Rashba-split image-potential state and unoccupied surface electronic structure of Re(0001)

Abstract

The influence of spin-orbit interaction on the unoccupied electronic structure of the Re(0001) surface is investigated by spin- and angle-resolved inverse photoemission and density-functional theory calculations. In the two high-symmetry azimuths $\overline{\mathrm{\ensuremath{\Gamma}}}\phantom{\rule{0.16em}{0ex}}\overline{\text{K}}$ and $\overline{\mathrm{\ensuremath{\Gamma}}}\phantom{\rule{0.16em}{0ex}}\overline{\text{M}}$, we identify transitions into $d$-derived bulk states as well as different types of surface states. The Rashba-type spin-split hole pocket around $\overline{\mathrm{\ensuremath{\Gamma}}}$ finds continuation in empty spin-split surface states for higher ${\mathbf{k}}_{\ensuremath{\parallel}}$, thereby forming $\mathsf{W}$-shaped states whose lower parts are partially occupied. A large energy gap below and above the vacuum energy around $\overline{\mathrm{\ensuremath{\Gamma}}}$ hosts image-potential-induced surface states. The $n=1$ member of the Rydberg-like series exhibits a free-electron-like $E({\mathbf{k}}_{\ensuremath{\parallel}})$ dispersion with an effective mass of ${m}^{*}/{m}_{e}=1.2\ifmmode\pm\else\textpm\fi{}0.1$. Careful spin-resolved measurements for several angles of electron incidence allow us to detect Rashba-type spin-dependent energy splittings of this state with a Rashba parameter of ${\ensuremath{\alpha}}_{\text{R}}=105\ifmmode\pm\else\textpm\fi{}33\phantom{\rule{0.16em}{0ex}}\text{meV}\phantom{\rule{0.16em}{0ex}}\AA{}$.