14
Ò. Iglesias and H. Kachkachi
Moreover, for a NM of diameter 2 nm, we may expect higher anisotropies. For
such materials, the effective macroscopic model has been shown to be valid for
K s /J ≤ 0.25 in a simple cubic lattice and K s /J ≤ 0.35 in a face-centred cubic
lattice [26].
In Sect. 1.3.1.1 we discuss the effect of surface anisotropy on the relaxation rate
in a study that has been made possible with the help of this effective model.
In the case of an assembly of magnetic NMs this model was used to compute the
magnetization, the static susceptibility and the ac susceptibility. More precisely, the
system studied is an ensemble of macrospins, each described with the help of the
effective macroscopic model, and mutually interacting via the long-ranged dipolar
coupling. In such a setup, it was possible to investigate the competition between
(intrinsic) surface effects and dipolar interactions [28, 33–35] and to provide (semi)analytical expressions for the observables mentioned above taking account of temperature, applied DC field, surface anisotropy and dipolar interactions. It is clear that
such analytical developments would not be possible for an assembly of NMs treated
as many-spin systems.
1.2.2.4 Strong Surface Anisotropy
When the conditions discussed above are not met, one has to deal with the Hamiltonian in (1.7) in its full generality with respect to the various energy contributions.
Thus, as discussed earlier, all techniques, both analytical and numerical, that are
usually applied to bulk systems for investigating the magnetic properties, have to be
adapted to nano-scaled systems. In particular, spin-wave theory, which is the study
of the fluctuations of local spins, has to be extended so as to include the fluctuations
of the net magnetic moment as well. This has been done in [2, 3] and references
therein. The choice of the computing method depends on the observable of interest
and on the approach considered. For equilibrium properties, the hysteresis loop and
the switching field, for instance, is computed by numerically solving the deterministic
Landau-Lifshitz equation without the Gaussian field. Within the macrospin approach,
this equation is in fact a system of two (three) coupled equations in the system of
spherical (Cartesian) coordinates, whereas within the many-spin approach this leads
to a system of 2N (3N ) coupled equations. However, in all cases the numerical
procedure is quite straightforward and uses standard routines such as the Euler,
Heun or Runge-Kutta methods [36, 37]. General magnetic properties of a many-spin
system with, in principle, an arbitrary set of physical parameters, are also accessible
to Monte Carlo simulations.
The classical Monte Carlo (MC) method based on the Metropolis algorithm is a
standard technique [38] used in principle to simulate equilibrium statistical properties taking averages over a sample of possible spin configurations. At difference from
atomistic simulations based on the Landau-Lifshitz equations, which are well suited
for dynamic studies because they give the time evolution of magnetic moments, in MC
simulations they are evolved through a sequence of MC steps with no real correspondence to real time. Even so, attempts have been made to establish a time quantification
Ò. Iglesias and H. Kachkachi
Moreover, for a NM of diameter 2 nm, we may expect higher anisotropies. For
such materials, the effective macroscopic model has been shown to be valid for
K s /J ≤ 0.25 in a simple cubic lattice and K s /J ≤ 0.35 in a face-centred cubic
lattice [26].
In Sect. 1.3.1.1 we discuss the effect of surface anisotropy on the relaxation rate
in a study that has been made possible with the help of this effective model.
In the case of an assembly of magnetic NMs this model was used to compute the
magnetization, the static susceptibility and the ac susceptibility. More precisely, the
system studied is an ensemble of macrospins, each described with the help of the
effective macroscopic model, and mutually interacting via the long-ranged dipolar
coupling. In such a setup, it was possible to investigate the competition between
(intrinsic) surface effects and dipolar interactions [28, 33–35] and to provide (semi)analytical expressions for the observables mentioned above taking account of temperature, applied DC field, surface anisotropy and dipolar interactions. It is clear that
such analytical developments would not be possible for an assembly of NMs treated
as many-spin systems.
1.2.2.4 Strong Surface Anisotropy
When the conditions discussed above are not met, one has to deal with the Hamiltonian in (1.7) in its full generality with respect to the various energy contributions.
Thus, as discussed earlier, all techniques, both analytical and numerical, that are
usually applied to bulk systems for investigating the magnetic properties, have to be
adapted to nano-scaled systems. In particular, spin-wave theory, which is the study
of the fluctuations of local spins, has to be extended so as to include the fluctuations
of the net magnetic moment as well. This has been done in [2, 3] and references
therein. The choice of the computing method depends on the observable of interest
and on the approach considered. For equilibrium properties, the hysteresis loop and
the switching field, for instance, is computed by numerically solving the deterministic
Landau-Lifshitz equation without the Gaussian field. Within the macrospin approach,
this equation is in fact a system of two (three) coupled equations in the system of
spherical (Cartesian) coordinates, whereas within the many-spin approach this leads
to a system of 2N (3N ) coupled equations. However, in all cases the numerical
procedure is quite straightforward and uses standard routines such as the Euler,
Heun or Runge-Kutta methods [36, 37]. General magnetic properties of a many-spin
system with, in principle, an arbitrary set of physical parameters, are also accessible
to Monte Carlo simulations.
The classical Monte Carlo (MC) method based on the Metropolis algorithm is a
standard technique [38] used in principle to simulate equilibrium statistical properties taking averages over a sample of possible spin configurations. At difference from
atomistic simulations based on the Landau-Lifshitz equations, which are well suited
for dynamic studies because they give the time evolution of magnetic moments, in MC
simulations they are evolved through a sequence of MC steps with no real correspondence to real time. Even so, attempts have been made to establish a time quantification