Table of Contents:
Skyrmions in Thin Films
We generate a crystal of skyrmions in two dimensions (2D) using a Heisenberg Hamiltonian including the ferromagnetic interaction ], the Dzyaloshinskii-Moriya interaction D, and an applied magnetic field H. The ground state (GS) is determined by minimizing the interaction energy. We show that the GS is a skyrmion crystal in a region of (D, H). The stability of this skyrmion crystalline phase at finite temperatures is shown by a study of the time-dependence of the order parameter using Monte Carlo simulations. We observe that the relaxation is very slow and follows a stretched exponential law. The skyrmion crystal phase is shown to undergo a transition to the paramagnetic state at a finite temperature.
We also study a 2D skyrmion system on fluctuating surfaces with periodic boundaiy conditions using a Monte Carlo simulation technique. In this model, not only spins but also lattice vertices are integrated into the partition function of the model. From the obtained results, we conclude that skyrmions are stable even on fluctuating surfaces, contrary to initial expectations.
The main results of this chapter have been recently published in Refs.  and . The reader is referred to those papers for more details.
Physics of Magnetic Thin Films: Theory and Simulation Hung T. Diep
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Introduction: Magnetic Field Effect, Excitations of Skyrmions
Skyrmions have been extensively investigated in condensed matter physics [37, 39, 113, 207] since its theoretical formulation by Skyrme  in the context of nuclear matter.
There are several mechanisms and interactions leading to the appearance of skyrmions in various kinds of matter. The most popular one is certainly the Dzyaloshinskii-Moriya (DM) interaction which was initially proposed to explain the weak ferromagnetism observed in antiferromagnetic Mn compounds. The phenomenological Landau-Ginzburg model introduced by I. Dzyaloshinskii  was microscopically derived by T. Moriya . This demonstration shows that the DM interaction comes from the second-order perturbation of the exchange interaction between two spins which is not zero only under some geometrical conditions of non-magnetic atoms found between them. The order of magnitude of DM interaction, D, is therefore, perturbation theory obliges, small. The explicit form of the DM interaction will be given in the next section. However, we can think that the demonstration of Moriya  is a special case and the general Hamiltonian may have the same form but with different microscopic origin.
The DM interaction has been shown to generate skyrmions in various kinds of crystals. For example, it can generate a ciystal of skyrmions in which skyrmions arrange themselves in a periodic structure [51,195,196,294]. Skyrmions have been shown to exist in crystal liquids [1,39,207]. A single skyrmion has also been observed . Existence of skyrmion crystals have been found in thin films [109, 370]. Direct observation of the skyrmion Hall effect has been realized . Artificial skyrmion lattices have been devised for room temperatures . Experimental observations of skyrmion lattices have been realized in MnSi in 2009 [24, 242] and in doped semiconductors in 2010 . At this stage, it should be noted that skyrmion crystals can also be created by competing exchange interactions without DM interactions [141,263]. So, mechanisms for creating skyrmions are multiple.
We note that spin wave excitations in systems with a DM interaction in the helical phase without skyrmions have been investigated by many authors [104, 237, 282, 328, 353, 373].
Applications of skyrmions in spintronics have been largely discussed and their advantages compared to early magnetic devices such as magnetic bubbles have been pointed out in a recent review by W. Kang et al. . Among the most important applications of skyrmions, let us mention skyrmion-based racetrack memory , skyrmion-based logic gates [377, 379], skyrmion-based transistor [183, 323, 378] and skyrmion-based artificial synapse and neuron devices [158, 210].
In this chapter, we study a skyrmion crystal created by the competition between the nearest-neighbor (NN) ferromagnetic interaction ] and the DM interaction of magnitude D under an applied magnetic field H. We show by Monte Carlo (MC) simulation that the skyrmion crystal is stable at finite temperatures up to a transition temperature Tc where the topological structure of each skyrmion and the periodic structure of skyrmions are destroyed.
The chapter is organized as follows. Section 14.2 is devoted to the description of the model and the method to determine the ground state (GS). It is shown that our model generates a skyrmion crystal with a perfect periodicity at temperature T = 0. The GS phase diagram in the space (D, H) is presented. The phase transition of the skyrmion crystal is studied in Section 14.3. Results showing the stability of the skyrmion crystal at finite T obtained from MC simulations are shown in Section 14.4. We show in this section that the relaxation of the skyrmions is very slow and follows a stretched exponential law. The stability of the skyrmion phase is destroyed at a phase transition to the paramagnetic state. In Section 14.5, we study the effect of the fluctuating lattice, namely the elastic effect, on the stability of the skyrmion crystal. We find that the skyrmion crystal is stable against the lattice deformation. Concluding remarks are given in Section 14.6.
Model and Ground State
The DM interaction energy between two spins S, and S; is written as
where D,j is a vector which results from the displacements of nonmagnetic ions located between S, and S;, for example, in Mn-O- Mn bonds in the historical papers [99, 240]. The direction of D/,y depends on the symmetry of the displacements .
Theoretical and experimental investigations on the effect of the DM interaction in various materials have been extensively carried out in the context of weak ferromagnetism observed in perovskite compounds (see references cited in Refs. [100, 316]). As said in the Introduction, the interest in the DM interaction goes beyond the weak ferromagnetism. It has been shown that the DM interaction is at the origin of topological skyrmions [2, 37, 103, 106, 150, 166, 213, 224, 242, 296, 313, 357, 370, 371] and new kinds of magnetic domain walls [144, 292]. The increasing interest in skyrmions results from the fact that skyrmions may play an important role in technological application devices [113,377].
In this chapter, we consider for simplicity the two-dimensional (2D) case where the spins are on a square lattice in the xy plane. We are interested in the stability of the skyrmion ciystal generated in a system of spins interacting with each other via a DM interaction and a symmetric isotropic Heisenberg exchange interaction in an applied field perpendicular to the xy plane. All interactions are limited to NN. The full Hamiltonian is given by
where the DM interaction and the exchange interaction are taken between NN on both x and у directions. Rewriting it in a convenient form, we have
For the i -th spin, one has
where the local-field components are given by
To determine the ground state (GS), we minimize the energy of each spin, one after another. This can be numerically achieved as the following. At each spin, we calculate its local-field components acting on it from its NN using the above equations. Next we align the spin in its local field, i.e., taking S* = H*//H* * *2 + Hf * *2 + Hf * *2 etc. The denominator is the modulus of the local field. In doing so, the spin modulus is normalized to be 1. As seen from Eq. (14.4), the energy of the spin S, is minimum. We take another spin and repeat the same procedure until all spins are visited. This achieves one iteration. We have to do a sufficient number of iterations until the system energy converges. For the skyrmion case, it takes about 1000 iterations to have the fifth-digit convergence. We have used random initial configurations and we observed that a number of them lead to metastable states. This is seen by comparing the energies of several thousands of initial configurations and examining the snapshots. A good GS has always a perfect hexagonal structure as shown below.
Note that the GS configuration does not depend on the system size if it is large enough. This size is from N = 50 for the value of D = 1 and H = 0.5 used in the example shown below.
An example of GS are displayed in Fig. 14.1 for the crystal size 50 x 50: A ciystal of skyrmions is seen using D = 1 and H = 0.5 (in unit of J = 1). Note that we use here an example of large D for clarity, but we will show later in the phase diagram that the skyrmion crystal exists in a much smaller region for small D.
Let us give some comments on Fig. 14.1:
In Fig. 14.2a we show a GS at H = 0 where domains of long and round islands of up spins separated by labyrinths of down spins are mixed. When H is increased, vortices begin to appear. The GS is a mixing of long islands of up spins and vortices as seen in Fig. 14.2b obtained with D = 1 and H = 0.25. This phase can be called "labyrinth phase” or "stripe phase.”
It is interesting to note that skyrmion crystals with texture similar to those shown in Figs. 14.1 and 14.2 have been experimentally observed in various materials [126, 242, 370, 372],
Figure 14.1 Ground state for D/J =1 and H/J = 0.5, a crystal of skyrmions is observed. Top: Skyrmion crystal viewed in the xy plane. Middle: a 3D view. Bottom: zoom of the structure of a single vortex. The value of Sz is indicated on the color scale. See text for comments.
but the most similar skyrmion crystal was observed in two- dimensional Feo.5Coo.5Si by Yu et al. using Lorentz transmission electron microscopy .
We have performed the GS calculation taking many values in the plane (Д H). The phase diagram is established in Fig. 14.3. Above the blue line is the field-induced ferromagnetic phase. Below the red line is the labyrinth phase with a mixing of skyrmions and rectangular domains. The skyrmion crystal phase is found in a
Figure 14.2 (Top) Ground state for D/J =1 and H /] = 0, a mixing of domains of long and round islands; (bottom) ground state for D/J = 1 and H /] = 0.25, a mixing of domains of long islands and vortices. We call these structures the "labyrinth phase."
narrow region between these two lines, down to infinitesimal D. In order to enlarge the stability region of skyrmions, we may need to include other kinds of interaction such as dipole-dipole interaction, anisotropies and/or other competing interactions. However, this is out of the scope of this work.
Although the effects of the temperature (Г) will be shown in the next section, we anticipate here the phase diagram shown in Fig. 14.4 in the space (T, H) for an overview before the detailed
Figure 14.3 Phase diagram in the (D, H) plane for size N = 100.
Figure 14.4 Phase diagram in the (Г, Я) plane for size N = 100. Phases I, II and III indicate the labyrinth phase, the skyrmion crystal phase and the field-induced ferromagnetic phase. The discontinued line separating phase I and III at high T is not a transition line.
data presentation given below. The definitions of phases I, И and III are given in the caption. Note that the discontinued line is not a transition line it schematically indicates a region where spins turned progressively to the field at high T.
In the following, we are interested in the phase transition of the skyrmion crystal. The stability of the skyrmion crystal phase at finite temperatures T is studied by calculating the relaxation time.