3-Monolayer of Pb on Si(111

Article Cite This: J. Phys. Chem. C XXXX, XXX, XXX−XXX

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Electron−Phonon Interaction in the 4/3-Monolayer of Pb on Si(111): Theory Versus He-Atom Scattering Experiments I. Yu. Sklyadneva,*,†,‡,§,∥ G. Benedek,†,⊥ R. Heid,‡ P. M. Echenique,†,# J. P. Toennies,¶ and E. V. Chulkov†,∥,#,∇ †

Donostia International Physics Center (DIPC), 20018 San Sebastián/Donostia, Basque Country, Spain Institut für Festkörperphysik, Karlsruher Institute für Technologie, P.O. Box 3640, D-76021 Karlsruhe, Germany § Institute of Strength Physics and Materials Science, 634021 Tomsk, Russia ∥ Tomsk State University, 634050 Tomsk, Russia ⊥ Dipartimento di Scienza dei Materiali, Universitá di Milano−Bicocca, 20125 Milano, Italy # 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 ¶ Max-Planck-Institut für Dynamik und Selbstorganisation, Bunsenstraße 10, 37073 Göttingen, Germany ∇ Saint Petersburg State University, Saint Petersburg 198504, Russia

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‡

ABSTRACT: The electron−phonon coupling parameter for a dense phase of a 4/3-monolayer of Pb on Si(111) is derived from the temperature dependence of He-atom scattering (HAS) reflectivity upon cooling from the high-temperature (1 × 1) to the low-temperature ( 3 × 3 R 30°) phase. The obtained value and the phonon dispersion curves of the Pb nanofilm measured with HAS are in excellent agreement with the first-principles calculations when the influence of the substrate and the interface on the interaction of electrons and phonons is taken into account. Consideration of this effect is obviously very important for such small thicknesses as a single Pb wetting layer. An overall strong coupling constant, λ = 0.84 ± 0.07, is found.

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surfaces and thin films through the temperature dependence of the Debye−Waller (DW) exponent, 2W(T).12,14,15 In this paper, the strength of e−ph interaction λ is obtained both for the low-temperature (low-T) 4/3 ML 3 × 3 R 30° and high-temperature (high-T) 1 × 1 phases9,16,17 of Pb on the Si(111) surface using a theoretical analysis of the experimental HAS data obtained during cooling from the annealing temperature of 700−120 K. We show that the DFPT calculation of the dense low-temperature phase,8 including self-consistently the spin−orbit coupling (SOC), yields both phonon dispersion curves and mass enhancement parameter λ in good agreement with the data measured with HAS for the wetting Pb layer at low temperature. To take into account the influence of the substrate on the electronic and vibrational properties of the wetting layer, the theoretical ab initio study considers a combined film-substrate system which obviously should be important for such small thicknesses.

INTRODUCTION

The presence of superconductivity in lead nanofilms grown on Si(111) up to a single wetting layer1−10 raises the question of the origin and strength of the electron−phonon (e−ph) interaction. With a decrease in the film thickness, the influence of the substrate and the interface on the interaction of electrons and phonons manifests itself more strongly and extra decay channels toward the substrate can be opened.4 To study these effects, lead deposited on Si with its well-defined interface due to the low mutual solubility11 is the optimal candidate. Such quasi-two-dimensional lead films have become an important model system for studying superconductivity and dynamical properties of metals in general as well as the e−ph interaction on the nanoscale. A previous combined He-atom scattering (HAS) and density functional perturbation theory (DFPT) study on the phonon dispersion curves of 3−7 ML Pb films on Cu(111)12,13 revealed that the inelastic HAS intensity for a specific phonon (Qν) of the film is proportional to its mode-selected electron−phonon coupling strength λQν. More recently, it was shown that the elastic HAS reflectivity provides reliable values of the global electron−phonon coupling strength (mass-enhancement parameter) λ for conducting © XXXX American Chemical Society

Received: October 16, 2018 Revised: November 30, 2018 Published: November 30, 2018 A

DOI: 10.1021/acs.jpcc.8b10081 J. Phys. Chem. C XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry C

Figure 1. (a) HAS specular intensity (in 106 counts/s with incident He-atom wavevector ki = 5.3 Å−1) measured during cooling of a 4/3 ML Pb film on Si(111) from the annealing temperature of 700 K down to 120 K.16 The decrease with time surface temperature from 700 K to the equilibrium thermostat temperature of 120 K is interrupted at a temperature To by a kink, signaling a transition from a high-T disordered (actually 1 × 1 in the diffraction pattern18) to the so-called striped-incommensurate ordered phase. The transition temperature To ≈ 340 K is much lower than the expected value for the coverage of the 4/3 ML of Pb (the inset shows the phase diagram taken from refs 18 and 19), thus assuming an initial 1 × 1 wetting layer coverage at 700 K with a disordered second layer of additional Pb atoms that act as a reservoir and gradually fit into the wetting layer on decreasing temperature (the process is qualitatively represented by a broken line on the inset). (b) HAS specular intensity, plotted as a function of the surface temperature, shows a slight difference in slope for the two phases. There is a good correspondence between the data for ki = 5.3 Å−1 (lozenges) and two low-T data points for ki = 6.6 Å−1 (filled circles).16

Figure 2. (a) HAS diffraction pattern of the 4/3 ML 3 × 3 R 30° Pb/Si(111) phase along the Si(111) [112̅] direction ([11̅0] for Pb(111)). The 30° rotation of the 4/3 ML Pb film with respect to the 1 × 1 Si(111) substrate (Si1 = the first layer, Si2 = the second layer) is shown in (b): Pb atoms sit either in a top or in a bridge position. This structure is the α-phase proposed in refs 9 and 19 (in ref 9 the labels α and β for the 3 × 3 phases have been interchanged with respect to ref 19). More stable phases recently found in the ab initio calculations of Ren et al.20 are obtained by rigidly shifting the Pb layer so as to bring the top Pb atoms on the T4 (T4-phase) or on the H3 (H3-phase) hollow positions, the latter being somewhat more stable. (c) Brillouin zones (BZs) of the 4/3 ML 3 × 3 R 30° Pb/Si(111) (smaller shadowed hexagons) are correspondingly rotated by 30° with respect to the larger Si(111) BZs. (d) Same as (a) for the Si(111) [11̅0] direction ([112̅] for Pb(111)).

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RESULTS AND DISCUSSION

deposition, corresponding to the completion of the Pb wetting layer. An increase in the reflectivity when the temperature is lowered is attributed to a decrease of thermal disorder, that is, to a decrease of the DW exponent 2W(T). The exponential decrease in surface temperature with time, associated with a constant cooling rate, from the initial temperature of 700 K to the equilibrium thermostat temperature of 120 K is interrupted at approximately 340 K by a kink. This fact is interpreted as being caused by a latent heat release during the transition from a disordered high-temperature (actually with 1 × 1 diffraction patterns18) to the well-ordered low-temperature 4/3 ML 3 × 3 R 30° phase. This transition was the subject of several experimental studies that established a rather complex phase diagram for the Pb wetting layer deposited on the Si(111) surface18−20,22−27 and which indicate the substrate reconstruction from its native 2 × 1 or stable 7 × 7 to the 1 × 1 structure. Here, we refer to Stepanovsky et al. study,18 which shows that the transition temperature of 380 K corresponds to an initial coverage of about 1.28 ML. However, the transition leads to a denser ordered phase of 4/3 = 1.33 ML Pb, which is consistent with a much greater compression of lead upon cooling than silicon. The HAS diffraction patterns shown in Figure 2a,d provide the periodicity of the Pb wetting layer and a clear evidence that

Experiment. In Figure 1a, the HAS specular intensity is shown for a 4/3 ML Pb film on Si(111), measured on cooling from an annealing temperature of 700−120 K for the incident He-atom wavevector ki = 5.3 Å−1. The details of the experiment are given in refs,16,17,21 as well as in the recent paper on the HAS dispersion curves of 1−6 ML Pb films grown on either Si(111) or Ge(111).9 Lead was deposited from a Knudsen cell located normal to the surface at a distance of 40 cm. The most uniform and reproducible films corresponding to maxima in HAS reflectivity are achieved at deposition rates in the range (0.2−0.5) ML/min: the highest reflectivity at each maximum indicates the best quality of the surface, that is, the lowest amount of defects.21 At these deposition rates, the base pressure of the scattering chamber of 5 × 10−11 mbar is kept constant, which indicates a Pb sticking coefficient equal to 1. The initial stage of layer growth was monitored by Auger electron spectroscopy (AES) to ascertain surface cleanliness to the limits of AES sensitivity. Auger signals from the contaminants measured with a cylinder mirror analyzer were not observed even after the deposition of several Pb layers. It is however a HAS reflectivity that provides the highest sensitivity to absolute coverage. Further, reflectivity data refer to the first maximum observed after starting Pb B


Article

The Journal of Physical Chemistry C

as a rough estimation because on the one hand, the surface Debye temperature of lead is considerably smaller (55 ± 10 K in refs 30 and 31) and on the other hand, it is expected to be considerably larger than for the Pb(111) surface because of a 5% contraction of the interatomic distances in the 4/3 ML 3 × 3 R 30° film. As will be seen below, the DFPT calculation for this film gives the highest phonon branch at about 12 meV compared with ∼9 meV in Pb(111)32 and bulk Pb.33 For the high-temperature 1 × 1 phase, where the Pb lattice has ac = 12.77 Å2 (corresponding to the 1 × 1 Si(111) unit cell), T = 340 K, and T′ = 700 K, λ = 0.75 is found. This value is substantially smaller than the mass enhancement parameter for the dense low-temperature 4/3 ML 3 × 3 R 30° phase; the main reason is apparently the lower density of Pb atoms in the high-T phase. For the 4/3 ML 3 × 3 R 30° Pb nanofilm, a series of phonon dispersion curves have also been obtained with timeof-flight HAS measurements in the [11̅0] ≡ Γ̅ K̅ Si direction at a surface temperature of 120 K (Figure 3, lozenges). Three

the phase measured at 120 K is the dense 4/3 ML 3 × 3 R 30° phase, as proposed, for example, by Yaguchi et al. (Figure 2(b) of ref 19), with the superstructure reciprocal lattice vectors in the Si(111) [11̅0] and [112̅] directions (Figure 2c). In this phase, one Pb atom in each unit cell is in the top position. However, recent ab initio calculations by Ren et al.20 have shown that two most stable phases are obtained by rigidly shifting of the Pb overlayer so as to have another atom in the T4 hollow site (Figure 2b, T4 phase) or in the H3 hollow site (H3 phase). The calculation neglecting the spin−orbit interaction shows that the most stable phase is T4 with an advantage of 25 meV over H3, whereas the calculation including the SOC gives the H3 phase as more stable than T4 by 12 meV.20 HAS data appear to confirm this theoretical result. The structure has no vertical mirror plane normal to the Γ̅ K̅ direction and therefore the diffraction amplitudes for opposite surface wavevector transfers show a slight asymmetry for both the substrate surface (Figure 2d, lines K̅ Si) and the Pb wetting layer (lines Γ̅ ). On the contrary, in the Γ̅ M̅ direction, where there is a normal mirror plane, the diffraction peaks are perfectly symmetric. However, the substrate and Pb layer asymmetries along Γ̅ K̅ will have the same orientation in the H3 phase and opposite orientation in the T4 phase. Because Figure 2d shows the same slight asymmetry of the diffraction peaks for both the substrate and the Pb layer, HAS data confirm that the Pb wetting layer grows in the H3 phase, which is consistent with the theoretical prediction.20 It can be seen that the HAS specular intensity for a constant incident wavevector ki = 5.3 Å−1 decreases with increasing temperature with slightly different slopes below and above the transition temperature (Figure 1b). There is also a good agreement between the data for ki = 5.3 Å−1 (lozenges) and the two low-temperature data points for ki = 6.6 Å−1 (filled circles), also reported in ref 16. As shown in ref 12, for a conductive film on a substrate that is assumed to be inert, the mass enhancement parameter can be derived from the T-dependence of the DW exponent, that is, from the variation with T of the logarithm of HAS reflectivity. For a 90°-scattering geometry, a single-layer film, and a temperature range above the film Debye temperature (as in this case), the value is defined as12 λ=

πϕ ln(T /T ′) ack i 2 kB(T ′ − T )

Figure 3. Calculated phonon dispersion curves of a 4/3 ML of Pb on Si(111) in the 3 × 3 R 30° structure (Figure 2c) superimposed on the calculated phonon spectra of a 60-layer Si(111) film in the ideal 1 × 1 structure (light broken lines) on the same wavevector scale. Phonon modes which are mainly localized on Pb atoms are shown by circles. The filled (brown) circles indicate Pb-localized modes with predominant SV polarization. The lowest Pb branches have mainly shear-horizontal (SH) polarization while the four highest branches are predominantly longitudinal (L). HAS experimental data are denoted by filled lozenges. The cross at 13.5 meV gives the RW energy for the H-passivated Si(111) 1 × 1 surface at the point as measured by HAS,35 while the arrows indicate the positions of the flat L and SV branches observed by HAS for 1 ML Pb/Cu(111).36

(1)

where ϕ is the surface work function (4.25 eV for lead ) and ac is the surface unit cell. For the 4/3 ML 3 × 3 R 30° phase, where the Pb lattice has ac = 9.58 Å2 (corresponding to a 5% linear contraction with respect to the Pb(111) surface unit cell of 10.62 Å2), T = 120 K, and T′ = 340 K, it is found that λ = 0.88. The low-temperature data measured with ki = 6.6 Å−1 (filled circles on Figure 1b) can also be used to derive λ, provided the temperature variation in the denominator of eq 1 is replaced by 28

kBΔT ⇒

kBθD θ Δ coth D 2 2T

distinct phonon branches are observed which are associated with: (a) the Si(111) Rayleigh wave (RW), which in the longwave limit is not affected by the Pb layer because of the increase in the penetration length; (b) quasi-shear-vertical (SV) modes of the Pb film; and (c) quasi-longitudinal (L) modes. Because the 3 × 3 unit cell contains 4 distinct Pb atoms, 12 phonon branches are expected for the film, 4 for each polarization. The spread of HAS data, corresponding to a typical HAS resolution of 1 meV, does not allow us to resolve different branches within each polarization. In any case, the observation of L modes with HAS intensities, which are generally comparable with the intensities of the SV modes, is indicative of an important contribution of the L modes to the electron−phonon interaction, as has already been estimated for Pb nanofilms grown on Cu(111).34

(2)

to take into account the actual thermal distribution when T is below the film Debye temperature θD. With the Debye temperature of lead, θD = 95 K at T = 85 K,29 it is found that λ = 0.93. Although this value is in good agreement with that obtained above in the range of 120−340 K, it should be noted that the use of the bulk Pb value for θD can only be considered C


Article

The Journal of Physical Chemistry C Theoretical Analysis. In Figure 3, the phonon dispersion curves of the 4/3 ML 3 × 3 R 30° Pb film on Si(111), calculated including spin−orbit interaction and detailed in ref 8, are superimposed on the calculated phonon dispersion curves of Si(111) in the ideal 1 × 1 structure (light broken lines) on the same wavevector scale to allow a detailed comparison with the actual HAS experimental data. To obtain a more reliable description of the modes, the substrate thickness was increased up to 60 layers by inserting bulk-like layers with force constants taken from an ab initio calculation of bulk Si. Because the phonon energies of lead are considerably smaller than the corresponding frequencies of the substrate, the phonon spectra are given up to 14 meV so as to show mainly the modes localized in the Pb film (circles). The modes with predominantly SV displacements of lead atoms (along the normal to the surface) are shown with filled circles. There are altogether 12 phonon branches for the Pb layer, as expected for the 4 Pb atoms in the 3 × 3 unit cell. The lowest phonons have mainly SH polarization (SH, spread, however, over in the 1−4 meV range); the intermediate modes are predominantly SV, falling mainly in the region 3−5.5. meV, while the four highest branches (7−12 meV) have prevalent longitudinal (L) polarization. Comparison of the present DFPT + SOC calculations with the HAS experimental data (Figure 3, filled lozenges) indicates a good agreement, which makes it possible to assign with confidence the observed modes to the calculated branches. The weak dispersion of the L and SV phonons indicates that Pb atoms interact more strongly with neighboring silicon atoms than with each other, despite the fact that the Pb adlayer is compressed by 5% compared to the bulk Pb(111) plane. This correlates with a strong covalent bonding observed between Pb and Si atoms at the interface7,25,26 where three of the four Pb atoms form a covalent bond with an underlying Si atom. In fact, the dispersionless branches at ∼5.5 and ∼9.5 meV agree quite well with the positions of similar flat branches of the epitaxial 1 ML Pb/Cu(111) (arrows in Figure 3) observed with HAS and reproduced well with the embedded atom method.36 This indicates that the larger density of the 4/ 3 ML Pb film on Si(111) does not significantly enhance the radial Pb−Pb force constants because of a partial transfer of the valence charge committed during the passivation of Si(111) dangling bonds. Moreover, the energy of the SV modes at Γ̅ (∼5 meV) is scaled in much the same way as the square-root mass ratio (MSi−H/MPb)1/2 to the energy of the RW mode at K̅ Si of the H-passivated Si(111) surface (cross in Figure 3),35 where the Si−H group oscillates practically along the vertical direction. This means that the Si−Pb force constant in the 4/3 ML film is approximately the same as the Si−H surface force constant with respect to the second layer Si(111) plane. However, the most important fact is that the L branch of 4/3 ML Pb/Si(111) is clearly observed with HAS, which gives an explicit indication of a relatively strong contribution to the total electron−phonon interaction. The calculated contribution of the longitudinal optical modes to the e−ph (Eliashberg) spectral function can be clearly seen in Figure 3(d,f) of ref 8 as a doubled peak between 8 and 10 meV and a separate peak at the highest frequency of 12 meV. The contribution of these phonons to the strength of e−ph interaction, defined as the spectral function weighted by 1/ω, where ω is the phonon frequency,37 is about 20%. The

momentum-averaged strength of e−ph coupling estimated theoretically at an energy corresponding to the Fermi level in the experiment gives a mass enhancement parameter λ = 0.84 ± 0.07, which is very close to the values extracted from the HAS measurements and given in the previous section. This also agrees with the value of λ = 0.93, extrapolated to the case of 4/3 ML from the HAS reflectivity data measured during the layer-by-layer growth of Pb films on Si(111).9

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SUMMARY In summary, the phonon dispersion curves and the mass enhancement parameter obtained from the scattering of helium atoms for the dense low-temperature 4/3 ML 3 × 3 R 30° phase of Pb/Si(111) are well reproduced in the ab initio DFPT calculation if the SOC and the effects of the Si substrate on the wetting Pb layer are taken into account. In particular, the prominence of two HAS dispersionless optical phonon branches about 9.5 and 5.5 meV is indicative of their major contribution to the total electron−phonon coupling and, therefore, of their important role in the persistent superconductivity of the 4/3 ML Pb/Si(111) system. The observed increase in the mass enhancement parameter during the phase transition from the high-temperature 1 ML 1 × 1 phase to the low-temperature 4/3 ML 3 × 3 R 30° phase is associated with the corresponding increase in the surface free-electron gas density. The present analysis provides further theoretical justification for using helium atom scattering reflectivity for direct determination of the electron−phonon coupling strength at conducting surfaces, in particular for 2D superconductors.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

I. Yu. Sklyadneva: 0000-0002-4651-8281 R. Heid: 0000-0002-2144-1417 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work has been supported by the University of the Basque Country (grants no. GIC07-IT-366-07 and no. IT-756-13), the Spanish Ministry of Science and Innovation (grants no. FIS2013-48286-C02-02-P, no. FIS2013-48286-C02-01-P, and no. FIS2016-75862-P), the Tomsk State University Academic D.I. Mendeleev Fund Program (grant no. 8.1.01.2017), and Saint Petersburg State University (project 15.61.202.2015).

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3-Monolayer of Pb on Si(111

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