Hierarchy and scaling behaviour of multi-rank domain patterns in ferroelectric K_0.9Na_0.1NbO₃ strained films.

The formation process of a ferroelectric multi-rank domain pattern in the thickness range of 7 to 52 nm is investigated for monoclinic K_0.9Na_0.1NbO₃ strained epitaxial films on (110) NdScO₃ substrates. Although the elastic strain energy density is degenerated for two pseudocubic orientations, a distinctive hierarchy of domain evolution is observed with exclusive in-plane a₁a₂ domains for very thin films and the retarded onset of a ferroelectric M_C phase at larger film thickness. This is accompanied by a thickness dependent transformation from stripe domains to a herringbone pattern and, eventually, for the thickest film, to a checkerboard-like structure. These transformations in the domain arrangement and width are correlated to energetic aspects as depolarization field and anisotropic strain relaxation in the film. While for the M_C domains plastic strain relaxation is throughout observed, the a₁a₂ domains show a two-step strain relaxation mechanism starting with an in-plane elastic shearing, which is followed by plastic lattice relaxation. Our results highlight a pathway for engineering and patterning of periodic ferroelectric domain structures.