Magnons in magnetic metamaterials: Theoretical analysis -- Presentazione poster by R. Zivieri - Conferenza internazionale

We present the results obtained by means of two different methods of calculation of spin-wave mode dispersion in magnonic crystals. One is called plane-wave and the other is a combined micromagnetic-dynamical matrix method. Band structures of all periodic composites, such as photonic, phononic, or magnonic crystals, are calculated by similar methods. One of the commonly used numerical techniques is the plane-wave method, popular because of its conceptual simplicity and applicability to any lattice type and scattering centre shape. In our calculation the spin-wave dispersion relation for magnonic crystal is determined from Landau-Lifshitz equation – the equation of motion of the space- and time-dependent magnetization vector. We assume the effective magnetic field acting on magnetic moments in the magnonic crystal to be a sum of three components: applied (external) magnetic field, exchange field (dependent on material parameters: exchange stiffness constant and saturation magnetization), and magnetostatic interaction field (derived from magnetostatic Maxwell’s equations). The plane-wave method require to apply Fourier transform to all periodic in space parameters in Landau-Lifshitz equation (exchange stiffness constant, saturation magnetization). Using the Bloch theorem for the dynamic functions (dynamic component of magnetisation, magnetostatic potential) we obtain an eigenproblem for spin-wave frequencies. We employ the plane-wave method to determine spin-wave spectra of three-dimensional magnonic crystals. The scattering centres considered are ferromagnetic spheroids distributed in sites of a cubic (sc, fcc, or bcc) lattice embedded in a matrix of a different material. We now give a description of the combined micromagnetic-dynamical matrix method. The particle is subdivided into N cells (square base prisms of lateral size d and height L). In each cell the magnetization assumed constant is defined by two polar angles θ and φ (canonical coordinates). The approach also works in the 3D case, by subdividing each prism in a multilayer stack. The magnetization in each cell is written as the sum of a static part plus a dynamical part. The Hamilton’s equations of motion are expressed in each cell in terms of the second derivatives of the energy with respect to the angular variations calculated at equilibrium. The solution of the system can be written in the form of eigensystem where the angular deviations are the eigenvectors and the spin mode eigenfrequencies. The method is generalized to 1D and 2D arrays of interacting particles assuming that the dynamic magnetization can be expressed in a Bloch form typical of periodic systems. The corresponding equations of motion depend also on the vector R defining the generic array. The described method can be applied to the calculation of spin modes in ferromagnetic particles of any shape and of size ranging from the nanometric to the submicrometric scale. As an application we use the dinamical matrix method to calculate band structure in 1D arrays of Permalloy circular dots with in-plane magnetization interacting via the dipolar coupling. Both the case with the external field and the static magnetization along and perpendicular to the array (in the array plane) are considered. The frequencies of the most representative spin modes are calculated for different values of the Bloch wave vector in the first 1D Brillouin zone. The band calculation is extended to 2D arrays of Permalloy dots. The dispersions of spin modes are determined along the high-symmetry directions of the first 2D Brillouin zone. -- Presentazione poster by R. Zivieri