This journal is© the Owner Societies 2014 Phys. Chem. Chem. Phys.

Cite this:DOI: 10.1039/c3cp54834a

Unveiling mode-selected electron–phononinteractions in metal films by heliumatom scattering

G. Benedek,*abc M. Bernasconi,b K.-P. Bohnen,d D. Campi,b E. V. Chulkov,aef

P. M. Echenique,ae R. Heid,d I. Yu. Sklyadnevaaf and J. P. Toenniesc

The quasi two-dimensional electron gas on a metal film can transmit to the surface even minute

mechanical disturbances occurring in the depth, thus allowing the gentlest of all surface probes, helium

atoms, to perceive the vibrations of the deepest atoms via the induced surface-charge density

oscillations. A density functional perturbation theory (DFPT) and a helium atom scattering study of the

phonon dispersion curves in lead films of up to 7 mono-layers on a copper substrate show that: (a) the

electron–phonon interaction is responsible for the coupling of He atoms to in-depth phonon modes;

and (b) the inelastic HAS intensity from a given phonon mode is proportional to its electron–phonon

coupling. The direct determination of mode-selected electron–phonon coupling strengths has great

relevance for understanding superconductivity in thin films and two-dimensional systems.

1. Introduction

The persistence of superconductivity in supported lead films asthin as only a few1–5 or even one single monolayer,6 as well asthe oscillations with thickness of the critical temperature(Tc(bulk) = 7.23 K) and of the upper critical field7,8 are strikingmanifestations of the electron–phonon interaction in reduceddimensions. Quantum-size oscillations, first observed by HeliumAtom Scattering (HAS) in the layer-by-layer growth of lead films,9 arealso manifested in electronic transport10,11 and photoemission12

properties, as well as in the interlayer distances,13–16 the islandheight distributions,17 the zone-centre phonon frequencies ofthe film18 and work functions.19

The coupling between the electrons and phonons of an ultra-thin metal film also comes into play in inelastic electron tunnellingspectroscopy20 via the phonon-induced modulation of the tunnel-ling barrier,21,22 in promoting inelastic photoemission,23,24

and in determining carrier lifetimes on metal surfaces.25,26

The electron–phonon (e–p) interaction, due to its comparativelylong range, can also couple sub-surface vibrations to electroncharge density oscillations at the surface.21,22 Of possible greatrelevance in surface chemistry is the fact that phonon-inducedsurface charge–density oscillations (SCDOs) can play an importantrole in phonon-induced reactive scattering, especially of openshell atoms and molecules at metal surfaces.27

For all the above phenomena the direct knowledge of the e–pinteraction for individual phonon modes is of basic fundamentalimportance. The current understanding of these phenomenarelies so far on the effective electron–phonon coupling constantl involving the integration over the entire phonon spectrum.28–30

As discussed in this Perspective, the direct measurement of e–pcoupling constants lQn for selected vibration modes of wavevectorQ and branch index n30 can answer a basic question in the theoryof superconductivity: which phonons are actually doing the job ofbinding Cooper pairs? A detailed knowledge of lQn may eventuallylead to the control on the nanometric scale of the direct conver-sion of phonons into electronic excitations and vice versa, andultimately to novel devices operating in the THz domain.

Ultra-thin metal films on stiffer substrates can indeed guidevarious kinds of sound waves, some travelling at the surface,some underneath, and some at the interface with the substrate.Not just the surface waves, presently exploited in surface acousticwave (SAW) devices, but also their sub-surface companions holdgreat promise for future novel electro- and opto-acoustic devices.Until recently the sub-surface phonons remained elusive to thecurrent surface probes, HAS and Electron Energy Loss Spectro-scopy (EELS). Thus it came as a surprise that the gentlest of all

a Donostia International Physics Centre (DIPC), Paseo Manuel de Lardizabal 4,

20018 Donostia/San Sebastian, Spain. E-mail: giorgio.benedek@unimib.itb Dipartimento di Scienza dei Materiali, Universita di Milano-Bicocca, Via Cozzi 53,

20125 Milano, Italyc Max Planck Institut fur Dynamik und Selbstorganisation, Bunsenstrasse 10,

37073 Gottingen, Germanyd Karlsruher Institut fur Technologie, Institut fur Festkorperphysik, P.O. Box 3640,

D-76021 Karlsruhe, Germanye Departamento de Fısica de Materiales and CFM (CSIC-UPV/EHU), Universidad del

Paıs Vasco/Euskal Herriko Unibertsitatea, E-20018 San Sebastian/Donostia, Spainf Tomsk State University, 634050, Tomsk, Russia

Received 15th November 2013,Accepted 2nd January 2014

DOI: 10.1039/c3cp54834a

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PERSPECTIVE

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surface probes, neutral helium atoms, can actually excite andmeasure the dispersion of sub-surface phonons of ultra-thinfilms of lead.30,31 More recently such ability of He atom spectroscopywas demonstrated for the Bi(111) surface, where the dispersioncurves of optical modes localized on the second bi-layer could bedistinctly measured.32 The present interest in the surface dynamicsof 6 sp metals is largely motivated by the important effects ofspin–orbit coupling (SOC)33–36 and the appreciable SOC con-tribution to the electron–phonon coupling strength.36

2. The quantum sonar effect

Former inelastic HAS measurements of Pb(111) ultra-thin films(3–8, 10 and 50 monolayers (ML)) grown on a Cu(111) substrate31

revealed a surprising fact: the number of observable dispersioncurves increases with the film thickness, much beyond thefew dispersion curves observed in the semi-infinite crystal. Thispeculiar aspect of HAS spectroscopy was already known sincethe early measurements on Na multi-layers on Cu(001).37 Therethe observed phonon frequencies were shown to correspond to theharmonics of an open-end organ pipe (organ-pipe modes), similarlyto the Sezawa waves, known in seismology for geological multi-layers.38 Annealed Pb(111) films grown on Cu(111) are stable andwell ordered for any thickness NL Z 3 ML, with a commensuratePb–Cu interface in a 3/4 ratio along both symmetry crystallo-graphic directions31 (Fig. 1). In the HAS diffraction patternsreported in ref. 31 for NL = 1, 3, 4, 5, 50 ML, the substrate satellitepeaks in the [11%2] direction are still visible at 5 ML, whereas in the[1%10] direction the surface corrugation is much smaller and nosubstrate satellite is visible for NL > 3.

Fig. 2 shows the HAS data for the dispersion curves of 3, 4, 5and 7 mono-layers (ML) of lead on a Cu(111) substrate, for asurface temperature of 95 K.31 The experimental phononbranches are labelled (ai, bi,. . .; i = 1, 2, . . .) following theconvention of ref. 31. A full set of dispersion curves has alsobeen measured for 6 ML, however only for an un-annealedsample39 (Fig. 3, left-hand ordinate scale). The comparativelylarge difference in the acoustic impedance between Cu and Pbensures that the observed phonons refer exclusively to the Pbfilm.40 Nevertheless already for 3 ML at least four distinctphonon branches are observed, six for 4 ML, seven for 5 ML,and eight for 7 ML. Above this thickness the branch number

starts decreasing and at 50 ML the most prominent features ofthe time-of-flight (TOF) spectra can be assigned to the ordinaryfour branches of sagittal polarization occurring at the (111)surface of fcc metals.22,39,41 Along the �G %M symmetry direction,where the sagittal plane is a mirror plane, a film of NL layershosts 2NL branches of sagittal polarization which can bedetected by HAS in the planar scattering geometry; in anyother direction, e.g., �G %X, there are however 2NL branches ofquasi-sagittal polarization. Thus it is no surprise from puresymmetry arguments that so many branches are seen usingHAS in thin films. However a simple physical analysis31 showsthat only two of these modes have a real surface character,whereas all the other modes have a negligible amplitude at thesurface layer and should not be observed according tothe theory of inelastic HAS based on atom–atom two-bodycollisions (impulsive model).42–45

An explanation was found with DFPT calculation of thephonon dispersion curves and of the related phonon-inducedelectron density oscillations (EDOs) for Pb(111) films of NL = 3 to7 ML supported on a rigid substrate. The study led to an entirelynew formulation of the theory of inelastic scattering of atomsfrom crystal surfaces. In order to keep the computational effortwithin a reasonable size, the rigid substrate was simulated by asingle additional monolayer of Pb(111) with a virtually infinitenuclear mass.30 The distance between the bottom layer of the Pbfilm and the substrate layer was reduced by 5% with respect tothe spacing found by equilibration in order to artificially repro-duce the force-constant stiffening expected at the Cu–Pb inter-face. The calculations were performed using the mixed-basisab initio simulation package based on the density-functionaltheory with a norm-conserving scalar-relativistic pseudo-potentialand the local density approximation for the exchange–correlationfunctional. Phonon frequencies and polarization vectors were thenobtained using DFPT in a mixed-basis approach,46 after an energyminimization performed for the NL + 1 film (thus including thesubstrate layer) by allowing for the variation of interlayer distancesin the films. The lateral lattice parameter inside the Pb(111) layerswas fixed at the theoretical bulk value, a = 9.20 a.u., which is 1.6%smaller than the experimental value of 9.35 a.u. The calculatedrelaxation showed a contraction of the outermost interlayer spacingrelative to the bulk distance of a few percents (between �2.7%for 4 ML and �5.4% for 5 ML). The phonon dispersion curveswere obtained by interpolation of a (12 � 12) Q point mesh forthe hexagonal surface Brillouin zone (SBZ).

The dispersion curves calculated for 3 to 7 ML along thesymmetry directions are compared with the experimental datain Fig. 2 (NL = 3, 4, 5, 7 ML) and Fig. 3 (NL = 6 ML). The best fitfor the un-annealed 6 ML film (Fig. 3) is obtained by plottingthe theoretical branches on a 7% compressed energy scale(right-hand ordinate). The corresponding stiffening of theexperimental phonon energies needed to have the best agree-ment with DFPT calculation is viewed as an effect of annealing,as we learn from the comparison between un-annealed andannealed 5 ML films shown in ref. 31. Thick lines correspond tomodes with prevalent shear-vertical (SV) polarization; brokenlines to modes with a shear-horizontal (SH) polarization, which

Fig. 1 A hexagonal lead monolayer on a Cu(111) substrate forms a commen-surate phase with 9 Pb atoms and 16 Cu atoms per unit cell.

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are forbidden by symmetry to HAS along �G %M (h112i azimuth).The modes with SV polarization are those which according to theconventional impulsive model42–45 should be detected usingHAS. The following analysis shows however that the inelasticHAS intensities exclusively depend on the amplitude of thesurface EDO’s and not on the surface atom-core displacementsas in the impulsive model. As seen in Fig. 1 and 2 the measureddispersion curves are comfortably associated with the calculatedones, notably with those of prevalent SV polarization.

The two upper branches e1 and e2 are of special interest.They correspond to a distinct doublet in the energy-gain spectra(Fig. 4), observed for NL = 4 to 8 ML. The e1 mode is missing for3 ML; it becomes more intense than e2 at 5 ML, then decreasesremaining however still visible up to 10 ML, at least in the h110idirection. At 50 ML only the e2 branch is observed.31 Both e1

and e2 have SV polarization at the �G-point (Q = 0). Fig. 5(a) shows atypical HAS energy gain spectrum for 4 ML measured along h112iat an incident angle of 36.51 in a 901 planar scattering geometryand an incident energy of 22 meV. The corresponding scan curve

Fig. 2 The dispersion curves of ultra-thin lead films of 3, 4, 5 and 7 monolayers (ML) grown on a Cu(111) substrate. The data points measured with Heatom scattering at a surface temperature of 95 K are superimposed to the dispersion curves calculated with DFPT for a rigid, infinitely heavy substrate: inthe high symmetry direction �G %M phonon polarizations are either sagittal (full lines: thicker for quasi-SV modes, thinner for quasi-L modes) or shear-horizontal (SH) (broken lines). Labels designate the Rayleigh wave (a1), the surface and interface bi-layer breathing modes (e2 and e1, respectively) andother quasi-SV modes of the film (a2, a3, s1, etc.).

Fig. 3 Same as Fig. 2 for an un-annealed 6 ML lead film. Note the differentordinate scale for the experimental and theoretical phonon energies.

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is plotted in Fig. 5(b) (folded into the positive quadrant of the SBZ)on top of the dispersion curves: the data points crossed by thescan curve correspond to the peaks observed in (a). Theoryshows that the e1 and e2 modes correspond at Q = 0 to breathingmodes of the bottom and top bi-layers of the film, respectively,as schematically represented in the inset (c). The mode e2 runsabove the bulk maximum phonon energy as an effect of thecontraction of the first atomic bi-layer distance.18 This featureis common to all films with NL Z 4: e1 is a sub-surface mode,which gets deeper for increasing NL, posing the question of themechanism which makes it observable by HAS.

The answer is in the calculation of the phonon-inducedsurface EDOs. Fig. 6 shows an example for 5 ML of four EDOsinduced by the modes e2, e1, r2 and r1 at the �G-point (Q = 0).The EDO amplitudes have been calculated for the frozenpatterns of the atom core displacements corresponding to thecalculated phonon quantum displacements (represented by thearrows of proportional length) uB(Q = 0, nl3) = �h/(2MeQ=0,n)

1/2eB(Q =0, nl3), where l3 labels the film planes, M is the atom mass of Pb,eQn and eB(Q, nl3) are the energy and the normalized eigenvector ofthe n-th phonon of wavevector Q (n = e1,2), respectively, and �h

the reduced Planck’s constant. The contours of the calculatedEDOs, represented on a logarithmic scale, correspond tothe values (in atomic units (a.u.)) +10�n (red lines) and �10�n

(blue lines) with 3n = 18, 17,. . ., 9. It appears that the topmostmode e2 has the largest atom displacement amplitudes at thefirst two surface layers, whereas the mode e1 is localized at thebottom of the film near the interface with the substrate.Similarly, the other two modes r2 and r1 have the largestdisplacement at the bottom and at the surface of the film,respectively. All modes, however, induce EDOs of comparableamplitudes at a distance zt = 3.5 meV from the surface atomplane corresponding to the turning point of He atoms for thekinematic conditions of present experiments (Fig. 6). Thisexplains why all the four modes are observed by HAS (Fig. 7),even those whose surface atom displacements are vanishinglysmall. It can be appreciated from Fig. 6 and 7(a) that thephonon peak intensities, as derived from a Gaussian fitafter subtraction of the multi-phonon (MP) background,roughly correspond to the respective EDO intensities at theturning point.

It is therefore the range of the electron–phonon interactionwhich determines how deep a sub-surface atomic displacementcan be detected by helium atom scattering. As seen in Fig. 6, theQ = 0 phonon displacements modulate the electronic density inthe direction normal to the surface, and it is via these wavesthat the He atoms can perceive atomic displacements severallayers beneath the surface (quantum sonar effect). All this leads

Fig. 4 The evolution with increasing film thickness (3 to 8 ML) of the surface (e2) and interface (e1) bilayer modes, measured along the two symmetrydirections at similar incident angles. For 3 ML only the e2 mode is observed.

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to a new formulation of inelastic HAS theory based on electron–phonon (e–p) interaction, opening the possibility of a directmeasurement of the mode-selected e–p coupling strengths ofmetal surfaces (mode-l spectroscopy).

3. Electron–phonon interaction theoryof inelastic HAS from metal surfaces

The simplest way to describe the interaction of a He atom witha metal surface is provided by the Esbjerg–Nørskov (EN)potential47

V(r) = An(r) (1)

proportional to the surface electron density n(r) at the classicalturning point, with A = 364 eV and a0

3 and a0 the Bohr’sradius.48 As discussed in ref. 49, the EN potential rests on alocal response approximation which holds quite well for Heatoms. Consider for a Pb film an occupied state of wave-function cKn(r) and energy EKn near the Fermi level, wherer R (R, z) is the electron position, K the parallel wave-vector ofthe electron and n labels the quantum-well mini-bands cross-ing the Fermi level (Fig. 8). Hereafter capital bold-face lettersindicate vectors parallel to the surface and z is the normalposition coordinate. To second-order perturbation theory, thetime (t)-dependent adiabatic perturbation of the electron den-sity dnKn(r, t) due to the modulation of the electron potential

Fig. 5 A HAS energy-gain spectrum (a) for a 4 ML film in the �G %M directionshowing distinctly all modes of quasi-SV polarization met by thescan curve for an incident angle of 36.51 and an incident energy of 22meV (b). The RW (a1) is observed also on the annihilation side at B2 meV.The small feature marked f-a1 corresponds to a RW mode folded into thesuperstructure BZ and indicates that the different periodicity of thesubstrate is still felt at the surface of a 4 ML film. Labels are as in Fig. 2.E is the diffused elastic peak. The peaks e2 and e1 correspond to thebreathing modes of the surface and interface bi-layers, respectively, asschematically shown in (c).

Fig. 6 Electron density oscillations (EDOs) induced by a quantum ofatomic displacements of the e2, e1, r2 and r1 phonon modes at zerowavevector of a 5 ML lead film on Cu(111). The contour lines are for equalEDO amplitudes �10�n atomic units (+ for red, � for blue lines) and arelabelled by the exponent n. The core displacement amplitudes are shownby vertical arrows. The heavy dark line at �14.1 Å indicates the location ofthe rigid substrate. Although the displacement fields of the modes e1 andr2 is mostly concentrated at the bottom near the substrate layer (heavyline), whereas the other modes e2 and r1 have large amplitudes at thesurface, the four modes induce almost equal EDOs at a distance zt = 3.5 Åabove the topmost surface atomic layer corresponding to the classicalHe-atom turning point for the kinematic conditions of the experiments.

Fig. 7 (a) The amplitudes of the highest four quasi-SV phonon modes of a5 ML lead film obtained from a Gaussian analysis after subtraction of the multi-phonon (MP) background for an incident angle of 391, an incident energy of22 meV in the h110i direction. The corresponding scan curve and its intersectionwith the phonon dispersion curves are shown in (b). Labels as in Fig. 2.

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energy,Pj

dV r� roj � ujðtÞ� �

, produced by the phonon displa

cements uj (t) of the ions at positions roj , is given by

dnKnðr; tÞ ¼ ~cKnðr; tÞ�� ��2� cKnðrÞj j2

¼X0K0n0;j

cKn�ðrÞcK0n0 ðrÞ

EKn � EK0n0

ðd3r0cKnðr0ÞcK0n0

�ðr0ÞdV r0 � roj � ujðtÞ� �

(2)

where ~cKn(r,t) is the perturbed wave-function, and the prime inthe sum excludes K = K0, and n = n0. The contribution of theelectron in the state (Kn) to the change in the electron densityinduced by a phonon of parallel wave-vector Q and branchindex n can be expressed as28

dnKn;QnðrÞ ¼Xn0

cKn�ðrÞcKþQn0 ðrÞ

EKn � EKþQn0gnn0 ðK;KþQ; nÞ (3)

where gnn0(K,K + Q;n) is the e–p coupling matrix. The latterinvolves the sum over the matrix elements hK,n|rV(r� ro

j )|K + Q, n0iof the j-th ion pseudo-potential gradient. In what follows thechoice of the electronic wavefunctions |K,ni in the factorizedform jKn(z) exp(iK�R) appears to be a sufficient approximation,

so that gnn0(K,K + Q;n) depends of K only via the z-dependentpart of the wavefunctions.

We will now consider an inelastic scattering process of aHe atom from an initial state |ii of energy Ei and wavevectorki = (Ki, kiz) into a final state hf| of energy Ef and wavevectorkf = (Kf, kfz). The inelastic scattering probability P(ki,kf) for one-phonon creation processes in the standard distorted wave Bornapproximation is given by (up to a constant factor):42

P ki; kfð Þ / kf

kizj j1þ nBEðDEÞ½ �

XQv

XKn

dVfiðKn;QnÞj j2d DE � eQvð Þ;

(4)

where nBE(DE) is the Bose–Einstein occupation number fora phonon of energy DE = eQn. The matrix element between theHe-atom initial and final states of the inelastic scattering potentialproduced by the density variation, eqn (3), is then given by

dVfiðKn;QnÞ ¼ A fh jdnKn;QnðrÞ ij i

¼ AXn0

fh jcKn�ðrÞcKþQn0 ðrÞ ij i

EKn � EKþQn0gnn0 ðK;KþQ; nÞ:

(5)

Fig. 8 (a, c) The calculated band structure of 5 and 7 monolayer (ML) lead films, respectively, is shown along the symmetry directions together with thecorresponding Fermi lines (b, d) at the Fermi level in the first surface Brillouin zone. There are 10 bands (labelled n = 0,1,. . .,9) cutting the Fermi level for5 ML and 14 (n = 0,1,. . .,13) for 7 ML.

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The factorization of the electronic and He-atom wavefunctionsjKn(z)eiK�R and wi,f(z)eiKi,f�R with respect to the coordinates parallel(R) and normal (z) to the surface yields

fh jcKn�ðrÞcKþQn0 ðrÞ ij i ¼ d Kf � Ki þQð Þ

�ð1�1

dzwi�ðzÞfKn

�ðzÞfKþQn0 ðzÞwfðzÞ

� d Kf � Ki þQð ÞI eQnð Þ: (6)

The He wave-functions wi,f(z) decay very rapidly as they penetratethe surface electron density beyond their respective classical turn-ing points zti and ztf. Moreover the wave-function jKn(z) decaysoutside approximately as e�kz, where k = (kj

2 + kF,n2 � kn

2)1/2;

kf is the wave-vector corresponding to the Pb work function(kf

2 D 1.4 �2) and kF,n is the Fermi wavevector in the K directionfor the n-th band. Due to the small phonon energy on theelectronic energy scale, kn D kF,n and k D kf. Thus the integralI(eQn) in eqn (6) can be taken in a first approximation to beindependent of K. This integral is however a slowly decreasingfunction of the energy transfer DE = eQn, due to the fact that thesmall difference zti � ztf between the initial and final turningpoints increases slowly with DE, thus reducing the overlap amongthe wave-functions in eqn (6).

With the approximation made in eqn (6), and by restrictingthe sum in eqn (5) over the final electronic states to thosefor which EKn � EK+Qn0 is equal to the phonon energy eQn,as done for ordinary electron–phonon scattering processes

Fig. 9 Examples of HAS energy-gain spectra for 5 ML (a) and 7 ML (c) in the h112i direction and the corresponding scan curves across the dispersioncurves (e, f). The HAS intensities, relative to that of the specular peak (I00), are compared with the calculated electron–phonon coupling constants lQn forthe modes of prevalent SV polarization (b, d). The labels are as in Fig. 2; the phonon branches of prevalent SV polarization in (e) and (f) have been colouredas the corresponding intensities in (b) and (d) and lQn curves in Fig. 10.

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(see Grimwall28), the sum over the squared e–p matrix elementsbecomes proportional to the e–p coupling constant lQn for eachindividual phonon (the mode-l) as defined in ref. 50:

XKn

Xn0

gnn0 ðK;KþQ; nÞ fh jcKn�ðrÞcKþQn0 ðrÞ ij i

����������2

ffi 1

2N EFð ÞeQn3I2 eQn

� �lQn ;

(7)

where N(EF) is the density of states at the Fermi energy. Notethat for the comparatively simple band structure of the Pb film

quantum well the condition EKn� EK+Qn0 = eQn for given K, n andQ can be fulfilled only by one value of n0 at a time (if any, seebelow Fig. 11), which ensures the proportionality of the l.h.m.of eqn (7) to lQn. By inserting eqn (5) to (7) into eqn (4) the HASinelastic scattering probability takes the simple form:

P ki; kfð Þ / f ðDEÞN EFð ÞXQn

lQnd DE � eQnð Þ; (8)

where

f ðDEÞ � kf

kizj jDE½1þ nðDEÞ�½IðDEÞA�2 (9)

is a slowly varying function of the energy transfer DE. Actually,due to a partial compensation between the decreasing function[1 + n(DE)]I2(DE) and the factor DE, the kinematical factor f (DE)is found to be almost independent of DE. Thus to a goodapproximation the HAS amplitudes for individual phonons areproportional to the respective e–p coupling constants lQn.

This conclusion is supported by the comparison in Fig. 9between the calculated coupling constants lQn and the corres-ponding energy-gain spectra measured at different incident anglesfor 5 and 7 ML films in the �G %M direction. Only the mode-l’s for themodes corresponding to the intersections of the scan curves withthe phonon branches of prevalent SV character (Fig. 9(e and f)) areplotted. The nice agreement between the observed spectra and themode-l distributions confirms that the kinematic factor, eqn (9), is

Fig. 10 The mode-selected electron–phonon coupling constants lQn formodes of quasi-SV polarization calculated as functions of the two-dimensional wavevector in the �G %M symmetry direction, for 3, 5 and 7 MLPb/Cu(111) films. The modes are labeled as in Fig. 2, the colours corre-spond to those in Fig. 9.

Fig. 11 Kohn anomalies observed in some dispersion curves as small dips(downward arrows) in the �G %M direction for a 5 ML Pb film (a) and in the �G %Kdirection for 3, 4, and 5 ML films (b). Possible inter-band nestings at theFermi levels corresponding to the anomalies for the 5 ML film are shown in(c). In the �G %K direction 3 - 0, 2 - 0 and other similar transitions may beassociated with the broad anomaly in a2; the transitions from bands 1 to 4to the same bands in the next SBZ may be associated with the anomaly ins1. In the �G %K direction the transitions between two parallel bands 5 may beassociated with the anomaly in a1.

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fairly constant. As shown in ref. 30, the calculations of P(ki,kf) withits actual pre-factor f (DE), the sum extended to all phonons (thusincluding the minor contribution of quasi-longitudinal modes) andthe instrumental peak half-widths are seen to reproduce very wellthe experimental profiles. On this basis it has been concluded thatinelastic HAS can be used for a direct measurement of mode-selectede–p coupling constants in conducting films, notably in super-conductors and other low-dimensional systems with strongmanifestations of the e–p interaction.

The calculated coupling constants lQn for 3, 5, and 7 ML areplotted in Fig. 10 as functions of Q for the phonon of prevalent SVcharacter, with the labels of Fig. 2 and the colour code of Fig. 9. Asseen in Fig. 10 the mode-l’s vary in an oscillatory manner with thewave-vector Q and the phonon index n, with maxima correspondingin general to some good nesting on the Fermi contours.

Some examples are illustrated in Fig. 11 for the 5 ML film inboth �G %M and �G %K directions, where they may be associated withpossible Kohn anomalies in the dispersion curves of a2 and s1

modes for �G %M, and of a1 and a2 modes for �G %K. The maxima inFig. 10 also indicate the specific phonons which mostly con-tribute to the superconducting properties of Pb films. Asmentioned in the Introduction, Pb films are superconductorsdown to one single layer, with a fairly constant Tc between 6and 6.7 K for 8 to 4 ML that drops below 4 ML (Fig. 12).

It is interesting to note that hlQni, the average of lQn over theSBZ and all phonon indices n, increases for decreasing

Fig. 12 Calculated effective e–p coupling constant l (K, r.h. ordinatescale) and corresponding superconducting critical temperature Tc ( , l.h.ordinate scale) for Pb ultra-thin films (NL = 3 to 7 ML) on a rigid Pbsubstrate layer. Tc is derived from McMillan’s formula for an effectiveelectron–electron coupling constant m* = 0.12. For comparison themeasured Tc for ultra-thin Pb films3 and for a Pb monolayer6 on Si(111)

substrates are also shown for comparison ( , , , l.h. ordinate scale).

Fig. 13 (a) The surface phonon dispersion curves of Bi(111) measured with inelastic HAS and calculated with DFPT, with no inclusion of the spin–orbitcoupling.32 The intensity of the colored areas provides the phonon density of sagittal modes (SV = quasi-shear vertical, L = quasi-longitudinal) projectedonto either the first surface bi-layer (SV1, L1) or the second surface bi-layer (actually 3rd atomic layer) (SV3, L3).32 The scan curve crossing the dispersioncurves along �G %M corresponds to the HAS energy-gain spectrum shown in (b) for an incident angle of 41.751 and energy of 16.2 meV. A comparison ismade with a HAS energy-gain spectrum for a 6 ML Pb(111) film on Cu(111) under similar kinematic conditions (c). Note that six atomic layers would be theperiodicity of Bi(111) in the normal direction.

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thickness: 0.045 for 7 ML, 0.067 for 5 ML and 0.074 for 3 ML.The effective e–p coupling constant (mass-enhancement para-meter) l = 3NL hlQni is therefore 0.95, 1.00 and 0.67 for 7, 5 and3 ML, respectively. The calculated l for NL = 3 to 7 ML is plotted inFig. 12 and compared with the corresponding superconductingcritical temperature Tc as derived from McMillan’s formula28 withan effective electron–electron coupling constant m* = 0.12.5 Thecomparison of the calculated with the measured Tc for ultra-thin Pbfilms3 and for a single Pb monolayer6 on Si(111) substrates shows asimilar behaviour, with the same relative maximum Tc at 4 ML and arapid drop for thinner films. The comparison with the previoustheoretical study of l and Tc calculated for free-standing Pb films5

shows important differences, which are exclusively due to compar-able differences between the phonon structures of supported andfree-standing films. This underlines the importance of possessing adetailed knowledge of the phonon structure for the prediction andanalysis of the superconducting properties in ultra-thin metal films.

As anticipated in Section 2, the calculated phonon-inducedEDOs alone provide qualitative information on the inelastic

HAS amplitudes. This is readily seen in the special case whenthe electronic wavefunctions inside the film quantum well canbe taken as plane waves also in the z direction, jKn(z) p

exp(iknz). In this case the matrix element of the pseudo-potential reduces to its Fourier transform V(qnn0), with qnn0 �(Q, kn � kn0), and the electron–phonon coupling matrix takesthe more familiar form28

gnn0 ðK;KþQ; nÞ ¼ �hffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2MeQn

p qnn0 � n qnn0nð ÞV qnn0ð Þ; (10)

where M is the atomic mass of lead,

n qnn0nð Þ �Xl

eðQn; lÞ exp i kn � kn0ð Þld½ � exp �12

qnn0 � ulj j2D E�

;

(11)

e(Qn, l) is the phonon eigenvector at the l-th layer and d is theaverage interlayer spacing. The last exponential factor accountsfor the Debye–Waller (DW) factor appropriate for the present

Fig. 14 (a) The HAS data points and the DFPT dispersion curves and projected phonon densities of Fig. 13(a) in the �G %M direction are shown separately foreach polarization and depth. The additional data belonging to the optical SV1, SV3 and L1 branches have been obtained by bound-state resonanceenhancement.52 (b) The same data are compared using a DFPT calculation including the spin–orbit coupling (SOC).

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scattering process, h. . .i being the thermal average. In thisplane-wave approximation for the quantum-well statesgnn0(K,K + Q;n) is actually independent of K, and the matrixelement hf|dnKn,Qn(r)|ii in eqn (5) becomes just the EDOinduced by the phonon (Q,n). Thus, within the limits of theabove approximations, eqn (4) and (5) imply that the inelasticHAS intensities are roughly proportional to the square of thecorresponding EDOs.

4. Bi(111)

The theory of inelastic HAS exposed in the previous section formetal surfaces can be extended to the surfaces of semimetals,e.g., Bi(111). The surface electronic structure at the Fermi level ischaracterized by electron and hole pockets along certain symmetrydirections and at symmetry points of the SBZ.51 The concentrationof charge in the reciprocal space largely contributes to the surfacecorrugation – a distinctive feature of semimetals with respect to thedensity-packed fcc metal surfaces, which appear in HAS as perfectlyflat. Although the Bi(111) is not densely packed and exhibits twodifferent hollow sites, the fcc and the hcp sites, HAS diffractiondata and ab initio simulations show an exact equivalence of the twosites, indicating a free-electron shielding of the actual crystallo-graphic structure.32 Thus the same mechanism of energy transmis-sion from the scattering He atom to the surface and subsurfacephonons described for Pb(111) films is also expected for Bi(111).

The surface phonon dispersion curves of Bi(111), recentlymeasured with HAS,32 are shown in Fig. 13(a) and compared

with the calculated contour plots of the surface-projectedphonon density of sagittal polarization.32 This calculation isbased on the DFPT with no inclusion of the spin–orbit couplingnor of dispersion forces. Fig. 14 shows the data of Fig. 13(b),plus additional data in the optical region obtained using thetechnique of bound-state resonance enhancement,52 separatelyplotted for the different phonon polarizations and localization.They are compared with both DFPT calculations without(Fig. 14(a)) and with spin–orbit coupling (SOC) corrections(Fig. 14(b)). The theoretical equilibrium lattice parameters ofbulk Bi are a = 5.54 Å and c = 11.79 Å by including SOC ora = 5.53 Å and c = 11.71 Å by neglecting SOC. The labels SVmand Lm indicate whether the phonon branches have a prevalentSV or L polarization and the largest amplitude on the m-thatomic layer: (m = 1,2) refers to the surface bi-layer, whereasm = 3 is the first layer of the second bi-layer. The SOC correctionis known to produce some softening,33,36 an effect confirmedby the present calculation, which apparently spoils the goodagreement with experiment obtained without SOC. It should benoted, however, that heavy elements have not only a largeSOC but also large dispersion forces, which have, in the caseof phonons, a stiffening effect via the contraction of theequilibrium lattice constant.

Unlike Pb(111), the phonon structure of the Bi(111) surfaceexhibits a large gap between acoustic and optical phononbranches (about 3 meV without SOC and 2.5 meV with SOC).A striking difference between the surface dynamics of the twoelements, which are first neighbours in the periodic table,appears however in the HAS response (Fig. 13(b and c)): the

Fig. 15 The calculated EDOs induced by a phonon displacement quantum for the most prominent sagittal surface modes of Bi(111) at Q = 0: (a) first-layer longitudinal (L1) mode; (b,c) third-layer longitudinal (L3) modes; (d,e) third-layer shear-vertical (SV3) modes; (f) first-layer shear-vertical (SV1)mode. The respective energies are indicated in parenthesis. The contours are for equal EDO amplitudes corresponding to the figures shown in (f) in unitsof 10�4 atomic units (positive for red, negative for blue lines, or vice versa).

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scattering amplitude from the Bi(111) optical mode SV1 iscomparatively far more intense than that from the e2 mode ofPb(111) films, although both are localized above the maximum bulkenergy and restricted within the first atomic bi-layer. Accordingto the present interpretation of inelastic HAS amplitudes, theoptical SV1 has a comparatively large mode-l, a fact which hasbeen associated32 with the observed enhancement of l observedwith ARPES at the Bi(111) surface53 and the appearance ofsuperconductivity in Bi crystallites and interfaces,54,55 whereasbulk Bi is not superconducting.

The DFPT calculation of the surface EDOs for the observedphonons at Q = 0 (L1, L3, and SV3 in the acoustic region,SV1 and SV3 in the optical region, Fig. 15) confirms that theoptical SV1 has the largest EDO. It is also remarkable that thesubsurface modes SV3 and L3 (at 4.3 meV) are predicted to havea comparable amplitude, as actually observed in Fig. 13(b),despite the fact that the third atomic layer, belonging to thesecond bi-layer, is separated from the second layer by theso-called van der Waals gap. This confirms however the ab initioband structure calculations which locate an appreciable amountof free charge in the inter-bilayer space, thus permitting theenergy transfer from the scattering He atom to the secondbi-layer. As seen from the scan curve in Fig. 13(a) correspondingto the kinematics of Fig. 13(b), there are actually two L3 modescrossed by the scan curve, as predicted by theory, though thelower one falls onto the strong L1 peak. The EDOs at the�G-point in Fig. 15 actually assign a much lower intensity to L3(3.8 meV) and an appreciable intensity to L1 (1.2 meV), thoughthe comparison of the intensities for the latter is not significantbecause the experimental peak occurs at finite Q and a muchlarger energy. It may nevertheless be concluded that the calculationof the EDOs provides a good guideline for the interpretation of HASspectra and a safe basis for deriving from HAS intensities informa-tion on mode-selected e–p interaction also for semimetal surfaces.Moreover the possibility of dividing out the SOC contribution to thee–p interaction discussed in ref. 36 suggests that inelastic HAScould represent an excellent tool for the investigation of topologicalinsulator surfaces. The recent HAS studies of Bi2Se3(111) andBi2Te3(111)56–58 move in this direction.

5. Conclusions

In summary the following conclusions can be drawn.(i) The fact that the phonon displacements can induce charge

density oscillations over fairly long distances allows the Heatoms, via their electronic closed shell, to probe the atomicmotion even several layers beneath the surface. In contrast,current surface phonon probes like EELS essentially interactonly with the atom cores: thus the probe electrons are practicallyinsensitive to the electronic charge oscillations and therefore tothe atomic displacements deeper than their penetration length.True mode-lambda spectroscopy can now be envisaged from thelarge improvement in resolution obtained in 3He inelasticscattering using the spin-echo technique (energy resolution20 neV, angular resolution 0.1 degree).59 An example of the

type of valuable information which might become accessible isthe identification of the phonons which are most efficient in theformation of Cooper pairs in BCS thin-film superconductors.

(ii) Once interface phonon waves become accessible, so as tobe generated and revealed by external probes via the associatedsurface charge density oscillations, this effect could be exploited inhypothetical interface acoustic wave (IAW) devices. The advantagewith respect to SAW devices would be that interfaces, onceprepared, are not affected by contamination or by any otherchemical or mechanical disturbance which may affect a freesurface. Shear-vertical waves propagating at the interface of twoelastic media are known in seismology as Stonely waves.60 Thepresent observation of the e1 phonon is likely to be the firstspectroscopic evidence of their atomic-scale counterpart.

(iii) In free-electron metals the phonon-induced surfaceEDOs, more than the motion of ions, modulate the tunnellingpotential in inelastic electron tunnelling spectroscopy. This facthas crucial implications for scanning tunnel spectroscopy (STS)and specifically in STM phonon imaging experiments,20 sincethe strongest coupling is not to phonons with the largestsurface ion displacements but to phonons which produce thelargest surface charge density oscillation. The same conclusionmay apply to the analysis of inelastic angle-resolved photo-emission experiments.22,23

(iv) It is finally observed that phonon-induced surface EDOsmay play a role in inelastic and reactive scattering of atoms andmolecules from metal surfaces,61 as well as in surface non-adiabatic processes.27,62–64

Acknowledgements

GB acknowledges financial support from the Basque Foundationfor Science (Ikerbasque, project ABSIDES) and the Alexander-vonHumboldt Foundation. The authors thank Prof. WolfgangErnst (Technische Universitat, Graz, Austria) and his coworkersfor useful discussions.

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