6
Ò. Iglesias and H. Kachkachi
where α = x, y, z. This subtle difference is responsible for much stronger thermal
fluctuations in the fbc model due to boundary effects. Indeed, the separation between
two successive values of the wavenumber k for fbc is smaller than that for pbc.
Therefore, more spin-wave modes are excited in the fbc system, thus leading to a
weaker magnetization.
Surface Effects
Surface effects are due to the breaking of crystal-field symmetry at the boundary of
the NM, and in reality they cannot be disentangled from boundary effects. In order
to study surface effects one has to resort to microscopic theories capable of distinguishing between different atomic environments and taking account of physical
parameters such as single-site surface anisotropy, exchange and dipolar interactions
(DI), in addition of course to the magneto-crystalline anisotropy and magnetic field.
Unfortunately, this leads to complex many-body problems which can only be efficiently dealt with, in general, using numerical approaches such as Monte Carlo
simulations or numerical solutions of the Landau-Lifshitz equation (see Sect. 1.2.2).
Shape Effects
Apart from the finite-size, boundary and surface effects, the NM shape is also a
distinctive property that has an influence not only on its magnetic properties but
also on the optical, plasmonic and electric properties of nanostructured systems. The
equilibrium states and hysteresis loops are clearly dependent on the NM’s shape
(see Sect. 1.3.2). Indeed, when the shape changes, the distribution of atomic spins
changes and the core-to-surface ratio is thereby modified, leading to a change in the
corresponding effective fields. There is even a drastic change in the distribution of
local fields that lead to new spin configurations and thereby to new magnetic states
with different macroscopic properties.
1.2 Basic Theoretical Models and Computing Tools
Building theoretical models for confined magnetic systems requires explicitly taking
into account the local atomic environment with all its specificities which are strongly
dependent on the size, shape and underlying material. On the other hand, the model
have to account for the macroscopic behavior of the net magnetic moment which
is rather sensitive to the external stimili such as heat and magnetic fields. Such a
behavior is exemplified by superparamagnetism, which is a fast shuttling motion of
the net magnetic moment between its energy minima. Low temperature magnetic
order within the NM has to incorporate exchange coupling between the magnetic
ions, the Zeeman coupling to the external field and the (local) magnetic anisotropy
energy. Apart from this, the spatial lattice in which the magnetic ions reside has to
reflect the real crystallographic structure of the material to be studied, since lattice
geometry may play an essential role in establishing the minimum energy configurations by inducing competing orders and frustration. For these reasons, a reasonable
Ò. Iglesias and H. Kachkachi
where α = x, y, z. This subtle difference is responsible for much stronger thermal
fluctuations in the fbc model due to boundary effects. Indeed, the separation between
two successive values of the wavenumber k for fbc is smaller than that for pbc.
Therefore, more spin-wave modes are excited in the fbc system, thus leading to a
weaker magnetization.
Surface Effects
Surface effects are due to the breaking of crystal-field symmetry at the boundary of
the NM, and in reality they cannot be disentangled from boundary effects. In order
to study surface effects one has to resort to microscopic theories capable of distinguishing between different atomic environments and taking account of physical
parameters such as single-site surface anisotropy, exchange and dipolar interactions
(DI), in addition of course to the magneto-crystalline anisotropy and magnetic field.
Unfortunately, this leads to complex many-body problems which can only be efficiently dealt with, in general, using numerical approaches such as Monte Carlo
simulations or numerical solutions of the Landau-Lifshitz equation (see Sect. 1.2.2).
Shape Effects
Apart from the finite-size, boundary and surface effects, the NM shape is also a
distinctive property that has an influence not only on its magnetic properties but
also on the optical, plasmonic and electric properties of nanostructured systems. The
equilibrium states and hysteresis loops are clearly dependent on the NM’s shape
(see Sect. 1.3.2). Indeed, when the shape changes, the distribution of atomic spins
changes and the core-to-surface ratio is thereby modified, leading to a change in the
corresponding effective fields. There is even a drastic change in the distribution of
local fields that lead to new spin configurations and thereby to new magnetic states
with different macroscopic properties.
1.2 Basic Theoretical Models and Computing Tools
Building theoretical models for confined magnetic systems requires explicitly taking
into account the local atomic environment with all its specificities which are strongly
dependent on the size, shape and underlying material. On the other hand, the model
have to account for the macroscopic behavior of the net magnetic moment which
is rather sensitive to the external stimili such as heat and magnetic fields. Such a
behavior is exemplified by superparamagnetism, which is a fast shuttling motion of
the net magnetic moment between its energy minima. Low temperature magnetic
order within the NM has to incorporate exchange coupling between the magnetic
ions, the Zeeman coupling to the external field and the (local) magnetic anisotropy
energy. Apart from this, the spatial lattice in which the magnetic ions reside has to
reflect the real crystallographic structure of the material to be studied, since lattice
geometry may play an essential role in establishing the minimum energy configurations by inducing competing orders and frustration. For these reasons, a reasonable