Nonvortical Rashba Spin Structure on a Surface with $C_{1h}$ Symmetry

A totally anisotropic peculiar Rashba-Bychkov (RB) splitting of electronic bands was found on the $\mathrm{Tl}/\mathrm{Si}(110)\text{−}(1\times 1)$ surface with $C_{1h}$ symmetry by angle- and spin-resolved photoelectron spectroscopy and first-principles theoretical calculation. The constant energy contour of the upper branch of the RB split band has a warped elliptical shape centered at a $k$ point located between $\overline{\mathrm{\Gamma }}$ and the edge of the surface Brillouin zone, i.e., at a point without time-reversal symmetry. The spin-polarization vector of this state is in-plane and points almost the same direction along the whole elliptic contour. This novel nonvortical RB spin structure is confirmed as a general phenomenon originating from the $C_{1h}$ symmetry of the surface.

Figure 1

(a) LEED pattern of $\mathrm{Tl}/\mathrm{Si}(110)\text{−}(1\times 1)$; the solid rectangle indicates the SBZ, and $\overline{\mathrm{\Gamma }}$, $\overline{X}$, and $\overline{X^{\prime }}$ are the symmetry points of the SBZ. (b) A $10\times 2.5{\mathrm{nm}}^2$ STM image of $\mathrm{Tl}/\mathrm{Si}(110)\text{−}(1\times 1)$ obtained at a sample bias of $-500\mathrm{mV}$. The rectangle corresponds to the size of $\mathrm{Si}(110)\text{−}(1\times 1)$. Top- and side-view atomic structures of (c) the ideal Si(110)-(11) surface and (d) the $\mathrm{Tl}/\mathrm{Si}(110)\text{−}(11)$ surface. The large and small circles indicate Tl and Si atoms, and the rectangles in the top view models show the ($1\times 1$) unit cell. The horizontal and vertical dashed lines in (c) and (d) indicate the mirror plane and the glide plane, respectively. The black circle in (c) shows the rotation center.

Figure 2

Spin-integrated photoemission band from $\mathrm{Tl}/\mathrm{Si}(110)\text{−}(1\times 1)$ measured at $h\nu =40\mathrm{eV}$. (a) and (b) are band structures along the $\overline{X^{\prime }}\text{−}\overline{\mathrm{\Gamma }}\text{−}\overline{X^{\prime }}$ and $\overline{\mathrm{\Gamma }}\text{−}\overline{X}$ directions, respectively. The two upper Tl induced bands on Si(110)surface obtained theoretically along $\overline{\mathrm{\Gamma }}\text{−}\overline{X^{\prime }}$ are superimposed in (a). The edges of the bulk valence band projection are indicated by dotted lines in (a) and (b). (c) Spin-integrated band structure along $\overline{\mathrm{\Gamma }}\text{−}\overline{X^{\prime }}$, enlargement of (a). (d)–(f) are characteristic energy contours, i.e., $k_x\text{−}{k}_y$ band mapping, at $E_{\mathrm{B}}=50$, 100, and 200 meV, whose energies correspond to the dotted lined in (c).

Figure 3

(a) Constant energy contour at $E_B=0.1\mathrm{eV}$ superimposed with the spin structure obtained theoretically. The arrows in (a) indicate the spin orientation. (b) Filled circles showing the $k$ points measured using SARPES. (c) and (d) are the SARPES spectra along $k_y$ at $k_x=-0.27{\AA }^{-1}$ and those along $k_x$ at $k_y=0$. The lowest spectrum in (d) shows the extracted spin polarization at $(k_x,k_y)=(-0.27{\AA }^{-1},0)$.