Magnetic oscillation modes in square-lattice artificial spin ice

Small amplitude dipolar oscillations are considered in artificial spin ice on a square lattice in two dimensions. The net magnetic moment of each elongated magnetic island in the spin ice is assumed to have Heisenberg-like dynamics and be influenced by shape anisotropies and nearest neighbor dipole interactions. The magnetic dynamics linearized around a ground state with four sublattices leads to an eigenvalue problem with four branches of magnetic spin wave modes. An energy analysis of the dipole motions enables an analytic solution while showing that the lowest frequency modes are antisymmetric with regard to their in-plane dynamic fluctuations in each four-island vertex. Although only the leading dipolar interactions are included, modes similar to these may be observable experimentally in artificial square spin ice.