Longitudinal dynamic susceptibility and relaxation time of superparamagnetic particles with cubic anisotropy: Effect of a biasing magnetic field

Abstract

The magnetic relaxation of single domain ferromagnetic particles with cubic magnetic anisotropy subjected to a strong uniform magnetic field is treated by averaging the Gilbert-Langevin equation for an individual particle, so that the system of linear differential-recurrence equations for the appropriate equilibrium correlation functions is derived. The solution of this system (in terms of matrix continued fractions) is obtained and the longitudinal relaxation time ${\ensuremath{\tau}}_{\ensuremath{\parallel}}$ of the magnetization and spectrum of the complex magnetic susceptibility ${\ensuremath{\chi}}_{\ensuremath{\parallel}}(\ensuremath{\omega})$ are evaluated. It is shown that the depletion effect discovered for uniaxial particles by Coffey et al. [Phys. Rev. B 51, 15 947 (1995)] and interpreted by Garanin [Phys. Rev. E 54, 3250 (1996)] also exists for particles having a cubic anisotropy. This effect consists of the drastic deviation of ${\ensuremath{\tau}}_{\ensuremath{\parallel}}$ from the inverse of the smallest nonvanishing eigenvalue $({\ensuremath{\lambda}}_{1}^{\ensuremath{-}1})$ of the Fokker-Planck equation in the low temperature limit starting from some critical values ${h}_{c}$ of the ratio bias field parameter/anisotropy barrier height parameter and is due to depletion of the upper (shallow) potential well involved into the relaxation process. For uniaxial particles the critical value is known to be ${h}_{c}\ensuremath{\approx}0.17.$ For cubic crystals it is shown that ${h}_{c}\ensuremath{\approx}0.3$ (for positive anisotropy constant, Fe-type) and ${h}_{c}\ensuremath{\approx}0.1$ (for negative anisotropy constant, Ni-type). However, it is also demonstrated, in contrast to uniaxial particles, that for cubic crystals there is an inherent geometric dependence of the complex susceptibility and the relaxation time on the damping parameter arising from the coupling of longitudinal and transverse relaxation modes.