Probing two topological surface bands of Sb${}_2$Te${}_3$ by spin-polarized photoemission spectroscopy

Using high-resolution spin- and angle-resolved photoemission spectroscopy, we map the electronic structure and spin texture of the surface states of the topological insulator Sb${}_2$Te${}_3$. In combination with density functional calculations (DFT), we directly show that Sb${}_2$Te${}_3$ exhibits a partially occupied, single spin-Dirac cone around the Fermi energy $E_{\mathrm{F}}$, which is topologically protected. DFT obtains a spin polarization of the occupied Dirac cone states of 80–90$\%$, which is in reasonable agreement with the experimental data after careful background subtraction. Furthermore, we observe a strongly spin-orbit split surface band at lower energy. This state is found at $E-E_{\mathrm{F}}\simeq -0.8$ eV at the $\overline{\Gamma }$ point, disperses upward, and disappears at about $E-E_{\mathrm{F}}=-0.4$ eV into two different bulk bands. Along the $\overline{\Gamma }-\overline{\mathrm{K}}$ direction, the band is located within a spin-orbit gap. According to an argument given by Pendry and Gurman in 1975, such a gap must contain a surface state, if it is located away from the high-symmetry points of the Brillouin zone. Thus, the novel spin-split state is protected by symmetry, too.

Figure 1

(a) STM image of cleaved Sb${}_2$Te${}_3$(0001); $V=-1$ V, $I=0.8$ nA, $375\times 375$ nm${}^2$. Inset: Line profile showing the step which corresponds to the height of one quintuple layer. (b) ARPES data of Sb${}_2$Te${}_3$(0001) along $\overline{\Gamma }-\overline{\mathrm{K}}$ at an incident photon energy $h\nu =50$ eV; Dirac cone is marked.

Figure 2

(a) $dI/dV$ spectrum recorded by STM on the cleaved Sb${}_2$Te${}_3$(0001) surface; $V_{\mathrm{stab}}=-0.5$ V, $I_{\mathrm{stab}}=1$ nA, $V_{\mathrm{mod}}=10$ mV; band edges of the valence band (VB) and conduction band (CB) are marked by lines, Fermi level $E_{\mathrm{F}}$ is marked. (b) Band structure in $\overline{\Gamma }-\overline{\mathrm{K}}$ direction as calculated by DFT including spin-orbit coupling; states resulting from a film calculation are shown as circles with the color (blue or red) indicating different spin directions and the size of colored circles marking the magnitude of the spin density (for absolute spin polarization values, see Fig. 4); shaded areas are projected bulk bands originating from a bulk calculation. (c) Sketch of the crystal structure of Sb${}_2$Te${}_3$; one QL is marked with different atoms in different colors as indicated. (d) ARPES data along $\overline{\Gamma }-\overline{\mathrm{K}}$ of the lower Dirac cone and bulk valence bands (BVB) marked; $h\nu =55$ eV; orange, dashed line is a guide to the eye from which the Fermi velocity $v_{\mathrm{F}}$ is deduced; white dashed lines mark positions of the energy distribution curves (EDCs) shown in Figs. 3a, 3b, 3c.

Figure 3

(a),(b) Spin-resolved energy distribution curves (EDCs) for the in-plane spin component perpendicular to ${\mathbf{k}}_{\parallel }$ recorded at $k_{\parallel }$ values as indicated and marked by dashed lines in Fig. 2d; different colors mark different spin directions; $h\nu =54.5$ eV. (c) Spin-resolved EDCs for the spin component parallel to ${\mathbf{k}}_{\parallel }$; $h\nu =54.5$ eV. (d) Resulting spin polarization perpendicular to the two different ${\mathbf{k}}_{\parallel }$ as marked. (e) Sketch of the lower Dirac cone with spin directions marked as deduced from spin-ARPES and in accordance with DFT.

Figure 4

Expectation values from DFT for the out-of-plane and in-plane component of the spin polarization along $\overline{\Gamma }-\overline{\mathrm{K}}$ for the Dirac cone and the Rashba-type surface state (Rashba). Upper and lower indicates the energies above and below the Dirac point and the energetically higher and lower Rashba band, respectively.

Figure 5

(a) ARPES dispersion of the Rashba-type surface state (Rashba SS) along $\overline{\Gamma }-\overline{\mathrm{K}}$ direction; $h\nu =22$ eV; band structure of the Rashba surface states from DFT is superimposed as blue and red lines; dashed lines mark the position of the EDCs in (b). (b) Spin-resolved EDCs (points in red and blue for the two spin directions perpendicular to ${\mathbf{k}}_{\parallel }$) at different momenta as indicated and marked by dashed lines in (a); fits of the peaks are shown as solid lines (red, blue); peak positions are marked by dashed lines and lead to the spin-splitting energies $\Delta E_{\mathrm{SO}}$ indicated; (c) calculated spin splitting (Theo) of the Rashba state in comparison with measured spin splitting (Exp). (d) Two-dimensional cut through the calculated local density of states for the Rashba state and the lower and upper part of the Dirac cone at $k_{\parallel }=0.06$ ${\AA }^{-1}$.

Figure 6

(a) ARPES data including the Rashba state at higher $|{\mathbf{k}}_{\parallel }|$; $h\nu =54.5$ eV; for better visibility, the derivative with respect to $E$ is shown; Dirac cone and Rashba-type surface state (Rashba SS) with superimposed $E\left({\mathbf{k}}_{\parallel }\right)$ curves of the Rashba states (colored lines) from DFT calculation are marked. Dashed boxes indicate the same area in (a) and (b) as well as the $|{\mathbf{k}}_{\parallel }|$ position of the EDC in (c). The width is given by the angular resolution of the experiment. (b) Bulk band structure along $\overline{\Gamma }-\overline{\mathrm{K}}$ calculated with (gray lines) and without (yellow lines) SO interaction; black lines mark the band edges with (dashed) and without (solid) SO interaction; with SO interaction, a gap opens near the center of the graph around $E=-0.5$ eV and $k_{\parallel }=\pm 0.26$ ${\AA }^{-1}$; ${\Gamma }^{(-)}$, ${\Gamma }^{(+)}$ mark the parity of the two bands at the $\Gamma$ point; (c) spin-resolved EDCs measured at the position of the SO gap ($|{\mathbf{k}}_{\parallel }|$ as marked); $h\nu =54.5$ eV; peak positions as determined from Lorentzian fits (solid lines) are indicated by dashed lines; the resulting $\Delta E_{\mathrm{SO}}$ is marked.

Figure 7

DFT band structure calculated for a slab without SO coupling. The surface state (red dots) still exists, but is only connected to the lower bulk band exhibiting ${\Gamma }^{(-)}$ character.

Figure 8

(a),(b) Spin-resolved energy distribution curves (EDCs) for the spin component perpendicular to $k_{\parallel }$ after subtraction of the Rashba state background [compare with Figs. 3a and 3b]. (c) Corresponding spin polarization as a function of energy. The subtraction of the background increases the averaged spin polarization from 20$\%$ to 27$\%$.

Figure 9

(a) High-resolution ARPES data of the Dirac cone [close-up view from Fig. 2d] at an incident photon energy $h\nu =55$ eV. (b) Data from (a) convolved with a Gaussian curve taking into account the energy resolution (FWHM: 100 meV) and momentum resolution (FWHM: 0.09 Å${}^{-1}$) of the spin detector. (c) ARPES data measured with the spin detector, $h\nu =54.5$ eV. (d)–(f) Constant-energy cuts through the Dirac cone along the green line in the ARPES data on the left. Constant energy cuts in (e) and (f) show a similar distribution. In (d), the result of the two-component fitting to the spectrum is shown as marked Dirac cone and BVB (bulk valance band). Ratio of the spectral weight of the Dirac cone $W_{\mathrm{DC}}$ to the total spectral weight $W_{\mathrm{total}}$ is given.

Figure 10

Root-mean-square error (RMSE) for different fits of the spectrum of Fig. 9d leading to different $W_{\mathrm{DC}}/W_{\mathrm{total}}$ and, thus, to different spin polarizations of the Dirac cone states $P_{\mathrm{DC}}$. The RMSE increases by a factor of 3 for a spin polarization below 80$\%$.