Magnetic reversal in nanoscopic ferromagnetic rings

We present a theory of magnetization reversal due to thermal fluctuations in thin submicron-scale rings composed of soft magnetic materials. The magnetization in such geometries is more stable against reversal than that in thin needles and other geometries, where sharp ends or edges can initiate nucleation of a reversed state. The two-dimensional ring geometry also allows us to evaluate the effects of nonlocal magnetostatic forces. We find a “phase transition,” which should be experimentally observable, between an Arrhenius and a non-Arrhenius activation regime as magnetic field is varied in a ring of fixed size.

Figure 1

Ferromagnetic annulus viewed from above, showing coordinates used in text.

Figure 2

The phase boundary between the two activation regimes in the $(\ell ,h)$ plane. In the shaded region the instanton state is the saddle configuration; in the unshaded region, the constant state.

Figure 3

Activation energy $\mathrm{\Delta }W$ for fixed $h=0.3$ as $\ell$ varies. The dot indicates the transition from constant to instanton saddle configuration.

Figure 4

Activation energy $\mathrm{\Delta }W$ for fixed $\ell =7$ as $h$ varies. The dot indicates the transition from instanton to constant saddle configuration.

Figure 5

The prefactor ${\Gamma }_0$ (in units of ${\tau }_0^{-1}$) vs $h$ when $\ell =7$ on the “constant saddle” side of the transition. The prefactor on the “instanton saddle” side of the transition acquires an additional temperature dependence, as discussed in the text.

Figure 6

Lowest eigenvalue ${\lambda }_{u,0}$ as a function of $h$ for $\ell =7$.

Figure 7

Total switching rate (in units of $s^{-1}$) vs $\beta =1∕k_{B}T$ (in units of ${\mathrm{K}}^{-1}$), at fields of (a) $60\mathrm{mT}$ (instanton saddle) and (b) $72\mathrm{mT}$ (constant saddle). Parameters used are $k=0.01$, $l=0.1$, $R=200\mathrm{nm}$, $R_1=180\mathrm{nm}$, $R_2=220\mathrm{nm}$, $M_0=8\times {10}^5\mathrm{A}∕\mathrm{m}$ (permalloy), $\alpha =0.01$, and $\gamma =1.7\times {10}^{11}{\mathrm{T}}^{-1}{\mathrm{s}}^{-1}$. Deviation of low-field switching rate in (a) from dashed line signals non-Arrhenius behavior.