$c$-axis oriented epitaxial $BaTiO3$ films on (001) Si

The strain due to thermal expansion mismatch between the silicon substrate and $BaTiO3$ film is calculated by assuming an equivalent cubic cell of the same volume as tetragonal $BaTiO3$ at temperatures below the bulk Curie temperature of $BaTiO3$⁠.

1 ML is defined as the concentration of atoms on the (001) surface of silicon, i.e., $6.78×1014atoms∕cm2$⁠.

The commensurate nature of (Ba,Sr)O films displaying such ideal RHEED patterns was confirmed by four-circle x-ray diffraction (the films were capped with aluminum to prevent degradation in air when removed from the MBE chamber).

It is not possible to distinguish whether the structural distortion of $Ba0.7Sr0.3TiO3$ buffer layer is due to thermal expansion mismatch induced tensile strain or due to strain-induced paraelectric-ferroelectric (or cubic-tetragonal) phase transition. Therefore, pseudocubic indexing of the $Ba0.7Sr0.3TiO3$ buffer layer is used throughout this manuscript.

The in-plane lattice spacing and therefore the lattice constant of the film are obtained from the synchrotron data using the following equation, $dfilmsinθfilm=dsubsinθsub$ (variation of Bragg’s law) and the known lattice spacing of the substrate (silicon, in our case). $dfilm$ and $dsub$ are the lattice spacings of the film and substrate (silicon), respectively, and $θfilm$ and $θsub$ are the Bragg peak positions (established using a Gaussian fit of the synchrotron data) of the film peak and substrate reference peak (Si 220), respectively.

In bulk, $Ba0.7Sr0.3TiO3$ is cubic at room temperature and hence has isotropic refractive index.