Abstract
Skyrmions are topologically protected winding vector fields characterized by a spherical topology^{1}. Magnetic skyrmions can arise as the result of the interplay of various interactions, including exchange, dipolar and anisotropy energy in the case of magnetic bubbles^{2,3,4} and an additional Dzyaloshinskii–Moriya interaction in the case of chiral skyrmions^{5}. Whereas the static and lowfrequency dynamics of skyrmions are already well under control^{6,7,8,9}, their gigahertz dynamical behaviour^{2} has not been directly observed in real space. Here, we image the gigahertz gyrotropic eigenmode dynamics of a single magnetic bubble and use its trajectory to experimentally confirm its skyrmion topology. The particular trajectory points to the presence of strong inertia, with a mass much larger than predicted by existing theories. This mass is endowed by the topological confinement of the skyrmion and the energy associated with its size change. It is thereby expected to be found in all skyrmionic structures in magnetic systems and beyond. Our experiments demonstrate that the mass term plays a key role in describing skyrmion dynamics.
Main
Skyrmion configurations are pervasive in nature and have been observed in a wide variety of research areas in physics^{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}. By definition, skyrmions are vector fields that can be continuously wrapped around a sphere and the number of times the sphere is covered during this continuous deformation is the skyrmion number N. It can be shown that skyrmions are always confined in all spatial directions (see Fig. 1 and Supplementary Information I). Magnetic skyrmions are promising candidates for currentdriven memory devices^{14,15} owing to their high mobility at ultralow current densities^{7}. The static manipulation of magnetic skyrmions (nucleation, stability, and annihilation) is now well under control (see refs 8, 15 and further references in Supplementary Information VII); however, their intrinsic highfrequency dynamics has thus far not been directly observed in real space, and previous experimental investigations have not allowed the determination of inertial effects^{6,16}. Furthermore, a number of existing theoretical descriptions using quasiparticle models of magnetic skyrmions neglect inertia^{6,17,18}. Inertial effects are visible only at frequencies comparable to or faster than those intrinsic to the system (for example, the Larmor frequency), which are in the gigahertz regime in the case of solidstate magnetism. Consistently, it is expected from numerical simulations that the gigahertz gyrotropic motion of skyrmionic spin systems exhibits the characteristics of inertia^{2,19} through the existence of two gyrotropic eigenmodes, one clockwise and one anticlockwise (massless skyrmions have just one gyrotropic eigenmode, see Supplementary Information II). However, only one of these modes has been observed so far^{16}. Here, we observe that the skyrmion motion is indeed composed of two eigenfrequencies, and we also use this effect to experimentally determine the mass of a skyrmionic magnetic quasiparticle. We observe that this mass is sizable and, in fact, much larger than masses found for other magnetic quasiparticles^{20,21,22}. Such a large mass, which significantly exceeds theoretical predictions^{19}, has a fundamental impact on the dynamical behaviour of magnetic skyrmions, especially in the technologically important case of magnetic thinfilm nanostructures, as we show in this work.
The presence of the inertial mass originates from the ability of the system to store energy internally during its motion^{2,19}; in our system this is achieved through a deformation in shape, a feature which we expect to be common to all confined (meta)stable structures. Such deformations are visible in simulations of bubbles^{2} as well as of chiral skyrmions^{11} (see Supplementary Information II). Inertia of domain walls, confined in one dimension and extended in the other, has been known for a long time^{23}. Domain wall inertia causes retardation, but does not change the trajectory of the wall dynamics^{24}. In contrast, as shown in this Letter, fully confined skyrmions exhibit an additional topological source of inertia that makes the effective mass very large in comparison to straight domain walls and induces fundamental changes to their trajectory.
The experiment is performed using timeresolved pump–probe Xray holography^{25,26} to image the gyrotropic trajectory of a skyrmionic magnetic bubble (a skyrmion stabilized by dipolar energies, a ‘skyrmion bubble’) in a submicrometre diameter magnetic disk (Fig. 2a) after a magnetic field gradient excitation. A static outofplane field of −120 mT is applied, resulting in a ground state with two bubbles. Both bubbles are several radii away from the edge of the disk; thus, interactions with this edge can be neglected. Rather, the bubbles are subjected to a local magnetic potential, which originates from static irregularities of the material structure of the magnetic film and from (possibly dynamic) interactions with the domain pattern in the disk. The excitation of this skyrmionic state is achieved by injecting bipolar current pulses into a microcoil enclosing the disk, generating an outofplane field pulse. The time evolution of the current pulse is plotted in Fig. 2b, and the field generated at the pulse peak is visualized in the inset of Fig. 2d. A positive current generates a field which is antiparallel to the negative static bias field, favouring the growth of the bubble domains and the nucleation of new reverse domains.
Selected images acquired at different times after the start of the pulse are presented in Fig. 2c. To generate the required field gradient to displace bubbles, we apply a positive field pulse to nucleate a third domain at an artificial notchshaped nucleation centre at the top of the disk, and we use the strong (and ultrafast) gradient of the stray field created by this domain to dynamically alter the potential landscape and subsequently displace one of the two bubbles (further referred to as ‘the bubble’) sufficiently for experimental observation. Field gradients of this magnitude (expected to be of the order of T μm^{−1})^{2} are otherwise experimentally inaccessible, in particular with rise times of only a few hundred picoseconds. After reversal of the field pulse, the third domain annihilates through the intermediate formation of an N = 0 bubble. The bubble subsequently returns to its equilibrium position. The other bubble in the disk does not move because it is pinned in a strong potential minimum. The interaction with the first bubble is therefore purely static, and the relaxation of this first bubble is thus fully analogous to the motion of a bubble in the static potential defined by the edge of a disk, which has been studied in previous theoretical work^{2,19}. The full video of the bubble motion and comprehensive discussions of further details of the magnetization dynamics in the disk can be found in Supplementary Information III.
The full trajectory of the bubble, plotted in Fig. 2d, forms a loop with clearly distinct excitation and relaxation paths. In particular, after t = 4.5 ns, the bubble moves on a spiralling trajectory, an unequivocal signature of the influence of the gyrocoupling vector and hence of a nonzero skyrmion number. In materials with preferred outofplane spin alignment, all fully confined objects (that is, domains that do not touch the sample boundary) are configurations with integer N; specifically, from micromagnetic simulations we deduce that the observed bubble has unity N because all N ≠ 1 configurations that could be expected for a bubble in this material show very complex trajectories (see ref. 2 and Supplementary Information IV). As we know the direction of the applied magnetic field, we also know the polarity of the bubble and thus its skyrmion number N = +1.
Our experiment constitutes the first direct experimental observation of gigahertz dynamics of a skyrmionic spin structure, which was made feasible as a result of our extremely high accuracy in tracking the centre of the bubble (of better than 3 nm). This outstanding accuracy for timeresolved experiments can be reached with Xray holography because the imaging process is intrinsically insensitive to drift^{25,26}. Isolated skyrmions can be imaged even if they are much smaller than the pixel resolution, which is 40 nm in our case. With prospective pixel resolutions of sub10 nm, imaging of all varieties of skyrmions is expected to be possible with Xray holography^{27}.
The observed gyrotropic trajectory is in excellent agreement with the recently proposed theoretical model assigning to the N = ±1 bubble an inertial mass. The theory predicts a secondorder differential equation of motion for the centre of mass R of the magnetization^{19}
Here, M is the effective inertial mass, G = (0,0, G) is the gyrocoupling vector, with G = −4πNt_{Co}M_{s}/γ (where t_{Co} is the total magnetic material thickness, M_{s} its saturation magnetization and γ = 1.76 × 10^{11}A s kg^{−1} the gyromagnetic ratio), D is the dissipation tensor and U is the magnetostatic potential, which we approximate to be parabolic ∂_{R}U = K R. The general solution of this equation is a superposition of two damped spiralling motions, one clockwise (CW) and one anticlockwise (ACW). The parameters of this general solution are extracted from a fit to the experimental data (see Supplementary Information IX for the fitting details).
The theoretical model given by equation (1) accurately describes the gyrotropic part of the observed trajectory R(t) = (X(t), Y (t)). This is visualized in Fig. 3, showing the projections X(t), Y (t) and Y (X) in a, b and c, respectively. A good fit to the data, visualized by a solid, coloured line in Fig. 3, is possible only with a globally simultaneous, coherent superposition of two gyrotropic modes: a CW higherfrequency mode and a ACW lowerfrequency mode. This confirms the presence of an inertial mass, because, without the mass term, the sense of gyration would be fixed by G (which is a constant of motion) and only one mode could be excited at a time. For comparison, the best fit of the trajectory of a massless bubble is illustrated in Fig. 3 by a grey line, showing that the mass term is necessary to describe the data. The excellent agreement with the theoretical model furthermore confirms the validity of the parabolic approximation for the potential used, see Supplementary Information V.
From the fit we obtain the frequencies ω_{1} = 1.00(13) GHz (ACW mode) and ω_{2} = −1.35(16) GHz (CW mode, as indicated by the minus sign). The frequencies of both modes are in the gigahertz regime, which is in line with numerical predictions in ref. 2. The same characteristic spectrum of one CW and one ACW mode, with very similar frequencies, has been predicted by numerical simulations for chiral skyrmion crystals, underlining the fundamental topological origin of the gyrotropic motion and of the presence of inertia^{11}. The inertial mass of the bubble is inversely proportional to the sum of the two frequencies^{19}, which allows us to derive a lower limit M_{LL} > 8 × 10^{−22} kg for the inertial mass with high confidence (5 σ) despite the relatively large experimental uncertainties for the frequencies. From the product of the frequencies we obtain the stiffness K = 3.5(16) × 10^{−3} N m^{−1} of the local magnetostatic potential.
The lower limit for the mass of the bubble corresponds to an areal mass density of 2.0(4) × 10^{−7} kg m^{−2}. Such large inertia has been reported so far only for crosstie domain walls^{28}. It is at least two orders of magnitude larger than any value previously reported for straight domain walls^{20,21,22}, and a factor of five larger than predicted by the theory in ref. 19 (for details, see Supplementary Information VI). Such a large mass is responsible for the enormous deviation of our measured trajectory from the best fit with a massless model (Fig. 3). The impact of the mass is also significant in other geometries, such as the recently suggested case of a skyrmion in a wire^{14,15}. Here, the mass helps to avoid the transversal expulsion of the skyrmion from the wire, thus leading to higher longitudinal displacements per pulse, as demonstrated in Supplementary Information VII.
The amplification of inertia in confined, lowdimensional spin structures by up to two orders of magnitude is known from Bloch lines and has been attributed to the local exchange energy^{6}. Here, we attribute the surprisingly large mass of magnetic skyrmions to an additional energy reservoir associated with a change of its size (breathing mode, see Supplementary Information II)—that is, to a collective or emergent source of inertia. Such a breathing mode is inherent to all structures that are of finite extent in thermodynamic equilibrium (in magnetism always objects with integer N topology), where energy can be stored by a change in the spatial extent. The breathing mode mass is associated with the N = 1 topology, because N = 1 structures are the only ones that exhibit the gyrotropic eigenmodes described by equation (1). We therefore call the mass, arising from the specific topology, a topological mass. The particularly large mass found in magnetic skyrmions is related to the strong dipolar energy of the breathing mode. The nonlocal breathing mode energy is not included in the Döring mass and has thus been neglected in previous theoretical models^{19}, but contributes significantly to the mass found here. The previously found large mass of crosstie walls can actually be attributed to their local skyrmionic spin structure^{13,29}. A broad range of simulations indicates that the mass density of a skyrmion is not modified by any environmental factor changing its spatial extent (see Supplementary Information V). Therefore, the large skyrmion mass found by us is expected to dominate the skyrmion motion independently of the device geometry. We anticipate that this topological mass provides an important possibility to tailor the dynamical behaviour of skyrmions and to facilitate their application in future devices.
Methods
The sample was prepared as described in refs 26, 30, with a Pt(2)/[Co_{68}B_{32}(0.4)/Pt(0.7)]_{30}/Pt(1.3) (thickness in nm) magnetic layer patterned onto a 550 nm disk with a notch at the top, a Au microcoil tightly around the disk (see Fig. 2a), and two reference holes with diameters of 80 nm and 55 nm. The magnetic imaging was performed using timeresolved Xray holography at the UE52SGM beamline at BESSY II in Berlin, Germany. Bipolar rectangular microwave pulses with a current of 52 mA were injected into the microcoil at a repetition rate of 1.25 MHz, synchronized to the probe Xrays (Fig. 2b). The resulting field, estimated using finite difference calculations, was roughly homogeneous in the outofplane direction (inset Fig. 2d), with a strength of B_{z} = −35 mT at the position of the bubble at maximum negative current, and small inplane components B_{x} = −1.5 mT and B_{y} = 6.5 mT. The residual field gradient was ∂_{x}B_{z} = 8.7 μT nm^{−1} and ∂_{y}B_{z} = −45 μT nm^{−1}, pushing the bubble towards the centre of the disk. The magnetic domain configuration was imaged at variable time delays t with a temporal resolution of 50 ps (standard deviation), determined by the duration of the synchrotron Xray pulses and their temporal jitter with respect to our electronics. For each time delay, the pump–probe experiment was repeated about 10^{9} times, corresponding to a total number of 6.4(5) × 10^{7} recorded photons of each helicity (positive and negative) per image. Images were recorded in a random nonchronological order.
In the reconstructed realspace magnetic image, the area of the disk was cropped using the topographic information in the holograms. The centre of the bubble was calculated by statistical means. By recording Fourier space information, the realspace reconstruction is inherently drift free. Therefore, we can determine the position of the centre of the bubble with an average precision of 2.3 nm, which is much better than the approximate spatial resolution of 40 nm (ref. 30).
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Acknowledgements
We thank V. Guzenko and the Laboratory for Micro and Nanotechnology at the Paul Scherrer Institut for access to the clean room, use of the Electron Beam Lithography system, and for support. C.M. thanks J. Raabe and S. Komineas for discussions. We thank the Department of Physics of the University of Hamburg for access to the PHYSnetComputing Center. This work was funded by the German Ministry for Education and Science (BMBF) through the projects MULTIMAG (13N9911) and MPSCATT (05K10KTB), the EU’s 7th Framework Programme MAGWIRE (FP7ICT20095 257707), the European Research Council through the Starting Independent Researcher Grant MASPIC (ERC2007StG 208162), the Mainz Center for Complex Materials (COMATT), the Swiss National Science Foundation (SNF) and the Deutsche Forschungsgemeinschaft (DFG).
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C.M., M.K. and S.E. conceived the experiment. F.B., M.S., C.M.G., C.A.F.V. and J.H.F. prepared the samples. F.B., C.M., M.S., C.M.G., J.G., C.v.K.S., J.M., B.P., S.S., A.B., M.F. and T.S. performed the dynamic imaging experiment. The data analysis was carried out by F.B. and B.K. along with discussions with C.M., C.A.F.V., M.K. and S.E. The manuscript was written by F.B. with input from C.M., C.A.F.V., M.K., S.E. and all other authors. Supervision was by H.J.M.S., M.K. and S.E.
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Büttner, F., Moutafis, C., Schneider, M. et al. Dynamics and inertia of skyrmionic spin structures. Nature Phys 11, 225–228 (2015). https://doi.org/10.1038/nphys3234
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DOI: https://doi.org/10.1038/nphys3234
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