Quasiantiferromagnetic $120°$ Néel state in two-dimensional clusters of dipole-quadrupole-interacting particles on a hexagonal lattice

The magnetostatic interactions of colloidal particles, “capped” with radially magnetized Co/Pt multilayers, are modeled. Motivated by experiment the particles are arranged in microscopic two-dimensional clusters on a hexagonal lattice and are free to rotate. The thermodynamically stable states of clusters containing up to 108 particles are calculated theoretically by means of Monte Carlo simulations in the framework of multipole expansion. It is shown analytically that radially magnetized hemispheres have higher-order multipole moments beyond the dipole. Depending on geometrical details also even order moments appear. The even order moments break the inversion symmetry of the magnetic potential of a single particle. For a specific mixing ratio of dipole and quadrupole moments, the experimentally observed antiferromagnetic $120°$ Néel state in the clusters is found.

Figure 1

(Color online) Properties of the capped particles. (a) Metallic layers are deposited on top of silica spheres by thermal evaporation and molecular beam epitaxy; (b) scanning electron microscope (SEM) image illustrates capped colloids—particles with metallic hemispheres (green) on top; and (c) sketch of capped particles in the parallel orientation that is used to normalize the interaction energy. The direction of magnetization stays normal to the surface of the metal film.

Figure 2

Magnetic colloidal clusters with 2D arrangements of macrospins, which resemble a $120°$ Néel state, observed for antiferromagnets on triangular lattices.

Figure 3

(Color online) Contour plot in the $x\text{−}z$ plane of the magnetic potential $\Phi$ (with arbitrary scaling) of a cap with ${\theta }_{\text{c}}=\pi /2$ and $\sigma \propto \text{cos}\theta$. Negative equipotential lines are white, while constant $\Phi \ge 0\text{A}$ are shown in black. The potential is evaluated from the multipoles up to order 32. The contour line with $\Phi =0\text{A}$—the almost horizontal one in the right-hand side of the figure—is shifted in direction of larger $z$ if ${\theta }_{\text{c}}$ decreases. The dipole cap is indicated by a red-green double line.

Figure 4

The relative difference between dipole-dipole and quadrupole-quadrupole energies as a function of $s$ for two spheres (with a diameter of one unit distance) pointing in the $z$ direction separated in the $x$ direction by one unit distance.

Figure 5

(Color online) Optimal angle $\alpha$ (see text) for a three-particle cluster as a function of quadrupole contribution $s$. The pure dipole case (solid arrows) corresponds to $\alpha =0\text{rad}$.

Figure 6

Monte Carlo simulations of 12-particle clusters and varying quadrupole contribution. The dipolar state $\left(s=0\right)$ shows the typical vortex state. In case of the pure quadrupole $\left(s=1\right)$ the equatorial plane is plotted as the parity is even and the up and down directions are degenerate. Hence, in this case it does not make sense to interpret the black caps as magnetic material. The cluster represents a section of the well-known pinwheel structure (Refs. 4, 5). For intermediate $s$ the $120°$ Néel structure appears similar to the ground state observed in experiments.

Figure 7

(Color online) Monte Carlo simulation of 108-particle clusters. In case of $s=0$ each sextant shows a collinear state, which is, therefore, called quasiferromagnetic. For $s=0.4$ almost the full area shows the $120°$ Néel state but at the edges some weakly bound spheres have deviating orientations. The three arrows in the lower right corner schematically show the high symmetry of the unit cell. Choosing this unit cell, the cell boundaries are given by the honeycomb lattice drawn on top of the simulated structure (dashed line). At $s=1$ the pinwheel state forms, for which—as in Fig. 6—only the equatorial plane is drawn.

Figure 8

Monte Carlo simulation of 108-particle clusters with $s=0.55$. In the left graph the state is shown using the standard representation of capped spheres, while in the right graph the orientation of the dipole is neglected, i.e., only the orientation of the quadrupole is shown via the white line of the sphere’s equatorial plane. While the left graph seems rather disordered the right one shows domains of a pinwheel state coexisting with herringbone structure domains.

Figure 9

(Color online) Monte Carlo simulation for a 108-particle cluster with $s=0.3$. The cluster shows several domains with a herringbone structure. The largest domain is highlighted.

Figure 10

The coefficients $A_l\left(\theta \right)$ as a function of $\theta$: $l=1$ (solid line), $l=2$ (dotted dashed), and $l=3$ (dotted). Note that for decreasing $\theta$ the influence of $A_2\left(\theta \right)$ increases relative to $A_1\left(\theta \right)$, i.e., the quadrupole becomes more important.