  Silicene is a honeycomb-structure silicon atoms, which shares many intriguing
properties with graphene. Silicene is expected to be a quantum spin-Hall
insulator due to its spin-orbit interactions. We investigate the electronic
properties of silicene nanodisks, which are silicene derivatives with closed
edge. In case of the simplest model of graphene nanodisks, the number of the
zero-energy modes is given by the lower bound of the Lieb theorem. They are
standing wave states.When the spin-orbit interaction is introduced and the
system becomes a topological insulator, they begin to propagate around a
nanodisk. Helical edge currents flow around the edge of a nanodisk though the
crystal momentum is ill-defined. We have newly found the counting rule of the
zero-energy states, and constructed the low-energy theory of zigzag triangular
silicene. We also show the validity of the bulk-edge correspondence in
nanodisks with rough edge by calculating the local probability amplitude and
current. Finally, the signatures of the topological phase transition in
silicene nanodisks under an external electric field are revealed.Our results
will be observable by means of STM experiments.
