Spin-orbit interaction and Dirac cones ind-orbital noble metal surface states

Abstract

Band splittings, chiral spin polarization, and topological surface states generated by spin-orbit interactions at crystal surfaces are receiving a lot of attention for their potential device applications as well as fascinating physical properties. Most studies have focused on $sp$ states near the Fermi energy, which are relevant for transport and have long lifetimes. Far less explored, though in principle stronger, are spin-orbit interaction effects within $d$ states, including those deep below the Fermi energy. Here, we report a joint photoemission and ab initio study of spin-orbit effects in the deep $d$-orbital surface states of a 24-layer Au film grown on Ag(111) and a 24-layer Ag film grown on Au(111), singling out a conical intersection (Dirac cone) between two surface states in a large surface-projected gap at the time-reversal symmetric $\overline{M}$ points. Unlike the often isotropic dispersion at $\overline{\ensuremath{\Gamma}}$ point Dirac cones, the $\overline{M}$ point cones are strongly anisotropic. An effective $\mathbf{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{p}$ Hamiltonian is derived to describe the anisotropic band splitting and spin polarization near the Dirac cone.