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

International Tables for Crystallography (2013). Vol. D, ch. 3.4, pp. 535-538

Section 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: Domain walls of finite thickness – continuous description

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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( ]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 ([link]) this leads to the condition [\eta(\xi)=-\eta(-\xi), \eqno(]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( ]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 ([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(]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(]This dependence, expressed in relative dimensionless variables [\xi/w] and [\eta/\eta_{0}], is displayed in Fig.[link].


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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 ([link]) is an odd function of [\xi], [\eta(-\xi)=\eta_{0}\tanh({-\xi}/w)=-\eta_{0}\tanh({\xi}/w)=-\eta(\xi), \eqno( ]and fulfils thus the condition ([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 ([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 ([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 ([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( ]where [2w] is the effective thickness of the wall [see equation ([link])] and the coefficients are defined in equation ([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  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.[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 ([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.[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( ]The first part of the equation corresponds to a state-exchanging operation [r_{12}^{\star}] (cf. point [r^{\star}{\rm A}] in Fig.[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[link]).


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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 ([link]) but equations ([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.[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}].


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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.[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[link]), [{\eta}^{\rm rev}(\xi)=-{\eta}(\xi).\eqno( ]

An example of an asymmetric side-reversible (A[\underline {\rm R} ]) wall is shown in Fig.[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( ]


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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(]

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( ]

Example  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[link] and Fig.[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.[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.[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 ([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 ([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.[link] by a dashed curve.


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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.[link] and[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]).


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