Lifetimes of Shockley electrons and holes at the Cu(111) surface

A theoretical many-body analysis is presented of the electron-electron inelastic lifetimes of Shockley electrons and holes at the (111) surface of Cu. For a description of the decay of Shockley states both below and above the Fermi level, single-particle wave functions have been obtained by solving the Schr\"odinger equation with the use of an approximate one-dimensional pseudopotential fitted to reproduce the correct bulk energy bands and surface-state dispersion. A comparison with previous calculations and experiment indicates that inelastic lifetimes are very sensitive to the actual shape of the surface-state single-particle orbitals beyond the $\bar\Gamma$ (${\bf k}_\parallel=0$) point, which controls the coupling between the Shockley electrons and holes.


I. INTRODUCTION
A variety of metal surfaces, such as the (111) surfaces of the noble metals, are known to support a partially occupied band of Shockley surface states with energies near the Fermi level, 1 whose dynamics have been the subject of long-standing interest. 2,3,4 In particular, the lifetimes of excited holes at the band edge (k = 0) of these surface states have been investigated with high resolution angle-resolved photoemission (ARP) 5,6,7,8 and with the use of the scanning tunneling microscope (STM). 9 STM techniques have also allowed the determination of the lifetimes of excited Shockley and image electrons over a range of energies above the Fermi level. 10,11 Many-body calculations of the electron-electron (e-e) inelastic lifetimes of excited holes at the surface-state band edge of the (111) surfaces of the noble metals Cu, Ag, and Au, which were based upon the G 0 W one-dimensional scheme that had been introduced by Chulkov et al. 12 to describe the lifetimes of imagepotential states, 13 showed considerable agreement with experiment. 14,15 These calculations were then extended to treat the case of excited surface-state and surfaceresonance electrons above the Fermi level. 16,17 In order to account approximately for the potential variation in the plane of the surface, the original one-dimensional model potential, which had been introduced to describe surface states at theΓ point, was modified along with the introduction of a realistic effective mass for the dispersion curve of both bulk and surface states. Within this model, however, all Shockley states have the same effective mass, so the projected band structure is not correct, especially at energies above the Fermi level, as shown in Fig. 1.
In this paper, we present a new approach that although at theΓ point is less sophisticated than the model used in Refs. 14,15,16,17 (i) basically reproduces the surfacestate probability density of Refs. 14,15,16,17 at the band edge of the surface-state band in Cu (111)  the merit that it reproduces, through the introduction of a k -dependent one-dimensional potential, the actual bulk energy bands and surface-state energy dispersion of Fig. 1, thereby allowing for a realistic description of the electronic orbitals beyond theΓ point. Adding the contribution from electron-phonon coupling, 18 which is particularly important at the smallest excitation energies, our calculations of the lifetime broadening of excited Shockley electrons and holes in Cu(111) indicate that (i) there is good agreement with experiment at the surfacestate band edge and (ii) at energies above the Fermi level the lifetime broadening is closer to experiment and very sensitive to the actual shape of the surface-state singleparticle orbitals beyond theΓ point. Let us consider a semi-infinite many-electron system that is translationally invariant in the plane of the surface (normal to the z axis). The decay rate or inverse lifetime of a quasiparticle (electron or hole) that has been added to the system in the single-particle state e iki·r ψ ki,Ei (z) of energy E i is obtained as follows (we use atomic units, i.e., e 2 =h = m e = 1) 19 where the ∓ sign in front of the integral should be taken to be minus or plus depending on whether the quasiparticle is an electron (E i > E F ) or a hole (E i ≤ E F ), respectively, E F is the Fermi energy, r and k i represent the position and wave vectors in the plane of the surface, and Σ(z, z ′ ; k i , E i ) is the nonlocal self-energy operator.
To lowest order in a series-expansion of the self-energy 20,21 and replacing the interacting Green function G(z, z ′ ; k, E) by its noninteracting counterpart, one finds the following expression for the imaginary part of the so-called G 0 W self-energy: Here, where z < (z > ) is the lesser (greater) of z and z ′ , and The functions ψ ± k,E (z) are solutions of the single-particle Schrödinger equation regular at ±∞, with V k (z) being a momentum-dependent one-dimensional effective potential that we fit to the projected surface band structure. We use where U k and V k are fitted to the bulk energy bands (which we have obtained from three-dimensional ab initio calculations), a s = 2.08Å represents the interlayer spacing, Φ = 4.94 eV is the experimentally determined work function, and the matching plane z k is chosen to give the correct surface-state dispersion represented in Fig. 1 by a thick solid line. 22 In the random-phase approximation (RPA), 23 the screened interaction W (z, z ′ ; q, E) is obtained from the knowledge of the noninteracting density-response function χ 0 (z, z ′ ; q, E) by solving the following integral equation (7) where v(z, z ′ ; q) represents the two-dimensional Fourier transform of the bare Coulomb interaction. The results presented below have been obtained by using in χ 0 (z, z ′ ; q, E) the eigenfunctions and eigenvalues of the one-dimensional Hamiltonian of Ref. 12 with all effective masses set equal to the free-electron mass. We have also used the eigenfunctions and eigenvalues of a singleparticle jellium-surface Kohn-Sham Hamiltonian (in the local-density approximation) with r s = 2.67, 24 and we have found that the surface-state lifetimes are rather insensitive to whether one or the other choice is employed. The abrupt step model potential of Eq. (6), which does not account for the image tail outside the surface, could not possibly be used to describe image states. However, Shockley surface states are known to be rather insensitive to the actual shape of the potential outside the surface; indeed, the model potential of Eq. (6) is found to yield a surface-state probability-density |ψ ki,Ei | 2 at the band edge of the Shockley surface-state band of Cu(111) (k i = 0) that is in reasonably good agreement with the more realistic surface-state probability density used in Refs. 14,15,16,17, as shown in Fig. 2. Both probabilitydensities coincide within the bulk, although our approximate probability-density appears to be slightly more localized near the surface, as expected. 25 Nevertheless, we find that decay rates of an excited hole atΓ based on the use of these two models to describe the wave function ψ ki,Ei entering Eq. (1) agree within less than 1 meV. Differences between our new calculations, which are based on the use of the k-dependent model potential of Eq. (6), and those reported previously, 14,15,16,17 are due to our more realistic description of the band structure and surface-state wave functions beyond theΓ point. First of all, we consider an excited hole at the band edge of the Shockley surface-state band of Cu(111), i.e., with E i = −0.44 eV and k i = 0 (see Fig. 1). The decay of this excited quasiparticle may proceed either through the coupling with bulk states (interband contribution) or through the coupling, within the surface-state band itself, with surface states of different wave vector k parallel to the surface (intraband contribution). In order to investigate the impact of the actual shape of the surfacestate wave functions with k = 0 on the decay of the surface-state hole atΓ, we have compared in Table I the decay rates that we have calculated either by using in Eq. (3) the actual k-dependent model wave function ψ k,E (full calculation) or by replacing all surface-state wave functions ψ k,E with k = 0 by that at theΓ point (approximate calculation). This comparison shows that the penetration of the actual k-dependent surface-state wave functions ψ k,E being larger than atΓ (compare the dashed and dotted lines of Fig. 1) yields a reduction in the decay rate from 33 meV to 19 meV, which is due to the fact that the coupling of the surface-state hole atΓ with actual surface states of different wave vector k (intraband contribution) is smaller than the coupling that would take place with surface-state orbitals that do not change with k . The difference between our predicted surface-state lifetime broadening of 19 meV and that re-ported before (τ −1 = 25 meV) 14,15 is entirely due to our more accurate description of (i) the projected band structure and (ii) the wave-vector dependence of the surfacestate wave functions ψ k,E (z) entering the evaluation of the Green function of Eq. (3).
We have also carried out a full calculation of the decay of an excited hole atΓ but replacing the actual surfacestate wave vector k f entering Eq. (2) by the wave vector that would correspond to a parabolic surface-state dispersion of the form dictated by the thin solid line of Fig. 1, and we have found that the linewidth is reduced (as expected, since the parabolic dispersion results in fewer final states) by less than ∼ 1 meV. However, if one further replaces our wave-vector dependent surface-state orbitals entering Eq. (2) by their less accurate counterparts used previously, 14,15 the lifetime broadening is increased considerably (from 19 to 25 meV), showing the important role that the actual coupling between initial and final states plays in the surface-state decay mechanism.
Our model, which correctly reproduces the behaviour of s-p valence states, does not account for the presence of d electrons with energies a few electronvolts below the Fermi level. The screening of d electrons is known to play a crucial role in the decay mechanism of bulk states. 26 However, in the case of Shockley holes, whose decay is dominated by intraband transitions that are associated with very small values of the momentum transfer, the screening of d electrons is expected to reduce the lifetime broadening only very slightly, 27 and will not be included in the present work. Adding to our estimated e-e linewidth of 19 meV the electron-phonon (e-ph) linewidth of 7 eV reported in Ref. 18, we find Γ total = 26 meV in close agreement with the experimentally measured linewidth of 24 meV, as shown in Table I. In Fig. 3, we show our full calculation (thick solid line) of the inelastic linewidth (Γ total = Γ e−e + Γ e−ph ) of excited Shockley holes and electrons in Cu(111) with energies E i below and above the Fermi level. Also shown are separate interband and intraband contributions to the linewidth (inset), the approximate calculation that we have carried out by replacing all surface-state wave functions with k = 0 by the surface-state wave function atΓ (thin solid line), and the calculations reported in Ref. 16  tween our full calculations and experiment shows that there is close agreement at the surface-state band edge (at E − E F = −0.44 eV) and there is also reasonable agreement at energies above the Fermi level. At energies where the surface-state band merges into the continuum of bulk states, however, our calculated linewidths are still too low, which might be a signature of the need of a fully three-dimensional description of the surface band structure. We also note that differences between the calculations reported here and those reported previously 16 indicate that inelastic lifetimes are very sensitive to the actual shape of the surface-state singleparticle orbitals beyond theΓ point. The linewidths reported here are smaller (larger) for excited holes (electrons) below (above) the Fermi level, thus bringing the theretical predictions closer to experiment.
In summary, we have presented a new G 0 W onedimensional scheme to calculate the inelastic lifetime broadening of excited Shockley electrons and holes in Cu(111), which is based on a realistic description of the projected bulk energy bands and the surface-state orbitals beyond theΓ point. Adding the contribution from electron-phonon coupling, 18 which is particularly important at the smallest excitation energies, our calculations indicate that there is reasonable agreement with experiment, especially at low excitation energies. The screening of d electrons, not included in this work, is expected to reduce the lifetime broadening only very slightly, at least at the hot-electron energies nearest to the Fermi level.