In the B20 chiral helimagnet MnGe, the 2011 discovery of a large, wide-temperature-range topological Hall effect (Kanazawa et al., Phys. Rev. Lett. 106, 156603) attributed the effect to a real-space emergent magnetic field from an anticipated short-period (3-6 nm) noncoplanar spin texture formed under applied field, but the limited powder neutron diffraction did not suffice to uniquely determine the magnetic modulation vector(s) or the real-space spin configuration. More than a decade of later work — small-angle neutron scattering, real-space electron microscopy, muon-spin rotation, and magnetotransport — has produced strong but partly conflicting evidence for the multi-q character (a magnetic structure built from several symmetry-related modulation wavevectors rather than from a single helical wavevector) of the field-induced state and delivered real-space imaging, yet a unique bulk structure (cubic versus orthorhombic hedgehog lattice, skyrmion lattice, or canonical helix) and the microscopic origin of the anomalous temperature-dependent helical period remain unresolved. The open problem is a cross-probe-consistent unique structure determination for bulk MnGe in zero and applied field, together with the mechanism of its period evolution and its temperature-field phase diagram.

Background

The topological Hall effect (THE) is an additional contribution to the Hall resistivity of a metal that appears when conduction electrons adiabatically follow a noncoplanar spin texture: the Berry phase accumulated along real-space electron trajectories acts as an emergent ("fictitious") magnetic field proportional to the scalar spin chirality χs=m⋅(∂xm×∂ym)\chi_s=\mathbf{m}\cdot(\partial_x\mathbf{m}\times\partial_y\mathbf{m}), where m(r)\mathbf{m}(\mathbf{r}) is the local magnetization direction. In cubic chiral magnets with the B20 crystal structure (space group P213P2_1 3), the absence of inversion symmetry produces a Dzyaloshinskii–Moriya interaction — an antisymmetric exchange term that twists ferromagnetic exchange into a long-wavelength helix whose period is set by the ratio of that term to the symmetric exchange. Under moderate magnetic fields, B20 magnets such as MnSi and FeGe form a two-dimensional skyrmion crystal, a lattice of vortex-like spin whirls each carrying an integer topological charge, which generates a robust THE. MnGe is the short-period member of this family: it orders below TN≈170 KT_N\approx170\ \mathrm{K}, its helical period is only about 3–6 nm (roughly an order of magnitude shorter than MnSi), and it shrinks with cooling, locking to a commensurate value (a period equal to a rational fraction of the atomic lattice constant) below roughly 30 K. Kanazawa et al. (2011) reported a large topological Hall resistivity in MnGe that is nearly temperature-independent below about 70 K, consistent with the emergent-field mechanism, and anticipated from limited powder neutron diffraction that a short-period noncoplanar structure is stabilized from the proper-screw (helical) state in a magnetic field. Because the field-induced diffraction peaks were few and weak, the authors could not uniquely determine the new magnetic modulation vector(s) or the full real-space spin configuration, nor the lattice symmetry and commensurability of the putative skyrmion crystal; direct real-space imaging (e.g., Lorentz transmission electron microscopy, which images magnetic textures in real space through the electron phase shift) or higher-resolution reciprocal-space mapping was called for; the microscopic origin of the temperature evolution of the helical period and its effect on the stability, density, and dynamics of the topological spin entities remained unclarified; and the temperature–field phase diagram required further experimental and theoretical work. Later work recast the expected texture in three-dimensional terms: a hedgehog lattice is a three-dimensional array of noncoplanar spin configurations whose magnetization vanishes at isolated points acting as emergent magnetic monopoles and antimonopoles, and MnGe became the prototype host of such 3D topological order. A multi-q (equivalently multi-k) magnetic structure is one in which several symmetry-related helical modulation wavevectors k\mathbf{k} coexist in the same phase, as opposed to the canonical single-q helix built from one wavevector; in B20 magnets a multi-q superposition can realize either the two-dimensional skyrmion crystal or, with three perpendicular arms as proposed for MnGe, the three-dimensional hedgehog lattice. Small-angle neutron scattering (SANS), which probes magnetic modulations of nanometre scale; muon-spin rotation (muSR), which samples the internal magnetic-field distribution via the precession of implanted spin-polarized muons; and spin-polarized scanning tunneling microscopy (SP-STM), which resolves surface spin order atom by atom, have since been applied, with partially conflicting conclusions, so the uniqueness question posed in 2011 survives in sharpened form.

Problem Statement

Determine the microscopic magnetic structure of bulk MnGe and the mechanism of its anomalous short, temperature-dependent period, as left open by the 2011 discovery of the large topological Hall effect. Concretely: (1) uniquely determine the field-induced noncoplanar magnetic texture of bulk MnGe — its magnetic modulation vector(s) (direction, magnitude, and commensurability, i.e. whether the period locks to a rational fraction of the atomic lattice constant), the lattice symmetry, and the full real-space spin configuration — discriminating among the candidate models that currently fit the available data equally well (multi-k cubic or orthorhombic hedgehog lattices, a skyrmion-lattice or skyrmion–antiskyrmion lattice, or a canonical single-q helix), in a way that is consistent across probes (neutron diffraction, muon-spin rotation, Lorentz-TEM or equivalent real-space imaging, and magnetotransport) and that reconciles the bulk with the thin-film/surface structures; (2) identify the microscopic origin of the evolution of the helical period from roughly 6 nm near the ordering temperature to roughly 3 nm at low temperature, including its commensurate lock-in below about 30 K, and establish how that evolution controls the stability, density, and dynamics of the topological spin entities (skyrmions, hedgehogs, emergent monopoles) under applied field; and (3) complete the temperature–field phase diagram of bulk MnGe — including any additional phases or phase boundaries associated with the noncoplanar texture, such as the field-induced cubic-to-rhombohedral structural transition of the hedgehog lattice and the static-to-dynamic crossover of the emergent monopoles — in a form consistent with the determined structure and period mechanism. These three parts share one objective: replacing the anticipated, model-consistent picture of the 2011 paper with a determined microscopic structure and mechanism; part (1) is the load-bearing unresolved question, part (2) supplies the exchange-level explanation of the length scale that defines that structure, and part (3) integrates both into the phase behavior that the topological Hall effect probes. Because the published probes do not yet single out one structure, part (1) must select a single candidate structure, reconcile it with the probes named above, and resolve the competition among the models from the published cross-probe results, without new measurements.

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