Temperature-pressure phase diagram and ferroelectric properties ofBaTiO3 single crystal based on a modified Landau potentialJ. J. Wang, P. P. Wu, X. Q. Ma, and L. Q. Chen Citation: J. Appl. Phys. 108, 114105 (2010); doi: 10.1063/1.3504194 View online: http://dx.doi.org/10.1063/1.3504194 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v108/i11 Published by the American Institute of Physics. Related ArticlesMisfit strain dependence of ferroelectric and piezoelectric properties of clamped (001) epitaxialPb(Zr0.52,Ti0.48)O3 thin films Appl. Phys. Lett. 99, 252904 (2011) Phase transitions in ferroelectric-paraelectric superlattices J. Appl. Phys. 110, 114109 (2011) Dielectric and nonlinear current–voltage characteristics of rare–earth doped CaCu3Ti4O12 ceramics J. Appl. Phys. 110, 094101 (2011) Scanning electro-optic microscopy of ferroelectric domain structure with a near-field fiber probe J. Appl. Phys. 110, 084117 (2011) Ferroelectric domains in epitaxial PbTiO3 films on LaAlO3 substrate investigated by piezoresponse forcemicroscopy and far-infrared reflectance J. Appl. Phys. 110, 084115 (2011) Additional information on J. Appl. Phys.Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors

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Temperature-pressure phase diagram and ferroelectric properties of BaTiO3single crystal based on a modified Landau potential

J. J. Wang,1,2 P. P. Wu,1,2 X. Q. Ma,2,a� and L. Q. Chen1

1Department of Materials Science and Engineering, Pennsylvania State University, University Park,Pennsylvania 16802, USA2Department of Physics, University of Science and Technology Beijing, Beijing 100083, China

�Received 1 July 2010; accepted 18 September 2010; published online 6 December 2010�

A modified eighth-order Landau potential was proposed for the BaTiO3 single crystal by takingaccount into the quantum mechanical effects at low temperature. While all existing thermodynamicpotentials for BaTiO3 fail to accurately describe the pressure dependence of ferroelectric transitiontemperatures, the temperature and hydrostatic pressure phase diagram constructed using themodified potential shows excellent agreement with experimental measurements by Ishidate, Abe,Takahashi, and Mori �Phys. Rev. Lett. 78, 2397 �1997��. On the basis of the new proposed Landaupotential, we calculated the dielectric coefficients, spontaneous polarizations, temperature-electricfield phase diagram, and piezoelectric coefficients, all in good agreement well with existingexperimental data. © 2010 American Institute of Physics. �doi:10.1063/1.3504194�

I. INTRODUCTION

As a classic ferroelectric, the thermodynamics ofBaTiO3 is the most extensively studied over half acentury1–7since Devonshire wrote down a sixth-order classicLandau potential to describe the thermodynamics of phasetransitions in BaTiO3.1 Bell and Cross reported a set of sixth-order Landau polynomial potential coefficients by assumingthree of them temperature-dependent. Li, et al.,7 proposed aneighth-order Landau potential for predicting the phase tran-sitions, domain structures, and other properties of BaTiO3 forboth thin films and bulk crystals.7–11 Finally, Wang, et al.,12

proposed another eighth-order Landau potential by arguingthat the second and higher-order coefficients should betemperature-dependent to describe adequately the thermody-namic behavior of BaTiO3. The classical Landau potentialshave been reasonably successful in describing the phase tran-sitions and ferroelectric properties of BaTiO3 above the so-called saturation temperature above which the quantum me-chanical effects can be neglected.13

The phase transitions and properties of a ferroelectriccan be modified by external fields such as biaxial or unixalstress, hydrostatic pressure, or electric field. The focus of thiswork is on the pressure dependence of transition tempera-ture, represented by the pressure-temperature phase diagram.Both first principle studies and experimental measurementshowed that the phase transitions of bulk BaTiO3 fromparaelectric phase to ferroelectric phase and from the parallelphase to ferroelectric phase should be first-order,14–17 thusthe phase boundaries can be discussed using the Clapeyronequation. On a pressure-temperature phase diagram, the pres-sure dependence of a transition temperature is described bythe Clapeyron relation, dp /dT=L /T�V=�S /�V, wheredp /dT is the slope of the two-phase coexistence curve, L thelatent heat, �S the entropy change, T the transition tempera-ture, �V the volume change. In the classical limit, �S isapproximately proportional to �V, hence the pressure depen-

dence of transition temperature is also approximatelylinear.18,19 However, at low temperature near 0 K, the quan-tum theory predicts: TC�p�� �pC− p�1/2, where TC is the tran-sition temperature, p is the hydrostatic pressure and pC is itsvalue at TC=0 K.19–21 Therefore, in order to accurately de-scribe the dependence of the transition temperatures on ex-ternal fields including those below the saturation tempera-ture, the effect of quantum fluctuation such as zero-pointvibration on vibrational mode of the lattice should be con-sidered.

The dielectric constants, polarizations, and phase transi-tions under hydrostatic pressure were experimentally studiedby Samara et al. in 1966,22 Decker and Zhao in 1989,23 andIshidate et al. in 1997.24 Samara et al. considered hydrostaticpressures up to 2.5GPa. It was shown that the transition tem-perature decreases with hydrostatic pressure and a triplepoint was obtained by classical extrapolation.22 A similarmeasurement was performed by Decker et al. for hydrostaticpressure up to 3.8 GPa. They showed that the cubic to tetrag-onal ferroelectric phase transition in BaTiO3 under hydro-static pressure was displacive and became less discontinuousas pressure was increased until a triple point was reached atabout 3.5GPa and −40 °C. Furthermore, it was demonstratedthat the peak values for the dielectric constants increasedwith hydrostatic pressure. A more recent and thorough mea-surement was conducted by Ishidate et al. determined theBaTiO3 temperature-pressure phase diagram for pressures upto 10 GPa and temperatures down to 4 K, resulting in arelatively complete pressure-temperature phase diagram in-cluding both the classical limit above the saturation tempera-ture and the quantum limit below the saturation temperature.They also obtained three critical pressures of 5.4, 6.0, and6.5 GPa at which the rhombohedral, orthorhombic, or tetrag-onal phase disappears, respectively, over the whole tempera-ture range. The pressure-temperature phase diagram obtainedby Zhong et al.15 using first principle calculations, althoughthere are quantitative differences, qualitatively agrees withIshidate’s experimentally measured phase diagram. Jorge etal. recalculated the pressure-temperature phase diagram ofa�Electronic mail: xqma@sas.ustb.edu.cn.

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BaTiO3 by considering the zero-point motion of ions andusing theoretically perfect hydrostatic pressure. The recalcu-lated phase diagram by first principles corroborated Ishi-date’s experimental results.25

To correctly describe the phase transitions at low tem-peratures using a phenomenological description, Salje et al.included the quantum mechanical effects in the classicalLandau potential.4–6,13,18 For example, Salje et al. employeda modified Landau-type expression for the Gibbs energy byincluding the saturation effects of order parameter to de-scribe the variation in the order parameter with temperaturefor As2O5, LaAlO3, CaCO3, NaNO3, Pb3�PO4�2, etc.13 andthe obtained results were in excellent agreement with experi-mental measurements. Salje et al. also computed thetemperature-pressure or temperature-composition phase dia-grams for SrTiO3, KH2PO4, KTaO3, and SbSI using Landautheory with quantum effects.18 Hayward et al. determined thepressure-temperature phase diagram of BaTiO3 based on theLandau description proposed by Salje et al.6 However, due tothe complexity of multiple phase transitions in BaTiO3, theirmodel was not able to predict the orthorhombic to rhombo-hedral phase transition. There has also been no systematictheoretical study on the physical properties of BaTiO3 undera hydrostatic pressure.6 Therefore, the main objective of thiswork is to analyze the effect of hydrostatic pressure on ferro-electric phase transitions as well as ferroelectric properties inBaTiO3. For this purpose, we modified the eighth-order ther-modynamic potential for BaTiO3 by connecting the classicaland quantum limits following the work of Salje et al.7

The paper is organized as follows. In Sec. II, we describethe modified Landau potential and the determination of thecoefficients. In Sec. III, a temperature-pressure phase dia-gram is constructed based on the new Landau potential co-efficients. We compared the predicted results with those ob-tained by experiments as well as with those obtained usingthe classical Landau potential without including the quantumeffects. In Sec. IV, the dielectric constants under a hydro-static pressure are calculated and are compared with existingexperimental data. In Sec. V, the phase transitions under anelectric field along �001�, �101�, and �111� directions arestudied. In Sec. VI, the electric field induced strains andpiezoelectric coefficients are discussed. Finally, we summa-rize the results in Sec. VII.

II. LANDAU THERMODYNAMIC POTENTIAL

Taking the cubic paraelectric phase as the reference, theGibbs free energy as a function of polarization under stresscan be expressed as

�G = fLGD −1

2s11��1

2 + �22 + �3

2� − s12��1�2 + �1�3

+ �2�3� −1

2s44��4

2 + �52 + �6

2� − Q11��1P12 + �2P2

2

+ �3P32� − Q12��1�P2

2 + P32� + �2�P1

2 + P32� + �3�P2

2

+ P12�� − Q44��4P2P3 + �5P1P3 + �6P2P1� , �1�

with

fLGD = �1�P12 + P2

2 + P32� + �11�P1

4 + P24 + P3

4� + �12�P12P2

2

+ P12P3

2 + P22P3

2� + �123P12P2

2P32 + �111�P1

6 + P26

+ P36� + �112�P1

2�P24 + P3

4� + P22�P1

4 + P34� + P3

2�P14

+ P24�� + �1111�P1

8 + P28 + P3

8� + �1122�P14P2

4 + P14P3

4

+ P24P3

4� + �1112�P16�P2

2 + P32� + P2

6�P12 + P3

2�

+ P36�P1

2 + P22�� + �1123�P1

4P22P3

2 + P12P2

4P32

+ P12P2

2P34� , �2�

where Pi is the ith component of polarization, �i ith compo-nent of applied stress in Voigt notation, s11, s12, and s44 theelastic compliance constants of cubic phase, and Q11, Q12,and Q44 the corresponding electrostrictive coefficients.

All the Landau potential coefficients in Eq. �2� are as-sumed to be temperature-independent except �1. The TS in-cluded in �1 is named the saturation temperature belowwhich the quantum mechanical effects are significant. Weuse the same saturation temperature �160 K� as in Salje etal.’s work.6 The main difference from the existing eighth-order Landau potential7 is the fact that �11, �12, �111, �112,and �123 in the modified potential are dependent on the ap-plied stress �see Table I for the specific dependencies�. Thestress dependences of �11, �12, �111, �112, and �123 representthe higher-order electrostrictive coupling between stress andpolarization. Such high-order electrostrictive coupling wasdiscussed as early as in the 1950s on the electric field in-duced strains �=bE2+dE4+ fE6+¯.26 Although the firstterm is sufficient for most purposes, the higher-order anhar-monic terms are necessary in order to account for the thermalexpansion. In the case of cubic BaTiO3, for instance, thedeparture of the oxygen potential wells from being truly har-monic is one of the important features of the theory, ex-plained by Devonshire and Slater.1,27Furthermore, the pres-ence of higher-order terms is also used to explain theexperimentally observed temperature dependence of the di-electric susceptibility.26 Similarly, one can also write the gen-eral polarization dependence of strain through, �=bP2

+dP4+ fP6+ . . .. Under high hydrostatic pressure, the higher-order coupling between stress and polarization is required toaccount for the accurate pressure-temperature phase diagram.

The coefficients obtained in this work along with thosefrom a number of previous works are shown in Table I. Itshould be noted that the existing thermodynamic potentialswere obtained by fitting the coefficients to the properties ofBaTiO3 under zero pressure. Therefore, the existing coeffi-cients may not be applicable for describing properties underhydrostatic pressure. In this work, �11, �111, and �1111 wereobtained using the properties of the tetragonal phase underpressure, including the cubic to tetragonal transition tem-perature, and the spontaneous polarization and the dielectricconstants of the tetragonal phase. Similarly, �12, �112, �1122,and �1112 were fitted to the properties of the orthorhombicphase, and �123 and �1123 to the properties of the rhomobhe-dral phase. While �1 is assumed to be linearly dependent ontemperature in the classical approximation,1,3,7 �1 is assumedto be a hyperbolic cotangent function of temperature to in-clude quantum effects as it was proposed by Salje et al.6,13,18

114105-2 Wang et al. J. Appl. Phys. 108, 114105 �2010�

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Since only the second-order coefficient in the Landaufree energy is temperature-dependent, the entropy change fora phase transition at the transition temperature can be ob-tained from �S= �Pproduct

2 �TC�− Pparent2 �TC�� /2�0C, where �0

is the permittivity of vacuum and C is the Curie–Weissconstant.28 Therefore, the entropy changes obtained in ourwork are 13 361 J m−3 K−1 for cubic to tetragonal phasetransition, 9237 J m−3 K−1 for tetragonal to orthorhombic,and 5531 J m−3 K−1 for orthorhombic to rhombohedral. Forcomparison, the entropy changes in Li et al.’s work are13 535 J m−3 K−1, 6640 J m−3 K−1, and 6971 J m−3 K−1,respectively.7 These values are in general agreement withexperimental values listed in Ref. 28: 13 098–13 644, 5898–9940, and 4357–7624 J m−3 K−1.28

III. TEMPERATURE-PRESSURE PHASE DIAGRAM

Under a hydrostatic pressure, the applied stress tensorssatisfy �1=�2=�3=−p, �4=�5=�6=0, and Eq. �1� becomes

�G = fLGD − �3

2s11 + 3s12�p2 + p�Q11 + 2Q12�

��P12 + P2

2 + P32� . �3�

The variation in polarization with temperature and pressurewas obtained and shown in Figs. 1�a� and 1�b�. Each phasetransition can easily be identified in Fig. 1 based on thejumps in polarization values. Application of a hydrostaticpressure decreases polarization at a given temperature. Asthe hydrostatic pressure increases, the rhomobohedral phasedisappears first at its critical pressure, followed by the disap-pearance of the orthorhombic phase at 5.8 GPa, and finally,the tetragonal phase at about 6.4 GPa.

With the coefficients listed in Table I, a temperature-pressure phase diagram was constructed and shown in Fig. 2.We also included the experimental data from Ishidate’swork24 and theoretical results calculated by using classicalLandau coefficients of Li et al. It can be seen from Fig. 2 that

the classical Landau potential coefficients of Li et al. pro-duced linear dependence of transition temperatures on thehydrostatic pressure for all three phase transitions, rhombo-hedral to orthorhombic, orthorhombic to tetragonal, and te-tragonal to cubic. While the transition temperatures obtainedusing the classical Landau potential coefficients of Li et al.show good agreements with experimentally measured valuesabove the saturation temperature, their linear dependence on

TABLE I. Coefficients of Landau potential in Eq. �2�, where T is temperature in kelvin, TS is the saturation temperature below which the quantum mechanicaleffects are not ignorable, and �1, �2, and �3 are stress along x, y, and z direction in unit of gigapascal.

Coefficients This Work Bell and Cross Li et al. Wang et al. Units

�1 5.0 � 105 � TS

��Coth�TS

T� − Coth� TS

390�

3.34�105

��T−381�4.124�105

��T−388�3.61�105

��T−391�V m C−1

�11

−1.154�108

��1+0.037��1+�2+�3��4.69�106��T−393�

−2.02�108 −2.097�108�1.83�109+4.0�106T V m5 C−3

�12

6.530�108

��1+0.037��1+�2+�3�� 3.230�108 7.974�108�2.24�109+6.7�106T V m5 C−3

�111

−2.106�109

��1+0.023��1+�2+�3���5.52�107��T−393�

+2.76�109 1.294�1091.39�1010−3.2�107T V m9 C−5

�112

4.091�109

��1+0.023��1+�2+�3�� 4.470�109 −1.950�109 −2.2�109 V m9 C−5

�123

−6.688�109

��1+0.023��1+�2+�3�� 4.910�109 −2.500�109 5.51�1010 V m9 C−5

�1111 7.590�1010 0.0 3.863�1010 4.84�1010 V m13 C−7

�1112 −2.193�1010 0.0 2.529�1010 2.53�1011 V m13 C−7

�1122 −2.221�1010 0.0 1.637�1010 2.80�1011 V m13 C−7

�1123 2.416�1010 0.0 1.367�1010 9.35�1010 V m13 C−7

FIG. 1. �Color online� �a�. Polarization of BaTiO3 vs temperature underdifferent hydrostatic pressures. �b�. Polarization of BaTiO3 vs hydrostaticpressure at different temperatures.

114105-3 Wang et al. J. Appl. Phys. 108, 114105 �2010�

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pressure is inconsistent with experiments at low tempera-tures. On the other hand, the phase boundaries obtained bythe new Landau coefficients agree well with experimentalmeasurements over the entire temperature range. It should bepointed out that above the saturation temperature TS, theslope of the phase boundary between the cubic and tetrago-nal phases is mainly decided by the electrostrictive coeffi-cients �Q11+2Q12�. In this work, we used Q11=0.11, Q12

=−0.045, and Q44=0.029.28,29 However, the higher-ordercouplings between stress and polarization have a significanteffect on the phase transitions at lower temperatures, i.e., onboth the orthorhombic/tetragonal and rhombohedral/orthorhombic phase boundaries.

Using the new coefficients, the zero pressure transitiontemperatures from cubic to tetragonal, tetragonal to ortho-rhombic, and orthorhombic to rhombohedral phase are 396K, 280 K, and 207 K, respectively, which are consistent withexisting experimental data.30–34 The critical pressures for te-tragonal, orthorhombic, and rhomobohedral phases are 5.1,5.8, and 6.4, which also agree well with experimental valuesof 5.4 GPa, 6.0 GPa and 6.5 GPa, respectively.24 The slopesfor the cubic/tetragonal, tetragonal/orthorhombic, andorthorhombic/rhombohedral phase boundaries above thesaturation temperature 160 K are �45 K/GPa, �28 K/GPa,and �18 K/GPa, respectively.

The modified Landau potential was obtained accordingto the ferroelectric properties of bulk BaTiO3 at zero stress.In order to study the phase transitions of BaTiO3 thin filmsunder equally biaxial or nonequally biaxial in-plane misfitstrains,35–37 One can use a Legendre transformation to trans-form the Gibbs-free energy as a function of stress into theHelmholtz free energy as a function of strain. TheHelmholtz-free energy can be used to study the phase tran-sitions of thin films and nanowires under strained boundaryconditions. In this work, we focus on the properties of bulkBaTiO3, and the phase transitions for BaTiO3 nanowires un-der different mechanical boundary conditions will be dis-cussed in next paper.

IV. DIELECTRIC CONSTANTS UNDER PRESSURE

Dielectric constant is one of the most important proper-ties for a ferroelectric, e.g., for capacitor applications. Wecalculated dielectric constants under hydrostatic pressure us-ing the modified thermodynamic potential. We focused onthe dielectric constants �c along the polar directions, namely,the �001� direction for the tetragonal phase, the �011� direc-tion for the orthorhombic phase, and the �111� direction forthe rhombohedral phase. The relative dielectric stiffnesses ij

are deduced from the second-order derivatives of the freeenergy function with respect to polarization,

ij = �0�2�G

�Pi � Pj, �4�

In the cubic phase, P1= P2= P3=0,

11 = 22 = 33 = 2�0��1 + pQ11 + 2pQ12� , �5�

12 = 13 = 23 = 0. �6�

In the tetragonal phase, P1= P2=0, P3�0,

11 = 22 = 2�0��1 + pQ11 + 2pQ12 + �12P32 + �112P3

4

+ �1112P36� , �7�

33 = 2�0��1 + pQ11 + 2pQ12 + 6�12P32 + 15�111P3

4

+ 28�1111P36� , �8�

12 = 13 = 23 = 0. �9�

In the orthorhombic phase, P1=0, P2= P3�0,

11 = 2�0��1 + pQ11 + 2pQ12 + 2�12P32 + 2�112P3

4

+ �123P34 + 2�1112P3

6 + 2�1123P36� , �10�

22 = 33 = 2�0��1 + pQ11 + 2pQ12 + 6�11P32 + 15�111P3

4

+ 28�1111P36 + �12P3

2 + 7�112P34 + 16�1112P3

6

+ 6�1122P36� , �11�

23 = 2�0�2�12P32 + 8�112P3

4 + 12�1112P36 + 8�1122P3

2� ,

�12�

12 = 13 = 0. �13�

In the rhombohedral phase, P1= P2= P3�0,

11 = 22 = 33 = 2�0��1 + pQ11 + 2pQ12 + 6�11P32

+ 15�111P34 + 28�1111P3

6 + �12P32 + 14�112P3

4

+ �123P34 + 32�1112P3

6 + 12�1122P36 + 8�1123P3

6� ,

�14�

12 = 13 = 23 = 2�0�2�12P32 + 8�112P3

4 + 2�123P34

+ 12�1112P36 + 8�1122P3

6 + 10�1123P36� . �15�

We make a coordinate transformation for each phase suchthat the polar direction is always along the �001� direction.The dielectric constant along the polar direction, �c, and that

FIG. 2. �Color online� Temperature-pressure phase diagram of BaTiO3. Theblue square, circle, and triangle indicate experimentally measured cubic-tetragonal phase boundary, tetragonal-orthorhombic phase boundary, andorthorhombic-rhombohedral phase boundary. The cyan solid line, dottedline, and dashed line indicate the cubic-tetragonal phase boundary,tetragonal-orthorhombic phase boundary, and orthorhombic-rhombohedralphase boundary calculated from classical Landau coefficients calculated.The red solid line, dotted line, and dashed line indicate phase boundariesobtained by this work.

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perpendicular to the polar direction, �a, are given byIn the cubic phase,

�a = �c =1

33. �16�

In the tetragonal phase,

�a =1

11, �c =

1

33. �17�

In the orthorhombic phase,

�a =1

11, �c =

1

23 + 33. �18�

In the rhombohedral phase,

�a =1

11 − 12, �c =

1

11 + 212. �19�

Figure 3 shows the variation in dielectric constants �c

with temperature under different hydrostatic pressures. Theresults agree well with the experimental measurements ofDecker and Zhao.23 The dielectric constant displays peaks ateach transition temperature. It is shown that the height of apeak increases with pressure, i.e., applied pressure enhancesthe dielectric response of a dielectric. It should be empha-sized that the high-order electrostrictive couplings, i.e., thefourth- and sixth-order couplings between polarization andstress, are critical for the increase in the dielectric constantpeaks with hydrostatic pressure.

Additional comparisons between theory and experimen-tal on the temperature-dependent dielectric constants, �a and�c are plotted in Fig. 4.38 It is shown that �c values showmuch better agreement with experiments than those of �a

although our theoretical results for �a qualitatively agreeswith the experimental data. However, it is noted that thereare significant differences among measured values for �a

from different sources. For example, in Merz’s experiment,�a :�c is about 20:1 while in Mason and Matthias’s experi-ment, the ratio is as high as 500:1.38,39 Therefore, more ac-curate experimental measurements are needed for a more re-liable comparison between theory and experiments for �a.

V. TEMPERATURE-ELECTRIC FIELD PHASEDIAGRAM UNDER PRESSURE

The phase transitions in BaTiO3 under an electric field inthe stress-free condition were investigated by several groupsfor domain engineering and for understanding the role ofelectric field in phase transition of ferroelectrics.7,40–44 Withthe presence of an applied external electric field, the Gibbs-free energy is rewritten as

�G = fLGD −1

2s11��1

2 + �22 + �3

2� − s12��1�2 + �1�3

+ �2�3� −1

2s44��4

2 + �52 + �6

2� − Q11��1P12 + �2P2

2

+ �3P32� − Q12��1�P2

2 + P32� + �2�P1

2 + P32� + �3�P2

2

+ P12�� − Q44��4P2P3 + �5P1P3 + �6P2P1� − �E1P1

+ E2P2 + E3P3� . �20�

Here, we consider electric fields along �001�, �101�, and�111� directions. Based on Eq. �20�, the temperature versuselectric fiend phase diagrams are constructed. These phasediagrams are shown in Fig. 5 in �a�, �b�, and �c� for theelectric field along �001�, �010�, or �111� direction, respec-tively. Because of the lack of experimental data, only phasetransition under an electric field along the �001� direction isconsidered for comparison with experiment. With an appliedfield along �001� direction, BaTiO3 undergoes a phase tran-sition from the tetragonal phase to the first type of mono-clinic phase labeled as M1, and then another from the M1

phase to the distorted rhombohedral phase M2.45It can beseen that the phase boundaries calculated in this work agreewell with Fesenko et al.’s experiment measurement.

VI. STRAINS AND PIEZOELECTRIC COEFFICIENTSUNDER ELECTRIC FIELD

The strains under hydrostatic pressure can be derivedfrom Eq. �20� by using the first-order derivatives of theGibbs-free energy with respect to stress,

FIG. 3. �Color online� Dielectric constants �c of BaTiO3 single crystal ver-sus temperature at different hydrostatic pressures, compared with Decker etal.’s experimental measurement result.

FIG. 4. �Color online� Dielectric constants �a and �c of BaTiO3 singlecrystal vs temperature at zero hydrostatic pressures, compared with Merz’sexperimental measurement from 90 to 400 K. �c is the dielectric constantalong the polar direction, which is along �001� direction in tetragonal phase,�011� direction in orthorhombic phase, and �111� direction in rhombohedralphase. In our work �a is along �100� direction in tetragonal and orthorhom-

bic phase, along �11̄0� in rhombohedral phase.

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e1 = − � fLGD

��1

�1=�2=�3=−p

+ Q11P12 + Q12�P2

2 + P32�

− s11p − 2s12p , �21�

e2 = − � fLGD

��1

�1=�2=�3=−p

+ Q11P22 + Q12�P1

2 + P32�

− s11p − 2s12p , �22�

e3 = − � fLGD

��1

�1=�2=�3=−p

+ Q11P32 + Q12�P2

2 + P12�

− s11p − 2s12p , �23�

e4 = Q44P2P3, �24�

e5 = Q44P1P3, �25�

e6 = Q44P2P1, �26�

where only e1, e2, and e3 are pressure-dependent. We em-ployed the elastic compliance constants s11 and s12, 9.07�10−12 and −3.186�10−12 m2 /N, from Devonshire’swork.1

Since some of the Landau potential coefficients arestress-dependent, the first term in e1, e2, and e3 are the firstderivative of Landau energy in Eq. �2� with respect to stressunder a hydrostatic pressure condition. The strain variationswith temperature under different hydrostatic pressures areshown in Fig. 6 where the normal strains e1, e2, and e3 areplotted. The different colors in Fig. 6 indicate strain varia-tions under different pressures, pink: zero hydrostatic pres-sure, dark: 2 GPa, and blue: 4 GPa. For each pressure, thereare four stages, indicating the four phases of BaTiO3: cubic:e1=e2=e3, tetragonal: e1=e2�e3, orthorhombic: e1�e2=e3,and rhombohedral phase: e1=e2=e3. It can be seen from Fig.6 that the strain curves are shifted to low left as hydrostaticpressure increases as a result of decrease in transition tem-peratures and strains under pressure.

A number of recent studies demonstrated that the piezo-electricity of BaTiO3 could be enhanced by applying an elec-tric field.41,42,46–48 For example, Wada et al.’s experimentshowed that with an electric field over 6 kV/cm along �111�direction, two continuous changes were observed in thestrain-field curve, indicating the discontinuities in the piezo-electric coefficients under electric field along �111� direction.

Theoretically, the piezoelectric coefficients can be calcu-lated from the first-order derivative of strains with respect toelectric field. The strain e�lmn� along any arbitrary directionwith respect to the pseudocubic cell can be calculatedthrough the following expression:

e�lmn� = e1l2 + e2m2 + e3n2 + 2e4mn + 2e5ln + 2e6lm , �27�

where e1, e2, e3, e4, e5, and e6 are obtained from Eqs.�21�–�26�, and l ,m ,n are directional cosines that satisfy l2

+m2+n2=1.49

FIG. 5. �Color online� Phase transitions of BaTiO3 single crystal underelectric field along �001�, �101�, and �111� directions at stress-free condition.�a� Temperature-E�001� phase diagram, the squares of navy color indicateexperimental data. �b� Temperature-E�101� phase diagram. �c� Temperature-E�111� phase diagram.

FIG. 6. �Color online� Elastic strains of BaTiO3 single crystal under hydro-static pressure of 0.0, 2.0, and 4.0 GPa. Counting from the right of thecurves, the first stage is cubic phase where the strain components satisfye1=e2=e3, the second stage is tetragonal phase where the strain componentssatisfy e1=e2�e3, the third stage is orthorhombic phase where the straincomponents satisfy e1�e2=e3 and the last stage is rhombohedral phasewhere the strain components satisfy e1=e2=e3.

114105-6 Wang et al. J. Appl. Phys. 108, 114105 �2010�

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We consider an electric field along �001�, �101�, or �111�directions. Piezoelectric coefficients were calculated alongthe field direction and other two orthogonal directions.Therefore, for the case of �001� direction, the three orthogo-nal directions are along �001�, �010�, and �100�; for �101�direction, they are �101�, �101̄�, and �010�; for �111� direc-

tion, they are �111�, �112̄�, and �11̄0�. Figure 7 shows thepiezoelectric coefficients d31, d32, and d33. From Fig. 7 it canbe seen that there are indeed two discontinuities for d33, d32,and d31 with the electric field along �111� direction, and onediscontinuity for d33, d32, and d31 with the electric field along�101� direction, and all the discontinuities appear at the criti-cal electric fields where a phase transition takes place. Ourcalculated d33 under zero pressure and room temperatureagrees well with the measured value by Wada et al. indicatedby two pentagons of navy blue color.41 Finally, there is nodiscontinuity for piezoelectric coefficients at room tempera-

ture by applying electric field along the �001� direction,which is consistent with the fact that there is no ferroelectricphase transition in this case.

VII. SUMMARY

A modified eighth-order Landau potential was proposedby introducing high-order electrocoupling effects and incor-porating the quantum mechanical effects at low temperaturefor a single crystal BaTiO3. It leads to a temperature-pressurephase diagram that reproduces experimental results over thewhole temperature range, overcoming the short-comings ofall existing thermodynamic potentials for BaTiO3. The spon-taneous polarization, dielectric constants, temperature-electric field phase diagram, and piezoelectric coefficientsobtained using the new Landau coefficients all agree verywell with existing experimental measurements.

ACKNOWLEDGMENTS

This work was partially sponsored by the Project-BasedPersonnel Exchange Program by the China ScholarshipCouncil and American Academic Exchange Service ��2008�3012� National Science Foundation of China under GrantNo. 50428101 and partially funded by NSF WMN Programunder Grant No. DMR0908718 and NIRT Program underGrant No. DMR-0507146. The computations were per-formed using the Cyberstar cluster at the Pennsylvania StateUniversity, funded by the National Science Foundationthrough Grant No. OCI-0821527.

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