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Model and Ground StateModelConsider a superlattice composed of alternate magnetic and ferroelectric films (see Fig. 15.2a). Both have the structure of simple cubic lattice of the same lattice constant, for simplicity. The Hamiltonian of this multiferroic superlattice is expressed as:
where H_{m} and Hf are the Hamiltonians of the ferromagnetic and ferroelectric subsystems, respectively, while H_{m}f is the Hamiltonian of magnetoelectric interaction at the interface between two adjacent films. We describe the Hamiltonian of the magnetic film with the Heisenberg spin model on a simple cubic lattice:
where S, is the spin on the z'th site, H is the external magnetic field, Jj > 0 the ferromagnetic interaction parameter between a spin and its nearest neighbors (NN) and the sum is taken over NN spin pairs. We consider//" > 0 to be the same, namely / ^{m}, for spins eveiywhere in the magnetic film. The external magnetic field H is applied along the zaxis which is perpendicular to the plane of the layers. The interaction of the spins at the interface will be given below. For the ferroelectric film, we suppose for simplicity that electric polarizations are Isinglike vectors of magnitude 1, pointing in the ±z direction. The Hamiltonian is given by
where P; is the polarization on the z'th lattice site, // >0 the interaction parameter between NN and the sum is taken over NN sites. Similar to the ferromagnetic subsystem we will take the same j/j = J t for all ferroelectric sites. We apply the external electric field E along the zaxis. We suppose the following Hamiltonian for the magnetoelectric interaction at the interface:
In this expression / ”/ D,y plays the role of the DM vector which is perpendicular to the xy plane. Using Eqs. (15.2)(15.3), one has
Now, let us define for our model
which is the DM interaction parameter between the electric polarization at the interface ferroelectric layer and the two NN spins Si and Sj belonging to the interface ferromagnetic layer. Hereafter, we suppose J™j = J ^{m}f independent of (/, j). Selecting R in the xy plane perpendicular to г, у (see Fig. 15.1) we can write R x rj'j = az e, y where e,_y = —_{/} = 1, a is a constant and z the unit vector on the z axis. It is worth at this stage to specify the nature of the DM interaction to avoid a confusion often seen in the literature. The term [S, x S_{y }changes its sign with the permutation of i and j, but the whole DM interaction defined in Eq. (15.2) does not change its sign because Dij changes its sign with the permutation as seen in Eq. (15.3). Note that if the whole DM interaction is antisymmetric then when we perform the lattice sum, nothing of the DM interaction remains in the Hamiltonian. This explains why we need the coefficient e,,y introduced above and present in Eq. (15.10) below. We collect all these definitions we write H_{m}f in a simple form
where the constant a is absorbed in J ^{m}f. As seen in Eq. (15.10), the coefficient of the interface coupling is proportional to We note that the DM interaction is taken only between NN spin. If we choose the DM vector D perpendicular to the xy plane then the DM interaction energy is minimum when the spins are in the xy plane because D is parallel to [S, x Sy]. One can choose any orientation for D but in that case to have the minimum energy the plane containing S, and S_{y} should be perpendicular to D: The spins are not in the xy plane, making the spin configuration analysis difficult. The superlattice and the interface interaction are shown in Fig. 15.2. A polarization at the interface interact with 5 spins on the magnetic layer according to Eq. (15.10), for example (see Fig. 15.2b):
Since we suppose P^. is a vector of magnitude 1 pointing along the z axis, namely its z component is P_{k} = ±1, we will use hereafter Pg for electric polarization instead of P_{k}. From Eq. (15.10), we see that the magnetoelectric interaction J ^{m}f favors a canted spin structure. It competes with the exchange interaction J of H_{m} which favors collinear spin configurations. Usually the magnetic or ferroelectric exchange interaction is the leading term in the Hamiltonian, so that in many situations the magnetoelectric effect is negligible. However, in nanofilms of superlattices the magnetoelectric interaction is crucial for the creation of noncollinear longrange spin order. Note that the hypothesis that P_{k} is in the z direction in order to have the polarization proportional to the DM vector [Eqs. (15.7) (15.10)]. The DM vector is taken in the z direction in order to have spins in the magnetic layers lying in the xy plane, in the absence of an applied field (see Sections 15.2.2.1 and 15.4). The polarization is in addition supposed of the Ising type since in this chapter, this assumption allows us to have the DM vector in a fixed direction z. The assumption is justified by the fact that in ferroelectric materials, if atoms are displaced in the same direction it gives rise to a spontaneous polarization in that direction as illustrated in Fig. 15.1b. Ground StateGround state in zero magnetic fieldLet us analyze the structure of the ground state (GS) in zero magnetic field. Since the polarizations are along the z axis, the interface DM interaction is minimum when Sj and Sj lie in the xy interface plane and perpendicular to each other. However, the ferromagnetic Figure 15.2 (a) The superlattice composed of alternately a ferroelectric layer indicated by F and a magnetic layer indicated by M; (b) A polarization Pi at the interface interacts with 5 spins in the magnetic layer. See text for expression. exchange interaction among the spins will compete with the DM perpendicular configuration. The resulting configuration is non collinear. We will determine it below, but at this stage, we note that the ferroelectric film has always polarizations along the z axis even when interface interaction is turned on. Let us determine the GS spin configurations in magnetic layers in zero field. If the magnetic film has only one monolayer, the minimization of H_{m}f in zero magnetic field is done as follows. By symmetry, each spin has the same angle в with its four NN in the xy plane. The energy of the spin S, gives the relation between в and J ^{m}
where в = 0,,y and care has been taken on the signs of sin0,_{;}y when counting NN, namely two opposite NN have opposite signs, and the opposite coefficient e_{/;}, as given in Eq. (15.11). Note that the coefficient 4 of the first term is the number of inplane NN pairs, and the coefficient 8 of the second term is due to the fact that each spin has 4 coupling DM pairs with the NN polarization in the upper ferroelectric plane, and 4 with the NN polarization of the lower ferroelectric plane (we are in the case of a magnetic monolayer). The minimization of E, yields, taking P^{z} = 1 in the GS and S = 1,
The value of в for a given is precisely what obtained by the numerical minimization of the energy. We see that when J ^{m}f *■ 0, one has в > 0, and when J ^{m}f >■ — oo, one has ]^{m}f я/2 as it should be. Note that we will consider here ]^{m}f < 0 so as to have в > 0. The above relation between the angle and J ^{m}f will be used in the last section to calculate the spin waves in the case of a magnetic monolayer sandwiched between ferroelectric films. In the case when the magnetic film has a thickness, the angle between NN spins in each magnetic layer is different from that of the neighboring layer. It is more convenient using the numerical minimization method called "steepest descent method" to obtain the GS spin configuration. This method consists in minimizing the energy of each spin by aligning it parallel to the local field acting on it from its NN. This is done as follows. We generate a random initial spin configuration, then we take one spin and calculate the interaction field from its NN. We align it in the direction of this field, and take another spin and repeat the procedure until all spins are considered. We go again for another sweep until the total energy converges to a minimum. In principle, with this iteration procedure the system can be stuck in a metastable state when there is a strong interaction disorder such as in spinglasses. But for uniform, translational interactions, we have never encountered such a problem in many systems studied so far. We use a sample size N x N x L. For most calculations, we select N = 40 and L= 8 using the periodic boundaiy conditions in the xy plane. For simplicity, when we investigate the effect of the exchange couplings on the magnetic and ferroelectric properties, we take the same thickness for the magnetic and ferroelectric films, namely Lg = L_{b} = 4 = L/2. Exchange parameters between spins and polarizations are taken as /= ] f = 1 for the simulation. For simplicity we will consider the case where the inplane and interplane exchange magnetic and ferroelectric interactions between nearest neighbors are both positive. All the results are obtained with J^{m} =J f = 1 for different values of the interaction parameter J ^{m}f. We investigated the following range of values for the interaction parameters J ^{m}f: From J ^{m}f = 0.05 to J ^{m}f = 6.0 with different values of the external magnetic and electric fields. We note that the steepest descent method calculates the real ground state with the minimum energy to the value J^{m}f = 1.25. After larger values, the angle в tends to n/2 so that all magnetic exchange terms (scalar products) will be close to zero, the minimum energy corresponds to the DM energy. Figure 15.3 shows the GS configurations of the magnetic interface layer for small values of J ^{m}f: —0.1, —0.125, —0.15. Such small values yield small values of angles between spins so that the GS configurations have ferromagnetic and noncollinear domains. Note that angles in magnetic interior layers are different but the GS configurations are of the same texture (not shown). For larger values of J ^{m}f, the GS spin configurations have periodic structures with no more mixed domains. We show in Fig. 15.4 examples where J ^{m}f = — 0.45 and —1.2. Several remarks are in order: (i) Each spin has the same turning angle в with its NN in both x and у direction. The schematic zoom in Fig. 15.4c shows that the spins on the same diagonal (spins 1 and 2, spins 3 and 4) are parallel. This explains the structures shown in Figs. 15.4a and 15.4b; Figure 15.3 GS spin configuration for weak couplings: J'"f = —0.1 (a), 0.125 (b), 0.15 (c), with Я = 0. (ii) The periodicity of the diagonal parallel lines depends on the value of в (comparing Fig. 15.4a and Fig. 15.4b). With a large size of N, the periodic conditions have no significant effects. 
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