Electronic states and spin-filter effect in three-dimensional topological insulator Bi₂Se₃ nanoribbons

We study the electronic band structure, density distribution, and transport of a Bi₂Se₃ nanoribbon. We find that the density distribution of the surface states is dependent on not only the shape and size of the transverse cross section of the nanoribbon, but also the energy of the electron. We demonstrate that a transverse electric field can eliminate the coupling between surface states on the walls of the nanoribbon, remove the gap of the surface states, and restore the quantum spin Hall effects. In addition, we study the spin-dependent transport property of the surface states transmitting from top and bottom surfaces (x-y plane) to the side surfaces (z-x plane) of a Bi₂Se₃ nanoribbon. We find that transverse electric fields can open two surface channels for spin-up and -down Dirac electrons, and then switch off one channel for the spin-up Dirac electron. Our results may provide a simple way for the design of a spin filter based on topological insulator nanostructures.