Nonlinear susceptibility of superparamagnets with a general anisotropy energy
J. L. Garcı
´
a-Palacios,* P. Jo
¨
nsson, and P. Svedlindh
Department of Materials Science, Uppsala University, Box 534, SE-751 21 Uppsala, Sweden
Received 13 July 1999; revised manuscript received 5 October 1999
The equilibrium nonlinear response of noninteracting superparamagnets with a general single-spin anisot-
ropy is investigated. Generalizing the results obtained for the simplest uniaxial anisotropy Garcı ´a-Palacios and
La ´zaro, Phys. Rev. B 55, 1006 1997, we derive a formula for the nonlinear susceptibility of spin ensembles
with randomly distributed anisotropy axes,
¯
(3)
, valid for any magnetic anisotropy with inversion symmetry.
The analysis of this expression reveals that: i
¯
(3)
is always negative, ii unlike the linear susceptibility,
¯
(3)
remains anisotropy dependent after the random axes average except for cubic anisotropy, for which
¯
(3)
is
equal to that of isotropic spins, iii the anisotropy always increases the magnitude of the nonlinear response,
and iv since this increase depends on temperature,
¯
(3)
deviates from the common
¯
(3)
T
-3
laws. The
general expression for
¯
(3)
is finally particularized to superparamagnets with competing uniaxial and cubic
anisotropies and superparamagnets with biaxial anisotropy arbitrary ‘‘shape’’ anisotropy, for which we study
the crossovers between the different regimes isotropic, discrete orientation, and plane rotator induced by the
magnetic anisotropy.
I. INTRODUCTION
The study of classical spin systems has shed much light
on the properties of their quantum counterparts and consti-
tutes, in addition, an important field of research in its own
right. Besides, there exist certain systems for which a de-
scription in terms of classical spins captures the essential
physics in certain ranges, for instance, molecular magnetic
clusters with high spin in their ground state ( S 10) and
magnetic nanoparticles ( S 10
2
–10
5
); both systems will
here be referred to as superparamagnets. Although we use
the language of magnetism, we could also include here sys-
tems as the so-called relaxor ferroelectrics, in which the net
polarization of small polar regions can reorient due to ther-
mal activation between several equienergetical orientations,
leading to a superparaelectric behavior.
Among the various experimental realizations of super-
paramagnets, some approximately consist of noninteracting
entities. The understanding of the properties of classical,
noninteracting systems is besides very important for the sub-
sequent study of their quantum, interacting counterparts. For
example, owing to an insufficient knowledge about some
properties of independent superparamagnets, it is not always
known from which ‘‘laws’’ the associated quantities depart
as a consequence of interspin interactions. Similar consider-
ations also apply to the study of quantum phenomena in
these systems; as complete a knowledge as possible of the
classical regime is a prerequisite for the study of, for in-
stance, quantum tunneling and coherence.
A. Magnetic anisotropy and extent of the equilibrium
„superparamagnetic… range
The single-spin anisotropy plays a fundamental role in the
behavior of superparamagnets. Nevertheless, the effects of
the anisotropy on the thermal-equilibrium properties of these
systems are sometimes overlooked because superparamag-
netism is restrictively ascribed to the temperature range in
which the heights of the energy barriers created by the mag-
netic anisotropy are lower than the thermal energy. Let us
briefly show the limitations of this view.
In the moderate-to-high barrier range, the characteristic
time for the rotation of a classical spin m
over the energy
barrier U can be written in the Arrhenius form
=
0
exp
U , 1.1
where =1/k
B
T and the pre-exponential term is weakly de-
pendent on temperature (
0
10
-7
–10
-8
s for molecular
magnetic clusters and
0
10
-10
–10
-12
s for magnetic
nanoparticles. Then, for a given measurement time t
m
, the
spins display their thermal-equilibrium response when the
condition of superparamagnetism, t
m
, is obeyed, which
corresponds to the temperature range:
ln t
m
/
0
U 0. 1.2
For instance, for ‘‘static’’ measurements ( t
m
1 –100 s) this
range is extremely wide (25 U 0), showing that the
mentioned ascription of superparamagnetism to the range in
which the thermal energy is larger than the anisotropy barri-
ers (1 U 0) is unduly restrictive.
The preceding considerations also entail that, without
leaving the superparamagnetic regime, there are ranges in
which U 1 isotropic behavior, U 1 intermediate
behavior, or U 1 discrete-orientation behavior. Thus,
common approaches such as the isotropic or the discrete-
orientation ones have a restricted range of validity for them-
selves, while, even with the combined use of both, the effects
of the crossover between the different ranges are lost.
B. Linear and nonlinear responses
One of the most informative tools to investigate the prop-
erties of spin systems is the analysis of its linear response.
This analysis could give, for instance, important information
about the symmetry and strength of the magnetic anisotropy
PHYSICAL REVIEW B 1 MARCH 2000-II VOLUME 61, NUMBER 10
PRB 61 0163-1829/2000/6110/67268/$15.00 6726 ©2000 The American Physical Society