Proton ordering model of phase transitions in hydrogen bonded ferrielectric type systems: the GPI crystal
A microscopic model based on the consideration of the proton ordering is proposed for describing the H-bonded ferroelectric crystalline systems with a complex structure of the hydrogen bond network. The model has been used for the investigation of thermodynamics and dielectric properties of the...
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| Zitieren: | Proton ordering model of phase transitions in hydrogen bonded ferrielectric type systems: the GPI crystal / I. Stasyuk, Z. Czapla, S. Dacko, O. Velychko // Condensed Matter Physics. — 2003. — Т. 6, № 3(35). — С. 483-498. — Бібліогр.: 22 назв. — англ. |
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irk-123456789-1207482017-06-13T03:03:08Z Proton ordering model of phase transitions in hydrogen bonded ferrielectric type systems: the GPI crystal Stasyuk, I. Czapla, Z. Dacko, S. Velychko, O. A microscopic model based on the consideration of the proton ordering is proposed for describing the H-bonded ferroelectric crystalline systems with a complex structure of the hydrogen bond network. The model has been used for the investigation of thermodynamics and dielectric properties of the GPI crystal. The symmetry analysis of the order parameters responsible for the mixed (ferro- and antiferroelectric) nature of ordering is performed within the model. The phase transition into the ferroelectric state is described. Changes in the dielectric susceptibility of the crystal are studied in the presence of the transverse external electric field acting along the c - axis. The results of measurements of temperature and field dependences of dielectric permittivity ε'c in the paraelectric phase are presented. The microscopic mechanism of the observed effects is discussed based on the comparison of theoretical results and experimental data. A conclusion is made about the significant role of the ionic groups connected by hydrogen bonds in the charge transfer. So they make an important contribution into the polarizability of the GPI crystal along the direction of H-bonded chains. Для опису сегнетоелектричних водневозв’язаних кристалічних систем із складною структурою сітки водневих зв’язків запропоновано мікроскопічну модель, яка базується на врахуванні впорядкування протонів. Її застосовано для дослідження термодинаміки і діeлектричних властивостей кристалу GPI. В рамках моделі дано симетричний аналіз параметрів порядку, відповідальних за змішаний (сегнето- і антисегнетоелектричний) характер впорядкувань. Описано фазовий перехід до сегнетоелектричного стану та вивчено зміни у діелектричній сприйнятливості кристалу при наявності зовнішнього поперечного електричного поля прикладеного вздовж c -осі. Наведено результат вимірювань температурних і польових залежностей діелектричної проникності ε'c у параелектричній фазі. На основі порівняння теоретичних результатів і даних експерименту обговорюється мікроскопічний механізм спостережуваних ефектів. Зроблено висновок про важливу роль іонних груп, з’єднаних водневими зв’язками, у переносі заряду. Вони дають вагомий внесок до поляризованості кристалу GPI у напрямку вздовж ланцюжків водневих зв’язків. 2003 Article Proton ordering model of phase transitions in hydrogen bonded ferrielectric type systems: the GPI crystal / I. Stasyuk, Z. Czapla, S. Dacko, O. Velychko // Condensed Matter Physics. — 2003. — Т. 6, № 3(35). — С. 483-498. — Бібліогр.: 22 назв. — англ. 1607-324X PACS: 64.60.Cn, 77.22.-d, 77.80.-e, 77.80.Bh DOI:10.5488/CMP.6.3.483 http://dspace.nbuv.gov.ua/handle/123456789/120748 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| language |
English |
| description |
A microscopic model based on the consideration of the proton ordering is
proposed for describing the H-bonded ferroelectric crystalline systems with
a complex structure of the hydrogen bond network. The model has been
used for the investigation of thermodynamics and dielectric properties of
the GPI crystal. The symmetry analysis of the order parameters responsible
for the mixed (ferro- and antiferroelectric) nature of ordering is performed
within the model. The phase transition into the ferroelectric state is
described. Changes in the dielectric susceptibility of the crystal are studied
in the presence of the transverse external electric field acting along the c -
axis. The results of measurements of temperature and field dependences
of dielectric permittivity ε'c
in the paraelectric phase are presented. The
microscopic mechanism of the observed effects is discussed based on the
comparison of theoretical results and experimental data. A conclusion is
made about the significant role of the ionic groups connected by hydrogen
bonds in the charge transfer. So they make an important contribution into
the polarizability of the GPI crystal along the direction of H-bonded chains. |
| format |
Article |
| author |
Stasyuk, I. Czapla, Z. Dacko, S. Velychko, O. |
| spellingShingle |
Stasyuk, I. Czapla, Z. Dacko, S. Velychko, O. Proton ordering model of phase transitions in hydrogen bonded ferrielectric type systems: the GPI crystal Condensed Matter Physics |
| author_facet |
Stasyuk, I. Czapla, Z. Dacko, S. Velychko, O. |
| author_sort |
Stasyuk, I. |
| title |
Proton ordering model of phase transitions in hydrogen bonded ferrielectric type systems: the GPI crystal |
| title_short |
Proton ordering model of phase transitions in hydrogen bonded ferrielectric type systems: the GPI crystal |
| title_full |
Proton ordering model of phase transitions in hydrogen bonded ferrielectric type systems: the GPI crystal |
| title_fullStr |
Proton ordering model of phase transitions in hydrogen bonded ferrielectric type systems: the GPI crystal |
| title_full_unstemmed |
Proton ordering model of phase transitions in hydrogen bonded ferrielectric type systems: the GPI crystal |
| title_sort |
proton ordering model of phase transitions in hydrogen bonded ferrielectric type systems: the gpi crystal |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2003 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/120748 |
| citation_txt |
Proton ordering model of phase transitions in hydrogen bonded ferrielectric type systems: the GPI crystal / I. Stasyuk, Z. Czapla, S. Dacko, O. Velychko // Condensed Matter Physics. — 2003. — Т. 6, № 3(35). — С. 483-498. — Бібліогр.: 22 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT stasyuki protonorderingmodelofphasetransitionsinhydrogenbondedferrielectrictypesystemsthegpicrystal AT czaplaz protonorderingmodelofphasetransitionsinhydrogenbondedferrielectrictypesystemsthegpicrystal AT dackos protonorderingmodelofphasetransitionsinhydrogenbondedferrielectrictypesystemsthegpicrystal AT velychkoo protonorderingmodelofphasetransitionsinhydrogenbondedferrielectrictypesystemsthegpicrystal |
| first_indexed |
2025-07-08T18:30:46Z |
| last_indexed |
2025-07-08T18:30:46Z |
| _version_ |
1837104572852600832 |
| fulltext |
Condensed Matter Physics, 2003, Vol. 6, No. 3(35), pp. 483–498
Proton ordering model of phase
transitions in hydrogen bonded
ferrielectric type systems: the GPI
crystal∗
I.Stasyuk 1 , Z.Czapla 2 , S.Dacko 2 , O.Velychko 1
1 Institute for Condensed Matter Physics
of the National Academy of Sciences of Ukraine,
1 Svientsitskii Str., 79011 Lviv, Ukraine
2 Institute of Experimental Physics, Wrocław University,
9 M.Borna Sq., 50–204 Wrocław, Poland
Received July 11, 2003
A microscopic model based on the consideration of the proton ordering is
proposed for describing the H-bonded ferroelectric crystalline systems with
a complex structure of the hydrogen bond network. The model has been
used for the investigation of thermodynamics and dielectric properties of
the GPI crystal. The symmetry analysis of the order parameters respon-
sible for the mixed (ferro- and antiferroelectric) nature of ordering is per-
formed within the model. The phase transition into the ferroelectric state is
described. Changes in the dielectric susceptibility of the crystal are studied
in the presence of the transverse external electric field acting along the c -
axis. The results of measurements of temperature and field dependences
of dielectric permittivity ε′
c
in the paraelectric phase are presented. The
microscopic mechanism of the observed effects is discussed based on the
comparison of theoretical results and experimental data. A conclusion is
made about the significant role of the ionic groups connected by hydrogen
bonds in the charge transfer. So they make an important contribution into
the polarizability of the GPI crystal along the direction of H-bonded chains.
Key words: ferroelectricity, hydrogen bond, phase transition, dielectric
susceptibility, electric field
PACS: 64.60.Cn, 77.22.-d, 77.80.-e, 77.80.Bh
1. Introduction
Ferroelectric crystals with hydrogen bonds have been the subject of investigations
for many years. The phase transitions in such systems take place at the decrease of
∗Dedicated to Professor Myroslav Holovko on the occasion of his 60-th birthday.
c© I.Stasyuk, Z.Czapla, S.Dacko, O.Velychko 483
I.Stasyuk et al.
temperature mainly to the ferroelectric or antiferroelectric phase and are connected
with the ordering of protons in hydrogen bonds (the proton on a bond is in a double
potential well). Due to the features of the lattice structure, spontaneous polarization
can appear in the direction which not necessarily coincides with the orientation of
hydrogen bonds. Such an effect can be caused by participation of the other elements
of structure (ionic groups) in the creation of spontaneous dipole moments (as it takes
place, for example, in the KDP crystal). At the same time, the situation is possible
when the hydrogen bond network is complex and the proton ordering produces
the dipole moments diversely oriented in various sublattices. It corresponds to the
ferrielectric state, where the total dipole moment is different from zero and arises
due to partial decompensation of certain components of separate dipole moments.
Dielectric properties of such objects are of interest because of the existing internal
correlation between contributions of protons into the crystal polarization in different
directions. One should expect the manifestation of peculiar effects connected with
the external electric field action. Similar to the field effect in magnetic systems, the
electric field can lead here to the reorientation of dipole moments and can induce
the corresponding phase transitions.
Among the H-bonded ferroelectric type systems, in which the above mentioned
situation takes place, one can consider the glycine phosphite, NH3CH2COOH3PO3
(abbreviated as GPI) crystal belonging to the group of recently synthesized crystals
with the hydrogen bond network possessing a rather complicated structure.
Their special feature is the presence of a chain-like arrangement of hydrogen
bonds sequentially linking the phosphite H(HPO3)
− anions. Besides, the glycini-
um cations [H3NCH2COOH]+ alternately interconnect the chains by the hydrogen
bonds. At room temperature the crystal is monoclinic (the space group P21/a). The
unit cell with dimensions a = 9.792 Å, b = 8.487 Å, c = 7.411 Å, β = 100.43◦
contains four molecular units [1]. The hydrogen bonds linking the phosphite groups
are symmetric (with hydrogens equally disordered over two sites). Investigation of
symmetry and structure of crystal in the ferroelectric phase (below temperature of
the second order phase transition Tc = 224.7 K) was recently performed by means
of X-ray structure analysis [2] and neutron diffraction study [3]. The space group of
this phase is P21. Spontaneous polarization appears along the b-axis and is perpen-
dicular to the direction of the chains of hydrogen bonds (the c-axis). Protons are
ordered at positions approximately equal to one of the two previous sites. Phosphite
tetrahedra are distorted with respect to their shape in the paraelectric phase.
Physical properties of GPI crystal were investigated experimentally in a number
of papers. The effect of deuteration on Tc was studied. The effect is significant: the
phase transition point shifts considerably to higher temperatures (T D
c − TH
c = 97 K
[4]). The dependence of Tc against deuterium contents is linear [5]. The measure-
ments of dielectric constant along the b-axis were performed. The temperature be-
havior of the εb component as well as the spontaneous polarization Ps were studied;
the conclusion was made that the phase transition can be regarded as the one close to
a tricritical [6]. The absolute value of Ps in the state of saturation (∼ 5×10−3 C/m2
[7]) is considerably smaller in comparison to other crystals of hydrogen bonded fam-
484
Hydrogen bonded systems: the GPI crystal
ily. A large anomaly of permittivity in the c-direction has also been detected; at
room temperature εc is higher than εb [7].
Apart from the structure investigations, the measurements of frequency depen-
dence of dielectric constant also lead to the conclusion that the phase transition
in GPI is of the order-disorder type [8,9]. However, the observed dynamics of the
soft relaxational mode connected with the proton flipping in the double potential
well is more complicated in GPI than in the other similar H-bonded ferroelectrics
(for example, BPI). The estimates of the corresponding activation energy and relax-
ation time imply that para-ferroelectric phase transition in GPI does not seem to
be strictly connected only with the movement of protons [9,10]. A reorientation and
deformation of the ionic group (phosphite anions) could also play an important role
(the corresponding structure changes can be extracted from the neutron scattering
data [3]).
On the other hand, the results of experiments with deuterated crystals (DGPI)
give a strong support to the proton ordering mechanism of the phase transition and
point to the important role of hydrogen bonds. Besides the large shift of Tc, one
can mention the “geometric” isotope effect (a linear change of the R0−0 distance in
hydrogen bond at the deuteration of crystal). There is a direct correlation between
∆R0−0 and δTc for partially deuterated GPI crystals [11], which is very similar to
the one for a variety of H-bonded ferroelectrics with phase transitions caused by the
ordering of protons.
One should also mention the high hydrostatic pressure studies of the transition
into ferroelectric phase in GPI. The effect of pressure is found to be significant. With
increase of pressure, Tc decreases with dTc/dp = −11.0 K/kbar [12]. Such a decrease
is comparable with the value dTc/dp = −11.8 K/kbar reported for the BPI crystal
[13], but is larger than for KDP-type ferroelectrics. The hydrostatic pressure effects
also the saturation value of spontaneous polarization. Sensitivity of GPI crystal to
the external stress was also studied by means of ultrasonic investigations of elastic
constants [14]. The high values of linear compressibility β were obtained along the
a and c-directions (β ∼ 0.16× 10−11 m2/N). It is a consequence of a weak coupling
due to the layered arrangement along the a-axis and the hydrogen bonding of ionic
groups along the c-axis. The observed temperature anomalies of elastic constants in
the vicinity of the Tc [15] were explained as a manifestation of the probably pseudo-
proper origin of ferroelectricity in GPI due to anharmonic interaction of the order
parameter with deformation of the crystal lattice.
Investigations of the IR and Raman spectra [10], the thermal expansion coeffi-
cients, spontaneous electrostriction and refraction indices measurements were also
performed [16]. Relatively large values of the thermal expansion coefficients, especial-
ly in the c-axis direction, give some more evidence of the existence of the noticeable
lattice anharmonicity.
Although from the present experimental data the conclusion can be made about
a significant role of hydrogen bonds and ordering of protons in the appearance of
ferroelectric phase in the GPI crystal, an appropriate microscopic model of tran-
sition into this state has not been considered so far. Such a model is proposed in
485
I.Stasyuk et al.
this work due to the information obtained last year about the changes in the struc-
ture (including the changes in distribution of protons on hydrogen bonds) at the
paraelectric-ferroelectric phase transition [2,3]. The symmetry analysis of the order
parameters is performed. The phase transition into ferroelectric phase with Ps ‖ b
at the presence of the external electric field acting along the c-direction is described
based on the model. Attention is paid to the shift of the transition temperature and
to the changes of dielectric constants in the c-direction. The results of measurements
of temperature and field dependences of dielectric permittivity ε′c in the paraelectric
phase are presented as well. The predictions of theoretical investigations are com-
pared with the experimental data. Microscopic mechanism of the observed effects is
discussed.
2. Proton ordering model
In our approach the description of the proton subsystem of the GPI crystal
will be based on consideration of the long-range and short-range interactions of
protons. We start from the structural data for localization of hydrogen bonds and
equilibrium positions of protons in paraelectric and ferroelectric phases [2,3]. There
are four translationally nonequivalent bonds in the unit cell of crystal (the 011−021
and 013−023 bonds as well as other two bonds connected with the mentioned ones
by the point symmetry operations). Their projections on the XZ and Y Z planes
are shown in figure 1 (Cartesian Y and Z-axes coincide with crystallographic axes b
and c, respectively, the cartesian X (or a′)-axis makes an angle of 100.43◦ with the
crystallographic axis c in the (a, c) plane). Directions connected with the ordering
of protons and corresponding to the prevailing occupancy of proton positions (nia >
nib) are indicated by arrows.
As can be seen, the dipole moments di arising due to the displacement of protons
along the bonds compensate mutually in pairs (d1 and d3; d2 and d4) in the low
temperature phase in Z and X-directions. At the same time, they sum up along the
Y -axis leading to the appearance of the spontaneous polarization. Due to the zig-
zag structure of the hydrogen bond network we have the ferrielectric type ordering,
when the ferroelectricity along b-axis is accompanied by the antiferroelectric-like
(antiparallel) arrangement of dipole moments of the neighbouring H-bonded chains
along the c-axis (the chains are oriented in this direction).
According to the data given in [2], in the fully ordered state
d1 = (−a,−b,−c), d2 = (d, e,−f), d3 = (a,−b, c), d4 = (−d, e, f).
(2.1)
where
a = 0.239q, b = 2.361q, c = 0.786q,
d = 1.221q, e = 1.279q, f = 1.568q.
(2.2)
The numerical coefficients are given in Å; q is an effective charge transferred along
the bond at the displacement of proton from one equilibrium position into another.
To describe the proton ordering we use the pseudospin variables Sz
i = 1
2
(nia − nib),
486
Hydrogen bonded systems: the GPI crystal
-2 -1 0 1 2 3 4 5
-1
0
1
2
3
4
5
6
(a) b
a
2 1
34
X
(Å
)
Z (Å)
-2 -1 0 1 2 3 4 5
-2
-1
0
1
2
3
4
5
6
(b)
b
a2 1
34
Y
(Å
)
Z (Å)
Figure 1. Equilibrium position of O- and H-atoms on hydrogen bonds in the unit
cell of GPI crystal in the paraphase (the directions of dipole moments connected
with the ordering of protons in the ferroelectric phase are shown by arrows): (a)
projection on the XZ (a′c) plane; (b) projection on the Y Z (bc) plane.
where niα is the occupation number of proton in the α position on the bond i,
and introduce the dipole moment operators D̂α
i = dα
i Sz
i . The pseudospin mean
values ηi = 〈Sz
i 〉 are equal to zero in the paraphase while in the ferroelectric phase
η1 = η3 6= 0, η2 = η4 6= 0.
The energy of proton subsystem can be written in this representation in the
following general form:
Ĥ = −
∑
nk
∑
α
EαDα
nk −
1
2
∑
nn′
kk′
∑
αβ
Ψkk′
αβ (nn′)Dα
nkD
β
n′k′ ≡ ĤS + ĤL , (2.3)
where n is the unit cell number, k is the number of sublattice, α = x, y, z. The effect
of an external electric field E is taken into account. The short-range interactions
between protons belonging to the neighbouring bonds in the H-bonded chains are
included in the expression (2.3):
ĤS = −1
2
b
∑
µ=a
∑
li
(
jSz
lµ,i1S
z
lµ,i2 + j ′Sz
lµ,i2S
z
lµ,i+1 1
)
. (2.4)
Here and below the bonds k = 1, . . . , 4 in the unit cell n are characterized by the
chain indices µ = a, b and, for the given chain with the index (l, m), by the cell
number i and the number of bond s = 1, 2 in the cell (l = 1, . . . , M , i = 1, . . . , L,
LM = N).
For a description of the long-range proton-proton interactions we use the mean
field approximation. In this approach the corresponding part of the Hamiltonian
487
I.Stasyuk et al.
(2.3) is
ĤL = U −
∑
µli
(
Hµ
1 Sz
lµ,i1 + Hµ
2 Sz
lµ,i2
)
, (2.5)
U = N
[1
2
j11(η
2
1 + η2
3) +
1
2
j22(η
2
2 + η2
4)
+
1
2
(j12 + j21)(η1η2 + η3η4) +
1
2
(j14 + j23)(η1η4 + η2η3)
+ j14η1η3 + j24η2η4
]
,
and the fields Hµ
1,2 acting on protons on the bonds s = 1, 2 are introduced
H
a(b)
1 = j11η1(3) + j12η2(4) + j13η3(1) + j14η4(2) ± aEx + bEy ± cEz ,
H
a(b)
2 = j12η1(3) + j22η2(4) + j23η3(1) + j24η4(2) ∓ dEx − eEy ± fEz , (2.6)
where jkk′ are the constants of the long-range interactions.
The self-consistent set of equations for the ηi mean values is obtained here with-
in the scheme based on exact consideration of short-range interactions. Such an
interaction can be quite large in GPI, since redistribution of protons on the bonds
is accompanied by deformation and reorientation of ionic groups connected by the
bonds [3]. The procedure applied below is well known and accepted for the quasi-
one-dimensional systems (see [17,18]) and corresponds to the two-particle cluster
approximation [19,20].
3. Phase transition in proton subsystem
We use the transfer-matrix method for the calculation of partition function and
other thermodynamic functions of H-bonded chains in GPI in the framework of the
model considered. In such a case a partition function of the proton subsystem can
be written as
Z = Tr e−βĤ = e−βUZM
a ZM
b , (3.1)
where
Zµ = (λµ)L, λµ =
A + B
2
+
[
(
A − B
2
)2
+ C
]1/2
,
A + B
2
= 2 cosh
βHµ
1
2
cosh
βHµ
2
2
cosh
β(j + j ′)
4
+ 2 sinh
βHµ
1
2
sinh
βHµ
2
2
sinh
β(j + j ′)
4
,
A − B
2
= 2 sinh
βHµ
1
2
cosh
βHµ
2
2
cosh
β(j + j ′)
4
+ 2 cosh
βHµ
1
2
sinh
βHµ
2
2
sinh
β(j + j ′)
4
,
C = 4 cosh2 βHµ
2
2
cosh2 β(j − j ′)
4
− 4 sinh2 βHµ
2
2
sinh2 β(j − j ′)
4
, (3.2)
488
Hydrogen bonded systems: the GPI crystal
λa,(b) is the maximum value of the transfer matrix for the a(b) chain and β = 1/Θ =
1/kBT . Respectively, the free energy per a unit cell is
F = U/N − Θ lnλa − Θ lnλb . (3.3)
The self-consistency parameters ηi are determined from the minimization proce-
dure
∂F
∂ηi
= 0, δ2F > 0, (3.4)
which reduces in our case to the following set of equations
1
N
∂U
∂ηi
−
b
∑
µ=a
Θ
λµ
(
∂λµ
∂Hµ
1
∂Hµ
1
∂ηi
+
∂λµ
∂Hµ
2
∂Hµ
2
∂ηi
)
= 0. (3.5)
These equations describe the continuous transition (of the second order) to the
phase with nonzero values of the η1 and η2 parameters and spontaneous polarization
along b-axis. In the presence of the external field Ez, the phase transition remains
and the order of this transition does not change (the field Ez acts along the H-
bonded chains perpendicularly to the ferroelectric axis). Such a conclusion follows
from the analysis of the free energy behaviour in the vicinity of a transition point.
The temperature of phase transition Tc is determined from the equation
∆(+) ≡ 1 − Θ(ϕ
(+)
1 Q11 + ϕ
(+)
4 Q22) − Θ(ϕ
(+)
2 + ϕ
(+)
3 )Q12
+ Θ2(Q11Q22 − Q2
12)(ϕ
(+)
1 ϕ
(+)
4 − ϕ
(+)
2 ϕ
(+)
3 ) = 0, (3.6)
which is obtained from the condition of the appearance of non-zero solutions of
equation (3.5). Here
Q11 =
1
4
β2S + βa
(+)
11 E2
z , Q22 =
1
4
β2S + βa
(+)
22 E2
z , Q12 =
1
4
β2R + βa
(+)
12 E2
z
(3.7)
and
R =
sinh 1
4
β(j + j ′)
cosh 1
4
β(j − j ′)
, S =
cosh 1
4
β(j + j ′)
cosh 1
4
β(j − j ′)
, (3.8)
ϕ
(±)
1 = j11 ± j13, ϕ
(±)
2 = ϕ
(±)
3 = j12 ± j14, ϕ
(±)
4 = j22 ± j24 , (3.9)
where a
(±)
ik are the parameters formed by the long-range interaction constants.
It can be seen that under the action of the field Ez the transition temperature
decreases. The shift ∆Tc = Tc|Ez=0 − Tc is proportional to E2
z at small values of
Ez. The effect takes place also in the absence of the short-range correlations. In
particular, in the case when ϕ
(+)
1 ϕ
(+)
4 ≈ ϕ
(+)
2 ϕ
(+)
3
kT 0
c =
1
4
(ϕ
(+)
1 + ϕ
(+)
4 ), ∆Tc = −k2
1ϕ
(+)
1 + k2
2ϕ
(+)
4
(
ϕ
(+)
1 + ϕ
(+)
4
)2 E2
z , (3.10)
489
I.Stasyuk et al.
0.000 0.005 0.010
0.08
0.09
0.10
~
~T
c
E
z
Figure 2. The field dependence of the paraelectric-ferroelectric phase transition
temperature calculated based on the proton ordering model (2.4) and (2.5).
where k1 and k2 are the constants connected with linear (with respect to field Ez)
contributions in the internal fields Hµ
1,2; T 0
c = Tc|Ez=0.
Respective changes take place in the behaviour of dielectric susceptibility χyy
along the ferroelectric axis. In the model, the contribution of the proton subsystem
is given by the expression
χyy =
1
ε0vc
∑
k
∂〈Dy
nk〉
∂Ey
∣
∣
∣
∣
Ey=0
, (3.11)
where ε0 is the electric permittivity of vacuum, vc is the volume of the unit cell.
As it follows from equations (3.5) at Ey → 0, Ez 6= 0, in the case of paraphase
(T > Tc)
χyy =
2Θ
vc∆(+)
{
b2Q11 + e2Q22 − 2beQ12
− Θ(Q11Q22 − Q2
12)[ϕ
(+)
4 b2 + ϕ
(+)
1 e2 + (ϕ
(+)
2 + ϕ
(+)
3 )be]
}
. (3.12)
In the external field the susceptibility χyy diverges at Tc while at T > Tc it decreases
proportionally to E2
z under a small enough field. This effect can be interpreted as a
result of the shift of Tc to the lower temperature side.
There are also changes in the dielectric susceptibility χzz ≡ χc under the effect
of the field Ez. In the model, it is given by the expression
χzz = χyy
∣
∣
∣
ϕ
(+)
i
→ϕ
(−)
i
; b→c, e→−f
(3.13)
formally obtained from (3.12) by substitution of parameters ϕi. Calculations show
that χzz also decreases proportionally to E2
z at T > Tc. The reason for such a be-
haviour may appear to be more complicated than in the case of the χyy susceptibility
(see discussion below).
490
Hydrogen bonded systems: the GPI crystal
0.095 0.100
0
100
200
~χ
yy
~T
~
~
~
E
z
= 0
E
z
= 0.003
E
z
= 0.006
0.095 0.100
20
25
30
35
40
T
c
~
~
~
~
E
z
= 0
E
z
= 0.003
E
z
= 0.006
~χ
zz
~T
Figure 3. The temperature dependence of (a) longitudinal χyy and (b) transverse
χzz susceptibility of GPI crystal within the proton ordering model.
The above mentioned can be illustrated by the results of numerical calculations.
In figure 2 the field dependence of Tc is shown calculated according to equation (3.6).
Respectively, in figure 3 the temperature dependences of χzz at different values of
the field Ez are presented.
In our numerical calculations and in figures 2 and 3 all energetic quantities were
taken in relation to j + j ′ (the sum of the short-range interaction constants) and
dipole moments were given in relation to µ0 = q
√
a2 + b2 + c2; respectively, the
dimensionless ratios were introduced. For example
Θ̃ =
Θ
(j + j ′)
, Ẽz = Ez
µ0
(j + j ′)
, χ̃αα = χαα
ε0vc(j + j ′)
µ2
0
.
The long-range interaction has been modelled by two parameters Φ and Φ̃ respon-
sible for the interaction along a separate chain and between chains, respectively:
j̃11 = (a2 + b2 + c2)(Φ + Φ̃), j̃21 = j̃12,
j̃12 = (−ad − be + cf)(Φ + Φ̃), j̃22 = (d2 + e2 + f 2)(Φ + Φ̃),
j̃13 = (−a2 + b2 − c2)Φ̃, j̃23 = j̃14,
j̃14 = (ad − be − cf)Φ̃, j̃24 = (−d2 + e2 − f 2)Φ̃,
where j̃ik = jikµ
2
0/(j+j ′). The values Φ = 0.053, Φ̃ = −0.031 were taken. This choice,
as well as the values j + j ′ = 0.204 eV, T̃ 0
c = 0.094, µ0 = 7× 10−30 C·m, q = 0.17e1,
correspond to the values of the observed critical temperature (T 0
c = 224.7 K [7]) and
saturated spontaneous polarization (Ps = 5 × 10−3 C/m2 [7]).
In this case, for susceptibility we have χ̃zz = 25−35 (figure 3b) at T > Tc, which
corresponds to the change of the true susceptibility from 7 to 10 (the coefficient
µ2
0/ε0vc(j + j ′) is equal to 0.28 at the chosen parameter values). To achieve an
1This value of the effective charge redistributed between the ends of the hydrogen bonds agrees
with the calculated one for O–H–O bond within the quantum chemical approach [21].
491
I.Stasyuk et al.
agreement with the experimental data (χzz ≈ ε′c ≈ 200 at T = 273 K) we have to
suppose that the z-components of the effective dipole moments, arising due to the
displacements of protons along the bonds under the field effect, are approximately
5–6 times larger than the initially taken values. The necessary result is obtained at
the increase of the c and f parameters.
In this case, the dimensionless value Ẽz = 1.5 × 10−3 corresponds to the field
Ez = 1 MV/m. The difference ∆T̃ = 0.002 at Ẽz = 0.006 corresponds to the shift
∆Tc = 4.5 K at the field Ez = 4.0 MV/m (see the experimental data below).
It should be mentioned that in the proton ordering model, which reduces in
the considered case to the four-sublattice Ising model, the phase transition into the
ordered phase is of the second order. It concerns both the transition into ferroelectric
phase (phase F ) with polarization along b-axis and the transition into the ordered
phase (hypothetical phase F ′) with the orientation of dipole moments parallel to c-
axis (and, respectively, antiparallel orientation with respect to b-axis). Such a phase
is not realized in GPI, but the example of phase transition with the substitution
of ferroelectric and antiferroelectric orderings is known in the literature; in this
connection one can mention the DMAGaS crystal [22].
Because of the mentioned property of the model the dielectric susceptibility χyy
decreases in paraphase under the field Ez. The orientation of the dipole moments
parallel to c-axis (induced by the field Ez), counteracts the appearance of ferroelec-
tric phase with Ps ‖ b. It leads to the decrease of the critical temperature Tc and,
as it was mentioned above, to the effective decrease of the χyy component. On the
other hand, the ordering induced along c-axis suppresses the polarization fluctua-
tions in this direction which results in the going down of the χzz component. Large
enough values of the component observed experimentally at T > Tc in the wide
temperature range and the increase of χzz at the decrease of T can be connected
with the proximity to the point of the virtual phase transition to the phase F ′ (the
latter could be realized in the GPI crystal in the absence of transition to the phase
F at Tc).
In addition, a certain role in the formation of the high polarizability of the crystal
in the direction of the H-bonded chains belongs to the short-range proton interac-
tions. This can be seen considering the expression for “nonperturbed” susceptibilities
in the paraphase (when the long range interaction is neglected)
χyy =
β
2ε0vc
[
(b2 + e2)S − 2beR
]
, χzz =
β
2ε0vc
[
(c2 + f 2)S + 2cfR
]
. (3.14)
Here, only the interaction between the neighbouring hydrogen bonds along a chain
is taken into account. The ratio χzz/χyy changes with temperature approaching to
maximum values at low temperatures; formally
χzz
χyy
∣
∣
∣
∣
T→0
=
(c + f)2
(b − e)2
. (3.15)
At high temperatures it goes to the value
χzz
χyy
∣
∣
∣
∣
T→∞
=
c2 + f 2
b2 + e2
. (3.16)
492
Hydrogen bonded systems: the GPI crystal
0.0 0.5 1.0
0
1
2
3
~
T
~
~
χ
zz
—
χ
yy
Figure 4. The ratio χ̃zz/χ̃yy as a function of temperature in the short-range
interaction approximation.
At T̃ 6 0.12(j+j ′), which corresponds to the condition T 6 285 K, χzz/χyy ∼ 3.0
(see figure 4, χzz/χyy = χ̃zz/χ̃yy). We see that due to the considered mechanism the
χzz component of susceptibility can about three times exceed the χyy component.
The observed ratio of the εc and εb permittivities at room temperature is much larger
(ε′c/ε
′
b ∼ 10 [6,7]). The long-range interaction, which favours the proton arrangement
characteristic of the hypothetic F ′ phase, can be the additional reason for such an
anomaly.
4. Experimental
Hereinbelow we present the results of the experimental investigations of the tem-
perature dependences of the dielectric permittivity ε′c = 1 + χzz in the presence of
the electric field Ez applied in the direction of the c-axis. In the absence of external
field the temperature behaviour of ε′c was measured earlier [7].
Single crystals of GPI were grown from saturated water solution of deuterated
polycrystals by slow evaporation method at temperature of 304 K. The dielectric
properties were investigated along the crystallographic c-axis. The capacity of the
samples was measured with a precision LCR-meter HP 4284 A at the measuring
field frequency equal to 10 kHz and amplitude of 4 V/cm. The relative electric
permittivity was measured as a function of temperature with various values of dc
electric field ranging from zero to 4 × 106 V/m. The rate of temperature changes
was equal to 0.5 K/min
The temperature dependences of relative electric permittivity ε′c for various val-
ues of dc field obtained from dielectric studies are presented in figure 5. At the
applied field equal to zero, a large value of permittivity is observed at room tem-
493
I.Stasyuk et al.
205 210 215 220 225 230 235
150
200
250
300
350
400
E=0
- 0.5MV/m
+1.0MV/m
-1.5MV/m
+2.0MV/m
-2.5MV/m
+3.0MV/m
+4.0MV/m
ε'
c
T (K)
-3.5MV/m
210 215 220 225
2
3
4
5
6
E=0
T (K)
E=1MV/m
E=3MV/m
E=2MV/m
E=4MV/m1/
ε' c
(1
0-3
)
Figure 5. The temperature depen-
dence of dielectric permittivity ε′c mea-
sured at different values of dc electric
field Ez.
Figure 6. The temperature depen-
dences of inverse dielectric permittivi-
ty ε′c
−1 at different values of dc electric
field Ez (experiment).
perature and its increase is observed on the cooling run. The maximum value is
noted at Tc while below this temperature the permittivity decreases. Temperature
dependences of permittivity under the electric field are similar to that observed at
zero field. However, as it can bee seen in figure 5, in the vicinity of phase transi-
tion the additional increase of permittivity is observed. The increase becomes higher
with the increase of electric field intensity. Besides, its position is shifted to lower
temperatures with an increase of the electric field intensity.
The results obtained from figure 5 are presented in the form of temperature
dependences of inverse permittivity in figure 6. The fulfillment of the Curie-Weiss law
is seen both in paraphase and below the phase transition region in the ferroelectric
phase.
The change of permittivity under the field in the vicinity of Tc has a form of the
jump-like increase. We can introduce, respectively, a jump of 1/ε′c (defined as ∆1, see
figure 7) in the middle point of the interval at which the transition from the straight
line for 1/ε′c in paraphase to the straight line representing the strong increase of ε′c
in ferroelectric phase takes place. Without the field, the jump ∆1 is equal to zero.
Assuming internal bias field Eb existence and introducing the field (±E − Eb) we
can notice that the value of the ∆1 increases linearly with (±E − Eb)
2 as it follows
from the experimental data (Eb value is small in comparison with the value of the
applied fields).
As a temperature Tc of the phase transition we assume the temperature that
corresponds to the jump ∆1. The position of the jump changes with the field Ec.
The shift of phase transition temperature obtained in such a way is presented in
figure 8. The shift is satisfactorily described by the parabolic function with the
maximum at the temperature T 0
c which is the point of the phase transition into
ferroelectric phase at the zero field (the slight asymmetry of the Tc versus Ez curve
is caused by the presence of the small bias field). Independent measurements of the
494
Hydrogen bonded systems: the GPI crystal
215 220 225 230
3.5
4.0
4.5
T
c
∆∆∆∆
1
T (K)
1/
ε' c
(1
0-3
)
E
z
=2MV/m
Figure 7. The jump of inverse dielectric permittivity ε′c
−1 in the point of phase
transition.
-4 -3 -2 -1 0 1 2 3 4
219
220
221
222
223
T=const
E
z
=const
T c (
K
)
E
z
(MV/m)
-4 -3 -2 -1 0 1 2 3 4
230
232
234
236
238
240
242
230.9K
ε'
c
E -E
corr
(MV/m)
Figure 8. The field dependence of the
phase transition point, taken from the
measurements at fixed values of electric
field Ez or temperature T .
Figure 9. The field dependence of rel-
ative dielectric permittivity ε′c at T =
230.9 K (experiment).
495
I.Stasyuk et al.
field dependences of ε′c at fixed temperatures (T 6 T 0
c ) reveal the anomalies similar
to the described above, at the field values which correspond to the curve in figure 8.
This fact confirms the existence of the paraelectric-ferroelectric transition at Ez 6= 0
and points to the possibility of transformation from one phase to another at change
of temperature or field Ez.
The field dependence of the relative permittivity ε′c measured in the paraphase
(T = 230.9 K) is illustrated by the curve given in figure 9. It was obtained using the
procedure of eliminating the hysteresis behaviour of the ε′c, caused by the internal
field changing its value with the same frequency as the frequency of the applied
field but shifted in phase. The effective field introduced as a difference of both fields
(Eeff = E − Ecorr) is used, and in that case the maximum value of permittivity
is reached at zero field. As a result, a quadratic dependence ε′c(Ec) is obtained; it
describes decrease of ε′c (and, respectively, the increase of ε′c
−1) with the field at
T > Tc. Such a behaviour of ε′c is characteristic for the whole temperature range in
paraelectric phase.
5. Discussion
The experimentally obtained and theoretically calculated temperature depen-
dences of direct (ε′c) and inverse (ε′c
−1) permittivity as well as their changes in the
presence of the field Ez are in a good mutual agreement. In particular, the described
within the model decrease of ε′c (increase of ε′c
−1) with the field in paraphase is in
a direct correspondence with the experimental curve in figure 9. The changes of χzz
at the field values from figure 3 (∆χzz ≈ 1.3 at Ez = 1.0 MV/m; ∆χzz ≈ 5.1 at
Ez = 2.0 MV/m; T = 239 K) agree with the quadratic dependence of χzz on Ez.
As it was mentioned, the quantitative agreement of the calculated values of
χzz with the data of measurements can be obtained by increase of the effective
dipole moments along the c-axis connected with the proton displacements. It is
a manifestation of an existence of a certain additional way of the charge transfer
(and, respectively, polarization), caused by the redistribution of protons between
equilibrium positions on bonds (an effective charge increases: q = 0.17e → q∗ =
(0.85−1.00)e). Most probably, such a mechanism consists in the participation of
the intermediate ionic groups (phosphite ions), which are shifted and distorted at
the proton ordering. Their contribution can be taken into account implicitly in our
model by the renormalization of the corresponding components of dipole moments
connected with hydrogen bonds.
The change of Tc under the field Ez calculated within the proton ordering model
is qualitatively the same as the measured one (figure 8). In both cases the shift
∆Tc is described by the parabolic function of the field. As it was mentioned above,
we can achieve a quantitative agreement between them using the optimal choice of
numerical values of the interaction constants in the model.
The performed theoretical consideration shows that the changes in the longitu-
dinal (with respect to the ferroelectric axis) susceptibility χyy and, respectively, in
permittivity ε′b should also take place under the field Ez. An experimental exami-
496
Hydrogen bonded systems: the GPI crystal
nation of the predicted dependences shown in figure 3a is a difficult task when the
dielectric measurements are used (it is connected with the principal problems arising
in the case of application of the crossed fields).
References
1. Averbuch-Pouchot M.-T. // Acta Crystallogr. C, 1993, vol. 49, p. 815–818.
2. Shikanai F.S., Komukae M., Czapla Z., Osaka T. // J. Phys. Soc. Japan, 2002, vol. 71,
p. 498–503.
3. Taniguchi H., Machida M., Koyano N. // J. Phys. Soc. Japan, 2003, vol. 72, p. 1111–
1117
4. Baran J., Śledź M., Jakubas R., Bator G. // Phys. Rev. B, 1997, vol. 55, p. 169–172.
5. Dacko S., Czapla. Z. // Ferroelectrics letters, 2000, vol. 27(1–2), p. 17–21.
6. Tchukvinskyi B., Cach R., Czapla Z., Dacko S. // Phys. Status Solidi (a), 1998,
vol. 165, p. 309–316.
7. Dacko S., Czapla Z., Baran J., Drozd M. // Phys. Lett. A, 1996, vol. 223, p. 217–220.
8. Tchukvinski R., Czapla Z., Sobiestianskas R., Brilingas A., Grigas J., Baran J. //
Acta Phys. Polonica A, 1997, vol. 92, p. 1191–1196.
9. Sobiestinskas R., Brilingas A., Czapla Z. // J. Korean Phys. Soc., 1998, vol. 32,
p. S397–S398.
10. Baran J., Bator G., Jakubas R., Śledź M. // J. Phys.: Condens. Matter, 1996, vol. 8,
p. 10647–10658.
11. Shikanai F., Yamasaki M., Komukae M., Osaka T. // J. Phys. Soc. Japan, 2003,
vol. 72, p. 325–329.
12. Yasuda N., Sakurai T., Czapla Z. // J. Phys.: Condens. Matter, 1997, vol. 9, p. L347–
L350.
13. Launer S., Maire L.E., Schaack G., Haussuhl S. // Ferroelectrics, 1992, vol. 135, p. 257.
14. Deepthy A., Bhat H.L., Alex A.V., Philip J. // Phys. Rev. B, 2000, vol. 62, p. 8752–
8758.
15. Furtak J., Czapla Z., Kityk A.V. // Naturforsch. Z., 1997, vol. 52a, p. 778–782.
16. Andriyevski B., Czapla Z., Romanyuk M., Myshchyshyn O. // Phys. Stat. Sol. (a),
2000, vol. 177, p. 575–582.
17. Kojyo N., Onodera Y. // J. Phys. Soc. Japan, 1988, vol. 57, p. 4391–4402.
18. Linares J., Spiering H., Varret F. // Eur. Phys. J. B, 1999, vol. 10, p. 271–275.
19. Levitskii R.R., Sorokov S.I. // Condens. Matter Phys., 1994, iss. 3, p. 79–115.
20. Sorokov S.I. // Condens. Matter Phys., 2002, vol. 5, No. 4(32), p. 617–623.
21. Stasyuk I.V., Sizonenko Yu.V., Stetsiv R.Ya. // J. Phys. Studies, 1998, vol. 2, No. 4,
p. 463–475.
22. Stasyuk I.V., Velychko O.V. // Phase transitions, 2001, vol. 73, p. 483–501.
497
I.Stasyuk et al.
Модель протонного впорядкування для фазових
переходів у системах сегнетіелектричного типу з
водневими зв’язками: кристал GPI
І.Стасюк 1 , З.Чапля 2 , С.Дацко 2 , О.Величко 1
1 Інститут фізики конденсованих систем НАН Укpаїни,
79011 Львів, вул. Свєнціцького, 1
2 Інститут експериментальної фізики, Університет м. Вроцлав,
Польща, 50–204 Вроцлав, пл. М.Борна, 9
Отримано 11 липня 2003 р.
Для опису сегнетоелектричних водневозв’язаних кристалічних сис-
тем із складною структурою сітки водневих зв’язків запропонова-
но мікроскопічну модель, яка базується на врахуванні впорядкуван-
ня протонів. Її застосовано для дослідження термодинаміки і ді-
електричних властивостей кристалу GPI. В рамках моделі дано си-
метричний аналіз параметрів порядку, відповідальних за змішаний
(сегнето- і антисегнетоелектричний) характер впорядкувань. Описа-
но фазовий перехід до сегнетоелектричного стану та вивчено зміни
у діелектричній сприйнятливості кристалу при наявності зовнішньо-
го поперечного електричного поля прикладеного вздовж c -осі. На-
ведено результат вимірювань температурних і польових залежнос-
тей діелектричної проникності ε′
c
у параелектричній фазі. На основі
порівняння теоретичних результатів і даних експерименту обгово-
рюється мікроскопічний механізм спостережуваних ефектів. Зроб-
лено висновок про важливу роль іонних груп, з’єднаних водневими
зв’язками, у переносі заряду. Вони дають вагомий внесок до поля-
ризованості кристалу GPI у напрямку вздовж ланцюжків водневих
зв’язків.
Ключові слова: сегнетоелектричність, водневий зв’язок, фазовий
перехід, діелектрична проникність, електричне поле
PACS: 64.60.Cn, 77.22.-d, 77.80.-e, 77.80.Bh
498
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