1 Single Nanomagnet Behaviour: Surface and Finite-Size Effects
31
the surface size. However, the peculiarities of the shape of the interface in a NM
depend on its size and, as a consequence, EB is also affected by finite-size effects.
As can be seen in panels (a) and (b) of Fig. 1.21, for increasing NP size, h C also
exhibits an increasing trend attributed to a higher proportion of interfacial core spins
that have to be reversed. For h eb , the tendency is the contrary, although with clear
oscillations that are in complete correspondence to the ones observed in M int attained
after the FC process [125, 130]. This demonstrates again the direct link between the
net magnetization component of the shell interfacial spins and the loop shifts. Finally,
notice that surface anisotropy also influences all this phenomenology. As seen on
panels (c) and (d) of Fig. 1.21, there is a minimal value of k Sh for the observation of
EB. On increasing k Sh above it, the bias field increases progressively as the proportion
of interfacial spins pinned during the hysteresis loop increases, and finally saturates.
In contrast, in the presence of EB, h C is reduced with respect to the low-anisotropy
case, but its value does not show appreciable variations with k Sh .
1.3.2.4 Effects of Surface Anisotropy on the Dynamics of NM
Availing ourselves of the compromise provided by the EOSP approach, i.e. a
macrospin capturing some of the intrinsic features of the NM, we can then investigate
the effects of surface anisotropy on the switching field. Accordingly, the relaxation
rate turns out to be a non monotonic function of the surface anisotropy constant
K s [27]. More precisely, owing to the variation of the energy barrier as a function of the surface anisotropy (see Fig. 1.22 left), the relaxation rate increases for
(small) increasing K s (see Fig. 1.22 right) since the (surface) quartic contribution to
anisotropy induces saddle points at the equator. As K s further increases, the quartic
anisotropy starts to dominate, inducing much deeper energy minima and thereby
much higher energy barriers, which finally makes the switching less likely.
Fig. 1.22 Left: Energyscape with increasing surface anisotropy and the corresponding energy
barrier. Right: Relaxation rate as a function of the surface parameter ζ = K 4 /K 2 (see (1.11)).
Source Reprinted with permission from [1]. Copyright (2020), Elsevier Books
31
the surface size. However, the peculiarities of the shape of the interface in a NM
depend on its size and, as a consequence, EB is also affected by finite-size effects.
As can be seen in panels (a) and (b) of Fig. 1.21, for increasing NP size, h C also
exhibits an increasing trend attributed to a higher proportion of interfacial core spins
that have to be reversed. For h eb , the tendency is the contrary, although with clear
oscillations that are in complete correspondence to the ones observed in M int attained
after the FC process [125, 130]. This demonstrates again the direct link between the
net magnetization component of the shell interfacial spins and the loop shifts. Finally,
notice that surface anisotropy also influences all this phenomenology. As seen on
panels (c) and (d) of Fig. 1.21, there is a minimal value of k Sh for the observation of
EB. On increasing k Sh above it, the bias field increases progressively as the proportion
of interfacial spins pinned during the hysteresis loop increases, and finally saturates.
In contrast, in the presence of EB, h C is reduced with respect to the low-anisotropy
case, but its value does not show appreciable variations with k Sh .
1.3.2.4 Effects of Surface Anisotropy on the Dynamics of NM
Availing ourselves of the compromise provided by the EOSP approach, i.e. a
macrospin capturing some of the intrinsic features of the NM, we can then investigate
the effects of surface anisotropy on the switching field. Accordingly, the relaxation
rate turns out to be a non monotonic function of the surface anisotropy constant
K s [27]. More precisely, owing to the variation of the energy barrier as a function of the surface anisotropy (see Fig. 1.22 left), the relaxation rate increases for
(small) increasing K s (see Fig. 1.22 right) since the (surface) quartic contribution to
anisotropy induces saddle points at the equator. As K s further increases, the quartic
anisotropy starts to dominate, inducing much deeper energy minima and thereby
much higher energy barriers, which finally makes the switching less likely.
Fig. 1.22 Left: Energyscape with increasing surface anisotropy and the corresponding energy
barrier. Right: Relaxation rate as a function of the surface parameter ζ = K 4 /K 2 (see (1.11)).
Source Reprinted with permission from [1]. Copyright (2020), Elsevier Books