Simple Cubic Helimagnetic Films
In the above section, we have presented the work performed for the case of a helimagnetic film with the BCC lattice structure . A similar study has been carried out for the case of a helimagnetic film with the simple cubic lattice .
We consider a thin film of simple cubic lattice of Nz layers described by the Hamiltonian
Figure 11.13 Angle ct... on across the film for /2//1 = —0.6, —0.5, —0.4, —0.35, —0.3 (from top) with Nz = 8.
where S, is the Heisenberg spin at the lattice site /, is the exchange interaction between two NN spins. As before, we have added an anisotropic term /,j along the spin-quantization axes z taken to be very small. This anisotropy is necessary to have a phase transition at a finite temperature in a thin film (according to the theorem of Mermin and Wagner, 2D systems of continuous spins with a short-range interaction do not have long-range ordering at finite Г).
The helimagnetic order results from the competition between NN ferromagnetic interaction (7i > 0) and the NNN aniferromag- netic exchange interaction (J2 < 0) in the z. direction. Let a, be the angle of a spin in the /-th layer made with the spin in the next layer. In the bulk case the helimagnetic structure is possible only for 1/2! > /1 /4- (see Section 3.4 and Ref. ).
In a film with a thickness, we have used the steepest descent method  to calculate the turn angle between spins of adjacent layers. The result is shown in Fig. 11.13 where we see the spin configurations near the two surfaces are strongly modified with respect to the bulk turn angle.
Using the GF method as described above we have calculated the spin wave frequency w versus kx = ky for various surface exchange interactions Js in the case of an 8-layer film with an anisotropy
Figure 11.14 Spin wave frequency spectrum versus k = kx= ky in the case where Nz = 8, J2 = — 1 and d = 0.1 for ]s/] = 1 (top, left), Js/Ji =0.6 (top, right) and ]s/) = 1.6 (bottom).
d/Ji = 0.1. We note the existence of acoustic surface modes which lie in the low-energy region for Js/Ji = 0.6 (see Fig. 11.14, middle) and optical surface modes which lie in the high-energy region for Js = 1.6//1 (see Fig. 11.14, bottom). Note that no such modes exist in the case Js/Ji = 1 (see Fig. 11.14, top).
The surface magnetization, which has a large value at T = 0 as seen in Fig. 11.15, crosses the interior layer magnetizations at Г = 0.6 to become smaller than interior magnetizations at higher temperatures. This crossover phenomenon is due to the competition between quantum fluctuations, which dominate the low-Г behavior, and the low-lying surface spin wave modes which strongly diminish the surface magnetization at higher T. Note that the second- layer magnetization makes also a crossover at T =0.6. Similar crossovers have been observed in quantum antiferromagnetic films , quantum superlattices  and the BCC case presented above.
Figure 11.15 (Color online) Layer magnetization as a function of T for J2/J1 = —0.7 with d/]i = ds/J i = 0.1, W2 = 8: Red circles, blue void circles, magenta squares and black void squares are magnetizations of the first, second, third and fourth layers, respectively.
Figure 11.16 Spin length at T = 0 versus ]2, with d = ds = 0.1, Nz = 8 (/( = 1): Black circles, void squares, black squares and void circles are data for spins in first, second, third and fourth layers, respectively.
We show the zero-point contraction for various values of ]2 (J1 = 1) in Fig. 11.16. When J2 increases, namely the antiferromagnetic interaction becomes stronger, we observe stronger contractions. Note that the contraction tends to zero when the spin configuration becomes ferromagnetic, namely J2 tends to -0.25.
Let us show the effect of the surface anisotropy in Fig. 11.17: Stronger ds makes larger surface magnetization and compensates the lack of a neighbor for surface spins.
Figure 11.17 Layemiagnetizationasfunctionof^effectofdsids//! = 0.2, with у2/Уl = —0.5, d/h = 0.1, Nz = 8. Red circles, green void triangles, blue triangles and magenta circles are magnetizations of the first, second, third and fourth layers, respectively.
Figure 11.18 Layer magnetizations as functions of T, effect of Js: ]s = 0.4. with У2/У1 = — 0.5, d/Ji = 0.1, Nz = 8. Red circles, green squares, blue void squares, and magenta void squares are magnetizations of the first, second, third and fourth layers, respectively.
We show now the effect of the surface interaction Js. Figure 11.18 shows the case of weak Js where a low surface magnetization is observed. This "soft surface" is due to the effect of the acoustic surface modes such as those seen in Fig. 11.14 (top, right). On the contrary, when the surface interaction is strong, we have the case of "hard” surface as seen in Fig. 11.19 for a strong Js [the optical
Figure 11.19 Layer magnetizations versus T, effect of ]s: )S/Ji = 1.6. У2/У1 = —0.5, d/J = 0.1, Nz = 8. The color code is given in the previous figure.
surface modes seen in Fig. 11.14 (bottom) are responsible for the hard surface].
We have studied in this chapter surface effects in a helimagnetic film of BCC and SC lattices with quantum Heisenberg spins. Note that the method presented above can be applied to any lattice structure, and the results found here are valid for general helimagnetic structures. Note also that the results have been shown for the case of spin S = 1/2 where quantum fluctuations are strong at low temperatures. Rare-earth elements Ho and Dy with helical structures along the c- axis and ferromagnetic in the basal planes which are very similar to the present model are expected to bear the same features as what has been shown above. However, at low temperatures, due to their larger spin amplitudes, S = 7/2 and 5/2 for Ho and Dy, quantum fluctuations are certainly weaker and the crossover may occur with smaller difference. Numerical applications of our formalism should be performed to get precise values for these cases.
In this chapter, the classical bulk ground-state spin configuration in a thin film is exactly calculated and is found to be strongly modified near the film surface. The surface spin rearrangement is found at the first four layers in our model, regardless of the bulk angle, namely the NNN interaction strength _/г- The spin wave excitation is calculated using a general GF technique for non-collinear spin configurations. The layer magnetization as a function of temperature and the transition temperature are shown for various interaction parameters. Among the striking features presented in the present chapter, let us mention
Let us make some comments on works of similar models. The work by Mello et al.  have treated almost the same model as ours using a hexagonal anisotropy which corresponds to the case of Dy in which the helical angle is ~ 60°. However, the authors of this work studied only the classical spin configuration at Г = 0. Rodrigues et al.  have used exactly the same model as Mello et al. but with application to the Ho case. They have used the mean- field estimation to establish the phase diagram in the space (T, H) (H: magnetic field) and shown that surface effects affect the phase diagram. The MC work of Cinti et al.  was based on a classical spin model with a Hamiltonian, very different from ours, including a 6-constant interaction (a kind of dipolar interaction) in the c-direction.
To conclude, let us emphasize that the general theoretical method proposed here allows us to study at a microscopic level surface spin waves and their physical consequences at finite temperatures in helimagnetic films with non-collinear spin configurations. It can be used in more complicated situations such as helimagnets with Dzyaloshinskii-Moriya interactions . We will show this case in Chapters 13-15.