The phase transitions character and microscopic models for Sn₂P₂S(Se)₆ ferroelectrics
The paper considers an example of concrete objects, i.e. ferroelectrics of Sn₂P₂S₆ family, for which, according to general estimations, a correlation between the shape of a phase diagram (the presence of a Lifshitz point and of a tricritical point) and a crossover “displacive to order/disorder” i...
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Vysochanskii, Yu. Drobnich, A. 2017-06-12T17:02:03Z 2017-06-12T17:02:03Z 2002 The phase transitions character and microscopic models for Sn₂P₂S(Se)₆ ferroelectrics / Yu. Vysochanskii, A. Drobnich // Condensed Matter Physics. — 2002. — Т. 5, № 4(32). — С. 669-683. — Бібліогр.: 26 назв. — англ. 1607-324X PACS: 64.60.Cn DOI:10.5488/CMP.5.4.669 https://nasplib.isofts.kiev.ua/handle/123456789/120678 The paper considers an example of concrete objects, i.e. ferroelectrics of Sn₂P₂S₆ family, for which, according to general estimations, a correlation between the shape of a phase diagram (the presence of a Lifshitz point and of a tricritical point) and a crossover “displacive to order/disorder” is observed. Simulations by means of microscopic models, which most adequately represent a dipole ordering in these crystals, are analyzed. A dimension dependence of a ferroelectric phase transition temperature for microcrystals of Sn₂P₂S₆ compound is considered. В статті досліджена кореляція між особливостями фазової діаграми (наявність на діаграмі точки Ліфшиця та трикритичної точки) та кросовером характеру фазових переходів від типу зміщення до типу лад-безлад на прикладі конкретних об’єктів, а саме сегнетоелектриків родини Sn₂P₂S₆. Аналізуються результати комп’ютерного моделювання з використанням мікроскопічних моделей, що найбільш адекватно відображають дипольне впорядкування в цих кристалах. Розглядається ефект розмірної залежності температури сег-нетоелектричного фазового переходу для мікрокристалів сполуки Sn₂P₂S₆. We appreciate Prof. R. Levitskii’s stimulating discussions and his support at the start of MC simulations for the models investigated. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics The phase transitions character and microscopic models for Sn₂P₂S(Se)₆ ferroelectrics Характер фазового переходу та мікроскопічні моделі для сегнетоелектриків Sn₂P₂S(Se)₆ В статті досліджена кореляція між особливост Article published earlier |
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The phase transitions character and microscopic models for Sn₂P₂S(Se)₆ ferroelectrics |
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The phase transitions character and microscopic models for Sn₂P₂S(Se)₆ ferroelectrics Vysochanskii, Yu. Drobnich, A. |
| title_short |
The phase transitions character and microscopic models for Sn₂P₂S(Se)₆ ferroelectrics |
| title_full |
The phase transitions character and microscopic models for Sn₂P₂S(Se)₆ ferroelectrics |
| title_fullStr |
The phase transitions character and microscopic models for Sn₂P₂S(Se)₆ ferroelectrics |
| title_full_unstemmed |
The phase transitions character and microscopic models for Sn₂P₂S(Se)₆ ferroelectrics |
| title_sort |
phase transitions character and microscopic models for sn₂p₂s(se)₆ ferroelectrics |
| author |
Vysochanskii, Yu. Drobnich, A. |
| author_facet |
Vysochanskii, Yu. Drobnich, A. |
| publishDate |
2002 |
| language |
English |
| container_title |
Condensed Matter Physics |
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Інститут фізики конденсованих систем НАН України |
| format |
Article |
| title_alt |
Характер фазового переходу та мікроскопічні моделі для сегнетоелектриків Sn₂P₂S(Se)₆ В статті досліджена кореляція між особливост |
| description |
The paper considers an example of concrete objects, i.e. ferroelectrics of
Sn₂P₂S₆ family, for which, according to general estimations, a correlation
between the shape of a phase diagram (the presence of a Lifshitz point
and of a tricritical point) and a crossover “displacive to order/disorder” is
observed. Simulations by means of microscopic models, which most adequately
represent a dipole ordering in these crystals, are analyzed. A dimension
dependence of a ferroelectric phase transition temperature for
microcrystals of Sn₂P₂S₆ compound is considered.
В статті досліджена кореляція між особливостями фазової діаграми (наявність на діаграмі точки Ліфшиця та трикритичної точки) та кросовером характеру фазових переходів від типу зміщення до типу лад-безлад на прикладі конкретних об’єктів, а саме сегнетоелектриків родини Sn₂P₂S₆. Аналізуються результати комп’ютерного моделювання з використанням мікроскопічних моделей, що найбільш адекватно відображають дипольне впорядкування в цих кристалах. Розглядається ефект розмірної залежності температури сег-нетоелектричного фазового переходу для мікрокристалів сполуки Sn₂P₂S₆.
|
| issn |
1607-324X |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/120678 |
| citation_txt |
The phase transitions character and microscopic models for Sn₂P₂S(Se)₆ ferroelectrics / Yu. Vysochanskii, A. Drobnich // Condensed Matter Physics. — 2002. — Т. 5, № 4(32). — С. 669-683. — Бібліогр.: 26 назв. — англ. |
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2025-11-25T22:33:24Z |
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2025-11-25T22:33:24Z |
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| fulltext |
Condensed Matter Physics, 2002, Vol. 5, No. 4(32), pp. 669–683
The phase transitions character and
microscopic models for Sn2P2S(Se)6
ferroelectrics
Yu.Vysochanskii, A.Drobnich
Institute for Solid State Physics and Chemistry,
Uzhgorod National University,
46 Pidgirna Str., 88000 Uzhgorod, Ukraine
Received August 30, 2002
The paper considers an example of concrete objects, i.e. ferroelectrics of
Sn2P2S6 family, for which, according to general estimations, a correlation
between the shape of a phase diagram (the presence of a Lifshitz point
and of a tricritical point) and a crossover “displacive to order/disorder” is
observed. Simulations by means of microscopic models, which most ade-
quately represent a dipole ordering in these crystals, are analyzed. A di-
mension dependence of a ferroelectric phase transition temperature for
microcrystals of Sn2P2S6 compound is considered.
Key words: ferroelectrics, displacive vs. order/disorder crossover, Lifshitz
point, incommensurate phase, microscopic models, Monte Carlo
simulations
PACS: 64.60.Cn
1. Introduction
Structure phase transitions (PT) obviously are classified, from experimental
point of view, as displacive type or order/disorder type. The observation of under-
damped soft mode is considered to be an experimental evidence of a displacive type
of transition, while in the case of order/disorder transition the transformation occurs
from the equal probable distribution of some atoms between several positions to a
more ordered configuration with a similar set of atomic positions. The self-consistent
phonon approximation, or pseudo-spin models, consecutively, can serve as a starting
scheme of the theoretical description in each case [1–4].
From microscopic point of view, however, a displacive type or order/disorder type
PT are considered to be two opposite limiting mathematical cases for the energetic
system, which is never realized in real systems. Real materials show intermediate
behavior, and their pertaining to one of the two limiting cases can be used as “zero”
approximation. The deviation from both of these limiting schemes is caused by
c© Yu.Vysochanskii, A.Drobnich 669
Yu.Vysochanskii, A.Drobnich
the peculiarities of the microscopic energy of a specific system. It is desirable to
find a certain “key” to the decoding, at least approximate, of the relation between
microscopic parameters of a real system with the character of the observed PT in the
system investigated. Herein we can systematize a nonuniversal behavior of crystals’
properties at the phase PT, i. e. their behavior off the critical region – in the most
convenient (for majority cases) temperature interval for experimental investigations.
Experimental data for the family of Sn2P2S(Se)6 crystals of uniaxial ferroelectrics
motivate the study of this system using different theoretical approaches. In a com-
position – temperature phase diagram of solid solutions Sn2P2(SexS1−x)6 of these
crystals, the Lifshitz point is present [5,6] at which the paraelectric, ferroelectric and
incommensurate (IC) phases meet. There is also a virtual tricritical point which to-
gether with long range dipole interaction determines a specific critical behavior of
thermodynamic properties in the region of phase transitions [7,8]. Furthermore, this
family displays the crossover effects from displacive to order/disorder type of phase
transition [9,10]. The investigations of Sn2P2S6 ceramics [11] have shown a decrease
of the PT temperature at lowering of the dimensions of microcrystals.
To compare different models used for the discussed ferroelectrics analysis is the
aim of this paper. Thereby, arguments were found for performing calculations by
Monte Carlo (MC) in a rather simple discrete Ising-like model of dimension depen-
dence for phase transitions.
2. Thermodynamic and dynamic properties of Sn2P2S(Se)6 fer-
roelectrics: illustration of polycritical points and displacive-
order/disorder crossover coincidence on the phase diagram
Figure 1. Phase diagram of the solid so-
lutions Sn2P2(SexS1−x)6. LP – Lifshitz
point; dashed line – second order phase
transitions; continuous line – first order
phase transitions [5,6].
From the set of experimental da-
ta it follows that in the Sn2P2S6 and
Sn2P2Se6 ferroelectrics, the phase tran-
sitions are placed in the region of
crossover from displacive type to or-
der/disorder type. Herein, by substitu-
tion of sulfur by selenium, the PT char-
acter is approached to order/disorder
type. Let’s see the experimental data
and general estimations based on them.
On the temperature-concentration
diagram for Sn2P2(SexS1−x)6 solid solu-
tions the Lifshitz point is reached (fig-
ure 1) – ferroelectric PT of the sec-
ond order (for Sn2P2S6 T0 ∼ 337 K) at
x > 0.28 splits into the second and in-
to the first order transitions, that lim-
it the incommensurate (IC) phase (for
670
Microscopic models for Sn2P2S(Se)6
Figure 2. Soft mode spectral line in the
ferroelectric phase of Sn2P2S6 at: 1 – 80 K;
2 – 233 K; 3 – 293 K; 4 – 313 K. Solid line
– experiment, dashed line – Lorentz con-
tour fitting, points – fitting by a response
function with account of an interaction be-
tween oscillator and relaxator [6].
Figure 3. Soft mode behavior for
the ferroelectric phase of Sn2P2Se6
from a neutron scattering data at the
(3 0 1) Brillouin zone center. The full
lines are damped harmonic oscillator
fits [13].
Sn2P2Se6 Ti ∼ 221 K, Tc ∼ 193 K). The modulation period of spontaneous polariza-
tion in the IC phase equals 12–15 primitive cells [5,6].
Figure 4. The model simulation of the
incommensurate instability on the neu-
tron scattering data for Sn2P2Se6. 1 –
235, 2 – 205, 3 – 180, 4 – 171 K [13].
For Sn2P2S6 and Sn2P2Se6 in the Ra-
man scattering spectra (figure 2) [6],
and in the neutron scattering spectra
(figures 3 and 4) [12,13], the soft op-
tical mode is observed. According to
the temperature dependence of a dielec-
tric susceptibility [8] in the paraelectric
phase of these compounds (figure 5) the
Curie-Weiss constant is in the range of
C = 0.5−0.7·105 K. Such characteristics
are peculiar for the displacive type PT.
At the same time, for Sn2P2S6 in the
Brillouin scattering [6] and in neutron
scattering [12] spectra, a central peak is
observed which testifies to the presence
671
Yu.Vysochanskii, A.Drobnich
Figure 5. The calculated concentrational evolution of dielectric constant temper-
ature dependence for the crystals Sn2P2(SexS1−x)6: 1− x = 1, 2 − 0.9, 3 − 0.8 [8]
(a) in the mean field approximation and (b) determined experimentally.
of a relaxation component of critical dynamics. This component is also observed in
a spectral response of the soft phonons both for Sn2P2S6 and Sn2P2Se6 (figures 2
and 3). Calorimetric studies [14] of both crystals reveal a large entropy of the phase
transitions: 8.6 K−1mol−1 for the ferroelectric-paraelectric transition in Sn2P2S6, and
8.5 and 1.5 K−1mol−1 for the ferroelectric-IC transition and IC phase-paraelectric
phase transitions, correspondingly, in case of Sn2P2Se6 (figure 6). These entropy
values are close to 2R ln 2 = 11.4 K−1mol−1 which corresponds to a positional dis-
ordering of Sn2+ cations.
Figure 6. The heat capacity and the transition entropy for Sn2P2Se6 as a function
of temperature [14].
672
Microscopic models for Sn2P2S(Se)6
Figure 7. The temperature dependen-
cies of the Debye-Waller factor for
Sn2P2Se6 (full circles) and Sn2P2S6
(open squares) from the 119Sn Moss-
bauer spectroscopy [15].
According to the investigations [15]
of Mossbauer spectra of 119Sn isotope,
the temperature dependence of a mean-
square amplitude of heat shifts of fer-
roactive Sn atoms under the heating
in the ferroelectric phase deviates from
a linear, and in paraelectric phase it
reaches 0.34 Å (figure 7). These data
confirm the speculation about the possi-
ble existence of a two-well potential for
Sn atoms in the paraelectric phase of
the crystals studied. The X-ray struc-
tural studies also confirm a substantial
increase of the mean-square shifts for
Sn atoms at the transition into para-
electric phase, both in Sn2P2S6 [16] and
Sn2P2Se6 [17,18] (figure 8).
Figure 8. The temperature dependencies of the parameters of isotropic atomic
heat displacements for Sn2+ cations in Sn2P2Se6 and Sn2P2S6 crystals on X-ray
data [17].
The comparison of the structures of paraelectric and ferroelectric phases for
Sn2P2Se6 crystals was made in [17] and [18] by means of a detailed analysis of
X-ray data. According to [18] the change of the crystal structure at ferroelectric-
paraelectric transition is due to the shift of the Sn atoms to centrosymmetric posi-
tions. Heat vibrations of Sn atoms in the paraelectric phase are very anharmonic.
However, the attempt made in [18] to describe the structure of paraelectric phase
based on the assumption of the presence of Sn atoms disordering in two positions,
failed to improve the quality of comparison of the X-ray diffraction data. In the
mean time, the X-ray diffraction data obtained in [17] had been analyzed more pre-
cisely in the model assuming Sn atoms disordering in two positions, separated by
673
Yu.Vysochanskii, A.Drobnich
0.34 Å, and the shift of the average distance between these positions separated by
0.31 Å and by 0.28 Å, correspondingly, for two Sn pairs relative to the positions of
Sn atoms in the ferroelectric phase at 173 K. Thus, direct data on the structure of
Sn2P2S6 and Sn2P2Se6 crystals testify to a substantially anharmonic potential for
Sn2+ cations in paraelectric phase. However, certain proofs of a two-well potential
have been obtained only for Sn2P2Se6.
While analysing the experimental data, it is convenient to start with the simplest
approximations. Such approximations serve as a background of the so-called Φ4
model [1–3]. This model can be generalized by including a many-component order
parameter, taking into account the interactions with different radii, and also, if it
is needed, long-range dipole-dipole interactions, by accounting for the interaction
with elastic deformations. The temperature behavior of Φ4 model is determined by
E0 and J parameters, that measure the depth of the potential well and the strength
of interaction between the neighbors. The behavior of the system is determined by
relation ε = J/E0: at ε � 1, the system approaches the order/disorder regime and
its behavior is adequate to the behavior of a lattice of local spins. At the opposite
border of large values of ε, the picture of the soft phonons becomes adequate and
the approximation of self-consistent phonons becomes exact at ε → ∞. In the mean
field approach, the crossover border was determined ε = 1/3 [3].
For the estimation of the PT type in specific crystals it is convenient to use the
empirical factor proposed in [3]:
R = (2∆S/kB)−0.5, (1)
which is determined by entropy and Boltsmann constant. The relation between the
factor R and the characteristic ε (figure 9) was determined by numerical calculation.
Figure 9. Temperature dependence of
the empiric factor R (was determined by
the relation (1)) in Φ4 model vs. the dis-
placive degree ε [3].
In the framework of the simplest
one-ion model [9] while classifying the
PT character in specific objects it is also
convenient to determine the relation be-
tween the heat energy kT0 (at tempera-
ture of phase transition T0) and the bar-
rier energy Um for a two-well potential of
a ferroactive particle. The change of the
energy Uf of interaction of the neigh-
boring elementary cells, which appears
at the flipping of the ferroactive cation
across the potential barrier within one of
these cells, is an important characteris-
tic. For PT of displacive type, kT0 > Um
and Uf > Um. For order/disorder type
PT, the opposite relations are expected:
kT0 < Um and Uf < Um.
As follows from the estimations performed in [9], that are based on the thermo-
dynamic and spectroscopic data, for Sn2P2S6 crystal, the energy of the barrier for
674
Microscopic models for Sn2P2S(Se)6
the normal coordinate of the soft phonon Um = 4 · 10−21 J, which is close to kT0 =
6 · 10−21 J at T0 = 337 K. Thus, the ferroelectric crystal Sn2P2S6 has the phase
transition in the region of crossover from displacive type to order/disorder type.
Available experimental data are not sufficient to perform similar estimations for
Sn2P2Se6 crystal. Not mentioning the possibility of using the relation (1) to estimate
the PT character in Sn2P2S6 and Sn2P2Se6 crystals. According to the calorimetric
data [14] for these crystals, the entropy of the transition from paraelectric phase to
ferroelectric phase equals 8.6 and 10 JK−1mol−1. Herein, the empiric factor R = 0.69
for Sn2P2S6 and R = 0.64 for Sn2P2Se6. From the dependence R vs. log ε (figure 9)
we can find that ε 6 1/3 for sulfide and selenide compounds. Remember that in
the mean-field approach, the border of crossover from displacive type of PT to
order/disorder type was determined ε ∼ 1/3 [3].
For Sn2P2Se6, compound the presence of the intermediate incommensurate phase
itself is probably more important for the actual soft mode vs. order/disorder behav-
ior. To consider this situation, a detailed analysis of the order/disorder vs. soft mode
crossover in the DIFFOUR (Discrete Frustrated Φ4) model was performed in papers
[19,20].
The crossover situation for Sn2P2S(Se)6 ferroelectrics actually implies that these
compounds should reveal a soft mode, which, however, does not soften completely,
which is really observed in the experiment (figures 2 and 3) [6,12,13]. In this regime,
the ratio r of slopes of the temperature dependencies of the soft mode frequency
could be high enough, depending on the actual form of the phonon dispersion. This
agrees with scattering data value r ∼ −7 [12,13] experimentally estimated from in-
elastic neutron, and is also responsible for the large value of the ratio of Curie-Weiss
constants, that is observed experimentally (figure 5) [8]. It can be also claimed that
the quartic coefficient of the Landau potential for ferroelectric Sn2P2S6 should be
considered as temperature dependent on the temperature scales comparable with
T0. The reduction of its value at T0 should be significant, so that the sixth-order
term in the Landau expansion may really play an important role in the vicinity of
T0 [3]. This suggests that the crossover behavior could be responsible for a non-
standard temperature dependence of the spontaneous polarization. Their saturation
is achieved faster than it would be expected for a usual second order displacive
phase transition. According to the simulations [9], an effective value of index β in a
dependence PS ∼ (T0 − T )β should be smaller than 0.5.
Hence, based on the review of the dynamic and thermodynamic properties of
Sn2P2S6 family ferroelectrics and taking into account their analysis in the frame-
work of simple models we can focus our attention on the presence of the following
interesting correlations. Approaching the Lifshitz point on the state diagram of the
solid solutions Sn2P2(SexS1−x)6 by increasing x to xLP = 0.28 determines the in-
crease of the temperature interval near the phase transition temperature where the
effect of strongly developed fluctuations of the order parameter is important. This
determines the deviation of the thermodynamic properties temperature behavior
from the one expected in the mean field approaches. Such deviation, particularly,
appears in the decreasing of the effective value of critical index β for temperature de-
675
Yu.Vysochanskii, A.Drobnich
pendence of the order parameter. At the rise of selenium concentration, the PT from
paraelectric phase to ferroelectric phase approaches the tricritical point at x = 0.6
[7,8]. This also assists in a considerable decrease of the named index β.
At the same time, it is known that at approaching the LP, the dispersion of
the rigidity for the order parameter fluctuations in the direction of modulation de-
creases, i.e., the intercell interactions decrease. On the other hand, the coefficient
at the invariant of the fourth power in the thermodynamic potential decreases at
approaching the tricritical point. Thereby, obviously, the coefficient at the addend
of the same power in the configuration energy function also decreases i.e., the ap-
pearance of local anharmonism changes. The latter fact determines the increase of
the potential barrier value E0 for the local two-well potential of the ferroactive par-
ticle. Thus, at approaching LP and TCP on the state diagram of the solid solutions
Sn2P2(SexS1−x)6, a simultaneous intercell interaction decreases and the height of
potential barrier in the local potential increases, which determines the change of
the phase transition character from displacive type to order-disorder type. Such a
crossover is connected with the deviation from simple Φ4 model. The deviation tes-
tify, as was shown by numerical analysis [3], that at a successive consideration of the
thermodynamic potential it is needed to take into account the nonlinearity in the
temperature dependence of the coefficient at the quadratic invariant, as well as the
presence of the temperature dependence for the coefficient at the quartic invariant.
Accounting for these dependencies also explains the decrease of the critical index
for the temperature dependence of the order parameter at the change of the second
order phase transition character in the direction to order/disorder type.
3. Simulations based on the microscopic models
Models based on the discrete structure of the lattice can provide a way to achieve
a physical understanding of the processes on the atomic level of the crystal, which
goes beyond a mere phenomenological description. Such microscopic models can
be handled analytically or numerically. The models, which have been closely stud-
ied with respect to the application to experimentally investigated substances, are
discussed in [4]. Each model variable – either continuous as in the case of the DIF-
FOUR models [4,19,20] or discrete as in the case of pseudo spin models [4] – usually
describes one degree of freedom per lattice site.
The experimental phase diagrams, however, are spanned by external quantities
like temperature, concentration, pressure or applied fields. Thus, it is necessary to
find a mapping between the experimental quantities and the theoretical parameters
in order to compare the experimental and the theoretical phase diagrams. In what
follows we will present a procedure for transforming a two-dimensional theoreti-
cal phase diagram from model parameters to temperature-concentration (pressure)
variables.
A generalization of the simplest possible discrete DIFFOUR model with the
Lifshitz point on the phase diagram was made in papers [19,20] with the aim to
explain the experimental data for Sn2P2(SexS1−x)6 crystals. Herein the phase dia-
676
Microscopic models for Sn2P2S(Se)6
gram for 0 6 x 6 1 and the temperature dependence of the modulation wave vector
for Sn2P2Se6 were calculated. In order to satisfactorily reproduce the experimental
data, a nonlinear coupling with nearest neighbors is taken into account [20]. The
incommensurability in this model arises due to the frustration between the nearest
neighbor and the next-nearest neighbor interactions. However, the tricriticality is
not taken into account in the generalized DIFFOUR model [19,20]. Of course, the
inclusion of a sixth order term to the potential energy that is needed to describe the
herein important tricritical fluctuations, determines no linear mapping of the calcu-
lated phase diagram on the experimental T−x diagram. Thereby, more experimental
data, in particular for the modulation wave vector in the interval 0.28 6 x 6 1, are
needed for this mapping.
In papers [10,21] to explain the specificity of the phase diagrams of Sn2P2S(Se)6
ferroelectrics there was proposed a dipole Ising model of the PT mechanisms using
the calculations according to the standard procedure of Monte Carlo [22]. This mod-
el was developed on the basis of structural data of these crystals in the paraelectic
phase. The basis of the model is as follows: a dipole, which is parallel to the crysta-
lographic axis a and is located in Sn2+ position in the paraelectic phase, corresponds
to each of the four atoms of ferroactive Sn2+ sublattice. To consider the dipole long
range interaction around the dipole, an area of simple form is singled out. Within
this area the interaction between the dipoles is calculated precisely. The interaction
with the other dipoles is considered by introducing a local field. The macroscopic
sample was taken as a limited Ising lattice with the imposed periodical boundary
conditions (Monte Carlo cell). For a better coincidence with the experiment, an ad-
ditional parameter was introduced to the basic model, i.e., correction for the short
range interaction that encapsulates other (except the dipole) short range forces. The
mentioned correction could be considered as a degree of deviation of a real object
from the dipole Ising model that is adequate to the case of an ideal order/disorder
PT. Using the calculations made by MC techniques it has been found that when
certain values of short range interaction coefficient for Sn2P2S6 and Sn2P2Se6 are
selected (the correction is lower for the latter compound, which shows a more pro-
nounced order/disorder character of the PT), the temperatures of PT’s are as fol-
lows: for Sn2P2S6 model T0 ∼ 10.3; for Sn2P2Se6 model Tc ∼ 8.7, Ti ∼ 9.7 (at the
effective temperatures). These data are in satisfactory agreement with the results of
the experimental diagram (figure 1). From the dipole model point of view, the pe-
culiarities of this diagram – the existence of incommensurate phase in Sn2P2Se6 and
its absence in Sn2P2S6 crystal – are based on the difference of geometric locations
of the ferroactive sublattice atoms.
We should note that the application of the dipole model of the Ising type provides
the PT modelling of the order/disorder type, whereas the real crystals demonstrate
a somewhat intermediate character of PT – between order/disorder and displacive
type. Furthermore, it was assumed that a statistical approach to the evaluation of
thermodynamic functions and a phase diagram of the system permits to disregard
the peculiarities of the dynamic motion of separate atoms.
Mean-field models were also used for the analysis of the properties of
677
Yu.Vysochanskii, A.Drobnich
Sn2P2(SexS1−x)6 ferroelectrics [7,8,23]. Taking into account the interactions between
spontaneous polarization and elastic deformations, as well as taking into account the
higher harmonics of the modulation of the polarization in the IC phase, the behavior
of the thermodynamic functions (spontaneous polarization, dielectric susceptibility,
heat capacity) along the phase diagram was described [7,8]. It was shown [23] that
the temperature dependence of the wave vector modulation can be correctly de-
scribed simultaneously with the anomalous heat capacity only when the nonlinear
properties of the order parameter distribution are taken into account.
For a type II incommensurate phase, that was determined by symmetry for
proper Sn2P2(SexS1−x)6 ferroelectrics, the PT instability appears to be off (but still
near) the Brillouin zone center. A coupling of transverse optic (TO) and transverse
acoustic (TA) phonon branches lies at the origin of this IC instability, that was
clearly demonstrated by neutron scattering data (figure 4) [13]. We wonder which
is more important for the mixed displacive-order/disorder type of the PT in the
crystals considered, a dipolar coupling among the displacements of Sn2+ cations,
or the TO-TA coupling? This question was considered in the paper [24] within a
model, which permits to comment on the physical relevance of both these approaches
to Sn2P2S6. It was concluded, that the origin of the IC instability and the value
of the wave vector modulation at Ti is indeed uniquely determined by the dipolar
interactions between Sn2+ cations. Anions [P2S(Se)
6
]4− have to be taken into account
only if the low frequency phonon dispersion (TO-TA coupling) is also to be described.
So, we could again conclude that the simplest dipole Ising-like models could be
used for the simulation of the phase diagram and of some thermodynamic proper-
ties of Sn2P2S6 type ferroelectrics. Further on we will present the ANNNI adapted
model for the simulation of the PT temperature dependence as the function of the
dimension of microcrystals. Such an analysis is quite interesting because according
to experimental data [11] on the dielectric properties of Sn2P2S6 ceramics at the
average dimensions of microcrystals about 1 µm, the ferroelectric PT temperature
is already about 325 K, instead of T0 ∼ 337 K for the bulk samples of Sn2P2S6.
The described numerical experiments were executed over the dipole model of
Sn2P2S(Se)6 crystals with four pseudo-dipoles in the primitive cell [10,21,25]. The
long range character of dipole interaction was taken into account under the clas-
sic scheme [25]. The calculations conducted have shown a good consent with the
phase diagram obtained experimentally for a case, when the sample of macroscopic
dimensions was considered. An essential drawback of this model is the failure at
attempting to simulate microscopic samples. At the removal of a Lorentz local field
by a ground state of the model we observe a modulated phase.
To study the PT’s in microcrystals, an ANNNI (Axial Next Nearest Neighbors
Ising) model is widely used [4,26]. In the framework of this model there were demon-
strated such effects as the relation of temperature of bulk PT between ferroelectrics
and paraelectric phases to the dimensions of the model lattices, the influence of a
surface of a microscopic sample on phase changes. The basic sense of returning to
ANNNI model in the given paper is binding the known abstract model to the actual
phase diagram of microcrystals such as Sn2P2S6.
678
Microscopic models for Sn2P2S(Se)6
Figure 10. The projection of the crystal structure of Sn2P2S6 in the ferroelectric
phase on the plane (010). The altitude of atoms is indicated in thousand shares
of a period. The arrows mark the atoms of the ferroelectric sublattice.
To adjust the ANNNI model in order to make it distinct from the cubic lattice
of dipoles, the scheme of interactions was changed. Hereinafter we shall call such
a model the ANNNI-adapted. We can obtain the coefficients of ANNNI-adapted
model by involving periodic boundary conditions and by comparing the resulting
phase diagram with the experimental phase diagram of Sn2P2S6 (figure 1). With the
coefficients obtained, we investigate the phase changes in microcrystal models with
the different dimension of a Monte Carlo cell and constant boundary conditions.
Figure 11. The phase diagram of
ANNNI-adapted model in the area of a
Lifshitz point at J0 = J1 = 1.
We shall link an effective dipole to
the molecular group of atoms Sn2P2S6
(four such groups are marked in fig-
ure 10 by ellipses), its electrical dipole
moment being directed along the axis a.
In the spirit of concurrent interactions in
the framework of ANNNI model we de-
fine the interactions in our model. They
are represented by coefficients: J0 > 0
– between dipoles located on an axis a,
J1 > 0 – between the nearest adjacent
dipoles, J2 < 0 – between the nearest
dipoles with negative interaction. For a
simplicity, let’s assume J0 = J1 = 1.
Unlike in the earlier papers [4] where
the phase diagram of ANNNI model on
679
Yu.Vysochanskii, A.Drobnich
a cubic lattice was obtained using the method of a mean field, we have done a more
correct binding of the coefficients of the adapted ANNNI model to the crystal lattice
of Sn2P2S6 compound. We have constructed its precise phase diagram in the area
of a Lifshitz point by a Monte Carlo method (figure 11).
It is necessary to focus on the boundary conditions used while performing the
calculations: to simulate the bulk crystal we used periodic boundary conditions
(PBC), while to simulate microcrystals, free boundary conditions were used.
The coefficients for Sn2P2S6 model could be evaluated by comparing the exper-
imental diagram of the solid solutions Sn2P2(SexS1−x)6 (figure 1) with the calcula-
tions depicted in figure 11, and using the proportional relation between J2 and x.
For x = 1 (Sn2P2Se6 compound), the width of the IC phase over temperature is
(Ti − Tc)/Ti ∼ 0.13 and x = 0.28 (Lifshitz point), the IC phase disappears. Herein
we can define J2= −0.92 for x = 0.
Figure 12. The results of MC calcula-
tions of polarization of a bulk sample
of Sn2P2S6 (MC cell with PBC) and of
microscopic samples (MC cell with free
boundary conditions) in the ANNNI-
adapted model.
Figure 13. The scheme of crystal struc-
ture of Sn2P2S6 in ANNNI-dipole model.
With the parameters obtained we
have performed a series of numerical
tests under the ANNNI-adapted mod-
el for different dimensions of the Monte
Carlo cell. The results are depicted in
figure 12. We can see the increase of
the calculated effective temperature Tef
of PT from ferroelectric to paraelectric
phase with the growth of this value on
saturation, defined by the models with
PBC, i.e. the models of a bulk sample of
Sn2P2S6. These results are in the qual-
itative consent with experimental data
[11].
ANNNI-adapted model has the main
severe drawback, i.e., a weak affection to
an actual lattice of such a complex crys-
tal as Sn2P2S6. To overcome this draw-
back the following short range model for
Sn2P2S6 was offered: the pseudo-dipoles
are arranged just as in the latter case,
the scheme of interactions as well as
in ANNNI model is constructed on the
nearest and on the next nearest neigh-
bors, but the coefficients of interactions
J1 < 0, J2 < 0, J3 > 0, J4 > 0 are calcu-
lated under the formula of dipole-dipole
interaction on the basis of the geometri-
cal data of the crystal lattice (figure 13).
Moreover, the positive correction ∆J on
the coefficient of interaction with the
680
Microscopic models for Sn2P2S(Se)6
nearest adjacent pseudo-dipole J3 is entered. Hereinafter, we shall call such a model
the ANNNI-dipole model.
Figure 14. The MC calculated phase di-
agram of the ANNNI-dipole model in the
area of the Lifshitz point.
Figure 15. The results of MC calcula-
tions of polarization temperature depen-
dence for the bulk sample of Sn2P2S6
(MC cell with PBC) and of microscop-
ic samples (MC cell with free boundary
conditions) in ANNNI-dipole model.
In this model there is only one cor-
rectional parameter – ∆J . To study its
effect on the phase diagram of the given
model, calculations were carried out by
the Monte Carlo method, the results of
which are shown in figure 14. In this di-
agram we can see the same three phases
i.e., ferroelectric, incommensurate and
paraelectric one, as well as the Lifshitz
point.
Applying the given model to the
Sn2P2S6 crystal is a more complex prob-
lem than for the previous ANNNI-
adapted model, because at transition
from Sn2P2S6 to Sn2P2Se6 lattice all pa-
rameters change. Now, we shall restrict
ourselves to a qualitative analysis of a
size effect for microcrystals far from the
Lifshitz point (for ∆J = 1.7). In fig-
ure 15, the results of these calculations
for MC cells of different dimensions with
periodic and free boundary conditions
are shown.
The character of the size effect ac-
cords with the one found in the above
mentioned before ANNNI-adapted mod-
el and qualitatively satisfies the exper-
imentally detected one. The proposed
ANNNI-dipole model, among the con-
sidered claims for a role of the most uni-
versal model, approaches the simulation
of both bulk and microscopic samples of
Sn2P2S6 and qualitatively describes the
size effect. Obviously, interesting exper-
imental properties of Sn2P2S(Se)6 ferroelectrics could be fully enough analyzed in
this model. Similar phenomena are characteristic of different ferroelectrics that ob-
viously make the considered models quite common and useful.
4. Acknowledgements
We appreciate Prof. R. Levitskii’s stimulating discussions and his support at the
start of MC simulations for the models investigated.
681
Yu.Vysochanskii, A.Drobnich
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682
Microscopic models for Sn2P2S(Se)6
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