Skyrmion Crystal Research Articles

Skyrmion crystal in the RKKY system on the two-dimensional triangular lattice

We study the ordering properties of the isotropic RKKY Heisenberg model on the two-dimensional (2D) triangular lattice by extensive Monte Carlo simulations to get insights into the chiral-degenerate skyrmion crystal (SkX) of metallic magnets. Our Hamiltonian contains only the spin-quadratic RKKY interaction derived from the spherical Fermi surface, containing neither the nesting nor the many-body interaction. The SkX phase is stabilized under applied fields where the frustration associated with the oscillating nature of the RKKY interaction and the emergent many-body interactions generated by thermal fluctuations play important roles. Replica symmetry breaking, reported in our recent study on the 3D RKKY model [Phys. Rev. B 104, 184432 (2021)], turns out to be absent in the present 2D model. Implications to the SkX formation mechanism are discussed.

Skyrmion crystals in centrosymmetric triangular magnets under hexagonal and trigonal single-ion anisotropy

We theoretically report an instability toward a skyrmion crystal in centrosymmetric magnets under hexagonal and trigonal single-ion anisotropy. The results are obtained for a minimal spin model with a crystal-dependent single-ion anisotropy on a triangular lattice by performing simulated annealing. By constructing magnetic phase diagrams in the presence of six different types of single-ion anisotropy while changing the amplitudes of external magnetic field and single-ion anisotropy in a systematic way, we find that the hexagonal and trigonal single-ion anisotropy becomes a source of the skyrmion crystal depending on the magnetic-field direction. We show that the skyrmion crystal is stabilized by the uniaxial-type or trigonal-type single-ion anisotropy under the out-of-plane magnetic field, while it is stabilized by the hexagonal-type inplane single-ion anisotropy under the inplane magnetic field. We also find various multiple-Q spin states depending on the types of the single-ion anisotropy.

Nematic and smectic stripe phases and stripe-SkX transformations

Based on the findings of stripe skyrmions and the metastability of a state of an arbitrary number of skyrmions, precisely controlled manipulation of stripe skyrmions in pre-designed structures and mutual transformation between helical states and skyrmion crystals (SkXs) are demonstrated in chiral magnetic films. As a proof of the concept, we show how to use patterned magnetic fields and spin-transfer torques (STTs) to generate nematic and smectic stripe phases, as well as "UST" mosaic from three curved stripe skyrmions. Cutting one stripe into many pieces and coalescing several skyrmions into one by various external fields are good ways to transform helical states and SkXs from each other.

Multiple Skyrmion Crystal Phases by Itinerant Frustration in Centrosymmetric Tetragonal Magnets

A skyrmion crystal (SkX) expressed as a multiple number of spiral modulations manifests itself not only in its peculiar magnetic texture but also in nontrivial transport properties originating from an emergent magnetic field. We here report our numerical results for multiple SkXs in a centrosymmetric tetragonal crystal system. By performing simulated annealing for an effective spin model for itinerant magnets, we find that three types of the SkXs with the skyrmion numbers of one and two, which are characterized by different superpositions of helices, are stabilized in the ground state, and are transformed by an external magnetic field. The essence of the emergent multiple SkXs is lied in itinerant frustration where exchange interactions are competed in momentum space due to the nature of itinerant electrons.

Transport properties of the electronic states in the gate voltage-modulated skyrmion crystal

As a result of the Hund's coupling, the band structure of the conducting electrons in the skyrmion crystal (SkX) shares similar topological properties with that of graphene, such as its cone-like shape, nonzero band Chern number, edge states, and etc. In this work, we rigorously demonstrate that the Klein tunneling phenomena is also shared by these two. We use the Green's function technique and calculated the transmission probability of the electrons tunneling through an electrostatic barrier in the SkX expressed by the double exchange model. Numerical results of the SkX reproduced the Dirac model obtained by linear fitting the two-dimensional band structure of the SkX.

Magnetic skyrmion crystal at a topological insulator surface

We consider a magnetic skyrmion crystal formed at the surface of a topological insulator. Incorporating the exchange interaction between the helical Dirac surface states and the periodic N\'eel or Bloch skyrmion texture, we obtain the resulting electronic band structure and discuss the constraints that symmetries impose on the energies and Berry curvature. We find substantive qualitative differences between the N\'eel and Bloch cases, with the latter generically permitting a multiband low energy tight-binding representation whose parameters are tightly constrained by symmetries. We explicitly compute the associated Wannier orbitals, which resemble the ringlike chiral bound states of helical Dirac fermions coupled to a single skyrmion in a ferromagnetic background. We construct a two-band tight-binding model with real nearest-neighbor hoppings which captures the salient topological features of the low-energy bands. Our results are relevant to magnetic topological insulators (TIs), as well as to TI-magnetic thin film heterostructures, in which skyrmion crystals may be stabilized.

Magnon band structure of skyrmion crystals and stereographic projection approach

Using semiclassical method combined with stereographic projection approach, we investigate the magnetic dynamics of the skyrmion crystal (SkX), formed in planar ferromagnet with both Dzyaloshinskii-Moriya interaction and uniform magnetic field. The topologically non-trivial ground state of SkX is described in stereographic projection by the complex valued function with simple poles at skyrmions' positions. We use the earlier proposed ansatz for this ground state function in the form of the sum of individual skyrmions. The dynamics follows from the second variation of the classical action. Numerical analysis yields the magnon band structure of tight-binding form in accordance with previously known results. There are two sets of bands, one set with a flat dispersion, topologically trivial and rapidly evolving with magnetic field. Another set is robust to magnetic field, characterized by pronounced dispersion and with the Berry curvature which may be sign-reversal in the Brillouin zone. The developed theory can be straightforwardly generalized for the analysis of magnetic dynamics in topological spin structures of other types.

Laser-Controlled Real- and Reciprocal-Space Topology in Multiferroic Insulators.

Magnetic materials in which it is possible to control the topology of their magnetic order in real space or the topology of their magnetic excitations in reciprocal space are highly sought after as platforms for alternative data storage and computing architectures. Here we show that multiferroic insulators, owing to their magnetoelectric coupling, offer a natural and advantageous way to address these two different topologies using laser fields. We demonstrate that via a delicate balance between the energy injection from a high-frequency laser and dissipation, single skyrmions-archetypical topological magnetic textures-can be set into motion with a velocity and propagation direction that can be tuned by the laser field amplitude and polarization, respectively. Moreover, we uncover an ultrafast Floquet magnonic topological phase transition in a laser-driven skyrmion crystal and we propose a new diagnostic tool to reveal it using the magnonic thermal Hall conductivity.

Mechanism of antisymmetric spin polarization in centrosymmetric multiple- Q magnets based on effective chiral bilinear and biquadratic spin cross products

We investigate how to engineer an antisymmetric spin-split band structure under spin density waves with finite ordering wave vectors in centrosymmetric systems without the relativistic spin-orbit coupling. On the basis of a perturbative analysis for the spin-charge coupled model in centrosymmetric itinerant magnets, we show that nonzero chiral-type bilinear and biquadratic spin cross products in momentum space under the magnetic orderings are related to an antisymmetric spin polarization in the electronic band structure. We apply the derived formula to the single-$Q$ cycloidal spiral and double-$Q$ noncoplanar states including the meron-antimeron and skyrmion crystals. Our results present a clue to realize a giant antisymmetric spin splitting driven by magnetic phase transitions in the centrosymmetric lattice structures without the spin-orbit coupling.

Skyrmion crystal and spiral phases in centrosymmetric bilayer magnets with staggered Dzyaloshinskii-Moriya interaction

We theoretically study a stabilization mechanism of the skyrmion crystal in centrosymmetric magnets with a bilayer structure. We show that the interplay between a layer-dependent staggered Dzyaloshinskii-Moriya interaction that arises from the absence of local inversion symmetry and the interlayer exchange interaction gives rise to a plethora of multiple-$Q$ states including the skyrmion crystal with a quantized topological number. By performing the simulated annealing for the bilayer triangular-lattice model under an external magnetic field, we demonstrate that the skyrmion forms the triangular-shaped crystals with different helicities in each layer owing to the staggered Dzyaloshinskii-Moriya interaction. Although the relative positions of the skyrmion core in each layer are different depending on the sign of the interlayer exchange interactions, the skyrmion crystal phases robustly appear under both ferromagnetic and antiferromagnetic interlayer interactions. We also find another two triple-$Q$ states with a uniform scalar chirality but without a quantized topological number in the low- and high-field regions. Especially, the low-field triple-$Q$ state exhibits the opposite sign of the scalar chirality to the skyrmion crystal, which is not found in the single-layer system. Our results indicate that the lack of local inversion symmetry in the lattice structure is another key ingredient to induce topological spin textures in centrosymmetric magnets.

Néel-type antiferromagnetic skyrmionic crystals on two-dimensional square lattices investigated with optimized quantum Monte Carlo method

The formations of individual antiferromagnetic (AF) skyrmions and AF skyrmionic lattices on two-dimensional (2D) magnets with square crystal structure are debatable in recent years, for only an isolated skyrmion can be generated in such systems if classical Monte Carlo (CMC) method is employed. For the sake, we apply here an optimized quantum Monte Carlo approach to a 2D square magnet where the AF Heisenberg exchange (HE) and Dzyaloshinskii–Moriya (DM) interactions co-exist. Consequently, the computing program converges to the equilibrium states with appreciable computational speed, and the results obtained in the last one iteration are able to accurately produce well symmetric and periodic AF skyrmionic lattices (SLs) at elevated temperatures when a considerably strong external magnetic field is exerted perpendicular to the 2D monolayer. Moreover, each of these AF SLs can be decomposed into two almost identical ferromagnetic (FM) SLs, and the distribution of topological charge density also forms symmetric lattice with the same periodicity as the AF SL, dividing the AF SL into several areas of distinct spin configurations. The reasons why the OQMC approach can work beyond CMC method are explained in the Discussion Section. • Optimized Monte Carlo simulating method is described and applied here. • Results obtained in last loop can produce symmetric and periodical spin textures. • AF skyrmion crystals are generated in 2D square lattices with the method. • Each AF skyrmion crystal can be decomposed into two identical FM sub-lattices. • The topological charge density forms lattice almost identical with the spin texture.

A theory of skyrmion crystal formation.

A generic theory about skyrmion crystal (SkX) formation in chiral magnetic thin films and its fascinating thermodynamic behaviours is presented. A chiral magnetic film can have many metastable states with an arbitrary skyrmion density up to a maximal value when the parameter κ, which measures the relative Dzyaloshinskii-Moriya interaction (DMI) strength, is large enough. The lowest energy state of an infinite film is a long zig-zag ramified stripe skyrmion occupying the whole film in the absence of a magnetic field. Under an intermediate field perpendicular to the film, the lowest energy state has a finite skyrmion density. This is why a chiral magnetic film is often in a stripy state at a low field and a SkX only around an optimal field when κ is above a critical value. The lowest energy state is still a stripy helical state no matter with or without a field when κ is below the critical value. The multi-metastable states explain the thermodynamic path dependences of the various metastable states of a film. The decrease of the κ value with the temperature explains why SkXs become metastable at low temperatures in many skyrmion systems. These findings open a new avenue for SkX manipulation and skyrmion-based applications.

Skyrmion-driven topological Hall effect in a Shastry-Sutherland magnet

The Shastry-Sutherland model and its generalizations have been shown to capture emergent complex magnetic properties from geometric frustration in several quasi-two-dimensional quantum magnets. Using an $sd$ exchange model, we show here that metallic Shastry-Sutherland magnets can exhibit a topological Hall effect driven by magnetic skyrmions under realistic conditions. The magnetic properties are modeled with competing symmetric Heisenberg and asymmetric Dzyaloshinskii-Moriya exchange interactions, while a coupling between the spins of the itinerant electrons and the localized moments describes the magnetotransport behavior. Our results, employing complementary Monte Carlo simulations and a novel machine learning analysis to investigate the magnetic phases, provide evidence for field-driven skyrmion crystal formation for an extended range of Hamiltonian parameters. By constructing an effective tight-binding model of conduction electrons coupled to the skyrmion lattice, we clearly demonstrate the appearance of the topological Hall effect. We further elaborate on the effects of finite temperatures on both magnetic and magnetotransport properties.

Locking of skyrmion cores on a centrosymmetric discrete lattice: Onsite versus offsite

A magnetic skyrmion crystal (SkX) with a swirling spin configuration, which is one of topological spin crystals as a consequence of an interference between multiple spin density waves, shows a variety of noncoplanar spin patterns depending on a way of superposing the waves. By focusing on a phase degree of freedom among the constituent waves in the SkX, we theoretically investigate a position of the skyrmion core on a discrete lattice, which is relevant with the symmetry of the SkX. The results are obtained for the double exchange (classical Kondo lattice) model on a discrete triangular lattice by the variational calculations. We find that the skyrmion cores in both two SkXs with the skyrmion number of one and two are locked at the interstitial site on the triangular lattice, while it is located at the onsite by introducing a relatively large easy-axis single-ion anisotropy. The variational parameters and the resultant Fermi surfaces in each SkX spin texture are also discussed. The different symmetry of the Fermi surfaces depending on the core position is obtained when the skyrmion crystal is commensurate with the lattice. The different Fermi-surface topology is directly distinguished by an electric probe of angle-resolved photoemission spectroscopy. Furthermore, we show that the SkXs obtained by the variational calculations are also confirmed by numerical simulations on the basis of the kernel polynomial method and the Langevin dynamics for the double exchange model and the simulated annealing for an effective spin model.

Phase shift in skyrmion crystals

The magnetic skyrmion crystal is a periodic array of a swirling topological spin texture. Since it is regarded as an interference pattern by multiple helical spin density waves, the texture changes with the relative phase shifts among the constituent waves. Although such a phase degree of freedom is relevant to not only magnetism but also transport properties, its effect has not been elucidated thus far. We here theoretically show that a phase shift in the skyrmion crystals leads to a tetra-axial vortex crystal and a meron-antimeron crystal, both of which show a staggered pattern of the scalar spin chirality and give rise to nonreciprocal transport phenomena without the spin-orbit coupling. We demonstrate that such a phase shift can be driven by exchange interactions between the localized spins, thermal fluctuations, and long-range chirality interactions in spin-charge coupled systems. Our results provide a further diversity of topological spin textures and open a new field of emergent electromagnetism by the phase shift engineering.

Controllable interskyrmion attractive interactions and resulting skyrmion-lattice structures in two-dimensional chiral magnets with in-plane anisotropy

We study inter-skyrmion interactions and stable spin configurations in a 2D chiral magnet with in-plane anisotropies of a tilted magnetic field and the magneto-crystalline anisotropy on a (011) thin film. We find that in both cases a small deformation of a skyrmion shape makes the inter-skyrmion interaction anisotropic, and that the skyrmions are weakly bounded along a certain direction due to an emergent attractive interaction. Furthermore, when the magneto-crystalline anisotropy is comparable to the Zeeman energy, skyrmions embedded in a uniform magnetization are tightly bound by creating a magnetic domain between them. The formation of the magnetic domain, and thus the strength of the inter-skyrmion interaction, can be controlled by the direction of an external magnetic field. The anisotropic interaction also affects the skyrmion alignment in the skyrmion crystal (SkX) phase. By employing the Monte Carlo simulation and the micromagnetic simulation, we obtain an elongated triangular lattice structure in the SkX phase. In particular, in the presence of a strong magneto-crystalline anisotropy, magnetic domains appear in the background of the lattice structure, and bimerons aligned on the domain walls form an elongated triangular lattice. We also find a parameter region that the SkX phase is stabilized due to the inter-skyrmion attraction.

Replica symmetry breaking in the RKKY skyrmion-crystal system

We study the RKKY Heisenberg model on a three-dimensional stacked-triangular lattice under magnetic fields by extensive Monte Carlo simulations to get insight into the chiral-degenerate symmetric skyrmion crystal (SkX) in centrosymmetric metallic magnets. The triple-$q$ SkX state and the double-$q$ states are realized, together with the single-$q$ state. We find an unexpected phenomenon of the replica-symmetry breaking (RSB) well-known in glassy systems, although the Hamiltonian and the ordered state are entirely regular. In the RSB SkX phase, the triple-$q$ SkX state macroscopically coexists with the single-$q$ state, in spite of the fact that these ordered states cannot be transformed via any Hamiltonian-symmetry operation. In the thermodynamic limit, the free energies of these states are degenerate whereas the free-energy barrier between the states diverges, breaking the ergodicity. A similar RSB is observed also in the RSB double-$q$ phase where the double-$q$ state macroscopically coexists with the single-$q$ state. Experimental implications are discussed.

Tuning scalar spin chirality in ultrathin films of the kagome-lattice ferromagnet Fe3Sn

Non-coplanar spin textures with finite scalar spin chirality can be artificially induced at surfaces and interfaces through the interfacial Dzyaloshinskii-Moriya interaction. However, stabilizing a proper magnetic skyrmion crystal via this route remains elusive. Here, using an epitaxial bilayer of platinum and geometrically frustrated kagome-lattice ferromagnet Fe3Sn, we show the possible formation of a two-dimensional skyrmion crystal under well-regulated Fe3Sn thickness conditions. Magnetization measurements reveal that the magnetic anisotropy is systematically varied from an inherent in-plane type to a perpendicular type with the thickness reduction. Below approximately 0.5 nm, we clearly detect a topological Hall effect that provides evidence for finite scalar spin chirality. Our topological Hall effect analysis, combined with theoretical simulations, not only establishes its interfacial Dzyaloshinskii-Moriya interaction origin, but also indicates the emergence of a stable skyrmion crystal phase, demonstrating the potential of kagome-lattice ferromagnets in spin chirality engineering using thin-film nanostructures.

Temperature-driven transition from skyrmion to bubble crystals in centrosymmetric itinerant magnets

Interplay between itinerant electrons and localized spins in itinerant magnets gives rise to a variety of noncoplanar multiple-Q spin textures, such as the skyrmion, hedgehog, meron, and vortex. We elucidate that another type of multiple-Q state consisting of collinear sinusoidal waves, a magnetic bubble crystal, appears at finite temperatures in a centrosymmetric itinerant electron system. The results are obtained for the classical Kondo lattice model with easy-axis single-ion anisotropy on a triangular lattice by a large-scale numerical simulation. We find that a finite-temperature topological phase transition between the skyrmion crystal and the bubble crystal occurs by changing the temperature. We obtain the minimal key ingredients for inducing the finite-temperature transition by analyzing an effective spin model where it is shown that the synergy between the multiple-spin interaction and magnetic anisotropy plays a significant role.

Resonant optical topological Hall conductivity from skyrmions

We study the high frequency Hall conductivity in a two-dimensional (2D) model of conduction electrons coupled to a background magnetic skyrmion texture via an effective Hund's coupling term. For an ordered skyrmion crystal, a Kubo formula calculation using the basis of skyrmion crystal Chern bands reveals a resonant Hall response at a frequency set by the Hund's coupling: $\hbar\omega_{\text{res}} \approx J_H$. A complementary real-space Kubo formula calculation for an isolated skyrmion in a box reveals a similar resonant Hall response. A linear relation between the area under the Hall resonant curve and the skyrmion density is discovered numerically and is further elucidated using a gradient expansion which is valid for smooth textures and a local approximation based on a spin-trimer calculation. We point out the issue of distinguishing this skyrmion contribution from a similar feature arising from spin-orbit interactions, as demonstrated in a model for Rashba spin-orbit coupled electrons in a collinear ferromagnet, which is analogous to the difficulty of unambiguously separating the d.c. topological Hall effect from the anomalous Hall effect. The resonant feature in the high frequency topological Hall effect is proposed to provide a potentially useful local optical signature of skyrmions via probes such as scanning magneto-optical Kerr microscopy.