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Spin Waves in Systems with DzyaloshinskiiMoriya InteractionIn this chapter, we study the magnetic properties of a system of quantum Heisenberg spins interacting with each other via a ferromagnetic exchange interaction J and an inplane Dzyaloshinskii Moriya interaction D. The noncollinear ground state due to the competition between J and D is determined. We employ a self consistent Green's function theory to calculate the spin wave spectrum and the layer magnetizations at finite T in two and three dimensions as well as in a thin film with surface effects. Analytical details and the validity of the method are shown and discussed. Numerical solutions are shown for realistic physical interaction parameters. Discussion on possible experimental verifications is given. The main results of this chapter have been published in Refs. [93,104]. IntroductionThe DzyaloshinskiiMoriya (DM) interaction was proposed to explain the weak ferromagnetism which was observed in antiferro 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 magnetic Mn compounds. The phenomenological LandauGinzburg model introduced by I. Dzyaloshinskii [99] was microscopically derived by T. Moriya [240]. The interaction between two spins S, and Sj is written as
where D,,y is a vector which results from the displacement of nonmagnetic ions located between S, and Sj, for example, in MnO Mn bonds. The direction of D,,y depends on the symmetry of the displacement [240]. The definition of D/,y is given below [see also Section 15.1). There have been a large number of investigations on the effect of the DM interaction in various materials, both experimentally and theoretically for weak ferromagnetism in perovskite compounds (see the references cited in Refs. [100, 316], for example). However, the interest in the DM interaction goes beyond the weak ferromagnetism: For example, it has been recently shown in various works that the DM interaction is at the origin of topological skyrmions [2, 37, 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 the electronic transport which is at the heart of technological application devices [113]. In this chapter, we are interested in the spin wave (SW) properties of a system of spins interacting with each other via a DM interaction in addition to the symmetric isotropic Heisenberg exchange interaction. The competition between these interactions gives rise to a noncollinear spin configuration in the ground state (GS). Unlike helimagnets where the helical GS spin configuration results from the competition between the ferromagnetic nearest neighbor (NN) and antiferromagnetic nextnearest neighbor (NNN) interactions [350, 369], the DM interaction favors the perpendicular spin configuration. This gives rise to a nontrivial SW behavior as will be seen below. Note that there has been a number of early works dealing with some aspects of the SW properties in DM systems [237,282,328,353,373]. Section 13.2 is devoted to the description of the model and the determination of the GS. Section 13.3 shows the formulation of our selfconsistent Green's function (GF) method. Section 13.4 shows results on the SW spectrum and the magnetization in two dimensions (2D) and three dimensions (3D). The case of thin films with free surfaces is shown in Section 13.5 where layer magnetizations at finite temperature (Г) and the thickness effect are presented. Discussion and experimental suggestion are made in Section 13.6. Model and Ground StateWe consider a thin film of simple cubic (SC) lattice of N layers stacked in they direction perpendicular to the film surface. For the reason which is shown below, we choose the film surface as a xz plane. The Hamiltonian is given by
where Ду and D,y are the exchange and DM interactions, respectively, between two Heisenberg spins S, and Sy of magnitude S = 1/2 occupying the lattice sites / and j. The SC lattice can support the DM interaction in the absence of the inversion symmetry [224, 240, 242] as in MnSi. The absence of inversion symmetry can be also achieved by the positions of nonmagnetic ions between magnetic ions. The vector D between two magnetic ions is defined as D;,y = Лг, л г у where г, is the vector connecting the nonmagnetic ion to the spin S, and r_{;} is that to the spin Sy, A being a constant. One sees that D/,y is perpendicular to the plane formed by r, and ry. For simplicity, let us consider the case where the inplane and interplane exchange interactions between NN are both ferromagnetic and denoted by Д and J±, respectively. The vector D,_{r}y is chosen by supposing the situation where nonmagnetic ions are positioned in the plane xz. With this choice, D, y is perpendicular to the xz plane and our model gives rise to a planar spin configuration, namely the spins stay in xz planes. There are no situations where the spins are out of plane. This simplifies our calculation. The DM interaction is supposed to be between NN in the plane with a constant D. Due to the competition between the exchange J term which favors the collinear configuration, and the DM term which favors the perpendicular one, we expect that the spin S, makes an angle 0/,y with its neighbor Sy. Therefore, the quantization axis of S, is not the same as that of Sy. Let us call the quantization axis of S, and its perpendicular axis in the xz plane. The third axis гц, perpendicular to the film surface, is chosen in such a way to make (/, гц, £,) an orthogonal direct frame. Writing S, and Sy in their respective local coordinates, one has We choose the vector D, у perpendicular to the xz plane, namely where e_{/(}y = +1(1) if j > i (j < i) for NN on the/ or £,• axis. Note that ejj = —e_{if}j. To determine the GS, the easiest way is to use the steepest descent method: We calculate the local field acting on each spin from its neighbors and we align the spin in its localfield direction to minimize its energy. Repeating this for all spins and iterating many times until the convergence is reached with a desired precision (usually at the sixth digit, namely at ~ 10^{6} per cents), we obtain the lowest energy state of the system (see Ref. [248]). Note that we have used several thousands of different initial conditions to check the convergence to a single GS for each set of parameters. Choosing D, y lying perpendicular to the spin plane (i.e., xz plane) as indicated in Eq. (13.7), we determine the GS as a function of D by the steepest descent method. An example is shown in Fig. 13.1 for в = я/6 (D = —0.577) with/„ = J± = 1. We see that each spin has the same angle with its four NN in the plane (angle between NN in adjacent planes is zero). In the present model, the DM interaction is supposed between the NN in the plane xz, so we see in Fig. 13.1 that in the GS the angle Figure 13.1 (a) The ground state is a planar configuration on the xz plane. The figure shows the case where в = тг/6 (D = —0.577) along the x and z axes, with /„ = J_{±} = 1, obtained by using the steepest descent method; (b) a zoom is shown around a spin with its nearest neighbors in the xz plane. between inplane NN is not zero. The spin configuration is planar, meaning that the turn angle is in the plane: Each spin turns an angle в with respect to its NN, and it is in both directions in the plane (see Fig. 13.1b). There is no helicoidal configuration. Note that the spin structure is different from that of MnSi where the turn angle is in a plane perpendicular to the screw axis. This case has been studied in our previous papers [89,102]. Note also that the helicalwave vector in the MnSi case is along the screw axis, while in our case, it lies along each of the two axes x and z of the xz plane (see Fig. 13.1b). Let us show the relation between в, D and The energy of the spin S, is written as
where в = 0,,y and care has been taken on the signs of sin0,,y and e,,y when counting NN, namely two opposite NN have opposite signs. The minimization of £, yields
The value of 0 for a given j is precisely what obtained by the steepest descent method. Figure 13.2 Local coordinates in the xz plane. The spin quantization axes of S, and Sj are £, and fy, respectively. Note that the perpendicular axes >}, and rjj coincide. Now, expressing the local frame of Sj in the local frame of S, , we have
so that
The DM term of Eq. (13.4) can be rewritten as Using Eq. (13.7), we have where we have replaced S^{x} = (S^{+} + S~)/2. Note that sin0,,y is always positive since for a NN on the positive axis direction, e,j = 1 and sin djj = sin0 where в is positively defined, while for a NN on the negative axis direction, e,,y = —1 and sin0,,y = sin(—в) = — sin 6. Note that for noncollinear spin configurations, the local spin coordinates allow one to use the commutation relations between spin operators of a spin which are valid only when the z spin component is defined on its quantification axis. This method has been applied for helimagnets [91,139, 286]. 
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