Toward room-temperature nanoscale skyrmions in ultrathin films

The study addresses whether ultrathin ferromagnetic films can host skyrmions that are simultaneously nanoscale (sub-10 nm) and stable at room temperature—an essential requirement for skyrmion-based information technologies. Prior experiments have observed very small skyrmions only at low temperatures, while room-temperature stability has been achieved for larger skyrmions. The central question is how to decouple size from stability so that small skyrmions can have long lifetimes under ambient conditions. The authors emphasize that thermal stability is governed by the Arrhenius law, with both the energy barrier and the pre-exponential factor (attempt frequency) determining the lifetime. Earlier theoretical work predominantly focused on energy barriers for collapse (radial, boundary escape, and asymmetric mechanisms) and often treated the prefactor as a constant, leading to conclusions that ultrasmall skyrmions in locally ferromagnetic materials cannot be stable at room temperature. However, the prefactor can vary widely due to entropic and dynamical effects linked to skyrmion internal modes, implying that assuming a fixed attempt frequency may be misleading. The authors propose a systematic analysis using atomistic spin modeling and harmonic transition state theory (HTST) to evaluate both barrier and prefactor, exploring whether tuning magnetic interactions (exchange J, Dzyaloshinskii–Moriya D, anisotropy K) can greatly enhance lifetime without changing skyrmion size.

The introduction surveys prior theoretical and experimental literature: experimental observations of skyrmions in various ultrathin films, including Pd/Fe on Ir(111) and Rh(111), and reports of room-temperature skyrmions mainly for larger sizes. Theoretical studies have mapped collapse mechanisms (radial, edge escape, asymmetric collapse in frustrated systems) and parameter dependencies of energy barriers using minimum energy path (MEP) methods. Büttner et al. analyzed phase diagrams with a universal zero-diameter skyrmion energy approximation, treating the prefactor as constant, concluding that ultrasmall skyrmions cannot be stable at room temperature in locally ferromagnetic systems. Other works highlighted the crucial role of internal skyrmion modes, entropy barriers, and the sensitivity of the attempt frequency to field and mechanism. Overall, the literature has underexplored the materials dependence of the Arrhenius prefactor, motivating this paper’s definitive HTST treatment of both barrier and prefactor.

• Model: A two-dimensional monolayer of classical spins on a hexagonal lattice described by an atomistic Hamiltonian E = E_ex + E_DM + E_ani, with nearest-neighbor Heisenberg exchange (J), interfacial Dzyaloshinskii–Moriya interaction (D) with in-plane DM vectors perpendicular to bonds, and effective out-of-plane uniaxial anisotropy (K) capturing magnetocrystalline and magnetostatic contributions. Zero external field is assumed. System size is 80 × 80 sites with periodic boundary conditions, containing a single skyrmion.
• Phase diagram and radius isolines: Trial single-skyrmion states were relaxed across a grid of reduced parameters (K/J, D/J) to map regions of ferromagnetic (FM), spin spiral (SS), and metastable isolated skyrmions (Sk). Skyrmion radius R was defined per Bogdanov–Hubert, and eight R-isolines (R = 5a to 12a) were obtained numerically by interpolation and refinement across fixed D values to maintain constant R within ±0.1a.
• Minimum energy paths (MEP): Geodesic Nudged Elastic Band (GNEB) with 10–20 images, periodic boundary conditions, and climbing-image refinement was used to find MEPs for radial collapse into the FM state and locate first-order saddle points (SPs) for each parameter set along R-isolines.
• Lifetime calculations (HTST): Harmonic Transition State Theory for magnetic systems (including Goldstone modes) provides τ = τ0 exp(ΔE/kBT). ΔE = E_SP − E_min is taken from GNEB. The prefactor τ0 = (A V_SP det H_SK) / (2π V_SK det' H_SP), where:
  • det H_SK and det' H_SP are products of Hessian eigenvalues at the skyrmion minimum and SP (negative and translational Goldstone mode eigenvalues omitted), computed using a projection-operator approach to account for the curved configuration manifold (product of 2-spheres) and Intel MKL eigensolvers.
  • A is a dynamical factor computed as Λ = s^T A^† H_SP A s, with s the MEP tangent at SP transformed into tangent-space coordinates via the local projector U.
  • V_SK and V_SP are Goldstone-mode volumes obtained from spatial integrals over translational modes, evaluated via finite differences and lattice translations.
• Spectral analysis: Eigenvalue spectra of the Hessian at the skyrmion minimum and at SP were computed to identify localized magnon modes within the anisotropy gap (E_gap ≈ 2K) and their evolution along R-isolines, linking spectral differences to entropy contributions in τ0.
• Material mapping: Effective parameter positions for experimentally studied ultrathin systems (e.g., Pd/Fe/Ir(111), Pd/Fe/Rh(111), Rh/Co/Ir(111)) were overlaid on the phase diagram for context.
• Units and conditions: Room-temperature lifetimes reported with T = 300 K, J = 10 meV (unless varied), and μ = 3 μB. Intrinsic precession time τ_int ≈ (γJ)^{-1} used for normalized lifetimes in some plots.

• Stability at fixed size can be enhanced: Along contours of constant skyrmion radius R (5a–12a), increasing K/J and D/J together transforms skyrmion profiles from arrow-like to bubble-like while keeping R fixed, substantially increasing lifetime τ.
• Energy barriers alone are insufficient: Collapse energy barriers ΔE increase along R-isolines but remain below ~10 J even for R = 12a. For J ≈ 10 meV, 10 J corresponds to ~4 kBT at 300 K, too small to ensure long lifetimes on its own.
• Prefactor dominates stabilization: The Arrhenius pre-exponential factor τ0 exhibits extreme material dependence, varying by up to ~20 orders of magnitude across the considered parameter space. τ0 grows strongly along R-isolines toward bubble-like profiles and also increases with R (more strongly at larger D/J).
• Entropy-driven mechanism: The dramatic increase in τ0 arises from a growing number of skyrmion-localized deformation modes within the magnon gap and their softening at the skyrmion state, which are not mirrored at the transition state. This yields a large ratio of Hessian determinants (skyrmion versus SP), i.e., a large entropy barrier.
• Quantitative examples: Two parameter sets on the same R = 6a isoline yield lifetimes differing by five orders of magnitude despite identical size. Using full HTST (varying τ0) predicts lifetimes up to and exceeding 10 years for nanoscale skyrmions at room temperature at sufficiently large reduced D and K, whereas a constant-τ0 approximation severely underestimates lifetimes.
• Size scale: For Ir(111)-lattice spacing, skyrmion diameters up to 24a correspond to ≈6.5 nm, placing the predicted long lifetimes within the sub-10 nm regime.
• Universal-energy approximation overestimates barriers: Approximating ΔE by the Belavin–Polyakov zero-diameter skyrmion energy difference significantly overestimates the SP energy versus MEP results across all cases studied, cautioning against this shortcut for lifetime prediction.

The enhancement of lifetime at fixed size is traced to spectral differences between the skyrmion state and the transition state. Bubble-like skyrmions (thin circular domain walls) support many localized deformation modes within the magnon gap that soften as K/J and D/J increase together, yielding a large entropy difference and hence large τ0. Larger skyrmions accommodate more azimuthal deformation modes, further sensitizing τ0 to material parameters and size. A domain-wall model provides qualitative insights: the number of localized harmonics N within the gap scales with perimeter over wall width and increases as the wall narrows (effectively larger K/J and D/J), while including DM interaction reproduces the observed softening trend under concerted increases of K and D. Practically, the pathway to room-temperature stability for nanoscale skyrmions is to engineer materials with relatively large reduced parameters D/J and K/J, which can be achieved either by increasing D and K or by decreasing effective J. Interface engineering in ultrathin films and multilayers (e.g., 4d/3d/5d stacks) allows independent tuning of J, D, and K via composition, stacking, and intermixing. Reported systems already approach the favorable regime, making sub-10 nm, zero-field, room-temperature skyrmion stabilization via entropy barriers realistic. While additional magnetic interactions (exchange frustration, higher-order exchange) can further impact both ΔE and τ0, the core principle—maximizing the difference in excitation spectra between the skyrmion and transition states—remains broadly applicable.

Using atomistic spin modeling and HTST, the paper demonstrates that sub-10 nm skyrmions in ultrathin ferromagnetic films can achieve room-temperature stability primarily through enormously large Arrhenius prefactors driven by entropy barriers, rather than through large collapse energy barriers. By concertedly increasing anisotropy and DMI (or effectively reducing exchange) at fixed skyrmion radius, skyrmions adopt bubble-like profiles with thin domain walls that host many softened localized deformation modes, dramatically increasing τ0 and the lifetime. The work clarifies the interplay between size and stability, guides materials design toward regimes of large K/J and D/J, and suggests interface engineering (in 4d/3d/5d stacks) as a viable route. Future directions include experimental exploration of tunable ultrathin film compositions to reach the predicted regime, and theoretical inclusion and engineering of additional interactions (frustration, higher-order exchange) to further enhance stability, all while leveraging the general principle of maximizing spectral differences between skyrmion and transition states.

• Zero-field and simplified Hamiltonian: The model omits explicit Zeeman terms and treats dipolar interactions through an effective anisotropy K; this may be less accurate for thicker films where stray fields are significant.
• Nearest-neighbor exchange: Only nearest-neighbor Heisenberg exchange is included; further-neighbor and higher-order exchange can alter both ΔE and τ0.
• TST assumptions: HTST assumes rare events and quasi-equilibrium in the initial basin; while justified for long-lived skyrmions, deviations could occur for very low barriers or strong driving.
• Saddle-point approximation: Reliance on MEP and harmonic expansion around minima and SP neglects anharmonic effects that might influence τ0 at elevated temperatures.
• Material parameter mapping: Effective parameters (especially J in frustrated systems) may not uniquely capture all excitations far from equilibrium, and achieving large D/J and K/J experimentally may be challenging though feasible via interface engineering.
• Finite-size and boundary effects: Simulations use a single skyrmion in a periodic 80×80 lattice; interactions with defects, boundaries, or other skyrmions are not included and could affect stability in devices.