Surface states and topological invariants in three-dimensional topological insulators: Application to ${\text{Bi}}_{1-x}{\text{Sb}}_x$

We study the electronic surface states of the semiconducting alloy bismuth antimony $\left({\text{Bi}}_{1-x}{\text{Sb}}_x\right)$. Using a phenomenological tight-binding model, we show that the Fermi surface for the 111 surface states encloses an odd number of time-reversal-invariant momenta (TRIM) in the surface Brillouin zone. This confirms that the alloy is a strong topological insulator in the (1;111) ${\mathbb{Z}}_2$ topological class. We go on to develop general arguments which show that spatial symmetries lead to additional topological structure of the bulk energy bands, and impose further constraints on the surface band structure. Inversion-symmetric band structures are characterized by eight ${\mathbb{Z}}_2$ “parity invariants,” which include the four ${\mathbb{Z}}_2$ invariants defined by time-reversal symmetry. The extra invariants determine the “surface fermion parity,” which specifies which surface TRIM are enclosed by an odd number of electron or hole pockets. We provide a simple proof of this result, which provides a direct link between the surface-state structure and the parity eigenvalues characterizing the bulk. Using this result, we make specific predictions for the surface-state structure for several faces of ${\text{Bi}}_{1-x}{\text{Sb}}_x$. We next show that mirror-invariant band structures are characterized by an integer “mirror Chern number” $n_{\mathcal{M}}$, which further constrains the surface states. We show that the sign of $n_{\mathcal{M}}$ in the topological insulator phase of ${\text{Bi}}_{1-x}{\text{Sb}}_x$ is related to a previously unexplored ${\mathbb{Z}}_2$ parameter in the $L$ point $\mathbf{k}\cdot \mathbf{p}$ theory of pure bismuth, which we refer to as the “mirror chirality” $\eta$. The value of $\eta$ predicted by the tight-binding model for bismuth disagrees with the value predicted by a more fundamental pseudopotential calculation. This explains a subtle disagreement between our tight-binding surface-state calculation and previous first-principles calculations of the surface states of bismuth. This suggests that the tight-binding parameters in the Liu-Allen model of bismuth need to be reconsidered. Implications for existing and future angle-resolved photoemission spectroscopy (ARPES) experiments and spin-polarized ARPES experiments will be discussed.

Figure 1

(Color online) (a) Crystal structure of Bi. (b) Three-dimensional (3D) Brillouin zone and its projection onto the (111) surface. Also displayed is the choice of coordinate system throughout the paper: $z$ is along the (111) direction, $y$ is along the $\overline{\Gamma }$ to $\overline{M}$ direction, and $O$ is a center of inversion.

Figure 2

Band evolution of interpolated tight-binding model using the parameters in Eqs. (2.4, 2.5).

Figure 3

(Color online) (a) Geometry for our surface-state calculations, which defines our coordinate system and specifies the spin directions of the ${\overline{\Sigma }}_1$ and ${\overline{\Sigma }}_2$ bands, which have mirror eigenvalues $+i$ and $-i$, respectively. (b) Brillouin zone for the (111) face of ${\text{Bi}}_{1-x}{\text{Sb}}_x$ with the electron pocket and six hole pockets predicted by our tight-binding calculation. (c) Surface band structure along the line between $\overline{\Gamma }$ and $\overline{M}$ predicted by the tight-binding model. The shaded regions are the bulk states projected onto the surface. (d) Schematic illustration of experimental surface band structure and Fermi surface probed by angle-resolved photoemission spectroscopy (Ref. 24). The top shows the Fermi surface in a slice of the Brillouin zone near $k_x=0$, and the bottom shows the surface-state dispersion. Compared with (c), there are two additional bands near $\overline{M}$.

Figure 4

(Color online) (a) Bi surface states between $\overline{\Gamma }$ and $\overline{M}$ calculated using tight-binding model. (b) Schematic picture of Bi bands in (a) in which Hartree effects raise the bands to accommodate charge neutrality. The crossing of ${\Sigma }_1$ and ${\Sigma }_2$ results in a Dirac point enclosed by a hole pocket. (c) Schematic picture without the crossing between ${\Sigma }_1$ and ${\Sigma }_2$, which resembles a first-principles calculation of surface states in Bi (Refs. 22, 25).

Figure 5

Two inequivalent inversion centers $c$ and $c^{\prime }$ in an inversion-symmetric crystal, which differ by half a lattice vector. The parity eigenvalues of Bloch state at momentum $k=\pi /R$ with inversion center chosen at $c$ and $c^{\prime }$ are different. Crystals terminated at $c$ and $c^{\prime }$ will have surface charges that differ by an odd integer.

Figure 6

(Color online) Schematic diagram showing which surface TRIM are enclosed by an odd number of electron or hole pockets for different faces of ${\text{Bi}}_{1-x}{\text{Sb}}_x$ predicted by the surface fermion parity in Table . (a)–(d) show the (111), ${\left(111\right)}^{\prime }$, (110), and (100) faces. The ${\left(111\right)}^{\prime }$ surface is a hypothetical surface cleaved in the middle of a bilayer.

Figure 7

(Color online) (a) A one-dimensional inversion-symmetric insulator cut at $z=0$ by replacing hopping amplitudes $t$ across $z=0$ by $\lambda t$. The fully cleaved crystal corresponds to $\lambda =0$. (b) Energy spectrum as a function of $\lambda$ between $-1$ and 1. The conduction and valence bands exchange a Kramers pair of states with opposite parities. [(c) and (d)] The bulk energy levels at $\lambda =\pm 1$. For $\lambda =-1$ (d) every state at $k$ has a partner at $-k$ with the same energy and opposite parity. For $\lambda =+1$ (c) the states at $k={\Gamma }_1$ and $k={\Gamma }_2$ are not paired.