24
Ò. Iglesias and H. Kachkachi
apart from the exchange interaction, it is necessary to include magnetocrystalline
anisotropy and to introduce the distinct surface Néel anisotropy term (1.10) for
spins close to the surface with reduced coordination as compared to bulk. While
the values of bulk anisotropy constants K C can be obtained indirectly by magnetic
measurements, the surface contribution K S is more difficult to evaluate. However,
for maghemite with K C 4.7 × 10
4 erg/cm
3 [92, 93], K S has been estimated as
K S 0.06 erg/cm
2 from Mössbauer experiments [94, 95]. Therefore, in the simulations for maghemite NM, we will vary the core anisotropy values in the range
k C = 0.01 − 1 K and those of the surface anisotropy in the range k S = 1 − 100 K.
We start again by examining the thermal dependence of the magnetization for
different diameters, now with a variable k S and fixed k C = 1K . For small surface
anisotropy (k S < 10), the demagnetizing process has a linear dependence with T
over a wide range of temperature for all NM sizes studied. This linear dependence is
similar to that found for the model with Ising spins [96] in Sect. 1.3.1.2 . However,
in that case, the linear dependence was blurred as the NM size increased and it
was limited to a narrower range of temperatures. For Heisenberg spins, instead,
this behavior is clearly observed for all the simulated D’s and extends up to the
ordering temperature, being more evident for the largest NM and for the core spins
(see Fig. 1.15). This linear behavior is in agreement with the variation predicted by
a surface spin wave theory and, therefore, is indicative of the effective 2D behavior
of the surface shell, that completely dominates the magnetic behavior of the NM
[97]. This is also in contrast with the results for FM NM (not shown, [98]) where,
due to the absence of frustration, the core contribution dominates as indicated by a
clear downward curvature of the M(T ) curves [98, 99]. Note also that the ordering
temperature, marked by appearance of a non-zero M z value, increases with the NM
size as in the Ising model, but its value is lower.
Next, we focus on the magnetic order reached after the cooling process. For the real
maghemite NM, the attained configurations are the result of the competition between
0.7
0.8
0.9
1
M
n
Surface
0
1 0
2 0
3 0
4 0
5 0
T (K)
0
0.1
0.2
0.3
0.4
M
z
k S = 1
k S = 5
k S = 10
k S = 20
k S = 50
k S = 100
0.5
0.6
0.7
0.8
0.9
1
M
n
Core
0
1 0
2 0
3 0
4 0
5 0
T (K)
0
0.1
0.2
0.3
0.4
0.5
M
z
0.7
0.8
0.9
1
M
n
Surface
0
1 0
2 0
3 0
4 0
5 0
T (K)
0
0.1
0.2
0.3
0.4
M
z
k S = 1
k S = 5
k S = 10
k S = 20
k S = 50
k S = 100
0.5
0.6
0.7
0.8
0.9
1
M
n
Core
0
1 0
2 0
3 0
4 0
5 0
T (K)
0
0.1
0.2
0.3
0.4
0.5
M
z
Fig. 1.15 Spherical NM (D = 3a for left panels and D = 6a right panels) with the real maghemite
AF interactions. Thermal dependence of the surface (left column) and core (right column) contributions to M z and to M n during a progressive cooling from a high T at a constant rate δT = −1 K.
Different curves correspond to different values of the surface anisotropy k S = 1, 5, 10, 20, 50, 100
with k C = 1. Reprinted from [100] Copyright (2004), with permission from Elsevier
Ò. Iglesias and H. Kachkachi
apart from the exchange interaction, it is necessary to include magnetocrystalline
anisotropy and to introduce the distinct surface Néel anisotropy term (1.10) for
spins close to the surface with reduced coordination as compared to bulk. While
the values of bulk anisotropy constants K C can be obtained indirectly by magnetic
measurements, the surface contribution K S is more difficult to evaluate. However,
for maghemite with K C 4.7 × 10
4 erg/cm
3 [92, 93], K S has been estimated as
K S 0.06 erg/cm
2 from Mössbauer experiments [94, 95]. Therefore, in the simulations for maghemite NM, we will vary the core anisotropy values in the range
k C = 0.01 − 1 K and those of the surface anisotropy in the range k S = 1 − 100 K.
We start again by examining the thermal dependence of the magnetization for
different diameters, now with a variable k S and fixed k C = 1K . For small surface
anisotropy (k S < 10), the demagnetizing process has a linear dependence with T
over a wide range of temperature for all NM sizes studied. This linear dependence is
similar to that found for the model with Ising spins [96] in Sect. 1.3.1.2 . However,
in that case, the linear dependence was blurred as the NM size increased and it
was limited to a narrower range of temperatures. For Heisenberg spins, instead,
this behavior is clearly observed for all the simulated D’s and extends up to the
ordering temperature, being more evident for the largest NM and for the core spins
(see Fig. 1.15). This linear behavior is in agreement with the variation predicted by
a surface spin wave theory and, therefore, is indicative of the effective 2D behavior
of the surface shell, that completely dominates the magnetic behavior of the NM
[97]. This is also in contrast with the results for FM NM (not shown, [98]) where,
due to the absence of frustration, the core contribution dominates as indicated by a
clear downward curvature of the M(T ) curves [98, 99]. Note also that the ordering
temperature, marked by appearance of a non-zero M z value, increases with the NM
size as in the Ising model, but its value is lower.
Next, we focus on the magnetic order reached after the cooling process. For the real
maghemite NM, the attained configurations are the result of the competition between
0.7
0.8
0.9
1
M
n
Surface
0
1 0
2 0
3 0
4 0
5 0
T (K)
0
0.1
0.2
0.3
0.4
M
z
k S = 1
k S = 5
k S = 10
k S = 20
k S = 50
k S = 100
0.5
0.6
0.7
0.8
0.9
1
M
n
Core
0
1 0
2 0
3 0
4 0
5 0
T (K)
0
0.1
0.2
0.3
0.4
0.5
M
z
0.7
0.8
0.9
1
M
n
Surface
0
1 0
2 0
3 0
4 0
5 0
T (K)
0
0.1
0.2
0.3
0.4
M
z
k S = 1
k S = 5
k S = 10
k S = 20
k S = 50
k S = 100
0.5
0.6
0.7
0.8
0.9
1
M
n
Core
0
1 0
2 0
3 0
4 0
5 0
T (K)
0
0.1
0.2
0.3
0.4
0.5
M
z
Fig. 1.15 Spherical NM (D = 3a for left panels and D = 6a right panels) with the real maghemite
AF interactions. Thermal dependence of the surface (left column) and core (right column) contributions to M z and to M n during a progressive cooling from a high T at a constant rate δT = −1 K.
Different curves correspond to different values of the surface anisotropy k S = 1, 5, 10, 20, 50, 100
with k C = 1. Reprinted from [100] Copyright (2004), with permission from Elsevier