Effect of an oblique magnetic field on the superparamagnetic relaxation time

Abstract

The effect of a constant magnetic field, applied at an angle P to the easy axis of magnetization, on the Neel relaxation time r of a single domain ferromagnetic particle (with uniaxial anisotropy) is studied by calculating the lowest nonvanishing eigenvalue k, (the escape rate) of the appropriate Fokker-Planck equation using matrix methods. The effect is investigated by plotting kl versus the anisotropy parameter n for various values of P, and the ratio h =g/2a t where g is the external field parameter and k& versus P 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 k& versus 1/I is symmetric about t/t=vrl4 in the range 0(t/1(m/2 and significant decrease in r with increasing 1/I is predicted for large ( and n The m. aximum decrease in r occurs at t/mI/4 whereupon r increases again to the /=0 value at P=rr/2. For rotation of M in three dimensions, the curve of X, versus P (0(~7r) is symmetric about~7r/2. Thus the maximum decrease in r again occurs at P= 7r/4 with maximum increase to a value exceeding that at /=0 (i.e., with the field applied along the polar axis with that axis taken as the easy axis), at t/r=rr/2 (field applied along the equator), the /=0 value being again attained at $=7r 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 k& calculated from the Fokker-Planck equation. Pfeiffer's approximate formula for the barrier height as a function of tr [H. Pfeiffer, Phys. Status Solidi 122, 377 (1990)]is shown to be in reasonable agreement with our results.

Transverse complex magnetic susceptibility of single-domain ferromagnetic particles with uniaxial anisotropy subjected to a longitudinal uniform magnetic field

“…When H 0 is at an arbitrary angle to that axis the theory becomes very much more complicated. 17 The results of such calculations of the transverse response are described in Ref. 32.…”

Effect of an oblique magnetic field on the superparamagnetic relaxation time. II. Influence of the gyromagnetic term

“…The reciprocal of 1 in the high barrier limit is approximately 3 the time of reversal of the magnetization ͑that is, the mean first passage time, Néel time, or time taken to climb the potential hill͒ over the potential barrier due to the crystalline anisotropy and the applied field. We have presented 12 both an exact numerical solution for 1 and an asymptotic estimate based on an extension 3,12,13 of the Kramers theory 14 of the thermally activated escape of particles over potential barriers to include spin relaxation. The numerical calculations we have hitherto presented, however, all proceed from the assumption that the dimensionless damping parameter a is so large that the effect of the gyromagnetic ͑precessional͒ term ͑which contains 1/a as a prefactor͒ on the Néel time may be ignored.…”

“…These lead to recurrence relations for the real and imaginary parts of the x l,m (t) as we shall describe later. As far as the asymptotic estimate of 1 is concerned the omission of the gyromagnetic term indicated at first that an axially symmetric approximation 12,15,16 might be made in the Kramers approach to the problem so reducing it to an effective one variable problem as only the reaction rate coordinate ͑colatitude͒ will now be involved. This, however, leads to asymptotic estimates which deviate appreciably from the numerical solution so that a more rigorous treatment including both reaction coordinates , must be given.…”

“…A preliminary discussion of the effect of an external uniform magnetic field, applied at an oblique angle to the easy axis of magnetization of a single-domain ferromagnetic particle, on the Néel or superparamagnetic relaxation time for uniaxial anisotropy has recently 12 been published by us. The calculation of the Néel relaxation time is in general accomplished by determining the smallest nonvanishing eigenvalue 1 of the appropriate Fokker-Planck equation.…”

Switching field distributions with spin transfer torques in perpendicularly magnetized spin-valve nanopillars

“…The evolution of the mean switching field versus field-sweep rate is sensitive to the nanomagnet's thermal stability factor ξ = E 0 /k B T , where E 0 is the barrier height at zero applied field, k B is the Boltzmann constant, and T = 300 K for our measurements. We assume an Arrheniustype law for thermal activation (H ) = 0 exp(−ξε η ), where ε = (1 − H/H c0 ) and η = 1.5 [27][28][29] determine the scaling of the thermal stability with field, H c0 is the switching field at zero temperature and we assume 0 = 1 GHz. Then, the cumulative probability that the nanomagnet does not switch under a magnetic field ramped linearly in time (dH/dt = v = const.)…”