Abrupt Rotation of the Rashba Spin to the Direction Perpendicular to the Surface

The polarization vector of the Rashba spin, which must be parallel to the two-dimensional (2D) plane in an ideal system, is found to change abruptly and definitely to the direction perpendicular to the surface at the $\overline{K}$ point of the Brillouin zone of a real hexagonal system, the $\mathrm{Tl}/\mathrm{Si}(111)\mathrm{\text{−}}(1\times 1)$ surface. This finding obtained experimentally by angle-resolved and spin-resolved photoemission measurements is fully confirmed by a first-principles theoretical calculation. We found that the abrupt rotation of the Rashba spin is simply understood by the 2D symmetry of the hexagonal structure.

Figure 1

(a) Schematic illustrations of the $\mathrm{Si}(111)\mathrm{\text{−}}(1\times 1)$ surface formed by the adsorption of 1 ML of Tl. The thick dashed lines indicate the unit cell of this surface. (b) SBZs and LEED pattern of the $\mathrm{Tl}/\mathrm{Si}(111)\mathrm{\text{−}}(1\times 1)$ surface. $\overline{\Gamma }$, $\overline{M}$, and $\overline{K}$ are the symmetry points of the ($1\times 1$) SBZ.

Figure 2

Band structures measured along the (a) $\overline{\Gamma }\mathrm{\text{−}}\overline{M}\mathrm{\text{−}}\overline{\Gamma }$ and (b) $\overline{\Gamma }\mathrm{\text{−}}\overline{K}\mathrm{\text{−}}\overline{M}$ directions. The hatched areas show the bulk band projection taken from Ref. 33. The horizontal dashed line at $E_B=0\mathrm{eV}$ indicates the Fermi level.

Figure 3

(a) SR-ARPES spectra of the electrons with $\overrightarrow{P}$ parallel to the surface and perpendicular to the $\overline{\Gamma }\mathrm{\text{−}}\overline{K}$ direction, and (b) spectra of the electrons with $\overrightarrow{P}$ perpendicular to the surface. ${\theta }_e=0°$ corresponds to $\overline{\Gamma }$, and $\overline{K}$ is between ${\theta }_e=32°$ and 34°. The insets indicate the $\overrightarrow{P}$ of the detected electrons.

Figure 4

(a) Spin-split surface band structures obtained experimentally (open and filled circles) and theoretically (solid lines) along the $\overline{\Gamma }\mathrm{\text{−}}\overline{K}$ direction. (b) Band structures around the $\overline{\Gamma }$ point. $k_0=0.2{\AA }^{-1}$ and $E_R=20\mathrm{meV}$ are obtained from the calculated spin-split bands. (c) The arrows show the $\overrightarrow{P}$ of one of the split bands, as derived from the normal RB effect. The alternating direction of $\overrightarrow{P}$ around $\overline{K}$ is discussed in the text.