Topological surface states scattering in antimony

In this work we study the topologically protected states of the Sb(111) surface by using ab initio transport theory. In the presence of a strong surface perturbation we obtain standing-wave states resulting from the superposition of spin-polarized surface states. By Fourier analysis, we identify the underlying two dimensional scattering processes and the spin texture. We find evidence of resonant transmission across surface barriers at quantum well state energies and evaluate their lifetimes. Our results are in excellent agreement with experimental findings. We also show that despite the presence of a step edge along a different high-symmetry direction, the surface states exhibit unperturbed transmission around the Fermi energy for states with near to normal incidence.

Figure 1

(a) Structure of antimony in the hexagonal setting. The atoms form a bilayer structure with the intralayer distance as $1.51$ Å and the interlayer distance as $2.25$ Å. Band structure for (b) 6- and (c) 12-bilayer-thick slabs along $\overline{K}-\overline{\Gamma }-\overline{M}$ directions. (d) Simulated ARPES from a semi-infinite slab. The distorted Dirac cone is found to comprise spin-polarized surface bands. Here and henceforth warmer colors represent higher PDOS (red represents largest values, blue lowest ones, with the color scale in between being linear). Spin-resolved ARPES along (e) $\overline{\Gamma }-\overline{M}$ and (f) $\overline{\Gamma }-\overline{K}$ directions showing the opposite spins of the two surface bands along the directions indicated by arrows in the inset of the figures. In this case, red and blue colors indicate up and down spins, respectively.

Figure 2

(a) A schematic of the setup showing one-bilayer-high surface perturbation on a 12-bilayer slab. (b) Top view of the slab; surface step runs along $y$ for the first setup and along $x$ for the second. (c) PDOS for surface atoms with a single step adjacent to a flat region for the first setup at $k_{\perp }=0$. The quantization of the energy levels in the step is clearly seen, with accompanying phase shifts in the adjacent flat region. (d) Transmission at $k_{\perp }=0$ with and without the step, indicating finite scattering due to the step. Average of transmission over all $k_{\perp }$ points is also shown. (e) PDOS for surface atoms averaged over all $k_{\perp }$. (f) Fourier transform of PDOS data in (e) over the flat region reveals the different allowed scattering wave vectors. The most prominent features, $q_A$ and $q_B$, are nearly linear with slopes equal to $1.1$ eV Å.

Figure 3

The allowed scattering wave vectors obtained from the Fourier transform of PDOS data at (a) $k_{\perp }b=0$, (b) $k_{\perp }b=0.16\pi$, and (c) $k_{\perp }b=0.50\pi$. The panels on the right show the corresponding band structures at the same $k_{\perp }$ for the 12-bilayer slab electrodes.

Figure 4

(a) PDOS at $k_{\perp }=0$ for surface atoms with a single step adjacent to a flat region, with the step extending along the $x$ direction in Fig. 2b. There is an energy region from $-60$ to 20 meV with no scattering and hence no standing-wave states. (b) Transmission at $k_{\perp }=0$, indicating the perfect transmission around $E_F$ even in the presence of a surface barrier, the defining feature of a topological state. Total transmission also shows minimal scattering in that energy range.