Topological Insulators in Three Dimensions

TL;DR

A tight binding model is introduced which realizes the WTI and STI phases, and its relevance to real materials, including bismuth is discussed.

Abstract

We study three dimensional generalizations of the quantum spin Hall (QSH) effect. Unlike two dimensions, where the QSH effect is distinguished by a single $Z_2$ topological invariant, in three dimensions there are 4 invariants distinguishing 16 "topological insulator" phases. There are two general classes: weak (WTI) and strong (STI) topological insulators. The WTI states are equivalent to layered 2D QSH states, but are fragile because disorder continuously connects them to band insulators. The STI states are robust and have surface states that realize the 2+1 dimensional parity anomaly without fermion doubling, giving rise to a novel "topological metal" surface phase. We introduce a tight binding model which realizes both the WTI and STI phases, and we discuss the relevance of this model to real three dimensional materials, including bismuth.

Figures (4)

• Figure 1: Schematic surface (or edge) state spectra as a function of momentum along a line connecting $\Lambda_{a}$ to $\Lambda_{b}$ for (a) $\pi_a\pi_b=-1$ and (b) $\pi_a\pi_b=+1$. The shaded region shows the bulk states. In (a) the TRP changes between $\Lambda_a$ and $\Lambda_b$, while in (b) it does not.
• Figure 2: Diagrams depicting four different phases indexed by $\nu_0 ; (\nu_1\nu_2\nu_3)$. (a) depicts $\delta_i$ at the TRIM $\Gamma_i$ at the vertices of the cube. (b) characterizes the $001$ surface in each phase. The surface TRIM $\Lambda_a$ are denoted by open (filled) circles for $\pi_a = \delta_{a1}\delta_{a2} = +1 (-1)$. They are projections of $\Gamma_{a1}$ and $\Gamma_{a2}$, which are connected by solid lines in (a). The thick lines and shaded regions in (b) indicate possible surface Fermi arcs which enclose specific $\Lambda_a$.
• Figure 3: Energy bands for (a) the model (\ref{['tbmodel']}) with $t=1$, $\lambda_{SO}= .125$. The symmetry points are $\Gamma = (0,0,0)$, $X = (1,0,0)$, $W = (1,1/2,0)$, $K=(3/4,3/4,0)$ and $L = (1/2,1/2,1/2)$ in units of $2\pi/a$. The dashed line shows the energy gap due to $\delta t_1 = .4$. (b) shows the phase diagram as a function of $\delta t_1$ and $\delta t_2$ (for bonds in the $111$ and $1\bar{1}\bar{1}$ directions) with phases indexed according to cubic Miller indices for ${\bf G}_\nu$. The shaded region is the STI phase.
• Figure 4: 2D band structures for a slab with a 111 face for the four phases in Fig. 3. The states crossing the bulk energy gap are localized at the surface. In the WTI (STI) phases there are an even (odd) number of Dirac points in the surface spectrum. The inset shows the surface Brillouin zone.