Boundary waves in ferromagnetically ordered two-dimensional arrays of magnetic dots

Abstract

The state of the art in modern nanolithography makes it possible to create regular arrays of magnetic superstructures with large numbers of nanodimen sional elements [1]. Of these systems, two dimen sional (2D) arrays of submicron magnetic particles (magnetic dots) on nonmagnetic substrates are of spe cial interest. The distances between these particles are much greater than the exchange length of the magnet; they can only interact by means of magnetic dipole interaction. These arrays have important practical applications in systems of high density magnetic recording [1, 2] and they are of interest for various new areas of study in the applied physics of magnetism, such as magnonics, spintronics, and physics of mag nonic crystals. It is also important to note that these systems represent a new implementation of dipole magnets, which are basic objects of the fundamental physics of magnetism (see, e.g., review [3]).It should be noted that all artificial superstructures are large, yet finite, systems. Therefore, it is expected that boundary elements play a large role in the forma tion of properties of these systems and, hence, an anal ysis of the role of boundaries in samples of finite 2D arrays is of considerable interest. The presence of boundaries in an arbitrary periodic structure implies that the system loses translational symmetry in the direction perpendicular to the boundary. This decrease in the symmetry allows us to expect the appearance of qualitatively new solutions that are localized at the boundary-solutions of the type of Tamm surface states [4,5]. These states are usually studied in systems with a small number of interacting neighbors, e.g., in a simple approximation of nearest neighbor interactions [5], where a local decrease in the coordination number at the boundary is an important factor. For dipole coupled systems, this effect is expected to be not as important as in systems with short range interactions. On the other hand, in view of the long range character of magnetic dipole interac tions, some other effects of finiteness of the system that are manifested by the presence of macroscopically inhomogeneous fields may become much more signif icant. Therefore, the issues concerning the character of the ground state and the surface localization of modes in finite systems with long range interactions are of special interest.For a wave localized on the boundary, the ampli tude of oscillations at a given site depends on a discrete argument (site number) and is described by the corre sponding matrix equation [5]. The presence of a boundary can be treated as some local perturbation. Since the rank of the matrix for a long range interac tion is large, the exact problem in this case was studied by numerical methods and analytic estimations were obtained using the nearest neighbor approximation. It was found that this simple approximation still reflects important features of the problem.Consider an array of magnetic particles in the form of a semi infinite square lattice in the xy plane and let the boundar...

Theoretical formalism for collective spin-wave edge excitations in arrays of dipolarly interacting magnetic nanodots

“…To further analyze the FMR absorption spectra and explain the differences between the theoretical and numerical values of the power absorption, we present in Fig. 5 the distribution of the microwave magnetization in the array for several values of the excitation frequency, calculated from (60). In Fig.…”

“…In Fig. 4(d) we also show (see red dotted line) the absorption spectrum found by the direct numerical solution of the linear inhomogeneous system of 1640 equations (60) for N = 820 dots. It is evident, that the direct numerical simulation agrees reasonably well with our theory, as the frequencies and heights of the absorption peaks are in good quantitative agreement.…”

“…Although equations (27) and (37) were obtained using purely analytical methods, further analytical solution of these equations (e.g., using the nearest neighbor approximation 60 ) is tedious and does not allow one to obtain qualitative analytical results in a closed form. The internal structure of the tensorsÊ κ (n) is complicated, and, moreover, the dispersion equation (27) leads to a generalized eigenvalue problem with an infinite block "Toeplitzlike" matrix, in which blocks standing on the main diagonal are not equal, becauseB n =B m when n = m in (28).…”