Figure 1

Calculated energies of the QWSs as a function of the overlayer thickness for the Pb/Cu(111) system. Results are shown for the calculations performed with (black dots) and without (open squares) inclusion of the image potential tail in the DFT potentials. Energies $E_n$ are referred to the vacuum level. The energy with respect to the Fermi level of the thick Pb film is $E_n+4.08$ eV, where $4.08$ eV is the work function of Pb. The horizontal shaded area denotes the set of states lying at energy $\approx$$-0.8$ eV.Reuse & Permissions

Figure 2

Electron densities of the hybridized QWS/ISs with energy $\approx$$-0.8$ eV with respect to the vacuum level (see horizontal shaded area in Fig. 1). Results for the 6, 12, and 18 ML-thick Pb films on Cu(111) substrate are shown as a function of $z$ coordinate perpendicular to the surface of the film. For each state, the energy $E$ and weight in the vacuum region $\mathrm{w}$ are indicated. The origin $z=0$ is placed at the image plane position and the vertical dotted line at $z=-1.23a_0$ denotes the jellium edge.Reuse & Permissions

Figure 3

Electron densities of the hybridized QWS/ISs with different energies approaching the vacuum level $E_V=0$. Results for the 15 ML-thick Pb film on Cu(111) substrate are presented as a function of $z$ coordinate perpendicular to the surface of the film and measured from the Pb image plane. The $z$-distance range is zoomed at the Pb/vacuum interface to show the IS contribution to the wave functions. The results are vertically offset according to the energy of the corresponding state with respect to the vacuum level denoted by a horizontal line. Dashed line represents the image potential and the vertical dotted line at $z=-1.23a_0$ denotes the jellium edge. For each state, the energy $E$ and weight in the vacuum region $\mathrm{w}$ are indicated.Reuse & Permissions

Figure 4

(a) Calculated many-body ${\gamma }_{e–e}^j$ decay rates of the states lying at $-0.8$ eV as a function of overlayer thickness. Open squares: many-body $GW$ results. Triangles: WPP results. Dots: decay rates calculated from the wave-function penetration. Solid line: fit to the $GW$ data with Eq. (15) (see main text for details). The horizontal dashed line at 130 meV denotes the theoretical bulk decay rate for the energy $-0.8$ eV. (b) Decay rates ${\gamma }_{e–e}^j$ of the hybridized QWS/ISs as calculated from the wave-function penetration with Eq. (12) for 9, 17, and 25 ML-thick Pb overlayers on Cu(111). Results are shown as a function of the energies of the states measured with respect to the vacuum level.Reuse & Permissions

Figure 5

Calculated many-body ${\gamma }_{e–e}^j$ decay rates of the hybrid QWS/ISs for overlayer thicknesses ranging from 1–28 ML. Results are shown as a function of the energy of the states as measured with respect to the vacuum level. Thick solid lines connect the decay rates for a few selected overlayer thicknesses (see the legend). Thin dashed line represents the linear fit to the dispersion of the decay rates of the QWSs obtained in an earlier work[45] without accounting for the image potential tail of the electron-surface interaction. Different symbols are explained in the legend. Penetration stands for the present calculations based on the wave-function penetration. These calculations account for the image potential tail of the electron-surface interaction. $GW$-LDA stands for the decay rates calculated within $GW$ approach and effective one-electron potential derived within the local density approximation of density functional theory where the image potential is not accounted for. $GW$-IS stands for the decay rates calculated within $GW$ approach and effective one-electron potential, corrected such that the image potential tail of the electron-surface interaction is accounted for. The vertical shadowed region represents the narrow energy window of the shadowed area of Fig. 1 at $\approx$$-0.8$ eV.Reuse & Permissions

Figure 6

Projected density of states (arbitrary units) at the Pb surface in Pb/Cu(111) system. Results are obtained with WPP method for different Pb overlayer thicknesses, as indicated above the corresponding lines. The limit of the infinite thickness is indicated as Pb metal surface. The data are vertically offset for clarity.Reuse & Permissions