International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

International Tables for Crystallography (2013). Vol. D, ch. 3.4, pp. 484-485

Section 3.4.1.1. Basic concepts

V. Janoveca* and J. Přívratskáb

aInstitute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, CZ-18221 Prague 8, Czech Republic, and bDepartment of Mathematics and Didactics of Mathematics, Technical University of Liberec, Hálkova 6, 461 17 Liberec 1, Czech Republic
Correspondence e-mail:  janovec@fzu.cz

3.4.1.1. Basic concepts

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It was demonstrated in Section 3.1.2[link] that a characteristic feature of structural phase transitions connected with a lowering of crystal symmetry is an anomalous behaviour near the transition, namely unusually large values of certain physical properties that vary strongly with temperature. In this chapter, we shall deal with another fundamental feature of structural phase transitions associated with symmetry lowering: the formation of a non-homogeneous, textured low-symmetry phase called a domain structure.

When a crystal homogeneous in the parent (prototypic, high-symmetry) phase undergoes a phase transition into a ferroic low-symmetry phase with lower point-group symmetry, then this ferroic phase is almost always formed as a non-homogeneous structure consisting of homogeneous regions called domains and contact regions between domains called domain walls. All domains have the same or the enantiomorphous crystal structure of the ferroic phase, but this structure has in different domains a different orientation, and sometimes also a different position in space. When a domain structure is observed by a measuring instrument, different domains can exhibit different tensor properties, different diffraction patterns and can differ in other physical properties. The domain structure can be visualized optically (see Fig. 3.4.1.1[link]) or by other experimental techniques. Powerful high-resolution electron microscopy (HREM) techniques have made it possible to visualize atomic arrangements in domain structures (see Fig. 3.4.1.2[link]). The appearance of a domain structure, detected by any reliable technique, provides the simplest unambiguous experimental proof of a structural phase transition.

[Figure 3.4.1.1]

Figure 3.4.1.1 | top | pdf |

Domain structure of tetragonal barium titanate (BaTiO3). A thin section of barium titanate ceramic observed at room temperature in a polarized-light microscope (transmitted light, crossed polarizers). Courtesy of U. Täffner, Max-Planck-Institut für Metallforschung, Stuttgart. Different colours correspond to different ferroelastic domain states, connected areas of the same colour are ferroelastic domains and sharp boundaries between these areas are domain walls. Areas of continuously changing colour correspond to gradually changing thickness of wedge-shaped domains. An average distance between parallel ferroelastic domain walls is of the order of 1–10 µm.

[Figure 3.4.1.2]

Figure 3.4.1.2 | top | pdf |

Domain structure of a BaGa2O4 crystal seen by high-resolution transmission electron microscopy. Parallel rows are atomic layers. Different directions correspond to different ferroelastic domain states of domains, connected areas with parallel layers are different ferroelastic domains and boundaries between these areas are ferroelastic domain walls. Courtesy of H. Lemmens, EMAT, University of Antwerp.

Under the influence of external fields (mechanical stress, electric or magnetic fields, or combinations thereof), the domain structure can change; usually some domains grow while others decrease in size or eventually vanish. This process is called domain switching. After removing or decreasing the field a domain structure might not change considerably, i.e. the form of a domain pattern depends upon the field history: the domain structure exhibits hysteresis (see Fig. 3.4.1.3[link]). In large enough fields, switching results in a reduction of the number of domains. Such a procedure is called detwinning. In rare cases, the crystal may consist of one domain only. Then we speak of a single-domain crystal.

[Figure 3.4.1.3]

Figure 3.4.1.3 | top | pdf |

Elastic hysteresis of ferroelastic lead phosphate Pb3(PO4)2 (Salje, 1990[link]). Courtesy of E. K. H. Salje, University of Cambridge. The dependence of strain on applied stress has the form of a loop. The states at the extreme left and right correspond to two ferroelastic domain states, steep parts of the loop represent switching of one state into the other by applied stress. The strain at zero stress corresponds to the last single-domain state formed in a field larger than the coercive stress defined by the stress at zero strain (the intersection of the loop with the axis of the applied stress). Similar dielectric hysteresis loops of polarization versus applied electric field are observed in ferroelectric phases (see e.g. Jona & Shirane, 1962[link]).

There are two basic types of domain structures:

  • (i) Domain structures with one or several systems of parallel plane domain walls that can be observed in an optical or electron microscope. Two systems of perpendicular domain walls are often visible (see Fig. 3.4.1.4[link]). In polarized light, domains exhibit different colours (see Fig. 3.4.1.1[link]) and in diffraction experiments splitting of reflections can be observed (see Fig. 3.4.3.9[link]). Domains can be switched by external mechanical stress. These features are typical for a ferroelastic domain structure in which neighbouring domains differ in mechanical strain (deformation). Ferroelastic domain structures can appear only in ferroelastic phases, i.e. as a result of a phase transition characterized by a decrease in the number of independent strain components (see Table 3.4.2.2[link]).

    [Figure 3.4.1.4]

    Figure 3.4.1.4 | top | pdf |

    Transmission electron microscopy image of the ferroelastic domain structure in a YBa2Cu3O7−y crystal (Rosová, 1999[link]). Courtesy of A. Rosová, Institute of Electrical Engineering, SAS, Bratislava. There are two systems (`complexes'), each of which is formed by almost parallel ferroelastic domain walls with needle-like tips. The domain walls in one complex are nearly perpendicular to the domain walls in the other complex.

  • (ii) Domain structures that are not visible using a polarized-light microscope and in whose diffraction patterns no splitting of reflections is observed. Special methods [e.g. etching, deposition of liquid crystals (see Fig. 3.4.1.5[link]), electron or atomic force microscopy, or higher-rank optical effects (see Fig. 3.4.3.3[link])] are needed to visualize domains. Domains have the same strain and cannot usually be switched by an external mechanical stress. Such domain structures are called non-ferroelastic domain structures. They appear in all non-ferroelastic phases resulting from sym­metry lowering that preserves the number of independent strain components (see Table 3.4.2.2[link]).

    [Figure 3.4.1.5]

    Figure 3.4.1.5 | top | pdf |

    Non-ferroelastic ferroelectric domains in triglycine sulfate (TGS) revealed by a liquid-crystal method. A thin layer of a nematic liquid crystal deposited on a crystal surface perpendicular to the spontaneous polarization is observed in a polarized-light microscope. Black and white areas correspond to ferroelectric domains with antiparallel spontaneous polarization. The typical size of the domains is of order of 1–10µm. Courtesy of M. Połomska, Institute of Molecular Physics, PAN, Poznań. Although one preferential direction of domain walls prevails, the rounded shapes of the domains indicate that all orientations of non-ferroelastic walls are possible.

Another important kind of domain structure is a ferroelectric domain structure, in which domains differ in the direction of the spontaneous polarization. Such a domain structure is formed at ferroelectric phase transitions that are characterized by the appearance of a new polar direction in the ferroic phase. Ferroelectric domains can usually be switched by external electric fields. Two ferroelectric domains with different directions of spontaneous polarization can have different spontaneous strain [e.g. in dihydrogen phosphate (KDP) crystals, two ferroelectric domains with opposite directions of the spontaneous polarization have different spontaneous shear strain], or two ferroelectric domains with antiparallel spontaneous polarization can possess the same strain [e.g. in triglycine sulfate (TGS) crystals].

The physical properties of polydomain crystals are significantly influenced by their domain structure. The values of important material property tensor components, e.g. permittivity, piezoelectric and elastic coefficients, may be enhanced or diminished by the presence of a domain structure. Owing to switching and detwinning phenomena, polydomain materials exhibit hysteresis of material properties. These features have important practical implications, e.g. the production of anisotropic ceramic materials or ferroelectric memories.

The domain structure resulting from a structural phase transition belongs to a special type of twinning referred to as transformation twinning (see Section 3.3.7.2[link] ). Despite this, the current terminology used in domain-structure studies is different. The main terms were coined during the first investigations of ferroelectric materials, where striking similarities with the behaviour of ferromagnetic materials led researchers to introduce terms analogous to those used in studies of ferromagnetic domain structures that had been examined well at that time.

Bicrystallography (see Section 3.2.2[link] ) provides another possible frame for discussing domain structures. Bicrystallography and domain-structure analysis have developed independently and almost simultaneously but different language has again precluded deeper confrontation. Nevertheless, there are common features in the methodology of both approaches, in particular, the principle of symmetry compensation (see Section 3.2.2[link] ), which plays a fundamental role in both theories.

In Chapter 3.1[link] , it is shown that the anomalous behaviour near phase transitions can be explained in the framework of the Landau theory. In this theory, the formation of the domain structures follows from the existence of several equivalent solutions for the order parameter. This result is a direct consequence of a symmetry reduction at a ferroic phase transition. It is this dissymmetrization which is the genuine origin of the domain-structure formation and which determines the basic static features of all domain structures.








































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