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

International Tables for Crystallography (2013). Vol. D, ch. 3.4, pp. 510-511

Section Switching of ferroic domain states

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: Switching of ferroic domain states

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We saw in Section[link] that all domain states of the orbit [G{\bf S}_1] have the same chance of appearing. This implies that they have the same free energy, i.e. they are degenerate. The same conclusion follows from thermodynamic theory, where domain states appear as equivalent solutions of equilibrium values of the order parameter, i.e. all domain states exhibit the same free energy [\Psi] (see Section 3.1.2[link] ). These statements hold under a tacit assumption of absent external electric and mechanical fields. If these fields are nonzero, the degeneracy of domain states can be partially or completely lifted.

The free energy [\Psi^{(k)}] per unit volume of a ferroic domain state [{\bf S}_k], [k=1,2,\ldots,n], with spontaneous polarization [{\bf P}_0^{(k)}] with components [P_{0i}^{(k)}], [i=1,2,3], and with spontaneous strain components [u_{0\mu}^{(k)}], [\mu=1,2,\ldots, 6, ] is (Aizu, 1972[link])[\eqalignno{\Psi^{(k)}&={\Psi}_0-P_{0i}^{(k)}E_{i}-u_{0\mu}^{(k)}\sigma_{\mu}- d_{i\mu}^{(k)}E_i\sigma_{\mu}-\textstyle{{1}\over{2}}{\varepsilon}_0{\kappa}_{ik}^{(k)}E_iE_k&\cr&\quad- \textstyle{{1}\over{2}}s_{\mu\nu}^{(k)}\sigma_{\mu}\sigma_{\nu}-\textstyle{{1}\over{2}} Q_{ik\mu}E_iE_{k}\sigma_{\mu}-\ldots, &(} ]where the Einstein summation convention (summation with respect to suffixes that occur twice in the same term) is used with [i,j=1,2,3] and [\mu,\nu=1,2,\ldots,6]. The symbols in equation ([link]) have the following meaning: [E_{i}] and [u_{\mu}] are components of the external electric field and of the mechanical stress, respectively, [d_{i\mu}^{(k)}] are components of the piezoelectric tensor, [{\varepsilon}_0{\kappa}_{ij}^{(k)} ] are components of the electric susceptibility, [s_{\mu\nu}^{(k)} ] are compliance components, and [Q_{ij\mu}^{(k)}] are components of electrostriction (components with Greek indices are expressed in matrix notation) [see Section 3.4.5[link] (Glossary), Chapter 1.1[link] or Nye (1985[link]); Sirotin & Shaskolskaya (1982[link])].

We shall examine two domain states [{\bf S}_1] and [{\bf S}_j ], i.e. a domain pair [({\bf S}_1,{\bf S}_j)], in electric and mechanical fields. The difference of their free energies is given by[\eqalignno{\Psi^{(j)}-\Psi^{(1)}&=-(P_{0i}^{(j)}-P_{0i}^{(1)})E_{i}- (u_{0\mu}^{(j)}-u_{0\mu}^{(1)}){\sigma}_{\mu}-(d_{i\mu}^{(j)}-d_{i\mu}^{(1)})E_i{\sigma}_{\mu}&\cr &\quad-\textstyle{{1}\over{2}}{\varepsilon}_0({\kappa}_{ik}^{(j)}-{\kappa}_{ik}^{(1)})E_iE_k-\textstyle{{1}\over{2}}(s_{\mu\nu}^{(j)}-s_{\mu\nu}^{(1)})\sigma_{\mu}\sigma_{\nu}&\cr &\quad-\textstyle{{1}\over{2}}(Q_{ik\mu}^{(j)}-Q_{ik\mu}^{(1)})E_iE_k\sigma_{\mu}-\ldots. &(} ]

For a domain pair [({\bf S}_1,{\bf S}_j)] and given external fields, there are three possibilities:

  • (1) [{\Psi}^{(j)}=\Psi^{(1)}]. Domain states [{\bf S}_1] and [{\bf S}_j] can coexist in equilibrium in given external fields.

  • (2) [{\Psi}^{(j)} \,\lt\, \Psi^{(1)}]. In given external fields, domain state [{\bf S}_j] is more stable than [{\bf S}_1]; for large enough fields (higher than the coercive ones), the state [{\bf S}_1] switches into the state [{\bf S}_j].

  • (3) [{\Psi}^{(j)}\,\gt\,\Psi^{(1)}]. In given external fields, domain state [{\bf S}_j] is less stable than [{\bf S}_1]; for large enough fields (higher than the coercive ones), the state [{\bf S}_j] switches into the state [{\bf S}_1].

A typical dependence of applied stress and corresponding strain in ferroelastic materials has a form of a elastic hysteresis loop (see Fig.[link]). Similar dielectric hysteresis loops are observed in ferroelectric materials; examples can be found in books on ferroelectric crystals (e.g. Jona & Shirane, 1962[link]).

A classification of switching (state shifts in Aizu's terminology) based on equation ([link]) was put forward by Aizu (1972[link], 1973[link]) and is summarized in the second and fourth columns of Table[link]. The order of the state shifts specifies the switching fields that are necessary for switching one domain state of a domain pair into the second state of the pair.

Another distinction related to switching distinguishes between actual and potential ferroelectric (ferroelastic) phases, depending on whether or not it is possible to switch the spontaneous polarization (spontaneous strain) by applying an electric field (mechanical stress) lower than the electrical (mechanical) breakdown limit under reasonable experimental conditions (Wadhawan, 2000[link]). We consider in our classification always the potential ferroelectric (ferroelastic) phase.

A closer look at equation ([link]) reveals a correspondence between the difference coefficients in front of products of field components and the tensor distinction of domain states [{\bf S}_1] and [{\bf S}_j ] in the domain pair [({\bf S}_1,{\bf S}_j)]: If a morphic Cartesian tensor component of a polar tensor is different in these two domain states, then the corresponding difference coefficient is nonzero and defines components of fields that can switch one of these domain states into the other. A similar statement holds for the symmetric tensors of rank two (e.g. the spontaneous strain tensor).

Tensor distinction for all representative non-ferroelastic domain pairs is available in the synoptic Table[link]. These data also carry information about the switching fields.


Aizu, K. (1972). Electrical, mechanical and electromechanical orders of state shifts in nonmagnetic ferroic crystals. J. Phys. Soc. Jpn, 32, 1287–1301.
Aizu, K. (1973). Second-order ferroic state shifts. J. Phys. Soc. Jpn, 34, 121–128.
Jona, F. & Shirane, G. (1962). Ferroelectric Crystals. Oxford: Pergamon Press.
Nye, J. F. (1985). Physical Properties of Crystals. Oxford: Clarendon Press.
Sirotin, Yu. I. & Shaskolskaya, M. P. (1982). Fundamentals of Crystal Physics. Moscow: Mir.
Wadhawan, V. K. (2000). Introduction to Ferroic Materials. The Netherlands: Gordon and Breach.

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