Anomalous finite size effects on surface states in the topological insulator ${\text{Bi}}_2{\text{Se}}_3$

We study how the surface states in the strong topological insulator ${\text{Bi}}_2{\text{Se}}_3$ are influenced by finite size effects and compare our results with those recently obtained for two-dimensional topological insulator HgTe. We demonstrate two important distinctions: (i) contrary to HgTe, the surface states in ${\text{Bi}}_2{\text{Se}}_3$ display a remarkable robustness towards decreasing the width $L$ down to a few nm thus ensuring that the topological surface states remain intact and (ii) the gapping due to the hybridization of the surface states features an oscillating exponential decay as a function of $L$ in ${\text{Bi}}_2{\text{Se}}_3$ in sharp contrast to HgTe. Our findings suggest that ${\text{Bi}}_2{\text{Se}}_3$ is suitable for nanoscale applications in quantum computing or spintronics. Also, we propose a way to experimentally detect both of the predicted effects.

Figure 1

(Color online) Suggested experimental setup: two-terminal geometry with the topological insulator ${\text{Bi}}_2{\text{Se}}_3$. The direction of the current carried by the surface states is shown by the arrows and the width of the sample is $L$. For $L\sim 2\hslash v_F/\left|M\right|$, finite size effects become important. Here, $\left|M\right|$ is the bulk charge-excitation gap.

Figure 2

(Color online) Plot of the surface/edge state energy dispersions versus the transverse momentum $k_{\perp }$ in the case of ${\text{Bi}}_2{\text{Se}}_3$ and HgTe. For ${\text{Bi}}_2{\text{Se}}_3$, we give only the result for $L=1000\text{nm}$ since the dispersion remains completely unchanged down to $L=150\text{nm}$, in contrast to HgTe where a gap opens up. The inset of ${\text{Bi}}_2{\text{Se}}_3$ shows the bulk bands and the surface states with a gap of $2\left|M\right|=0.56\text{eV}$ opening at the $\Gamma$ point.

Figure 3

(Color online) Plot of the finite size induced gap $\Delta$ between the surface states at $k_{\perp }=0$ for ${\text{Bi}}_2{\text{Se}}_3$ as a function of the width $L$. In contrast to HgTe where the decay with $L$ is purely exponential, an oscillatory pattern is superimposed on the decay for ${\text{Bi}}_2{\text{Se}}_3$.

Figure 4

(Color online) Plot of the two-terminal conductance at $T=20\text{K}$ and $T=200\text{K}$ as a function of $\mu$ and $L$.