  Silicene is a monolayer of silicon atoms forming a two-dimensional honeycomb
lattice, which shares almost every remarkable property with graphene. The low
energy structure of silicene is described by Dirac electrons with relatively
large spin-orbit interactions due to its buckled structure. The key observation
is that the band structure is controllable by applying the 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 demonstrate
that a topological phase transition occurs from a topological insulator to a
band insulator with the increase of the 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
applicable also to germanene, that is a two-dimensional honeycomb structure of
germanium.
