Abrupt Topological Transitions in the Hysteresis Curves of Ferromagnetic Metalattices

When a metal is confined to the interstices of an inert colloidal crystal, the intrinsic order parameter(s) of electronic and magnetic phenomena within the metal interact with the structural order parameter of the surrounding (and confining) colloidal crystal. If the magnetic stiffness length is comparable to the colloidal lattice constant, the interplay of competing interactions stabilizes multiple topologically distinct magnetic phases separated by sharp transitions in the hysteresis curves. The colloidal confinement also induces substantial coercivity in metals that are perfectly soft in the bulk.

Figure 1

Left: The tetrahedral and octahedral interstitial spaces of an fcc colloidal lattice (with spheres centered at the vertices) are filled with ferromagnetic material. Intersections of necks with the cell edges are in light gray. Right: The primitive infiltrant cell within a single fcc cell, with colloidal spheres on each vertex. The necks between filled sites penetrate each face. Arrows show magnetization directions in the necks for a moderately strong external field pointing upward, along (001).

Figure 2

Top: The coercivity as a function of colloidal sphere diameter $D$ with the external field in the easy direction, (001). At $D\ll {\ell }_{\mathrm{m}\mathrm{a}\mathrm{g}}$, the system behaves as a hard magnet. For $D>{\ell }_{\mathrm{m}\mathrm{a}\mathrm{g}}$, the system better accommodates gradual changes in magnetization and behaves like a soft magnet. The coercivity peaks near $D\sim {\ell }_{\mathrm{m}\mathrm{a}\mathrm{g}}$. Bottom left: Hysteresis loops for an external field along (001). At $D=30\mathrm{n}\mathrm{m}$, the magnetization aligns parallel to the field. At $D=200\mathrm{n}\mathrm{m}$ ($D>{\ell }_{\mathrm{m}\mathrm{a}\mathrm{g}}$), the reversal traverses several stages of local energy minima. Bottom right: The magnetization along a noneasy axis (111). The reversal is triggered at a smaller external field, is softer, and passes through more complex local minima.

Figure 3

(a) Layered ferromagnetism at $D=90\mathrm{n}\mathrm{m}$ with $\mathbf{H}$ along $(001)$ ($A_2$ of Fig. 2). Layered ferromagnetic domains form normal to (001). (b) The magnetic ring along (001) inside an octahedral site, showing only a single cell of the supercell ($B_2$ of Fig. 2). This magnetization pattern minimizes the magnetostatic energy. (c) Layered ferromagnetism at $D=30\mathrm{n}\mathrm{m}$ with $\mathbf{H}$ along $(111)$ ($D_2$ of Fig. 2). The layer planes are normal to (111) and the magnetization within the domains is along one of the easy-axis directions.