Skyrmion Lattice Research Articles

Multiple low-temperature skyrmionic states in a bulk chiral magnet

Magnetic skyrmions are topologically protected nanoscale spin textures with particle-like properties. In bulk cubic helimagnets, they appear under applied magnetic fields and condense spontaneously into a lattice in a narrow region of the phase diagram just below the magnetic ordering temperature, the so-called A-phase. Theory, however, predicts skyrmions to be locally stable in a wide range of magnetic fields and temperatures. Our neutron diffraction measurements reveal the formation of skyrmion states in large areas of the magnetic phase diagram, from the lowest temperatures up to the A-phase. We show that nascent and disappearing spiral states near critical lines catalyze topological charge changing processes, leading to the formation and destruction of skyrmionic states at low temperatures, which are thermodynamically stable or metastable depending on the orientation and strength of the magnetic field. Skyrmions are surprisingly resilient to high magnetic fields: the memory of skyrmion lattice states persists in the field polarized state, even when the skyrmion lattice signal has disappeared. These findings highlight the paramount role of magnetic anisotropies in stabilizing skyrmionic states and open up new routes for manipulating these quasi-particles towards energy-efficient spintronics applications.

A sum rule of uniaxial anisotropy and external magnetic field for formation of Néel-type skyrmion lattices in two-dimensional ferromagnets

It is generally believed that the perpendicular magnetic anisotropy (PMA) plays an important role in stabilizing skyrmion lattices (SkL) in two-dimensional (2D) magnetic systems in which both Heisenberg exchange and Dzyaloshinskii–Moriya interactions co-exist, and the skyrmion sizes in SkLs are mainly determined by the strengths of these two intrinsic interactions. To investigate the details, we employ here a quantum computational approach we develop in recent years to simulate the Néel-type skyrmion lattices formed on a 2D PdFe/Ir(1 1 1)-like film. From our simulated results, we find that: within an external magnetic field applied normal to the film plane, the PMA is indeed able to help induce Néel-type SkLs in a wider field range; however, to stabilize the SkLs, the PMA cannot be too strong, the strengths of the external magnetic field and the maximal PMA must satisfy a sum rule since the effective perpendicular magnetic field generated by these two interactions cannot exceed a largest value. We also notice that the periodical boundary condition imposed on the FM system in simulations is able to facilitate SkL formations, and it can also modify the skyrmion size in a certain extend.

Optically Driven Collective Spin Excitations and Magnetization Dynamics in the Néel-type Skyrmion Host GaV_{4}S_{8}.

GaV_{4}S_{8} is a multiferroic semiconductor hosting magnetic cycloid (Cyc) and Néel-type skyrmion lattice (SkL) phases with a broad region of thermal and magnetic stability. Here, we use time-resolved magneto-optical Kerr spectroscopy to show the coherent generation of collective spin excitations in the Cyc and SkL phases. Our micromagnetic simulations reveal that these are driven by an optically induced modulation of uniaxial anisotropy. Our results shed light on spin dynamics in anisotropic materials hosting skyrmions and pave a new pathway for the optical manipulation of their magnetic order.

Nonlinear transport, dynamic ordering, and clustering for driven skyrmions on random pinning

Using numerical simulations, we examine the nonlinear dynamics of skyrmions driven over random pinning. For weak pinning, the skyrmions depin elastically, retaining sixfold ordering; however, at the onset of motion there is a dip in the magnitude of the structure factor peaks due to a decrease in positional ordering, indicating that the depinning transition can be detected using the structure factor even within the elastic depinning regime. At higher drives the moving skyrmion lattice regains full ordering. For increasing pinning strength, we find a transition from elastic to plastic depinning that is accompanied by a sharp increase in the depinning threshold due to the proliferation of topological defects, similar to the peak effect found at the elastic to plastic depinning transition in superconducting vortex systems. For strong pinning and strong Magnus force, the skyrmions in the moving phase can form a strongly clustered or phase separated state with highly modulated skyrmion density, similar to that recently observed in continuum-based simulations for strong disorder. As the Magnus force is decreased, the density phase separated state crosses over to a dynamically phase separated state with uniform density but with flow localized in bands of motion, while in the strongly damped limit, both types of phase separated states are lost. In the strong pinning limit, we find highly nonlinear velocity-force curves in the transverse and longitudinal directions, along with distinct regions of negative differential conductivity in the plastic flow regime. The negative differential conductivity is absent in the overdamped limit. The Magnus force is responsible for both the negative differential conductivity and the clustering effects, since it causes faster moving skyrmions to partially rotate around slower moving or pinned skyrmions.

Erratum: Edge states in a two-dimensional honeycomb lattice of massive magnetic skyrmions [Phys. Rev. B 98 , 180407(R) (2018)

Received 23 February 2019DOI:https://doi.org/10.1103/PhysRevB.99.099901©2019 American Physical SocietyPhysics Subject Headings (PhySH)Research AreasChiralityEdge statesSkyrmionsTopological insulatorsTechniquesLandau-Lifschitz-Gilbert equationMicromagnetic modelingCondensed Matter, Materials & Applied Physics

Microscopic theory of electric polarization induced by skyrmionic order in GaV4S8

The lacunar spinel GaV$_{4}$S$_{8}$ was recently suggested to be a prototype multiferroic material hosting skyrmion lattice states with a sizeable polarization $\boldsymbol{P}$ coupled to magnetic order. We explain this phenomenon on the microscopic level. On the basis of density functional theory, we construct an effective model describing the behavior of magnetically active electrons in a weakly coupled lattice formed by molecular orbitals of the (V$_{4}$S$_{4}$)$^{5+}$ clusters. By applying superexchange theory combined with the Berry-phase theory for $\boldsymbol{P}$, we derive a compass model relating the energy and polarization change with the directions of spins $\boldsymbol{e}_{i}$ in magnetic bonds. We argue that, although each skyrmion layer is mainly formed by superexchange interactions in the same plane, the spin-dependence of $\boldsymbol{P}$ arises from the stacking misalignment of such planes in the perpendicular direction, which is inherent to the lacunar spinel structure. We predict a strong competition of isotropic, $\sim \boldsymbol{e}_{i}\boldsymbol{e}_{j}$, and antisymmetric, $\sim \boldsymbol{e}_{i} \times \boldsymbol{e}_{j}$, contributions to $\boldsymbol{P}$ that explains the experimentally observed effect.

Interaction of Skyrmions and Pearl Vortices in Superconductor-Chiral Ferromagnet Heterostructures.

We investigate a hybrid heterostructure with magnetic skyrmions (Sk) inside a chiral ferromagnet interfaced by a thin superconducting film via an insulating barrier. The barrier prevents electronic transport between the superconductor and the chiral magnet, such that the coupling can occur only through the magnetic fields generated by these materials. We find that Pearl vortices (PV) are generated spontaneously in the superconductor within the skyrmion radius, while anti-Pearl vortices (PV[over ¯]) compensating the magnetic moment of the Pearl vortices are generated outside of the Sk radius, forming an energetically stable topological hybrid structure. Finally, we analyze the interplay of skyrmion and vortex lattices and their mutual feedback on each other. In particular, we argue that the size of the skyrmions will be greatly affected by the presence of the vortices, offering another prospect of manipulating the skyrmionic size by the proximity to a superconductor.

Topological transitions among skyrmion- and hedgehog-lattice states in cubic chiral magnets

Manipulating topological spin textures is a key for exploring unprecedented emergent electromagnetic phenomena. Whereas switching control of magnetic skyrmions, e.g., the transitions between a skyrmion-lattice phase and conventional magnetic orders, is intensively studied towards development of future memory device concepts, transitions among spin textures with different topological orders remain largely unexplored. Here we develop a series of chiral magnets MnSi1−xGex, serving as a platform for transitions among skyrmion- and hedgehog-lattice states. By neutron scattering, Lorentz transmission electron microscopy and high-field transport measurements, we observe three different topological spin textures with variation of the lattice constant controlled by Si/Ge substitution: two-dimensional skyrmion lattice in x = 0–0.25 and two distinct three-dimensional hedgehog lattices in x = 0.3–0.6 and x = 0.7–1. The emergence of various topological spin states in the chemical-pressure-controlled materials suggests a new route for direct manipulation of the spin-texture topology by facile mechanical methods.

Topological Insulators in Twisted Transition Metal Dichalcogenide Homobilayers

We show that moiré bands of twisted homobilayers can be topologically nontrivial, and illustrate the tendency by studying valence band states in ±K valleys of twisted bilayer transition metal dichalcogenides, in particular, bilayer MoTe_{2}. Because of the large spin-orbit splitting at the monolayer valence band maxima, the low energy valence states of the twisted bilayer MoTe_{2} at the +K (-K) valley can be described using a two-band model with a layer-pseudospin magnetic field Δ(r) that has the moiré period. We show that Δ(r) has a topologically nontrivial skyrmion lattice texture in real space, and that the topmost moiré valence bands provide a realization of the Kane-Mele quantum spin-Hall model, i.e., the two-dimensional time-reversal-invariant topological insulator. Because the bands narrow at small twist angles, a rich set of broken symmetry insulating states can occur at integer numbers of electrons per moiré cell.

Exploring optical resonances of nanoparticles excited by optical Skyrmion lattices

Recently, optical Skyrmion lattices (OSLs) have been realized in evanescent electromagnetic fields. OSLs possess topologically stable field configurations, which promise many optics and photonics applications. Here, we demonstrate that OSLs can serve as versatile structured optical near-fields to assist with studies of a variety of photonic modes in nanoparticles. We firstly show that OSL is capable of selectively exciting electric and magnetic multipole modes by placing a nanoparticle at different positions in the lattice. We then disclose that OSLs can efficiently excite some intriguing resonant modes, including toroidal and plasmonic dark modes, in dielectric or metal nanoparticles. Our results may enhance understanding of the interaction between OSLs and nanoparticles and find applications associated with precise control over resonant modes in nanostructures.

The Helical Magnet MnSi: Skyrmions and Magnons

Since the late 1970s, MnSi has played a major role in developing the theory of helical magnets in non-centrosymmetric materials showing the Dzyaloshinsky-Moriya interaction (DMI). With a long helimagnetic pitch of 175 Å as compared to the lattice d-spacing of 4.55 Å, it was ideal for performing neutron studies, especially as large single crystals could be grown. A (B-T)-phase diagram was measured, and in these studies, under the application of a field of about 180 mT perpendicular to the scattering vector Q, a so-called A-phase in the B-T phase diagram was found and first interpreted as a re-orientation of the magnetic helix. After the surprising discovery of the skyrmion lattice in the A-phase in 2009, much interest arose due to the rigidity of the skyrmionic lattice, which is only loosely bound to the crystal lattice, and therefore only relatively small current densities can already induce a motion of this lattice. A very interesting approach to even better understand the complex structures in the phase diagram is to measure and model the spin excitations in MnSi. As the helimagnetic state is characterized by a long pitch of about 175 Å, the associated characteristic excitations form a band structure due to Umklapp scattering and can only be observed at very small Q with energies below 1 meV. Similarly, the excitations of the skyrmion lattice are very soft and low-energetic. We investigated the magnons in MnSi in the whole (B,T)-phase diagram starting in the single-k helimagnetic state by applying a small magnetic field, B = 100 mT. This way, the complexity of the magnon spectrum is significantly reduced, allowing for a detailed comparison of the data with theory, resulting in a full theoretical understanding of the spin system of MnSi in all its different magnetic phases.

Skyrmions in anisotropic magnetic fields: strain and defect driven dynamics

ABSTRACTMagnetic skyrmions are particle-like, topologically protected magnetization entities that are promising candidates for information carriers in racetrack-memory schemes. The transport of skyrmions in a shift-register-like fashion is crucial for their embodiment in practical devices. Recently, we demonstrated experimentally that chiral skyrmions in Cu2OSeO3 can be effectively manipulated by a magnetic field gradient, leading to a collective rotation of the skyrmion lattice with well-defined dynamics in a radial field gradient. Here, we employ a skyrmion particle model to numerically study the effects of resultant shear forces on the structure of the skyrmion lattice. We demonstrate that anisotropic peak broadening in experimentally observed diffraction patterns can be attributed to extended linear regions in the magnetic field profile. We show that topological (5-7) defects emerge to protect the six-fold symmetry of the lattice under the application of local shear forces, further enhancing the stability of proposed magnetic field driven devices.

Unpinning the skyrmion lattice in MnSi: Effect of substitutional disorder

By employing magnetization and small angle neutron scattering (SANS) measurements, we have investigated the behavior of the skyrmion lattice (SKL) and the helical order in MnSi0.992Ga0.008. Our results indicate that the order of the SKL is sensitive to the orientation of an applied magnetic field with respect to the crystal lattice and small variations in the sequence of temperature and applied magnetic field changes. The disorder caused by the substitution of the heavier element Ga for Si is sufficient to reduce the pinning of the SKL to the underlying crystalline lattice. This reduces the propensity for the SKL to be aligned with the crystal lattice. This tendency is most evident when the applied field is not well oriented with respect to the high symmetry axes of the crystal resulting in disorder in the long range SKL while maintaining sharp radial order. We have also investigated the effect of substituting heavier elements into MnSi on the reorientation process of the helical domains with field cycling in MnSi0.992Ga0.008 and Mn0.985Ir0.015Si. A comparison of the reorientation process in these materials with field reduction indicates that the substitution of heavier elements on either Mn or Si sites creates a higher energy barrier for the reorientation of the helical order and for the formation of domains.

Precipitating ordered skyrmion lattices from helical spaghetti and granular powders

Magnetic skyrmions have been the focus of intense research due to their potential applications in ultra-high density data and logic technologies, as well as for the unique physics arising from their antisymmetric exchange term and topological protections. In this work we prepare a chiral jammed state in chemically disordered (Fe, Co)Si consisting of a combination of randomly-oriented magnetic helices, labyrinth domains, rotationally disordered skyrmion lattices and/or isolated skyrmions. Using small angle neutron scattering, (SANS) we demonstrate a symmetry-breaking magnetic field sequence which disentangles the jammed state, resulting in an ordered, oriented skyrmion lattice. The same field sequence was performed on a sample of powdered Cu2OSeO3 and again yields an ordered, oriented skyrmion lattice, despite relatively non-interacting nature of the grains. Micromagnetic simulations confirm the promotion of a preferred skyrmion lattice orientation after field treatment, independent of the initial configuration, suggesting this effect may be universally applicable. Energetics extracted from the simulations suggest that approaching a magnetic hard axis causes the moments to diverge away from the magnetic field, increasing the Dzyaloshinskii-Moriya energy, followed subsequently by a lattice re-orientation. The ability to facilitate an emergent ordered magnetic lattice with long-range orientation in a variety of materials despite overwhelming internal disorder enables the study of skyrmions even in imperfect powdered or polycrystalline systems and greatly improves the ability to rapidly screen candidate skyrmion materials.

Unique Helical Magnetic Order and Field-Induced Phase in Trillium Lattice Antiferromagnet EuPtSi

Magnetic transition phenomena in cubic chiral antiferromagnet EuPtSi with $T_{\rm N}$=4.0~K were investigated by means of single crystal neutron diffraction. At 0.3~K in the ground state, magnetic peaks emerge at positions represented by an ordering vector ${q}_{1}$=$(0.2, 0.3, 0)$ and its cyclic permutation. Upon heating, an additional magnetic peak splitting with hysteresis was uncovered at around $T^*_{\rm N}{\sim}$2.5~K, indicating the presence of a first-order commensurate-incommensurate transition with ${q}^*_{1}$=$(0.2, 0.3, {\delta})$ (${\delta}_{\rm max}{\simeq}$0.04) at $T^*_{\rm N}$. A half-polarized neutron scattering experiment for polarization parallel to the scattering vector revealed that polarization antiparallel to the scattering vector has stronger intensity in both magnetic phases. This feature clarifies the single chiral character of the helical structure with moments lying perpendicular to the ordering vector in both ordered states. Under a vertical magnetic field of 1.2~T for ${B}{\parallel}$[1,1,1] at 1.9~K entering into the so-called $A$ phase, magnetic peaks form characteristic hexagonal patterns in the equatorial scattering plane around nuclear peaks. An ordering vector ${q}_{A}{\simeq}({\pm}0.09, {\pm}0.20, {\mp}0.28)$ of the $A$-phase has similar periodic length as $q_{1}$, and could be the hallmark of a formation of skyrmion lattice in EuPtSi.

Deciphering structural and magnetic disorder in the chiral skyrmion host materials Co x Zn y Mn z (x + y+ z = 20).

Co x Zn y Mn z (x + y + z = 20) compounds crystallizing in the chiral β-Mn crystal structure are known to host skyrmion spin textures even at elevated temperatures. As in other chiral cubic skyrmion hosts, skyrmion lattices in these materials are found at equilibrium in a small pocket just below the magnetic Curie temperature. Remarkably, Co x Zn y Mn z compounds have also been found to host metastable nonequlibrium skyrmion lattices in a broad temperature and field range, including down to zero field and low temperature. This behavior is believed to be related to disorder present in the materials. Here, we use neutron and synchrotron diffraction, density functional theory calculations, and dc and ac magnetic measurements to characterize the atomic and magnetic disorder in these materials. We demonstrate that Co has a strong site preference for the diamondoid 8c site in the crystal structure, while Mn tends to share the geometrically frustrated 12d site with Zn, due to its ability to develop a large local moment on that site. This magnetism-driven site specificity leads to distinct magnetic behavior for the Co-rich 8c sublattice and the Mn on the 12d sublattice. The Co-rich sublattice orders at high temperatures (compositionally tunable between 210 and 470 K) with a moment around 1μ B /atom and maintains this order to low temperature. The Mn-rich sublattice holds larger moments (about 3μ B ) which remain fluctuating below the Co moment ordering temperature. At lower temperature, the fluctuating Mn moments freeze into a reentrant disordered cluster-glass state with no net moment, while the Co moments maintain order. This two-sublattice behavior allows for the observed coexistence of strong magnetic disorder and ordered magnetic states such as helimagnetism and skyrmion lattices.

Stability of the skyrmion lattice near the critical temperature in cubic helimagnets

The phase diagram of cubic helimagnets near the critical temperature is obtained from a Landau-Ginzburg model, including fluctuations to gaussian level. The free energy is evaluated via a saddle point expansion around the local minima of the Landau-Ginzburg functional. The local minima are computed by solving the Euler-Lagrange equations with appropriate boundary conditions, preserving manifestly the full nonlinearity that is characteristic of skyrmion states. It is shown that the fluctuations stabilize the skyrmion lattice in a region of the phase diagram close to the critical temperature, where it becomes the equilibrium state. A comparison of this approach with previous computations performed with a different approach (truncated Fourier expansion of magnetic states) is given.

Transformation between meron and skyrmion topological spin textures in a chiral magnet.

Crystal lattices with tetragonal or hexagonal structure often exhibit structural transitions in response to external stimuli1. Similar behaviour is anticipated for the lattice forms of topological spin textures, such as lattices composed of merons and antimerons or skyrmions and antiskyrmions (types of vortex related to the distribution of electron spins in a magnetic field), but has yet to be verified experimentally2,3. Here we report real-space observations of spin textures in a thin plate of the chiral-lattice magnet Co8Zn9Mn3, which exhibits in-plane magnetic anisotropy. The observations demonstrate the emergence of a two-dimensional square lattice of merons and antimerons from a helical state, and its transformation into a hexagonal lattice of skyrmions in the presence of a magnetic field at room temperature. Sequential observations with decreasing temperature reveal that the topologically protected skyrmions remain robust to changes in temperature, whereas the square lattice of merons and antimerons relaxes to non-topological in-plane spin helices, highlighting the different topological stabilities of merons, antimerons and skyrmions. Our results demonstrate the rich variety of topological spin textures and their lattice forms, and should stimulate further investigation of emergent electromagnetic properties.

Relaxation Dynamics of Zero-Field Skyrmions over a Wide Temperature Range.

The promise of magnetic skyrmions in future spintronic devices hinges on their topologically enhanced stability and the ability to be manipulated by external fields. The technological advantages of nonvolatile zero-field skyrmion lattice (SkL) are significant if their stability and reliability can be demonstrated over a broad temperature range. Here, we study the relaxation dynamics including the evolution and lifetime of zero-field skyrmions generated from field cooling (FC) in an FeGe single-crystal plate via in situ Lorentz transmission electron microscopy (L-TEM). Three types of dynamic switching between zero-field skyrmions and stripes are identified and distinguished. Moreover, the generation and annihilation of these metastable skyrmions can be tailored during and after FC by varying the magnetic fields and the temperature. This dynamic relaxation behavior under the external fields provides a new understanding of zero-field skyrmions for their stability and reliability in spintronic applications and also raises new questions for theoretical models of skyrmion systems.

Edge states in a two-dimensional honeycomb lattice of massive magnetic skyrmions

We study the collective dynamics of a two-dimensional honeycomb lattice of magnetic skyrmions. By performing large-scale micromagnetic simulations, we find multiple chiral and non-chiral edge modes of skyrmion oscillations in the lattice. The non-chiral edge states are due to the Tamm-Shockley mechanism, while the chiral ones are topologically protected against structure defects and hold different handednesses depending on the mode frequency. To interpret the emerging multiband nature of the chiral edge states, we generalize the massless Thiele's equation by including a second-order inertial term of skyrmion mass as well as a third-order non-Newtonian gyroscopic term, which allows us to model the band structure of skrymion oscillations. Theoretical results compare well with numerical simulations. Our findings uncover the importance of high order effects in strongly coupled skyrmions and are helpful for designing novel topological devices.