Reversal time of the magnetization of single-domain ferromagnetic particles with mixed uniaxial and cubic anisotropy

The reversal time of the magnetization of single-domain ferromagnetic particles is estimated for mixed uniaxial and cubic anisotropy energies possessing nonparaboloidal saddles and well bottoms or either. The calculation generalizes the existing adaptation of the Kramers escape rate theory to fine ferromagnetic particles with nonaxially symmetric magnetocrystalline-Zeeman energies, originally based on the paraboloidal approximation for the energy near its stationary points, yielding in addition a simple universal Kramers turnover formula (based on the Mel’nikov-Meshkov depopulation factor) for the reversal time valid for all values of the damping. The asymptotic solution is compared with the appropriate numerically exact solution of the corresponding Fokker-Planck equation for the probability density function of magnetization orientations. The predictions of the generalized nonparaboloidal stationary point turnover formula agree with the numerical solution for a wide range of damping and other parameters characterizing the mixed anisotropy.

Figure 1

(Color online) Three-dimensional (3D) plot of the mixed anisotropy potential for $\zeta =1$ (a: flat saddles), 0 (b: uniaxial anisotropy), and $-1$ (c: flat wells).

Figure 2

The normalized reversal time of the magnetization $\tau /{\tau }_0$ as a function of the potential shape parameter $\zeta$ for $\alpha =10$ and $\sigma =10$ and 15. Filled circles: numerical calculation from Eq. (32), solid lines: universal Mel’nikov-Meshkov formula (10), dashed lines: Eq. (33), dotted lines: Eq. (35).

Figure 3

The normalized reversal time of the magnetization $\tau /{\tau }_0$ as a function of the damping parameter $\alpha$ for $\zeta =-1$ and $\zeta =1$. Filled circles: numerical calculation from Eq. (32). Solid lines: universal Mel’nikov-Meshkov formula (10). Dashed lines: Eqs. (30, 23). Dotted line: uniaxial Eq. (35). The predictions of TST, viz., $\tau \sim {\left(8{\Gamma }_{ij}^{\text{TST}}\right)}^{-1}$, are also shown for comparison.