Crystal symmetry, ferroelectric

No ferroelectricity is possible when the dipoles in the crystal compensate each other due to the crystal symmetry. All centrosymmetric, all cubic and a few other crystal classes are... 

Ferroelectrics. Among the 32 crystal classes, 11 possess a centre of symmetry and are centrosymmetric and therefore do not possess polar properties. Of the 21 noncentrosymmetric classes, 20 of them exhibit electric polarity when subjected to a stress and are called piezoelectric one of the noncentrosymmetric classes (cubic 432) has other symmetry elements which combine to exclude piezoelectric character. Piezoelectric crystals obey a linear relationship P,- = gijFj between polarization P and force F, where is the piezoelectric coefficient. An inverse piezoelectric effect leads to mechanical deformation or strain under the influence of an electric field. Ten of the 20 piezoelectric classes possess a unique polar axis. In nonconducting crystals, a change in polarization can be observed by a change in temperature, and they are referred to as pyroelectric crystals. If the polarity of a pyroelectric crystal can be reversed by the application on an electric field, we call such a crystal a ferroelectric. A knowledge of the crystal class is therefore sufficient to establish the piezoelectric or the pyroelectric nature of a solid, but reversible polarization is a necessary condition for ferroelectricity. While all ferroelectric materials are also piezoelectric, the converse is not true for example, quartz is piezoelectric, but not ferroelectric. 

Ferroelectricity depends on temperature. Above 0c ferroelectric behavior is lost and the material becomes paraelectric. The change from the ferroelectric to the non-ferroelectric state is accompanied either by a change in crystal symmetry (e.g., as in BaTiOs) or by an order-disorder transition such as in the organic ferroelectric compound triglycine sulfate (TGS). 

In general a ferroelectric perovskite single crystal will be composed of a roughly equal number of domains oriented in all the equivalent directions allowed by the crystal symmetry. The overall polarisation of the crystal will be zero. The application of an electric field will cause a polarisation switch and lead to a classical hysteresis loop in which the important values are P, (the remanent or residual polarisation when the electric field is reduced to zero), and E, (the coercive field, which is the reverse field required to reduce the polarisation to zero). Extrapolation of the high-field portion of the curve to =0 gives the value of the spontaneous polarisation P (Figure 6.9a). 

Ferroelectricity can arise in a number of ways other than that described previously, in which dipoles are generated in a crystal structure and combine to give a spontaneous polarisation if the crystal symmetry permits. These alternatives have been called improper ferroelectricity (perhaps better extrinsic ferroelectricity, see Lines and Glass 2001). In essence, improper ferroelectricity is ferroelectricity which is not due to the normal polarisation of the stmcture but arises from other interactions. Improper ferroelectricity has been considered to be rare in bulk materials and is a weak effect, usually rather difficult to detect, but the creation of artificial superlat-fices and studies of layered perovskites have changed this and now improper ferro-electrics are becoming widely studied. 

It is known that the crystal symmetry defines point symmetry group of any macroscopic physical property, and this symmetry cannot be lower than corresponding point symmetry of a whole crystal. The simplest example is the spontaneous electric polarization that cannot exist in centrosymmetric lattice as the symmetry elements of polarization vector have no operation of inversion. We remind that inversion operation means that a system remains intact when coordinates x, y, z are substituted by —x, —y, —z. If the inversion center is lost under the phase transition in a ferroic at T < 7), Tc is the temperature of ferroelectric phase transition or, equivalently, the Curie temperature), the appearance of spontaneous electrical polarization is allowed. Spontaneous polarization P named order parameter appears smoothly... 

An important aspect of the enhanced intrinsic response of ferroelectrics is anisotropy, the direction dependence of properties. By symmetry, ferroelectric crystals are anisotropic with respect to dielectric, mechanical, and electromechanical properties, and this issue is essential when designing devices that exploit the highest material responses by correctly orienting single crystals [8, 33-35]. The crystal axes, along which the highest material response occurs, may not coincide with the polar directions of spontaneous polarization for a given ferroelectric phase. This is the case... 

BaTIOj (LB Number IA-IO). BaTiOs is the most extensively studied ferroelectric crystal. It is ferroelectric below about 123 C, where the crystal symmetry changes from cubic to tetragonal. Further phase transitions take place from tetragonal to orthorhombic at about 5 and to rhombohedral at about —90 °C (Figs. 4.5-18 to 4.5-20). It is believed that the ferroelectric transition... 

For some applications, such as in multiferroic materials based devices, it is the crystal symmetry of the multiferroic what matters. In these materials the presence or not of a centre of symmetry is crucial for the observation of ferroelectricity. With this regard, there are cases, as in some AMnOs perovskites (A=Y or Dy), in which the synthetic route determines whether an orthorhombic compound with a centre of symmetry (i.e. non ferroelectric) or a hexagonal phase without the centre (i.e. ferroelectric) is formed (Carp et al., 2003 Dho et al., 2004). Consequently, preparative conditions have to be carefully selected in order to obtain crystal phases with the adequate structure. The use of more than one synthesis method is thus worth trying in all cases. 

Figure 6. The hierarchy of dielectric materials. All are of course dielectrics in a broad sense. To distinguish between them we limit the sense, and then a dielectric without special properties is simply called a dielectric if it has piezoelectric properties it is called a piezoelectric, if it further has pyroelectric but not ferroelectric properties it is called a pyroelectric, etc. A ferroelectric is always pyroelectric and piezoelectric, a pyroelectric always piezoelectric, but the reverse is not true. Knowing the crystal symmetry we can decide whether a material is piezoelectric or pyroelectric, but not whether it is ferroelectric. A pyroelectric must possess a so-called polar axis (which admits no inversion). If in addition this axis can be reversed by the application of an electric field, i.e., if the polarization can be reversed by the reversal of an applied field, the material is called ferroelectric. Hence a ferroelectric must have two stable states in which it can be permanently polarized.

Depending on the composition and crystal symmetry the hydrogen-bond dipoles may be oriented to form a ferroelectric structure. Such an arrangement b more probable Cor odd polyamides than for even ones. For an illustratioo. see Ref. 6 (p. 403). 

I. Dahl and S.T. Lagerwall, Elastic and flexoelectric properties of chiral smectic-C phase and symmetry considerations on ferroelectric liquid-crystal cells, Ferroelectrics, 58, 215-243 (1984). 

As witli tlie nematic phase, a chiral version of tlie smectic C phase has been observed and is denoted SniC. In tliis phase, tlie director rotates around tlie cone generated by tlie tilt angle [9,32]. This phase is helielectric, i.e. tlie spontaneous polarization induced by dipolar ordering (transverse to tlie molecular long axis) rotates around a helix. However, if tlie helix is unwound by external forces such as surface interactions, or electric fields or by compensating tlie pitch in a mixture, so tliat it becomes infinite, tlie phase becomes ferroelectric. This is tlie basis of ferroelectric liquid crystal displays (section C2.2.4.4). If tliere is an alternation in polarization direction between layers tlie phase can be ferrielectric or antiferroelectric. A smectic A phase foniied by chiral molecules is sometimes denoted SiiiA, altliough, due to the untilted symmetry of tlie phase, it is not itself chiral. This notation is strictly incorrect because tlie asterisk should be used to indicate the chirality of tlie phase and not tliat of tlie constituent molecules. 

The most important materials among nonlinear dielectrics are ferroelectrics which can exhibit a spontaneous polarization PI in the absence of an external electric field and which can spHt into spontaneously polarized regions known as domains (5). It is evident that in the ferroelectric the domain states differ in orientation of spontaneous electric polarization, which are in equiUbrium thermodynamically, and that the ferroelectric character is estabUshed when one domain state can be transformed to another by a suitably directed external electric field (6). It is the reorientabiUty of the domain state polarizations that distinguishes ferroelectrics as a subgroup of materials from the 10-polar-point symmetry group of pyroelectric crystals (7—9). 

It should be noted that, whereas ferroelectrics are necessarily piezoelectrics, the converse need not apply. The necessary condition for a crystal to be piezoelectric is that it must lack a centre of inversion symmetry. Of the 32 point groups, 20 qualify for piezoelectricity on this criterion, but for ferroelectric behaviour a further criterion is required (the possession of a single non-equivalent direction) and only 10 space groups meet this additional requirement. An example of a crystal that is piezoelectric but not ferroelectric is quartz, and ind this is a particularly important example since the use of quartz for oscillator stabilization has permitted the development of extremely accurate clocks (I in 10 ) and has also made possible the whole of modern radio and television broadcasting including mobile radio communications with aircraft and ground vehicles. 

Crystals with one of the ten polar point-group symmetries (Ci, C2, Cs, C2V, C4, C4V, C3, C3v, C(, Cgv) are called polar crystals. They display spontaneous polarization and form a family of ferroelectric materials. The main properties of ferroelectric materials include relatively high dielectric permittivity, ferroelectric-paraelectric phase transition that occurs at a certain temperature called the Curie temperature, piezoelectric effect, pyroelectric effect, nonlinear optic property - the ability to multiply frequencies, ferroelectric hysteresis loop, and electrostrictive, electro-optic and other properties [16, 388],... 

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