**2. Formulation of piezoelectric constants**

surface acoustic wave devices, and actuators. Only the material having no inversion symmetry shows piezoelectricity. For example, **Figure 1** shows schematic illustration of the piezoelectric effect. Positions of positively charged ion (cation) and negatively charged ion (anion) are represented as plus and minus symbols. **Figure 1a** shows the paraelectric phase, where ions are orderly located with inversion symmetry. On the other hand, ions are slightly displaced by δ with respect to those in paraelectric phase, as shown in **Figure 1b**. Such small displacement

Because ferroelectric phase is energetically more stable than paraelectric phase under low

ture, ferroelectric properties are disappeared since paraelectric phase becomes more stable than ferroelectric one. **Figure 1c** shows the schematic illustration of the principal of piezoelectricity, where external stress (red-colored arrows) increases the ionic displacement and resultant polarization. In this case, external stress increases the spontaneous polarization by

polarization with respect to the external field. More detailed and comprehensive description

First-principles calculation based on density functional theory (DFT [2, 3]) has been widely utilized as the computational method to predict the electronic properties of material under the ground state. Ideally, required information to conduct the first-principles calculation is only the crystal structure, including atomic species and position of periodic/nonperiodic structure unit. The most significant advantage of first-principles calculation is its predictability. Since King-Smith and Vanderbilt showed the theoretical methodology to calculate change in polarization per unit volume Δ*P* [4], dielectric and piezoelectric properties of wide range of materials in which electronic correlations are not too strong [5–7] have been accurately predicted. The derivative of total energy determines various properties. For example, determined forces, stresses, dipole moment (first-order derivatives), dynamical matrix, elastic constants, dielectric and piezoelectric constants (second-order derivative), nonlinear dielectric susceptibility, phonon–phonon interaction and Grüneisen parameters (third-order derivative), and

**Figure 1.** Ionic configuration of (a) paraelectric phase and (b) ferroelectric phase. (c) Ionic displacement according to the

along the ionic displaced direction.

is frequently referred as the spontaneous polarization. Above Curie tempera-

. Therefore, piezoelectric constant is defined as the derivative of the spontaneous

induces microscopic polarization *P*<sup>s</sup>

2 Perturbation Methods with Applications in Science and Engineering

of piezoelectricity is reviewed by Martin [1].

temperature, *P*<sup>s</sup>

Δ*P*<sup>s</sup> = *P*<sup>s</sup> ' −*P*<sup>s</sup>

external force.

Formulation and calculation methodologies to obtain response properties of materials in the framework of DFPT have been developed in a step-by-step manner, because degrees of freedom by perturbations of atomic displacement, homogeneous electric fields, and strain are often strongly coupled. For example, piezoelectricity affects elastic and dielectric properties. Therefore, special care must be paid for the calculation of coupled properties. In 2005, Hamann et al. demonstrated that elastic and piezoelectric tensors can be efficiently calculated by treating homogeneous strain within the framework of DFPT [31]. At the same time, Wu et al. systematically formulated response properties with respect to displacement, strain, and electric fields [32]. In this section, we will briefly introduce how piezoelectric properties are formulated in the framework of DFPT. In each formulation, Einstein implied-sum notation is used. Cartesian directions {*x*, *y*, *z*} are represented as *α* and *β*. Subscription of *j* and *k* = 1, …, 6 is the standard Voigt notation (represents directions of *xx*, *yy, zz, yz, zx*, and *xy*). The subscripts *m* and *n* are the degrees of freedom in the cell. They range from 1 to 3*i*, where *i* is the number of irreducible atoms because each atom has three degree of freedom along *x*, *y*, and *z* directions.

Total energy of material under perturbation of atomic displacement *u*, electric field *σ*, and strain *η,* E(*u*,*σ*,*η*), is defined as follows:

$$E(\mu, \sigma, \eta) = \frac{1}{\Omega\_0} [E^0 - \Omega \sigma \cdot \mathbf{P}] \tag{1}$$

Thus, proper piezoelectric constant can be obtained by Eqs. (7) and (9):

∂ *σα*∂ *η<sup>j</sup>* |*u* + \_\_\_1 Ω0

On the other hand, the internal-strain term of the piezoelectric stress constant *e*α*<sup>j</sup>*

strain. In this expression, the meaning of the piezoelectric stress constant, i.e., *ej*

usually measured. It can be obtained from piezoelectric strain constant *e*α*<sup>j</sup>*

/∂ *η<sup>j</sup>*

*Γnj* = *Λmj* (*K*<sup>−</sup>1)

*Zm* (*K*<sup>−</sup>1)

Here, the first and second terms on the right-hand side in Eq. (10) are the clamped-ion term and internal-strain term, respectively. The former shows the electronic contribution ignoring the atomic relaxation effect, and the latter shows the ionic contribution including the response of the atomic displacement to the strain. The Born effective charge *Zmα*, force-constant matrix *Kmn*, and internal-strain tensor *Λnj* are the second derivatives of the energy with respect to the displacement and electric field, pairs of displacements, and displacement and strain, respectively. The internal-strain term of the piezoelectric stress constants can be further decomposed into the individual atomic contributions when the above second-derivative tensors are

described by the following equation, using the Born effective charge *Z*αβ and displacement *u*<sup>β</sup>

*<sup>j</sup>* = *Z*

of the change in polarization induced by the external strain, is much more visible than in Eq.

Because the subscript *n* in *Γnj* indicates the degrees of freedom, *Γnj* can be decomposed into the individual atomic components, which also enables to calculate individual contribution of

Here, piezoelectric *e* constant defined as Eq. (9) is frequently referred as "piezoelectric strain constant." On the other hand, it is much more natural and easy to control the stress (electric field) than to control the strain in any case. In this case, the piezoelectric strain constant *d*α*<sup>j</sup>*

*d<sup>j</sup>* = *sjk e<sup>k</sup>* (13)

where *sjk* is the elastic compliance, which is given by the inverse matrix of the elastic con-

<sup>∂</sup> *<sup>u</sup>* \_\_\_*<sup>β</sup>* ∂ *η<sup>j</sup>*

shows the response of the first-order atomic displacement to the first-order

is implicitly calculated as a displacement-response internal-

*mn* (12)

*mn Λnj* (10)

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

Density Functional Perturbation Theory to Predict Piezoelectric Properties

is frequently

is a measure

(11)

5

is

using the following

*<sup>e</sup><sup>j</sup>* <sup>=</sup> <sup>∂</sup><sup>2</sup> \_\_\_\_\_\_ *<sup>E</sup>*

of each atom in the calculation cell:

*e*̂

each atom for total piezoelectric constant.

fully obtained.

where <sup>∂</sup> *<sup>u</sup><sup>β</sup>*

relation:

stants *C*jk.

/∂ *η<sup>j</sup>*

(9). In the DFPT formalism, <sup>∂</sup> *<sup>u</sup><sup>β</sup>*

strain tensor *Γ* as follows [32]:

where *E*<sup>0</sup> is the total energy of material under the ground state, Ω<sup>0</sup> is volume of the unit cell (smallest repeat unit of crystal), Ω is deformed volume of the unit cell, and P is the electric polarization. Following response functional tensor can be obtained by second-order differential of Eq. (1):

$$\text{Force constant matrix:} K\_{uu} = \left. \Omega\_0 \frac{\partial^2 E}{\partial u\_n \partial u\_n} \right|\_{\rho\_{\parallel}} \tag{2}$$

$$\text{Camped - ion term of electric susceptibility:} \overline{\chi}\_{a\theta} = -\frac{\partial^2 E}{\partial \sigma\_a \partial \sigma\_\beta} \bigg|\_{\text{a.p.} } \tag{3}$$

$$\text{Camped - ion term of elastic tensor:} \overline{\mathbb{C}}\_{\mathbb{A}} = -\frac{\partial^2 E}{\partial \eta\_{\rangle} \partial \eta\_{\natural}} \bigg|\_{\mu, \rho} \tag{4}$$

$$\text{Born effective charge tensor:} Z\_{ma} = -\Omega\_0 \frac{\partial^2 E}{\partial \left. u\_m \partial \sigma\_a \right|\_\eta} \Big|\_\eta \tag{5}$$

$$\text{Force} - \text{response internal} - \text{strain tensor:} \boldsymbol{\Lambda}\_{\eta \dot{\eta}} = -\boldsymbol{\Omega}\_0 \left. \frac{\partial^2 E}{\partial \boldsymbol{u}\_n \partial \eta\_{\dot{\eta}}} \right|\_{\sigma} \tag{6}$$

$$\text{Changed - ion piezoelectric tensor:} \newline \overline{\boldsymbol{\sigma}}\_{\boldsymbol{\sigma}\boldsymbol{\rangle}} = \left. \frac{\partial^2 \boldsymbol{E}}{\partial \sigma\_a \partial \eta\_{i\_j}} \right|\_{\boldsymbol{\mu}} \tag{7}$$

Clamped-ion term is a frozen quantity, which indicates that atomic coordinates are not allowed to relax as the homogeneous electric field or strain. Therefore, dynamical term should be added into the clamped-ion term in order to obtain proper response properties.

Simplest and physically well-understandable piezoelectric constant can be expressed as follows:

$$e\_{\left|q\right>} = \frac{\partial P\_a}{\partial \eta\_j} \tag{8}$$

In this expression, it is easily understood that piezoelectric *e* constant *e*αj is a measure of the change in polarization induced by the external strain. As the atomic positions are changed according to the strain, change of the polarization includes both electronic contribution (clamped-ion term) and dynamical contribution (internal-strain term). The internal-strain term of piezoelectric constant is represented as follows:

$$\left. \partial\_{\left. \left. \right|\_{0}} \right| = \frac{1}{\Omega\_{0}} Z\_{mn} \left\{ K^{-1} \right\}\_{mn} \Lambda\_{n\rangle} \tag{9}$$

Thus, proper piezoelectric constant can be obtained by Eqs. (7) and (9):

*<sup>E</sup>*(*u*, *<sup>σ</sup>*, *<sup>η</sup>*) <sup>=</sup> \_\_\_1

4 Perturbation Methods with Applications in Science and Engineering

Force constant matrix:*Kmn* = Ω<sup>0</sup>

where *E*<sup>0</sup>

follows:

Ω0

est repeat unit of crystal), Ω is deformed volume of the unit cell, and P is the electric polarization. Following response functional tensor can be obtained by second-order differential of Eq. (1):

Clamped-ion term is a frozen quantity, which indicates that atomic coordinates are not allowed to relax as the homogeneous electric field or strain. Therefore, dynamical term should

Simplest and physically well-understandable piezoelectric constant can be expressed as

∂ *η<sup>j</sup>*

In this expression, it is easily understood that piezoelectric *e* constant *e*αj is a measure of the change in polarization induced by the external strain. As the atomic positions are changed according to the strain, change of the polarization includes both electronic contribution (clamped-ion term) and dynamical contribution (internal-strain term). The internal-strain

*Zm* (*K*<sup>−</sup>1)

be added into the clamped-ion term in order to obtain proper response properties.

*<sup>j</sup>* <sup>=</sup> \_\_\_1 Ω0

is the total energy of material under the ground state, Ω<sup>0</sup>

Clamped <sup>−</sup> ion term of electric susceptibility:χ¯ <sup>=</sup> <sup>−</sup> <sup>∂</sup><sup>2</sup> \_\_\_\_\_\_ *<sup>E</sup>*

Clamped <sup>−</sup> ion term of elastic tensor:*C*¯*jk* <sup>=</sup> <sup>−</sup> <sup>∂</sup><sup>2</sup> \_\_\_\_\_\_ *<sup>E</sup>*

Born effective charge tensor:*Zm* = −Ω<sup>0</sup>

Clamped − ion piezoelectric tensor:*e*

*<sup>e</sup><sup>j</sup>* <sup>=</sup> <sup>∂</sup> *<sup>P</sup>*\_\_\_*<sup>α</sup>*

term of piezoelectric constant is represented as follows:

*e*̂

Force − response internal − strain tensor:*Λmj* = −Ω<sup>0</sup>

[*E*<sup>0</sup> − Ω*σ* ∙ P] (1)

<sup>∂</sup> *σα*<sup>∂</sup> *σβ*|*<sup>u</sup>*,*<sup>η</sup>*

<sup>∂</sup> *<sup>η</sup><sup>j</sup>* <sup>∂</sup> *<sup>η</sup>k*|*<sup>u</sup>*,*<sup>σ</sup>*

∂<sup>2</sup> \_\_\_\_\_\_ *E* ∂ *um* ∂ *η<sup>j</sup>* |*σ*

*mn Λnj* (9)

∂<sup>2</sup> \_\_\_\_\_\_ *E* <sup>∂</sup> *um* <sup>∂</sup> *σα*|*<sup>η</sup>*

¯*<sup>j</sup>* <sup>=</sup> <sup>∂</sup><sup>2</sup> \_\_\_\_\_\_ *<sup>E</sup>* ∂ *σα*∂ *η<sup>j</sup>* |*u*

∂<sup>2</sup> \_\_\_\_\_\_ *E* <sup>∂</sup> *um* <sup>∂</sup> *un*|*<sup>σ</sup>*,*<sup>η</sup>*

is volume of the unit cell (small-

(2)

(3)

(4)

(5)

(6)

(7)

(8)

$$\left. e\_{a\dagger} \right| = \left. \frac{\partial^2 E}{\partial \sigma\_a \partial \eta\_{\dagger}} \right|\_{\mu} + \frac{1}{\Omega\_o} Z\_{ma} \left( \mathbf{K}^{-1} \right)\_{mn} \Lambda\_{\eta} \tag{10}$$

Here, the first and second terms on the right-hand side in Eq. (10) are the clamped-ion term and internal-strain term, respectively. The former shows the electronic contribution ignoring the atomic relaxation effect, and the latter shows the ionic contribution including the response of the atomic displacement to the strain. The Born effective charge *Zmα*, force-constant matrix *Kmn*, and internal-strain tensor *Λnj* are the second derivatives of the energy with respect to the displacement and electric field, pairs of displacements, and displacement and strain, respectively. The internal-strain term of the piezoelectric stress constants can be further decomposed into the individual atomic contributions when the above second-derivative tensors are fully obtained.

On the other hand, the internal-strain term of the piezoelectric stress constant *e*α*<sup>j</sup>* is frequently described by the following equation, using the Born effective charge *Z*αβ and displacement *u*<sup>β</sup> of each atom in the calculation cell:

$$\text{è}\_{\text{eq}} = Z\_{\text{eq}} \frac{\partial \, u\_{\text{\beta}}}{\partial \, \eta\_{\text{\beta}}} \tag{11}$$

where <sup>∂</sup> *<sup>u</sup><sup>β</sup>* /∂ *η<sup>j</sup>* shows the response of the first-order atomic displacement to the first-order strain. In this expression, the meaning of the piezoelectric stress constant, i.e., *ej* is a measure of the change in polarization induced by the external strain, is much more visible than in Eq. (9). In the DFPT formalism, <sup>∂</sup> *<sup>u</sup><sup>β</sup>* /∂ *η<sup>j</sup>* is implicitly calculated as a displacement-response internalstrain tensor *Γ* as follows [32]:

$$
\Gamma\_{\mathfrak{u}} = \Lambda\_{\mathfrak{u}} \left( \mathbf{K}^{-1} \right)\_{\mathfrak{u}\mathfrak{u}} \tag{12}
$$

Because the subscript *n* in *Γnj* indicates the degrees of freedom, *Γnj* can be decomposed into the individual atomic components, which also enables to calculate individual contribution of each atom for total piezoelectric constant.

Here, piezoelectric *e* constant defined as Eq. (9) is frequently referred as "piezoelectric strain constant." On the other hand, it is much more natural and easy to control the stress (electric field) than to control the strain in any case. In this case, the piezoelectric strain constant *d*α*<sup>j</sup>* is usually measured. It can be obtained from piezoelectric strain constant *e*α*<sup>j</sup>* using the following relation:

$$d\_{a\rangle} = s\_{\parallel k} e\_{ak} \tag{13}$$

where *sjk* is the elastic compliance, which is given by the inverse matrix of the elastic constants *C*jk.

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 precisely gives piezoelectric properties of ferroelectric materials.

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

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 fer-

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

high-pressure synthesis [38], and some of them were quenchable phase. For example, LiNbO3

. One of the notable properties of LiNbO3

means that LiO6

3*c* phase.

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

is its high-curie tem-

perovskite is

7

. The "strained

polyhedron are



are not so much superior as

and NbO6

Density Functional Perturbation Theory to Predict Piezoelectric Properties

perovskite and possible polarization directions. (b) Representative unstable

along downward direction of *c*-axis with respect to those in paraelectric *R*¯

compared with Pb-based perovskites. Crystal structure of piezoelectric *AB*O3

Cubic lattice is symmetric and usually high-temperature phase, same as LiNbO<sup>3</sup>

perature (~1400 K). However, piezoelectric properties of LiNbO3

perovskite structure" expression for LiNbO3

**Figure 3.** (a) Crystal structure of cubic *AB*O3

vibrational mode of cubic *AB*O3

based on the cubic structure (of *Pm3m* symmetry), shown in **Figure 3a**.

pressure. Many compounds have found to be possible to form LiNbO<sup>3</sup>

perovskite showing as arrows.

roelectric nature of LiNbO3
