Spin-orbit interaction and Dirac cones in $d$-orbital noble metal surface states

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 $\overline{\Gamma }$ point Dirac cones, the $\overline{M}$ point cones are strongly anisotropic. An effective $\mathbf{k}·\mathbf{p}$ Hamiltonian is derived to describe the anisotropic band splitting and spin polarization near the Dirac cone.

Figure 1

DFT band structure of bulk fcc Au in the LDA (black) and LDA+$U$ (orange) approximations. The value of the Hubbard correction, $U=1.5$ eV, was chosen to shift the center of gravity of the $d$ bands into agreement with experiment.

Figure 2

Au(111) surface: DFT electronic band structure along the path $\overline{\Gamma KM\Gamma }$ with surface states highlighted in red according to the amplitude of the state on the top two surface layers; shaded areas show the surface-projection of bulk bands. The box surrounds the Dirac cone to be described later.

Figure 3

Dirac cone in surface states of the Au film/Ag(111) imaged along high symmetry lines: (a),(b) ARPES data; (c),(d) DFT calculations (colored lines, red = surface amplitude) with ARPES data in background. (e),(f),(g) Constant energy cuts at binding energies $E-E_D=(-0.14,0.0,+0.1)$ eV. Dotted lines indicate where the bands intersect the constant energy plane. Spectra were measured at a photon energy of 80 eV.

Figure 4

(left) Warping of the Dirac cone shown schematically; (right) two-dimensional Brillouin zone and choice of reciprocal lattice vectors ${\vec{b}}_1$ and ${\vec{b}}_2$.

Figure 5

Decomposition of the surface state according to layer and $d$ orbital. Inset: spin polarization.

Figure 6

Calculated spin polarization of the upper branch (upper panels) and lower branch (lower panels) of the Dirac cone surface states on an ideal Au(111) surface. The left panels plot the in-plane spin polarization $(S_x,S_y)$ with the background color showing its modulus (red = maximal, blue = minimal). The right panels show $S_z$ as a color density plot (red = positive, blue = negative); $S_z/\sqrt{S_x^2+S_y^2}$ ranges between $(-0.75,+0.75)$ for the upper branch and $(-0.85,+0.85)$ for the lower branch. The plotted region of the surface Brillouin zone corresponds to the gray shaded region in Fig. 4.

Figure 7

Ag(111) surface: DFT electronic band structure along the path $\overline{\Gamma KM\Gamma }$ with surface states highlighted in red according to the amplitude of the state on the top two surface layers; shaded areas show the surface-projection of bulk bands.

Figure 8

Dirac cone in surface states of a Ag film/Au(111): ARPES (background) and DFT (colored lines, red = surface amplitude) bands are shown along $\overline{\Gamma }\overline{M}\overline{\Gamma }$ (left panels) and $\overline{K}\overline{M}\overline{K}$ (middle panels) high symmetry lines and along constant energy cuts (bottom panels). Dotted lines indicate where the bands intersect the constant energy plane. Spectra were measured at a photon energy of 80 eV and DFT bands were shifted down rigidly by 1.29 eV.

Figure 9

Calculated spin polarization of the upper branch (upper panels) and lower branch (lower panels) of the $\overline{M}$ point Dirac cone of the deep surface states on the Ag(111) surface. The left panels plot the in-plane spin polarization $(S_x,S_y)$ with the background color showing its modulus (red = maximal, blue = minimal). The right panels show $S_z$ as a color density plot (red = positive, blue = negative); $S_z/\sqrt{S_x^2+S_y^2}$ ranges between $(-0.65,+0.65)$ for the upper branch and $(-0.75,+0.75)$ for the lower branch. The plotted region of the surface Brillouin zone is shown in red in the inset.

Figure 10

Calculated spin polarization of the inner branch (left panel) and outer branch (right panel) of the Dirac cone of the $L$-gap states at $\Gamma$ on the Ag(111) surface. The $z$ component of spin is shown by the background color (red = positive, blue = negative).