Three-dimensional magnetic-flux-closure patterns in mesoscopic Fe islands

We have investigated three-dimensional magnetization structures in numerous mesoscopic $\mathrm{Fe}∕\mathrm{Mo}\left(110\right)$ islands by means of x-ray magnetic circular dichroism combined with photoemission electron microscopy (XMCD-PEEM). The particles are epitaxial islands with an elongated hexagonal shape with length of up to $2.5\mu \mathrm{m}$ and thickness of up to $250\mathrm{nm}$. The XMCD-PEEM studies reveal asymmetric magnetization distributions at the surface of these particles. Micromagnetic simulations are in excellent agreement with the observed magnetic structures and provide information on the internal structure of the magnetization which is not accessible in the experiment. It is shown that the magnetization is influenced mostly by the particle size and thickness rather than by the details of its shape. Hence these hexagonal samples can be regarded as model systems for the study of the magnetization in thick, mesoscopic ferromagnets.

Figure 1

(Color online) (a) Atomic force microscopy of a Fe island, about $1\mu \mathrm{m}$ long. (b) Schematic presentation of a typical particle geometry. The angles $\alpha =45°$ and $\beta =35.26°$ are fixed parameters that result from the monocrystalline structure of the islands. The shape is uniquely defined by the points $A_1$, $A_2$, $B_1$, $B_2$ and the thickness $t$. Note that this set of parameters is subject to various constraints. For instance, the edges $(A_1,A_2)$ and $(B_1,B_2)$ are parallel to each other (the $x$ direction).

Figure 2

Example of an experimental observation of the particle shape and the magnetization structure. (a) Two Fe islands, incidentally located next to each other, imaged with LEEM. (b) The XMCD-PEEM image displays the in-plane magnetization component parallel to the incident beam. The beam direction is sketched by the white arrow. Due to grazing incidence of the photons (74° with respect to the plane normal) the back side of the island is shadowed; thus it appears dark in the XMCD-PEEM image.

Figure 3

(Color online) Simple magnetic-flux-closure patterns in thin-film elements. (a), (c) Landau and diamond structure in a rectangle, respectively; (b), (d) modified Landau and diamond structure, respectively, in a hexagonal particle. In the hexagonal element, the closure domains are subdivided in two regions separated by an additional domain wall (dashed line).

Figure 4

(Color online) Difference between idealized and observed magnetic domain structure. (a) LEEM image of the sample shape. XMCD-PEEM image of the magnetic domain structure. (c) Schematic domain structure for the given shape according to the Van den Berg scheme. (d) Approximate shape of the magnetic domains according to the experimental observation.

Figure 5

(Color) To avoid discretization errors in the simulation of the three-dimensional magnetic domain structure, the sample is first reduced to a tenth of its size. By using cycles of micromagnetic energy minimization and three-dimensional adaptive mesh refinement, the sample is gradually increased to its real size (a). After this procedure, a finite-element mesh with regions of strongly different discretization density is obtained (b). The finite elements are particularly small in regions of strong magnetic inhomogeneities (vortices, domain walls). Owing to the mesh refinement, even the nanometric vortex core with magnetization perpendicular to the surface is fully resolved, as can be seen in the red spot in panel (c). Throughout the paper, unless otherwise specified, the color coding used to represent a component of the reduced magnetization $\mathit{m}$ is chosen according to the bar on the bottom of panel (a).

Figure 6

Experimental and simulated data for five self-assembled $\mathrm{Fe}∕\mathrm{Mo}\left(110\right)$ islands of different size and shape (a)–(e). First row: The shape of the particle is obtained by LEEM imaging. The particle thickness is derived from the width of the (001) facets, which appear as a dark border around the hexagonal top surface, by using the known inclination angles of the inclined facets (cf. Fig. 1). The sample’s approximate lateral dimension and thickness is displayed on top of each column. Second row: One magnetization component in the surface plane is imaged as grey scale with XMCD-PEEM. In sample (a), the contrast refers to the magnetization component parallel to the long edge while in the other samples the in-plane magnetization component perpendicular to the long edge is displayed. Notice that only the top surface, i.e., the light grey internal hexagon in the first row, is imaged. Third row: Micromagnetic simulation results. To compare the results with the XMCD-PEEM experiments, the same magnetization component is displayed in grey scale. The sample shape was modeled according to the LEEM images and the model explained in Fig. 1.

Figure 7

(Color) Magnetization distribution on different cross sections. The example shown here refers to the particle displayed in Fig. 6b. Figures (a) and (d) show the top view on the three-dimensional sample with lateral facets. (b),(e) Cross section through the middle of the sample at $z=t∕2$; (c),(f) magnetization on the bottom surface of the sample, $z=0$. (g) $m_z$ component on the cross section through the middle of the sample parallel to the $xz$ plane. (h)–(j) Asymmetric Bloch wall: magnetization components on a cross section at midlength, parallel to the $yz$ plane.

Figure 8

(Color) Simulation of four different configurations in a sample of identical shape. The shape and the size of the sample is the same as in Fig. 6d. Although topologically the structure is very similar in all cases (Landau structure with asymmetric Bloch wall), the resulting magnetic patterns are significantly different. In the structures (a) and (b) the Bloch component of the internal wall, $m_z$, is positive, whereas in (c) and (d) it is negative. The structures (a) and (c) have the same direction of the magnetization in the Néel caps, $m_y$, which is opposite to the orientation in the structures (b) and (d).

Figure 9

(Color) Geometry dependence of the magnetization structure in a three-dimensional island. (a) Perspective view on the simulated magnetization structure of the particle shown in Fig. 6b. (b) Simulated magnetic domain structure in a particle of same size and thichkness, but with symmetric in-plane shape and vertical facets. (c) Even if the hexagonal shape is replaced by a smooth elliptical boundary, the magnetization structure remains almost unchanged, with the same asymmetric characteristics.

Figure 10

(Color online) In a relatively large and particularly thin island, identifiable as such in a LEEM image (a) by the thin dark boundary (the lateral facets), a “diamond state” magnetization structure was found. The diamond state is characterized by two vortices with opposite vorticity and a diamond-shaped central domain, which is clearly recognizable on the right panel (b).

Figure 11

(Color) (a)–(c) $z$, $x$, and $y$ component of the magnetization of a simulated magnetic domain structure with undulated central domain wall. (f) $z$ component of the magnetization on a cross section of constant $y$. The alternating red-blue regions indicate switches of the Bloch component. (d) LEEM image and XMCD-PEEM image (e) of a sample in an apparently similar magnetization state. The size of the island is $1500\mathrm{nm}\times 620\mathrm{nm}\times 130\mathrm{nm}$.