^{1,a)}, N. Mikuszeit

^{1}, E. Y. Vedmedenko

^{1}and H. P. Oepen

^{1}

### Abstract

The influence of tilted edges on the magnetostatic properties of uniformly magnetized thin rectangular elements is studied. To calculate the magnetostatic energy, the Poisson equation is solved. The shape of the magnetic element is approximated by horizontally assembled thin cuboids and the solutions of Rhodes and Rowlands [Proc. Leeds Phil. Soc.6, 191 (1954)] are utilized. A second approach is the straightforward integration of the Poisson equation taking into account the trapezoidal shape of the side faces due to the tilted edges. For an adequate number of cuboids, both methods agree very well. It is found that the shape anisotropy of a single magnetic element with tilted edges is reduced compared to that of an ideal cuboid. For a two element system the shape anisotropy competes with the magnetostatic interaction favoring a magnetization orientation parallel to the connecting line of the elements. If the elements are oriented in-line with their short axes, the easy magnetization axis switches at a critical distance between the elements. This distance increases when the elements have tilted edges.

Financial support from the Deutsche Forschungsgemeinschaft (SFB 668, Project Nos. A11 and A12) is acknowledged.

I. INTRODUCTION

II. METHODS

III. RESULTS AND DISCUSSION

A. Anisotropy of a single element

B. Interaction of two magnetic elements

IV. SUMMARY

### Key Topics

- Anisotropy
- 38.0
- Magnetic anisotropy
- 25.0
- Surface charge
- 9.0
- Demagnetization
- 7.0
- Double layers
- 6.0

## Figures

Sketch of a magnetic element with tilted edges.

Sketch of a magnetic element with tilted edges.

Sketch of the edge of a magnetic element. The tilted edge is defined by the inclination angle or the edge broadening . The shape of the element is approximated by slices of thin cuboids with decreasing base area.

Sketch of the edge of a magnetic element. The tilted edge is defined by the inclination angle or the edge broadening . The shape of the element is approximated by slices of thin cuboids with decreasing base area.

Comparison of different stray field energies for a magnetic element magnetized along its long axis as a function of the number of slices used for the method of Rhodes and Rowlands. The stray field energies were calculated by solving the Poisson equation with the method of Rhodes and Rowlands and by straightforward integration taking into account the tapered shape of the element. The edge broadening is varied. Nominal dimensions of the element: .

Comparison of different stray field energies for a magnetic element magnetized along its long axis as a function of the number of slices used for the method of Rhodes and Rowlands. The stray field energies were calculated by solving the Poisson equation with the method of Rhodes and Rowlands and by straightforward integration taking into account the tapered shape of the element. The edge broadening is varied. Nominal dimensions of the element: .

Shape anisotropy of magnetic elements with tilted edges as a function of the inclination angle broadening in units of the anisotropy of the corresponding perfect magnetic element, respectively .

Shape anisotropy of magnetic elements with tilted edges as a function of the inclination angle broadening in units of the anisotropy of the corresponding perfect magnetic element, respectively .

Shape anisotropy of a magnetic element with tilted edges as a function of the short axis length in units of the shape anisotropy of a perfect magnetic element. The edge broadening is varied while the aspect ratio and the height are fixed.

Shape anisotropy of a magnetic element with tilted edges as a function of the short axis length in units of the shape anisotropy of a perfect magnetic element. The edge broadening is varied while the aspect ratio and the height are fixed.

Configurations of two elements either in-line (a) or parallel to their long axes (b) with different magnetization orientations.

Configurations of two elements either in-line (a) or parallel to their long axes (b) with different magnetization orientations.

Anisotropy of a system of two element system in-line. The anisotropy is given as a function of the distance for different edge broadenings as indicated in the figure and normalized by the values for a perfect single element. The horizontal lines mark the single element anisotropies of given edge broadening. (, , and )

Anisotropy of a system of two element system in-line. The anisotropy is given as a function of the distance for different edge broadenings as indicated in the figure and normalized by the values for a perfect single element. The horizontal lines mark the single element anisotropies of given edge broadening. (, , and )

Anisotropy of a system of two perfect elements arranged parallel to their long axes as a function of their distance in units of the anisotropy of a perfect element with the same dimensions. The aspect ratio is taken as parameter by varying and constant . The height of the elements is 25 nm.

Anisotropy of a system of two perfect elements arranged parallel to their long axes as a function of their distance in units of the anisotropy of a perfect element with the same dimensions. The aspect ratio is taken as parameter by varying and constant . The height of the elements is 25 nm.

The distance of the two element system at which the easy axis is degenerate. The switching is plotted as a function of the aspect ratio of the single element for different edge broadenings. The parameters and are fixed.

The distance of the two element system at which the easy axis is degenerate. The switching is plotted as a function of the aspect ratio of the single element for different edge broadenings. The parameters and are fixed.

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