Valence fluctuations in Sn(Pb)₂P₂S₆ ferroelectrics
The valence fluctuations which are related to the charge disproportionation of phosphorous ions P⁴⁺+P⁴⁺→P³⁺+P⁺⁵ are the origin of ferroelectric and quantum paraelectric states in Sn(Pb)₂P₂S₆ semiconductors. They involve recharging of SnPS ₃ or PbPS ₃ structural groups which can be represented as hal...
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irk-123456789-1293432018-01-19T03:04:28Z Valence fluctuations in Sn(Pb)₂P₂S₆ ferroelectrics Yevych, R. Haborets, V. Medulych, M. Molnar, A. Kohutych, A. Dziaugys, A. Banys, Ju. Vysochanskii, Yu. Квантовые эффекты в полупpоводниках и диэлектриках The valence fluctuations which are related to the charge disproportionation of phosphorous ions P⁴⁺+P⁴⁺→P³⁺+P⁺⁵ are the origin of ferroelectric and quantum paraelectric states in Sn(Pb)₂P₂S₆ semiconductors. They involve recharging of SnPS ₃ or PbPS ₃ structural groups which can be represented as half-filled sites in the crystal lattice. Temperature–pressure phase diagram for Sn ₂P ₂S ₆ compound and temperature-composition phase diagram for (Pb ySn ₁– y) ₂P ₂S ₆ mixed crystals, which include tricritical points and where a temperature of phase transitions decrease to 0 K, together with the data about some softening of low energy optic phonons and rise of dielectric susceptibility at cooling in quantum paraelectric state of Pb₂P₂S₆ are analyzed by GGA electron and phonon calculations and compared with electronic correlations models. The anharmonic quantum oscillators model is developed for description of phase diagrams and temperature dependence of dielectric susceptibility. 2016 Article Valence fluctuations in Sn(Pb)₂P₂S₆ ferroelectrics / R. Yevych, V. Haborets, M. Medulych, A. Molnar, A. Kohutych, A. Dziaugys, Ju. Banys, Yu. Vysochanskii // Физика низких температур. — 2016. — Т. 42, № 12. — С. 1477-1486. — Бібліогр.: 39 назв. — англ. 0132-6414 PACS: 63.20.D–, 63.20.Ry, 71.20.Nr, 77.22.–d http://dspace.nbuv.gov.ua/handle/123456789/129343 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Квантовые эффекты в полупpоводниках и диэлектриках Квантовые эффекты в полупpоводниках и диэлектриках |
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Квантовые эффекты в полупpоводниках и диэлектриках Квантовые эффекты в полупpоводниках и диэлектриках Yevych, R. Haborets, V. Medulych, M. Molnar, A. Kohutych, A. Dziaugys, A. Banys, Ju. Vysochanskii, Yu. Valence fluctuations in Sn(Pb)₂P₂S₆ ferroelectrics Физика низких температур |
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The valence fluctuations which are related to the charge disproportionation of phosphorous ions P⁴⁺+P⁴⁺→P³⁺+P⁺⁵ are the origin of ferroelectric and quantum paraelectric states in Sn(Pb)₂P₂S₆ semiconductors. They involve recharging of SnPS ₃ or PbPS ₃ structural groups which can be represented as half-filled sites in the crystal lattice. Temperature–pressure phase diagram for Sn ₂P ₂S ₆ compound and temperature-composition phase diagram for (Pb ySn ₁– y) ₂P ₂S ₆ mixed crystals, which include tricritical points and where a temperature of phase transitions decrease to 0 K, together with the data about some softening of low energy optic phonons and rise of dielectric susceptibility at cooling in quantum paraelectric state of Pb₂P₂S₆ are analyzed by GGA electron and phonon calculations and compared with electronic correlations models. The anharmonic quantum oscillators model is developed for description of phase diagrams and temperature dependence of dielectric susceptibility. |
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Article |
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Yevych, R. Haborets, V. Medulych, M. Molnar, A. Kohutych, A. Dziaugys, A. Banys, Ju. Vysochanskii, Yu. |
| author_facet |
Yevych, R. Haborets, V. Medulych, M. Molnar, A. Kohutych, A. Dziaugys, A. Banys, Ju. Vysochanskii, Yu. |
| author_sort |
Yevych, R. |
| title |
Valence fluctuations in Sn(Pb)₂P₂S₆ ferroelectrics |
| title_short |
Valence fluctuations in Sn(Pb)₂P₂S₆ ferroelectrics |
| title_full |
Valence fluctuations in Sn(Pb)₂P₂S₆ ferroelectrics |
| title_fullStr |
Valence fluctuations in Sn(Pb)₂P₂S₆ ferroelectrics |
| title_full_unstemmed |
Valence fluctuations in Sn(Pb)₂P₂S₆ ferroelectrics |
| title_sort |
valence fluctuations in sn(pb)₂p₂s₆ ferroelectrics |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2016 |
| topic_facet |
Квантовые эффекты в полупpоводниках и диэлектриках |
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http://dspace.nbuv.gov.ua/handle/123456789/129343 |
| citation_txt |
Valence fluctuations in Sn(Pb)₂P₂S₆ ferroelectrics / R. Yevych, V. Haborets, M. Medulych, A. Molnar, A. Kohutych, A. Dziaugys, Ju. Banys, Yu. Vysochanskii // Физика низких температур. — 2016. — Т. 42, № 12. — С. 1477-1486. — Бібліогр.: 39 назв. — англ. |
| series |
Физика низких температур |
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2025-07-09T11:10:57Z |
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| fulltext |
Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 12, pp. 1477–1486
Valence fluctuations in Sn(Pb)2P2S6 ferroelectrics
R. Yevych1, V. Haborets1, M. Medulych1, A. Molnar1, A. Kohutych1, A. Dziaugys2,
Ju. Banys2, and Yu. Vysochanskii1
1Institute for Solid State Physics and Chemistry, Uzhgorod National University
54 Voloshyn Str., Uzhgorod 88000, Ukraine
2Physics Faculty, Vilnius University, 9/3, Sauletekio Al., Vilnius 10222, Lithuania
E-mail: vysochansrii@gmail.com
Received May 11, 2016, published online October 24, 2016
The valence fluctuations which are related to the charge disproportionation of phosphorous ions
P4+ + P4+ ® P3+ + P+5 are the origin of ferroelectric and quantum paraelectric states in Sn(Pb)2P2S6 semi-
conductors. They involve recharging of SnPS3 or PbPS3 structural groups which can be represented as half-
filled sites in the crystal lattice. Temperature–pressure phase diagram for Sn2P2S6 compound and tempera-
ture-composition phase diagram for (PbySn1–y)2P2S6 mixed crystals, which include tricritical points and
where a temperature of phase transitions decrease to 0 K, together with the data about some softening of low
energy optic phonons and rise of dielectric susceptibility at cooling in quantum paraelectric state of Pb2P2S6
are analyzed by GGA electron and phonon calculations and compared with electronic correlations models.
The anharmonic quantum oscillators model is developed for description of phase diagrams and temperature
dependence of dielectric susceptibility.
PACS: 63.20.D– Phonon states and bands, normal modes, and phonon dispersion;
63.20.Ry Anharmonic lattice modes;
71.20.Nr Semiconductor compounds;
77.22.–d Dielectric properties of solids and liquids.
Keywords: mixed valency, quantum paraelectrics, ferroelectrics.
Introduction
For Sn2P2S6 ferroelectrics, as follows from the first prin-
ciples calculations [1], the spontaneous polarization is de-
termined by three-well local potential for the order parame-
ter fluctuations. Continuous phase transition can be desc-
ribed as second order Jahn–Teller (SOJT) effect which is
related to the stereoactivity of Sn2+ cations placed inside of
sulfur ions polyhedron [1,2]. For the Sn2P2S6 crystal lattice
(Fig. 1) in mechanism of the ferroelectricity, the hybridiza-
tion of Sn 5s orbitals and S 3p orbitals is a main driving
force. Thermodynamics of such transition can be described
in well-known Blume–Emery–Griffith (BEG) model [3]
with two order parameters (dipolar and quadrupolar) and
three possible values of pseudospin: –1, 0, +1. The most
important feature of BEG model is a presence of tricritical
point (TCP) on states diagram [3,4] — while the decrease of
temperature of phase transition by pressure or by alloying in
mixed crystals, the transition becomes of first order. Indeed,
below tricritical “waterline temperature” about 220–240 K
and at compression above 0.6 GPa, the ferroelectric transi-
tion in Sn2P2S6 evolves to first order [5,6]. Also, at tin by
lead substitution in the (PbySn1–y)2P2S6 mixed crystals, for
y > 0.3 and below 220 K, the hysteresis which is related to
the paraelectric and ferroelectric phase coexistence appears
that give evidence about a discontinuous character of the
phase transition [7].
One more feature of discussed T–P and T–y phase dia-
grams is finite range of the ferroelectric phase existence.
Under compression, the temperature of ferroelectric transi-
tion decreases to 110CT K at 1.2 GPa in Sn2P2S6 crystal
[6]. The observed ( )CT P dependence at linear extrapola-
tion reaches 0 K near the pressure of 1.5 GPa. For the
(PbySn1–y)2P2S6 mixed crystals the paraelectric phase be-
comes stable till 0 K above y ~ 0.7 [8]. Comparison of
( )CT P and ( )CT y dependencies (Fig. 2) shows, that sta-
bility of paraelectric state in Pb2P2S6 compound can be
similar to behavior of Sn2P2S6 crystal under a pressure of
2.2 GPa. With increasing pressure the elastic energy in-
creases what make unfavorable the SOJT effect and ferroe-
lectric state disappears in Sn2P2S6 compound. At transition
© R. Yevych, V. Haborets, M. Medulych, A. Molnar, A. Kohutych, A. Dziaugys, Ju. Banys, and Yu. Vysochanskii, 2016
mailto:vysochansrii@gmail.com
R. Yevych, V. Haborets, M. Medulych, A. Molnar, A. Kohutych, A. Dziaugys, Ju. Banys, and Yu. Vysochanskii
from Sn2P2S6 to more ionic Pb2P2S6 compound at normal
pressure, the SOJT effect weakens because Pb 6s orbitals
have energy about 1 eV lower than in the case of Sn 5s
orbitals. This rise of energy distance between S 3p orbitals
and Pb 6s orbitals determines suppression of Pb2+ cations
stereoactivity [2].
So, according to the experimental data [6,8] the para-
electric ground state can be stable at P > 1.5 GPa for Sn2P2S6,
and at y > 0.7 and normal pressure for (PbySn1–y)2P2S6 mixed
crystals. At 0 K, the quantum critical point (QCP) can be
reached by variation of pressure or composition [9]. Early the
QCP have been obviously found in the quantum paraelectric
SrTiO3 at increase of 18O isotope concentration only [10].
Here the ferroelectric phase appears almost continuously at
near 45% content of 18O, and for SrTi 18O3 the ferroelectric
phase appears below C 25 K.T ≈ At cooling to QCP the
reciprocal dielectric susceptibility is proportional to 2 ,T
what is the most important criterion of the quantum critical
behavior of susceptibility. But for the most experimental
observations, the ferroelectric or magnetic ordering appears
by first order transition and 2T behavior of the related sus-
ceptibilities is observed only in a some finite temperature
interval. Such transformation of QCP into the first order
phase transition at = 0T K can be explained by an interac-
tion between different order parameters that break different
symmetry elements of paraelectric phase [9,10].
For (PbySn1–y)2P2S6 mixed crystals with compositions
0.61y ≈ and 0.65,y ≈ which are closed to the transition
at zero temperature from polar phase (y < 0.7) to
paraelectric one (y > 0.7), the dielectric susceptibility
demonstrates the quantum critical behavior 1 2T−χ
in
vicinity of the first order transitions with C 35T ≈ K and
20 K, respectively [8]. It is interesting to understand the
nature of quantum paraelectric state at high pressures in
Sn2P2S6 compound or at high lead concentration in
(PbySn1–y)2P2S6 mixed crystals.
Appearance of spontaneous polarization at low temper-
atures can be mostly related to electronic correlations. Ear-
ly in the extended Falikov–Kimbal (EFK) model it was
shown [11–13], that hybridization between orbitals of lo-
calized carriers, at the top of valence band, and itinerant
electrons, at the bottom of conduction band, can destroy
the crystal lattice symmetry center. In this case the sponta-
neous polarization in the ground state appears together
with excitonic condensate [12]. The EFK model can be
related to Hubbard model [14] which also take into account
the spins of fermions. It is interesting that the last one is
usually converted into anisotropic XYZ Heisenberg model,
and for special set of parameters — into isotropic Heisen-
berg XXZ model [14]. It should be noted that isotropic
Heisenberg model can be equivalent to BEG model [15],
and in fermionic presentation the BEG model is also relat-
ed to EFK model [11].
Up to now nobody found experimental evidences about
electronic ferroelectricity. A long history of so-called vi-
bronic theory of ferroelectric phase transitions [16] stimu-
lates searching of the electronic origin of crystal lattices
spontaneous polarization. The vibronic models are mostly
connected to Jahn–Teller effect [17] and they are based on
electron-phonon interaction. Obviously the Jahn–Teller
effect can be leading at description of compounds with
ferroelectric phase transitions at high temperatures. Both,
SOJT effect and models of electronic correlations, should
be obviously involved for description of ferroelectric sys-
tems with three-well local potential. For this case of two
Fig. 1. (Color online) Structure of Sn2P2S6 type crystals in
paraelectric (P21/n) phase with shown shifts of cation atoms in
ferroelectric (Pn) phase.
Fig. 2. (Color online) Temperature-pressure phase diagram for
Sn2P2S6 crystal and temperature-composition phase diagram for
(PbySn1–y)2P2S6 mixed crystals. The tricritical points are loca-
ted near 220 K for pressure P » 0.6 GPa [6] or for concentra-
tion y » 0.3 [7]. For Pb2P2S6 crystal the calculated temperature-
pressure phase diagram is shown. The metastable region in
ground state (T = 0 K) with possible coexistence of paraelectric
and ferroelectric phases is shaded.
1478 Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 12
Valence fluctuations in Sn(Pb)2P2S6 ferroelectrics
order parameters (dipolar and quadrupolar), at lowering of
second order transition temperature (by compression or by
variation in chemical composition) the TCP is reached.
Further the line of first order transitions goes down to 0 K.
Here the quantum paraelectric state is evidently realized
and possibility of electronic correlation weak effects can
be checked. Of course, a some phonon contribution to the
electronic correlations is expected, what is accounted in the
Holstein-Hubbard type models [18,19]. The relaxations of
paired in a real space Anderson's electrons [15,20,21] or
phononic Kondo screening [22–24] can also be involved in
some approaching for description of phonon-like excita-
tions in the low-temperature quantum paraelectric state.
In this paper we present the results of dielectric suscepti-
bility and Raman scattering studies of Pb2P2S6 crystals.
Some growth of dielectric susceptibility and softening of the
lowest energy phonon optic mode are observed at cooling
from room temperature till 20 K in the paraelectric phase of
Pb2P2S6. Experimental data on T–P phase diagram of
Sn2P2S6 and on T–y phase diagram of (PbySn1–y)2P2S6 are
compared and involved for explanation of unusual behavior
of paraelectric phase of Pb2P2S6 crystal. The electron and
phonon spectra of Pb2P2S6 crystal are calculated in GGA
approximation of Density Functional Theory and compared
with experimental data. For this crystal the transition from
paraelectric state into ferroelectric one under negative pres-
sure is modeled by first-principles calculations.
Taking into account the ideas about valence skipping
[25] and charge disproportionation [26], the electronic
origin of possible spontaneous polarization of Pb2P2S6
crystal lattice is analyzed in the frame of EFK model [12].
Also the approximations of Hubbard model and related
BEG model [15] are used for consideration of simplified
picture of half-filled lattice with sites that incorporate the
SnPS3 atomic groups with 27 valence electrons. The min-
imal energetic model is proposed that considers the 26 va-
lence electrons on hybridized bonding orbitals of SnPS3
structure units as core and only one electron is considered
at every site. The 4 4 3 5P P P P+ + + ++ → + charge dispro-
portionation process is energetically favorable for P4+
phosphorus ions valence skipping and 1 13 3s s+ →
2 03 3s s→ + recharging. The on-site Coulomb repulsion
CU acts in opposite way to this disproportionation process
with energy gain dispU . The orbitals hybridization compli-
cates this picture, but also induce acentric state. We esti-
mate the energy of disproportionation which is main origin
of acentric ground state at condition of suppressed SOJT
effect. At C disp| | 2gU U E≈ ≈ ≈ eV, where gE is a ener-
gy gap, the EFK model was used for estimation of elec-
tronic contribution to dielectric susceptibility. The calcu-
lated electron spectra are compared with parameters of
minimal BEG like energetic model that is derived from
Hubbard Hamiltonian [15] and agrees with observed T–P
and T–y diagrams of (PbySn1–y)2P2S6 ferroelectrics. Final-
ly, the quantum anharmonic oscillators models, that is
based on statistics of phonon-like excitations in the lattice
with three-well local potential, is proposed for description
of T–P and T–y state diagrams of (PbySn1–y)2P2S6 ferroe-
lectrics and explanation of their dielectric susceptibility
growth at cooling in the quantum paraelectric state.
1. Experimental data
The dielectric susceptibility and Raman scattering were
investigated for Pb2P2S6 crystals that was grown by vapor
transport method. Temperature dependence of dielectric
susceptibility was investigated by Bridge scheme at fre-
quencies range 104–108 Hz [7]. The Raman scattering was
excited by He-Ne laser and analyzed by DFS-24 spectro-
meter with 1 cm–1 frequency resolution. The spectra were
fitted by using a Voight profile shape.
The temperature dependence of real part of dielectric
susceptibility is shown at Fig. 3 for different frequencies.
The susceptibility increases about ten percepts at cooling
in the region of three hundred Kelvins without frequency
dispersion in the studied diapason.
According to temperature evolution of Raman spectra
(Fig. 4), the lowest energy optic mode softens about 2 cm–1
at cooling from room temperature to 79 K. At temperature
decrease, the low-frequency spectral tail and damping of
optical phonons growth also. Such behavior coincides with
earlier found [27] increase of Pb2P2S6 crystal lattice an-
harmonicity according to the rise of Gruneisen parameter at
cooling on the data of Brillouin scattering by acoustic pho-
nons. The observed rise of dielectric susceptibility about
10% at cooling correlates with a lowering of optic mode
frequency by 1 cm–1.
2. Calculation of energetic spectra
For characterization of Pb2P2S6 paraelectric phase and
to search a possible origin of its instability, the electron
and phonon spectra were calculated in GGA approach of
Fig. 3. (Color online) Temperature dependence of Pb2P2S6 die-
lectric susceptibility at different frequencies.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 12 1479
R. Yevych, V. Haborets, M. Medulych, A. Molnar, A. Kohutych, A. Dziaugys, Ju. Banys, and Yu. Vysochanskii
DFT. The monoclinic elementary cell (Fig. 1) contains two
formula units and belong to P21/c space group [28,29].
Structure of valence zone (Fig. 5) correlates with earlier
[2] described for Sn2P2S6 analog. But here the contribution
of Pb2+ cations 6s orbitals into the density of states at the
top of valence band is smaller as in the case of Sn2+ 5s
orbitals presence. This fact reflects smaller stereoactivity
of Pb2+ cations in more ionic Pb2P2S6 lattice. Such peculi-
arity is evidently determined by bigger energy distance
between sulfur 3p orbitals and lead 6s orbitals. Indeed, the
Pb 6s orbitals are placed mostly below –8 eV — by 1 eV
lower in compare with energetic position of Sn 5s orbitals
in Sn2P2S6 crystal.
It is important to mention that for both edges of valence
band and conductivity band, the orbitals of sulfur, phos-
phorous and lead atoms are presented. Obviously the elec-
tronic structure of Pb2P2S6 compound can be considered as
bounding and antibounding counterparts of PbPS3 atomic
groups orbitals.
The electron spectra calculations in GGA approach are
in good enough agreement for the energy gap value - cal-
culated one (about 2.2 eV) is closed to 2.45 eV gap accord-
ing to optic absorption measurements at 4.2 K [30].
In GGA approach the phonon spectra for Pb2P2S6 crystal
lattice were calculated also. The phonon branches and partial
phonon densities of states are presented at Fig. 6. Long-wave
phonon frequencies are in good agreement with experimental
data what give evidence about adequate GGA approach for
the energetic spectra analysis. Below 70 cm–1, the transla-
tions of Pb ions relatively to P2S6 structural groups mostly
contribute into phonon eigenvectors as follows from the par-
Fig. 4. (Color online) Raman scattering spectra for low-energy optic modes in Pb2P2S6 crystals at 293 K (a) and 79 K (b), and tempera-
ture dependence of spectral lines frequencies (c) and damping (d). Labels 1–4 denote the individual peaks.
Fig. 5. (Color online) The electron band structure and partial
electron densities of states for Pb2P2S6 crystal.
1480 Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 12
Valence fluctuations in Sn(Pb)2P2S6 ferroelectrics
tial phonon densities of states. Above 150 cm–1 the internal
vibrations of (P2S6)4– anions dominate and in higher fre-
quency interval 530–590 cm–1 mostly phosphorous atoms
participate in the phonon eigenvectors.
The LO–TO splitting has the biggest values for the
translational lattice polar modes with frequencies 43, 58
and 70 cm–1. It is remarkable that big LO–TO splitting
also was found for the (P2S6)4– internal valence vibrations
with frequencies 563, 566 and 578 cm–1.
Adequate calculations of electronic and phonon spectra
of Pb2P2S6 crystal at normal pressure permit to suppose a
possibility for obtaining of reasonable calculation results at
negative pressure. It was found that the transition into fer-
roelectric phase occurs near –1.95 GPa (Fig. 7). It is im-
portant that the acentric deviation appears already at pres-
sure about –1.7 GPa. The ground state of Pb2P2S6 crystal
at normal pressure is similar to the Sn2P2S6 ground state at
2.2 GPa (Fig. 2).
Some similarity of Sn2P2S6 compound T–P diagram at
pressures 0 2.2− GPa with calculated states diagram for
Pb2P2S6 crystal in pressure range from –2 to 0 GPa permits
us to suppose that temperature behavior of Pb2P2S6 crystal
(at normal pressure) could be similar to Sn2P2S6 tempera-
ture behavior at 2.2 GPa. This behavior can be determined
by some polar fluctuations, like in quadrupole phase ac-
cording to MC simulations [31], and some growth of die-
lectric susceptibility at cooling can be expected. Let's try to
found some explanation for quantum paraelectric behavior
in Pb2P2S6 crystal.
3. Discussion of spontaneous polarization origin
Earlier [1,2], for Sn2P2S6 ferroelectrics the stereo-
activity of Sn2+ cations, which are placed inside sulfur
octahedron, was only considered as origin of local dipoles
that exist already in paraelectric phase, because tin cations
occupy general position in the monoclinic elementary cell.
This stereoactivity is determined by covalency between tin
atomic orbitals and molecular orbitals of P2S6 groups, and
can be considered as second order Jahn–Teller effect,
which involve phosphorous orbitals also. At weaking of
SOJT effect by pressure, or at tin by lead substitution, an-
other origins of crystal acentricity can be observed. The
first of them is disproportionation of P4+ cations:
4 4 3 5P P P P .+ + + ++ → + This process is also related to
recharging of surrounding sulfur anions that is accounted
by polarizability of nearest space [26].
The energy of disproportionation can be determined
as disp vac pol= .U U U− Here vac 5 4=U I I− is the differ-
rence in ionization potentials for P5+ and P4+ ions in
Fig. 6. (Color online) Calculated phonon spectra (a) and partial
phonon densities of states (b) for Pb2P2S6 crystal.
Fig. 7. Calculated acentric deviation induced by negative com-
pression of Pb2P2S6 crystal lattice.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 12 1481
R. Yevych, V. Haborets, M. Medulych, A. Molnar, A. Kohutych, A. Dziaugys, Ju. Banys, and Yu. Vysochanskii
vacuum. The polarization energy is determined by Born
formula [26] 2
pol = ( /2 )(1 / ),U e R+
∞− γ ε where =γ
01 ( /( ))(1 / ).R R R+ + −
∞= − + − ε ε Here the phenomeno-
logical parameters are included — anions and cations
radii R− and ,R+ respectively, and dielectric constants
0ε and ∞ε .
The value vac 13.1U ≈ eV was estimated by using the
ionization energies 5 = 62.74I eV for P 5+ cations and
4 = 49.63I eV for P4+ cations [32]. The value 0.82γ ≈
was found for next set of parameters: S2–anion radius
2.03R− ≈ Å, according S–P interatomic distance [28],
and P4+ cation radius 0.42R+ ≈ Å; dielectric constants
0 50ε ≈ and 7.84∞ε ≈ [33]. At such parameters, the po-
larization energy p 15.1olU ≈ − eV have been found and,
sequently, disp 2U ≈ − eV.
Negative value of disprU reflects the valence skipping
effect that favors 3s0 and 3s2 electronic configuration of P5+
and P3+ cations instead of 3s1 configuration of P4+ ions.
The phosphorous ions charge disproportionation, or va-
lence ordering process, can be viewed as a lattice of An-
derson electron pairs [20] which is stabilized by polarizing
of surrounding sulfur polyhedrons. The energy to displace
each S atom by x can be written as [15] =abE
2 /2 ( )a bCx gx q q= − − where a bq q− is the charge differ-
ence between the two nearest neighbor phosphorous atoms,
g and C are coupling constants. The Hubbard type Ham-
iltonian [15] can be applied for description of electrons
hopping in band with contribution of phosphorous orbitals.
Such Hamiltonian contains the intra-site and inter-site
Coulomb interactions. The first one is determined by CU
constant. The inter-site interaction for simplicity can be
presented by the short-range interaction 2 /6e aα ε only,
where α is Madelung constant, a is lattice parameter, e
is charge of electron and ε is dielectric permittivity.
At condition of small band width limit in the spin
presentation the Hubbard model can be reflected onto the
BEG model [4,15]:
2
< >
= .i i j
i ij
H m J m m∆ +∑ ∑ (1)
Here parameters are derived from Hubbard model of An-
derson's electron pairs — 2
C= ( 6 / )/2U g C∆ − and
2 2= / /6 .J g C e a+ α ε Pseudospin variable im has values
+1, 0, –1 and can be are related to P3+, P4+ and P5+ states
of phosphorous cations. General view of calculated for
BEG model phase diagram (Fig. 8) correlates with experi-
mental observations. Indeed, under compression the lattice
stiffness constant C increases, lattice period a lowers and
following the on-site energy ∆ rises and the intersite in-
teraction energy J remains almost unchanged.
At tin by lead substitution, the stiffness decrease a little
[27], but the lattice period demonstrates some rise [28]. So
again, the intercell interaction J remains almost un-
changed. But in lead compound the ionicity is bigger than
in the case of Sn2P2S6, and the onsite Coulomb repulsion
CU is obviously higher. At this the on-site energy ∆ also
growth at Sn by Pb substitution. Such influence of chemi-
cal composition on the system ground state qualitatively
agrees with derived in Hubbard approximation [34] phase
diagram where phase transition from paraelectric phase
into state with dipole ordering is governed by difference
between energies of filled orbitals of two type cations in
the crystal lattice. In our case, according to GGA calcula-
tions (Fig. 5), the difference between P 3p orbitals energy
and Sn 5s or Pb 6s orbitals energies increases nearly 1 eV
at transition from Sn2P2S6 to Pb2P2S6. Evidently the
growth of ionicity stabilizes the paraelectric phase what
agrees with experimental observation.
Finally, we can estimate some parameters by using the
following characteristics: the second order phase transition
temperature ( 0 337T ≈ K) for Sn2P2S6 crystal, the coordi-
nates of tricritical points ( 220TCPT ≈ K) on T–y and T–P
diagrams, the composition 0.7y ≈ or pressure 1.5 GPaP ≈
at which the phase transition temperature goes down to zero.
By comparing the experimental diagram (Fig. 1) with calcu-
lated one (Fig. 8), it can be found that 110 KJ ≈ for consid-
ered ferroelectrics. At almost constant value of intersite inter-
action energy (parameter J), the on-site energy parameter ∆
mostly determines the global temperature-pressure-compo-
sition phase diagram. The ∆ parameter has small value and
is governed by balance of two significant characteristics —
the on-site Coulomb repulsion CU and disproportionation
energy dispU . The first one have the value above 2 eV ac-
cording to ab-initio GGA+U calculations [35] that explain
observed optic gap width 2.4gE ≈ eV for Sn2P2S6 crystal.
The disproportionation energy dispU can be compared with
ratio 26 /g C and equals about –2 eV. Obviously, for
Sn2P2S6 crystal a some small value of ∆ can be found
Fig. 8. The phase transition temperature as a function of ∆/J cal-
culated in the mean-field approximation on the BEG model (1).
Solid lines denote second order and dashed lines first order transi-
tions and the dot, a tricritical point [15].
1482 Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 12
Valence fluctuations in Sn(Pb)2P2S6 ferroelectrics
(Fig. 8) and 0∆ ≈ for some pressure P or lead content y .
At 0∆ ≈ or for condition C dispU U≈ , the valence fluctua-
tions are strongly developed, and possibly, they are still
enough strong at rise of ∆ value to 3J and phase transition
temperature CT going down to 0 K. So, the valence or
charge fluctuations can contribute to growth of dielectric
susceptibility for (PbySn1–y)2P2S6 crystals with > 0.7y and
for Pb2P2S6 compound.
Recent calculations in Hubbard model show [34] that in
paraelectric ground state, the charge density at different ca-
tions (phosphorous and lead in our case) strongly depends
on temperature. This peculiarity can be evidently related to
the temperature dependence of dielectric susceptibility.
Fluctuations of Anderson's 2e electron pairs as origin of
dielectric susceptibility growth at cooling in wide tempera-
ture interval can be compared with Kondo screening phe-
nomena [21–24]. At incoherent 2e pairs fluctuations the
susceptibility increases with cooling as 1,T −
but below
some temperature the coherence of electron pairs fluctua-
tions can appear and susceptibility reaches some constant
value. The model of Kondo effect with involving of pho-
non channel definitely predicts [24] a rise of dielectric sus-
ceptibility at the system cooling.
At low enough temperatures, near 0 K, electronic corre-
lations themselves can induce the system acentricity. The
electronic ferroelectricity has been predicted [12,13] as
result of hybridization of itinerant electron wave function
with hole at the top of valence band and appearance of
excitonic condensate. The excitonic order parameter corre-
lates with acentric space distribution of the charge density,
as it is follows from the EFK model. For ground state in
paraelectric phase at approaching to transition into polar
phase, by variation of itinerant electrons energy level fE ,
the dielectric susceptibility is expected to growth according
relation [12]:
2
C
arccoth
2
=
arccoth
f
z
f
E
WN
E
W U
W
µ
ε
Ω
−
, (2)
where zµ is the interband dipole matrix element, Ω is the
volume of unit cell, N is the number of sites, 2W is the
bandwidth. It was taken = 0.02W eV and = 2fE eV
according to estimations which were made from GGA cal-
culations (Fig. 5).
As was mentioned above, for Sn2P2S6 crystals at nor-
mal pressure very delicate energetic balance C dispU U≈
serves for development of charge fluctuations. Moreover,
the energy gap between valence and conductivity gap
g 2.45E ≈ eV is also comparable with estimates values
2≈ eV for CU and disp| |U . Such conditions permit to
expect possibility of electronic origin for increased at cool-
ing dielectric susceptibility in paraelectric phase. Estima-
tion of dielectric susceptibility for Pb2P2S6 at = 0T K
within EFK model (2) gives value 100ε ≈ for the
interband transition dipole moment 3 Å.eµ ≈ At heating
the susceptibility will decrease because of thermal disor-
dering of the local electric dipoles.
4. Anharmonic quantum oscillators model
Now we propose to use the AQO model [36] that con-
sider phonon-like bosonic excitations. These excitations
can be regarded as incorporation of electron-phonon inter-
action in generalized Holstein–Hubbard model [18] or in
Bose–Hubbard model [37].
In the AQO model, crystal is representing as one di-
mensional system of anharmonic oscillators which interact
via quadratic interaction term. So, Hamiltonian of such a
system is
( )= ( ) ( ) ,i i ij i j
i ij
H T x V x J x x+ +∑ ∑ (3)
where ( )iT x and ( )iV x are operators of kinetic and poten-
tial energy, respectively, ix is a i oscillator's displace-
ment, ijJ are coupling constants between i and j oscilla-
tors. In the mean-field approach, the last term of (3) can be
replaced by another one — i
i
J x x〈 〉∑ where x〈 〉 is aver-
age position of all other oscillators in the lattice. In this
case, the Hamiltonian (3) can be represented as a sum of
independent single-particle Hamiltonians:
eff eff= , = ( ) ( ) .i i i i i
i
H H H T x V x J x x+ + 〈 〉∑ (4)
Such effective particle is an oscillator under the influence
of linear symmetry-breaking field which is calculated self-
consistently. Solving Schrödinger equation with Hamilto-
nian (4) one obtain a set of eigen energies { }nE of levels
and its wave functions { ( )}n xΨ which are used for self-
consistent calculation of average expectation value x〈 〉 . In
our calculations we used matrix representation form for
position and momentum operators to solve Schrödinger
equation [38].
It should be noted that temperature enters in Hamiltoni-
an (4) indirectly through Boltzmann distribution for the
occupation of the levels. Really, from definition of x〈 〉 it
follows
= ,n n
n
x p x〈 〉 ∑ (5)
where exp ( / )n np E kT− is the occupation number for
thn level, k is a Boltzmann constant, =nx
* */n n n nx dx dx= Ψ Ψ Ψ Ψ∫ ∫ is average value of displacement
for thn level.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 12 1483
R. Yevych, V. Haborets, M. Medulych, A. Molnar, A. Kohutych, A. Dziaugys, Ju. Banys, and Yu. Vysochanskii
Obviously, in the paraelectirc phase for average
desplacement, which is proportional to a order parameter,
one obtain = 0.x〈 〉 On the other hand, below phase transi-
tion temperature cT we have 0.x〈 〉 ≠ So, one can model a
changes on phase diagram at ionic substitution or pressure
by varying coupling constant or shape of potential energy.
Here we use a transformation of the local three-well poten-
tial ( )iV x (Fig. 9) as function of pressure according to the
results of [31]. In present calculations the mass of oscilla-
tor was equal to tin atomic mass. Moreover, it is possible
to calculate a dielectric response of such the system of os-
cillators using following relation [39]:
2
eff
0
( , )( , ) = 1 ,
1 ( , )
e TT
V J T
Π ω
ε ω + ε − Π ω
(6)
where
2( ) | | | |
( , ) = ,
p p x
T
i
α β α β
αβ
α β
− 〈Ψ Ψ 〉
Π ω
ω − ω + ω − ωγ∑
and effective charge of oscillator effe and its volume V
can be used as fitting parameters.
Examples of the QAO model solutions are shown at
Fig. 10(a). Calculated in this model state diagram is shown
on Fig.11. In addition to the paraelectric and ferroelectric
phases, it also contain a coexistence region for stable and
metastable solutions. This diagram reflects the main fea-
tures of experimental diagram. For Sn2P2S6 crystal under
compression, the second order phase transition temperature
lowers from 337 K (at normal pressure) and the tricritical
point is reached near 0.6 GPa and at 220 K. The coexist-
ence region of ferroelectric and metastable solutions is
delayed till 1.5 GPa. For higher pressure, the paraelectric
phase is stabilized. The Pb2P2S6 compound at normal pres-
sure has quantum paraelectric ground state. By negative
pressure below –1.7 GPa the crystal lattice acentricity can
be induced.
The dielectric susceptibility temperature dependence
in quantum paraelectric Pb2P2S6 have been calculated in
the AQO model and compared with experimental data
(Fig. 10(b). The dielectric susceptibility increase at cooling
from room temperature to 20 K obviously reflects strong
anharmonicity of the local potential in this compound.
5. Conclusions
The valence fluctuations play important role in the nature
of ferroelectric and quantum paraelectric states in
Sn(Pb)2P2S6 semiconductors. The charge disproportionation
of phosphorous ions 4 4 3 5P P P P+ + + ++ → + can be related
to recharging of SnPS3 (or PbPS3) structural groups. This
approximation permits to consider a simplified model of the
crystal lattice as set of the half-filled sites. Experimental
Fig. 9. (Color online) The shape of local potential for Sn2P2S6
crystal at different pressures [31].
Fig. 10. (Color online) The examples of spontaneous polarization temperature dependence for AQO system for different values of cou-
pling constant J at pressure P = 1.5 GPa (a); calculated (continuous line) and experimental (points) temperature dependence of dielectric
susceptibility for Pb2P2S6 quantum paraelectric (b).
1484 Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 12
Valence fluctuations in Sn(Pb)2P2S6 ferroelectrics
temperature-pressure phase diagrams for Sn2P2S6 crystal
and temperature-composition one for (PbySn1–y)2P2S6
mixed crystals with tricritical point and with decrease of
phase transitions lines to 0 K, together with the data about
some softening of low energy optic phonons and rise of die-
lectric susceptibility at cooling in quantum paraelectric state
of Pb2P2S6 crystal, are analyzed by first principles electron
and phonon calculations and compared with electronic cor-
relations models. The anharmonic quantum oscillators mod-
el is developed for description of temperature-pressure-
composition phase diagram shape. Temperature dependence
of dielectric susceptibility is also described in this model.
The chemical bonds covalence obviously complicates pic-
ture of the charge disproportionation which can be presented
as 4 4 (4 ) (4 )P P P P ,x x+ + − + + ++ → + and parameter x is in-
teresting for determination at the further development of
electronic correlation models.
Authors acknowledge Prof. Ihor V. Stasyuk for fruitful
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1486 Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 12
Introduction
1. Experimental data
2. Calculation of energetic spectra
3. Discussion of spontaneous polarization origin
4. Anharmonic quantum oscillators model
5. Conclusions
|