Theory of surface energies and crystal shapes of topological crystalline insulators

Understanding equilibrium crystal shapes makes a substantial contribution to controlling the properties of materials. Relationships between topology and crystal shapes are important for applications of topological materials to dissipationless electronics, spintronics, and quantum computers. However, very little is known about the crystal shapes of the topological materials. Here we show that the surface energy of glide-symmetric topological crystalline insulators (TCI) depends on the parity of the Miller index of the surface in a singular way. This singular surface energy of the TCI affects the equilibrium crystal shapes, resulting in emergence of the unique crystal facets of the TCI. This singular dependence of the topological surface states is unique to the TCI protected by the glide symmetry in contrast to a TCI protected by a mirror symmetry. In addition, we show that such singular surface states of the TCI protected by the glide symmetries can be realized in KHgSb by performing first-principles calculations. Our results provide the basis for designs and manipulations of crystal facets by utilizing symmetry and topology.