2 Interparticle Interactions: Theory and Mesoscopic Modeling
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interactions are present, and (b) on bi-magnetic nanoparticles with Co core/CoO
shell morphology.
2.2.1.1 The Model
Despite the importance of interparticle interactions, atomic scale Monte Carlo
approaches have been inadequate to simulate assemblies of nanoparticles with
core/shell or core/surface morphology due to the prohibitively large computational
requirements, since each nanoparticle includes a few hundreds up to a few thousands
of spins. For this purpose, we have developed a simple mesoscopic model to simulate
the magnetic properties of assemblies of magnetic nanoparticle, taking into account
their internal structure, namely core/shell or core/surface morphology. Our mesoscopic method was based on the reduction of the amount of simulated spins to the
minimum number necessary to describe the magnetic structure of the particles and
on the introduction of the adequate exchange and anisotropy parameters between the
different spin regions inside the nanoparticle. Our modeling is multi-scale because
the magnetic moments are evaluated using data from our atomistic simulations of the
core/shell nanoparticles [38], with the appropriate rescaling [21]. Then, we integrate
them properly into the mesoscopic model going in this way from the atomic scale to
the mesoscopic modeling.
Our model goes beyond the classical model of coherent rotation of a particle’s
magnetization of Stoner–Wohlfarth [33] in which each nanoparticle is described by
a classical spin vector (s i ). Here, each nanoparticle in the assembly is described by
a set of three classical spin vectors one for the core and two for the two sublattices
of the AFM or ferrimagnetic (FiM) shell or surface. The values of the different
parameters in the simulation are set on the basis of their bulk values, if they exist,
and their modifications are established considering the nanoparticles’ morphology
(e.g., reduced symmetry and reduced size) using a mean field approach and the data
from atomistic simulations wherever it is possible. We have to note here that “surface”
describes a layer of thickness approximately one lattice constant of the same material
as that of the core and very high anisotropy while the term “shell” corresponds to a
thicker layer that includes also the surface layer and it consists of a different material
than that of the core. The core/shell model includes also interface effects.
In what follows, we consider an assembly of N bi-magnetic nanoparticles with FM
core/AFM shell morphology or of N ferrimagnetic nanoparticles with core/surface
morphology. These nanoparticles are spherical in shape with diameter d and they are
located randomly at the nodes of a simple cubic lattice of lattice constant, a, inside a
box of edge length Lα. Each nanoparticle is described by a set of three classical spin
vectors, one for the core s 1i and two for the shell or the surface s 2i and s 3i i = 1,
…, N (total number of particles), with magnetic moments m n = M n V n /M s V, n = 1
stands for the core and n = 2,3 for the “up” and “down” shell or surface sublattices
of the nanoparticle, respectively. V is the particle volume and M S its saturation
magnetization. V n and M n are the volume and the saturation magnetization of the
core, the “up” and the “down” shell or surface sublattice spins.
43
interactions are present, and (b) on bi-magnetic nanoparticles with Co core/CoO
shell morphology.
2.2.1.1 The Model
Despite the importance of interparticle interactions, atomic scale Monte Carlo
approaches have been inadequate to simulate assemblies of nanoparticles with
core/shell or core/surface morphology due to the prohibitively large computational
requirements, since each nanoparticle includes a few hundreds up to a few thousands
of spins. For this purpose, we have developed a simple mesoscopic model to simulate
the magnetic properties of assemblies of magnetic nanoparticle, taking into account
their internal structure, namely core/shell or core/surface morphology. Our mesoscopic method was based on the reduction of the amount of simulated spins to the
minimum number necessary to describe the magnetic structure of the particles and
on the introduction of the adequate exchange and anisotropy parameters between the
different spin regions inside the nanoparticle. Our modeling is multi-scale because
the magnetic moments are evaluated using data from our atomistic simulations of the
core/shell nanoparticles [38], with the appropriate rescaling [21]. Then, we integrate
them properly into the mesoscopic model going in this way from the atomic scale to
the mesoscopic modeling.
Our model goes beyond the classical model of coherent rotation of a particle’s
magnetization of Stoner–Wohlfarth [33] in which each nanoparticle is described by
a classical spin vector (s i ). Here, each nanoparticle in the assembly is described by
a set of three classical spin vectors one for the core and two for the two sublattices
of the AFM or ferrimagnetic (FiM) shell or surface. The values of the different
parameters in the simulation are set on the basis of their bulk values, if they exist,
and their modifications are established considering the nanoparticles’ morphology
(e.g., reduced symmetry and reduced size) using a mean field approach and the data
from atomistic simulations wherever it is possible. We have to note here that “surface”
describes a layer of thickness approximately one lattice constant of the same material
as that of the core and very high anisotropy while the term “shell” corresponds to a
thicker layer that includes also the surface layer and it consists of a different material
than that of the core. The core/shell model includes also interface effects.
In what follows, we consider an assembly of N bi-magnetic nanoparticles with FM
core/AFM shell morphology or of N ferrimagnetic nanoparticles with core/surface
morphology. These nanoparticles are spherical in shape with diameter d and they are
located randomly at the nodes of a simple cubic lattice of lattice constant, a, inside a
box of edge length Lα. Each nanoparticle is described by a set of three classical spin
vectors, one for the core s 1i and two for the shell or the surface s 2i and s 3i i = 1,
…, N (total number of particles), with magnetic moments m n = M n V n /M s V, n = 1
stands for the core and n = 2,3 for the “up” and “down” shell or surface sublattices
of the nanoparticle, respectively. V is the particle volume and M S its saturation
magnetization. V n and M n are the volume and the saturation magnetization of the
core, the “up” and the “down” shell or surface sublattice spins.