Advanced Solid State Physics

PbTiO 3 ,KNbO 3 andLiNbO 3.
Fig. 138 shows the phase diagram of PZT. The area labeledPCmeans in there the crystal is paraelec-
tric with a cubic crystal structure.FEilabels areas with ferroelectric behaviour and a rhombohedral
structure andFTferroelectric areas with tetragonal structure.

Figure 138: Phase diagram of PZT

14.4.8 Polarization

As one could see in the previous sections, all the mentioned material properties are in some sort
related to each other and because of that we want to express the polarization in mathematical terms.
Basically, the polarization can be computed by taking the derivative of the free energyGwith respect
to the electric fieldEi(see chapter 8.2 Statistical Physics):

Pi=

∂G

∂Ei

Now, one can build the total derivation of the polarization as follows:

dPi=

∂Pi

∂σkl

dσkl+

∂Pi

∂Ek

dEk+

∂Pi

∂Hk

dHk+

∂Pi

∂T

dT (256)

Here,σklis one element of the strain tensor,His the magnetic field andT the temperature. The
first term describes the piezoelectric effect, the second term the electric polarization, the third the
magneto-electric polarization and the fourth the pyroelectric effect.

The strain is also a very interesting property to look at, because as the polarization it also reveals
much of a material’s behaviour. Again, we compute the total derivation of the elementi,jof the
strain-tensor:

dij=

∂ij

∂σkl

dσkl+

∂ij

∂Ek

dEk+

∂ij

∂Hk

dHk+

∂ij

∂T

dT (257)

We identify the first term as the one ruling the elastic deformation, the second term describes the
reciprocal piezoelectric effect, the third one the reciprocal piezomagnetic effect and the fourth the
thermal expansion.