Surface Dynamics of the Wetting Layers and Ultrathin Films on a

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Surface Dynamics of the Wetting Layers and Ultrathin Films on a Dynamic Substrate: (0.5–4) ML Pb/Cu(111) Galina G. Rusina, Svetlana D. Borisova, Sergey V. Eremeev, Irina Yu. Sklyadneva, Evgueni V. Chulkov, Giorgio Benedek, and Peter Toennies J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.6b05857 • Publication Date (Web): 06 Sep 2016 Downloaded from http://pubs.acs.org on September 12, 2016

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The Journal of Physical Chemistry

Surface Dynamics of the Wetting Layers and Ultrathin Films on a Dynamic Substrate: (0.5–4) ML Pb/Cu(111) G. G. Rusina,†,‡ S. D. Borisova,†,‡ S. V. Eremeev,†,‡ I. Yu. Sklyadneva,∗,¶ E. V. Chulkov,¶,§ G. Benedek,¶,∥ and J.P. Toennies⊥ Institute of Strength Physics and Materials Science, 634021, Tomsk, Russia, Tomsk State University, 634050 Tomsk, Russia, Donostia International Physics Center (DIPC), 20018 San Sebastián/Donostia, Basque Country, Spain, Departamento de Física de Materiales UPV/EHU, Centro de Física de Materiales CFM - MPC and Centro Mixto CSIC-UPV/EHU, 20080 San Sebastián/Donostia, Basque Country, Spain, Dipartimento di Scienza dei Materiali, Universitá di Milano-Bicocca, 20125 Milano, Italy, and Max-Planck-Institut für Dynamik und Selbstorganisation, Am Fassberg 17, 37077 Göttingen, Germany E-mail: [email protected]

∗ To

whom correspondence should be addressed of Strength Physics and Materials Science, 634021, Tomsk, Russia ‡ Tomsk State University, 634050 Tomsk, Russia ¶ Donostia International Physics Center (DIPC), 20018 San Sebastián/Donostia, Basque Country, Spain § Departamento de Física de Materiales UPV/EHU, Centro de Física de Materiales CFM - MPC and Centro Mixto CSIC-UPV/EHU, 20080 San Sebastián/Donostia, Basque Country, Spain ∥ Dipartimento di Scienza dei Materiali, Universitá di Milano-Bicocca, 20125 Milano, Italy ⊥ Max-Planck-Institut für Dynamik und Selbstorganisation, Am Fassberg 17, 37077 Göttingen, Germany † Institute

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Abstract The growth of Pb ultra-thin films on Cu(111) has been long studied in connection with electronic quantum-size effects and for the different temperature-dependent growth kinetics. At low temperature the formation of a wetting layer (1 monolayer (ML)), is followed by an instability of the 2 ML film and a regular layer-by-layer growth is then only observed for more than two monolayers. The 2 ML film was, however, shown to be stabilized by alloying Pb with 20% Tl. This work presents a theoretical study of the dynamics of the wetting layer as well as for 2 ML Pb0.8 Tl0.2 , 3 ML and 4 ML Pb on Cu(111) in the 4×4 commensurate phase, for which detailed inelastic Helium atom scattering (HAS) spectra have been measured. The present calculations based on the embedded atom method (EAM) include the dynamics of the substrate. Besides leading to a detailed interpretation of the HAS experimental data, the present results are compared with a previous density-functional perturbation theory (DFPT) study for 3 to 7 ML Pb on a rigid substrate. The comparison reveals the role played by the substrate dynamics at the smallest thicknesses, despite the large mass and stiffness differences between Pb and Cu. Also the different thermal expansion of the film with respect to the substrate is shown to cause appreciable anomalies in the temperature and thickness dependence of the phonon dispersion curves.

Introduction Thin metal films grown on metal surfaces are of great interest for serving as model systems in the study of materials with reduced dimensionality. In particular thin lead films have attracted much attention because of sizeable quantum-size oscillations occurring in the layer-by-layer growth. This effect was first observed by He-atom scattering (HAS) and attributed to interference with the quantum-well states. 1,2 The latter modulate the electron density of states at the Fermi level and the electron-phonon (e − p) interaction, 3 which causes oscillations with thickness in various structural and mechanical properties such as the interlayer distances, 2,4,5 the island height distribution, 6,7 the zone-center surface phonon frequencies. 8,9 More important are the quantum-size


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effects on electronic properties such as the oscillations in the electronic transport, 10 excited electron lifetimes, 11–13 the work function and photoemission, 14,15 the electron-phonon coupling, 16,17 the superconducting critical temperature and the upper critical field 16,18–21 and other electronic characteristics. 22,23 It was recently demonstrated for ultra-thin Pb films on Cu(111) that the HAS inelastic intensities provide a direct measurement of mode-selected e − p coupling strengths for individual surface and sub-surface phonons (mode-lambda spectroscopy), 24,25 thus allowing to assess which phonons are actually relevant in superconductivity. The discovery of superconductivity in a single lead monolayer on silicon 26 raises intriguing questions about the e − p coupling in quasi-two-dimensional systems. To describe correctly the electron-phonon coupling it is necessary to know the phonon spectrum of the film and the layer distribution of the density of vibrational modes. Usually, because of a high computational demand, the substrate is not included in the calculation, and the phonon spectrum is calculated either for free-standing films 16,17 or for a rigid substrate. 24 However, the structure of the interface can play an important and sometimes decisive role in the morphology and properties of the growing film. Structural investigations using low-energy electron diffraction (LEED) revealed that in the first study of growth (submonolayer coverage) of lead on low-index Cu surfaces a surface alloy with c(4×4) superstructure is formed. 27 Nagl et al. using scanning tunneling microscopy (STM) also observed a Pb-Cu surface alloy. 28 At room and higher temperatures the surface alloying were observed on all low-index Cu surfaces for submonolayer of Pb. 28,29 The LEED, STM and Auger electron spectroscopy (AES) studies 27,28,30,31 showed that at full monolayer lead coverage a quasihexagonal close-packed superstructure is formed on the Cu(111) surface which consists of nine Pb atoms accommodated in the 4×4 supercell containing 16 Cu atoms per layer. Once the first Pb layer (wetting layer) is formed further deposition of lead follows different routes depending on temperature. While at low temperature a layer-by-layer Frank-van-der-Merwe growth regime is observed, at room temperature the Stranski-Krastanov layer-plus-island growth occurs where islands of different size and height are formed above the wetting layer. 32,33 The monitoring by specular He-atom scattering of the low-temperature growth process shows indeed



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that Pb films grow on the Cu(111) surface first by the formation of the wetting layer (1 monolayer (ML)), followed by an unstable 2 ML film and a regular layer-by-layer growth already from 3 ML on. 1,2,34,35 The 2 ML film was, however, shown to be stabilized by alloying Pb with 20% Tl. 35,36 The present theoretical study of the dynamics of the wetting layer and of the 2 ML Pb0.8 Tl0.2 , as well as of 3 ML and 4 ML Pb films on Cu(111) in the 4×4 commensurate phase, originates from previous measurements of inelastic HAS spectra performed at the MPI-DSO in Göttingen, of which only part has been published, 9,34,35 while others, still waiting for a thorough theoretical analysis, are here reported for the first time. There is a special interest in analysing the HAS inelastic spectra of metal surfaces since the HAS intensity from a specific phonon is directly proportional to its electron-phonon coupling. 24,25 Thus the observation by HAS of a selected number of dispersion curves provides in principle a way to pinpoint those phonons which mostly contribute to the total electron-phonon coupling strength. The ability of HAS to detect subsurface phonons as deep as the e − p interaction range makes it a valuable tool for ultra-thin films, where the film-substrate interface phonons can be detected as well. Because of the fact the present EAM analysis has been extended to the interface phonons. Another important feature specifically related to the interface structure arises from the experimental HAS data at different surface temperatures of 95 – 100 K and 140 K, notably some anomalous anharmonic effects for certain surface phonons which can be related to the different thermal expansion of the film with respect to that of the substrate. For a correct description of the distribution of the phonon density of states in the film, it is important to take into account the emerging structure of the interface and the interaction between the film atoms and the atoms of the first substrate layer. In the present theoretical study the atomic relaxation and the phonon dispersion curves of the films with the respective projected densities have been calculated in the framework of the embedded atom method (EAM), where the Cu(111) substrate dynamics is duly taken into account.


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Helium atom scattering spectra The experiments have been performed with two Helium-atom scattering apparatus. 9,34–36 Both are equipped with high pressure helium jet sources to provide high intensity atom beams with a velocity resolution of about ∆v/v = 1 %. 37,38 The surfaces are initially characterized with LEED and AUGER and the target chamber residual gas pressures is below about 5× 10−11 mbar. It is well established that for substrate temperatures below 200 K lead grows on Cu(111) layer-by-layer according to the Frank-van-der-Merwe growth regime. 1,30 As reported by Zhang and confirmed by Braun 34–36 HAS-monitored growth experiments for substrate temperatures ranging from 60 to 300 K, after the formation of the p(4×4) wetting layer, only a poorly ordered second layer can be grown, followed however by well-ordered thicker films from 3 ML on. A stable bilayer could however be obtained through the co-evaporation of lead with a 20% thallium onto a substrate at 95 K. 35,36 The great similarity of Pb and thallium solids (lattice distances and atomic weights differing by about 1.5% and melting absolute temperatures by 4%) permits to consider the 2 ML-Pb0.8 Tl0.2 bilayer as a good approximation to a 2 ML Pb films, at least for the lattice dynamics. It is important to note that for a deposition of the wetting layer at 300 K the interatomic distances of the resulting p(4×4) structure turn out to be contracted by 2.6% with respect to the ideal Pb monolayer, 30 whereas a subsequent annealing at 700 K leads to an incommensurate structure where the interatomic distances are equal to those of the Pb(111) surface. As expected, the contraction tends however to disappear with increasing thickness. As discussed below, the different thermal expansion of lead and copper causes relevant anomalies in the temperature dependence of phonon frequencies for the 1 ML film, which however disappear for the thicker films. A few exemplary HAS energy-gain spectra for 1 ML Pb/Cu(111) are shown in Figure 1 and Figure 2 for a surface temperature of 100 K and 140 K, respectively, for similar sequences of incident angles in both ⟨112⟩ and ⟨110⟩ symmetry directions, and nearly equal incident energies. Further energy-gain spectra for 2 ML–Pb0.8 Tl0.2 bilayer, 36 and for 3 and 4 ML films can be found in Refs. 9,34,35. In Figure 1 and Figure 2, RW labels the Rayleigh wave, L the longitudinal resonance, Z and X other folded modes which have been assigned to prevalent z or x polarization, 5 ACS Paragon Plus Environment


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Figure 1: Energy gain spectra of He scattered from 1 ML Pb/Cu(111) surface at 100 K for different incident angles θi and the same incident energy Ei = 22 meV in the (left) and (right) directions. 35 RW – Rayleigh wave; L – longitudinal resonance; Z(X) other folded modes with assigned z-(x-) polarization; the peak at zero energy is due to diffused elastic scattering and provides information on the surface quality.


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Figure 2: Energy gain spectra of He scattered from 1 ML Pb/Cu(111) surface at 140 K for different incident angles θi , and the same incident energy Ei = 23.3 meV in the (left) and (right) directions. 34 RW – Rayleigh wave; L – longitudinal resonance; Z(X) other folded modes with assigned z-(x-) polarization; the peak DE at zero energy is due to diffused elastic scattering and provides information on the surface quality.



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Figure 3: The dispersion curve of the Rayleigh wave (RW) for 1 ML Pb/Cu(111) measured by HAS at 140 K (open circles), represented on the Pb(111) Brillouin zone, is compared to that for the 0.5 ML Pb/Cu(111) (full diamonds and interpolating eye-guideline). 34 The slopes of the Cu(111) RW are indicated by straight lines. The kinks in the 0.5 ML dispersion (red arrows) correspond to symmetry points of the 0.5 ML Brillouin zone (BZ). The softening of the 1 ML RW modes with respect to those for the 0.5 ML reflects the lateral compressive stress occurring at the commensurate full coverage, but also the increased loading effect of the full Pb wetting layer with respect to the half-filled one. respectively. The peak at zero energy is due to diffused elastic scattering and provides information on the surface quality. The dispersion curve of the RW measured for the 1ML Pb film at 140 K is plotted in Figure 3 along the symmetry directions of the Pb(111) Brillouin zone (BZ) (open circles), together with that for a 0.5 ML Pb film at the same temperature (full diamonds and interpolating guideline). The comparison indicates a substantial softening of the RW modes for a full coverage with respect to the RW modes at half a coverage. This is easily interpreted as an effect of the in-plane compressive stress, which occurs in the commensurate 4×4 phase of the 1 ML film, but not in the less dense 0.5 ML phase. Note that at large wavelengths (small wavevectors Q) the slope of the RW tends to that of the substrate (straight lines) since the corresponding penetration length increases as 1/Q, thus gradually reducing the loading effect of the film. Thus the softening of the 1 ML branch with respect to that for 0.5 ML can also be interpreted as due to a larger loading effect of the full Pb wetting layer with respect to the half-filled one. The analysis of the thermal stress effects discussed


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below would favor the second interpretation. On the other hand for Q corresponding to the zone boundaries of the 0.5 ML Brillouin zone (red arrows), the 0.5 ML dispersion curve of the RW exhibits a kink and starts separating from the dispersion curve for 1 ML. This also means that beyond these points the penetration length has shortened down to the scale of the Pb-Pb distances of the 0.5 ML film, and the dispersion curves of both films deviate from that of the substrate, tending at the zone boundaries to energies comparable to those of Pb(111). In the next Section the structure and the phonon dispersion curves of the 1 to 4ML films, calculated in the framework of the embedded atom method for the commensurate 4×4 phase, provide a detailed interpretation of the main features observed in the HAS experiments.

Theoretical analysis Calculation details The calculations are performed using the embedded atom method (EAM) to construct interatomic interaction potentials for Cu–Cu and Pb–Pb. 39 Parameters of the method are determined by fitting to experimental data such as the equilibrium lattice constant, elastic constants, sublimation energy, and vacancy formation energy of the pure bulk metals. The Cu–Cu interatomic potentials were successfully applied before to the calculation of phonons on the clean metal surfaces 40,41 and on the surfaces covered by adsorbates. 40,42,43 The interaction between Cu and Pb atoms is described by a pair potential constructed in the form proposed by Johnson. 44 A two-dimensional periodic slab consisting of 31 atomic layers of Cu(111) is used for calculating the structure and phonons. For a monolayer of Pb, lead atoms are arranged according to the p(4×4) superstructure on both sides of the slab. Actually the approximate 3-to-4 lattice mismatch of the (111) planes of lead and copper allows for a commensurate growth of ultra-thin Pb films where three Pb atoms are placed on four Cu atoms so that a 3×3 Pb supercell is formed on the 31-layer Cu(111)-(4×4) substrate (Figure 4(a,b)). This structure, which is found to be compressed compared to an ideal Pb(111) overlayer by 2.6 %, has therefore 9 ACS Paragon Plus Environment


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a)

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b)

[112]

y

fcc

hcp 7

8

9

top 4 1

5 2

6 3

[110]

x

c)

d)

5 1

2

3

6

4

1

4

[112] MCu

[111] x

[110]

5 4

MPb

z

9 6

4

7

8

M

7

[110] K

K Pb K Cu

Figure 4: The commensurate structure of one p(3×3)-Pb monolayer on a p(4×4)-Cu(111) substrate. (a) A hard sphere model (top view). Pb atoms are numbered by 1 to 9. The white rhombus marks the unit cell. (b) Top view of the interface structure: the 16 Cu atoms of the top substrate layer are distributed in the unit cell over five inequivalent positions depending on the positions with respect to the NN lead atom (large dotted circles). The corresponding equilibrium relaxations in the vertical z = [111] direction of Pb atoms (labelled as in (a)) and top-layer Cu atoms (same grey scale as in (b)) for the four Cu-atom rows in the x direction are shown in (c). This commensurate phase implies a 2.6% compression of the Pb monolayer, which is accommodated by a pronounced outward relaxation of Pb atom 5 (“adatom”).(d) The irreducible parts of the surface Brillouin zones ¯ Cu , Γ¯ M ¯ Cu ), Pb(111) (Γ¯ K ¯ Pb , Γ¯ M ¯ Pb ), and for the p(3×3)-Pb and p(4×4)-Cu(111) for Cu(111) (Γ¯ K ′ ′ ¯ ¯ ¯ ¯ superstructure (ΓK , ΓM )


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a unit cell containing nine Pb atoms and sixteen of the smaller Cu atoms per layer. As appears in Figure 4(c)) the compression of the Pb monolayer is accommodated by a pronounced outward relaxation of the Pb atoms “5”, which form a sort of adatom superlattice with a fairly weak coupling to the substrate. In the following calculations for 2, 3 and 4 MLs the fcc(111) surface stacking sequence (ABC) with the same monolayer periodicity is adopted. Although it is expected that by increasing the film thickness the Pb surface rapidly tends to its own periodicity, it is however noted that HAS spectra from the 4 ML film can still show the weak features of the RW folded into the BZ of the supercell (see Figure 5(a) of Ref. 25). To obtain the equilibrium configuration of the system at zero temperature the atomic positions of both adsorbates and substrate atoms were relaxed using a standard molecular-dynamics technique based on the constructed EAM interaction potentials.

Atomic structure and relaxation The optimized atomic structure of the clean 31-layer Cu(111) slab (with no overlayer) differs slightly from the ideal one. The two outermost interlayer spacings exhibit a small contraction relative to the bulk distance: ∆12 = −1.05% and ∆23 = −0.1%. These values agree well with both the experimental data obtained by medium-energy ion scattering, 45 ∆12 = −1.0 ± 0.4% and ∆23 = −0.2 ± 0.4%, and the ab initio result, 46 ∆12 = −0.9 ± 0.4%. The bond length between the nearest Pb–Pb neighbors (NN) is equal to 3.408 Å which is 2.6% shorter than the NN distance in bulk Pb and agrees with the LEED data 27 for the reported commensurate p(4×4) superstructure (2.65%). Then the adatoms together with Cu substrate atoms were allowed to move according to the calculated forces until the equilibrium positions were achieved. A hard sphere model and the interface of the calculated Cu(111)-p(4×4)-Pb structure are shown in Figure 4 (a) and (b), respectively. As appears in Figure 4 (b) four of the nine Pb atoms in the unit cell are symmetrically inequivalent: atom 1 sits on top of a Cu atom, atoms 5 and 9 occupy hcp and fcc hollow sites, respectively, while the other six Pb atoms sit in asymmetrical but equivalent bridge positions. The three Pb atoms each are surrounded by six NN lead atoms lying at displaced 11 ACS Paragon Plus Environment


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top, hcp and fcc hollow positions so that a pseudo-hexagonal structure is formed. Upon relaxation, the NN Pb–Pb distance changes by less than 1% and varies between 3.368 Å and 3.428 Å. The number of symmetry inequivalent Cu atoms is five per layer. In the top substrate layer, the atoms turn out to be differently coordinated to neighboring lead adatoms and, therefore, can be classified according to the position of the NN lead atom. The first group (1-Cu) is formed by the atoms lying under on-top Pb atoms, the second (2-Cu) and the third groups (3-Cu) are formed by the Cu atoms lying under Pb at hcp- and fcc-sites, respectively, and the fourth (4-Cu) and the fifth (5-Cu) groups consist of the Cu atoms which are under Pb at a displaced top-site position and at a bridge-site, respectively (see Figure 4 (b)). Table 1: Vertical displacements of atoms with respect to an average plane and buckling amplitudes, (∆max z), in the Pb adlayer and in the first three substrate layers of Cu: Cu1−3 . The numbering of Pb atoms is according to Figure 4(a). Positive (negative) values denote outward (inward) displacements relative to the layer average plane. To visualize each row contains Cu atoms lying under the corresponding Pb atom. Pb adlayer 1 5 9 2,3,4 6,7,8

group Cu1 Cu2 Vertical displacements (in Å)

-0.021 +0.027 +0.014 -0.0004

(1-Cu) (2-Cu) (3-Cu) (4-Cu) (5-Cu)

-0.185 +0.075 +0.067 -0.071 +0.062

Cu3

-0.090 -0.042 +0.066 +0.016 +0.056 +0.045 -0.022 -0.017 +0.017 +0.014

Buckling amplitude ∆max z (in Å) Calc. Expt. (a)

a c

0.048 0.07 b 0.20

0.260 0.25 0.24

0.156 0.13 0.18

0.087 0.07 0.09

EMT, Ref. 28; (b) STM, Ref. 28,31; (c) LEED, Ref. 31.

Table 1 shows the relaxed vertical displacements of the atoms. Since the displacement depends on the atom position the layers exhibit buckling. The reconstruction of the copper substrate extends as deep as three atomic layers. The calculated buckling amplitudes of the Cu layers agree well with the values determined by means of LEED. 31 The corrugation of the Pb layer, as seen in Figure 4 (c), is mainly due to the prominence of atoms 5 and 9, and is well reflected in the sharp 3/4 and 12 ACS Paragon Plus Environment

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¯ azimuth. 9 Note that the 1/4 satellite peaks observed in the HAS diffraction patters along the [121] large inequivalence of the Cu atomic rows marked by stars in Figure 4 (b), resulting from the calculations and responsible for the large buckling of the Cu layers induced by the Pb adsorption, also accounts for the (quasi-) absence of the 1/2 satellite in the HAS diffraction pattern. 9 The calculated corrugation amplitude of 0.024 Å (defined as one half of the maximum-to-minimum buckling of 0.048 Å given in Table 1) is twice as large as the HAS corrugation amplitude for the semi-infinite Pb(111) surface (0.012 Å), 9 the difference being therefore due to the buckling of the monolayer. The theoretical result is also fairly consistent with the STM experimental data 28,31 (corrugation amplitude ∼0.035 Å) whereas the LEED analysis 31 gives also for the Pb ML a buckling of 0.20 Å (corrugation amplitude 0.1 Å), which is quite larger than HAS, STM and theoretical data. Pb−Cu Since the layers are strongly buckled, the average interlayer spacings were defined: d12 Cu−Cu Cu−Cu = 2.48 Å, d23 = 2.08 Å and d34 = 2.09 Å. In the substrate, the average vertical distances Cu = 2.087 Å. The value of the average spacing are very close to the value for the bulk copper, dbulk

between the lead and top substrate layer is very close to the experimental one (2.44 Å). 31 The calculated bond lengths between copper atoms are in the range of 2.48–2.58 Å which is rather close to the value in bulk Cu (2.556 Å). Most bond lengths between lead and the NN substrate atoms (2.85–2.93 Å) are found to be a bit smaller (3–5%) than the atomic hard-sphere radius (3.02 Å). This does not hold for two Pb/Cu pairs, top Pb/Cu-1 and distorted Pb/Cu-4, where the NN distances (2.63–2.69 Å) are smaller by 11–13% compared to the hard-sphere model. The same features were observed experimentally. 31 With increasing film thickness the character of the relaxation in the substrate is not changed. The distance between the surface and subsurface lead layers is always contracted. For a 4 ML film, the relaxation of the lead films are ∆12 = −1.3% and ∆23 = +0.6%, relative to the ideal Pb(111) interlayer spacing. The value of ∆12 is smaller compared to the contraction of the outermost distance (∆12 = −3.5 ± 1%) obtained experimentally for the Pb(111) surface 47 due to the conserved lateral compression (by 2.6%) of lead layers. The value of buckling amplitude in the surface Pb layer, 0.003Å, is consistent with a smooth surface observed in the experiment. 1



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Phonons: 1 ML Pb/Cu(111)-p(4×4) The two-dimensional (2D) Brillouin zone (BZ), corresponding to the p(4×4) structure, is 16 times smaller than that of the conventional (1×1) Cu(111) unit cell, and 9 times smaller than that of Pb(111) clean surface (see Figure 4 (d)). This implies that different points of the (1×1) BZ are folded into the same points of the smaller p(4×4) BZ according to the following scheme:   1  ΓM, M, 34 ΓK → Γ¯  2   1 1 1 ¯′ (4 × 4) 2 KM, 4 ΓK, 2 ΓK, K → K      1 ΓM, 3 ΓM, 3 KM, 3 ΓK → M ¯ ′, 4 4 4 8 For a straightforward comparison of the calculated dispersion curves with the experimental HAS data it is convenient to plot both within the reduced BZ. Due to the large number of backfolded branches and the complexity of the folding scheme, the experimental data points are more conveniently compared to the branches which have a significant intensity projected onto the film layer. The calculated phonon local densities of states (LDOS) for 1 ML of Pb on Cu(111) are presented in Figure 5 for (a) the Pb adlayer and (b) the first substrate layer immediately below the lead layer. Figure 5 (c) shows for comparison the LDOS for the topmost layer of the clean Cu(111) surface, in order to illustrate the effect of the Pb layer on the substrate phonon spectrum. The latter shows features quite typical for the (111) fcc metal surface which is characterized by two prominent peaks. The one at lower energies is determined to a considerable degree by the Rayleigh wave (RW) with mainly SV atomic displacements. The calculated vibrational energies for the RW at the ¯ (13.2 meV) and K ¯ (14.4 meV) agree with both experimental data 48–50 high-symmetry points M (13.24 and 14.06 meV, respectively) and with the results of first-principle calculations (13.5 and 15.3 meV). 50,51 A critical comparison between EAM results for Cu(111), the multiple-expansion method and DFPT is in the review by Benedek et al. 52 The higher energy peak is related to in¯ point, the surface localized mode is found at 27.8 plane polarized gap phonon modes. At the M meV while the experiment 48 gives 26.1 meV. The effect of the surface on the surface phonons is 14 ACS Paragon Plus Environment

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Figure 5: 1 ML Pb/Cu(111)-p(4×4) system: calculated total phonon local densities of states (LDOS) for (a) the Pb adlayer and (b) the first substrate layer (S), compared with that of (c) the surface layer of the clean Cu(111) surface. LDOS’s for in-plane longitudinal and shear-horizontal (L+SH) components, and for the shear-vertical (SV) component of the vibrational amplitude are presented separately. The RW peak corresponding to the zone-boundary RW energy of the clean surface is shifted down to 10 meV in panel (b) due to the loading effect of the Pb layer, with a strong increase of the L component. Similarly the zone-boundary longitudinal resonance (LR) of the clean surface, also shifted down by the loading effect, acquires a SV component.



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Figure 6: (a) Wavevector-resolved phonon LDOS and (b) experimental and calculated dispersion ¯ ′ symmetry direction for 1 ML of lead on Cu(111). (c) and (d) are the same curves along the Γ¯ M ′ ¯ symmetry direction. In (a) and (c) the LDOS are from top to bottom for L, SH and SV for the Γ¯ K polarizations. They are compared with the HAS experimental dispersion curves measured at 100 and 140 K in panel(b) and (d) where the full lines correspond to the LDOS maxima for the sagittal components L and SV (capital letters label corresponding modes in (a) and (b)) The dashed line curve in (c) and (d) shows the predicted longitudinal resonance of the Cu substrate (L(Cu)), which extends up to 12 meV. Although no feature is found in the LDOS for L(Cu), some experimental points are seen to correspond to L(Cu) via the quantum sonar effect. 25 The same can be said for ¯ which is clearly observed in the experiment but missing in the LDOS in the lowest SV branch Γ¯ A, both directions.


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confined to the first two atomic layers. 53 Upon lead adsorption the vibrations of Cu atoms change markedly, though the surface phonon modes which are inherent to the clean surface remain, including the RW mode, which is however shifted down to 10 meV due to the loading effect of the Pb layer, with a strong increase of the L component. Similarly the zone-boundary longitudinal resonance (LR) of the clean surface 50 is also shifted down by the loading effect, acquiring a SV component. The large mixing of SV and L components of substrate the RW and LR modes produced by the Pb overlayer is essentially due to the loss of planar inversion symmetry of most surface Cu atoms in their p(4×4) unit cell. The alterations due to the inequivalent positions of Cu atoms which are differently coordinated to neighboring lead adatoms also explain the splitting of the 27 meV peak in the LDOS of in-plane polarized vibrations (the zone-boundary S2 mode resonance 50 ). In this case the energy is much above the maximum of the Pb spectrum (Figure 5(a)): the Pb atoms remain silent, thus producing no loading effect nor polarization mixing, but add force constants to the 1st-layer Cu atoms, thus shifting upward the Cu phonon spectrum edge (a small peak at ∼31 meV). ¯′ Figure 6 (a) shows momentum-resolved phonon LDOS (MR LDOS) calculated along the Γ¯ M symmetry direction for 1 ML of lead on Cu(111), for the shear-vertical (SV), shear-horizontal (SH) and longitudinal (L) polarizations. The densities are normalized to unity. They consist of a set of ridges whose maxima depict the complex network of dispersion curves originating from the large number of atoms in the surface unit cell. The dispersion curves for sagittal modes, encompassing SV and L components, are plotted in panel (b) with their extremities labelled by capital letters so as to show the correspondence with the ridges of panel (a). The comparison with the experimental data measured at 100 and 140 K shows an overall good agreement. The HAS data at 100 K are apparently less dispersed than those at 140 K and distinctly resolve in both symmetry directions ¯ However the calculated flat branch DE of SV the two optical branches starting from C and D at Γ. polarization appears to be about 10% below the experimental one. This also hold for the modes of L polarization around C. The 140 K data for optical modes are somewhat softer and in slightly better agreement with calculation. On the other hand there is a good agreement between theory



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and experiment for the folded acoustic SV branch corresponding to the Rayleigh wave (RW in Figure 1 and Figure 2) and running from 0 to A, then from A to D. For this branch the calculated LDOS is very weak as compared to experimental observation, showing a clear example of the quantum sonar effect. 25 Thus the Pb atomic displacements in the Rayleigh wave are quite small, being appreciable only at the zone boundary, whereas the displacements of the substrate atoms are comparatively large, and so are the associated surface charge density oscillations which cause the inelastic scattering of He atoms. Note that in general in a clean surface the lowest surface acoustic branch is the RW wave and has a quasi-SV polarization. The calculation shows however that for a Pb ML on copper, the lowest acoustic branch is SH in both directions with an almost constant DOS from the zone boundary (B) down to Q = 0. ¯ ′ also holds for the direction Γ¯ K ¯ ′ (Figure 6 (c,d)). There is an overall All what is said for Γ¯ M satisfactory agreement in both symmetry directions between EAM calculations and HAS data, which allows to unravel the complex set of experimental branches and safely assign them to various phonons of the Pb overlayer. A few HAS data-points above 7 meV are seen to correspond to the longitudinal resonance of the Cu substrate (dashed line), which apparently does not produce any contribution to the DOS projected onto the Pb monolayer. This is interpreted as another example of the quantum sonar effect. 25

Phonons: 2 ML Pb0.8 Tl0.2 /Cu(111)-p(4×4) ¯ direction The HAS dispersion curves of 2 ML Pb0.8 Tl0.2 /Cu(111) measured at 100 K in the Γ¯ K have been reported in Ref. 2. Figure 7 (a) shows two examples of HAS energy-gain spectra measured at two different incident angles Θi and incident energy Ei = 22 meV, and Figure 7 (b) the dispersion curves after folding into the p(4×4) Brillouin zone. Indexed greek letters show the correspondence between HAS peaks in panel (a) and the phonon branches in panel (b). In this new analysis a few weak features have been identified as associated to the an additional optical branch

ε2 , discussed in the following comparison with the EAM calculations. Since the masses and force constants of bulk Pb and Tl are quite similar, a reasonable approx-


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imation to the dynamics of 2 ML of the Pb0.8 Tl0.2 alloy on Cu(111), also forming a 4×4 superstructure, is that of a monoatomic bilayer with EAM force constants and masses suitably averaged over the two different atomic species. Although this model is sufficient to provide a stable bilayer with real phonon frequencies, the following analysis shows that the substitution of Pb with Tl also yields some reduction of the compressive stress, so that the additional change of force constants should be considered.

¯′ Figure 7: HAS experimental data 2 for the 2 ML Pb0.8 Tl0.2 /Cu(111)-p(4×4) at 100 K along the Γ¯ K symmetry direction: (a) two examples of energy-gain spectra measured at two different incident angles Θi and incident energy Ei = 22 meV; (b) the set of the dispersion curves after folding into the p(4×4) Brillouin zone. Indexed greek letters show the correspondence between HAS peaks in panel (a) and the phonon branches in panel (b) and with the EAM theoretical branches in Figure 8. ¯ ′ direction are shown The calculated LDOSs projected onto the first surface layer along the Γ¯ M for the three displacement components in Figure 8 (a). The comparison with the LDOS in the same direction projected onto the second (interface) Pb layer, shown in Figure 8 (c), allows to learn about the degree of localization of the modes observed with HAS. For example the branch ε1 is localized on the surface layer with an L polarization and has no amplitude on the 2-nd interface layer, except around the zone center where it is however SV. On the contrary the optical branch ε2 , also with a L polarization, is localized on the 2-nd layer, with a complex dispersion similar to that of ε1 but about 1.5 meV above. These are the embryos 19 ACS Paragon Plus Environment


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Figure 8: Wavevector-resolved phonon LDOS projected on the surface plane, calculated along the ¯ ′ and (b) Γ¯ K ¯ ′ symmetry directions, and (c) Γ¯ K ¯ ′ LDOS projected on the interface layer for (a) Γ¯ M 2 ML Pb0.8 Tl0.2 /Cu(111)-p(4×4). The L, SH and SV polarizations are shown from left to right panels in (a), (b) and (c). Indexed greek letters show the correspondence of some salient modes with the experimental data in Figure 7.


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of the surface and interface branches ε1 and ε2 observed by HAS in thicker Pb films 24,25 (see next Subsection). Also the branch β1 of SV polarization has a very strong intensity in the second layer and a weaker one in the surface layer around the zone center. On the other hand its L component acquires a strong intensity. It appears that also for a bilayer the lowest acoustic branch has a SH polarization. The RW is slightly above, ending at about 3 meV at the zone boundary. It is interesting that the ¯ ′ but rapidly fade away in SV components of the RW are fairly strong at the zone boundary M approaching the zone center, while its L component survives down to Q = 0. This is at odds with the RW in clean surfaces, where the polarization is elliptical with a prevalent SV character over the entire zone. ¯ ′ symmetry direction are shown in Figure 8(b) for The calculated first layer LDOSs in the Γ¯ K ¯ K ¯ ′ direction are the three polarizations. The branches marked by indexed greek letters in the Γassociated with the corresponding experimental branches shown in Figure 7. For comparison the ¯ ′ direction are shown in FigLDOS projected on the second (interface) layer for the same Γ¯ K ure 8(c). As expected the interface LDOSs bear the signature of the complex network due to the substrate dispersion curves, which give instead a negligible intensity on the surface LDOS. The most relevant feature is however the extension up to 11.5 meV of the interface LDOS spectrum for the in-plane components L and SH (just as the 1 ML LDOS), whereas the surface LDOS extends up to only about 9.5 meV. This is clearly accounted for by the larger reduced mass of the Pb-Pb pair with respect to that of the Cu-Pb pair which determines the energy of the highest interface modes. Nevertheless the surface LDOS for SV polarization shows weak signatures of the interface ¯ ′ direction (Figure 8(a)). optical branches above 10 meV, as also seen in the Γ¯ M Another important feature appearing in the interface LDOS is the increasing prominence of the RW SV component, which instead rapidly fades out in the surface LDOS at small wavevectors. As already commented for the 1 ML case, the subsurface oscillation associated with the RW produces however intense surface charge density oscillations, which permits to observe the RW branch by HAS down to near the zone center. This is clearly seen in the HAS dispersion curves for the 2



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¯ ′ direction, 2 and plotted in ML Pb0.8 Tl0.2 /Cu(111)-p(4×4) film measured at 100 K along the Γ¯ K Figure 7 after folding into the reduced surface BZ. As expected the number of calculated phonon branches of the film are far more than those which can be resolved in HAS experiments, despite its comparatively large resolution. To establish a correspondence between the observed phonon branches and the most prominent theoretical ones, the same indexed greek letters of Figure 7 are ¯ ′ . The used in Figure 8(b,c) to label the corresponding phonons at the symmetry points Γ¯ and M five experimental branches identified in Figure 8(b,c) as α1 , β1 , δ1 , ε1 , and ε2 , are not the most prominent LDOS features, but should probably have a large e − p interaction. The agreement of the calculated phonon energies at the symmetry points with the HAS data is reasonable, but the ¯ ′ for the SV mode α1 (3.8 meV) is considerable stiffer than the experimental energy calculated at M value (3.0 meV), while that for the L mode β1 (4.7 meV) is somewhat softer than in experiment (5.2 meV). This indicates that modeling the substitution of Pb with Tl atoms with a simple weighed average over the masses and force constants of Pb and Tl lattices, as assumed in the model, is not sufficient, because the lattice periodicity imposed by the substrate is unchanged and the compressive stress is reduced due to the smaller size of Tl ions.

Phonons: 3 ML and 4 ML Pb/Cu(111)-p(4×4) The phonon dispersion curves of the 3 and 4 ML Pb films on Cu(111) measured with HAS at a temperature of 95 K have been reported in Ref. 9 and analyzed theoretically with a force constant model. A subsequent DFPT analysis for a rigid substrate model permitted to link the observed inelastic HAS intensities from single surface modes to the respective electron-phonon coupling strengths. 24,25 When the substrate dynamics is included, the actual commensurate p(4×4) structure needs to be considered and the experimental dispersion curves are conveniently folded into the p(4×4) Brillouin zone. The phonon branches measured at 140 K along the symmetry directions are shown in Figure 9 for 3 ML and Figure 10 for 4 ML, together with the previously reported branches at 95 K. 31 The indexed greek letters label the branches according to the conventions of Ref. 9 (in some case different from those used in Ref. 24,25). The comparison between the data at


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two different temperatures permits to identify interesting anomalies in the temperature dependence, as will be discussed below.

Figure 9: Left panels: The HAS dispersion curves for the 3 ML Pb(111)/Cu(111) measured at 95 K 35 (black triangles) and 140 K 34 (red points) are plotted after folding into the p(4×4) BZ for ¯ ′ and the Γ¯ K ¯ ′ directions. Right panels: Wavevector-resolved phonon EAM calculated both the Γ¯ M LDOS projected on the surface plane for the two orthogonal L and SV displacement components ¯ ′ (above) and Γ¯ K ¯ ′ (below). Color codes as in Figure 8. along the symmetry directions Γ¯ M For an interpretation of the HAS data the EAM calculated dispersion curves for the 3 and 4 ML are reported aside in Figure 9 and Figure 10, respectively, and represented with their wavevectorresolved LDOS projected onto the first surface layer for the SV and L components along the symmetry directions. The SH components are not requested for the analysis of HAS data and have been omitted. The calculated LDOSs form a dense set of branches, hard to compare to experi23 ACS Paragon Plus Environment


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Figure 10: Same as Figure 9 for the 4 ML Pb(111)/Cu(111) film.


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ment, especially in view of the fact that the phonon observed by HAS are those that produce the largest surface charge density oscillations and not necessarily those with the largest LDOS. Nevertheless an assignment of the observed branches to the calculated ones can be done at a good level of confidence, as reported in Table 2 (see also Figure 11) for the observed phonons at 95 K at the symmetry points. As expected the lower energy part of the surface-projected spectrum receives the largest contribution from the SV modes, while the higher energy part, besides the acoustic L surface resonance, includes optical branches of prevalent L polarization. A better insight is offered by the LDOSs projected onto the bottom (interface) Pb plane for the 3 and 4 ML films (Figure 12). The comparison with the HAS data reveals some interesting features. One concerns the dispersion of the lowest α1 mode, starting at Q = 0 slightly below 1 meV and more evident in the interface LDOS. The fact that this energy is finite and about the same for 4 and 3 ML, and also for 2 ML (Figure 7), indicates that the α1 branch, evolving at larger Q and thickness into the Pb(111) RW, actually is at Q = 0 a SV interface mode, possibly the one known in the continuum elastic limit as Stonely wave. 55 Another intriguing feature concerns the polarization of the highest optical modes. The ε1 mode, having a strong L character in the first plane, at the interface shares a weaker amplitude between the L and SV components. On the contrary the ε2 mode, only having a weak SV intensity in the first layer, is a strong L resonance at the interface in both 3 and 4 ML films. The DFPT calculations previously reported for 3 ML on a rigid substrate, 24,25 and represented on the extended BZ, predict for both ε1 and ε2 modes a dominant SV polarization up to 2/3 of the BZ, then a conversion to a dominant L polarization up to the BZ boundary due to strong hybridization with the L acoustic branch. Due to the folding imposed by the substrate geometry the L portions of the two optical branches now start from Q = 0 at energies in agreement with HAS data. A detailed comparison between a DFPT calculation including spin-orbit coupling (SOC) for 3 ML and the surface and interface LDOS calculated with EAM for the SV and L components, as well as the whole set of HAS data, is shown in Figure 13 after folding into the reduced BZ. This allows to appreciate another important effect of substrate dynamics, concerning the localization



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Table 2: Comparison of the phonon energies (in meV) calculated at the symmetry points of the p(4×4) BZ by EAM including the substrate dynamics, with the calculated DFPT energies (with SOC for 3 ML) for a rigid substrate and with the experimental energies for 3 and 4 ML Pb/Cu(111)-p(4×4). In the last three columns the corresponding energies for the semi-infinite Pb(111) calculated ab-initio with LDA, including (SOC) or excluding (noSOC) spin-orbit coupling, and measured with HAS. The indexed greek letters labelling the experimental branches at the symmetry points are those used in Figs. 9-12 and help showing the correspondence with the calculated branches. Modea

EAM

Γ¯ α1 Γ¯ γ1 Γ¯ τ1 ,ρ1 Γ¯ ′ α1 Γ¯ ′ σ1 ,β2 Γ¯ β2 ,β1 Γ¯ ′ τ1 ,σ1 ,β1 Γ¯ ′ δ1 ¯ Γ¯ ′ ε1 Γ, ¯ Γ¯ ′ ε2 Γ, ¯ ′ α1 M ¯ ′ α1⊥ M ¯ ′ σ1 ,α2 ,σ2 M ¯ ′ α1 M ¯ ′ β1 ,σ1 M ¯ ′ γ1 M ¯ ′ δ1 ,σ1 ,ρ1 M ¯ ′ ε1 M ¯ ′ ε1⊥ M ¯ ′ ε2 M ¯ ′ α1⊥ K ¯ ′ α2 K ¯ ′ α1 K ¯ ′ γ1 ,σ1 K ¯ ′ β1 K ¯ ′ δ1 ,γ1 K ¯ ′ ε1⊥ K ¯ ′ ε2 K

0.8 2.2 2.9 3.9 5.5 – 6.8 8.1 8.7 10.0 2.5 2.7 3.4 4.5 4.0 5.2 6.3 8.6 9.3 10.1 2.6 – 4.1 4.5 5.5 6.4 9.4 –

3 ML DFPT Exp.c SOCb 95 K 0.9 ≈1.0 2.6i 2.3 i 3.8 2.7 3.8 3.5 5.0 5.6 – – 6.5 6.8 8.0 8.2 8.4 8.5 9.7,9.8 10.1,9.7g 2.6 2.4 2.6 2.9 4.1 3.6 4.4 4.5 4.2 4.2 5.3 5.6 6.0 6.2 8.8 8.9 9.5 9.2 9.8 – 2.2 2.3 – – 3.8 3.7 4.6 4.5 5.4 5.3 6.4 6.5 9.2 9.5 – –

EAM

4 ML DFPTd

Exp.c

LDAe

0.7 1.6 2.8 3.8 5.2 6.1 6.4 8.1 8.4 9.1,9.8 2.3 3.0 3.9 4.3 4.8 5.7 6.1 8.7 9.0 9.8, 10.4 2.4 3.2 3.9 4.4 5.5 6.0,6.4 8.3,9.1 10.2

0.8 1.6 2.5 3.5 5.6 5.6 6.4 8.0 8.9,9.5 10.0 2.4 3.0 3.8 4.2 5.0 5.7 6.3 8.3 9.2 – 2.2 3.0 3.8 4.2 5.4 6.3,7.0 8.3 10.0

95 K ≈0.7 – – 3.5 5.5 6.1 6.4 ≈8.0j 8.5 9.2,9.8 2.1 2.8 3.8 4.4 4.9 – 6.0 8.6 9.2 9.8h 2.2 2.8 3.6 4.3 5.4 6.4 8.7, 9.2 9.9h

SOC 0 – – 3.7 – – – – – 9.0 – – 3.1 4.6 – – – 7.5 9.0 9.6 – – 3.7 4.5 5.3 6.1 7.9 9.1

Pb(111) LDAe noSOC 0 – – 4.3 – – – – – 9.8 – – 4.0 5.4 – – – 8.3 9.8 10.4 – – 4.3 5.1 6.0 6.8 8.6 9.7

Exp.f 95 K 0 – – 3.3,3.5 5.3 – 6.2 8.0 9.0,9.4 – 2.0 2.9 3.3 4.0 4.8 7.3 5.9,6.1 9.2,9.5 9.3 – 1.8 2.3 4.0 5.0 5.8 6.4 9.2 –

Γ¯ refers to the zone center, Γ¯ ′ to modes which are folded into the zone center. Indexed greek letter labels as in the original Ref. 9. Since the folding yields several avoided crossings and mode mixing, the folded modes may be associated with more than one unfolded branch and more labels are indicated. The present labels may occasionally differ from those adopted in Ref. 25 on the basis of similarities in polarization and localization resulting from DFPT ¯ M ¯ ′ direction): DFPT with SOC on a rigid substrate; (c) Present calculations; (b) Present work (Figure 13 for the Γevaluation of data in Refs. 34,35; (d) From Ref. 25: DFPT calculation without SOC on a rigid substrate; (e) From Ref. 54; (f ) From Ref. 9; (g) In Ref. 25 two different sets of data from Refs. 34 and 35 have been plotted for 3 ML and this 26 (i) These modes are strongly stiffened by the rigid substrate; (j) mode has been labelled ε2 ; (h) 140ACS K data; Paragon Plus Environment ¯ K ¯ ′ direction. Extrapolated from the Γ(a)

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4 ML

3 ML

'

Phonon energy (meV)

10

'

8

' '

6

'

4

'

2 0 '

Phonon energy (meV)

10

' '

8 '

6

' '

4

' '

2

' '

10

Phonon energy (meV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


' '

8

'

6

' '

4

' '

2

' EAM

DFPT

Expt.

EAM

DFPT

Expt.

Figure 11: Graphical representation of the data reported in Table 2 for a comparison of the HAS data for 3 and 4 ML films to the calculated EAM and DFPT phonon energies.



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Figure 12: Comparison of the HAS dispersion curves for 3 ML (above) and 4 ML (below) Pb films on Cu(111) along the symmetry direction of the reduced BZ with the LDOS projected onto the interface plane calculated with EAM for SV and L polarizations. The projection on the interface plane yields, besides the intense phonon modes of the film, a complex network of weaker dispersion curves of the substrate slab. The superposition of the experimental HAS dispersion curves at 95 and 140 K shows anomalous temperature effects for certain modes (red labels): the anomaly consists in an increase with temperature of the phonon energy instead of the ordinary anharmonic softening observed for the other branches. The anomalies are attributed to the different thermal expansion of the film and the substrate.


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of the ε1 and ε2 modes: while on a rigid substrate ε1 was localized at the interface and ε2 at the surface, now the interface branch runs above the surface-localized branch. In any case, despite their L character, both modes produce a large oscillation of the surface charge density, which explains their large HAS inelastic amplitudes. Moreover the observation of the interface mode by HAS up to about 10 ML films 25 confirms its strong e − p nteraction and its relevant role in the superconductivity of supported ultrathin Pb films. A quantitative comparison of the present EAM results with experiment and DFPT calculations for 3 (with SOC) and 4 ML (no SOC) and for the semi-infinite Pb(111) surface, with and without spin-orbit SOC, is given in Table 2 and graphically illustrated in Figure 11.

Figure 13: Comparison between EAM calculations including the substrate dynamics with a DFPT calculation with spin-orbit coupling (SOC) (center) for the 3 ML-Pb/Cu(111) film on a rigid sub¯ ′ direction. The EAM LDOS projected onto strate, and with HAS data (blue symbols) along the Γ¯ M the top and interface planes are shown on the left panels for SV polarization, and on the right panels for L polarization. The indexed greek letters label the experimental branches at the symmetry points and show the correspondence with the calculated branches. In general Table 2 (Figure 11) shows that some experimental branches are better reproduced by the DFPT calculations, others by the EAM, depending on which of the EAM and rigid-substrate approximations is less important. The SOC yields some softening of the phonon frequencies with 29 ACS Paragon Plus Environment


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respect to those calculated without the SOC. This general effect can be clearly appreciated by comparing the columns LDA-SOC and LDA-noSOC for Pb(111) in Table 2. Also for 3 ML the DFPT-SOC phonons energies however show in general some softening with respect to the EAM energies, with notable exceptions for the lowest modes where the effect of substrate rigidity prevails. The rigid substrate model cannot account for the lowest surface acoustic branches of the layer when their penetration length exceeds the film thickness. Also the modes with the largest amplitude at the interface Pb layer are expected to be strongly modified when the substrate dynamics is allowed. As seen in Figure 13 the folded DFPT ε2 branch is rather flat and does not account for the ¯ ′ . This is in clear disagreement observed small dispersion from almost 10 meV at Γ¯ to 9 meV at M also with the EAM LDOS projected on the top layer, but is very well reproduced by the EAM LDOS for L polarization projected on the interface layer. The same holds for the 4 ML film for which the ε2 dispersion curve is well reproduced by an intense interface branch (not shown). On the other hand the strong folded RW branch of Cu(111) above 10 meV (EAM-L interface panel) is not seen with HAS – a fact which can be explained by the modest electron-phonon coupling of the Cu(111) substrate which is insufficient to produce detectable charge density oscillations at the surface across three Pb layers. The multiple folding of the phonon branches produced by the matching of the substrate-to-film 4-to-3 periods yields a sequence of fairly flat branches and therefore to a set of corresponding peaks in the total LDOS as well as in the LDOS integrated along a given symmetry directions. The total ¯′ phonon density of states projected on the first Pb layer and the LDOS integrated along the the Γ¯ K direction for 3 ML and 4 ML Pb/Cu(111) calculated with the EAM are plotted in Figure 14(a,c) and Figure 14(b,d), respectively. Their most prominent features are associated with the zone-boundary high-symmetry points and are therefore labelled with the same indexed greek letters of Figure 9 and Figure 10 and Table 2. The LDOS themselves indicate that in both 3 and 4 ML films the phonon branches observed below 6 meV have a prominent SV polarization, those above 8 meV are essentially longitudinal, whereas


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the observed branches in the range 6 to 7 meV have a mixed character. The comparison of the total ¯ ′ direction, Figure 14(c) and (d), permits to qualitatively LDOS with the one projected onto the Γ¯ K ¯ ′ and the M ¯ ′ symmetry points. More specifically, referring to assess the contributions from the K the labels adopted in Ref. 9, the phonon corresponding to the peaks α1 and α2 are mainly SV, those labelled β1 and σ1 have both SV and in-plane components, while the upper optical branches

ε1 and α2 , yielding only weak features in the z-LDOS but strong peaks in the (x,y)-LDOS, have essentially a longitudinal (L) polarization. As shown in Ref. 25, these modes, despite the in-plane motion of the surface atoms, produce a large surface charge density oscillations and therefore a comparatively large intensity in inelastic HAS spectra. 0.4 0.3

3 ML

4 ML

(a)

Pb (xy) Pb (z)

(b)

Pb (xy) Pb (z) ?

0.2 0.1

0.4

a1

0.3

g1

0.2 0.1 0.0 0

(c) GK [110] a2 a1

GK [110]

g1

e1

e2

b1 d1

(d)

e1

b1

?

LDOS (arb. units)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


g1

b2

e2

d1 3

6

9 ?

12

15

3

6

9

12

15

Energy (meV)

Figure 14: The total local phonon densities of states calculated for (a,c) 3 ML and (b,d) 4 ML of lead on Cu(111). Panels (a) and (b) show the total LDOS for the shear-vertical (z) and in-plane (xy) ¯ ′ symmetry direction motion of surface lead atoms. Panels (c) and (d) present the LDOS in the Γ¯ K also for the surface lead atoms. The greek labels have the same meaning as in the previous figures.

Temperature effects The comparison between HAS data at 95 and 140 K shown in Figure 12 for both 3 and 4 ML allows to discern some interesting effects related to the larger thermal expansion of Pb layers with 31 ACS Paragon Plus Environment


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respect to that of Cu layers. As seen in Figure 6 (b) the observed 0-A branch at 100 K is somewhat softer than that at 140 ¯ ′ direction, which would indicate a negative anharmonic K at intermediate wavevectors in the Γ¯ M ¯ ′ direction (Figure 6 (d)). In the absence of buckling effect, whereas no anomaly is seen in the Γ¯ K as described in Figure 4 (c) the thermal expansion would produce an isotropic compressive stress and therefore a similar temperature effect in the two symmetry directions, whereas the buckling has the effect to accommodate the thermal stress more efficiently in the x than in the y direction. Note that the softening of the 1 ML RW with respect to that for 0.5 ML is anisotropic at small Q, ¯ direction, apparently at odds with the thermal effect. This would favor with no effect in the Γ¯ M the loading effect as the main cause of the above RW softening for increasing coverage from 0.5 to 1 ML, rather than the increase of the compressive surface stress. Also for 3 ML (Figure 12 (a)) and 4 ML (Figure 12 (b)) the increase of temperature from 95 to 140 K yields an anomalous stiffening for some branches of SV polarization, especially the lowest one (α1 ) and, at least for 3 ML, a regular softening for the upper modes with a prevalent longitudinal polarization. The effect observed for 1 ML is however reversed for the lowest α1 ¯ ′ direction for branch. Now the anomalous stiffening for increasing temperature occurs in the Γ¯ K both 3 and 4 ML (Figure 12: anomalous branches are encircled), while no effect is seen in the ¯ ′ direction. This change of direction suggests a difference between the thermal stress acting Γ¯ M on the wetting layer and the one affecting the film overlayers. The buckling of the wetting layer, ¯ ′ direction than in the described in Figure 4, may better accommodate the surface stress in the Γ¯ K ¯ ′ direction, but the situation is apparently reversed when further layers are deposited. Γ¯ M More intriguing are the clear anomalies concerning the upper branches (marked by indexed greek letters), observed for 4 ML but not for 3 ML. There is a remarkable stiffening with temperature of the highest interface branch ε2 and probably, but to a minor extent, also for the surface ¯ ′ point and the σ1 branch near optical branch ε1 . Also the upper α1 and δ1 branches towards the K ¯ ′ direction show a sizeable stiffening at higher temperature. It is noted the zone center in the Γ¯ M that the branches δ1 , σ1 and ε2 are mostly L and more intense at the interface, whereas ε1 is also


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mostly L but more intense at the surface. Thus it may be concluded that the longitudinal polarization is responsible for the anomalous temperature effect at the interface, where the different thermal expansion of the two materials causes the largest stress in the film. On the other hand in the 3 ML film these effects are not observed. Only δ1 seems to be slightly anomalous, while the

σ1 , ε1 and ε2 branches show a regular softening at higher temperature. This different behavior may be related to the oscillating relaxation of the interlayer distance with the number of monolayers suggesting a bilayer organization of the layers near the surface. Thus the unpaired interface Pb layer of the 3 ML film can better accommodate to the substrate than in the 4 ML film.

Conclusion In summary it has been shown that the sensitivity of helium atom scattering to both surface and subsurface phonons in conducting materials permits to unravel the complex dynamics of ultrathin films and their interface with the substrate. The perturbing effects of the substrate structure and dynamics, often disregarded in a first analysis of overlayer spectra, have been shown to have important consequences on the energy and polarization of interface phonons. Since the HAS intensities are a measure of their electron-phonon coupling strength, their specific role in the superconductivity of supported ultrathin films can in principle be inferred from HAS spectroscopy. Three major effects are shown to arise from a substrate dynamically coupled to the film and characterized by a larger (but finite) stiffness and a different lattice constant than those of the film: i) The lock-in of the film and substrate periodicities into a commensurate phase yields a folding of the phonon dispersion curves into a smaller surface Brillouin zone with several avoided crossings. HAS spectroscopy is sensitive to the substrate periodicity even for the 3 and 4 ML films, thus providing data-points for most folded branches. ii) The mixing of film and substrate modes, especially at the interface, and the folding into the reduced BZ, strongly affect their polarization, turning for example the optical modes, which are predicted to be mostly SV on a rigid flat substrate, into mostly longitudinal modes. Despite



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their prevalent longitudinal displacements and localization at the buried interface these modes are detected by HAS thanks to their large electron-phonon coupling (quantum sonar effect). 25 iii) The different thermal expansion of the film and the substrate causes an anomalous (negative) anharmonic effect, i.e., a stiffening of some phonon energies for increasing temperature. The effect is attributed to an increase with temperature of the surface stress in the film, and as expected, affects mainly the longitudinal interface modes. However a different behavior is noted between the single wetting layer and the film overlayers for what concerns thermal stress orientation. iv) More specifically the EAM structure analysis reveals that for 1 and 2 ML coverage the known Pb/Cu(111)-p(4×4) structure exhibits a considerable corrugation of the film surface and of the substrate atomic planes. For one lead monolayer a dynamically stable structure is formed with a low-energy stretching mode. This mode results from a strong coupling of the monolayer phonons with the low-energy shear-vertical vibrations of the top substrate atoms. It appears that the longitudinal component of the Pb atom displacements, rather than the shear-vertical one, is coupled to the RW mode of the substrate. For 2 ML the pure lead film is found to be unstable with imaginary frequencies in the phonon spectrum, in agreement with experiment. Starting at 3 ML the film surface becomes smooth and the relaxation corresponds to that of a thick film. The relaxation of the top substrate layer is found to be determined by the interface lead layer and decreases to less than 1 % as the lead film thickness increases. Overall the dynamics of the Pb wetting monolayer on Cu(111) calculated with the EAM for a non-rigid surface is found to agree reasonably well with HAS experiments although the peculiar flat optical branch at about 6 meV is calculated to be about 10 % softer than in experiment. Equally satisfactory is the agreement between the calculation and experiment for 2 ML of the alloy Pb0.8 Tl0.2 /Cu(111)-p(4×4) which, unlike the 2 ML Pb film, is known to be stable and for which HAS data are available. The calculation also shows that about half of the surface phonons are vertically polarized and agree reasonably well with the experimentally observed modes. As in the experiment, highfrequency SV optical modes, known as organ-pipe modes 56 are found at Q = 0. Interestingly


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their energies were found to be in reasonable agreement also with the results of first-principles calculations for free-standing lead films. The calculations for 3 and 4 ML have been directly compared to the previous ab-initio DFPT calculations for a rigid Cu(111) substrate. The comparison for 3 ML of the EAM results with a DFPT calculation including spin-orbit coupling shows reasonable correspondence so as to allow to single out the role of substrate dynamics for both the lowest branch, associated with the substrate RW, and the highest optical branch. The HAS observation of this branch, which is confined at the interface Pb layer, confirms the importance of the quantum sonar effect, which allows HAS to detect deep subsurface modes via the electron-phonon interaction. 24,25

Acknowledgement The work was carried out at the financial support by the grant (8.1.05.2015) from The Tomsk State University Academic D.I. Mendeleev Fund Program, RFBR grant (15-02-02717-a), Fundamental Research Program of the State Academies of Sciences for 2013 – 2020.

References (1) Hinch, B. J.; Koziol, C.; Toennies, J. P.; Zhang, G. Evidence for Quantum Size Effects Observed by Helium Atom Scattering during the Growth of Pb on Cu(111). Europhysics Letters 1989, 10, 341. (2) Braun, J.; Toennies, J. Observation of a Quantum Size Effect in the Surface Electron Density of Thin Lead Films. Surface Science 1997, 384, L858 – L864. (3) Materzanini, G.; Saalfrank, P.; Lindan, P. J. D. Quantum Size Effects in Metal Films: Energies and Charge Densities of Pb(111) Grown on Cu(111). Phys. Rev. B 2001, 63, 235405. (4) Jia, Y.; Wu, B.; Weitering, H. H.; Zhang, Z. Quantum Size Effects in Pb Films From First Principles: The Role of the Substrate. Phys. Rev. B 2006, 74, 035433.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(5) Calleja, F.; Vázquez de Parga, A. L.; Anglada, E.; Hinarejos, J. J.; Miranda, R.; Yndurain, F. Crystallographic and Electronic Contribution to the Apparent Step Height in Nanometer-Thin Pb(111) Films Grown on Cu(111). New Journal of Physics 2009, 11, 123003. (6) Otero, R.; Vázquez de Parga, A. L.; Miranda, R. Observation of Preferred Heights in Pb Nanoislands: A Quantum Size Effect. Phys. Rev. B 2002, 66, 115401. (7) Schackert, M.; Märkl, T.; Jandke, J.; Hölzer, M.; Ostanin, S.; Gross, E. K. U.; Ernst, A.; Wulfhekel, W. Local Measurement of the Eliashberg Function of Pb Islands: Enhancement of Electron-Phonon Coupling by Quantum Well States. Phys. Rev. Lett. 2015, 114, 047002. (8) Yndurain, F.; Jigato, M. P. First Principles Calculation of Localized Surface Phonons and Electron-Phonon Interaction at Pb(111) Thin Films. Phys. Rev. Lett. 2008, 100, 205501. (9) Braun, J.; Ruggerone, P.; Zhang, G.; Toennies, J. P.; Benedek, G. Surface Phonon Dispersion Curves of Thin Pb Films on Cu(111). Phys. Rev. B 2009, 79, 205423. (10) Jałochowski, M.; Hoffman, M.; Bauer, E. Quantized Hall Effect in Ultrathin Metallic Films. Phys. Rev. Lett. 1996, 76, 4227–4229. (11) Kirchmann, P. S.; Rettig, L.; Zubizarreta, X.; Silkin, V. M.; Chulkov, E. V.; Bovensiepen, U. Quasiparticle Lifetimes in Metallic Quantum-Well Nanostructures. Nature Physics 2010, 6, 1735. (12) Chulkov, E. V.; Kliewer, J.; Berndt, R.; Silkin, V. M.; Hellsing, B.; Crampin, S.; Echenique, P. M. Hole Dynamics in a Quantum-Well State at Na/Cu(111). Phys. Rev. B 2003, 68, 195422. (13) Díez Muiño, R.; Sánchez-Portal, D.; Silkin, V. M.; Chulkov, E. V.; Echenique, P. M. TimeDependent Electron Phenomena at Surfaces. PNAS 2011, 108, 971. (14) Kirchmann, P. S.; Wolf, M.; Dil, J. H.; Horn, K.; Bovensiepen, U. Quantum Size Effects in Pb/Si(111) Investigated by Laser-Induced Photoemission. Phys. Rev. B 2007, 76, 075406. 36 ACS Paragon Plus Environment

Page 36 of 42

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(15) Kim, J.; Qin, S.; Yao, W.; Niu, Q.; Chou, M. Y.; Shih, C.-K. Quantum Size Effects on the Work Function of Metallic Thin Film Nanostructures. PNAS 2010, 107, 12761–12765. (16) Brun, C.; Hong, I.-P.; Patthey, F.; Sklyadneva, I. Y.; Heid, R.; Echenique, P. M.; Bohnen, K. P.; Chulkov, E. V.; Schneider, W.-D. Reduction of the Superconducting Gap of Ultrathin Pb Islands Grown on Si(111). Phys. Rev. Lett. 2009, 102, 207002. (17) Sklyadneva, I. Y.; Heid, R.; Bohnen, K.-P.; Echenique, P. M.; Chulkov, E. V. Mass Enhancement Parameter in Free-Standing Ultrathin Pb(111) Films: The Effect of Spin-Orbit Coupling. Phys. Rev. B 2013, 87, 085440. (18) 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.; et al, Superconductivity Modulated by Quantum Size Effects. Science 2004, 306, 1915–1917. (19) Eom, D.; Qin, S.; Chou, M.-Y.; Shih, C. K. Persistent Superconductivity in Ultrathin Pb Films: A Scanning Tunneling Spectroscopy Study. Phys. Rev. Lett. 2006, 96, 027005. (20) Ozer, M. M.; Thompson, J. R.; Weitering, H. H. Hard Superconductivity of a Soft Metal in the Quantum Regime. Nature Phys. 2006, 2, 173 – 176. (21) Qin, S.; Kim, J.; Niu, Q.; Shih, C.-K. Superconductivity at the Two-Dimensional Limit. Science 2009, 324, 1314–1317. (22) Ogando, E.; Zabala, N.; Chulkov, E. V.; Puska, M. J. Self-Consistent Study of Electron Confinement to Metallic Thin Films on Solid Surfaces. Phys. Rev. B 2005, 71, 205401. (23) Yu, D.; Scheffler, M.; Persson, M. Quantum Size Effect in Pb(100) Films: Role of Symmetry and Implications for Film Growth. Phys. Rev. B 2006, 74, 113401. (24) Sklyadneva, I. Y.; Benedek, G.; Chulkov, E. V.; Echenique, P. M.; Heid, R.; Bohnen, K.-P.; Toennies, J. P. Mode-Selected Electron-Phonon Coupling in Superconducting Pb Nanofilms Determined from He Atom Scattering. Phys. Rev. Lett. 2011, 107, 095502. 37 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(25) Benedek, G.; Bernasconi, M.; Bohnen, K.-P.; Campi, D.; Chulkov, E. V.; Echenique, P. M.; Heid, R.; Sklyadneva, I. Y.; Toennies, J. P. Unveiling Mode-Selected Electron-Phonon Interactions in Metal Films by Helium Atom Scattering. Phys. Chem. Chem. Phys. 2014, 16, 7159 – 7172. (26) Zhang, T.; Cheng, P.; Li, W.-J.; Sun, Y.-J.; Wang, G.; Zhu, X.-G.; He, K.; Wang, L.; Ma, X.; Chen, X.; et al, Superconductivity in One-Atomic-Layer Metal Films Grown on Si(111). Nature Phys. 2010, 6, 104 – 108. (27) Henrion, J.; Rhead, G. LEED Studies of the First Stages of Deposition and Melting of Lead on Low Index Faces of Copper. Surface Science 1972, 29, 20 – 36. (28) Nagl, C.; Schmid, M.; Varga, P. Inverse Corrugation and Corrugation Enhancement of Pb Superstructures on Cu(111) and (110). Surface Science 1996, 369, 159 – 168. (29) Nagl, C.; Platzgummer, E.; Haller, O.; Schmid, M.; Varga, P. Surface Alloying and Superstructures of Pb on Cu(100). Surface Science 1995, 331-333, Part A, 831 – 837, Proceedings of the 14th European Conference on Surface Science. (30) Meyer, G.; Michailov, M.; Henzler, M. LEED Studies of the Epitaxy of Pb on Cu(111). Surface Science 1988, 202, 125 – 141. (31) Müller, S.; Prieto, J. E.; Rath, C.; Hammer, L.; Miranda, R.; Heinz, K. Surfactant-Induced Surface Restructuring: (4×4)-Pb/Cu(111). Journal of Physics: Condensed Matter 2001, 13, 1793. (32) Pimpinelli, A.; Villain, J. Physics of Crystal Growth; Cambridge University Press, 1998. (33) Rusina, G. G.; Borisova, S. D.; Chulkov, E. V. Atomic Structure and Phonons of a Pb Ultrathin Film on the Al(100) Surface. JETP Letters 2014, 100, 237–241. (34) Zhang, G. Ph. D. Dissertation; University of Göttingen; Max-Planck-Institut für Strömungsforschung Report, 1991. 38 ACS Paragon Plus Environment

Page 38 of 42

Page 39 of 42

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(35) Braun, J. Ph. D. Dissertation; University of Göttingen; Max-Planck-Institut für Strömungsforschung Report, 1997. (36) Braun, J.; Toennies, J. P. A Helium Atom Scattering Study of Ultrathin Lead Films Grown on Cu(111). Surface Science 1996, 368, 226 – 231. (37) Toennies, J. P.; Winkelmann, K. Theoretical Studies of Highly Expanded Free Jets: Influence of Quantum Effects and a Realistic Intermolecular Potential. J. Chem Phys. 1977, 66, 3965. (38) Brusdeylins, G.; Meyer, H. D.; Toennies, J. P.; Winkelmann, K. Production of Helium Nozzle Beams with Very High Speed Ratios, in 10th Symposium of Rarefied Gas Dynamics, J.L. Potter, Editor.; AIAA, New York, Aspen, 1977. (39) Foiles, S. M.; Baskes, M. I.; Daw, M. S. Embedded-Atom-Method Functions for the fcc Metals Cu, Ag, Au, Ni, Pd, Pt, and Their Alloys. Phys. Rev. B 1986, 33, 7983–7991. (40) Rusina, G. G.; Chulkov, E. V. Phonons on the Clean Metal Surfaces and in Adsorption Structures. Russian Chemical Reviews 2013, 82, 483. (41) Sklyadneva, I.; Rusina, G.; Chulkov, E. Vibrational States on Vicinal Surfaces of Al, Ag, Cu and Pd. Surface Science 1998, 416, 17 – 36. (42) Rusina, G. G.; Eremeev, S. V.; Echenique, P. M.; Benedek, G.; Borisova, S. D.; Chulkov, E. V. Vibrations of Alkali Metal Overlayers on Metal Surfaces. Journal of Physics: Condensed Matter 2008, 20, 224007. (43) Eremeev, S. V.; Sklyadneva, I. Y.; Echenique, P. M.; Borisova, S. D.; Benedek, G.; Rusina, G. G.; Chulkov, E. V. Electron-Phonon Coupling in a Sodium Monolayer on Cu(111). Surface Science 2007, 601, 4553–4556. (44) Johnson, R. A. Alloy Models With the Embedded-Atom Method. Phys. Rev. B 1989, 39, 12554–12559.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(45) Chae, K. H.; Lu, H. C.; Gustafsson, T. Medium-Energy Ion-Scattering Study of the Temperature Dependence of the Structure of Cu(111). Phys. Rev. B 1996, 54, 14082–14086. (46) Bohnen, K. P.; Ho, K. M. Structure and Dynamics at Metal Surfaces. Surface Science Reports 1993, 19, 99–120. (47) Li, Y. S.; Jona, F.; Marcus, P. M. Multilayer Relaxation of a Pb(111) Surface. Phys. Rev. B 1991, 43, 6337–6341. (48) Mohamed, M. H.; Kesmodel, L. L.; Hall, B. M.; Mills, D. L. Surface Phonon Dispersion on Cu(111). Phys. Rev. B 1988, 37, 2763–2765. (49) Harten, J. P., U. Toennies; Wöll, C. Helium Time-of-Flight Spectroscopy of Surface-Phonon Dispersion Curves of the Noble Metals. Faraday Discuss. Chem. Soc. 1985, 80, 137–140. (50) Chis, V.; Hellsing, B.; Benedek, G.; Bernasconi, M.; Chulkov, E. V.; Toennies, J. P. Large Surface Charge Density Oscillations Induced by Subsurface Phonon Resonances. Phys. Rev. Lett. 2008, 101, 206102. (51) Heid, R.; Bohnen, K.-P. Ab Initio Lattice Dynamics of Metal Surfaces. Physics Reports 2003, 387, 151–213. (52) Benedek, G.; Bernasconi, M.; Chis, V.; Chulkov, E.; Echenique, P. M.; Hellsing, B.; Toennies, J. P. Theory of Surface Phonons at Metal Surfaces: Recent Advances. Journal of Physics: Condensed Matter 2010, 22, 084020. (53) Borisova, S. D.; Rusina, G. G.; Eremeev, S. V.; Benedek, G.; Echenique, P. M.; Sklyadneva, I. Y.; Chulkov, E. V. Vibrations in Submonolayer Structures of Na on Cu(111). Phys. Rev. B 2006, 74, 165412. (54) Sklyadneva, I. Y.; Heid, R.; Bohnen, K.-P.; Echenique, P. M.; Chulkov, E. V. Surface Phonons on Pb(111). Journal of Physics: Condensed Matter 2012, 24, 104004.


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(55) Stoneley, R. Elastic Waves at the Surface of Separation of Two Solids. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character 1924, 106, 416–428. (56) Benedek, G.; Ellis, J.; Reichmuth, A.; Ruggerone, P.; Schief, H.; Toennies, J. P. Organ-Pipe Modes of Sodium Epitaxial Multilayers on Cu(001) Observed by Inelastic Helium-Atom Scattering. Phys. Rev. Lett. 1992, 69, 2951–2954.



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Surface Dynamics of the Wetting Layers and Ultrathin Films on a

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