Ferroelectric/piezoelectric crystal symmetry

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. 

Eleven acentric crystal classes are chiral, i.e., they exist in enantiomorphic forms, whereas ten are polar, i.e., they exhibit a dipole moment. Only five (1,2, 3, 4, and 6) have both chiral and polar symmetry. All acentric crystal classes except 432 possess the same symmetry requirements for materials to display piezoelectric and SHG properties. Both ferroelectricity and pyroelectricity are related to polarity a ferroelectric material crystallizes in one of ten polar crystal classes (1, 2, 3,4, 6, m, mm2, 3m, 4mm, and 6mm) and possesses a permanent dipole moment that can be reversed by an applied voltage, but the spontaneous polarization (as a function of temperature) of a pyroelectric material is not. Thus all ferroelectric materials are pyroelectric, but the converse is not true. 

Like solid ferroelectrics, the ferroelectric liquid crystals, particularly the FLCPs, show a pyroelectric effect and a piezoelectric effect and are capable of switching polarization direction (dielectric hysteresis). Moreover, they can switch propagating or reflected polarized light. Finally, the polar symmetry of the phase leads to nonlinear optical properties of the FLCPs such as second-harmonic generation, the Pockels effect, and the Kerr effect. These physical properties of the ferroelectric LC polymers are discussed in the following sections. 

Ferroelectric materials are a subclass of pyro- and piezoelectric materials (Fig. 1) (see Piezoelectric Polymers). They are very rarely foimd in crystalline organic or polymeric materials because ferroelectric hysteresis requires enough molecular mobility to reorient molecular dipoles in space. So semicrystalline poly(vinylidene fluoride) (PVDF) is nearly the only known compoimd (1). On the contrary, ferroelectric behavior is very often observed in chiral liquid crystalline materials, both low molar mass and poljuneric. For an overview of ferroelectric liquid crystals, see Reference 2. Tilted smectic liquid crystals that are made from chiral molecules lack the symmetry plane perpendicular to the smectic layer structure (Fig. 2). Therefore, they develop a spontaneous electric polarization, which is oriented perpendicular to the layer normal and perpendicular to the tilt direction. Because of the liquid-like structure inside the smectic layers, the direction of the tilt and thns the polar axis can be easily switched in external electric fields (see Figs. 2 and 3). 

Ferroelectricity was discovered in Rochelle salt in 1921. A ferroelectric crystal exhibits a spontaneous polarization P, in a certain temperature range and the direction of P, can be reversed by an external electric field. From a physical point of view, ferroelectric crystals are those crystalline compounds, which possess one or more ferroelectric phases. The ferroelectric phase is a particular state exhibiting spontaneous polarization, which can be reversed by an external field. A reversal ofpolarization is considered as a special case of the polarization reorientation. From a crystallographic point of view, ferroelectricity can be foimd in polar crystals. A polar crystal is a piezoelectric crystal (without center of symmetry) crystal whose point-group symmetry has a unique rotational axis, but does not have any mirror plane perpendicular to this axis. Along a unique rotational axis, the atomic arrangement at one end is different from that at the other (opposite end). Therefore, they display spontaneous polarization. Polar crystal, which can be found in ten point groups, are 1, 2, m, mm2,4, 4 mm, 3, 3 m, 6, 6 mm. 

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.

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],... 

Finally, ferroelectricity is manifest in asymmetrical crystals producing domains of spontaneous polarization whose polar axis direction can be reversed in an electric field directed opposite the total dipole moment of the lattice. The two (or more) directions can coexist in a crystal as domain structures comprising millions of unit cells which contain the same electric orientation. The symmetry elements are temperature sensitive in ferroelectric materials [27]. At a particular temperature called the Curie Point the values of the piezoelectric coefficients reach particularly high values. Above the Curie Point the crystal transformation is to a less polar form and the ferroelectric nature disappears. 

There is a structural requirement for ferroelectricity. There are a total of 32 different symmetry point groups, 21 of which do not possess a center of symmetry. Ferro-electrics are part of a small subgroup of noncentrosym-metric crystals. Related properties are piezoelectricity and pyroelectricity. Dielectrics belonging to all but one of the groups of noncentrosymmetric crystals are piezoelectric. Pyroelectric crystals form a further subgroup of 10 types of crystal having especially low symmetry as shown in Table 31.5. 

The spontaneously polarized crystal is anisotropic and has lower symmetry than the nonpolarized one. Ferroelectric materials below the Curie temperature are also always piezoelectric, because the polarized sample has no center of symmetry. If the nonpolarized crystal has the center of symmetry, the piezoelectricity of the sample vanishes above the Curie temperamre. All ferroelectrics below the Curie temperature also always show pyroelectric behavior. 

Among the piezoelectric (and non-ferroelectric) crystals the cubic (synunetry 23) sillenite Bii2SiO20 and isomorphous Bii2GeO20 and tetragonal (symmetry Amm) bariiun germanium titanate Ba2Ge2Ti08 and isomorphous fresnoite Ba2Si2TiOg seems to be perspective crystal materials. 

It is important to recognize that a useful piezoelectric effect is defined macroscopically. Each unit cell has to contribute constructively in order for the macroscopic effect to occur. It is the global symmetry that determines the macroscopic piezoelectric effect. For example, a piezoelectric ceramic containing randomly oriented crystal grains has no piezoelectric effect even though the symmetry of each unit cell allows piezoelectricity. A net polarization in the material is a sufficient but not a necessary condition for the presence of piezoelectricity for example, quartz is one of the popular piezocrystals without polarization. The existence of a polarization, however, does make the piezoelectric effect much more pronounced. In fact, the best piezoelectric materials are all ferroelectric materials. Most importantly, the hydrostatic piezoelectric effect belongs uniquely to polar materials. 

Pyroelectricity is a property inherent in crystals with unique polar axes that are, consequently, without a center of symmetry. However, as opposed to piezoelectricity not all symmetry groups (point groups) lacking a center of symmetry are pyroelectric, whereas all pyroelectrics are also piezoelectrics. Most pyroelectrics belong to the hnear dielectrics as the polarization is linearly dependent on the electric field. However, some-for example, KDP and Seignette salt-are nonhnear pyroelectrics-that is, they are ferroelectrics. 

Piezoelectric, pyroelectric and ferroelectric materials are often discussed simultaneously, owing to their interrelationship with each other at the crystalline structure level. For a crystalline structure to exhibit piezoelectricity, there should be no symmetry at the inversion centre for point group(s) (Tilley, 2006). A piezoelectric material can show both pyroelectricity (generation of electric charge on a crystal by change of temperature) and ferroelectricity (a property of certain materials that have a spontaneous electric polarisation). The relationship between different types of materials is shown in Figure 9.1. Ferroelectric materials are known to have superior piezoelectric properties over their non-feiroelectric counterparts. 

The helical smectic C state has the point symmetry (< 22), illustrated in Fig. 19, which does not permit a polar vector. It is therefore neither pyroelectric nor ferroelectric. Nor can it, of course, be piezoelectric, which is also easily realized after a glance at Fig. 14 if we apply a pressure or tension vertically, i.e. across the smectic layers (only in this direction can the liquid crystal sustain a strain), we may influence the pitch of the helix but no macroscopic po-... 

Piezoelectric effects were also studied in ferroelectric columnar liquid crystals, which have the same Ci symmetries as of the SmC materials. Piezoelectrical effects were observed also on various biological systems, in lyotropic liquid crystals and in membranes. ... 

All crystalline materials may be categorized into 32 crystallographic point groups. Of the 21 classes that lack a centre of symmetry, 20 produce an electric dipole (i.e. polarization) when mechanically stressed. These materials are termed piezoelectric, the lack of centrosymmetry being a necessary condition to allow movement of the positive and negative ions in order to produce a dipole. Ten of these classes possess a permanent dipole and will respond to changes in temperature as well as stress. These are defined as pyroelectric. These classes can be subdivided further into ferroelectric crystals, in which the dipole moments of the individual crystalline units can be reversed by application of an electric field. 

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