The Case of a Thin Film: Spin Wave Spectrum, Layer Magnetizations
In the 2D and 3D cases shown above, there is no need at Г = 0 to use a small anisotropy d. However, in the case of thin films shown below, due to the lack of neighbors at the surface, the introduction of a DM interaction destabilizes the spectrum at long wavelength к = 0. Depending on в, we have to use a value for dn larger or equal to a "critical value” dc to avoid imaginary SW energies at к = 0. The critical value dc is shown in Fig. 13.7 for a four-layer film. Note that at the perpendicular configuration в = тг/2, no SW excitation is possible: SW cannot propagate in a perpendicular spin configuration since the wave-vectors cannot be defined.
We show now a SW spectrum at a given thickness N. There are 2N energy values half of them are positive and the other half negative (left and right precessions): F, (i = 1,, 2 N). Figure 13.8 shows the case of a film of 8 layers with J„ = J± = 1 for a weak and a strong value of D (small and large в). As in the 2D and 2D cases, for strong D, E is proportional to к at small к (cf. Fig. 13.8b). It is noted that this behavior concerns only the first mode. The upper modes remain in the к2 behavior.
Figure 13.7 Value dc above which the SW energy £(k = 0) is real as a function of в (in radian), for a four-layer film. Note that no spin wave excitations are possible near the perpendicular configuration в = n/2. See text for comments.
Figure 13.8 Spin wave spectrum £(k) versus к = kx = kz for a thin film of 8 layers: (а) в = тт/6 (in radian) (b) 9 = тг/3, using d = dc for each case (dt = 0.012 and 0.021, respectively). Positive and negative branches correspond to right and left precessions. Note the linear-/; behavior at low к for the large в case. See text for comments.
Figure 13.9 Eight-layer film: layer magnetizations M versus temperature T for (а) в = n/b (radian), (b) в = тг/3, with d = 0.1. Red circles, blue void circles, green void triangles and magenta squares correspond, respectively, to the first, second, third and fourth layer.
Figure 13.9 shows the layer magnetizations of the first four layers in an eight-layer film (the other half is symmetric) for several values of в. In each case, we see that the surface layer magnetization is smallest. This is a general effect of the lack of neighbors for surface spins even when there is no surface-localized SW as in the present simple-cubic lattice case [78, 87].
Figure 13.10 Spin length S0 at T = 0 of the first 4 layers as a function of в, for N = 8, d = 0.1. Red circles, blue void circles, green void triangles and magenta squares correspond, respectively, to the first, second, third and fourth layer.
The spin length at T = 0 for a eight-layer film is shown in Fig. 13.10 as a function of в. One observes that the spins are strongly contracted with large в.
Let us touch upon the surface effect in the present model. We know that for the simple cubic lattice, if the interactions are the same everywhere in the film, then there is no surface localized modes, and this is true with DM interaction (see spectrum in Fig. 13.8) and without DM interaction (see Ref. ). In order to create surface modes, we have to take the surface exchange interactions different from the bulk ones. Low-lying branches of surface modes which are "detached” from the bulk spectrum are seen in the SW spectrum shown in Fig. 13.11a with J* = 0.5, ]]_ = 0.5. These surface modes strongly affect the surface magnetization as observed in Fig. 13.11b: The surface magnetization is strongly diminished with increasing T. The role of surface-localized modes on the strong decrease of the surface magnetization as T increases has already been analyzed more than 40 years ago .
We show now the effect of the film thickness in the present model. The case of thickness N = 12 is shown in Fig. 13.12a with в = тг/6 where the layer magnetizations versus T are shown in details. The gap at к = 0 due to d is shown in Fig. 13.12b as a
Figure 13.11 Surface effect: (a) spin wave spectrum E(k) versus к = kx = kz for a thin film of 8 layers: в = л/Ь, d = 0.2, /,f = 0.5, J[ = 0.5, the gap at к = 0 is due to d. The surface-mode branches are detached from the bulk spectrum, (b) Layer magnetizations versus T for the first, second, third and fourth layer (red circles, green void circles, blue void circles and magenta filled squares, respectively). See text for comments.
function of the film thickness N for d = 0.1 and в = я/6, at T = 0. We see that the gap depends not only on d but also on the value of the surface magnetization which is larger for thicker films. The transition temperature Tc versus the thickness N is shown in Fig. 13.12c where one observes that Tc tends rapidly to the bulk value (3D) which is ~ 2.82 for d = 0.1.
Figure 13.12 Twelve-layer film: (a) Layer magnetizations versus T for в = л/6 and anisotropy d = 0.1. Red circles, blue squares, green void squares magenta circles, void turquoise triangles and brown triangles correspond, respectively, to first, second, third, fourth, fifth and sixth layer, (b) Gap at к = 0 as a function of film thickness N ford = л/6, d = 0.1, at T = 0.1, (c) Critical temperature Tc versus the film thickness N calculated with 9 = л/6 and d = 0.1 using Eq. (13.37). Note that for infinite thickness (namely 3D), Tc ~ 2.8 for d = 0.1.
Discussion and Experimental Suggestion
Several results found above can be experimentally verified. Let us confine ourselves in the case of thin films in this section, although the following discussion applies also in the 2D and 3D crystals.
The first striking aspect is the very particular effect of the DM interaction in the SW spectrum: We have seen in Fig. 13.8b that the first mode is linear in к for small k, but not the upper modes. This is not the case of ferromagnetic and antiferromagnetic interactions in thin films: For a ferromagnetic interaction, all modes have the k2 behavior at small к , and for antiferromagnetic interactions all modes have the linear-к behavior. Thus, the DM interaction affects only the first mode, and the J -term in the Hamiltonian, Eq. (13.3), maintains the ferromagnetic behavior for higher-energy modes in the SW spectrum. Experiments which measure spin wave spectra such as neutron scattering  and spin wave resonance can verify if the first mode in the SW spectrum shown in Fig. 13.8b is linear in к for large DM interactions. For SW higher-energy modes, there has been an experimental breakthrough with the spin polarized electron energy loss spectroscopy (SPEELS): This technique allows us to detect very high-energy surface magnons, up to 240 meV. It has been proved to be very efficient to probe the dispersion of magnons in the ultrathin ferromagnetic films . We believe that using such high-efficient experimental means, the effect of the DM interaction shown in this chapter can be verified.
The second striking feature is the surface effect on the SW spectrum and the surface magnetization shown, respectively, in Figs. 13.11a and 13.11b. These can be experimentally verified with the above-mentioned techniques. Note that the slope of the first SW mode allows us to deduce the interaction parameter; therefore, the DM interaction strength D. Note that neutron scattering can be used for a complete determination of the magnetic properties of a system at finite temperatures .
The third interesting result is the dependence of the critical temperature Tc on the film thickness shown in Fig. 13.12c. This can be completely determined by neutron scattering and SPEELS which measure the magnetization as a function of temperature of films as thin as a few atomic layers [216, 275].
We believe that, in spite of cumbersome mathematical details shown in Section 13.3, numerical results coming out from the formulation shown in Sections 13.4 and 13.5 are very physically meaningful. We expect, therefore, experimental verifications on materials showing a DM interaction such as those mentioned in Refs. [100,113,144,316].
By a self-consistent Green’s function theory, we obtain the expression of the spin wave dispersion relation in 2D and 3D as well as in a thin film. Due to the competition between ferromagnetic interaction J and the perpendicular DM interaction D, the GS is non-linear with an angle в which is shown to explicitly depend on the ratio D/J. The spectrum is shown to depend on в and the layer magnetization is calculated self-consistently as a function of temperature up to the critical temperature Tc.
We have obtained new and interesting results. In particular we have showed that (i) the spin wave excitation in 2D and 3D crystals is stable at T = 0 with the non-collinear spin configuration induced by the DM interaction D without the need of an anisotropy, (ii) in the case of thin films, we need a small anisotropy d to stabilize the spin wave excitations because of the lack of neighbors at the surface, (iii) the spin wave energy E depends on D, namely on в: At the long wavelength limit, E is proportional to k2 for small D but E is linear in к for strong D, in 2D and 3D as well as in a thin film, (iv) quantum fluctuations are inhomogeneous for layer magnetizations near the surface, (v) unlike in some previous works, spin waves in systems with asymmetric DM interactions are found to be symmetric with respect to opposite propagation directions.
We have suggested some experiments to verify the above results in Section 13.6.