Theory of Spin Waves in a Thin Ferromagnetic Film with a Periodic System of Circular Antidots. Solutions that Correspond to the Crystal Band Theory
Abstract
The paper continues study of dipole-exchange spin waves in a two-dimensional magnonic crystal (a thin ferromagnetic film with a periodic system of circular antidots) started by the author in the previous paper. The proposed model considers the magnetic dipole-dipole interaction, the exchange interaction and the anisotropy effects. An improved method of obtaining the dispersion relation as well as the values’ spectra of the frequencies and wavenumbers for the investigated spin waves – the method based on using the Bloch-type solutions of the Landau-Lifshitz equation for the spin waves together with the Born–von Karman boundary conditions – is proposed. Exploitation of the above-mentioned method essentially extends the area of application of the obtained results compared to the previous paper. Newly obtained spectral characteristics are shown to exhibit crystal-type band structure with band gaps. The wave vector component that correspond to the wave propagating orthogonally to the film plane is shown to have narrow allowed bands, so its values' spectrum is near discrete.
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Introduction
Spin waves in nanosystems become an actual and promising topic of research because of their numerous applications - both current and prospective - in different fields of technology. These applications include mostly new devices for data storage, transfer and processing [1-4]. Such applications require precise theoretical models of excitation and propagation of spin waves in nanosystems of different configurations, thus causing these models to be extensively developed recently.
Prospective materials for applications in spin-wave technologies include, in particular, magnonic crystals [5,6] - composite materials whose magnetic properties change periodically along one, two or three directions. They are known to exhibit unique magnetic properties [5] making them prospective for creating novel magnonic devices [5,6]. As a result, spin waves in magnonic crystals of different configurations are studied extensively, both theoretically and experimentally [7-9].
Because of the parameters' periodicity, magnonic crystals often exhibit properties similar to those observed in crystals, such as appearance of crystal-like band structure in the spin waves' spectrum (see, e.g., [8]). Therefore, elements of crystals theory can be used in a theory of spin waves in such nanosystems in order to refine corresponding models and, therefore, obtain more precise results.
The paper extends theoretical study of dipole-exchange spin waves in a two-dimensional magnonic crystal (a thin ferromagnetic film with a two-dimensional periodic system of circular antidots) started by the author in the previous paper [10].The magnetic dipole-dipole interaction, the exchange interaction and the anisotropy effects are considered. Unlike in the previous paper, periodicity of the system is taking into account by applying the Bloch theorem and using the Bloch-type solutions of the Landau-Lifshitz equations for a spin wave together with Born–von Karman boundary condition. As a result, a refined dispersion relation and the wave vector components' spectrum of such waves is obtained. Analysis shows that a crystal-type band structure with band gaps appears in the resulting spectral characteristics.
Conclusion
Therefore, the paper extends the study of the dipole-exchange linear spin waves in a thin ferromagnetic film with a twodimensional periodic system of identical circular antidots started by the author in the previous paper [10]. The film is assumed to be composed of the uniaxial "easy axis"-type ferromagnet, with the axis of easy magnetization directed orthogonally to the film plane. For such waves, the differential equation for the magnetic potential in the magnetostatic approximation is written. The equation is solved for the case when either the external magnetic field is strong enough or the film is thin enough (l<<2(a–R)) to ignore the inhomogeneity of the equilibrium magnetization and magnetic field - and, additionally, the antidots are far enough from each other, so the minimum distance between them is much bigger than the exchange length.
Unlike the previous paper [10], the solution of the Landau-Lifshitz equations for the above-described spin wave in this paper is sought in the form of a two-dimensional function of Bloch type (for the in-plane wave propagation). For such solution, the dispersion relation and the relation between planar and longitudinal wavenumbers are obtained. Then, the crystal solid state formalism is used to obtain the values' spectra of the wave vector components and (after combining with the abovementioned relations) of the spin waves' frequencies. The obtained results are refined near the edge of the Brillouin zones using the electronic band theory.
It is shown that the values' spectrum of the longitudinal wave numbers has band gaps and is near discrete. It is also shown that a band structure - which is analogous to the electronic band structure of a crystal solid - is inherent for the investigated spin waves.