Real-space Lorentz transmission electron microscopy of a thin Fe0.5Co0.5Si film (Yu et al., Nature 465, 901 (2010)) revealed a two-dimensional skyrmion crystal (SkX) occupying a far larger region of the magnetic-field-temperature (B-T) phase diagram than strictly two-dimensional Monte Carlo simulations predict, and left four linked issues open: identifying and quantitatively modelling the cause of the enlargement (pinning by crystalline imperfections versus finite-thickness interlayer coupling), determining the full three-dimensional magnetization profile of individual skyrmions, delivering direct experimental tests of the predicted unconventional magnetotransport, and generalizing simulations to include finite thickness, demagnetization, realistic anisotropies and defect landscapes. Later literature has delivered major parts of this programme: electron holography and vector-field electron tomography now reconstruct the 3D skyrmion texture, current-driven skyrmion flow has been imaged at ultralow current density, and topological Hall signals have been measured in several thin-film chiral magnets; the enhanced thin-film stability is qualitatively attributed to dimensionality and chiral surface twists. What remains open is the quantitative core: a mechanism-resolved model that reproduces the measured enlarged SkX phase boundaries of Fe0.5Co0.5Si-type films including the defect landscape, quantifies the relative contributions of thickness/surface-twist, demagnetization and disorder pinning, and establishes a quantitative, same-specimen link between real-space skyrmion textures and magnetotransport signatures.

Background

In non-centrosymmetric cubic (B20-structure) chiral magnets such as MnSi, FeGe and Fe1-xCoxSi, the competition between ferromagnetic exchange and the Dzyaloshinskii-Moriya interaction (DMI) - a chiral spin coupling arising from the lack of inversion symmetry - produces helical spin order with a nanometre-to-tens-of-nanometres pitch. A skyrmion is a whirl-like, topologically protected spin texture in which the magnetization wraps a sphere once; in these materials the core spins point opposite to the applied field and the in-plane spins swirl around the core (Bloch type). A skyrmion crystal (SkX, also called skyrmion lattice) is a hexagonal periodic array of these whirls. In bulk crystals the SkX is thermally stabilized only in a narrow pocket of the temperature-field (T-B) phase diagram just below the magnetic ordering temperature, interleaved between helical, conical (spiral aligned with the field) and saturated ferromagnetic phases.

Yu et al. (Nature 465, 901 (2010)) imaged a thin (001) Fe0.5Co0.5Si film by Lorentz transmission electron microscopy (LTEM): in-focus/defocused imaging of the electron beam deflected by in-plane magnetization. Using the transport-of-intensity equation (TIE) - a phase-retrieval relation linking intensity variations through a defocus series to the phase shift of the electron wave - they reconstructed quantitative maps of the in-plane magnetization and observed a 90-nm-pitch hexagonal SkX at 50-70 mT. Two findings framed the open issues. First, the experimental B-T phase diagram agreed qualitatively with Monte Carlo simulations of a strictly two-dimensional discretized DMI-exchange Hamiltonian, but the measured SkX region was substantially larger, extending to the lowest measured temperatures; the paper suggested pinning by crystalline imperfections (absent from the simulations) and finite film thickness with inter-layer ferromagnetic coupling increasing the effective spin stiffness as candidate causes, without deciding between them. Second, the TIE reconstruction of single defocus images fixes only the in-plane magnetization; the out-of-plane (film-normal) component is not directly determined, so the full three-dimensional magnetization profile of the thin-film skyrmions was unspecified. The paper also anticipated that the observed topological spin texture should generate unconventional magnetotransport - the topological Hall effect, an extra Hall voltage produced by the emergent magnetic field that conduction electrons experience when flowing through a non-coplanar spin texture, and low-current-density skyrmion flow - but delivered no transport measurement.

Subsequent work established the generic phenomenology: two-dimensional skyrmion crystals in thin plates of FeGe (Yu et al., Nat. Mater. 10, 106 (2011)) and MnSi (Tonomura et al., Nano Lett. 12, 1673 (2012)) occupy wide T-B regions, unlike their bulk counterparts. Theory attributed this enhancement to dimensionality - in strictly 2D models the conical phase does not compete with the skyrmion lattice, whereas 3D Monte Carlo simulations recover the narrow bulk pocket (Buhrandt and Fritz, Phys. Rev. B 88, 195137 (2013)) - and to chiral surface twists: DM-gradient surface energy twists the magnetization near the free surfaces of finite-thickness films and stabilizes skyrmion lattices over broad field ranges (Meynell et al., Phys. Rev. B 90, 014406 (2014); Leonov et al., Phys. Rev. Lett. 117, 087202 (2016)). On the experimental side, electron holography - a technique recovering the electron phase shift itself rather than its intensity derivative - visualized the magnetic flux and three-dimensional structure of the skyrmion lattice in Fe0.5Co0.5Si foils of stepped thickness (Park et al., Nat. Nanotechnol. 9, 337 (2014)); holographic vector-field electron tomography later reconstructed the 3D magnetic texture of skyrmion tubes at sub-10-nm resolution (Wolf et al., Nat. Nanotechnol. 17, 250 (2022)), and soft X-ray laminography - a computed-tomography geometry suited to extended flat specimens - resolved the full 3D spin texture of a multilayer skyrmion (Raftrey et al., Sci. Adv. 10, eadp8615 (2024)). On the transport side, current-driven skyrmion flow was imaged in a FeGe microdevice at current densities below 100 A/cm^2 (Yu et al., Nat. Commun. 3, 988 (2012)), topological Hall anomalies were reported for epitaxial MnSi films (Li et al., Phys. Rev. Lett. 110, 117202 (2013); contested by Monchesky et al., Phys. Rev. Lett. 112, 059701 (2014)) and for strained Fe0.7Co0.3Si epilayers (thin epitaxial films) (Porter et al., arXiv:1312.1722), and the topological Hall effect was used to monitor skyrmion dynamics in MnSi nanowires (Liang et al., Nat. Commun. 6, 8217 (2015)). Disorder pinning of skyrmions at lattice defects and surfaces has been imaged and modelled (Li et al., Nano Lett. 17, 1395 (2017); Milde et al., Phys. Rev. B 100, 024408 (2019); Gruber et al., Nat. Commun. 13, 3102 (2022)), and metastable skyrmion lifetimes in Fe0.5Co0.5Si have been measured (Wild et al., Sci. Adv. 3, e1701704 (2017)).

Problem Statement

Establish a quantitative, mechanism-resolved theory-experiment account of the two-dimensional skyrmion crystal in thin films of the B20 chiral magnet Fe0.5Co0.5Si (and comparable thin-film chiral magnets), as posed by the open issues of Yu et al. (Nature 465, 901 (2010)). The common scientific objective is a complete quantitative description of the thin-film SkX state - why it is stable, what its three-dimensional structure is, and how it manifests in magnetotransport - resting on four jointly linked deliverables:

  1. Identify and quantitatively model the cause of the experimentally observed enlargement of the SkX region in the B-T phase diagram relative to strictly 2D Monte Carlo simulations. The source paper's candidate mechanisms - pinning by crystalline imperfections versus finite sample thickness producing inter-layer ferromagnetic coupling that increases the effective spin stiffness - should be discriminated and quantified; later work has refined the finite-thickness branch into dimensionality (absence of the conical competitor in 2D) and chiral surface twists, so the open target is a model that includes thickness, surface twists, demagnetization (the sample's magnetostatic self-field), realistic anisotropy and the defect landscape and reproduces the measured phase boundaries of the Fe0.5Co0.5Si film, with a quantified attribution of the relative contributions of these mechanisms.

  2. Specify the full three-dimensional magnetization profile of individual skyrmions in the thin film, including the out-of-plane (film-normal) component that single-defocus TIE analysis of Lorentz TEM images cannot determine.

  3. Deliver direct experimental tests of the predicted unconventional transport phenomena associated with the SkX and establish quantitative links between the observed real-space SkX textures - including defects and pinning - and magnetotransport signatures such as the topological Hall effect and current-driven skyrmion flow, on the same specimen or device.

  4. Generalize the simulations beyond the idealized 2D, pinning-free, simplified-anisotropy model used in the source paper so as to reach quantitative agreement with the measured phase diagrams and textures.

Later literature has already delivered substantial parts of deliverables 2 and 3 (three-dimensional reconstructions by electron holography and vector-field electron tomography; real-space observation of current-driven skyrmion flow at ultralow current density; topological Hall measurements in several thin-film chiral magnets, with the skyrmion assignment contested in some cases) and qualitative mechanisms for deliverable 1. The open scientific target is therefore the surviving core: a mechanism-resolved, quantitative model-experiment reconciliation of the enlarged SkX stability region in Fe0.5Co0.5Si-type thin-film chiral magnets - reproducing the measured B-T phase boundaries with the real defect landscape and quantifying the relative roles of thickness/surface twist, demagnetization, anisotropy and disorder pinning - together with the quantitative same-specimen correlation between real-space skyrmion texture and magnetotransport response.

References

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