20
Ò. Iglesias and H. Kachkachi
This expression fits nicely the MC data with D 0 = (1.86 ± 0.03)a being a microscopic length scale (in this case, it is roughly twice the cell parameter), and a critical
exponent ν = 0.49 ± 0.03, which seems to indicate a mean field behaviour [59].
This result can be ascribed to the high coordination of the O and T sublattices. The
fitted curve is drawn in Fig. 1.12 where deviations from scaling are appreciable
for the smallest diameters, for which corrections to the finite-size scaling in (1.13)
may be important [57]. Thus, these results discard any important surface effects on
the ordering temperature and are consistent with spin-wave calculations [60] and
old MC simulations [61]. Similar finite-size effects have been found in fine NM
[62] of MnFe0 4 , but with a surprising increase of T c (D) as D decreases, which has
been attributed to surface effects due to the interactions with the NM coating. More
recently, other experimental [63–68] and theoretical [69–72] studies have reported
similar scaling laws, although with different values of the scaling exponents depending on the NM composition and spin lattice.
The effects of the NM size can also be appreciated in the thermal dependence
of the magnetization of an individual NM when going from the high temperature
paramagnetic phase through T C , the ordering temperature. Finite-size effects show
up as changes in the M(T ) law with NM size D as compared to bulk behavior. To
study the effects of a free surface and of finite size on the magnetization of the NM, we
compare in Fig. 1.12a the results for four NM diameters (D = 3a, 4a, 6a, 8a, 10a, 14a,
symbols) with those corresponding to N = 14 with pbc (representing the behavior
of the bulk). The main feature observed is the reduction of the total magnetization
M T otal with respect to the pbc case (continuous line) due to the lower coordination
of the spins at the surface, which hinders ferrimagnetic order at finite temperatures.
Figure 1.12c, d clearly show the role played by the surface (blue dashes) and the
core (red dashes) in establishing the magnetic order. On one hand, independently
of the NM size, the core tends to a perfect ferrimagnetic order at low T (marked
by M = 1/3), progressively departing from the bulk behavior as T approaches T c ,
this finite-size effect being more important as the NM size decreases. However, the
surface magnetization does not reach perfect ferrimagnetic order at T = 0 even for
D = 8 due to the reduced coordination of the spins. For this reason, a rapid thermal
demagnetization is observed leading to M Sur f that significantly departs from the bulk
behavior.
It is worthwhile to note that for all the diameters studied, there is a temperature
range in which this demagnetization process is linear, this range being wider as the
NM size decreases. In this linear regime, the NM demagnetization becomes dominated by surface effects since the core and surface behaviors are strongly correlated.
Linear demagnetization is indicative of the effective 3D-2D dimensional reduction
of the surface shell and has previously been observed in thin film systems [73, 74]
and in simulations of rough FM surfaces [75]. M Total is always strongly dominated
by the surface contribution, progressively tending to the bulk behavior as the NM
size is increased.
In Fig. 1.12b, we show the size dependence of the M Total at different temperatures.
All the curves follow a quasi-linear behavior with 1/D except for very small NM
sizes (D = 3). This is consistent with the existence of a surface layer of constant
Ò. Iglesias and H. Kachkachi
This expression fits nicely the MC data with D 0 = (1.86 ± 0.03)a being a microscopic length scale (in this case, it is roughly twice the cell parameter), and a critical
exponent ν = 0.49 ± 0.03, which seems to indicate a mean field behaviour [59].
This result can be ascribed to the high coordination of the O and T sublattices. The
fitted curve is drawn in Fig. 1.12 where deviations from scaling are appreciable
for the smallest diameters, for which corrections to the finite-size scaling in (1.13)
may be important [57]. Thus, these results discard any important surface effects on
the ordering temperature and are consistent with spin-wave calculations [60] and
old MC simulations [61]. Similar finite-size effects have been found in fine NM
[62] of MnFe0 4 , but with a surprising increase of T c (D) as D decreases, which has
been attributed to surface effects due to the interactions with the NM coating. More
recently, other experimental [63–68] and theoretical [69–72] studies have reported
similar scaling laws, although with different values of the scaling exponents depending on the NM composition and spin lattice.
The effects of the NM size can also be appreciated in the thermal dependence
of the magnetization of an individual NM when going from the high temperature
paramagnetic phase through T C , the ordering temperature. Finite-size effects show
up as changes in the M(T ) law with NM size D as compared to bulk behavior. To
study the effects of a free surface and of finite size on the magnetization of the NM, we
compare in Fig. 1.12a the results for four NM diameters (D = 3a, 4a, 6a, 8a, 10a, 14a,
symbols) with those corresponding to N = 14 with pbc (representing the behavior
of the bulk). The main feature observed is the reduction of the total magnetization
M T otal with respect to the pbc case (continuous line) due to the lower coordination
of the spins at the surface, which hinders ferrimagnetic order at finite temperatures.
Figure 1.12c, d clearly show the role played by the surface (blue dashes) and the
core (red dashes) in establishing the magnetic order. On one hand, independently
of the NM size, the core tends to a perfect ferrimagnetic order at low T (marked
by M = 1/3), progressively departing from the bulk behavior as T approaches T c ,
this finite-size effect being more important as the NM size decreases. However, the
surface magnetization does not reach perfect ferrimagnetic order at T = 0 even for
D = 8 due to the reduced coordination of the spins. For this reason, a rapid thermal
demagnetization is observed leading to M Sur f that significantly departs from the bulk
behavior.
It is worthwhile to note that for all the diameters studied, there is a temperature
range in which this demagnetization process is linear, this range being wider as the
NM size decreases. In this linear regime, the NM demagnetization becomes dominated by surface effects since the core and surface behaviors are strongly correlated.
Linear demagnetization is indicative of the effective 3D-2D dimensional reduction
of the surface shell and has previously been observed in thin film systems [73, 74]
and in simulations of rough FM surfaces [75]. M Total is always strongly dominated
by the surface contribution, progressively tending to the bulk behavior as the NM
size is increased.
In Fig. 1.12b, we show the size dependence of the M Total at different temperatures.
All the curves follow a quasi-linear behavior with 1/D except for very small NM
sizes (D = 3). This is consistent with the existence of a surface layer of constant