Role of the intrinsic surface state in the decay of image states at a metal surface

The role of the intrinsic surface state ( n (cid:53) 0) in the decay of the ﬁrst image state ( n (cid:53) 1) at the (cid:126) 111 (cid:33) surface of copper is investigated. Inelastic linewidths are evaluated from the knowledge of the imaginary part of the electron self energy, which we compute, within the GW approximation of the many-body theory, by going beyond a free-electron description of the metal surface. Single-particle wave functions are obtained by solving the Schro¨dinger equation with a realistic one-dimensional model potential, and departure of the motion along the surface from free-electron behavior is considered through the introduction of the effective mass. The decay of the ﬁrst image state of Cu (cid:126) 111 (cid:33) into the intrinsic surface state is found to result in a linewidth that represents 40% of the total linewidth. The dependence of linewidths on the momentum of the image state parallel to the surface is also investigated. (cid:64) S0163-1829


I. INTRODUCTION
The presence of an electron in front of a solid surface redistributes the charge in the solid. As a consequence, an attractive potential is induced, which far from the surface approaches the long-range classical image potential V im (z) ϭϪg s /4z ͓z being the distance from the surface, g s ϭ(⑀ s Ϫ1)/(⑀ s ϩ1), and ⑀ s the static bulk-dielectric constant; for a metal, g s ϭ1]. If the bulk-band structure projected onto the surface presents an energy gap near the vacuum level, an electron located in front of the surface cannot propagate into the solid. Therefore, the electron may be trapped in the vacuum well, and an infinite series of Rydberg-like states appears, which converges, for zero parallel momentum, towards the vacuum energy. These so-called image states 1-3 are localized in the vacuum region of the surface, the penetration of the first (nϭ1) image wave function into the solid varying typically between 4% and 22%. 4 As a result, image states are almost decoupled from bulk-electron scattering, and they are much longer lived than bulk excitations: the lifetime of bulk electrons with energies of 4 eV above the Fermi level is approximately one order of magnitude smaller than that of the first image state. Furthermore, the lifetime of higher-order image states (nу1) has been predicted 1 to scale asymptotically with n 3 , which makes the series to be resolvable.
During the last decade the linewidth of image states has been measured by inverse photoemission, 5,6 two-photon photoemission, [7][8][9] and time-resolved two-photon photoemission. [10][11][12][13][14] Recently, time-resolved two-photon photoemission has been used in combination with the coherent excitation of several quantum states, and the lifetime of the first six image states on the ͑100͒ surface of copper has been accurately determined. 15 Theoretical calculations of the linewidth of image states were first reported in Refs. 16 and 17, within a many-body free-electron description of the metal surface and with the use of simplified models to approximate both initial and final electronic states and, also, the screened-Coulomb interaction. Later on, the decay of the first image state on the ͑111͒ surfaces of copper and nickel metals to the nϭ0 crystalinduced surface state was calculated, 18 in terms of Auger transitions, with the use of a three-band model to describe the surface band structure. In Ref. 18, hydrogeniclike states with no penetration into the solid were used to describe the image-state wave functions, a simplified parametrized form was used for the surface-state wave functions, and screening effects were neglected. Self-consistent calculations of the linewidths of image states on copper surfaces have been reported recently, 19 and good agreement with experimentally determined decay times has been found. In Ref. 19, the linewidths of image states were computed, within the ͑GW͒ approximation of the many-body theory, 20 by going beyond a free-electron description of the metal surface. Single-particle wave functions were obtained by solving the Schrödinger equation with a realistic one-dimensional model potential, 21 and the screened interaction was evaluated in the randomphase approximation ͑RPA͒. 22 In this paper we focus our attention on the role that the crystal-induced surface state (nϭ0) plays in the relaxation of the first image state at the ͑111͒ surface of copper, which we find to represent 40% of the total linewidth. We present self-consistent calculations along the lines of Ref. 19, and we also consider simplified models for both the electronic wave functions and the screened-Coulomb interaction, showing that a detailed description of these quantities is of crucial importance in the understanding of the origin and magnitude of linewidths of image states. We account for potential variation parallel to the surface through the introduction of the effective mass, and we find that the linewidth of the first image state of Cu͑111͒ is 20% smaller than in the case of free-electron behavior along the surface. Finally, we investigate the dependence of interband 23 linewidths on the momentum of the image state parallel to the surface k ʈ . Our results indicate that for image state total energies lying below the top of the gap, the linewidth of the first image state of Cu͑111͒ is increased with k ʈ 0 up to 20%.

II. THEORY
We assume translational invariance in the plane parallel to the surface, which is taken to be normal to the z axis, and we evaluate the inelastic linewidth of the image state 1 (z)e ik ʈ •r ʈ with energy E 1 ϭ⑀ 1 ϩk ʈ 2 /(2m 1 ) ͑Ref. 24͒ ͑we use atomic units throughout, i.e., e 2 ϭបϭm e ϭ1), as the projection of the imaginary part of the electron self-energy ⌺(r,rЈ;E 1 ), over the state itself: where ⌺(z,zЈ,k ʈ ;E 1 ) represents the two-dimensional Fourier transform of ⌺(r,rЈ;E 1 ).
In the GW approximation, 20 the self-energy is obtained by just keeping the first term of the expansion in the screened interaction (W). Then, after replacing the Green function by the zero-order approximation, one finds Im ⌺͑z,zЈ,k ʈ ;E 1 ͒ϭ ͚ where the sum is extended over a complete set of final states is the effective mass, which accounts for the departure of the motion along the surface from free-electron behavior, E F is the Fermi energy, and W ind (z,zЈ,q ʈ ,E) represents the twodimensional Fourier transform of the induced part of the screened interaction.
In particular, if only transitions into the crystal-induced nϭ0 surface state 0 (z)e i(k ʈ ϩq ʈ )•r ʈ with energy E 0 ϭ⑀ 0 ϩ(k ʈ ϩq ʈ ) 2 /(2m 0 ) are considered, one finds and introducing this contribution to Im ⌺(z,zЈ,k ʈ ;E 1 ) into Eq. ͑1͒ one finds Three different models have been used for the evaluation of the screened interaction W. First, the specular-reflection model ͑SRM͒ of Ritchie and Marusak 25 has been considered. In this model, bulk electrons are assumed to be specularly reflected at the surface, the interference between the ingoing and outgoing waves being neglected, and the electronic charge density abruptly terminates at the surface (zϭ0), which we choose to be located half a lattice spacing beyond the last atomic layer. Within this simplified model, 26 also called semiclassical infinite barrier model the screened interaction is obtained in terms of the wave-vector and frequencydependent bulk-dielectric function, which we evaluate in the RPA. Second, for the vacuum contribution to the linewidth (zϾ0, zЈϾ0) the surface response function suggested by Persson and Zaremba 27 ͑PZ͒ has been used. Finally, the screened interaction has been evaluated, as in Ref. 19, by solving the RPA integral equation for the density-response function of inhomogeneous media in terms of the eigenfunctions of the one-electron effective Hamiltonian. These eigenfunctions have been computed by solving the Schrödinger equation with the realistic one-dimensional model potential suggested in Ref. 21. This model potential uses as parameters the width and position of the energy gap at the ⌫ point (k ʈ ϭ0) and, also, the binding energies of both the nϭ0 crystal-induced surface state at ⌫ and the first image state.
For the evaluation of nϭ0 and nϭ1 surface states, we have first used simplified models for the wave functions inside and outside the solid. In the vacuum side of the surface (zϾ0), nϭ1 and nϭ0 states have been approximated by a parametrized 1s-like hydrogenic wave function and a mere exponential, respectively, which have been matched to a decaying wave function in the crystal band gap (zϽ0) obtained within a nearly-free-electron two-band model: 17 Here, nϭ0 and nϭ1 correspond to the crystal-induced surface state and the first image state, respectively, G is the limit of the Brillouin zone in the direction normal to the surface, and where E gap and ⑀ n represent the energy gap and the energy of the n surface state with respect to the midgap, respectively.
The phase shift ␦ n is given by The image state on Cu͑111͒ is located right at the top of the gap (␦ 1 Ӎ0.9ϫ/2), both the hydrogeniclike wave function in the vacuum (zϾ0) and the decaying s-like wave function in the bulk (zϽ0) having, therefore, nodes at the surface (zϭ0). The nϭ0 crystal-induced surface state on Cu͑111͒ is located at the bottom of the gap (␦ 0 Ӎ0.2 ϫ/2); thus, it is described by a p-like wave function in the bulk. These approximate wave functions ͑AWF͒ are exhibited in Fig. 1, together with the corresponding wave functions that we obtain by solving the Schrödinger equation with the one-dimensional potential of Ref. 21 ͓model wave functions ͑MWF͔͒. Both wave functions, AWF and MWF, coincide within the bulk, but the hydrogeniclike wave function for the nϭ1 image state appears to be less localized near the surface than our model wave function. The nϭ0 and nϭ1 surface states on Cu͑111͒ have binding energies ͑measured with respect to the vacuum level͒ of 0.83 and 5.32 eV, respectively. The nϭ1 probability density has a maximum at 4.3 a.u. outside the crystal edge (zϭ0). The penetration into the bulk of nϭ0 and nϭ1 surface states is found to be, at the ⌫ point, of 74.5% and 22.1%, respectively.

III. RESULTS AND DISCUSSION
The results of our calculations for the linewidth ⌫ s , coming from the decay of the nϭ1 image state into the nϭ0 intrinsic surface state on Cu͑111͒ are presented in Table I, with the momentum of the image electron parallel to the surface k ʈ set equal to zero. Here, the linewidth has been split as follows: where ⌫ vac , ⌫ sol , and ⌫ inter represent vacuum, bulk, and interference contributions, respectively, as obtained by confining the integrals in Eq. ͑4͒ to either vacuum (zϾ0,zЈ Ͼ0), bulk (zϽ0,zЈϽ0), or vacuum-bulk (z Ͻ Ͼ 0,zЈ Ͼ Ͻ 0) coordinates. First we show our full RPA calculations, in which the screened interaction is obtained on the basis of oneelectron eigenfunctions computed from the realistic onedimensional model potential of Ref. 21  I. Calculated linewidth ͑in meV͒ coming from the decay of the nϭ1 image state on Cu͑111͒ into the nϭ0 intrinsic surface state, as obtained within three different models for the description of the surface response and two different models for the description of both initial and final wave functions ͑see text͒. RPA accounts for our full RPA realistic calculation of the screened interaction, SRM for the simplified specular reflection model of Ritchie and Marusak ͑Ref. 25͒, and PZ for the vacuum side surface response suggested by Persson and Zaremba ͑Ref. 27͒. MWF accounts for the wave functions obtained by solving the Schrödinger equation with the realistic one-dimensional model potential of Ref. 21, and AWF for the approximate model described in the text. The effective mass of the nϭ1 image state has been set equal to the free-electron mass, and for the nϭ0 surface state we have used either m 0 ϭ1 or m 0 ϭ0.42. The momentum of the image electron parallel to the surface is set equal to zero.
Zaremba 32 for the screened interaction, we have used the MWF nϭ0 and nϭ1 surface-state wave functions as well as the simplified models ͑AWF͒ described in the previous section, with m 0 ϭ1. Total linewidths ⌫, as obtained from the decay of the n ϭ1 image state on Cu͑111͒ into any final state with energy E f ,E F рE f рE 1 , ͓see Eqs. ͑1͒ and ͑2͔͒, are presented in Table II. Here, our full RPA calculations are shown, with all wave functions computed from the one-dimensional model potential of Ref. 21 ͑MWF͒. Realistic values of the effective mass of final states have been considered, according to the experiment or to ab initio band-structure calculations. As for the nϭ0 surface state, we have used m 0 ϭ0.42, as in Table I, and for bulk states we have chosen to increase the effective mass from our computed value 33,34 of m f ϭ0.22 at the bottom of the gap to m f ϭ1 at the bottom of the valence band.
Our full RPA calculations indicate that the decaying rate of the nϭ1 image state into the nϭ0 crystal-induced surface state results for m 0 ϭ1 in a linewidth of 16 meV, while use of the more realistic effective mass m 0 ϭ0.42 leads to a linewidth of 12 meV. With the use of either the free-electron mass or more realistic effective masses for both bulk and crystal-induced surface states, ⌫ s approximately represents 40% of the total linewidth, ⌫ϭ37 meV (m f ϭ1) or ⌫ ϭ29 meV (m f 1). The more realistic result of ⌫ ϭ29 meV for the total linewidth is in good agreement with the experimentally measured lifetime 35 of 22Ϯ3 fs at 25 K. 13,14 Within the vacuum side of the surface the nϭ1 image state couples dominantly to the nϭ0 surface state ͑this coupling approximately represents 90% of the total ⌫ vac linewidth͒; however, the coupling of image states with all bulk crystal states occurring through the bulk penetration plays an important role and cannot, therefore, be neglected if one is to accurately describe the lifetime of image states.
We note that simplified jellium models for the evaluation of the screened interaction lead to unrealistic results for the contribution of the surface state to the linewidth of image states. 36 First, we compare our full RPA calculations ͑see Table I͒ with the results we obtain, also with use of our model initial and final wave functions, when our realistic screened interaction is replaced by that obtained within the specular reflection model and the model of Persson and Zaremba. Bulk contributions to the linewidth are approximately well described within the specular reflection model, small differences resulting from an approximate description, within this model, of the so-called begrenzung effects. As the approximate treatment of Ritchie and Marusak 25 ignores the quantum-mechanical details of the surface, this model fails to describe both vacuum and interference contributions to the linewidth. These quantum-mechanical details of the surface are approximately taken into account within the jellium model of Ref. 27, thus resulting in a better approximation for the vacuum contribution to the linewidth. Discrepancies between vacuum contributions obtained within this model ͑PZ͒ and our more realistic full RPA calculations 36 appear as a result of the jellium model of Ref. 27 being accurate provided q ʈ /q F and /E F Ӷ1(q F is the Fermi momentum, i.e., E F ϭq F 2 /2). In order to investigate the dependence of ⌫ s on the details of both nϭ1 and nϭ0 wave functions, we present in Table  I calculations, within SRM and PZ models for the screened interaction, in which our realistic wave functions are replaced by the simplified models ͑AWF͒ described in the previous section. As the hydrogeniclike wave function used to describe the nϭ1 image state on the vacuum side of the surface presents an image-state charge-gravity center localized further away from the surface than our more realistic model wave function, both vacuum and interference contributions to the linewidth are largely underestimated within this approximation. Furthermore, we note that the linewidth is highly sensitive to the details of the image-state wave functions. This is a consequence of the critical behavior of the imaginary part of the nonlocal self-energy coupling points near the surface, as we will discuss below. Now we focus on our full RPA calculation of the total linewidth of the nϭ1 image state on Cu͑111͒ ͑see Table II͒, with all effective masses set equal to the free-electron mass. We show in Fig. 2͑b͒ separate contributions to the linewidth, ⌫, coming from the decay into the various f bulk crystal states, ⌫ f , such that ⌫ϭ ͚ f ⌫ f ϩ⌫ s . ͑9͒ Figure 2͑a͒ exhibits the bulk band structure projected onto the ͑111͒ surface of copper. The arrows indicate the available phase space in the decay of the nϭ1 image state at the ⌫ point (k ʈ ϭ0) into the unoccupied portion of the nϭ0 surface state and a generic f bulk state, which are represented by their characteristic ⑀ 0 ϩq ʈ 2 /2 and ⑀ f ϩq ʈ 2 /2 parabolic dispersions, respectively. ⌫ vac , ⌫ sol , and Ϫ⌫ inter contributions to ⌫ f are represented, together with the total contribution ⌫ f as a function of ⑀ f . The lower edge, at the ⌫ point, of the energy gap projected onto the Cu͑111͒ surface lies below the Fermi level (E g ϽE F ) and, consequently, the decay from the (k ʈ ϭ0) image state occurs through finite-parallel momentum transfer. Hence, as the coupling of the image state with the crystal occurring through the tails of bulk states outside the crystal is expected to be dominated by vertical transitions (q ʈ Ӎ0), vacuum contributions to the ⌫ f linewidth are very small, especially for those bulk states located at the bottom of the valence band ͑decay into these states is only allowed for large values of the momentum transfer; also, their vacuum  Table I, the effective mass and the momentum parallel to the surface of the nϭ1 image state have been set equal to the free-electron mass and equal to zero,respectively. As for final ͑bulk and intrinsic surface͒ states, we have used either m f ϭ1 or the realistic effective masses (m f ϭ1) described in the text. Contributions to the linewidth from decay into bulk states ⌫Ϫ⌫ s are displayed in parentheses.
penetration is small͒. Actually, the coupling of the nϭ1 image state with all bulk states taking place at the vacuum side results in a linewidth of only 5 meV, which approximately represents 10% of the total ⌫ vac linewidth. 37 A realistic description of motion along the surface can be approximated by introduction of the effective mass, as described above. The effective mass of all final states with energies E f ,E f рE f рE 1 , is found to be smaller than the free-electron mass, thus both ⌫ s and ⌫ being about 20% smaller than in the case of free-electron behavior along the surface ͑see Tables I and II͒. This is the result of two competing effects: First, there is the effect of the decrease of the available phase space, which is easily found to scale as ͱm.
Second, as the effective mass decreases the decay from the image state occurs, for a given energy transfer, through smaller parallel momentum transfer, which may result in both enlarged and diminished screened interactions, depending on momentum and energy transfers. This is illustrated in Fig. 3, where the impact of the introduction of the effective mass on the evaluation of the various contributions ͑vacuum, bulk, and interference͒ to both ⌫ f and ⌫ s ͓see Eq. ͑9͔͒ is exhibited through the percentage ratio as a function of the final-state energy ⑀ f . If the screened interaction were independent of the parallel momentum transfer, all ratios would scale as ͱm, which is represented in Fig. 3 as R ⌬q ʈ ϭ100ͱm f . Instead, as the parallel momentum transfer decreases the screened interaction is predominantly larger, which results in the ratio R ⌫ f to be larger than R ⌬q ʈ , especially in the case of vacuum contributions to the linewidth, which are expected to be dominated by vertical transitions. When the decay from the image state may occur through very small parallel momentum transfer q ʈ ͓this is the case of final states that are just below the bottom of the gap and also the case of the nϭ0 surface state͔, a decrease in q ʈ may result in a diminished screened interaction ͑see Fig. 4͒, thus R ⌫ f being slightly smaller than R ⌬q ʈ for these states. Now, we analyze the behavior of the imaginary part of the image-electron self-energy, which results in vacuum and interference contributions to the linewidth to be comparable in magnitude and opposite in sign ͑see Table I͒. Figure 5 shows full RPA ͑solid line͒ and SRM ͑dashed-dotted line͒ calculations of ϪIm ⌺ s (z,zЈ,k ʈ ϭ0,E 1 ) ͓see Eq. ͑3͔͒ for Cu͑111͒ with all effective masses set equal to the free-electron mass, together with the nϭ1 image-state wave function, as a function of the zЈ-coordinate and for zϭ7.4 a.u.. We note that the probability for electron-hole pair creation ͑the dominant channel for the decay of these states is provided by this pro-cess͒ is underestimated within the SRM.
The coupling between electronic states is well known to be maximum, within the bulk, at the position of the electron. Nevertheless, as the electron moves into the vacuum the maximum of the imaginary part of the electron self-energy Ratio R ⌫ f of Eq. ͑10͒, as a function of the final state energy. The effective mass and the momentum parallel to the surface of the nϭ1 image state have been set equal to the free-electron mass and equal to zero, respectively. As for final ͑bulk and intrinsic surface͒ states, we have used the realistic effective masses (m f 1) described in the text. Here, R ⌬q ʈ ϭ100ͱm f . stays ͑see Fig. 5͒ near the surface (zϭ0), as demonstrated, within a jellium model of the surface, by Deisz et al. 38 Hence, for any given value of zϾ0, main vacuum and interference contributions to the linewidth are determined ͓see Eq. ͑1͔͒ by the specific shape of the image state wave function in regions A and B of Fig. 5. As the image-state on Cu͑111͒ is located right at the top of the energy gap and the corresponding wave function has, therefore, a node at the surface (zӍ0), an inspection of Fig. 5 leads us to the conclusion that vacuum and interference contributions to the linewidth are comparable in magnitude and opposite in sign. On Cu͑100͒ the image state is located close to the center of the gap, and the corresponding wave function has a node at zӍ1.3 a.u. On this surface, total vacuum and interference contributions to the linewidth are still opposite in sign, though interference contributions coming from the decay into states at the bottom of the valence band are now positive due to their minor vacuum penetration. If the image state were located at the bottom of the gap, matching at the surface would occur at maximum amplitude and the total interference contribution might be positive, as the sign of the image-state wave function would be the same on both sides just around the surface (zϭ0).
Finally, we investigate the dependence of interband 23 linewidths on the momentum of the image electron parallel to the surface k ʈ . First, we use the MWF image-state wave function evaluated at the ⌫ point and introduce, within this model, the dependence on k ʈ of the quasiparticle self-energy, as indicated in Eq. ͑2͒. We find that for image-state total energies lying below the top of the gap (k ʈ Ӎ0.11 a.u.), the linewidth of the first image state of Cu͑111͒ increases less than 4%. Second, we also account for the change of the z-dependent initial wave function along the dispersion curve of the image state, by solving the Schrödinger equation for a one-dimensional model potential that we build following Ref. 21 with various values of k ʈ : 0.06, 0.09, and 0.10 a.u.. The penetration into the bulk of the nϭ1 image state with these values of k ʈ is found to vary from 22.1% at k ʈ ϭ0 to 22.6%, 24.1%, and 26.2% at k ʈ ϭ0.06, 0.09, and 0.10 a.u., respectively. Our results, as obtained with all effective masses set equal to the free-electron mass, are presented in Table III. Though the penetration of the imagestate wave function increases with k ʈ , the amplitude of this wave function on the bulk side and near the jellium edge decreases, thus the absolute value of both bulk and interference contributions to the linewidth decreasing with k ʈ . Nevertheless, the total overlap between image-state and final wave functions becomes more efficient as k ʈ increases, which results in larger values of the total linewidth. Also, it is interesting to notice that especially sensitive to the variation of the momentum of the image state parallel to the surface is the contribution to the linewidth from damping into the nϭ0 surface state. The sign appearing in regions A and B accounts for the sign of the product between the imaginary part of the self-energy and the image-state wave function. Full circles represent the atomic positions in the ͑111͒ direction. The geometrical electronic edge (zϭ0) has been chosen to be located half an interlayer spacing beyond the last atomic layer. All effective masses have been set equal to the free-electron mass.

IV. SUMMARY
We have reported calculations of the inelastic broadening of the first image state at the ͑111͒ surface of copper, and we have investigated, in particular, the role that the intrinsic crystal-induced surface state plays in the decay of this image state. We have presented self-consistent RPA calculations, by going beyond a free-electron description of the metal surface. We have also considered simplified models for both the electron wave functions and the screened-Coulomb interaction, showing that a detailed description of these quantities is of crucial importance in the understanding of the origin of linewidths of image states. We have accounted for potential variation parallel to the surface through the introduction of the effective mass.
We have analyzed the origin and magnitude of the various contributions to the linewidth. Though the dominant contribution to the decay of the first image state into the crystalinduced surface state comes from the coupling between image and surface states within the vacuum part of the surface, it appears to be approximately canceled out by the contribution from the interference between bulk and vacuum coordinates. For the vacuum contribution to the decaying rate into the intrinsic surface state, we have found that it approximately represents 90% of the total vacuum contribution. We also conclude that the coupling of image states with all bulk crystal states occurring through the bulk penetration plays an important role in the determination of lifetimes, and that this penetration cannot be neglected if one is to accurately describe the lifetime of image states.
We have found, within our full RPA scheme, that in the case of k ʈ ϭ0 the decaying rate of the first image state on Cu͑111͒ into the intrinsic surface state results, with all effective masses set equal to the free-electron mass, in a linewidth of 16 meV, while use of more realistic effective masses leads to a linewidth of 12 meV. With the use of either the free-electron mass or more realistic effective masses for both bulk and crystal-induced surface states, ⌫ s approximately represents 40% of the total linewidth, ⌫ϭ37 meV (m f ϭ1) or ⌫ϭ29 meV (m f 1). The more realistic result of ⌫ϭ29 meV for the total linewidth is in good agreement with recent experimental results reported in Ref. 13.
We have investigated the dependence of interband linewidths on the momentum of the image electron parallel to the surface, showing that for image-state total energies lying below the top of the gap the linewidth of the first image state increases with k ʈ up to 20%. We conclude that this increase appears mainly as a consequence of the change of the z-dependent initial wave function with k ʈ , and our results indicate that the contribution to the linewidth from damping into the nϭ0 surface state is responsible for the dependence of the total linewidth with the momentum parallel to the surface.  35 The lifetime is related to the linewidth ⌫ of Eqs. ͑1͒ and ͑4͒ by ⌫ϭបϭ660 meV fs. 36 We find that the impact of the band structure in the evaluation of the screened interaction is not large, and differences between our full RPA calculations, on the one hand, and the SRM and PZ results, on the other hand, appear as a consequence of simplifications introduced, within a jellium model of the surface, in both SRM and PZ approaches. 37 Notice that both ⌫ vac and ⌫ inter contributions to the linewidth present a maximum for energies just below the bottom of the energy gap. This is due to the fact that the vacuum penetration of the bulk states increases as one goes up in energy and then decreases just below the gap due to the stronger bulk character of these states reproducing the bottom of the gap. This maximum appears here as a cusp because of the relatively thin slab used in the present calculation. This sharp feature will be significantly smoother for thicker films.