**3. Introduction of target material**

In this chapter, we selected LiNbO3 as a target material to show the predictability of DFPT calculation. LiNbO3 is one of ferroelectric materials and widely used as surface acoustic wave (SAW) and optical waveguide elements. Crystal structure of LiNbO3 , which belongs to the space group of *R3c*, is frequently referred as "strained perovskite structure." Schematic illustrations of crystal structure of *AB*O3 perovskite and LiNbO3 are shown in **Figure 2**.

Crystal structures shown in the present chapter was visualized by using VESTA software [35]. Curie temperature of LiNbO3 is quite high and ranges from 1140 [36] to 1210°C dependent on the quality of sample (variation of Li/Nb relation can shift Curie temperature [37]). Below Curie temperature, ferroelectric phase with *R3c* symmetry (crystal structure can be classified into 230 types of space group according to the symmetry group) shown in **Figure 2a** is stable. Paraelectric phase with *R*¯ 3*c* symmetry, shown in **Figure 2b**, becomes stable above Curie temperature. In the paraelectric phase, it can be seen that both Li and oxygen is positioned with

**Figure 2.** Crystal structures of (a) ferroelectric phase with *R3c* symmetry and (b) paraelectric phase with *<sup>R</sup>*¯ 3*c* symmetry LiNbO3 . Yellow green-, green-, and red-colored balls represent Li, Nb, and oxygen atoms, respectively. Bonding structures between Nb and surrounding oxygen atoms are represented as green-colored polyhedron. Two orthogonal crystallographic directions are shown as both *a*- and *c*-axes.

the same height along *c*-axis, and the position of Nb is just the center between two oxygen layers along *c*-axis. On the other hand, both Li and Nb are shifted in ferroelectric *R3c* phase along downward direction of *c*-axis with respect to those in paraelectric *R*¯ 3*c* phase.

Those formulations are implemented in specific first-principles simulation packages such as ABINIT [33] and Vienna ab initio simulation package (VASP) [34], and piezoelectric constants can be calculated on a daily basis. From the next section, we will show how DFPT calculation

space group of *R3c*, is frequently referred as "strained perovskite structure." Schematic illus-

perovskite and LiNbO3

Crystal structures shown in the present chapter was visualized by using VESTA software [35].

on the quality of sample (variation of Li/Nb relation can shift Curie temperature [37]). Below Curie temperature, ferroelectric phase with *R3c* symmetry (crystal structure can be classified into 230 types of space group according to the symmetry group) shown in **Figure 2a** is stable.

perature. In the paraelectric phase, it can be seen that both Li and oxygen is positioned with

**Figure 2.** Crystal structures of (a) ferroelectric phase with *R3c* symmetry and (b) paraelectric phase with *<sup>R</sup>*¯

crystallographic directions are shown as both *a*- and *c*-axes.

. Yellow green-, green-, and red-colored balls represent Li, Nb, and oxygen atoms, respectively. Bonding structures between Nb and surrounding oxygen atoms are represented as green-colored polyhedron. Two orthogonal

as a target material to show the predictability of DFPT

is quite high and ranges from 1140 [36] to 1210°C dependent

3*c* symmetry, shown in **Figure 2b**, becomes stable above Curie tem-

are shown in **Figure 2**.

, which belongs to the

3*c* symmetry

is one of ferroelectric materials and widely used as surface acoustic wave

precisely gives piezoelectric properties of ferroelectric materials.

(SAW) and optical waveguide elements. Crystal structure of LiNbO3

**3. Introduction of target material**

6 Perturbation Methods with Applications in Science and Engineering

In this chapter, we selected LiNbO3

trations of crystal structure of *AB*O3

Curie temperature of LiNbO3

Paraelectric phase with *R*¯

LiNbO3

calculation. LiNbO3

Due to the different bonding nature between Li-O and Nb-O, atomic positions of Li and Nb are off-centered within oxygen layers along *c*-axis. This structural characteristic is the ferroelectric nature of LiNbO3 . One of the notable properties of LiNbO3 is its high-curie temperature (~1400 K). However, piezoelectric properties of LiNbO3 are not so much superior as compared with Pb-based perovskites. Crystal structure of piezoelectric *AB*O3 perovskite is based on the cubic structure (of *Pm3m* symmetry), shown in **Figure 3a**.

Cubic lattice is symmetric and usually high-temperature phase, same as LiNbO<sup>3</sup> . The "strained perovskite structure" expression for LiNbO3 means that LiO6 and NbO6 polyhedron are largely rotated with respect to the cubic perovskite structure. However, because of the simple atomic configuration of cubic structure, atoms can be displaced along various directions and change crystalline symmetry as shown in **Figure 3a**. Crystalline lattice is vibrated (referred as phonon) under finite temperature. Some lattice vibrations along specific directions are unstable. This specific phonon is called as soft mode with imaginary frequency. In such case, atoms are displaced along unstable phonon mode to lower the total energy. For example, cooperative atomic displacement along [001] direction shown in **Figure 3b** (referred as *Γ*15 mode) changes symmetry from cubic to tetragonal (of *P4mm* symmetry), which leads polarization along [001] direction. Thus, polarization direction of perovskite is not restricted and allowed to be changed. This characteristic rotational polarization direction is favorable for piezoelectricity because grains in polycrystalline material are oriented along various directions. Thus, careful controlling of crystal structure is essential to obtain superior piezoelectric properties.

The most convenient way to control and drastically change the crystal structure is imposing high pressure. Many compounds have found to be possible to form LiNbO<sup>3</sup> -type structure under the high-pressure synthesis [38], and some of them were quenchable phase. For example, LiNbO3 -

**Figure 3.** (a) Crystal structure of cubic *AB*O3 perovskite and possible polarization directions. (b) Representative unstable vibrational mode of cubic *AB*O3 perovskite showing as arrows.

type structured ZnSbO3 was successfully synthesized [39] under high pressure, and improvement of the spontaneous polarization is suggested by enhancement of the covalency of Sn site from first-principles simulation [40]. Moreover, high-pressure synthesized research on LiNbO<sup>3</sup> type structure is now extended to more complex compounds such as oxynitrides [41, 42].

calculated by the direct method, in which the strain-displacement relation of each ion was

On the basis of cubic *Pm3m* phase, lattice instability analysis was performed by phonon calculation utilizing phonopy code [47]. Force constant matrix shown in Eq. (2) was constructed by DFPT calculation implemented in VASP code combined with supercell approach. Supercell was constructed by using unit cell so that orthogonal three axes of the supercell exceed 10 Å. Note that although supercell is not required in DFPT approach, the present VASP code imple-

Some experimentally measured values are also shown in **Table 1**. All properties are confirmed to be well reproduced by calculation. In a technological importance, 33 components are the

*C*33, and *ε*33 are especially well reproduced. It should be mentioned here that chemical com-

used for experiment is congruent and includes Li vacancy. On the other

**Calculated value Experimental value**

in ferroelectric phase are summarized in **Table 1**.

Density Functional Perturbation Theory to Predict Piezoelectric Properties

http://dx.doi.org/10.5772/intechopen.76827

9

is polarization direction. Calculated values of *e*33,

.

explicitly calculated.

position of LiNbO3

Elastic constant (GPa)

Dielectric constant

values.

Piezoelectric stress constant (C/m2

ments perturbation at the zone center.

Calculated piezoelectric properties of LiNbO3

most important because *C*-axis of LiNbO<sup>3</sup>

**5. Calculated piezoelectric properties of LiNbO3**

hand, calculation was performed by using stoichiometric LiNbO3

*e*<sup>15</sup> 3.73 3.655 ± 0.022 [48], 3.7 [49] *e*<sup>22</sup> 2.51 2.407 ± 0.015 [48], 2.5 [49] *e*<sup>31</sup> 0.21 0.328 ± 0.032 [48], 0.2 [49] *e*<sup>33</sup> 1.69 1.894 ± 0.054 [48], 1.3 [49]

*C*<sup>11</sup> 190.7 198.86 ± 0.033 [48], 203 [49] *C*<sup>12</sup> 58.3 54.67 ± 0.04 [48], 53 [49] *C*<sup>13</sup> 62.4 67.99 ± 0.55 [48], 75 [49] *C*<sup>14</sup> 13.5 7.83 ± 0.02 [48], 9 [49] *C*<sup>33</sup> 220.0 234.18 ± 0.75 [48], 245 [49] *C*<sup>44</sup> 49.2 59.85 ± 0.01 [48], 60 [49]

*ε*<sup>11</sup> 40.6 44.9 ± 0.4 [48], 44 [49] *ε*<sup>33</sup> 24.1 26.7 ± 0.3 [48], 29 [49]

**Table 1.** Piezoelectric constant, elastic constant, and dielectric constant calculated by DFPT and experimentally measured

)

The crystal structure of *AB*O3 compound is generally determined by the balance between the ionic radius of *A* and *B* element, which is frequently referred as tolerance factor. Due to the small size of the Li ion with respect to the tolerance factor of LiNbO3 , this compound cannot form stably the popular perovskite structure under the ambient condition. On the other hand, we predicted the crystal structures of high-pressure phase of LiNbO<sup>3</sup> [43], which were not completely elucidated by experimental study [44]. Revealed structures are NaIO3 -type structure (*Pnma*) as room temperature high-pressure phase and apatite-like structure (*P*63 /*m*) as hightemperature high-pressure phase. It should be noted that the NaIO<sup>3</sup> -type structure is closely related with the popular GdFeO3 -type perovskite structure. The only difference between these structures is that A-site position and B-site position are inter-exchanged. Therefore, there seems to be a possible way to connect the perovskite structure and LiNbO3 -type structure.

In our previous theoretical study on high-pressure phase, analysis was mainly concerned with phase transition mechanism only from the viewpoint of subgroup symmetry and energy barrier [43]. It will be instructive to deal with this phase transition phenomenon from the viewpoint of lattice instability as discussed in the field of the ferroelectric instability analysis. In the following section, we will show investigation on the potential piezoelectric properties of LiNbO3 with various hypothetical crystal structures by the method of DFPT, and possible phase transition mechanism will be discussed from the viewpoint of soft mode.
