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Recently, the quantum spin-Hall edge channels of two-dimensional colloidalnanocrystals of the topological insulator Bi$_2$Se$_3$ were observed directly.Motivated by this development, we reconsider the four-band effective modelwhich has been traditionally employed in the past to describe thin nanosheetsof this material. Derived from a three-dimensional $\boldsymbol{k}\boldsymbol{\cdot} \boldsymbol{p}$ model, it physically describes the top andbottom electronic surface states that become gapped due to the material's smallthickness. However, we find that the four-band model for the surface statesalone, as derived directly from the three-dimensional theory, is inadequate forthe description of thin films of a few quintuple layers and even yields anincorrect topological invariant within a significant range of thicknesses. Toaddress this limitation we propose an eight-band model which, in addition tothe surface states, also incorporates the set of bulk bands closest to theFermi level. We find that the eight-band model not only captures most of theexperimental observations, but also agrees with previous first-principlescalculations of the $\mathbb{Z}_{2}$ invariant in thin films of varyingthickness. Moreover, we demonstrate that the topological properties of thinBi$_2$Se$_3$ nanosheets emerge as a result of an intricate interplay betweenthe surface and bulk states. In particular, the surface bands of the eight-bandmodel differ drastically from their counterparts in their four-band model, withthe missing topology of the latter restored by the newly added bulk bands.

Topology of Bi$_2$Se$_3$ nanosheets