Figure 1

Dispersion of the Cu(111) Shockley surface state (thick solid line), as obtained from three-dimensional ab initio calculations. Shaded areas represent areas outside the band gap, where bulk states exist. The thin solid and dotted lines represent approximate energy dispersions of the Shockley surface state and the bottom of the projected band gap, respectively, as obtained from $\epsilon +k^2∕2m$ with $\epsilon =-0.44\mathrm{eV}$ and $m=0.42$ in the case of the surface state (thin solid line) and $\epsilon =-1.09\mathrm{eV}$ and $m=0.22$ for the bottom of the projected band gap (thin dotted line).Reuse & Permissions

Figure 2

Probability-density ${\mid {\psi }_{k,E}\mid }^2$ of the Shockley surface state at the center of the surface Brillouin zone of Cu(111), as obtained within two different models: the model presented here (dashed line) and the model reported in Ref. 12 (solid line). The dotted line represents the probability-density that we obtain at $k=0.2\mathrm{a.u.}$. Full circles represent the atomic positions of Cu in the (111) direction. The geometrical (jellium) electronic edge $(z=0)$ has been chosen to be located half an interlayer spacing beyond the last atomic layer.Reuse & Permissions

Figure 3

Scaled total linewidth ${\Gamma }_{\text{total}}∕{(E_i-E_F)}^2$ of Shockley surface-state electrons and holes in Cu(111) (including the $e\text{−}\mathrm{ph}$ contribution ${\Gamma }_{e\text{−}\mathrm{ph}}=7\mathrm{meV}$) versus the surface-state energy $E_i$, as obtained from Eqs. (1, 2, 3, 4, 5, 6, 7) (thick solid line), from Eqs. (1, 2, 3, 4, 5, 6, 7) but replacing all surface-state wave functions with $\mathbf{k}\ne 0$ by those at the $\overline{\Gamma }$ point (thin solid line), and from Ref. 16 (thin dashed line). The dotted line represents the energy-independent $e\text{−}\mathrm{ph}$ contribution ${\Gamma }_{e\text{−}\mathrm{ph}}=7\mathrm{meV}$. The dotted line with squares represents the STM measurements reported in Ref. 10, multiplied by a factor of 2 to correct for an error in the phase coherence length used in that work. The solid circle represents the experimentally determined linewidth of an excited hole at the $\overline{\Gamma }$ point, as reported in Ref. 14. Separate unscaled interband $\left({\Gamma }_{\mathrm{inter}}\right)$ and intraband $\left({\Gamma }_{\mathrm{intra}}\right)$ contributions to the total linewidth $({\Gamma }_{\text{total}}={\Gamma }_{\mathrm{intra}}+{\Gamma }_{\mathrm{inter}}+{\Gamma }_{e\text{−}\mathrm{ph}})$ are represented in the inset by dashed and dotted lines, respectively.Reuse & Permissions