The anomalous Hall effect in Mn₃Sn is large but confined to the basal plane — the component of Hall conductivity along the c-axis is negligible. The coplanar 120° spin arrangement that produces the in-plane Hall signal has vanishing Berry curvature along the out-of-plane direction. A material with giant Hall conductivity in two dimensions but not the third is useful only in oriented thin films.
The paper shows that enriching the Mn content drives a transition from the coplanar spin configuration to a non-coplanar one tilted toward the c-axis. The tilt is mediated by four-spin ring exchange interactions — a higher-order magnetic coupling that becomes relevant when extra Mn atoms occupy interstitial sites. The non-coplanar configuration has nonzero Berry curvature in all three directions, producing anomalous Hall conductivity over the full three-dimensional space.
The basal-plane component reaches approximately −468 Ω⁻¹cm⁻¹; the c-axis component, previously zero, emerges at −229 Ω⁻¹cm⁻¹ with light self-doping. The full-space effect is achieved without external fields, strain, or heterostructure engineering — only by adjusting the Mn:Sn ratio.
The structural lesson: dimensionality of a transport response is not fixed by the material's symmetry class but by the spin configuration's geometry. The same crystal structure with a coplanar spin arrangement gives a 2D Hall effect; with a non-coplanar arrangement, a 3D Hall effect. The missing dimension was not absent from the material — it was absent from the magnetic order. Tilting the spins adds a spatial direction to the Berry curvature.