Some understandings of ferroelectricity

Some conceptional paradox with ferroelectricity (FE).

Figure 1

1) As shown in Fig. 1, there is a one-dimensional system that consists equally N anions with charge -q and N cations with charge +q, separated by distance a. This system definitely has spontaneous permanent dipole moment p=Nqa, but why it is not FE?

Note that normally FE is defined for bulk (infinite) system, in which case not so many systems have FE. However, for a finite non-metallic system, spontaneous dipole moments are not rare at all. Figure 1 is actually a good example. Then the paradox is that the real systems are finite anyway, how come they are not normally FE?

The answer has to do with the imperfectness of the real system. One can see in Fig. 1, the direction of the dipole moment depends on the type of the atoms on the surface, which is random for a real system. For example, we don't normally have a NaCl sample that has a surface only with Na atoms and the opposite surface with only Cl atoms. This randomness will eventually cancel out these "spontaneous dipole moments", making the total dipole moment of the sample negligibly zero.

Figure 2

2) The system shown in Fig. 2 is trully FE.

For this system in panel A, the dipole moment is p=Nqa. But if the surface atoms change to panel B, the dipole moment is p=-Nqb. Therefore the randomness of the surface can not cancel the dipole moment, which on average is p=Nq(a-b). This is why the system is FE.

3) To summarize the cases we discussed above, to determine if a system is FE. We have to test if the system (infinite) is central-symmetric. In other words, whether you can find an unit cell that has an inversion symmetry. For the system in Fig. 1, it is central symmetric. But for system in Fig. 2, it is not. That's why it is FE.