Energetics and geometry of 90° domain structures in epitaxial ferroelectric and ferroelastic films

Abstract

Statics of 90° domain structures formed in a thin tetragonal film epitaxially grown on a cubic substrate is studied theoretically on the basis of a rigorous solution of the associated elasticity problem. Inhomogeneous internal stresses existing in the strained polydomain epitaxy are calculated by the method of fictitious dislocations distributed along domain boundaries and the film/substrate interface. The calculation properly takes into account the influence of the film free surface on the field of internal stresses. For both possible periodic laminar structures, the energies of elastic strain fields existing in the film and the substrate are evaluated. These results are used to compute the equilibrium geometric parameters of periodic domain structures that minimize the total internal energy of a polydomain epitaxy. The dependencies of the equilibrium parameters on the film thickness and the relative misfit strain between the substrate and film are described. By comparing the minimum internal energies associated with various domain configurations, a diagram of equilibrium domain patterns is developed. In the range of relative misfit strains lower than some critical value, two threshold film thicknesses exist, separating three stability regions of domain patterns which correspond to a monovariant film and two periodic domain structures. In the rest of the diagram, a pattern consisting of equivalent domains with the tetragonal axes parallel to the interface is stable, irrespective of the film thickness.

Thickness scaling of ferroelastic domains in PbTiO3 films on DyScO3

“…6,7 It is known that there is a quadratic dependence, W / d 1=2 , of the domain width (W) with the crystal thickness (b). In ferroelastic domains (typically 90 domains), this b 1=2 dependence is an approximation in the regime of d ) W. 2 For smaller thicknesses, a linear dependence has been predicted by Pertsev and Zembilgotov (P&Z), 8 but it has not been experimentally confirmed yet.…”

“…21). For the fast-grown samples, the experimental domain size does not follow the predicted evolution, since it does scale neither as d 1=2 (Ref. 2) nor linearly with d. 8 Instead, fitting the data in Figure 3 for the fast-grown samples results in an exponent n ¼ 0:68 6 0:03 ðn % 2=3Þ. Our experimental results exhibit thus a behavior in between the classic Roytburd's square-root law and Pertsev's linear law.…”

“…4 Domain formation in ferroelectrics has been largely studied. 2,3,[5][6][7][8][9][10][11][12][13][14][15] In order to adapt to the substrate and to locally minimize the mismatch strain or the depolarizing field, the domains form in a periodic manner. 6,7 It is known that there is a quadratic dependence, W / d 1=2 , of the domain width (W) with the crystal thickness (b).…”

Fractal Dimension and Size Scaling of Domains in Thin Films of MultiferroicBiFeO3

“…In very thin films the interaction between the two opposite surfaces [35], dead layers [36,37] or ferroelastic effects [38][39][40] can induce a departure from LLK, often in the form of a divergence in domain size for very small thickness. On the other hand, experimentally, for perovskite ferroelectrics the LLK holds down to smaller thickness than those studied here (Fig.…”

Domain annihilation due to temperature and thickness gradients in single-crystal BaTiO3

“…37 In an attempt to understand the reason for changes in domain period with increasing temperature, a semiempirical model was considered, based on previous theoretical treatments of ferroelastic 90 • domains in thin films. 34,[38][39][40][41] Even though the BaTiO 3 imaged in the current work was a bulk single crystal, the expressions developed for thin films become increasingly relevant as film thickness tends to infinity. In this case, the terms in the originally developed expressions associated with the mismatch stress between film and substrate are replaced by the spontaneous strain in the ferroelastic itself.…”