Long-Time Simulation Of Spin Dynamics Of Superparamagnetic Particles

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1 Poster 21 Long-Time Simulation Of Spin Dynamics Of Superparamagnetic Particles P. B.Visscher and Xiaoguang Deng Department of Physics and Astronomy The University of Alabama Supported by DOE grant No. DE-FG02-98ER45714 and NSF-MRSEC DMR ,

2 Motivation This work is motivated by the recent synthesis of very finegrained, high-coercivity media by the annealing of very small (4-6 nm) super-paramagnetic particles of FePt (Sun et al, IBM), FePd (Nikles&Chen, UA) and related materials. Before annealing, these particles are very mobile and form 2D colloidal crystalline arrays; the uniformity of these arrays is critical to the performance of the resulting annealed media. Our objective is to understand the formation ( self-assembly ) of these arrays. We believe that superparamagnetic fluctuations may be critical to the annealingout of defects in these arrays. Thus we need to simulate the magnetization dynamics of paramagnetic particles over very long time scales. FeCoPt particle array, courtesy M. Chen

3 Long-time Simulation of Superparamagnetic Dynamics This is a very difficult problem because of the disparity of time scales: if we follow precession of the magnetization in detail on a nanosecond time scale (as one would in a micromagnetic calculation) we can t model long-time diffusion of the particles. This problem has been solved in the past by a quasistatic approximation to Landau-Lifshitz dynamics in which the magnetization is assumed to relax instantly to the minimum of the Stoner-Wohlfarth energy. However, this leaves out superparamagnetic fluctuations, which are very important in this problem. We have tested several solutions to this problem. One possibility is to use Arrhenius-like jump dynamics.

4 Arrhenius ( jump ) dynamics Assumes M lies at minimum of Stoner-Wohlfarth energy M jumps between minima with probability ωexp(- E/k B T) where E(H, M) is the energy barrier and ω is an attempt frequency. Problem: what is ω? Practical problem: jumps send a discontinuous shock through the system that is difficult to deal with numerically. So we have tried other methods.

5 Averaging out precession Since the fast dynamics (precession) is hard to follow in detail, one possibility is to assume that the component of M transverse to the local field H averages to zero (even before it decays due to the LL damping.) The problem with this is that when the local field is small, precession is slow and the component does not average to zero. One has to make some arbitrary assumptions about the low-field dynamics. We have found a method that effectively averages out fast precession, but still treats slow precession correctly.

6 Large- t simulation In principle, the magnetization evolves according to the Landau- Lifshitz equation dm γα = γ M H dt M s M ( M H) where H includes the external field, a random thermal field, and the magnetostatic fields of the neighbors of this particle. We can divide a neighbor s field into a slowly varying part and (if the neighbor is subject to a large field) a rapidly varying part. The latter will affect the motion greatly only if the frequencies match (resonant energy transfer). We assume that in a superparamagnetic system with large fluctuations, resonant transfer can be neglected, and we replace the field by its slowly varying part. Specifically, we assume the field is constant and equal to its average value over the time interval [t, t+ t].

7 Using the Kikuchi exact solution Assuming the field to be constant over [t, t+ t], we can 1 solve the LL equation exactly (Kikuchi 1956): To calculate the fields acting on the neighbors 0 of this particle, we need its average -1 0 magnetization <M> over the interval. <M z > -0.5 M init 1 can be calculated analytically; the transverse Kikuchi trajectory -1 components cannot be integrated in closed (α=0.1) form but we have used a parameterized approximation whose error is less than 1%, far less than the other uncertainties in the problem. 0.5 M final The key advantage of this method is that it is accurate for both very small and very large fields: in the latter case the precession is rapid and the transverse magnetization averages almost to zero. H

8 Simulation of Particle Motion The procedure for modeling particle motion is similar to that we have used previously for acicular particles. Our superparamagnetic results so far are for spherical particles, but the code is designed to handle cylindrical particles with hemispherical end caps, of arbitrary aspect ratio (or distribution of aspect ratios) we have simply set the aspect ratio to unity. Thus we can also model the effects of polydispersity in both size and shape. Particle motions are followed using Newton s laws: total force F and total torque N on each particle are computed, and the motion (change in position r and angular momentum L are computed from dr(t)/dt = F(t)/m & dl(t)/dt = N(t) using a Verlet algorithm.

9 Summary We have developed an algorithm for following superparamagnetic dynamics over the long time scales of self-assembly, without following the precession in detail. This is done by using an exact constant-field solution to evolve the system over a time step that can be long compared to the precession period.