Electrical constraints of twin interfaces

Hahn, Th.; Klapper, H.

doi:10.1107/97809553602060000644

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

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International Tables for Crystallography (2006). Vol. D. ch. 3.3, pp. 430-432

Section 3.3.10.3. Electrical constraints of twin interfaces

Th. Hahn^a ^* and H. Klapper^b

^a Institut für Kristallographie, Rheinisch–Westfälische Technische Hochschule, D-52056 Aachen, Germany, and ^bMineralogisch-Petrologisches Institut, Universität Bonn, D-53113 Bonn, Germany
Correspondence e-mail: hahn@xtal.rwth-aachen.de

3.3.10.3. Electrical constraints of twin interfaces

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As mentioned before, the mechanical compatibility of twin boundaries is a necessary but not a sufficient criterion for the occurrence of stress-free low-energy twin interfaces. An additional restriction occurs in materials with a permanent (spontaneous) electrical polarization, i.e. in crystals belonging to one of the ten pyroelectric crystal classes which include all ferroelectric materials. In these crystals, domains with different directions of the spontaneous polarization may occur and lead to `electrically charged boundaries'.

3.3.10.3.1. Merohedral twins

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Of particular significance are merohedral twins with polar domains of antiparallel spontaneous polarization $[\pm {\bf P}]$ (180° domains). The charge density $[\rho]$ at a boundary between two twin domains is given by $[\rho = \pm 2 P_n,]$ where [P_n] is the component of the polarization normal to the boundary. The interfaces with positive charge are called `head-to-head' boundaries, those with negative charge `tail-to-tail' boundaries. Interfaces parallel to the polarization direction are uncharged ( [P_n = 0] ) (Fig. 3.3.10.2).

Figure 3.3.10.2 | top | pdf |

Boundaries B–B between 180° domains (merohedral twins) of pyroelectric crystals. (a) Tail-to-tail boundary. (b) Head-to-head boundary. (c) Uncharged boundary ( [P_n = 0] ). (d) Charged zigzag boundary, with average orientation normal to the polar axis. The charge density is significantly reduced. Note that the charges at the boundaries are usually compensated by stray charges of opposite sign.

The electrical charges on a twin boundary constitute an additional (now electrostatic) energy of the twin boundary and are `electrically forbidden'. Only boundaries parallel to the polar axes are `permitted'. This is in fact mostly observed: practically all 180° domains originating during a phase transition from a paraelectric parent phase to the polar (usually ferroelectric) daughter phase exhibit uncharged boundaries parallel to the spontaneous polarization. Uncharged boundaries have also been found in inversion growth twins obtained from aqueous solutions, such as lithium formate monohydrate and ammonium lithium sulfate . Both crystals possess the polar eigensymmetry [mm2] and contain grown-in inversion twin lamellae (180° domains) parallel to their polar axis.

`Charged' boundaries, however, may occur in crystals that are electrical conductors. In such cases, the polarization charges accumulating along head-to-head or tail-to-tail boundaries are compensated by opposite charges obtained through the electrical conductivity . This compensation may lead to a considerable reduction of the interface energy. Note that the term `charged' is often used for boundaries of head-to-head and tail-to-tail character, even if they are uncharged due to charge compensation.

Examples

(1) Lithium niobate, LiNbO₃, exhibits a phase transition from $[{\bar 3}2/m]$ to between 1323 and 1473 K (depending on the Li/Nb stoichiometry). Crystals are grown from the melt ( K) by Czochralski pulling along the trigonal axis [001] in the paraelectric phase. They transform into the ferroelectric polar phase when cooled below the Curie temperature. The crystals are electrically conductive at high temperatures and can be poled by an electric field parallel to the polar axis. By applying an alternating rectangular voltage between seed crystal and melt, a sequence of 180° domains is formed during the subsequent transition. The domain boundaries follow the curved growth front (crystal–melt interface) and have alternating head-to-head and tail-to-tail character (Räuber, 1978).
(2) Orthorhombic polar potassium titanyl phosphate, KTiOPO₄ (KTP), exhibits a para- to ferroelectric phase transition ( $[mmm \Longleftrightarrow mm2]$ ) and a considerable conductivity of potassium ions. In this material, head-to-head and tail-to-tail boundaries are common. Sometimes strongly folded, charged zigzag boundaries occur, which contain large segments of faces nearly parallel to the spontaneous polarization (Scherf et al., 1999). The average orientation of these boundaries is roughly normal to the polar axis (Fig. 3.3.10.2d), but their charge density is considerably reduced by the zigzag geometry.
(3) Head-to-head and tail-to-tail twin boundaries are also found in crystals grown from aqueous solutions. In such cases, the polarization charges are compensated by the opposite charges present in the electrolytic solution. An interesting example is hexagonal potassium lithium sulfate KLiSO₄ (point group 6) which exhibits, among other types of twins, anti-polar domains of inversion twins. The twin boundaries often have head-to-head or tail-to-tail character and frequently coincide with the growth-sector boundaries (Klapper et al., 1987).

3.3.10.3.2. Non-merohedral twins

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Charged and uncharged boundaries may also occur in non-merohedral twins of pyroelectric crystals. In this case, the polar axes of the two twin domains 1 and 2 are not parallel. The charge density $[\rho]$ of the boundary is given by $[\rho = P_n(2) - P_n(1),]$ with [P_n(1)] and [P_n(2)] the components of the spontaneous polarization normal to the boundary. An example of both charged and uncharged boundaries is provided by the growth twins of ammonium lithium sulfate with eigensymmetry [m2m] . These crystals exhibit, besides the inversion twinning mentioned above, growth-sector twins with twin laws `reflection plane (110)' and `twofold twin axis normal to (110)'. (Both twin elements would constitute the same twin law if the crystal were centrosymmetric.) The observed and permissible composition plane for both laws is (110) itself. As is shown in Fig. 3.3.10.3, the (110) boundary is charged for the reflection twin and uncharged for the rotation twin. Both cases are realized for ammonium lithium sulfate. The charges of the reflection-twin boundary are compensated by the charges contained in the electrolytic aqueous solution from which the crystal is grown. On heating (cooling), however, positive (negative) charges appear along the twin boundary.

Figure 3.3.10.3 | top | pdf |

Charged and uncharged boundaries B–B of non-merohedral twins of pseudo-hexagonal NH₄LiSO₄. Point group [m2m] , spontaneous polarization P along twofold axis [010]. (a) Twin element mirror plane (110): electrically charged boundary of head-to-head character. (b) Twin element twofold twin axis normal to plane (110): uncharged twin boundary (`head-to-tail' boundary).

3.3.10.3.3. Non-pyroelectric acentric crystals

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Finally, it is pointed out that electrical constraints of twin boundaries do not occur for non-pyroelectric acentric crystals. This is due to the absence of spontaneous polarization and, consequently, of electrical boundary charges. This fact is apparent for the Dauphiné and Brazil twins of quartz: they exhibit boundaries normal to the polar twofold axes which are reversed by the twin operations.

Nevertheless, it seems that among possible twin laws those leading to opposite directions of the polar axes are avoided. This can be explained for spinel twins of cubic crystals with the sphalerite structure and eigensymmetry $[{\bar 4}3m]$ . Two twin laws, different due to the lack of the symmetry centre, are possible:

(i) twofold twin rotation around [111],
(ii) twin reflection across the plane (111).

In the first case, the sense of the polar axis [111] is not reversed, in the second case it is reversed. All publications on this kind of twinning, common in III–V and II–VI compound semiconductors (GaAs, InP, ZnS, CdTe etc.), report the twofold axis along [111] as the true twin element, not the mirror plane (111); this was discussed very early on in a significant paper by Aminoff & Broomé (1931).

References

Aminoff, G. & Broomé, B. (1931). Strukturtheoretische Studien über Zwillinge I. Z. Kristallogr. 80, 355–376.Google Scholar

Klapper, H., Hahn, Th. & Chung, S. J. (1987). Optical, pyroelectric and X-ray topographic studies of twin domains and twin boundaries in KLiSO₄. Acta Cryst. B43, 147–159.Google Scholar

Räuber, A. (1978). Chemistry and physics of lithium niobate. In Current topics in materials science, Vol. 1, edited by E. Kaldis, pp. 548–550 and 585–587. Amsterdam: North Holland.Google Scholar

Scherf, Ch., Hahn, Th., Heger, G., Ivanov, N. R. & Klapper, H. (1999). Imaging of inversion twin boundaries in potassium titanyl phosphate (KTP) by liquid-crystal surface decoration and X-ray diffraction topography. Philos. Trans. R. Soc. London Ser. A, 357, 2651–2658.Google Scholar

International Tables for Crystallography (2006). Vol. D. ch. 3.3, pp. 430-432