Abstract
In this section we introduce the basic concepts required to discuss structural phase transitions in crystals on the basis of a symmetry approach. The latter is in fact closely connected with the Landau theory of phase transitions, to which we come later in this chapter. However, for the reader who is more oriented toward domain properties without studying the nature of phase transitions themselves, it may be practical to become acquainted with the symmetry approach in the first place. The analysis of domain states on the basis of symmetry gives essential information on the number of domain states and on how they can be distinguished. It is this information that forms the background of any considerations about domain reorientation processes, about domain walls, as well as about properties and applications of multidomain samples.
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Notes
- 1.
Coupling a parent together with something distorted or low symmetry is somewhat ungraceful. Indeed Wadhawan’s (1982) daughter phase would be more elegant.
- 2.
Since these oscillations occur around the equilibrium positions that are temperature dependent, it would be more rigorous to call such oscillation quasi-harmonic.
- 3.
Spontaneous polarization at T TR may appear discontinuously. This is so for transitions of the first order, as discussed in detail in Sect. 2.3. Then the corresponding component of p at T TR is represented by a δ-function and this must be respected in integral (2.1.4). In real experiments, every component of P emerges continuously, be it because of even slightly inhomogeneous distribution of temperature in the measured sample or because a phase front between the two phases travels across the sample. Thus integrating the electric current gives a correct information on P S.
- 4.
In case of the first-order phase transition the reader should consider a remark similar to that connected with Eq. (2.1.4).
- 5.
If the elastic compatibility problem between the parent and ferroelastic phase is considered, within the framework of natural spontaneous strain approach, zero spontaneous strain should be ascribed to the parent phase.
- 6.
It is amusing how lavishly we throw around the prefix ferro, although nothing here has to do with iron. But this goes well back in history (Fousek, 1994). The Schrödinger’s proposal of the concept of a ferroelectric material was forgotten by the time Rochelle salt (Seignette salt) was discovered as the first ferroelectric material. Then the concept Seignette electric became common, to be later replaced first by the term ferro-dielectric and then ferroelectric. Quoting Megaw (1957) “… perhaps the real reason for its [i.e. of the term Seignette-electricity] rejection … is its failure to fit comfortably into the English language. As an adjective, >ferroelectric< is euphonius, while >Seignette-electric< grates on the ear.” Here the Russians seem to stick more to facts than to sounds: their ‘segniettoelektrichestvo’ survives and no one pretends to deal with iron when investigating barium titanate.
- 7.
In the chapter devoted to the thermodynamic theory we shall point out that there may be reasons for further and finer classification which includes proper ferroelectrics in contrast to pseudoproper or weak ferroelectrics. But this is of little importance for understanding domains on the basis of symmetry.
- 8.
As one can notice, the pair of notions of full and partial is very close to that of proper and improper; however, a comprehensive analysis show that these pairs are not identical (Janovec et al., 1975). For equitranslational transitions proper–improper and full–partial are not exactly identical concepts: here all proper ferroelectrics (ferroelastics) are necessarily full and all partial ferroelectrics (ferroelastics) are necessarily improper. However, there exist species which are full ferroelectrics (ferroelastics) in Aizu’s wording but may represent materials which must be classified as improper ferroelectrics (ferroelastics). The species no. 032 (see Table B.1 or C.1), i.e., \(m\overline 3 m - P\varepsilon ds - 1\), provides an example. Here the order parameter can transform as a polar vector, as indicated by the symbol P x , P y , P z in the fourth column of Table C.1. Clearly, in this case the material is proper ferroelectric. Alternatively (Janovec et al., 1975), the order parameter can transform as a third-rank tensor and polarization arises as a secondary effect, as indicated by the symbol (P x , P y , P z ) in Table C.1. In this case the material is an improper ferroelectric. In either case, this species is a full ferroelectric since all domain states can be distinguished by polarization.
- 9.
We remark that our twinning operations are what in Aizu’s analyses of domain states (or simply states in his terminology) has been referred to as F operations (Aizu, 1969, 1970b, 1972), a term used by other authors as well (Wadhawan, 1982).
- 10.
This interesting situation may have many practical consequences; some are connected to structures of domain walls, others relate to macroscopic phenomena.
- 11.
In everyday language, researchers in ferroelectricity often refer to γ pairs as to γ domains. But the plural is essential; sometimes one can see that just one domain is designated as a 90° domain or 180° domain, a ridiculous specification.
- 12.
This relates to the case of proper ferroelectrics only; see Sect. 2.1 for definition of proper and improper ferroelectrics.
- 13.
In BaTiO3, the theory should be slightly modified making allowance for the first-order phase transition.
- 14.
One comes across in the literature the use of condition (2.2.10) with matrix \(\Delta _{ij} \) defined as the difference between the squares of the spontaneous strain tensors in the domains. This condition is not justified unless it leads to results identical to those derived with the use of \(\Delta _{ij} \) defined by Eq. (2.2.11).
- 15.
In the paper of Sapriel (1975) the S walls have been renamed to W′ walls. Here we shall adhere to the original notation.
- 16.
Or, obviously, any strain, which differs from it by a tensor identical for all phases of the ferroic.
- 17.
Note that the phenomenologically introduced quantity P s exactly corresponds to spontaneous polarization introduced via the pyroelectric coefficient in Sect. 2.1.2 since, first, the temperature derivative of P s defined according to Eq. (2.3.8) equals the newly acquired component of the pyroelectric coefficient (by definition), second, the value of the spontaneous polarization is opposite for two of the domain states.
- 18.
We recall that natural spontaneous strain at a given temperature was defined as a deformation that should be imposed on the unit cell of the parent phase (real or extrapolated to this temperature) to get the unit cell of the ferroic phase. In the used thermodynamic approach the thermal expansion of the parent unit cell is described by the strain \(\varepsilon _n = - (\partial \Phi _0 /\partial \sigma _n )_P \). Thus one sees that the strain difference given by Eq. (2.3.25) does correspond to natural spontaneous strain.
- 19.
The factor of 2 multiplying Q 44 is introduced in this energy to respect the Voigt notation, as defined in Landolt–Bornstein (1993), where Q ijkl = Q mn for n = 1,2,3 and 2Q ijkl = Q mn for n = 4, 5, 6. In many papers where this energy is used (see, e.g., Haun et al., 1987), the factor in front of Q 44 is omitted; therefore, their Q 44 is twice the Q 44 defined in the textbooks.
- 20.
The stability condition in this case can be formulated as the requirement that all eigenvalues of the matrix \(\frac{{\partial ^2 \Phi }}{{\partial P_i \partial P_j }}\) calculated at the state be positive.
- 21.
The information on β and γ coefficients for BaTiO3 available in the literature is contradictory, e.g., the coefficients given in the classical book of Jona and Shirane (1962) correspond to a situation where the 4mm phase is always energetically favorable compared to the mm2 and 3m phases.
- 22.
For simplicity we are not discussing here the purely electronic contribution to the polarization; the incorporation of this contribution into the consideration would affect neither its logic nor the final conclusions.
- 23.
It is useful to note that, for the considered problem, this potential does not reach the minimum at equilibrium; however, as far as we are interested in the equation of state any potential can be employed for its derivation.
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Tagantsev, A.K., Cross, L.E., Fousek, J. (2010). Fundamentals of Ferroic Domain Structures. In: Domains in Ferroic Crystals and Thin Films. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-1417-0_2
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