We studied several types of flat lattices with direct exchange and Dzyaloshinskii-Moriya interaction between spins: a honeycomb lattice with 3 nearest neighbours (NN), a square lattice with 4 NN and a hexagonal or triangular lattice with 6 NN. 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 diagrams (Hz, D) for the different lattices were plotted. The various states of the systems under observation were visualised and the boundaries between the different phases were defined as a spiral, a skyrmion and others. The data from the numerical experiments will be used in further studies to determine the model parameters of the systems for the formation of a stable skyrmion state and the development of methods for controlling skyrmions in a magnetic film.
I. INTRODUCTION
Skyrmions are only formed in magnets in which spin interactions create a magnetic structure with chiral symmetry, such as a vortex. It was previously predicted that such vortices arise when different types of interactions between spins are competing.1,2 The energy of the direct exchange interaction, which contributes to the collinear alignment of neighbouring spins and controls the ordering in the system, is given by the Heisenberg exchange. Another interaction, favouring the perpendicular orientation of neighbouring spins, exists in some Heisenberg magnets, in which the electrons have a strong spin-orbit coupling; this is known as the Dzyaloshinskii-Moriya interaction (DMI).3,4 This is the reason why the spins in the Heisenberg magnet deviate from a parallel orientation, which leads to a chiral structure.5
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 nanometres in size, very stable and can be driven by pulses of spin-polarized currents. The racetrack memory (RM), a new type of data storage, in which information bits are moved along the nano-tracks, is considered to be a potential candidate for promising high-density data storage devices, as an alternative to conventional hard disk drives (HDDs). Thus, topological magnetic objects, such as skyrmions, can be used to store 0 and 1 i.e. the main elements of digital data.6–8 At the 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.9,10
Possible approaches to the implementation of track memory based on skyrmions have been described theoretically.11 Therefore, the creation, detection and control of individual skyrmions have become especially relevant topics in connection with the possible implementation of physical devices based on skyrmions in spintronics. For the development of control methods for magnetic skyrmions in a magnetic nanostrip, it is necessary to conduct a detailed analysis of the simulation parameters and the correlations between them in order to select the optimal parameters for further studies of magnetic skyrmions.
In our paper, the conditions for the nucleation and stable existence of magnetic skyrmions in two-dimensional magnetic films were considered in the frame of the classical Heisenberg model. We studied several types of flat lattices with DMI and the direct exchange between spins:
a honeycomb lattice with 3 nearest neighbours (NN),
a square lattice with 4 NN,
a hexagonal or triangular lattice with 6 NN.
In real materials, a magnet is a continuous magnetic structure, the internal interactions in which lead to the formation of skyrmions. To simulate the magnetic states is made a discretisation of surface by uniform grid, wherein each grid point is regarded as a point magnet. In theory, the discretisation should not affect the magnetic properties of the system, since this is a conditional abstraction necessary for performing calculations. However, in this work we showed that the geometry of the lattices affected the magnetic structure of skyrmions and changed the phase space.
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 DMI and the external magnetic field (Hz). Based on these data, the phase diagrams (Hz, D) for different lattices were plotted. It is known that, one of the features of the practical application of skyrmions is that they are usually stable in a narrow region of low temperatures and external magnetic fields. Therefore, based on the data from the phase diagrams, we can select appropriate parameters that significantly affect on the properties of spin systems and lead to the formation of various phases that provide different functionalities for use in spintronics.
II. MODEL AND ALGORITHMS
Monte Carlo methods, such as the Metropolis12 or Wang-Landau algorithms,13 are not only actively used to study various physical systems14–17 but also continue to actively develop and improve18,19 due to the development of supercomputers. The Metropolis algorithm, being a Monte Carlo method, is a general method suitable for high-performance computing, for studying the thermodynamic properties of substances consisting of interacting individual particles and dynamics in external magnetic fields.20–23 In this study, we used the Metropolis algorithm for the simulation of magnetic skyrmions in the frame of the classical Heisenberg model, taking into account direct short-range exchange and Dzyaloshinsky-Moriya interaction.
A. Heisenberg model
The Heisenberg model is one of the mathematical models of statistical physics used to study phase transitions and critical points. We used a lattice Hamiltonian consisting of the Heisenberg exchange Hamiltonian (HJ) and the DM interaction Hamiltonian (HD) (see formulas below).24,25 We studied the honeycomb lattice with (3 nearest neighbours (NN)), the square lattice (4 NN) and the hexagonal or triangular (6 NN) flat lattice with periodic boundary conditions.
where is a spin, J is the ferromagnetic short-range exchange constant, D is the DM interaction constant, Hh is the Hamiltonian describing the direct exchange interaction between neighboring spins and HZ is the external magnetic field.
The Dzyaloshinskii-Moriya interaction was originally a model for describing weak ferromagnetic interaction.3 Dzyaloshinskii presented a model and introduced an asymmetric term based on the theory of symmetries. Moriya later discovered that the interaction is based on the spin-orbit coupling.4 Due to the influence of this interaction, the spins in the Heisenberg magnet deviate from a parallel orientation, which leads to a chiral structure.5
B. Metropolis algorithm
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 (E1 > E2), otherwise it is accepted with the probability:
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 and also we performed five independent calculation cycles for computing averages. C++ and Rust programming languages were used for software development, providing possibilities for the independent calculation of the properties of the spin systems. We used dimensionless quantities in J units for simulation and count of the MC-steps was 107 per one step of temperature or external magnetic field. The software has been verified for the Heisenberg model.26–28
C. Neural network for states classification
In this study, 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 athe CNN29 and to classify our spin systems to different phases. The main problem in machine learning for classification of magnetic phases is how to relate the states of the input neurons of the network to the specific magnetic configuration of spin system.
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 recognizing images, the CNN accepts them in the RGB format as a three-dimensional matrix. In our case, the CNN 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 on the training dataset to highlight the features inherent in one or another spin configuration. Our CNN consists of next layers (main ones) and see Figure 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 on the basis of Monte Carlo simulation data for training the neural network in state recognition.
Convolutional layer
When neurons are connected to only a few neurons in the next layer, the layer is said to be a 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, then the values are summed up and written to the feature map. In our network, we used 3 convolutional layers, each of which consists of 32, 64 and 128 neurons, respectively. The last layer (skipped on the Figure 1) is followed by a flatten layer of the same dimension. The convolution kernel size was 3 × 3.
Pooling layer for reducing the dimensions of the data. The pooling core size was 2 × 2.
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. We used two such layers: 128 and 16 neurons, respectively. 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.
III. RESULTS AND DISCUSSIONS
Consider the process of nucleation of skyrmions in two-dimensional structures. One of the conventional method is to compute the skyrmion number, which is evaluated to keep track of the skyrmion creation process.30,31 However, it does not indicate the mixed states of the spin systems very well, depending on the simulation parameters, e.g. a spiral-skyrmion phase, therefore, we use the convolutional neural network in our work.
The neural network was trained on a limited set of configurations belonging to ferromagnetic, skyrmion and other states, and was subsequently able to recognise such states, including the transition regions between different phases. We studied different phases that appeared depending on the magnitude of the Dzyaloshinskii-Moriya interaction D from 0 to 2 with step 0.1 and the external magnetic field Hz from 0 to 2 with step 0.1 at fixed temperature T = 0.01J for the MC simulation. Based on these data, the phase diagrams (Hz, D) for different lattices were plotted as 20 × 20 grids with smoothing effect. For each point we carried out five Monte Carlo runs. For a stable skyrmion phase (Sk), parameters from a rather narrow range are necessary, see the phase diagrams on the Figures Figures (2-4).
(Hz, D) phase diagram for the honeycomb lattice. The inner panel demonstrates skyrmions at different Hz and D: a) (0.1; 0.4), b) (0.3; 1.0), c) (1.3; 1.4).
(Hz, D) phase diagram for the honeycomb lattice. The inner panel demonstrates skyrmions at different Hz and D: a) (0.1; 0.4), b) (0.3; 1.0), c) (1.3; 1.4).
(Hz, D) phase diagram for the square lattice. The inner panel demonstrates skyrmions at different Hz and D: a) (0.8; 1.4), b) (2.0; 2.2).
(Hz, D) phase diagram for the square lattice. The inner panel demonstrates skyrmions at different Hz and D: a) (0.8; 1.4), b) (2.0; 2.2).
(Hz, D) phase diagram for the hexagonal or triangular lattice. The inner panel demonstrates skyrmions at different Hz and D: a) (0.5; 0.7), b) (1.3; 1.3).
(Hz, D) phase diagram for the hexagonal or triangular lattice. The inner panel demonstrates skyrmions at different Hz and D: a) (0.5; 0.7), b) (1.3; 1.3).
As can be seen from inner panels of the figures, the size and structure of the skyrmions depend on values of Hz, D and a type of lattice. Additionally, we can conclude from our MC simulation that skyrmions on the square lattice were more stable with regard to temperature fluctuation than on the other lattices (for the square lattice, skyrmions were observed to T < 0.8J; for the other two lattices T < 0.1J). For the honeycomb lattice at simulation parameters Hz = 1.3, D = 1.4 we observed the smallest skyrmions in our research, see the inner panel (c) on Figure 2. Each spin is represented by an arrow. The colour of the arrows is a function of the coordinate z between the direction of the spin down − 1 (blue) and up + 1 (red).
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 phase (S)
In the absence of a magnetic field, DM interaction and direct exchange form stripes. Spins remain perpendicular to each other, but turn around, transforming the system into a strip domain, thereby forming spirals.
Spiral-skyrmion phase (SS)
With increasing magnetic field strength, some spins in the spirals begin to rotate against the magnetic field, which leads to the local formation of skyrmions.
Skyrmion phase (Sk)
Following 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.
Skyrmion-ferromagnetic phase (SF)
The system begins to transition to a ferromagnetic state. The remaining skyrmions are reduced in quantity and size, and randomly distributed throughout the system.
Ferromagnetic phase (F)
Skyrmions completely disappear and all spins are aligned with the magnetic field.
IV. CONCLUSION
In the frame of the classical two-dimensional Heisenberg model, a spin system with direct short-range exchange was modelled, and a study of its competition with the Dzyaloshinskii-Moriya interaction was carried out. Due to the direct exchange interaction, the neighbouring spins of the system are collinearly aligned, in turn, the Dzyaloshinskii-Moriya interaction contributes to the deviation of the spins from parallel orientation. As a result, competition appears 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.
We studied several types of flat lattices with direct exchange and the Dzyaloshinskii-Moriya interaction between spins: a honeycomb lattice with 3 nearest neighbours (NN), a square lattice with 4 NN, a hexagonal or triangular lattice with 6 NN. For each of these lattices, the processes of nucleation of magnetic skyrmions (along with other spin states) were investigated dependent on the magnitude of the external magnetic field and the DMI using the Metropolis algorithm. Following the MC simulation, 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 diagrams (Hz, D) for different lattices were plotted. Given the above, in this work we showed that the geometry of the lattices affected the magnetic structure of skyrmions and changed the phase space.
Based on our results, we concluded that skyrmions on square lattice were more stable with regard to temperature fluctuation than on other lattices (for the square lattice, skyrmions were observed to T < 0.8J, for other two lattices T < 0.1J). For the honeycomb lattice at simulation parameters Hz = 1.3, D = 1.4 we observed the smallest skyrmions in our research.
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 for potential practical application in magnetic carriers operating on the new principles of reading/writing bits of information.
ACKNOWLEDGMENTS
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) (#2267-21).
DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding author upon reasonable request.