Hexagonal Warping Effects in the Surface States of the Topological Insulator ${\mathrm{Bi}}_2{\mathrm{Te}}_3$

A single two-dimensinoal Dirac fermion state has been recently observed on the surface of the topological insulator ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ by angle-resolved photoemission spectroscopy. We study the surface band structure using $k·p$ theory and find an unconventional hexagonal warping term, which is the counterpart of cubic Dresselhaus spin-orbit coupling in rhombohedral structures. We show that this hexagonal warping term naturally explains the observed hexagonal snowflake Fermi surface. The strength of hexagonal warping is characterized by a single parameter, which is extracted from the size of the Fermi surface. We predict a number of testable signatures of hexagonal warping in spectroscopy experiments on ${\mathrm{Bi}}_2{\mathrm{Te}}_3$. We also explore the possibility of a spin-density wave due to strong nesting of the Fermi surface.

Figure 1

(i) Snowflakelike Fermi surface of the surface states on 0.67% Sn-doped ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ observed in ARPES. (ii) A set of constant energy contours at different energies. From Y. L. Chen et al., Science 325, 178 (2009). Reprinted with permission from AAAS.

Figure 2

(a) Constant energy contour of $H(\overrightarrow{k})$. $k_x$ and $k_y$ axis are in the unit of $\sqrt{v/\lambda }$. (b) Constant energy contour at $E=1.2E^{*}$ is superimposed on the Fermi surface of ${\mathrm{Bi}}_2{\mathrm{Te}}_3$.

Figure 3

Illustration of scattering processes due to a point defect that causes the oscillation of LDOS. In a given direction $\widehat{x}$ along $\Gamma M$, the oscillation is dominated by scattering between stationary points marked by dots, where the Fermi velocity is parallel to $\widehat{x}$. (a) A convex constant energy contour has a single pair of stationary points at $\overrightarrow{k}$ and $-\overrightarrow{k}$. (b) A nonconvex constant energy contour has three pairs of stationary points. Intrapair scatterings in (a) and (b) are forbidden by time-reversal symmetry. But interpair scatterings in (b), for example, those between $k_2$ and $k_3$, are allowed. Therefore, the LDOS oscillation at leading order is absent in (a) but exists in (b).