Paraelectric state

The ferroelectricity usually disappears above a certain transition temperature (often called a Curie temperature) above which the crystal is said to be paraelectric this is because thermal motion has destroyed the ferroelectric order. Occasionally the crystal melts or decomposes before the paraelectric state is reached. There are thus some analogies to ferromagnetic and paramagnetic compounds though it should be noted that there is no iron in ferroelectric compounds. Some typical examples, together with their transition temperatures and spontaneous permanent electric polarization P, are given in the Table. 

Above a temperature called the Curie temperature, Tc, ferroelectric behavior is lost, and the material is said to be in the paraelectric state in which it resembles a normal insulator. 

At high temperatures, ferroelectric materials transform to the paraelectric state (where dipoles are randomly oriented), ferromagnetic materials to the paramagnetic state, and ferroelastic materials to the twin-free normal state. The transitions are characterized through order parameters (Rao Rao, 1978). These order parameters are characteristic properties parametrized in such a way that the resulting quantity is unity for the ferroic state at a temperature sufficiently below the transition temperature, and is zero in the nonferroic phase beyond the transition temperature. Polarization, magnetization and strain are the proper order parameters for the ferroelectric. 

Fig. 6.17 The position of Li+ and M (Nb5+, Ta5+) ions relative to the O2 planes (horizontal lines) in LiNbOj and LiTaOj in the two ferroelectric states and the paraelectric state.

When a single crystal in the paraelectric state transforms to a ferroelectric state of lower symmetry, stresses induced by dimensional changes can be relieved by domain twinning. For the tetragonal, pseudocubic BaTiOa, there are two possible modes of twinning 180° twins and 90° twins, with (100) and (101) respectively as composition planes. For the highly elongated structures considered in this paper, it may be expected that 180° twins could have (100)ss, (010)ss or (001)Ss as composition planes, but that 90° twins would be restricted to (110)ss as composition plane. 

The fact that an applied field can cause the polarisation to alter its direction implies that the atoms involved make only small movements and that the energy barrier between the different states is low. With increasing temperature the thermal motion of the atoms will increase, and eventually they can overcome the energy barrier separating the various orientations. Thus at high temperatures the distribution of atoms becomes statistical and the crystal behaves as a normal dielectric and no longer as a polar material. This is referred to as the paraelectric state. The temperature at which this occurs is known as the Curie temperature, Tc, or the transition temperature. The relative permittivity often rises to a sharp peak in the neighbourhood of Tb. 

The temperature dependence of the relative permittivity of many ferroelectric crystals in the paraelectric state can be described fairly accurately by a relationship called the Curie-Weiss law ... 

Figure 6.10 Temperature dependence of ferroelectricity (a) first-order and second-order transitions to the high-temperature paraelectric state (b) the variation of relative permittivity across the Curie temperature...

Figure 6.11 Curie behaviour of an ideal ferroelectric in the paraelectric state...

Ceramic PLZT has a number of structures, depending upon composition, and can show both the Pockels (linear) electro-optic effect in the ferroelectric rhombohedral and tetragonal phases and the Kerr (quadratic) effect in the cubic paraelectric state. Because of the ceramic nature of the material, the non-cubic phases show no birefringence in the as-prepared state and must be poled to become useful electro-optically (Section 6.4.1). PMN-PT and PZN-PT are relaxor ferroelectrics. These have an isotropic structure in the absence of an electric field, but this is easily altered in an applied electric field to give a birefringent electro-optic material. All of these phases, with optimised compositions, have much higher electro-optic coefficients than LiNb03 and are actively studied for device application. 

Ma, W., Cross, L.E. Flexoelectric polarization of barium strontium titanate in the paraelectric state. Appl. Phys. Lett. 81, 3440-3442 (2002)... 

Figure 18.16 Relaxation of the ferroelectric hysteresis loops in hard (doped with 1.0atom% Fe) PZT (58/42) ceramics, (a) For different cooling rates from the paraelectric state (b) The...

Figures 1 and 2 respectively show the temperature dependence of the relative permittivity and loss tangent of relaxor ferroelectric PLZT (9.5/65/35). As the temperature increases from -60°C to 100°C, the relative permittivity generally increased due to the unfreezing of domains. Between 0°C and 10°C, a broad peak can be seen in the lower frequency curves. This peak corresponds to the diffuse phase transition in this relaxor ceramic from the ferroelectric to the paraelectric state (also called the relaxor phase). Further heating continued to increase the relative dielectric permittivity until a maximum was achieved, at which point, the crystal s structure became cubic. This maximum in the permittivity, which is frequency dependent, occurs at the Curie temperature. Evidence of these phase transitions can also be seen in the loss tangent graph in figure 2.

In the absence of any constraints, the direction of Ps rotates from one smectic layer to the next, with a period equal to the smectic C pitch, and so the average polarization for a sample would be zero. However, surface treatment or application of a field can cause the helix to untwist, resulting in a permanently polarized sample. The spontaneous polarization arises from a preferred alignment of molecular dipole components which are perpendicular to the molecular long axis, but it behaves differently from the ferroelectric and ferromagnetic polarization characterised for crystals. The liquid crystalline ferroelectric phases identified so far are improper ferroelectrics, since the spontaneous polarization results from a symmetry constraint, whereas in proper ferroelectrics the polarization results from dipole -dipole interactions. The Curie-Weiss law for proper ferroelectrics predicts a second order phase transition at the Curie temperature from the high temperature paraelectric state to a permanently polarized ferroelectric state ... 

Gq(T) being the free energy of the paraelectric state. For small deformations, m , the interaction potential between the ions, can be considered as harmortic ... 

In the paraelectric state, T < T, the relative permittivity obeys the Curie-Weiss law (see section 11.2.2) and ... 

The explicit functional relationship between dielectric constant and electric field for a ferroelectric in the paraelectric state is derived by K. M. Johnson in 1961 based on Slater s and Devonshire s work [3,40,41]. In the light of the phenomenological theory of Devonshire [41], the free energy F is expanded as [3]... 

B T) is a phenomenological constant, and is the dielectric constant of vacuum. The dielectric constant follows Curie-Weiss law above the Curie temperature in the paraelectric state. Since the dielectric constant of BST S (T,E) we can express s, T,E) as [10]... 

Furthermore, it is also noticed, in Figure 5, that the symmetry of C-V curves is destroyed after the films electric-annealed. For the films annealed at +200 V, the negative part of the C-V curve exhibits slightly stronger tuning property than the positive part, whereas the films annealed at -200 V shows the opposite phenomenon. Such asymmetry of the C-V characteristics might also result from the appearance of the nano-polar-regions in the paraelectric state, since the similar asymmetries observed in ferroelectric films due to the hysteresis effects. 

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