**6. Piezoelectric properties of perovskite-LiNbO3**

Next, we will show how piezoelectric properties are affected by crystal structure, while chemical composition is kept as LiNbO3 . Various hypothetical crystal structures common for perovskite-type structure were constructed, and their energetic stabilities were examined by calculating enthalpy *H* = *U* + *PV* (*U* is total energy obtained by first-principles calculation, *P* is external pressure, and *V* is equilibrium volume under pressure *P*) as a function of external pressure. Imposing high pressure is most convenient method to modify crystal structure and find unexpected stable phase. The following eight types of phases were considered:

Cubic, *Pm-3 m*; tetragonal, *P4mm*; and rhombohedral, *R-3 m*.

LiNbO3 -ferroelectric phase, *R3c*, and LiNbO3 -paraelectric phase, *R-3c*.

Orthorhombic, *Amm2* and *Cmmm*, and high-pressure phase, *P63/m.*

where names of space groups are used to distinguish each structure. Crystal structure of each phase is shown in **Figure 4a**. Polyhedra shown in **Figure 4a** correspond to Nb-centered bonding structure of Nb-O bondings. **Figure 4b** shows the enthalpy difference of each phases as a function of external pressure. Here, external pressure is assumed to be isotropic. Standard of enthalpy was set to be the enthalpy of most stable *R3c* phase under ambient condition. At the positive (compressive) pressure region, *P63 /m* phase becomes stable above 21 GPa, which is close to the experimental phase transition pressure of 25 GPa [43]. Details of the phase transition behavior under high pressure are theoretically investigated in our previous work [44]. Unfortunately, *P63 /m* phase is highly symmetric and shows no piezoelectricity. At the negative (expansive) pressure region, enthalpy difference becomes smaller as there is an increase of negative pressure except for *R*¯ <sup>3</sup>*c* and *P63 /m* phases.

**Figure 4.** (a) Schematic illustration of eight types of perovskite-structured LiNbO<sup>3</sup> and their space groups. (b) Enthalpy differences of each phase measured from the enthalpy of *R3c* phase as a function of pressure.

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

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 mechanical stability conditions of rhombohedral symmetry:

$$\text{C}\_{11} + \text{C}\_{12} \ge 0, \text{C}\_{33} \ge 0, \text{(C}\_{11} + \text{C}\_{12})^\* \text{C}\_{33} \ge 2 \text{C}\_{13'} \text{C}\_{11} - \text{C}\_{12} \ge 0, \text{C}\_{44} \ge 0, \text{(C}\_{11} - \text{C}\_{12})^\* \text{C}\_{44} \ge 2 \text{C}\_{14} \tag{14}$$

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

**Decomposed** *e***33' (C/m2**

ity of LiNbO3

LiNbO3

Unfortunately, *P63*

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

10 Perturbation Methods with Applications in Science and Engineering

rior piezoelectric properties of perovskite *AB*O3

chemical composition is kept as LiNbO3

**6. Piezoelectric properties of perovskite-LiNbO3**

Cubic, *Pm-3 m*; tetragonal, *P4mm*; and rhombohedral, *R-3 m*.

Orthorhombic, *Amm2* and *Cmmm*, and high-pressure phase, *P63/m.*


positive (compressive) pressure region, *P63*

of negative pressure except for *R*¯

**) Born effective charge** *Z***33 (e) Displacement-response internal-strain constant** *Γ***<sup>33</sup>**

is mainly dominated by displacement of Li. Born effective charge indicates a

materials [50], the present study of decom-

. Various hypothetical crystal structures common for

*/m* phase becomes stable above 21 GPa, which is


*/m* phase is highly symmetric and shows no piezoelectricity. At the nega-

Li Nb O Li Nb O Li Nb O 0.1 0.05 0.16 1.03 6.77 −2.6 0.67 −0.05 −0.21

.

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 piezoelectric-

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 supe-

position of piezoelectric constant shows that coupling degree between external strain and atomic displacement is also indispensable to understand the piezoelectric properties.

Next, we will show how piezoelectric properties are affected by crystal structure, while

perovskite-type structure were constructed, and their energetic stabilities were examined by calculating enthalpy *H* = *U* + *PV* (*U* is total energy obtained by first-principles calculation, *P* is external pressure, and *V* is equilibrium volume under pressure *P*) as a function of external pressure. Imposing high pressure is most convenient method to modify crystal structure and

where names of space groups are used to distinguish each structure. Crystal structure of each phase is shown in **Figure 4a**. Polyhedra shown in **Figure 4a** correspond to Nb-centered bonding structure of Nb-O bondings. **Figure 4b** shows the enthalpy difference of each phases as a function of external pressure. Here, external pressure is assumed to be isotropic. Standard of enthalpy was set to be the enthalpy of most stable *R3c* phase under ambient condition. At the

close to the experimental phase transition pressure of 25 GPa [43]. Details of the phase transition behavior under high pressure are theoretically investigated in our previous work [44].

tive (expansive) pressure region, enthalpy difference becomes smaller as there is an increase

*/m* phases.

<sup>3</sup>*c* and *P63*

find unexpected stable phase. The following eight types of phases were considered:

**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 boundary [52]. The same as *P4mm* phase of LiNbO3 , we revealed that ZnO also showed anomalously large piezoelectric constant just before phase transition [30].


Li displacement. Li displacement along <001>, <011>, and <111> directions induces tetragonal, orthorhombic, and rhombohedral phase transition from cubic phase. **Figure 6a** clearly shows that tetragonal phase transition from *Pm3m* phase to *P4mm* phase is the most energetically advantageous. **Figure 6b** shows the phonon dispersion curve of cubic *Pm3m* phase of LiNbO3

Density Functional Perturbation Theory to Predict Piezoelectric Properties

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

Horizontal axis corresponds to sampling path along high symmetric reciprocal point (*q*-point). Within the whole Brillouin zone of reciprocal space, unstable phonon modes with imaginary phonon frequencies are observed. Here, imaginary phonon frequency is represented as nega-

unstable and considered to show phase transition in accordance with specific phonon mode of imaginary frequency (referred as soft mode). Thus, modulated structures were constructed by imposing atomic displacement along normal modes at each symmetric *q*-points. Modulated structures were structurally relaxed, and their space group and energy change from cubic *P4mm* phase were investigated. Summary of such modulated structures are shown in **Table 4**. At Γ point, tetragonal phase transition along with *Γ*15 soft mode of cubic phase shown in **Figure 3b** shows energy gain of −0.422 eV/formula unit (f.u.). On the other hand, it was found that modulation at R point gives more stable energy gain of −0.682 eV/f.u. In this case, *R*25 soft

**Figure 6.** (a) Energy change of *Pm3m* phase as a function of li displacement along <001>, <011>, and <111> directions. (b)

"Structure" indicates Bravais lattice of modulated structure from *P4mm* phase. Space group of the relaxed modulated

**Table 4.** Summary of imaginary phonon frequency at each symmetric *q*-point in *P4mm* phase of LiNbO3

*q***-point Frequency (THz) Structure Space group Energy gain (eV/f.u.)**

Γ −7.22 Tetragonal *P4mm* −0.422 X −5.38 Orthorhombic *Pmma* −0.220 M −6.56 Orthorhombic *Pmma* −0.125 R −5.51 Rhombohedral *R-3c* −0.682

tive value for convenience. Therefore, cubic *Pm3m* phase of LiNbO3

mode induces phase transition from cubic *Pm3m* phase to *R*¯

Phonon dispersion curve of *Pm3m* phase.

structure and energy gain is also shown.

corresponding structural phase transition.

.

13

and

is thermodynamically

3*c* phase shown in **Figure 2b**.

**Table 3.** Piezoelectric constant, elastic constant, and dielectric constant of *R3c*, *P4mm*, *R3m*, and *Amm2* phases calculated by DFPT.

**Figure 5.** (a) Piezoelectric stress constant *e*33 and elastic constant *C*33 and (b) piezoelectric strain constant *d*33 of *P4mm* phase as a function of pressure and corresponding volume of unit cell.

Finally, we would like to show phase transition path between cubic perovskite structure and LiNbO3 structure. **Figure 6a** shows the energy change of *Pm3m* phase as a function of Li displacement along <001>, <011>, and <111> directions. *Pm3m* phase is paraelectric phase. Because ferroelectricity and piezoelectricity of LiNbO3 are dominated by off-centering and displacement of Li, respectively, phase transition from *Pm3m* phase is also expected to be occurred by Li displacement. Li displacement along <001>, <011>, and <111> directions induces tetragonal, orthorhombic, and rhombohedral phase transition from cubic phase. **Figure 6a** clearly shows that tetragonal phase transition from *Pm3m* phase to *P4mm* phase is the most energetically advantageous. **Figure 6b** shows the phonon dispersion curve of cubic *Pm3m* phase of LiNbO3 . Horizontal axis corresponds to sampling path along high symmetric reciprocal point (*q*-point). Within the whole Brillouin zone of reciprocal space, unstable phonon modes with imaginary phonon frequencies are observed. Here, imaginary phonon frequency is represented as negative value for convenience. Therefore, cubic *Pm3m* phase of LiNbO3 is thermodynamically unstable and considered to show phase transition in accordance with specific phonon mode of imaginary frequency (referred as soft mode). Thus, modulated structures were constructed by imposing atomic displacement along normal modes at each symmetric *q*-points. Modulated structures were structurally relaxed, and their space group and energy change from cubic *P4mm* phase were investigated. Summary of such modulated structures are shown in **Table 4**. At Γ point, tetragonal phase transition along with *Γ*15 soft mode of cubic phase shown in **Figure 3b** shows energy gain of −0.422 eV/formula unit (f.u.). On the other hand, it was found that modulation at R point gives more stable energy gain of −0.682 eV/f.u. In this case, *R*25 soft mode induces phase transition from cubic *Pm3m* phase to *R*¯ 3*c* phase shown in **Figure 2b**.

**Figure 6.** (a) Energy change of *Pm3m* phase as a function of li displacement along <001>, <011>, and <111> directions. (b) Phonon dispersion curve of *Pm3m* phase.


**Figure 5.** (a) Piezoelectric stress constant *e*33 and elastic constant *C*33 and (b) piezoelectric strain constant *d*33 of *P4mm*

Finally, we would like to show phase transition path between cubic perovskite structure and

ment of Li, respectively, phase transition from *Pm3m* phase is also expected to be occurred by

 structure. **Figure 6a** shows the energy change of *Pm3m* phase as a function of Li displacement along <001>, <011>, and <111> directions. *Pm3m* phase is paraelectric phase. Because

are dominated by off-centering and displace-

*R3c P4mm R3m Amm2*

phase as a function of pressure and corresponding volume of unit cell.

ferroelectricity and piezoelectricity of LiNbO3

LiNbO3

Piezoelectric stress constant (C/m2

Elastic constant (GPa)

Dielectric constant

by DFPT.

)

12 Perturbation Methods with Applications in Science and Engineering

*e*<sup>15</sup> 3. 73 1.14 5.10 0.64 *e*<sup>22</sup> 2.51 — 1.19 *e*<sup>31</sup> 0.21 0.46 0.24 0.80 *e*<sup>33</sup> 1.69 3.28 1.92 2.99

*C*<sup>11</sup> 190.7 297.2 203.0 321.8 *C*<sup>12</sup> 58.3 48.9 169.0 91.6 *C*<sup>13</sup> 62.4 77.7 90.6 92.8 *C*<sup>14</sup> 13.5 — −42.1 — *C*<sup>33</sup> 220.0 157.7 206.6 176.4 *C*<sup>44</sup> 49.2 39.5 −29.6 32.5

*ε*<sup>11</sup> 40.6 56.2 36.6 28.2 *ε*<sup>33</sup> 24.1 12.3 16.2 13.3

**Table 3.** Piezoelectric constant, elastic constant, and dielectric constant of *R3c*, *P4mm*, *R3m*, and *Amm2* phases calculated

"Structure" indicates Bravais lattice of modulated structure from *P4mm* phase. Space group of the relaxed modulated structure and energy gain is also shown.

**Table 4.** Summary of imaginary phonon frequency at each symmetric *q*-point in *P4mm* phase of LiNbO3 and corresponding structural phase transition.

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**Figure 7.** Phase transition path between cubic perovskite structure (*Pm3m*) and LiNbO3 -structure. Unit cell of LiNbO<sup>3</sup> is enclosed with blue lines.

**Figure 7** shows schematic illustration of phase transition mechanism from cubic perovskite structure to LiNbO3 structure. On the contrary to the result of **Figure 6a**, *R*25 soft mode is represented as rotation of NbO6 polyhedra. Then, *Γ*15 soft mode of *R*¯ 3*c* phase leads *R3c* phase, which is ground state of LiNbO3 . Although the present study shows that perovskite-structured LiNbO3 is thermodynamically unstable while its piezoelectricity is excellent, it can be possible to control phase transition behavior by dopant substitution.

## **7. Summary and conclusion**

In this chapter, we briefly introduced sophisticated method of density functional perturbation theory. DFPT can effectively calculate the second derivative of the total energy with respect to the atomic displacement within the framework of first-principles calculation. By using DFPT method, we can predict piezoelectric constants, dielectric constants, elastic constants, and phonon dispersion relationship of any given crystal structure. Moreover, we showed our established computational technique to decompose piezoelectric constants into each atomic contribution, which enable us to gain deeper insights to understand the piezoelectricity of material. By using LiNbO3 as a model material, we showed the predictability of DFPT for piezoelectric properties. In addition, we showed that superior piezoelectric properties are hidden in perovskite-structured LiNbO<sup>3</sup> . Structural relationship and possible phase transition path between LiNbO3 structure and perovskite structure were discussed and concluded that perovskite-structured LiNbO<sup>3</sup> is thermodynamically unstable. Further studies are expected to control relative phase stability between perovskite and LiNbO3 structure by dopant substation and solid solution.
