Open access peer-reviewed chapter

Electronic Structures of Tetragonal ABX3: Role of the B-X Coulomb Repulsions for Ferroelectricity and Piezoelectricity

By Kaoru Miura and Hiroshi Funakubo

Submitted: April 24th 2012Reviewed: August 8th 2012Published: November 19th 2012

DOI: 10.5772/52187

Downloaded: 1979

1. Introduction

Since Cohen proposed an origin for ferroelectricity in perovskites (ABX3) [1], investigations of ferroelectric materials using first-principles calculations have been extensively studied [2-20]. Currently, using the pseudopotential (PP) methods, most of the crystal structures in ferroelectric ABX3 can be precisely predicted. However, even in BaTiO3, which is a well-known ferroelectric perovskite oxide with tetragonal structure at room temperature, the optimized structure by the PP methods is strongly dependent on the choice of the Ti PPs as illustrated in Fig. 1; preparation for Ti 3s and 3p semicore states in addition to Ti 3d, 4s, and 4p valence states is essential to the appearance of the tetragonal structure. This is an important problem for ferroelectricity, but it has been generally recognized for a long time that this problem is within an empirical framework of the calculational techniques [21].

Figure 1.

Illustration of the choice of Ti 3s and 3p states in pseudopotentials.

It is known that ferroelectric state appears when the long-range forces due to the dipole-dipole interaction overcome the short-range forces due to the Coulomb repulsions. Investigations about the relationship between the Ti-O Coulomb repulsions and the appearance of ferroelectricity in ATiO3 (A = Ba, Pb) were reported both theoretically and experimentally. Theoretically, Cohen first proposed the hybridization between Ti 3d state and O 2p state (Ti 3d-O 2p) as an origin for ferroelectricity in BaTiO3 and PbTiO3 [1]. On the other hand, we investigated [20] the influence of the Ti-Oz Coulomb repulsions on Ti ion displacement in tetragonal BaTiO3 and PbTiO3, where Oz denotes the O atom to the z-axis (Ti is displaced to the z-axis). Whereas the hybridization between Ti 3d state and Oz 2pz state stabilize Ti ion displacement, the strong Coulomb repulsions between Ti 3s and 3pz states and O 2pz states do not favourably cause Ti ion displacement. Experimentally, on the other hand, Kuroiwa et al. [22] showed that the appearance of ferroelectric state is closely related to the total charge density of Ti-O bonding in BaTiO3. As discussed above, investigation about a role of Ti 3s and 3p states is important in the appearance of the ferroelectric state in tetragonal BaTiO3, in addition to the Ti 3d-O 2p hybridization as an origin of ferroelectricity [1].

It seems that the strong B-X Coulomb repulsions affect the most stable structure of ABX3. It has been well known that the most stable structure of ABX3 is closely related to the tolerance factor t,

t = rA+rX2 (rB+rX)  , E1

where rA, rB, and rX denote the ionic radii of A, B, and X ions, respectively [23]. In general ferroelectric ABX3, the most stable structure is tetragonal for t ≳ 1, cubic for t ≈ 1, and rhombohedral or orthorhombic for t ≲ 1. In fact, BaTiO3 with t = 1.062 shows tetragonal structure in room temperature. However, recently, BiZn0.5Ti0.5O3 (BZT) with t = 0.935 was experimentally reported [24] to show a tetragonal PbTiO3-type structure with high c/a ratio (1.211). This result is in contrast to that of BiZn0.5Mg0.5O3 (BMT) with t = 0.939, i.e., the most stable structure was reported to be the orthorhombic or rhombohedral structure [25, 26]. Several theoretical papers of BZT have been reported [4-6], but the role of the Zn-O Coulomb repulsions in the appearance of the tetragonal structure has not been discussed sufficiently.

Piezoelectric properties in ABX3 are also closely related to the crystal structure. Investigations of the relationship between piezoelectric properties and the crystal structure of ABX3 by first-principles calculations have been extensively studied [2-19]. Moreover, phenomenological investigations of the piezoelectric properties have been also performed [27, 28]. However, it seems that the piezoelectric properties in the atomic level have not been sufficiently investigated. Therefore, further theoretical investigation of the relationship between piezoelectric properties and the crystal structure of ABX3, especially the B-X Coulomb repulsions, should be needed.

Recently, we investigated the roles of the Ti-O Coulomb repulsions in the appearance of a ferroelectric state in tetragonal BaTiO3 by the analysis of a first-principles PP method [11-15]. We investigated the structural properties of tetragonal and rhombohedral BaTiO3 with two kinds of Ti PPs, and propose the role of Ti 3s and 3p states for ferroelectricity. We also investigated the role of the Zn-O Coulomb repulsions in the appearance of a ferroelectric state in tetragonal BZT [10, 13]. Moreover, we also investigated the structural, ferroelectric, and piezoelectric properties of tetragonal ABX3 and discussed the piezoelectric mechanisms based on the B-X Coulomb repulsions [12, 14, 15, 18, 19].

In this chapter, based on our recent papers and patents [10-19], we discuss a general role of B-X Coulomb repulsions for the appearance of the ferroelectric state in ABX3. Then, we also discuss the relationship between the B-X Coulomb repulsions and the piezoelectric properties of tetragonal ABX3.

2. Methodology

The calculations for ABX3 were performed using the ABINIT code [29], which is one of the norm-conserving PP (NCPP) methods. Electron-electron interaction was treated in the local-density approximation (LDA) [30]. Pseudopotentials were generated using the OPIUM code [31]:

  1. In BaTiO3, 5s, 5p and 6s electrons for Ba PP, and 2s and 2p electrons for O PP were treated as semicore or valence electrons, respectively. Moreover, in order to investigate the role of Ti 3s and 3p states, two kinds of Ti PPs were prepared: the Ti PP with 3s, 3p, 3d and 4s electrons treated as semicore or valence electrons (Ti3spd4s PP), and that with only 3d and 4s electrons treated as valence electrons (Ti3d4s PP). In both PPs, the differences between the calculated result and experimental one are within 1.5 % of the lattice constant and within 10 % of the bulk modulus in the optimized calculation of bulk Ti. The cutoff energy for plane-wave basis functions was set to be 50 Hartree (Hr). The number of atoms in the unit cell was set to be five, and a 6×6×6 Monkhorst-Pack k-point mesh was set in the Brillouin zone of the unit cell. Positions of all the atoms were optimized within the framework of the tetragonal (P4mm) or rhombohedral (R3m) structure.

  2. In BZT and BMT, 5d, 6s, and 6p electrons for Bi PP, and 2s and 2p electrons for O PP were treated as semicore or valence electrons, respectively. Moreover, in order to investigate the roles of Zn and Ti 3s and 3p states, and Mg 2s and 2p states, two types of PPs were prepared: the PPs with only Zn and Ti 3d and 4s states, and Mg 3s states, considered as valence electrons (Case I), Zn and Ti 3s, 3p, 3d, and 4s states, and Mg 2s, 2p, and 3s states considered as semicore or valence electrons (Case II). The cutoff energy for plane-wave basis functions was set to be 70 Hr for Case I and 110 Hr for Case II. A 4×4×4 Monkhorst-Pack k-point mesh was set in the Brillouin zone of the unit cell. The calculated results can be discussed within 0.02 eV per formula unit (f.u.) using the above conditions. The present calculations were performed for the monoclinic, rhombohedral, and A-, C-, and G-type tetragonal structures. The number of atoms in the unit cell was set to be 10 for the rhombohedral and monoclinic structures, and 20 for the A-, C-, and G-type tetragonal structures. Positions of all the atoms were optimized within the framework of the rhombohedral (R3), monoclinic (Pm), and tetragonal (P4mm) structures.

  3. Relationship between the B-X Coulomb repulsions and the piezoelectric properties in tetragonal ABX3 is investigated. The pseudopotentials were generated using the opium code [31] with semicore and valence electrons (e.g., Ti3spd4s PP), and the virtual crystal approximation [32] were applied to several ABX3.

Spontaneous polarizations and piezoelectric constants were also evaluated, due to the Born effective charges [33]. The spontaneous polarization of tetragonal structures along the [001] axis, P3, is defined as

 P3 =kecΩZ33*(k)u(k)3  ,E2

where e, c, and Ω denote the charge unit, lattice parameter of the unit cell along the [001] axis, and the volume of the unit cell, respectively. u3(k) denotes the displacement along the [001] axis of the kth atom, and Z33*(k) denotes the Born effective charges [33] which contributes to the P3 from the u3(k).

The piezoelectric e33 constant is defined as

  e3j=(P3η3)u+ kecΩZ33*(k)u(k)3ηj        (j=3, 1),E3

where e and Ω denote the charge unit and the volume of the unit cell. P3 and c denote the spontaneous polarization of tetragonal structures and the lattice parameter of the unit cell along the [001] axis, respectively. u3(k) denotes the displacement along the [001] axis of the kth atom, and Z33*(k) denotes the Born effective charges which contributes to the P3 from the u3(k). η3 denotes the strain of lattice along the [001] axis, which is defined as η3 ≡ (cc0)/c0; c0 denotes the c lattice parameter with fully optimized structure. On the other hand, η1 denotes the strain of lattice along the [100] axis, which is defined as η1 ≡ (aa0)/a0; a0 denotes the a lattice parameter with fully optimized structure. The first term of the right hand in Eq. (3) denotes the clamped term evaluated at vanishing internal strain, and the second term denotes the relaxed term that is due to the relative displacements.

The relationship between the piezoelectric d33 constant and the e33 one is

  d33j=16s3jE ×(e3j)T  ,  E4

where s3jE denotes the elastic compliance, and ``T ” denotes the transposition of matrix elements. The suffix j denotes the direction-indexes of the axis, i.e., 1 along the [100] axis, 2 along the [010] axis, 3 along the [001] axis, and 4 to 6 along the shear directions, respectively.

3. Results and discussion

3.1. Ferroelectricity

3.1.1. Role of Ti 3s and 3p states in ferroelectric BaTiO3

Figures 2(a) and 2(b)show the optimized results for the ratio c/a of the lattice parameters and the value of the Ti ion displacement (δTi) as a function of the a lattice parameter in tetragonal BaTiO3, respectively. Results with arrows are the fully optimized results, and the others results are those with the c lattice parameters and all the inner coordination optimized for fixed a. Note that the fully optimized structure of BaTiO3 is tetragonal with the Ti3spd4s PP, whereas it is cubic with the Ti3d4s PP. This result suggests that the explicit treatment of Ti 3s and 3p semicore states is essential to the appearance of ferroelectric states in BaTiO3.

Figure 2.

Optimized calculated results in tetragonal BaTiO3. Results with arrows are the fully optimized results [11].

The calculated results shown in Fig. 2 suggest that the explicit treatment of Ti 3s and 3p semicore states is essential to the appearance of ferroelectric states in BaTiO3. In the following, we investigate the role of Ti 3s and 3p states for ferroelectricity from two viewpoints.

One viewpoint concerns hybridizations between Ti 3s and 3p states and other states. Figure 3(a) and 3(b) shows the total density of states (DOS) of tetragonal BaTiO3 with two Ti PPs. Both results are in good agreement with previous calculated results [7] by the full-potential linear augmented plane wave (FLAPW) method. In the DOS with the Ti3spd4s PP, the energy ``levels", not bands, of Ti 3s and 3p states, are located at -2.0 Hr and -1.2 Hr, respectively. This result suggests that the Ti 3s and 3p orbitals do not make any hybridization but only give Coulomb repulsions with the O orbitals as well as the Ba orbitals. In the DOS with the Ti3d4s PP, on the other hand, the energy levels of Ti 3s and 3p states are not shown because Ti 3s and 3p states were treated as the core charges. This result means that the Ti 3s and 3p orbitals cannot even give Coulomb repulsions with the O orbitals as well as the Ba orbitals.

Figure 3.

Total density of states (DOS) of fully optimized tetragonal BaTiO3 with the Ti3spd4s PP (solid line) and cubic BaTiO3 with the Ti3d4s PP (red dashed line) [11].

Another viewpoint is about the Coulomb repulsions between Ti 3s and 3px (y) states and Ox (y) 2s and 2px (y) states in tetragonal BaTiO3. Figure 4(a) and 4(b) show two-dimensional electron-density contour map on the xz-plane. These are the optimized calculated results with a fixed to be 3.8 Å, and the electron density in Fig. 4(a) is quantitatively in good agreement with the experimental result [22]. The electron density between Ti and Ox ions in Fig. 3(a) is larger than that in Fig. 4(b), which suggests that Ti ion displacement is closely related to the Coulomb repulsions between Ti 3s and 3px (y) states and Ox (y) 2s and 2px (y) states; the Ti-O Coulomb repulsion is an important role in the appearance of the ferroelectric state in BaTiO3.

The present discussion of the Coulomb repulsions is consistent with the previous reports. A recent soft mode investigation [8] of BaTiO3 shows that Ba ions contribute little to the appearance of Ti ion displacement along the [001] axis. This result suggests that Ti ion displacement is closely related to the structural distortion of TiO6 octahedra. In the present calculations, on the other hand, the only difference between BaTiO3 with the Ti3spd4s PP and with the Ti3d4s PP is the difference in the expression for the Ti 3s and 3p states, i.e., the explicit treatment and including core charges. However, our previous calculation [20] shows that the strong Coulomb repulsions between Ti 3s and 3pz states and Oz 2s and 2pz states do not favor Ti ion displacement along the [001] axis. This result suggests that the Coulomb repulsions between Ti 3s and 3px (y) states and Ox (y) 2s and 2px (y) states would contribute to Ti ion displacement along the [001] axis, and the suggestion is consistent with a recent calculation [9] for PbTiO3 indicating that the tetragonal and ferroelectric structure appears more favorable as the a lattice constant decreases.

Figure 4.

Two-dimensional electron-density contour map on the xz-plane for tetragonal BaTiO3: (a) with the Ti3spd4s PP, and (b) with the Ti3d4s PP. The optimized calculated results with a fixed to be 3.8 Å are shown in both figures. The electron density increases as color changes from blue to red via white. Contour curves are drawn from 0.4 to 2.0 e/Å3 with increments of 0.2 e/Å3 [11].

Considering the above investigations, we propose the mechanism of Ti ion displacement as follows: Ti ion displacement along the z-axis appears when the Coulomb repulsions between Ti 3s and 3 px (y) states and Ox (y) 2s and 2 px (y) states, in addition to the dipole-dipole interaction, overcome the Coulomb repulsions between Ti 3s and 3pz states and Oz 2s and 2pz states. An illustration of the Coulomb repulsions is shown in Fig. 5(a). In fully optimized BaTiO3 with the Ti3spd4s PP, the Ti ion can be displaced due to the above mechanism. In fully optimized BaTiO3 with the Ti3d4sPP, on the other hand, the Ti ion cannot be displaced due to the weaker Coulomb repulsions between Ti and Ox (y) ions. However, since the Coulomb repulsion between Ti and Oz ions in BaTiO3 with the Ti3d4s PP is also weaker than that in BaTiO3 with the Ti3spd4s PP, the Coulomb repulsions between Ti and Ox (y) ions in addition to the log-range force become comparable to the Coulomb repulsions between Ti and Oz ions both in Ti PPs, as the a lattice parameter becomes smaller. The above discussion suggests that the hybridization between Ti 3d and Oz 2s and 2pz stabilizes Ti ion displacement, but contribute little to a driving force for the appearance of Ti ion displacement.

It seems that the above proposed mechanism for tetragonal BaTiO3 can be applied to the mechanism of Ti ion displacement in rhombohedral BaTiO3, as illustrated in Fig. 5(b). The strong isotropic Coulomb repulsions between Ti 3s and 3px (y, z) states and Ox (y, z) 2s and 2px (y, z) states yield Ti ion displacement along the [111] axis. On the other hand, when the isotropic Coulomb repulsions are weaker or stronger, the Ti ion cannot be displaced and therefore it is favoured for the crystal structure to be cubic.

Figure 5.

Illustrations of the proposed mechanisms for the Coulomb repulsions between Ti 3s and 3p states and O 2s and 2p states in BaTiO3: (a) anisotropic Coulomb repulsions between Ti 3s and 3px (y) states and Ox (y) 2s and 2px (y) states, and between Ti 3s and 3pz states and Oz 2s and 2pz states, in the tetragonal structure. (b) isotropic Coulomb repulsions between Ti 3s and 3px (y, z) states and Ox (y, z) 2s and 2px (y, z) states, in the rhombohedral structure [11].

Let us investigate the structural properties of rhombohedral BaTiO3. Figures 6(a) and 6(b) show the optimized results of the 90-α degree and δTi as a function of fixed volumes of the unit cells in rhombohedral BaTiO3, respectively, where α denotes the angle between two lattice vectors. In these figures, α denotes the angle between two crystal axes of rhombohedral BaTiO3, and δTi denotes the value of the Ti ion displacement along the [111] axis. Results with arrows are the fully optimized results; Vopt denote the volume of the fully optimized unit cell with the Ti3spd4s PP. The other results are those with all the inner coordination optimized with fixed volumes of the unit cells. The proposal mechanisms about the Coulomb repulsions seem to be consistent with the calculated results shown in Fig.6: For V/Vopt ≲ 0.9 or ≳ 1.3, the isotropic Coulomb repulsions are weaker or stronger, and the Ti ion cannot be displaced along the [111] axis and therefore the crystal structure is cubic for both Ti PPs. For 0.9 ≲ V/Vopt ≲ 1.3, on the other hand, the isotropic Coulomb repulsions are strong enough to yield Ti ion displacement for both Ti PPs. However, since the magnitude of the isotropic Coulomb repulsion is different in the two Ti PPs, the properties of the 90-α degree and δTi are different quantitatively.

Figure 6.

Optimized calculated results as a function of the fixed volumes of the unit cells in rhombohedral BaTiO3: (a) 90-α degree and (b) δTi to the [111] axis. Blue lines correspond to the results with the Ti3spd4s PP, and red lines correspond to those with the Ti3d4s PP. Vopt denote the volume of the fully optimized unit cell with the Ti3spd4s PP. Results with arrows are the fully optimized results, and the other results are those with all the inner coordination optimized for fixed volumes of the unit cells [11].

3.1.2. Role of Zn 3s, 3p and 3d states in ferroelectric BiZn0.5Ti0.5O3

As discussed in Sec. 3.1.1, the Coulomb repulsions between Ti 3s and 3px (y) states and Ox (y) 2s and 2px (y) states have an important role in the appearance of the ferroelectric state in tetragonal BaTiO3. In this subsection, we discuss the role of Zn 3d (d10) states in addition to 3s and 3p states for ferroelectricity in tetragonal BZT.

Table 1 shows a summary of the optimized results of BZT in Cases I and II. ΔEtotal denotes the difference in total energy per f.u. between the rhombohedral and other structures. Although the lattice constant in each structure except the rhombohedral one seems to be quantitatively similar in both cases, properties of ΔEtotal are different. In Case I, the rhombohedral structure is the most stable, which is in disagreement with the experimental result [24]. In Case II, on the other hand, the monoclinic structure, which is the ``pseudo-C-type-tetragonal'' structure, is the most stable. Unfortunately, this result seems to be in disagreement with the experimental result [24], but is in good agreement with the recent calculated result [6]. Note that the magnitude of ΔEtotal in Case II is markedly smaller than that in Case I. In contrast to BZT, the rhombohedral structure is the most stable structure in both cases in BMT, which is consistent with the experimental result [26].

Figures 7(a) and 7(b) show two-dimensional electron density contour maps of the C-type tetragonal BZT in Cases I and II, respectively. The Coulomb repulsion of Zn-Ox in Case II is larger than that in Case I, and the Coulomb repulsion favorably causes Zn ion displacement to Oz in Case II. This result is consistent with Sec. 3.1.1. In contrast to the properties of Zn-O bonding, the inner coordination of the Ti ion is similar in both cases, although the electron densities are markedly different. This result suggests that the Coulomb repulsion magnitude of Ti-Oz is the same as that of Ti-Ox in small Ti-O bonding (≈ 1.8 Å), in both Cases I and II. Figures 7(c) and 7(d) show two-dimensional electron density contour maps of the C-type tetragonal BMT in Cases I and II, respectively. Although the electron densities in both cases are markedly different, the inner coordination of the Mg ion are similar. This result suggests that the Coulomb repulsion between Mg and O is not strong sufficiently for inducing Mg ion displacement even in Case II.

Structurea (Å)c (Å)c/aα (deg.)ΔE total (eV/f.u.)
A-type Tetra.3.7484.5791.222900.316
C-type Tetra.3.6814.7841.299900.240
G-type Tetra.3.7254.5741.228900.158
Monoclinic3.7354.7411.269β = 91.50.193
Rhombohedral5.560159.930
Experiment [24]3.8224.6281.21190---
(a)
Structurea (Å)c (Å)c/aα (deg.)ΔE total (eV/f.u.)
A-type Tetra.3.7114.6621.256900.135
C-type Tetra.3.6704.7891.305900.091
G-type Tetra.3.6844.6981.275900.047
Monoclinic3.7264.7401.272β = 91.1-0.021
Rhombohedral5.590159.900
Experiment [24]3.8224.6281.21190---
(b)

Table 1.

Summary of the optimized results of BZT in (a) Case I and (b) Case II. a and c denote the lattice parameters, and α and β denote angles between two lattice axes. ΔEtotal denotes the difference in total energy per f.u. between the rhombohedral and other structures [10].

Finally in this subsection, we discuss the difference in the electronic structures between the C-type tetragonal and the monoclinic BZT. Figures 8(a) and 8(b) show the electron density contour maps of the C-type tetragonal BZT and that of the monoclinic BZT in Case II, respectively. This result suggests that the strong Coulomb repulsion between Zn and Oz causes the small Zn ion displacement in the [110] direction in the monoclinic BZT, which makes the Coulomb repulsion of Zn-Oz weaker than that in the C-type tetragonal BZT. As a result, this small Zn ion displacement makes the monoclinic BZT more stable than the C-type tetragonal structure.

Figure 7.

Two-dimensional electron density contour maps of monoclinic (a) BZT in Case I, (b) BZT in Case II, (c) BMT in Case I, and (d) BMT in Case II. The electron density increases as color changes from blue to red via white. Contour curves are drawn from 0.2 to 2.0 e/Å3 with increments of 0.2 e/Å3 [10].

Figure 8.

Two-dimensional electron density contour maps of BZT in Case II (a) C-type tetragonal and (b) monoclinic. The electron density increases as color changes from blue to red via white. Contour curves are drawn from 0.2 to 2.0 e/Å3 with increments of 0.2 e/Å3 [10].

3.2. Piezoelectricity

3.2.1. Role of the Ti-O Coulomb repulsions in tetragonal piezoelectric SrTiO3 and BaTiO3

As discussed in Sec. 3.1, the Coulomb repulsions between Ti 3s and 3px (y) states and Ox (y) 2s and 2px (y) states have an important role in the appearance of the ferroelectric state in tetragonal BaTiO3. In this subsection, we discuss the role of the Ti-O Coulomb repulsions for piezoelectric SrTiO3 and BaTiO3.

Figures 9(a) shows the optimized results for cccub as a function of the a lattice parameters in tetragonal SrTiO3 and BaTiO3, where ccub denotes the c lattice parameter in cubic SrTiO3 and BaTiO3, respectively. These results are the fully optimized results and the results with the c lattice parameters and all the inner coordination optimized for fixed a. The fully optimized parameters of SrTiO3 (a = 3.84 Å: cubic) and BaTiO3 (a = 3.91 Å and c = 4.00 Å: tetragonal) are within 2.0 % in agreement with the experimental results in room temperature. Figure 9(b) shows the evaluated results for P3 as a function of the a lattice parameters in tetragonal SrTiO3 and BaTiO3, where P3, which is evaluated by Eq. (2), denotes the spontaneous polarization along the [001] axis. Note that the tetragonal and ferroelectric structures appear even in SrTiO3 when the fixed a lattice parameter is compressed to be smaller than the fully-optimized a lattice parameter. As shown in Figs. 9(a) and 9(b), the tetragonal and ferroelectric structure appears more favorable as the fixed a lattice parameter decreases, which is consistent with previous calculated results [9, 11]. The results would be due to the suggestion discussed in the previous section that the large Coulomb repulsion of Ti-O bonding along the [100] axis (and the [010] axis) is a driving force of the displacement of Ti ions along the [001] axis, i.e., the large Coulomb repulsion along the [100] axis (and the [010] axis) is essential for the appearance of the tetragonal structure.

Figure 9.

Optimized calculated results as a function of a lattice parameters in compressive tetragonal SrTiO3 and BaTiO3: (a) c – ccub and (b) P3, i.e., spontaneous polarization along the [001] axis [12].

In the following, we use cccub as a functional parameter, because cccub is closely related to η3. Figures 10(a) and 10(b) shows the piezoelectric properties of e33 and e31 as a function of cccub in tetragonal SrTiO3 and BaTiO3. The value cccub is optimized value as shown in Fig. 9(a) and e33 and e31 are evaluated values in their optimized structures. Note that e33 become larger at cccub ≈ 0, especially in SrTiO3. These properties seem to be similar to the properties arond the Curie temperatures in piezoelectric ABO3; Damjanovic emphasized the importance of the polarization extension as a mechanism of larger piezoelectric constants in a recent paper [28]. Contrary to e33, on the other hand, the changes in e31 are much smaller than the changes in e33, but note that e31 shows negative in SrTiO3 while positive in BaTiO3.

Figure 10.

Evaluated piezoelectric constants as a function of c – ccub in optimized tetragonal SrTiO3 and BaTiO3: (a) e33 and (b) e31 [12].

As expressed in Eq. (3), e3j is the sum of the contributions from the clamped term and the relaxed term. However, it has been generally known that the contribution to e3j from the clamped term is much smaller than that from the relaxed term; in fact, the absolute values of the e33 clamped terms are less than 1 C/m2 in both SrTiO3 and BaTiO3. We therefore investigate the contributions to the relaxed term of e33 and e31 in detail. As expressed in Eq. (3), the relaxed terms of e3j are proportional to the sum of the products between the Z33* (k) and ∂u3 (k)/∂ηj (j = 3 or 1) values. Let us show the evaluated results of Z33* (k), ∂u3(k)/∂η3, and ∂u3(k)/∂η1 in the following. Figures 11(a) and 11(b) show the Z33* (k) values in SrTiO3 and BaTiO3, respectively. Properties of the Z33* (k) values are quantitatively similar in both SrTiO3 and BaTiO3. Therefore, the difference in the properties of e33 and e31 between SrTiO3 and BaTiO3 must be due to the difference in the properties of ∂u3(k)/∂ηj. Figures 12(a) and 12(b) show the ∂u3(k)/∂η3 values in SrTiO3 and BaTiO3, respectively. In these figures, Ox and Oz denote oxygen atoms along the [100] and [001] axes, respectively. Clearly, the absolute values of ∂u3(k)/∂η3 are different in between SrTiO3 and BaTiO3. On the other hand, Figs. 13(a) and 13(b) show the ∂u3(k)/∂η1 values in SrTiO3 and BaTiO3, respectively. The absolute values of ∂u3(k)/∂η1, especially for Ti, Ox, and Oz are different in between SrTiO3 and BaTiO3. As a result, the quantitative differences in e33 and e31 between SrTiO3 and BaTiO3 are due to the differences in the contribution of the ∂u3(k)/∂η3 and ∂u3(k)/∂η1 values, respectively.

Figure 11.

Evaluated Born effective charges Z33* (k) as a function of c – ccub: (a) SrTiO3 and (b) BaTiO3. Ox and Oz denote oxygen atoms along the [100] axis and the [001] axis, respectively [12].

Figure 12.

Evaluated values of ∂u3(k)/∂η3 as a function of c – ccub: (a) SrTiO3 and (b) BaTiO3 [12].

Figure 13.

Evaluated values of ∂u3(k)/∂η1 as a function of c – ccub: (a) SrTiO3 and (b) BaTiO3 [12].

Let us discuss the reasons of the quantitative differences in e33 between SrTiO3 and BaTiO3. Figure 14(a) shows the difference between the Ti-Oz distance (RTi-Oz) and the sum of the rTi and rOz (rTi + rOz) along the [001] axis as a function of cccub. Note that RTi-Oz is smaller than rTi + rOz in both SrTiO3 and BaTiO3. However, the difference in absolute value between RTi-Oz and rTi + rOz in SrTiO3 is smaller than the difference in BaTiO3 for 0 ≲ cccub ≲ 0.20. This result suggests that the Ti-Oz Coulomb repulsion along the [001] axis in SrTiO3 is smaller than that in BaTiO3 and that therefore the Ti ion of SrTiO3 can be displaced more easily along the [001] axis than that of BaTiO3. This would be a reason why the absolute values of ∂u3(k)/∂η3 of Ti and Oz ions in SrTiO3 are larger than that in BaTiO3. Figure 14(b) shows the difference between the A-Ox distance (RA-Ox) and the sum of rA and rOx (rA + rOx) on the (100) plane as a function of cccub, where the values of the ionic radii are defined as Shannon's ones [23]. Note that RA-Ox is smaller than rA + rOx in both SrTiO3 and BaTiO3. However, the difference in absolute value between RA-Ox and rA + rOx in SrTiO3 is much smaller than the difference in BaTiO3 for 0 ≲ cccub ≲ 0.20. This result suggests that the Sr-Ox Coulomb repulsion on the (100) plane in SrTiO3 is much smaller than the Ba-Ox Coulomb repulsion in BaTiO3 and that therefore Sr and Ox ions of SrTiO3 can be displaced more easily along the [001] axis than Ba and Ox ions of BaTiO3. This would be a reason why the absolute values of ∂u3(k)/∂η3 of Sr and Ox ions in SrTiO3 are larger than those of Ba and Ox ions in BaTiO3.

Figure 14.

Evaluated values as a function of c – ccub in optimized tetragonal SrTiO3 and BaTiO3: (a) difference between the Ti-Oz distance (RTi-Oz) and rTi + rOz. (b) difference between the A-Ox distance (RA-Ox) and rA + rOx. RA-Ox and RTi-Oz in ATiO3 are also illustrated. Note that all the ionic radii are much larger and that A and Ti ions are displaced along the [001] axis in actual ATiO3 [12].

Finally, in this subsection, we discuss the relationship between ∂u3(k)/∂η3 and cccub in detail. Figure 15(a) shows the differences in the total energy (ΔEtotal ) as a function of u3(Ti). In this figure, the properties of SrTiO3 with η3 = 0.011, SrTiO3 with η3 = 0.053 and fully optimized BaTiO3 as a reference, are shown. Calculations of Etotal were performed with the fixed crystal structures of previously optimized structures except Ti ions. Clearly, the magnitude of u3(Ti) at the minimum points of the ΔEtotal and the depth of the potential are closely related to the spontaneous polarization P3 and the Curie temperature (TC), respectively. On the other hand, e33 seems to be closely related to the deviation at the minimum points of the ΔEtotal. Figure 15(b) shows illustrations of ΔEtotal curves with deviations at the minimum points of the ΔEtotal values, corresponding to the ΔEtotal curves of SrTiO3 in Fig. 15(a). Clearly, as η3 becomes smaller, the deviated value at the minimum point of the ΔEtotal values becomes smaller, i.e., the Ti ion can be displaced more favourably. On the other hand, as shown in Fig. 12(a), the absolute value of ∂u3(Ti)/∂η3 becomes larger as η3 becomes smaller.

Therefore, the Ti ion can be displaced more favourably as the deviated value at the minimum point of the ΔEtotal values becomes smaller. The relationship between e33 and ∂u3(Ti)/∂η3 is discussed in Sec. 3.2.3.

Figure 15.

a) ΔEtotal as a function of u3(Ti) in tetragonal SrTiO3 and BaTiO3. (b) Illustration of the ΔEtotal curves in tetragonal SrTiO3 with η3 = 0.011 and SrTiO3 with η3 = 0.053 with deviations at the minimum point of ΔEtotal [14].

3.2.2. Proposal of new piezoelectric materials

The previous discussion in Sec. 3.2.1 suggests that the piezoelectric properties of e33 are closely related to the B-X Coulomb repulsions in tegtragonal ABX3. In the viewpoint of the change of the B-X Coulomb repulsions, we recently proposed new piezoelectric materials [16, 17], i.e., BaTi1-xNixO3 and Ba(Ti1-3zNb3z)(O1-zNz)3.

It has been known that BaNiO3 shows the 2H hexagonal structure as the most stable structure in room temperature. Moreover, the ionic radius of Ni4+ (d6) with the low-spin state in 2H BaNiO3 is 0.48 Å, which is much smaller than that of Ti4+ (d0), 0.605 Å, in BaTiO3. Therefore, due to the drastic change in the (Ti1-xNix)-O Coulomb repulsions in tetragonal BaTi1-xNixO3, the e33 piezoelectric values are expected to be larger than that in tetragonal BaTiO3, especially around the morphotropic phase boundary (MPB). Figure 16(a) shows the total-energy difference ΔEtotal between 2H and tetragonal structures of BaTi1-xNixO3 as a function of x. The most stable structure changes at x ≈ 0.26. Figure 16(b) shows cccub as a function of x. The cccub value shows 0 around x = 0.26, which suggests the appearance of the MPB, i.e., the e33 piezoelectric value shows a maximum at x ≈ 0.26.

Figure 16.

a) ΔEtotal (total-energy difference between 2H and tetragonal structures), and (b) c – ccub of the tetragonal structure, as a function of x in BaTi1-xNixO3 [16]. For or 0.26 ≤ x ≦ 1, the tetragonal structure is not the most stable one.

Figure 17.

(a) ΔEtotal (total-energy difference between cubic and tetragonal structures), and (b) c – ccub, as a function of x in Ba(Ti1-3zNb3z)(O1-zNz)3 [17].

Another proposal is tetragonal Ba(Ti1-3zNb3z)(O1-zNz)3, which consists of BaTiO3 and BaNbO2N [17]. Due to the change of (Ti1-3zNb3z)-(O1-zNz) Coulomb repulsions, the e33 piezoelectric values are expected to be larger than that in tetragonal BaTiO3. Recent experimental paper reported that the most stable structure of BaNbO2N is cubic in room temperature [34]. Contrary to the experimental result, however, our calculations suggest that the tetragonal structure will be more stable than the cubic one, as shown in Fig. 17(a). Figure 17(b) shows cccub as a function of x. The cccub value shows almost 0 at x ≈ 0.12. Although the MPB does not appear in tetragonal Ba(Ti1-3zNb3z)(O1-zNz)3, the e33 piezoelectric values are expected to show a maximum at x ≈ 0.12.

3.2.3. Piezoelectric properties of in tetragonal ABX3

In the following, we discuss the role of the B-X Coulomb repulsions in piezoelectric ABX3.

Figures 18(a) and 18(b) show the piezoelectric properties of e33 as a function of the value cccub in tetragonal ABX3, where ccub denotes the c lattice parameter in cubic ABX3; cccub is a closely related parameter to η3. For ABX3, SrTiO3, BaTiO3 and PbTiO3 with the c lattice parameter and all the inner coordination optimized for fixed a, and BaTi1-xNixO3 (0 ≦ x ≦ 0.05), Ba(Ti1-3zNb3z)(O1-zNz)3 (0 ≦ z ≦ 0.125), Ba1-ySryTiO3 (0 ≦ y ≦ 0.5), BaTi1-xZrxO3 (0 ≦ x ≦ 0.06), and BiM’O3 (M’ = Al, Sc) with fully optimized, were prepared [15]. Note that e33 becomes larger as cccub becomes smaller and that the trend of e33 is almost independent of the kind of A ions. Moreover, note also that e33 of BaTi1-xNixO3 and that of Ba(Ti1-3zNb3z)(O1-zNz)3 show much larger values than the other ABX3.

Figure 18.

e33 as a function of c – ccub for different scales [15].

Let us discuss the relationship between ∂u3(k)/∂η3 and cccub in BaTi1-xNixO3 and BaTiO3 in the following. Figures 19(a) and 19(b) show the ∂u3(k)/∂η3 values. In these figures, Ox and Oz denote oxygen atoms along the [100] and [001] axes, respectively. Clearly, the absolute values of ∂u3(k)/∂η3 in BaTi1-xNixO3 are much larger than those in BaTiO3. Moreover, in comparison with Fig.18, properties of e33 are closely related to those of ∂u3(k)/∂η3. Figure 20(a) shows the difference between RB-Oz and rB + rOz along the [001] axis, and Fig. 20(b) shows the difference between RA-Ox and rA + rOx on the (100) plane for several ABO3, as a function of cccub. Clearly, the difference between RB-Oz and rB + rOz is closely related to e33 shown in Fig. 18, rather than the difference between RA-Ox and rA + rOx. Moreover, note that the difference in absolute value between RB-Oz and rB + rOz in BaTi1-xNixO3 is much smaller than that in BaTiO3. This result suggests that the (Ti1-xNix)-Oz Coulomb repulsion along the [001] axis in BaTi1-xNixO3 is much smaller than the Ti-Oz Coulomb repulsion in BaTiO3 and that therefore Ti1-xNix ion of BaTi1-xNixO3 can be displaced more easily along the [001] axis than Ti ion of BaTiO3. This must be a reason why the absolute value of ∂u3(k)/∂η3 of Ti1-xNix and Oz ions in BaTi1-xNixO3 is larger than those in BaTiO3.

Figure 19.

∂u3(k)/∂η3 as a function of c – ccub : (a) BaTi1-xNixO3 and (b) BaTiO3 [15].

Figure 20.

Evaluated values of in optimized tetragonal BaTi1-xNixO3, BaTiO3, and several ABO3, as a function of c – ccub: (a) RB-Oz - (rB + rOz), and (b) RA-Ox – (rA + rOx) [15].

Figure 21(a) shows ΔEtotal as a function of the displacement of the Ti1-xNix ions with fixed crystal structures of fully-optimized BaTi1-xNixO3. Calculations of Etotal were performed with the fixed crystal structures of previously optimized structures except Ti1-xNix ions. The deviated value at the minimum point of ΔEtotal, i.e., (ΔEtotal)/∂u3(Ti1-xNix), becomes smaller as x becomes larger. Moreover, both e33 and ∂u3(Ti1-xNix)/∂η3 become larger as x becomes larger, as shown in Figs. 18 and 19. This result is consistent with the result of SrTiO3 shown in Fig. 15(a).

Let us discuss the above reasons in the following. (⊿Etotal)/∂u3(Ti1-xNix) can be written as

  (ΔEtotalu3(Ti1xNix))= (ΔEtotalη3) ×(u3(Ti1xNix)η3)1.  E5

As shown in Fig. 21(b), Etotal)/ ∂η3 is almost constant, and therefore, Etotal)/∂u3(Ti1-xNix) is almost proportional to (∂u3(Ti1-xNix)/∂η3)-1, i.e.,

(ΔEtotalu3(Ti1xNix))  (u3(Ti1xNix)η3)1. E6

On the other hand, according to Eq. (3), e33 becomes larger as ∂u3(Ti1-xNix)/∂η3 becomes larger. This is a reason why e33 becomes larger as (ΔEtotal)/∂u3(Ti1-xNix) becomes smaller. This result is consistent with the result of SrTiO3 discussed in Sec. 3.2.1.

Figure 21.

(a) ΔEtotal as a function of u3(Ti1-xNix) in BaTi1-xNixO3. Results with x = 0.05, 0.04, and 0 are shown. Dashed lines denote guidelines of ∂(ΔEtotal)/∂u3(Ti1-xNix) for each x. (b) ΔEtotal as a function of η3 for BaTi1-xNixO3 and BaTiO3 [15].

Finally, we comment on the difference in the properties between e33 and d33 in tetragonal ABX3. Figures 22(a) and 22(b) show the piezoelectric properties of d33 as a function of cccub. Note that the trend of d33 is closely dependent on the kind of A ions. This result is in contrast with the trend of e33 as shown in Fig. 18. As expressed in Eq. (4), d33 is closely related to the elastic compliance s3jE as well as e3j. In fact, the absolute value of s3jE in BiBX3 or PbBX3 is generally larger than that in ABX3 with alkaline-earth A ions. This result must be due to the larger Coulomb repulsion of Bi-X or Pb-X derived from 6s electrons in Bi (Pb) ion.

Figure 22.

d33 as a function of c – ccub for different scales [15].

Figure 23.

(a) Illustration of the relationship between the B-X Coulomb repulsions and the ferroelectric and piezoelectric states in tetragonal ABX3. (b) Illustration of the relationship between e33 and the deviation. P3 and TC denote the spontaneous polarization and the Curie temperature, respectively.

4. Conclusion

We have discussed a general role of the B-X Coulomb repulsions for ferroelectric and piezoelectric properties of tetragonal ABX3, based on our recent papers and patents. We have found that both ferroelectric state and piezoelectric state are closely related to the B-Xz Coulomb repulsions as well as the B-Xx ones, as illustrated in Fig. 23(a). Moreover, as illustrated in Fig. 23(b), we have also found that e33 is closely related to the deviation at the minimum point of the ΔEtotal.

Acknowledgments

We thank Professor M. Azuma in Tokyo Institute of Technology for useful discussion. We also thank T. Furuta, M. Kubota, H. Yabuta, and T. Watanabe for useful discussion and computational support. The present work was partly supported by the Elements Science and Technology Project from the Ministry of Education, Culture, Sports, Science and Technology.

© 2012 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Kaoru Miura and Hiroshi Funakubo (November 19th 2012). Electronic Structures of Tetragonal ABX3: Role of the B-X Coulomb Repulsions for Ferroelectricity and Piezoelectricity, Advances in Ferroelectrics, Aimé Peláiz Barranco, IntechOpen, DOI: 10.5772/52187. Available from:

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