Magnetic Ground State of Single and Coupled Permalloy Rectangles

We have studied the magnetic domain structure in Permalloy rectangles that reveal flux-closure domain configurations. Arrays with varying spacing between the rectangles are investigated by scanning electron microscopy with polarization analysis as well as by micromagnetic simulation. In contrast to general expectation, rectangles in the flux-closure Landau state show significant coupling and form a magnetic pattern of common chirality. The coupling is due to the stray field that originates from small changes of the magnetization alignment, which is sensitive to the exact shape and the separation of the rectangles.

Figure 1

SEMPA image of an array of well separated Permalloy rectangles (a). The image exhibits the magnetic structure obtained by one polarization component. The polarization sensitive axis is parallel to the horizontal edge. The dimensions of the rectangles are $1\mu \mathrm{m}\times 2\mu \mathrm{m}\times 23\mathrm{nm}$. The rectangles show either the Landau state or the diamond state, only one structure is in the $\mathsf{S}$ state. (b)–(e) Sketch of the Landau state (b), diamond state (c), $\mathsf{S}$ state (d), and $\mathsf{C}$ state (e). “$\mathsf{S}$ state” and “$\mathsf{C}$ state” refer to the flux lines through the rectangle having shapes similar to the letters $\mathsf{S}$ and $\mathsf{C}$, respectively.

Figure 2

Magnetic structure of Permalloy rectangles obtained by SEMPA. (a) and (b) give the images obtained via the two polarization sensitive axes oriented perpendicular to each other, parallel to the edges of the images. Both images have been measured simultaneously. (c) SEM image of a single Permalloy structure. In the lower left corner, a zoom into one edge is plotted. The inset displays an AFM line profile of the edge of the structure, which was capped by 2 nm Pt.

Figure 3

Frequency of magnetization orientation as a function of angle. The angle is given with respect to the short edge of the Permalloy rectangle. The experimental data for the diamond- and the Landau structure is shown.

Figure 4

Frequency of magnetization orientation as a function of angle for a single Landau structure obtained by simulations [15]. The black curve (i) gives the result for a perfect rectangle with SP1 parameters [2]. The gray plot (ii) additionaly includes edge roughness with a period of 30 nm and a peak-to-peak amplitude of 20 nm. The green curve (iii) gives the result for SP1 including inclined edges with an inclination length of 35 nm.

Figure 5

SEMPA micrograph of an array of coupled Permalloy rectangles. (a) and (b) give the two polarization sensitive axes oriented perpendicular to each other, as indicated by the arrows. Both images have been measured simultaneously.

Figure 6

Magnetization orientation of one single element from the array of Fig. 5. (a) shows a color (arrow) plot of the magnetic microstructure. The magnetization orientation was calculated from the SEMPA images. For easier understanding, arrows are given which show the direction of magnetization. (b) is the frequency of magnetization versus angle for the element shown in (a).