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Submitted on 1 Jan 1988
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LOW-FREQUENCY SUSCEPTIBILITY OF
SUPERPARAMAGNETS
E. Gray, R. Cywinski
To cite this version:
JOURNAL DE PHYSIQUE
Colloque C8, Supplement au no 12, Tome 49, d6cembre 1988
LOW-FREQUENCY SUSCEPTIBILITY OF SUPERPARAMAGNETS
E. M. Gray (I) and R. Cywinski (2)
(I) Division of Science and Technology, Grifith University, Nathan 4 1 11, Australia (2) Department of Physics, Reading University, Reading RG6 BAH, G.B.
Abstract. - We used a log-normal distribution of activation energies to model the alternating susceptibility of a su- perparamagnet without interactions. Scaling relationships facilitating the calculation of measurable quantities without knowing the interdependence of particle volume, moment and anisotropy energy were found. A peaked temperature - dependent susceptibility and log - w frequency response were obtained.
Several models of superparamagnetic relaxation have been published (eg. [I-31) incorporating various distributions of the activation energy, but they are not of wide applicability. In [I], the dynamic part of the susceptibility was approximated by a sharp cut off a t wr = 1, leading t o an anomalous maximum suscepti- bility which did not describe the experimental data. The model of [3] gives a small deviation of the suscep- tibility peak from Arrhenius behaviour, which appears insufficient t o account for the strong non-Arrhenius be- haviour of classic spin glasses [4]. In the light of these findings, we were interested to determine explicitly the influence of the width of the distribution of activation energy, which was not previously done. Our results show clearly (in the frequency domain) the crossover from exponential t o logarithmic time response as the distribution broadens, while leading t o negligible devi- ation from Arrhenius behaviour.
For our purposes, a superparamagnet is an ensem- ble of non-interacting giant moments, with two stable states separated by an energy barrier having uniaxial symmetry. We assume t h a t the magnetic moment of a superparamagnetic fine particle is constant in the range of temperatures investigated. The blocked sus- ceptibility a t
.O K due t o rounding of the potential well
is not included. Its value is kT (sin2 0) /6E, [5] (0 = angle between field and anisotropy axis), which for submegahertz frequencies amounts t o less than 2 % of the peak contribution by an individual superpara- magnetic particle t o the total susceptibility [6].The relaxation of an ensemble of N magnetic mo- ments, p, over a single energy barrier leads by stan- dard arguments to a time-dependent magnetisation of the form
M (t) = u (t) Nvp tanh /3 [ I - exp (-t/r)],
We now determine the behaviour of the susceptibil- ity as a function of temperature. According to equa- tion (1) the relaxation time increases rapidly with de- creasing temperature. When the relaxation time be- comes greater than the time available t o make the mea- surement, the measured susceptibility will rapidly de- crease t o zero, i.e., the system will block. In terms of frequency, equation (2) shows that lx(w)l begins to decrease rapidly when wr = 1. Thus we define the blocking temperature,
Tb,
by wer (Tb) = 1, where we is the angular frequency at which the experiment is carried out or, by re-arranging equation (I),This is Arrhenius' law for a superparamagnet. From equations (1) and (2), the modulus of the frequency- dependent low-field susceptibility is
-112
x
..
= xo {I+
u2.i exp(s)
}
As T approaches Tb from above, there is an abrupt but continuous freezing out of the susceptibility.With the assumptions of small field and low fre- quency, the dynamic susceptibility of a system of mo- ments with distributed anisotropy energy is
where X o = p 2 ~ , / 3 k ~ for randomly-aligned anisotropy axes. For reasons outlined in [6], we use the log-normal form for the distribution of particle vol- umes and assume that the anisotropy energy is related to the particle volume by
where, if W
1
and W J, are the probabilities of up- E K v1IX.and down-transitions over a barrier of height E o
>>
(4),LT, .r = (W
1.
+W11-1
is the relaxation time, readily Likely values of X are 1 for volume energy and 3/2 forfound t o be surface energy. The distribution of energies is then
T = 70 exp (Eo/kT). (1)
P ( E ' ) = 1 1 ln2
(E')
u (t) is the unit step function. Fourier transformingthe time-dependent magnetisation gives the complex
frequency-dependent susceptibility where E' = E/E, E being the mode of the energy dis- XO tribution and uv the standard deviation of the volume
x
(w) =~ f .
(2) distribution.C8 - 1852 JOURNAL DE PHYSIQUE
To evaluate X o in equation (3) we need to relate p2 t o E'. We consider two distinct cases, namely, coher- ent precipitate particles with uniform magnetisation, and magnetic clusters in which the magnetic bonds are generated by a random walk, leading to p = Np, (coherent) or p = ~ ' ' ~ p , (random), where N is the number of spins in the cluster, each having a magnetic moment p,. Using equation (4) gives
p2 = pz ( ~ 0 8 ) ~ E'", (6) where No is the number of spins per volume, B is the mode of the volume distribution and J = 1 for random clusters or 2 for coherent clusters.
Inserting equations (1, 5 and 6) into (3) we obtain the frequency-dependent susceptibility:
XL
dE' I1
+
iu[-
exp (ao$)I
ln2 (E')
xEf7 exP
(-T)
,
(7)where y = A[ - 1, T' = T I T , a0 = E l k T and
a = av/X. is the blocking temperature correspond- ing t o the modal energy barrier, E. Ignoring the com- bination of a random cluster with anisotropy propor-
,
tional to surface area, y =
i,
we take the cases y = 0, 1, 2. Equation (7) was evafuated by numerical meth- ods for T' = 0.1, 0.2,..., 3.0,
y = 0 , 1, 2 a n d a =0.2, 0.4,...,
1.4. The answers were normalised to 1 at T' = 1. A peaked temperature-dependent susceptibil- ity was, found. The peak broadens as a increases and moves t o higher temperatures. For y>
0 the peak occurs at very high temperatures relative to T . I t s de- pendence on y was found to bePlots of the temperature-dependent susceptibility on reduced axes (x/xpeak against T/Tpeak) superimpose for y = 0, 1 and 2. This is illustrated in figure 1. The same scaling occurs for frequency response plots of
x
( w )/xo
vs. TITpeak. This very desirable result oc- curs, apparently, because the log-normal distribution retains the same form whether distributed over length, area or volume. The difficulty of guessing X and [ is thus avoided when comparing predictions with exper- imental results.Figure 2 shows the frequency response at T = for y = 0. As a increases above zero, the amplitude of the susceptibility tends to vary linearly with the loga- rithm of frequency, analogous to the well-known log -
t
decay of systems with broadly-distributed activation energies.
Finally, it was found that Tpeak obeys Arrhenius' law almost perfectly [6]
(T
obeys it perfectly but is inaccessible). With knowledge of y and a, therefore,E
can in principle be found from an Arrhenius plot.Fig. 1. - Universal plot (independent of y) of the modu- lus of the alternating susceptibility versus temperature at indicated values of a. we = rad S-'
Fig. 2. - Frequency dependence of the modulus of the alternating susceptibility at T =
iT
and indicated values of a. Note log-
w behaviour at high u.[I] Gittleman, J. I., Ables, B. and Bozowski, S., Phys.
Rev.
B. 9 (1974) 3891.[2] Rechenberg, H. R., Bieman, L. H., Huang, F. S. and de Graaf, A.
M.,
J. Appl. Phys. 49 (1978) 1638.[3] Khater, A., Ferre, J. and Meyer, P., J. Phys. C
20 (1987) 1857.
[4] Tholence, J.-L., Solid State Commun. 35 (1980) 113.
[5] Chikazumi, S., Physics of Magnetism (New York, Wiley) Chapter 13 (1964).
[6] Gray, E. M. and Cywinski, R., submitted to J.
