Real-space Lorentz transmission electron microscopy in a (001) thin plate of the chiral-lattice magnet Co8Zn9Mn3 revealed a square lattice of merons and antimerons (half-skyrmions with topological number ±1/2) emerging from a helical state and transforming continuously into a hexagonal skyrmion lattice under a modest magnetic field at room temperature, with the net topological number per lattice unit conserved. The interpretation rests on Fresnel-image simulations, and the originating work left a precise follow-up agenda that remains open: (1) the exact three-dimensional magnetic configuration of the square meron lattice, including core polarities and out-of-plane components, is not uniquely determined by Fresnel-mode Lorentz TEM alone; (2) the microscopic origin and quantitative role of sample strain and anisotropic deformation in producing deformed vortex shapes and selecting core polarities are undetermined; (3) the energetic landscape, metastability and dynamical pathways governing the helix–square-meron-lattice–skyrmion-lattice transitions, including the magnitude, temperature dependence and rates of topological-number-changing events versus continuous deformations in realistic samples, are not quantified; (4) the emergent electromagnetic responses (transport signatures, collective dynamics) specific to the meron/antimeron lattice and their dependence on anisotropy and strain are uncharacterized. No resolution of this follow-up agenda is known for the chiral-magnet meron lattice itself.

Background

In chiral magnets—cubic crystals lacking inversion symmetry, such as the β-Mn-type alloy Co8Zn9Mn3 (space groups P4132/P4332)—the Dzyaloshinskii–Moriya interaction (DMI), an antisymmetric exchange coupling arising from the broken inversion symmetry, twists ferromagnetic exchange into helical spin order with nanometer-scale period. Under a magnetic field, these materials host skyrmions: two-dimensional whirl-like spin textures with integer topological number N=±1N=\pm1, where N=14π∫n⋅(∂xn×∂yn) dx dyN=\frac{1}{4\pi}\int \mathbf{n}\cdot(\partial_x\mathbf{n}\times\partial_y\mathbf{n})\,dx\,dy for the unit magnetization n=M/∣M∣\mathbf{n}=\mathbf{M}/|\mathbf{M}|. A meron (or antimeron, with reversed swirl sense) is a 'half-skyrmion' carrying fractional topological number N=±12N=\pm\tfrac{1}{2}, with the magnetization wrapping only half of the unit sphere. Theory (Lin, Saxena and Batista, 2015) predicted that sufficient easy-plane (in-plane) magnetic anisotropy in a quasi-two-dimensional chiral magnet stabilizes a square lattice of alternating merons and antimerons—a double-q\mathbf{q} state built from two orthogonal helices—instead of the conventional triple-q\mathbf{q} hexagonal skyrmion lattice (a state superposing three equivalent helices at 120° within the plane), and that the two lattice forms could be interconverted. Yu et al. (Nature 564, 95–98, 2018) verified this experimentally: in a (001) thin plate of Co8Zn9Mn3, imaged by Lorentz transmission electron microscopy (a defocused-contrast TEM technique sensitive to the in-plane magnetization; in 'Fresnel' mode the sample is defocused so that the magnetic induction produces intensity contrast, and in-plane magnetization maps are reconstructed via the transport-of-intensity equation, which relates the defocus-induced intensity gradient along the beam direction to the divergence of the in-plane magnetic induction), a square meron/antimeron lattice (sq-ML) formed from the helical state after low-field cooling near 295 K and, under an increasing out-of-plane bias field (~20 to ~60 mT), deformed continuously through a deformed-square lattice into a hexagonal skyrmion lattice (hex-SkL), with the net topological number per structural unit conserved at Ntotal=−1N_\text{total}=-1 (the fractional topological numbers of the merons and antimerons within one square lattice unit sum algebraically to one integer skyrmion equivalent). On cooling, the topologically protected skyrmion lattice survived while the square meron lattice relaxed into non-topological in-plane spin helices near 120 K, demonstrating different topological stabilities of the fractionalized and integer textures. The sq-ML interpretation, however, was established by comparing observed Fresnel contrasts with simulations built from superposed proper-screw helices (helices whose spins rotate in the plane perpendicular to their propagation wavevector): because the inverse problem of recovering the full three-dimensional magnetization from a single projection-contrast image is ill-posed, agreement with simulation supports but does not uniquely prove the sq-ML configuration, and the authors noted correlations between sample strain and anisotropic vortex deformation, cooling-field-dependent metastability in the (T,B)(T,B) phase diagram, and the anticipation of emergent electromagnetic properties—each of which defines an unresolved follow-up question addressed here.

Problem Statement

For the square meron/antimeron lattice (sq-ML) and its field- and temperature-driven transformation into the hexagonal skyrmion lattice (hex-SkL) in (001) thin plates of the chiral-lattice magnet Co8Zn9Mn3, establish a quantitative and experimentally unambiguous physical account of the state and the transformation, specifically: (1) determine the exact three-dimensional magnetic configuration of the sq-ML—including unambiguous assignment of vortex core polarities and out-of-plane magnetization components—by means beyond Fresnel-mode Lorentz TEM imaging and its simulations, so that alternative magnetic configurations projected to similar contrasts are definitively excluded; (2) identify the microscopic origin and quantify the role of sample strain and anisotropic deformation in producing the elongated or deformed vortex shapes and in selecting core polarities, including how these combine with composition-dependent in-plane magnetocrystalline anisotropy; (3) characterize the energetic landscape, metastability and dynamical pathways controlling the transitions among the helical, sq-ML and hex-SkL states—including the magnitude and temperature dependence of the barriers for changing the local fractional topological numbers, and the mechanisms and rates of topological-number-changing events versus continuous deformations in realistic samples; and (4) characterize experimentally the emergent electromagnetic responses (e.g., transport signatures such as topological-Hall-type signals—an emergent Hall voltage arising when conduction electrons traverse the fictitious magnetic field associated with a noncoplanar spin texture—or spin-current signals, and collective dynamics) specific to the square meron/antimeron lattice and to its transformation into the skyrmion lattice, including their dependence on anisotropy and strain. These four deliverables share one objective: converting the observationally inferred sq-ML state and its conserved-topological-charge transformation into a controlled, quantitatively understood and functionally characterized physical process in the chiral magnet, each component constraining the same underlying spin configuration, energy landscape and electronic response. These four sub-questions (1)–(4) are jointly constitutive of that single objective.

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