Skyrmions in magnetic nanostructures — Computational Science and Data Science

Summary of our recent Scientific Reports paper on stability, hysteretic behaviour, and reversal mechanism of skyrmionic textures in confined helimagnetic nanostructures.

Energy minimisation leads system to Skyrmion configuration -- without applied field

Marijan Beg, Rebecca Carey, Weiwei Wang, David Cortes-Ortuno, Mark Vousden, Marc-Antonio Bisotti, Maximilian Albert, Dmitri Chernyshenko, Ondrej Hovorka, Robert L. Stamps, and Hans Fangohr
Ground state search, hysteretic behaviour, and reversal mechanism of skyrmionic textures in confined helimagnetic nanostructures
Scientific Reports 7, 17137 (2015)
(Online: journal arXiv.org)

The system in this study

Thin film helimagnetic FeGe disk sample with thickness 10 nm and radius d.
The system's total energy contains the following contributions: symmetric exchange, bulk DMI, demagnetisation energy, and Zeeman energy terms.
Uniaxial anisotropy energy was neglected in this study because it is assumed that the simulated material is isotropic and also to determine whether the magnetocrystalline anisotropy is a crucial mechanism allowing the stability of skyrmionic textures.
Full three-dimensional model which does not assume magnetisation invariance in the out-of-film direction and includes a full computation of demagnetisation energy.

Geometry 2

Thin film disk sample with thickness 10 nm and diameter d in uniform external magnetic field applied in the positive z-direction.

Ground state search in confined helimagnetic nanostructures

So far, skyrmions in helimagnetic materials had to be stabilised using an external magnetic field. For practical applications of skyrmions in high-density, low-power data storage and information processing devices, this requirement is an obstacle, because these devices are hard to construct and engineer, volatile, and consume more energy. Here, we study the stability of skyrmionic textures in confined helimagnetic nanostructures and demonstrate that skyrmions can be the energetic ground state in helimagnetic materials in absence of an external magnetic field which is of great significance for their practical application.

We carry out the following (simulated) experiment:

Relax the system from multiple different initial states, including random states.
Compute the energies of final relaxed states.
Select the state with the lowest energy and identify it as the ground state for that parameter space point.

We repeat this procedure at all parameter space points, where we vary the disk diameter d and external magnetic field H.

We identify that skyrmionic textures emerge in confined helimagnetic nanostructures in absence of both external magnetic field and magnetocrystalline anisotropy, which is in contrast to bulk and infinite thin film samples. Skyrmionic textures occur as incomplete Skyrmion (iSk) and isolated Skyrmion (Sk). In the case of an incomplete Skyrmion, the magnetisation field does have a full rotation along the disk diameter, required for the magnetisation to cover the entire sphere and the skyrmion to be present in the sample. In the isolated skyrmion state, a complete magnetisation rotation is present in the sample along the disk diameter, and the magnetisation covers the sphere at least once.

Figure 2 in the paper. (a) The scalar value for the thin film disk sample with thickness 10 nm as a function of disk diameter d and external out-of-plane magnetic field H (as shown in an inset). (b) Two identified ground states: incomplete Skyrmion (iSk) and isolated Skyrmion (Sk) magnetisation configurations at single phase diagram points together with their out-of-plane magnetisation component profiles along the horizontal symmetry line.

Apart from identifying the ground states, we also show the regions of metastability for other magnetisation configurations observed in this study. Similar to the iSk and Sk states, we also observed another radially symmetric state at large disk diameters and at low external magnetic field values and refer to it as the Target (T) state, where the magnetisation rotation along the disk diameter occurs at least twice. Three different helical states were also observed and they differ in the number of half-periods that can fit in the disk diameter, plus additional magnetisation twist at the boundaries due to the specific boundary conditions.

Figure 1 in the paper. The phase diagram with regions where different states are in equilibrium together with magnetisation configurations and out-of-plane magnetisation component along the horizontal symmetry line corresponding to different regions in the phase diagram.

The role of demagnetisation

Because the stability of skyrmionic textures in helimagnetic samples at zero external magnetic field and in absence of magnetocrystalline anisotropy is surprising, we also conduct additional studies in order to determine what might be the stabilising mechanism that allows the skyrmionic textures to be the ground state.

We repeat the study, but this time ignoring the demagnetisation energy contributions (setting the magnetostatic energy artificially to zero). We notice that in this case, although the isolated skyrmion is still the equilibrium state, it is not the ground state at zero external magnetic field: without demagnetisation, the helical states have lower total energies.

Importance of 2d versus 3d simulation model

We also examine how the finite thickness of the sample affects the stability of iSk and Sk states. More precisely, we study whether energitically it is important if the magnetisation can change along the z-direction (i.e. along the thickness of the thin film). For infinitely large thin films Rybakov et al. [PRB 87, 094424 (2013)] showed this to be important.

We repeat the same study, but approximate the thin film sample with a two dimensional mesh. We observe that the region of helical states stability expands even further. These findings suggest that both demagnetisation energy and variation of magnetisation along the film thickness play a crucial role in the stability of skyrmionic textures. Apart from that, we show that approximating the thin film helimagnetic samples using two-dimensional meshes is not generally justified.

Figure 5 in the paper. The ground state as a function of disk sample diameter d and external magnetic field H computed for the ground state at every phase space point in absence of demagnetisation energy contribution for (a) a 3d mesh and (c) for a 2d mesh. In order to better resolve the boundaries of the Helical (H) state region, the skyrmion number S is shown in (b,d). (e) The magnetisation configurations of three identified ground states as well as the out-of-plane magnetisation component along the horizontal symmetry line.

Hysteretic behaviour

Because we show that both iSk and Sk are the ground state at zero external magnetic field, one can assume that there are two energetically equivalent states at zero field: skyrmionic texture with core orientation up and down. Because of that we study whether the skyrmionic texture core orientation can be changed using an external magnetic field which is crucial for data imprint. Because of that we simulate hysteretic behaviour, from which we conclude that the skyrmion orientation can be changed using an external magnetic field.

Figure 6 in the paper. The average out-of-plane magnetisation component hysteretic dependence on the external out-of-plane magnetic field H for 10 nm thin film disk samples for (a) incomplete Skyrmion (iSk) magnetisation configuration in 80 nm diameter sample and (b) isolated Skyrmion (Sk) magnetisation configuration in 150 nm diameter sample. (c) The magnetisation states and out-of-plane magnetisation component profiles along the horizontal symmetry lines for positive and negative iSk and Sk core orientations from H=0 in the hysteresis loop, both in presence and in absence of demagnetisation energy (demagnetisation-based shape anisotropy).

Since the magnetocrystalline anisotropy is not present in our simulation model, it can be assumed that the the hysteretic behaviour originates from the demagnetisation-based shape anisotropy. Accordingly, we repeat the hysteresis study, but this time ignoring the demagnetisation energy contribution. We observe that hysteresis is still present, which suggests the existence of Dzyaloshinskii-Moriya based shape anisotropy.

Reversal mechanism

In the hysteresis study, we observed that skyrmionic texture orientation can be changed using an external magnetic field. Now, we examine what is the mechanism that facilitates the reversal of skyrmionic textures. Therefore, we relax the system at -210 mT and then abruptly change the external magnetic field to -250 mT. In that process, we simulate the magnetisation dynamics and observe that the skyrmionic texture reversal occurs via Bloch point occurrence and propagation. In this particular case we observe the Bloch point propagating from the bottom to the top boundary of the sample. One can assume that two different propagation directions of a Bloch point (upwards and downwards) are energetically equivalent and that this process is stochastic. Because of that, we change the Gilbert damping value and repeat the reversal simulation. In this case the Bloch point changes its propagation direction.

Summary

Systematic study of equilibrium states in helimagnetic nanostructures
Skyrmionic textures are the ground state in absence of both external magnetic field and magnetocrystalline anisotropy.
Skyrmionic textures emerge as incomplete Skyrmion (iSk) and isolated Skyrmion (Sk) ground states.
Other states were also observed with higher energies that iSk and/or Sk states (Figure 1 in the paper).
Skyrmionic textures undergo hysteretic behaviour when their orientation is changed using an external magnetic field.
Hysteretic behaviour remains present even in absence of magnetocrystalline anisotropy and demagnetisation-based shape anisotropy, suggesting the existence of Dzyaloshinskii-Moriya-based shape anisotropy.
The reversal of skyrmionic textures is facilitated via Bloch point occurrence and propagation.

Future work

Magnetisation dynamics is an active field of research - if you are interested to pursue a PhD in computational modelling of such systems, please get in touch.

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