Coulomb impurities in two-dimensional topological insulators

Introducing a powerful method, we obtain the exact solutions for a Coulomb impurity in two-dimensional infinite and finite topological insulators. The level order and zero-energy degeneracy of the spectra are found to be quite different between topological trivial and nontrivial phases. For quantum dots of topological insulator, the variation of the edge and Coulomb states with dot size, Coulomb potential, and magnetic field are clearly shown. It is found that for small dots the edge states can be strongly coupled with the Coulomb states and for large dots the edge states are insensitive to the Coulomb fields but sensitive to the magnetic fields.

Figure 1

The Coulomb states of $H_{\uparrow }$ as a function of $\alpha$ for an infinite 2DBI (a) and 2DTI (b). The signals $a^{\pm }, b^{\pm }$, and $c^{\pm }$ stand for the Coulomb states ${(0,\pm 1/2)}_{\uparrow }, {(1,\pm 1/2)}_{\uparrow }$, and ${(0,\pm 3/2)}_{\uparrow }$, respectively. The dash-dotted lines stand for the states $(0,j)$ of gapped graphene. The dashed lines in panel (b) show the edge states $(\uparrow ,j)$ for a 2DTI quantum dot with $R=500\mathrm{nm}$.

Figure 2

The levels of Coulomb states $a^{+}$ and edge states $e^{+}$ of $H_{\uparrow }$ under different $\alpha$ as a function of $R$ for 2DTI (a) and 2DBI (b). The red and blue dot lines are the levels of $a^{+}$ for gapped graphene. The radial density distribution ${\rho }_r$ of $a^{+}$ and $e^{+}$ states for $\alpha =1.2$ at $R=34\mathrm{nm}$ and $100\mathrm{nm}$ (c) and at $R=44\mathrm{nm}$ (d) in 2DTI quantum dots.

Figure 3

The energy levels of Coulomb states ${(n,\frac{1}{2})}_{\uparrow }$ and edge states $e^{+}$ in a 2DTI dot as a function of $\alpha$ with $R=50\mathrm{nm}$ (a) and $R=30\mathrm{nm}$ (b). The red dashed lines are the corresponding Coulomb levels in an infinite 2DTI. (c) The values of $\gamma =\langle \widehat{r}\rangle /R$ for $a^{+}$ and $e^{+}$ as a function of $\alpha$. (d) The wave functions of the states $e^{+}$ respectively for $R=30\mathrm{nm}$ and $R=50\mathrm{nm}$ when $\alpha ={\alpha }_c-0.1$, as labeled by the dots in panel (c).

Figure 4

The energy levels of Coulomb and edge states in a 2DTI quantum dot with $B=1\mathrm{T}$ as a function of $R$ at fixed $\alpha =0.3$ for $j=\frac{1}{2}, H_{\uparrow }$ (a) and $j=-\frac{1}{2}, H_{\downarrow }$ (b). The levels as a function of $\alpha$ at fixed $R=50\mathrm{nm}$ for $j=\frac{1}{2}, H_{\uparrow }$ (c) and $j=-\frac{1}{2}, H_{\downarrow }$ (d). Both levels of $H_{\uparrow }$ (black lines) and $H_{\downarrow }$ (red lines) as a function of $B$ at fixed $R=150\mathrm{nm}$ for $\alpha =0.6$ (e) and $\alpha =1.3$ (f). The dashed lines are levels for $B=0$ in panels (a)–(d).