  Understanding the stability of thin film nanomagnets with perpendicular
magnetic anisotropy (PMA) against thermally induced magnetization reversal is
important when designing perpendicularly magnetized patterned media and
magnetic random access memories. The leading-order dependence of magnetization
reversal rates are governed by the energy barrier the system needs to surmount
in order for reversal to proceed. In this paper we study the reversal dynamics
of these systems and compute the relevant barriers using the string method of
E, Vanden-Eijnden, and Ren. We find the reversal to be often spatially
incoherent; that is, rather than the magnetization flipping as a rigid unit,
reversal proceeds instead through a soliton-like domain wall sweeping through
the system. We show that for square nanomagnetic elements the energy barrier
increases with element size up to a critical length scale, beyond which the
energy barrier is constant. For circular elements the energy barrier continues
to increase indefinitely, albeit more slowly beyond a critical size. In both
cases the energy barriers are smaller than those expected for coherent
magnetization reversal.
