Topology of Bi_2Se_3 nanosheets

Recently, the quantum spin-Hall edge channels of two-dimensional colloidal nanocrystals of the topological insulator Bi_2Se_3 were observed directly. Motivated by this development, we reconsider the four-band effective model which has been traditionally employed in the past to describe thin nanosheets of this material. Derived from a three-dimensional k·p model, it physically describes the top and bottom electronic surface states that become gapped due to the material's small thickness. However, we find that the four-band model for the surface states alone, as derived directly from the three-dimensional theory, is inadequate for the description of thin films of a few quintuple layers and even yields an incorrect topological invariant within a significant range of thicknesses. To address this limitation we propose an eight-band model which, in addition to the surface states, also incorporates the set of bulk bands closest to the Fermi level. We find that the eight-band model not only captures most of the experimental observations, but also agrees with previous first-principles calculations of the ℤ_2 invariant in thin films of varying thickness. Moreover, we demonstrate that the topological properties of thin Bi_2Se_3 nanosheets emerge as a result of an intricate interplay between the surface and bulk states. In particular, the surface bands of the eight-band model differ drastically from their counterparts in their four-band model, with the missing topology of the latter restored by the newly added bulk bands.