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# Semi-Infinite Solids

One examines the case of a semi-infinite crystal. One calculates the spin wave spectrum and shows the existence of surface modes in this section. The simplest way to do is to use the method of equation of motion described in Chapter 3. To give a simple example, one considers the same system illustrated there, namely a semiinfinite ferrimagnetic crystal of body-centered cubic lattice but with the inclusion of a surface. One uses the same Hamiltonian (3.98) and the same equations of motion (3.99) with the same hypothesis

)2 = J2B = у2. Note that this system is a body-centered cubic antiferromagnet if = Sb. Now, for a semi-infinite crystal, one uses the following Fourier transforms only in the xy plane which is still periodic:

where кц is a wave vector parallel to the surface, n and n' denote respectively the indices of the layers to which the spins / and m belong, BZ stands for the first Brillouin zone in the xy plane.

To simplify the presentation, we take the body-centered cubic lattice with the surface plane (001). We suppose that the sublattice of A spins (t) occupies the planes of even indices and the sublattice of В spins (T) takes the planes of odd indices as indicated in Fig. 8.2.

The equations of motion for S,~ and S+ give two following coupled equations:

For the first two layers, we have

Figure 8.2 Semi-infinite crystal of body-centered cubic lattice with a (001) surface (side view). The sublattices | and f are denoted by black and white circles, respectively. The surface has index n = 1.

where we have used, at T = 0, < 5^ >~ 5л = (1 — a)S, < SB > — SB = —(1 + a)5 with |a| < 1 and 5 a constant. The other notations used in (8.3)-(8.6) are

a being the lattice constant. The factor yi(k||) couples the nearest neighbors belonging to the adjacent planes while у2(кц) connects the neighbors belonging to the same sublattice (namely, next- nearest neighbors, by distance).

We take the following forms for the bulk spin wave amplitudes

U2n and U2n+1:

where kz is the real wave vector in the z direction. Replacing these in (8.3) and (8.4), we obtain the following secular equation for a nontrivial solution:

We deduce the dispersion relation for bulk modes

We are now interested in finding the surface modes. We look for solutions of the form

Figure 8.3 Spin wave spectrum of a semi-infinite ferrimagnet versus kx = ky in the case where e = 0, a = —1/3 (top) and a = 1/3 (bottom). Surface spin wave branches are indicated by MS. The hachured bands are the bulk continuum. The upper limit of each band corresponds to kz = л /a and the lower limit to kz = 0.

where ф is a real factor defined by фп = e-*2'10 where k2 is the imaginary part of kz. For a decaying wave, ф < 1. ф is called "decay factor." Replacing these amplitudes in (8.3)—(8.6) we obtain a system of coupled equations. Surface modes correspond to solutions ф < 1. We examine a particular case where kx = ky = 0. In this case, the following solution for a surface mode is found:

Figure 8.4 Variation of the amplitude of the surface mode near the surface. kx = ky = 0, e = 0, a = —1/3. Surface spins are В spins (for convenience, В spins are drawn as up spins).

We see that in order to have ф < 1, we should have a < 0 for the case kx = ky = 0. For kx, ky Ф 0, there exists for a < 0 and a > 0 a surface spin wave branch in the spectrum as shown in Fig. 8.3.

Note: In the case of an antiferromagnet, we just put a = 0 in the above equations. There is no surface mode for kx = ky = 0.

The gap at kx = ky = 0 in the ferrimagnet is proportional to a. We show in Fig. 8.3 the spin wave spectrum versus kx = ky.

We show in Fig. 8.4 the spatial variation of the amplitude of the surface mode at kx = ky = 0 with a = - 1/3. The decay factor calculated by (8.22) is equal to 0.5.

We discuss now the effect of the interaction between the next-nearest neighbors. The interaction between nearest neighbors /1 is antiferromagnetic [/i>0 as seen in the definition of the Hamiltonian (3.98)]. If /2 is ferromagnetic, i.e., e < 0, the magnetic order is antiferromagnetic between the two sublattices. On the other hand, if/2 is antiferromagnetic (>0), there is a competition between /1 and/2-When |/2| is large enough, the collinear antiferromagnetic configuration is no more stable for e = ^ > ec. The determination of ec with a surface is more complicated than in the bulk case because there may exist a non-uniform spin configuration near the

Figure 8.5 Non-collinear spin configuration near the surface (side view) for € > €c, with a > 0. Surface spins are В spins (for convenience, В spins are drawn as up spins).

surface. In the present case, the critical value ec depends also on a. We show in Fig. 8.5 the spin configuration near the surface for a value ofe > ec.

# Spin Wave Theory in Ferromagnetic Films

We show in this section that localized surface spin wave modes affect strongly thermodynamic behaviors of ferromagnetic thin films. In particular, we will show that low-lying localized modes diminish the surface magnetization and the Curie temperature with respect to the bulk ones. These quantities depend, of course, on the surface interaction parameters and the film thickness.

The method which can cover correctly a large region of temperature is no doubt the Green’s function method (cf. Chapter 4).

We shall use here that method to study properties of thin films from T = 0 up to the phase transition.

We consider a thin film of NT layers with the Heisenberg quantum spin model. The Hamiltonian is written as

where is positive (ferromagnetic) and D/; > 0 denotes an exchange anisotropy. When D,; is very large with respect to J,j, the spins have an Ising-like behavior. The factor 2 in front of the terms is used for historical reasons [see (1.57)].

## Method

The Green’s function method has been formulated in detail in Chapter 4. It is useful to summarize here the main steps in its application to thin films.

We define one Green’s function for each layer, numbering the surface as the first layer. We write next the equation of motion for each of Green’s functions. We obtain a system of coupled equations. We linearize these equations to reduce higher-order Green's functions by using the Tyablikov decoupling scheme. We are then ready to make the Fourier transforms for all Green’s functions in the xy planes. We obtain a system of equations in the space (k*y, to) where kx>, is the wave vector parallel to the xy plane and to the spin wave frequency (pulsation). Solving this system we obtain Green's functions and to as functions of kxy. Using the spectral theorem, Eq. (4.39), we calculate the layer magnetization.

Let us define the following Green's function for two spins S, and

s r-

The equation of motion of G,,y(t, t') is written as

where [• • • ] is the boson commutator and (• • •) the thermal average in the canonical ensemble defined as

with p = l/kBT.

The commutator of the right-hand side of Eq. (8.25) generates functions of higher orders. In a first approximation, we can reduce these functions with the help of the Tyablikov decoupling [43, 346] as follows:

We obtain then the same kind of Green's function defined in Eq. (8.24). As the system is translation invariant in the xy plane, we use the following Fourier transforms:

where со is the spin wave pulsation (frequency), k*y the wave vector parallel to the surface, R, the position of the spin at the site /, n and ri are respectively the indices of the planes to which / and j belong (n = 1 is the index of the surface plane). The integration on k*y is performed within the first Brillouin zone in the xy plane. Let Д be the surface of that zone. Equation (8.25) becomes

where the factors (1 - 5„д) and (1 - SlbNr) are added to ensure that there are no C„ and Bn terms for the first and the last layer. The coefficients An, B„ and C„ depend on the crystalline lattice of the film. We give here some examples:

### Film of stacked triangular lattices

where the following notations have been used:

• (0 Jn, D„ are the interactions in the layer n,
• (ii) Jn,n±i and D„,x are the interactions between a spin in the

layer n and a spin in the layer (n ± 1). Of course, Ai,»-i =

• 0 if n = 1, and Jn,n+i, Дьл+i = 0 if n = NT,
• (iii) Yk = [2 cos[kxa) + 4 cos(/txo/2) cos[kycis/3/2)/C
• (iv) C = 6 is the coordination number in the xy plane.

### Film of simple cubic lattice

where C = 4 and yk = |[cos(/cxa) + cosf/c^a)].

### Film of body-centered cubic lattice

where yk = cos[kxa/2) cos[kya/2).

We go back to Eq. (8.29). Writing it for n = 1, 2, • • • , NT, we obtain a system of NT equations which can be rewritten in a matrix form

where u is a column matrix whose n-th element is 2S„,„- < S* >.

For a given kY>. the spin wave dispersion relation Л<а(кА>,) can be obtained by solving the secular equation det|M| = 0. There are Nr eigenvalues fuoj (/' = 1, ■■■, Nr) for each kxy. It is obvious that со, depends on all (Sft) contained in the coefficients A„, B„ and Cn.

To calculate the thermal average of the magnetization of the layer n in the case where S = we use the following relation: (see Chapter 4):

where {S~S+) is given by the following spectral theorem [see (4.39)]:

e being an infinitesimal positive constant. Equation (8.40) becomes

where Green’s function gn n is obtained by the solution ofEq. (8.39):

|M|„ is the determinant obtained by replacing the n-th column of |M| by u.

To simplify the notations we put /;&>, = £, and hw = E in the following. By expressing

we see that E, (/ = 1, • • ■ , NT) are the poles of Green’s function. We can, therefore, rewrite g,h as

where /„(£,) is given by

Replacing Eq. (8.45) in Eq. (8.42) and making use of the following identity:

we obtain

where n = 1, ■ ■ ■ , Nr-

As < Szn > depends on the magnetizations of the neighboring layers via £,(/ = 1, • ■ • , Nr), we should solve by iteration the equations (8.48) written for all layers, namely for n = 1, ■ ■ ■ , Nr, to obtain the layer magnetizations at a given temperature T.

The critical temperature can be calculated in a self-consistent manner by iteration, letting all < S* > tend to zero.

In the case where the surface parameters are not different from the bulk ones, it is not exaggerated to calculate the critical temperature by supposing that all layer magnetizations are equal to a unique average value M which is to be determined self- consistently. The value of M is defined from < 5,^ > of Eq. (8.48) as follows:

Replacing all < S* > in the matrix elements by M, we see that Лп /п(£;) = 2M by using Eq. (8.46). We deduce

When T -*■ Tc, M -> 0. We can then make an expansion of the exponential of the denominator of Eq. (8.50). We obtain

We notice that all matrix elements An, Bn and C„ are proportional to M in the above hypothesis of uniform layer magnetization. We see that £, is proportional to M. The right-hand side of Eq. (8.51), therefore, does not depend on M.

## Results

To calculate numerically the above equations, one must first determine the first Brillouin zone according to the lattice structure. For the iteration process, one starts in general at a very low T with input values of < Sj; > close to the spin amplitude. Next, one uses the solutions of < S% > as inputs for a temperature not far from the previous T in order to facilitate the convergence. Of course, if we use the hypothesis of uniform layer magnetizations, i.e., < S,t >= M for all n, then there is only one solution M to find at a given T.

At each T, using input values for < Sj; >, one calculates the eigenvalues of the spin wave energy £, by solving det|M| = 0 for each value of kxy. Using the values so obtained of £, one calculates the output values of < Szn > (n = 1, • • • , NT). If the output values are equal to the input values within a given precision, one stops the iteration. In general, a few iterations suffice at low T for a solution with a precision of 1%. For a temperature close to Tc, one needs a few dozens to a few hundreds of iterations. We show here the results of a few cases for comparison.

### Spin wave spectrum

We show in Figs. 8.6 and 8.6 the spin wave spectra of ferromagnetic films of simple cubic and body-centered cubic lattices with a thickness Nt = 8. For simplicity, we suppose all exchange interactions are identical and equal to J. Also, all anisotropy constants D are equal to 0.01/. We see in Figs. 8.6 and 8.6 that there is no surface mode in the simple cubic case while there are two branches of surface modes in the body-centered lattice. Two branches result from the interaction of the surface modes coming from the two surfaces of the film. If the thickness is thick enough (longer than the penetration length of the surface mode), then two branches are degenerate due to symmetrical surface conditions. This is not the case for NT = 8 shown in Fig. 8.7.

Figure 8.6 Magnon spectrum E = hw of a ferromagnetic film with a simple cubic lattice versus к = kx = ky for Nr = 8 and D/J = 0.01. No surface mode is observed for this case.

Figure 8.7 Magnon spectrum E = hw of a ferromagnetic film with a body- centered cubic lattice versus к = kx = ky for Nr = 8 and D/J = 0.01. The branches of surface modes are indicated by MS.

Figure 8.8 Ferromagnetic films of simple cubic lattice (left) and body- centered cubic lattice (right): magnetizations of the surface layer (lower curve) and the second layer (upper curve), with Nt =4,0 = 0.01/,/ = 1.

### Layer magnetizations

Figure 8.8 shows the results of the layer magnetizations for the first two layers in the cases considered above with NT = 4. One observes that the layer magnetization at the surface is smaller than that of the second layer. This difference is larger in the case where there exists a surface mode as in the body-centered cubic lattice because surface modes are localized at the surface, making a larger deviation for surface spins.

One also observes that the critical temperature is strongly decreased in the case where surface modes of low-lying energy exist (acoustic surface modes).

# Antiferromagnetic Films

We can adapt the Green’s function method presented in Chapter 4 for bulk antiferromagnets to the case of antiferromagnetic thin films. We consider the following Hamiltonian:

where /,у > 0 and D/; > 0. We divide the lattice into two sublattices: sublattice A contains f spins and sublattice В i spins. We define the following Green's functions:

where j and j' belong to sublattice A, and i to sublattice B.

## Films of Simple Cubic Lattice

We write the equations of motion for and then we use

the Tyablikov decoupling [43] and the following Fourier transforms:

where we use the same notations as in the above ferromagnetic case. We obtain

where n, ri = 1, • ■ • , NT and the following notations have been used:

• ]i,j = J for all pairs (/', j),
• d = Dj j/J for all pairs (/, j) except when (/, j) are both on the surface (n = 1 and n = NT) where d = ds,
• E = fuo/J,
• de = d{ 1 - 5lr„)(l - 8N.nn) + ds[8hn + 8Nrin) (this complicated notation was used in order to include all cases in the same formula),
• Yk = cos(/cxa) cos(/f,,a) (for convenience, we used the distance between the two neighbors of the same sublattice in the xy plane equal to 2a and kx and ky oriented along the axes of one sublattice).

It is noted that in Eqs. (8.57)-(8.58) we have changed the sign of the average values of В spins (< S* >-> — < S* >) so that all < S* > are positive in Eqs. (8.57)-(8.58).

## Films of Body-Centered Cubic Lattice

In the case of a film of body-centered cubic lattice with a (001) surface, we suppose that f sublattice and ), sublattice occupy even- and odd-index layers, respectively. Using the following Fourier transforms:

we obtain

where n, ri = 1, ■ ■ ■ , N with N = Л/г /2 (Af=number of layers in each sublattice), df = d(l - <5i,„)(l - <\$«,„) + s(Si,n + Sw>„), yk = cos(/cxa/2) cos(/q,o/2). As before, we have redefined < S2n±1 >->■ — < Sln±1 > of В spins.

The next steps of the calculation are the same as in the ferromagnetic case: We calculate spin wave energy eigenvalues E, by solving det|M| = 0. The layer magnetization is calculated by Eq. (8.48). The value of the spin in the layer n at T = 0 is calculated by (see Chapter 3)

where the sum is performed over negative values of £, (for positive values the Bose-Einstein factor is equal to 0 at Г = 0).

The numerical results for S = 1/2, NT = 4 and d = 0.01 are Sf = 0.44075, 5f = 0.41885. We conclude that the zero- point spin contraction (cf. Chapter 3) is smaller for surface spins. This is not surprising because zero-point fluctuations are due to the antiferromagnetic interaction acting on a spin: The weaker the antiferromagnetic interaction, the smaller the zero-point spin contraction. Due to the lack of neighbors, the antiferromagnetic local field acting on a surface spin is weaker than that on an interior spin [87].

The Neel temperature TN is obtained in the same manner as for Tc. We have the following NT coupled equations to solve by iteration:

where n = 1, • • • , NT- The uniform layer-magnetization approximation presented above for ferromagnetic films [see (8.49)-(8.51)] yields for the antiferromagnetic case the following expression:

Table 8.1 Values of Neel temperature Tn and Curie temperature Tc for thicknesses NT = 2 and 20, calculated with several values of anisotropy d

 0.01 0.1 0.2 0.01 0.1 0.2 1.37 1.91 2.25 1.86 2.02 2.14 1.40 1.93 2.27 2.03 2.10 2.14

We show in Table 8.1 the values of TN and Tc calculated for films of simple cubic lattice with two thicknesses Nr = 2 and NT = 20. We see that the antiferromagnetic film has the critical temperature lightly but systematically smaller than that of the ferromagnetic film, except when there is a strong anisotropy which suppresses more or less antiferromagnetic fluctuations (last column). The critical temperature increases with increasing thickness as expected. It reaches the bulk value at a few dozens of layers.

# Conclusion

We have shown in this chapter how to calculate the spin wave spectrum and properties due to the spin waves in thin ferromagnetic and antiferromagnetic films with several lattice structures. The aim of this chapter is to provide basic theoretical methods to deal with simple models of clean surfaces. Physical results obtained here such as origin of low surface magnetization and low critical temperature are explained by the existence of localized surface spin wave modes which depends on the lattice structure, the surface orientation, the surface exchange interaction, etc. Many aspects of these results remain in real magnetic films where surface conditions are much more complicated. We did not show here all types of surface modes. In general, we can have acoustic surface modes which lie in the low- energy region below the bulk band, and optical surface modes which lie in the high energy region above the bulk band. The low-lying acoustic modes lower the surface magnetization and the critical temperature as can be seen by examining the formulas shown above. The low surface magnetization corresponds what was called in the 1980s "magnetically dead surface." On the other hand, high- energy optical surface modes make the surface magnetization larger than the bulk one. This situation occurs when the surface exchange interactions are much larger than the bulk values. The surface in this case was called "hard surface” [374].

So far, we have studied thin films with collinear spin ordering. In the following chapters, we consider the effect of the frustration in thin films. The situation in frustrated thin films is more complicated because of the non-collinear spin configuration.

# Problems

Problem 1. Surface spin wave modes:

Calculate the surface spin wave modes in the case of a semi-infinite ferromagnetic crystal of body-centered cubic lattice for kx = ky = 0, л/a in using the method presented in Section 8.3.

Problem 2. Critical next-nearest-neighbor interaction:

Calculate the critical value of e defined in Section 8.3 for an infinite crystal.

Problem 3. Uniform magnetization approximation:

Show that with the hypothesis of uniform layer- magnetization [Eq. (8.51)], the energy eigenvalue £, is proportional to M.

Problem 4. Multilayers: critical magnetic field

One considers a system composed of three films A, В and C, of Ising spins with respective thicknesses Ni, N2 and N3. The lattice sites are occupied by Ising spins pointing in the ±z direction perpendicular to the films. The interaction between two spins of the same film is ferromagnetic. Let /1, J2 and /3 be the magnitudes of these interactions in the three films. One supposes that the interactions at the interfaces A — В and В — C are antiferromagnetic and both equal to )s. One applies a magnetic field along the z direction. Determine the critical field above which all spins are turned into the field direction. For simplicity, consider the case Ji = J2 = J3.

Problem 5. Mean-field theory of thin films:

Calculate the layer magnetizations of a 3-layer film by the mean-field theoiy (cf. Chapter 2). One supposes the Ising spin model with values ±1/2 and a ferromagnetic interaction J for all pairs of nearest neighbors.

Problem 6. Holstein-Primakoff method:

Using the Holstein-Primakoff method of Chapter 3 for a semi-infinite crystal with the Heisenberg spin model, write the expression which allows us to calculate the surface magnetization as a function of temperature. Show that a surface mode of low energy (acoustic surface mode) diminishes the surface magnetization.

Problem 7. Frustrated surface: surface spin rearrangement

Consider a semi-infinite system of Heisenberg spins composed of stacked triangular lattices. Suppose that the interaction between nearest neighbors ] is eveiywhere ferromagnetic except for the spins on the surface: They interact with each other via an antiferromagnetic interaction Js. Determine the ground state of the system as a function Of Js/J-

Problem 8. Ferrimagnetic film:

Write the equations of motion for a four-layer ferrimagnetic film of body-centered cubic lattice, using the model and the method presented in Section 8.3. Consider the cases kx = ky = 0, tt/q. Solve numerically these equations to find surface and bulk magnons.

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