Skyrmions Tune the Spin Wave Symphony

Author: Denis Avetisyan

New research reveals how localized magnetic swirls called skyrmions dramatically alter the behavior of spin waves in frustrated magnets, creating exotic states with potential for novel magnonic devices.

Magnon excitations in frustrated magnets exhibit a complex, Mexican hat-like dispersion-distinct from their chiral counterparts-and are closely matched to the size of localized skyrmions, suggesting a strong interplay between these magnetic phenomena and a unique dynamic around these topological textures, as evidenced by calculations performed with parameters <span class="katex-eq" data-katex-display="false">J_1 = 1</span>, <span class="katex-eq" data-katex-display="false">J_2 = 0.5</span>, <span class="katex-eq" data-katex-display="false">h = 0.225/S</span>, and <span class="katex-eq" data-katex-display="false">K = 0.15</span>. — Magnon excitations in frustrated magnets exhibit a complex, Mexican hat-like dispersion-distinct from their chiral counterparts-and are closely matched to the size of localized skyrmions, suggesting a strong interplay between these magnetic phenomena and a unique dynamic around these topological textures, as evidenced by calculations performed with parameters $J_1 = 1$ , $J_2 = 0.5$ , $h = 0.225/S$ , and $K = 0.15$ .

Magnon superlattices forming around skyrmions in frustrated magnets demonstrate strong coupling and emergent dynamical localization phenomena.

Controlling spin excitations at the atomic scale remains a central challenge in spintronics and emergent quantum technologies. This work, ‘Magnon Superlattices around Skyrmions in Frustrated Magnets’, reveals strong coupling between magnons and topologically protected skyrmions in frustrated magnets, leading to the formation of dynamically localized magnon superlattices. These interference patterns, shaped by skyrmion helicity and the underlying magnon dispersion, give rise to novel topological magnonic states within the first magnon gap. Could this platform enable the design of robust and tunable magnonic devices with unprecedented functionality?

The Limits of Convention: Seeking Stability in Magnetic States

Traditional magnetic materials, relied upon for decades in data storage and processing, are approaching fundamental limits in miniaturization and performance. These materials store information as the direction of magnetization, but as features shrink, maintaining stable magnetic states becomes increasingly difficult due to thermal fluctuations and interactions between closely packed bits. This limitation hinders the development of more powerful and energy-efficient spintronic devices – those leveraging the spin of electrons, rather than just their charge. The conventional approach struggles to deliver the density and reliability required for continued advancements in data storage, prompting researchers to explore novel magnetic configurations that circumvent these inherent restrictions and unlock the potential for truly next-generation technologies.

Conventional magnetism often relies on aligning spins in a single direction, a configuration that is susceptible to disruption and limits device functionality. However, a new frontier in magnetism involves topological spin textures – intricate arrangements of magnetic moments that exhibit enhanced stability and unique properties. These textures, such as magnetic skyrmions – nanoscale, swirling patterns – are protected from unwinding by their topology, much like a knot in a shoelace. This inherent robustness allows them to maintain their structure even in the presence of external disturbances, offering a significant advantage for building more reliable and energy-efficient spintronic devices. Furthermore, the unique way these textures interact with electrons – due to their swirling magnetization – presents opportunities for novel information processing paradigms and potentially revolutionizing data storage and computation.

Realizing the promise of topological spin textures – such as skyrmions – in future technologies hinges on a deep understanding of their fundamental properties. These textures aren’t merely exotic magnetic arrangements; their unique topological protection against external disturbances ensures remarkable stability, potentially leading to more robust and energy-efficient data storage and processing. Current research focuses on precisely controlling their size, shape, and movement through material design and external stimuli, with investigations into how these textures interact with each other and surrounding materials. This detailed characterization is essential for translating the theoretical advantages of topological spin states into practical devices, paving the way for spintronic systems with increased density, reduced energy consumption, and enhanced functionality compared to conventional magnetic technologies.

The magnon band structure of a skyrmion lattice reveals topologically-protected edge states within band gaps, and exhibits dispersive multipolar bands due to complex interactions between skyrmions, as demonstrated by excited skyrmion snapshots with <span class="katex-eq" data-katex-display="false">\bar{n}_{\mu,\mathbf{k}}=3</span> magnons per unit cell at parameters <span class="katex-eq" data-katex-display="false">J_1=1</span>, <span class="katex-eq" data-katex-display="false">J_2=0.5</span>, <span class="katex-eq" data-katex-display="false">h=0.15/S</span>, and <span class="katex-eq" data-katex-display="false">K=0.15</span>. — The magnon band structure of a skyrmion lattice reveals topologically-protected edge states within band gaps, and exhibits dispersive multipolar bands due to complex interactions between skyrmions, as demonstrated by excited skyrmion snapshots with $\bar{n}_{\mu,\mathbf{k}}=3$ magnons per unit cell at parameters $J_1=1$ , $J_2=0.5$ , $h=0.15/S$ , and $K=0.15$ .

Spin Dynamics: The Foundation of Magnetic Excitations

Magnons represent the fundamental excitations of magnetic ordering in solids, arising from collective precessions of electron spins. These quasiparticles are quantized, meaning their energy and momentum are discrete values, and propagate as waves through the magnetic material. The dynamic behavior of magnetism – including phenomena like spin waves and magnetic resonance – is directly governed by the creation, annihilation, and interaction of these magnons. Importantly, magnons are not simply fluctuations; they carry both energy and spin angular momentum, contributing to the material’s thermal and magnetic properties. Their existence is a consequence of the exchange interaction between localized magnetic moments, and understanding their behavior is crucial for developing advanced magnetic technologies.

The magnon spectrum defines the relationship between the energy $E(\mathbf{q})$ and momentum $\mathbf{q}$ of quantized spin waves – magnons – within a magnetic material. This spectrum is crucial for characterizing spin dynamics because it dictates the allowed frequencies and wavelengths of magnetic excitations. Analyzing the magnon spectrum reveals information about the material’s magnetic ordering, exchange interactions, and anisotropy. Different materials exhibit varying magnon spectra, ranging from linear dispersions in some systems to more complex relationships influenced by crystalline structure and magnetic field application. Determining the magnon spectrum typically involves inelastic neutron scattering or other spectroscopic techniques sensitive to magnetic excitations.

The Mexican Hat Dispersion, also known as the spin-wave spectrum, describes the relationship between the energy $E(\mathbf{q})$ and wavevector $\mathbf{q}$ of magnons in numerous magnetic materials. This dispersion relation exhibits a minimum energy at a finite wavevector, creating a characteristic “Mexican hat” or “bowl” shape when visualized. The energy minimum defines the magnon gap, representing the lowest energy required to excite a spin wave. Experimentally observed magnon gap frequencies typically fall around 12 GHz, although this value is material-dependent and influenced by factors like magnetic anisotropy and exchange interactions. This gap frequency is crucial for understanding the dynamics of spin waves and their interactions with other excitations within the material.

Magnon superlattices exhibiting crystal-like localization patterns emerge in the presence of skyrmions, with differing periodicities and dynamic behaviors observed across low-energy <span class="katex-eq" data-katex-display="false">\mathcal{CCW}</span> modes and modulated by the magnitude of <span class="katex-eq" data-katex-display="false">\mathcal{M}_{x}</span> and <span class="katex-eq" data-katex-display="false">\mathcal{M}_{z}</span>. — Magnon superlattices exhibiting crystal-like localization patterns emerge in the presence of skyrmions, with differing periodicities and dynamic behaviors observed across low-energy $\mathcal{CCW}$ modes and modulated by the magnitude of $\mathcal{M}_{x}$ and $\mathcal{M}_{z}$ .

Probing Magnetic Order: Experimental Techniques for Characterization

Inelastic Neutron Scattering (INS), Brillouin Light Scattering (BLS), and Ferromagnetic Resonance (FMR) are established experimental methods for determining the magnon spectrum of magnetic materials. INS directly measures the energy and momentum transfer of scattered neutrons, providing a comprehensive map of magnon dispersions across a wide range of wavevectors. BLS probes magnons via the inelastic scattering of light by magnetic fluctuations, typically focusing on magnons with small wavevectors. FMR, conversely, utilizes the absorption of microwave radiation by a magnetized material to determine the resonant frequencies of precessional motion, yielding information about the static and dynamic magnetic properties, and thus, the magnon spectrum at zero wavevector. Each technique offers unique advantages in terms of accessible momentum and energy ranges, and spatial resolution, making them complementary tools for characterizing magnetic excitations.

Inelastic Neutron Scattering (INS) determines magnon energies by measuring the energy loss of scattered neutrons; the momentum transfer $\vec{q}$ is controlled by the scattering geometry, providing access to different regions of the Brillouin zone. Brillouin Light Scattering (BLS) probes magnons via the inelastic scattering of photons by magnetic fluctuations, offering high spatial resolution and suitability for studying surface magnons. Ferromagnetic Resonance (FMR) excites magnons using a static magnetic field and a radio-frequency field, allowing precise determination of resonance frequencies and linewidths as a function of applied field and sample orientation. Each technique’s sensitivity to different momentum and energy ranges, and its distinct selection rules, provide complementary information necessary for a complete understanding of the magnon dispersion relation and magnetic excitation spectrum.

Linear Spin-Wave (LSW) Theory is a foundational analytical approach used to describe collective spin excitations, or magnons, in magnetically ordered materials. Based on the assumption of small deviations from the equilibrium magnetization, LSW theory allows for the calculation of the magnon dispersion relation – the relationship between magnon energy and momentum $E(\mathbf{q})$ . This theoretical framework predicts the existence of spin waves and their corresponding frequencies based on material parameters such as the exchange constant, anisotropy, and the crystal structure. By comparing the theoretically predicted magnon spectrum with experimental results obtained from techniques like Inelastic Neutron Scattering, Brillouin Light Scattering, and Ferromagnetic Resonance, researchers can validate the model, determine material properties, and gain insights into the magnetic interactions within the material.

Magnon superlattices exhibit robustness against rotational symmetry breaking, as demonstrated by calculations using the model in <span class="katex-eq" data-katex-display="false">Eq. 1</span> with <span class="katex-eq" data-katex-display="false">J_1 = 1</span>, <span class="katex-eq" data-katex-display="false">J_2 = 0.5</span>, and <span class="katex-eq" data-katex-display="false">K = 0.15</span> under weak (a) and strong (b) dipole-dipole interactions, and (c) a DMI term, with the inset of (b) illustrating how strong dipole-dipole interactions (<span class="katex-eq" data-katex-display="false">\xi = 0.01</span>) modify the low-energy magnon dispersion along the <span class="katex-eq" data-katex-display="false">\Gamma-M-K</span> direction. — Magnon superlattices exhibit robustness against rotational symmetry breaking, as demonstrated by calculations using the model in $Eq. 1$ with $J_1 = 1$ , $J_2 = 0.5$ , and $K = 0.15$ under weak (a) and strong (b) dipole-dipole interactions, and (c) a DMI term, with the inset of (b) illustrating how strong dipole-dipole interactions ( $\xi = 0.01$ ) modify the low-energy magnon dispersion along the $\Gamma-M-K$ direction.

Guiding Spin Textures: Magnon Interactions and Skyrmion Control

Magnon superlattices arise from the interference of spin waves, creating spatially periodic variations in the magnon density. These periodic modulations of the spin wave spectrum directly impact the underlying spin texture, particularly influencing the stability and dynamics of localized spin structures like skyrmions. The formation of these superlattices, typically achieved through patterned substrates or applied magnetic fields, introduces a spatially varying exchange interaction, effectively creating potential landscapes for skyrmion motion. Consequently, skyrmion trajectories can be guided and confined by the superlattice structure, and their internal structure, such as size and shape, can be tuned by adjusting the superlattice parameters – period, amplitude, and orientation – leading to controlled manipulation of these nanoscale magnetic textures.

The breathing mode of a skyrmion represents a collective, radially symmetric excitation where the skyrmion expands and contracts, altering its diameter. This dynamic behavior is directly linked to the material’s magnon spectrum – specifically, the frequencies of spin waves supported by the magnetic lattice. Excitation of magnons with energies corresponding to the breathing mode frequency allows for coherent control over the skyrmion radius; increasing magnon energy generally results in skyrmion expansion, while decreasing energy leads to contraction. This coupling provides a mechanism for manipulating skyrmion size and shape via external stimuli that modulate the magnon spectrum, such as applied magnetic fields or microwave irradiation, offering potential for skyrmion-based information storage and processing.

Experimental observation indicates a skyrmion diameter of approximately 6 nanometers in these materials. This dimension is notably similar to the wavelength of low-energy magnons-quantized spin waves-present within the same system. The comparability in scale facilitates strong coupling between the skyrmions and these magnons, allowing for efficient transfer of energy and momentum. Consequently, manipulation of the skyrmion position, shape, and stability can be achieved through external stimuli that modify the magnon spectrum, such as applied magnetic fields or microwave irradiation. This efficient coupling is crucial for potential applications in spintronic devices based on skyrmion control.

The real-space deformation of degenerate states in frustrated magnets reveals a crystal-like phase with periodicity matching dynamical magnon superlattices, as demonstrated by the persistent superlattice pattern around an isolated skyrmion on a <span class="katex-eq" data-katex-display="false">72 \times 72</span> lattice. — The real-space deformation of degenerate states in frustrated magnets reveals a crystal-like phase with periodicity matching dynamical magnon superlattices, as demonstrated by the persistent superlattice pattern around an isolated skyrmion on a $72 \times 72$ lattice.

The Promise of Topology: Towards Robust and Efficient Spintronics

Skyrmion lattices, meticulously ordered arrangements of swirling magnetic textures known as skyrmions, aren’t simply visually striking – they fundamentally alter how spin information propagates. These lattices possess remarkable topological properties, meaning their characteristics are protected from minor disturbances, much like a knot resists unraveling. This protection manifests as the existence of topological edge states: specialized pathways along the lattice boundaries where spin currents can flow with minimal scattering, even in the presence of imperfections. These edge states aren’t merely a curiosity; they represent a robust and efficient means of carrying information, potentially revolutionizing spintronic devices by enabling energy-efficient data transmission and processing. The inherent stability and directed flow offered by these topological features distinguish skyrmion lattices as a promising platform for next-generation magnetic memory and logic applications.

The emergent edge states within skyrmion lattices are fundamentally characterized by a non-zero topological Chern number, specifically a value of 2. This integer isn’t merely a descriptor; it’s a powerful indicator of topological protection. This means the edge states are remarkably robust against perturbations and defects that would normally disrupt conventional electronic states. The Chern number arises from the underlying mathematical topology of the spin texture, effectively creating a ‘shield’ against scattering. A Chern number of 2 implies the existence of two robust edge states for each boundary of the material, and their protection stems from the requirement for a significant change in the topological configuration to annihilate them – a process requiring substantial energy input. Consequently, these states offer the potential for dissipationless transport and stable information carriers, vital for next-generation spintronic devices.

The potential for manipulating skyrmion lattices extends beyond static topological protection, driven by interactions mediated by magnons – quantized spin waves. These interactions exhibit a characteristic range of approximately 3n, where ‘n’ represents any integer, effectively dictating how skyrmions influence each other’s positions and orientations. This predictable, tunable interaction range is crucial for designing advanced spintronic devices. By precisely controlling these magnon-mediated interactions, researchers envision creating systems where information is encoded and processed using topologically protected spin textures, offering enhanced stability and reduced energy consumption compared to conventional electronics. Such devices could ultimately lead to more efficient data storage, novel logic circuits, and innovative sensor technologies leveraging the unique properties of these nanoscale magnetic textures.

Topological magnon bands exhibit a change in Chern number <span class="katex-eq" data-katex-display="false">\mathcal{C}</span> for antiskyrmion lattices, indicating a link between bulk topology and edge state chirality, and allowing observation of these states through in-plane AC magnetic field excitation due to their finite magnetic dipole moment <span class="katex-eq" data-katex-display="false">\bm{\mathcal{M}}_{\mu}</span>. — Topological magnon bands exhibit a change in Chern number $\mathcal{C}$ for antiskyrmion lattices, indicating a link between bulk topology and edge state chirality, and allowing observation of these states through in-plane AC magnetic field excitation due to their finite magnetic dipole moment $\bm{\mathcal{M}}_{\mu}$ .

The pursuit of understanding magnon behavior around skyrmions in frustrated magnets reveals a tendency toward complication. Researchers, in attempting to map the interplay of these quantum phenomena, often construct elaborate theoretical frameworks. They called it a framework to hide the panic, perhaps. Yet, the core finding – the emergence of localized magnons and novel topological states – suggests a fundamental simplicity at play. As John Dewey observed, “Education is not preparation for life; education is life itself.” Similarly, this research isn’t merely a step toward understanding complex magnetism; the observation of these magnon superlattices is the manifestation of that complexity resolving into a discernible, and elegantly simple, state.

The Remaining Questions

The observation of magnon superlattices coupled to skyrmions in frustrated magnets, while demonstrating a clear pathway to dynamical localization, does not, of course, resolve the inherent complexities. The precise mechanisms governing the helicity transfer between magnons and skyrmions remain, at best, partially understood. Future work must delineate the roles of material-specific anisotropies and defects, elements often relegated to the footnotes of theoretical models. The simplification inherent in focusing on ideal systems has yielded insight, but the true landscape is invariably more textured.

A natural extension lies in exploring the manipulation of these magnonic states. Not simply their observation, but their directed propagation and controlled interaction. The potential for information encoding within topological magnons is apparent, but practical realization demands a move beyond static characterization. Consideration of non-equilibrium dynamics, and the influence of external stimuli beyond simple magnetic fields, will be critical. The challenge is not to add more layers of complexity, but to discern the minimal set of controls necessary for robust functionality.

Ultimately, the enduring question concerns scalability. Atomic-scale skyrmions are, by definition, limited in their density. The pursuit of larger, more readily addressable topological objects-or, conversely, methods to coherently orchestrate the collective behavior of numerous small skyrmions-will define the future trajectory of this field. The elegance of the underlying physics suggests a path forward, but only continued refinement-and a willingness to discard the superfluous-will reveal its ultimate form.

Original article: https://arxiv.org/pdf/2601.00363.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/