32
Ò. Iglesias and H. Kachkachi
Compared to the effect of thermal fluctuations on the switching field, which is a
simple scaling law, the effect of surface anisotropy depends on the direction of the
applied field. This leads to a flattening of the switching field curve [27]. In the case
of an assembly of magnetic NMs, this model is used to compute the ac susceptibility
and to investigate the competition between (intrinsic) surface effects and dipolar
interactions within an assembly of NM [28, 33–35].
1.3.2.5 Magnetic Excitations
When the conditions stated at the end of Sect. 1.2.2.2 regarding the limit of validity
of the effective models are no longer satisfied, one has to adopt the full many-spin
approach discussed earlier with the Hamiltonian in (1.7), and resort to fully numerical
approaches, such as the Monte Carlo simulation method and/or the numerical solution
of the Landau-Lifshitz equation, in order to study the equilibrium and dynamical
properties of the NM. Among the results obtained for the equilibrium behavior, using
the extended Monte Carlo approach that integrates both global and local spin rotations
[2, 3], it has been shown that due to surface anisotropy the magnetization saturation
requires relatively very strong magnetic fields (∼ 10 T). In addition, as was discussed
earlier, the hysteresis loop exhibits various jumps which account for a (cluster-wise)
switching of groups of atomic spins, and thus showing that the magnetization reversal
is not a coherent mechanism as in the Stoner-Wohlfarth model. Regarding the manyspin approach to the magnetization reversal, it was shown in [131, 132] that, under
specific conditions, second-generation spin waves can develop within the NM which,
through their coupling to the uniform mode, destabilize the latter and ultimately
induce the magnetization switching. More precisely, it was shown that a box-shaped
NM exhibit an exponential spin-wave instability in the case of a uniaxial anisotropy
and a linear spin-wave instability for a random anisotropy, with the exponential
instability leading to a faster relaxation than the linear instability.
We have also studied surface effects on ferromagnetic resonance in magnetic
nanocubes [133]. The numerical method used consists in linearizing the LandauLifshitz equation around the equilibrium state of the system, thus leading to an eigenvalue problem whose solution renders the excitation spectrum. For a box-shaped NM,
the results were also compared to those of the generalized spin-wave theory [2, 3]. We
computed the absorbed power as a function of the excitation frequency and showed
that it is possible to attribute the different contributions of the surface and those of the
core spins to the various peaks obtained by our calculations. In particular, the lowenergy peak, corresponding to the k = 0 mode, consists of equal contributions from
the surface and core spins. Furthermore, in the case of less symmetric box-shaped
samples with Néel surface anisotropy, we observe an elliptic precession of the spins
whose signature could be seen in a parametric resonance experiment. For 8 nm iron
nanocubes, we show that the absorbed power spectrum should exhibit a low-energy
peak around 10 GHz, typical of the uniform mode, followed by other low-energy
features that couple to the uniform mode but with a stronger contribution from the
surface. There are also high-frequency exchange-mode peaks around 60 GHz.
Ò. Iglesias and H. Kachkachi
Compared to the effect of thermal fluctuations on the switching field, which is a
simple scaling law, the effect of surface anisotropy depends on the direction of the
applied field. This leads to a flattening of the switching field curve [27]. In the case
of an assembly of magnetic NMs, this model is used to compute the ac susceptibility
and to investigate the competition between (intrinsic) surface effects and dipolar
interactions within an assembly of NM [28, 33–35].
1.3.2.5 Magnetic Excitations
When the conditions stated at the end of Sect. 1.2.2.2 regarding the limit of validity
of the effective models are no longer satisfied, one has to adopt the full many-spin
approach discussed earlier with the Hamiltonian in (1.7), and resort to fully numerical
approaches, such as the Monte Carlo simulation method and/or the numerical solution
of the Landau-Lifshitz equation, in order to study the equilibrium and dynamical
properties of the NM. Among the results obtained for the equilibrium behavior, using
the extended Monte Carlo approach that integrates both global and local spin rotations
[2, 3], it has been shown that due to surface anisotropy the magnetization saturation
requires relatively very strong magnetic fields (∼ 10 T). In addition, as was discussed
earlier, the hysteresis loop exhibits various jumps which account for a (cluster-wise)
switching of groups of atomic spins, and thus showing that the magnetization reversal
is not a coherent mechanism as in the Stoner-Wohlfarth model. Regarding the manyspin approach to the magnetization reversal, it was shown in [131, 132] that, under
specific conditions, second-generation spin waves can develop within the NM which,
through their coupling to the uniform mode, destabilize the latter and ultimately
induce the magnetization switching. More precisely, it was shown that a box-shaped
NM exhibit an exponential spin-wave instability in the case of a uniaxial anisotropy
and a linear spin-wave instability for a random anisotropy, with the exponential
instability leading to a faster relaxation than the linear instability.
We have also studied surface effects on ferromagnetic resonance in magnetic
nanocubes [133]. The numerical method used consists in linearizing the LandauLifshitz equation around the equilibrium state of the system, thus leading to an eigenvalue problem whose solution renders the excitation spectrum. For a box-shaped NM,
the results were also compared to those of the generalized spin-wave theory [2, 3]. We
computed the absorbed power as a function of the excitation frequency and showed
that it is possible to attribute the different contributions of the surface and those of the
core spins to the various peaks obtained by our calculations. In particular, the lowenergy peak, corresponding to the k = 0 mode, consists of equal contributions from
the surface and core spins. Furthermore, in the case of less symmetric box-shaped
samples with Néel surface anisotropy, we observe an elliptic precession of the spins
whose signature could be seen in a parametric resonance experiment. For 8 nm iron
nanocubes, we show that the absorbed power spectrum should exhibit a low-energy
peak around 10 GHz, typical of the uniform mode, followed by other low-energy
features that couple to the uniform mode but with a stronger contribution from the
surface. There are also high-frequency exchange-mode peaks around 60 GHz.