Direct Measurement of the Out-of-Plane Spin Texture in the Dirac-Cone Surface State of a Topological Insulator

We have performed spin- and angle-resolved photoemission spectroscopy of ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ and present the first direct evidence for the existence of the out-of-plane spin component on the surface state of a topological insulator. We found that the magnitude of the out-of-plane spin polarization on a hexagonally deformed Fermi surface of ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ reaches maximally 25% of the in-plane counterpart, while such a sizable out-of-plane spin component does not exist in the more circular Fermi surface of $\mathrm{TlBi}{\mathrm{Se}}_2$, indicating that the hexagonal deformation of the Fermi surface is responsible for the deviation from the ideal helical spin texture. The observed out-of-plane polarization is much smaller than that expected from the existing theory, suggesting that an additional ingredient is necessary for correctly understanding the surface spin polarization in ${\mathrm{Bi}}_2{\mathrm{Te}}_3$.

Figure 1

(a) Schematic picture of the Dirac-cone SS with a simple helical spin texture. (b) ARPES intensity at $E_F$ of an $n$-type ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ crystal around the $\overline{\Gamma }$ point plotted as a function of two-dimensional wave vector $\mathbf{k}$, highlighting the snowflakelike FS. The ARPES intensity is integrated within $\pm 20\mathrm{meV}$ with respect to $E_F$ and folded by taking into account the crystal symmetry. The definitions of the FS angle $\varphi$ and the IP up-spin vector ($\varphi$ up) are also indicated. (c) Near-$E_F$ ARPES intensity measured along the $\overline{\Gamma }\overline{M}$ cut. (d),(e) SR-EDCs for the IP ($\varphi$) and OP ($z$) components, respectively, measured at various $\mathbf{k}$ points indicated by open circles in (b). (f),(g) IP and OP spin polarizations $P_{\varphi }$ and $P_z$ at points $A$, $D$, and $C$.

Figure 2

(a),(b) Experimental spin polarization along the $x$ and $z$ axes, $P_x$ and $P_z$, of the SS along the $k_x=0$ cut, obtained by integrating the SR-ARPES intensity at $E_{\mathrm{B}}=0.08–0.12\mathrm{eV}$. (c) Schematic picture of the $z$ component of the spin vector as a function of $\mathbf{k}$ in the SS of ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ determined from the present ARPES experiment. The length of the arrow corresponds to the magnitude of $P_z$. The data are symmetrized by taking into account the crystal symmetry. The intensity with gradual shading (red and blue) shows the calculated $P_z$ value based on the theory in Ref. 6.

Figure 3

(a),(b) Comparison of the FS between $\mathrm{TlBi}{\mathrm{Se}}_2$ and ${\mathrm{Bi}}_2{\mathrm{Te}}_3$. Solid lines are the simulated FS using the theory in Ref. 6. (c),(d) SR-EDCs measured along the $\overline{\Gamma }\overline{K}$ line ($\varphi =0°$) for $\mathrm{TlBi}{\mathrm{Se}}_2$ and ${\mathrm{Bi}}_2{\mathrm{Te}}_3$, respectively. (e),(f) Corresponding $E_{\mathrm{B}}$ dependences of $P_z$. (g) Calculated $P_z$ value along the $k_x=0$ cut as a function of $k_y$ for $\mathrm{TlBi}{\mathrm{Se}}_2$ and ${\mathrm{Bi}}_2{\mathrm{Te}}_3$.