We studied a square lattice of spins in frame of Heisenberg model with direct exchange and Dzyaloshinskii-Moriya interaction. For the analysis of data obtained during the Monte Carlo simulation, a convolutional neural network was used for the recognition of different phases of the spin system, which was dependent on simulation parameters such as DMI and external magnetic field (Hz). Based on these data, the phase diagram (Hz,D) was plotted. The various states of the systems under observation were visualized, and the boundaries between the different phases were defined as spirals, skyrmions and others. We proposed the controlling method for movement of skyrmions using by Monte Carlo simulation. Monte Carlo method, convolutional neural network, magnetic skyrmion, Heisenberg model, VI International Conference Information Technologies and High-Performance Computing (ITHPC-2021),
Spintronics or magnetic electronics is continually evolving, and new promising materials for new
storage and processing data devices are emerging. Skyrmions are attractive candidates for information
carriers in a new type of non-mechanical magnetic medium - a racetrack memory - because they are
only a few nanometers in size, very stable, and can be driven by pulses of spin-polarized currents. At
a fundamental level, skyrmions are model systems for topologically protected spin structures and can
be regarded as an analogue of topologically protected states, emphasizing the role of topology in the
formation of complex states of condensed matter [
In our paper, the conditions for the nucleation and stable existence of magnetic skyrmions in twodimensional magnetic films were considered in the frame of the classical Heisenberg model. For computer simulation, we used the Metropolis algorithm. For the analysis of the data obtained during the Monte Carlo simulation, a convolutional neural network (CNN) was used for the recognition of different phases of the spin system, depending on simulation parameters such as DzyaloshinskiiMoriya interaction and the external magnetic field (Hz). Based on these data, the phase diagram (Hz,D) was plotted. Also, we proposed the controlling method for the movement of skyrmions.
2020 Copyright for this paper by its authors.
In 1960 Dzyaloshinskii presented a model to describe weak ferromagnetism [
= ( + + ) + , = − ∑ ⃗ ⋅ ( ⃗ + ̂ + ⃗ + ̂ ) − ∑ ⃗ − ∑ ⃗|⃗⃗⃗ |2, = − ∑ ⃗ × ⃗ + ̂ ⋅ ̂ + ⃗ × ⃗ + ̂ ⋅ ̂, (1) (2) (3) where ⃗ is the atomic spin, is the value of ferromagnetic short-range exchange interaction, is the value of DMI, Hz – an external magnetic field and a magnetic anisotropy coefficient is HA. 2.2.
The Metropolis algorithm is used to determine the global minimum. The main idea is to uniformly sample the state space with a given distribution probability. At each iteration of the sample, the configuration of the system changes due to a change in the orientation of a randomly selected spin. The configuration is accepted and becomes the initial one for the next step if the new energy value is greater than the previous one ( 1 > 2); otherwise, it is accepted with the probability: ( → ) = ( (( )) , 1) (4)
Due to this, the algorithm avoids getting stuck in local minima. Convergence is achieved after
passing a given number of Monte Carlo steps until the moment when the standard deviation reaches a
specified minimum, depending on the problem being solved [
We used configurations of spin systems obtained at different simulation parameters for the training
and subsequent classification of them in a neural network. To date, the most accurate analysis results
are demonstrated by neural networks based on convolutional architecture. We used the TensorFlow
library to create a convolutional neural network [
In our research, we have reduced the problem of determining the phases of spin systems to the problem of image classification - in fact, to the main problem area in which neural networks are used. For recognising images, CNN accepts them in the RGB format as a three-dimensional matrix. In our case, the convolutional neural network received as input a three-dimensional array representing the components of a three-dimensional spin in the frame of the Heisenberg model.
Following this, the convolutional neural network learned, using the training dataset, to highlight the features inherent in one or another spin configuration. Our CNN consists of next layers (main ones), see Figure 1: 1. Input layer
Input data (configurations of spins), each of the neurons (spins) of which is assigned an initial random weight. The components of a three-dimensional vector were fed to the network input (i.e., the components of Heisenberg spin). The dataset was prepared using Monte Carlo simulation data for training the neural network in state recognition. 2. Convolutional layer with 3×3 filter
When neurons are connected to only a few neurons in the next layer, the layer is said to be convolutional. The convolutional layer acts as a filter that discards the least informative parts of the input data. Each layer has filters (i.e., matrices with weight values). When the filter moves along the matrix of the previous layer, each filter element is multiplied by the value of the neuron, and the values are summed up and written to the feature map.
3. Pooling layer for reducing the dimensions of the data 4. Fully connected layer
Fully connected layers are used for classification. All layers before the fully connected layer are used to highlight various features that are fed to the input of the classifier. This layer can also be used as the final (output) CNN layer, the result of which is the probability of the input configuration of spins belonging to a certain class.
We studied different phases that appeared depending on the magnitude of the DzyaloshinskiiMoriya interaction D and the external magnetic field Hz at fixed temperature T, see Figure 2. The convolutional neural network was used to analyze the data obtained from the Monte Carlo simulations for the recognition of the different phases of the spin system, dependent on the simulation parameters.
In a magnetic film, with an increase of the magnetic field strength and DMI, various phases were observed for the flat Heisenberg spin systems: Spiral (Sp), Spiral-skyrmion (SpSk) Skyrmion (Sk), Skyrmion-ferromagnetic (SkF) and Ferromagnetic (FM) phases, see Figure 3. In Skyrmion phase, due to the alignment of the stripes against the magnetic field, stable skyrmions are formed in the system. In these skyrmions, the spins of the nucleus are directed against the magnetic field. In this study, skyrmions of the Bloch type were formed.
We could also “push” the skyrmion from one side of the sample to another using the increasing of a magnetic anisotropy. This is a rather precise method, see Figure 4. In frame of this numerical experiment, we have an anisotropy gradient from 0.9 on the left side of the sample to 0.1 on the other side. We have an incline plane of anisotropy gradient. And we increase the value of anisotropy step by step for pushing the skyrmion. In the physical experiment, it is possible to control the anisotropy applying the voltage to the sample.
In the frame of the classical two-dimensional Heisenberg model, a spin system with direct shortrange exchange was modelled, and a study of its competition with the Dzyaloshinskii-Moriya interaction was carried out. Due to the direct exchange interaction, the neighboring spins of the system are collinearly aligned and, in turn, the Dzyaloshinskii-Moriya interaction contributes to the deviation of the spins from parallel orientation. As a result, competition results between collinear and noncollinear alignments of spins, which leads to the transition of the system of spins from a ferromagnetic to a spiral ground state. In the presence of an external magnetic field, stable topological structures - magnetic skyrmions - are generated in such systems.
In this paper, we proposed a method for manipulation of the position of a skyrmion using by a control of anisotropy gradient.
We performed MC simulation and the convolutional neural network was used for the recognition of the different phases of the spin systems, depending on the simulation parameters. For the visualisation and analysis of research data, the phase diagram (Hz,D) was plotted.
The data obtained in the numerical experiments will be used in our further studies to determine the model parameters of the system for the formation of a stable skyrmion state, both in the form of individual skyrmions and skyrmion lattices and for the development of methods for controlling skyrmions in magnetic stripes.
This work was supported in frame of joint program "Mikhail Lomonosov" of the Ministry of
Science and Higher Education of Russia and German Academic Exchange Service (DAAD)
(#229821). The studies were carried out using the resources of the Center for Shared Use of Scientific
Equipment "Center for Processing and Storage of Scientific Data of the Far Eastern Branch of the
Russian Academy of Sciences" [