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

International Tables for Crystallography (2013). Vol. D, ch. 3.4, pp. 533-535

Section Non-ferroelastic domain twins and domain walls

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: Non-ferroelastic domain twins and domain walls

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Compatibility conditions impose no restriction on the orientation of non-ferroelastic domain walls. Any of the non-ferro­elastic domain pairs listed in Table[link] can be sectioned on any crystallographic plane p and the sectional group [\overline{\sf J}_{1j}] specifies the symmetry properties of the corresponding twin and domain wall. The analysis can be confined to one representative orientation of each class of equivalent planes, but a listing of all possible cases is too voluminous for the present article. We give, therefore, in Table[link] only possible symmetries [{\sf T}_{1j}] and [\overline{\sf J}_{1j}] of non-ferro­elastic domain twins and walls, together with their classification, without specifying the orientation of the wall plane p.

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Symmetries of non-ferroelastic domain twins and walls

[{\sf T}_{1j}][{\sf \overline{J}}_{1j} ]Classification
1 1 AI
1 [\underline{\bar1}] [{\rm A\underline{R}}]
[\underline2 ] [{\rm A\underline{R}}]
[2^{\star} ] [{\rm AR}^{\star}]
[m^{\star}] [{\rm AR}^{\star}]
[{\bar1}^\star] [{\bar1}^\star ] SI
[\underline2/m^{\star} ] SR
[2^{\star}/\underline{m} ] SR
[2 ] [2m^{\star}m^{\star} ] [{\rm AR}^{\star}]
[\underline2^{\star} ] [\underline2^{\star} ] SI
[\underline2^{\star}/m^{\star} ] SR
[2^{\star}\underline2^{\star}\underline2 ] SR
[\underline2^{\star}\underline{m}m^{\star} ] SR
m m AI
[\underline2/m ] [{\rm A\underline{R}}]
[2^{\star}mm^{\star} ] [{\rm AR}^{\star}]
[\underline{m}^{\star} ] [\underline{m}^{\star} ] SI
[2^{\star}/\underline{m}^{\star} ] SR
[\underline{m}^{\star}m^{\star}\underline2 ] SR
[\underline2^{\star}/m ] [\underline2^{\star}/m ] SI
[m\underline{m}m^{\star} ] SR
[2/\underline{m}^{\star} ] [2/\underline{m}^{\star} ] SI
[m^{\star}m^{\star}\underline{m}^{\star} ] SR
[4^{\star}/\underline{m}^{\star} ] SR
[2\underline2^{\star}\underline2^{\star} ] [2\underline2^{\star}\underline2^{\star} ] SI
[\underline{m}m^{\star}m^{\star} ] SR
[4^\star {\underline 2} {\underline 2}^\star ] SR
[\underline{\bar4}\underline2^{\star}m^{\star} ] SR
[\underline{m}^{\star}m\underline2^{\star} ] [\underline{m}^{\star}m\underline2^{\star} ] SI
[\underline{m}^{\star}mm^{\star}] SR
[mm\underline{m}^{\star} ] [mm\underline{m}^{\star} ] SI
[4^{\star}/\underline{m}^{\star}m^{\star}m ] SR
[4 ] [4m^{\star}m^{\star}] [{\rm AR}^{\star}]
[\underline{\bar4}^{\star} ] [\underline{\bar4}^{\star}\underline2m^{\star} ] SR
[4/\underline{m}^{\star} ] [4/\underline{m}^{\star} ] SI
[4/\underline{m}^{\star}m^{\star}m^{\star} ] SR
[4\underline2^{\star}\underline2^{\star} ] [4\underline2^{\star}\underline2^{\star} ] SI
[4/\underline{m}m^{\star}m^{\star} ] SR
[\underline{\bar4}^{\star}\underline2^{\star}m ] [4^{\star}/\underline{m}m^{\star}m] SR
[4/\underline{m}^{\star}mm ] [4/\underline{m}^{\star}mm ] SI
[3 ] [3m^{\star} ] [{\rm AR}^{\star}]
[6^{\star} ] [{\rm AR}^{\star}]
[\underline{\bar3}^{\star} ] [\underline{\bar3}^{\star} ] SI
[\underline{\bar3}^{\star} m^{\star} ] SR
[6^{\star}/\underline{m} ] SR
[3m ] [6^{\star}mm^{\star} ] [{\rm AR}^{\star}]
[3\underline2^{\star} ] [3\underline2^{\star} ] SI
[\underline{\bar3}m^{\star} ] SR
[6^{\star}\underline2\underline2^{\star} ] SR
[\underline{\bar6}\underline2^{\star}m^{\star} ] SR
[\underline{\bar3}^{\star}m ] [\underline{\bar3}^{\star}m ] SI
[6^{\star}/\underline{m}mm^{\star} ] SR
[6 ] [6m^{\star}m^{\star} ] [{\rm AR}^{\star}]
[\underline{\bar6}^{\star} ] [\underline{\bar6}^{\star} ] SI
[6^{\star}/\underline{m}^{\star} ] SR
[\underline{\bar6}^{\star}\underline2m^{\star} ] SR
[6/\underline{m}^{\star} ] [6/\underline{m}^{\star} ] SI
[6/\underline{m}^{\star}m^{\star}m^{\star} ] SR
[6\underline2^{\star}\underline2^{\star} ] [6\underline2^{\star}\underline2^{\star} ] SI
[6/\underline{m}m^{\star}m^{\star} ] SR
[\underline{\bar6}^{\star}m\underline2^{\star} ] [\underline{\bar6}^{\star}m\underline2^{\star} ] SI
[6^{\star}/\underline{m}^{\star}mm^{\star} ] SR
[6/\underline{m}^{\star}mm ] [6/\underline{m}^{\star}mm ] SI

Non-ferroelastic domain walls are usually curved with a slight preference for certain orientations (see Figs.[link] and[link]). Such shapes indicate a weak anisotropy of the wall energy [\sigma], i.e. small changes of [\sigma] with the orientation of the wall. The situation is different in ferroelectric domain structures, where charged domain walls have higher energies than uncharged ones.

A small energetic anisotropy of non-ferroelastic domain walls is utilized in producing tailored domain structures (Newnham et al., 1975[link]). A required domain pattern in a non-ferroelastic ferroelectric crystal can be obtained by evaporating electrodes of a desired shape (e.g. stripes) onto a single-domain plate cut perpendicular to the spontaneous polarization [{\bf P}_0]. Subsequent poling by an electric field switches only regions below the electrodes and thus produces the desired antiparallel domain structure.

Periodically poled ferroelectric domain structures fabricated by this technique are used for example in quasi-phase-matching optical multipliers (see e.g. Shur et al., 1999[link], 2001[link]; Rosenman et al., 1998[link]). An example of such an engineered domain structure is presented in Fig.[link].


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Engineered periodic non-ferroelastic ferroelectric stripe domain structure within a lithium tantalate crystal with symmetry descent [\bar 6\supset 3 ]. The domain structure is revealed by etching and observed in an optical microscope (Shur et al., 2001[link]). Courtesy of Vl. Shur, Ural State University, Ekaterinburg.

Anisotropic domain walls can also appear if the Landau free energy contains a so-called Lifshitz invariant (see Section[link] ), which lowers the energy of walls with certain orientations and can be responsible for the appearance of an incommensurate phase (see e.g. Dolino, 1985[link]; Tolédano & Tolédano, 1987[link]; Tolédano & Dmitriev, 1996[link]; Strukov & Levanyuk, 1998[link]). The irreversible character of domain walls in a commensurate phase of crystals also containing (at least theoretically) an incommensurate phase has been confirmed in the frame of phenomenological theory by Ishibashi (1992[link]). The incommensurate structure in quartz that demonstrates such an anisotropy is discussed at the end of the next example.

Example Domain walls in the α phase of quartz.  Quartz (SiO2) undergoes a structural phase transition from the parent [\beta] phase (symmetry group [6_{z}2_x2_y]) to the ferroic [\alpha] phase (symmetry [3_{z}2_x]). The [\alpha ] phase can appear in two domain states [{\bf S}_1] and [{\bf S}_2 ], which have the same symmetry [F_1 =F_2 =3_z2_x]. The symmetry [J_{12}] of the unordered domain pair [\{{\bf S}_1,{\bf S}_2\}] is given by [J_{12}^{\star}=] [3_z2_x \cup 2_y^*\{3_z2_x\} =] [6_z^*2_x2_y^*].

Table[link] summarizes the results of the symmetry analysis of domain walls (twins). Each row of the table contains data for one representative domain wall [{\bf W}_{12}({\bf n}_{12})] from one orbit [G{\bf W}_{12}({\bf n}_{12})]. The first column of the table specifies the normal [{\bf n}] of the wall plane p, further columns list the layer groups [\widehat{\sf F}_{12} ], [{\sf T}_{12} ] and [{\overline{\sf J}}_{12}] that describe the symmetry properties and classification of the wall (defined in Table[link]), and [n_W] is the number of symmetry-equivalent domain walls [cf. equation ([link])].

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Symmetry properties of domain walls in α quartz

[|{\bf P}| \not=|{\bf P^\prime}|, \ P^\prime_{i} \not= -P_{i}, \ i = x,y,z ].

[{\bf n}][\widehat{\sf F}_{12} ][{\sf T}_{12} ][\overline{\sf J}_{12} ]Classification[n_W][{\bf P}({\bf W}_{12}) ][{\bf P}({\bf W}_{21}) ]
[[001] ] [3_z ] [3_z\underline{2}^*_{y} ] [6^*_z\underline{2}_{x}\underline{2}^*_{y} ] SR 2    
[[100] ] [2_{x} ] [2_{x}\underline{2}^*_{y}\underline{2}^*_z ] [2_{x}\underline{2}^*_{y}\underline{2}^*_z ] SI 3    
[[010] ] 1 [\underline{2}^*_z ] [\underline{2}_{x}2^*_{y}\underline{2}^*_z ] SR 6 [0,0,P_z ] [0,0,-P_z ]
[[0vw] ] 1 1 [\underline{2}_x ] [{\rm A}\underline{\rm R}] 12 [P_x,P_y,P_z ] [P_x,-P_y,-P_z ]
[[u0w] ] 1 [\underline{2}^*_{y} ] [\underline{2}^*_{y} ] SI 6 [0,P_y,0 ] [0,-P^\prime_y,0 ]
[[uv0] ] 1 [\underline{2}^*_z ] [\underline{2}^*_z ] SI 6 [0,0,P_z ] [0,0,P^\prime_z ]
[[uvw] ] 1 1 1 AI 12 [P_x,P_y,P_z ] [P^\prime_{x},P^\prime_{y},P^\prime_z ]

The last two columns give possible components of the spontaneous polarization P of the wall [{\bf W}_{12}({\bf n})] and the reversed wall [{\bf W}_{21}({\bf n})]. Except for walls with normals [001] and [100], all walls are polar, i.e. they can be spontaneously polarized. The reversal of the polarization in reversible domain walls requires the reversal of domain states. In irreversible domain walls, the reversal of [{\bf W}_{12} ] into [{\bf W}_{21}] is accompanied by a change of the polarization P into P′, which may have a different absolute value and direction different to that of P.

The structure of two domain states and two mutually reversed domain walls obtained by molecular dynamics calculations are depicted in Fig.[link] (Calleja et al., 2001[link]). This shows a projection on the ab plane of the structure represented by SiO4 tetrahedra, in which each tetrahedron shares four corners. The threefold symmetry axes in the centres of distorted hexagonal channels and three twofold symmetry axes (one with vertical orientation) perpendicular to the threefold axes can be easily seen. The two vertical dotted lines are the wall planes p of two mutually reversed walls [[{\bf S}_{1}[010]{\bf S}_{2}]=] [{\bf W}_{12}[010]] and [[{\bf S}_{2}[010]{\bf S}_{1}] =] [{\bf W}_{21}[010]]. In Table[link] we find that these walls have the symmetry [{\sf T}_{12}[010] =] [{\sf T}_{21}[010]=] [\underline{2}_x2_y^{\star}\underline{2}_z^{\star}], and in Fig.[link] we can verify that the operation [\underline{2}_x] is a `side-reversing' operation [\underline{s}_{12}] of the wall (and the whole twin as wall), operation [2_y^{\star}] is a `state-exchanging operation' [r_{12}^{\star}] and the operation [\underline{2}_z^{\star} ] is a non-trivial `side-and-state reversing' operation [\underline{t}_{12}^{\star} ] of the wall. The walls [{\bf W}_{12}[010]] and [{\bf W}_{21}[010] ] are, therefore, symmetric and reversible walls.


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Microscopic structure of two domain states and two parallel mutually reversed domain walls in the [\alpha] phase of quartz. The left-hand vertical dotted line represents the domain wall [{\sf W}_{12}], the right-hand line is the reversed domain wall [{\sf W}_{12}]. To the left of the left-hand line and to the right of the right-hand line are domains with domain state [{\bf S}_1], the domain between the lines has domain state [{\bf S}_2]. For more details see text. Courtesy of M. Calleja, University of Cambridge.

During a small temperature interval above the appearance of the [\alpha ] phase at 846 K, there exists an incommensurate phase that can be treated as a regular domain structure, consisting of triangular columnar domains with domain walls (discommensurations) of negative wall energy [\sigma] (see e.g. Dolino, 1985[link]). Both theoretical considerations and electron microscopy observations (see e.g. Van Landuyt et al., 1985[link]) show that the wall normal has the [[uv0]] direction. From Table[link] it follows that there are six equivalent walls that are symmetric but irreversible, therefore any two equivalent walls differ in orientation.

This prediction is confirmed by electron microscopy in Fig.[link], where black and white triangles correspond to domains with domain states [{\bf S}_1] and [{\bf S}_2], and the transition regions between black and white areas to domain walls (discommensurations). To a domain wall of a certain orientation no reversible wall appears with the same orientation but with a reversed order of black and white. Domain walls in homogeneous triangular parts of the structure are related by 120 and 240° rotations and carry, therefore, parallel spontaneous polarizations; wall orientations in two differently oriented blocks (the middle of the right-hand part and the rest on the left-hand side) are related by 180° rotations about the axis [2_x ] in the plane of the photograph and are, therefore, polarized in antiparallel directions (for more details see Saint-Grégoire & Janovec, 1989[link]; Snoeck et al., 1994[link]). After cooling down to room temperature, the wall energy becomes positive and the regular domain texture changes into a coarse domain structure in which these six symmetry-related wall orientations still prevail (Van Landuyt et al., 1985[link]).


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Transmission electron microscopy (TEM) image of the incommensurate triangular ([3-q] modulated) phase of quartz. The black and white triangles correspond to domains with domain states [{\bf S}_1] and [{\bf S}_2 ], and the transition regions between black and white areas to domain walls (discommensurations). For a domain wall of a certain orientation there are no reversed domain walls with the same orientation but reversed order of black and white; the walls are, therefore, non-reversible. Domain walls in regions with regular triangular structures are related by 120 and 240° rotations about the z direction and carry parallel spontaneous polarizations (see text). Triangular structures in two regions (blocks) with different orientations of the triangles are related e.g. by [2_x] and carry, therefore, antiparallel spontaneous polarizations and behave macroscopically as two ferroelectric domains with antiparallel spontaneous polarization. Courtesy of E. Snoeck, CEMES, Toulouse and P. Saint-Grégoire, Université de Toulon.


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