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

(3), e 3 j is the sum of the contributions from the clamped term and the relaxed term. However, it has been generally known that the contribution to e 3 j from the clamped term is much smaller than that from the relaxed term; in fact, the absolute values of the e 33 clamped terms are less than 1 C/m 2 in both SrTiO 3 and BaTiO 3 . We therefore investigate the contributions to the relaxed term of e 33 and e 31 in detail. As expressed in Eq. (3), the relaxed terms of e 3 j are proportional to the sum of the products between the Z 33* ( k ) and ∂ u 3 ( k )/ ∂ η j ( j = 3 or 1) values. Let us show the evaluated results of Z 33* ( k ), ∂ u 3 ( k )/ ∂ η 3 , and ∂ u 3 ( k )/ ∂ η 1 in the following. Figures 11(a) and 11(b) show the Z 33* ( k ) values in SrTiO 3 and BaTiO 3 , respectively. Properties of the Z 33* ( k ) values are quantitatively similar in both SrTiO 3 and BaTiO 3 . Therefore, the difference in the properties of e 33 and e 31 between SrTiO 3 and BaTiO 3 must be due to the difference in the properties of ∂ u 3 ( k )/ ∂ η j . Figures 12(a) and 12(b) show the ∂ u 3 ( k )/ ∂ η 3 values in SrTiO 3 and BaTiO 3 , respectively. In these figures, O x and O z denote oxygen atoms along the [100] and [001] axes, respectively. Clearly, the absolute values of ∂ u 3 ( k )/ ∂ η 3 are different in between SrTiO 3 and BaTiO 3 . On the other hand, Figs. 13(a) and 13(b) show the ∂ u 3 ( k )/ ∂ η 1 values in SrTiO 3 and BaTiO 3 , respectively. The absolute values of ∂ u 3 ( k )/ ∂ η 1 , especially for Ti, O x , and O z are different in between SrTiO 3 and BaTiO 3 . As a result, the quantitative differences in e 33 and e 31 between SrTiO 3 and BaTiO 3 are due to the differences in the contribution of the ∂ u 3 ( k )/ ∂ η 3 and ∂ u 3 ( k )/


Introduction
Since Cohen proposed an origin for ferroelectricity in perovskites (ABX 3 ) [1], investigations of ferroelectric materials using first-principles calculations have been extensively studied [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. Currently, using the pseudopotential (PP) methods, most of the crystal structures in ferroelectric ABX 3 can be precisely predicted. However, even in BaTiO 3 , which is a wellknown 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].  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 ATiO 3 (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 BaTiO 3 and PbTiO 3 [1]. On the other hand, we investigated [20] the influence of the Ti-O z Coulomb repulsions on Ti ion displacement in tetragonal BaTiO 3 and PbTiO 3 , where O z denotes the O atom to the z-axis (Ti is displaced to the z-axis). Whereas the hybridization between Ti 3d state and O z 2p z state stabilize Ti ion displacement, the strong Coulomb repulsions between Ti 3s and 3p z states and O 2p z 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 BaTiO 3 . As discussed above, investigation about a role of Ti 3s and 3p states is important in the appearance of the ferroelectric state in tetragonal BaTiO 3 , 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 ABX 3 . It has been well known that the most stable structure of ABX 3 is closely related to the tolerance factor t, r r t r r where r A , r B , and r X denote the ionic radii of A, B, and X ions, respectively [23]. In general ferroelectric ABX 3 , the most stable structure is tetragonal for t ≳ 1, cubic for t ≈ 1, and rhombohedral or orthorhombic for t ≲ 1. In fact, BaTiO 3 with t = 1.062 shows tetragonal structure in room temperature. However, recently, BiZn 0.5 Ti 0.5 O 3 (BZT) with t = 0.935 was experimentally reported [24] to show a tetragonal PbTiO 3 -type structure with high c/a ratio (1.211). This result is in contrast to that of BiZn 0.5 Mg 0.5 O 3 (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][5][6], but the role of the Zn-O Coulomb repulsions in the appearance of the tetragonal structure has not been discussed sufficiently.

iii.
Relationship between the B-X Coulomb repulsions and the piezoelectric properties in tetragonal ABX 3 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 ABX 3 .
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, P 3 , is defined as ( ) 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. u 3 (k) denotes the displacement along the [001] axis of the kth atom, and Z 33 * (k) denotes the Born effective charges [33] which contributes to the P 3 from the u 3 (k).
The piezoelectric e 33 constant is defined as where e and Ω denote the charge unit and the volume of the unit cell. P 3 and c denote the spontaneous polarization of tetragonal structures and the lattice parameter of the unit cell along the [001] axis, respectively. u 3 (k) denotes the displacement along the [001] axis of the kth atom, and Z 33 * (k) denotes the Born effective charges which contributes to the P 3 from the u 3 (k). η 3 denotes the strain of lattice along the [001] axis, which is defined as η 3 ≡ (c -c 0 )/c 0 ; c 0 denotes the c lattice parameter with fully optimized structure. On the other hand, η 1 denotes the strain of lattice along the [100] axis, which is defined as η 1 ≡ (a -a 0 )/a 0 ; a 0 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 d 33 constant and the e 33 one is where s 3j E denotes the elastic compliance, and ``T " denotes the transposition of matrix elements. The suffix j denotes the direction-indexes of the axis, i. 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 (δ Ti ) as a function of the a lattice parameter in tetragonal BaTiO 3 , 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 BaTiO 3 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 BaTiO 3 .
The relationship between the piezoelectric d33 constant and the e33 one is 6

Results and discussion
3.1. Ferroelectricity

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. 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.  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 BaTiO 3 . In the following, we investigate the role of Ti 3s and 3p states for ferroelectricity from two viewpoints.  Another viewpoint is about the Coulomb repulsions between Ti 3s and 3p x (y) states and O x (y) 2s and 2p x (y) states in tetragonal BaTiO 3 . 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 O x 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 3p x (y) states and O x (y) 2s and 2p x (y) states; the Ti-O Coulomb repulsion is an important role in the appearance of the ferroelectric state in BaTiO 3 .
The present discussion of the Coulomb repulsions is consistent with the previous reports. A recent soft mode investigation [8] of BaTiO 3 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 TiO 6 octahedra. In the present calculations, on the other hand, the only difference between BaTiO 3 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 3p z states and O z 2s and 2p z states do not favor Ti ion displacement along the [001] axis. This result suggests that the Coulomb repulsions between Ti 3s and 3p x (y) states and O x (y) 2s and 2p x (y) states would contribute to Ti ion displacement along the [001] axis, and the suggestion is consistent with a recent calculation [9] for PbTiO 3 indicating that the tetragonal and ferroelectric structure appears more favorable as the a lattice constant decreases. 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 p x (y) states and O x (y) 2s and 2 p x (y) states, in addition to the dipole-dipole interaction, overcome the Coulomb repulsions between Ti 3s and 3p z states and O z 2s and 2p z states. An illustration of the Coulomb repulsions is shown in Fig. 5(a). In fully optimized BaTiO 3 with the Ti3spd4s PP, the Ti ion can be displaced due to the above mechanism. In fully optimized BaTiO 3 with the Ti3d4sPP, on the other hand, the Ti ion cannot be displaced due to the weaker Coulomb repulsions between Ti and O x (y) ions. However, since the Coulomb repulsion between Ti and O z ions in BaTiO 3 with the Ti3d4s PP is also weaker than that in BaTiO 3 with the Ti3spd4s PP, the Coulomb repulsions between Ti and O x (y) ions in addition to the log-range force become comparable to the Coulomb repulsions between Ti and O z ions both in Ti PPs, as the a lattice parameter becomes smaller. The above discussion suggests that the hybridization between Ti 3d and O z 2s and 2p z 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. 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.

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 (d 10 ) 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

Role of Zn 3s, 3p and 3d states in ferroelectric BiZn 0.5 Ti 0.5 O 3
As discussed in Sec. 3.1.1, the Coulomb repulsions between Ti 3s and 3p x (y) states and O x (y) 2s and 2p x (y) states have an important role in the appearance of the ferroelectric state in tetragonal BaTiO 3 . In this subsection, we discuss the role of Zn 3d (d 10 ) 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. ΔE total 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 ΔE total 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 ΔE total 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].

Role of the Ti-O Coulomb repulsions in tetragonal piezoelectric SrTiO 3 and BaTiO 3
As discussed in Sec. 3.1, the Coulomb repulsions between Ti 3s and 3p x (y) states and O x (y) 2s and 2p x (y) states have an important role in the appearance of the ferroelectric state in tetragonal BaTiO 3 . In this subsection, we discuss the role of the Ti-O Coulomb repulsions for piezoelectric SrTiO 3 and BaTiO 3 .  Figure 9(b) shows the evaluated results for P 3 as a function of the a lattice parameters in tetragonal SrTiO 3 and BaTiO 3 , where P 3 , which is evaluated by Eq. (2), denotes the spontaneous polarization along the [001] axis. Note that the tetragonal and ferroelectric structures appear even in SrTiO 3 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  In the following, we use c -ccub as a functional parameter, because c -ccub is closely related to η3. Figures 10(a) and 10(b) shows the piezoelectric properties of e33 and e31 as a function of c -ccub in tetragonal SrTiO3 and BaTiO3. The value c -ccub 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 c -ccub ≈ 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.  In the following, we use c -c cub as a functional parameter, because c -c cub is closely related to larger at c -c cub ≈ 0, especially in SrTiO 3 . These properties seem to be similar to the properties arond the Curie temperatures in piezoelectric ABO 3 ; Damjanovic emphasized the importance of the polarization extension as a mechanism of larger piezoelectric constants in a recent paper [28]. Contrary to e 33 , on the other hand, the changes in e 31 are much smaller than the changes in e 33 , but note that e 31 shows negative in SrTiO 3 while positive in BaTiO 3 . In the following, we use c -ccub as a functional parameter, because c -ccub is closely related to η3. Figures 10(a) and 10(b) shows the piezoelectric properties of e33 and e31 as a function of c -ccub in tetragonal SrTiO3 and BaTiO3. The value c -ccub 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 c -ccub ≈ 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. 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/m 2 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. As expressed in Eq. (3), e 3j is the sum of the contributions from the clamped term and the relaxed term. However, it has been generally known that the contribution to e 3j from the clamped term is much smaller than that from the relaxed term; in fact, the absolute values of the e 33 clamped terms are less than 1 C/m 2 in both SrTiO 3 and BaTiO 3 . We therefore investigate the contributions to the relaxed term of e 33 and e 31 in detail. As expressed in Eq. (3), the relaxed terms of e 3j are proportional to the sum of the products between the Z 33 * (k) and ∂u 3 (k)/∂η j (j = 3 or 1) values. Let us show the evaluated results of Z 33 * (k), ∂u 3 (k)/∂η 3 , and ∂u 3 (k)/ ∂η 1 in the following. Figures 11(a)    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 c -ccub. 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 ≲ c -ccub ≲ 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 c -ccub, 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 ≲ c -ccub ≲ 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.    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 c -ccub. 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 ≲ c -ccub ≲ 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 c -ccub, 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 ≲ c -ccub ≲ 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    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 c -ccub. 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 ≲ c -ccub ≲ 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 c -ccub, 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 ≲ c -ccub ≲ 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 Let us discuss the reasons of the quantitative differences in e 33 between SrTiO 3 and BaTiO 3 . Figure 14(a) shows the difference between the Ti-O z distance (R Ti-Oz ) and the sum of the r Ti Advances in Ferroelectrics and r Oz (r Ti + r Oz ) along the [001] axis as a function of c -c cub . Note that R Ti-Oz is smaller than r Ti + r Oz in both SrTiO 3 and BaTiO 3 . However, the difference in absolute value between R Ti-Oz and r Ti + r Oz in SrTiO 3 is smaller than the difference in BaTiO 3 for 0 ≲ c -c cub ≲ 0.20. This result suggests that the Ti-O z Coulomb repulsion along the [001] axis in SrTiO 3 is smaller than that in BaTiO 3 and that therefore the Ti ion of SrTiO 3 can be displaced more easily along the [001] axis than that of BaTiO 3 . This would be a reason why the absolute values of ∂u 3 (k)/∂η 3 of Ti and O z ions in SrTiO 3 are larger than that in BaTiO 3 . Figure 14(b) shows the difference between the A-O x distance (R A-Ox ) and the sum of r A and r Ox (r A + r Ox ) on the (100) plane as a function of c -c cub , where the values of the ionic radii are defined as Shannon's ones [23]. Note that R A-Ox is smaller than r A + r Ox in both SrTiO 3 and BaTiO 3 . However, the difference in absolute value between R A-Ox and r A + r Ox in SrTiO 3 is much smaller than the difference in BaTiO 3   Finally, in this subsection, we discuss the relationship between ∂u3(k)/∂η3 and c -ccub 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. Finally, in this subsection, we discuss the relationship between ∂u 3 (k)/∂η 3 and c -c cub in detail. Figure 15(a) shows the differences in the total energy (ΔE total ) as a function of u 3 (Ti). In this figure, the properties of SrTiO 3 with η 3 = 0.011, SrTiO 3 with η 3 = 0.053 and fully optimized BaTiO 3 as a reference, are shown. Calculations of E total were performed with the fixed crystal structures of previously optimized structures except Ti ions. Clearly, the magnitude of u 3 (Ti) at the minimum points of the ΔE total and the depth of the potential are closely related to the spontaneous polarization P 3 and the Curie temperature (T C ), respectively. On the other hand, e 33 seems to be closely related to the deviation at the minimum points of the ΔE total . Figure 15(b) shows illustrations of ΔE total curves with deviations at the minimum points of the ΔE total values, corresponding to the ΔE total curves of SrTiO 3 in Fig. 15(a). Clearly, as η 3 becomes smaller, the deviated value at the minimum point of the ΔE total 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 ∂u 3 (Ti)/∂η 3 becomes larger as η 3 becomes smaller.
Therefore, the Ti ion can be displaced more favourably as the deviated value at the minimum point of the ΔE total values becomes smaller. The relationship between e 33 and ∂u 3 (Ti)/∂η 3 is discussed in Sec. 3.2.3. 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 c -ccub 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.

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 Ni 4+ (d 6 ) with the low-spin state in 2H BaNiO3 is 0.48 Å, which is much smaller than that of Ti 4+ (d 0 ), 0.605 Å, in

Proposal of new piezoelectric materials
The previous discussion in Sec. 3.2.1 suggests that the piezoelectric properties of e 33 are closely related to the B-X Coulomb repulsions in tegtragonal ABX 3 . In the viewpoint of the change of the B-X Coulomb repulsions, we recently proposed new piezoelectric materials [16,17], i.e., BaTi 1-x Ni x O 3 and Ba(Ti 1-3z Nb 3z )(O 1-z N z ) 3 .
It has been known that BaNiO 3 shows the 2H hexagonal structure as the most stable structure in room temperature. Moreover, the ionic radius of Ni 4+ (d 6 ) with the low-spin state in 2H BaNiO 3 is 0.48 Å, which is much smaller than that of Ti 4+ (d 0 ), 0.605 Å, in BaTiO 3 . Therefore, due to the drastic change in the (Ti 1-x Ni x )-O Coulomb repulsions in tetragonal BaTi 1- x Ni x O 3 , the e 33 piezoelectric values are expected to be larger than that in tetragonal BaTiO 3 , especially around the morphotropic phase boundary (MPB). Figure 16(a) shows the total-en-ergy difference ΔE total between 2H and tetragonal structures of BaTi 1-x Ni x O 3 as a function of x. The most stable structure changes at x ≈ 0.26. Figure 16(b) shows c -c cub as a function of x.
The c -c cub value shows 0 around x = 0.26, which suggests the appearance of the MPB, i.e., the e 33 piezoelectric value shows a maximum at x ≈ 0.26.
(a) b Figure 16. a) ΔE total (total-energy difference between 2H and tetragonal structures), and (b) cc cub of the tetragonal structure, as a function of x in BaTi 1-x Ni x O 3 [16]. For or 0.26 ≤ x ≦ 1, the tetragonal structure is not the most stable one. Another proposal is tetragonal Ba(Ti 1-3z Nb 3z )(O 1-z N z ) 3 , which consists of BaTiO 3 and BaN-bO 2 N [17]. Due to the change of (Ti 1-3z Nb 3z )-(O 1-z N z ) Coulomb repulsions, the e 33 piezoelectric values are expected to be larger than that in tetragonal BaTiO 3 . Recent experimental paper reported that the most stable structure of BaNbO 2 N 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 c -c cub as a function of x. The c -c cub value shows almost 0 at x ≈ 0.12. Although the MPB does not appear in tetragonal Ba(Ti 1-3z Nb 3z )(O 1-z N z ) 3 , the e 33 piezoelectric values are expected to show a maximum at x ≈ 0.12.

Piezoelectric properties of in tetragonal ABX 3
In the following, we discuss the role of the B-X Coulomb repulsions in piezoelectric ABX 3 .   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 c -ccub. 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 20(a) shows the difference between R B-Oz and r B + r Oz along the [001] axis, and Fig. 20(b) shows the difference between R A-Ox and r A + r Ox on the (100) plane for several ABO 3 , as a function of c -c cub . Clearly, the difference between R B-Oz and r B + r Oz is closely related to e 33 shown in Fig. 18 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 c -ccub. 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.    Fig. 15(a).
Let us discuss the above reasons in the following. ∂(⊿E total )/∂u 3 (Ti 1-x Ni x ) can be written as 1 total total 3 1 As shown in Fig. 21(b), ∂(ΔE total )/ ∂η 3 is almost constant, and therefore, ∂(ΔE total )/∂u 3 (Ti 1-x Ni x ) is almost proportional to (∂u 3 (Ti 1-x Ni x )/∂η 3 ) -1 , i.e., On the other hand, according to Eq. (3), e 33 becomes larger as ∂u 3 (Ti 1-x Ni x )/∂η 3 becomes larger. This is a reason why e 33 becomes larger as ∂(ΔE total )/∂u 3 (Ti 1-x Ni x ) becomes smaller. This result is consistent with the result of SrTiO 3 discussed in Sec. 3.2.1.   Note that the trend of d 33 is closely dependent on the kind of A ions. This result is in contrast with the trend of e 33 as shown in Fig. 18. As expressed in Eq. (4), d 33 is closely related to the elastic compliance s 3j E as well as e 3 j . In fact, the absolute value of s 3j E in BiBX 3 or PbBX 3 is generally larger than that in ABX 3 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.
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.

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.  (a) (b) Figure 22. d 33 as a function of c -c cub for different scales [15].
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].

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.

Conclusion
We have discussed a general role of the B-X Coulomb repulsions for ferroelectric and piezoelectric properties of tetragonal ABX 3 , based on our recent papers and patents. We have found that both ferroelectric state and piezoelectric state are closely related to the B-X z Coulomb repulsions as well as the B-X x ones, as illustrated in Fig. 23(a). Moreover, as illustrated in Fig. 23(b), we have also found that e 33 is closely related to the deviation at the minimum point of the ΔE total .