MEDIO DE TRANSMISIÓN

The analysis shown in the previous section indicates that the samples that exhibit superparamagnetic relaxation at a practical temperature follow a Vogel-Fulcher

law, scaling to In Tm against l/(T - To). However, the Vogel-Fulcher law was

never intended to be anything other than a phenomenological description of superparamagnetic relaxation with non-negligible intergranular interactions.

The problem is not simply that o f an assembly o f single-domain grains interacting with each other; the problem is also that all these grains undergo spontaneous magnetic reversal. If one also considers the distribution o f grain sizes, and hence moments and fluctuation times, this makes for an extremely difficult computational problem. So clearly, a grain-level treatment of the intergranular interaction problem is beyond the scope o f this thesis. The bulk effects o f intergranular interactions on the frequency behaviour with temperature can however be studied.

The problem of intergranular interactions in superparamagnetic systems has received a great deal of attention lately [Dormann et al., various; Morup and Tronc, 1994], and it is by no means a solved problem at this time. Two models stand out, one suggested by Morup and Tronc, and one suggested by Dormann, Bessais and Fiorani (DBF), which was referred to in chapter 2. The Morup model deals only with weakly interacting systems, whereas the DBF model deals with weak and medium-strength intergranular interactions. Notably neither o f these models are applicable in the blocked state, since they have been explicitly designed to deal with the superparamagnetic, fluctuating state.

The DBF model considers an energy barrier that is the sum o f the intrinsic grain anisotropy energy and the anisotropy energy due to intergranular interactions. This is a valid assumption for uniaxial anisotropy, but possibly not for cubic anisotropy.

As discussed in chapter 2, the DBF model [Dormann et al., 1988, 1997] obtains the total energy barrier for a superparamagnetic system o f inhomogeneous blocking as

Eb = ^Bo + n\a\M nrVL[a\M mVlkT\ (7.10)

where Ebq is the energy barrier for a non-interacting grain, and the second term

represents the intergranular interactions. This considers only nearest-neighbour

(NN) interactions. Z is a Langevin function coth a - Ma, with a = [a\M^nrVlkT\.

In this n\ is the average number o f NN, a\ ~ Cy/V2 where Cv is the volume

concentration o f magnetic grains in the sample. For a weakly interacting system this becomes

EB = Em + rn(axM\,VflOkT) (7,11)

and for medium-strength interactions one obtains

Eb ^ Ebq — n\kT + n\{a\M^nrF)^ (7.12)

For strong interactions the system enters a collective state, with spin glass-like homogeneous freezing. For systems this dense the relaxation time ceases to obey any modified Arrhenius-Néel relation and instead follows a power law

T=To[rg/(Tf-Tg)r (7.13)

where Tg is the glass transition temperature and Tf is the freezing temperature. The

exponent v depends on the law governing the phase transition. This spin glass

relation adequately describes the collective state grain assemblies, though it is difficult to determine the value o f v. For all these cases, the DBF model predicts

an decrease in the energy barrier Eb with applied field, whereas the Morup model

predicts an increase with field. Thus the DBF model, though developed for uniaxial rather than cubic anisotropy, seems to agree with the results obtained in this study.

gives an Eg o f 9.51 x 10’^’ J and a To ~ 10'^^ s, after Tq and size distribution corrections, which is about an order o f magnitude above the normal superparamagnetic tq, which usually varies from 10'^ to 10'*^ s [Jiang and Morup,

1997]. The Eb values are more realistic [Dormann et al., various, 1999]. For Fe2oCui9Ag6i, an Eg o f 2.2 x 10'*^ J and a tq o f -10'^"^ s is obtained, which is much

too high and indicated that this sample is approaching the collective state [Dormann, Fiorani, Cherkaoui, et al., 1999].

The Néel-Arrhenius relation is also valid for relaxation in an applied field provided the intergranular interactions are not too strong [Spinu et al., 1999]. At 500 Oe one can still see scaling as an approximate trend, but there is no appreciable scaling at IkOe for Fe2oCui9Ag6i (figure 7.2).

I n r v s I / ( T ^ _ - T ) A g,, In r v s 1/(T-T )

0 0049 0.005 0.0051 0 0052 0 0053 0 0054 0 0055 0 0056

1 / ( T - T ) ( T = 2 0 K ) ( K ' )

Figure 7.1. Zero-field Vogel-Fulcher plots o f Fei2Cuo9Ag79 and Fe2oCui9Ag6i,

showing AC susceptibility blocking temperatures and the calculated DC magnetisation blocking temperature.

Tq is not known for Fe2oCuo9Ag7i or Fe2 6CuogAg6 6. These samples are

strongly interacting and not superparamagnetic at zero field, so the Néel- Arrhenius relation should probably not apply here [Spinu et al., 1999]. Fe2oCuo9Ag7i at 500 Oe applied field scales well with no correction applied

A . I " r v s l/ ( T - T In r vs I/(T -T ) H = 500 Oe H = 1 kOe 1 / ( T - T ) (K - ‘) ^ H = 5 0 (T 6 e H = IkO e B (uncom 1 / T ( K ') A g,, In r vs l/T , B (uncorrtctcd) Intentionally blank 1 / T ( K ')

Figure 7.2. //dc = 500 Oe and IkOe Vogel-Fulcher plots of Fe2oCui9Ag6i,

Fc2oCuo9Ag7i and Fe2 6CuosAg6 6, showing AC susceptibility blocking temperatures

and the DC magnetisation blocking temperature.

Figure 7.3 shows a fit to equation 7.9 to peak temperatures obtained in

ZFC/FC DC magnetisation experiments on the related samples Fci2Cuo9Ag7 9,

Fci9Cui3Ag68 and Fc2oCui9Ag6i at seven different applied fields. It has been

magnetic material, since the mean grains sizes have been shown to be

approximately the same for all three samples. The parameter Tr = Tq!Tb{^) from

equation (7.9) is about 0.3 for all three samples, meaning the relationship between T o and the zero-field blocking temperature J b( 0 ) is the same in all these alloys. Tq scales with this concentration, 28, 49 and 89 K for Fe-Cu atomic fractions 20.6, 31.7 and 39.32, leading to volume fractions 0.127, 0.243 and 0.309 respectively. This directly relates the magnitude of intergranular interactions to concentrations

in the FexCuxAgi.2x systems.

T vs. H