Reorganization of a topological surface state: Theory for ${\mathrm{Bi}}_2{\mathrm{Te}}_3$(111) covered by noble metals

The electronic structure of ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ covered by noble metals is investigated by first-principles calculations. The Dirac surface state of the topological insulator ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ hybridizes with the $sp$ states of the noble metal, which gives rise to strong reorganization of the surface electronic structure. Striking features of the modified Dirac surface state are (i) the introduction of new Dirac points within the fundamental band gap of ${\mathrm{Bi}}_2{\mathrm{Te}}_3$, (ii) a very weak dispersion, and (iii) a multisheeted Fermi surface which results in an orientation-dependent number of conducting channels in the fundamental band gap of ${\mathrm{Bi}}_2{\mathrm{Te}}_3$. Our findings have impact for spin-dependent surface transport.

Figure 1

Reorganization of the electronic structure at an interface of a topological insulator and a noble metal (schematic). (a) For the completely separated surfaces, there are three surface states: the Dirac surface state (DSS) of the topological insulator (represented by straight lines) and the spin-degenerate $sp$ surface states of the noble metal (curved lines; the spin orientation is distinguished by solid and dashed lines). The DSS forms Dirac cones $D_{\Gamma }$ and $D_{\mathrm{M}}$ at the time-reversal invariant momenta $\overline{\Gamma }$ and $\overline{\mathrm{M}}$, respectively. (b) For the interface (coupled surfaces), the spin degeneracy of the $sp$ surface states is lifted; they hybridize with the Dirac state forming new Dirac points and causing dispersion. (c) As panel (b) but with a crossing due to dispersion of the surface state along $\overline{\Gamma }\text{−}\overline{\mathrm{M}}$.

Figure 2

Total energy of $\mathrm{Au}/{\mathrm{Bi}}_2{\mathrm{Te}}_3$(111) versus Au-Te distance. The vertical lines indicate the largest distance used (5.4 Å) and the equilibrium distance (1.2 Å), respectively.

Figure 3

Illustration of the spin-dependent hybridization of a nearly free electron band (with sinusoidal dispersion) and a linearly dispersing topological surface state. The coupling strength $\kappa$ increases from left to right, as indicated in each panel. Original Dirac points of the topological surface state are labeled ${\mathrm{DP}}_i,i=1,2,3$, whereas the new Dirac points that are brought about by the hybridization are labeled ${\mathrm{NDP}}_i,i=1,2$. Dirac points belonging together are connected by straight lines.

Figure 4

Reorganization of the electronic structure of $\mathrm{Au}/{\mathrm{Bi}}_2{\mathrm{Te}}_3$. (a) Surface electronic structure for a large Au-Te distance of $5.4\text{Å}$. The stronger the localization of the electronic states in the surface layers, the darker (red) the color scale [shown in (b)]. The Dirac point of ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ at $\overline{\Gamma }$ is denoted $D_{\Gamma }$. The Au-derived $sp$ states show up as highly dispersing bands bridging the fundamental band gap of ${\mathrm{Bi}}_2{\mathrm{Te}}_3$. (b) As (a) but for the equilibrium Au-Te distance of $1.2\text{Å}$. The Dirac point at $\overline{\mathrm{M}}$ is denoted $D_{\mathrm{M}}$ (cf. Fig. 1). The dashed lines are guides to the eye, indicating the connection to $D_{\Gamma }$. The inset shows the two-dimensional Brillouin zone. (c) Energies of the Dirac points $D_{\Gamma }$ and $D_{\mathrm{M}}$, shown in (b), versus Au-Te distance. $E_{\mathrm{D}}$ is the energy of $D_{\Gamma }$ without the gold layer. (d) Geometry of $\mathrm{Au}/{\mathrm{Bi}}_2{\mathrm{Te}}_3$ (top view), with Au, Bi, and Te atoms colored yellow, purple, and green, respectively.

Figure 5

Spin-resolved surface electronic structure of $\mathrm{Au}/{\mathrm{Bi}}_2{\mathrm{Te}}_3$ at the $\mathrm{Au}\text{−}{\mathrm{Bi}}_2{\mathrm{Te}}_3$ equilibrium distance. (a) “Rashba” component of the spin polarization (within the surface plane and perpendicular to $\vec{k})$. (b) Out-of-plane component of the spin polarization.

Figure 6

Fermi surface of $\mathrm{Au}/{\mathrm{Bi}}_2{\mathrm{Te}}_3$ at the equilibrium Au-Te distance $\left(1.2\text{Å}\right)$. (a) Spin texture of the bands crossing the Fermi surface: the vector size and orientation represent the in-plane component, while the color represents the out-of-plane projection. (b) The two Fermi sheets are distinguished by color (blue and red); regions occupied by both sheets are given in purple.

Figure 7

Surface electronic structure of $\mathrm{Au}/{\mathrm{Bi}}_2{\mathrm{Te}}_3$(111) for selected $\mathrm{Au}\text{−}{\mathrm{Bi}}_2{\mathrm{Te}}_3$ distances (indicated in each panel). $D_{\Gamma }$ and $D_{\mathrm{M}}$ mark Dirac points of the topologically protected surface states. The yellow-red color scale of each band indicates the localization of the associated wave function in the surface region (yellow: weakly localized; red: strongly localized).

Figure 8

Spin-resolved surface electronic structure of a ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ $2\times 2$ supercell covered by a Au monolayer on top. The Au supercell is $3\times 3$; cf. the sketch on the right. The color scheme indicates the component of the spin polarization perpendicular to ${\vec{k}}_{\parallel }$ (i. e., the Rashba component of the spin polarization).

Figure 9

Surface-projected band structures of ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ with two to six Au layers (a)–(e) or one layer of Cu (f). The color scheme indicates the component of the spin polarization perpendicular to ${\vec{k}}_{\parallel }$ (i. e., the Rashba component of the spin polarization).

Figure 10

Surface electronic structure of $\mathrm{Au}/{\mathrm{Bi}}_2{\mathrm{Te}}_3$(111) with spin-orbit coupling turned off. Color coding as in Fig. 7.