We discuss in details the electron scattering pattern on skyrmion-like magnetic textures in two-dimensional geometry. The special attention is focused on analyzing the scattering asymmetry, which is a precursor of the topological Hall effect. We present analytical results valid for the limiting regimes of strong and weak coupling, we also describe the numerical scheme that gives access to the exact solution of the scattering problem. Based on the numerical computations we investigate the properties of the asymmetric scattering for an arbitrary magnitude of the interaction strength and the topology of a magnetic texture. We analyze in details the conditions when the topological charge of a magnetic texture is indeed essential for the appearance of the Hall response. We also describe how the electron scattering asymmetry is modified due to an additional short-range impurity located inside a magnetic skyrmion.
Theory of interference-induced quantum corrections to conductivity is developed for two dimensional systems with chiral spin textures including skyrmions. The effect of exchange interaction between electrons and spin textures on weak localization of electronic waves is studied. The spin dephasing rates are calculated as functions of the spin texture size. The anomalous magnetoresistance is shown to be governed by the size and magnetization spatial distribution of the spin textures. The effect of average magnetization-induced spin splitting on weak localization is analyzed. The sign-alternating…
Here we show that the zero point energy associated to the quantum spin fluctuations of a non- collinear spin texture produce Casimir-like magnetic fields. We study the effect of these Casimir fields on the topologically protected non-collinear spin textures known as skyrmions. We calculate the zero point energy, to the lowest order in the spin wave expansion, in a Heisenberg model with Dzyalonshinkii-Moriya interactions chosen so that the classical ground state displays skyrmion textures, that disappear upon application of a strong enough magnetic field. Our calculations show that the Casimir …
Exotic spin textures viz. chiral domain wall, vortices, skyrmion, skyrmionium, etc. have recently emerged as active field of research because of their potential applications in high density data storage technology and logic gate computing. Magnetic skyrmionium is a skyrmion like soliton, which carries zero topological quantum number. Skyrmioniums are superior to conventional skyrmions in ferromagnets due to their negligible skyrmion hall effect and higher velocity. The physical properties of both skyrmion and skyrmionium have been investigated rigorously in ferromagnetic systems. Recent observ…
We carry out a throughout study of the topological Hall and topological spin Hall effects in disordered skyrmionic systems: the dimensionless (spin) Hall angles are evaluated across the energy band structure in the multiprobe Landauer-Büttiker formalism and their link to the effective magnetic field emerging from the real space topology of the spin texture is highlighted. We discuss these results for an optimal skyrmion size and for various sizes of the sample and found that the adiabatic approximation still holds for large skyrmions as well as for few atomic size-nanoskyrmions. Finally, we te…
We introduce a new class of isolated magnetic skyrmions emerging within tilted ferromagnetic phases of polar magnets with easy-plane anisotropy. The asymmetric magnetic structure of these skyrmions is associated with an intricate pattern of the energy density, which exhibits positive and negative asymptotics with respect to the surrounding state with a ferromagnetic moment tilted away from the polar axis. Correspondingly, the skyrmion-skyrmion interaction has an anisotropic character and can be either attractive or repulsive depending on the relative orientation of the skyrmion pair. We invest…
We develop a general theory to classify magnetic skyrmions and related spin textures in terms of their magnetoelectric multipoles. Since magnetic skyrmions are now established in insulating materials, where the magnetoelectric multipoles govern the linear magnetoelectric response, our classification provides a recipe for manipulating the magnetic properties of skyrmions using applied electric fields. We apply our formalism to skyrmions and anti-skyrmions of different helicities, as well as to magnetic bimerons, which are topologically, but not geometrically, equivalent to skyrmions. We show th…
The quantum anomalous Hall effect has been theoretically predicted and experimentally verified in magnetic topological insulators. In addition, the surface states of these materials exhibit a hedgehog-like "spin" texture in momentum space. Here, we apply the previously formulated low-energy model for Bi$_2$Se$_3$, a parent compound for magnetic topological insulators, to a slab geometry in which an exchange field acts only within one of the surface layers. In this sample set up, the hedgehog transforms into a skyrmion texture beyond a critical exchange field. This critical field marks a transi…
The anomalous Hall effect is caused by magnetic textures such as skyrmions. We derive an analytical formula of the Hall conductivity on the surface of a topological insulator up to third order in magnetization, $\boldsymbol{M}(\boldsymbol{x})$, based on a perturbative approach. We identify the magnetic textures that contribute to the Hall conductivity up to third order in magnetization and second order in spatial differentiation. We treat magnetization as a perturbation to calculate the Hall conductivity for each magnetic texture based on the linear response theory. Furthermore, we estimate th…
A large enough piece of ferromagnet is usually not magnetized uniformly, but develops a magnetization texture. In thin films these textures can be doubly-periodic. Such are the well known magnetic bubble domains and the recently observed "skyrmion" magnetization textures in MnSi. In this paper we develop a theory of periodic magnetization textures, based on complex calculus to answer the question -- is there a difference between those two textures even if they seem to carry the same topological winding number (or topological charge) ? We find that such difference exists, facilitated by a diffe…
The Reservoir Computing (RC) paradigm posits that sufficiently complex physical systems can be used to massively simplify pattern recognition tasks and nonlinear signal prediction. This work demonstrates how random topological magnetic textures present sufficiently complex resistance responses for the implementation of RC as applied to A/C current pulses. In doing so, we stress how the applicability of this paradigm hinges on very general dynamical properties which are satisfied by a large class of physical systems where complexity can be put to computational use. By harnessing the complex res…
Magnetic skyrmions are nontrivial spin textures which resist external perturbations, being promising candidates for the next generation recording devices. Nevertheless, a major challenge in realizing skyrmion-based devices is the stabilization of ordered arrays of these spin textures under ambient conditions and zero applied field. Here, we demonstrate for the first time the formation and stabilization of magnetic skyrmions on arrays of self-assembled hexagonal nanodomes taking advantage of the intrinsic properties of its curved geometry. Magnetic force microscopy images from the arrays of 100…
Ever increasing demand of skyrmion manipulation in nanodevices has brought up interesting research to 8 understand the stabilization of these topologically protected chiral structures. To understand the actual shape 9 and size of skyrmion observed experimentally, we have performed micromagnetic simulations to investigate 10 skyrmion stabilization in presence of random anisotropy in magnetic thin film system. Previous experimental 11 reports of skyrmion imaging in thin films depicts that the skyrmion shape is not perfectly circular. Here we 12 show via simulations that the shape of a skyrmion c…
In non-centrosymmetric magnets the chiral Dzyaloshinskii-Moriya exchange stabilizes Skyrmion-strings as excitations which may condense into multiply modulated phases. Such extended Skyrmionic textures are determined by the stability of the localized "solitonic" Skyrmion cores and their geometrical incompatibility which frustrates regular space-filling. We present numerically exact solutions for Skyrmion lattices and formulate basic properties of the Skyrmionic states.
Skyrmions are topological solitons that emerge in many physical contexts. In magnetism, they appear as textures of the spin-density field stabilized by different competing interactions and characterized by a topological charge that counts the number of times the order parameter wraps the sphere. They can behave as classical objects, when the spin texture varies slowly on the scale of the microscopic lattice of the magnet. However, the fast development of experimental tools to create and stabilize skyrmions in thin magnetic films has lead to a rich variety of textures, sometimes of atomistic si…
Recently proposed spintronic devices use magnetic skyrmions as bits of information. The reliable detection of those chiral magnetic objects is an indispensable requirement. Yet, the high mobility of magnetic skyrmions leads to their stochastic motion at finite temperatures, which hinders the precise measurement of the topological numbers. Here, we demonstrate the successful training of artificial neural networks to reconstruct the skyrmion number in confined geometries from time-integrated, dimensionally reduced data. Our results prove the possibility to recover the topological charge from a t…
Electrons moving through a noncoplanar magnetic texture acquire a Berry phase, which can be described as an effective magnetic field. This effect is known as the topological Hall effect and has been observed in topological spin textures. Motivated by recent experimental realizations, here we study the Hall effect in a nontopological magnetic texture known as a target skyrmion. We start from a simplified semiclassical picture and show that the Hall signal is a nonmonotonic function of both the electronic energy and target skyrmion radius. That observation carries over to the fully quantum mecha…
A magnetic skyrmion isusually refers to a twisted spin texture surrounded by uniformly aligned out-of-plane spinsin the background of a uniformly magnetized state. The invariance of the magnetic skyrmion conserves its topological charge under any continuous transformations of the spin textures, leads to which represents the robustness of a magnetic skyrmion their texture against external perturbations, making it ideal to use skyrmions as . Such a behaviour is required for an ideal information carriers. To date, most magnetic skyrmion studies have been performedfocused on in perpendicularly mag…
We report that in cylindrical nanorods skyrmion chains, i.e., three-dimensional spin textures, emerge dynamically, as revealed by micromagnetic simulations. The skyrmion-chain state occurs when the diameter of the rod is larger than the helical pitch length of the material, and the number of skyrmions on the chain is proportional to the length of the nanorod. This finding provides a deeper understanding of the interplay between geometry and skyrmionic symmetry, and shows how spatial confinement can stabilize spontaneous topological spin textures.
We propose that topological spin currents flowing in topologically non-trivial magnetic textures, such as magnetic skyrmions and vortices, produce an intrinsic non-adiabatic torque of the form ${\bf T}_t\sim [(\partial_x{\bf m}\times\partial_y{\bf m})\cdot{\bf m}]\partial_y{\bf m}$. We show that this torque, which is absent in one-dimensional domain walls and/or non-topological textures, is responsible for the enhanced non-adiabaticity parameter observed in magnetic vortices compared to one-dimensional textures. The impact of this torque on the motion of magnetic skyrmions is expected to be cr…
We study skyrmions in magnetic thin films with structural inversion asymmetry perpendicular to the film plane. We determine the magnetization texture of a single skyrmion and its dependence on the strength of the Dzyaloshinskii-Moriya interaction relative to the magnetostatic energy. Furthermore, we construct a phenomenological model that describes the interaction between the motion of skyrmions and electric currents to lowest order in spin-orbit coupling. We estimate the experimental verifiable velocities for current-driven motion of skyrmion textures based on available results obtained from …
Electron spin-momentum locking together with background magnetic textures can significantly alter the electron transport properties. We theoretically investigate the electron transport at the interface between a topological insulator and a magnetic insulator with magnetic skyrmions on the top. In contrast to the conventional topological Hall effect in normal metals, the skyrmions yield an additional contribution to the anomalous Hall conductivity even in the absence of in-plane magnetic texture, arising from the phase factor characteristic to Dirac electrons acquired at skyrmion boundary.
In multiferroic materials, the microscopic magnetic and electric textures are strongly correlated with each other by magnetoelectric (ME) coupling. Therefore, topological electric dipole textures, such as ferroelectric (FE) dipole skyrmions and periodic FE dipole crystals, are expected to be induced if ferromagnetic (FM) skyrmions and FM skyrmionic crystals (or lattices so as to be abbreviated as SLs for convenience) can be stabilized. In the present work, a quantum computational approach is utilized to simulate the topological textures of FE SLs in a two-dimensional (2D) multiferroic system. …
We provide a derivation of a spin Skyrmion number classification for two-dimensional topological superconductors constructed from ferromagnetic and Rashba spin-orbit coupled semiconductor-superconductor heterostructures. We show that in the non-trivial topological phase, characterized by a non-zero Chern number, there is always a topological spin texture in the occupied bands represented by a Skyrmion number. The Skyrmion number has the advantage of being both physically intuitive and directly measurable using spin-sensitive band structure imaging techniques. In addition, we show that the Skyr…
Magnetic skyrmions have the potential to provide solutions for low-power, high-density data storage and processing. One of the major challenges in developing skyrmion-based devices is the skyrmions' magnetic stability in confined helimagnetic nanostructures. Through a systematic study of equilibrium states, using a full three-dimensional micromagnetic model including demagnetisation effects, we demonstrate that skyrmionic textures are the lowest energy states in helimagnetic thin film nanostructures at zero external magnetic field and in absence of magnetocrystalline anisotropy. We also report…
We report on analytic and numerical studies of spin textures in quantum Hall systems using a long-wavelength effective action for the magnetic degrees of freedom derived previously. The majority of our results concern skyrmions or solitons of this action. We have constructed approximate analytic solutions for skyrmions of arbitrary topological and electric charge and derived expressions for their energies and charge and spin radii. We describe a combined shooting/relaxational technique for numerical determination of the skyrmion profiles and present results that compare favorably with the anal…
Symmetry breaking together with strong spin-orbit interaction give rise to many exciting phenomena within condensed matter physics. A recent example is the existence of chiral spin textures, which are observed in magnetic systems lacking inversion symmetry. These chiral spin textures, including domain walls and magnetic skyrmions, are both fundamentally interesting and technologically promising. For example, they can be driven very efficiently by electrical currents, and exhibit many new physical properties determined by their real-space topological characteristics. Depending on the details of…
Topological spin textures in an itinerant ferromagnet, SrRuO$_3$ is studied combining Hall transport measurements and numerical simulations. We observe characteristic signatures of the Topological Hall Effect associated with skyrmions. A relatively large thickness of our films and absence of heavy metal layers make the interfacial Dzyaloshinskii-Moriya interaction an unlikely source of these topological spin textures. Additionally, the transport anomalies exhibit an unprecedented robustness to magnetic field tilting and temperature. Our numerical simulations suggest that this unconventional be…
An ideal layered $\hat{x}$-polarized antiferromagnet (AFM) between two antialigned $\pm \hat{z}$ polarized ferromagnetic (FM) contacts transmits no current due to a $π$ phase difference of the matrix elements coupling the spin degenerate states to the two FM contacts. The ratio of the two matrix elements coupling the two spin degenerate $\hat{x}$ AFM states to a $\hat{z}$ FM contact is determined by the exchange energy and the eigenstate energy. Inserting a normal metal layer or tunnel barrier layer between one FM contact and the AFM alters this phase difference, and, due to the unequal weight…
The helimagnet (HM) and skyrmion lattice (SL) are topologically nontrivial magnetic states. Their spin texture gives rise to finite topological magnetic field and Lorentz force. As a demonstration of the emergent electrodynamics besides the Hall effect, the transmission of electrons within a waveguide (WG) embedded with a HM/SL layer is shaped by the topological spin texture. In this work, we investigated the conductance properties in the HM/SL-film-embedded electron WG and found that under translation of the HM/SL layer the conductance contour as a function of the layer center demonstrates ne…
We consider the interplay between magnetic skyrmions in an insulating thin film and the Dirac surface states of a 3D topological insulator (TI), coupled by proximity effect. The magnetic texture of skyrmions can lead to confinement of Dirac states at the skyrmion radius, where out of plane magnetization vanishes. This confinement can result in charging of the skyrmion texture. The presence of bound states is robust in an external magnetic field, which is needed to stabilize skyrmions. It is expected that for relevant experimental parameters skyrmions will have a few bound states that can be tu…
Planar Interfaces for Transmission of Chiral Spin Textures
Theoretical Division, Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室理论部）；Center for Memory and Recording Research, University of California, San Diego（加州大学圣地亚哥分校记忆与记录研究中心）
Magnetic skyrmions are two-dimensional non-collinear spin textures characterised by an integer topological number. They commonly crystallise at low temperatures in bulk noncentrosymmetric ferromagnets where the lack of inversion symmetry leads to an antisymmetric component of the exchange interaction. Recently, stable room-temperature skyrmions were reported in stacks of thin magnetic films where antisymmetric exchange results from broken symmetry at the interface. Determining the spin structure of these technologically-relevant skyrmions in the presence of external magnetic fields has remaine…
We investigate the impact of the local spin texture on the differential conductance by scanning tunneling microscopy. In the focus is the previously found non-collinear magnetoresistance, which originates from spin mixing effects upon electron hopping between adjacent sites with canted magnetic moments. In the present work it is studied with lateral resolution both for the zero magnetic field spin spiral state as well as for individual magnetic skyrmions at different magnetic field values. We analyze in detail the response of the differential conductance and find different dependencies of peak…
A type of vortex spin texture with skyrmionic core and a series of circle spin stripes was obtained in thin nanodisk of the chiral magnets with Dzyaloshinskii-Moriya interaction and uniaxial anisotropy by means of micromagnetic approach and Monte Carlo simulation. The size of skyrmionic core can be modulated continuously by controlling the disk size. Moreover, in some certain values of the disk size, this vortex state may be spontaneous ground state even without the help of the external magnetic field and thermal fluctuation. In addition, the uniaxial anisotropy is able to stabilize this vorte…
We investigate an instability toward a square-lattice formation of magnetic skyrmions in centrosymmetric layered systems. By focusing on a bilayer square-lattice structure with the inversion center at the interlayer bond instead of the atomic site, we numerically examine the stability of the square skyrmion crystal based on an effective spin model with the momentum-resolved interaction in the ground state through the simulated annealing. As a result, we find that a layer-dependent staggered Dzyaloshinskii-Moriya interaction built in the lattice structure becomes the origin of the square skyrmi…
We study the polarization dynamics of a spatially expanding polariton condensate under nonresonant linearly polarized optical excitation. The spatially and temporally resolved polariton emission reveals the formation of non-trivial spin textures in the form of a quadruplet polarization pattern both in the linear and circular Stokes parameters, and an octuplet in the diagonal Stokes parameter. The continuous rotation of the polariton pseudospin vector through the condensate due to TE-TM splitting exhibits an ordered pattern of half-skyrmions associated with a half-integer topological number. A …
Quantization of topological charges determines the various topological spin textures that are expected to play a key role in future spintronic devices. While the magnetic skyrmion with a unit topological charge Q has been extensively studied, spin textures with other integer valued have not been verified well so far. Here, we report the real-space image, creation, and manipulation of a type of multi Q three-dimensional skyrmionic texture, where a circular spin spiral ties a bunch of skyrmion tubes. We define these objects as skyrmion bundles, and show they have arbitrarily integer values Q fro…
We present a consistent theory of the topological Hall effect (THE) in 2D magnetic systems with disordered array of chiral spin textures, such as magnetic skyrmions. We focus on the scattering regime when the mean-free path of itinerant electrons exceeds the spin texture size, and THE arises from the asymmetric carrier scattering on individual chiral spin textures. We calculate the resistivity tensor on the basis of the Boltzmann kinetic equation taking into account the asymmetric scattering on skyrmions via the collision integral. Our theory describes both the adiabatic regime, when THE arise…
The intrinsic magnetism observed in two-dimensional (2D) van der Waals (vdW) materials provides a unique opportunity for exploring the 2D topological magnetic textures, in particular skyrmionic magnetic textures (SMTs) including skyrmion and its topological equivalents. Since the experimental discovery of skyrmions in the 2D vdW materials and their heterostructures, a critical challenge lies in the control of these SMTs to translate their intriguing features into spintronic applications. Here, we review the recent experimental and theoretical progresses on the modulations of SMTs in 2D vdW mon…
In cubic noncentrosymmetric ferromagnets uniaxial distortions suppress the helical states and stabilize Skyrmion lattices in a broad range of thermodynamical parameters. Using a phenomenological theory for modulated and localized states in chiral magnets, the equilibrium parameters of the Skyrmion and helical states are derived as functions of the applied magnetic field and induced uniaxial anisotropy. These results show that due to a combined effect of induced uniaxial anisotropy and an applied magnetic field Skyrmion lattices can be formed as thermodynamically stable states in large interval…
The recent experimental confirmation of a transformation between meron and skyrmion topological spin textures in the chiral magnet Co$_8$Zn$_9$Mn$_3$ [S.-Z. Lin et al., Phys. Rev. B 91, 224407 (2015); X. Z. Yu et al., Nature 564, 95 (2018)] confirms that the skyrmion crystals discovered in 2009 [S. Mühlbauer et al., Science 323, 915 (2009)] are just the tip of the iceberg. Crystals of topological textures, including skyrmions, merons, vortices, and monopoles, can be stabilized by combining simple physical ingredients, such as lattice symmetry, frustration, and spin anisotropy. The current chal…
Skyrmions are nanoscale spin configurations with topological properties that hold great promise for spintronic devices. Here, we establish their Néel texture, helicity, and size in Ir/Fe/Co/Pt multilayer films by constructing a multipole expansion to model their stray field signatures and applying it to magnetic force microscopy (MFM) images. Furthermore, the demonstrated sensitivity to inhomogeneity in skyrmion properties, coupled with a unique capability to estimate the pinning force governing dynamics, portends broad applicability in the burgeoning field of topological spin textures.
When a foreign atom is placed on a surface of a metal, the surrounding sea of electrons responds screening the additional charge leading to oscillations or ripples. On surfaces, those electrons are sometimes confined to two-dimensional surface states, whose spin-degeneracy is lifted due to the Rashba effect arising from the spin-orbit interaction of electrons and the inversion asymmetric environment. It is believed that at least for a single adatom scanning tunneling microscopy measurements are insensitive to the Rashba splitting i.e. no signatures in the charge oscillations will be observed. …
Integrating topologically stabilized magnetic textures such as skyrmions as nanoscale information carriers into future technologies requires the reliable control by electric currents. Here, we uncover that the relevant skyrmion Hall effect, which describes the deflection of moving skyrmions from the current flow direction, acquires important corrections owing to anisotropic spin-orbit torques that alter the dynamics of topological spin structures. Thereby, we propose a viable means for manipulating the current-induced motion of skyrmions and antiskyrmions. Based on these insights, we demonstra…
High-resolution tunneling electron spin transport properties (longitudinal spin current (LSC) and spin transfer torque (STT) maps) of topologically distinct real-space magnetic skyrmionic textures are reported by employing a 3D-WKB combined scalar charge and vector spin transport theory in the framework of spin-polarized scanning tunneling microscopy (SP-STM). For our theoretical investigation metastable skyrmionic spin structures with various topological charges ($Q=-3,-2,-1,0,1,2$) in the (Pt$_{0.95}$Ir$_{0.05}$)/Fe/Pd(111) ultrathin magnetic film are considered. Using an out-of-plane magnet…
Increasing amounts of information force the continuous improvement of information storage and processing technologies, further device miniaturization, and their efficiency increase. Magnetic skyrmions, topological quasiparticles, and the smallest stable magnetic textures possess intriguing properties and potential for data storage applications. Hybrid nanostructures with elements of different magnetization orientations can offer additional advantages for developing skyrmion-based spintronic and magnonic devices. We show that an Néel-type skyrmion confined within a nanodot placed on top of a fe…
