Appl Phys A (2011) 104:1085–1089
DOI 10.1007/s00339-011-6375-3
Theoretical predictions of morphotropic phase boundary
in (1 − x)Na1/2Bi1/2 TiO3 -xBaTiO3 by first-principle calculations
Yang Deng · Ru-Zhi Wang · Li-Chun Xu · Hui Fang ·
Xiaodong Yang · Hui Yan · Paul K. Chu
Received: 23 August 2010 / Accepted: 11 March 2011 / Published online: 7 April 2011
© Springer-Verlag 2011
Abstract A morphotropic phase boundary (MPB) in a solid
solution of (1 − x)Na1/2 Bi1/2 TiO3 -xBaTiO3 (0 ≤ x ≤ 1.0)
(NBT-BT) is directly demonstrated by first-principle calculations. The results show that in the NBT-BT system,
a structural transition from rhombohedral to tetragonal occurs when x is changed from 0.06 to 0.12, and the MPB
appears at compositions of around 0.05 ≤ x ≤ 0.07. It can
be verified by the stronger hybridization orbital of p-d based
on the analysis of the partial density of states (PDOS) in
the Ti–O atoms. The theoretical predictions agree well with
experimental observations which indicate that the best ferroelectric properties can be attained in the vicinity of around
0.05 ≤ x ≤ 0.07 in the MPB of the NBT-BT system. Another tetragonal-cubic phase transition which needs further
experimental verification is also found from the system with
compositions of 0.475 < x < 0.65.
1 Introduction
Environmental friendly Na1/2 Bi1/2 TiO3 (NBT) and related
materials which possess outstanding piezoelectric properY. Deng · R.-Z. Wang () · L.-C. Xu · H. Fang · H. Yan
Laboratory of Thin Film Materials, College of Materials Science
and Engineering, Beijing University of Technology, Beijing
100124, China
e-mail: wrz@bjut.edu.cn
Fax: +86-10-67392445
X. Yang
Department of Electrical and Computer Engineering, University
of Florida, Gainesville, FL 32611, USA
R.-Z. Wang · P.K. Chu
Department of Physics and Materials Science, City University of
Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong, China
ties close to the morphotropic phase boundary similar to
the PZT systems [1, 2] are promising substitutes for leadbased piezoelectric ceramics on account of their potential
electromechanical properties [3]. It is well known that the
best ferroelectric materials for piezoelectric applications are
solid solutions with compositions in the vicinity of the
morphotropic phase boundary (MPB) [4] where the crystal
structure changes abruptly, and the electromechanical properties are optimal [5]. Consequently, it is necessary to investigate the MPB of the NBT-based solid solutions such as
the dual system of (1 − x)Na1/2 Bi1/2 TiO3 -xBaTiO3 (NBTBT), which can reduce costs greatly and expand the utility
of piezoelectric materials. A number of studies [3, 6, 7] have
in fact been carried out to improve the properties of NBT-BT
near the MPB compositions. Several studies have been conducted by using X-ray diffraction, dielectric measurements,
and Raman spectroscopy, and the MPB is suggested to be
between x = 0.055 to 0.07 [3, 8]. However, few theoretical
investigations have hitherto been performed on the MPB in
the NBT-BT system. First-principle methods based on the
density functional theory (DFT) have emerged to be a powerful tool in investigating the properties of ferroelectric system theoretically [9, 10], and in this letter, we theoretically
confirm the existence of the MPB in the NBT-BT system to
establish the theoretical framework to explain and corroborate the experimental observations [8, 11].
2 Computational model
The first-principle calculations are conducted using planewave pseudopotential method and density functional theory (DFT) [3, 12]. Since the virtual crystal approximation (VCA) [13] has been successfully applied to a number of materials from semiconductor alloys to perovskite
1086
Fig. 1 A typical structural model of the ABO3 -type perovskite with
Na, Bi, and Ba all occupying the A-site based on the relationship of
Na(1 − x)Na1/2 Bi1/2 -xBa resulting from the application of the idea of
the VCA, and Ti occupying B-site
solid solutions such as PZT [14], BST [15], or PSN [16]
and LNTO [10], we have also adopted the VCA to simplify
the structural model as shown in Fig. 1. In our calculations,
we use the CASTEP computer code [17]. The generalized
gradient approximation [18] (GGA) is used with the ultrasoft pseudopotential [19], and a plane-wave cutoff energy of
380 eV is adopted. The first Brillouin zone (FBZ) is sampled
with a k-point mesh corresponding to (6, 6, 6) in the FBZ of
the five-atom unit cell for the self-consistent calculation of
the total energy and for the calculation of density of states.
To check the accuracy of our results, we have increased the
energy cutoff from 380 to 500 eV and increased the number of k points from (6, 6, 6) to (8, 8, 8). In the geometry relaxation, the total energy of the system was converged
<1 × 10−4 eV, the displacement of atoms <1 × 10−3 Å,
and the residual bulk stress <0.05 GPa. The atomic positions have been relaxed until the maximum component of
the force on any atom is smaller than 0.03 eV/Å.
3 Results and discussion
The original structure of the trigonal phase is optimized by
calculation, and the lattice constant of NBT (x = 0) is only
2.8% bigger than the experimental value [9]. The change
in the lattice parameters with different compositions of BT
in NBT-BT is calculated at first. As shown in the inset of
Fig. 2, there are two distinct peaks in the curve when x is
changed from 0 to 1. For the peak of x = 0.125 in Fig. 2
which shows the composition range of 0 < x < 0.125, the
volume (or lattice constant) increases monotonically with x,
which is related to the ion radius of Ba2+ of 0.135 nm
[larger than the average radius (0.0955 nm) of A-site ions
(Na+ 0.095 nm, Bi3+ 0.096 nm)]. The structure shows
the tetragonal phase completely in the composition range
of x > 0.125. It is probably a consequence of phase transition (it has already be confirmed that the first peak is
Y. Deng et al.
Fig. 2 Lattice constants of the NBT-BT solid solution system as a
function of x (0 < x < 0.35). The insert depicts an expanded view of
the lattice constants at compositions with x ranging from 0 to 1
rhombohedral-tetragonal phase transition due to the Minimum Energy, which will be further confirmed in the Fig. 4).
It has been suggested that the volume effect induces phase
transition which can be verified by experiments [3, 8]. However, the phase transition should lead to a region with both
the rhombohedral and tetragonal phases, from which the
tetragonal phase begins to emerge when x < 0.125. An inflexion point is also observed at around x = 0.06 in Fig. 2,
after which tetragonal phase appears. In general, the MPB
is characterized by two competing, coexisting phases in the
solid solutions. At the same time, the energy difference between the rhombohedral and tetragonal phases is very small,
and the two phases exist in competition in the composition
range from x = 0.06 to 0.125 (detailed analysis shown in
Fig. 4). Therefore, in light of the present observations and
analysis, it can be inferred that the MPB of the NBT-BT
solid solution is around x = 0.06 and the rhombohedral and
tetragonal phases may coexist in the composition range from
x = 0.06 to 0.125. In comparison with previous results, our
predictions are in good agreement with, for example, those
obtained by Takenaka et al. [3] pointing out x = 0.055 located in the surrounding area of MPB in the NBT-BT system
and those by Chu et al. [11] reporting the maximum value
of the piezoelectric constant at x = 0.06.
To further investigate and confirm that the MPB in the
NBT-BT system is at around x = 0.06, the formation enthalpy [20, 21] (H ) is analyzed. Here, the enthalpy change
is associated with the formation of the solid solution from its
constituents. The formation enthalpy (H ) of the NBT-BT
system is calculated in the following:
H = ENBT-BT − xEBT − (1 − x)ENBT ,
(1)
where ENBT-BT , EBT , and ENBT are the total energies of
the relaxed systems NBT-BT, BT, and NBT, respectively.
Theoretical predictions of morphotropic phase boundary in (1 − x)Na1/2 Bi1/2 TiO3 -xBaTiO3
Fig. 3 Formation enthalpy of the NBT-BT solid solution system as a
function of x
Fig. 4 The total difference between rhombohedral and tetragonal
phase structure (EP4 − ER3 ) as a function of x
H , which is the formation enthalpy of the NBT-BT system, determines the stability of the solid solution system.
The system is unstable and prone to segregation or phase
change when the formation enthalpy is too high. That is to
say, a smaller formation enthalpy renders the system more
stable. Figure 3 shows the change in the formation enthalpy with increasing composition x. When the composition of the BT is small (x < 0.06), the NBT-BT system retains the stability from the original structure. However, the
slope of the formation enthalpy changes significantly in the
range of 0.05 < x < 0.07. Consequently, the phase structure in the dual system of NBT-BT may be altered. In other
words, the rhombohedral phase coexists with the tetragonal
phase when x > 0.05, and it transforms into the tetragonal
phase completely when x > 0.12, as suggested by our calculations. Analysis of the formation enthalpy indicates that
x = 0.05–0.07 is the morphotropic phase boundary (MPB)
of the rhombohedral-tetragonal structure, which is also consistent with the analysis of the lattice parameters.
1087
In order to further support our conclusions of MPB at
0.05 < x < 0.07, the structural properties of the tetragonal (P4mm) and cube (Pm3m) phases are also calculated by
the same method as that of the rhombohedral (R3c/R3m)
phases. It is clear in Fig. 4 that, when x < 0.05, the pseudocubic (rhombohedral) is the optimal structure; the total energy of rhombohedral structure is close to that of the tetragonal phase at 0.05 < x < 0.08, and it may indicate that two
phases coexist in MPB; the energy discrepancy between the
rhombohedral and tetragonal phase structure is more obvious at 0.07 < x < 0.10, while the rhombohedral structure is
more stable (Minimum Energy); the tetragonal phase structure is in the highest flight when x > 0.12. These calculations further validate the previous analysis of MPB.
In order to further clarify the micro-mechanism of MPB,
we plot the partial density of states (PDOS) of the Ti 3d and
O 2p orbitals for the various x values in the (1 − x)NBTxBT system. As shown in Fig. 5, the O 2p and Ti 3d orbital
electrons in the vicinity of the Fermi level move to the Fermi
level for x = 0, 0.05, 0.07, and 0.15. The hybridization of the
Ti 3d and O 2p states is weak, and these states mostly determine the nature of the available states around the Fermi level
at which the Ti–O octahedral state gives rise to the regular
octahedron at x = 0 [9]. The Ti–O band rises near the Fermi
level as x increases, and when x = 0.15, the LDOS of the
O 2p state is clearly different from those in other compositions. This means that the configuration when x = 0.15 is
composed of the tetragonal phase.
At an energy of about −3.0 eV, a resonant peak emerges
from samples with x = 0.05 and 0.07. Hybridization of the
Ti 3d and O 2p states is much greater for the composition
around 5%–7% compared with other compositions. It forms
the so-called p-d hybridization orbital. As a result of the p-d
hybridization orbital, the Ti–O octahedral is tilted, and this
is similar to that reported by Frantti [4]. A large hybridization orbital of p-d is one of the factors forming the MPB
in piezoelectric perovskites. In additon, because of the difference in the A-site cations for various x values, the relative position between the B-site (Ti) cation and O anions
around the B-site has changed. Hence, displacement of the
Ti cation occurs along the c-axis relative to the site of O
ions, consequently increasing the TiO6 -octahedra distortion
and producing a separation of core valence charge in the
c-axis crystal direction. An electric dipole moment reflecting the characteristics of the excellent ferroelectric tetragonal phase is produced. At compositions of around 5%–7%,
the O 2p and Ti 3d states represent stronger p-d hybridization, and it reduces the short-range repulsive force between
the atoms and promotes formation of the ferroelectric tetragonal phase. The calculated results are in good agreement
with those produced by Cohen [22].
Our analysis reveals more interesting results. Figure 2
(inset) shows an additional peak at x = 0.475 and Fig. 3
1088
Y. Deng et al.
Fig. 5 PDOS of the Ti–O
atoms in compositions of x = 0,
x = 0.05, x = 0.07, and
x = 0.15
shows another peak at x = 0.65, and the region of 0.475 <
x < 0.65 may be another phase transition region. The results show that the phase structures tend to transform into
the cubic phase at x > 0.475 and complete transform into
the cubic phase at x > 0.65. Therefore the tetragonal-tocubic phase transition occurs in the composition range of
0.475 < x < 0.65. Although similar experimental results
have been reported [23] showing that the tetragonal-to-cubic
phase transition occurs at x = 0.5 in the NBT-BT system,
direct verification requires further experiments.
4 Conclusions
In conclusion, the morphotropic phase boundary (MPB) in
the solid solution (1 − x)Na1/2 Bi1/2 TiO3 -xBaTiO3 (0 ≤
x ≤ 1.0) (NBT-BT) system is predicted based on virtual
crystal approximation (VCA) by first-principle calculations.
The results show that in the NBT-BT system, a phase transition from rhombohedral to tetragonal occurs when x is
changed from 0.06 to 0.12, and the MPB appears for x values between 0.05 and 0.07. It arises from the stronger hybridization orbital of p-d according to the analysis of the
partial density of states (PDOS) in the Ti–O atoms. Our theoretical predictions are consistent with experimental results
showing that the best ferroelectric properties can be attained
when the x value is between 0.05 and 0.07 in the MPB
of the NBT-BT system. Moreover, it is predicted that the
tetragonal-to-cubic phase transition occurs in the composition range of 0.475 < x < 0.65, but more experiments are
needed for verification.
Acknowledgements The work was financially supported by the National Natural Science Foundation of China (NSFC) (Grant
No. 11074017), the IHLB (Grant No. PHR201007101), the Beijing
Nova Program (Grant No. 2008B10), the Beijing Natural Science
Foundation (Grant No. 1102006), the City University of Hong Kong
Strategic Research Grant (SRG) (Grant No. 7008009), and the Scientific Research Foundation for the Returned Overseas Chinese Scholars
of the Ministry of Education.
References
1. L.A. Schmitt, K.A. Schonau, R. Theissmann, H. Fuess, H. Kungl,
M.J. Hoffmann, J. Appl. Phys. 101, 074107 (2007)
2. R. Theissmann, L.A. Schmitt, J. Kling, R. Schierholz, K.A.
Schonau, H. Fuess, M. Knapp, H. Kungl, M.J. Hoffmann, J. Appl.
Phys. 102, 079902 (2007)
3. T. Takenaka, K. Maruyama, K. Sakata, Jpn. J. Appl. Phys. Part 1.
Regul. Pap. Short Not. Rev. Pap. 30, 2236 (1991)
4. J. Frantti, Y. Fujioka, J. Zhang, S.C. Vogel, Y. Wang, Y. Zhao,
R.M. Nieminen, J. Phys. Chem. B 113, 7967 (2009)
5. M. Ahart, M. Somayazulu, R.E. Cohen, P. Ganesh, P. Dera, H.K.
Mao, R.J. Hemley, Y. Ren, P. Liermann, Z.G. Wu, Nature 451, 545
(2008)
6. K. Pengpat, S. Hanphimol, S. Eitssayeam, U. Intatha, G. Rujijanagul, T. Tunkasiri, in 3rd International Conference on Materials for Advanced Technologies (ICMAT-2005)/9th International Conference on Advanced Materials (ICAM 2005) (Springer,
Berlin, 2005), p. 301
7. Y. Hosono, K. Harada, Y. Yamashita, Jpn. J. Appl. Phys. 40, 5722
(2001)
8. D. Rout, K.S. Moon, V.S. Rao, S.J.L. Kang, J. Ceram. Soc. Jpn.
117, 797 (2009)
9. J. Zhou, W.W. Peng, D. Zhang, X.Y. Yang, W. Chen, in Symposium on Frontiers in Computational Materials Science Held at the
International Conference on Materials for Advanced Technologies
(Elsevier, Amsterdam, 2007), p. 67
10. G. Geneste, J.M. Kiat, C. Malibert, Phys. Rev. B 77, 052106
(2008)
11. B.J. Chu, D.R. Chen, G.R. Li, Q.R. Yin, J. Eur. Ceram. Soc. 22,
2115 (2002)
12. A.N. Soukhojak, H. Wang, G.W. Farrey, Y.M. Chiang, in
Williamsburg Workshop on Ferroelectrics 99 (Pergamon/Elsevier,
Amsterdam, 1999), p. 301
13. L. Bellaiche, D. Vanderbilt, Phys. Rev. B 61, 7877 (2000)
14. N.J. Ramer, A.M. Rappe, Phys. Rev. B 62, 743 (2000)
15. P. Ghosez, D. Desquesnes, X. Gonze, K.M. Rabe, in AspenCenter-for-Physics Winter Workshop on Fundamental Physics of
Theoretical predictions of morphotropic phase boundary in (1 − x)Na1/2 Bi1/2 TiO3 -xBaTiO3
Ferroelectrics, ed. by R.E. Cohen (Amer. Inst. Physics, New York,
2000), p. 102
16. R. Haumont, B. Dkhil, J.M. Kiat, A. Al-Barakaty, H. Dammak,
L. Bellaiche, Phys. Rev. B 68, 014114 (2003)
17. M.D. Segall, P.J.D. Lindan, M.J. Probert, C.J. Pickard, P.J. Hasnip,
S.J. Clark, M.C. Payne, J. Phys., Condens. Matter 14, 2717 (2002)
18. J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77, 3865
(1996)
1089
19. D. Vanderbilt, Phys. Rev. B 41, 7892 (1990)
20. A. Shigemi, T. Wada, Jpn. J. Appl. Phys. 43, 6793 (2004)
21. A. Zoroddu, F. Bernardini, P. Ruggerone, V. Fiorentini, Phys. Rev.
B 64, 045208 (2001)
22. R.E. Cohen, Nature 358, 136 (1992)
23. J. Suchanicz, J. Kusz, H. Bohm, Mater. Sci. Eng. B, Solid-State
Mater. Adv. Technol. 97, 154 (2003)