Mott scattering at the interface between a metal and a topological insulator

We compute the spin-active scattering matrix and the local spectrum at the interface between a metal and a three-dimensional topological band insulator. We show that there exists a critical incident angle at which complete (100%) spin-flip reflection occurs and the spin-rotation angle jumps by $\pi$. We discuss the origin of this phenomena, and systematically study the dependence of spin-flip and spin-conserving scattering amplitudes on the interface transparency and metal Fermi-surface parameters. The interface spectrum contains a well-defined Dirac cone in the tunneling limit and smoothly evolves into a continuum of metal-induced gap states for good contacts. We also investigate the complex band structure of ${\text{Bi}}_2{\text{Se}}_3$.

Figure 1

(Color online) (a) Scattering geometry at a M-TI interface. (b) Schematic band structure of the metal (modeled by ${\widehat{H}}_{\text{M}}$) and topological insulator.

Figure 2

(Color online) The complex band structure of topological insulator described by ${\widehat{H}}_{\text{TI}}\left(\mathbf{k}\right)$ for $k_y=0$, $k_x=0.02$ (left) and 0.04 (right). $E$ is measured in electron volt and $k$ in per angstrom. Subgap states with complex $k_z$ represent evanescent waves. The topology of real lines (Ref. 25) changes as $k_x$ is increased.

Figure 3

(Color online) The magnitudes (upper panel) and the phases (lower panel) of the spin-flip amplitude $f$ and spin-conserving amplitude $g$ versus the incident angle $\theta$. $E=0.1\text{eV}$ and $E_F=0.28\text{eV}$. ${\left|g\right|}^2+{\left|f\right|}^2=1$. $\text{arg}\left(g\right)$ and $\text{arg}\left(f\right)$ are shifted upward by $\pi$ for clarity.

Figure 4

(Color online) The spectral function $N\left(E,k_x,k_y=0\right)$ at the interface of metal and topological insulator. Left: good contact, $J=t_{\text{M}}$, showing the continuum of metal-induced gap states. Right: poor contact with low transparency, $J=0.2t_{\text{M}}$, showing well-defined Dirac spectrum as on the TI surface. $t_{\text{M}}=0.18\text{eV}$, ${\mu }_{\text{M}}=-4t_{\text{M}}$, $a$ is lattice spacing.

Figure 5

(Color online) The spin-conserving reflection amplitude $\left|g\right|$ and spin-rotation angle $\alpha$ versus the incident angle $\theta$ for increasing contact transparency, $J/t_{\text{M}}=0.25$, 1, 1.5, and 2 (from left to right). $t_{\text{M}}=0.18\text{eV}$, ${\mu }_{\text{M}}=-4t_{\text{M}}$, $E=0.05\text{eV}$, and $k_y=0$. ${\left|f\right|}^2=1-{\left|g\right|}^2$.