Epitaxially strained SnTiO$_{3}$ at finite temperatures

Combining effective Hamiltonian and direct $ab$ $initio$ computation, we obtain the phase diagram of SnTiO$_{3}$ with respect to epitaxial strain and temperature, demonstrating the complex features of the phase diagram and providing insight into the epitaxially strained SnTiO$_{3}$, a presumably simple perovskite. In the phase diagram, two triple points are found, which may be exploited to to achieve high-performance piezoelectric effects. On the other hand, despite the inclusion of the degree of freedoms related to oxygen octahedron tilting, it is found that ferroelectric displacements dominate the structural phases over the whole strain misfit range. Finally, it is shown that the SnTiO$_{3}$ can be converted from hard to soft ferroelectrics with epitaxial strain.