Figure 1: Theoretical analysis of heteroepitaxial system.

Tetragonal ferroelectric grown on orthorhombic (pseudo-cubic) substrate. (a) Illustration of allowed domain orientations. The polarization direction (indicated by coloured arrows) in the tetragonal films is along the c axis of its unit cell. a-Domains have one a axis of the unit cell perpendicular to the surface of the film and the polar axis lying in the plane of the film, parallel to the []_S or [001]_S orientation of the pseudo-cubic orthorhombic substrate (orthorhombic notation of Miller indices, the subscript s denotes the substrate). (b) Stripe domain geometry in terms of the film thickness h and the coherency factor γ. The solid curves (red—online) correspond to a constant width of the a-domains w_d; the numbers at the curves give the width w_d measured in units of h₀ given by equation (5) of the main text. The upper dashed curve shows the limit of the applicability of the approximation used in the calculation, w_d/h<1/2. The area below the dotted curve corresponds to the c-monovariant. (c) Schematic of temperature dependence of lattice constants of the tetragonal film clamped to the anisotropic substrate: a and c—lattice parameters of the film, a_s1 and a_s2—long and short axis of the anisotropic substrate. T_C depicts the Curie temperature, shifted by the clamping. At temperatures T₁ and T₂, the misfit strains along the long and short pseudo-cubic axes of the substrate change the sign, respectively. At temperatures T_1F and T_2F, the formation of a-domains (in the c-domain matrix) normal to the long and short pseudo-cubic axes (easy and hard domains) becomes energetically favourable. (d) Phase-field calculated cross-hatched pattern. The model assumes periodic boundary conditions (see Methods). (d–g) Illustrate meta-stability of a-domain crossing. A small variation of simulation parameters like temperature, film thickness or domain wall energy (used in this case) dictates whether the a-domain crossing is stabilized or not. Here the increase of domain wall energy leads to disappearance of one type of a-domains. The scale bar is 50 nm.