Dual Topological Character of Chalcogenides: Theory for ${\mathrm{Bi}}_2{\mathrm{Te}}_3$

A topological insulator is realized via band inversions driven by the spin-orbit interaction. In the case of ${\mathcal{Z}}_2$ topological phases, the number of band inversions is odd and time-reversal invariance is a further unalterable ingredient. For topological crystalline insulators, the number of band inversions may be even but mirror symmetry is required. Here, we prove that the chalcogenide ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ is a dual topological insulator: it is simultaneously in a ${\mathcal{Z}}_2$ topological phase with ${\mathcal{Z}}_2$ invariants $({\nu }_0;{\nu }_1{\nu }_2{\nu }_3)=(1;000)$ and in a topological crystalline phase with mirror Chern number $-1$. In our theoretical investigation we show in addition that the ${\mathcal{Z}}_2$ phase can be broken by magnetism while keeping the topological crystalline phase. As a consequence, the Dirac state at the (111) surface is shifted off the time-reversal invariant momentum $\overline{\mathrm{\Gamma }}$; being protected by mirror symmetry, there is no band gap opening. Our observations provide theoretical groundwork for opening the research on magnetic control of topological phases in quantum devices.

Figure 1

Variation of a Dirac surface state’s dispersion with respect to the topological phase (schematic): without magnetic field (left, $\mathrm{B⃗}=0$: dual topological insulator), with magnetic field perpendicular to a mirror plane (center, $\mathrm{B⃗}\perp$ mirror plane: topological crystalline insulator), and with magnetic field within the mirror plane (right, $\mathrm{B⃗}\parallel$ mirror plane: conventional insulator). The dispersions calculated from Fu’s model [20] are shown in perspective view. The inset displays the surface Brillouin zone. The $\overline{M}\text{−}\overline{\mathrm{\Gamma }}\text{−}\overline{M}$ direction lies within a mirror plane.

Figure 2

Left: perspective view of ${\mathrm{Bi}}_2{\mathrm{Te}}_3$. Each of the quintuple layers shown consists of two Bi atoms (large magenta spheres) and three Te atoms (small brown spheres). The shaded green area depicts the $yz$ mirror plane of the crystal ($z$ along [111]). Right: Dirac surface state in ${\mathrm{Bi}}_2{\mathrm{Te}}_3(111)$, obtained from tight-binding calculations. The spectral density of the topmost quintuple layer is shown as color scale (in states per eV) along a $\overline{K}\text{−}\overline{\mathrm{\Gamma }}\text{−}\overline{M}$ path of the two-dimensional Brillouin zone (inset). The yellow rectangle highlights the ($E$, $k$) area addressed in Fig. 3.

Figure 3

Dispersion of the Dirac state at ${\mathrm{Bi}}_2{\mathrm{Te}}_3(111)$ for different magnetic configurations. Only a small part of the two-dimensional Brillouin zone (indicated in Fig. 2) is displayed (top row: $k_x$; bottom row: $k_y$; the respective zeroes are indicated by vertical dash-dotted lines). Without magnetic field [$\mathrm{B⃗}=0$, (a) and (e)], ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ is in its dual topological phase (Dirac point at ${\mathrm{k⃗}}_{\parallel }=0$). For an in-plane magnetic field $\mathrm{B⃗}\parallel \mathrm{x⃗}$ [(b) and (f)] it stays in its topological crystalline phase; the inset in (f) zooms into the Dirac point shifted along $k_y$. For both $\mathrm{B⃗}\parallel \mathrm{y⃗}$ [(c) and (g)] and $\mathrm{B⃗}\parallel \mathrm{z⃗}$ [(d) and (h)] the topological phase is trivial: there is no Dirac point [cf. the zoom into the tiny band gap in (c)]. The color scale, in states per eV, displays the spectral density of the topmost quintuple layer. In regions in which the Dirac state hybridizes with bulk electronic states, the spectral density becomes blurred.

Figure 4

Effect of a magnetic field on the dispersion of a Dirac state, obtained from a model calculation (schematic; see text). Constant energy contours are calculated (a) without magnetic field ($\mathrm{B⃗}=0$), (b) with magnetic field perpendicular to a mirror plane ($\mathrm{B⃗}\parallel \mathrm{x⃗}$), and (c) with magnetic field within the mirror plane ($\mathrm{B⃗}\parallel \mathrm{y⃗}$). The $k_x$ ($k_y$) axis is along $\overline{K}\text{−}\overline{\mathrm{\Gamma }}\text{−}\overline{K}$ ($\overline{M}\text{−}\overline{\mathrm{\Gamma }}\text{−}\overline{M}$).