Ferroelectric Materialson 24/2/2012 & Updated on Wednesday 27th of June 2018 at 06:28:11 PM
Theory of Ferroelectric MaterialsThe 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) Px → component of polarization vector, x Py → component of polarization vector, y Pz → component of polarization vector, z αi, αij, αijk → coefficients should be constant with the crystal symmetry. α0 > 0, α111> 0 → for all ferroelectrics α11< 0 → ferroelectrics with first order transition α0 > 0 → ferroelectrics with second order transition
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 = ± Ps Ps → spontaneous polarization By simplifying, eliminating the negative root and substitute α11 = 0 we get,
Polarization and Hysteresis LoopFirst 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 TemperatureThe 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 (TC). Most of these materials above Tc 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 constant ε∞ → ε at temperature, T >> TC A → Constant TC → Curie point T → Temperature χ → Susceptibility CC → Curie constant of the material The dielectric constant and temperature characteristic of a ferroelectric material are represented below.
Examples of Ferroelectric Materials
- Lead Zirconate Titanate (PZT)
- Triglycine Sulphate
- Lithium tantalite etc.
Application of Ferroelectric Materials
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