Eur. Phys. J. B (2012) 85: 308
https://doi.org/10.1140/epjb/e2012-30630-0

Regular Article

Electronic structure of three-dimensional topological insulators: thickness dependence and warping effect

¹ School of Physics and Electronic Information, Huaibei Normal University, Huaibei 235000, P.R. China
² National Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, P.R. China

^a e-mail: luhongyan2006@gmail.com

Received: 9 June 2012
Received in final form: 27 July 2012
Published online: 10 September 2012

Abstract

We theoretically study the electronic structure of a three-dimensional topological insulator using a 3D lattice model with a warping term included. Thickness dependence of the electronic structure and the shape of constant-energy contour at different energy are both investigated in this model. We demonstrate that an energy gap opens in the surface state when the thickness is below six quintuple layers and disappears for thicker topological insulators. With the increasing of energy, the shape of constant-energy contour of the surface state band gradually evolves from a point at Dirac point to a circle with increasing volume, then to a hexagon, and finally to a hexagram. We clarify that the critical thickness of six quintuple is not modified by the warping term and the appearance of a hexagram is not influenced by the finite size effect. These phenomena agree well with recent angle-resolved photoemission spectroscopy experiments.