2 Interparticle Interactions: Theory and Mesoscopic Modeling
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interactions is a great challenge, since it would allow controlling equilibrium and nonequilibrium magnetization dynamics of exchange coupled nanoparticles systems and
the fine tuning of their anisotropy. In what follows, we present our study on the influence of interparticle interaction on the exchange bias effect of a dilute assembly of
Co nanoparticles embedded in a granular Mn matrix (5% volume fraction of Co
particles in Mn matrix) using the three-spin mesoscopic model developed in ref [32].
Our simulations provide evidence that this interplay leads to a collective superspin
glass behavior of the system.
2.2.3.1 The Model
For the numerical study of the Cobalt nanoparticles embedded in the Mn matrix, we
have used the Monte Carlo simulation technique and the Metropolis algorithm. We
consider an assembly of Co nanoparticles randomly placed on the nodes of a simple
cubic lattice with lattice characteristic lengths L x , L y , L z , with L x = L y = L z = 10α.
The parameter α is defined as the smallest interparticle distance.
Each Co particle has spherical shape and diameter D. The magnetic particles are
single domain and we represented each of them with a three-dimensional classical
unit spin vectors ˆ
s i . Their magnetic moments have magnitude m i = M S V i where M S
is the saturation magnetization and V i = π D
3 /6 is the particle’s volume.
There is experimental evidence that the texture of the antiferromagnetic Mn matrix
is granular and its magnetic behavior is typical of an assembly of uncompensated
antiferromagnetic nanoparticles [31]. The Co particles were distributed at the lattice
sites with occupation probability p Co . From the experimental measurements, we know
that the metal volume fraction of Co in the assembly is x Co = 5% so the occupation
probability is p Co = (6/π) x Co ~ 10% and the total number of Co nanoparticles is N Co
= p Co × (N x × N y × N z ) where N x = L x /α, N y = L y /α and N z = L z /α. A randomly
distributed uniaxial easy axis ê i was assigned to each particle for the anisotropy
vector. Each Mn grain has a small magnetic moment due to the uncompensated spins.
To simulate the matrix, we assigned to each empty lattice site, i.e., non-occupied by
Co nanoparticles, unit spin vectors with magnetic moment of a small magnitude
m Mn = 0.1m Co and a weak random uniaxial anisotropy K Mn = 0.1 K Co . The Co
particles interact via long-range dipolar forces and via exchange forces when they are
sufficiently close. The exchange forces between two Co particles are ferromagnetic,
yielding positive exchange constants (j CoCo = 1). Due to the fact that occupation
probability is much smaller that the percolation threshold (p Co = 10% < p C ≈ 31%)
very few Co particles have other Co nanoparticles as nearest neighbors [15, 34]. In
Fig. 2.10, a 2D schematic representation of the nanoparticles system is given.
The Mn magnetic moments interact via long-range dipolar forces with all the other
magnetic moments (Mn or Co). Furthermore, they interact via exchange forces with
their nearest neighbors. To represent the antiferromagnetic character of the matrix,
we have set the exchange constant between two Mn moments negative (j MnMn = −
0.1). The exchange coupling between a Co nanoparticle and its nearest Mn grains is
taken ferromagnetic j CoMn = 0.3.
57
interactions is a great challenge, since it would allow controlling equilibrium and nonequilibrium magnetization dynamics of exchange coupled nanoparticles systems and
the fine tuning of their anisotropy. In what follows, we present our study on the influence of interparticle interaction on the exchange bias effect of a dilute assembly of
Co nanoparticles embedded in a granular Mn matrix (5% volume fraction of Co
particles in Mn matrix) using the three-spin mesoscopic model developed in ref [32].
Our simulations provide evidence that this interplay leads to a collective superspin
glass behavior of the system.
2.2.3.1 The Model
For the numerical study of the Cobalt nanoparticles embedded in the Mn matrix, we
have used the Monte Carlo simulation technique and the Metropolis algorithm. We
consider an assembly of Co nanoparticles randomly placed on the nodes of a simple
cubic lattice with lattice characteristic lengths L x , L y , L z , with L x = L y = L z = 10α.
The parameter α is defined as the smallest interparticle distance.
Each Co particle has spherical shape and diameter D. The magnetic particles are
single domain and we represented each of them with a three-dimensional classical
unit spin vectors ˆ
s i . Their magnetic moments have magnitude m i = M S V i where M S
is the saturation magnetization and V i = π D
3 /6 is the particle’s volume.
There is experimental evidence that the texture of the antiferromagnetic Mn matrix
is granular and its magnetic behavior is typical of an assembly of uncompensated
antiferromagnetic nanoparticles [31]. The Co particles were distributed at the lattice
sites with occupation probability p Co . From the experimental measurements, we know
that the metal volume fraction of Co in the assembly is x Co = 5% so the occupation
probability is p Co = (6/π) x Co ~ 10% and the total number of Co nanoparticles is N Co
= p Co × (N x × N y × N z ) where N x = L x /α, N y = L y /α and N z = L z /α. A randomly
distributed uniaxial easy axis ê i was assigned to each particle for the anisotropy
vector. Each Mn grain has a small magnetic moment due to the uncompensated spins.
To simulate the matrix, we assigned to each empty lattice site, i.e., non-occupied by
Co nanoparticles, unit spin vectors with magnetic moment of a small magnitude
m Mn = 0.1m Co and a weak random uniaxial anisotropy K Mn = 0.1 K Co . The Co
particles interact via long-range dipolar forces and via exchange forces when they are
sufficiently close. The exchange forces between two Co particles are ferromagnetic,
yielding positive exchange constants (j CoCo = 1). Due to the fact that occupation
probability is much smaller that the percolation threshold (p Co = 10% < p C ≈ 31%)
very few Co particles have other Co nanoparticles as nearest neighbors [15, 34]. In
Fig. 2.10, a 2D schematic representation of the nanoparticles system is given.
The Mn magnetic moments interact via long-range dipolar forces with all the other
magnetic moments (Mn or Co). Furthermore, they interact via exchange forces with
their nearest neighbors. To represent the antiferromagnetic character of the matrix,
we have set the exchange constant between two Mn moments negative (j MnMn = −
0.1). The exchange coupling between a Co nanoparticle and its nearest Mn grains is
taken ferromagnetic j CoMn = 0.3.