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SemiInfinite SolidsOne examines the case of a semiinfinite 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 bodycentered 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} = J_{2}^{B} = у_{2}. Note that this system is a bodycentered cubic antiferromagnet if = Sb. Now, for a semiinfinite 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 bodycentered 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 Semiinfinite crystal of bodycentered 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, < S_{B} > — S_{B} = —(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 у_{2}(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 U_{2n} and U_{2n+1}: where k_{z} 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 semiinfinite ferrimagnet versus k_{x} = k_{y} 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 k_{z} = л /a and the lower limit to k_{z} = 0. where ф is a real factor defined by ф^{п} = e^{}*^{2}'^{10} where k_{2} is the imaginary part of k_{z}. 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 k_{x} = k_{y} = 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. k_{x} = k_{y} = 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 k_{x} = k_{y} = 0. For k_{x}, k_{y} 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 k_{x} = k_{y} = 0. The gap at k_{x} = k_{y} = 0 in the ferrimagnet is proportional to a. We show in Fig. 8.3 the spin wave spectrum versus k_{x} = k_{y}. We show in Fig. 8.4 the spatial variation of the amplitude of the surface mode at k_{x} = k_{y} = 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 nextnearest 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 = ^ > e_{c}. The determination of e_{c} with a surface is more complicated than in the bulk case because there may exist a nonuniform spin configuration near the Figure 8.5 Noncollinear 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 e_{c} depends also on a. We show in Fig. 8.5 the spin configuration near the surface for a value ofe > e_{c}. Spin Wave Theory in Ferromagnetic FilmsWe show in this section that localized surface spin wave modes affect strongly thermodynamic behaviors of ferromagnetic thin films. In particular, we will show that lowlying 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 N_{T} 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 Isinglike behavior. The factor 2 in front of the terms is used for historical reasons [see (1.57)]. MethodThe 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 higherorder 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 k_{x>}, 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 k_{xy}. 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/k_{B}T. The commutator of the righthand 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  S_{lbNr}) are added to ensure that there are no C„ and B_{n} terms for the first and the last layer. The coefficients A_{n}, B„ and C„ depend on the crystalline lattice of the film. We give here some examples: Film of stacked triangular latticeswhere the following notations have been used:
layer n and a spin in the layer (n ± 1). Of course, Ai,»i ^{=}
Film of simple cubic latticewhere C = 4 and y_{k} = [cos(/c_{x}a) + cosf/c^a)]. Film of bodycentered cubic latticewhere y_{k} = cos[k_{x}a/2) cos[k_{y}a/2). We go back to Eq. (8.29). Writing it for n = 1, 2, • • • , N_{T}, we obtain a system of N_{T} equations which can be rewritten in a matrix form
where u is a column matrix whose nth element is 2S„,„ < S* >. For a given k_{Y>}. the spin wave dispersion relation Л<а(к_{А>},) can be obtained by solving the secular equation detM = 0. There are Nr eigenvalues fuoj (/' = 1, ■■■, Nr) for each k_{xy}. It is obvious that со, depends on all (Sf_{t}) contained in the coefficients A„, B„ and C_{n}. 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 g_{n n} is obtained by the solution ofEq. (8.39):
M„ is the determinant obtained by replacing the nth column of M by u. To simplify the notations we put /;&>, = £, and hw = E in the following. By expressing
we see that E, (/ = 1, • • ■ , N_{T}) 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 < S^{z}n > 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 selfconsistent 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 *■ T_{c}, 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 A_{n}, B_{n} and C„ are proportional to M in the above hypothesis of uniform layer magnetization. We see that £, is proportional to M. The righthand side of Eq. (8.51), therefore, does not depend on M. ResultsTo 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 detM = 0 for each value of k_{xy}. Using the values so obtained of £, one calculates the output values of < S^{z}n > (n = 1, • • • , N_{T}). 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 T_{c}, one needs a few dozens to a few hundreds of iterations. We show here the results of a few cases for comparison. Spin wave spectrumWe show in Figs. 8.6 and 8.6 the spin wave spectra of ferromagnetic films of simple cubic and bodycentered 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 bodycentered 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 N_{T} = 8 shown in Fig. 8.7. Figure 8.6 Magnon spectrum E = hw of a ferromagnetic film with a simple cubic lattice versus к = k_{x} = k_{y} for N_{r} = 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 = k_{x} = k_{y} 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 magnetizationsFigure 8.8 shows the results of the layer magnetizations for the first two layers in the cases considered above with N_{T} = 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 bodycentered 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 lowlying energy exist (acoustic surface modes). Antiferromagnetic FilmsWe 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 LatticeWe 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, • ■ • , N_{T} and the following notations have been used:
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 BodyCentered Cubic LatticeIn the case of a film of bodycentered cubic lattice with a (001) surface, we suppose that f sublattice and ), sublattice occupy even and oddindex 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), d_{f} = d(l  <5i,„)(l  <$«,„) + s(Si,_{n} + S_{w>}„), y_{k} = cos(/c_{x}a/2) cos(/q,o/2). As before, we have redefined < S2_{n±1} >>■ — < Sl_{n±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 detM = 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 BoseEinstein factor is equal to 0 at Г = 0). The numerical results for S = 1/2, N_{T} = 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 zeropoint fluctuations are due to the antiferromagnetic interaction acting on a spin: The weaker the antiferromagnetic interaction, the smaller the zeropoint 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 T_{N} is obtained in the same manner as for T_{c}. We have the following N_{T} coupled equations to solve by iteration: where n = 1, • • • , N_{T} The uniform layermagnetization 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 T_{c} for thicknesses N_{T} = 2 and 20, calculated with several values of anisotropy d
We show in Table 8.1 the values of T_{N} and T_{c} calculated for films of simple cubic lattice with two thicknesses Nr = 2 and N_{T} = 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. ConclusionWe 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 lowlying 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 noncollinear spin configuration. ProblemsProblem 1. Surface spin wave modes: Calculate the surface spin wave modes in the case of a semiinfinite ferromagnetic crystal of bodycentered cubic lattice for k_{x} = k_{y} = 0, л/a in using the method presented in Section 8.3. Problem 2. Critical nextnearestneighbor 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, N_{2} 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 = J_{2} = J_{3}. Problem 5. Meanfield theory of thin films: Calculate the layer magnetizations of a 3layer film by the meanfield 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. HolsteinPrimakoff method: Using the HolsteinPrimakoff method of Chapter 3 for a semiinfinite 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 semiinfinite 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 J_{s}. Determine the ground state of the system as a function Of Js/J Problem 8. Ferrimagnetic film: Write the equations of motion for a fourlayer ferrimagnetic film of bodycentered cubic lattice, using the model and the method presented in Section 8.3. Consider the cases k_{x} = k_{y} = 0, tt/q. Solve numerically these equations to find surface and bulk magnons. 
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