Landau level states on a topological insulator thin film

We analyze the four-dimensional Hamiltonian proposed to describe the band structure of the single-Dirac-cone family of topological insulators in the presence of a uniform perpendicular magnetic field. Surface Landau level (LL) states appear, decoupled from the bulk levels and following the quantized energy dispersion of a purely two-dimensional surface Dirac Hamiltonian. A small hybridization gap splits the degeneracy of the central $n=0$ LL, with dependence on the film thickness and the field strength that can be obtained analytically. Explicit calculations of the spin and charge densities show that surface LL states are localized within approximately one quintuple layer from the surface termination. Some new surface-bound LLs are shown to exist at a higher Landau level index.

Figure 1

(a) Surface energy spectra without magnetic field when ${\varepsilon }_k=0$ (red) and ${\varepsilon }_k\ne 0$ (blue). (b) Bulk and surface energy dispersions in the absence of magnetic field and ${\varepsilon }_k=0$. $L_z/l_z$ $=$ 3000 was used.

Figure 2

LL energies for $L_z/l_z=3\times {30}^3$ and $H_{\mathrm{phys}}$ $=$ 10 T, showing both surface (sky blue squares) and bulk states (black squares). Three surface branches are labeled (I)–(III) with analytic fits shown as red solid curves to $\sqrt{2n}/l_H$ (branch I) and $\sqrt{A\pm \delta -\mathit{Bn}}$ (branches II and III). $A$ and $B$ values are derived in the Appendix and a small offset $\delta$ is used to fit the numerical results. When $H_{\mathrm{phys}}$ $=$ 10 T, then $A=0.918,\delta =0.068,B=0.037$, respectively.

Figure 3

Hybridization gap energies for $H_{\mathrm{phys}}=0$, 10, and 50 T with varying thickness $L_z$.

Figure 4

th-LL wave-function density$n$ ${\rho }_{n,c}(z)$ and the spin density ${\rho }_{n,s}(z)$ for $L_z=60$ Å.