Thin line of a Rashba-type spin texture: Unoccupied surface resonance of Tl/Si(111) along $\overline{\Gamma }\overline{M}$

The Tl/Si(111)-$\left(1\times 1\right)$ surface features outstanding properties with a unique spin pattern in momentum space. Along $\overline{\Gamma }\overline{K}$ $\left(\overline{\Gamma }{\overline{K}}^{\prime }\right)$, surface states with rotating spin had been observed. Here, the focus is put on the unoccupied surface electronic structure along the $\overline{\Gamma }\overline{M}$ high-symmetry direction. Spin- and angle-resolved inverse-photoemission experiments with sensitivity to the in-plane and out-of-plane components of the spin-polarization vector are conducted with the help of our recently developed rotatable spin-polarized electron source (ROSE). In agreement with our theoretical findings, we identify a surface resonance with giant spin-orbit-induced spin splitting, which exhibits an exclusive Rashba-type spin texture, i.e., a pure in-plane spin polarization perpendicular to $\overline{\Gamma }\overline{M}$, in compliance with the mirror-plane symmetry of the $\overline{\Gamma }\overline{M}$ direction. Notably, for slight deviations from the high-symmetry line, this constraint is lifted and additional spin-polarization components emerge. This highlights the importance of a correct alignment of the experiment when investigating samples with spin textures that are more complex than in simple Rashba systems or in ferromagnets.

Figure 1

(a) Structural model of the Tl/Si(111)-$\left(1\times 1\right)$ surface. The vertical mirror planes ${\sigma }_v$ are indicated by thin lines. (b) Corresponding reciprocal lattice points (crosses) with two adjacent surface Brillouin zones and respective high-symmetry points (dots). SR-IPE spectra were taken along the $\overline{\Gamma }\overline{M}\overline{\Gamma }$ direction. The spin polarization of the electron beam determines the spin sensitivity of the measurement. Arrows represent spin polarization perpendicular (red and blue) and parallel (purple and orange) to $\overline{\Gamma }\overline{M}\overline{\Gamma }$, referred to as in-${\mathrm{plane}}_{\perp }$ and in-${\mathrm{plane}}_{\parallel }$ spin polarization, respectively. Out-of-plane spin polarization parallel (antiparallel) to the surface normal is illustrated by purple (orange) markers.

Figure 2

(a), (b) SR-IPE spectra of Tl/Si(111)-$\left(1\times 1\right)$ along $\overline{\Gamma }\overline{M}\overline{\Gamma }$ with sensitivity to the in-${\mathrm{plane}}_{\perp }$ spin-polarization direction (red up- and blue down-pointing triangles). (c) SR-IPE spectra along $\overline{\Gamma }\overline{M}\overline{\Gamma }$ with sensitivity to the out-of-plane/in-${\mathrm{plane}}_{\parallel }$ spin-polarization direction (purple up-pointing and orange down-pointing triangles). $\theta$ and $\phi$ denote the angle of electron incidence and the sample azimuth, respectively.

Figure 3

(a) $E\left({\mathbf{k}}_{\parallel }\right)$ dispersion plot derived from spectra taken along $\overline{\Gamma }\overline{M}\overline{\Gamma }$ with sensitivity to the in-${\mathrm{plane}}_{\perp }$ spin-polarization direction. The gray-shaded area represents the calculated projected bulk band structure. The Rashba energy $E_R$ and the shift $k_0$ of the band minimum away from $\overline{M}$ are determined to $E_R\approx 200$ meV and $k_0\approx 0.17$ Å${}^{-1}$. (b) Calculated band structure including spin-orbit coupling. Spin-polarized states within the surface region are illustrated as circles. The size of the circles is proportional to the spin polarization within the outermost five layers. A correction of the band gap has been applied according to quasiparticle calculations and measurements presented in Ref. [15].

Figure 4

Charge distribution plots of the surface resonances $S_3$ (left-hand side) and $S_4$ (right-hand side) at three-fourths of the $\overline{\Gamma }\overline{M}$ line (see the dotted line in Fig. 3). Red dots represent Tl atoms. Si atoms in the drawing plane are shown as black dots, otherwise as white dots.

Figure 5

(a) Illustration of the SR-IPE measurement direction at an angle $\phi$ with respect to $\overline{\Gamma }\overline{M}$ (blue line). The blue dotted line represents a curve with constant $k_{\parallel }$ $\left(\theta ={40}^{\circ }\right)$ but varying $\phi$ for an constant energy of $E\approx 1$ eV. (b) SR-IPE measurements of Tl/Si(111)-$\left(1\times 1\right)$ with sensitivity to the out-of-plane/in-${\mathrm{plane}}_{\parallel }$ spin-polarization direction for $\phi =8^{\circ }$ and $\phi =-8^{\circ }$. (c) Left-hand side: Plot of the observed spin asymmetry at $E_1$ and $E_2$ [see (b)] for $\theta ={40}^{\circ }$ as a function of the sample azimuth $\phi$ and (right-hand side) calculated out-of-plane (solid line) and in-${\mathrm{plane}}_{\parallel }$ (dotted line) spin-polarization components of $S_3$ for $k_{\parallel }=0.85$ Å${}^{-1}$ and varying $\phi$.