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Spin Waves in Zero FieldWe have shown in the previous section Monte Carlo results for the phase transition in our superlattice model. Here let us show theoretically spin waves (SW) excited in the magnetic film in zero field, in some simple cases. The method we employ is the Green's function technique for noncollinear spin configurations which has been shown to be efficient for studying lowГ properties of quantum spin systems such as helimagnets [89] and systems with a DM interaction [104]. In this section, we consider the same Hamiltonian supposed in Eqs. (15.4)(15.10) but with quantum spins of amplitude 1/2. As seen in the previous section, the spins lie in the xy planes, each on its quantization local axis lying in the xy plane (quantization axis being the < axis, see Fig. 15.15). Figure 15.15 The spin quantization axes of S, and S, are and respectively, in the xy plane. Expressing the spins in the local coordinates, one has where the i and j coordinates are connected by the rotation
where 0,y = в, — 6j being the angle between S, and Sy. As we have seen above, the GS spin configuration for one monolayer is periodically noncollinear. For twolayer magnetic film, the spin configurations in two layers are identical by symmetry. However, for thickness larger than 2, the interior layer have angles different from that on the interface layer. It is not our purpose to treat that case though it is possible to do so using the method described in Ref. [104]. We rather concentrate ourselves in the case of a monolayer in this section. In the following, we consider the case of spin onehalf 5 = 1/2. Expressing the total magnetic Hamiltonian Нм = H_{m} + H_{m}f in the local coordinates [104]. Writing Sy in the coordinates (f,, /?,, /,), one gets the following exchange Hamiltonian from Eqs. (15.4) (15.10):
where D = J^{m}f P^{z}. Note that P^{z} = 1 in the GS. At finite T we replace P^{z} by < P^{z} >. In the above equation, we have used standard notations of spin operators for easier recognition when using the commutation relations in the course of calculation, namely
where we understand that Sf is in fact Sp and so on. Note that the sinus terms of H_{m}, the 3rd line of Eq. (15.18), are zero when summed up on opposite NN unlike the sinus term of the DM Hamiltonian H_{m}f, Eq. (15.10) which remains thanks to the choice of the DM vectors for opposite directions in Eq. [104]. MonolayerIn two dimensions (2D) there is no longrange order at finite temperature (Г) for isotropic spin models with shortrange interaction [231]. Therefore, to stabilize the ordering at finite T, it is useful to add an anisotropic interaction to stabilize the magnetic long range ordering when the wave vector vanishes (see Fig. 15.16a, for instance). It is known that in 2D or in very thin films, the integrands to calculate < M > at a finite T [see Eq. (15.36)] diverges as kdk/k^{2} (since E => k^{2} for the ferromagnetic mode when к => 0) while in 3D this is k^{2}dk/k^{2} as к => 0 (no problem of divergence). In MC simulations shown in the previous section, the statistical average was done using stochastic random configurations generated by statistical probability (no possible mode of к = 0). So, we do not encounter such a mathematical divergence as in the SW calculation. We use the following anisotropy between S, and Sy which stabilizes the angle determined above between their local quantization axes Sf and SJ:
where Kjj is supposed to be positive, small compared to J ^{m}, and limited to NN. Hereafter we take Kj,j = К for NN pair in the xy plane, for simplicity. The total magnetic Hamiltonian Нм is finally given by (using operator notations)
We now define the following two doubletime Green’s functions in the real space:
The equations of motion of these functions read
For the H_{m} and H_{a} parts, the above equations of motion generate terms such as SfSf; SJ J5> and <£ S^Sf; SJ ». These functions can be approximated by using the Tyablikov decoupling to reduce to the abovedefined G and F functions:
The last expression is due to the fact that transverse spin wave motions < Sf > are zero with time. For the DM term, the commutation relations ['H, S*] give rise to the following term:
This leads to the following type of Green's function:
Note that we have used defined в positively. The above equation is thus related to G and F functions [see Eq. (13.24)]. We use the following Fourier transforms in the xy plane of the G and F Green’s functions:
where the integral is performed in the first xy Brillouin zone (BZ) of surface Д and w is the SW frequency. Let us define the SW energy as E = hco in the following. For a monolayer, we have after the Fourier transforms where A and В are
where the reduced anisotropy is d = K/]^{m} and у = (cos k_{x}a + cosk_{y}a)/2, k_{x} and k_{y} being the wave vector components in the xy planes, a the lattice constant. The SW energies are determined by the secular equation
where ± indicate the left and right SW precessions. We see that (1) if в = 0, we have В and the last two terms of A are zero. We recover then the ferromagnetic SW dispersion relation
where Z = 4 is the coordination number of the square lattice (taking d = 0), (2) if в = n, we have A = 8/ ^{m} < S^{z} > and В = 8J^{m} < S^{z} > y. We recover then the antiferromagnetic SW energy
(3) in the presence of a DM interaction, we have 0 < cos в < 1 (0 < в < л/2). If d = 0, the quantity in the square root of Eq. (15.33) is always > 0 for any в. It is zero at у = 1. We do not need an anisotropy d to stabilize the SW at Г = 0. If d Ф 0 then it gives a gap at у = 1. We show in Fig. 15.16 the SW energy calculated from Eq. (15.33) for в = 0.3 radian (~ 17.2°) and 1 radian (~ 57.30°). The spectrum is symmetric for positive and negative wave vectors and for left and right precessions. Note that for small values of в (i.e., small D) E is proportional to k^{2} at low к (cf. Fig. 15.16a), as in ferromagnets. However, for strong в, E is proportional to к as seen in Fig. 15.16b. This behavior is similar to that in antiferromagnets [87]. The change of behavior is progressive with increasing в, no sudden transition from k^{2} to к behavior is observed. In the case of 5 = 1/2, the magnetization is given by (see technical details in chapter 8):
where for each к one has ±£, values. Figure 15.16 Spin wave energy E(к) versus /r (/c = k_{x} = k_{z}) for (а) в = 0.3 radian and (b) в = 1 in 2D at T = 0. See text for comments. Since Ej depends on S^{z}, the magnetization can be calculated at finite temperatures selfconsistently using the above formula. It is noted that the anisotropy d avoids the logarithmic divergence at к = 0 so that we can observe a longrange ordering at finite T in 2D. We show in Fig. 15.17 the magnetization M (=< S^{z} >) calculated by Eq. (15.36) for using d = 0.001. It is interesting to observe that M depends strongly on в: At high T, larger в yields stronger M. However, at T = 0 the spin length is smaller for larger в due to the socalled spin contraction in antifer romagnets [see Eq. (8.61), Section 8.5]. As a consequence, there is a crossover of magnetizations with different в at low T as shown in Fig. 15.17. The spin length at Г = 0 is shown in Fig. 15.18 for several в. Figure 15.17 (a) Spin length M =< S^{z} > versus temperature T for a 2D sheet with в = 0.175 (radian) (magenta void squares), в = 0.524 (blue filled squares), в = 0.698 (green void circles), в = 1.047 (black filled circles); (b) Zoom at low T to show magnetization crossovers. BilayerWe note that for magnetic bilayer between two ferroelectric films, the calculation similar to that of a monolayer can be done. By symmetry, spins between the two layers are parallel, the energy of a spin on a layer is
where there are 4 inplane NN and one parallel NN spin on the other layer. The interface coupling is with only one polarization instead of two (see Eq. (15.12)) for a monolayer for comparison. The minimum energy corresponds to tan# = —]^{m}f/]^{m}. The calculation by Green’s functions for a film with a thickness is straightforward: Writing Green’s functions for each layer and making Fourier transforms in the xy planes, we obtain a system Figure 15.18 Spin length at temperature T = 0 for a monolayer versus в (radian). of coupled equations. For the details, the reader is referred to Ref. [89]. For a bilayer, the SW energy is the eigenvalues of the following matrix equation:
where
where E = hw and M (E) is given by with Figure 15.19 Spin wave energy E versus к = k_{x} = k_{y} at T = 0 for a bilayer with 9 = 0.6 radian. Note that by symmetry, one has < Sf >=< Sf >. We show in Fig. 15.19 the SW spectrum of the bilayer case for a strong value в = 0.6 radian. There are two important points:
In conclusion of this section, we emphasize that the DM interaction affects strongly the SW mode at k* 0. Quantum fluctuations in competition with thermal effects cause the crossover of magnetizations of different в: In general, stronger в yields stronger spin contraction at and near T = 0 so that the corresponding spin length is shorter. However, at higher T, stronger в means stronger J ^{m}f which yields stronger magnetization. It explains the crossover at moderate T. 
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