# Theory of domain-wall motion in magnetic films and platelets

## Abstract

We calculate dynamical properties of plane and cylindrical magnetic domain walls in a uniaxial film or platelet whose plane is perpendicular to the easy axis. First, we calculate the internal structure of a freely moving wall to determine the nonlinear velocity-momentum relation. Using a Bloch-line approximation to the kinetic wall energy, we find that stray magnetic fields emanating from surface poles destabilize the structure at a critical velocity Vp, above which uniform motion is not possible. For a plane wall, we have Vp=24γA/hK1/2, where γ is the gyromagnetic ratio, A is the exchange stiffness, h is the film thickness, and K is the anisotropy. This velocity is usually much less than the critical velocity for bulk wall motion derived by Walker. Combination of the velocity-momentum relation with the conservation of momentum provides a simple method of calculating the time-dependent velocity in the presence of a small drive field and small viscous damping. For very small drive, the terminal velocity equals that given by the conventional linear-mobility theory. However, for small drives exceeding a critical value corresponding to Vp, a periodic cycle of Bloch-line generation, motion, and annihilation takes place at a frequency approaching the Larmor frequency corresponding to the driving field. This is accompanied by a synchronous nonsinusoidal oscillation of the wall, with a mean forward velocity tending to an asymptote V0. Our relation V 0=7.1γA/hK1/2 agrees within a factor of 3 with the onset of nonlinear behavior observed by means of bubble collapse in certain low-loss garnet specimens. Partial support is also given by data for hexaferrite platelets. For radial motion of cylinders with finite diameter D, numerically computed values of V0 are larger but never exceed this expression by more than 17%. The critical velocity Vp also varies little for D/h>2, but varies rapidly and attains large values for small D/h. © 1973 American Institute of Physics.