Two-dimensional Chiral Magnets Research Articles

Non-Lifshitz invariants corrections to Dzyaloshinskii-Moriya interaction energy

We study the continuum limit of two-dimensional chiral magnets in which Dzyaloshinskii-Moriya interaction (DMI) is due to the interplay between a smooth magnetic texture and spin-orbit coupling. The resulting free-energy density of the system contains linear terms in the spatial gradient of the magnetic texture, which mark an instability of the system towards the formation of nontrivial magnetic orders such as skyrmions or chiral domain walls. We perform a microscopic analysis of DMI tensors responsible for this contribution to free energy based on a Berry phase formulation in the mixed space of momentum and position, and reveal that they exhibit non-Lifshitz invariants features. In particular, a perturbation theory shows in the case of Rashba spin-orbit interactions the presence of non-Lifshitz invariants to third order in the small spin-orbit interaction and fourth order in the small exchange coupling. The higher-order terms may even lead to an enhancement of DMI interaction at strong spin-orbit coupling due to divergences in the density of states at the bottom of the conduction band. Finally, we also study the DMI free energy generated from Rashba spin-orbit interaction in different symmetry groups.

Skyrmion and meron ordering in quasi-two-dimensional chiral magnets

We revisit the well-known theoretical phase diagram for a two-dimensional chiral magnet with an easy-axis uniaxial anisotropy and an applied magnetic field. A particular emphasis is given to merons---quanta of the phase transition between the cycloid and the hexagonal skyrmion lattice, which possess fractional topological charges $\ifmmode\pm\else\textpm\fi{}1/2$. A localized state of two coupled merons with opposite topological charges exhibits the nonreciprocal current-driven dynamics: it experiences no skyrmion Hall effect, but due to the asymmetry of the attracting intermeron potential gives rise to asymmetric current-velocity characteristics. We also address the problem of skyrmion ordering into hexagonal and square skyrmion crystals. In an entire field range, the square lattice turns out a saddle point state between differently oriented hexagonal lattices. Merons separating elongated skyrmions were also shown to form a local energy minimum by forming a hexagonal lattice shadowing the ordinary lattice of skyrmions. In addition, we focus on the anisotropy-driven transformations of modulated phases and their isolated counterparts, which exist as detached branches of solutions on both sides from the critical anisotropy value. Moreover, a family of isolated skyrmions with both rotational senses can be found near this anisotropy value. Our results are useful from both the fundamental point of view, since they complement previous findings, and from the point of view of applications, since they provide guidelines for developing meron-based spintronic devices.

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.

Deep learning approach for image classification of magnetic phases in chiral magnets

Chiral magnets have been vastly studied in the last years due to their potential applications as information carrying devices. In this work, it was studied the influence of the magnetocrystalline anisotropy term on the magnetic properties of two-dimensional chiral magnets, by means of Monte Carlo simulations. On the obtained magnetic arrangements, a Deep Learning algorithm based on computer vision techniques, was used for the phase recognition in each sample and their later classification. In this way, the low-temperature phase diagrams were built for a given set of anisotropy constants. It was observed that the skyrmion phase can be stabilized by tuning the magnetocrystalline anisotropy constant; in particular, the skyrmion region in the low-temperature magnetic phase diagrams can be increased by using an easy-plane anisotropy term. The machine learning algorithm described in this work can be easily extended to real-world applications, in the analysis and classification of magnetic force or Lorentz transmission electron microscopy images.

SU(4) description of bilayer skyrmion-antiskyrmion pairs

The antiferromagnetic coupling and entanglement between skyrmion lattices are treated in magnetic bilayer systems. We first formulate the problem of large bilayer skyrmions using the -theory. We have considered bilayer skyrmions under the presence of Dzyaloshinskii-Moriya (DMI) and Zeeman interactions confined in a two-dimensional chiral magnet such as Fe0.5Co0.5Si. We parametrize bilayer skyrmions using the SU(4) representation, and represent each skyrmion and antiskyrmion using the Schmidt decomposition. The reduced density matrices for skyrmion and antiskyrmion have been calculated. The conditions for maximal, partial entanglement and separable bilayer skyrmions are presented. Our results can be used for generating entanglement in systems with a large number of spins.

Solid-liquid transition of skyrmions in a two-dimensional chiral magnet

We study the melting of skyrmions in a two-dimensional Heisenberg chiral magnet with bi-axial Dzyaloshinskii--Moriya interactions. These topological excitations may form at zero temperature a triangular crystal with long-range positional order. However, we show using large-scale Monte Carlo simulations that at small finite temperature, the skyrmions rather form a typical two-dimensional solid: Positional correlations decay with distance as power laws while the orientational correlations remain finite. At higher temperature, we observe a direct transition from this two-dimensional solid to a liquid with short-range correlations. This differs from generic two-dimensional homogeneous particle systems, where a hexatic phase is realized between the solid and the liquid.

Effects of second neighbor interactions on skyrmion lattices in chiral magnets

In this paper we investigate the influences of the second neighbor interactions on a skyrmion lattice in two-dimensional chiral magnets. Such a system contains the exchange and the Dzyaloshinskii–Moriya for the spin interactions and therefore, we analyse three situations: firstly, the second neighbor interaction is present only in the exchange coupling; secondly, it is present only in the Dzyaloshinskii–Moriya coupling. Finally, the second neighbor interactions are present in both exchange and Dzyaloshinskii–Moriya couplings. We show that such effects cause important modifications to the helical and skyrmion phases when an external magnetic field is applied.

Enhanced Stability of the Magnetic Skyrmion Lattice Phase under a Tilted Magnetic Field in a Two-Dimensional Chiral Magnet.

The magnetic skyrmion is a topologically stable vortex-like spin texture that offers great promise as information carriers for future spintronic devices. In a two-dimensional chiral magnet, it was generally considered that a tilted magnetic field is harmful to its formation and stability. Here we investigated the angular-dependent stability of magnetic skyrmions in FeGe nanosheets by using high-resolution Lorentz transmission electron microscopy (Lorentz TEM). Besides the theoretically predicted destruction of skyrmion lattice state by an oblique magnetic field as the temperature closes to its magnetic Curie temperature Tc ∼ 278 K, we also observed an unexpected reentry-like phenomenon at the moderate temperatures near the border between conical and skyrmion phase, Tt ∼ 240 K. This behavior is completely beyond the theoretical prediction in a conventional two-dimensional (2D) system. Instead, a three-dimensional (3D) model involving the competition between conical phase and skyrmions is likely to play a crucial role.

Quantum skyrmions in two-dimensional chiral magnets

We study the quantum mechanics of magnetic skyrmions in the vicinity of the skyrmion-crystal to ferromagnet phase boundary in two-dimensional magnets. We show that the skyrmion excitation has an energy dispersion that splits into multiple bands due to the combination of magnus force and the underlying lattice. Condensation of the skyrmions can give rise to an intermediate phase between the skyrmion crystal and ferromagnet: a quantum liquid, in which skyrmions are not spatially localized. We show that the critical behavior depends on the spin size $S$ and the topological number of the skyrmion. Experimental signatures of quantum skyrmions in inelastic neutron-scattering measurements are also discussed.

Scattering of high-energy magnons off a magnetic skyrmion

We discuss the scattering of high-energy magnons off a single magnetic skyrmion within the field-polarized ground state of a two-dimensional chiral magnet. For wavevectors larger than the inverse skyrmion radius, krs ≫ 1 the magnon scattering is dominated by an emerging magnetic field whose flux density is essentially determined by the topological charge density of the skyrmion texture. This leads to skew and rainbow scattering characterized by an asymmetric and oscillating differential cross section. We demonstrate that the transversal momentum transfer to the skyrmion is universal due to the quantization of the total emerging flux while the longitudinal momentum transfer is negligible in the high-energy limit. This results in a magnon-driven skyrmion motion approximately antiparallel to the incoming magnon current and a universal relation between current and skyrmion-velocity.

Angular dependence of the spin textures in two-dimensional chiral magnets

The angular dependence of spin textures in thin helimagnetic films is investigated by a Monte Carlo simulation. When an external field is applied at an angle relative to the film normal, we find that the skyrmion states with broken axis-symmetric structure are able to persist over a wide range of angles by changing the spin orientation. In addition, the uniaxial anisotropy is able to stabilize the distorted skyrmions. This behavior reflects the robust topological stability of skyrmion states in helimagnets and favors their application in spintronic devices.

Enhanced Stability of Skyrmions in Two-Dimensional Chiral Magnets with Rashba Spin-Orbit Coupling

Recent developments have led to an explosion of activity on skyrmions in three-dimensional (3D) chiral magnets. Experiments have directly probed these topological spin textures, revealed their nontrivial properties, and led to suggestions for novel applications. However, in 3D the skyrmion crystal phase is observed only in a narrow region of the temperature-field phase diagram. We show here, using a general analysis based on symmetry, that skyrmions are much more readily stabilized in two-dimensional (2D) systems with Rashba spin-orbit coupling. This enhanced stability arises from the competition between field and easy-plane magnetic anisotropy and results in a nontrivial structure in the topological charge density in the core of the skyrmions. We further show that, in a variety of microscopic models for magnetic exchange, the required easy-plane anisotropy naturally arises from the same spin-orbit coupling that is responsible for the chiral Dzyaloshinskii-Moriya interactions. Our results are of particular interest for 2D materials like thin films, surfaces, and oxide interfaces, where broken surface-inversion symmetry and Rashba spin-orbit coupling naturally lead to chiral exchange and easy-plane compass anisotropy. Our theory gives a clear direction for experimental studies of 2D magnetic materials to stabilize skyrmions over a large range of magnetic fields down to T=0.

Skyrmion lattice in a two-dimensional chiral magnet

We develop a theory of the magnetic field-induced formation of Skyrmion crystal state in chiral magnets in two spatial dimensions, motivated by the recent discovery of the Skyrmionic phase of magnetization in thin film of ${\text{Fe}}_{0.5}{\text{Co}}_{0.5}\text{Si}$ and in the A phase of MnSi. Ginzburg-Landau functional of the chiral magnet rewritten in the ${\text{CP}}^{1}$ representation is shown to be a convenient framework for the analysis of the Skyrmion states. Phase diagram of the model at zero temperature gives a sequence of ground states, helical $\text{spin}\ensuremath{\rightarrow}\text{Skyrme}$ $\text{crystal}\ensuremath{\rightarrow}\text{ferromagnet}$, as the external field $B$ increases, in good accord with the thin-film experiment. In close analogy with Abrikosov's derivation of the vortex lattice solution in type-II superconductor, the ${\text{CP}}^{1}$ mean-field equation is solved and shown to reproduce the Skyrmion crystal state.

Quantized spin currents in two-dimensional chiral magnets.

Two-dimensional quantum magnets in which reflection and time-reversal symmetries are broken may be classified by integers which reflect the topology of the ground state. An analogy with the quantum Hall effect exists: these integers are Chern numbers which describe a quantized spin current response to an inhomogeneous external magnetic field. Implications for spontaneous chiral symmetry breaking are discussed.