# Ferroelectric Materials

The first discovered

P

P

P

α

α

α

α

For investigating different phenomenon and domain formation in ferroelectrics, these equations are used in phase field model. Usually, it is used by adding some terms such as an elastic term, a gradient term and an electrostatic term to this free energy equation. Using finite difference method, the equations are solved subject to the constraints of Linear elasticity and Gauss’s law.

For a cubic to tetragonal phase transition of a spontaneous polarization of a ferroelectric can be obtained from the expression for free energy It has a character of dual well potential with double energy minima at P = ± P

P

ε

A → Constant

T

T → Temperature

χ → Susceptibility

C

The dielectric constant and temperature characteristic of a

**ferroelectric material**is Rochelle salt by Valasek in 1921. These materials can produce spontaneous polarization. By inverting the direction of applied electrical field, the direction of polarization of these materials can be inverted or changed (figure 1). This is called switching. It can also maintain the polarisation even once the field is removed. These materials have some similarities over ferromagnetic materials which reveal permanent magnetic moment. The hysteresis loop is almost same for both materials. Since, there are similarities; the prefix is same for both the materials. But all the ferroelectric material must not have Ferro (iron). All the**ferroelectric materials**exhibit piezoelectric effect. The opposed properties of these materials are seen in antiferromagnetic materials.## Theory of Ferroelectric Materials

The free energy of ferroelectric material based on Ginburg-Landau theory without electric field and any applied stress can be written as Taylor expansion. It is written in terms of P (order parameter) as (if sixth order expansion is used)P

_{x}→ component of polarization vector, xP

_{y}→ component of polarization vector, yP

_{z}→ component of polarization vector, zα

_{i}, α_{ij}, α_{ijk}→ coefficients should be constant with the crystal symmetry.α

_{0}> 0, α_{111}> 0 → for all ferroelectricsα

_{11}< 0 → ferroelectrics with first order transitionα

_{0}> 0 → ferroelectrics with second order transitionFor investigating different phenomenon and domain formation in ferroelectrics, these equations are used in phase field model. Usually, it is used by adding some terms such as an elastic term, a gradient term and an electrostatic term to this free energy equation. Using finite difference method, the equations are solved subject to the constraints of Linear elasticity and Gauss’s law.

For a cubic to tetragonal phase transition of a spontaneous polarization of a ferroelectric can be obtained from the expression for free energy It has a character of dual well potential with double energy minima at P = ± P

_{s}P

_{s}→ spontaneous polarization By simplifying, eliminating the negative root and substitute α_{11}= 0 we get,## Polarization and Hysteresis Loop

First we take a dielectric material and a peripheral electric field is given, then we can see that the polarization will be always directly proportional to the applied field which is represented in figure 2. Next; when we polarise a paraelectric material, we get a non linear polarization. However it is a function of field as shown in figure 3. Next, we take a**ferroelectric material**and electric field is given to it. We get a non linear polarization. It also exhibit nonzero spontaneous polarization without a peripheral field. We can also see that by inverting the direction of applied electrical field, the direction of polarization can be inverted or changed.Thus, we can say that, the polarization will depend on the present as well as the previous condition of electric field. The hysteresis loop is obtained as in figure 4.

## Curie Temperature

The properties of these materials exist only below a definite phase conversion temperature. Above this temperature, the material will become paraelectric materials. That is, loss in spontaneous polarization. This definite temperature is called Curie temperature (T_{C}). Most of these materials above T_{c}will loss the piezoelectric property as well. The variation of dielectric constant by means of temperature in the non polar, paraelectric state is shown by Curie-Weiss law as given below ε → Dielectric constantε

_{∞}→ ε at temperature, T >> TCA → Constant

T

_{C}→ Curie pointT → Temperature

χ → Susceptibility

C

_{C}→ Curie constant of the materialThe dielectric constant and temperature characteristic of a

**ferroelectric material**are represented below.## Examples of Ferroelectric Materials

- BaTiO
_{3} - PbTiO
_{3} - Lead Zirconate Titanate (PZT)
- Triglycine Sulphate
- PVDF
- Lithium tantalite etc.

## Application of Ferroelectric Materials

- Thermistors
- Oscillators
- Non-volatile memory
- Filters
- Capacitors
- Light deflectors
- Transchargers
- Electro-optic materials
- Modulators
- Piezoelectrics
- Display etc

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