2 Interparticle Interactions: Theory and Mesoscopic Modeling
51
2.2.2.2 Results and Discussion
(a) Effect of the clustering on Fe nanoparticle assemblies
Here, the role of the interparticle interactions and of the system’s morphology in the
magnetic behavior of dense assemblies of Fe nanoparticles is studied. To model the
system, we consider N identical magnetic particles (grains) with spherical shape and
diameter d = 2.6 nm. The magnetic particles are single domain, and we represent each
of them as a three-dimensional classical unit spin vector with magnetic moment m i (i
= 1, …, N). To each particle, a uniaxial easy axis is assigned, randomly distributed and
its anisotropy constant equals to 1 in dimensionless units. In our study, the Fe particles
produced by femtosecond pulsed laser deposition (fsPLD) + UV systematically show
a disk-like shape, giving rise to a higher magnetic anisotropy, enhanced by shape
and surface contributions. Also, the average saturation magnetization M s of the NP
is expected to have a value smaller than the bulk iron value (M s,Fe = 1.7 × 10
6 A/m),
due to surface effects. Taking into account the above considerations, we expect that
the dipolar interaction strength for the NPs is much smaller than the iron value for a
spherical NP with d = 2.6 nm (g Fe ~ 0.6). So here, we consider g = 0.1 [28].
We divide the lattice in eight areas with size L x ×L y × L z = 6 × 4 × 6 each and a
different particle concentration p i in each one of them, but under the constraint that
the total concentration will be p = 0.5. We consider that each particle interacts with
exchange forces of the same strength with a nearest neighbor, if they both belong
in the same area. More specifically, p i takes the values 0.50, 0.80, 0.30, 0.40, 0.70,
0.60, 0.40 and 0.30 in each of the eight areas, respectively. In the denser areas (p i
≥ p), the intra-cluster exchange interaction is j inter = 1.0 and in the more diluted
ones (p i < p) is j inter = 8.0 because, in this case, the whole cluster is considered
to represent a bigger isolated particle. The exchange interaction strength between
neighboring particles in different clusters is taken j inter = 0.1. In general, we assume a
small intercluster exchange constant which allows the cluster moments to be initially
randomly oriented.
To study the magnetic behavior of the system, we calculate numerically [28] the
virgin curve (VC), where we plot the normalized magnetization as function of the
field, and the ZFC/FC magnetization curves, where the normalized magnetization is
plotted as a function of the temperature. The calculated quantity is the normalized
magnetization along the field direction, which is the z-axis direction,
M z /M s =
1
N M s V
N
i=1
m iz =
1
N
N
i=1
s iz
(2.2)
The simulation of the VC starts from the zero field at a given temperature. In the
initial configuration, the spins are randomly oriented. The field is increased gradually until the assembly magnetization reaches saturation (M z /spin ~ 1). The simulations are repeated at two different temperatures. The simulations of the ZFC/FC
magnetization versus temperature curves are also performed.
51
2.2.2.2 Results and Discussion
(a) Effect of the clustering on Fe nanoparticle assemblies
Here, the role of the interparticle interactions and of the system’s morphology in the
magnetic behavior of dense assemblies of Fe nanoparticles is studied. To model the
system, we consider N identical magnetic particles (grains) with spherical shape and
diameter d = 2.6 nm. The magnetic particles are single domain, and we represent each
of them as a three-dimensional classical unit spin vector with magnetic moment m i (i
= 1, …, N). To each particle, a uniaxial easy axis is assigned, randomly distributed and
its anisotropy constant equals to 1 in dimensionless units. In our study, the Fe particles
produced by femtosecond pulsed laser deposition (fsPLD) + UV systematically show
a disk-like shape, giving rise to a higher magnetic anisotropy, enhanced by shape
and surface contributions. Also, the average saturation magnetization M s of the NP
is expected to have a value smaller than the bulk iron value (M s,Fe = 1.7 × 10
6 A/m),
due to surface effects. Taking into account the above considerations, we expect that
the dipolar interaction strength for the NPs is much smaller than the iron value for a
spherical NP with d = 2.6 nm (g Fe ~ 0.6). So here, we consider g = 0.1 [28].
We divide the lattice in eight areas with size L x ×L y × L z = 6 × 4 × 6 each and a
different particle concentration p i in each one of them, but under the constraint that
the total concentration will be p = 0.5. We consider that each particle interacts with
exchange forces of the same strength with a nearest neighbor, if they both belong
in the same area. More specifically, p i takes the values 0.50, 0.80, 0.30, 0.40, 0.70,
0.60, 0.40 and 0.30 in each of the eight areas, respectively. In the denser areas (p i
≥ p), the intra-cluster exchange interaction is j inter = 1.0 and in the more diluted
ones (p i < p) is j inter = 8.0 because, in this case, the whole cluster is considered
to represent a bigger isolated particle. The exchange interaction strength between
neighboring particles in different clusters is taken j inter = 0.1. In general, we assume a
small intercluster exchange constant which allows the cluster moments to be initially
randomly oriented.
To study the magnetic behavior of the system, we calculate numerically [28] the
virgin curve (VC), where we plot the normalized magnetization as function of the
field, and the ZFC/FC magnetization curves, where the normalized magnetization is
plotted as a function of the temperature. The calculated quantity is the normalized
magnetization along the field direction, which is the z-axis direction,
M z /M s =
1
N M s V
N
i=1
m iz =
1
N
N
i=1
s iz
(2.2)
The simulation of the VC starts from the zero field at a given temperature. In the
initial configuration, the spins are randomly oriented. The field is increased gradually until the assembly magnetization reaches saturation (M z /spin ~ 1). The simulations are repeated at two different temperatures. The simulations of the ZFC/FC
magnetization versus temperature curves are also performed.