Topological phase transition in a topological crystalline insulator induced by finite-size effects

We study electronic states and topological invariants of (001) films of the topological crystalline insulator (TCI) PbₓSn_1-xTe. Gapless surface Dirac cones on bulk TCIs become gapped in thin films due to the finite-size effect, which is the hybridization between the states on the top and bottom surfaces. We clarify that the TCI film has a strong finite-size effect as compared to three-dimensional topological insulators such as Bi₂Se₃. Moreover, the energy gap oscillates with the thickness of film. The oscillation stems from topological phase transitions in two dimensions. The obtained data of the topological invariants and energy gap serve as guide to TCI-device applications.