Finite-size energy gap in weak and strong topological insulators

The nontrivialness of a topological insulator (TI) is characterized either by a bulk topological invariant or by the existence of a protected metallic surface state. Yet, in realistic samples of finite size, this nontrivialness does not necessarily guarantee the gaplessness of the surface state. Depending on the geometry and on the topological indices, a finite-size energy gap of different nature can appear, and, correspondingly, exhibit various scaling behaviors of the gap. The spin-to-surface locking provides one such gap-opening mechanism, resulting in a power-law scaling of the energy gap. Weak and strong TIs show different degrees of sensitivity to the geometry of the sample. As a noteworthy example, a strong TI nanowire of a rectangular-prism shape is shown to be more gapped than that of a weak TI of precisely the same geometry.

Figure 1

The phase diagram of the Wilson-Dirac-type effective tight-binding Hamiltonian given in Eqs. (1) and (2). Notice the anisotropy of our hopping parameters [see Eqs. (3)]. In each of the strong (STI) and weak (WTI) topological insulator phases, together with the nature of the specific phase, the four ${\mathbb{Z}}_2$ indices ${\nu }_j$ ($j=0,1,2,3$) and the winding number $N_3$ are shown, as $N_3({\nu }_0,{\nu }_1{\nu }_2{\nu }_3)$ [See Eq. (A4) for the definition of $N_3$]. The solid lines representing the phase boundaries correspond to closing of the bulk energy gap.

Figure 2

Surface wave function in the rectangular prism geometry [Eq. (24)]; WTI phase ($m_{2z}/m_{2\parallel }=0.2$) with $N_z$ even. (a) The square of the wave function, ${\left|\psi (z,x)\right|}^2$, with $N_z=20$, $N_x=20$, and $k_y=0$ plotted in the $(z,x)$ plane. Spin and orbital indices are summed over; $A_{\perp }=A_{\parallel }=1$. (b) ${\left|\psi (x,y,z)\right|}^2$ is plotted in the 3D $(x,y,z)$ space. The front, upper, and right surfaces correspond, respectively, to the ones normal to $(-1,0,0)$, $(0,1,0)$, and $(0,0,1)$. Fixed boundary condition (FBC) in the $z$ and $x$ directions. PBC in the $y$ direction.

Figure 3

Plots of the surface wave function in the rectangular-prism geometry analogous to Fig. 2. Case of $N_z$ odd ($N_z=19$). WTI phase. $N_x=20$, $k_y=0$, $A_{\perp }=A_{\parallel }=1$. FBC in the $z$ and $x$ directions. PBC in the $y$ direction.

Figure 4

Plots of the surface wave function in the STI case ($m_{2z}/m_{2\parallel }=0.3$); plots similar to Figs. 2 and 3. Here, the surface wave function is extended over all four facets of the prism.

Figure 5

Even/odd feature in the finite-size energy gap (case of WTI). The mass parameters are on the (red) line $m_0/m_{2\perp }=-1$ of the phase diagram (Fig. 1), (a) slightly below (WTI-A case) and (b) above (STI case) the phase boundary at $m_{2\parallel }/m_{2\perp }=1/4$. The gap is plotted as a function $N_z$. (a) In the WTI-A case: $m_{2z}/m_{2\parallel }=0.2$, $E_0=E_0\left(N_z\right)$ shows an even/odd feature, and for $N_z$, even the gap scales as $\sim$${(N_z+1)}^{-1}$. (b) In the STI case: $m_{2\perp }/m_{2\parallel }=0.3$, a weak even/odd feature for small $N_z$ is washed out as $N_z$ increases, and the gap scales as $\sim$${(N_z+N_x)}^{-1}$. $N_x=20$. $A_{\perp }=A_{\parallel }=1$.

Figure 6

Shape of the surface wave function: tight-binding model vs $k·p$ approximation. ${\left|\psi \left(z\right)\right|}^2$, the squared amplitude of the surface state wave function at $x=1$ (and in the case of $k_z=0$), is plotted for the case of $N_z$ even ($N_z=30$, blue points). A continuous red curve is the prediction of $k·p$ theory [cf. Eq. (31)]. As in Fig. 2, the mass parameters are chosen to be $m_0/m_{2\parallel }=-1$, $m_{2\perp }/m_{2\parallel }=0.2$; other parameters are also set as in Fig. 2.

Figure 7

Surface wave function in the presence of disorder. Comparison of the WTI and STI cases: (a) $m_{2z}/m_{2\parallel }=0.2$ vs (b) $m_{2z}/m_{2\parallel }=0.3$. Here, the simulation is done for a system of size $N_x\times N_y\times N_z=10\times 10\times 10$; i.e., $N_z$ is even.

Figure 8

Finite-size energy gap in the rectangular-prism geometry plotted as a function of the “width” $N_x$. Comparison between the WTI (blue points) and STI (red points) regimes in the case of prism thickness $N_z$ odd ($N_z=9$). The logarithm of the energy gap $E_0$ is plotted vs $N_x$ for demonstrating that $E_0=E_0\left(N_x\right)$ decays exponentially, showing actually an exponentially damped oscillation in the WTI phase. The corresponding solutions of Eq. (B10) are a pair of complex numbers (see main text for details). The model parameters employed are also given there.

Figure 9

Size dependence of $E_0=E_0\left(N_x\right)$ in the case of $N_z$ odd ($N_z=9$). A plot similar to Fig. 8, but in the case of model parameters, yielding, as solutions for $\rho$ in Eq. (B10), two real solutions given in the main text. The data points for the WTI and STI cases are shown, respectively, in blue and red.

Figure 10

Size dependence of $E_0=E_0\left(N_x\right)$ in the case of $N_z$ even ($N_z=10$). The mass and velocity parameters are chosen to be the same as in the case of Fig. 8. Here, the vertical axis for $E_0$ is in the linear scale. For $N_z$ even, $E_0\left(N_x\right)$ shows at most a power-law decay, whether the system is in the STI or WTI phase (see Table ). The data points for the WTI and STI cases are as shown before, respectively, as blue and red filled circles.

Figure 11

The winding number $N_3$ [given in Eq. (A4)], evaluated on a horizontal or vertical line in Fig. 1. In (a), $m_{2\perp }/m_{2\parallel }$ is fixed at the isotropic point ($m_{2\perp }/m_{2\parallel }=1$), with $m_0/m_{2\parallel }$ being varied, while in the remaining panels, $m_0/m_{2\parallel }$ is fixed (b) at $m_0/m_{2\parallel }=-1$ and (c) at $m_0/m_{2\parallel }=-5$ with $m_{2\perp }/m_{2\parallel }$ being varied. The lines are shown in the same color in the phase diagram (see Fig. 1).