Electron transmission through atomic steps of Bi${}_2$Se${}_3$ and Bi${}_2$Te${}_3$ surfaces

Transmission properties of surface-state electrons through steps of Bi${}_2$Se${}_3$ and Bi${}_2$Te${}_3$ surfaces are theoretically studied with a tight-binding method. Transmission spectra of Bi${}_2$Se${}_3$ as a function of the incident angle are simply explained in terms of potential scattering of a massless Dirac Hamiltonian. However, those of Bi${}_2$Te${}_3$ are not similarly explained, and perfect reflection appears at an angle. The mechanism of perfect reflection is discussed in terms of the massless Dirac Hamiltonian with the hexagonal-warping term. It is found that evanescent waves generated by the hexagonal-warping term are the origin of the perfect reflection, which is similar to bilayer graphene.

Figure 1

Atomic structure of steps.

Figure 2

Step structures. (a) Step I with single QL height, (b) step I with double QL height, (c) step II with single QL height, and (d) the same as (a) with a slip in the reverse direction.

Figure 3

Total transmission probability through a 1QL step of Bi${}_2$Se${}_3$ thin films as a function of energy. Probability is calculated for normal incidence to the step line. Solid and dotted lines show transmission probability and the number of transmission channels, respectively. The band structures of thin films without steps are also shown on the right side.

Figure 4

Total transmission probability through steps of a Bi${}_2$Se${}_3$ thin film as a function of the wave number parallel to the step line. Energy is $0.2$ eV. (a) Step I with single QL height, (b) step I with double QL height, (c) step II with single QL height, and (d) step I with single QL height and reverse slip. Solid and dotted lines show transmission probability and the number of transmission channels, respectively. Broken lines show the transmission probability calculated by using a massless Dirac Hamiltonian.

Figure 5

Amplitude distribution of wave functions near surface steps of Bi${}_2$Se${}_3$. The wave functions are calculated for the normal incidence. Open circles show the positions of atoms.

Figure 6

Total transmission probability through a 1 QL step of Bi${}_2$Te${}_3$ thin films as a function of energy. Probability is calculated for normal incidence to the step line. Solid and dotted lines show transmission probability and the number of transmission channels, respectively. The band structures of thin films without steps are also shown on the right side.

Figure 7

Total transmission probability through steps of a Bi${}_2$Te${}_3$ thin film as a function of the wave number parallel to the step line. Energy is $0.1$ eV. (a) Step I with single QL height, (b) step I with double QL height, (c) step II with single QL height, and (d) step I with single QL height and reverse slip. Solid and dotted lines show transmission probability and the number of transmission channels, respectively.

Figure 8

Amplitude distribution of wave functions near surface steps of Bi${}_2$Te${}_3$. The parallel wave number is $0.048$ Å${}^{-1}$. The transmission probability is zero at this wave number in the spectrum of Fig. 7a. Open circles show the positions of atoms.

Figure 9

Band structure of a stepped Bi${}_2$Te${}_3$ thin film. Energy bands are shown as a function of the wave number parallel to the step line. $a$ is the lattice constant.

Figure 10

Squared amplitude of the wave function of the edge state localized at surface steps of Bi${}_2$Te${}_3$. Open circles show the position of atoms.

Figure 11

Decay constants of generalized Bloch states of Bi${}_2$Se${}_3$ (a) and Bi${}_2$Te${}_3$ (b) thin films as a function of the wave number parallel to the step line. The thickness of the thin films is 5 QLs. Energy is 0.2 and 0.1 eV for Bi${}_2$Se${}_3$ and Bi${}_2$Te${}_3$, respectively. The states shown by thick and thin lines are doubly and fourfold degenerated, respectively. Energy bands of the Bi${}_2$Se${}_3$ (c) and Bi${}_2$Te${}_3$ (d) thin films along lines parallel to $\overline{\Gamma }$-$\overline{\mathrm{M}}$ for fixed wave vectors perpendicular to $\overline{\Gamma }$-$\overline{\mathrm{M}}$. $\overline{\Gamma }$-$\overline{\mathrm{M}}$ is perpendicular the step line. $k_{\perp }$ shows the norm of the perpendicular wave vectors and corresponds to the wave number in (a) and (b). The points at the left and right edges of bands correspond to $\overline{\Gamma }$ and the middle of $\overline{\Gamma }$-$\overline{\mathrm{M}}$ line when they are projected along the perpendicular direction.

Figure 12

(a) Transmission probability as a function of wave number $k_x$. Solid and dotted lines show the probability with and without the hexagonal-warping term, respectively. The inset shows constant-energy lines of the Dirac cone at energy of 0.2 eV. Solid and dotted lines show the Dirac cones with and without the hexagonal-warping term, respectively. (b) Wave function at $k_x=0.0483$ Å${}^{-1}$. Solid line shows total squared amplitude of the wave function. Dotted and broken lines show the components with up and down spins, respectively.

Figure 13

Decay constants of the massless Dirac Hamiltonian with the hexagonal-warping term. The decay constants are shown as a function of wave number $k_x$ for energy $E=0.2$ eV.

Figure 14

Bulk band structures of Bi${}_2$Se${}_3$ (a) and Bi${}_2$Te${}_3$ (b). Solid and dotted lines show the band structures obtained by the tight-binding and density-functional methods, respectively.

Figure 15

Band structures of Bi${}_2$Se${}_3$ (a) and Bi${}_2$Te${}_3$ (b) (111) thin films calculated by the tight-binding method. The thickness of the thin films is 20 QLs.