Relaxation Processes in Small Particles

Amikam Aharoni,     Department of Electronics, Weizmann Institute of Science, 76100 Rehovoth, Israel

Abstract

The superparamagnetic behaviour of small ferromagnetic particles is reviewed, with a particular emphasis on ignored or forgotten theories which may explain some observed anomalies.

I INTRODUCTION

There have not been many changes in the use or the understanding of the concept of superparamagnetism since Jacobs and Bean [1] reviewed it almost 30 years ago. The measuring techniques have improved during that time, and have become more accurate and more sophisticated, but their analysis and interpretation still use the same theory and the same crude approximations as in that review. It is not that new theoretical approaches have not been published, or that the published ones were too difficult or too complicated to use. These theories were just ignored.

It is the main purpose of this paper to try and revive the forgotten theories, which will hopefully encourage their use by some of the workers in this field, or even lead to developing more sophisticated generalisations of them. However, it will not be assumed that the reader is already familiar with the older, simplified approach of Néel, which will be described in an even more simplified form in the next section.

II NÉEL’S THEORY

Rather large ferromagnets are known to be subdivided into domains, with a complicated magnetisation structure. But a sufficiently small ferromagnetic particle is a “single domain” [2], which can always be considered (at least approximately) to be magnetised to saturation. When such a particle is magnetised in some direction, it remains that way, and does not change the direction, because of an energy barrier, which is made of anisotropy energy, and as such is proportional to the particle’s volume. Therefore, if the volume is small, the barrier can be small enough for thermal fluctuations to flip the direction of the magnetisation back and forth.

Néel [3] considered an ensemble of such particles, which are put in a very large field, then taken out of the field. He argued that one should then observe a remanent magnetisation which decays in time, as the thermal fluctuations keep changing the directions of the particles’ magnetisation. A decay is usually exponential to a first order, so that if t is the time, one should observe

(1)

where is a parameter, called the relaxation time.

Using the (classical) probability of jumping over an energy barrier, Néel came up with the relation

(2)

where K is the anisotropy constant, V is the particle’s volume, kB is Boltzmann’s constant, and T is the absolute temperature. The pre-exponential factor, f0, is a constant, which Néel estimated [1, 3] to be of the order of 109 sec−1, only in more recent years it has become more customary to take it as 1010 sec−1.

Eq. (2) is for a uniaxial anisotropy, but can also be adapted, with a slight modification [1], to the case of a cubic anisotropy. In order to illustrate its meaning, the following values have been computed from this equation, using the known anisotropy constants for iron or cobalt, and using f0 = 109 sec−1, and T = 300°K: For a sphere of Fe with radius R = 115 Å, the relaxation time is 0.1 sec, while for R = 150 Å, this time becomes 109 sec. For a sphere of Co with R = 36 Å, the time is = 0.1 sec, and for R = 44 Å it is = 6 × 105 sec.

In both cases it is thus seen that above a certain (rather small) R, the time it takes to observe a change is very large, so that for all practical purposes the magnetisation is stable and does not change with time. A similar region can be found for all other ferromagnets, and in this region nothing has to be changed in the conventional theory of ferromagnetism. The theoretical experiment of Néel, where the change of Mr can be observed, is limited to a rather small range of sizes, which is not always possible to obtain experimentally, in which is of the same order as the time it takes to do the measurement. This phenomenon, which has been observed in many systems, and was given the name “magnetic after-effect”, or “magnetic viscosity”, will be discussed in section V.

For still smaller particles, can be so small that several flips may occur during the time it takes to measure the magnetisation (which is usually taken to be about 102 seconds). Therefore, its measured average value will be zero in a zero applied field. It is quite easy to generalise this argument and find that an ensemble of such particles will behave in a non-zero field the same as an ensemble of paramagnetic atoms: There is no hysteresis, but there is a saturation when all the particles are aligned (which theoretically exists in all paramagnets, even though in some cases it takes a very large field to reach that state). All the theoretical treatments of paramagnetism fit this case without any change, only with the quantitative difference that the spin number, S, is of the order of, say, 104, whereas the more conventional paramagnets have S of the order of 1. Néel gave this phenomenon the name “superparamagnetism”.

The phenomenon of superparamagnetism has been observed [1] in many experiments. Thus, e.g. Yatsuya et al [4] measured the magnetisation of Fe particles whose diameter was about 25 Å (with most probably a relatively thick oxide layer on the surface) in different applied fields, H, and at different temperatures, T. The particles were dispersed in oil so that there was hardly any interaction between them. The measurements superimposed to one curve of M vs H/T, without hysteresis, and without any remanence, as befits paramagnetic materials. As was also the case in all previous experiments [1], these particles became ferromagnetic at very small T, as is obvious from Eq. (2). Moreover, unlike previous experiments [1] in which a wide distribution of particle sizes led to a superposition of different Langevin functions in the M (H/T) plot, Fig. 6 of Yatsuya et al [4] is essentially a pure Langevin function, which means a very narrow size distribution. It does not even seem to call for an adjustment of the Curie temperature of the superparamagnetic region to a different value than that of the bulk, as was the case [5] in other measurements. It would have been even more interesting if it were possible to measure just one particle, but this has not been done yet.

Most measurements in more recent years are done by the Mössbauer effect, for which the “time of measurement” is essentially the cycle of a Larmor precession, which is about 10−8 sec. It is possible to see in this kind of measurement the dramatic change from stable ferromagnets to paramagnets when the average particle size is decreased, like e.g. in Fig. 3 of Shinjo [6]. In other experiments, the same sample is measured at different temperatures, and it can be seen, like in Fig. 3 of Mørup [7], how the behaviour changes from that of stable ferromagnets at the lower temperatures, to that of superparamagnets at the higher temperatures. This is the behaviour which can be qualitatively expected from Eq. (2).

These, and other, experiments as well as some that will be presented in this workshop, agree with the general picture of Néel, but it should be noted that the agreement is mainly qualitative. Actually, from the linewidth of the Mössbauer effect one can obtain the value of the relaxation time, , and compare quantitatively the theory to the experiment. Not many workers tried to do it this way, and in the cases which were tried the agreement is not good, as will be discussed in the next section.

The same theory of Néel has also been used [8, 9] for estimating the probability of thermally activated release of a domain wall from pinning sites. All it takes is to replace KV in Eq. (2) by the appropriate energy barrier, if it is known what that barrier is. It should be noted, however, that Brown’s criticism of Néel’s theory applies even more strongly to this application, in which the change of the magnetisation is certainly not by a simple rotation, in one jump.

III BROWN’S THEORY

Brown has argued that Néel’s theory is oversimplified because it considers only a single jump from one energy minimium the other, and does not allow the magnetisation vector to spend some of the time in between these two energy minima before jumping. It also ignores the probability of a jump back to the original minimum. In his first attempt to take these effects roughly into account [10], Brown came up with the same relation as in Eq. (2) here, only with

(3)

for a uniaxial...