin force,the dynamics of skyrmions becomes equivalent to that of classical particles in an externalmagnetic field. In this case, the famous Bohr-van Leeuwen theorem [4, 5], which forbidsorbital magnetism of classical particles in thermal equilibrium, precludes the spontaneousrotation of skyrmions, in agreement with our numerical results.Next we discuss how a nonzero temperature gradient can induce a persistent rotation.One possible scenario is that the gradient dT /dr modifies the radial distribution of skyrmions5esulting in a non-vanishing radial force − h dU/dr i , which according to equation (3) givesrise to a nonzero angular velocity. This, however, is forbidden, as the Bohr-van Leeuwentheorem can be generalized to the case of local equilibrium with a spatially inhomogeneoustemperature T ( r ). What is required, is a nonequilibrium state with a heat flow forminga heat engine, in which an amount of heat Q transferred from the high-temperature side( T = T ) is partly transformed into work W , while the remaining heat Q = Q − W isabsorbed on the low-temperature side ( T = T < T ). The ratchet rotation requires W > η = W/Q is less than 1 − T /T .The underlying microscopic mechanism involves the flow of the spin energy. There aretwo possible heat carriers: skyrmions and magnons. The dimensional analysis of the ratchetrotation frequency ν due to the thermal motion of skyrmions gives ν ∼ R (cid:0) − dTdr (cid:1) (in the J = ¯ h = k B = a = 1 units), which is too small to explain the results of numerical sim-ulations. Magnons, on the other hand, are much more efficient in driving the rotation ofskyrmions through the spin transfer torque [27]. Importantly, a skyrmion induces an effec-tive magnetic field h z = − m · ( ∂ x m × ∂ y m ) with the total flux R d r h z = 4 πQ , which exertsthe Lorentz force on magnons [28] (Supplementary Information). The skew scattering ofmagnons off skyrmions gives rise to the Topological Magnon Hall Effect: in addition to thethermally-driven magnon current in the radial direction, J magnon r = κ magnon xx (cid:0) − dTdr (cid:1) , there isa current J magnon θ = κ magnon xy (cid:0) − dTdr (cid:1) in the direction transverse to the temperature gradientcorresponding to the counterclockwise rotation of the magnon gas.Figure 4 displays the result of simulations for the magnon current density. We show areal-space map of the magnon current density at a selected time in Fig. 4a where the arrowspoint in the current directions while their lengths represent the current amplitudes. In thismap, the currents may seem to flow in random directions, but we find a net positive valueof the time-averaged quantity h J magnon θ i ≡ N P ′ i ( r i × J magnon i ) / | r i | shown in Fig. 4b. Herethe vector r i connects the center of the disk and the i th site, and the summation P ′ i goesover the sites with | r i | >
20. The positive sign indicates that the magnon current flowsin the counterclockwise direction. The magnitude of the transverse magnon current in ournumerical simulations, J magnon θ ∼ − , which for − dT /dr ∼ − corresponds to a ratherlarge magnon Hall conductivity, κ magnon xy ∼ [29].The skew magnon scattering off skyrmions exerts a reaction force on skyrmions in thenegative θ -direction. The reaction force appears in the right-hand side of Eqs.(3), e.g., the6orce F y equals a product of the flux of the effective magnetic field 4 πQ and the magnoncurrent J magnon x . From equations of motion (3), we obtain the estimate for the rotation rateof SkX (Supplementary Information) ν ∼ − J magnon θ πR ∼ − × − , (4)for J magnon θ ∼ − . The minus sign corresponds to clockwise rotation of skyrmions. Equation(4) shows that the experimentally observed clockwise rotation of skyrmions is a consequenceof the anticlockwise rotation of magnons. The estimate (4) is in good quantitative agreementwith the numerical result for the SkX rotation rate, − × − .We thus showed that the magnon current induced by the temperature gradient is deflectedby the emergent magnetic field of skyrmions, which in turn gives rise to the rotation of SkXthrough the spin-transfer torque. Since the sign of J magnon is governed by the sign of dT /dr ,the rotation of SkX should be reversed upon the sign reversal of the temperature gradient,which is indeed what we find in our simulation (see Fig. 4c and Supplementary Movie 6).Also the rotation direction becomes reversed upon the sign reversal of magnetic field B z butnot upon the sign reversal of the DM parameter D as seen in Fig. 4d because the formerchanges the sign of J magnon but the latter does not. This shows that skyrmion-magnoninteractions and thermal spin fluctuations provide a key to understanding of the observedchiral rotation of skyrmions.The proposed physics behind the observed rotation is distinct from the Skyrmion Halleffect discussed in a recent paper by Kong and Zang [27]. They theoretically proposed thata longitudinal skyrmion motion due to the magnon current along the thermal gradient isaccompanied by a small transverse motion due to the Gilbert damping. This SkyrmionHall effect necessarily requires the longitudinal motion. In our case, however, the skyrmionmotion in the radial direction (parallel to the temperature gradient) is forbidden due tothe geometrical confinement. Hence, no Skyrmion Hall effect is possible and the topologicalmagnon Hall effect is the only source of the observed skyrmion rotation. The reaction forcefrom the magnon current deflected by the effective magnetic field of the topological skyrmiontexture drives the peculiar chiral motion.Note that the time scale of the rotation is microseconds in the simulation for J ∼ T decreases, and vanishes when ∆ T =0. In addition, the rotation is lesspronounced and its rate is lower in a larger-sized disk, indicating the absence of rotation inthe thermodynamic limit.To summarize, we have found experimentally and explained theoretically that micron-sized skyrmion crystals behave as the Feynman’s ratchet. In the presence of the radialmagnon flow, skyrmions exhibit persistent rotation in the direction determined by the signof the topological charge of the skyrmions. The physical origin of this unusual phenomenoncan be traced back to the chiral nature of spin dynamics. Our finding shows how thermalspin fluctuations can be harnessed to control topological spin textures by irradiating themwith light and electrons. The manipulation of skyrmions with magnons can be used to buildall-spin memory and logic devices with low dissipation losses by replacing the charge currentby magnon current especially in the insulating magnets [30]. Methods Summary
The single crystal samples of MnSi were grown by the floating zone technique, whilethose of Cu OSeO were grown by the chemical vapor transport method. For the real-spaceimaging of spin textures, their thin specimens with thickness of ∼
50 nm were prepared bymechanical polishing and subsequent argon-ion thinning with an acceleration voltage of 4 kV.All experiments were performed with a transmission electron microscope (JEM2100F, JEOL)at an acceleration voltage of 200 kV. Images of the SkX were obtained at the over-focusedLorentz-TEM mode. The movies of Lorentz-TEM image were taken with the exposure timeof 50 ms and the frame rate of 18 fps (frame per second). A liquid Helium cooling holderwas utilized to investigate the T dependence, by which T at the specimen can be controlledfrom 6 K to 300 K. The electron beam strength is 2 . × Am − (3 . × Am − ) forthe MnSi (Cu OSeO ) specimen. The magnetic field of B =175 mT (65 mT) parallel to theelectron beam was applied, and the measurement was performed at 8 K (40 K) for MnSi8Cu OSeO ). [1] Gardner, M. The New Ambidextrous Universe (Freeman, 1990).[2] Feynman, R. P. The Feynman Lectures on Physics, Vol. 1. Chapter 46 (Addison-Wesley, 1963).[3] Skyrme, T. H. R., A unified field theory of mesons and baryons. Nucl. Phys. , 556-569(1962).[4] Bohr, N. Studier over Metallernes Elektrontheori (Kobenhavns Universitet, 1911).[5] van Leeuwen, H. J. Problemes de la theorie electronique du magnetisme. Journal de Physiqueet le Radium 2 (12): 361-377 (1921).[6] Bogdanov, A. N. & Yablonskiˆı, D. A. Thermodynamically stable “vortices” in magneticallyordered crystals: The mixed state of magnets. Sov. Phys. JETP , 101-103 (1989).[7] Bogdanov, A. & Hubert, A. Thermodynamically stable magnetic vortex states in magneticcrystals. J. Mag. Mag. Mat. , 255-269 (1994).[8] R¨oßler, U. K., Bogdanov, A. N. & Pfleiderer, C. Spontaneous skyrmion ground states inmagnetic metals.
Nature , 797-801 (2006).[9] M¨uhlbauer, S. et al. Skyrmion lattice in a chiral magnet.
Science , 915-919 (2009).[10] Tonomura, A., et al. Real-Space Observation of Skyrmion Lattice in Helimagnet MnSi ThinSamples.
Nano Lett. , 1673-1677 (2012).[11] Pfleiderer, C. et al. Skyrmion lattices in metallic and semiconducting B20 transition metalcompounds. J. Phys. Condens. Matter , 164207 (2010).[12] M¨unzer, A. et al. Skyrmion lattice in the doped semiconductor Fe − x Co x Si.
Phys. Rev. B ,041203(R) (2010).[13] Yu, X. Z. et al. Real-space observation of a two-dimensional skyrmion crystal. Nature ,901-904 (2010).[14] Yu, X. Z. et al. Near room-temperature formation of a skyrmion crystal in thin-films of thehelimagnet FeGe.
Nature Mater. , 106-109 (2010).[15] Seki, S., Yu, X. Z., Ishiwata, S., Tokura, Y. Observation of skyrmions in a multiferroic material. Science , 198-201 (2012).[16] Adams, T. et al. Long-wavelength helimagnetic order and skyrmion lattice phase in Cu OSeO . Phys. Rev. Lett. , 237204 (2012).
17] Seki, S. et al. Formation and rotation of skyrmion crystal in the chiral-lattice insulatorCu OSeO . Phys. Rev. B , 220406 (2012).[18] Yi, S. D., Onoda, S., Nagaosa, N. & Han, J. H. Skyrmions and anomalous Hall effect in aDzyaloshinskii-Moriya spiral magnet. Phys. Rev. B , 054416 (2009).[19] Bak, P. & Jensen, M. H. Theory of helical magnetic structures and phase transitions in MnSiand FeGe. J. Phys. C , L881-L885 (1980).[20] Jonietz, F. et al. Spin transfer torques in MnSi at ultralow current densities. Science ,1648-1651 (2010).[21] Everschor, K. et al. Rotating skyrmion lattices by spin torques and field or temperaturegradients.
Phys. Rev. B , 054432 (2012).[22] Yu, X. Z. et al. Skyrmion flow near room temperature in an ultralow current density. NatureCommun. , 988 (2012).[23] Iwasaki, J., Mochizuki, M. & Nagaosa, N. Universal current-velocity relation of skyrmionmotion in chiral magnets. Nature Commun. Phys. Rev. , 1677-1686(1963).[25] Kubo, R. & Hashitsume, N. Brownian motion of spins.
Prog. Theor. Phys. Suppl. , 210-220(1970).[26] Garc´ıa, J. L. & L´azaro, F. J. Langevin-dynamics study of the dynamical properties of smallmagnetic particles. Phys. Rev. B , 14937-14958 (1998).[27] Kong, L. & Zang, J. Dynamics of an Insulating Skyrmion under a Temperature Gradient. Phys. Rev. Lett. , 067203 (2013).[28] Nagaosa, N. & Tokura, Y. Emergent electromagnetism in solids.
Phys. Scr. T , 014020(2012).[29] van Hoogdalem, K. A., Tserkovnyak, Y. & Loss, D. Magnetic texture-induced thermal Halleffects.
Phys. Rev. B , 024402 (2013).[30] Kruglyak, V. V., Demokritov, S. O. & Grundler, D. Magnonics. J. Phys. D: Appl. Phys. ,260301 (2010). cknowledgements The authors thank A. Rosch, M. Ichikawa, Y. Matsui, Y. Ogimoto, and E. Saito for discus-sions. XZY is grateful K. Nishizawa and T. Kikitsu for providing a transmission electronmicroscope (JEM2100F). This research was in part supported by JSPS KAKENHI (GrantNumbers 24224009, 25870169, and 25287088), by the Funding Program for World-LeadingInnovative R&D on Science and Technology (FIRST Program), Japan, and by G-COE Pro-gram “Physical Sciences Frontier” from MEXT Japan. M. Mostovoy was supported byFOM grant 11PR2928 and the Niels Bohr International Academy. JZ is supported by theTheoretical Interdisciplinary Physics and Astrophysics Center and by the U.S. Departmentof Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineeringunder Award DEFG02-08ER46544.
Author Contributions
M. Mochizuki carried out the numerical simulations and analyzed the simulation data. XYcarried out the Lorentz transmission electron microscopy measurement and analyzed theexperimental data. SS carried out the crystal growth of Cu OSeO . NK carried out thecrystal growth of MnSi. The whole work has been lead by NN and YT. The results werediscussed and interpreted by M. Mochizuki, XY, WK, JZ, M. Mostovoy, YT, and NN. M.Mochizuki, M. Mostovoy, YT, and NN wrote the draft. Competing Financial Interests
The authors declare that they have no competing financial interests.11 igure Legends
Fig.1 | Experimentally observed Skyrmions and unidirectional rotation ofmicro-scale skyrmion-crystal domains in MnSi. a,
Schematic figure of skyrmion inchiral-lattice magnets. b, Real-space Lorentz-TEM image of single skyrmion in MnSi. Thecolour map and arrows represent in-plane components of magnetizations. c, Real-spaceLorentz-TEM image of the static skyrmion crystal in MnSi in which skyrmions are packedto form a triangular lattice. d, Over-focused Lorentz-TEM image of the skyrmion crystal inMnSi. e-l,
Time profiles of Fourier transforms of temporally changing magnetic structuresin the skyrmion crystal observed in MnSi by the Lorentz TEM, which show a clockwise rotation. Open circles and solid lines are guides for the eyes.
Fig.2 | Setup of the numerical simulation. a,
Magnetic configuration of the skyrmionmicrocrystal confined in a circular-shaped disk at T =0 where the in-plane magnetizationcomponents at sites ( i x , i y ) are indicated by arrows when mod( i x , 2)=mod( i y , 2)=0, whiledistribution of the magnetization z -axis components, m zi , is shown by a color map. b, Schematics of induced thermal gradient by the electron-beam irradiation in the LorentzTEM experiment.
Fig.3 | Simulated thermally driven rotation of the skyrmion microcrystal.a-d,
Snapshots of simulated temporally changing distribution of the magnetization z -axiscomponents m zi at t =1.9 × ( a ), t =2.6 × ( b ), t =3.3 × ( c ), and t =4.0 × ( d ),which show a clockwise rotation. Here the time unit is ¯ h/J . Also shown in d is thetrajectory of a selected skyrmion indicated by rectangles. e-l, Time evolution of Fouriertransforms of the magnetic structure in the reciprocal space − . π < k x < . π and − . π < k y < . π for m =1, B z /J = − . α G =0.01, k B T /J =0.1, and k B ∆ T /J =0.006at selected times between t =1.90 × and t =4.35 × with constant time intervals of∆ t =0.35 × , which also show clockwise rotation. Fig.4 | Simulation of the magnon current density. a,
Spatial distribution ofthe magnon current density J magnon i at a selected time where lengths of the arrows areproportional to the local amplitudes of J magnon . b, Time evolution of the quantity12 N P ′ i ( r i × J magnon i ) / | r i | , which shows a finite and positive component of J magnon i in thecircumferential direction. Here r i is the vector connecting the disk center and the i th site,and the summation P ′ i runs over the sites with | r i | >