Electronic standing waves on the surface of the topological insulator Bi${}_2$Te${}_3$

A line defect on a metallic surface induces standing waves in the electronic local density of states (LDOS). Asymptotically far from the defect, the wave number of the LDOS oscillations at the Fermi energy is usually equal to the distance between nesting segments of the Fermi contour, and the envelope of the LDOS oscillations shows a power-law decay as moving away from the line defect. Here, we theoretically analyze the LDOS oscillations close to a line defect on the surface of the topological insulator Bi${}_2$Te${}_3$, and identify an important preasymptotic contribution with wave-number and decay characteristics markedly different from the asymptotic contributions. The calculated energy dependence of the wave number of the preasymptotic LDOS oscillations is in quantitative agreement with the result of a recent scanning tunneling microscopy experiment [Phys. Rev. Lett. 104, 016401 (2010)].

Figure 1

(a) and (b) Surface-state conduction-band dispersion of Bi${}_2$Te${}_3$ along the $\Gamma M$ (solid line) and $\Gamma K$ (dashed line) directions of the surface Brillouin zone. (c) and (d) Surface-state conduction-band density of states of Bi${}_2$Te${}_3$. For (a) and (c), Eq. (2) was used with parameter values[14] $v_0=2.55$ eVÅ, $\lambda =250$ eVÅ${}^3$, and $\alpha ,\gamma =0$. For (b) and (d), the parameter values $v_0=3.5\mathrm{eV}\AA$, $\lambda =150\mathrm{eV}{\AA }^3$, $\alpha =21{\AA }^2$, and $\gamma =-19.5\mathrm{eV}{\AA }^2$ were used. Red crosses (blue points) represent the measured dispersion along $\Gamma M$ ($\Gamma K$) (data taken from Ref. 11). The zero of energy corresponds to the Dirac point of the spectrum. Energy intervals overlapping with the bulk valence band (BVB) and bulk conduction band (BCB) are also shown.

Figure 2

(a) An incident wave from the $x>0$ region (solid arrow) is partially reflected (dashed arrow) and transmitted (dotted arrow) at a line defect (e.g., an atomic terrace) on the surface of Bi${}_2$Te${}_3$. (b) Hexagonally warped constant-energy contour in reciprocal space, corresponding to energy $E=330$ meV above the Dirac point. Incident and reflected wave vectors from (a) are also shown.

Figure 3

(a) Position-dependent contribution $\delta \rho (E,x)$ (solid line) to the LDOS at energy $E=330$ meV in the vicinity of a line defect (black points). Red solid and blue dashed lines are fits of the functions $f_1$ and $f_2$ (see text), respectively, to the data. ${\rho }_0=628{\mathrm{meV}}^{-1}\mu {\mathrm{m}}^{-2}$. (b) Constant-energy contour at the same energy, and relevant scattering wave vectors in reciprocal space. Thick blue (thin red) pieces of the contour correspond to left-moving (right-moving) plane waves. The union of the thick blue pieces form ${\Gamma }_E^{\left(R\right)}$ in Eq. (6). (c) Dominant wave number $2k_{\mathrm{fit}}$ of $\delta \rho (E,x)$ (open circles), and characteristic wave numbers $k_{\mathrm{nest}}$ (red points), $k_{\mathrm{v}}$ (green diagonal crosses), and $2k_{\Gamma \mathrm{M}}$ (blue crosses) of the constant-energy contour, as functions of energy $E$. (d) Magnitude of reflection amplitude $\left|r\right|\equiv |r_{k,q^{\prime }}|$, (e) magnitude of spinor overlap $|{\chi }^{†}\chi |\equiv |{\chi }_{k,q^{\prime }}^{†}{\chi }_{-k,q^{\prime }}|$, (f) magnitude of the inverse of the parallel-to-defect group velocity component $v_{\parallel }\left(k\right)$ (in units of ${10}^{-6}\mathrm{s}/\mathrm{m}$), and (g) the product of the above three quantities (in units of ${10}^{-6}\mathrm{s}/\mathrm{m}$), as functions of perpendicular-to-defect wave number component $k$.

Figure 4

(a) Numerically obtained LDOS oscillations (points) at energy $E=330$ meV for the case when the line defect is perpendicular to the $\Gamma K$ direction, and the fit of $f\left(x\right)=A\sin (kx+\varphi )/x^{1/2}$ with fitting parameters $A$, $k$, and $\varphi$ (solid red line). (b) Constant energy contour and the relevant wave numbers $k_{\mathrm{nest}}$ and $2k_{\Gamma K}$. The wave number of the oscillation in (a) is given by $k_{\mathrm{nest}}^{\Gamma K}$, as the oscillations with $2k_{\Gamma K}$ are fast decaying ($\propto x^{-3/2}$) due to the absence of backscattering.

Figure 5

Comparison of the discrete Fourier transforms of (a) the computed LDOS oscillation $\delta \rho \left(x\right)$ shown in Fig. 3a, (b) the function $f_2\left(x\right)=A_2\sin (2k_{\mathrm{fit}}x+{\varphi }_2)e^{-x/L}$, where $A_2$, $k_{\mathrm{fit}}$, ${\varphi }_2$, $L$ are obtained from fitting $f_2$ to $\delta \rho$, (c) $A_2\sin (2k_{\mathrm{fit}}x+{\varphi }_2)x^{-3/2}$, and (d) $A_2\sin (2k_{\mathrm{fit}}x+{\varphi }_2)x^{-1/2}$.