# Green’s Function Method

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 c-axis, 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 xy-plane perpendicular to the c-axis. *Q *denotes — *в,.*

Note that in the laboratory coordinate system, namely in the film coordinates, the *2* direction coincides with the c-direction 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 Non-Collinear Magnets

We define the following two double-time 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 higher-order functions, for example, ~< *Sf,* S“(t')»>, etc., we

obtain the following general equations for non-collinear magnets:

## BCC Helimagnetic Films

In 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 in-plane Fourier transforms:

where *u>* is the spin wave frequency, *k _{xy}* denotes the wave-vector parallel to

*xy*planes and R, is the position of the spin at the site /.

*Hi, nj*and

*n*are, respectively, the z-component indices of the layers where the sites R/, R

_{k}_{;}and R/< belong to. The integral over

*k*is performed in the first Brillouin zone (

_{xy}*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 three-dimensional 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 c-axis),

*у*=

*cos[k*(a: lattice constant). We see that

_{x}a/2)cos[k_{y}a/2)cos[k_{z}a/2)*hw*is zero when

*A*= ±B, namely at

*k*=

_{x}*k*=

_{y}*k*0

_{z}=*(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 in-plane 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 det|M| = 0, we obtain the spin wave spectrum *u> *of the present system: For each value (/c_{x}, *k _{y}),* there are 2

*N*eigenvalues of

_{z }*u>*corresponding to two opposite spin precessions as in antiferromagnets (the dimension of det |M| is 2

*N*x 2

_{z}*N*Note that the above equation depends on the values of <

_{z}).*S* > (n =*1,...,

*N*Even at temperature

_{z}).*T =*0, these z-components are not equal to 1/2 because we are dealing with an antiferromagnetic system where fluctuations at

*T*= 0 give rise to the so-called 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 (2*n* — 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*

*,,*

*2*

*N*are poles of

_{Z},*д*

_{П:П}. со,**(k**can be obtained by solving |M| = 0. In this case,

_{XJ},)*g*can be expressed as

_{n},_{n}

where *D _{2n}-i (со,*

**(k**is

_{xy}))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 n-th 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*depends on the magnetizations of the neighboring layers via &>, (/ = 1, • • • ,

^{z}>*2*

*N*we should solve by iteration the equations (11.34) written for all layers, namely for

_{Z}),*n*= 1,...,

*N*to obtain the magnetizations of layers 1, 2, 3,...,

_{z},*N*at a given temperature

_{z }*T.*Note that by symmetiy, < Sf >=<

*S*>, < >=<

^{Z}N*S*Sf >=<

^{z}Ni__{l}>, <*S*

*>, and so on. Thus, only*

^{z}Ni__{2}*N*

_{z}/2self-consistent 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 Bose-Einstein 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 right-hand side of Eq. (11.34) where pc = (kBTc)-1, we have by putting {5^) = 0 on the left-hand side,}

There are *N _{z}* such equations using Eq. (11.34) with

*n*=

*1,, N*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

_{z}.*T*The method to do this will be shown below.

_{c}.# Spin Waves: Results from the Green’s Function Method

Let us take *]* = 1, namely ferromagnetic interaction between NN. We consider the helimagnetic case where the NNN interaction *)г *is negative and |/_{2}| > *J.* The non-uniform 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**,..., **2**N _{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*we calculate the output

_{z }*(S*)[n*=

*1,..., N*

_{z}). Using this output set as input, we calculate again (

*S?*= 1,...,

_{t})(n*N*until the input and output are identical within a desired precision

_{z})*P.*Numerically, we use a Brillouin zone of 100

^{2}wave-vector 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*only a few iterations suffice to get

_{c},*P <*1%. Near

*T*several dozens of iteration are needed to get the convergence. We show below our results.

_{c},## Spectrum

We calculated the spin wave spectrum as described above for each a given *Jz/J-* The spin wave spectrum depends on the temperature via the temperature-dependence of layer magnetizations. Let us show in Fig. 11.3 the spin wave frequency *w* versus *k _{x}* =

*k*in the case of an 8-layer film where _/

_{y}_{2}//i = —1-4 at two temperatures

*T*= 0.1 and Г = 1.02 (in units of Д/

*k*= 1). Some remarks are in order:

_{B}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*8 and

_{z}=*d*= 0.1. The surface branches are indicated by s.

- (i) There are 8 positive and 8 negative modes corresponding two opposite spin precessions. Unlike ferromagnets, spin waves in antiferromagnets and non-collinear spin structures have opposite spin precessions which describe the opposite circular motion of each sublattice spins [87]. The negative sign does not mean spin wave negative energy, but it indicates just the precession contrary to the trigonometric sense.
- (ii) Note that there are two degenerate acoustic surface branches lying at low energy on each side. This degeneracy comes from the two symmetrical surfaces of the film. These surface modes propagate parallel to the film surface but are damped from the surface inward.
- (iii) As
*T*increases, layer magnetizations decrease (see below), reducing, therefore, the spin wave energy as seen in Fig. 11.3 (bottom). - (iv) If the spin magnitude 5
*Ф*1/2, then the spectrum is shifted toward higher frequency since it is proportional to*S.* - (v) Surface spin wave spectrum (and bulk spin waves) can be experimentally observed by inelastic neutron scattering in ferromagnetic and antiferromagnetic films [36, 374]. To our knowledge, such experiments have not been performed for helimagnets.