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Skyrmions in SuperlatticesIn this chapter, we study the effects of DzyaloshinskiiMoriya (DM) magnetoelectric coupling between ferroelectric and magnetic layers in a superlattice formed by alternate magnetic and ferroelectric films. Magnetic films are films of simple cubic lattice with Heisenberg spins interacting with each other via an exchange J and a DM interaction with the ferroelectric interface. Electrical polarizations of ±1 are assigned at simple cubic lattice sites in the ferroelectric films. We determine the groundstate (GS) spin configuration in the magnetic film. In zero field, the GS is periodically noncollinear and in an applied field H perpendicular to the layers, it shows the existence of skyrmions at the interface. Using the Green’s function (GF) method we study the spin waves (SW) excited in a monolayer and also in a bilayer sandwiched between ferroelectric films, in zero field. We show that the DM interaction strongly affects the long wavelength SW mode. We calculate also the magnetization at low temperature T. We use next Monte Carlo simulations to calculate various physical quantities at finite temperatures such as the critical temperature, the layer magnetization and the layer polarization, as functions of the interface magnetoelectric DM coupling and the applied magnetic field. Phase transition to the disordered phase is studied in detail. Physics of Magnetic Thin Films: Theory and Simulation Hung T. Diep Copyright © 2021 Jenny Stanford Publishing Pte. Ltd. ISBN 9789814877428 (Hardcover), 9781003121107 (eBook) www.jennystanford.com The main results of this chapter have been published in Refs. [317,318]. IntroductionNonuniform spin structures, which are quite interesting by themselves, became the subject of close attention after the discovery of electrical polarization in some of them [97]. The existence of polarization is possible due to the inhomogeneous magnetoelectric effect, namely that electrical polarization can occur in the region of magnetic inhomogeneity. It is known that the electric polarization vector is transformed in the same way as the combination of the magnetization vector and the gradient of the magnetization vector, meaning that these values can be related by the proportionality relation. In Ref. [241], it was found that in a crystal with cubic symmetiy the relationship between electrical polarization and inhomogeneous distribution of the magnetization vector has the following form:
here у is the magnetoelectric coefficient, and y_{e} the permittivity. In noncollinear structures, the microscopic mechanism of the coupling of polarization and the relative orientation of the magnetization vectors is based on the interaction of DzyaloshinskiiMoriya [61,180, 316]. The corresponding term in the Hamiltonian is
where Sj is the spin of the /th magnetic ion, and D,,y is the DzyaloshinskiiMoriya vector. The vector D,,_{;} is proportional to the vector product R x r_{f},j of the vector R which specifies the displacement of the ligand (for example, oxygen) and the unit vector Tj j along the axis connecting the magnetic ions i and j (see Fig. 15.1a). We write
Thus, the DzyaloshinskiiMoriya interaction connects the angle between the spins and the magnitude of the displacement of Figure 15.1 (a) Schema of DzyaloshinskiiMoriya interaction, spins are in the xy plane; (b) microscopic mechanisms of creation of electric spontaneous polarization P due to displacements of atoms (red) in the same direction z. nonmagnetic ions. In some micromagnetic structures all ligands are shifted in one direction, which leads to the appearance of macroscopic electrical polarization (see Fig. 15.1b). By nature, this interaction is a relativistic amendment to the indirect exchange interaction, and is relatively weak [284]. In the case of magnetically ordered matter, the contribution of the DzyaloshinskiiMoriya interaction to the free energy can be represented as Lifshitz antisymmetric invariants containing spatial derivatives of the magnetization vector. In analogy, the vortex magnetic configuration can be stable via Skyrme mechanism [37]. Skyrmions were theoretically predicted more than twenty years ago as stable micromagnetic structures [41]. The idea came from nuclear physics, where the elementary particles were represented as vortex configurations of continuous fields. The stability of such configurations was provided by the "Skyrme mechanism"—the components in Lagrangians containing antisymmetric combinations of spatial derivatives of field components [325]. For a long time, skyrmions have been the subject only of theoretical studies. In particular, it was shown that such structures can exist in antiferromagnets [42] and in magnetic metals [296]. In the latter case, the model included the possibility of changing the magnitude of the magnetization vector and spontaneous emergence of the skyrmion lattice without the application of external magnetic field. A necessaiy condition for the existence of skyrmions in bulk samples was the absence of an inverse transformation in the crystal magnetic symmetiy group. Diep et al. [92] have studied a ciystal of skyrmions generated on a square lattice using a ferromagnetic exchange interaction and a DzyaloshinskiiMoriya interaction between nearestneighbors under an external magnetic field. They have shown that the skyrmion crystal has a hexagonal structure which is shown to be stable up to a temperature T_{c} where a transition to the paramagnetic phase occurs and the dynamics of the skyrmions at Г < T_{c} follows a stretched exponential law. In Ref. [296] it was shown that the most extensive class of candidates for the detection of skyrmions includes the surfaces and interfaces of magnetic materials, where the geometiy of the material breaks the central symmetry and, therefore, can lead to the appearance of chiral interactions similar to the DzyaloshinskiiMoriya interaction. In addition, skyrmions are twodimensional solitons, the stability of which is provided by the local competition of shortrange interactions exchange and DzyaloshinskiiMoriya interactions [92,184]. The idea of using skyrmions in memory devices nowadays is reduced to the information encoding using the presence or absence of a skyrmion in certain area of the material. A numerical simulation of the creation and displacement of skyrmions in thin films was carried out in Ref. [302] using a spinpolarized current. The advantage of skyrmions with respect to the domain boundaries in such magnetic memory circuits (e.g., racetrack memory, see Ref. [266]) is the relatively low magnitude of the currents required to move the skyrmions along the "track.” For the first time, skyrmions were experimentally detected in the MnSi helimagnet [242]. Below the Curie temperature in MnSi spins are aligned in helicoidal or conical structure [the field was applied along the [100] axis), depending on the magnitude of the applied magnetic field. Similar experimental results were obtained for the compound Fei__{x}Co_{x}Si, x = 0.2 [243]. The investigation of Feo.5Coo.5Si made it possible to take the next important step in the study of skyrmions—to directly observe them using Lorentz electron microscopy [371]. The dependence of the stability of the skyrmion lattice on the sample thickness of FeGe was studied in more detail in Ref. [370]. Studies have confirmed that the thinner was the film, the greater was the "stability region” of skyrmions. Skyrmions as the most compact isolated micromagnetic objects are of great practical interest as memory elements [184]. The stability of skyrmions [92] can make the memory on their basis nonvolatile, and low control currents will reduce the cost of rewriting compared to similar technologies based on domain boundaries. In Refs. [313] and [315], magnetic and electrical properties of the skyrmion lattice were studied in the multiferroic Cu_{2}0Se0_{3}. It has been shown that that energy consumption can be minimized by using the electric field to control the micromagnetic structures. It is worth noting that the multiferroics BaFei2*o.osSc*Mgo.osOi9 may also have a skyrmion structure [295, 372]. The manipulations with skyrmions were first demonstrated in the diatomic PdFe layer on the iridium substrate, and the importance of this achievement for the technology of information storing is difficult to overestimate: It makes possible to write and read the individual skyrmions using a spinpolarized tunneling current [293]. In Ref. [283], the possibility of the nucleation of skyrmions by the electric field by means of an inhomogeneous magnetoelectric effect was established. Recent studies are focused on the interfaceinduced skyrmions. Therefore, the superstructures naturally lead to the interaction of skyrmions on different interfaces, which has unique dynamics compared to the interaction of the sameinterface skyrmions. In Ref. [190], a theoretical study of two skyrmions on twolayer systems was carried using micromagnetic modeling, as well as an analysis based on the Thiele equation, which revealed a reaction between them, such as the collision and a bound state formation. The dynamics sensitively depends on the sign of DM interaction, i.e., the helicity, and the skyrmion numbers of two skyrmions, which are well described by the Thiele equation. In addition, the colossal spintransfertorque effect of bound skyrmion pair on antiferromagnetically coupled bilayer systems was discovered. In Ref. [226] the study of the Thiele equation was carried for current induced motion in a skyrmion lattice through two soluble models of the pinning potential. We consider in this chapter a superlattice composed of alternate magnetic films and ferroelectric films. The aim is to propose a new model for the coupling between the magnetic film and the ferroelectric film by introducing a DMlike interaction. It turns out that this interface coupling gives rise to noncollinear spin configurations in zero applied magnetic field and to skyrmions in a field H applied perpendicularly to the films. Using the Green's function method, we study spin wave excitations in zero field of a monolayer and a bilayer. We find that the DM interaction affects strongly the longwavelength mode. Monte Carlo simulations are carried out to study the phase transition of the superlattice as functions of the interface coupling strength. Section 15.2 is devoted to the description of our model and the determination of the groundstate spin configuration with and without applied magnetic field. Section 15.3 shows the results obtained by Monte Carlo simulations for the phase transition in the system as a function of the interface DM coupling. In Section 15.4, we show the results of spin wave spectrum obtained by the Green's function technique in zero field for a monolayer and a bilayer. Section 15.5 shows the effect of the frustration on the skyrmion structure. The frustration is introduced by adding the competing interactions J _{x} and J_{2} between nearest neighbors (NN) and next nearest neighbors (NNN). 
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