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Green’s Function MethodTable of Contents:
Let us define the local spin coordinates as follows: the quantization axis of spin S, is on its axis which lies in the plane, the гц axis of S, is along the caxis, and the axis forms with »;,• and £,■ axes a direct trihedron. Since the spin configuration is planar, all spins have the same axis. Furthermore, all spins in a given layer are parallel. Let I,, гц and be the unit vectors on the local гц, £,) axes. We use the following local transformation which has been used for the first time in Ref. [139] and described in Section 3.4:
We have [see Fig. 11.2)
where cos = cos(0, — 0_{;}) is the angle between two spins / and j. Figure 11.2 Local coordinates in a xyplane perpendicular to the caxis. Q denotes — в,. Note that in the laboratory coordinate system, namely in the film coordinates, the 2 direction coincides with the cdirection or the i) axis perpendicular to the film surface, while the x and у directions are taken to be the BCC crystal axes in the film plane. Replacing these into Eq. (11.11) to express Sy in the (f;, i)j, £,) coordinates, then calculating S, Sy, we obtain the following exchange Hamiltonian from (11.2): General Formulation for NonCollinear MagnetsWe define the following two doubletime Green’s functions in the real space:
We need these two functions because the equation of motion of the first function generates functions of the second type, and vice versa. These equations of motion are where the spin operators and their commutation relations are given by
Expanding the commutators in Eqs. (11.15)(11.16), and using the Tyablikov decoupling scheme [346] for higherorder functions, for example, ~< Sf, S“(t')»>, etc., we obtain the following general equations for noncollinear magnets: BCC Helimagnetic FilmsIn the case of a BCC thin film with a (001) surface, the above equations yield a closed system of coupled equations within the Tyablikov decoupling scheme [346]. For clarity, we separate the sums on NN interactions and NNN interactions as follows:
For simplicity, except otherwise stated, all NN interactions [}k,k'< h,k) are taken equal to (Ji, /1) and all NNN interactions are taken equal to J_{2} in the following. Furthermore, let us denote, in the film coordinates defined above, the Cartesian components of the spin position R, by (f,, m,, n,). We now introduce the following inplane Fourier transforms: where u> is the spin wave frequency, k_{xy} denotes the wavevector parallel to xy planes and R, is the position of the spin at the site /. Hi, nj and n_{k} are, respectively, the zcomponent indices of the layers where the sites R/, R_{;} and R/< belong to. The integral over k_{xy} is performed in the first Brillouin zone (BZ) whose surface is Д in the xy reciprocal plane. For convenience, we denote n, = 1 for all sites on the surface layer, n, = 2 for all sites of the second layer and so on. Note that for a threedimensional case, making a 3D Fourier transformation of Eqs. (11.19)—(11.20) we obtain the spin wave dispersion relation in the absence of anisotropy:
where
where Z = 8 (NN number), Z_{c} = 2 (NNN number on the caxis), у = cos[k_{x}a/2)cos[k_{y}a/2)cos[k_{z}a/2) (a: lattice constant). We see that hw is zero when A = ±B, namely at k_{x} = k_{y} = k_{z} = 0 (y = 1) and at k_{z} = 2в along the helical axis. The case of ferromagnets (antiferromagnets) with NN interaction only is recovered by putting cos 6 = 1 (1) [78]. Let us return to the film case. We make the inplane Fourier transforms Eqs. (11.21)(11.22) for Eqs. (11.19)(11.20). We obtain the following matrix equation: where M (w) is a square matrix of dimension (2A/_{Z} x 2iV_{z}), h and u are the column matrices which are defined as follows:
where, taking h = 1 hereafter,
where where n = 1, 2, • • • , N_{z}, d = /i//i, and
In the above expressions, 0„,„±i the angle between a spin in the layer n and its NN spins in layers n ± 1, etc., and у = cos (^) cos • Solving detM = 0, we obtain the spin wave spectrum u> of the present system: For each value (/c_{x}, k_{y}), there are 2N_{z }eigenvalues of u> corresponding to two opposite spin precessions as in antiferromagnets (the dimension of det M is 2N_{z} x 2N_{z}). Note that the above equation depends on the values of < S* > (n = 1,..., N_{z}). Even at temperature T = 0, these zcomponents are not equal to 1/2 because we are dealing with an antiferromagnetic system where fluctuations at T = 0 give rise to the socalled zero point spin contraction [87]. Worse, in our system with the existence of the film surfaces, the spin contractions are not spatially uniform as will be seen below. So the solution of det M = 0 should be found by iteration. This will be explicitly shown hereafter. The solution for g„_{fn} is given by
where M_{2}„_i is the determinant made by replacing the (2n — l)th column of M by u given by Eq. (11.25) [note that occupies the (2n — l)th line of the matrix h]. Writing now
we see that ш/ (k_{X}y), i = 1,, 2N_{Z}, are poles of д_{П:П}. со, (k_{XJ},) can be obtained by solving M = 0. In this case, g_{n},_{n} can be expressed as
where D_{2n}i (со, (k_{xy})) is Next, using the spectral theorem which relates the correlation function {Sj'S'f') to Green's function [383], we have
where e is an infinitesimal positive constant and /8 = (/cgT)^{1}, кв being the Boltzmann constant. Using Green's function presented above, we can calculate self consistently various physical quantities as functions of temperature T. The magnetization (S^{z}) of the nth layer is given by
Replacing Eq. (11.29) in Eq. (11.32) and making use of the following identity:
we obtain
where n = 1,..., N_{z}. As < S^{z} > depends on the magnetizations of the neighboring layers via &>, (/ = 1, • • • , 2N_{Z}), we should solve by iteration the equations (11.34) written for all layers, namely for n = 1,..., N_{z}, to obtain the magnetizations of layers 1, 2, 3,..., N_{z }at a given temperature T. Note that by symmetiy, < Sf >=< S^{Z}N >, < >=< S^{z}Ni__{l} >, < Sf >=< S^{z}Ni__{2} >, and so on. Thus, only N_{z}/2 selfconsistent layer magnetizations are to be calculated. The value of the spin in the layer n at T = 0 is calculated by where the sum is performed over N_{z} negative values of &>, (for positive values the BoseEinstein factor is equal to 0 at Г = 0). The transition temperature T_{c} can be calculated in a self consistent manner by iteration, letting all < S* > tend to zero, namely &>, *■ 0. Expanding e^{/3' — 1 > f5ca>i on the righthand side of Eq. (11.34) where pc = (kBTc)1, we have by putting {5^) = 0 on the lefthand side,} There are N_{z} such equations using Eq. (11.34) with n = 1,, N_{z}. Since the layer magnetizations tend to zero at the transition temperature from different values, it is obvious that we have to look for a convergence of the solutions of the equations Eq. (11.36) to a single value of T_{c}. The method to do this will be shown below. Spin Waves: Results from the Green’s Function MethodLet us take ] = 1, namely ferromagnetic interaction between NN. We consider the helimagnetic case where the NNN interaction )г is negative and /_{2} > J. The nonuniform GS spin configuration across the film has been determined in Section 11.2 for each value of p = Jz/Ji Using the values of 0„,n±i and 6>„,„_{±}2 to calculate the matrix elements of M, then solving det M = 0 we find the eigenvalues o>, (/ = 1,..., 2N_{Z}) for each k_{xy} with an input set of (S*)(n = 1,..., N_{z}) at a given T. Using Eq. (11.34) for n = 1,..., N_{z }we calculate the output (S*)[n = 1,..., N_{z}). Using this output set as input, we calculate again (S?_{t})(n = 1,..., N_{z}) until the input and output are identical within a desired precision P. Numerically, we use a Brillouin zone of 100^{2} wavevector values, and we use the obtained values (S*) at a given T as input for a neighboring T. At low T and up to T_{c}, only a few iterations suffice to get P < 1%. Near T_{c}, several dozens of iteration are needed to get the convergence. We show below our results. SpectrumWe calculated the spin wave spectrum as described above for each a given Jz/J The spin wave spectrum depends on the temperature via the temperaturedependence of layer magnetizations. Let us show in Fig. 11.3 the spin wave frequency w versus k_{x} = k_{y} in the case of an 8layer film where _/_{2}//i = —14 at two temperatures T = 0.1 and Г = 1.02 (in units of Д/k_{B} = 1). Some remarks are in order: Figure 11.3 Spectrum E = hw versus к = k_{x} = k_{y} for /2/У1 = —1.4 at T = 0.1 (top) and T = 1.02 (bottom) for N_{z} = 8 and d = 0.1. The surface branches are indicated by s.

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