152
I. Gudyma and A. Maksymov
Since the manifestation of first-order phase transition is size-dependent for the
studies in the current sections, we choose the reference system size L = 8, which is
also optimal to observe evident functional dependence on fluctuation strength. Other
parameters of the system, the energy gap Δ between LS and HS state, the degeneracy
ratio g, and the number of MC steps used in simulations are kept as in Sect. 4. We
focused our analysis on the influence of fluctuation strength ε on manifestation of
hysteresis, as well as on the role of size effects in fluctuationless system and in the
system with nonzero fluctuations. The results are shown in Fig. 3.
At the beginning, the nature of cooperativity of the molecules on the surface is considered ferromagnetic (Fig. 3a–c) with coupling of surface’s molecules J
s
= 0.5J
b .
On the next step, the impact of antiferromagnetic nature of surface is investigated
by changing the sign of coupling for surface molecules to opposite (J
s
= −0.5J
b )
(Fig. 3d–f). We notice that the interface coupling between the surface and the bulk
part of nanocrystal J
bs is chosen to be the same by absolute value as J
s for both
cases.
The influence of fluctuations on temperature transition is shown in Fig. 3a, d
for the system with ferromagnetic and antiferromagnetic interactions on surfaces,
respectively. For the model (6) considered here, only the influence of Gaussian white
fluctuations described by statistical conditions (4) has been analyzed. The system
size was chosen in such a way that it provides the clearly observed hysteresis in the
absence of fluctuations; therefore, as one can see, from both cases, the increase of
fluctuation strength narrows the width of hysteresis. However, the changing of hysteresis width is not the same for both cases which speaks about different sensitivities
of nanocrystal with ferromagnetic and antiferromagnetic surfaces on action of fluctuations. Such a behavior matches well the classical consideration about destructive
role of noise in the systems with hysteresis. The keystone difference between the
model reported in paper [12], for which the enlargement of hysteresis with increase
of fluctuation is observed, and the model studied here is the way of introducing the
external randomness. For the transition curves shown in Fig. 3, the external fluctuations were updated at each temperature step, whereas for the results reported in
paper [12] the external randomness is updated just after the spin-flip. Physically, such
a setup determines the nature of fluctuations: the changing of fluctuations after spinflip relates their nature rather to the structural disorder of the lattice, in comparison
to the latter case in which the fluctuations have thermal origin.
Besides the narrowing of hysteresis loop due to the increase of fluctuation strength,
from Fig. 3a, b, one can also conclude that the reaching of saturation values of magnetization takes place for stronger control field (here temperature) for fully ordered HS
state and weaker one for fully ordered LS configuration in the system with nonzero
fluctuation in comparison to the fluctuationless model. The effect of incomplete transition for the model with antiferromagnetic surface reported in work [23] that favor
the remanent magnetization in the system expectedly is also observed in this case,
but now one can analyze the role of fluctuations into incompleteness of temperature
transition curves. We notice that, since the magnetization of the system is determined
by the fractions of HS molecules n H S , the remanent magnetization is inherent only
to the incomplete transition from HS to LS phase, whereas the incompleteness of LS
I. Gudyma and A. Maksymov
Since the manifestation of first-order phase transition is size-dependent for the
studies in the current sections, we choose the reference system size L = 8, which is
also optimal to observe evident functional dependence on fluctuation strength. Other
parameters of the system, the energy gap Δ between LS and HS state, the degeneracy
ratio g, and the number of MC steps used in simulations are kept as in Sect. 4. We
focused our analysis on the influence of fluctuation strength ε on manifestation of
hysteresis, as well as on the role of size effects in fluctuationless system and in the
system with nonzero fluctuations. The results are shown in Fig. 3.
At the beginning, the nature of cooperativity of the molecules on the surface is considered ferromagnetic (Fig. 3a–c) with coupling of surface’s molecules J
s
= 0.5J
b .
On the next step, the impact of antiferromagnetic nature of surface is investigated
by changing the sign of coupling for surface molecules to opposite (J
s
= −0.5J
b )
(Fig. 3d–f). We notice that the interface coupling between the surface and the bulk
part of nanocrystal J
bs is chosen to be the same by absolute value as J
s for both
cases.
The influence of fluctuations on temperature transition is shown in Fig. 3a, d
for the system with ferromagnetic and antiferromagnetic interactions on surfaces,
respectively. For the model (6) considered here, only the influence of Gaussian white
fluctuations described by statistical conditions (4) has been analyzed. The system
size was chosen in such a way that it provides the clearly observed hysteresis in the
absence of fluctuations; therefore, as one can see, from both cases, the increase of
fluctuation strength narrows the width of hysteresis. However, the changing of hysteresis width is not the same for both cases which speaks about different sensitivities
of nanocrystal with ferromagnetic and antiferromagnetic surfaces on action of fluctuations. Such a behavior matches well the classical consideration about destructive
role of noise in the systems with hysteresis. The keystone difference between the
model reported in paper [12], for which the enlargement of hysteresis with increase
of fluctuation is observed, and the model studied here is the way of introducing the
external randomness. For the transition curves shown in Fig. 3, the external fluctuations were updated at each temperature step, whereas for the results reported in
paper [12] the external randomness is updated just after the spin-flip. Physically, such
a setup determines the nature of fluctuations: the changing of fluctuations after spinflip relates their nature rather to the structural disorder of the lattice, in comparison
to the latter case in which the fluctuations have thermal origin.
Besides the narrowing of hysteresis loop due to the increase of fluctuation strength,
from Fig. 3a, b, one can also conclude that the reaching of saturation values of magnetization takes place for stronger control field (here temperature) for fully ordered HS
state and weaker one for fully ordered LS configuration in the system with nonzero
fluctuation in comparison to the fluctuationless model. The effect of incomplete transition for the model with antiferromagnetic surface reported in work [23] that favor
the remanent magnetization in the system expectedly is also observed in this case,
but now one can analyze the role of fluctuations into incompleteness of temperature
transition curves. We notice that, since the magnetization of the system is determined
by the fractions of HS molecules n H S , the remanent magnetization is inherent only
to the incomplete transition from HS to LS phase, whereas the incompleteness of LS