Spin-polarization limit in Bi${}_{\mathbf{2}}$Te${}_{\mathbf{3}}$ Dirac cone studied by angle- and spin-resolved photoemission experiments and ab initio calculations

The magnitude of electron spin polarization in topologically protected surface states is an important parameter with respect to spintronics applications. In order to analyze the warped spin texture in Bi${}_2$Te${}_3$ thin films, we combine angle- and spin-resolved photoemission experiments with theoretical ab initio calculations. We find an in-plane spin polarization of up to $\sim$45$\%$ in the topologically protected Dirac cone states near the Fermi level. The Fermi surface of the Dirac cone state is warped and shows an out-of-plane spin polarization of $\sim$15$\%$. These findings are in quantitative agreement with dedicated simulations which find electron density of the Dirac cone delocalized over the first quintuple layer with spin reversal occurring in the surface atomic layer.

Figure 1

(a) Photoemission map of the 40 nm Bi${}_2$Te${}_3$ film taken at 15 K using the He i ($h\nu \approx 21.22$ eV) excitation along the $\overline{\Gamma K}$ direction, with the overlaid DFT in-plane calculation [see Fig. 3 (a)], shifted by 0.45 eV to achieve the best match to the experiment. Blue and red solid circles indicate the spin direction, whereas the size of the symbols is related to the weight of the states at the surface. (b) Constant energy surfaces for the Fermi level (top), $E_B=2.2$ eV, and $E_B=3.0$ eV. Each map was separately normalized for optimized contrast. Arrows in the $E_B=2.2$ eV map indicate the two example symmetric positions on the ring, related to the $\delta$ band [see panel (a)], for which the direction of the polarization vector is reversed due to its helical nature.

Figure 2

Spin-polarized data taken (a) near the Fermi level and (b) at higher binding energies on selected k-space locations along the $\overline{\Gamma K}$ direction on the 40 nm Bi${}_2$Te${}_3$ film. Here, blue and red solid lines show smoothed $I_L$ and $I_R$ intensities, respectively. States being described within the text are indicated by letters $\alpha$ to $\epsilon$. The top row indicates the in-plane spin-vector component intensities, whereas the out-of-plane intensities are plotted in the lower rows. The deduced spin asymmetries $P_x$ and $P_z$ are plotted below the corresponding intensity plots with standard deviations given by vertical error bars, whereas the solid line represents smoothed data. Furthermore, panel (b) shows the He i data integrated over the SPLEED entrance (the black curve) and the theoretical prediction for the in-plane spin polarization in surface-related eigenvalues at $|k_{\parallel }|=0.18$ Å${}^{-1}$ shifted by 0.45 eV to achieve the best match to the experiment (the bar graph). (c) Experimental three-dimensional illustration of the band structure of a Bi${}_2$Te${}_3$ thin film over the full valence band region, indicating the $\mathbf{\text{k}}$-space volumes A and B which are integrated in the spin-polarized experiment.

Figure 3

(a) Simulated spin-polarized spectral weight of the 6 QL Bi${}_2$Te${}_3$ slab along the $\overline{\Gamma K}$ direction. The size of the circles gives the absolute spin polarization of the states for the in-plane $P_x$ and out-of-plane $P_z$ components. Herein, the photoelectron mean free path has been taken into account in reference to the experiment. Two states, namely $\alpha$ and $\delta$ analyzed in the further drawings, are indicated by arrows. Panels (b) and (e) show charge, and (c) and (f) spin densities plotted in real space for the $\alpha$ and $\delta$ states at $|k_{\parallel }|=0.18$ Å${}^{-1}$, respectively. Panels (d) and (g) show charge and spin densities for those states with respect to the distance from the surface, integrated over the $(x,y)$ plane. Arrows in (d) and (g) denote spin reversals taking place for each state. See the text for details. The Dirac-cone state has a smaller polarization, due to the polarization inversion above the topmost Te atom (local spin reversal). The in-plane polarization value, integrated taking into account the photoemission information depth, is about 45$\%$, in contrast to the surface state $\delta$ that has 88$\%$.