Ferroelectric domain states

Janovec, V.; Přívratská

doi:10.1107/97809553602060000645

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. 3.4, p. 457

Section 3.4.2.2.2. Ferroelectric domain states

V. Janovec^a ^* and J. Přívratská^b

^a Department of Physics, Technical University of Liberec, Hálkova 6, 461 17 Liberec 1, 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

3.4.2.2.2. Ferroelectric domain states

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Ferroelectric domain states are defined as states with a homogeneous spontaneous polarization; different ferroelectric domain states differ in the direction of the spontaneous polarization. Ferroelectric domain states are specified by the stabilizer $[I_G({\rm P}_s^{(1)})]$ of the spontaneous polarization $[{\rm P}_s^{(1)}]$ in the first principal domain state $[{\bf S}_1]$ [see equation (3.4.2.16)]: $[F_1\subseteq C_1 \equiv I_G({\rm P}_s^{(1)}) \subseteq G. \eqno(3.4.2.31) ]$ The stabilizer [C_1] is one of ten polar groups: 1, 2, 3, 4, 6, m, [mm2] , [3m] , [4mm ] , [6mm] . Since [F_1] must be a polar group too, it is simple to find the stabilizer [C_1] fulfilling relation (3.4.2.31).

The number [n_e] of ferroelectric domain states is given by $[n_e=[G:C_1]=|G|:|C_1|. \eqno(3.4.2.32)]$ If the polar group [C_1] does not exist, we put [n_e=0] . The number [n_e] of ferroelectric domain states is given for all ferroic phase transitions in the eighth column of Table 3.4.2.7.

The number [d_a] of principal domain states compatible with one ferroelectric domain state (degeneracy of ferroelectric domain states) is given by $[d_e=[C_1:F_1]=|C_1|:|F_1|. \eqno(3.4.2.33) ]$

The product of [n_e] and [d_e] is equal to the number n of all principal domain states [see equation (3.4.2.19)], $[n_ed_e=n. \eqno(3.4.2.34)]$ The degeneracy [d_e] of ferroelectric domain states can be calculated for all ferroic phase transitions from the ratio of the numbers n and [n_e] that are given in Table 3.4.2.7.

According to Aizu (1969, 1970a), we can again recognize three possible cases (see also Table 3.4.2.3):

(i) Full ferroelectrics: All principal domain states differ in spontaneous polarization. In this case, , i.e. , ferroelectric domain states are identical with principal domain states.
(ii) Partial ferroelectrics: Some but not all principal domain states differ in spontaneous polarization. A necessary and sufficient condition is $[1\,\lt\, n_e\,\lt \,n]$ , or equivalently, $[F_1 \subset C_1 \subset G]$ . Ferroelectric domain states are degenerate secondary domain states with degeneracy $[n\,\gt \,d_e\,\gt\, 1]$ . In this case, the phase transition $[G\supset F_1]$ can be classified as an improper ferroelectric one (see Section 3.1.3.2 ).
(iii) Non-ferroelectrics: No principal domain states differ in spontaneous polarization. There are two possible cases: (a) The parent phase is polar; then and . (b) The parent phase is non-polar; in this case a polar stabilizer does not exist, then we put .

The classification of full-, partial- and non-ferroelectrics and ferroelastics is given for all Aizu's species in Aizu (1970a).

This classification for all symmetry descents is readily available from the numbers n, [n_a] , [n_e] in Table 3.4.2.7. One can conclude that partial ferroelectrics are rather rare.

Example 3.4.2.3. Domain structure in tetragonal perovskites. Some perovskites (e.g. barium titanate, BaTiO₃) undergo a phase transition from the cubic parent phase with $[G=m\bar3m]$ to a tetragonal ferroelectric phase with symmetry [F_1=4_xm_ym_z] . The stabilizer [A_1 =] Hol $[(4_xm_ym_z)\cap m3m =]$ [m_xm_ym_z] . There are [n_a =] [ |m3m|: |m_xm_ym_z| =] 3 ferroelastic domain states each compatible with [d_a =] [|m_xm_ym_z|:|4_xm_ym_z| =] 2 principal ferroelectric domain states that are related e.g. by inversion $[\bar1]$ , i.e. spontaneous polarization is antiparallel in two principal domain states within one ferroelastic domain state.

A similar situation, i.e. two non-ferroelastic domain states with antiparallel spontaneous polarization compatible with one ferroelastic domain state, occurs in perovskites in the trigonal ferroic phase with symmetry [F=3m] and in the orthorhombic ferroic phase with symmetry $[F_1=m_{x\bar y}2_{xy}m_z ]$ .

Many other examples are discussed by Newnham (1974, 1975), Newnham & Cross (1974a,b), and Newnham & Skinner (1976).

References

Aizu, K. (1969). Possible species of `ferroelastic' crystals and of simultaneously ferroelectric and ferroelastic crystals. J. Phys. Soc. Jpn, 27, 387–396.Google Scholar

Aizu, K. (1970a). Possible species of ferromagnetic, ferroelectric and ferroelastic crystals. Phys. Rev. B, 2, 754–772. Google Scholar

Newnham, R. E. (1974). Domains in minerals. Am. Mineral. 59, 906–918.Google Scholar

Newnham, R. E. (1975). Structure–property relations. Berlin: Springer.Google Scholar

Newnham, R. E. & Cross, L. E. (1974a). Symmetry of secondary ferroics I. Mater. Res. Bull. 9, 927–934.Google Scholar

Newnham, R. E. & Cross, L. E. (1974b). Symmetry of secondary ferroics II. Mater. Res. Bull. 9, 1021–1032.Google Scholar

Newnham, R. E. & Skinner, D. P. Jr (1976). Polycrystalline secondary ferroics. Mater. Res. Bull. 11, 1273–1284.Google Scholar

International Tables for Crystallography (2006). Vol. D. ch. 3.4, p. 457