Oscillatory crossover from two-dimensional to three-dimensional topological insulators

We investigate the crossover regime from three-dimensional topological insulators ${\text{Bi}}_2{\text{Te}}_3$ and ${\text{Bi}}_2{\text{Se}}_3$ to two-dimensional topological insulators with quantum spin Hall effect when the layer thickness is reduced. Using both analytical models and first-principles calculations, we find that the crossover occurs in an oscillatory fashion as a function of the layer thickness, alternating between topologically trivial and nontrivial two-dimensional behavior.

Figure 1

(Color online) The energy level versus the thickness of the quantum well is shown for (a) $A_1=0\text{eV}\cdot \text{Å}$, (b) $A_1=1.1\text{eV}\cdot \text{Å}$. Other parameters are taken from Ref. 5. The shaded region indicates the regime for QSH states. The blue dashed line in (b) shows how the crossing between $|E_1\left(H_1\right)⟩$ and $|H_2\left(E_2\right)⟩$ is changed to anticrossing when $A_1$ is nonzero. In (c), the density of $|S_1^{+}⟩$ ($|S_2^{-}⟩$ has the same density) is plotted for $A_1=1.1\text{eV}\cdot \text{Å}$. In (d), the critical thickness $d_{cn}\left(n=1,2,\cdots \right)$ is plotted as a function of $A_1$. QSH states appear in the shaded region.

Figure 2

(Color online) The band gap and the total parity are plotted as the function of the number of the QLs for (a) ${\text{Bi}}_2{\text{Se}}_3$ and (b) ${\text{Bi}}_2{\text{Te}}_3$. The calculation is based on TB model constructed by MLWF from first-principles calculation. The results from the fully first-principles calculation are shown in table (c) for $1\sim 5$ QLs. Good agreement between the TB model calculation and first-principles calculation is found.

Figure 3

(Color online) Left: the energy dispersion of 2D thin film is plotted for (a) 2QLs ${\text{Bi}}_2{\text{Te}}_3$ and (c) 3QLs ${\text{Bi}}_2{\text{Se}}_3$. Right: the energy dispersion of the semi-infinite film with the edge along $A1$ direction is plotted for (b) 2QLs ${\text{Bi}}_2{\text{Te}}_3$ and (d) 3QLs ${\text{Bi}}_2{\text{Se}}_3$. The inset of (a) shows the 2D BZ and that of (c) is the top view of 2D thin film with two in-plane lattice vectors $A1$ and $A2$. The 1D edge is indicated by the blue line along $A1$ direction. The insets of (b) and (d) are the zoom-in of the energy dispersion near the bulk gap.