Abstract
This chapter is devoted to the properties of domain walls and let us stress first that here we have in mind a static situation in a ferroic sample. Problems connected with domain walls which are not in static or quasistatic equilibrium will be dealt with separately in Chap. 8. An important aspect of walls has been already discussed in Chap. 2, namely their orientation from the point of view of electrical and mechanical compatibilities of neighboring domains. There we considered walls as infinitely thin two-dimensional objects. Now we focus on its internal structure and properties governed by this structure. In Chap. 4 we have pointed out that several domain-imaging methods provided information about the wall thickness; in Sect. 6.1 we wish to describe some of these observations in more detail. In this section, we will also present available experimental data on thickness and surface tension of domain walls.
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- 1.
This is not the case for the materials where the minimal energy of the crystal corresponds to a spatially modulated state. These materials are not considered in this book.
- 2.
Explicit expressions for \(c'_{ij}\) and \(q'_{ij}\) can be found in the paper by Fouskova and Fousek (1975).
- 3.
This is a good approximation in materials with ‘normal’ values of the soft-mode effective charge (of the order of the charge electronic). In ferroelectrics with anomalously small values of this charge, like weak ferroelectrics (see Sect. 2.3.6), this approximation fails.
- 4.
Actually, the P z component may also appear in the wall. The stability of the available solutions for 90° walls with respect to the appearance of this component can be investigated following the approach outlined in Sect. 6.2.1.
- 5.
Cao and Barsch (1990) give a much smaller value for t w because of a lost numerical factor (Tagantsev et al., 2001a).
- 6.
The problem is that the optical displacements of the ions (or internal strains) are by definition defined to within an arbitrary constant (see, e.g., a detailed discussion of Tagantsev, 1991). This makes difficult separation between the optical and acoustical displacements (which enter the deformation tensor) on the two sides of the wall. Without this separation, one cannot tell what mechanical off-set corresponds to a given profile of atomic displacements. In Meyer and Vanderbilt (2002), this separation has been postulated and mechanical offset of 0.04 Å across the wall has been calculated. However, the interpretation of this result is not clear because of the apparent ambiguity of the aforementioned separation.
- 7.
Here and thereafter the use of sign ‘\(\cong\)’ means that we are dealing with rough order-of-magnitude estimates. The exact relations can contain numerical factors varying from a few tenths to a few units.
- 8.
Here, for simplicity, we consider the case of a domain wall with the isotropic surface tension. If the surface tension is anisotropic, obviously for the optimal bulges \(L_1 \ne L_2\). In this case, however, \(L_1 \propto L_2\) and one can readily show that the roughness exponent will be the same as in the case of the isotropic surface tension.
- 9.
Estimate (6.4.15) corresponds to the estimate of the depolarizing energy associated with a bulge on the wall used by Miller and Weinreih (1960).
- 10.
This estimate can be obtained, for example, using the results on the oscillation spectrum of ferroelastic walls (Nechaev and Roshchupkin, 1989).
- 11.
The conclusion about suppression of thermal roughening for ferroelastic wall obtained in this paper is correct; however, the calculations have been performed for the correlation function qualitatively corresponding to a wall in a ferroelectric/nonferroelastic.
- 12.
The result for the RB case is not rigorously justified.
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Tagantsev, A.K., Cross, L.E., Fousek, J. (2010). Domain Walls at Rest. In: Domains in Ferroic Crystals and Thin Films. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-1417-0_6
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DOI: https://doi.org/10.1007/978-1-4419-1417-0_6
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