Collective modes for an array of magnetic dots with perpendicular magnetization

The dispersion relations of collective oscillations of the magnetic moment of magnetic dots arranged in square-planar arrays and having magnetic moments perpendicular to the array plane are calculated. The presence of the external magnetic field perpendicular to the plane of array, as well as the uniaxial anisotropy for single dot are taken into account. The ferromagnetic state with all the magnetic moments parallel and chessboard antiferromagnetic state are considered. The dispersion relation yields information about the stability of different states of the array. There is a critical magnetic field below which the ferromagnetic state is unstable. The antiferromagnetic state is stable for small enough magnetic fields. Here the dispersion relations for collective modes for two phases, ferromagnetic and chessboard antiferromagnetic, within the whole Brillouin zone, are calculated. Nonstandard behavior of the mode frequencies on the wave vector is present for many cases. As the value of the wave vector approaches zero, for both phases a nonanalytic behavior of the mode frequency is found. For ferromagnetic state, the center of the Brillouin zone corresponds to a nonparabolic minimum, common to that is known for continuous thin films. For antiferromagnetic state, the saddle point with nonanalytic dependence of the components of the wave vector is located at small values of the wave vector. Nontrivial Van Hove anomalies are also found for both ferromagnetic and antiferromagnetic states.