ARTICLE pubs.acs.org/JPCA
Phonon-Induced Surface Charge Density Oscillations in Quantum Wells: A First-Principles Study of the (2 2)-K Overlayer on Be(0001) V. Chis Department of Physics, University of Central Florida, Orlando, Florida 32816, United States Donostia International Physics Center (DIPC), 20018 Donostia - San Sebastian, Spain
G. Benedek* Dipartimento di Scienza dei Materiali, Universita di Milano-Bicocca, 20125 Milano, Italy Donostia International Physics Center (DIPC), 20018 Donostia - San Sebastian, Spain ABSTRACT: Density functional perturbation theory has been applied to study the surface vibrations of (2 2)-K monolayer on the Be(0001) surface. We present the full phonon dispersion curves along the high symmetry directions of the surface Brillouin zone (SBZ) together with the layer-projected phonon density of states and the phonon-induced surface charge density oscillations at Γ and M for the alkali SV and L modes. Surprisingly, at the M point, the L-phonon displacements produce a more pronounced perturbation on the surface charge density than the SVphonon displacements. These results apparently solve the long-standing question regarding helium atom scattering (HAS) experiments performed on the similar system (2 2)-K on graphite, where the alkali SV phonon mode is not observed. Moreover, this result confirms the previous finding that HAS from free-electron metal surfaces probes directly the phonon-induced charge density oscillations and the related electronphonon interaction.
1. INTRODUCTION Recent inelastic helium atom scattering (HAS) studies of metal surfaces14 revealed a complex dynamic response of the surface charge density to phonon displacements. Besides determining HAS intensities4 and the carrier lifetimes at metal surfaces,5,6 such manifestation of electronphonon (e-p) interaction has relevant implications in other surface properties such as ultrathin layer superconductivity,711 inelastic electron tunneling (IETS),12,1 inelastic photoemission (IPES),13 time-resolved second harmonic generation,14,15 and phonon-assisted chemical reactions.16,17 The recent observation of superconductivity in thin Pb overlayers on Si(111) down to one single monolayer18 shows alone the relevance of e-p interaction in two dimensions. These effects are found to be pronounced in supported metal multilayers characterized by free-electron quantum well (QW) states.3,4 A special case is represented by alkali overlayers, due to their ability in enhancing certain surface reactions and field emission.19,20 On a more fundamental side, in these systems, the coupling of the overlayer phonons to electronic transitions between states of the two-dimensional electron gas (2DEG) allows the study of the effects of a quasi-2D electronphonon interaction.17,21,22 2DEG associated with a potassium layer adsorbed on Be(0001) is presently attracting much interest of theoreticians also for the occurrence of collective electronic excitations of acoustic type2325 and their involvement in photoemission26,27 as a natural follow-up to the recent discovery r 2011 American Chemical Society
of surface acoustic plasmons in Be(0001).28 The investigation of the surface phonon structure and e-p interaction of K/Be(0001) appears as a necessary complement to these studies. Former calculations based on the embedded-atom method of the phonon dispersion curves of alkali overlayers on metal surfaces2931 while capturing the essential role of free electrons in mediating the interatomic force constants are not suitable, however, to directly correlate the mode-selected e-p coupling constants to the surface band structure. More recently, the surface dynamics and friction of K on Cu(001) has been investigated by means of 3 He spinecho spectroscopy and a first-principle calculation.32 In this paper, we report on a density-functional perturbation theory (DFPT) calculation for a potassium monolayer on Be(0001) of the phonon dynamics and the dynamic electron density oscillations induced by some selected overlayer and surface phonons. The chosen system provides a test bench for studying the interaction of a 2DEG with individual overlayers and surface phonons. Besides the interest illustrated above, this study was motivated by the fact that the calculated mode-selected e-p coupling constants λQ v can now be directly measured with inelastic atom scattering.4 While no HAS study of K/Be(0001) is Special Issue: J. Peter Toennies Festschrift Received: January 13, 2011 Revised: March 26, 2011 Published: April 08, 2011 7242
dx.doi.org/10.1021/jp200373h | J. Phys. Chem. A 2011, 115, 7242–7248
The Journal of Physical Chemistry A
ARTICLE
Figure 1. Three high-symmetry configurations of the (2 2) K/Be(0001) with potassium atoms at (a) fcc hollow, (b) hcp hollow, or (c) on-top positions. The Be atoms in the first layer are marked by a red spot. The left inset shows the Cartesian symmetry directions; the right inset shows the symmetry points and directions in the irreducible part of the surface Brillouin zone.
available yet, the present results shed light on the so-far unexplained observation of an unusual selective coupling of HAS to longitudinal phonons in the system K/graphite.3335
2. DFPT CALCULATIONS AND RESULTS The dynamical properties of the (2 2)-K/Be(0001) surface have been calculated by applying first-principles density functional perturbation theory (DFPT) with the package QUANTUM-ESPRESSO36 and norm-conserving pseudopotentials37 within the local-density approximation for the exchange-correlation energy functional. The electron wave functions were expanded in plane waves with an energy cutoff of 22 Ryd. The irreducible Brillouin zone was sampled over a Monkhorst-Pack grid of 10 10, resulting in 52 k-points. Brillouin zone integration was performed with the smearing technique,38 with a smearing width of 20 mRyd. The atomic arrangement of (2 2)-K/ Be(0001) was obtained by minimizing the total energy with respect to the positions of the K and Be atoms using a nine atomic layers thick Be slab as a substrate. The supercell contains four Be atoms in each layer and one K atom on the vacuum sides of the cell. To make surface-to-surface interactions through the vacuum region negligible, the surfaces were separated by 35 Å of vacuum. The Be(0001) substrate was relaxed until the resulting forces acting on an atom was less than 25 meV/Å. The relaxed structure results in an expansion of the first and second interlayer distance by 3.2% and 0.93%, respectively, while the relaxation field of deeper layers is negligible. Such a large expansion of the first interlayer distance is in agreement with that of a previous calculation (3.8%),39 but smaller than that from a LEED study (5.8%).40 As for the first interlayer distance, the present second interlayer separation is also smaller than found in the previous calculation (2.2%).39 Based on the optimized structure for the Be(0001) substrate, three different K adsorption sites were investigated. The on-top and the two inequivalent hollow sites (Figure 1) were compared with respect to total energy after having fully relaxed all atoms in a two step procedure. In the first step the adsorbed K atom was allowed to relax freely but the structure optimized for the Be substrate was kept rigid. This resulted in the following order of preference: (a) fcc hollow, (b) hcp hollow, (c) on-top, with a 12 meV lower energy for the fcc hollow site (Figure 1). In the second step, all atoms are allowed to relax. The result is that the on-top site is now favored by 15 meV per K atom. With the atom in this site there is, in the second step, an inward relaxation for both the K atom and the uppermost Be layer such that the KBe distance is reduced by 2.6%.
Figure 2. Surface phonon dispersion curves of (2 2)K/B(0001) calculated with DFPT along the high-symmetry directions of the surface Brillouin zone.
Furthermore, the adatom induces a rumpling of the Be surface layer. The on-top atom resides 2.9 Å above the Be atom and this distance is 0.13 Å larger than the distance to the plane of the surrounding Be atoms in the top-most layer. The adsorption of a K monolayer gives a work function change and a redistribution of the charge compared to that of the constituents. The calculated work function of Be(0001) is 4.9 eV and is reduced by 2.6 eV when the K layer has been adsorbed. The charge density redistribution is obtained by summation of the charge density distribution of a free-standing K monolayer and the Be(0001) slab and subtraction of the sum from the charge density distribution for the overlayer system. Accumulation of charge is found in the interface region which originates from depletion of charge within the K layer, though some additional charge is contributed from the first and second Be layers. 2.1. Dispersion Curves. The dynamical matrix of the system was evaluated on a 3 3 mesh of q-points in the surface Brillouin zone (BZ). The Fourier transform of this matrix provides the real-space interatomic force constants of the system which extend up to the third nearest neighbor atoms within the alkali overlayer. The phonon dispersion curves were evaluated in 206 q-points along the high symmetry directions of the surface BZ. The phonon dispersion curves of the (2 2)-K/Be(0001) system calculated with DFPT are shown in Figure 2. The dispersion curves are characterized by three low-energy branches that are assigned to the alkali overlayer. The lowest of the three modes in the ΓM direction has a shear-horizontal (SH) polarization, the second lowest one has a longitudinal (L) character and the third one is a shear-vertical (SV) mode, the next one is predominantly longitudinal (L), whereas the highest one has a predominant shear-vertical (SV) polarization. These modes are 7243
dx.doi.org/10.1021/jp200373h |J. Phys. Chem. A 2011, 115, 7242–7248
The Journal of Physical Chemistry A
ARTICLE
Figure 3. Contour plots of the spectral intensities of the longitudinal (L), shear-horizontal (SH), and shear-vertical (SV) components of the surface modes projected onto the alkali overlayer. The upper part of the spectrum with the substrate modes is not shown since the projection of these modes on the substrate coordinates gives negligible intensities.
Figure 4. Folding of the Be(0001) RW surface (light gray area) as derived from Lazzeri and De Gironcoli DFPT calculation45 from the irreducible part ΓM0 K0 of the (1 1) surface BZ to the irreducible part ΓMK of the (2 2) surface BZs. As an effect of the (1 1) f (2 2) folding, the original RW branch yields in both ΓM and ΓK symmetry directions three branches (one acoustic and two optical) and degenerate pairs of modes at Γ and along the line mk. The RW surface is folded into four sheets within the irreducible zone ΓMK (the “acoustic” sheet Γmk and the “optical” sheets mkγ, γkm0 , γm0 k0 ).
strongly localized on the K overlayer, as appears from the contour plots of the respective spectral intensities projected onto the alkali overlayer shown in Figure 3 in the low energy region of the modes. As appears from Figure 3, the L and SH modes form a degenerate pair at the Γ-point (2.23 meV), as required by symmetry. Along the symmetry directions of the surface BZ the L and SH modes follow each other up to the M-point, where they have the largest separation in energy (L: 8.23 meV and SH: 5.24 meV). At the K-point (7.57 meV) they form again a degenerate pair, and exchange their polarizations in the KΓ direction. The alkali SV mode shows almost no dispersion along the surface BZ symmetry lines, (11.2 meV at Γ, 13.8 meV at M, and 14 meV at K). Near the Γ-point, the SV mode appears as a weak resonance due to hybridization with several modes within the continuum of bulk bands. Previous experimental4143 and theoretical4446 studies of the surface phonons of the clean Be(0001) surface have reported several surface localized phonon modes, some more intense than others. The Rayleigh wave (RW) of the clean surface was found to be located well below the bulk continuum at the high
Figure 5. Phonon dispersion curves of (2 2)-K/Be(0001) with their spectral intensities projected onto the first Be layer for longitudinal (L1), shear-horizontal (SH1), and shear-vertical (SV1) polarizations.
symmetry points of the surface BZ, precisely at 39 meV at M and 43 meV at K. The K overlayer doubles the periodicity of the Be substrate leading to a (2 2) unit cell repetition with respect to that of the substrate, and the surface BZ is four times smaller than that of Be(0001). Thus, the original Be(0001) phonon dispersion curves are folded into the smaller surface BZ. Figure 4 shows the folding of the Be(0001) RW surface (light gray area) as derived from Lazzeri and De Gironcoli DFPT calculation:45 ΓM0 K0 and KMK are the irreducible parts of the (1 1) and (2 2) surface BZs, respectively. The RW in the ΓM direction (thick black line) is folded at m and the point γ at M0 is cast into the zone center; similarly, the RW in the ΓK direction (thick gray line) is folded at k, but now the kk0 segment is folded along the KM boundary of the (2 2) surface BZ and the middle point m0 occurs at the M point. Thus, as an effect of the (1 1) f (2 2) folding, the original RW branch yields in both symmetry directions three branches (one acoustic and two optical) and degenerate pairs of modes at Γ and along the line mk. These degeneracies can be lifted by the addition of the K layer due to the double periodicity of the perturbing potential. The perturbed dispersion curves for phonons with SV, SH, and L polarizations with their spectral intensities projected onto the first (SV1, SH1, L1), second (SV2, SH2, L2) and third (SV3, SH3, L3) layer of the Be substrate are plotted in Figures 5, 6 and 7, respectively. As appears in Figure 5 and in agreement with the folding scheme of Figure 4, the spectrum of the SV modes 7244
dx.doi.org/10.1021/jp200373h |J. Phys. Chem. A 2011, 115, 7242–7248
The Journal of Physical Chemistry A
Figure 6. Same as Figure 5 for the longitudinal (L2), shear-vertical (SH2), and shear-vertical (SV2) components projected onto the second Be layer.
projected onto the first Be layer is dominated by two branches originating from the folding of the RW. They are almost degenerate at Γ with energies of 44 and 45 meV and weakly dispersed in both ΓM and ΓK directions. Were the Be(0001) surface unaffected by the adsorbed alkali overlayer, the backfolded RW in the ΓM direction would have a dispersion extending from 32 to 39 meV in the (2 2) surface BZ. Actually the dispersion curve of the substrate RW, after a strong noncrossing hybridization with the potassium SV mode (Figure 3), the folded branch raises from 36 meV at M to 44 meV at theΓ point, very close to its continuation as a flat branch at 45 meV. Thus, the degeneracy at Γ and M points is lifted by a small amount of about 1 meV. The increase of energy of the folded RW with respect to that of the clean Be surface is due to the fact that the K adatoms, which are much heavier than Be atoms, do not move in the short-wave portion of the RW, thus, contributing additional force constants acting on the topmost Be atoms. Moreover, the decrease by ∼0.13 Å of the spacing of the Be top bilayer produced by the K overlayer yields a further stiffening of the first Be interlayer force constant. These two combined effects raise the RW energy at the (1 1) M point, now Γ, from 39 to 44 meV. As appears in Figures 6(SV2) and 7(SV3), the spectral intensities corresponding to SV modes projected onto the second and third Be layers around 45 meV are much smaller than the intensities at the first layer of the folded RW (Figure 5(SV1)). This means that the two folded RW branches as sharp SV resonances. In the Be spectral region the L modes show the most prominent features in the second layer around
ARTICLE
Figure 7. Same as Figure 5 for the longitudinal (L3), shear-vertical (SH3) and shear-vertical (SV3) components projected onto the third Be layer.
50 meV, notably at the M point (Figure 6(L2)). As already observed for Cu(111),1,2 this resonance originates from the longitudinal resonance found in most metal surfaces. In addition around the M point there is another strong L resonance at 53 meV, which has a large amplitude at the first as well as at the third layer. The two L resonances at 50 and 53 meV can be associated with the two sublattices of the hcp Be structure. At higher energy, especially above 70 meV, there are other intense branches that are also seen in SV3 and SH3 (Figure 7) and have therefore a less pronounced surface character. All of these features are better appreciated by inspecting the phonon density of states (DOS) for the SV and L components projected onto the K overlayer and the first two Be layers, plotted in Figure 8 for some selected wavevectors along the ΓM direction. It is interesting to compare these results with the Hannon et al. electron-energy loss spectroscopy (EELS) data for the clean Be(0001) surface.43 In the EELS experiments a weak intensity mode with SH components is reported at the M-point of the (1 1) BZ at an energy of 50 meV. In the present calculations this mode would be folded back into the zone center of the (2 2) BZ (Figure 4). As seen in Figure 6(SV2), this mode is indeed found at Γ slightly above 50 meV. Also, the two resonances of L and SH polarization predicted at the K-point, having similar intensities and energy of ∼63 meV, match very well the EELS modes at the points K and K/2 of the (1 1) BZ. Such a good agreement suggests that the addition of the K overlayer produces a negligible effect on these EELS resonances, 7245
dx.doi.org/10.1021/jp200373h |J. Phys. Chem. A 2011, 115, 7242–7248
The Journal of Physical Chemistry A
Figure 8. Phonon density of states (DOS) of (2 2)-K/Be(0001) projected onto the K, Be1, and Be2 layers for SV (left) and L (right) polarizations and different wavevectors along the ΓM symmetry direction.
the reason being that they are mostly localized on the second layer and the K atoms are not at all involved in such high-energy resonances due to their large mass. This also indicates, however, that the K overlayer does not affect the interlayer shear force constants of the Be layers. 2.2. Surface Charge-Density Oscillations. The adiabatic charge density oscillations (CDOs) associated with some particular surface phonons have been calculated in the frozen phonon scheme as follows. The atomic position of the l-th atom for a phonon of wavevector Q and branch index j is set at X l = X(0)l þ p/(2MlεQ j)1/2 el(Qj), where X(0)l is the l-th atom equilibrium position, Ml is its mass, and εQj and el(Qj) are the phonon energy and normalized eigenvector. The CDO is then extracted by taking the difference between the charge density for the displaced atoms and the charge density at equilibrium. Results for the alkali L and SV modes with frozen-phonon displacements for wavevectors at the Γ and M points are shown in Figure 9. The contour lines are only plotted for values of the CDOs in the region of interest, that is, between þ104 and 104 a.u., The contour lines correspond to CDO values in units of 104 a.u. ranging from (1 to (1/128, each step corresponding to a factor 1/2. Red (blue) lines correspond to positive (negative) modulations. In ordinary HAS experiments, with incident energies in the range 2030 meV, He atoms are repelled from the surface at about 3 Å from the first atomic layer. Thus, the maximum phonon-induced CDOs perceived by He atoms plotted in Figure 9 are about 0.02 104 a.u for the Γ-point SV mode at 11.2 meV, 0.01 104 a.u for both the SV and L modes at the M point (13.8 and 8.23 meV, respectively), whereas it is much smaller for the Γ-point L mode at 2.23 meV. It should be noted that the actual matrix elements of the scattering potential giving the inelastic HAS amplitudes depend on the space modulation of the CDOs in the direction of the parallel momentum transfer ΔK. Thus, the 11.2 meV mode shows a flat CDO, which ensures a large coupling to a scattering process with ΔK = 0, as expected for the creation (annihilation) of a phonon at Γ. Similarly, a process involving a phonon at M requires a corresponding CDO space modulation. Thus, the L(M) mode at 8.23 meV, which shows a change of sign in the CDO in the region where the
ARTICLE
scattering takes place, is expected to give a substantially more intense HAS signal than the SV(M) mode at 13.8 meV, which does not change sign in that region. The ability of HAS to detect longitudinal modes, despite the fact that the static surface at the He-atom turning point distance looks completely flat and the He atoms do not hit the surface atoms directly, has intrigued theoreticians for many years. Before the advent of DFPT calculations, various phenomenological models, involving the electronic degrees of freedom and specifically addressing the problem of HAS intensities, have been suggested. A summary discussion is found in ref 1. In light of the recent DFPT calculations, effective phenomenological approaches were the multipole expansion (ME) method47,48 and the embedded atom (EA) method.4951 Worth mentioning here is the fact that the EA method proves quite efficient for the calculation of the dynamics52 and the electronphonon interaction53 of complex alkali overlayer structures on transition metal surfaces for which reliable DFPT calculations may still be too computer-intensive. The present DFPT results show that the longitudinal in-plane displacements of the surface atoms cause a normal modulation of the 2DEG density and therefore a coupling to the He atoms. An interesting question is whether the 2DEG is affected as well by the substrate phonons, in view of the fact that the overlayer and substrate dynamics are largely decoupled. As an example, the CDOs for the pair of SV resonant modes at Γ of energy 44 and 45 meV originating from the folding of the RW have been calculated and displayed in Figure 10. The most interesting feature is the comparatively large response of the 2DEG in the overlayer region despite the fact that the K ions are practically at rest. Something similar was already reported for Cu(111)2 and for ultrathin Pb films on Cu(111),4 as an evidence that He atoms can detect subsurface phonons via the induced CDOs, which can be transmitted, like Friedel oscillations, at distances substantially larger than the interplanar distance. Here, however, we deal with a 2DEG that is dynamically modulated by substrate phonons, which keep the overlayer atoms practically at rest.
3. DISCUSSION The CDO induced by the alkali SV mode at Γ shows quite a peculiar property as the K atom oscillates about its equilibrium position. This mode actually is a weak resonance with a uniform displacement field propagating inside the Be substrate and a strong enhancement at the K layer that, in particular, generates no modulation of the surface charge density. For an outward vertical motion of the K atoms, the valence charge donated to the 2DEG is called back, so as to give a positive spot in front of the ion at the expense of the delocalized electron gas charge, which shows a uniform depletion, with practically no space modulation in the low density region. Of course the inward displacement of the K atoms yields the opposite change in the 2DEG density. Because the K atom is in the on-top position, a downward motion of the K ion attracts electronic charge between the two ions to reinforce the KBe bonding and to better screen the positive ion charges. Also, at the M point, the SV modes produces a modest spatial modulation of the CDO, almost vanishing at the K plane, with no change of sign. The small modulation in space of the 2DEG density all over the BZ may explain the very small dispersion for the K-ion SV branch (Figure 3(SV)). On the contrary, the 2DEG responds dramatically to the L motion of the K atoms. Because the L motion of the K ions is 7246
dx.doi.org/10.1021/jp200373h |J. Phys. Chem. A 2011, 115, 7242–7248
The Journal of Physical Chemistry A
ARTICLE
Figure 9. Contour plots of the charge density oscillations (CDOs) in the surface region, including the K adsorbed layer and the three first substrate Be layers, as functions of normal (z) and parallel (x) coordinates (cfr. Figure 1), induced by frozen-phonon displacements of the low-energy modes of the K layer with SV and L polarizations at the Γ and M symmetry points. The insets indicate the respective energies (cfr. Figure 3(L,SV)). The contour lines correspond to CDO values in units of 104 a.u. ranging from (1 to (128, each step corresponding to a factor 1/2. Red (blue) lines correspond to positive (negative) modulations. The equilibrium distance (2.82 Å) between the K layer and the Be1 layer is averaged over the rumpling (not shown).
Γ to M is associated with an appreciable dispersion of this mode (Figure 3(L)). The present analysis may shed some light on the intriguing case of the inelastic HAS data for the (2 2) alkali overlayers on graphite,3335 where only the longitudinal overlayer branch was observed, besides its avoided crossing with the substrate RW. It should be noted that the overlayer density on the honeycomb structure of graphite is smaller than on the closed-packed Be(0001) surface. This fact combined with the larger inertness of graphite suggests that the effects described above should be emphasized for the graphite substrate. In particular, the CDO associated with the SV alkali mode should remain rather small all over the BZ, whereas the alkali L branch, while less dispersed, should be characterized by a comparatively large CDO for wavevectors beyond the region of hybridization with the RW, thus, giving the only relevant feature in HAS time-of-flight spectra.
Figure 10. CDOs for the doublet of SV modes at Γ originating from the folding of the substrate RW.
opposite to the L motion of the first Be layer, the charge density is compressed on one side and is squeezed outward, compensated by the charge depletion on the other side. The strongly modulated outward CDO lobes occurring at M are due to the part of the quantum-well wave function provided by the Be layer, while the smaller lobes near the K ion and following its motion are due to the part of the wave function provided by the K ion itself. The L mode of the K ion at Γ is instead strongly localized, almost decoupled from the substrate, and no modulation of the surface charge density is produced above the K overlayer. The important change in the charge density response in moving from
’ ACKNOWLEDGMENT We thank Profs. Pedro M. Echenique and Evgueni V. Chulkov (DIPC, Donostia-San Sebastian, Spain) and Prof. Renee D. Diehl (Penn State University) for useful discussion. G.B. gratefully acknowledges the support of IKERBASQUE (Project ABSIDES). ’ REFERENCES (1) Benedek, G.; Bernasconi, M.; Chis, V.; Chulkov, E.; Echenique, P. M.; Hellsing, B.; Toennies, J. P. J. Phys.: Condens. Matter 2010, 22, 084020. (2) Chis, V.; Hellsing, B.; Benedek, G.; Bernasconi, M.; Chulkov, E. V.; Toennies, J. P. Phys. Rev. Lett. 2008, 101, 206102; 2009, 103, 069902. 7247
dx.doi.org/10.1021/jp200373h |J. Phys. Chem. A 2011, 115, 7242–7248
The Journal of Physical Chemistry A (3) Braun, J.; Ruggerone, P.; Zhang, G.; Toennies, J. P.; Benedek, G. Phys. Rev. B 2009, 79, 205423. (4) Sklyadneva, I. Y.; Benedek, G.; Chulkov, E. V.; Echenique, P. M.; Heid, R.; Bohnen, K.-P.; Toennies, J. P., manuscript in process. (5) Kliever, J.; Berndt, R.; Chulkov, E. V.; Silkin, V. M.; Echenique, P. M.; Crampin, S. Science 2000, 288, 1399. (6) Sklyadneva, I. Y.; Heid, R.; Bohnen, K.-P.; Chulkov, E. V. New J. Phys. 2009, 11, 103038. (7) Guo, Y.; Zhang, Y.-F.; Bao, X.-Y.; Han, T.-Z.; Tang, Z.; Zhang, L.-X.; Zhu, W.-G.; Wang, E. G.; Niu, Q.; Qiu, Z. Q.; Jia, J.-F.; Zhao, Z.-X.; Xue, Q.-K. Science 2004, 306, 1915. (8) Bao, X.-Y.; Zhang, Y.-F.; Wang, Y.; Jia, J.-F.; Xue, Q.-K.; Xie, X. C.; Zhao, Z.-X. Phys. Rev. Lett. 2005, 95, 247005. € (9) Ozer, M. M.; Thompson, J. R.; Weitering, H. H. Nat. Phys. 2006, € 2, 173. Ozer, M. M.; Jia, J.; Zhang, Z.; Thompson, J. R.; Weitering, H. H. Science 2007, 316, 1594. (10) Eom, D.; Qin, Q.; Chou, M.-Y.; Shih, C.-K. Phys. Rev. Lett. 2006, 96, 027005. Qin, S.; Kim, J.; Niu, Q.; Shih, C.-K. Science 2009, 324, 1314. (11) Brun, C.; Hong, I.-P.; Patthey, F.; Yu. Sklyadneva, I; Heid, R.; Echenique, P. M.; Bohnen, K. P.; Chulkov, E. V.; Schneider, W.-D. Phys. Rev. Lett. 2009, 102, 207002. (12) Gawronski, H.; Mehlhorn, M.; Morgenstern, K. Science 2008, 319, 930. (13) Arafune, R.; Hayashi, K.; Ueda, S.; Uehara, Y.; Ushioda, S. Phys. Rev. Lett. 2005, 95, 207601. Arafune, R.; Yamamoto, M. Q.; Takagi, N.; Kawai, M. Phys. Rev. B 2009, 80, 073407. (14) Fuyuki, M.; Watanabe, K.; Matsumoto, Y. Phys. Rev. B 2006, 74, 195412. (15) Watanabe, K.; Inoue, K.-I.; Nakai, I. F.; Fuyuki, M.; Matsumoto, Y. Phys. Rev. B 2009, 80, 075404. (16) Kroes, G.-J. Science 2008, 321, 794. Diaz, C.; Pijper, E.; Olsen, R. A.; Busnengo, H. F.; Auerbach, D. J.; Kroes, G. J. Science 2009, 326, 832. (17) Petek, H.; Ogawa, S. Annu. Rev. Phys. Chem. 2002, 53, 507. (18) Zhang, T.; Cheng, P.; Li, W.-J.; Sun, Y.-J.; Wang, G.; Zhu, X.-G.; He, K.; Wang, L.; Ma, X.; Chen, X.; Wang, Y.; Liu, Y.; Lin, H.-Q.; Jia, J. F.; Xue, Q.-K. Nat. Phys. 2010, 6, 104. (19) Diehl, R. D.; McGrath, R. Surf. Sci. Rep 1996, 23, 43; J. Phys. Cond. Matter 1997, 9, 951. (20) Politano, A.; Chiarello, G.; Benedek, G.; Chulkov, E.V.; Echenique, P. M.; Surf. Sci. Rep. 2011, to be published. (21) Nojima, A.; Yamashita, K.; Hellsing, B. Phys. Rev. B 2008, 78 035417. (22) Bovensiepen, U.; Petek, H.; , Wolf, M. In Dynamics at Solid State Surfaces and Interfaces: Current Developments; Matsumoto, Y., Watanabe, K., Eds.; Wiley-VCH Verlag GmbH & Co. KgaA: New York, 2011; Vol. 1, on-line Oct 2010. (23) Silkin, V. M.; Pitarke, J. M.; Chulkov, E. V.; Echenique, P. M. Phys. Rev. B 2005, 72, 115435. (24) Silkin, V. M.; Chulkov, E. V.; Echeverry, I. P.; Echenique, P. M. Phys. Status Solidi B 2010, 247, 1849. (25) Echeverry, J. P.; Chulkov, E. V.; Silkin, V. M. Phys. Status Solidi C 2010, 7, 2640. (26) Algdal, J.; Balasubramanian, T.; Chis, V.; Hellsing, B.; Lindgren, S.-Å; Wallden, L. Phys. Rev. B 2008, 78, 085102. (27) Silkin, V. M.; Hellsing, B.; Wallden, L.; Echenique, P. M.; Chulkov, E. V. Phys. Rev. B 2010, 81, 113406. (28) Diaconescu, B.; Pohl, K.; Vattuone, L.; Savio, L.; Hofmann, Ph.; Silkin, V. M.; Pitarke, J. M.; Chulkov, E. V.; Echenique, P. M.; Farias, D.; Rocca, M. Nature 2007, 448, 57. (29) Borisova, S. D.; Rusina, G. G.; Eremeev, S. V.; Benedek, G.; Echenique, P. M.; Sklyadneva, I. Y.; Chulkov, E. V. Phys. Rev. B 2006, 74, 165412. (30) Eremeev, S. V.; Yu. Sklyadneva, I.; Echenique, P. M.; Borisova, S. D.; Benedek, G.; Rusina, G. G.; Chulkov, E. V. Surf. Sci. 2007, 601, 4553–4556. (31) Rusina, G. G.; Eremeev, S. V.; Echenique, P. M.; Benedek, G.; Borisova, S. D.; Yu. Sklyadneva, I.; Chulkov, E. V. J. Phys.: Condens. Matter 2008, 20, 224007.
ARTICLE
(32) Hedgeland, H.; Kole, P. R.; Davies, H. R.; Jardine, A. P.; Alexandrowicz, G.; Allison, W.; Ellis, J.; Fratesi, G.; Brivio, e G. P. Phys. Rev. B 2009, 80, 125426. (33) Cui, J.; White, J. D.; Diehl, R. D.; Annett, J. F.; Cole, M. W. Surf. Sci. 1992, 279, 149. (34) Cui, J.; White, J. D.; Diehl, R. D. Surf. Sci. Lett. 1993, 293, L841. (35) White, J. D.; Cui, J.; Strauss, M.; Diehl, R. D.; Ancillotto, F.; Toigo, F. Surf. Sci. 1994, 307, 1134. (36) Giannozzi, P.; Baroni, S.; Bonini, N.; Calandra, M.; Car, R.; Cavazzoni, C.; Ceresoli, D.; Chiarotti, G. L.; Cococcioni, M.; Dabo, I.; Dal Corso, A.; de Gironcoli, S.; Fabris, S.; Fratesi, G.; Gebauer, R.; Gerstmann, U.; Gougoussis, C.; Kokalj, A.; Lazzeri, M.; Martin-Samos, L.; Marzari, N.; Mauri, F.; Mazzarello, R.; Paolini, S.; Pasquarello, A.; Paulatto, L.; Sbraccia, C.; Scandolo, S.; Sclauzero, G.; Seitsonen, A. P.; Smogunov, A.; Umari, P.; Wentzcovitch, e. R. M. J. Phys.: Condens. Matter 2009, 21, 395502http://arxiv.org/abs/0906.2569. (37) Use has been made of the pseudopotentials Be.pz-n-vbc.UPF and K.pz-n-vbc.UPF from http://www.quantum-espresso.org (38) Marzari, N.; Vanderbilt, D.; De Vita, A.; Payne, M. C. Phys. Rev. Lett. 1999, 82, 3296. (39) Feibelman, P. J. Phys. Rev. B 1992, 46, 2532. (40) Davis, H. L.; Hannon, J. B.; Ray, K. B.; Plummer, E. W. Phys. Rev. Lett. 1992, 68, 2632. (41) Hannon, J. B.; Plummer, E. W. J. Electron Spectrosc. Relat. Phenom. 1993, 65/65, 683. (42) Plummer, E. W.; Hannon, J. B. Prog. Surf. Sci. 1994, 46, 149. (43) Hannon, J. B.; Mele, E. J.; Plummer, E. W. Phys. Rev. B 1996, 53, 2090. (44) Lazzeri, M.; de Gironcoli, S. Phys. Rev. Lett. 1998, 81, 2096. (45) Lazzeri, M.; de Gironcoli, S. Surf. Sci. 1998, 402, 715. (46) de Gironcoli, S.; Lazzeri, M. Science and Supercomputing at CINECA; CINECA: Bologna, 2001. (47) Jayanthi, C. S.; Bilz, H.; Kress, W.; Benedek, G. Phys. Rev. Lett. 1987, 59, 795. (48) Kaden, C.; Ruggerone, P.; Toennies, J. P.; Benedek, G. Nuovo Cimento D 1992, 14, 627. Kaden, C.; Ruggerone, P.; Toennies, J. P.; Zhang, G.; Benedek, G. Phys. Rev. B 1992, 46, 13509. Benedek, G.; Ellis, J.; Luo, N. S.; Reichmuth, A.; Ruggerone, P.; Toennies, J. P. Phys. Rev. B 1993, 48, 4917. (49) Daw, M. S.; Baskes, M. I. Phys. Rev. B 1984, 29, 6443. (50) Nelson, J. S.; Sowa, E. C.; Daw, M. S. Phys. Rev. Lett. 1988, 61 1977. (51) Nelson, J. S.; Daw, M. S.; Sowa, E. C. Phys. Rev. B 1989, 40, 1465. (52) Borisova, S. D.; Rusina, G. G.; Eremeev, S. V.; Benedek, G.; Echenique, P. M.; Sklyadneva, I. Y.; Chulkov, E. V. Phys. Rev. B 2006, 74, 165412. (53) Eremeev, S. V.; Sklyadneva, I. Y.; Echenique, P. M.; Borisova, S. D.; Benedek, G.; Rusina, G. G.; Chulkov, E. V. Surf. Sci. 2007, 601, 4553–4556.
7248
dx.doi.org/10.1021/jp200373h |J. Phys. Chem. A 2011, 115, 7242–7248
