Anomalous change in the material moduli of thin films of barium titanate

A method is proposed to determine the moduli of thin ferroelectric films subjected to forced deformation due to the mismatch between the crystal lattice sizes of the film and substrate materials and the difference of their thermal expansion coefficients. Models of single-crystal films of barium titanate are studied using the eighth-degree Landau thermodynamic potential. It is shown that in the considered range of forced strain of the barium titanate film, three main states exist: a tetragonal phase ($c$ phase) of 4 mm symmetry with spontaneous polarization directed along the normal to the film plane; an orthorhombic phase ($aa$ phase) of 2 mm symmetry with polarization oriented along the diagonal plane of the film; monoclinic phase ($r$ phase) of $C_\mathrm{m}$ symmetry with the intermediate direction of polarization. All phases are separated by the lines of transitions of the second kind. Dependences of the elastic, electric and piezoelectric moduli on the magnitude of forced strain are plotted. It is shown that at the phase boundaries there is an anomalous change in the constants of the ferroelectric film and in the region of existence of the $r$ phase, some moduli reach extreme values.