4
Ò. Iglesias and H. Kachkachi
properties are drastically modified owing to the fact that finite-size, boundary and
surface effects then play a major role. Roughly, finite-size and boundary effects are
due to the nanometric size and shape of the system, while surface effects emerge
from the symmetry breaking of the crystal structure at the boundaries of the nanoobject. As such, before addressing the study of nano-scaled magnetic systems, call
them nanomagnets (NM), we ought to distinguish between these effects and try to
assess their separate contributions, at least from the standpoint of theory. In reality
all these effects are intertwined together and it is not possible to single out the impact
of each effect on the macroscopic observable, for the simple reason that the surface
is intimately connected with boundary, shape and size.
Accordingly, in this short review we will attempt to cover, with no pretension to
be exhaustive, the main results of previous works on these effects. We first recall a
simple comparison and clear distinction between finite-size, boundary and surface
effects. Next, we will proceed through chosen examples to illustrate each one of
these effects on the main physical observables which are relevant in the context of
nanomagnetism [1]. We will then consider the magnetization reversal, relaxation
time and blocking temperature and the (quasi-)equilibrium properties of the NMs
such as the hysteresis loop, the “critical temperature”, spin configuration and, of
course, the magnetization itself. We will also discuss, where necessary, the main
computing methods (analytical and numerical) employed.
1.1.1 Finite-Size Versus Boundary Effects
As discussed above, when dealing with confined magnetic systems, such as a NM,
one should distinguish, at least from a theoretical point of view, between finitesize, boundary, and surface effects. For instance, for a cube with simple cubic (sc)
lattice (see Fig. 1.1 left) with periodic boundary conditions (pbc), there is only one
environment (crystal field) with coordination number z = 6.
In this case, the temperature behavior of the magnetization is marked by the wellknown M ∼ 1/
√ N tail in the critical region, where N is the total number of spins
in the NM (Fig. 1.1, right). In the case of more realistic free boundary conditions
(fbc), a cube with sc structure shows four different environments with z = 3, 4, 5, 6
(see Fig. 1.2 left). In this case, in addition to the previous finite-size effects, one
is faced with boundary effects. These induce stronger fluctuations that suppress the
magnetization of the system (see Fig. 1.2 right). Considering both cases of pbc and
fbc allows for a separate estimation of the related effects. Now, if the boundary of a
system with fbc is endowed with a surface anisotropy, which is indeed a consequence
of boundary defects, we may then speak of surface effects, in addition to the finite-size
and boundary effects (see below).
For both pbc and fbc, it can be shown [2] that the magnetization can be written
in a simple form. At low temperature and zero field, M (when normalized) deviates
from unity, its saturation value, according to
Ò. Iglesias and H. Kachkachi
properties are drastically modified owing to the fact that finite-size, boundary and
surface effects then play a major role. Roughly, finite-size and boundary effects are
due to the nanometric size and shape of the system, while surface effects emerge
from the symmetry breaking of the crystal structure at the boundaries of the nanoobject. As such, before addressing the study of nano-scaled magnetic systems, call
them nanomagnets (NM), we ought to distinguish between these effects and try to
assess their separate contributions, at least from the standpoint of theory. In reality
all these effects are intertwined together and it is not possible to single out the impact
of each effect on the macroscopic observable, for the simple reason that the surface
is intimately connected with boundary, shape and size.
Accordingly, in this short review we will attempt to cover, with no pretension to
be exhaustive, the main results of previous works on these effects. We first recall a
simple comparison and clear distinction between finite-size, boundary and surface
effects. Next, we will proceed through chosen examples to illustrate each one of
these effects on the main physical observables which are relevant in the context of
nanomagnetism [1]. We will then consider the magnetization reversal, relaxation
time and blocking temperature and the (quasi-)equilibrium properties of the NMs
such as the hysteresis loop, the “critical temperature”, spin configuration and, of
course, the magnetization itself. We will also discuss, where necessary, the main
computing methods (analytical and numerical) employed.
1.1.1 Finite-Size Versus Boundary Effects
As discussed above, when dealing with confined magnetic systems, such as a NM,
one should distinguish, at least from a theoretical point of view, between finitesize, boundary, and surface effects. For instance, for a cube with simple cubic (sc)
lattice (see Fig. 1.1 left) with periodic boundary conditions (pbc), there is only one
environment (crystal field) with coordination number z = 6.
In this case, the temperature behavior of the magnetization is marked by the wellknown M ∼ 1/
√ N tail in the critical region, where N is the total number of spins
in the NM (Fig. 1.1, right). In the case of more realistic free boundary conditions
(fbc), a cube with sc structure shows four different environments with z = 3, 4, 5, 6
(see Fig. 1.2 left). In this case, in addition to the previous finite-size effects, one
is faced with boundary effects. These induce stronger fluctuations that suppress the
magnetization of the system (see Fig. 1.2 right). Considering both cases of pbc and
fbc allows for a separate estimation of the related effects. Now, if the boundary of a
system with fbc is endowed with a surface anisotropy, which is indeed a consequence
of boundary defects, we may then speak of surface effects, in addition to the finite-size
and boundary effects (see below).
For both pbc and fbc, it can be shown [2] that the magnetization can be written
in a simple form. At low temperature and zero field, M (when normalized) deviates
from unity, its saturation value, according to