Nonlinear stochastic relaxation dynamics in spin-crossover solid-state compounds
A study of dynamic of spin-crossover solid-state compound has been carried out. The investigated macroscopic phenomenological model for molecular spin-crossover complexes with optical control parameter has been extended to the case of noise action. The noise-driven phase transition was observed....
Збережено в:
| Дата: | 2010 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2010
|
| Назва видання: | Semiconductor Physics Quantum Electronics & Optoelectronics |
| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/118740 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Nonlinear stochastic relaxation dynamics in spin-crossover solid-state compounds / Iu.V. Gudyma, A.Iu. Maksymov // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2010. — Т. 13, № 4. — С. 357-362. — Бібліогр.: 20 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-118740 |
|---|---|
| record_format |
dspace |
| spelling |
nasplib_isofts_kiev_ua-123456789-1187402025-06-03T16:26:27Z Nonlinear stochastic relaxation dynamics in spin-crossover solid-state compounds Gudyma, Iu.V. Maksymov, A.Iu. A study of dynamic of spin-crossover solid-state compound has been carried out. The investigated macroscopic phenomenological model for molecular spin-crossover complexes with optical control parameter has been extended to the case of noise action. The noise-driven phase transition was observed. Also, ascertained was the role of additive noise as a main factor for suppressing the potential barrier height. 2010 Article Nonlinear stochastic relaxation dynamics in spin-crossover solid-state compounds / Iu.V. Gudyma, A.Iu. Maksymov // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2010. — Т. 13, № 4. — С. 357-362. — Бібліогр.: 20 назв. — англ. 1560-8034 PACS 05.45.-a, 42.65.Pc, 75.30.Wx https://nasplib.isofts.kiev.ua/handle/123456789/118740 en Semiconductor Physics Quantum Electronics & Optoelectronics application/pdf Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
A study of dynamic of spin-crossover solid-state compound has been carried
out. The investigated macroscopic phenomenological model for molecular spin-crossover
complexes with optical control parameter has been extended to the case of noise action.
The noise-driven phase transition was observed. Also, ascertained was the role of
additive noise as a main factor for suppressing the potential barrier height. |
| format |
Article |
| author |
Gudyma, Iu.V. Maksymov, A.Iu. |
| spellingShingle |
Gudyma, Iu.V. Maksymov, A.Iu. Nonlinear stochastic relaxation dynamics in spin-crossover solid-state compounds Semiconductor Physics Quantum Electronics & Optoelectronics |
| author_facet |
Gudyma, Iu.V. Maksymov, A.Iu. |
| author_sort |
Gudyma, Iu.V. |
| title |
Nonlinear stochastic relaxation dynamics in spin-crossover solid-state compounds |
| title_short |
Nonlinear stochastic relaxation dynamics in spin-crossover solid-state compounds |
| title_full |
Nonlinear stochastic relaxation dynamics in spin-crossover solid-state compounds |
| title_fullStr |
Nonlinear stochastic relaxation dynamics in spin-crossover solid-state compounds |
| title_full_unstemmed |
Nonlinear stochastic relaxation dynamics in spin-crossover solid-state compounds |
| title_sort |
nonlinear stochastic relaxation dynamics in spin-crossover solid-state compounds |
| publisher |
Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| publishDate |
2010 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/118740 |
| citation_txt |
Nonlinear stochastic relaxation dynamics in spin-crossover
solid-state compounds / Iu.V. Gudyma, A.Iu. Maksymov // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2010. — Т. 13, № 4. — С. 357-362. — Бібліогр.: 20 назв. — англ. |
| series |
Semiconductor Physics Quantum Electronics & Optoelectronics |
| work_keys_str_mv |
AT gudymaiuv nonlinearstochasticrelaxationdynamicsinspincrossoversolidstatecompounds AT maksymovaiu nonlinearstochasticrelaxationdynamicsinspincrossoversolidstatecompounds |
| first_indexed |
2025-12-01T05:09:05Z |
| last_indexed |
2025-12-01T05:09:05Z |
| _version_ |
1850281298308890624 |
| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2010. V. 13, N 4. P. 357-362.
PACS 05.45.-a, 42.65.Pc, 75.30.Wx
Nonlinear stochastic relaxation dynamics in spin-crossover
solid-state compounds
Iu.V. Gudyma, A.Iu. Maksymov
Chernivtsi National University, Department of General Physics, 58012 Chernivtsi, Ukraine,
E-mail: yugudyma@gmail.com
Abstract. A study of dynamic of spin-crossover solid-state compound has been carried
out. The investigated macroscopic phenomenological model for molecular spin-crossover
complexes with optical control parameter has been extended to the case of noise action.
The noise-driven phase transition was observed. Also, ascertained was the role of
additive noise as a main factor for suppressing the potential barrier height.
Keywords: optical bistability, additive noise, multiplicative noise, noise-driven
dynamics.
Manuscript received 10.10.10; accepted for publication 02.12.10; published online 30.12.10.
1. Introduction
With increasing needs in computing enormous volume
of information, investigation of new molecular materials
which may satisfy the requirements in the field of
informational technology becomes actual. One of the
most promising bistable materials is spin-crossover
coordination complexes that possess two stable states:
paramagnetic high-spin (HS) and diamagnetic low-spin
(LS) states [1]. There is a class of inorganic complexes
of chemical elements with electron
configuration and central transition metal ion located in
octahedral ligand field. Under the influence of
environmental perturbation, namely: light irradiation,
magnetic field, temperature, noise influence and others,
there possible is the transition between the states of spin-
crossover compounds. The transition provokes a drastic
change in properties of spin-crossover complexes, in
particular, magnetic properties due to spin
rearrangement between sublevels of 3d-orbitals. Thus,
there exist many experimental works where bistability in
magnetic susceptibility is shown [2].
74 dd −
One of the most important developments in
researches of spin crossover was that the equilibrium
existing between high-spin and low-spin species could
be perturbed by laser irradiation. A light irradiation of a
spin-crossover system in the solid state at low
temperature induces partial or complete conversion of a
low-spin state to a high-spin metastable one with
virtually infinitely long lifetime. This solid state effect
became known as the LIESST effect (light induced
excited spin state trapping) [3]. This effect is the basis of
different light-induced bistability effects that provoke
appearance of optical (light induced optical hysteresis,
shortly LIOH) and thermal hysteresis (light induced
thermal hysteresis, shortly LITH). A light-induced
bistability in a spin transition solids leading to thermal
and optical hysteresis has been well studied in [4, 5].
Until now, many experiments were performed to
study the process and physics of the transition from one
state to another, but there is still no clarity in this
problem. Here, we propose to represent the model of
spin-crossover complexes as a macroscopic
phenomenological model written in terms of relaxation
rate with light-induced phase transition. To fuller satisfy
the condition of physical reality, we have taken into
account environmental action on the system dynamics as
additive and multiplicative Gaussian distributed noises.
The environmental random noise influencing on
spin-crossover solids can be represented as an
interaction with a heat bath. This model may be
described by macroscopic kinetic stochastic differential
equation with a deterministic term possessing slow
degree of freedom and random term with fast degree of
freedom. Dynamics of a system being in contact with an
environment playing a role of the heat bath is based on
the concept of the Langevin equation. Such assumption
is possible because the condition of heat bath does not
depend on spin configuration. Detailed theoretical
analysis of the states of spin-crossover compounds under
cross-correlated noises was made in [6].
On the assumption of possible technological usage
of these materials as data storage systems or display
systems, it is important to know the lifetime of
© 2010, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
357
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2010. V. 13, N 4. P. 357-362.
metastable state or transition time from one state to
another. Due to environmental perturbation and size
limitation in any real system, the lifetime of a metastable
state is not infinitely long. In this work, we also
estimated the time of decay of the metastable state in a
mean field approach by defining the mean first passage
time (MFPT) for this metastable state. The estimation of
MFPT was made in Kramers-like approximation by
using a reflecting boundary condition at the minimum of
the metastable state and absorption boundary condition
at maximum of the potential barrier height that separates
the stable and metastable states. A simple but very
general approach to the classical MFPT problem was
offered by Pontryagin et al. [7], which has been
extended by us for investigation MFPT of spin-crossover
solids.
© 2010, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
2. Model
Here we regard the macroscopic phenomenological
model of spin-crossover compounds in terms of a
relaxation rate in the mean field approach. With this aim,
it is necessary to admit one of the states of spin-
crossover compound as a metastable state, and the other
as a stable one. In our investigation, we did not focused
on specific spin-crossover compound, but due to more
widespread spin-crossover compounds with LS state as a
ground one, we have considered that paramagnetic HS
state is metastable. For the HS fractions , the static
relaxation equation first investigated by Hauser with the
Mainz group [8-10] is as follows:
Hn
( ) ( )
⎥
⎦
⎤
⎢
⎣
⎡
−= ∞ Tk
nEkTnk
B
Ha
HHL exp, , (1)
with
( ) ( ) ( ) HBaHa nTkTEnE α+= 0 , (2)
where is the high-temperature asymptotic of the
relaxation rates at the end of the process ( );
is the activation energy; is the Boltzmann
constant; is the temperature, and is the self-
acceleration factor of relaxation defined by the
cooperative coefficient and depending linearly on the
relative atomic concentration of the spin-crossover
element and is proportional to the inverse temperature
[11]. For simplicity, the activation energy has been
equaled to zero, so the relaxation equation (1) takes the
form:
∞k
0→Hn
( )0aE Bk
T ( )Tα
( ) ( )[ HHHL nTkTnk α−= ∞ exp, ] . (3)
The general form of macroscopic master equation
in condition of light irradiation can be expressed in
terms of photoexcitation and relaxation high-spin
fraction flows from HS to LS state and vice versa [4]:
relexc
H
dt
dn
Φ−Φ= . (4)
The form of photoexcitation and relaxation excΦ
relΦ -flows is as follows: Hn
( )
( ,exp
,10
HHrel
Hexc
nkn
nI
α−=Φ
−ω=Φ
∞ ) (5)
where ω0I is the probability per time unit for a LS
molecule to switch to the HS state, with the beam
intensity and a proportionality factor that includes
the absorption cross-section of the optically active
element (here the bulk absorbance is neglected). By
introducing a new variable and taking into
account a denotation
0I ω
∞=τ kt
∞ω=β kI /0 , it is easy to obtain the
final macroscopic master equation of the described
model with non-dimensional quantities:
( ) ( HHH
H nnn
d
nd
α−−−β=
τ
exp1 ) . (6)
From the stationary condition of the system when
relexc Φ=Φ , which is realized at the spin-crossover
point, one can estimate the possibility of existence of all
theoretical realizable stable and unstable states. A
comprehensive analysis of static and dynamical non-
equilibrium kinetics of spin-crossover compounds was
made by Gudyma in [12] and Varret in [13].
3. Macroscopic stationary dynamics
of spin-crossover compounds
Any real system being in contact with environment may
be modeled as a system that interacts with a heath bath.
To more accurately describe our model, we must take
into consideration an environmental noise influence. In
other words, the control parameter from Eq. (6) is not
constant, but fluctuates in time. Besides, it is necessary
to take into account the direct action of heat bath on the
system. Thus, it is need to assume that
β
( ) ( )tt ξ+β=β in
Eq. (6), which gives the following stochastic differential
equation in Stratonovich interpretation:
( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ),1
exp1ѓА
τντξτντξ
α
τ
++=+−+
+−−−=
HHH
HHH
H
ngnfn
nnn
d
dn
(7)
where the noise correlation functions are:
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ),2
,2
,2ѓМ
2
2
ττδχσετντξ
ττδετντν
ττδστξτ
−′=′
−′=′
−′=′
(8)
where σ and ε are the intensity of multiplicative noise
and additive noise, respectively, which correspond to the
influence of internal and external fluctuation on the
system; χ is the correlation coefficient between additive
and multiplicative noises. In a general case, external
358
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2010. V. 13, N 4. P. 357-362.
environmental fluctuations can influence on the internal
fluctuations but, for simplicity, we did not take this
effect into account. The analysis of states in spin-
crossover solids under cross-correlated noises was made
in the paper [6], which is based on the mathematical
tools developed in [14].
The Eq. (7) can be reduced to the equivalent
stochastic differential equation with one effective
multiplicative noise [16]:
( ) ( ) (τΞ+=
τ HH
H nGnf
d
nd
© 2010, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
) (9)
with the condition
( ) ( ) ( τ−τ′δ=τ′Ξτ 2Ξ ) . (10)
The effective noise amplitude ( )HnG is
determined as follows:
( ) ( ) ( ) 222 121σ ε+−χσε+−= HHH nnnG . (11)
For nonlinear stochastic differential equations, one
can obtain the Fokker-Planck equation corresponding to
the Langevin equation in the following form [15]:
( ) ( ) ( ) ( ) ( )HH
H
HH
H
H nPnG
n
nPnf
n
nP
2
2
2
1,
∂
∂
+
∂
∂
=
τ∂
τ∂ .
(12)
By solving the Fokker-Planck equation (12), we
found the probability of system location at the phase
space point. The stationary solution of this equation has
the Gibbs-Boltzmann form:
( ) ( )effHst UNnP −= exp , (13)
where is the normalization constant and N
( ) ( ) ( )
( )[ ]∫
∂
∂
−
−=
Hn
H
H
H
HH
eff nG
nG
n
nGnf
U
0
2 (14)
represents the effective dynamical system potential. The
extremes of this potential correspond to stationary fixed
points of system dynamics. Bistable range of system
with corresponding dynamic potentials at spinodal points
obtained from Eq. (6) with stationary equilibrium
condition and Eq. (14) are consequently shown in Fig. 1.
The intersection point between the curves excΦ
(photoexcited -flow marked as dotted curve in the
plots) and (relaxed -flow marked as a dashed
curve) from Fig. 1 represent the possible steady and
unsteady states, thus we defined the range of bistability.
The effective potential (marked as a solid curve on
all plots from Fig. 1) at the spinodal points (the critical
points of phase transition) between monostable-bistable
range and bistable-monostable range are reflected in
Fig. 1a and Fig. 1c, consequently. Shown in this figure is
light-induced optical hysteresis (LIOH) where light
irradiation in the first approach is the main control
parameter. Analysis showed that noise acting also may
lead to transition from one state to another. Non-
stationary overall studies of noise action on the system
are described in the following section.
Hn
relΦ Hn
effU
a
b
c
Fig. 1. Light-induced bistable range with the effective
dynamical potential : (a), (c) indicate the spinodal points
of one state – two states transition; (b) is the realization of
bistable state.
effU
359
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2010. V. 13, N 4. P. 357-362.
4. Noise driven dynamics of spin-crossover
compounds
It is generally known that the more suitable to find
solution of numerically ordinary differential equations
similar to Eq. (6) are the second-order Runge-Kutt type
methods. One of the main requirements in this case is
that the functions appearing in the equation are
differentiable to some required order. But, unfortunately,
if using the white-noise functions as in Eqs (7) and (8),
even for a single realization of the white-noise term,
these functions are highly irregular and non-
differentiable. So, the only non-rigorous approach via a
series of delta-functions spread all over the real axis can
be realized. To solve the presented stochastic differential
equation in the non-stationary case, we need to use more
powerful computational methods. These methods may
be developed from the second-order Runge-Kutt method
with suitable modification based on application of
integral algorithms described in [17]. It results in the
Milshtein method:
( ) ( ) ( )( ) ( )
( )( ) ( )( ) ( )( ){ } ( )( )
[ ],
2
1
2/3
2
2/1
hO
unG
n
nGnfh
unGhnn
H
H
HH
HHH
+
+⎥
⎦
⎤
⎢
⎣
⎡
∂
∂
++
++=′
ττττ
ττττ
(15)
where ; is the order of a time
step in the system dynamics.
( ) ( )∫
′
Ξ=
τ
τ
τ dssuh 2/1 h
We got the non-stationary solution of Eq. (7) from
explicit two-stage second-order Runge-Kutt method well
known as the Heun or explicit trapezoidal method
[17, 18]:
( )( ) ( ) ( )( )
( ) ( ) ( )( ) ( )( )[
( )
© 2010, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
]
( )( ) ( )( )[ .,,
2
,,
2
,,,τ
2/1
2/1
ynGnGuh
ynfnfhnn
nGuhnhfy
HH
HHHH
HH
+ττ′+ττ
τ
+
++ττ′+ττ+τ=τ′
τττ+τ=
]
(16)
The Eq. (16) is obtained from the Euler equation
with stochastic process in its turn derived from the
Milshtein method (15), if ( )( ){ } 0=
∂
∂ τH
H
nG
n
(it
corresponds to an additive noise case) [17]. The Euler
equation resulting from (16) is as follows:
( ) ( ) ( )( ) ( )( ) ( ) [ ]2/32/1τ hOtuhnGnfhnn HHHH +τ+τ+τ=′ .
(17)
The derived Euler equation in midpoint quadratures
has the following form:
( ) ( ) ( )( ) ( )( )[ ]
( ) ( )( ) ( )( )[ ].,,
2
,,
2
2/1
τ′τ′+ττ
τ
+
+τ′τ′+ττ+τ=τ′
HH
HHHH
nGnGuh
nfnfhnn
(18)
By replacing ( )τ′Hn on the right-handside in the
later equation by the predictor of the Euler method (17)
again, we arrive to the Heun equation (16). Trajectories
from Eq. (16) represent the solution of Eq. (9) with
stochastic fluctuation in time of HS fractions.
We carried out the simulation of HS fraction
dynamic for 100 trajectories with 50000 time step and
obtained the distribution of probability density functions
(pdf) of HS fraction for a non-stationary case. Shown in
Fig. 2 are the distributions of pdf in stationary and non-
stationary cases under noise action. From the plots, we
observe a qualitative difference in action of additive
noise and that of multiplicative noise: while the increase
in multiplicative noise intensity leads to the first-order
phase transition, the increase in additive noise intensity
only suppresses the potential barrier height, which
makes HS and LS states to be unrecognized and is
undesirable for sensor devices.
Shown in Fig. 3 are the examples of stochastic
trajectories for values of additive noise intensity
001.0=σ and 18.0=σ . The later value corresponds to
spinodal point between bistable-monostable regions. The
stochastic trajectories in Fig. 3 are only illustration of the
processes that take place in a noise-driven system. In
practice, it is impossible to determine the one isolated
trajectory, due to time-scale of occurred processes. So,
one can only detect the average value of stochastic
fluctuations, which is important to know for designing
stable spin-crossover sensor devices.
5. Lifetime of the metastable state
in the mean field approach
As we have seen above, the dynamic of the HS fraction in
spin-crossover compounds under environmental noise
action is modeled as fluctuation in time of Brownian
particle. After the pioneering contribution by Kramers,
well described in [19], the escape problem of Brownian
particle became an important subject of studies in
nonlinear stochastic systems. The most common analysis
of metastable state decay is based on the MFPT technique
in a Kramer-like approximation described in general
aspects by Pontryagin in [7]. The escape time from a
metastable state in the stationary case was studied in
[12, 20]. The general expression for MFPT is as follows:
( ) ( )
( )
( )[ ]∫ ∫
ψ
ψ
=→
b
n
x
a
eff
rand
H
H
xG
x
x
dxnaT 2 , (19)
where
( ) ( )
( )[ ] ⎥
⎦
⎤
⎢
⎣
⎡
′
′
′
=ψ ∫
z
a
eff
rand
zd
zG
zfz 2
det2exp . (20)
Here, we supplemented theoretically obtained
MFPT by computing simulation. The results are shown
360
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2010. V. 13, N 4. P. 357-362.
in Fig. 4. Shown in the figure is estimation of MFPT
between stationary and non-stationary cases (see the
normalized time values), so the intersection of time
curves is not necessary to take into consideration.
a
b
c
d
Fig. 2. Additive (a), (b), (c) and multiplicative (d) noise actions
on the probability density function distribution. Simulation was
carried out for 100 trajectories and 50000 time steps for each
trajectory.
a
b
Fig. 3. The examples of stochastic trajectories for
multiplicative noise 001.0=σ (a) and (b). The
values of additive noise in both cases have the same magnitude
18.0=σ
1.0=ε .
Fig. 4. Normalized MFPT for given values of the additive
noise intensity. The value of multiplicative noise is constant
( 1.0=σ ).
© 2010, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
MFPT in Fig. 4 is corresponded by the bistable
potential in Fig. 1c with a reflecting boundary condition
at the point 16.0=a , and absorption boundary condition
at the point 4.0=b . These points indicate the range of
metastable state. Given in Table 1 are the values of
MFPT for the stationary case. From the type of change
361
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2010. V. 13, N 4. P. 357-362.
in values in Table 1, we can conclude that the increase in
the additive noise intensity decreases the lifetime of
metastable states. This fact is not favorable for
manufacturing data storage systems.
Table 1. Computed stationary values of . ( )HnaT →
Hn T (a → nH) for
ε = 0.01
T (a → nH) for
ε = 0.3
0.16 690.85 39.47
0.2 671.03 38.02
0.3 501.91 27.1
0.4 109.95 5.58
© 2010, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
6. Concluding remarks
Using the Fokker-Planck equation and its solutions, we
studied the noise-dependence properties of the critical
behavior of the spin-crossover compound based on
extended phenomenological model for spin-crossover
compound kinetics in the noise case action. The
investigated bistable range may be induced by light
irradiation that leads to occurrence of light-induced
optical hysteresis. Non-equilibrium dynamical
investigation in terms of heat bath firstly developed in
[12] has been extended here to the case of two noises
action. Dynamical study of the presented noisy model
has been made in the frame of Heun algorithm also
known as explicit trapezoidal method. The probability
density function distribution for stationary and non-
stationary dynamics has been obtained. Also, we have
shown that the multiplicative noise influence may lead to
transition from the metastable to stable state. Action of
additive noise is reduced to suppression of the potential
barrier height. Increasing the noise intensity provokes a
decrease in the decay time of metastable states.
References
1. Spin Crossover in Transition Metal Compounds 1,
2, 3, Eds. P. Gütlich, H.A. Goodwin. Springer-
Verlag, Berlin, 2004.
2. J.A. Real, A.B. Gaspar, M.C. Muñoz, P. Gütlich,
V. Ksenofontov and H. Spiering, Bipyrimidine-
bridged dinuclear iron(II) spin crossover
compounds, in: Spin Crossover in Transition Metal
Compounds 1, Eds. P. Gütlich, H.A. Goodwin,
p. 167-193. Springer-Verlag, Berlin, 2004.
3. S. Decurtins, P. Gütlich, C.P. Köhler, H. Spiering and
A. Hauser, Light-induced excited spin state trapping
in a transition-metal complex: The hexa-1-propyl-
tetrazole-iron (II) tetrafluoroborate spin-crossover
system // Chem. Phys. Lett. 105(1), p. 1-4 (1984).
4. A. Desaix, O. Roubeau, J. Jeftic, J.G. Haasnoot,
K. Boukheddaden, E. Codjovi, J. Linarès, M. No-
guès, and F. Varret, Light-induced bistability in spin
transition solids leading to thermal and optical hy-
steresis // Europ. Phys. J. B 6(2), p. 183-193 (1998).
5. C. Enachescu, R. Tanasa, A. Stancu, G. Chastanet,
J.-F. Létard, J. Linares, and F. Varret, Rate-
dependent light-induced thermal hysteresis of
[Fe(PM-BiA)2(NCS)2] spin transition complex // J.
Appl. Phys. 99(8), 08J504 (2006).
6. Iu.V. Gudyma, A.Iu. Maksymov, Theoretical
analysis of the states of spin-crossover solids under
cross-correlated noises // Physica B 405(11),
p. 2534-2537 (2010).
7. L.S. Pontryagin, A.A. Andronov, A.A. Vitt //
Zhurnal Eksp. Theor. Fiziki 3, p. 165 (1933) (in
Russian); English transl. in: Noise in Nonlinear
Dynamical Systems, Eds. F. Moss and P.V.E.
McClintock. Cambridge University Press,
Cambridge, 1989.
8. A. Hauser, P. Gutlich and H. Spiering, High-spin →
low-spin relaxation kinetics and cooperative effects
in the [Fe(ptz)6](BF4)2 and [Zn1-xFex(ptz)6](BF4)2
(ptz = 1-propyltetrazole) spin-crossover systems //
Inorg. Chem. 25(23), p. 4245-4248 (1986).
9. A. Hauser, Intersystem crossing in the
[Fe(ptz)6](BF4)2 spin crossover system (ptz =1-
propyltetrazole) // J. Chem. Phys. 94(4), p. 2741-
2749 (1991).
10. A. Hauser, Cooperative effects on the HS→LS
relaxation in the [Fe(ptz)6](BF4)2 spin-crossover
system // Chem. Phys. Lett. 192(1), p. 65-70 (1992).
11. F. Varret, K. Boukheddaden, J. Jeftic and
O. Roubeau, A macroscopic approach to the light-
induced instability of cooperative photo-switchable
systems // Mol. Cryst. Liquid Cryst. 335, p. 561-
572 (1999).
12. Yu. Gudyma and O. Semenko, Non-equilibrium
kinetics in spin-crossover compounds // Phys.
status solidi (b) 241(2), p. 370-376 (2004).
13. C. Enachescu, J. Linarès and F. Varret, Comparison
of static and light-induced thermal hysteresis of a
spin-crossover solid, in a mean-field approach //
J. Phys.: Condens. Matter 13(11), p. 2481-2497
(2001).
14. Yu. Gudyma and B. Ivans’kii, Behavior of
asymmetric bistable system under influence of
cross-correlated noises // Mod. Phys. Lett. B
20(20), p. 1233-1239 (2005).
15. N.G. van Kampen, Stochastic Processes in Physics
and Chemistry. North-Holland, Amsterdam, 1980.
16. D.-j. Wu, L. Cao, S.-z. Ke, Bistable kinetic model
driven by correlated noises: Steady-state analysis //
Phys. Rev. E 50(4), p. 2496-2502 (1994).
17. M.S. Miguel and R. Toral, Stochastic effects in
physical systems, in: Instabilities and
Nonequilibrium Structures VI, Eds. E. Tirapegui,
J. Martínez and R. Tiemann, p. 35-130. Kluwer
Academic Publ., Dordrecht, 2000.
18. U.M. Ascher and L.R. Petzold, Computer Methods
for Ordinary Differential Equations and Differential
Algebraic Equations. SIAM, Philadelphia, 1998.
362
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2010. V. 13, N 4. P. 357-362.
19. P. Hanggi, P. Talkner and M. Borkovec, Reaction-
rate theory: fifty years after Kramers // Rev. Mod.
Phys. 62(2), p. 251-341 (1990).
20. Iu. Gudyma, A. Maksymov, and C. Enachescu,
Decay of a metastable high-spin state in spin-
crossover compounds: mean first passage time
analysis // Europ. Phys. J. B (2010), (in press).
© 2010, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
363
|