Gapped Surface States in a Strong-Topological-Insulator Material

A three-dimensional strong-topological insulator or semimetal hosts topological surface states which are often said to be gapless so long as time-reversal symmetry is preserved. This narrative can be mistaken when surface state degeneracies occur away from time-reversal-invariant momenta. The mirror invariance of the system then becomes essential in protecting the existence of a surface Fermi surface. Here we show that such a case exists in the strong-topological-semimetal ${\mathrm{Bi}}_4{\mathrm{Se}}_3$. Angle-resolved photoemission spectroscopy and ab initio calculations reveal partial gapping of surface bands on the ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ termination of ${\mathrm{Bi}}_4{\mathrm{Se}}_3(111)$, where an 85 meV gap along $\overline{\mathrm{\Gamma }}\overline{K}$ closes to zero toward the mirror-invariant $\overline{\mathrm{\Gamma }}\overline{M}$ azimuth. The gap opening is attributed to an interband spin-orbit interaction that mixes states of opposite spin helicity.

Figure 1

Calculated band structure for the ${\mathrm{Bi}}_2{\mathrm{Se}}_3$-terminated slabs of ${\mathrm{Bi}}_4{\mathrm{Se}}_3$ plotted along the $\overline{M}\overline{\mathrm{\Gamma }}\overline{K}$ path in the surface Brillouin zone. The size of the circular plotting markers indicates the contribution from the surface layer. Shaded regions indicate the projection of bulk electrons. The red circle contains the mirror-symmetry-protected crossing of TSS and the red double arrow indicates the corresponding TSS avoided crossing gap $\mathrm{\Delta }$. The blue circle contains a TSS crossing protected by both TRS and MS.

Figure 2

Electron structure given by the model Hamiltonian for parameters stated in the text: band structure near the anticrossing point along $\overline{\mathrm{\Gamma }}\overline{K}$ ($k_y=0$) direction (a). The color scale [(a), inset] indicates the spin polarization in $\widehat{y}$ direction, while marker size indicates the degree of $z$ spin polarization. Constant energy contours (CECs) of the upper Dirac branch (b). The color scale [(b), inset] indicates the energy $E$ of each contour measured with respect to the Dirac point. The momentum-space contour of the gap minimum overlayed onto the CEC at the Lifshitz transition energy is shown in (c).

Figure 3

ARPES electron structure near the center of the first surface Brillouin zone: the Fermi surface (a),(b), and constant energy contour at $-185\mathrm{meV}$ (c). Band structure parallel to $\overline{\mathrm{\Gamma }}\overline{M}$ direction for $k_x=0$ (d) and $k_x=0.21{\AA }^{-1}$ (i). Band structure parallel to $\overline{\mathrm{\Gamma }}\overline{K}$ direction at $k_y=-0.255{\AA }^{-1}$ (e), $k_y=0.255{\AA }^{-1}$ (f), and $k_y=0$ (h). Energy distribution curves at the degeneracy point $(k_x,k_y)=(0,-0.255){\AA }^{-1}$ (g) and the saddle point of the lower Dirac branch $(k_x,k_y)=(0.21,0){\AA }^{-1}$ (j). The momentum-space locations for panels (d),(e) and (h),(i) are indicated by lines overlaid onto the Fermi surface in (a). The gray scale varies from white to black following the minimum-to-maximum photoemission intensity within each image individually.