Journal
J. Am. Ceram. Soc., 91 [6] 1748–1752 (2008) DOI: 10.1111/j.1551-2916.2008.02297.x r 2008 The American Ceramic Society
Comprehensive Linkage of Defect and Phase Equilibria through Ferroelectric Transition Behavior in BaTiO3-Based Dielectrics: Part 1. Defect Energies Under Ambient Air Conditions Soonil Leew and Clive A. Randall Center for Dielectric Studies, Materials Research Institute, The Pennsylvania State University, University Park, Pennsylvania 16802
Zi-Kui Liu Phase Research Laboratory, Department of Materials Science and Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802
equilibrium approaches into harmony, and then, in turn, to relate these trends back to intrinsic dielectric properties. To understand the relationship between defect structures and electrical properties in materials, it is important to estimate, theoretically and experimentally, the energetics of point defects and understand their distributions. The objective of this paper is to address the defect formation energies at different stoichiometries in high pure and equilibrated BaTiO3 through the solubility limit data. Figure 1 shows the revised BaO–TiO2 phase diagram completed with solubility range around the BaTiO3.21 In this investigation, we will use the solubility data obtained under various T and Ba/ Ti variables to determine self-consistent data of the underlying defect chemistry of the BaTiO3 system.
Defect and phase equilibria have been investigated via the ferroelectric phase transition behavior of pure and equilibrated nonstoichiometric BaTiO3 powder samples. Through fabricating the BaTiO3 materials under highly controlled conditions to preserve the equilibrium conditions with respect to Ba/Ti ratio, annealing temperature (T), and oxygen partial pressure (PO2), systematic variations in the phase transition temperature can be noted with respect to Ba/Ti ratio and T. From the data extracted, we can then determine solubility limits. Equilibrating the defect reactions at the solubility limits provides a direct approach to identify and calculate the defect energetics. The phase transition temperature decreased with increasing concentration of the TiO2 partial-Schottky defects (BaTi1dO32d) and the BaO partial-Schottky defects (Ba1dTiO3d), and showed discontinuous changes in the two-phase region. The formation enthalpy and entropy for the partial-Schottky defect reactions was evaluated to be 2.3270.1 eV and 10.1570.7 kB for the BaO partial-Schottky defect, and 2.8970.1 eV and 8.071.5 kB for the TiO2 partial-Schottky defects equilibrated under air annealing conditions.
II. Experimental Procedure BaTiO3 powders with different stoichiometries were fabricated with a modified citrate approach, based upon the earlier work of Pechini and others.2225 Details of this method are found in earlier references, but a brief summary of the method is shown in Fig. 2. In producing these powders, a number of important points are considered to control the experimental conditions: (1) All materials (e.g., Ti-isopropoxide, BaCO3, citric acid, etc.) were over 99.99% purity to limit foreign acceptor issues, (2) equilibrium conditions: the samples were annealed at fixed conditions for over 30 h in air to reach equilibrium. These times were determined by contrasting the differential Scanning Calorimeter (DSC: DSC2920, TA Instruments, New Castle, DE) peaks as a function of annealing time after samples were quenched to room temperature, as described earlier by Lee et al.20 The experimental annealing times agreed with estimates of times using the diffusion coefficient of Ba vacancies reported by Wernicke,13 and (3) controlling the degrees of freedom: CO2 through the formation of BaCO3 had to be carefully removed to avoid formation of additional phases that would perturb the system from the intended equilibrium conditions. Powders were contained in platinum foils, which were held in small alumina cylinders and then suspended by a wire in the hot zone of a vertical tube furnace. The alumina cylinder was then quenched by dropping the suspended sample to room temperature within a few seconds after the annealing process. As it is difficult to control the Ba/Ti ratio to a high degree in small batches, we also compare the X-ray fluorescence (XRF: PW2400, Philips, Eindhoven, the Netherlands) analysis with commercial standards, so the uncertainty of Ba/Ti in these experiments is 70.001. Quenched samples were analyzed by DSC to determine the paraelectric–ferroelectric transition as a function of g (Ba/Ti ratio), and X-ray diffraction (XRD: Scintag3
I. Introduction BaTiO3 was discovered over 60 years ago and is still of interest to both the academic and industrial communities. From an application perspective, it is a major electroceramic material used in both multilayer ceramic capacitors and also in positive temperature coefficient resistors. There are, however, still a number of hotly contested scientific issues with this material, including size effect of the phase transformation,17 mixing of order– disorder and displacive origins of the paraelectric–ferroelectric transition,811 and phase and defect equilibrium.1219 Recently, we have investigated the link between phase transition behavior and stoichiometry annealed at various equilibrium conditions.20,21 From these systematic trends, we have identified the solubility limits of the single-phase nonstoichiometry BaTiO3 fabricated at various temperatures in ambient air conditions and quenched to room temperature. The quenching enables the equilibrium conditions to be ‘‘frozen’’. With the increase in computational materials science activities, more quantitative data are necessary to bring the defect chemistry and phase Q. X. Jia—contributing editor
Manuscript No. 23743. Received September 14, 2007; approved December 4, 2007. Article was invited by guest editors from the Electronics Division. This study is based upon work supported by the National Science Foundation, as part of the Center for Dielectric Studies under Grant no. 0628817. w Author to whom correspondence should be addressed. e-mail:
[email protected]
1748
June 2008 1700 ~1625°
Table I. Solubility Limits Around Stoichiometric BaTiO3 Under Ambient Air Conditions
Hex. BT S.S.
1600
Cub. BT S.S. +Liq.
1400
Temperature (1C)
~1320°
1300 ~1150°
~1110°
1100 1000 900 800 45 BaO
50
55
60
65
70
75
80 TiO2
Fig. 1. Pseudo-binary phase diagram of BaO–TiO2 system in air.21
Model X2, 40 mA, 45 kV, Scintag Inc., Cupertino, CA) was performed to identify phase change with g factor.
III. Results and Discussion This study attempts to evaluate the formation energies for defect reactions in BaTiO3 by using the solubility regime around stoichiometric BaTiO3, shown in Table I. This approach provides a direct method to calculate the defect formation energies and tests for self-consistent data for the controversial defect-phase equilibrium. As shown in Fig. 1, we proposed a modified pseudo-binary phase diagram of the BaO–TiO2 system, especially for solubility regime around stoichiometric BaTiO3 equilibrated at different temperatures. The basic reactions for the solid solution (SS) on Ti- and Ba-rich sides are as follows: BaTiO3
Baa Tib Oaþ2b
0.98370.001r 0.96770.001r 0.98270.001r
Ba excess
g g g
r1.00570.001 r1.01070.001 r1.01170.001
The numbers here for Ba/Ti (g*) have been rounded off relative to the compositions used in an earlier paper.21
mol% TiO2
X 00 TiO2 ! TiX Ti þ 2OO þ VBa þ VO
Ti excess
1200 1320 1400
~1250°
BaTi O
Temperature (°C )
g (Ba/Ti ratio)
Liquid
1500
1200
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Linkage of Defect and Phase Equilibria in BaTiO3-Based Dielectrics
(1)
Baa Tib Oaþ2b
BaTiO3
X X 0000 ! aBaX Ba þ bTiTi þ ða þ 2bÞOO þ kVTi þ 2kVO
(4) where k 5 ab (k 5 0 at a 5 b). Ba2TiO4 (a 5 2 and b 5 1) and Ba1.054Ti0.946O2.946 (a 5 1.054 and b 5 0.946) were the secondary phases around stoichiometric BaTiO3 on Ba-rich side. In both cases, the BaO and TiO2 partial-Schottky defect reactions can be considered as the SS mechanism of TiO2 and BaO into BaTiO3, respectively. We further assume that the material is ionic and ignore the possibility of noninteger valence states. Equations (2) and (4) present the equilibria between SS and secondary phases. At the equilibrium condition, DG 5 0, assuming that the thermodynamic activity of each defect is a simple function of its concentration when interactions between defects do not become significant, the equilibrium constants for the defect chemical reactions are given by 00 KTiO2 ¼ ½VBa ½VO 00 V Þ 00 V Þ DsðVBa DhðVBa 0 O O ¼ KTiO exp exp 2 kB kB T 0000 ½VO 2 KBaO ¼ ½VTi DsðVTi0000 2VO Þ DhðVTi0000 2VO Þ 0 ; exp ¼ KBaO exp kB kB T
(5)
(6)
BaTiO3
X X 00 ! aBaX Ba þ bTiTi þ ða þ 2bÞOO þ mVBa þ mVO
(2) where m 5 ba (m 5 0 at a 5 b). BaTi2O5 (a 5 1 and b 5 2) and Ba6Ti17O40 (a 5 6 and b 5 17) were confirmed as secondary phases around stoichiometric BaTiO3 on Ti-rich side. BaTiO3
X 0000 BaO ! BaX Ba þ OO þ VTi þ 2VO
(3)
where [ ] denotes the defect concentrations, which can be expressed by both number fraction (dimensionless) and cm3; here the number fraction was used. KTiO2 and KBaO are the equi0 librium constants for Ti- and Ba-rich cases, respectively. KTiO 2 0 and KBaO are a preexponential, assuming unity for a simple vacancy, but not equal to one for point defect clusters. The partial-Schottky defects consisting of pairs of point defects can be regarded as a simple vacancy defect. Ds and Dh represent the entropy (eV/K) and enthalpy (eV) for each partial-Schottky 0000 defect reaction. For simplification, V 00BaV O and V Ti 2V O will be replaced by BOPS and TOPS, respectively.
(1) Partial-Schottky Defect Reactions From a thermodynamic point of view, the phase boundary in a phase diagram implies that at the boundary the two phases (M and N) are in equilibrium. Thus, the chemical potential of each component i is the same N mM i ¼ mi
Fig. 2. Experimental Procedure of a modified citrate process for nonstoichiometric BaTiO3 powder synthesis.
(7)
As shown in Fig. 3, the chemical potential of TiO2 and BaO SS components, mSS TiO2 and mBaO at point A, in nonstoichiometric BaTiO3 in the SS regime on the Ba-rich side should be the same 2 TiO4 as the chemical potential of TiO2 and BaO components, mBa TiO2 Ba2 TiO4 and mBaO at point B, in the Ba2TiO4 phase. Similarly, on the Ti-rich side, the chemical potential of BaO and TiO2 compoSS nents, mSS BaO and mTiO2 at point C, in nonstoichiometric BaTiO3 in the SS regime should be the same as the chemical potential of
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2 O5 2 O5 BaO and TiO2 components, mBaTi and mBaTi at point D, in BaO TiO2 BaTi2O5 or Ba6Ti17O40 phases. With a fixed PO2, this modeling considers metal vacancies. The equations are on the Ti-rich side,
mSS Ba
Vol. 91, No. 6
Journal of the American Ceramic Society—Lee et al.
¼
2 O5 mBaTi Ba
and
mSS Ti
¼
2 O5 mBaTi : Ti
00 ðdBa ¼ ½VBa SSTi- rich Þ
gsl-Ba ¼
(8)
At the solubility line with the equilibrium conditions of Eqs. (1) and (2), the activities of the Ba and Ti components are fixed. Therefore, assuming an ideal solution, the Ba-vacancy concentration caused by Ti excess in BaTiO3 does not change in the two-phase region. So, the Ba-vacancy concentration at the sol00 SS=BaTi2 O5 SSTi rich , is constant in the two-phase region. ubility limit, ½VBa The Ba-vacancy concentration in the solid solution region can be expressed by g ¼ 1 dBa
At the solubility limit, Eq. (14) can be rewritten as
(9)
SS=BaTi O5
SS=Ba TiO4
0000 2 ½VTi SSBa rich -
According to Eqs. (10) and (11), at the solubility limit, the gsl-Ti ð< 1 : Ti-richÞ value is: DsBOPS DhBOPS exp (12) gsl-Ti ¼ 1 exp 2kB 2kB T Similarly, on the Ba-rich side (13)
At the solubility limit with the equilibrium conditions of Eqs. (3) 0000 SS=Ba2 TiO4 and (4), the Ti-vacancy concentration, ½VTi SSBa rich , is constant in the two-phase region. The Ti-vacancy concentration in the solid solution region can be expressed by 1 1 dTi
0000 ðfor dTi ¼ ½VTi SSBa- rich Þ
DsTOPS DhTOPS exp ¼ 41=3 exp 3kB 3kB T
(16)
According to Eqs. (15) and (16), at the solubility limit, the gsl-Ba ð>1 : Ba-richÞ value is: ð17Þ
(10)
is the Ba/Ti ratio at the solubility limit on the Ti-rich side. From Eqs. (1) and (2), the defect reaction will be saturated at the equilibrium condition at the solubility limit. Therefore, from Eq. (5) the Ba-vacancy concentration is DsBOPS DhBOPS 00 SS=BaTi2 O5 ½VBa : (11) exp SSTi rich ¼ exp 2kB 2kB T
g ¼
(15)
;
DsTOPS DhTOPS 1 gsl-Ba ¼ 1 41=3 exp exp 3kB 3kB T
where gsl- Ti
Ba2 TiO4 Ba2 TiO4 mSS and mSS Ti ¼ mTi Ba ¼ mBa
1
where g sl-Ba is the Ba/Ti ratio at the solubility limit on the Barich side. From Eq. (6) the titanium vacancy concentration at the solubility limit is:
At the solubility limit, Eq. (9) can be rewritten as 00 SSTi rich 2 gsl-Ti ¼ 1 ½VBa -
1 0000 SS=Ba2 TiO4 ½VTi SSBa- rich
(14)
As shown in Fig. 4, the theoretical calculation from Eqs. (12) and (17) for the solubility regime agrees well with the solubility regime given by the experimental data obtained via phase transition temperature and XRD approaches, Table II, described earlier.20,21 The formation energies for partialSchottky defect reactions in BaTiO3 were evaluated from the solubility regime around stoichiometric BaTiO3 under ambient conditions (in air) to be 2.3270.1 eV and 10.1570.7 kB for the BaO partial-Schottky defect, and 2.8970.1 eV and 8.0671.5 kB for the TiO2 partial-Schottky defect, noting that if the K0 is not unity, it affects the values of the calculated entropies.
(2) Full-Schottky Defect Reactions Solubility on both Ba and Ti-rich sides indicates that both Ti and Ba metal vacancies are possible. In turn, this is an important evidence for the existence of a full-Schottky defect reaction. This has been a controversial concept surrounding the existence of defects in BaTiO3.26,27 The full-Schottky defect reaction consists of a Ba vacancy, a Ti vacancy, and three O vacancies together 00 0000 BaTiO3 2VBa þ VTi þ 3VO
(18)
The relationship between the defect concentration and the free energy can be expressed as 1500 in air
BaTiO3
1400 1320°C Solubility Regime (a>b)
Temp. (°C )
1300
a
1200
900
Fig. 3. A schematic of the Gibbs free energy curve with BaO and TiO2 components showing the solubility regime.
Ba-rich
Ti-rich
1100 1000
b
∆h
∆h
=2.32 ±0.1eV
=2.89 ± 0.1eV
Theoretical solubility line (Ba-rich) Theoretical solubility line (Ti-rich) Empirical data (Ba-rich) Empirical data (Ti-rich)
0.96
0.97
0.98 0.99 1.00 g* (Ba/Ti ratio)
1.01
1.02
Fig. 4. Comparison of experimentally determined solubility limits (closed circles) with thermodynamic theory in the temperature range of 14001–12001C, continuous lines.
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Linkage of Defect and Phase Equilibria in BaTiO3-Based Dielectrics Table II. Defect Energies for Schottky Defect Reactions Under Ambient Air Conditions
Conditions
In air
Defect model
BaO-partial-Schottky (BOPS) TiO2-partial-Schottky (TOPS) Full-Schottky on Ti-rich (FST) Full-Schottky on Ba-rich (FSB)
00 0000 KFS ¼ ½VBa ½VTi ½VO 3 DsFS DhFS 0 exp ¼ KFS exp kB kB T
00 VBa 0000 VTi 00 VBa 00 VBa
þ þ þ þ
VO 2VO 0000 VTi þ 0000 VTi þ
Enthalpy, Dh (eV)
Entropy, DS (eV/K)
2.3270.1 2.8970.1 3.3370.2 3.4870.2
10.1570.7 kB 8.0671.5 kB Assuming 10 kB Assuming 10 kB
3VO 3VO
Substituting Eq. (26) into Eq. (25) provides (19)
where KFS is the equilibrium constant for full-Schottky defect reactions, DSFS stands for the nonconfigurational entropy associated with lattice strains and changes in vibrational frequencies accompanying the full-Schottky defect, and DhFS is formation enthalpy for the full-Schottky defect. Assuming the existence of the full-Schottky defect, the formation energies can be calculated from the partial-Schottky defect energies derived through solubility data. One can consider two cases for the full-Schottky defect energy. First of all, note that at the solubility limit-line, the chemical potentials satisfy Eqs. (20) and (21).
DsBOPS DhBOPS x ¼ exp exp 2kB 2kB T DsFST DhFST 3=5 exp exp þ3 5kB 5kB T
(27)
Therefore, 33 DsBOPS DhBOPS exp 6 6 exp 2kB 77 2kB T 61 6 77 4 3=5 4 DsFST DhFST 55 exp exp þ3 5kB 5kB T gsl-Ti ¼ : DsFS DhFS 3=5 exp exp 13 5kB 5kB T 2
2
BaTi2 O5 FS 2 O5 FS mSS ¼ mBaTi and mSS ðTi-rich sideÞ Ba ¼ mBa Ti Ti
(20)
Ba2 TiO4 FS FS 2 TiO4 ¼ mBa and mSS ðBa-rich sideÞ mSS Ba ¼ mBa Ti Ti
(21)
(28)
where SSFS represents the SS containing full-Schottky defects, Ba1x Ti1y O3ðxþ2yÞ ð1 x 1 yÞ for Ti-rich side and Ba1x Ti1y O3ðxþ2yÞ ð1 x 1 yÞ for Ba-rich side. The lefthand side is a chemical potential of Ba and Ti atoms in the SS region with the full-Schottky defects, and the right-hand side represents the chemical potential of Ba and Ti atoms in the second phases. The partial-Schottky defect reaction could be dominant near the solubility limits and does not change in the two-phase regions. If some solubility existed, x is not equal to y. For the Ti-rich side of the solubility limit, the BaO partial-Schottky defect overwhelms the full-Schottky defect, therefore
For the Ba-rich side at the solubility limit, the TiO2 partialSchottky defect overwhelms the full-Schottky defect. Therefore:
1 x 1 y ðx; y 1Þ
(22)
x y ¼ d at ðBa1d TiO3d ÞSS
(23)
where d is the net Ba-vacancy concentration in Ti-rich compositions. One can then consider the same condition with Eqs. (8), (10), and (11) used to calculate the partial-Schottky defect energy. The Ba- and Ti-vacancy concentration in the solid solution region can be expressed by 1x ¼ g ð0 < g < 1Þ 1y
(24)
where g is the Ba/Ti ratio. The net Ba-vacancy concentration for Ti-rich samples is: 00 net d ¼ ½VBa Ti- rich ¼ x y DsBOPS DhBOPS ¼ exp exp 2kB 2kB T
(25)
Now, y is the Ti-vacancy concentration in full-Schottky defect reaction, therefore, from Eq. (19) 0000 FS ¼ 33=5 exp y ¼ ½VTi
DsFS DhFS exp 5kB 5kB T
(26)
1 x 1 y ðx; y 1Þ;
(29)
y x ¼ g at ðBaTi1g O32g ÞSS
(30)
where g is the net Ti-vacancy concentration in Ba-rich compositions. From Eqs. (13), (15), (16), and (21) 1x ¼ g ðg > 1Þ: 1y
(31)
The net Ti-vacancy concentration by Ba excess is: 0000 net Ba- rich ¼ y x g ¼ ½VTi DsTOPS DhTOPS ¼ 22=3 exp exp 3kB 3kB T
(32)
where x is the Ba-vacancy concentration in full-Schottky defect reaction, and 00 x ¼ ½VBa FS ¼ 33=5 exp
DsFS DhFS exp 5kB 5kB T
Substituting Eq. (33) into Eq. (32) DsTOPS DhTOPS exp y ¼ 22=3 exp 3kB 3kB T DsFSB DhFSB exp þ 33=5 exp 5kB 5kB T
(33)
(34)
Therefore, incorporation of both full and partial-Schottky reactions is possible, via
1752
gsl-Ba
Journal of the American Ceramic Society—Lee et al. 3
DsFSB DhFSB 3=5 exp 13 exp 5kB 5kB T 33 ¼2 2 Ds Dh TOPS TOPS 2=3 exp exp 62 77 6 61 6 2kB 2kB T 77 4 3=5 4 DsFSB DhFSB 55 exp exp þ3 5kB 5kB T (35)
This approach allows calculation of the full-Schottky defect energy, assuming both partial and full-Schottky defects coexist. The result obtained is 3.3370.2 eV for the full Schottky on the Ti-rich side and 3.4870.2 eV for full Schottky on the Ba-rich side. This is very important in understanding SS with synthesis in the presence of different Ba and Ti activities below the solubility limits.
IV. Summary and Conclusions From the ferroelectric phase transition temperature variation, the solubility regimes under equilibrium at ambient conditions were determined. Following evaluation of the solubility regime in terms of defect chemistry, this research obtained defect formation energies for partial-Schottky and full-Schottky defect reactions (see Table II): 2.3270.1 eV for BaO partial Schottky, 2.8970.1 eV for TiO2 partial Schottky, 3.3370.2 eV for full Schottky on the Ti-rich side, and 3.4870.2 eV for full Schottky on the Ba-rich side. Also determined were the entropies for the BaO and TiO2 partial-Schottky defects as 10.1570.7 and 8.0671.5 kB, respectively. Collectively, the research established a novel modeling to evaluate the energetics of defects with selfconsistent data linking the phase equilibria and defect chemistry under various conditions, utilizing the degrees of freedom of the system, Ba/Ti, and T at fixed Po2’s. In the second paper, all these degrees of freedom will be phenomenologically modeled.28
Acknowledgments The authors wish to acknowledge CDS members and NSF I/UCRC program for support on this topic. The authors have had many stimulating discussions with many members, particularly with Dr. Hirokazu Chazono, Dr. Ian Burn, and Dr. Mike Chu. Ferro Corporation and Edward Rast are acknowledged for making key Ba/Ti ratio measurements with XRF. The authors are grateful to Dr. Dimitry Filimonov for helping the synthesis and calorimetry protocols. One of authors (Dr. Zi-Kui Liu) would like to thank the partial support of the NSF grant of DMR-0510180.
References 1
K. Uchino, E. Sadanaga, and T. Hirose, ‘‘Dependence of the Crystal Structure on Particle Size in Barium Titanate,’’ J. Am. Ceram. Soc., 72 [8] 1555–8 (1989). 2 W. L. Zhong, Y. G. Wang, P. L. Zhang, and B. D. Qu, ‘‘Phenomenological Study of the Size Effect on Phase Transition in Ferroelectric Particles,’’ Phys. Rev. B, 50 [2] 698–703 (1994).
Vol. 91, No. 6
W. Y. Shih, W. H. Shih, and I. A. Aksay, ‘‘Size Dependence of the Ferroelectric Transition of Small BaTiO3 Particles: Effect of Depolarization,’’ Phys. Rev. B, 50 [21] 15575–85 (1994). 4 S. Wada, T. Suzuki, and T. Noma, ‘‘The Effect of the Particle Sizes and the Correlational Sizes of Dipoles Introduced by the Lattice Defects on the Crystal Structure of Barium Titanate Fine Particles,’’ Jpn. J. Appl. Phys., 34 [9B] 5368–79 (1995). 5 D. McCauley, R. E. Newnham, and C. A. Randall, ‘‘Intrinsic Size Effects in a Barium Titanate Glass–Ceramic,’’ J. Am. Ceram. Soc., 81 [4] 979–87 (1998). 6 T. Ando, H. Matsuda, K. Miyazawa, and M. Kuwabara, ‘‘Grain-Size Effects on the Structure and Properties of BaTiO3 Fabricated by Sol–Gel Method,’’ Trans. Mater. Res. Soc. Jpn., 26 [1] 95–98 (2001). 7 Z. Zhao, V. Buscaglia, M. Viviani, M. T. Buscaglia, L. Mitoseriu, A. Testino, M. Nygren, M. Johnsson, and P. Nanni, ‘‘Grain-Size Effects on the Ferroelectric Behavior of Dense Nanocrystalline BaTiO3 Ceramics,’’ Phys. Rev. B, 70 [2] 1–8 (2004). 8 F. Jona and G. Shirane, Ferroelectric Crystals. Pergamon, New York, 1962. 9 M. E. Lines and A. M. Glass, Principles and Applications of Ferroelectrics and Related Materials. Oxford University Press, Oxford, 1979. 10 R. E. Newnham, Structure-Property Relations. Springer-Verlag, New York, 1975. 11 W. Zhong, D. Vanderbilt, and K. M. Rabe, ‘‘Phase Transition in BaTiO3 from First Principles,’’ Phys. Rev. Lett., 73 [13] 1861–5 (1994). 12 A. M. J. H. Seuter, ‘‘Defect Chemistry and Electrical Transport Properties of Barium Titanate,’’ Philips Res. Rep. Suppl., 3, 1–89 (1974). 13 R. Wernicke, ‘‘Defect Chemistry and Electrical Conductivity of Doped Barium Titanate Ceramics: Part 4. The Kinetics of Equilibrium Restoration in Barium Titanate Ceramics,’’ Philips Res. Rep., 31, 526–43 (1976). 14 N. G. Eror and D. M. Smyth, ‘‘Nonstoichiometric Disorder in SingleCrystalline Barium Titanate (&) at Elevated Temperatures,’’ J. Solid State Chem., 24 [3–4] 235–44 (1978). 15 N. H. Chan, R. K. Sharma, and D. M. Smyth, ‘‘Nonstoichiometry in Undoped Barium Titanate,’’ J. Am. Ceram. Soc., 64 [9] 556–62 (1981). 16 G. V. Lewis and C. R. A. Catlow, ‘‘Defect Studies of Doped and Undoped Barium Titanate Using Computer Simulation Techniques,’’ J. Phys. Chem. Solids, 47 [1] 89–97 (1986). 17 Y. Tsur and C. A. Randall, ‘‘Point Defect Concentrations in Barium Titanate Revisited,’’ J. Am. Ceram. Soc., 84 [9] 2147–9 (2001). 18 R. K. Sharma, N. H. Chan, and D. M. Smyth, ‘‘Solubility of TiO2 in BaTiO3,’’ J. Am. Ceram. Soc., 64 [8] 448–51 (1981). 19 Y. H. Hu, M. P. Harmer, and D. M. Smyth, ‘‘Solubility of BaO in BaTiO3,’’ J. Am. Ceram. Soc., 68 [7] 372–6 (1985). 20 S. Lee, Z.-K. Liu, M. H. Kim, and C. A. Randall, ‘‘Influence of Nonstoichiometry on Ferroelectric Phase Transition in BaTiO3,’’ J. Appl. Phys., 101 [5] 054119/1–8 (2007). 21 S. Lee, Z.-K. Liu, and C. A. Randall, ‘‘Modified Phase Diagram for the Barium Oxide-Titanium Dioxide System for the Ferroelectric Barium Titanate,’’ J. Am. Ceram. Soc., 90 [8] 2589–94 (2007). 22 M. P. Pechini, ‘‘Method of Preparing Lead and Alkaline Earth Titanates and Niobates and Coating Method using same to form a Capacitor’’; US Patent 3,330,697, 1967. 23 P. P. Phule and S. H. Risbud, ‘‘Review: Low-Temperature Synthesis and Processing of Electronic Materials in the BaO–TiO2 System,’’ J. Mater. Sci., 25 [2B] 1169–83 (1990). 24 J. H. Choy, Y. S. Han, J. T. Kim, and Y. H. Kim, ‘‘Citrate Route to UltraFine Barium Polytitanates with Microwave Dielectric Properties,’’ J. Mater. Sci., 5 [1] 57–63 (1995). 25 D. S. Filimonov, Z.-K. Liu, and C. A. Randall, ‘‘Synthesis and Thermal Stability of a New Barium Polytitanate Compound, Ba1.054Ti0.946O2.946,’’ Mater. Res. Bull., 37, 467–73 (2002). 26 D. M. Smyth, The Defect Chemistry of Metal Oxides. Oxford University Press, New York, pp. 253–82, 2000. 27 T. Kolodiazhnyi and A. Petric, ‘‘Analysis of Point Defects in Polycrystalline BaTiO3 by Electron Paramagnetic Resonance,’’ J. Phys. Chem. Solids, 64 [6] 953–60 (2003). 28 S. Lee, Z.-K. Liu, and C. A. Randall, ‘‘Comprehensive Linkage of Defect and Phase Equilibria Through Ferroelectric Transition Behavior in BaTiO3-Based Dielectrics: Part 2. Defect Modeling under Low Oxygen Partial Pressure Conditions,’’ J. Am. Ceram. Soc. (2008). &
