Experimental search for one-dimensional edge states at surface steps of the topological insulator ${\mathrm{Bi}}_2{\mathrm{Se}}_3$: Distinguishing between effects and artifacts

The results of a detailed study of the topological insulator ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ surface-state energy structure in the vicinity of surface steps using scanning tunneling microscopy and spectroscopy methods are presented. An increase in the chemical potential level $\mu$ near the step edge is observed. The value of the increase $\delta \mu \sim 0.1$ eV is found to correlate with the step height. The effect is caused by redistribution of electron wave functions between the outer and inner edges of surface steps, as known for normal metals. The smaller value of the chemical potential shift and its larger characteristic length of $\sim 10$ nm reflect specifics of the helical surface states. This increase is accompanied by enlargement of the normalized differential tunneling conductance in the helical surface-state energy region and thereby produces the illusion of the appearance of edge states. We show that the enlargement is reproduced in the framework of the tunneling model taking into account the tunneling gap transparency change when the chemical potential moves away from the Dirac point.

Figure 1

STM images of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ surfaces with steps of different heights: (a) Group I sample with 2QL step; the inset shows the atomic resolution image. (b) Group II sample with 1QL and 2QL steps. In both cases $V_t=-0.4$ V, $I_t=100$ pA, $T=5$ K.

Figure 2

Typical differential tunneling conductance curves on the ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ (111) surface away from defects. The bulk valence band (VB), bulk conduction band (CB), and surface states (SS) are separated by vertical lines. The arrow points to the Dirac point identified as the minimum of the differential conductance curve. $V_1$ corresponds to (a) $\mathrm{\Delta }V=0.7$ V for group I samples and (b) 0.5 V for group II samples (see text for details). Set point $V_t=-0.4$ V, $I_t=100$ pA, $T=5$ K. Solid lines show $dI/dV$ curves calculated by using Eqs. (3) and (5) [19] with the model DOS shown in the inset in Fig. 10.

Figure 3

(a) Profile along a line crossing the 1-QL step on the ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ (group I) surface. (b) The respective sets of $dI/dV$ curves, positions of VB and CB edges, and $V_0$, shown by dashed white lines (pay attention to the logarithmic color scale). (c) $V_D$ and $V_0$ (c). (d) ${[(dI/dV)/G_t]}_{V=V_0}$ and ${[(dI/dV)/G_t]}_{V=V_D}$ along the line. $I\text{−}V$ curves were collected at $V_t=-0.4$ V, $I_t=100$ pA, $T=5$ K.

Figure 4

(a) Profile along a scan line crossing the 1-QL and 2-QL steps on the ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ (group II) surface. (b) The respective sets of $dI/dV$ curves, positions of VB and CB edges, and $V_0$, shown by dashed white lines ( pay attention to the logarithmic color scale). (c) $V_D$ and $V_0$. (d) ${[(dI/dV)/G_t]}_{V=V_0}$ and ${[(dI/dV)/G_t]}_{V=V_D}$ along the scan line. $I\text{−}V$ curves were collected at $V_t=-0.4$ V, $I_t=100$ pA, $T=5$ K.

Figure 5

Distributions of (a) $V_0$ and (b) ${[(dI/dV)/G_t]}_{V=V_0}$ calculated from a $100\times 100$ array of $I\text{−}V$ curves collected on the ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ surface shown in Fig. 1. Set point: $V_t=-0.4$ V, $I_t=100$ pA, $T=5$ K.

Figure 6

Distributions of (a) $V_0$ and (b) ${[(dI/dV)/G_t]}_{V=V_0}$ (b) on the ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ surface shown in Fig. 1. Set point: $V_t=-0.4$ V, $I_t=100$ pA, $T=5$ K.

Figure 7

Correlation between the step height and the band bending in all studied samples of groups I (circles) and II (triangles).

Figure 8

(a) Wave-function amplitudes and envelopes normalized by their maxima in various models. Black dotted line: ab initio calculations [23] at $k=0.2{\mathrm{nm}}^{-1}$; red dashed line: Eq. (1) with the data from Ref. [22] at $k=0.4{\mathrm{nm}}^{-1}$; blue solid line: Eq. (1) with the data from Ref. [22] extracted from ab initio calculations of Ref. [4] at $k=0.4{\mathrm{nm}}^{-1}$. (b) Respective set of $n_s\left(Z\right)$.

Figure 9

Energy diagrams for different chemical potential positions. Bulk DOS is omitted for simplicity.

Figure 10

A set of $dI/dV$ curves obtained in accordance with Eqs. (3) and (5) for the same energy spectra ${\rho }_s$ with different positions of the chemical potential level measured from the Dirac point $\mu =0,0.05,0,1,0.15,0,2,0.25,0.3$ eV, ${\varphi }_s=5.5$ eV, ${\varphi }_t=5.0$ eV, $z=1$ nm.

Figure 11

Effect of the chemical potential shift on ${[(dI/dV)/G_t]}_{V=V_0}$. The solid line corresponds to the model density of states shown in the inset in Fig. 10. Experimental points correspond to the data in Figs. 5 and 6. See text for details. Horizontal lines show the chemical potential variation range of the respective data set.