Engineered Unique Elastic Modes at a ${\mathrm{BaTiO}}_3/(2\times 1)\text{−}\mathrm{Ge}(001)$ Interface

The strong interaction at an interface between a substrate and thin film leads to epitaxy and provides a means of inducing structural changes in the epitaxial film. These induced material phases often exhibit technologically relevant electronic, magnetic, and functional properties. The $2\times 1$ surface of a Ge(001) substrate applies a unique type of epitaxial constraint on thin films of the perovskite oxide ${\mathrm{BaTiO}}_3$ where a change in bonding and symmetry at the interface leads to a non-bulk-like crystal structure of the ${\mathrm{BaTiO}}_3$. While the complex crystal structure is predicted using first-principles theory, it is further shown that the details of the structure are a consequence of hidden phases found in the bulk elastic response of the ${\mathrm{BaTiO}}_3$ induced by the symmetry of forces exerted by the germanium substrate.

Figure 1

(a) High angle annular dark field image of 2.5-uc $\mathrm{BTO}/\mathrm{Ge}$ along the Ge[100] zone axis. (b) Superimposed on the TEM image taken along the Ge[110] zone axis is a model structure of the system determined from surface x-ray diffraction measurements. The red arrows indicate the rumpling of the interfacial Ba along the [001] axis. (c) DFT-predicted ground-state structure of the 2.5-uc $\mathrm{BTO}/\mathrm{Ge}$ system.

Figure 2

(a) Integer-order (logarithmic vertical scale) and (b) superstructure (linear vertical scale) crystal truncation rods measured (blue circles) for a 2.5-uc BTO film grown on Ge(001) using molecular beam epitaxy. The model structure of the $\mathrm{BTO}/\mathrm{Ge}$ interface shown in Fig. 1 is derived from the fits (red lines) to the measured data.

Figure 3

Force constant matrix eigenmode dispersion curves in reciprocal space for bulk cubic ${\mathrm{BaTiO}}_3$. The colors correspond to the atomic content of each mode: red, green, blue correspond to amplitudes of motion of Ba, Ti, O atoms, respectively, following Ghosez et al. [44]. Solid circles superimposed on the dispersion curves express the decomposition of atomic displacements at the interface onto modes with $\mathrm{\Gamma }$ and $X$ symmetry. The radii of the circles are proportional to the magnitude of the projections of the atomic displacements in the 2.5-uc-thick BTO slab onto the eigenmodes. (a) The projections of the displacements of the first BaO and ${\mathrm{TiO}}_2$ layers adjacent to the interface. (b) The projections for the first ${\mathrm{TiO}}_2$ layer and the second BaO layer above it. The theoretical ground state of the $\mathrm{BTO}/\mathrm{Ge}$ system is reproduced in the inset with the numbering of the atomic layers and the color-coding of the atom types.

Figure 4

(a) Atomic motions for force constant matrix eigenmode contributing the most to the displacements of the 2.5-uc-thick BTO slab. The mode is soft at the $X$ point so that it involves antiparallel displacements of planar O and Ti. (b) Lowest energy stiff acoustic mode at $X$ where the Ti and O atoms move in the same direction.