Influence of electric fields on dielectric properties of GPI ferroelectric
Using modified microscopic model of GPI by taking into account the piezoelectric coupling with strains εi in the frames of two-particle cluster approximation, the components of polarization vector and static dielectric permittivity tensor of the crystal at applying the external transverse electri...
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| Cite this: | Influence of electric fields on dielectric properties of GPI ferroelectric / I.R. Zachek, R.R. Levitskii, A.S. Vdovych, I.V. Stasyuk // Condensed Matter Physics. — 2017. — Т. 20, № 2. — С. 23706: 1–17. — Бібліогр.: 22 назв. — англ. |
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irk-123456789-1570022019-06-20T01:26:07Z Influence of electric fields on dielectric properties of GPI ferroelectric Zachek, I.R. Levitskii, R.R. Vdovych, A.S. Stasyuk, I.V. Using modified microscopic model of GPI by taking into account the piezoelectric coupling with strains εi in the frames of two-particle cluster approximation, the components of polarization vector and static dielectric permittivity tensor of the crystal at applying the external transverse electric fields E₁ and E₃ are calculated. An analysis of the influence of these fields on thermodynamic characteristics of GPI is carried out. A satisfactory quantitative description of the available experimental data for these characteristics has been obtained at a proper choice of the model parameters Використовуючи модель GPI, модифiковану шляхом врахування п’єзоелектричного зв’язку з деформацiями εi в наближеннi двочастинкового кластера, розраховано компоненти вектора поляризацiї та тензора статичної дiелектричної проникностi кристала при прикладаннi зовнiшнiх поперечних електричних полiв E₁ i E₃. Проведено аналiз впливу цих полiв на дiелектричнi характеристики GPI. При належному виборi параметрiв теорiї отримано задовiльний кiлькiсний опис наявних експериментальних даних для цих характеристик. 2017 Article Influence of electric fields on dielectric properties of GPI ferroelectric / I.R. Zachek, R.R. Levitskii, A.S. Vdovych, I.V. Stasyuk // Condensed Matter Physics. — 2017. — Т. 20, № 2. — С. 23706: 1–17. — Бібліогр.: 22 назв. — англ. 1607-324X PACS: 77.22.-d, 77.22.Ch, 77.22.Ej, 77.65.-j, 77.80.Bh DOI:10.5488/CMP.20.23706 arXiv:1707.00619 http://dspace.nbuv.gov.ua/handle/123456789/157002 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Using modified microscopic model of GPI by taking into account the piezoelectric coupling with strains εi
in
the frames of two-particle cluster approximation, the components of polarization vector and static dielectric
permittivity tensor of the crystal at applying the external transverse electric fields E₁ and E₃ are calculated. An
analysis of the influence of these fields on thermodynamic characteristics of GPI is carried out. A satisfactory
quantitative description of the available experimental data for these characteristics has been obtained at a
proper choice of the model parameters |
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Zachek, I.R. Levitskii, R.R. Vdovych, A.S. Stasyuk, I.V. |
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Zachek, I.R. Levitskii, R.R. Vdovych, A.S. Stasyuk, I.V. Influence of electric fields on dielectric properties of GPI ferroelectric Condensed Matter Physics |
| author_facet |
Zachek, I.R. Levitskii, R.R. Vdovych, A.S. Stasyuk, I.V. |
| author_sort |
Zachek, I.R. |
| title |
Influence of electric fields on dielectric properties of GPI ferroelectric |
| title_short |
Influence of electric fields on dielectric properties of GPI ferroelectric |
| title_full |
Influence of electric fields on dielectric properties of GPI ferroelectric |
| title_fullStr |
Influence of electric fields on dielectric properties of GPI ferroelectric |
| title_full_unstemmed |
Influence of electric fields on dielectric properties of GPI ferroelectric |
| title_sort |
influence of electric fields on dielectric properties of gpi ferroelectric |
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Інститут фізики конденсованих систем НАН України |
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2017 |
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http://dspace.nbuv.gov.ua/handle/123456789/157002 |
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Influence of electric fields on dielectric properties of GPI ferroelectric / I.R. Zachek, R.R. Levitskii, A.S. Vdovych, I.V. Stasyuk // Condensed Matter Physics. — 2017. — Т. 20, № 2. — С. 23706: 1–17. — Бібліогр.: 22 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT zachekir influenceofelectricfieldsondielectricpropertiesofgpiferroelectric AT levitskiirr influenceofelectricfieldsondielectricpropertiesofgpiferroelectric AT vdovychas influenceofelectricfieldsondielectricpropertiesofgpiferroelectric AT stasyukiv influenceofelectricfieldsondielectricpropertiesofgpiferroelectric |
| first_indexed |
2025-07-14T09:21:04Z |
| last_indexed |
2025-07-14T09:21:04Z |
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| fulltext |
Condensed Matter Physics, 2017, Vol. 20, No 2, 23706: 1–17
DOI: 10.5488/CMP.20.23706
http://www.icmp.lviv.ua/journal
Influence of electric fields on dielectric properties of
GPI ferroelectric
I.R. Zachek1, R.R. Levitskii2, A.S. Vdovych2, I.V. Stasyuk2
1 Lviv Polytechnic National University, 12 Bandera St., 79013 Lviv, Ukraine
2 Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine,
1 Svientsitskii St., 79011 Lviv, Ukraine
Received April 20, 2017, in final form May 23, 2017
Using modified microscopic model of GPI by taking into account the piezoelectric coupling with strains εi in
the frames of two-particle cluster approximation, the components of polarization vector and static dielectric
permittivity tensor of the crystal at applying the external transverse electric fields E1 and E3 are calculated. An
analysis of the influence of these fields on thermodynamic characteristics of GPI is carried out. A satisfactory
quantitative description of the available experimental data for these characteristics has been obtained at a
proper choice of the model parameters.
Key words: ferroelectrics, electric field, polarization, dielectric permittivity, phase transition
PACS: 77.22.-d, 77.22.Ch, 77.22.Ej, 77.65.-j, 77.80.Bh
1. Introduction
One of the actual problems in physics of ferroelectric materials is the study of the effects that appear
under the action of an external electric field. It can be a powerful tool for purposeful control of their
physical characteristics. The effects of the action of external fields depend both on the intensity and the
type of such an action, and on the properties of the materials. The application of an electric field is a very
important instrument for the investigation of ferroelectric materials with a complex spatial arrangement
of the local effective dipole moments. Consequently, phase transitions with different order parameters
connected with each other can take place in these materials. In particular, it appears possible to influence
this system by means of an electric field, which is perpendicular to a spontaneous polarization, and to
study the changes of polarization and the other dielectric properties.
One of the most interesting examples of a crystal sensitive to an electric field effect is the glycinium
phosphite (GPI), which belongs to ferroelectric materials with hydrogen bonds [1, 2]. At the room
temperature this crystal has a monoclinic structure (space group P21/a) [3]. The hydrogen bonds between
the tetrahedra HPO3 form infinite chains along the crystallographic c-axis (figure 1). There are two types
of hydrogen bonds with the length ∼ 2.48 Å and ∼ 2.52 Å [3–5]. The ordering of protons on these bonds
[4, 5] causes an antiparallel orientation of the components of dipole moments of the equivalent hydrogen
bonds along the crystallographic axes a and c in the neighbouring chains. However, the changes of the
distances between ions in the tetrahedra HPO3 and the parallel ordering of the corresponding components
of dipole moments along the b-axis in the chains causes a total dipole moment along this axis.
Consequently, at the temperature 225 K the crystal passes to the ferroelectric state (space group P21)
with a spontaneous polarization perpendicular to the chains of hydrogen bonds. It is necessary to note
that the phase transition in GPI is closely connected with the short- and long-range interactions within
these chains and between them.
The study of the effect of deuteration on Tc witnesses in favour of the proton ordering mechanism of
a phase transition due to a strong isotopic shift of the transition temperature (TD
c − TH
c = 97 K [6]).
This work is licensed under a Creative Commons Attribution 4.0 International License . Further distribution
of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
23706-1
https://doi.org/10.5488/CMP.20.23706
http://www.icmp.lviv.ua/journal
http://creativecommons.org/licenses/by/4.0/
I.R. Zachek, R.R. Levitskii, A.S. Vdovych, I.V. Stasyuk
Figure 1. The lattice structure of glycinium phosphite crystal [3].
The results of measuring the frequency dependence of dielectric permittivity [7, 8] also testify that
the phase transition in this crystal is of the order-disorder type. It should be also mentioned that the data
obtained based on the slow neutron scattering investigation, indicate the reorientations and deformations
of the ionic groups (phosphite ions). The revealed temperature anomalies of elastic constants near Tc [9]
manifest an important role of deformation processes in a phase transition in GPI.
Highly important are the investigations of transverse electric fields effects on the physical character-
istics of GPI. A crystal seems to be quite special in this respect. The experiment, carried out in [10, 11],
showed a unique sensitivity to a transverse field Ez . As it was established, such a field, applied to the
crystal in ferroelectric phase (at T < T0
c ), is capable of reorienting the local dipole moments that are
connected with protons on hydrogen bonds and with adjacent ionic glycine groups. Consequently, at
some critical field Ec
z there occurs a phase transition, at which a spontaneous polarization along OY -axis
disappears and only the component Pz remains. Such an effect resembles the well known spin-flop
transition in antiferromagnetics under the action of an external magnetic field. On the other hand, as
was shown in [10, 11], under the action of the field Ez there occurs a decrease of critical temperature
of ferroelectric phase transition proportionally to E2
z . The existence of considerable (and increasing with
the field) anomalies of transverse dielectric permittivity εzz in the region of transition at Ez , 0 was
revealed.
An explanation of the discovered effects was given in [10] and [11, 12] based on the phenomenological
Landau theory and within the microscopic model approach, respectively. However, it failed to achieve
a full quantitative description of the observed temperature and field behaviour of εzz , inasmuch as the
reasons of a smeared character of such dependences remain unclear.
In the present work we continue the study of the transverse field effect, based on a microscopic
description within the model of a deformed crystal [13]. We supplement the approach, applied in [11, 12],
by taking into account the lattice strains and piezoelectric coupling. At the same time, our goal is to
consider the wider range of phenomena connected with the action of transverse fields Ez and Ex on a
ferroelectric phase transition and on dielectric and piezoelectric characteristics of GPI crystal.
2. The model
We consider a system of protons in GPI, localised on O–H. . .O bonds, which form zigzag chains along
the c-axis of a crystal. Dipole moments ®dq f (q is a number of a primitive cell, f = 1, . . . , 4) are ascribed
to the protons on the bonds. In the ferroelectric phase, the dipole moments compensate each other ( ®dq1
with ®dq3, ®dq2 with ®dq4) in directions Z and X, and simultaneously supplement each other in the direction
Y , creating a spontaneous polarization. Vectors ®dq f are oriented at some angles to crystallographic axes
and have longitudinal and transverse components along the b-axis (figure 2).
23706-2
Influence of electric fields
3
4
1
2
AB
a
c
I
II
II
I
3
4
A
1
2
B
a
b
c
a
b
c
I
III
II
Figure 2. (Color online) Orientations of vectors ®dq f in the primitive cell in the ferroelectric phase.
Pseudospin variables
σq1
2 , . . . ,
σq4
2 describe the changes connected with reorientation of the dipole
moments of the base units: dq f = µ f
σq f
2 . Mean values 〈σ2 〉 =
1
2 (na − nb) are connected with the
differences in the occupancy of the two possible molecular positions, na and nb.
Herein below for convenience we often use the notations 1, 2 and 3 instead of x, y and z for components
of vectors and tensors. The Hamiltonian of a proton subsystem of GPI, which takes into account the short-
range and long-range interactions and the applied electric fields E1, E2, E3 along positive directions of
the Descartes axes OX, OY and OZ , consists of the “seed” and pseudospin parts. The “seed” energy
Useed corresponds to the heavy ion sublattice and does not depend explicitly on the configuration of the
proton subsystem. The pseudospin part describes short-range Ĥshort and long-range ĤMF interactions of
protons near tetrahedra HPO3, as well as the effective interaction with the electric fields E1, E2 and E3.
Therefore,
Ĥ = NUseed + Ĥshort + ĤMF , (2.1)
where N is the total number of primitive cells.
The Useed corresponds to the “seed” energy, which includes the elastic, piezoelectric and dielectric
parts, expressed in terms of electric fields Ei (i = 1, 2, 3) and strains εi and ε j ( j = i + 3). Parameters
cE0
ii′
(T), cE0
i5 (T), cE0
46 (T), cE0
j j
(T), e0
ii′
, e0
i j
, χε0
ii
, χε0
31 (i′ = 1, 2, 3) correspond to the so-called “seed” elastic
constants, piezoelectric stresses and dielectric susceptibilities, respectively, v is the volume of a primitive
cell:
Useed = v
[
1
2
3
∑
i,i′=1
cE0
ii′ (T)εiεi′ +
1
2
6
∑
j=4
cE0
j j (T)ε
2
j +
3
∑
i=1
cE0
i5 (T)εiε5 + cE0
46 (T)ε4ε6
−
3
∑
i=1
e0
2iεiE2 − e0
25ε5E2 − e0
14ε4E1 − e0
16ε6E1 − e0
34ε4E3 − e0
36ε6E3
−
1
2
χε0
11 E2
1 −
1
2
χε0
22 E2
2 −
1
2
χε0
33 E2
3 − χε0
31 E3E1
]
. (2.2)
The Hamiltonian of short-range interactions is
Ĥshort = −2w
∑
qq′
(σq1
2
σq2
2
+
σq3
2
σq4
2
) (
δ ®Rq
®Rq′
+ δ ®Rq+
®Rc, ®Rq′
)
. (2.3)
23706-3
I.R. Zachek, R.R. Levitskii, A.S. Vdovych, I.V. Stasyuk
In (2.3), σq f is the z-component of pseudospin operator that describes the state of the f -th bond
( f = 1, 2, 3, 4), in the q-th cell. The first Kronecker delta corresponds to the interaction between protons
in the chains near the tetrahedra HPO3 of type “I” (figure 2), where the second one near the tetrahedra
HPO3 of type “II”, ®Rc is the lattice vector along OZ-axis. Contributions into the energy of interactions
between protons near tetrahedra of different types, as well as the mean values of the pseudospins 〈σq f 〉,
which are related to tetrahedra of different types, are equal.
Parameter w, which describes the short-range interactions within chains, is expanded linearly into
series over strains εi , ε j :
w = w0
+
3
∑
i=1
δiεi +
6
∑
j=4
δjε j . (2.4)
Mean field Hamiltonian ĤMF of the long-range dipole-dipole interactions and indirect (through the
lattice vibrations) interactions between protons, taking into account that Fourier transforms of interaction
constants Jf f ′ =
∑
q′
Jf f ′(qq′) at ®k = 0 are linearly expanded:
Jf f ′ = J0
f f ′ +
∂Jf f ′
∂εi
εi = J0
f f ′ +
3
∑
i=1
ψ f f ′iεi +
6
∑
j=4
ψ f f ′ jε j , (2.5)
can be written as:
ĤMF = NH0
+ Ĥs , (2.6)
where
H0
=
1
8
J0
11(η
2
1 + η
2
3) +
1
8
J0
22(η
2
2 + η
2
4) +
1
4
J0
13η1η3 +
1
4
J0
24η2η4 +
1
4
J0
12(η1η2 + η3η4) +
1
4
J0
14(η1η4
+ η2η3) +
1
8
(
3
∑
i=1
ψ11iεi +
6
∑
j=4
ψ11 jε j
)
(η2
1 + η
2
3) +
1
8
(
3
∑
i=1
ψ22iεi +
6
∑
j=4
ψ22 jε j
)
(η2
2 + η
2
4)
+
1
4
(
3
∑
i=1
ψ13iεi +
6
∑
j=4
ψ13 jε j
)
η1η3 +
1
4
(
3
∑
i=1
ψ24iεi +
6
∑
j=4
ψ24 jε j
)
η2η4
+
1
4
(
3
∑
i=1
ψ12iεi +
6
∑
j=4
ψ12 jε j
)
(η1η2 + η3η4) +
1
4
(
3
∑
i=1
ψ14iεi +
6
∑
j=4
ψ14 jε j
)
(η1η4 + η2η3), (2.7)
Ĥs = −
∑
q
(
H1
σq1
2
+H2
σq2
2
+H3
σq3
2
+H4
σq4
2
)
, (2.8)
and η f = 〈σq f 〉. In (2.8) the notations are used:
H1 =
1
2
J11η1 +
1
2
J12η2 +
1
2
J13η3 +
1
2
J14η4 + µ
x
13E1 + µ
y
13E2 + µ
z
13E3 ,
H2 =
1
2
J22η2 +
1
2
J12η1 +
1
2
J24η4 +
1
2
J14η3 − µx24E1 − µ
y
24E2 + µ
z
24E3 ,
H3 =
1
2
J11η3 +
1
2
J12η4 +
1
2
J13η1 +
1
2
J14η2 − µx13E1 + µ
y
13E2 − µz13E3 ,
H4 =
1
2
J22η4 +
1
2
J12η3 +
1
2
J24η2 +
1
2
J14η1 + µ
x
24E1 − µ
y
24E2 − µz24E3. (2.9)
In (2.9) µx,y,z13 = µ
x,y,z
1 = µ
x,y,z
3 , µx,y,z24 = µ
x,y,z
2 = µ
x,y,z
4 are the effective dipole moments per one
pseudospin.
The two-particle cluster approximation is used for calculation of thermodynamic and dielectric
characteristics of GPI. In this approximation, thermodynamic potential is given by:
G = NUseed + NH0 − kBT
∑
q
[
2 ln Sp e−βĤ
(2)
q −
4
∑
f=1
ln Sp e−βĤ
(1)
q f
]
, (2.10)
23706-4
Influence of electric fields
where Ĥ(2)
q , Ĥ(1)
q f
are two-particle and one-particle Hamiltonians:
Ĥ(2)
q = −2w
(σq1
2
σq2
2
+
σq3
2
σq4
2
)
−
y1
β
σq1
2
−
y2
β
σq2
2
−
y3
β
σq3
2
−
y4
β
σq4
2
, (2.11)
Ĥ(1)
q f
= −
ȳ f
β
σq f
2
. (2.12)
Here:
y f = β(∆ f +Hf ), ȳ f = β∆ f + y f . (2.13)
The symbols ∆ f are the effective fields created by the neighboring bonds from outside of the cluster. In
the cluster approximation, the fields ∆ f can be determined from the self-consistency condition, which
states that the mean values of the pseudospins 〈σq f 〉 calculated with the two-particle and one-particle
Gibbs distribution, respectively, should coincide. That is,
Spσq f e−βĤ
(2)
q
Sp e−βĤ
(2)
q
=
Spσq f e
−βĤ
(1)
q f
Sp e−βĤ
(1)
q f
. (2.14)
Hence, based on (2.14) taking into account (2.11) and (2.12) we obtain
η1,3 =
1
D
(
sinh n1 ± sinh n2 + a2 sinh n3 ± a2 sinh n4 + a sinh n5 + a sinh n6 ∓ a sinh n7 ± a sinh n8
)
= tanh
ȳ1,3
2
,
η2,4 =
1
D
(
sinh n1 ± sinh n2 − a2 sinh n3 ∓ a2 sinh n4 ∓ a sinh n5 ± a sinh n6 + a sinh n7 + a sinh n8
)
= tanh
ȳ2,4
2
,
D = cosh n1 + cosh n2 + a2 cosh n3 + a2 cosh n4 + a cosh n5 + a cosh n6 + a cosh n7 + a cosh n8 ,
(2.15)
where
a = exp
[
−
1
kBT
(
w0
+
3
∑
i=1
δiεi +
6
∑
j=4
δjε j
)]
,
n1 =
1
2
(y1 + y2 + y3 + y4), n2 =
1
2
(y1 + y2 − y3 − y4),
n3 =
1
2
(y1 − y2 + y3 − y4), n4 =
1
2
(y1 − y2 − y3 + y4),
n5 =
1
2
(y1 − y2 + y3 + y4), n6 =
1
2
(y1 + y2 + y3 − y4),
n7 =
1
2
(−y1 + y2 + y3 + y4), n8 =
1
2
(y1 + y2 − y3 + y4).
Taking into consideration (2.15), we exclude the parameters ∆ f and write the relations
y1 =
1
2
ln
1 + η1
1 − η1
+ βν11η1 + βν12η2 + βν13η3 + βν14η4 +
β
2
(µx13E1 + µ
y
13E2 + µ
z
13E3),
y2 = βν12η1 +
1
2
ln
1 + η2
1 − η2
+ βν22η2 + βν14η3 + βν24η4 +
β
2
(−µx24E1 − µ
y
24E2 + µ
z
24E3),
y3 = βν13η1 + βν14η2 +
1
2
ln
1 + η3
1 − η3
+ βν11η3 + βν12η4 +
β
2
(−µx13E1 + µ
y
13E2 − µz13E3),
y4 = βν14η1 + βν24η2 + βν12η3 +
1
2
ln
1 + η4
1 − η4
+ βν22η4 +
β
2
(µx24E1 − µ
y
24E2 − µz24E3),
where νf f ′ =
Jf f ′
4 .
23706-5
I.R. Zachek, R.R. Levitskii, A.S. Vdovych, I.V. Stasyuk
3. Dielectric characteristics of GPI
To calculate the dielectric, piezoelectric and elastic characteristics of the GPI, we use the thermody-
namic potential per one primitive cell obtained in the two-particle cluster approximation:
g =
G
N
= Useed + H0 − 2
(
w0
+
3
∑
i=1
δiεi +
6
∑
j=4
δjεi
)
−
1
2
kBT
4
∑
f=1
ln(1 − η2
f ) − 2kBT ln D
+ 2kBT ln 2. (3.1)
Minimizing the thermodynamic potential with respect to the strains εi , ε j , we have obtained equations
for the strains:
0 = cE0
l1 ε1 + cE0
l2 ε2 + cE0
l3 ε3 + cE0
l5 ε5 − e0
2lE2 −
2δl
υ
+
2δl
vD
Mε −
ψ11l
8v
(η2
1 + η
2
3) −
ψ13l
4v
η1η3
−
ψ22l
8v
(η2
2 + η
2
4) −
ψ24l
4v
η2η4 −
ψ12l
4v
(η1η2 + η3η4) −
ψ14l
4v
(η1η4 + η2η3), (l = 1, 2, 3, 5)
0 = cE0
44 ε4 + cE0
46 ε6 − e0
14E1 − e0
34E3 −
2δ4
υ
+
2δ4
vD
Mε −
ψ114
8v
(η2
1 + η
2
3) −
ψ134
4v
η1η3
−
ψ224
8v
(η2
2 + η
2
4) −
ψ244
4v
η2η4 −
ψ124
4v
(η1η2 + η3η4) −
ψ144
4v
(η1η4 + η2η3),
0 = cE0
46 ε4 + cE0
66 ε6 − e0
16E1 − e0
36E3 −
2δ6
υ
+
2δ6
vD
Mε −
ψ116
8v
(η2
1 + η
2
3) −
ψ136
4v
η1η3
−
ψ226
8v
(η2
2 + η
2
4) −
ψ246
4v
η2η4 −
ψ126
4v
(η1η2 + η3η4) −
ψ146
4v
(η1η4 + η2η3), (3.2)
where
Mε = 2a2 cosh n3 + 2a2 cosh n4 + a cosh n5 + a cosh n6 + a cosh n7 + a cosh n8.
Differentiating the thermodynamic potential over the fields Ei we get the expressions for polariza-
tions Pi
P1 = e0
14ε4 + e0
16ε6 + χ
ε0
11 E1 + χ
ε0
31 E3 +
1
2v
[µx13(η1 − η3) − µx24(η2 − η4)],
P2 = e0
21ε1 + e0
22ε2 + e0
23ε3 + e0
25ε5 + χ
ε0
22 E2 +
1
2v
[µ
y
13(η1 + η3) − µ
y
24(η2 + η4)],
P3 = e0
34ε4 + e0
66ε6 + χ
ε0
33 E3 + χ
ε0
31 E1 +
1
2v
[µz13(η1 − η3) + µ
z
24(η2 − η4)]. (3.3)
Diagonal components of the static isothermic dielectric susceptibilities of mechanically clamped crystal
GPI are given by:
χε11 = χε0
11 +
1
2υ∆
[µx13(∆
χx
1 − ∆
χx
3 ) − µx24(∆
χx
2 − ∆
χx
4 )], (3.4)
χε22 = χε0
22 +
1
2υ∆
[µ
y
13(∆
χy
1 + ∆
χy
3 ) − µ
y
24(∆
χy
2 + ∆
χy
4 )], (3.5)
χε33 = χε0
33 +
1
2υ∆
[µz13(∆
χz
1 − ∆
χz
3 ) + µc24(∆
χz
2 − ∆
χz
4 )]. (3.6)
Here, the ratio
∆
χα
f
∆
=
(
∂η f
∂Eα
)
εl
has the meaning of the local pseudospin susceptibility, which describes the reaction of the f -th order
parameter to the external electric field Eα at constant strains. Explicit expressions for quantities introduced
here are given in the appendix [formulae (A.1) and (A.2)].
23706-6
Influence of electric fields
Based on (3.2), we have obtained expressions for isothermic coefficients of piezoelectric stress e2 j of
GPI:
e2l =
(
∂P2
∂εl
)
E2
= e0
2l +
µ
y
13
2v∆
(∆e1l + ∆
e
3l) −
µ
y
24
2v∆
(∆e2l + ∆
e
4l). (3.7)
Here, the ratio
∆
e
f l
∆
=
(
∂η f
∂εl
)
E2
describes the reaction of the f -th order parameter on the strain εl at constant external fields [see the
appendix, formula (A.3)].
4. Comparison with the experimental data
To calculate the temperature and field dependences of dielectric and piezoelectric characteristics of
GPI, we have to determine the values of the following parameters:
• parameter of short-range interactions w0;
• parameters of long-range interactions ν0±
f
( f = 1, 2, 3);
• deformational potentials δi, ψ±
f i
( f = 1, 2, 3; i = 1, . . . , 6);
• effective dipole moments µx13; µx24; µy13; µy24; µz13; µz24;
• “seed” dielectric susceptibilities χε0
ii
, χε0
31 (i = 1, 2, 3);
• “seed” coefficients of piezoelectric stress e0
2i , e0
25, e0
14, e0
16, e0
34, e0
36;
• “seed” elastic constants cE0
ii′
, cE0
j j
, cE0
i5 , cE0
46 (i = 1, 2, 3; i′ = 1, 2, 3; j = 4, 5, 6).
To determine the above listed parameters, we use the measured temperature dependences for the set
of physical characteristics of GPI, namely Ps(T) [14], εσ11, εσ33 [1], d21, d23 [15], as well as the dependence
of phase transition temperature Tc(p) [16] on hydrostatic pressure.
The volume of primitive cell of GPI is the υH = 0.601·10−21 cm3 [5].
Numerical analysis shows that thermodynamic characteristics depend on the two linear combinations
of long-range interactions ν0+
= ν0+
1 + 2ν0+
2 + ν
0+
3 and ν0−
= ν0−
1 + 2ν0−
2 + ν
0−
3 and practically do not
depend (deviation < 0.1%) on separate values of the ν0±
f
at given ν0+ and ν0−. The optimal values of
these combinations are ν0+/kB = 10.57 K, ν0−/kB = −0.8 K; as concrete values of the ν0±
f
we use
ν̃0+
1 = ν̃
0+
2 = ν̃
0+
3 = 2.643 K, ν̃0−
1 = ν̃
0−
2 = ν̃
0−
3 = 0.2 K, where ν̃0±
f
= ν0±
f
/kB.
Since the phase transition in the GPI is of the second order, from the condition of nullification of
the inverse longitudinal dielectric susceptibility (3.5) we can obtain the equation ∆(Tc) = 0 for phase
transition temperature. This equation connects the parameter of short-range interactions w0 with the
parameters of long-range interactions ν0+
1 , ν0+
2 and ν0+
3 . From this equation at ®E = 0 and at the given ν0+
1 ,
ν0+
2 , ν0+
3 and other parameter values, we obtain the value of the short-range parameter w0. Its optimal
value is w0
= 820 K. The optimal values of deformational potentials δj , which are coefficients of linear
expansion of the parameter w0 over the strains ε j [see (2.4)], are as follows: δ̃1 = 500 K, δ̃2 = 600 K,
δ̃3 = 500 K, δ̃4 = 150 K, δ̃5 = 100 K, δ̃6 = 150 K; δ̃i = δi/kB.
For parameters ψ±
f i
, similarly to the ν0±
f
, the 6 linear combinations ψ+
i
= ψ+1i + 2ψ+2i + ψ
+
3i and 6
combinations ψ−
i
= ψ−
1i + 2ψ−
2i + ψ
−
3i are important. Thermodynamic characteristics practically do not
depend (deviation < 0.1%) on separate values of the ψ±
f i
at given ψ+
i
and ψ−
i
. The optimal values of
23706-7
I.R. Zachek, R.R. Levitskii, A.S. Vdovych, I.V. Stasyuk
the ψ±
f i
, are as follows: ψ̃+
f 1 = 87.9 K, ψ̃+
f 2 = 237.0 K, ψ̃+
f 3 = 103.8 K, ψ̃+
f 4 = 149.1 K, ψ̃+
f 5 = 21.3 K,
ψ̃+
f 6 = 143.8 K, ψ̃−
f i
= 0 K, where ψ̃±
f i
= ψ±
f i
/kB.
Effective dipole moments in the paraelectric phase are equal to ®µ13 = (0.4, 4.02, 4.3) · 10−18 esu·cm,
®µ24 = (−2.3, −3.0, 2.2) · 10−18 esu·cm. In the ferroelectric phase, the y-component of the first dipole
moment is µy13ferro = 3.82 · 10−18 esu·cm; X-ray investigation [4] determined the coordinates of atoms in
the primitive cell of GPI. The calculated displacements of the protons, which we marked as 1 and 2, relative
to the centers of hydrogen bonds in ferroelectric phase are equal to ∆®r1 = (−0.016,−0.495,−0.160) Å,
∆®r2 = (0.389, 0.383,−0.147) Å. The obtained dipole moments are not proportional to the corresponding
proton displacements. This means that in addition to the proton displacements, the phosphite and glycine
groups also take part in forming the effective dipole moments.
For the “seed” coefficients of piezoelectric stress, dielectric susceptibilities and elastic constants, the
following values are obtained:
e0
21 = e0
22 = e0
23 = e0
25 = e0
14 = e0
16 = e0
34 = e0
36 = 0.0 esu
cm2 ; χε0
11 = 0.1, χε0
22 = 0.403, χε0
33 = 0.5,
χε0
31 = 0.0; c0E
11 = 26.91 · 1010 dyn
cm2 , cE0
12 = 14.5 · 1010 dyn
cm2 , cE0
13 = 11.64 · 1010 dyn
cm2 ,
cE0
15 = 3.91 · 1010 dyn
cm2 , cE0
22 = [64.99 − 0.04(T − Tc)] · 1010 dyn
cm2 , cE0
23 = 20.38 · 1010 dyn
cm2 ,
cE0
25 = 5.64 · 1010 dyn
cm2 , cE0
33 = 24.41 · 1010 dyn
cm2 , cE0
35 = −2.84 · 1010 dyn
cm2 , cE0
55 = 8.54 · 1010 dyn
cm2 ,
cE0
44 = 15.31 · 1010 dyn
cm2 , cE0
46 = −1.1 · 1010 dyn
cm2 , cE0
66 = 11.88 · 1010 dyn
cm2 .
In [10], the phase transition temperature of the GPI crystal was Tc = 222 K. Explaining the experimen-
tal data [10] we suppose that all interactions in this crystal are proportional to the interactions in the crystal
with Tc = 225 K. Thus, w0(222 K) = kw0(225 K), ν0±
f
(222 K) = kν0±
f
(225 K), δi(222 K) = kδi(225 K),
ψ±
f i
(222 K) = kψ±
f i
(225 K), where k = 0.987 ≈ 222/225. Besides, the y-components of the dipole mo-
ments are the same in paraelectric and ferroelectric phases, that is µy13ferro = µ
y
13para = 3.82 ·10−18 esu·cm;
and z-component µz13 = 4.2 · 10−18 esu·cm. All other parameters are taken the same as for the crystal
with Tc = 225 K.
Now, let us look at the results obtained in this paper for temperature and field dependences of physical
characteristics of the GPI crystal at different values of strength of the electric fields E1 and E3. Numerical
calculations of dielectric characteristics of the GPI are carried out for the strength of the fields from 0 up
to ±4 MV/m.
Temperature dependences of the order parameters at different values of the fields E1 or E3 are
presented in figures 3 and 4. At zero fields, the mean values of pseudospins are η1 = η3, η2 = η4 in the
ferroelectric phase, and η1 = η2 = η3 = η4 = 0 in the paraelectric phase.
200 210 220 230
−0.1
0
0.2
0.4
0.6
0.8
T, K
η
1,2
(E
1
)
1 3 5 5’ 3’
200 210 220 230
0
0.2
0.4
0.6
0.8
1
5 3 1
T, K
η
3,4
(E
1
)
5’ 3’
Figure 3. The temperature dependences of the order
parameters η f of the GPI crystal at different values
of the electric field E1 (MV/m): 0.0 — 1; 2.0 — 3;
−2.0 — 3’; 4.0 — 5; −4.0 — 5’.
200 210 220 230
−0.2
0
0.2
0.4
0.6
0.8
T, K
η
1,2
(E
3
)
1 2 3 4 5
5’
3’
200 210 220 230
−0.2
0
0.2
0.4
0.6
0.8
1
T, K
η
3,4
(E
3
)
1 2 3 4 5
5’
3’
Figure 4. The temperature dependences of the order
parameters η f of the GPI crystal at different values
of the electric field E3 (MV/m): 0.0 — 1; 1.0 — 2;
2.0 — 3; 3.0 — 4; 4.0 — 5; −2.0 — 3’; −4.0 — 5’.
23706-8
Influence of electric fields
0 2 4 6 8 10 12
219
220
221
222
223
224
225
T
c
, K
E
3
2, (MV/m)2
1
3
Figure 5. The dependences of the phase transition
temperature Tc of GPI crystal on the electric fields
E1 (1) and E3 (3).
Figure 6. The dependences of the phase transition
temperature Tc of GPI crystal on the squares of the
electric fields E1 (1) and E3 (3).
The electric field E1 > 0 slightly splits the mean values of pseudospins in the ferroelectric phase, and
fairly strongly in the paraelectric phase. In the paraelectric phase, η1 = η2 < 0, η3 = η4 > 0. An increase
of the field E1 leads to a decrease of the η1, η2 and to an increase of η3, η4 parameters. In the case of
E1 < 0, in the paraelectric phase η1 = η2 > 0, η3 = η4 < 0.
Applying the electric field E3 > 0 also leads to a splitting of the mean values of pseudospins, but
much stronger than in the case of the field E1. Here, η1 = η2 > 0, η2 = η4 < 0 in the paraelectric phase.
An increase of the field E3 in the ferroelectric phase leads to an increase of η1, η2 and to a decrease of
η3, η4 parameters. At E3 < 0, in the paraelectric phase η1 = η2 < 0, η2 = η4 > 0.
The dependences of the phase transition temperature Tc of GPI crystal on the electric fields E1 and
E3, and on the squares of these fields are presented in figures 5 and 6, respectively. With an increase of
the fields E1 and E3, the phase transition temperatures Tc decrease, especially for the field E3.
It is shown that the dependences Tc(E1,3) are close to quadratic in the fields (see [10]), and at the
fields up to 4 MV/m, they can be written as:
Tc(E1) = Tc − kT1 E2
1 , Tc(E3) = Tc − kT3 E2
3 ,
where kT1 = 0.025 Km2/MV2, kT3 = 0.3325 Km2/MV2.
In figure 7 there are presented the temperature dependences of the components of polarization Pi of
GPI crystal at different values of the field E1, and in figure 8 — at different values of the field E3.
200 210 220 230
−0.2
−0.1
0
0.1
0.2
0.3
T, K
P
1
, 10−6 C / cm2
3 4 5
1
3’ 4’ 5’
200 210 220
0
0.1
0.2
0.3
T, K
P
2
, 10−6 C / cm2
1
3
5
3’
5’
200 210 220 230
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
T, K
P
3
, 10−6 C / cm2
1
2 3 4 5
3’ 4’ 5’
Figure 7. The temperature dependences of the components of polarization P1, P2, P3 of GPI crystal at
different values of the field E1 (MV/m): 0.0 — 1; 1.0 — 2; 2.0 — 3; 3.0 — 4; 4.0 — 5; −2.0 — 3’;
−3.0 — 4’; −4.0 — 5’; ◦ are the experimental data [14].
23706-9
I.R. Zachek, R.R. Levitskii, A.S. Vdovych, I.V. Stasyuk
200 210 220 230
−1
−0.5
0
0.5
1
T, K
P
1
, 10−6 C / cm2
1
2 3 4 5
5’ 4’ 3’ 2’
200 210 220
0
0.1
0.2
0.3
0.4
1
2
3
4
T, K
P
2
, . 10−6 C / cm2
5
2’
3’
4’
5’
200 210 220 230
−1
−0.5
0
0.5
1
T, K
P
3
, 10−6 C / cm2
2 3
1
5
4
5’ 4’
3’
2’
Figure 8. The temperature dependences of the components of polarization P1, P2, P3 of GPI crystal at
different values of the field E3 (MV/m): 0.0 — 1; 1.0 — 2; 2.0 — 3; 3.0 — 4; 4.0 — 5; −1.0 —2’;
−2.0 — 3’; −3.0 — 4’; −4.0 — 5’; ◦ are the experimental data [14].
0 1 2 3
−3
−2
−1
0
1
2
3
x 10
−3 P
i
(E
1
), 10−6 C/cm2
E
1
, MV/m
P
2
P
1
P
3
P
2
1
2
2
1
2
1
0 1 2 3
−4
−2
0
2
4
6
8
10
x 10
−3 P
i
(E
3
), 10−6 C/cm2
E
3
, MV/m
P
3
P
1
P
2
1
2
2
1
2
1
Figure 9. The dependences of polarizations P1, P2, P3 of GPI crystal on the fields E1 and E3 at different
temperatures T (K): 215 — 1; 230 — 2.
With an increase of strength of the electric field E1, the spontaneous polarization P2 slightly decreases,
but polarization P1 induced by the field increases. Polarization P3 induced by the field E1 is negative,
and in magnitude it is three times larger than the P1. At the field E1 < 0, the sign of polarizations P1 and
P3 is opposite, and the magnitude of the P2 also decreases.
However, an increase of the field E3 leads to a decrease of spontaneous polarization P2 and to an
increase of the polarization P3; besides, the P3(E3) increases more appreciably than in the case of P1(E1).
The temperature dependence of the negative polarization P1(E3) induced by the field E3 is analogous to
the P3(E1) and the value of the P1(E3) is almost equal to the value of the P3(E1). It is necessary to note
that the effect of the field E3 < 0 on the components of polarization is qualitatively similar to the effect
of the field E1 > 0 on them. The dependences of polarizations P1, P2, P3 of GPI crystal on the fields E1
and E3 at different temperatures T are presented in figure 9.
Changes in the temperature dependences of the components of static dielectric permittivities εii =
1+4πχii of GPI crystal under the action of transverse electric fields E1 and E3 are shown in figures 10–12.
Values of the permittivities ε11(E1), ε33(E1) slightly increase in the ferroelectric phase and slightly
decrease in the paraelectric phase. The action of the field E3 is much stronger. The temperature depen-
dences of the ε11(E3) and ε33(E3) have jumps at the phase transition point, which rise with an increase of
the field E3 and shift to the lower temperatures. Changes in signs of the fields do not influence the values
of permittivities.
23706-10
Influence of electric fields
200 210 220
0
5
10
15
20
25
30
T, K
ε
11
(E
1
)
1
3
5
200 205 210 215 220 225
0
10
20
30
40
T, K
ε
11
(E
3
)
1 2 3 4 5
Figure 10. The temperature dependences of the static dielectric permittivity ε11 of GPI crystal at
different values of the fields E1 and E3 (MV/m): 0.0 — 1; 1.0 — 2; 2.0 — 3; 3.0 — 4; 4.0 — 5; △ are
the experimental data [1].
222 224 226
0
1000
2000
3000
4000
5000
6000
T, K
ε
22
(E
1
)
1 3 5
216 220 224 228
0
1000
2000
3000
4000
5000
6000
T, K
ε
22
(E
3
)
1 3 4 5
Figure 11. The temperature dependences of the static dielectric permittivity ε22 of GPI crystal at different
values of the fields E1 and E3 (MV/m): 0.0 — 1; 1.0 — 2; 2.0 — 3; 3.0 — 4; 4.0 — 5; ◦ are the experimental
data [17].
200 210 220 230
0
50
100
150
200
250
300
T, K
ε
33
(E
1
)
13
5
200 205 210 215 220 225 230
0
100
200
300
400
500
ε
33
(E
3
)
T, K
1 2
3 4 5
Figure 12. The temperature dependences of the static dielectric permittivity ε33 of GPI crystal at
different values of the fields E1 and E3 (MV/m): 0.0 — 1; 1.0 — 2; 2.0 — 3; 3.0 — 4; 4.0 — 5; △ are
the experimental data [1].
The jumps of the permittivities at the phase transition point ∆ε11(E1,3) and ∆ε33(E1,3) are nearly
proportional to the squares of the strengths of the fields E1 and E3 (figure 13):
23706-11
I.R. Zachek, R.R. Levitskii, A.S. Vdovych, I.V. Stasyuk
0 2 4 6 8 10 12
0
2
4
6
8
10
E2
1
, (MV/m)2
∆ε
11
(E
1
), ∆ε
33
(E
1
)
1
2
0 2 4 6 8 10 12
0
50
100
150
200
250
1
2
E
3
2, (MV/m)2
∆ε
11
(E
3
), ∆ε
33
(E
3
)
Figure 13. The dependences of the jumps of the permittivities ε11(1) and ε33(2) of GPI crystal on the
squares of the electric fields E1 and E3.
∆ε11(E1) = kε
11E2
1 , ∆ε11(E3) = kε
13E2
3 ,
∆ε33(E1) = kε
31E2
1 , ∆ε33(E3) = kε
33E2
3 ,
where the coefficients are k11 = 0.064 Km2/MV2, k13 = 1.0 Km2/MV2, k31 = 0.725 Km2/MV2,
k33 = 12.5 Km2/MV2.
The temperature dependences of the coefficients of piezoelectric stress e2i at different values of the
electric fields E1 and E3 are presented in figures 14 and 15. An increase of the field E1 leads to a slight
increase of piezomoduli e2i . The splitting of the temperature dependences of e2i is much stronger in the
case of field E3.
The results of an experimental investigation of the static dielectric permittivity ε33 of GPI crystal at
different values of the field E3 are presented in [10, 11]. The phase transition temperature for this case was
222 K, but the field dependence of the Tc is similar to the crystal with Tc = 225 K. Therefore, having made
the above mentioned changes of the model parameters, we consider it possible to explain the experimental
data. The calculated temperature dependences of the static direct ε33 and inverse ε−1
33 permittivities of
GPI crystal at different values of the field E3 as well as the experimental data are presented in figure 16.
It is shown that at the phase transition temperature, theoretical curves ε33(T) have a sharp jump whose
magnitude increases with an increase of the field. However, the experimental curves ε33(T) are smooth,
as in the case of a smeared phase transition.
In order to consider the reason of such a behaviour of permittivity ε33, there was carried out a
calculation of this component assuming that together with the applied field E3 there also appears an
internal field E2. As it turned out, one can achieve a satisfactory description of the temperature dependence
215 220 225
0
1
2
3
4
5
6
x 10
5
T, K
e
21
(E
1
)esu/cm2
1
3
5
215 220 225
0
2
4
6
8
x 10
5
1
3
5
T, K
e
22
(E
1
)esu/cm2
215 220 225
0
1
2
3
4
5
6
x 10
5
1
3
5
T, K
e
23
(E
1
)esu/cm2
215 220 225
0
1
2
3
4
5
6
x 10
5
1
3
5
T, K
e
25
(E
1
)esu/cm2
Figure 14. The temperature dependences of the coefficients of piezoelectric stress e2i of GPI crystal at
different values of the electric field E1 (MV/m): 0.0 — 1; 2.0 — 3; 4.0 — 5.
23706-12
Influence of electric fields
215 220 225
0
1
2
3
4
5
6
x 10
5
1
3 5
T, K
e
21
(E
3
),esu/cm2
2
4
215 220 225
0
2
4
6
8
x 10
5
T, K
e
22
(E
3
)esu/cm2
1
3 5 4
2
215 220 225
0
1
2
3
4
5
6
x 10
5
T, K
e
23
(E
3
)esu/cm2
1
5
2
3 4
215 220 225
0
1
2
3
4
5
6
x 10
5
T, K
e
25
(E
3
)esu/cm2
1
5
2
3 4
Figure 15. The temperature dependences of the coefficients of piezoelectric stress e2i of GPI crystal at
different values of the electric field E3 (MV/m): 0.0 — 1; 1.0 — 2; 2.0 — 3; 3.0 — 4; 4.0 — 5.
200 210 220 230
0
100
200
300
400
500
T, K
ε
33
(E
3
)
1 2 3 4 5
200 210 220 230
2
3
4
5
6
7
8
x 10
−3 ε
33
−1(E
3
)
T, K
1
2
3
4
5
Figure 16. The temperature dependences of the static direct ε33 and inverse ε−1
33 permittivities of GPI
crystal at different values of the field E3 (MV/m): 0.0 — 1; 1.0 — 2; 2.0 — 3; 3.0 — 4; 4.0 — 5; symbols
▽, △, ♦, �, ⊳ are the experimental data [10, 11].
of ε33, assuming E2 ∼ 0.05E3 (figure 17). Such a component of the field E2 could appear due to an
incomplete reorientational relaxation of the glycine groups (which manifests itself during measurements
in the hysteresis behaviour of ε33); one cannot exclude the possibility of some deflection of the applied
transverse field from the OZ-axis during the experiment (about 2.86°). Nevertheless, if the effect is
200 205 210 215 220 225 230
50
100
150
200
250
300
350
400
ε
33
(E
3
, E
2
)
T, K
1
2 3 4 5
200 205 210 215 220 225 230
2
3
4
5
6
7
8
x 10
−3
T, K
ε
33
−1 (E
3
,E
2
)
1
2
3
4
5
Figure 17. The temperature dependences of the static direct ε33 and inverse ε−1
33 permittivities of GPI
crystal at different values of the field E3 (MV/m): 0.0 — 1; 1.0 — 2; 2.0 — 3; 3.0 — 4; 4.0 — 5 and the
field E2 = E3/20; symbols ▽, △, ♦, �, ⊳ are the experimental data [10, 11].
23706-13
I.R. Zachek, R.R. Levitskii, A.S. Vdovych, I.V. Stasyuk
connected with the character and peculiarities of internal fields in GPI crystal, the problem needs an
additional study.
It also concerns the role of glycine groups in the phase transition in GPI in the presence of external
fields. Their deformation and reorientation is significant at the transition to the ferroelectric phase and
manifests itself, for example, in the experiment on Raman scattering [18] or at simulations of lattice
dynamics [19]. At the same time, it was shown that the mechanism of the phase transition is connected
with the proton ordering on hydrogen bonds.
It should be mentioned that the attempt to describe the behaviour of the inverse transverse dielectric
permittivity at different electric fields within the phenomenological approach by means of Landau
expansions was also done in [20, 21]. The authors explain the smeared minimum of the inverse permittivity
below the transition temperature supposing that the phase transition is of the first order one, close to the
tricritical point. They qualitatively describe the experimental data [11], but quantitatively only at low
fields. Such a supposition was based on their experimental data for GPI [20], which noticeably differ
from the obtained ones in the majority of other measurements. This can be connected with the unlike
properties of the crystals grown at different conditions [21].
5. Conclusions
Based on the proposed model of a deformed crystal, the calculation of dielectric characteristics of the
crystal GPI in the presence of electric fields E1 and E3 is carried out. The obtained temperature and field
dependences show that the effect of field E3 on these characteristics is much more important than the
effect of field E1. At an increase of the field, the transition temperatures Tc(E1) and Tc(E3) decrease almost
as square of the field strengths. The magnitude of the jumps of permittivities ε11 and ε33 increases at the
phase transition temperature according to the same law. Electric fields E1 and E3 cause polarizations P1
and P3; their temperature dependences are analyzed in the work.
The shape of anomalies of piezoelectric moduli in the region of a phase transition in the presence of
transverse fields is analyzed. The obtained theoretical dependences have a character of predictions and
can urge the subsequent experimental investigations.
At the same time, it is necessary to note that the ability of GPI crystal to reorientate the local dipole
moments and to change the orientation of the polarization vector by means of phase transition under
reachable values of electric fields is unique. We do not know any analogues among the ferroelectric
crystals with hydrogen bonds.
Due to specific properties of GPI, special attention during investigations is also paid to possible
applications of the crystal in thin film structures [22]; the role of impurities that introduce internal fields
causing the appearance of pyroelectricity is studied [21].
In our opinion, an important role is played by glycine ions that relatively easily change their orien-
tations, exhibiting some inertia. Taking into account their relaxational dynamics, one could significantly
supplement the comprehension of the mechanisms of external fields effect on dielectric properties of GPI.
Acknowledgement
The authors are indebted to Prof. Z. Czapla for helpful discussions and useful comments.
A. Parameters determining the local pseudospin susceptibilities with
respect to electric fields and strains
The notations introduced in equations (3.4)–(3.6) are as follows:
23706-14
Influence of electric fields
∆ =
�
�
�
�
�
�
�
�
�
2D − ̹11 −̹12 −̹13 −̹14
−̹21 2D − ̹22 −̹23 −̹24
−̹31 −̹32 2D − ̹33 −̹34
−̹41 −̹42 −̹43 2D − ̹44
�
�
�
�
�
�
�
�
�
,
∆
χα
1 =
�
�
�
�
�
�
�
�
�
̹
χα
1 −̹12 −̹13 −̹14
̹
χα
2 2D − ̹22 −̹23 −̹24
̹
χα
3 −̹32 2D − ̹33 −̹34
̹
χα
4 −̹42 −̹43 2D − ̹44
�
�
�
�
�
�
�
�
�
, ∆
χα
3 =
�
�
�
�
�
�
�
�
�
2D − ̹11 −̹12 ̹
χα
1 −̹14
−̹21 2D − ̹22 ̹
χα
2 −̹24
−̹31 −̹32 ̹
χα
3 −̹4
−̹41 −̹42 ̹
χα
4 2D − ̹44
�
�
�
�
�
�
�
�
�
,
∆
χα
2 =
�
�
�
�
�
�
�
�
�
2D − ̹11 ̹
χα
1 −̹13 −̹14
−̹21 ̹
χα
2 −̹23 −̹24
−̹31 ̹
χα
3 2D − ̹33 −̹34
−̹41 ̹
χα
4 −̹43 2D − ̹44
�
�
�
�
�
�
�
�
�
, ∆
χα
4 =
�
�
�
�
�
�
�
�
�
2D − ̹11 −̹12 −̹13 ̹
χα
1
−̹21 2D − ̹22 −̹23 ̹
χα
2
−̹31 −̹32 2D − ̹33 ̹
χα
2
−̹41 −̹42 −̹43 ̹
χα
4
�
�
�
�
�
�
�
�
�
,
(A.1)
where
̹f 1 = ̹f ;11ϕ
+
1 + ̹f ;12βν
+
2 + ̹f ;13ϕ
−
1 + ̹f ;14βν
−
2 , ( f = 1, 2, 3, 4);
̹f 3 = ̹f ;11ϕ
+
3 + ̹f ;12βν
+
2 − ̹f ;13ϕ
−
3 − ̹f ;14βν
−
2 ,
̹f 2 = ̹f ;12ϕ
+
2 + ̹f ;11βν
+
2 + ̹f ;14ϕ
−
2 + ̹f ;13βν
−
2 ,
̹f 4 = ̹f ;12ϕ
+
4 + ̹f ;11βν
+
2 − ̹f ;14ϕ
−
4 − ̹f ;13βν
−
2 ,
̹
χx
f
= ̹f ;13βµ
x
13 + ̹f ;15βµ
x
24 , ̹
χy
f
= ̹f ;11βµ
y
13 + ̹f ;12βµ
y
24 , ̹
χz
f
= ̹f ;13βµ
z
13 + ̹f ;14βµ
z
24 ,
ϕ±1,3 =
1
1 − η2
1,3
+ βν±1 =
1
1 − η2
1,3
+
β
4
(J11 ± J13),
ϕ±2,4 =
1
1 − η2
2,4
+ βν±3 =
1
1 − η2
2,4
+
β
4
(J22 ± J24), βν±2 =
β
4
(J12 ± J14),
ν±1 = ν
0±
1 +
(
3
∑
i=1
ψ±
1iεi ±
6
∑
j=4
ψ±
1 jε j
)
, ν0±
1 =
1
4
(J0
11 ± J0
13); ψ±
1i =
1
4
(ψ11i ± ψ13i),
ν±2 = ν
0±
2 +
(
3
∑
i=1
ψ±
2iεi ±
6
∑
j=4
ψ+2 jε j
)
, ν0±
2 =
1
4
(J0
12 ± J0
14); ψ±
2i =
1
4
(ψ12i ± ψ14i),
ν±3 = ν
0±
3 +
(
3
∑
i=1
ψ±
3iεi ±
6
∑
j=4
ψ±
3 jε j
)
, ν0±
3 =
1
4
(J0
22 ± J0
24); ψ±
3i =
1
4
(ψ22i ± ψ24i),
̹1,3;11 = (lc1+3 + lc5+6) − η1,3(l
s
1+3 + ls5+6), ̹1,3;12 = (lc1−3 ∓ lc7−8) − η1,3(l
s
1−3 + ls7+8),
̹1,3;13 = ±(lc2+4 + lc7+8) − η1,3(l
s
2+4 − ls7−8), ̹1,3;14 = (±lc2−4 − lc5−6) − η1,3(l
s
2−4 − ls5−6),
̹2,4;11 = (lc1−3 ∓ lc5−6) − η2,4(l
s
1+3 + ls5+6), ̹2,4;12 = (lc1+3 + lc7+8) − η2,4(l
s
1−3 + ls7+8),
̹2,4;13 = (±lc2−4 − lc7−8) − η2,4(l
s
2+4 − ls7−8), ̹2,4;14 = (±lc2+4 ± lc5+6) − η2,4(l
s
2−4 − ls5−6),
̹1,3;15 = (∓lc2−4 + lc5−6) − η1,3(−ls2−4 + ls5−6), ̹2,4;15 = ∓(lc2+4 + lc5+6) + η2,4(−ls2−4 + ls5−6),
lc1±3 = cosh n1 ± a2 cosh n3; lc2±4 = cosh n2 ± a2 cosh n4;
lc5±6 = a cosh n5 ± a cosh n6; lc7±8 = a cosh n7 ± a cosh n8;
ls1±3 = sinh n1 ± a2 sinh n3; ls2±4 = sinh n2 ± a2 sinh n4;
ls5±6 = a sinh n5 ± a sinh n6; ls7±8 = a sinh n7 ± a sinh n8. (A.2)
23706-15
I.R. Zachek, R.R. Levitskii, A.S. Vdovych, I.V. Stasyuk
The notations introduced in equations (3.7) are as follows:
∆
e
1l =
�
�
�
�
�
�
�
�
�
̹
e
1l −̹12 −̹13 −̹14
̹
e
2l 2D − ̹22 −̹23 −̹24
̹
e
3l −̹32 2D − ̹33 −̹34
̹
e
4l −̹42 −̹43 2D − ̹44
�
�
�
�
�
�
�
�
�
, ∆e3l =
�
�
�
�
�
�
�
�
�
2D − ̹11 −̹12 ̹
e
1l −̹14
−̹21 2D − ̹22 ̹
e
2l −̹24
−̹31 −̹32 ̹
e
3l −̹4
−̹41 −̹42 ̹
e
4l 2D − ̹44
�
�
�
�
�
�
�
�
�
,
∆
e
2l =
�
�
�
�
�
�
�
�
�
2D − ̹11 ̹
e
1l −̹13 −̹14
−̹21 ̹
e
2l −̹23 −̹24
−̹31 ̹
e
3l 2D − ̹33 −̹34
−̹41 ̹
e
4l −̹43 2D − ̹44
�
�
�
�
�
�
�
�
�
, ∆e4l =
�
�
�
�
�
�
�
�
�
2D − ̹11 −̹12 −̹13 ̹
e
1l
−̹21 2D − ̹22 −̹23 ̹
e
2l
−̹31 −̹32 2D − ̹33 ̹
e
2l
−̹41 −̹42 −̹43 ̹
e
4l
�
�
�
�
�
�
�
�
�
,
̹
e
f l = β(ψ
+
1l̹f ;11 + ψ
+
2l̹f ;12)(η1 + η3) + β(ψ
+
2l̹f ;11 + ψ
+
3l̹f ;12)(η2 + η4)
+ β(ψ−
1l̹f ;13 + ψ
−
2l̹f ;14)(η1 − η3) + β(ψ
−
2l̹f ;13 + ψ
−
3l̹f ;14)(η2 − η4) + 2βδl(ρ f ;1 + ρ f ;2),
ψ±
1l =
1
4
(ψ11l ± ψ13l), ψ±
2l =
1
4
(ψ12l ± ψ14l), ψ±
3l =
1
4
(ψ22l ± ψ24l),
ρ1,3;1 = −2(ls3±4 − η1,3lc3+4), ρ1,3;2 = −ls5+6 ± ls7−8 + η1,3(l
c
5+6 + lc7+8),
ρ2,4;1 = 2(ls3±4 + η2,4lc3+4), ρ2,4;2 = ±ls5−6 − ls7+8 + η2,4(l
c
5+6 + lc7+8),
ls3±4 = a2 sinh n3 ± a2 sinh n4. (A.3)
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23706-16
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Influence of electric fields
Вплив електричних полiв на дiелектричнi властивостi
сегнетоелектрика GPI
I.Р. Зачек1, Р.Р. Левицький2, А.С. Вдович2, I.В. Стасюк2
1 Нацiональний унiверситет “Львiвська полiтехнiка”, вул. С. Бандери, 12, 79013 Львiв, Україна
2 Iнститут фiзики конденсованих систем НАН України, вул. Свєнцiцького, 1, 79011 Львiв, Україна
Використовуючи модель GPI, модифiковану шляхом врахування п’єзоелектричного зв’язку з деформацiя-
ми εi в наближеннi двочастинкового кластера, розраховано компоненти вектора поляризацiї та тензора
статичної дiелектричної проникностi кристала при прикладаннi зовнiшнiх поперечних електричних по-
лiв E1 i E3. Проведено аналiз впливу цих полiв на дiелектричнi характеристики GPI. При належному
виборi параметрiв теорiї отримано задовiльний кiлькiсний опис наявних експериментальних даних для
цих характеристик.
Ключовi слова: сегнетоелектрики, електричне поле, поляризацiя, дiелектрична проникнiсть, фазовий
перехiд
23706-17
Introduction
The model
Dielectric characteristics of GPI
Comparison with the experimental data
Conclusions
Parameters determining the local pseudospin susceptibilities with respect to electric fields and strains
|