The 2020 discovery of a square skyrmion lattice (SkL) of roughly 2 nm period in the centrosymmetric tetragonal magnet GdRu2Si2 established that skyrmions can form without either inversion-symmetry breaking (and hence without Dzyaloshinskii–Moriya interaction) or a geometrically frustrated lattice, and proposed itinerant-electron-mediated four-spin interactions as a plausible stabilization mechanism. The originating authors explicitly left open the quantitative evaluation of these interactions from the real electronic structure, the quantitative separation of the topological Hall signal in the skyrmion phase, the full three-dimensional magnetic structure, the generality of the mechanism across the RM2X2 (ThCr2Si2-type) family, and the observation of isolated skyrmions. Later work has firmly established Fermi-surface nesting as the origin of the magnetic modulation vectors and partially mapped the zero-field double-Q structures, but the operative multi-Q/skyrmion selection mechanism remains contested (inelastic-neutron magnon analysis finds no evidence for higher-order exchange), no first-principles evaluation of the four-spin coupling coefficients exists, the single-crystal topological Hall decomposition is still missing, and no isolated skyrmion has been observed. The problem therefore remains open.

Background

A magnetic skyrmion is a nanoscale, particle-like, topologically non-trivial swirl of spins, and a skyrmion lattice (SkL) is a periodic array of such objects; because of their small size and topological stability they are discussed as candidate high-density information bits. In nearly all early skyrmion hosts the lattice lacks inversion symmetry, so an antisymmetric Dzyaloshinskii–Moriya interaction (DMI) winds the spins, and the few centrosymmetric examples known before 2020 relied on geometrically frustrated (e.g. triangular-lattice) exchange. The DMI route fixes a triangular, hexagonal SkL largely irrespective of the crystal, and yields skyrmion sizes of tens of nanometres. In 2020, Khanh and co-workers reported resonant X-ray scattering (RXS, a diffraction technique sensitive to magnetic order through the Gd absorption edge) and Lorentz transmission electron microscopy (LTEM, real-space imaging of in-plane magnetization) evidence for a square SkL with a 1.9 nm period in GdRu2Si2, a centrosymmetric tetragonal ThCr2Si2-type (122-type) intermetallic whose square Gd layers are manifestly not geometrically frustrated. The compound belongs to the large RM2X2 family (R a rare earth, M a transition metal, X = Si, Ge, Sn), the same structure as the 122 iron-based superconductors, which permits extensive chemical tuning. Its magnetic phases were labelled phase I (the zero-field incommensurate phase, showing the symmetry-related propagation vectors of about (0.22, 0, 0) and (0, 0.22, 0) and interpreted in the original paper as domains of a single-Q screw, i.e. helical, structure), phase II (the square SkL, a double-Q, i.e. two-wave-vector, superposition with higher harmonics at 2Q1, 2Q2 and Q1+Q2, carrying one skyrmion of topological charge −1 — a winding-number invariant of the spin texture — per magnetic unit cell), and further field-induced phases, with first-order, hysteretic phase boundaries. Because GdRu2Si2 is centrosymmetric, DMI is forbidden; instead the modulation vectors were attributed to the Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction, the indirect exchange between localized 4f moments mediated by conduction electrons, whose sign and strength oscillate with wave vector and reflect Fermi-surface (FS) nesting, i.e. parallel sheet-like portions of the conduction-electron Fermi surface connected by the ordering wave vector. The selection of the double-Q skyrmion state over a single-Q helix was proposed to come from four-spin interactions (a quartic term in the spin Hamiltonian, expressible in reciprocal space through coupling coefficients for the wave vectors Q1, Q2, Q3, Q4) also mediated by the itinerant electrons, assisted by easy-axis (uniaxial) magnetic anisotropy; this mechanism had earlier been proposed for Fe monolayers on Ir(111). Conduction electrons traversing a skyrmion lattice acquire a Berry phase equivalent to an emergent magnetic field, producing a topological Hall (THE) contribution to the Hall resistivity on top of the ordinary and anomalous Hall terms, so magnetotransport is a direct probe of the skyrmion density; in GdRu2Si2, however, large anomalous Hall and magnetoresistive signals in the same field-temperature region complicate any decomposition.

Problem Statement

Establish the microscopic origin and the full physical characterization of the nanometric square skyrmion lattice (SkL) in the centrosymmetric tetragonal magnet GdRu2_2Si2_2, and determine whether this stabilization route is generic across the RM2_2X2_2 (ThCr2_2Si2_2-type) family of rare-earth intermetallics. As posed by the discovery paper and still only partially answered, this comprises five linked facets: (1) Mechanism — determine, by quantitative evaluation from the real electronic structure of GdRu2_2Si2_2 (Fermi-surface nesting, the wave-vector-dependent spin susceptibility, and the magnitudes and signs of the itinerant-electron-mediated four-spin coupling coefficients in reciprocal space), whether those interaction terms are sufficient to select the double-Q skyrmion order over competing single-Q helical and topologically trivial double-Q states or whether an alternative selection mechanism is required, and reconcile the outcome with the currently conflicting experimental evidence on the role of higher-order exchange; (2) Emergent transport — quantitatively separate and determine the topological Hall contribution of the phase II skyrmion lattice from the ordinary and anomalous Hall terms in single-crystal GdRu2_2Si2_2, whose large anomalous Hall and magnetoresistive signals currently obstruct the decomposition; (3) Magnetic structure — determine the full three-dimensional spin structure of the ordered phases, including interlayer modulation components, the complete higher-harmonic content, and the hysteretic field-temperature behaviour at microscopic resolution; (4) Generality — establish by chemical tuning (substitution on the R, M and X sites, doping, and electronic-structure control) how ubiquitous and how tunable nanometric skyrmion lattices are within the RM2_2X2_2 family (R a rare earth, M a transition metal, X = Si, Ge, Sn); comparison with other centrosymmetric rare-earth intermetallics is optional context and not part of this facet's required scope; (5) Single particles — experimentally identify and manipulate an isolated skyrmion in these centrosymmetric intermetallics. These five facets are the mandatory components of one overarching objective: a validated microscopic theory of skyrmion stabilization without inversion-symmetry breaking or geometric frustration, together with its observable structural, transport and single-particle consequences.

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