Module 5: Nonlinear Dielectrics
  Ferroelectric Ceramics
 


5.3.6 Case – II: First Order Transition

Another situation to consider is that when a < 0, b < 0 but c > 0.  What this means is that free energy vs polarization plot has three equal minima, one for P = 0 and the other two for P ≠ 0 at the same temperature i.e. at the same value of ‘a’ at a temperature T = T0, Curie temperature, which is now more than T0. This gives rise to the following free energy vs polarization plot.

Figure 5.9 Free energy vs polarization schematic plot for a first order phase transition

The most important feature of this phase transition is that polarization i.e. the order parameter drops from P ≠ 0 to zero discontinuously at T = Tc and is called as first order phase transition. This is also very clearly demonstrated by a discontinuity in the reciprocal of dielectric susceptibility as seen below. For example, solid-liquid phase transition is a first order phase transition while among various ferroelectrics, barium titanate is a fine example of first order transition among ferroelectrics.   

Figure 5.10 Polarization and reciprocal susceptibility plot for a first order phase transition

In order to compute the non-zero polarization (P0) and susceptibility at the transition, the value of free energy (5.13) for P = 0 and P = P0 must be equal at T = Tc i.e.

(5.25)

On the other hand, field E must also be zero for the polarization to be spontaneous i.e.

(5.26)

The polarization and susceptibility at Tc are obtained by solving two equations and are given as

(5.27)
and
(5.28)

The fact that there are three minima at T=Tc is reflected in whether the Tc is approached while heating or cooling. More specifically, the material will be in other of the two non-zero polarization states if is heated from an initial temperature that is lower than Tc whereas, if it is cooled from a temperature higher than Tc, the sample will be in paraelectric state. This results in thermal hysteresis when these materials are thermally cycled across Tc.

If you are interested in further reading about the phase transitions in ferroelectrics, refer to the following texts:

  • Principles and Applications of Ferroelectrics and Related Materials, M. Lines and A. Glass, Clarendon Press, Oxford

  • Solid State Physics, A.J. Dekker, Macmillan Publishing

  • Ferroelectric Crystals, F. Jona and G. Shirane, Dover Publishing