Chapter 1
Fundamentals of Ferroelectricity and Ferroelectric Materials

Yan Zhang1, Xubing Lu2, and Jun-ming Liu3*

1 School of Optoelectronic Engineering, Henan Normal University, Xinxiang, Henan, China

2 Guangdong Provincial Key Laboratory of Optical Information Materials and Technology and Institute for Advanced Materials, South China Academy of Advanced Optoelectronics, South China Normal University, Guangzhou, Guangdong, China

3 Laboratory of Solid State Microstructures and Innovation Center of Advanced Microstructures, Nanjing University, Nanjing, Jiangsu, China

*Corresponding author: liujm@nju.edu.cn

1.1 Ferroelectrics

Ferroelectrics are a special class of dielectrics whose ferroelectricity arises from the cooperative interaction of neighboring dipoles, exhibiting spontaneous polarization that can be reoriented under the influence of an external electric field. Ferroelectrics share many similarities with ferromagnets (magnetic materials with permanent magnetic moments) in terms of phase transitions and response to external fields. Historically, the term "ferroelectric" was coined by analogy with "ferromagnetic." Figure 1.1a schematically illustrates the unique properties and classification of polarized materials as ferroelectrics. Typically, the interaction between ferroelectric dipoles and the lattice structure directly influences the orientation and strength of the dipoles, i.e., the magnitude of spontaneous polarization. When the dipoles are arranged in a regular pattern within the ferroelectric material, those at the boundaries are unable to continue aligning or interacting, resulting in the accumulation of some uncompensated charges on the material's surface. Mechanical force and temperature are common external stimuli that alter the lattice structure of ferroelectric materials, thereby changing the surface charge distribution. The surface charge generated by an applied external stress is referred to as the piezoelectric effect, while the change in spontaneous polarization in response to temperature variations is known as the pyroelectric effect. Unlike the polarization response induced by stress, the polarization associated with the pyroelectric effect occurs spontaneously and is retained within the structure in the form of permanent dipoles. Ferroelectric polarization is similar to pyroelectric effect-associated polarization but differs in that it can be reversed by the application of an electric field. Figure 1.1b illustrates the reversible thermodynamic interactions among mechanical, thermal, and electric fields. The physical effects are associated with primary variables and their related ones, and a change in any one variable induces changes in the others. For piezoelectric materials, mechanical stress induces internal dielectric displacement, while applying an electric field results in material deformation. The interactions between electrical and mechanical variables define the converse and direct piezoelectric effects, as represented by the linear constitutive Equations (1.1) and (1.2). Here, , , , and represent mechanical stress, electric displacement, strain, and electric field, respectively, while , , and denote the piezoelectric coefficients, elastic constants, and dielectric permittivity under constant strain, respectively:

Figure 1.1 (a) Relationship between ferroelectric, magnetic, multiferroic, and magnetoelectric materials. (b) Diagram of the interrelationship between thermal, mechanical, and electrical properties of materials. Here, denotes the entropy, and denotes the temperature.

Polarization switching is a fundamental feature of ferroelectric materials, where the domains can be reoriented to different polarization states under the application of an electric field or mechanical stress. The polarization behavior of ferroelectric materials is characterized by measuring their polarization response to an electric field (i.e., dielectric displacement), which reflects both the spontaneous polarization and its reorientation under the influence of the electric field. In uniaxial ferroelectrics, the polarization direction can be switched within a 180 ° range under the application of an electric field. In biaxial ferroelectrics, both the electric field and high stress can alter polarization, with the switching angle depending on the symmetry of the crystal. For ferroelectric materials, the mechanical strain and dielectric displacement responses can be divided into a linear component and a residual component (remanent polarization), with their relationships expressed by Equations (1.3) and (1.4):

Ferroelectric materials are typically described using linear constitutive equations for their piezoelectric properties, but this approach overlooks the inherent nonlinear characteristics of ferroelectrics. As shown in Equations (1.5) and (1.6), the polarization response of ferroelectric materials under an electric field exhibits pronounced nonlinear behavior, highlighting that ferroelectrics generally exhibit superior piezoelectric performance compared to non-ferroelectric materials. Specifically, the nonlinear effects of polarization switching are typically manifested as a characteristic butterfly-shaped hysteresis loop.

The emergence of spontaneous polarization in a crystal establishes a unique polar direction that is asymmetric with respect to all other directions in the crystal. The entire crystal exhibits polarity along this direction, with one end being positively polarized and the other negatively polarized. A unit cell of a ferroelectric crystal can be viewed as an equivalent dipole, characterized by a permanent dipole moment, with one end having a layer of positive bound charge and the other having a layer of negative bound charge. The electric field generated by the bound charge, which opposes polarization within the crystal, is known as the depolarization field. Under mechanical constraints, the strain induced by spontaneous polarization increases the strain energy, making a uniformly polarized state unstable. This instability drives the formation of multiple small regions, or domains, each containing electric dipoles aligned in the same direction, but with varying orientations across domains. The boundaries between these domains are called domain walls. The formation of domains minimizes the crystal's electrostatic and strain energies, while the presence of domain walls introduces additional domain wall energy. The stable configuration of electric domains is determined by the condition that minimizes the total free energy, which is the result of a balance between electrostatic, strain, and domain wall energies.

The hysteresis loop reflects the movement of domain walls and changes in the domain structure of ferroelectric materials, representing the evolution of regions with aligned spontaneous polarization, as shown in Figure 1.2. The characteristics of the hysteresis loop are typically described by parameters such as saturation polarization , remnant polarization , and coercive field . Under weak electric fields, ferroelectric polarization depends linearly on the electric field, where reversible domain wall motion dominates. The hysteresis loop becomes nonlinear when the applied electric field exceeds , accompanied by irreversible nucleation of new domains, domain wall motion, and rapid reorientation and alignment of polarization. The contribution of induced polarization from the enhanced electric field further increases the total polarization until saturation is reached. The crystal retains a macroscopic polarized state even after the electric field is removed. Polarization weakens and reverses direction when the electric field is reversed, eventually tending toward saturation in the opposite direction. The hysteresis loop in ferroelectric crystals is typically observed within a specific temperature range. When the temperature exceeds a critical threshold, known as the Curie temperature , the material undergoes a phase transition to a paraelectric state, and spontaneous polarization vanishes.

Figure 1.2 Arrangement of electric domains and corresponding polarization hysteresis loops, the arrowheads denote assigned electric domains.

1.2 Material Structure and Theory

1.2.1 Structural Evolution

The phase transition of ferroelectric material is a special structural one accompanied by atomic-scale symmetry breaking. The paraelectric phase exhibits higher symmetry, whereas the ferroelectric phase has lower symmetry during the transition. Therefore, ferroelectricity is a material property that exists only in crystals lacking an inversion center, placing specific constraints on the symmetry of the point group to which the ferroelectric crystal belongs. According to symmetry requirements, ferroelectric phases are confined to 10 polar point groups within the 21 non-centrosymmetric groups within the 32 crystallographic point groups. The polar point groups include , , , , , , , , , and . Furthermore, due to the temperature-dependent nature of atomic configurations, ferroelectric crystals also exhibit pyroelectricity, and universally fall within the category of piezoelectric materials owing to inherent asymmetry-derived characteristics. To summarize, pyroelectricity is a common characteristic of all materials exhibiting...