Effect of an oblique magnetic field on the superparamagnetic relaxation time

The effect of a constant magnetic field, applied at an angle psi the easy axis of magnetization, on the Neel relaxation time tau of a single domain ferromagnetic particle (with uniaxial anisotropy) is studied by calculating the lowest nonvanishing eigenvalue lambda(1) (the escape rate) of the appropriate Fokker-Planck equation using matrix methods. The effect is investigated by plotting lambda(1) versus the anisotropy parameter alpha for various values of psi, and the ratio h=xi/2 alpha, where xi is the external field parameter and lambda(1) versus psi for various h values (for rotation of the magnetization vector M both in a plane and in three dimensions). If M rotates in a plane the curve of lambda(1) versus psi is symmetric about psi=pi/4 in the range 0 < psi < pi/2 and significant decrease in tau with increasing psi is predicted for large xi and alpha. The maximum decrease in tau occurs at psi=pi/4 whereupon tau increases again to the psi=0 value at psi=pi/2. For rotation of M in three dimensions, the curve of lambda(1) versus psi, (0 < psi < pi) is symmetric about psi=pi/2. Thus the maximum decrease in tau again occurs at psi=pi/4 with maximum increase to a value exceeding that at psi=0 (i.e., with the field applied along the polar axis with that axis taken as the easy axis), at psi=pi/2 (field applied along the equator), the psi=0 value being again attained at psi=pi. The results are shown to be consistent with the behavior predicted by the Kramers theory of the rate of escape of particles over potential barriers. This theory when applied to the potential barriers for the equatorial orientation of the field for rotation in three dimensions yields a simple approximate formula for the escape rate which is in reasonable agreement with the exact lambda(1) calculated from the Fokker-Planck equation. Pfeiffer's approximate formula for the barrier height as a function of alpha [H. Pfeiffer, Phys. Status Solidi 122, 377 (1990)] is shown to be in reasonable agreement with our results.