Bulk and surface topological indices for a skyrmion string: current-driven dynamics of skyrmion string in stepped samples.

The magnetic skyrmion is a topological magnetic vortex, and its topological nature is characterized by an index called skyrmion number which is a mapping of the magnetic moments defined on a two-dimensional space to a unit sphere. In three-dimensions, a skyrmion, i.e., a vortex penetrating though the magnet naturally forms a string, which terminates at the surfaces of the magnet or in the bulk. For such a string, the topological indices, which control its topological stability are less trivial. Here, we study theoretically, in terms of numerical simulation, the dynamics of current-driven motion of a skyrmion string in a film sample with the step edges on the surface. In particular, skyrmion-antiskyrmion pair is generated by driving a skyrmion string through the side step with an enough height. We find that the topological indices relevant to the stability are the followings; (1) skyrmion number along the developed surface, and (2) the monopole charge in the bulk defined as the integral over the surface enclosing a singular magnetic configuration. As long as the magnetic configuration is slowly varying, the former is conserved while its changes is associated with nonzero monopole charge. The skyrmion number and the monoplole charge offer a coherent understanding of the stability of the topological magnetic texture and the nontrivial dynamics of skyrmion strings.

Real-Space Observation of Topological Defects in Extended Skyrmion-Strings.

Exotic topological spin textures such as emergent magnetic monopole/anti-monopoles (hedgehog/anti-hedgehog) in the metastable extended skyrmion-strings attract much attention to the fundamental physics owing to their novel electromagnetic properties. However, the direct imaging of such spin textures is lacking. Here, we report the real-space observation of emergent magnetic monopoles involved in extended skyrmion-strings by Lorentz transmission electron microscopy (TEM) in combination with micromagnetic simulations. The in-plane extended skyrmion-strings are observed directly by Lorentz TEM to accompany the topological hedgehog-like defect, where the skyrmion-string terminates or merges with another skyrmion-string, as well as the surface-related defects where skyrmion-string bends 90° and ends on the surface. We also demonstrate the transformation of a metastabilized lattice of out-of-plane short skyrmion-strings into an in-plane array of extended skyrmion-strings by tuning the magnitude of oblique fields in a room-temperature helimagnet, revealing the stability of such topological spin textures and the possibility to control them.

Controlled transformation of skyrmions and antiskyrmions in a non-centrosymmetric magnet.

Control of topological spin textures in magnetic systems may enable future spintronic applications. Magnetic field pulses can switch the vortex polarity1 or the winding number of magnetic bubbles2. Thermal energy can reverse the helicity of skyrmions3 and induce the transformation between meron and skyrmion by modifying the in-plane anisotropy4,5. Among the various topological spin textures, skyrmions6,7 and antiskyrmions8-10 are nanometric spin-whirling structures carrying integer topological charges (N) of -1 and +1 (refs. 7,11,12), respectively, and can be observed in real space8,13. They exhibit different dynamical properties under current flow14-18, for example, opposite signs for the topological Hall effect. Here we observe, in real space, transformations among antiskyrmions, non-topological (NT) bubbles and skyrmions (with N of +1, 0 and -1, respectively) and their lattices in a non-centrosymmetric Heusler magnet, Mn1.4Pt0.9Pd0.1Sn, with D2d symmetry. Lorentz transmission electron microscopy images under out-of-plane magnetic fields show a square lattice of square-shaped antiskyrmions near the Curie temperature and a triangular lattice of elliptically deformed skyrmions with opposite helicities at lower temperatures. The clockwise and counter-clockwise helicities of the skyrmions originate from Dzyaloshinskii-Moriya interactions with opposite signs along the [100] and [010] directions, respectively. A variation of the in-plane magnetic field induces a topological transformation from antiskyrmions to NT-bubbles and to skyrmions, which is accompanied by a change of the lattice geometry. We also demonstrate control of the helicity of skyrmions by variations of the in-plane magnetic field. These results showcase the control of the topological nature of spin configurations in complex magnetic systems.

Nonlocal accumulation, chemical potential, and Hall effect of skyrmions in Pt/Co/Ir heterostructure.

Magnetic skyrmion is a swirling topological spin texture behaving as an individual particle. It shows a gyro-motion similarly to that of a charged particle under a magnetic field, being led to the transverse shift to the electric current, i.e., skyrmion Hall effect. With the open boundaries of a sample, this results in an accumulation of skyrmions on one side and their depletion on the other side. Here we demonstrate experimentally that this effect propagates non-locally over tens of micrometers even where the electric current is absent, when the narrow wires bridge bar-shaped Pt/Co/Ir heterostructure thin film systems. This nonlocality can be understood in terms of the "chemical potential" gradient for the skyrmion bubble induced by the skyrmion Hall effect in the nonequilibrium steady state under the electric current. The present result shows that the skyrmion Hall effect acts as the skyrmion pump and the thermodynamic concepts can be applied to the aggregate of skyrmion bubbles.

Dynamics of skyrmion in disordered chiral magnet of thin film form.

Magnetic skyrmion is a topological spin texture characterized by the mapping from the two dimensional real space to the unit sphere. It is realized in chiral magnets under an external magnetic field in the plane perpendicular to it. In thin film samples, which are most relevant to the applications, the thickness of the system parallel to the magnetic field is finite, and a skyrmion turns into a skyrmion string, which is often assumed to be a straight rod. There are phenomena related to the internal degrees of freedom along the string, e.g., the monopole and anti-monopole creation/annihilation, corresponding to the change in the skyrmion number. However, the role of this finite thickness in the topological stability and dynamics has not been explored yet. Here we study theoretically the current-driven dynamics of a skyrmion string under disorder potential by systematically changing the thickness of the sample to reveal the dynamical phase diagram in the plane of current density and thickness. We found the three regions, i.e., (i) pinned skyrmion string, (ii) moving depinned skyrmion string, and (iii) annihilation of skyrmion string, for thin and thick limits while (iii) is missing in the intermediate case. This indicates that there is the optimal range of thickness for the topological stability of skyrmion string enhanced compared with a two-dimensional skyrmion. This result provides a way to design and control skyrmions in thin films and interfaces of finite thickness.

Theory of current-driven skyrmions in disordered magnets.

An emergent topological particle in magnets, skyrmion, has several unique features distinct from the other magnetic textures such as domain wall, helical structure, and vortex. It is characterized by a topological integer called skyrmion number N sk , which counts how many times the directions of the magnetic moments wrap the unit sphere. This N sk gives the chiral nature of the skyrmion dynamics, and leads to the extremely small critical current density j c for the current-driven motion in terms of spin transfer torque effect. The finite j c indicates the pinning effect due to the disorder such as impurities and defects, and the behaviors of skyrmions under disorder have not been explored well theoretically although it is always relevant in real systems. Here we reveal by a numerical simulation of Landau-Lifshitz-Gilbert equation that there are four different skyrmion phases with the strong disorder, i.e., (A) pinned state, (B) depinned state, (C) skyrmion multiplication/annihilation, and (D) segregation of skyrmions, as the current density increases, while only two phases (A) and (B) appear in the weak disorder case. The microscopic mechanisms of the new phases (C) and (D) are analyzed theoretically. These results offer a coherent understanding of the skyrmion dynamics under current with disorder.

Current-induced viscoelastic topological unwinding of metastable skyrmion strings.

In the MnSi bulk chiral magnet, magnetic skyrmion strings of 17 nm in diameter appear in the form of a lattice, penetrating the sample thickness, 10-1000 µm. Although such a bundle of skyrmion strings may exhibit complex soft-matter-like dynamics when starting to move under the influence of a random pinning potential, the details remain highly elusive. Here, we show that a metastable skyrmion-string lattice is subject to topological unwinding under the application of pulsed currents of 3-5 × 106 A m-2 rather than being transported, as evidenced by measurements of the topological Hall effect. The critical current density above which the topological unwinding occurs is larger for a shorter pulse width, reminiscent of the viscoelastic characteristics accompanying the pinning-creep transition observed in domain-wall motion. Numerical simulations reveal that current-induced depinning of already segmented skyrmion strings initiates the topological unwinding. Thus, the skyrmion-string length is an element to consider when studying current-induced motion.

Theory of skyrmions in bilayer systems.

Skyrmion is an emergent particle consisting of many spins in magnets, and has many nontrivial features such as (i) nano-scale size, (ii) topological stability, (iii) gyrodynamics, and (iv) highly efficient spin transfer torque, which make skyrmions the promising candidate for the magnetic devices. Earlier works were focusing on the bulk or thin film of Dzyaloshinskii-Moriya (DM) magnets, while recent advances are focusing on the skyrmions induced by the interfaces. Therefore, the superstructures naturally leads to the interacting skyrmions on different interfaces, which has unique dynamics compared with those on the same interface. Here we theoretically study the two skyrmions on bilayer systems employing micromagnetic simulations as well as the analysis based on Thiele equation, revealing the reaction between them such as the collision and bound state formation. The dynamics depends sensitively on the sign of DM interactions, i.e., helicities, and skyrmion numbers of two skyrmions, which can be well described by Thiele equation. Furthermore, we have found the colossal spin-transfer-torque effect of bound skyrmion pair on antiferromagnetically coupled bilayer systems.

Interface-driven topological Hall effect in SrRuO3-SrIrO3 bilayer.

Electron transport coupled with magnetism has attracted attention over the years. Among them, recently discovered is topological Hall effect (THE), originating from scalar spin chirality, that is, the solid angle subtended by the spins. THE is found to be a promising tool for probing the Dzyaloshinskii-Moriya (DM) interaction and consequent magnetic skyrmions. This interaction arises from broken inversion symmetry and hence can be artificially introduced at interface; this concept is lately verified in metal multilayers. However, there are few attempts to investigate such DM interaction at interface through electron transport. We clarified how the transport properties couple with interface DM interaction by fabricating the epitaxial oxide interface. We observed THE in epitaxial bilayers consisting of ferromagnetic SrRuO3 and paramagnetic SrIrO3 over a wide region of both temperature and magnetic field. The magnitude of THE rapidly decreases with the thickness of SrRuO3, suggesting that the interface DM interaction plays a significant role. Such interaction is expected to realize a 10-nm-sized Néel-type magnetic skyrmion. The present results established that the high-quality oxide interface enables us to tune the effective DM interaction; this can be a step toward future topological electronics.

Theory of antiskyrmions in magnets.

Skyrmions and antiskyrmions are swirling topological magnetic textures realized as emergent particles in magnets. A skyrmion is stabilized by the Dzyaloshinskii-Moriya interaction in chiral magnets and/or a dipolar interaction in thin film magnets, which prefer the twist of the magnetic moments. Here we show by a numerical simulation of the Landau-Lifshitz-Gilbert equation that pairs of skyrmions and antiskyrmions are created from the helix state as the magnetic field is increased. Antiskyrmions are unstable and disappear immediately in chiral magnets, whereas they are metastable and survive in dipolar magnets. The collision between a skyrmion and an antiskyrmion in a dipolar magnet is also studied. It is found that the collision depends on their relative direction, and the pair annihilation occurs in some cases and only the antiskyrmion is destroyed in the other cases. These results indicate that the antiskyrmion offers a unique opportunity to study particles and antiparticles in condensed-matter systems.

Photodrive of magnetic bubbles via magnetoelastic waves.

Precise control of magnetic domain walls continues to be a central topic in the field of spintronics to boost infotech, logic, and memory applications. One way is to drive the domain wall by current in metals. In insulators, the incoherent flow of phonons and magnons induced by the temperature gradient can carry the spins, i.e., spin Seebeck effect, but the spatial and time dependence is difficult to control. Here, we report that coherent phonons hybridized with spin waves, magnetoelastic waves, can drive magnetic bubble domains, or curved domain walls, in an iron garnet, which are excited by ultrafast laser pulses at a nonabsorbing photon energy. These magnetoelastic waves were imaged by time-resolved Faraday microscopy, and the resultant spin transfer force was evaluated to be larger for domain walls with steeper curvature. This will pave a path for the rapid spatiotemporal control of magnetic textures in insulating magnets.

Creation of skyrmions and antiskyrmions by local heating.

Heating a system usually increases entropy and destroys order. However, there are also cases where heating gives a system the energy to overcome the potential barrier to reach a state with a nontrivial ordered pattern. Whether heating can manipulate the topological nature of the system is especially important. Here, we theoretically show by microsimulation that local heating can create topological magnetic textures, skyrmions, in a ferromagnetic background of chiral magnets and dipolar magnets. The resulting states depend sharply on intensity and spot size of heating, as well as the interaction to stabilize the skyrmions. Typically, the creation process is completed within 0.1 ns and 10 nm at the shortest time and smallest size, and these values can be longer and larger according to the choice of system. This finding will lead to the creation of skyrmions at will, which constitutes an important step towards their application to memory devices.

Colossal spin transfer torque effect on skyrmion along the edge.

We study by the micromagnetic simulations the skyrmion motion along the edge driven by the current transverse to it. We found that (i) the velocity is enhanced by the factor of ∼ 1/α (α: the Gilbert damping) with the maximum value determined only by the confining force from the edge, (ii) the inertia appear due to the confining potential with the coordinate perpendicular to the edge playing the role of the kinetic momentum, and (iii) the collision between the two skyrmions is almost elastic without causing any internal distortions.

Possible method to observe the breathing mode of a magnetic domain wall in the Josephson junction.

A magnetic domain wall (DW) behaves as a massive particle with elasticity. Sliding and oscillation of the DW have been observed experimentally, whereas vibration of a width in the DW, "breathing mode", has not been measured so far. We theoretically propose how to observe the breathing mode by the Josephson junction having a ferromagnetic layer between superconducting electrodes. The current-voltage (I-V) curve is calculated by an equivalent circuit of the resistively shunted junction model. The breathing mode is identified by stepwise structures in the I-V curve, which appear at the voltages V = n (h/2e)ω with the fundamental constant h/e, integer number n and the frequency of the breathing mode ω.