1 Single Nanomagnet Behaviour: Surface and Finite-Size Effects
15
of MC steps in some particular cases [39, 40] showing that this simulation technique
can also be used to qualitatively understand dynamic magnetization processes such as
magnetic relaxation or hysteresis. In brief, at each MC step, one selects a single spin
from the lattice either randomly or sequentially and a change of its orientation is proposed, repeating the sequence a number of times equal to the total number of spins.
At each selection, a new trial orientation of the spin, the corresponding energy change
E and Boltzmann probability p((E, T ) = exp(−E/K B T ) are computed. Then
the new spin orientation is accepted if either E < 0 or p((E, T ) > r [r ∈ (0, 1)
being a uniform random number], otherwise the trial is rejected and the initial spin
orientation is kept. For Ising spins, there is one option to change the sign of the spin
variable with probability 1/2, but for Heisenberg spins, the new trial step can be chosen in different ways as long as detailed balance condition is met. Instead of using
as trial moves random directions on a sphere, it turns out to be more convenient to
perform trials inside a cone around the current spin direction whose aperture can be
tuned, in order to keep a high trial acceptance rate [41–43]. It has been noticed that
it is of crucial importance to use a combination of methods (random, inside a cone
and spin flip) when simulating NM with inhomogeneous properties such as surface
anisotropy or NM with core/shell structure (see Sects. 1.3.2.1 and 1.3.2.3).
In order to account for the superparamagnetic behavior of the NM net magnetic
moment the MC method was extended in [3] by including global rotations of the
net magnetic moment, in addition to the usual (local) rotations of the atomic spin.
The semi-analytical expressions for the magnetisation in terms of temperature and
magnetic field were derived it was shown that there are three field regimes separated
by two critical values of the magnetic field, namely the one that suppresses the global
rotation and the higher one that suppresses spin waves.
The dynamics of a many-spin NM can in principle be tackled by solving the set
of coupled (stochastic) Landau-Lifshitz equations written for atomic spins [44, 45].
Indeed, solving the (stochastic) Landau-Lifshitz equation with a Langevin (thermal)
field is a very versatile technique that can deal with multi-variate systems and is
thus well suited for investigating equilibrium and dynamic behavior of such manyspin systems. However, this method inherently includes time consuming subroutines
that are necessary for (i) generating sequences of arrays of stochastic numbers and
(ii) computing averages over sufficiently large ensembles of time spin trajectories.
The other technique would consists in solving the Fokker-Planck equation but this
requires writing a hierarchy of equations that depend on the energy potential. This
means that this procedure is somewhat model-dependent. Moreover, this technique
is limited in practice to a small number of degrees of freedom, since otherwise this
hierarchy becomes too cumbersome to write and rather costly to solve numerically.
As such, and as far as NMs are concerned, this technique has been applied to a
maximum of two coupled magnetic moments [46]. In conclusion, the study of the
dynamics of NMs in the many-spin approach can only be done efficiently using the
Landau-Lifshitz equation, even though this remains a tremendous numerical task.
15
of MC steps in some particular cases [39, 40] showing that this simulation technique
can also be used to qualitatively understand dynamic magnetization processes such as
magnetic relaxation or hysteresis. In brief, at each MC step, one selects a single spin
from the lattice either randomly or sequentially and a change of its orientation is proposed, repeating the sequence a number of times equal to the total number of spins.
At each selection, a new trial orientation of the spin, the corresponding energy change
E and Boltzmann probability p((E, T ) = exp(−E/K B T ) are computed. Then
the new spin orientation is accepted if either E < 0 or p((E, T ) > r [r ∈ (0, 1)
being a uniform random number], otherwise the trial is rejected and the initial spin
orientation is kept. For Ising spins, there is one option to change the sign of the spin
variable with probability 1/2, but for Heisenberg spins, the new trial step can be chosen in different ways as long as detailed balance condition is met. Instead of using
as trial moves random directions on a sphere, it turns out to be more convenient to
perform trials inside a cone around the current spin direction whose aperture can be
tuned, in order to keep a high trial acceptance rate [41–43]. It has been noticed that
it is of crucial importance to use a combination of methods (random, inside a cone
and spin flip) when simulating NM with inhomogeneous properties such as surface
anisotropy or NM with core/shell structure (see Sects. 1.3.2.1 and 1.3.2.3).
In order to account for the superparamagnetic behavior of the NM net magnetic
moment the MC method was extended in [3] by including global rotations of the
net magnetic moment, in addition to the usual (local) rotations of the atomic spin.
The semi-analytical expressions for the magnetisation in terms of temperature and
magnetic field were derived it was shown that there are three field regimes separated
by two critical values of the magnetic field, namely the one that suppresses the global
rotation and the higher one that suppresses spin waves.
The dynamics of a many-spin NM can in principle be tackled by solving the set
of coupled (stochastic) Landau-Lifshitz equations written for atomic spins [44, 45].
Indeed, solving the (stochastic) Landau-Lifshitz equation with a Langevin (thermal)
field is a very versatile technique that can deal with multi-variate systems and is
thus well suited for investigating equilibrium and dynamic behavior of such manyspin systems. However, this method inherently includes time consuming subroutines
that are necessary for (i) generating sequences of arrays of stochastic numbers and
(ii) computing averages over sufficiently large ensembles of time spin trajectories.
The other technique would consists in solving the Fokker-Planck equation but this
requires writing a hierarchy of equations that depend on the energy potential. This
means that this procedure is somewhat model-dependent. Moreover, this technique
is limited in practice to a small number of degrees of freedom, since otherwise this
hierarchy becomes too cumbersome to write and rather costly to solve numerically.
As such, and as far as NMs are concerned, this technique has been applied to a
maximum of two coupled magnetic moments [46]. In conclusion, the study of the
dynamics of NMs in the many-spin approach can only be done efficiently using the
Landau-Lifshitz equation, even though this remains a tremendous numerical task.