A topological insulator and helical zero mode in silicene under an inhomogeneous electric field

Silicene is a monolayer of silicon atoms forming a two-dimensional (2D) honeycomb lattice and shares almost all the remarkable properties of graphene. The low-energy structure of silicene is described by Dirac electrons with relatively large spin-orbit interactions owing to its buckled structure. A key observation is that the band structure can be controlled by applying an electric field to a silicene sheet. In particular, the gap closes at a certain critical electric field. Examining the band structure of a silicene nanoribbon, we show that a topological phase transition occurs from a topological insulator to a band insulator with an increase of electric field. We also show that it is possible to generate helical zero modes anywhere in a silicene sheet by adjusting the electric field locally to this critical value. The region may act as a quantum wire or a quantum dot surrounded by topological and/or band insulators. We explicitly construct the wave functions for some simple geometries based on the low-energy effective Dirac theory. These results are also applicable to germanene, which is a 2D honeycomb structure of germanium.