International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2013 
International Tables for Crystallography (2013). Vol. D, ch. 3.4, pp. 535538
Section 3.4.4.6. Domain walls of finite thickness – continuous description^{a}Institute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, CZ18221 Prague 8, Czech Republic, and ^{b}Department of Mathematics and Didactics of Mathematics, Technical University of Liberec, Hálkova 6, 461 17 Liberec 1, Czech Republic 
A domain wall of zero thickness is a geometrical construct that enabled us to form a twin from a domain pair and to find a layer group that specifies the maximal symmetry of that twin. However, real domain walls have a finite, though small, thickness. Spatial changes of the structure within a wall may, or may not, lower the wall symmetry and can be conveniently described by a phenomenological theory.
We shall consider the simplest case of a one nonzero component of the order parameter (see Section 3.1.2 ). Two nonzero equilibrium homogeneous values of and of this parameter correspond to two domain states and . Spatial changes of the order parameter in a domain twin with a zerothickness domain wall are described by a steplike function for and for , where is the distance from the wall of zero thickness placed at .
A domain wall of finite thickness is described by a function with limiting values and : If the wall is symmetric, then the profile in one halfspace, say , determines the profile in the other halfspace . For continuous fulfilling conditions (3.4.4.23) this leads to the condition i.e. must be an odd function. This requirement is fulfilled if there exists a nontrivial symmetry operation of a domain wall (twin): a side reversal combined with an exchange of domain states results in an identical wall profile.
A particular form of the wall profile can be deduced from Landau theory. In the simplest case, the dependence of the domain wall would minimize the free energy where are phenomenological coefficients and T and T_{c} are the temperature and the temperature of the phase transition, respectively. The first three terms correspond to the homogeneous part of the Landau free energy (see Section 3.2.1 ) and the last term expresses the energy of the spatially changing order parameter. This variational task with boundary conditions (3.4.4.23) has the following solution (see e.g. Salje, 1990, 2000b; Ishibashi, 1990; Strukov & Levanyuk, 1998) where the value w specifies one half of the effective thickness of the domain wall and is given by This dependence, expressed in relative dimensionless variables and , is displayed in Fig. 3.4.4.6.

Profile of the onecomponent order parameter in a symmetric wall (S). The effective thickness of the wall is . 
The wall profile expressed by solution (3.4.4.26) is an odd function of , and fulfils thus the condition (3.4.4.24) of a symmetric wall.
The wall thickness can be estimated from electron microscopy observations, or more precisely by a diffuse Xray scattering technique (Locherer et al., 1998). The effective thickness [see equation (3.4.4.26)] in units of crystallographic repetition length A normal to the wall ranges from to , i.e. is about 10–100 nm (Salje, 2000b). The temperature dependence of the domain wall thickness expressed by equation (3.4.4.27) has been experimentally verified, e.g. on LaAlO_{3} (Chrosch & Salje, 1999).
The energy of the domain wall per unit area equals the difference between the energy of the twin and the energy of the singledomain crystal. For a one nonzero component order parameter with the profile (3.4.4.26), the wall energy is given by (Strukov & Levanyuk, 1998) where is the effective thickness of the wall [see equation (3.4.4.27)] and the coefficients are defined in equation (3.4.4.25).
The order of magnitude of the wall energy of ferroelastic and nonferroelastic domain walls is typically several millijoule per square metre (Salje, 2000b).
Example 3.4.4.3. In our example of a ferroelectric phase transition , one can identify with the component of spontaneous polarization and with the axis y. One can verify in Fig. 3.4.4.6 that the symmetry of the twin with a zerothickness domain wall is retained in the domain wall with symmetric profile (3.4.4.26): both nontrivial symmetry operations and transform the profile into an identical function.
This example illustrates another feature of a symmetric wall: All nontrivial symmetry operations of the wall are located at the central plane of the finitethickness wall. The sectional group of this plane thus expresses the symmetry of the central layer and also the global symmetry of a symmetric wall (twin). The local symmetry of the offcentre planes is equal to the face group of the the layer group (in our example ).
The relation between a wall profile of a symmetric reversible (SR) wall and the profile of the reversed wall is illustrated in Fig. 3.4.4.7, where the the dotted curve is the wall profile of the reversed wall. The profile of the reversed wall is completely determined by the the profile of the initial wall, since both profiles are related by equations The first part of the equation corresponds to a stateexchanging operation (cf. point in Fig. 3.4.4.7) and the second one to a sidereversing operation (point in the same figure). In a symmetric reversible wall, both types of reversing operations exist (see Table 3.4.4.3).

Profiles of the onecomponent order parameter in a symmetric wall (solid curve) and in the reversed wall (dotted curve). The wall is symmetric and reversible (SR). 
In a symmetric irreversible (SI) wall both initial and reversed wall profiles fulfil symmetry condition (3.4.4.24) but equations (3.4.4.30) relating both profiles do not exist. The profiles and may differ in shape and surface wall energy. Charged domain walls are always irreversible.
A possible profile of an asymmetric domain wall is depicted in Fig. 3.4.4.8 (full curve). There is no relation between the negative part and positive part of the wall profile . Owing to the absence of nontrivial twin operations, there is no central plane with higher symmetry. The local symmetry (sectional layer group) at any location within the wall is equal to the face group . This is also the global symmetry of the entire wall, .

Profiles of the onecomponent order parameter in an asymmetric wall (solid curve) and in the reversed asymmetric wall (dotted curve). The wall is asymmetric and statereversible (). 
The dotted curve in Fig. 3.4.4.8 represents the reversedwall profile of an asymmetric statereversible (AR) wall that is related to the initial wall by stateexchanging operations (see Table 3.4.4.5),
An example of an asymmetric sidereversible (A) wall is shown in Fig. 3.4.4.9. In this case, an asymmetric wall (full curve) and reversed wall (dotted curve) are related by sidereversing operations :

Profiles of the onecomponent order parameter in an asymmetric wall (solid curve) and in the reversed asymmetric wall (dotted curve). The wall is asymmetric and sidereversible (A). 
In an asymmetric irreversible (AI) wall, both profiles and are asymmetric and there is no relation between these two profiles.
The symmetry of a finitethickness wall with a profile is equal to or lower than the symmetry of the corresponding zerothickness domain wall, . A symmetry descent can be treated as a phase transition in the domain wall (see e.g. Bul'bich & Gufan, 1989a,b; Sonin & Tagancev, 1989). There are equivalent structural variants of the finitethickness domain wall with the same orientation and the same energy but with different structures of the wall,
Domainwall variants – twodimensional analogues of domain states – can coexist and meet along line defects – onedimensional analogues of a domain wall (Tagancev & Sonin, 1989).
Symmetry descent in domain walls of finite thickness may occur if the order parameter has more than one nonzero component. We can demonstrate this on ferroic phases with an order parameter with two components and . The profiles and can be found, as for a onecomponent order parameter, from the corresponding Landau free energy (see e.g. Cao & Barsch, 1990; Houchmandzadeh et al., 1991; Ishibashi, 1992, 1993; Rychetský & Schranz, 1993, 1994; Schranz, 1995; Huang et al., 1997; Strukov & Levanyuk, 1998; Hatt & Hatch, 1999; Hatch & Cao, 1999).
Let us denote by the symmetry of the profile and by the symmetry of the profile . Then the symmetry of the entire wall is a common part of the symmetries and ,
Example 3.4.4.4. In our illustrative phase transition , the order parameter has two components that can be associated with the x and y components and of the spontaneous polarization (see Table 3.1.3.1 and Fig. 3.4.2.2). We have seen that the domain wall of zero thickness has the symmetry . If one lets relax and keeps (a socalled linear structure), then (see Fig. 3.4.4.2 with ). If the last condition is lifted, a possible profile of a relaxed is depicted by the full curve in Fig. 3.4.4.10. If both components and are nonzero within the wall, one speaks about a rotational structure of domain wall. In this relaxed domain wall the spontaneous polarization rotates in the plane (001), resembling thus a Néel wall in magnetic materials. The even profile has the symmetry . Hence, according to (3.4.4.34), the symmetry of a relaxed wall with a rotational structure is . This is an asymmetric statereversible (AR) wall with two chiral variants [see equation (3.4.4.33)] that are related by and ; the profile of the second variant is depicted in Fig. 3.4.4.10 by a dashed curve.
Similarly, one gets for a zerothickness domain wall perpendicular to z the symmetry . For a relaxed domain wall with profiles and , displayed in Figs. 3.4.4.6 and 3.4.4.10 with , one gets , and . The relaxed domain wall with rotational structure has lower symmetry than the zerothickness wall or the wall with linear structure, but remains a symmetric and reversible (SR) domain wall in which spontaneous polarization rotates in a plane (001), resembling thus a Bloch wall in magnetic materials. Two chiral righthanded and lefthanded variants are related by operations and . This example illustrates that the structure of domain walls may differ with the wall orientation.
We note that the stability of a domain wall with a rotational structure and with a linear structure depends on the values of the coefficients in the Landau free energy, on temperature and on external fields. In favourable cases, a phase transition from a symmetric linear structure to a less symmetric rotational structure can occur. Such phase transitions in domain walls have been studied theoretically by Bul'bich & Gufan (1989a,b) and by Sonin & Tagancev (1989).
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