Figure 3.

Illustration of the spin-allowed nesting condition in (a), giving rise to the sharp CDW satellite peaks at $\pm {\mathbf{g}}_{\mathrm{CDW}}$ around the $\left(00\right)$-reflection in (b). HAS diffraction scans of Bi${}_2$Te${}_2$Se(111) along the $\overline{\mathsf{\Gamma }\mathrm{M}}$ direction at room (295 K) and low (118 K) temperature in (b), show two sharp peaks at wavevectors $\pm {\mathbf{g}}_{\mathrm{CDW}}$, indicative of a surface CDW. While, at room temperature, the two diffraction peaks have about the same intensity, at $118$K the peak intensity at $-{\mathbf{g}}_{\mathrm{CDW}}$ is almost doubled at the expense of the peak at $+{\mathbf{g}}_{\mathrm{CDW}}$, which is vanishing. (a) The contours of the Dirac surface states at the Fermi level, derived from Nurmamat et al. surface band structure calculations [25] for a Fermi level $0.43$eV above the Dirac point (see Figure 1c), form a hexastar (red and blue branches for states with spin-up and spin-down normal to the surface, respectively), with a clear nesting wavevector ${\mathbf{g}}_{\mathrm{CDW}}$ connecting parallel contours of equal spin. No such nesting occurs in the $\overline{\mathsf{\Gamma }\mathrm{K}}$ direction.