Excitation spectra of magnetic bubble lattices

Abstract

The excitation spectra of a hexagonal lattice of magnetic bubbles is calculated using the methods of lattice dynamics. The bubbles are allowed one internal zero-mode radial degree of freedom and two translational degrees of freedom of the center of mass. This results in three branches of free oscillation, one optical and two acoustical. The equations of motion of the system are obtained from a Lagrangian with a Rayleigh dissipative function. The Fourier transform of these equations yields a secular determinant of order three corresponding to the three branches. The secular equation is a polynomial equation of sixth degree in the complex frequency. This is solved for the directions kx and ky. The interbubble potential is approximated by a dipole-dipole potential, and restricted to nearest neighbors. The magnetostatic bubble self-energy is replaced by the analytic approximation of Josephs and Callen. For a close-packed lattice with no coupling between the radial and translational degrees of freedom, the optical and acoustical branches cross. The inclusion of coupling has drastic effects on the dispersion relationships. The acoustical branches consist of admixtures of translational and radial oscillations. Because of damping, a critical wave number exists below which no real part of the frequency exists. Changes in the acoustical branches can be brought about by changing the external magnetic field.