Preface

Chapter 8 **Perturbation Method for Solar/Infrared Radiative Transfer in a**

**Optical Properties 141**

**VI** Contents

**Scattering Medium with Vertical Inhomogeneity in Internal**

Yi-Ning Shi, Feng Zhang, Jia-Ren Yan, Qiu-Run Yu and Jiangnan Li

In this book, we aim to present the recent developments and applications of the perturba‐ tion theory for treating problems in applied mathematics, physics and engineering. The eight chapters presented in this book are written by 22 authors from 8 countries: Japan, Chi‐ na, Algeria, France, Russia, Mexico, USA and Canada.

Each chapter is independent and self-contained, providing a contemporary overview of the perturbation methods that are used in theoretical and applied sciences. The reference list at the end of each chapter provides the reader a selected list of journal papers, books and con‐ ference proceedings. The chapters can be summarized as follows: In the first chapter, a com‐ putational technique is developed to predict the piezoelectric properties of materials using the density functional perturbation theory (DFPT). In the next chapter, the development of a sliding-mode perturbation observer-based control scheme for voltage source converter based high voltage direct current systems is described. In the third chapter, a multiplication operation is introduced and via this operation, it is allowed to give the Carleman operator the form of a multiplication operator. In the same chapter, a formal perturbation theory of Carleman operators is also established. The next chapter is devoted to optimal perturbation techniques and various types of optimal perturbation techniques, namely optimal determin‐ istic perturbation theory, optimal stochastic perturbation and simultaneous stochastic per‐ turbation methods are introduced to demonstrate the efficiency of perturbation methods in predictability of dynamical systems that arise in atmospheric and oceanographic sciences. In the fifth chapter, a discussion of the nonlinear parametric systems is presented and the con‐ ditions of motions of existence in the resonance zones are put forward. In the sixth chapter, mechanical perturbations strategy is applied at the working electrode during one-step elec‐ trodeposition process and the results are compared to the standard one-step electrodeposi‐ tion. In the next chapter, an approximate analytical solution is constructed for the wellbore pressure via the method of matched asymptotic expansions applied to the one-dimensional saturation convection-dispersion equation. The solutions to this type of nonlinear equations is of great importance in fluid mechanics and especially in petroleum engineering. In the last chapter, a new inhomogeneous scheme based on perturbation methods to solve the solar/ infrared radiative transfer (SRT/IRT) problem is developed. This chapter contains significant and applicable information in meteorological sciences.

The book is intended to reach to researchers, scientists and postgraduate students in aca‐ demia as well as in industry and published as an open access book in order to significantly increase the reach and impact of the information that is contained in the book.

> **Dr. İlkay Bakırtaş** Istanbul Technical University Department of Mathematics Istanbul, Turkey

**Chapter 1**

**Provisional chapter**

**Density Functional Perturbation Theory to Predict**

**Density Functional Perturbation Theory to Predict** 

DOI: 10.5772/intechopen.76827

Among the various computational methods in materials science, only first-principles calculation based on the density functional theory has predictability for unknown material. Especially, density functional perturbation theory (DFPT) can effectively calculate the second derivative of the total energy with respect to the atomic displacement. By using DFPT method, we can predict piezoelectric constants, dielectric constants, elastic constants, and phonon dispersion relationship of any given crystal structure. Recently, we established the computational technique to decompose piezoelectric constants into each atomic contribution, which enable us to gain deeper insights to understand the piezoelectricity of material. Therefore, in this chapter, we will introduce the computational framework to predict piezoelectric properties of polar material by means of DFPT and details of decomposition technique of piezoelectric constants. Then, we will show some

**Keywords:** density functional perturbation theory, ferroelectricity, piezoelectricity,

In this chapter, we will introduce how recent computational techniques can successfully predict response properties, represented as piezoelectricity, by means of perturbation method. Piezoelectricity is the polarization change in response to external mechanical force. Inversely, if electrical field is applied to piezoelectric material, mechanical strain is induced (inverse piezoelectric effect). Therefore, piezoelectric materials are widely used as vibrational censors,

case studies to predict and discover new piezoelectric material.

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

**Piezoelectric Properties**

**Piezoelectric Properties**

Toshiharu Ohnuma

Toshiharu Ohnuma

**Abstract**

Kaoru Nakamura, Sadao Higuchi and

Kaoru Nakamura, Sadao Higuchi and

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

first-principles calculation

**1. Introduction**

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

#### **Density Functional Perturbation Theory to Predict Piezoelectric Properties Density Functional Perturbation Theory to Predict Piezoelectric Properties**

DOI: 10.5772/intechopen.76827

Kaoru Nakamura, Sadao Higuchi and Toshiharu Ohnuma Kaoru Nakamura, Sadao Higuchi and Toshiharu Ohnuma

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

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

#### **Abstract**

Among the various computational methods in materials science, only first-principles calculation based on the density functional theory has predictability for unknown material. Especially, density functional perturbation theory (DFPT) can effectively calculate the second derivative of the total energy with respect to the atomic displacement. By using DFPT method, we can predict piezoelectric constants, dielectric constants, elastic constants, and phonon dispersion relationship of any given crystal structure. Recently, we established the computational technique to decompose piezoelectric constants into each atomic contribution, which enable us to gain deeper insights to understand the piezoelectricity of material. Therefore, in this chapter, we will introduce the computational framework to predict piezoelectric properties of polar material by means of DFPT and details of decomposition technique of piezoelectric constants. Then, we will show some case studies to predict and discover new piezoelectric material.

**Keywords:** density functional perturbation theory, ferroelectricity, piezoelectricity, first-principles calculation

## **1. Introduction**

In this chapter, we will introduce how recent computational techniques can successfully predict response properties, represented as piezoelectricity, by means of perturbation method. Piezoelectricity is the polarization change in response to external mechanical force. Inversely, if electrical field is applied to piezoelectric material, mechanical strain is induced (inverse piezoelectric effect). Therefore, piezoelectric materials are widely used as vibrational censors,

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 induces microscopic polarization *P*<sup>s</sup> along the ionic displaced direction.

so on. Thus, first-principles calculation has been made use of calculating the perturbed total energy of materials because of its accuracy. Although perturbations were made by hand up to the early 1980s, sophisticated methodology of density functional perturbation theory (DFPT) was proposed in 1987 by Baroni et al. [5]. They showed general formulation of total energy change with respect to atomic displacement and opened the way to efficiently compute the energy derivative with respect to the perturbation [5]. DFPT can compute response properties directly arising from the perturbations of strain, atomic displacement, and electric field by making use of linear response theory [8–11]. Numbers of ferroelectric materials are theoretically investigated on the origin of their ferroelectric properties (including piezoelectricity and dielectric properties) by using DFPT. Because of technological importance, such theo-

Density Functional Perturbation Theory to Predict Piezoelectric Properties

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

retical researches have been focused on Pb-based perovskite material (e.g., PbTiO<sup>3</sup>

[26, 27], uniaxial stress for SrHfO3

pic stress for PbTiO3

*y*, and *z* directions.

strain *η,* E(*u*,*σ*,*η*), is defined as follows:

constants within the framework of DFPT.

**2. Formulation of piezoelectric constants**

and their solid solution [12–15]) because they have excellent piezoelectric properties and are widely applied for actuators. However, due to the restriction of hazardous substance (RoHS) directive, researches on lead-free ferroelectric materials have gathered great attraction. By taking advantage of the predictability of DFPT, various lead-free ferroelectric oxide and nitride materials were theoretically investigated on their piezoelectric properties [16–25]. Moreover, DFPT calculations showed that piezoelectricity can be greatly enhanced by imposing isotro-

AlN-GaN solid solution alloy [29], and two-dimensional epitaxial strain for doped ZnO [30]. As latterly explained, those enhancements of piezoelectric constant are thought to be closely related to the phase transition. In the next section, we will show the definition of piezoelectric

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*,

Total energy of material under perturbation of atomic displacement *u*, electric field *σ*, and

, PbZrO3

[28], uniaxial and biaxial strain for

,

3

Because ferroelectric phase is energetically more stable than paraelectric phase under low temperature, *P*<sup>s</sup> is frequently referred as the spontaneous polarization. Above Curie temperature, 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 Δ*P*<sup>s</sup> = *P*<sup>s</sup> ' −*P*<sup>s</sup> . Therefore, piezoelectric constant is defined as the derivative of the spontaneous polarization with respect to the external field. More detailed and comprehensive description of piezoelectricity is reviewed by Martin [1].

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 external force.

so on. Thus, first-principles calculation has been made use of calculating the perturbed total energy of materials because of its accuracy. Although perturbations were made by hand up to the early 1980s, sophisticated methodology of density functional perturbation theory (DFPT) was proposed in 1987 by Baroni et al. [5]. They showed general formulation of total energy change with respect to atomic displacement and opened the way to efficiently compute the energy derivative with respect to the perturbation [5]. DFPT can compute response properties directly arising from the perturbations of strain, atomic displacement, and electric field by making use of linear response theory [8–11]. Numbers of ferroelectric materials are theoretically investigated on the origin of their ferroelectric properties (including piezoelectricity and dielectric properties) by using DFPT. Because of technological importance, such theoretical researches have been focused on Pb-based perovskite material (e.g., PbTiO<sup>3</sup> , PbZrO3 , and their solid solution [12–15]) because they have excellent piezoelectric properties and are widely applied for actuators. However, due to the restriction of hazardous substance (RoHS) directive, researches on lead-free ferroelectric materials have gathered great attraction. By taking advantage of the predictability of DFPT, various lead-free ferroelectric oxide and nitride materials were theoretically investigated on their piezoelectric properties [16–25]. Moreover, DFPT calculations showed that piezoelectricity can be greatly enhanced by imposing isotropic stress for PbTiO3 [26, 27], uniaxial stress for SrHfO3 [28], uniaxial and biaxial strain for AlN-GaN solid solution alloy [29], and two-dimensional epitaxial strain for doped ZnO [30]. As latterly explained, those enhancements of piezoelectric constant are thought to be closely related to the phase transition. In the next section, we will show the definition of piezoelectric constants within the framework of DFPT.
