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Magnetic nanoparticles offer a broad spectrum of magnetization reversal processes and respective magnetic states, such as onion, horseshoe, or vortex states as well as various states including domain walls. These states can be correlated with stable intermediate states at remanence, enabling new quaternary memory devices storing two bits in one particle. The stability of these intermediated states was tested with respect to shape modifications, variations in the anisotropy axes, and rotations and fluctuations of the external magnetic field. In our micromagnetic simulations, 6 different stable intermediate states were observed at vanishing magnetic field in addition to the remanence state. The angular region of approx. 5°–12° between nanoring and external magnetic field was identified as being most stable with respect to all modifications, with an onion state as technologically best accessible intermediate state to create quaternary memory devices.

Magnetic nanoparticles have gained the interest of several research groups due to their unique magnetic properties, enabling their possible application in magnetic storage media, sensors, MRAMs, spin valves, and so on [

Particularly “square rings” of different shapes and materials have shown interesting magnetic properties. In several situations, hysteresis loops with steps on both sides occur in theoretical and experimental investigations, allowing creating “quaternary” memory systems with four stable states at vanishing external magnetic field and thus the possibility of storing two bits in one position [

In this paper, phase diagrams of magnetic states are presented for “ideal” and “real” square rings, varying the magnetocrystalline anisotropy axes between a random state, as in a typical sputtered sample, and two epitaxial states, as they could be gained by molecular beam epitaxy (MBE) [

For modelling, the micromagnetic simulator OOMMF (Object Oriented Micromagnetic Framework) [

Nanostructures of two different geometries were chosen for simulation (Figure

Ring shapes used for simulations with square (a) and diagonal corner cuts (b).

As material, iron (Fe) was chosen which showed ideal properties to reach the aforementioned intermediates states [^{3}, Gilbert damping constant

To avoid statistical variations which would superpose the effects due to interactions between the neighboring nanorings, the influence of thermal fluctuations of temperatures above zero was ignored. It can be assumed that any temperature above zero will further decrease the field regions in which all nanorings show the same behavior.

The anisotropy was on the one hand assumed to be uniform on the macroscale, that is, having randomly distributed axes in each cell, to create a sputtered nanoring with polycrystalline structure. In other simulations, the anisotropy axes were fixed at 0°/90° as well as at ±45° to create both extremal cases of epitaxially grown nanorings.

The external magnetic field was applied parallel to the sample plane. The sample orientation was varied between 0° (external magnetic field parallel to the

Figure

Sample hysteresis loop with the main magnetization states identified as onion and horseshoe states. Lilac marks depict the coercive field

Possible magnetic states for not-too-large angles: horseshoe 1 (a), onion (b), and horseshoe 2 (c).

While these states are also visible in the “realistic” shape with diagonal corner cuts, other states occur exclusively in the “ideal” nanorings with square corner cuts with randomly distributed anisotropy axes, that is, in a “sputtered” nanoring. These states are depicted in Figure

Possible magnetic states for larger angles, simulated in a nanoring with square corner cuts and randomly distributed anisotropy axes: one 360° domain wall (a), one 360° domain wall (b), and two 360° domain walls (c).

In Figure

Phase diagrams of minor loops via onion states, simulated for nanorings with square corner cuts (a) and diagonal corner cuts (b). The fields

To evaluate the reason for this difference as well as the occurrence of other possible magnetic states, Figure

Phase diagrams simulated for nanorings with square corner cuts, assuming epitaxial anisotropy orientation along 0° (a), 45° (c), and random anisotropy axes (b). The colors of the texts correspond to those in Figure

In case of a random distribution of the anisotropy axes, that is, a sputtered sample, most of the smaller angles up to approx. 17° reverse magnetization in the aforementioned steps, that is, from positive saturated onion via first horseshoe, intermediate onion, and second horseshoe to negative saturated onion. Nevertheless, for some angles this process works without horseshoe states. In this special nanoring, additional domain wall states may occur for larger angles (cf. Figure

It should be mentioned that the sputtered sample shows more deviations from a clear correlation between sample angle and coercive fields due to the arbitrary distribution of anisotropy axes. This effect results in small variations of the simulated values for repeated simulations.

In Figure

Phase diagrams simulated for nanorings with diagonal corner cuts, assuming epitaxial anisotropy orientation along 0° (a), 45° (c), and random anisotropy axes (b). The colors of the texts correspond to those in Figure

In general, it can be stated that the nanoring with diagonal corner cuts shows a more reliable, less arbitrary spectrum of magnetization reversal mechanisms. Due to their statistic character, the domain wall states in the “sputtered” sample with square corner cuts are only of academic interest. From the technical point of view, those angular regions are best suited which

avoid instabilities, for example, arising from symmetry axes, such as 0°, or from changes in the magnetization reversal processes, such as around 17°–20°;

are similar for patent shape and diagonal corner cuts;

are similar for sputtered and both epitaxial samples;

are similar for neighboring angles;

have large differences between

have large differences between

This means that a sputtered sample in an orientation around 5°–10° should be preferred in technological applications, such as quaternary magnetic storage devices.

To conclude, we have given an overview of different magnetization reversal processes and magnetic states occurring in square nanorings grown by sputtering or epitaxially from iron. Using micromagnetic simulations, we have shown the variety of magnetic states which can be reached in such nanostructures, depending on the exact shape, anisotropy orientation, and external magnetic field direction. While some processes exhibited a statistical nature and could not be repeated, an angular region around 5°–10° between external field and nanoring side has proven to reverse magnetization via a stable intermediate onion state which can be used for technological purposes, for example, in storage devices or MRAM applications.

The authors declare that there are no conflicts of interest regarding the publication of this paper.