170
M. L. Fdez-Gubieda et al.
an external magnetic field in a coherent way. Therefore the physical problem can be
implemented by using a single particle approach, the Stoner-Wohlfarth model, where
the position of the magnetization is given by counterbalancing between the anisotropy
energy, E ani , and the Zeeman contribution. Being the energy density E(θ, ϕ, H ) in
spherical coordinates:
E(θ, ϕ, H ) = E ani − μ 0 M H 0 ( ˆ
u H · ˆ
u m )
(7.3)
where ˆ
u H and ˆ
u m are the unit vectors that define the direction of the applied magnetic
field and the magnetization, respectively.
There are several sources to the anisotropy energy: the magnetocrystalline contribution of the magnetite phase, E c , which retains the cubic symmetry of the lattice
and produces 8 equivalent easy axes above the Verwey transition directed along the
111 crystallographic directions; shape anisotropy of magnetosome due to the morphology, and the intra-chain dipolar interactions. The two last terms are typically of
uniaxial nature and can be expressed as an effective uniaxial anisotropy K uni (see
Fig. 7.9a, b). As recently reported by Orue et al. [53, 57], due to compromise effects
of shape anisotropy and dipolar interactions between magnetosomes, the effective
magnetic moment of individual magnetosomes is tilted out of the [111] crystallographic easy axis of magnetite (see Fig. 7.9a, b). Then, the expression 7.3 can be
written as:
E(θ, ϕ, H ) = E c + K uni [1 − ( ˆ
u u · ˆ
u m )
2
] − μ 0 M H 0 ( ˆ
u H · ˆ
u m )
(7.4)
Note that an uniaxial anisotropy constant as small as 3 kJ/m
3 [53] is enough to
overcome the small magnetocrystalline contribution of the magnetite (K c = −11
kJ/m
3 ).
The hysteresis loops have been calculated following a dynamical approach in
which the single domain magnetization can switch between the available energy
minima states, at a rate determined by a Boltzmann factor (exp(−
V E
k B T
)), where E
is the energy density barrier between each pair of minima states as it is well explained
in [40, 58, 59]. Aiming to achieve the best match between experiment and theory,
K c and K uni have been adjusted at each temperature in each simulation. As shown in
Fig. 7.8, this model accurately reproduces the experimental hysteresis loops and the
thermal evolution of the coercive field and the reduced remanence magnetization.
Figure 7.9 displays the thermal evolution obtained for K c and K uni . From 300 K
down to the Verwey temperature T V , the tendency of K c reproduces the values and
trend reported in the bibliography for bulk single-crystalline magnetite (at 300 K,
K c = −11 kJ/m
3 ) [60]. K uni remains constant (11–12 kJ/m
3 ) down to T V , suggesting that the shape anisotropy and the strength of magnetic interactions are basically
temperature-independent in the whole studied temperature range. Below T V , the
magnetocrystalline anisotropy changes from cubic to essentially uniaxial along the
100 directions of the original cubic structure. Therefore, below T V , the effective
anisotropy is purely uniaxial and results from the competition between the magnetocrystalline uniaxial anisotropy, the shape anisotropy and the dipolar interaction
M. L. Fdez-Gubieda et al.
an external magnetic field in a coherent way. Therefore the physical problem can be
implemented by using a single particle approach, the Stoner-Wohlfarth model, where
the position of the magnetization is given by counterbalancing between the anisotropy
energy, E ani , and the Zeeman contribution. Being the energy density E(θ, ϕ, H ) in
spherical coordinates:
E(θ, ϕ, H ) = E ani − μ 0 M H 0 ( ˆ
u H · ˆ
u m )
(7.3)
where ˆ
u H and ˆ
u m are the unit vectors that define the direction of the applied magnetic
field and the magnetization, respectively.
There are several sources to the anisotropy energy: the magnetocrystalline contribution of the magnetite phase, E c , which retains the cubic symmetry of the lattice
and produces 8 equivalent easy axes above the Verwey transition directed along the
111 crystallographic directions; shape anisotropy of magnetosome due to the morphology, and the intra-chain dipolar interactions. The two last terms are typically of
uniaxial nature and can be expressed as an effective uniaxial anisotropy K uni (see
Fig. 7.9a, b). As recently reported by Orue et al. [53, 57], due to compromise effects
of shape anisotropy and dipolar interactions between magnetosomes, the effective
magnetic moment of individual magnetosomes is tilted out of the [111] crystallographic easy axis of magnetite (see Fig. 7.9a, b). Then, the expression 7.3 can be
written as:
E(θ, ϕ, H ) = E c + K uni [1 − ( ˆ
u u · ˆ
u m )
2
] − μ 0 M H 0 ( ˆ
u H · ˆ
u m )
(7.4)
Note that an uniaxial anisotropy constant as small as 3 kJ/m
3 [53] is enough to
overcome the small magnetocrystalline contribution of the magnetite (K c = −11
kJ/m
3 ).
The hysteresis loops have been calculated following a dynamical approach in
which the single domain magnetization can switch between the available energy
minima states, at a rate determined by a Boltzmann factor (exp(−
V E
k B T
)), where E
is the energy density barrier between each pair of minima states as it is well explained
in [40, 58, 59]. Aiming to achieve the best match between experiment and theory,
K c and K uni have been adjusted at each temperature in each simulation. As shown in
Fig. 7.8, this model accurately reproduces the experimental hysteresis loops and the
thermal evolution of the coercive field and the reduced remanence magnetization.
Figure 7.9 displays the thermal evolution obtained for K c and K uni . From 300 K
down to the Verwey temperature T V , the tendency of K c reproduces the values and
trend reported in the bibliography for bulk single-crystalline magnetite (at 300 K,
K c = −11 kJ/m
3 ) [60]. K uni remains constant (11–12 kJ/m
3 ) down to T V , suggesting that the shape anisotropy and the strength of magnetic interactions are basically
temperature-independent in the whole studied temperature range. Below T V , the
magnetocrystalline anisotropy changes from cubic to essentially uniaxial along the
100 directions of the original cubic structure. Therefore, below T V , the effective
anisotropy is purely uniaxial and results from the competition between the magnetocrystalline uniaxial anisotropy, the shape anisotropy and the dipolar interaction