ЖУРНАЛ ФIЗИЧНИХ ДОСЛIДЖЕНЬ

т. 23, № 3 (2019) 3702(8 с.)

JOURNAL OF PHYSICAL STUDIES

v. 23, No. 3 (2019) 3702(8 p.)

EFFECT OF STRESSES σ5, σ6 AND ELECTRIC FIELD E3 ON

THERMODYNAMIC CHARACTERISTICS OF GPI FERROELECTRIC

A. S. Vdovych1, R. R. Levitskii1, I. R. Zachek2

1Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine

1, Svientsitskii St., Lviv, UA–79011, Ukraine2Lviv Polytechnic National University

12, Bandery St., Lviv, UA–79013, Ukraine

(February 11, 2019; in final form — March 19, 2019)

The model of glycine phosphite (GPI) ferroelectric, modified by taking into account the piezo-electric coupling between the ordering structure elements and the lattice strains εj , is used for theinvestigation into the effects appearing under shear stresses σ5, σ6 and electric field E3. Withinthe framework of the two particle cluster approximation, the components of the polarization vec-tor and the static dielectric permittivity tensor of a mechanically clamped crystal, and also theirpiezoelectric and thermal characteristics are calculated. The simultaneous effects of shear stress σ5

and field E3 as well as the effects of stress σ6 and field E3 on the phase transition and physicalcharacteristics of the crystal are studied.

Key words: ferroelectrics, phase transition, dielectric permittivity, piezoelectric coefficients,shear stress effect, electric field effect.

DOI: https://doi.org/10.30970/jps.23.3702

PACS number(s): 77.22.−d, 77.22.Ch, 77.22.Ej, 77.65.−j, 77.80.Bh

I. INTRODUCTION

Investigation into the effects appearing under mechan-ical stress and external electric field is one of actual prob-lems in the physics of ferroelectric materials, especial-ly regarding the glycine phosphite crystal (GPI) thatbelongs to the ferroelectric compounds with hydrogenbonds [1].

Refs. [2,3] describe an experimental study of the effectof the transverse electric field E3 on the transverse dielec-tric permittivity ε33 of GPI crystal. It is shown that theelectric field E3 decreases the phase transition tempera-ture proportionally to E2

3 . Considerable (and increasingwith the field) anomalies of the permittivity ε33 wererevealed in the phase transition region at E3 6= 0. Anexplanation of the revealed effects is presented in [2–4]based on phenomenologic theory as well as within themicroscopic approach using the proposed proton modelof GPI. But a complete quantitative description of theexperimental data was not achieved.

In [5], based on the proton model [3], a model of de-formed crystal is proposed, which takes into account thepiezoelectric coupling of the proton and lattice subsys-tems. Using this model, the effect of electric fields E1

and E3 on the dielectric and piezoelectric properties ofGPI was investigated. The abovementioned experimen-tal data [3] for the temperature dependence of ε33 inthe presence of field E3 were quantitatively correctly de-scribed.

The model of GPI [5] was generalised in [7] to the caseof the reduction of symmetry in the presence of the shearstrains ε4 and ε6. Another special case was studied: theeffect of shear stresses on a GPI crystal in the absenceof any electric field. It was shown that the shear stress

σ5 does not change the thermodynamic characteristicsqualitatively, but the shear stresses σ4 or σ6 lead to theappearance of the components of a spontaneous polar-ization along OX and OZ axes in the ferroelectric phase,and to the appearance of a divergence at the temperatureTc in trasverse permittivities ε11 and ε33.

In the present paper, based on model [7], we investigatethe general case of a simultaneous effect of the transverseelectric field E3 and the shear stresses σ6 and σ6 on thephase transition and on the dielectric and piezoelectriccharacteristics of GPI. Also we demonstrate, how the re-sults previously obtained in [5] are reproduced within themodel generalised in [7].

II. THE MODEL OF GPI CRYSTAL

Models [5,7] consider the system of protons in GPI, lo-calised on O–H. . . O bonds between the phosphite groupsHPO3, which form zigzag chains along the crystallo-graphic c-axis of the crystal (see Fig. 1). For a better un-derstanding of the model, only the phosphite groups areshown in the figure. Dipole moments dqf (f = 1, . . . , 4)are ascribed to the protons on the bonds. In the fer-roelectric phase, the dipole moments compensate eachother (dq1 with dq3, dq2 with dq4) in directions Z andX (X ⊥ (b, c), Y ‖ b, Z ‖ c), and simultaneously supple-ment each other in direction Y , creating a spontaneouspolarization.

Pseudospin variablesσq1

2, . . . ,

σq4

2describe the reori-

entation of the dipole moments of the base units: dqf =µf

σqf

2. Mean values 〈σ

2〉 = 1

2(na − nb) are connected

with the differences in the occupancy of the two possiblemolecular positions, na and nb.

This work may be used under the terms of the Creative Commons Attribution 4.0 International License. Further distri-

bution of this work must maintain attribution to the author(s) and the title of the paper, journal citation, and DOI.3702-1

A. S.VDOVYCH, R. R. LEVITSKII, I. R. ZACHEK

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���

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Fig. 1. Orientations of vectors dqf in the primitive cell in ferroelectric phase [5, 7].

Hereinafter for convenience we use notations 1, 2 and 3instead of x, y and z for components of vectors and ten-sors. The Hamiltonian of the proton subsystem of GPI,which takes into account the short-range and long-rangeinteractions, and also the applied machanical stressesand electric fields E1, E2, E3 along the positive direc-tions of the Cartesian axes X , Y and Z, can be writtenin such a way [7]:

H = NUseed + Hshort + Hlong + HE , (2.1)

where N is the total number of primitive cells. Theterm Useed in (2.1) is the “seed” energy, which relatesto the heavy ion sublattice and does not explicitly de-pend on the configuration of the proton subsystem. It in-cludes elastic, piezolectric and dielectric parts, expressed

in terms of electric fields Ei (i=1, 2, 3) and strains εj

(j=1,. . . ,6):

Useed = v

(

1

2

6∑

j,j′=1

cE0jj′ (T )εjεj′

−

3∑

i=1

6∑

j=1

e0ijεjEi −

3∑

i,i′=1

1

2χε0

ii′EiEi′

)

. (2.2)

Parameters cE0jj′ (T ), e0ij , χ

ε0ii′ are the so called “seed” elas-

tic constants, “seed” coefficients of piezoelectric stressesand “seed” dielectric susceptibilities, respectively; v is thevolume of a primitive cell. Matrices cE0

jj′ , e0ij , χε0ii′ are giv-

en by:

cE0jj′ =

cE011 cE0

12 cE013 0 cE0

15 0

cE012 cE0

22 cE023 0 cE0

25 0

cE013 cE0

23 cE012 0 cE0

35 0

0 0 0 cE044 0 cE0

46

cE015 cE0

25 cE035 0 cE0

55 0

0 0 0 cE046 0 cE0

66

,

e0ij =

0 0 0 e014 0 e016

e021 e022 e023 0 e025 0

0 0 0 e034 0 e036

,

χε0ii′ =

χε011 0 χε0

13

0 χε022 0

χε013 0 χε0

33

.

(2.3)

In the paraelectric phase, all coefficients e0ij ≡ 0.

Other terms in (2.1) describe the pseudospin part ofthe Hamiltonian. In particular, the second term in (2.1)is the Hamiltonian of short-range interactions:

Hshort = −2∑

qq′

(

w1

σq1

2

σq2

2+ w2

σq3

2

σq4

2

)

×(

δRqRq′+ δRq+Rc,Rq′

)

. (2.4)

In (2.4), σqf is the z-component of the pseudospin op-erator that describes the state of the f -th bond (f =1, 2, 3, 4) in the q-th cell. The first Kronecker delta corre-sponds to the interaction between protons in the chainsnear the tetrahedra HPO3 of type “I” (see Fig. 1), whilethe second one near the tetrahedra HPO3 of type “II”,Rc is the lattice vector along OZ-axis. The contribu-tions into the energy of interactions between the protonsnear tetrahedra of different types, as well as the mean

3702-2

EFFECT OF STRESSES σ5, σ6 AND ELECTRIC FIELD E3 ON THERMODYNAMIC CHARACTERISTICS. . .

values of the pseudospins ηf = 〈σqf 〉, which are relatedto the tetrahedra of different types, are identical.

Parameters w1, w2, which describe the short-range in-teractions within the chains, are expanded linearly intoseries over strains εj :

w1,2 = w0 +∑

l

δlεl ± δ4ε4 ± δ6ε6, (l = 1, 2, 3, 5). (2.5)

The third term in (2.1) describes the long-range dipole-dipole interactions and indirect (through the lattice vi-brations) interactions between protons which are takeninto account in the mean field approximation:

Hlong =1

2

∑

qq′

ff′

Jff ′(qq′)〈σqf 〉

2

〈σq′f ′〉

2

−∑

qq′

ff′

Jff ′(qq′)〈σq′f ′〉

2

σqf

2. (2.6)

Fourier transforms of interaction constants Jff ′ =∑

q′

Jff ′(qq′) at k = 0 are linearly expanded over strains

εj :

J 11

33

= J011 +

∑

l

ψ11lεl ± ψ114ε4 ± ψ116ε6, J13 = J013 +

∑

l

ψ13lεl + ψ134ε4 + ψ136ε6,

J 12

34

= J012 +

∑

l

ψ12lεl ± ψ124ε4 ± ψ126ε6, J 14

23

= J014 +

∑

l

ψ14lεl ± ψ144ε4 ± ψ146ε6, (2.7)

J 22

44

= J022 +

∑

l

ψ22lεl ± ψ224ε4 ± ψ226ε6, J24 = J024 +

∑

l

ψ24lεl + ψ244ε4 + ψ246ε6.

In model [5], there is sign “+” instead of “±” in ex-pressions (2.5) and (2.7), that is w1 = w2, J11 = J33,J12 = J34, J14 = J23, J22 = J44.

As a result, (2.6) can be written as:

Hlong = NH0 −∑

q

4∑

f=1

Hf

σqf

2, (2.8)

where

H0 =

4∑

f,f ′=1

1

8Jff ′ηfηf ′ , Hf =

4∑

f ′=1

1

2Jff ′ηf ′ . (2.9)

The fourth term in (2.1) describes the interactions ofpseudospins with the external electric field:

HE = −∑

qf

µfEσqf

2. (2.10)

Here, µ1 = (µx13, µ

y13, µ

z13), µ3 = (−µx

13, µy13,−µ

z13),

µ2 = (−µx24,−µ

y24, µ

z24), µ4 = (µx

24,−µy24,−µ

z24) are the

effective dipole moments per one pseudospin.In [7], using Hamiltonian (2.1) and the two-particle

cluster approximation, the thermodynamic potential ofGPI was obtained in the presence of mechanical stressesand electric fields. On the basis of this thermodynamicpotential, the expressions for the dielectric, piezoelectricand thermal characteristics were derived for any direc-tion of the stress or field. In the present paper, we usethe expressions obtained in [7] for the calculation of thesimultaneous effect of the shear stress σ5 or σ6 togetherwith the electric field E3 on the thermodynamic charac-teristics of GPI. We do not show the explicit expressionsfor these characteristics in the present paper because oftheir complicated nature.

III. THE RESULTS OF NUMERICAL

CALCULATIONS

For the numerical calculation of the dielectric andpiezoelectric characteristics of GPI, we have used theparameters determined in [7] and earlier in [5] from thecondition of agreement of the calculated characteristicswith experimental data. They are enumerated below.

• The parameter of the short-range interactionsw0/kB=800 K (820 K in [5]);

• The parameters of the long-range interactions ap-pear in the expressions obtained in [5, 7] in theform of such combinations ν0±

1 = 14(J0

11 ± J013),

ν0±2 = 1

4(J0

12 ± J014), ν

0±3 = 1

4(J0

22 ± J024); they are

equal to: ν0+1 = ν0+

2 = ν0+3 =3.065 K (2.643 K in

[5]), ν0−1 = ν0−

2 = ν0−3 =0.05K (0.2K in [5]), where

ν0±f = ν0±

f /kB.

• The optimal values of the deformational potentials:δ1=500 K, δ2=600 K, δ3=500 K, δ4=150 K,δ5=100 K, δ6=150 K; δi=δi/kB;the parameters ψii′j also appear in the expres-sions obtained in [5, 7] in the form of combina-tions ψ±

1j = 14(ψ11j ± ψ13j), ψ

±

2j = 14(ψ12j ± ψ14j),

ψ±

3j = 14(ψ22j±ψ24j). These combinations are equal

to: ψ+f1 = 93.6 K, ψ+

f2 = 252.5 K, ψ+f3 = 110.7 K,

ψ+f4 = ψ+

f6 = ψ−

f4=ψ−

f6=79.5 K, ψ+f5 = 22.7 K,

3702-3

A. S.VDOVYCH, R. R. LEVITSKII, I. R. ZACHEK

ψ−

f1=ψ−

f2=ψ−

f3=ψ−

f5=0 K, where ψ±

fi =ψ±

fi/kB. In

[5] the parameters ψ±

fi were: ψ+f1 = 87.9 K, ψ+

f2 =

237.0 K, ψ+f3 = 103.8 K, ψ+

f4 = 149.1 K, ψ+f5 =

21.3 K, ψ+f6 = 143.8 K, ψ−

fi=0 K.

• The components of effective dipole mo-ments in the paraelectric phase are equal toµx

13=0.4·10−18 esu·cm; µy13=4.05·10−18 esu·cm

(4.02·10−18 esu·cm in [5]); µz13=4.2·10−18 esu·cm

(4.3·10−18 esu·cm in [5]); µx24=2.3·10−18 esu·cm;

µy24=3.0·10−18 esu·cm; µz

24=2.2·10−18 esu·cm. Inthe ferroelectric phase, the y-component of the firstdipole moment is µy

13ferro = 3.82 ·10−18 esu·cm; theother components are such as in the paraelectricphase.

• The volume of the primitive cell of GPI is υ =0.601·10−21 cm3.

• The “seed” coefficients of piezoelectric stress e0ij ,

“seed” dielectric susceptibilities χε0ij and “seed” elas-

tic constants cE0ij are obtained as follows: e0ij =

0.0 esucm

2;

χε011 = 0.1, χε0

22= 0.403, χε033 = 0.5, χε0

31 = 0.0;

c0E11 = 269.1 kbar, cE0

12 = 145 kbar,

cE013 = 116.4 kbar, cE0

15 = 39.1 kbar,

cE022 = (649.9− 0.4(T − Tc)) kbar,

cE023 = 203.8 kbar, cE0

25 = 56.4 kbar,

cE033 = 244.1 kbar, cE0

35 = −28.4 kbar,

cE055 = 85.4 kbar, cE0

44 = 153.1 kbar,

cE046 = −11 kbar, cE0

66 = 118.8 kbar.

Now, let us dwell upon the obtained results. Figure2 presents the temperature dependences of spontaneouspolarization P2, inverse longitudinal dielectric permittiv-ity ε−1

22 , transverse permittivities ε11 and ε33, molar heatcapacity Cp, piezoelectric coefficients e2i, d2i, h2i and g2i

of the GPI crystal calculated in [5] (dash-dotted lines),as well as corresponding temperature dependences cal-culated later in [7] taking into account the change ofsymmetry in the presence of the shear strains ε4 or ε6(solid lines).

As one can see from Fig. 2, taking into account thechange of symmetry and the corresponding correction ofthe theory parameters have weakly influenced the ther-modynamic characteristics of the GPI crystal in the ab-sence of any stresses and fields. The results of the inves-tigation into the effects of the hydrostatic and uniaxialpressure on the thermodynamic characteristics of GPI,which are calculated using the earlier parameters [12,13]and the corrected parameters [7], do not much differ ei-ther.

160 180 200 2200

0.1

0.2

0.3

0.4

0.5

T, K

P2, 10−6 C / cm2

210 220 2300

10

20

30

40

50

60

T, K

(ε22

− ε220 )−1

10−3

140 180 220 2600

50

100

150

200

250

300

T, K

εii

ε33

ε11

150 200 250100

120

140

160

180

200C

p, J/(mol K)

T, K

220 222 2240

2

4

6

8

10x 105

T, K

e2i

, esu / cm2

e21

e23

e22

e25

210 215 220 2250

0.2

0.4

0.6

0.8

1x 10−6

T, K

d2i

, esu / dyn

d21

d23

d22

d25

210 215 220 2250

2

4

6

8

10

12

14x 104

T, K

h2i

, dyn / esu

h21

h25h

23

h22

210 215 220 2250

1

2x 10−7

T, K

g2i

, cm2 /esu

g23

g21

g22

g25

Fig. 2. (Color online). The temperature dependences of spontaneous polarization P2 (• – [8]), inverse longitudinal dielectricpermittivity ε−1

22(• – [9]), transverse permittivities ε11, ε33 (N, N – [1]), molar heat capacity Cp (• – [10]), piezoelectric

coefficients e2i, d2i (d21 – • [11], d23 – • [11]), h2i, g2i of GPI crystal.

3702-4

EFFECT OF STRESSES σ5, σ6 AND ELECTRIC FIELD E3 ON THERMODYNAMIC CHARACTERISTICS. . .

Now let us consider how to change the thermodynam-ic characteristics of the GPI crystal at the simultaneousapplication of the shear stress σ5 or σ6 and the electricfield E3. It should be noted that experimental data forε22 obtained by different authors slightly differ from eachother (see [5]). I is quite likely that this is connected withthe different quality of the samples. In our model, thisdisagreement means different effective dipole momentsµy

13 in different samples. We assume that there can exista sample with the same µy

13 in the ferro- and paraelectricphases. Therefore, for the sake of simplicity, hereinafterwe consider a case of the GPI sample with the same val-ue of the effective dipole moment in the paraelectric andferroelectric phase µy

13ferro=µy13para=3.82·10−18 esu·cm.

Also we must note that in all further figures the mainnumber 5 or 6 means the direction of the applied stress(σ5 or σ6, respectively), the upper index shows the valueof the stress (kbar), and the lower index shows the valueof the electric field strength (MV/m). For example, thenotation 5−2

4 means the shear stress σ5 = −2 kbar andelectric field E3 = 4 MV/m applied simultaneously.

Figure 3 shows the dependences of the phase transi-tion temperature Tc in the GPI crystal on the electricfield E3 at different values of the shear stress σ5.

−4 −2 0 2

x 106

210

214

218

222

226

Tc, K

E3, MV/m

51

5−1

1 52

5−2

Fig. 3. The dependences of the phase transition tempera-ture Tc in GPI crystal on the electric field E3 at differentvalues of the shear stress σ5. Curve 1 corresponds to thecase of σj = 0, another curves correspond to the stressesσ5 = ±1,±2 kbar.

In the absence of the stress, temperature Tc dependson field E3 by the law Tc(E3) ∼ −E2

3 (curve 1 in Fig. 3).On the other hand, at zero field, temperature Tc linearlyincreases with stress σ5 because of the strengthening ofthe interactions between pseudospins under this stress(see [7]). Since stress σ5 does not change the symmetryof the crystal, the curve Tc(E3) qualitatively does notchange under this stress, but only shifts in temperature(Fig. 3, curves 5±1, 5±2 at σ5 = ±1,±2 kbar, respective-ly).

190 210 2300

0.1

0.2

0.3

0.4

P2, 10−6 C/cm2

T, K

520

500

504

514

524

522

521

200 220 2400

2

4

6

8

10

12

14

524

504

522

T, K

ε−122

500

520

514

10−2

190 210 2300

100

200

300

400

500

500

520

504

524

T, K

ε33

514

522 52

1

200 220 2402

4

6

8

10

524

T, K

∆Cp, J/mol K

500

504

514

520

522

200 220 240

0

1

2

3

4

5

6

7x 105

524

T, K

e21

, esu/cm2

504

514

520

522

500

180 200 220 2400

1

2

3

4

5

6x 104

504

514

524

T, K

h21

, dyn/esu

500

522

520

Fig. 4. The temperature dependences of polarization P2 (• are taken from [8]), inverse longitudinal dielectric permittivityε−1

22, transverse dielectric permittivity ε33 (N – [1]), proton contribution to molar heat capacity ∆Cp, longitudinal piezoelectric

coefficients e21, h21 of GPI crystal at different values of the stress σ5 and the electric field E3.

The temperature dependences of the thermodynamiccharacteristics at different values of stress σ5 and electricfield E3 are presented in Figs. 4 and 5. On these figures

curves 500 correspond to the case of a zero stress and field.

In the special case of a zero stress but a nonzero field E3

(curves 504 at σ5 = 0 kbar and E3 = 4 MV/m), the curves

3702-5

A. S.VDOVYCH, R. R. LEVITSKII, I. R. ZACHEK

of temperature dependences are shifted to lower temper-atures in relation to curves 50

0. Besides, as one can seefrom Fig. 4, permittivity ε33 in the ferroelectric phasesignificantly increases in the presence of field E3. As wasshown in [6], the decrease of Tc and increase in ε33 in theferroelectric phase are connected with the partial disor-

dering in the chain “B” (Fig. 1) under the action of fieldE3. One more peculiarity is the change of sign of thetransverse piezoelectric coefficients e1j , e3j , h1j and h3j

near Tc (see Fig. 5) in field E3, which is connected withalmost complete disordering of protons in the chain “B”near Tc, as was shown in [14].

140 180 220−5

−4

−3

−2

−1

0

1

2

3x 104 e

1j. esu/cm2

T,K

500

520

522

504

524

140 180 220

−5

0

5

10

15

20x 104

T, K

e3j

, esu/cm2

500

520 52

2

504

524

140 180 220−4

−3

−2

−1

0

1

2x 104

T, K

h1j

. dyn/esu

500 52

0

522

504

524

100 140 180 220−1.5

−1

−0.5

0

0.5

1

1.5x 104

T, K

h3j

, dyn/esu

520

500

522

504

514

524

Fig. 5. The temperature dependences of transverse piezoelectric coefficients e1j = e14 = e16, e3j = e34 = e36, h1j = h14 = h16

and h3j = h34 = h36 of GPI crystal at different values of the stress σ5 and the electric field E3.

180 220 2600

0.1

0.2

0.3

0.4

604

614

600

60.24

624 62

0

T, K

P2, 10−6 C/cm2

621 62

2

180 200 220 2400

2

4

6

8

10

12

14

604

621

624

614

T, K

ε−122

620

60.24

600

622

10−2

180 2200

100

200

300

400

500

600

620 60

4

624

T, K

ε33

614 62

2

621 60.2

4

200 220 2402

3

4

5

6

7

8

9

600

620

604

624

T, K

∆Cp, J/mol K

614 60.2

4

621

622

200 220 240

0

1

2

3

4

5

6

7x 105

624

T, K

e21

, esu/cm2

604

614

620

622

600

60.24

621

180 200 220 2400

1

2

3

4

5

6x 104

604

614

624

T, K

h21

, dyn/esu

600

622

620

60.24

621

Fig. 6. The temperature dependences of polarization P2 (• – [8]), inverse longitudinal dielectric permittivity ε−1

22, transverse

dielectric permittivity ε33 (N – [1]), proton contribution to molar heat capacity ∆Cp, longitudinal piezoelectric coefficients e21,h21 of GPI crystal at different values of the stress σ6 and the electric field E3.

Stress σ5 at the constant field E3 leads to a shiftof the curves of the temperature dependences of theabovementioned thermodynamic characteristics to high-er temperatures. That is, curves 52

0 (2 kbar,0 MV/m)are shifted to higher temperatures in relation to curves500 (0 kbar,0 MV/m); curves 52

4 are shifted in relation to504 and so on.

In the case of the application of the shear stress σ6 ina zero field (curves 62

0 in all the figures), the symmetry ofthe crystal reduces, and the two sublattices (the chains“A” and “B”) become nonequivalent (see [7]). As a re-sult, the interactions between pseudospins in chain “A”become stronger, but in chain “B” they become weak-er. Strengthening of interactions in a sublattice at some

3702-6

EFFECT OF STRESSES σ5, σ6 AND ELECTRIC FIELD E3 ON THERMODYNAMIC CHARACTERISTICS. . .

value of stress σ6 causes a phase transition to the ferro-electric phase and increases temperature Tc. Therefore,curves P2(T ), ε22(T ), e21(T ), h21(T ), ∆Cp(T ) shift tohigher temperatures (Fig. 6). Since chains “A” and “B”become nonequivalent, then the polarizations of bothsublattices in the XZ plane do not compensate each oth-er, similarly as in ferrimagnets. As a result, there appearthe components of spontaneous polarization P1 and P3

(Fig. 7) in the XZ plane, and the transverse permit-tivity ε33(T ) (Fig. 6, curve 62

0) looks like a longitudinalcomponent of dielectric permittivity.

160 200 240

−0.2

−0.1

0

0.1

0.2

0.3

T, K

P1, 10−6 C/cm2

614 62

4

620

604

622 62

1

160 200 2400

0.2

0.4

0.6

0.8

1

T, K

P3, 10−6 C/cm2

604

614

620

621

624

622

Fig. 7. The temperature dependences of polarizations P1

and P3 of GPI crystal at different values of the stress σ6 andthe electric field E3.

140 180 220−5

−4

−3

−2

−1

0

1

2

3x 104 e

1j. esu/cm2

T,K

600

620 62

1

622

604 61

4 62

4

140 180 220

−5

0

5

10

15

20x 104

T, K

e3j

, esu/cm2

600

620

621

622

604 61

4 62

4

140 180 220−4

−3

−2

−1

0

1

2x 104

T, K

h1j

. dyn/esu

600

620

621

622

60461

4624

100 140 180 220−1

−0.5

0

0.5

1

1.5x 104

T, K

h3j

, dyn/esu

620

600

622

604 62

4 61

4

621

Fig. 8. The temperature dependences of piezoelectric coef-ficients e1j , e3j , h1j i h3j of GPI crystal at different values ofthe stress σ6 and the electric fields E3.

It is necessary to note that in the presence of stress σ6

and in a zero field, components P1 6= 0 and P3 6= 0 onlyin the ferroelectric phase (Fig. 7, curves 62

0), whereas inthe other special case, σ6 = 0 and E3 6= 0, componentsP1 6= 0 and P3 6= 0 at all temperatures (curves 60

4).As one can see from Fig. 8 (curves 62

0), the piezo-electric coefficients e1j(T ), e3j(T ) tend to infinity inpoint Tc. This is because under a nonzero stress σ6 thesmall changes in strains dε4, dε6 are accompanied bya change in temperature dTc and by a shift of curvesP1(T ) and P3(T ) to higher temperatures. Inasmuch asdPi/dT → ∞ near the phase transition temperature,then dPi/dε4 → ∞, dPi/dε6 → ∞.

Since all the components Pα 6= 0 under stress σ6 inthe ferroelectric phase, field E3, which is applied addi-tionally to stress σ6, contains a component parallel to thetotal polarization (the longitudinal component E

∗3 ‖ P).

This longitudinal component leads to the smearing of thephase transition (curves 62

1, 622, 61

4, 624, 60.2

4 ).

IV. CONCLUSIONS

The reducing of the symmetry of interactions un-der shear stresses ε4 and ε6 practically does not affectthe components of the spontaneous polarization vector,dielectric permittivity tensor, longitudinal piezoelectriccoefficients, elastic constants, and molar heat capacity.However, it allows to calculate also the transverse piezo-electric coefficients.

Dielectric properties of GPI depend on the way of theapplication of mechanical stresses and fields. The appli-cation of only the electric field E3 in the absence of me-chanical stresses decreases temperature Tc, increases per-mittivity ε33 in the ferroelectric phase, and also inducesthe components of polarization P1 and P3 in the wholetemperature range. The shear stress σ5, which is appliedadditionally to field E3, qualitatively does not changethe temperature dependences of thermodynamic charac-teristics, but only linearly increases the phase transitiontemperature.

The application of only the shear stress σ6 to the crys-tal reduces the symmetry of the crystal, and chains “A”and “B” become nonequivalent. Consequently, in the fer-roelectric phase, besides component P2, the componentsof polarization P1 and P3 appear in the XZ plane. Thatis, the total spontaneous polarization vector is nonper-pendicular to the OZ axis. Therefore, field E3, whichis applied additionally to stress σ6, contains the compo-nent parallel to the total polarization (the longitudinalcomponent). This longitudinal component leads to thesmearing of the phase transition.

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ВПЛИВ НАПРУГ σ5, σ6 ТА ЕЛЕКТРИЧНОГО ПОЛЯ E3 НА ТЕРМОДИНАМIЧНI

ХАРАКТЕРИСТИКИ СЕГНЕТОЕЛЕКТРИКА GPI

А. С. Вдович1, Р. Р. Левицький1, I. Р. Зачек2

1Iнститут фiзики конденсованих систем НАН України

вул. Свєнцiцького, 1, Львiв, 79011, Україна2Нацiональний унiверситет “Львiвська полiтехнiка”

вул. С. Бандери 12, 79013, Львiв, Україна

Для дослiдження ефектiв, що виникають пiд дiєю зсувних напруг σ5, σ6 та електричного поля E3, вико-

ристано модифiковану модель сегнетоелектрика фосфiту глiцину (GPI) через врахування п’єзоелектричного

зв’язку структурних елементiв, якi впорядковуються, з деформацiями εj . В наближеннi двочастинкового

кластера розраховано вектори поляризацiї та компоненти тензора статичної дiелектричної проникностi ме-

ханiчно затиснутого кристала, їхнi п’єзоелектричнi та тепловi характеристики. Дослiджено одночасну дiю

напруги σ5 i поля E3, а також напруги σ6 i поля E3 на фазовий перехiд та фiзичнi характеристики кристала.

3702-8