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. 535-538

Section 3.4.4.6. Domain walls of finite thickness – continuous description

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

3.4.4.6. Domain walls of finite thickness – continuous description

| top | pdf |

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 [\eta ] of the order parameter (see Section 3.1.2[link] ). Two nonzero equilibrium homogeneous values of [-\eta_{0}] and [+\eta_{0}] of this parameter correspond to two domain states [{\bf S}_1] and [{\bf S}_2]. Spatial changes of the order parameter in a domain twin [({\bf S}_1|{\bf n}|{\bf S}_2) ] with a zero-thickness domain wall are described by a step-like function [\eta(\xi)=-\eta_0] for [\xi \,\lt\, 0] and [\eta(\xi)=+\eta_0 ] for [\xi\,\gt\,0], where [\xi] is the distance from the wall of zero thickness placed at [\xi=0].

A domain wall of finite thickness is described by a function [\eta(\xi) ] with limiting values [-\eta_{0}] and [\eta_{0} ]: [\lim_{\xi \rightarrow -\infty} \eta(\xi) = -\eta_{0}, \quad \lim_{\xi \rightarrow +\infty} \eta(\xi) = \eta_{0}.\eqno(3.4.4.23) ]If the wall is symmetric, then the profile [\eta(\xi)] in one half-space, say [\xi \,\lt\, 0], determines the profile in the other half-space [\xi\,\gt\,0]. For continuous [\eta(\xi)] fulfilling conditions (3.4.4.23[link]) this leads to the condition [\eta(\xi)=-\eta(-\xi), \eqno(3.4.4.24)]i.e. [\eta(\xi)] must be an odd function. This requirement is fulfilled if there exists a non-trivial symmetry operation of a domain wall (twin): a side reversal [(\xi \rightarrow -\xi)] combined with an exchange of domain states [[\eta(\xi) \rightarrow -\eta(\xi)]] results in an identical wall profile.

A particular form of the wall profile [\eta(\xi)] can be deduced from Landau theory. In the simplest case, the dependence [\eta(\xi)] of the domain wall would minimize the free energy [\int^{\infty}_{-\infty} \left(\Phi_{0} + \textstyle{{1}\over{2}}\alpha(T-T_c)\eta^{2} + \textstyle{{1}\over{4}}\beta\eta^4 + \textstyle{{1}\over{2}}\delta\left({{d^{2}\eta}\over{d\xi^{2}}}\right)^{2}\right)\,{\rm d}\xi, \eqno(3.4.4.25) ]where [\alpha,] [\beta,] [\delta] are phenomenological coefficients and T and Tc 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[link] ) and the last term expresses the energy of the spatially changing order parameter. This variational task with boundary conditions (3.4.4.23[link]) has the following solution (see e.g. Salje, 1990[link], 2000[link]b; Ishibashi, 1990[link]; Strukov & Levanyuk, 1998[link]) [\eta(\xi) = \eta_{0}\tanh({\xi}/{w}), \eqno(3.4.4.26)]where the value w specifies one half of the effective thickness [2w] of the domain wall and is given by [w = \sqrt{2\delta/\alpha(T_c-T)}. \eqno(3.4.4.27)]This dependence, expressed in relative dimensionless variables [\xi/w] and [\eta/\eta_{0}], is displayed in Fig. 3.4.4.6[link].

[Figure 3.4.4.6]

Figure 3.4.4.6 | top | pdf |

Profile of the one-component order parameter [\eta(\xi)] in a symmetric wall (S). The effective thickness of the wall is [2w].

The wall profile [\eta(\xi)] expressed by solution (3.4.4.26[link]) is an odd function of [\xi], [\eta(-\xi)=\eta_{0}\tanh({-\xi}/w)=-\eta_{0}\tanh({\xi}/w)=-\eta(\xi), \eqno(3.4.4.28) ]and fulfils thus the condition (3.4.4.24[link]) of a symmetric wall.

The wall thickness can be estimated from electron microscopy observations, or more precisely by a diffuse X-ray scattering technique (Locherer et al., 1998[link]). The effective thickness [2w] [see equation (3.4.4.26[link])] in units of crystallographic repetition length A normal to the wall ranges from [2w/A=2] to [2w/A=12], i.e. [2w] is about 10–100 nm (Salje, 2000[link]b). The temperature dependence of the domain wall thickness expressed by equation (3.4.4.27[link]) has been experimentally verified, e.g. on LaAlO3 (Chrosch & Salje, 1999[link]).

The energy [\sigma] of the domain wall per unit area equals the difference between the energy of the twin and the energy of the single-domain crystal. For a one nonzero component order parameter with the profile (3.4.4.26[link]), the wall energy [\sigma] is given by (Strukov & Levanyuk, 1998[link]) [\sigma = \int^{\infty}_{-\infty} \left[\Phi(\eta(\xi)) - \Phi(\eta_{0})\right]\,{\rm d}\xi = {{2\sqrt{2\delta}}\over{3\beta}}[\alpha(T_c-T)]^{3/2}, \eqno(3.4.4.29) ]where [2w] is the effective thickness of the wall [see equation (3.4.4.27[link])] and the coefficients are defined in equation (3.4.4.25[link]).

The order of magnitude of the wall energy [\sigma] of ferroelastic and non-ferroelastic domain walls is typically several millijoule per square metre (Salje, 2000[link]b).

Example 3.4.4.3.  In our example of a ferroelectric phase transition [4_z/m_zm_xm_{xy}\supset 2_xm_ym_z ], one can identify [\eta] with the [P_1] component of spontaneous polarization and [\xi] with the axis y. One can verify in Fig. 3.4.4.6[link] that the symmetry [{\sf T}_{12}[010]=\underline{2}_z^{\star}/m_z ] of the twin [({\bf S}_1[010]{\bf S}_2)] with a zero-thickness domain wall is retained in the domain wall with symmetric profile (3.4.4.26[link]): both non-trivial symmetry operations [\underline{2}_z^{\star} ] and [\underline{\bar1}^{\star}] transform the profile [\eta(y) ] into an identical function.

This example illustrates another feature of a symmetric wall: All non-trivial symmetry operations of the wall are located at the central plane [\xi=0 ] of the finite-thickness wall. The sectional group [{\sf T}_{12}] 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 off-centre planes [\xi\neq 0] is equal to the face group [\widehat{\sf F}_{12}] of the the layer group [{\sf T}_{12} ] (in our example [\widehat{\sf F}_{12}=\{1,m_z\}]).

The relation between a wall profile [\eta(\xi)] of a symmetric reversible (SR) wall and the profile [{\eta}^{\rm rev}(\xi) ] of the reversed wall is illustrated in Fig. 3.4.4.7[link], where the the dotted curve is the wall profile [{\eta}^{\rm rev}(\xi)] of the reversed wall. The profile [{\eta}^{\rm rev}(\xi)] of the reversed wall is completely determined by the the profile [{\eta}(\xi)] of the initial wall, since both profiles are related by equations [{\eta}^{\rm rev}(\xi)=-{\eta}(\xi)={\eta}(-\xi).\eqno(3.4.4.30) ]The first part of the equation corresponds to a state-exchanging operation [r_{12}^{\star}] (cf. point [r^{\star}{\rm A}] in Fig. 3.4.4.7[link]) and the second one to a side-reversing operation [\underline{s}_{12} ] (point [\underline{s}{\rm A}] in the same figure). In a symmetric reversible wall, both types of reversing operations exist (see Table 3.4.4.3[link]).

[Figure 3.4.4.7]

Figure 3.4.4.7 | top | pdf |

Profiles of the one-component order parameter [\eta(\xi)] 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[link]) but equations (3.4.4.30[link]) relating both profiles do not exist. The profiles [{\eta}(\xi)] and [{\eta}^{\rm rev}(\xi)] 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[link] (full curve). There is no relation between the negative part [{\eta}({\xi}) \,\lt\, 0] and positive part [{\eta}({\xi})\,\gt\,0] of the wall profile [{\eta}({\xi})]. Owing to the absence of non-trivial twin operations, there is no central plane with higher symmetry. The local symmetry (sectional layer group) at any location [\xi] within the wall is equal to the face group [\widehat{\sf F}_{12} ]. This is also the global symmetry [{\sf T}_{12}] of the entire wall, [{\sf T}_{12}=\widehat{\sf F}_{12}].

[Figure 3.4.4.8]

Figure 3.4.4.8 | top | pdf |

Profiles of the one-component order parameter [\eta(\xi)] in an asymmetric wall (solid curve) and in the reversed asymmetric wall (dotted curve). The wall is asymmetric and state-reversible ([{\rm AR}^{\star}]).

The dotted curve in Fig. 3.4.4.8[link] represents the reversed-wall profile of an asymmetric state-reversible (AR[^{\star}]) wall that is related to the initial wall by state-exchanging operations [r_{12}^{\star}\widehat{\sf F}_{12}] (see Table 3.4.4.5[link]), [{\eta}^{\rm rev}(\xi)=-{\eta}(\xi).\eqno(3.4.4.31) ]

An example of an asymmetric side-reversible (A[\underline {\rm R} ]) wall is shown in Fig. 3.4.4.9[link]. In this case, an asymmetric wall (full curve) and reversed wall (dotted curve) are related by side-reversing operations [\underline{s}_{12}\widehat{\sf F}_{12} ]: [{\eta}^{\rm rev}(\xi)={\eta}(-\xi). \eqno(3.4.4.32) ]

[Figure 3.4.4.9]

Figure 3.4.4.9 | top | pdf |

Profiles of the one-component order parameter [\eta(\xi)] in an asymmetric wall (solid curve) and in the reversed asymmetric wall (dotted curve). The wall is asymmetric and side-reversible (A[\underline{\rm R} ]).

In an asymmetric irreversible (AI) wall, both profiles [{\eta}(\xi)] and [{\eta}^{\rm rev}(\xi)] are asymmetric and there is no relation between these two profiles.

The symmetry [{\sf T}_{12}(\eta)] of a finite-thickness wall with a profile [\eta(\xi)] is equal to or lower than the symmetry [{\sf T}_{12}] of the corresponding zero-thickness domain wall, [{\sf T}_{12}\supseteq {\sf T}_{12}(\eta) ]. A symmetry descent [{\sf T}_{12}\supset {\sf T}_{12}(\eta)] can be treated as a phase transition in the domain wall (see e.g. Bul'bich & Gufan, 1989a[link],b[link]; Sonin & Tagancev, 1989[link]). There are [n_{W(\eta)}] equivalent structural variants of the finite-thickness domain wall with the same orientation and the same energy but with different structures of the wall, [n_{W(\eta)}=[{\sf T}_{12}:{\sf T}_{12}(\eta)]=|{\sf T}_{12}|:|{\sf T}_{12}(\eta)|. \eqno(3.4.4.33)]

Domain-wall variants – two-dimensional analogues of domain states – can coexist and meet along line defects – one-dimensional analogues of a domain wall (Tagancev & Sonin, 1989[link]).

Symmetry descent in domain walls of finite thickness may occur if the order parameter [\eta] has more than one nonzero component. We can demonstrate this on ferroic phases with an order parameter with two components [\eta_1 ] and [\eta_2]. The profiles [\eta_1(\xi)] and [\eta_2(\xi) ] can be found, as for a one-component order parameter, from the corresponding Landau free energy (see e.g. Cao & Barsch, 1990[link]; Houchmandzadeh et al., 1991[link]; Ishibashi, 1992[link], 1993[link]; Rychetský & Schranz, 1993[link], 1994[link]; Schranz, 1995[link]; Huang et al., 1997[link]; Strukov & Levanyuk, 1998[link]; Hatt & Hatch, 1999[link]; Hatch & Cao, 1999[link]).

Let us denote by [{\sf T}_{12}(\eta_1)] the symmetry of the profile [\eta_1(\xi)] and by [{\sf T}_{12}(\eta_2)] the symmetry of the profile [\eta_2(\xi)]. Then the symmetry of the entire wall [{\sf T}_{12}(\eta) ] is a common part of the symmetries [{\sf T}_{12}(\eta_1)] and [{\sf T}_{12}(\eta_2)], [{\sf T}_{12}(\eta)={\sf T}_{12}(\eta_1) \cap {\sf T}_{12}(\eta_2). \eqno(3.4.4.34) ]

Example 3.4.4.4.  In our illustrative phase transition [4_z/m_zm_xm_{xy}] [\supset] [2_xm_ym_z], the order parameter has two components [\eta_1, \eta_2] that can be associated with the x and y components [P_1] and [P_2] of the spontaneous polarization (see Table 3.1.3.1[link] and Fig. 3.4.2.2[link]). We have seen that the domain wall [[{\bf S}_1[010]{\bf S}_2]] of zero thickness has the symmetry [{\sf T}_{12}=\underline{2}_z^{\star}/m_z]. If one lets [\eta_1(y)] relax and keeps [\eta_2(y)=0] (a so-called linear structure), then [{\sf T}_{12}(\eta_1)=\underline{2}_z^{\star}/m_z ] (see Fig. 3.4.4.2[link] with [\xi=y]). If the last condition is lifted, a possible profile of a relaxed [\eta_2(y)] is depicted by the full curve in Fig. 3.4.4.10[link]. If both components [\eta_1(y)] and [\eta_2(y)] 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 [\eta_2(-y)=\eta_2(y)] has the symmetry [{\sf T}_{12}(\eta_2)=m_x^{\star}2_y^{\star}m_z ]. Hence, according to (3.4.4.34[link]), the symmetry of a relaxed wall with a rotational structure is [{\sf T}_{12}(\eta) =] [\underline{2}_z^{\star}/m_z ] [\cap] [m_x^{\star}2_y^{\star}m_z=] [\{1,m_z\}]. This is an asymmetric state-reversible (AR[^{\star}]) wall with two chiral variants [see equation (3.4.4.33[link])] that are related by [\underline{\bar 1}^{\star}] and [\underline{2}_z^{\star} ]; the profile [\eta_2(y)] of the second variant is depicted in Fig. 3.4.4.10[link] by a dashed curve.

[Figure 3.4.4.10]

Figure 3.4.4.10 | top | pdf |

A profile of the second order parameter component in a degenerate domain wall.

Similarly, one gets for a zero-thickness domain wall [[{\bf S}_1[001]{\bf S}_2] ] perpendicular to z the symmetry [{\sf T}_{12}=\underline{2}_y^{\star}/m_y ]. For a relaxed domain wall with profiles [\eta_1(z)] and [\eta_2(z)], displayed in Figs. 3.4.4.6[link] and 3.4.4.10[link] with [\xi=z], one gets [{\sf T}_{12}(\eta_1)=\underline{2}_y^{\star}/m_y ], [{\sf T}_{12}(\eta_2)] [=m_x^{\star}\underline{2}_y^{\star}m_z] and [{\sf T}_{12}(\eta)=\{1,\underline{2}_y^{\star}\}]. The relaxed domain wall with rotational structure has lower symmetry than the zero-thickness 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 right-handed and left-handed variants are related by operations [m_z] and [\underline{\bar 1}^{\star} ]. 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[link],b[link]) and by Sonin & Tagancev (1989[link]).

References

Bul'bich, A. A. & Gufan, Yu. M. (1989a). Inevitable symmetry lowering in a domain wall near a reordering phase transition. Sov. Phys. JETP, 67, 1153–1157.
Bul'bich, A. A. & Gufan, Yu. M. (1989b). Phase transition in domain walls. Ferroelectrics, 172, 351–359.
Cao, W. & Barsch, G. R. (1990). Landau–Ginzburg model of interphase boundaries in improper ferroelastic perovskites of [D{_{4h}^{18}}] symmetry. Phys. Rev. B, 41, 4334–4348.
Chrosch, J. & Salje, E. K. H. (1999). Temperature dependence of the domain wall width in LaAlO3. J. Appl. Phys. 85, 722–727.
Hatch, D. M. & Cao, W. (1999). Determination of domain and domain wall formation at ferroic transitions. Ferroelectrics, 222, 1–10.
Hatt, R. A. & Hatch, D. M. (1999). Order parameter profiles in ferroic phase transitions. Ferroelectrics, 226, 61–78.
Houchmandzadeh, B., Lajzerowicz, J. & Salje, E. K. H. (1991). Order parameter coupling and chirality of domain walls. J. Phys. Condens. Matter, 3, 5163–5169.
Huang, X. R., Jiang, S. S., Hu, X. B. & Liu, W. J. (1997). Theory of twinning structures in the orthorhombic phase of ferroelectric perovskites. J. Phys. Condens. Matter, 9, 4467–4482.
Ishibashi, Y. (1990). Structure and physical properties of domain walls. Ferroelectrics, 104, 299–310.
Ishibashi, Y. (1992). Domain walls in crystals with incommensurate phases. II. J. Phys. Soc. Jpn, 61, 357–362.
Ishibashi, Y. (1993). The 90°-wall in the tetragonal phase of BaTiO3-type ferroelectrics. J. Phys. Soc. Jpn, 62, 1044–1047.
Locherer, K. R., Chrosch, J. & Salje, E. K. H. (1998). Diffuse X-ray scattering in WO3. Phase Transit. 67, 51–63.
Rychetský, I. & Schranz, W. (1993). Antiphase boundaries in Hg2Br2 and KSCN. J. Phys. Condens. Matter, 5, 1455–1472.
Rychetský, I. & Schranz, W. (1994). Ferroelastic domain walls in Hg2Br2 and KSCN. J. Phys. Condens. Matter, 6, 11159–11165.
Salje, E. K. H. (1990). Phase Transitions in Ferroelastic and Co-elastic Crystals, 1st ed. Cambridge University Press.
Salje, E. K. H. (2000b). Ferroelasticity. Contemp. Phys. 41, 79–91.
Schranz, W. (1995). Domains and interfaces near ferroic phase transitions. Key Eng. Mater. 101102, 41–60.
Sonin, E. B. & Tagancev, A. K. (1989). Structure and phase transitions in antiphase boundaries of improper ferroelectrics. Ferroelectrics, 98, 291–295.
Strukov, B. A. & Levanyuk, A. P. (1998). Ferroelectric Phenomena in Crystals. Berlin: Springer.
Tagancev, A. R. & Sonin, E. B. (1989). Linear singularities and their motion in improper ferroelectrics. Ferroelectrics, 98, 297–300.








































to end of page
to top of page