**5. Calculated piezoelectric properties of LiNbO3**

type structured ZnSbO3

The crystal structure of *AB*O3

8 Perturbation Methods with Applications in Science and Engineering

related with the popular GdFeO3

**4. Computational methodology**

of LiNbO3

was successfully synthesized [39] under high pressure, and improve-

compound is generally determined by the balance between the



, this compound cannot


/*m*) as high-

[43], which were not com-



ment 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>

ionic radius of *A* and *B* element, which is frequently referred as tolerance factor. Due to the

form stably the popular perovskite structure under the ambient condition. On the other hand,

structures is that A-site position and B-site position are inter-exchanged. Therefore, there

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

First-principles calculation was performed by using VASP code [34]. Interactions between ion and electron were treated by projector augmented wave (PAW) method [45]. PBEsol functional [46] was used to approximate exchanges and correlate interactions of electrons, which can be used to reproduce the lattice constants of various materials [45]. Precise calculation on the lattice constant is essential to predict piezoelectric properties because they depend on volume of unit cell Ω as shown in Eq. (1). The kinetic energy cutoff for plane waves was set at 500 eV, and the k-point mesh was set at ~0.03/Å intervals to obtain the converged total energy at less than 0.1 meV/atom. Before calculating the piezoelectric constants, atomic positions and cell parameters were optimized until the forces on each atom and cell converged at below 5 × 10−4 eV/Å. Since VASP does not directly calculate Eq. (12), we added routine to calculate displacementresponse internal-strain tensor *Γnj* and decompose piezoelectric constants into each atom. The sum of the decomposed piezoelectric constants was confirmed to accurately reproduce the total piezoelectric constants. Careful convergence tests with a higher energy cutoff and denser *k*-point mesh showed that the numerical accuracy of the calculated *Γnj* was less than 0.01. It was confirmed that this error does not influence our discussion and conclusion. Moreover, it was confirmed that the values of *Γnj* obtained by the DFPT method were consistent with those

with various hypothetical crystal structures by the method of DFPT, and possible

type structure is now extended to more complex compounds such as oxynitrides [41, 42].

small size of the Li ion with respect to the tolerance factor of LiNbO3

we predicted the crystal structures of high-pressure phase of LiNbO<sup>3</sup>

temperature high-pressure phase. It should be noted that the NaIO<sup>3</sup>

seems to be a possible way to connect the perovskite structure and LiNbO3

phase transition mechanism will be discussed from the viewpoint of soft mode.

pletely elucidated by experimental study [44]. Revealed structures are NaIO3

(*Pnma*) as room temperature high-pressure phase and apatite-like structure (*P*63

Calculated piezoelectric properties of LiNbO3 in ferroelectric phase are summarized in **Table 1**. 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 most important because *C*-axis of LiNbO<sup>3</sup> is polarization direction. Calculated values of *e*33, *C*33, and *ε*33 are especially well reproduced. It should be mentioned here that chemical composition of LiNbO3 used for experiment is congruent and includes Li vacancy. On the other hand, calculation was performed by using stoichiometric LiNbO3 .


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


**Table 2.** Decomposed piezoelectric constants of LiNbO3 .

Thus, Li vacancy is considered to have negligible influence on the piezoelectric properties. Decomposed ionic contribution of piezoelectric strain constant *e*33 is summarized in **Table 2**. Although the Born effective charge of Nb is larger than its formal charge +5e, displacementresponse internal-strain constant of Nb is negative value. This indicates that piezoelectricity of LiNbO3 is mainly dominated by displacement of Li. Born effective charge indicates a degree of polarization induced by atomic displacement and dominated by the change in the orbital hybridization. Although anomalously large Born effective charge is crucial for superior piezoelectric properties of perovskite *AB*O3 materials [50], the present study of decomposition of piezoelectric constant shows that coupling degree between external strain and atomic displacement is also indispensable to understand the piezoelectric properties.

Imposing negative pressure can be achieved by solid solution with parent phase of larger lattice constant. At −6 GPa, *P4mm* phase becomes stable, while *R3m* and *Amm2* phases become stable at −9 GPa. However, bond breaking occurs in Nb-O bonding above −6 GPa for *P4mm* phase. The same bond breaking occurs in *Amm2* and *R3m* phases at −11GPa and −14 GPa,

Within the eight phases shown in **Figure 4a**, only *P4mm*, *R3m*, *R3c*, and *Amm2* phases show piezoelectricity. Piezoelectric stress constant, elastic constant, and dielectric constant of *P4mm*, *R3m*, and *Amm2* phases are compared with those of *R3c* phase in **Table 3**. Various piezoelectric properties are observed by each phase. Especially for *P4mm* and *Amm2* phases, high *e*<sup>33</sup> and relatively low *C*33 values are predicted, which are advantageous for large piezoelectric strain constant *d*33. On the other hand, *R3m* phase was found to be unstable because following

*C*<sup>11</sup> + *C*<sup>12</sup> > 0, *C*<sup>33</sup> > 0, (*C*<sup>11</sup> + *C*12)<sup>∗</sup> *C*<sup>33</sup> > 2 *C*13, *C*<sup>11</sup> − *C*<sup>12</sup> > 0, *C*<sup>44</sup> > 0, (*C*<sup>11</sup> − *C*12)<sup>∗</sup> *C*<sup>44</sup> > 2 *C*14 (14)

**Figure 5a** and **b** show piezoelectric properties of *P4mm* phase as a function of pressure and corresponding volume of unit cell. Dotted lines indicate zero pressure states. Piezoelectric stress constant *e*33 of *P4mm* phase shows parabolic behavior and maximum value at zero pressure state. On the other hand, elastic constant *C*33 of *P4mm* phase continuously decreases as volume increases, because orbital hybridization of Nb-O bonding along polarization direction decreases as bond length increases. At the pressure of −6 GPa, *C*33 of *P4mm* phase shows almost zero value. This indicates that Nb-O bonding is broken. Piezoelectric stress constant *d*33 shown in **Figure 5b** increases as volume, because of increase of elastic compliance. Especially at the pressure of −5 GPa just before bond breaking, *d*33 shows maximum value or approximately 1000 pC/N.

This giant piezoelectric constant is almost comparable to that of PZT material [51]. Giant piezoelectric constant is understood as a result of phase instability in morphotropic phase

, we revealed that ZnO also showed anom-

and their space groups. (b) Enthalpy

Density Functional Perturbation Theory to Predict Piezoelectric Properties

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

11

respectively. Thus, those phase transitions occur just before bond breaking.

**Figure 4.** (a) Schematic illustration of eight types of perovskite-structured LiNbO<sup>3</sup>

differences of each phase measured from the enthalpy of *R3c* phase as a function of pressure.

mechanical stability conditions of rhombohedral symmetry:

boundary [52]. The same as *P4mm* phase of LiNbO3

alously large piezoelectric constant just before phase transition [30].

are broken because of *C*<sup>44</sup> < 0.
