Spin wave dispersion in ferromagnetic films with a sinusoidal magnetization distribution

In 3D Magnonics[1,2], the vertical interaction between different layers can induce a non-uniform magnetic distribution in the associated ferromagnetic film layer, and in turn modify the propagation properties of spin-waves. This can be achieved, for example, by defining a layer upon the film with an artificial spin ice geometry (e.g., in a canted antiferromagnetic configuration as in Fig. 1-a) or a magnetic domain structure with magnetization pointing in different directions, or even misaligned ferroelectric domains determining a magnetic anisotropy in the ferromagnetic film by the inverse magnetoelastic effect. By means of micromagnetic simulations, carried out by mumax3 software [3], we studied the case of a sinusoidal magnetization distribution in a thin film (Fig. 1-b). This is the simplest possible periodic perturbation of the uniform magnetization state. By Fourier analysis[4], the spin-wave dispersion relations were obtained, together with the mode spatial profiles. We also studied the case in which the sinusoidal magnetization is found after the relaxation of a uniform state (at 0.1 T) to an applied weaker sinusoidal magnetic field of variable amplitude: this case is particularly interesting, since the sinusoidal field implements a possible vertical interaction induced by an overlayer of appropriate geometry. Finally, we studied the effects on the film magnetization of an overlayer with a canted ferromagnetic distribution (Fig. 1-c): since the x-component of the sinusoidal bias field changes sign every half lattice constant, the final relaxed magnetization is always in a state similar to Fig. 1-b, but with a marked asymmetry, in the amplitude, between the two halves of the primitive cell, which reflects also in the spin-wave spatial profile. In all the cases, we calculate the spin-wave dispersions and discuss the possible occurrence of magnonic frequency gaps and flat dispersions due to localized excitations (Fig. 1-d) as a function of the amplitude of the sinusoidal magnetization. Such a system, where just playing with the magnetic distribution in the overlayer it is possible to change the spin-wave dynamics, can be very useful in magnonics, either for information delivery and interferometric devices, insofar as spin-wave modes can be tuned from propagating to stationary and vice-versa, or for signal filtering, by varying the frequency gap and the number of localized modes.