Rashba spin splitting of $L$-gap surface states on Ag(111) and Cu(111)

Spin-resolved band structures of $L$-gap surface states on Ag(111) and Cu(111) are investigated by spin- and angle-resolved photoelectron spectroscopy (SARPES) with a vacuum-ultraviolet laser. The observed spin textures of the Ag(111) and Cu(111) surface states agree with that expected by the conventional Rashba effect. The Rashba parameter of the Ag(111) surface state is estimated quantitatively and is $80\%$ of that of Cu(111). The surface-state wave function is found to be predominantly of even mirror symmetry with negligible odd contribution by SARPES using a linearly polarized light. The results are consistent with our theoretical calculations for the orbital-resolved surface state.

Figure 1

(a) Constant energy laser-ARPES intensity map at the Fermi level ($E_{\mathrm{F}}$) of the Ag(111) surface state recorded with the $p$-polarized light. (b),(c) Spin-polarization and photoelectron intensity images along the $\overline{\mathrm{\Gamma }}\overline{K}$ (b) and $\overline{\mathrm{\Gamma }}\overline{M}$ (c) directions in the surface Brillouin zone of the Ag(111) surface measured by laser-SARPES. A color scale gives both amplitude of the spin polarization and the photoelectron intensity. (d) Surface Brillouin zone of the face-centered-cubic (111) surface and definition of the spin-polarization direction. (e),(f) Three-dimensional spin-resolved EDCs at $k_1$ and $k_2$ shown in (c). Red and blue curves represent the positive (negative) spin-polarization direction depicted in (d). (g) Emission angle dependence of the spin-resolved EDCs of the Ag(111) surface state along $\overline{\mathrm{\Gamma }}\overline{M}$, where the spin detector is arranged to detect the $y$ component of the spin polarization. Red and blue curves represent the positive (negative) spin polarization defined in (d). (h) Red filled and blue open triangles represent peak positions obtained from the spin-resolved EDCs in (g). Red and blue curves represent fitting results with free-electron-like parabolas.

Figure 2

(a) Spin-polarization and photoelectron intensity images of the Cu(111) surface state along $\overline{\mathrm{\Gamma }}\overline{M}$ measured by laser-SARPES. The spin detector is arranged to be sensitive to the $y$ component of the spin polarization [see Fig. 1 for the definition of the spin-polarization direction]. (b) Peak positions of the spin-resolved EDCs together with the laser-ARPES intensity image. Red and blue symbols correspond to mutually opposite spin directions in the $y$ direction. In the inset, the constant energy ARPES intensity map at $E_{\mathrm{F}}$ is displayed. The constant energy image is obtained by the summation of the photoelectron intensity within a 2-meV energy window centered at $E_{\mathrm{F}}$. (c),(d) Three-dimensional spin-resolved EDCs at $k_5$ and $k_6$ shown in (a). Red and blue curves represent the positive (negative) spin-polarization direction defined in Fig. 1.

Figure 3

(a),(b) (Upper panel) Spin-resolved EDCs and (lower panel) the spin polarizations for the $y$-spin component at the wave vectors of $k_5$ shown in Fig. 2 measured with (a) $p$ and (b) $s$ polarizations. (c) Schematic drawing of the experimental geometry and the definition of the angle $\theta$ as an angle between the electric-field vector of the incident light and the light incident plane. The $p$ polarization ($s$ polarization) corresponds to $\theta =0^{\circ }$ and ${180}^{\circ }$ ($\theta ={90}^{\circ }$). The light incident plane is parallel to the mirror plane of the crystal. The angle between the light and the analyzer was fixed to ${50}^{\circ }$. (d) Orbital projection in the mirror plane for the Cu(111) surface state at $E_{\mathrm{F}}$. Filled (opened) bars represent the even (odd) orbitals. The total weight of the even (odd) orbital contribution is 51.5 (5.5). (e) Photoelectron intensity at $k_6$ shown in Fig. 2 as a function of $\theta$.