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SelfConsistent Green’s Function Method: FormulationThe GF method has been developed for collinear [78, 79] and non collinear surface spin configurations in thin films [89,102, 248,249] and in magnetoferroelectric superlattices [317]. Let us briefly recall here the principal steps of calculation and give the results for the present model. In the following, we consider the case of spin one half S = 1/2. Expressing the Hamiltonian in the local coordinates, we obtain As said in the previous section, the spins lie in the xz planes, each on its quantization local zaxis (Fig. 13.2). Note that unlike the sinus term of the DM Hamiltonian, Eq. (13.15), the sinus terms of H_{e}, the third line of Eq. (13.16), are zero when summed up on opposite NN (no e,,y to compensate). The third line disappears, therefore, in the following. At this stage, it is very important to note that the standard commutation relations between spin operators S^{z} and S^{±} are defined with z as the spin quantization axis. In noncollinear spin configurations, calculations of SW spectrum using commutation relations without paying attention to this are wrong. It is known that in two dimensions (2D) there is no longrange order at finite temperature (Г) for isotropic spin models with short range interaction [231]. Thin films have small thickness; therefore, to stabilize the ordering at finite T, it is useful to add an anisotropic interaction. We use the following anisotropy between S, and S, which stabilizes the angle determined above between their local quantization axes Sf and S^{z}:
where is supposed to be positive, small compared to and limited to NN. Hereafter we take = /_{x} for NN pair in the xz plane, for simplicity. As it turns out, this anisotropy helps stabilize the ordering at finite T in 2D as discussed. It helps also stabilize the SW spectrum at Г = 0 in the case of thin films but it is not necessary for 2D and 3D at Г = 0. The total Hamiltonian is finally given by
We define the following two doubletime GF’s in the real space:
The equations of motion of these functions read For the He and H_{a} parts, the above equations of motion generate terms such as SfSf; SJ J5> and <£ Sfsf; 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 SW motions < Sf > are zero with time. For the DM term, the commutation relations ['H, Sf] give rise to the following term:
which leads to the following type of GFs:
Note that we have replaced e_{if}j sin#,,y by sin# where в is positive. The above equation is related to G and F functions [see Eq. (13.24)]. The Tyablikov decoupling scheme [43, 346] neglects higherorder functions. We now introduce the following inplane Fourier transforms g_{ni}„< and f_{n n}’ of the G and F Green's functions:
where the integral is performed in the first xz Brillouin zone (BZ) of surface Д, со is the spin wave frequency, n and ri are the indices of the layers along the c axis to which R, and R_{;} belong (n = 1 being the surface layer, n = 2 the second layer and so on). We finally obtain the following matrix equation: where M (£) is a square matrix of dimension (2 N x 2N), h and u are the column matrices which are defined as follows:
where E = hw and M (E) is given by with
where n = 1, 2,, N,d„ = li/J„, у = (cos k_{x}a + cos k_{z}a)/2, k_{x} and k_{z} denote the wavevector components in the xz planes, a the lattice constant. Note that (i) if n = 1 (surface layer) then there are no n — 1 terms in the matrix coefficients, (ii) if n = N then there are no n + 1 terms. Besides, we have distinguished the inplane NN interaction /„ from the interplane NN one J±. In the case of a thin film, the SW eigenvalues at a given wave vector к = (k_{x}, k_{z}) are calculated by diagonalizing the matrix 13.31. Taking S = 1/2, the layer magnetization of the layer n is given by (see technical details in Section 8.4 of Chapter 8 and in Ref. [87]): where n = 1,..., N, and n — l)th column of M by u at E,. The layer magnetizations can be calculated at finite temperatures selfconsistently using the above formula. The numerical method to carry out this task has been described in details in Refs. [89]. One can summarize here: (i) Using a set of trial values (inputs) for {S*) (n = 1,, N), one diagonalizes the matrix to find spin wave energies E, which are used to calculate the outputs (S%) (n = 1,..., N) by using Eq. (13.35); (ii) using the outputs as inputs to iterate the equations; (iii) if the output values are the same as the inputs within a precision (usually at 0.001%), the iteration is stopped. The method is thus self consistent. The value of the spin in the layer n at T = 0 is calculated by (see Section 8.4 and [87, 89]): where the sum is performed over N negative values of Е, (for positive values, the BoseEinstein factor in Eq. (13.35) is equal to 0 at T = 0). The transition temperature T_{c} can be calculated by letting (S?_{t}) on the lefthand side of Eq. (13.35) to go to zero. The energy E, tends then to zero, so that we can make an expansion of the exponential at T = T_{c}. We have Two and Three Dimensions: Spin Wave Spectrum and MagnetizationConsider just one single xz plane. The above matrix is reduced to two coupled equations:
where A„ is given by (13.32) but without J_{±} term for the 2D case considered here. Coefficients B_{n} and C„ are given by (13.33) and (13.34) with C„ = 0. The poles of the GF are the eigenvalues of the SW spectrum which are given by the secular equation
where ± indicate the left and right SW precessions. Several remarks are in order: (i) If # = 0, we have B„ = 0 and the last three terms of A_{n} are zero. We recover then the ferromagnetic SW dispersion relation
where Z = 4 is the coordination number of the square lattice (taking d_{n} = 0). (ii) If# = n, we have A_{n} = 8/„ < S* >, B_{n} = —8/„ < S% > y. We recover then the antiferromagnetic SW dispersion relation
(iii) in the presence of a DM interaction, we have 0 < cos# < 1 (0 < # < Л/2). If d_{n} = 0, the quantity in the square root of Eq. (13.39) is always > 0 for any #. It is zero at у = 1. The SW spectrum is, therefore, stable at the longwavelength limit. The anisotropy d_{n} gives a gap at у = 1. As said earlier, the necessity to include an anisotropy has a double purpose: It permits a gap and stabilizes a longrange ordering at finite T in 2D systems. Figure 13.3 Spin wave spectrum £(/c) versus к = k_{x} = k_{z} for (а) в = 0.524 radian and (b) в = 1.393 in two dimensions at T = 0.1. Positive and negative branches correspond to right and left precessions. A small d (= 0.001) has been used to stabilized the ordering at finite T in 2D. See text for comments. Figure 13.3 shows the SW spectrum calculated from Eq. (13.39) for в = 30° (л/6 radian) and 80° (1.396 radian). The spectrum is symmetric for positive and negative wave vectors and for left and right precessions. Note that for small в (i.e., small D) E is proportional to k^{2} at low к (cf. Fig. 13.3a), as in ferromagnets. However, as в increases, we observe that E becomes linear in к as seen in Fig. 13.3b. This is similar to antiferromagnets. The change of behavior is progressive with increasing в, we do not observe a sudden transition from к^{2} to к behavior. This feature is also observed in three dimensions (3D) and in thin films as seen below. It is noted that, thanks to the existence of the anisotropy d, we avoid the logarithmic divergence at к = 0 so that we can observe a longrange ordering at finite T in 2D. We show in Fig. 13.4 the magnetization M (=< S^{z} >) calculated by Eq. (13.35) for one layer using d = 0.001. It is interesting to observe that M depends strongly Figure 13.4 Magnetizations M versus temperature T for a monolayer (2D) 0 = 0.175 (radian), 0 = 0.524,0 = 0.698,0 = 1.047 (void magenta squares, green filled squares, blue void circles and filled red circles, respectively). A small d (= 0.001) has been used to stabilized the ordering at finite T in 2D. See text for comments. on 0: 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 [87] calculated by Eq. (13.36). As a consequence, there is a crossover of magnetizations obtained with different в at low T as shown in Fig. 13.4. Let us study the 3D case. The crystal is periodic in three directions. We can use the Fourier transformation in they direction, namely g_{n}± = g_{n}e^{±lk}>^{a} and f_{n±} = f„e^{±,k}y^{a}. The matrix (13.30) is reduced to two coupled equations of g and / functions, omitting index n,
where The spectrum is given by
If cos в = 1 (ferromagnetic), one has 6 = 0. By regrouping the Fourier transforms in three directions, one obtains the 3D ferromagnetic dispersion relation E = 2Z < S^{z} > (1 — y^{2}) where у = [cos(/f_{x}o) + cos(k_{y}a) + cos(k_{z}a)]/3 and Z = 6, coordination number of the simple cubic lattice. Unlike the 2D case where the angle is inside the plane so that the antiferromagnetic case can be recovered by setting cos в = 1 as seen above, one cannot use the above formula to find the antiferromagnetic case because in the 3D formulation it was supposed a ferromagnetic coupling between planes, namely there is no angle between adjacent planes in the above formulation. The same consideration as in the 2D case treated above shows that for d = 0 the spectrum E > 0 for positive precession and E < 0 for negative precession, for any в. The limit E = 0 is at у = 1 (к = 0). Thus there is no instability due to the DM interaction. Using Eq. (13.45), we have calculated the 3D spectrum. This is shown in Fig. 13.5 for a small and a large value of в. As in the 2D case, we observe E oc к when к —>■ 0 for large в. Main properties of the system are dominated by the inplane DM behavior. Figure 13.5 Spin wave spectrum E (/r) versus к = k_{x} = k_{z} for в = тг/6 (red circles) and в = л/3 (blue circles) in three dimensions at T = 0.1, with d = 0. Note the linearк behavior at low к for the large value of в (inset). See text for comments. Figure 13.6 (a) Magnetization M versus temperature T fora 3D crystal в = 0.175 (radian), 0 = 0.524,0 = 0.785,0 = 1.047 (red circles, green squares, blue triangles and void magenta circles, respectively), with d = 0. Inset: Zoom showing the crossover of magnetizations at low T for different 0, (b) The spin length So at T = 0 versus 0. See text for comments. Figure 13.6a displays the magnetization M versus T for several values of в. As in the 2D case, when в is not zero, the spins have a contraction at Г = 0: A stronger 0 yields a stronger contraction. This generates a magnetization crossover at low T shown in the inset of Fig. 13.6a. The spin length at T = 0 versus в is displayed in Fig. 13.6b. Note that the spin contraction in 3D is smaller than that in 2D. This is expected since quantum fluctuations are stronger at lower dimensions. 
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