Low-energy EELS investigation of surface electronic excitations on metals

Low-energy EELS investigation of surface electronic excitations on metals

M. Rocca Centro di Fisica delle Superfici e delle Basse Temperature del CNR and INFM, Dipartimento di Fisica, via Dodecaneso 33, 1-16146 Genova, Italy

ELSEVIER Amsterdam-Lausanne-New York-Oxford-Shannon-Tokyo

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M. Rocca/Surface Science Reports 22 (1995) 1-71

Contents

1. Introduction 2. Surface plasmons: overview of the fundamental concepts 2.1. What are surface plasma waves and Mie resonances 2.2. Surface plasmon dispersion on jellium 2.3. Surface plasmon damping 3. Experimental aspects of EELS measurements of surface plasmons 3.1. High-energy EELS 3.2. Low-energy EELS 3.2.1. Dipole and impact scattering 3.2.2. Low-energy EELS spectrometers 3.2.3. Measurement of surface plasmons 3.2.4. Artifacts associated with finite angular acceptance 3.2.5. Effects connected to surface roughness 4. Electronic excitations on simple metals 4.1. Alkali and alkaline-earth metal surfaces 4.2. Thin alkali overlayers on AI 4.3. Mie resonance on alkali metal clusters 4.4. Overlayer plasmon in K chains on Si(001) 4.5. Surface plasmon dispersion on AI 5. Surface electronic excitations in the presence of filled d-bands 5.1. Surface plasmon dispersion and Mie resonance shift for In and Hg 5.2. The special case of Ag 5.2.1. Surface plasmon dispersion and Mie resonance shift 5.2.1.1. Low-Miller-index surfaces 5.2.1.2. Ag clusters 5.2.1.3. Interpretation of the positive surface plasmon dispersion 5.2.1.4. Recent theoretical efforts 5.2.2. Surface plasmon damping 5.2.3. Effect of crystal temperature 6. Surface electronic excitations of transition metals: Pd(ll0) 7. Synopsis and conclusions 8. Symbols and acronyms

3 5 5 9 11 12 12 13 13 16 17 19 21 21 23 31 35 36 37 39 39 40 42 42 50 50 52 56 61 63 65 66

Acknowledgement References

67 67

surface science reports ELSEVIER

Surface Science Reports 22 (1995) 1-71

Low-energy EELS investigation of surface electronic excitations on metals M. Rocca Centro di Fisica delle Superfici e delle Basse Temperature del CNR and INFM, Dipartimento di Fisica, via Dodecaneso 33, 1-16146 Genova, Italy

Manuscript receivedin final form 7 February 1995

Abstract

Recent progress in experimental and theoretical investigations of surface electronic excitations of metals is reviewed with an emphasis on surface plasmon dispersion. The experimental methods applied to these studies are critically discussed, highlighting present limitations and possible future developments. Available data on the dependence of surface plasmon frequency, dispersion and damping on crystal temperature and of surface plasmon damping on parallel momentum and energy are collected. Surface plasmon dispersion and Mie resonance shift as a function of cluster size are compared when possible. The implications of surface plasmon dispersion for the position of the centroid of the induced charge at the surface and in general for the optical surface properties are presented and discussed.

1. Introduction

The study of the dynamical screening properties at surfaces has always attracted considerable attention because of the large number of physical properties influenced by them [ 1-3 ]. These include e.g.: the nature of Van der Waals forces [4] and the energy transfer in gas-surface interaction [5]; the damping of surface vibrational modes [6] and the de-excitation of adsorbed molecules [7] ; the surface photoelectric effect [ 1,8-11]; the dispersion of surface plasmons [ 1,12,13]; and in general the surface optical properties [ 1,10]. As an example to illustrate the importance of the electronic excitation spectrum let us consider the case of a photoemission study (photoemission spectroscopy (PES)), which led to wrong conclusions although being based on very reasonable assumptions: the adsorption of C1 on G e ( l l l ) and on Si(111) [ 14,15]. For the first substrate no photoemission from the Pz bond of C1 with the substrate was observed, while it was clearly present in the spectra recorded for C1 on Si( 111 ). On the basis of the relevant selection rules for s- and p-polarized light, the authors suggested that CI occupies two different adsorption sites: the atop position on Si and the four-fold hollow site on Ge. The matrix 0167-5729/95/$29.00 (~) 1995 Elsevier Science B.V. All rights reserved SSDI 0167-5729 ( 95 ) 00004-6

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M. Rocca/Surface Science Reports 22 (1995) 1-71

elements describing the transmission of the electrons through the surface might, however, cause the intensity of a peak to vanish. This was indeed (unfortunately) the case for C1 adsorption on Ge, as one of the same authors had to recognize later, when he investigated the Extended X-ray Appearance Fine Structure (EXAFS) and the Near-Edge X-ray Appearance Fine Structure (NEXAFS) of the same systems [ 16]. EXAFS contains information about the nearest-neighbor distance of C1 and NEXAFS on the bonds of C1 with the substrate. The spectra were nearly identical for the two systems, showing unambiguously that C1 occupies the atop site on both substrates. The same information could have been extracted from PES if the surface electronic excitation spectrum of that system would have been known. Surface electronic excitations and in particular surface plasmons have been studied for more than three decades, The first prediction of the existence of surface plasmons is due to Ritchie in 1957 [ 12], followed by Stem and Ferrell [ 17 ] in 1960 using a macroscopic dielectric treatment. Their existence was eventually verified by Powell and Swan [ 18] in 1959 for the metal/vacuum interface of A1 and Mg with electron energy-loss spectroscopy (EELS). Surface plasmons on metal surfaces were then studied by several groups. Theorists approached the problem either from the macroscopic hydrodynamic model or from more complicated microscopic quantum-mechanical calculations. Bennett [ 13 ] was the first to propose that a negative dispersion of the surface plasmon should be expected because the electron density varies smoothly across the metal/vacuum interface. Eventually this prediction was supported also by other theories [ 19], while other models favored still a positive sign for the dispersion [20,21 ]. Experimentally, surface plasmons were studied optically on corrugated surfaces or on surfaces with a grating structure, which act as couplers between the surface plasmon field and the photons, and with high-energy (~50 keV) angle-resolved electron energy-loss spectroscopy either transmitted through or reflected off thin films [22,23]. The results were, however, ambiguous: either no dispersion or a positive dispersion of the surface plasmon were reported. The ambiguity (see e.g. Refs. [ 24,25 ] ) was ascribed to the poor quality of the investigated surfaces, which might have been rough and covered with contaminants. Low-Miller-index surfaces of aluminum single crystals were eventually investigated with inelastic low-energy (~100 eV) electron diffraction [26]. In apparatuses for LEED studies surface properties are at a premium so that the experiment is performed in ultrahigh vacuum and facilities to control the status of the surface are readily available. Moreover, contrary to transmission experiments, which are limited to thin films, well defined single-crystal surfaces could be investigated. An anisotropy of the surface plasmon dispersion with respect to the different crystal faces was found, whereby AI(001) showed a negative and AI(111) a positive initial slope. The accuracy of the results was, however, questioned and no further experiments were reported with this method. A major development in theory took place when Feibelman demonstrated theoretically the link between surface plasmon dispersion and position of the centroid of the screening charge at the surface plasma frequency in a sophisticated, microscopic calculation. For a realistic potential the centroid of the induced charge is in the low-density spill-out region outside the geometric surface (defined by the jellium edge), where the electron gas is more compressible. At small transferred momentum a linear negative surface plasmon dispersion is hence expected. Quantitative results for the magnitude of the dispersion for the different electron densities were given [ 1,27,28], which were later confirmed by other authors (see e.g. Ref. [ 3 ] ). Angle-resolved low-energy (~10 eV) electron energy-loss spectroscopy (EELS) was first applied

M. Rocca/Surface Science Reports 22 (1995) 1-71

5

to study surface electronic excitations on semiconductor and insulator surfaces [29-34]. The interest was focused on interband transitions, to determine the energy gap between valence and conduction band at the surface, and on the properties of the two-dimensional free-electron gas at the Schottky inversion layer at the surface of a semiconductor. Distinct energy-loss peaks at low energy were oberved in-specular, which were ascribed to collective excitations of free carriers. The dispersion of such losses with impact energy was attributed to a free-carrier depletion layer at the surface [35] and depending on the system was either negative [36,37] or positive [38]. Other groups investigated with low-energy EELS the energetic broadening of the electron beam specularly reflected off semiconductor surfaces [39] or thin metal films [40]. The broadening is caused by multiple excitation of low-frequency surface plasmons connected to metallic surface properties. Surface plasmon dispersion on metals (Ag and thin Ag film on S i ( l l l ) ) were first investigated in Marseille by Layet and coworkers [41,42] in 1987. The potential importance of this work was, however, not recognized and little relevance was given to those results. A systematic investigation of surface plasmon properties on metal surfaces was started independently a couple of years later by the group of Plummer and by my group in Genova. These studies not only confirmed unambiguously Feibelman's forecast of a negative dispersion on alkali metal films [43], but disclosed a new very promising research field. Multipole modes were observed [44], which had been predicted by theory years before [ 13,45,46], and on Ag surfaces a positive dispersion was reported, which is anisotropic with respect to crystallographic plane and direction [47-50]. Work on this subject is still in progress, especially on the theoretical interpretation of the results [51,52]. Major reviews on the theory of surface electronic excitations were written at the beginning of the 1980's by Feibelman [ 1 ] and Gumhalter [53]. Major experimental reviews appeared so far only for high-energy EELS experiments in transmission through thin films [22,23]. In 1993 the notes were published of a lecture which I held on low-energy EELS investigations of surface plasmons at a summer school held at the ICTP in Trieste [54]. This review will focus on the physics which can be accessed by studying surface plasmon dispersion. It has the following outline: The fundamental concepts on surface electronic excitations are introduced in Section 2 with the theoretical forecast for single-metal surfaces. Limits and possibilities of the applied experimental methods are discussed in Section 3; a brief review of the fundamental concepts of electron scattering off surfaces is given, too. The existing experimental data base is finally presented in Section 4 for simple metals and in Sections 5 and 6 for d-electron metals with filled and open d-shells, respectively. Possible future developments in the field are outlined in the conclusions.

2. Surface plasmons: overview of the fundamental concepts 2.1. What are surface plasma waves and Mie resonances

A surface plasmon is a self-sustaining oscillation of the electron gas at the surface characterized by an exponential decay of the potential • associated to it, both toward the vacuum and toward the bulk, and by an oscillatory behavior along the surface, as shown in Fig. 1 [55]. • can be described by: (I) ( r ) = (I~0 e iqll'rll e -qll Iz I,

(l)

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M. Rocca/Surface Science Reports 22 (1995) 1-71

where rll defines the position vector in the surface plane, qll is the momentum of the surface plasmon and z the distance normal to the surface from the truncation plane of the crystal, half a lattice spacing away from the outermost ion cores. The condition for the existence of such a wave can be easily derived from classical electrodynamics, obtaining the field Ez by differentiating Eq. (1) with respect to the z direction and computing it just above and just below the surface plane: Ez (z = 0 +) = ~0qll e'q"'rll, Ez ( z = 0 - )

=--dPOqllelqll'rll.

(2a) (2b)

Ez is thus discontinuous across the surface. As no unbalanced surface charge can exist in a metal the z component of the electric displacement vector D z = EE z must, however, be continuous across the

metal/vacuum interface, i.e. its divergence must vanish. The continuity of D, or X7 • D = 0, can only be satisfied when the bulk dielectric constant E(to) is: E(OJsp) = - 1 .

(3)

Eq. (3) defines the surface plasma frequency tOsp, which is hence determined by the bulk properties of the solid. For a free-electron gas E(w) reads: E ( t o ) = 1 - - O)p/(.O 2 2,

(4)

where wp = x / ~ m e o

(5)

is the plasma frequency in the bulk (occurring at E(OJp) = 0) for an electron gas of density n, electron mass m and charge e, while ~0 is the permittivity constant of vacuum. Substituting Eq. (5) into Eq. (4) one obtains for the surface plasma frequency: (.Osp = O.)p/V/2.

(6)

Apart from the surface plasmon mode further collective excitations can exist at a surface: the so-called multipole surface plasmons [ 13,45 ]. They are characterized by a multipolar distribution of the induced charge also in the direction normal to the surface. In particular the dipolar mode occurs at (.Omp= 0.8¢0p. The induced charge density of surface plasmon and dipole surface plasmon are shown in Fig. 2. Multipolar modes are responsible for the observed enhancement of the second harmonic generation signal at simple metal surfaces when the laser frequency gets close to 0.4tOp [57].

Fig. 1. Surface charge distribution and electric field lines of a surface plasmon of wavevector qll-

M. Rocca/Surface Science Reports 22 (1995) 1-71

8

(b) M u l t i p o l e M o d e

x3

rs = 5 q = 0.15 k F

/./-.\. //'/' ¢t.,O °_ 60 c" (D

- -

\.\

R e 5 n (z Im 5 n (z~)

4

/

\

/

E) 2 -(3 Q) o 'o

-20

\

/

~

-2

7

\

lasm°n~x

I -16

I -12

I -8

I -4

z (Bohr

\\\.

0

I 4

I 8

I

12

units)

Fig. 2. Induced charge density distribution 8n(to, z) calculated for rs = 5 ( a ) at the frequency o f the multipole plasmon and ( b ) at the frequency o f the surface plasmon (from Ref. [ 5 6 ] , used with permission).

If the surface is not flat Eq. (3) has to be modified. For spherical metal particles classical electrodynamics allows for a surface excitation at the Mie frequency given by [58]" E ( t O M ) --~ - - 2 .

(7)

The ratio reads then I / x / 3 for simple metal clusters. For non-spherical cluster shapes tOM/tOp depends on the form of the cluster. For compact clusters it ranges from 1 / v ~ to l / x / 2 [59]. In the presence of d-electrons the simple relation between surface plasmon and bulk plasmon frequency in Eq. (6) does not hold anymore as E has now a free-electron contribution Ef and a boundelectron contribution Eb. Surface plasmons are expected whenever E(tOsp) = Ef(tOsp ) -q-Eb(tOsp ) ---~--1. Moreover, the dielectric function is no longer real, but has an imaginary part, E(tO) = e I (tO) -[-iE2(tO), due to the presence of decay channels connected to the presence of interband transitions, a mechanism which is particularly efficient in the presence of d-bands. For this reason surface plasmon losses are in general quite broad features, apart from a few remarkable exceptions. The presence of ie2 (tO) has the consequence that the resonance condition is slightly shifted from the frequency determined by Eq. 2

~-~ © 7"

I ....

I ....

0

2Ii

m~r ©

I ....

Ag

-4

i

2

, I ....

4

I ....

6 energy

I ....

8 (eV)

10

Fig. 3. Dielectric function ~(to) for A g versus energy as measured by optical means, according to Palik [ 6 0 ] .

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M. Rocca/Surface Science Reports 22 (1995) 1-71

(3); a peak in the loss spectra is then expected when the energy-loss function has a maximum, i.e. for maxima of Im[ l/E] in the bulk and of Im[ 1/(E + 1) ] at the surface. E(to) can be a steep function of to near this value, so that surface plasmon and bulk plasmon may be very close in frequency as is indeed the case for Ag (see Fig. 3) [60]. Moreover, E(to) can cross the 0 and - 1 values more than once (as is e.g. the case for Nb [61]), giving possibly rise to several plasmon and surface plasmon losses. Plasmons will then occur when Re[E(to)] has a positive first derivative in the relevant range of to. This might be the case also for Ag according to Palik [60], while older, but often quoted, optical measurements by Ehrenreich and Philipp [62] (see Fig. 4) exclude it. As shown in Fig. 3 the resonance condition for the electron plasma is met for Ag according to optical measurements at 3.78 eV for the bulk. At the surface a peak is expected at 3.68 eV according ,

T

,

,

,

4

2

-2

I

-..~°

~I (e~pt

-4

-6

-B

-I0

I I I I

-12

o

2

.i

; E,

~

,'o

ev

Fig. 4. Same as Fig. 3, but according to Ehrenreich and Philipp (from Ref. [621, used with permission).

M. Rocca/Surface Science Reports 22 (1995) 1-71

6z,O~

7BO~

1

9

900~,

L y

Fig. 5. EEL spectrum recorded with swift electrons (Ei = 50 keV) for thin Ag foils of different thickness at room temperature. Bulk and surface plasmons are clearly resolved, at htOp = 3.78 eV and htosp = 3.63 eV as e2((o) nearly vanishes in this frequency range (from Ref. [65], used with permission).

to Ref. [60] and at 3.65 eV according to Ref. [63]. Transmission experiments of swift electrons through thin films confirmed these values showing sharp losses at 3.63 and 3.78 eV [65], as shown in Fig. 5. Both values are heavily displaced from those expected for a free-electron gas which, for the density of Ag, r~ = 3, should resonate at htop = 9.05 eV and htOsp = 6.4 eV. The plasmon frequency is shifted by the presence of the d ~ s interband transition, whose threshold lies at 3.98 eV at 300 K [64] ). A broad and weak loss was observed in transmission at 8 eV [66] and at the surface at 7.1 eV [71 ]. These features are due to the low value of e (to) in this frequency region and are often referred to as remnants of the free-electron excitations. Although so near in energy, surface and bulk plasmons appear well resolved in the experiment as e2 (to) nearly vanishes in the relevant frequency range, thus giving rise to an extremely sharp structure [(AtOp/tOp) ~ (Ato~p/tosp) = 2 × 10-2]. Ag is therefore the ideal candidate to study the properties of electronic excitations with high-resolution EELS.

2.2. Surface plasmon dispersion on jellium The link between surface plasmon dispersion and the position of the centroid of the induced charge was derived theoretically for a half infinite free-electron gas (jellium model) already in the 1970's by different authors [27,67,68]. The dispersion relation is accordingly linear for small qll:

where the proportionality factor is given by the difference between dx (O~p), the frequency-dependent centroid of the fluctuating surface charge at romp:

M. Rocca/Surface Science Reports 22 (1995) 1-71

10

nIz/

dir(~) Fig. 6. Position of the centroid of the dynamical screening charge with respect to the geometric surface (half a lattice spacing from the outermost nuclei) for electric fields polarized parallel, dll, and vertical to the surface, d±. dll coincides with the geometric surface plane for a fiat surface, n(z) and n+(z) represent the positive jellium background and the negative charge density. ~b indicates the electric potential associated to the surface plasmon field at two different qll values. The field penetration into the surface is proportional to qll" 4,' corresponds thus to a smaller value of qll than ~b.

f dz z 8n(Wsp,z)

(9)

d± (tOsp) = f dz 8n (tOsp, z ) ' and dll (tOsp), the response to fields parallel to the surface, which within RPA is frequency independent and coincides with the edge of the static electron density no(z):

f dZ Z d~nO(z) dll =

(10)

f dZ d~nO(z) dEi coincides generally with the truncation plane of the geometric surface half a lattice spacing away from the outermost crystal plane (see Fig. 6). Due to the larger compressibility of the electron gas at lower densities, d±(to) lies outside the crystal in the low-density spill-out region for nearly freeelectron metals. The d quantities were first computed with a microscopical approach by Feibelman [11. A pictorial view of the link between the position of the centroid of the charge and the dispersion of the surface plasmon was given by Feibelman [28]. As shown in Eq. (1) the penetration of the potential associated with the perturbation depends exponentially on its wavevector. One can accordingly define an effective charge density neff seen by the excitation. If d ± is outside the surface, as was demonstrated to be the case for jellium, neff decreases with qlt as the perturbation becomes more and more localized at the surface. The plasma frequency will then decrease, according to Eq. (5), as wp ~ v,-~-~. For small qll the decrease is linear. The dependence of d± - dll on o~ was computed for simple metals within the jellium approximation [ 1,3,69]. As shown in Fig. 7, d±-dll is positive for oJ << Wp and becomes negative beyond ~o = 0.8Wp, i.e. beyond the surface multipole plasmon frequency ¢-Omp.Feibelman [27] and Kempa and Schaich [21] computed d±(o~sp ) - d l l for different shapes of the surface barrier, demonstrating that it is always positive in the presence of electron spill-out and it depends strongly on the form of the surface potential barrier, d ± (O~p) - d l l can he negative only when the electrons are confined artificially to the

M. Rocca/Surface Science Reports 22 (1995) 1-71

I1

---?.

"10 0

E

o

oo/OOp

~o

o

co/cop

1.o

Fig. 7. Real and imaginary parts of (d±(to) - dll (to)) as calculated within RPA and LDA for different jellium densities (from Ref. [69], used with permission).

solid by an infinite work function as it is the case for the unphysical infinite barrier model. This result holds true also for the more sophisticated stabilized jellium models [70] which imply in general a decrease of the image plane distance from the jellium edge compared to ordinary jellium [72]. By analogy'with surface plasmon frequency the Mie resonance energy is predicted to shift with cluster size as this excitation can be regarded as the superposition of surface plasmons of different wavelengths excited at its surface. Assuming a spherical shape, with radius R, the maximum wavelength, A, of the surface plasmons supported by the cluster is limited by its circumference 27rR. Apart from minor effects connected to the compact shape of the cluster, the 1/R dependence of tOM is then linked to the qll dependence of the surface plasmon dispersion [48] and is hence determined by the spilling out of the electronic charge at the surface of the cluster as was first proposed by Apell and Ljungbert [73]. The shift of the Mie frequency is accordingly given by: AtO tOM

3 Re dr(toM) 2 R '

(11)

where dr(tOM) is the centroid of the induced charge measured with respect to the jellium edge. In the limit R ~ c~, dr becomes d± of the plane surface.

2.3. Surface plasmon damping By analogy with the bulk plasmon case, the lifetime of the surface plasmon is limited by its decay into single particle excitations. One usually divides the decay processes into the direct Landau

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M. Rocca/Surface Science Reports 22 (1995) 1-71

mechanism which involves a single sp-band or direct d ~ sp transitions and the indirect Drude mechanism in which the inter- and intraband transitions are mediated by the exchange of reciprocal lattice vectors or of phonons [74]. Jellium theory [ 1 ] includes obviously only the Landau mechanism involving sp-electrons; it predicts an infinite lifetime at qll = 0 and a linear growth of the loss line with qlb corresponding to the growing phase space into which the surface plasmon can decay. The Drude mechanism causes a finite line width even at qll = 0. The elastic mechanism is then generally dominant. Electron-phonon interaction received therefore less attention and was directly addressed, to my knowledge, only in a minor theoretical paper on volume plasmons [75]. In real systems the lifetime of the surface plasmon will be further limited by scattering against crystallographic imperfections.

3. Experimental aspects of EELS measurements of surface plasmons The study of surface electronic excitations is based either on optical methods or on the inelastic scattering of electrons. When applying optical methods a coupling between the surface plasmon field and the electromagnetic field is necessary, which is generated either by a grating or by the attenuated total reflection method. These methods are, however, limited to quite large wavelengths at which retardation effects are important and give rise to so-called surface plasmon-polaritons. These phenomena were discussed e.g. by Maradudin [76] and fall outside the scope of the present review. Larger wavevectors can only be achieved by inelastic scattering of electrons. Most of the work with this technique was performed by the group of Raether with high-energy electrons transmitted through thin films [23], while low-energy electrons were used only recently. 3.1. High-energy EELS High-energy EELS was applied to surface plasmon dispersion already in the 1960's and has the advantage that also bulk excitations can be probed with the same apparatus thus allowing for a complete investigation of the system. As shown in Fig. 8 electrons penetrating a solid transfer to the electrons of the solid a momentum hq = h(ki - ks) and an energy Eloss = Ei - Es. The projection of q onto the surface of the film, qfl' determines the wavevector of the surface excitation. Electrons deflected to a scattering angle 0s correspond to the momentum transfer hql I = hks sin 0s ~ hksOs, as 0s is generally small. The energy dispersion relation is then obtained by measuring E~oss versus 0s. An electron energy resolution of 50 meV was employed to study plasmon excitations on Ag [ 65 ]. Larger half-widths of the order of some tenths of eV were, however, more common, as they can be obtained without monochromatizing the incident beam. Angular resolutions of (2-3) x 1 0 - 4 rad were achieved for the detector. At Ei = 50 keV this corresponds to a qll resolution of (2-3) x 10 -2/~-~. The major difficulty is connected with the dominant dipole scattering mechanism, which causes the intensity of surface losses to decrease like 0; -3 (contrary to bulk losses whose intensity falls like 0~-2). qll values larger than 0.3/~,-~ could hence not be investigated. EELS with swift electrons was originally developed for the investigation of bulk properties so that no facilities were present in the experimental apparatuses to check the status of cleanliness and order of the surface. In fact, little or no attention was paid in these early experiments to the preparation of the surface of the sample; in most cases the sample was not even in ultrahigh-vacuum environment!

M. Rocca/Surface Science Reports 22 (1995) 1-71

13

kel film •:

~e|

~p

(

....'i

~.

W$

fL k~

Fig. 8. High-energy electron energy-loss experiment: the electrons impinge onto a thin foil and lose energy and momentum to surface electronic excitations [22]. 3.2. Low-energy EELS

Low-energy electrons penetrate little into a crystal and are therefore ideally suited for surface investigations. Low-energy EELS was first applied in the form of inelastic low-energy electron diffraction (ILEED) [26] using a retarding field analyzer to sort out the inelastically from the quasi-elastically scattered electons, i.e. from the unresolved contributions due to elastic diffraction and to phonon excitation. The method was applied to single-crystal A1 surfaces. The investigation suggested a puzzling anisotropy between the (111 ) and the (001) surface which showed respectively a positive and a negative dispersion. The authors concluded however soon that the results were not as conclusive as they would have liked them to be as a large ambiguity for the choice of linear and quadratic coefficients was possible. The only sure information they could draw was therefore that the dispersion on AI(001 ) was "flatter" than on AI( 111 ). Angle-resolved low-energy EELS was applied to study surface plasmon dispersion only recently. In apparatuses designed for low-energy electron scattering surface properties are generally at a premium so that facilities to control surface order and cleanliness, such as LEED and Auger electron spectroscopy, are generally available [ 77 ]. Reproducible data on surface plasmon dispersion and damping could so be collected. 3.2.1. Dipole and impact scattering Prior to discussing the details and the limits of low-energy EELS let us recall the important concepts used to describe the limiting cases occurring in inelastic electron-surface interaction [78,79]: dipole and impact scattering. Dipole or Coulomb scattering corresponds to inelastic events mediated by the long-range Coulomb

M. Rocca/SurfaceScienceReports22 (1995)1-71

14

Fig. 9. Possible loss events in dipole scattering. The loss is caused by long-range electric fields and may take place either before or after the impact with the surface while the electron is on its way toward the surface or away from it. forces originating from oscillating electric charges. The loss may already take place far away from the surface, so that the inelastic event can be divided into the loss process, occurring while the electron is approaching or leaving the surface, and the elastic reflection off the surface, which includes diffraction and multiple scattering (see Fig. 9). Two major cases are possible, corresponding to events for which the loss takes place on the impinging trajectory (loss before reflection) and to events for which the loss takes place on the way back toward the vacuum (loss after reflection). The amplitudes of these two processes add up and interfere. A semiclassical theory for dipole scattering was developed by Lucas [80,81], while a complete quantum-mechanical approach was given by Mills [82]. Following Mills [78,82] the differential scattering cross section d2S/dtodfl, in the solid angle df~ and in the frequency range do), is given by: d2S dto dl)

m2e2v2

ks e(qll, to)

2rr2h 5 COS 0 i ki x

q~

v±qll(Rs q- Ri) q- i(Ri - Rs) (to - vii" qll) 2

,

(12)

[v2 ql~ + (to - vii "qll )2]2" where m and e are the mass and electric charge of the electron, vii and v± are the parallel and normal components of the velocity of the impinging electron with respect to the surface and P (qll, to) is the surface loss function, which contains the surface information. Ri and Rs are the complex reflectivity amplitudes at the initial and final energy, relevant for loss after reflection and loss before reflection, respectively. A useful expression for P (qll, to) was derived for a three-layer model description of the crystal (vacuum, surface layer, bulk) and reads: -1 ) + 1 ' P(qll, to) = 2hqllrr [1 + n ( t o ) ] Im g(qll,~,7

(13)

where g is the effective dielectric function of the system defined as:

g(qll'to)='s(to)[ lq-A(to) e - A(to) 2 q " We-2qlt '- w and

(14)

M. Rocca/Surface Science Reports 22 (1995) 1-71 Eb(ca) -- Es(ca) A = eb(ca) + Es(ca)"

15

(15)

n(ca) = 1 / ( e ~/kBr - 1) is the Bose-Einstein factor and Es is the dielectric function in the surface layer of thickness W and Eb is the dielectric function in the bulk. In the limit W ~ 0 Eqs. ( 1 3 ) - ( 1 5 ) describe the bare surface. The surface plasmon loss intensity scales then as 1/x/-&i. If the elastic reflectivity R is constant over the energy range of the experiment, the loss cross section is proportional to R 2, i.e. to the specular intensity. The surface reflectivity, which contains the multiple-scattering problem, can thus be eliminated by normalizing the loss intensity to the specular elastic intensity. This procedure is usual for vibrational losses but should be avoided for electronic excitations as the energy loss is usually large and one cannot assume R to be the same at E~ and Es. Inserting Eq. (13) into Eq. (12) one can see that the inelastic cross section vanishes at qll = 0. The divergence for 0 i ~ 90 ° in Eq. (12) is removed when taking the image potential into account [81 ]. The maximum of the inelastic intensity occurs for ca = vllqll, where the denominator of Eq. (12) has a minimum. This condition, called surfing condition, corresponds to the interaction of the electron with partial waves of phase velocity ca/qll equal to vii. If we define by O the deviation of the trajectory of the inelastic electron from the specular direction, then for hca << E~ and O << 1 the denominator can be arranged to read: 2 2

v±qll + (ca -

vii. qll) 2 -- 4E2(02 + O~) cos 2 0i,

(16)

with Oe = hca/2Ei. Eq. (16) highlights the angular dependence of dipole scattering and its concentration in a lobe with half-width Oe. Dipole scattering dominates therefore at small momentum transfer (small-angle scattering). When sweeping the detector across the empty cone of dipole scattering one therefore expects two maxima in the loss intensity, with a principal maximum centered around the surfing condition and a minor maximum at positive qll- If the detector is kept fixed and the monochromator is swept angularly, as is the case for most commercial EELS, the minor lobe will appear at negative qll. When measuring electronic excitations the energy loss, Elope, is often comparable to Ei so that care must be taken in applying the above theory. An extension of Mill's theory to large losses was developed recently by Hall and Mills [ 83 ]. The above-discussed events of loss before reflection and reflection before loss, correspond to the terms with Ri2 and R~ in Eq. (12). Ri and Rs can now differ significantly in magnitude as well as in phase. The minimum of the inelastic cross section may therefore be displaced away from qll = 0 [83] as was indeed verified experimentally [47] (see below). An alternative explanation for the displacement of the minimum off qll = 0 invokes the different self-energy of the impinging and of the scattered electron [84]. In this case, however, two minima, at positive and negative qll, are predicted to occur, contrary to the experimental result where a single minimum is observed, either at positive or at negative qll. As surface plasmons are mainly excited by dipole scattering this shift provides a good opportunity to perform precise measurements also near qll "~ 0. In impact scattering the loss takes place directly at the impact of the electron with the surface ion cores and the electrons are hence scattered over the whole solid angle. The differential probability to lose energy in the solid angle dl) then reads [78]: dS m ACOS2 OsEi ({nqllt} s [ f ( k ~ , ki; {R}) [{nq, t}i) 2, dl'] = 2"/r2h 2 ~0"0S-'-~i

(17)

16

M. Rocca/Surface Science Reports 22 (1995) 1-71

where (nqllt } is the ensemble of the occupation numbers of the modes t with wavevector qll before

(i) and after (s) the inelastic event. A is the area of the crystal over which coherent scattering takes place. The scattering amplitude f ( k s , ki; {R}) describes the amplitude of the wave scattered off an ensemble of atoms located at {R} and contains the multiple-scattering effects. It is inversely proportional to ~ so that Eq. (17) is obviously independent of A. As one can see, impact scattering is more important at larger Ei and more normal incidence. In low-energy EELS investigations of electronic exitations El is often comparable to E~oss. No clear distinction can therefore be drawn between dipole and impact scattering. The angular dependence of the inelastic intensity associated to surface plasmon excitations follows, however, the dipolar lobe structure surprisingly well. Large-angle impact scattering, on the other hand, turned out to be important for interband transitions.

3.2.2. Low-energy EELS spectrometers The energy-loss spectra of electronic excitations were recorded with EEL spectrometers of different design. In our group we used self-built spectrometers with cylindrical deflector analyzers, CDA127 °, either single or double pass. In such electrostatic devices the trajectories of the electrons present a stigmatic focus at an angle of 127 ° from the entrance slit for an ideal field, i.e. without the fringing fields introduced by the presence of the equipotential slit plates. In real devices the angle at which focusing occurs is therefore somewhat smaller and can nowadays be optimized by means of ray-tracing programs [85]. Other groups used commercial spectrometers: Contini and Layet [41] employed an EELS with hemispherical monochromator and analyzer (spherical deflector analyzer, SDA180 °) built by Riber; Plummet and coworkers [43,44] used double-pass CDA127 ° spectrometers built by Leybold and L-K Technologies. All these instruments were designed for vibrational spectroscopy and are therefore not optimized for the study of electronic excitations. They are characterized by angular acceptances of 1° to 2 °. In order to achieve a high-enough current to measure surface plasmons the energy resolution had to be degraded in all cases to 15-20 meV. Although different instruments were used the values reported by the different laboratories for the surface plasmon frequency on Ag agree within ten meV. Let us describe our single-pass spectrometer as a prototype of EELS. The other designs differ with respect to the nature and the number of monochromatizing elements but the general scheme for the construction is similar. As shown in Fig. 10 the electron beam is generated by a cathode (K) and focused on the entrance slit of the monochromator ( l ) by the triple-lens system (A) and by the repeller (R). The monochromator is of the cylindrical deflector type. Electrons within a narrow energy window can pass through the exit slit. The energy window AEi is determined by the width of entrance and exit slits, by the mean radius of the device and by the voltage difference applied to the internal and external electrodes of the cylindrical condenser. The beam is then focused on the sample (S) by the double-lens system (B1 and B2). The reflected electrons are then focused on the entrance slit of the analyzer (5) by another lens system and analyzed in energy by passing through another cylindrical condenser. The signal is then recorded by a channeltron (C). The angular resolution a of such a device is 1.5 °, as results from an angular acceptance of 1° for both monochromator and analyzer. Due to the low cross section of the surface plasmon a monochromatic current of s o m e 1 0 - 9 A in the direct beam is necessary to have a large-enough count rate also off-specular. This goal is easily achieved at the reasonable resolution of 15-20 meV, depending on the effort put into the tuning of the spectrometer.

M. Rocca/Surface Science Reports 22 (1995) 1-71

EELS

5

4

B4B3

17

3

2

B2B1

B

C

R

K

C I

I

I

5

I

1

I

om

Fig. 10. EEL spectrometer of CDA127 ° design [77].

When measuring electronic excitations, energy resolution is rarely the limiting parameter in the measurements as, at least for metals, the natural width of the losses is of the order of several tens of meV. The quality of the spectra is on the other hand given by the angular acceptance of the spectrometer. This parameter was actually maximized in the existing designs in order to gain in monochromatic current. A qtl resolution comparable to the one obtained with high-energy electrons was thus available only when working below 20 eV and for grazing scattering conditions. Dedicated EELS design would hence be necessary to improve the experimental investigation of electronic excitations. Work in this direction is in progress in several laboratories. An interesting possibility is to apply the spot profile analysis commonly used in LEED to the inelastic signal. A qll resolution of 10-3 ~ - l should then be easily achievable [86].

3.2.3. Measurement of surface plasmons A spectrum recorded with our spectrometer for Ag(001) at Ei = 105 eV is reported in Fig. 11. As one can see, one single energy-loss peak is present, which presents a tail on the high-energyloss side. The asymmetry of the loss peak is partially due to artifacts connected to integration in momentum space, which will be discussed in the following, and partially to excitations present on the high-energy-loss side. Such contributions to the loss intensity may be due either to the excitation of the volume plasmon or of electron-hole pairs. At lower impact energy the peak appears more

M. Rocca/ Surface Science Reports 22 (1995) 1-71

18 03 ..,J

C 3

~g(O01) 3

8.8°of~ <100>

q.(3.78)=O.O66A

9

a

-t

EI=IOS.OeV 81 = 8 s = 4 4 . 7 °

..O (_

>-

CO Z LId F-Z I-"4

4.0

5.0 ENERGY

B .0

7.0

LOSS

(eV)

8.0

Fig. 11. EEL spectrum recorded at Ei = 105 eV over a wide range of Elo~s. The asymmetry of the loss peak is enhanced by the large qll integration under these experimental conditions.

symmetric and one can reasonably assume that the loss is dominated by the surface plasmon. The dispersion relation is obtained by plotting the maximum of the loss intensity, E~oss, versus qllTwo limiting cases can be distiguished: (a) The angular acceptance, a, of the EELS is smaller than the width of the dipole cone, as it is the case for the geometry of Fig. 12. One can then sweep through the dipole cone by changing either the angle of incidence, 0i, or the angle of scattering, 0s. qll is then determined from Eloss, 0i and 0s, by applying the conservation laws for energy and momentum: Eloss = Ei - Es,

(18a)

hqll

(18b)

= ki sin 0i - ks sin 0s,

where ki and ks are the wavevector of the incident and of the scattered electron, respectively, so that:

(o,o)

qll'A'ql I Fig. 12. Scattering geometry of the EELS experiment, on the basis of the Ewald-sphere description of the scattering process. indicates the angular acceptance of the EEL spectrometer. Its translation in reciprocal space determines the integration window AqlI.

M. Rocca/Surface Science Reports 22 (1995) 1-71 q ll - ~2~--~i (sin Oi - ~ / 1 - E]o~s/ Ei sin O~)

19

(19)

In this limit the experimental accuracy is determined by a [87,88]. As one can see in Fig. 12, a is translated into an integration over momentum space, whose width Aql I can be obtained by differentiating Eq. ( 19): Aql I - ~

cos0i +

--

cos

(20)

Eq. (20) shows that the effect of integration is smaller for grazing geometry and low El. For Ag such a situation is typically verified for Ei < 20 eV. Frequency and intensity of the losses, measured for Ag(001) at Ei = 10.5 eV and 0s = 86.26 ° and analyzed according to Eqs. (18) and (19) are shown in Fig. 13. The bar indicates the integration window in qll space. (b) a is larger than the width of the dipole cone. All inelastically scattered electrons are then collected and qll depends only on the value of Eloss and 0 i. Only spectra recorded in-specular should then be analyzed. Aql I is then determined by the width of the dipole lobe which shrinks with Ei. For Ag valuable conditions are obtained for Ei above 200 eV and grazing incidence. The error in qll is generally smaller than Aql I and can be inferred by the spread in the experimental data. Nowadays spectrometers are characterized by an angular acceptance ce of the order of 1°-2 °. For Ei = 50 eV and for Oi = Os = 600 and assuming a = 1° for monochromator and analyzer as appropriate for our EELS, the integration in reciprocal space spans over 0.08 /~-l. As one can see in Fig. 13, this portion of reciprocal space is comparable with a momentum range over which surface plasmon dispersion takes place. The situation would be even worse for the geometry and impact energy of the spectrum of Fig. 11. For 0~ ~ 60 ° but for Ei < 20 eV the integration window, Aql I, reduces to 0.02 ,~-~ and becomes acceptable. Optimal conditions for Ag, where Eio~s ~ 4 eV, are reached for Ei "( 10 eV and 0s ~ 70 °. The severity of the constraints due to integration over qll was not recognized immediately. Only part of the early data of Contini and Layet for A g ( l l l ) [41] was recorded e.g. at sufficiently low Ei to be valuable. The most favorable conditions for the measurement are also determined by the available inelastic intensity. Within dipole scattering, this quantity is proportional to elastic reflectivity so that it is generally advisable to search first for its maxima and then let them coincide either with Ei or with E~. The range of qll over which the investigation can be carried out is limited both for high-energy and low-energy EELS by the signal-to-noise ratio to ~0.3/~-1. At low energy the dipolar cone is larger but the effect is compensated by the larger angular acceptance of the EELS.

3.2.4. Artifacts associated with finite angular acceptance The intensity of the losses, reported versus transferred momentum qll in Fig. 13b, nicely describes the double-lobe structure expected for dipole scattering. At low Ei dipole scattering is more efficient than impact scattering especially for surface plasmons which are characterized by a large dynamical dipole moment. Looking at Fig. 13a one can see, however, that life is not as easy as one could wish: the dispersion of the loss with qll is not symmetric with respect to qll = 0, being linear with qLI for qll < 0, and nearly quadratic for qll > 0! The correct behavior is the one observed at negative qLI as demonstrated by measurements recorded under other scattering conditions [87]. So what is wrong for qlL < 0? EEL spectra, recorded under such conditions, are reported in Fig. 14. As one can see,

M. Rocca/ Surface Science Reports 22 (1995) 1-71

20

(eV)

X Ag (001) ,100) • X E-10.5eV • ~6.26 o

4.0

ENERGY LOSS 3.9

(a)

-

3.8

I -0.2

3.7

I .0.1

~,

o.1

0

(~-,)

WAVEVECTOR O (c/sic) 60 LOSS

--0-

INTENSITY/ 40

(b) 20

el .0.2

%•

I

I -0.1

i

0

(~")

o.1

WAVEVECTOR Q Fig. 13. Surface plasmon dispersion (a) and loss intensity (b) for Ag(001). The corresponding spectra are shown in Fig. 14. Note the anomaly for positive values of qll, which is connected to an artifact due to the finite angular acceptance of the spectrometer and the occurrence of LEED fine structures for the specular beam.

/ilk.. "81"7 / Ik. q , , = O 114 / iJ '~.

P,g (0011 <100> E i = l O . 5 0 eV 88=86.26 °

CO t--

/t. .

z

/ >I-I----t

O9 Z LLI I--Z

I

I 3.2S

I 3,50 ENERGY

I 3,75

LOSS

I 4.00

I 4.25

4,7B

(eV)

Fig. 14. Anomalous EEL spectra recorded for the positive qll conditions of Fig. 13 [88].

M. Rocca/Surface Science Reports 22 (1995) 1-71

21

within experimental uncertainty the loss frequency remains constant above a certain qll, while the loss peak broadens. The anomaly is due to the rapid variation with energy and angle of the elastic reflectivity under LEED fine structure (LFS) conditions [89] for the specular beam. As the EEL spectra are integrated over AqlI the large inelastic intensity at qll = 0.1 /~-i contributes to the side wings of the spectra recorded at more normal incidence, while little or no intensity corresponds to the nominal qll value. LFS conditions can cause also spurious structures associated with the rapidly varying elastic reflectivity [90] or with emission of electrons trapped into the image potential states [91]. Such structures are superposed to the loss spectra but can be recognized as the apparent E~oss shifts when Ei is varied. Something similar occurs near qll = 0 where dipole scattering intensity vanishes. In Fig. 13 the inelastic cross section decreases particularly rapidly for the positive-qlI side. Integration over reciprocal space involves then an upward shift for the maximum of the surface plasmon loss as the high-ql I side contributes more than the low-ql I side in the loss spectra. Fortunately for other experimental conditions the minimum of dipole scattering is shifted away from qll = 0 thus allowing for accurate measurements also near vanishing momenta [47]. Such conditions are shown e.g. for Ag(001) in Fig. 15. The displacement of the minimum toward positive qll is accidental; for A g ( l l 0 ) in fact a displacement toward negative qll was observed for most of the measurements.

3.2.5. Effects connected to surface roughness Surface roughness causes a mixing of excitations corresponding to different qll values, which is reflected in a broadening of the surface plasmon and in a shift of its frequency. Under extreme roughness conditions Kretschmann et al. [92] predicted a splitting of the loss line, as was indeed observed experimentally by Krtz et al. [93] for a thick Ag(111) film in an electrochemical-cell environment. Roughness causes resonant light emission as it provides a coupling between the surface plasmon field and the electromagnetic field of light. The distribution of the emitted light allows then the determination of the roughness parameters. The amplitude of these effects is remarkable only when performing the experiment in air or in an electrochemical cell and is negligible for experiments performed in ultrahigh vacuum on single-crystal surfaces under ordinary ultrahigh-vacuum conditions. A broadening of the loss line and finally a splitting of the surface plasmon might, however, be expected at very high temperatures if the surface undergoes a roughening transition. The possible broadening and frequency shift induced by roughness must be borne in mind when comparing measurements performed on thick films with those recorded on well ordered single-crystal surfaces.

4. Electronic excitations on simple metals

Surface electronic excitations on simple metals bear a particular importance since these systems can be described within the jellium appoximation and theory can produce reliable predictions for frequency and dispersion of the surface plasmon. The dispersion should be negative because of the link between surface plasmon dispersion and position of the centroid of the induced charge at the surface plasma frequency [ 1 ]. The verification of this prediction constituted, however, a challenge to experimentalists for more than one decade.

22

M. Rocca/ Surface Science Reports 22 (1995) 1-71 i

i

i

(DL.) ~uJ 750 co-.. ~(_~

r'""&~

dTsoo

(.

I--t.U ~--4 U) U)

~z t~m250 Lt.FHZ C21~

bl 0

>-

~-

1S

#

Z W

~ zc3 ~--~I..t.l to

10 /

ou

5

o3 .J n

0

I

I

-0.10

I

I

0

0.10

N ~ V E V E C T O R ~ , [ ~-~]

>

I F--0.2 z

o z oo ~: ._J n

~

O

-

O

O

-

0.1 d)

0

3.8

t.u 7 u.l

z o z co _./ cl

3.7 c) I

-0.10

I

I

0

I

0.10

NP, V E V E C T O R q,,[ ~ - q Fig. 15. EEL intensity of the surface plasmon for Ag(001)(100) for different scattering conditions: ( o ) El = 16.00 eV, 8s = 81.6°; ( A ) Ei = 16.00 eV, 0s = 60.0°; ( x ) Ei = 10.50 eV, 0s = 86.2 °. The bars indicate the window in momentum space over which the experiment is integrated due to the angular acceptance of the EELS. For the data denoted by ( o ) also the FWHM of the losses is reported. Note the correlation of the anomalous width with the minimum in inelastic intensity.

M. Rocca/Surface Science Reports 22 (1995) 1-71

23

4.1. Alkali and alkaline-earth metal surfaces

Swift electron investigations [94,25,95] verified accurately the prediction of the 1/x/~ ratio of surface plasmon versus volume plasmon frequency for simple metals. A negative dispersion with qll was reported for Mg [25] but the data were not considered conclusive as a positive dispersion was observed for AI. The behavior of surface plasmon dispersion was settled only recently when low-energy EELS was applied by Tsuei and coworkers to study alkali metal films [43,44,56,96,97]. The first experiment was performed on a thick film of K deposited on AI(111). Later Na, Cs, Mg and Li were investigated, too. One major difficulty in the experiment is connected with the quality of the film, which was inferred from the angular distribution of the specularly reflected elastic beam of electrons. Typical values of the angular FWHM of 2.5 ° were achieved. This value exceeds, however, still the angular acceptance (1.5 °) of the EELS, indicating that a considerable residual roughness is present even for the best films. The dispersion coefficient of the surface plasmon is probably not affected too much, while the lifetime of the collective excitation is probably heavily reduced. The measured FWHM of the energy losses are in fact always considerably larger than for the corresponding bulk values. Typical spectra recorded for K at different Oi are shown in Fig. 16. In-specular only one peak is present which is assigned to the surface plasmon as its frequency is roughly tOp/X/~. When moving the monochromator off-specular this loss shifts in frequency because of dispersion and two further losses show up roughly at 0.8tOp and at O.)p. They are therefore attributed to the excitation of the multipole surface plasmon I

I

I

I

I

K Surface Plasmon Dispersion = 15 eV e, = 600

°

~"

.

MM

BP

I

I

.I

/i

{-

-

,/"

5

SP

r 20

52 o x 4

,\ - , t - - ~ - S o e c u l a r

~ 25

ENERGY

t 30

i 35

-

-

i 40

i 45

LOSS (eV)

Fig. ]6. EEL spectra for a thick K film recorded at different incidence angles showing the surface plasmon (SP), the

multipole mode (MM) and the bulk plasmon (BE) excitations (from Ref. [56], used with permission).

M. Rocca/Surface Science Reports 22 (1995) 1-71

24

Table 1 Surface-plasmon frequency and bulk plasmon frequency for simple metals

AI Mg Li

Na K Rb Cs

hWsp (eV)

Ahwsp (eV)

hoJp (eV)

AhoJp (eV)

~'M.Osp/]:~Op

10.3 7.38 4.28 3.98 2.73 2.46 1.99

,-,3 1.2 0.5 0.6 0.2

15.1 10.4 7.12 5.72 3.72 3.41 2.90

0.5 0.7 2.2 0.4 0.3 0.6 1.2

0.69 0.71 0.60 0.70 0.73 0.72 0.69

[56] [97] [97] [56] [56] [24] [56]

0.45

[102] [94] [102] [24] [24] [24] [24]

[94] [94] [94] [94] [94] [24] [24]

and of the volume plasmon. This was the first direct observation of multipole surface modes in a scattering experiment, previous evidence being given from photoyield measurements on K [98] and A1 [8]. A comparison of the EEL spectra and photoyield data is shown in Fig. 17. The angular dependence of the inelastic intensity is reported in Fig. 18. As one can see, the surface plasmon loss is peaked in the specular direction and is thus due to dipole scattering. The multipole mode on the contrary is more intense for steeper incidence and the relevant loss mechanism is hence non-dipolar. At steeper Oi the incident electrons penetrate deeper into the surface so that the excitation of volume plasmons becomes more probable explaining the presence of the loss at top. No minimum is present of the inelastic cross section near qll = 0 as one would have expected for dipole scattering (see discussion in Section 3.2.1). Its absence is probably connected to the quite large AqlI of the losses which averages out sharp features in the cross section. In Table 1 the measured values of htOsp and htos for simple metals are collected and compared with the corresponding bulk values. As one can see, the 1 / v ~ ratio is quite well respected for all materials, except for Li for which band structure effects are known to be non-negligible and cause a downward shift of the plasmon frequency already in the bulk. For this material the conduction band is still parabolic but has a large effective mass: m* = 1.64m. The band effects are evident in Fig. 19 where for Li and Mg other peaks in addition to the surface plasmon appear in the EEL spectra. For Li interband transitions are in fact expected to start already at Eios~= 3 eV [99]. The dispersion relation versus qll is reported in Fig. 20 for the thick amorphous films of K and Na [43] and in Figs. 21 and 22 for the crystalline films of L i ( l l 0 ) and Mg(0001) [97]. As one can Table 2 Surface plasmon energy and dispersion coefficients for simple metals as resulting from a polynomial fit

hWsp (eV) AI Mg Li

Na K Rb Cs

10.3 7.38 4.28 3.98 2.73 2.46 1.99

[56] [97] [97] [56] [56] [24] [56]

A (eVA)

B (eV]k2)

C (eV]k 3)

D (eV/~4)

-3.02 -0.87 - 1.56 -0.955

9.78 2.11 2.47 0.09

-0.44 0.48

47.9 48.4

-0.44

2.8

17.2

The data for AI are too dispersed to allow extraction of a reliable dispersion relation. No data are available for Rb.

M. Rocca/Surface Science Reports 22 (1995) 1-71

25

(a) K

J

:

06

oo

0.7

08

09 (O/COp

I 0

11

12

(b) j ~ AI

o o oO~Oq;~"~ce49o j 06

I 07

i 08

I 09 (0/O)p

"x~,

%o~ P~oc I 10

~3 11

I 1.2

Fig. 17. Comparison of EELS and photoyield data for (a) K and (b) A1. Photoyield is enhanced at the multipole plasmon frequency (from Ref. [56], used with permission). 104

-D +

103

Peak Intensity

Profile - K

Elastic (/102 ) Surface Plasmon Multipole Mode

//~ / \

.......KF

/

F~=15eV 0-600

1

s-

¢t. v

q~t-0

.m u~ t--

E

"'"', []

[] []

.."

"'-., "'-..,

10 ~

o~ 13..

10 ~ ............. "'" i

45

i

i

i l'l

i

50

I

]

55

I

n

I

I

I

60

I

n

a

e

I

65

I

n

n

70

Incident Angle (deg) Fig. 18. EEL intensities recorded for a K thick film: ( ) elastic peak, (E3) surface plasmon, ( + ) multipole mode, ( .... ) kinematical factor. A similar behavior was observed for Na and Cs (from Ref. [56], used with permission).

26

M. Rocca/Surface Science Reports 22 (1995) 1-71

Table 3 d± (~Osp) for simple metals as evaluated from experimentapplying Eq. (8) for the surface plasmon and Eq. (11 ) for Mie resonance, and real and imaginaryparts of d± (COsp)determinedtheoreticallywithin RPA and LDA

rs AI Mg Li Na K Rb Cs

2.02 2.66 3.25 3.93 4.86 5.20 5.62

d±(cOsp) (/~k) 0.82 0.48 0.73 -1-0.07 0.73 + 0.06

[97] [97] [ 100] [ 100]

0.88 -t- 0.12 [ 100]

dr(tOM) (/~)

0.33 [ 113] 0.045 [ 114]

dRPA (/~)

dLDA (/~)

0.4+ 1.0i 0.6+1.1i 0.8+1.1i 0.9+0.8i 0.7+0.6i 0.6+0.5i 0.5+0.4i

0.6+ 1.2i 1.0+1.6i 1.4+1.7i 1.9+1.5i 2.2+1:0i 2.2+0.9i 2.2+0.7i

The theoretical values are taken from Fig. 7 [69] at COspby interpolatinglinearly for the rs value. For Li d±(cOsp) is given at the experimentallydeterminedratio of Acosp= 0.6~Opas the deviation from 1/v~ is considerable,rs values are given in a.u. see, in accord with theory, the surface plasmon frequency disperses initially to lower values, while the multipole plasmon shifts linearly upward. To be more quantitative the data for the surface plasmon dispersion were fitted with a polynomial function, whose best-fit coefficients are reported in Table 2. The first term is always negative, i.e. according to Eq. (8), d±(oJsp) -dll(COsp) is positive and the centroid of the induced charge lies outside the geometric surface in the low-density spill-out region. A comparison with the theoretical outcome of RPA and LDA is shown in Fig. 25 and summarized in Table 3. The agreement of the RPA calculations for a jellium of appropriate density (rs values) with experiment might look puzzling in view of the failure of the jellium to describe correctly the plasmon dispersion in the bulk, where observable deviations from theory were reported: e.g. for A1 a small azimuthal anisotropy of the dispersion was observed [102] while for Cs (rs = 5.6) the initial dispersion is negative instead of positive [ 103]. Such effects are due to the band structure of the metal, which could be included only recently into the theoretical models [ 104]. The main reason for the success in describing the surface is that the dispersion of the bulk plasmon is totally a property of the dynamics of the interacting electron system and current Fermi liquid theories do not treat exchange and correlation effects properly. On the other hand, the properties of the normal modes at the surface depend principally on the ground-state charge-density profile which is correctly described by jellium. The experimental dispersion coefficients for alkali metals were recovered also by a sum rule approach [ 105], within which the linear slope of the dispersion at small qll is determined by the diffuseness coefficient of the surface. The dispersion of the damping of the surface plasmons is shown in Fig. 22b for Mg and in Fig. 23 for K [56]. The other metals are not shown as the spread in the data points is even larger than for Mg and K. According to Tsuei et al. [56] the FWHM grows quadratically for all metals. In my opinion this result is, however, not definitive as inspection of the raw data for K could indicate also a weak linear initial growth followed by a more rapid growth at larger qll, while for Mg the FWHM could even decrease initially with qll- Such behavior is quite puzzling as a linear increase would be expected on the basis of the general argument of the linear growth of the phase space available for decay into electron-hole pairs. The data for Mg are, however, at the limit of acceptability as they are integrated over AqlI = 0.15 /~-1 so that the negative initial dispersion should not be considered as definitively proven. A negative initial dispersion of the FWHM was observed on the other hand

M. Rocca/Surface Science Reports 22 (1995) 1-71

I

(~

(a)

I

t

27

l

Mg ( O O 1 ) Ein c = 3Q eV @inc = @scat = 5 0 °

>., 4J

!p~ ~

IB

r @ 4a E

i

(b)

L

:

I

i

Li(110)

t

I

Einc= 3QeV

..

= escot = 6 0 °

Q) 4J

0

2

l

I

4

6

i

8

liO

1'2

Energy toss (eV) Fig. 19. Electron energy-loss spectra for (a) Mg(0001) and (b) Li(110). The main loss peaks are ascribed to an interband transition (IB) and to the surface plasmon (SP) (from Ref. [97], used with permission). 2.76 2.74

(a)

400 i(b

Dispersion of K Surface Plasmon /

/ Dispersion of Na Surface Plasmon/

272

~" 3'95 F ' * , . ~

270

~ 3.90

o ..a 268

~2.66

/

2 64 262

I

"1

0.0

0.1

0 1

0.2

0.3

0.0

0.1

0.2 q il( ~-1 )

0.3

q it(A- ) Fig. 20. Dispersion of the surface plasmon loss for thick films of (a) K and (b) Na. Solid curve: best fit through the data points (from Ref. [43], used with permission).

M. Rocca/Surface Science Reports 22 (1995) 1-71

28

4.5.; . . . . . . . .

....

,

....

,

,

....

i ' ~ ' 'l

....

,

....

L[(110) 4.4

o

4~r

4,0

.'".':, 0.0

, . . . . . . . . . . 0.2 0.3

0.1 qll (~,-1)

Fig. 21. Surface plasmon dispersion for crystalline L i ( l l 0 ) . Assuming an angular integration over 2.5 ° as in Ref. [56] AqlI = 0.12 ~k- l . Solid curve: best fit through the data points. Dotted and dashed lines: theoretical forecast according to RPA and LDA (from Ref. [97], used with permission). 7.8

(a)

Mg (0001)

mZ6r o

~

7.4

22

7.O

:(b) 2.O

,,,"

-

,,,"

/

/ t

•

,,'

: ~

1.5

z h

/

-

i"

!

/"

-~t/./,

/

•

// ,,'"

~

~.o

z /

Q.5

," / /" / //

/ / / z/ /'/ //"

j, 0.0

. . . . . . . .

0.0

I

. . . .

i

0.1

. . . .

I

. . . .

t

. . . .

0.2

,

. . . .

,

. . . .

i

. . . .

0.3

~

....

0.4 qll

(~-I)

Fig. 22. (a) Dispersion of surface plasmon loss frequency and (b) width for M g ( 0 0 0 l ) . The data were recorded at Ei = 30 eV and Os = 50°; assuming an angular integration over 2.5 ° AqlI = 0.15 ~ - 1 . In view of this large value the maximum of the loss width near qll = 0 could also be an artifact. Solid curve: best fit through the data points. Dotted and dashed lines: theoretical forecast according to RPA and LDA (from Ref. [97], used with permission).

29

M. Rocca/Surface Science Reports 22 (1995) 1-71

0.8-- Surfac~ 0.6

>~ v -r

0.4

0.2 o

o o ..-'"

I

I

0.0

0.1

0.2

q ll(~ 1)

0.3

Fig. 23. Dispersion of surface plasmon loss line width for K. Solid line: fit to the data points. Dashed line: RPA prediction (from Ref. [431, used with permission).

also for A g ( l l 0 ) ( l l 0 ) [135] and A g ( l l l ) [50], for which cases, however, the band structure is definitely more important than for Mg. The measured widths on all alkali films are much larger than for the bulk plasmon (jellium theory would even predict zero width at qll = 0). This could be indicative of that the observed damping is dominated by decay channels other than Landau damping and that the quadratic dependence of the loss width is connected to the roughness of the film. The multipole mode was clearly resolved for the lowest-density metals, i.e. K, Na and Cs. Its absence for low rs values is not surprising as the multipole mode is expected to be more strongly damped at larger electron densities [56]. For Mg (rs -- 2.66) only a shoulder appears in the EEL spectra which was evidenced through a fit procedure as shown in Fig. 24. The experimentally

I ' ' '

,

I

. . . .

I

. . . .

I

"

~

Surface 31asmon

Mg (0001)

Emc = 3 0 e V Multipole s u r f a c e plasmon

O,n c = 5 0 ° Omc = 62 °

E

4

6

8 10 E n e r g y Loss (eV)

12

Fig. 24. EEL spectrum for Mg(0001) showing the deconvolution into contributions corresponding to the excitation of surface plasmon, multipole mode and bulk plasmon (from Ref. [97], used with permission).

30

M. Rocca/Surface Science Reports 22 (1995) 1-71

determined dispersion curves are shown in Fig. 25 and compared with RPA and LDA. The data are summarized in Table 4. Note again the excellent agreement of the measured frequency of the multipole mode with LDA theory.

0.96

0.96 (a) K

[

RPA

0.92-

RPA , ; , - ' ~ . ~

3.5 - - 5.0

0.88

0.88 ~

o_ 0.84 ~ $

m

m

(

0.80t

0.80 (D c LU

I

(b) Na

0.92 - - ) •.°..-~ -2 °..•••'°° . ~

o_ 0.84 3

I

I

Z'"--30

<

0.76 --

< 0.76 [ -

RPA °. .. °', o.-'"

0.72 -0.68 0.64 0.0

- - 4.5

c ,,~

0.72 ~--

LDA

RPA .........

0.68

[]O 0 ~ ........... I 0.1

,,, , ,,-:I:"

I 0.2

-

'\',1

0.0

0.3

4.0

[]

o.641

2.5

-

~

0.1

q (A -1)

LDA

q (A-1)

I 0.2

I 0.3

0.96 ..-" "" R P A T

(c) Cs

0.92 .)-:-.- y .......... 088

~'& 3

-

I

-28o

0.84,

m t~

~s

0.80 (9 c LU

0.76

--

~<

- -

RPA

0.72 i-r~ 0

2.25

.

064' 0.0

....... 6

8

--"

.-'"'" []

[]

~

"

'

0.1

0.2

' 0.3

q (A 4)

Fig. 25. Dispersion of surface plasmon and multipole plasmon mode for (a) K, (b) LDA and RPA theory. Solid curves: LDA theory; dashed curves: RPA theory; dotted dispersion according to LDA theory. At small qll agreement is better with RPA, at should, however, not be overrated in view of the simplicity of the theoretical models, properties from LDA (from Ref. [56], used with permission).

Na and (c) Cs and comparison with line: initial slopes of surface plasmon large qll with LDA; these differences both of which derive the ground-state

M. Rocca/Surface Science Reports 22 (1995) 1-71

31

Table 4 Multipole plasmon frequency, loss width and dispersion for simple metals (references: low-energy electron energy loss spectroscopy (EELS) [56] and photoyield (PY) [98])

A1 Mg Li Na K

Rb Cs

h~omp (eV)

h~mp/(hr, op)

Aht.Omp ( e V )

12.5 PY 8.7 5:0.3 EELS

0.83 0.84

3.27 PY 2.3 4- 0.25 EELS

0.811 + 0.007 0.84 4- 0.005 0.84 0.828 -t- 0.007

1.23 + 0.06 0.68 ± 0.18 0.41 0.64 -t- 0.07

4.67 -F 0.04 EELS 3.20 -t- 0.02 EELS 2.84 PY 2.40 EELS

Disp. " (eV ~ )

EELS EELS PY EELS

3.37 2.47 1.0

Linear dispersion coefficient of Ogmp(qlI).

4.2. Thin alkali overlayers on Al

Plummer and coworkers [ 107,108] investigated the dependence of the electronic loss associated with an alkali overlayer as a function of its thickness. The results are shown in Fig. 26 for the loss frequency and in Fig. 27 for the half-width of the losses. A discontinuity occurs as soon as the first monolayer is completed, which corresponds to the transition between an atomic-like behavior (loss due to an interband transition) and a collective behavior (loss due to surface or interface plasmon). This transition affects also the dispersion relation for the loss frequencies as shown in Fig. 28 for K and Na bilayers on A1 [109]. Similar results were reported earlier by Jostell [ 110] for K, Na and Rb on Ni(001 ) with ILEED. The phenomenon was explained by Liebsch [ 111 ] who computed the response function for different 3.0 Macroscop[c~Na/AI response

4.0

K/AI

~

I C-----'4~

response

/

3.5

2.5

/

o

po

>h C~

L 3,0

p o

o

p' ~,odO0

0

2.0 . . .. . 0.0

.

.

.

.

0.5

Theory plosmon-like ~ o ghost-peak /

Experiment

• plasmon-llke o atom;c-like

/

c

uJ 2,5

.

Mocroscop

i

2.0

i

1.0

,

,

~ ±

I

1.5

,

i

~ f

0 0

~o 0 ,

i

,

,

i

~t

0

o

/

1,5 2.0 0.0 0.5 Coverage eML I

,

! i

Theory , plosmon-llke 0 ghost-peak ,

i

Experiment • plosmon-like o atomlc-like

0 ,

~

,

i

tO

~

,

,

,

I

. . . . . .

~.5

i

2.0

Fig. 26. Loss energy versus coverage of the electronic excitation observed for Na/AI(111) (left panel) and K / A I ( I I 1) (right panel). The peak positions were extracted from EELS data recorded at Ei = 30 eV and from microscopic response theory. The dashed line is a guide to the eye. Theory predicts a weak intensity for the gost peak which escaped observation in experiment. The solid line indicates the peak position for a macroscopic response model (from Ref. [ 108 ], used with permission).

32

M. Rocca /Surface Science R e p o r t s 2 2 (1995) 1-71

ZX

Na/AI @

2.0 e# v "r

A

&

1.5

J •

Oo

• Experiment z~ Theory 0.0

0.5

1.0

1.5

2.0

OML

Coverage

Fig. 27. Half-width at half-maximum of the loss peak for Na/AI( 111 ). The theoretical points (A) were shifted upward by 1 eV in order to match with the extrapolated value of experiment ( o ) at a coverage of 2 monolayers. Note the narrowing of the loss with increasing coverage (from Ref. [ 108], used with permission). 2.9

4.2,

I (a)

)0

I (b) Na- AI

AI

K-

0

C=2

C=2 0

/

4.01 "

2.8

/

i

i

3.8

0

00

2.7

0

0

,

o •L%

0.0

0

/

0

3.6

0 0 1

,

2.6

o

I

q(I~-~)

0

I

0.0

0.2

t~ / , 0.2 q (,~-')

O

I 0.4

Fig. 28. Measured dispersion of electronic excitations for K and Na double layers on A1. (V]) Negative qll, ( o ) positive qll" The dashed curve indicates the result for a thick film (from Ref. [ 109], used with permission). T 152 (x 0.2)

1oE 5

0

:t,,ji ~21;/

// ",

.";"

............. 7 2

4

6

8

co ( e V )

Fig. 29. Calculated surface response function for different Na coverages, c, on AI. The vertical bars indicate the values of the work function for submonolayer coverage (from Ref. [ 111 ], used with permission).

33

M. Rocca/Surface Science Reports 22 (1995) 1-71

overlayer thicknesses. He found, in accord with Ref. [ 108], that the loss function is dominated by the threshold excitation at low coverage (see Fig. 29). The maximum of the broad loss feature is thereby determined by the value of the work function of the system which changes with coverage. At larger coverage only the multipole and the bulk plasmon mode are present at qll = 0; no surface plasmon mode of the overlayer is present as the overlayer is too thin to support it, while the substrate would like to oscillate at its own surface plasma frequency. When moving away from qLI = 0, however, the penetration depth of the surface plasmon diminishes so that this mode can be sustained again by the film. The bulk and the multipole losses evolve then toward the losses which one would expect for an infinitely thick film. Interestingly, as shown in Fig. 30 for K and Na overlayers on A1, the dispersion curves match in such a way that the bulk loss transforms into the multipole mode, while the multipole mode goes over into the surface plasmon. Three coverage regions with distinct excitation mechanisms can thus be identified: (i) For thicknesses larger than the double layer (c = 2) the spectra are dominated at small qll by the bulk (tap) and by the multipole mode (tamp) of the adlayer. In fact at qll = 0 the only monopole surface plasmon would be determined by the condition that the dielectric function of the substrate equals - 1 . In this case (the substrate is A1) this condition would be verified at 1 1 eV. At larger qll the two modes turn into the multipole mode and into the monopole surface plasmon, respectively. The transition is completed when the electrostatic coupling between the adsorbate/vacuum and the adsorbate/substrate interfaces becomes negligible. Due to the hybridization between these modes the multipole mode does not disperse upward as on the clean surface, but downward and finally turns into the ordinary surface plasmon at large-enough qll .

(ii) For a monolayer the ground-state electron density does not show anymore a plateau in the overlayer region. The modes are heavily damped and overlap giving rise to a broad and asymmetric loss peak. (iii) For c < 1 the density profile of the overlayer begins to resemble that of the substrate with only '

i

'

I

~t t,,0 '

'

I

'

(a) K - AI

~P ~

I ' (b) N a - AI

_

-

~s )p

0.0

0.2

q (k-')

~4

>

(Om . . . . .

-co

1

3

3

a) s ~.-~.

I

c=2 ......

co

..................... _2.'3__-

1

i~ -:Z?L-;...............1/4 ;)~.... ,

0.0

I

J

I

0.1 0.2 q (/~-1)

I/4

J

,

0.0

I

,

I

0.1 0.2 q (A-l)

,

0.3

Fig. 30. Calculated dispersion of electronic excitations for K and Ha overlayerson AI at different coverages c. Solid line c = 2, dotted line c = 1, dash-dotted line c < 1, dashed line c = oo (i.e, collective modes of clean thick K or Na). The arrows indicate the work function for submonolayercoverage (from Ref. [ 111], used with permission).

34

M. Rocca/Surface Science Reports 22 (1995) 1-71

~(22) ,

6o~

'1

t.O

20

0

0

';

hv(eV~

0

i

huleV}

/.

hvleV)

100

50

0

o (,~2) t,O01 3001

2001 1001 0

IL

0

8ooi

1

2

3

\ Li5oo

60(]

/.00

200

Fig. 31. Measured absorption cross sections at the Mie resonance for Li + clusters. Note the shift of the resonance toward higher energy with cluster size (red-shift with respect to l/R) (from Ref. [ 113], used with permission).

M. Rocca/Surface Science Reports 22 (1995) 1-71

35

a weak amplification of the outer tail. The volume and the multipole excitation thus disappear, while for low-enough coverages (¼ < c < l) the loss mechanism is governed by electronic transitions from the Fermi energyto the vacuum level. The loss function is thus dominated by a peak near the value of the work function ~. The theory was then further refined by Liebsch and Ishida describing the ground-state properties of the system by a first-principles method [ 112].

4.3. Mie resonance on alkali metal clusters Electronic excitation spectra were recently investigated by Br6chignac et al. on mass-selected alkali metal clusters of Li + [ 113] and K + [ 114] ions. The data show a well defined absorption peak in correspondence with the Mie resonance, whose position depends on the size of the cluster (see Fig. 31 for Li ÷ clusters). As expected the frequency increases with cluster size (red-shift of the Mie frequency with I / R ) in accord with a positive value of dr(toM) in Eq. (11). The value extrapolated for infinite cluster size converges indeed to hto = 3.55 + 0.1 eV, the value expected from the condition e(hto) = - 2 . Applying Eq. (11) one obtains dr(to~) = 0.50/k for Li + which should be compared to d:_ = 0.48/~ for the flat surface [97]. The agreement is surprisingly good. However, for K ÷ ions Br6chignac et al. [ 114] find a much weaker dispersion with 1/R than one would expect from theory [122,123] and from the dispersion of the surface plasmon (see Fig. 32). Applying Eq. (11) one would obtain in this case dr(toM) = 0.045 /~, which is more than one order of magnitude smaller than the d± value determined from surface plasmon dispersion. The reason for the discrepancy is unclear and might be connected to a larger (and size dependent) effective radius of the cluster.

h~ {eV) 2.5,

0(3

o',1

o'2

o',3

o'.s

n-~/3 Fig. 32. Resonance energy versus the inverse of the cluster size for K. The dash-dotted line and the solid line indicate the variation of the classical value of the Mie resonance with and without charge spill-out. (*) and ([3) are calculated values from Refs. [ 122,123]. Dots indicate the experimental values of Br6chignac et al. [ 114] and for n = oo the surface plasmon frequency as measured by Tsuei et al. [43]. n denotes the number of atoms in the cluster (from Ref. [114], used with permission).

36

M. Rocca/ Surface Science Reports 22 (1995) 1-71

4.4. Overlayer plasmon in K chains on Si(O01) K adatoms on Si(001 )2 × 1 are known to form parallel linear chains along the (110) azimuth of the substrate (see Fig. 33a). The dispersion relation for the overlayer plasmon was measured by Aruga et al. [ 115] and is indicative of that the K chains have the characteristics of a one-dimensional metal. In fact, in this configuration the interatomic distance between adjacent K atoms is 3.84/~ along the chains, i.e. ~,,17% shorter than in bulk K, while across the chains the spacing is too long to form delocalized electronic bands. The EELS data were recorded for 0i = 45.5 ° and Ei = 43.8 eV along (110) and Ei -- 39.8 eV along (100). Such conditions are quite extreme in terms of AqlI even for the small reported angular acceptance of 0.5 ° . The authors were helped by the presence of an anomalously large angular distribution of the inelastic intensity, whose origin was, however, not investigated. The data are reproduced in Fig. 33b. As one can see, the plasmon disperses upward with qll and the initial dispersion is linear and can be fitted with to(qll ) = to(0) + aql I. The dispersion relation is anisotropic with respect to the crystal azimuth reflecting the anisotropic nature of the system. Contrary to surface plasmon dispersion, however, the initial slope is positive. Such behavior is probably intimately related to the low dimensionality of the system. No quantitative theoretical interpretation of this behavior is presently available. Tsukada et al. [ 116] found a mode which roughly coincides with the experimental loss frequency. The forecasted dispersion is, however, quadratic in qll.

(b) [

'

I

2.d-- o - ~ \

(a)

~,

7.68,~.

'

I

;

;

I

I

I

[11o] [1oo]

'

/

"l

/

[100]

,o ?olo °o'.3

0.2

o

J

I

o.;

,

I

o.2

0.3

Fig. 33. (a) Linear chain arrangement of K atoms (solid circles) adsorbed on the Si(001)2 x 1 surface (o) at saturation coverage and (b) ovedayer plasmon dispersion with qll" The inset shows the azimuthal dependence of the initial dispersion coefficient a versus azimuthal angle ~b. Open and filled symbols in (b) denote data points recorded at positive and negative qll, respectively. The thin solid lines represent the theoretical dispersion relation and the modes found by Ref. [ 116]. The agreement is only qualitative and theory could not reproduce the positive sign of the dispersion relation (from Ref. [ 115], used with permission).

M. Rocca/Surface Science Reports 22 (1995) 1-71

37

4.5. Surface plasmon dispersion on Al The satisfying agreement between theory and experiment fails unexpectedly for A1. AI( 111 ) was investigated by Tsuei et al. [56] and his data are shown in Fig. 34. The dispersion is not symmetric with respect to qll = 0 as one would expect. The artifact is connected to the experimental difficulty in determining the position of the maximum of a loss, whose FWHM is several eV as in such circumstances a low-energy experiment may be strongly affected by the energy dependence of the elastic reflectivity. The measured dispersion of the maxima in the loss spectra is reported in Fig. 35. As one can see, the data points are strongly scattered but show no evident linear and negative dispersion with qll contrary to the interpretation given in Ref. [ 56]. Globally the data could be better fitted by a quadratic curve without linear term. A similar situation is observed for AI(001) [117]. High-energy EELS data (see Fig. 36) favored a quadratic dispersion, too [95]. A departure from the negative dispersion predicted by jellium theory would imply that lattice effects are more important at the surface of AI than has been thought so far. Theoretical work in this direction was performed by Schaich [ 118 ] and, although not directly aimed at surface plasmon dispersion by Fleszar et al. [ 119]. Another promising approach to the problem could be an extension of microscopic calculations of optical properties such as that performed by Del Sole and coworkers [ 120] for Si( 111 )7 x 7. Note that in Fig. 34 the volume plasmon loss shows up also for the smaller angles of incidence, while the multipole plasmon mode is absent. This latter feature is connected to the low rs value of A1 (r~ = 2). The existence of the multipole mode is on the other hand demonstrated by the enhancement

i--

a,-

>iZ iii

_z

5

10

15

20

E N E R G Y L O S S (eV)

Fig. 34. Low-energy EEL spectra recorded for AI( l 11 ) at different angles of impact (from Ref. [ 100], used with permission).

M. Rocca/Surface Science Reports 22 (1995) 1-71

38

I

'

'

'

'

t

AI

'

'

'

Surface

'

I

'

'

'

Plesmon

'

I

'

'

'

'

I

Dispersion

16

Z~ z~

A

A&

15

14

oo

o3 0 -I

13

nLU

[i] SP (111) & BP (111) ........... S P - T E L S (polycrys.) -S P - I L E E D (100)

[] [] D

12

Z 14,1

11 [2

r-i

[]

QD []

[]

1o

E3 []

9

I i -1.0

t

I

'

I . . . . -O.S

I . . . . -0.0

I . . . . O.S

I 1.0

q, (K ')

Fig. 35. Dispersion of the surface and of the bulk plasmon loss on AI( 11 l ) . The initial dispersion is not negative as one would have expected for a simple metal. The data points are, however, strongly scattered. The dotted line is the result of high-energy EELS [95], the continuous line the result of ILEED [26] 'from Ref. [ 100], used with permission). ,O'[mrad] 1.0

2.0

3,0

4.0

I

F

f

l

12.0

T i I I t E

o o

I

T

/

I

o

T I

I

. ×

t

××~j~,~

o

i

i

0.1

0.2

/

I /

I

_

I

1

1o.o

~

V o/ i

I I

~

"

a.

f 0.3

5,[~ -1]

i

i

0.4

O.5

Fig. 36. Dispersion of the surface plasma loss measured with swift electrons for an A1 film; ( x ) Ref. [25], ( A ) Ref. [121], ( o ) Ref. [95]. As for the low-energy EELS data shown in Fig. 34 a global positive dispersion is evident (from Ref. [95], used with permission).

M. Rocca/Surface Science Reports 22 (1995) 1-71

39

in the photoemission spectra at 0.8top [8]. The sensitivity to the bulk plasmon is on the other hand surprising in view of the small penetration depth at the energy employed in this experiment. Moreover, no significant intensity at top is observed for Ag surfaces for spectra recorded at similar Ei.

5. Surface electronic excitations in the presence of filled d-bands

Noble and in general d-electron metals present quite strongly damped features in the electronic excitation spectra because of the presence of efficient decay channels connected with the presence of interband transitions which involve the d-bands. For the same reason tOp and tOsp are strongly displaced with respect to the values appropriate for a free-electron gas.

5.1. Surface plasmon dispersion and Mie resonance shift for In and Hg Among the materials with a filled d-band, surface plasmon dispersion was investigated with swift electrons for an In film [95] and with low-energy EELS for an Hg film [ 106]. For the latter material also optical data for the Mie resonance exist [ 126]. For In the surface plasmon dispersion was reported to be positive as shown in Fig. 37. Swift electron measurements are less reliable than low-energy EELS data which are regretfully not available for clean In surfaces. The specificity of In can therefore not be considered as definitively proven. For this material the d-bands lie about 17 eV below the Fermi energy and are not expected to play such a significant role as they do for Ag, especially as htosp lies at the much lower value of 8.7 eV and satisfies quite well the l / x / 2 ratio with htop indicating a free-electron-gas behavior. For the Mie resonance on Hg clusters no dependence on cluster size was found, while the FWHM increases strongly with 1/R as shown in Fig. 38. The EELS data show a negative initial dispersion 1.0

2.0

~y~ [ m r o d ] 3.0

4.0

9.5

9.0

............. Lg
.

o_

:oOo:o°

o _.

o o'v°~°o °ol

0~0 O0 0

8.5 O0

8.0

I 0.1

I 0.2

I 0.3

l 0.4

l 0.5

kl I (.~ -1)

Fig. 37. Dispersion of the surface plasma loss measured with swift electrons for an in film [95]. The dashed line refers to the value for the surface plasmon frequency derived from optical-reflection experiments by Koyama et al. [ 124] and by high-energy EELS by Schilling [ 125] (from Ref. [95], used with permission).

40

M. Rocca/Surface Science Reports 22 (1995) 1-71 (a)

7

>

................

I1"11 .

~

.

•

.

.

.

~_-~t-i-]~

.

.

100

.

.

.

.

m

±

5

g5 .-+J

i/i O a4

//

3

g-3

2

19

1 0.0 ,._2.0 f >

I

i

v__v....

I

0.2

i

I

0.4

(b)

',

=

~

I

0.6

I

I

N_1/3 0.8 T / ~ "

1.5[-

T U-1,0 0,5 0.0 0.0

i

I

0.1

~

I

0.2

i

I

0.3

t

I

0.4

,

I

N_l/3 0.5

Fig. 38. (a) Mie resonance energy versus inverse of cluster size for Hg [126]. Full symbols refer to singly and open symbols to doubly charged clusters. The resonance energy equals within experimental error the value calculated for the Mie plasmon. The extra peaks observed for small cluster size are due to interband transitions. The 5d electrons cause a red-shift of tom from 6.12 to 5.73 eV. (b) FWHM of the resonance versus inverse cluster size [126], The relaxation time is inversely proportional to the electric resistivity which is strongly temperature dependent for Hg. The point at N = oo refers to the bulk value for the Mie resonance and is larger than the values observed for the clusters. The data point with a horizontal error bar is taken from Ref. [ 127] (from Ref. [ 126], used with permission). for surface plasmon dispersion as well as for the F W H M [ 106]. The linear coefficient for surface plasmon dispersion is weaker than expected for a free-electron metal of the density of Hg. Both to(qll) and AtO(qll ) could be reproduced within a sophisticated model based on stabilized jellium [ 7 0 ] and mimicking the d-electrons with a polarizable medium [52] which was also employed to describe surface plasmon dispersion on A g (see Section 5.2.1.4).

5.2. The special case of Ag A m o n g the noble metals A g constitutes an important exception to the strong damping rule and was therefore studied thoroughly. For this material the presence of d-bands reduces oJp from 9.05 eV (as expected for a free-electron gas of density rs = 3, appropriate for A g ) to 3.78 eV. At this frequency

M. Rocca/Surface Science Reports 22 (1995) 1-71

41

1.8 1.6 1.4

X

×

1.2 0 X

0.8 X

.0

X ©

0

0 X

0 ©

0.2

0

O

×

X 0

3.2

X

©

X

I

I

J

I

I

i

i

3.3

3.4

3.5

3.6 energy [eV]

3.7

3.8

3.9

4

Fig. 39. Electron energy-loss functions Im[ 1/e(tO)] and Im[ 1/( 1 + e(tO))] for Ag according to the optical data reported by Palik [60].

E2(to) is very small. According to optical data by Winsemius et al. [63] the volume plasmon should have a FWHM of 32 meV at 90 K and 45 meV at room temperature. For the surface plasmon optical data predict 72 meV at 90 K and 119 meV at room temperature [63]. A FWHM of 75 + 15 meV was indeed reported for the bulk plasmon [ 65 ]. The result for bulk and surface energy-loss functions, Im[1/E(to)] and Im[1/(1 + e(to))], derived from the optical data reported by Palik [60] at 300 K is shown in Fig. 39. Well separated peaks appear for bulk and surface plasmon, although with the somewhat broader FHWM of 152 and 169 meV, respectively. The optical data for the surface plasmon frequency and FWHM are summarized in Table 5. Table 5 Comparison of the optical values for tOp and tOsp for Ag according to Palik, Winsemius et al., and Ehrenreich and Philipp Ref.

Palik [ 60 ] Winsemius et al. [63] Winsemius et al. [63] Ehrenreich and Philipp [ 62 ]

T (K)

90 295

Bulk loss

Surface loss

top (eV)

Ahtop (eV)

tosp (eV)

Ahto,p (eV)

3.80 3.80 3.78 3.74

0.169 0.032 0.072 0.055

3.68 3.66 3.65 3.59

0.152 0.045 0.119 0.180

M. Rocca/Surface Science Reports 22 (1995) 1-71

42

r

i

,

1

i

i

i

/

/ 4.5

/

v

/

//

3

+.. +

4.0 •

~"

3,5

3.0

I

0

I 0.1

I

i 0 2

i

q (~-1)

i 0.3

J 0.4

Fig. 40. Dispersion of the volume plasmon in Ag. ( x ) EELS [66], ( - - - ) optical measurements [141], and ( - - - )

top(q) = tOp(0) + ot(h/m)q 2 with a = 0.8. The use of this formula is not justified for Ag, but the data indicate that the dispersion is quadratic with q as is the case for simple metals of comparable electron density (from Ref. [66], used with permission). In the bulk the dispersion behavior of the plasmon is adequately described by the usual quadratic form: tOp(q) = tOp(0) + o t ( h / m ) q 2, with tOp = 3.78 and a = 0.84-0.1 as shown in Fig. 40 [140]. In this case, however, a has no obvious meaning as it had for simple metals.

5.2.1. Surface plasmon dispersion and Mie resonance shift 5.2.1.1. Low-Miller-index surfaces. EELS spectra recorded for the low-Miller-index Ag surfaces for grazing 0s and low Ei and at different 0i are reported in Figs. 41-43. As one can see, a single and well defined loss is present which is superposed to a weakly structured background due to the excitation of electron-hole pairs. The data were recorded at very low Ei in order to reduce Aql t except for A g ( l l 0 ) along (001), where the elastic reflectivity is extremely low below 16 eV. The losses recorded on the different faces coincide with respect to frequency and width for qll = 0 and disperse toward larger Eloss values and broaden with qll" No significant intensity is present at the volume plasma frequency although the experimental resolution should be sufficient to resolve such contribution (volume and surface plasmons were indeed observed and resolved with swift electrons, see Fig. 5, while in Fig. 34 the loss intensity for A1 at top is comparable to the one at tosp). The threshold for e-h pair excitations lies at 3.86 eV for sp ~ sp and at 3.98 eV for d ~ sp at room temperature [63], i.e. only 0.16 eV above tosp and is responsible for the high-frequency tail in the loss spectra, which is particularly evident for Ag(110) (Fig. 42). When the crystal is dosed with

M. Rocca/Surface Science Reports 22 (1995) 1-71

43

5 9 [ i00) < 0 0 1 > E.:iG.OOeV

8~=81,G pl ,=

°

3.73eV

sls~

-]

3 .70

co

<

>H CO

~

- - 0

3 .72 .0149

5 .v4 H-

0.0234

Z

3.75 0.0408

I

3 .6 ENERGY

I

3 .8 4 o0 LOSS l e V I

Fig. 41. EEL spectra recorded at Ei --- 16.00 eV at low impact energy for different qll for Ag(001 ) along (100). The desired momentum transfer was obtained by rotating the EELS monochromator out of the specular direction, qll is evaluated at the maximum of the energy loss.

oxygen the surface plasmon loss becomes fainter and disappears already at 6% 02 coverage as shown in Fig. 44 for molecular adsorption on Ag(110). The dramatic effect is connected with the strong charge transfer from the surface to the chemisorbed 02 molecule (0.9-1.4 electron per molecule according to Refs. [ 128] and [ 129], respectively). At smaller exposures the frequency of the surface Ag(110) <110> -

'

",

"

"

- a)

'

i 3.84

j~-

I

.

Ag(llO) <001>

.

. . Ei = 8.13

o, =

eV

b)

76.4 °

3.941

q,

I

'Et

17.2

eV

"

0 / ' - ' ~ u , = 80.0 °

c~

-0.218 ~-t-

m

.d

/\

q, : - 8 . o o 3 , - ,

v

-0.145

~-t

b

b r-, i

]

3.71

3.71

<001> qll = 0 . 0 2 1

, 3.5 energy

I 4 loss (eV)

~t-:

q{I :

, 4.5

3.5

0.018

4

energy

.~-i

r 4.5

loss (eV)

Fig. 42. Same as Fig. 41 for the two high-symmetry directions on Ag(110). For this surface the interband transition responsible for the high-frequency tail is much stronger than for Ag(001 ). Its contribution to the intensity is larger on the high-loss side and causes a shift of the loss frequency, which had therefore to be corrected in order to recover ~oe. (a) (1 TO) direction; (b) (001) direction. Along (001) no measurements could be recorded below 16 eV as the elastic reflectivity (to which the inelastic intensity is proportional for dipole scattering) is too low.

44

M. Rocca/Surface Science Reports 22 (1995) 1-71

i

i

i

I

I

Ag(111)

i

i

i

i

i

i

i

OOK__

I

i

'

i

I

i

[3.64 b) 300K

13.88

(D ° ~.~

¢~~-0.160

~-1

.6

~

v

o e...~

50

13.79

CD

E i = 15 [3.74

-

A

13.77

_

-0.104

~

eV

O. = 63.8 ° qll =

0.009 ~.-~

0

105

~-i

q l l = _ 0 . 0 3 4 ,~-t

[

I

4.5

4.0

,

13.71

~ I

3.5

=--

I

i

4.0

I

I

I

I

4.5

energy loss (eV) Fig. 43. EEL spectra recorded for A g ( l l l ) (a) at 100 K and (b) at room temperature. Cooling the crystal causes an increase of the surface plasmon frequency, while the dispersion is affected little.

%--

Ag(110)

¢ /'¢.

<001> .d /

>,

f

cD

I

3.2

I

3.7

I

4.2

e n e r g y loss (eV) Fig. 44. Electronic excitations on Ag(ll0){001} for (a) the clean surface and (b) after adsorption of 02 (6% coverage of chemisorbed molecules). The experimental conditions are: El = 14.68 eV, 8i = 62.15 °, Os = 80.0°; the loss is at 3.81 eV and corresponds to qll = 0.0726 /~-1; the crystal is at T = 85 K. As one can see, most of the inelastic intensity at the surface plasmon frequency and above is quenched after oxygen adsorption, indicating its surface origin. The bulk contribution amounts to about half of the loss intensity at Eloss = 4.2 eV. No peak is present at tOp even after removing the surface contribution to the loss spectrum.

M. Rocca/Surface Science Reports 22 (1995) 1-71

45

Ag(110) <110> I

' E i '-- 1 8 . 4 7 'eV • 8, 76.2 ° -

13.74 t~

El 0, 0i qll

v

I 3.71

= 18.47 eV = 76.2 ° = 59.9 ° =-0.004 A -I

E I = 8.13 eV 8, = 76.4 ° qll = 0 . 0 2 1 A - l

~o . ,...~

i

3.5

L

i

i

I

i

i

4 e n e r g y loss (eV)

i

i

4.5

Fig. 45. Interbandtransition and surfaceplasmonloss for Ag(110) under differentexperimentalconditions.In (a) only the interband transition contributesto the inelastic intensity;in (b) both contributionsare present,while in (c) the spectrumis dominated by the surface plasmonloss. plasmon is hardly affected by the presence of the adsorbate; only the loss intensity associated with the excitation of the surface plasmon decreases, indicating the local nature of the charge transfer. The inelastic background is also affected by oxygen exposure, proving its surface origin. No well defined peak is present at tOp even after oxygen adsorption. We have therefore to conclude that the contribution of the volume plasmon to the inelastic intensity is negligible also for the clean surface. This result can be rationalized by thinking that the electric field connected to bulk plasmons is confined within the solid by the screening charge so that the impinging electrons can interact with it only for the time spent subsurface, which is relatively short compared to the flight time to and from the surface. Losses corresponding to volume plasmons were observed on the other hand in EELS spectra recorded with low-energy electrons for off-specular measurements on alkali metal films (see Fig. 16) and on AI(111) (see Fig. 34). At the moment there is no obvious explanation for the lower sensitivity to the bulk plasmon of Ag. The tail due to interband transitions is more intense in the EELS spectra recorded for the corrugated Ag(110) surface. As shown in Fig. 45 the relative intensity of tail and surface plasmon depends on Ei and on T indicating that, as the surface plasmon is mainly excited by dipole scattering, the tail has a strong non-dipolar contribution, which will dominate for scattering conditions in which dipole scattering is weak as e.g. around vanishing momentum transfer. This dependence allows each particular excitation to be tuned on and optimal conditions to be found for measuring the surface plasmon. The tail is in general more intense at large Ei in accord with its being excited by impact scattering. Its larger weight in the spectra recorded for Ag(110) with respect to Ag( 111 ) and Ag(001 ) indicates that corrugation plays a role in enhancing the excitation of interband transitions. Another broad maximum in the inelastic intensity was predicted at Eloss = 7.17 eV [ 130] (see Fig. 46) and effectively observed at 7.1 eV for Ag(001) [71]. This loss corresponds to a region where e2 is falling with increasing energy with E] remaining small. It does not appear in our lowenergy EELS spectra (as in Fig. 11 ) because although its integrated intensity is comparable with the surface plasmon loss, it is spread over a much larger Eloss range and disappears in the background.

M. Rocca /Surface Science Reports 22 (1995) 1-71

46

"~

80

~- 0.2 -~

60 ~

X ~u 0.1

.--''"

....---r''"T"'"

0.0 0

20

40

.....

,

40

0

60 E n e r g y (eV)

BO

100

Fig. 46. Surface electronic excitation spectrum of Ag at qll = 0 as calculated by Lovrid and Gumhalter using a semiempirical approach. Solid and dashed lines are the result for the sum rule describing the condition of overall charge neutrality and particle conservation (f-sum rule) of the system (from Ref. [ 130], used with permission).

The bulk counterpart of this loss appears at 7.8 eV [ 131 ] and shows a quadratic dispersion. It was therefore assigned to a collective excitation by Bornemann et al. [ 131 ]. Such mode was proposed to be associated to a splitting of the volume plasmon induced by the influence of 4d core polarization on the 5sp plasmon excitation. This effect is shown in Fig. 47 for different materials. The splitting would accordingly take place also at the surface but be smaller in magnitude. For the set of measurements shown in Figs. 41, 42a and 43 the surface plasmon loss is dominant, the loss peak is relatively symmetric and the plasmon energy can be determined by a gaussian fit to its ~0

I

I

I

I

I

i

p 1b \

30

\1111 /111

//~"P /

/ff////-~-

1a

/2/

GI

~20

//

imm

//

I

I0 i I~ "~/

Pd

/

-

-

I

I

I

I

I

Ag

Cd

In

Sn

Sb

Fig. 47. Energies of bulk collective excitations (E) and 4d binding energies (EF -- E40) versus the corresponding metal. (IS]) upper (label a) and lower edge (label b) of the 4d-band; ( o ) experimental values of bulk collective excitations; ( × ) plasmon energies calculated within the homogeneous electron gas approximation. Lines (1-3) are guides to the eye (from Ref. [ 131 ], used with permission).

Rocca/Surface Science Reports 2 2

M.

(1995)

47

1-71

Ag(001) <100>

Ag ( 0 0 1 ) < 1 0 0 > 8.8 ° off

3.90

4.1

/ 3.85 ;:> ©

3.80

b)

4.0

~>

/ °

_

,-, 3.9

c-,

3 "~

3.75

3.8

3.70

3.7 ....

,

....

0.025

L ....

0.05

....

0.075

....

, ....

0.1

0

0.125

I ....

0.05

J ....

0.1

L ....

0.15

j ....

02

J,,,

0.25

03

qll ('~ 1)

qll (~ 1) Ag(001) <110> 4.1

,---, a> ~D

4.0

-

y

c)

3.9 '~

3.8

3.7

.... 0

I ....

0.05

I .... 0.1

I ....

0.15

I .... 0.2

I ....

0.25

0.3

qr (.~- 1) Fig. 48. Surface plasmon dispersion along different directions on A g ( 0 0 1 ) . T h e bars indicate the integration windows in reciprocal space for the different experimental conditions: ( a ) 295 K (100): ( 0 ) Ei = 1 0 . 5 - 1 7 eV, Os = 7 5 ° - 9 0 ° ; 295 K: ( o ) E~ = 16.55 eV, 8s = 81.2°; 125 K: ( x ) Ei = 16.55 eV, 8s = 81.2°; data are available up to-qll = 0 . 3 / k - t (not reported here). They follow the linear dispersion indicated by the solid line. ( b ) 295 K 8.8 ° off (100}: ( . ) Ei = 38.5 or 76 eV, 8~ = 7 5 ° - 8 5 ° ; ( 0 ) Ei = 1 1 6 - 1 2 7 eV, O, = 8 0 ° - 8 5 ° ; ( o ) Ei = 20 eV, O, = 7 5 ° - 9 0 ° ; ( x ) El = 217 eV, Os = 4 5 0 - 6 0 °. ( c ) 295 K (110}: ( × ) Ei = 248 eV, 8s = 4 1 . 6 ° ; ( 0 ) E~ = 217 eV, Os = - 6 9 . 4 ° ; ( x ) Ei = 130 eV, 0s = 69.4°; ( * ) E~ = 80 eV, 8., = 68.4°; ( o ) Ei = 36 eV, O~ = 73.2 °. The best-fit parameters of the curves are reported in Table 6.

maximum. For Ei above ,-,12 eV on Ag(110), as for instance for the set of measurements reported in Fig. 42b, also the interband transition is intense. The loss frequency was then determined assuming a convolution of the plasmon loss with a function simulating the contribution of the interband transitions to the loss. The form of this function can be inferred from spectra such as the one in Fig. 45a. In case of the data reported in Fig. 42b the energy correction is of the order of 20 meV with respect to the maximum of the energy-loss peak. The measured surface plasmon frequencies are reported versus qll in Fig. 48 for Ag(001) [47] [87], in Fig. 49 for A g ( l l 0 ) [48,133] and in Fig. 50 for A g ( l l l ) . The bars on the data points represent Aql I evaluated according to Eq. (20) and should not be confused with the experimental indetermination in qll. As one can see, the slope of the surface plasmon dispersion is positive for vanishing qlt for all crystallographic faces. Quadratic terms are nearly absent for Ag(001) and for

M. Rocca/Surface Science Reports 22 (1995) 1-71

48

....

[ ....

I ....

i ....

I ....

I ....

4.0

4.0

,o ,

~)

x

:

,

b)

o

>

>

@

'

>< '

'

'~'

0

v

+.6"

t~

3 ;~

'

~'÷o

~'"

3.8

Ag(llO)

.%

#.o

3.8

_

....

0.05

i ........ 0.1

i ....

0.15

0.2

)

~;'~""

<001 >

,,,i 0

3

I,,,

0.25

.... 0

0.3

<110>

I .... 0.05

I .... 0.1

0.15

qll

q,

i ....

(1-1)

i .... 0.2

i .... 0.25

0.3

Fig. 49. Surface plasmon dispersion on A g ( l l 0 ) along: (a) (001) ( o ) Ei = 16.94 eV, 0s = 80.0°; (I-]) Ei = 14.52 eV, 8s = 78.7°; ( 0 ) Ei = 21.76 eV, 0s = 82.2°; ( x ) Ei = 14.83 eV, 6s = 82.5°; ( + ) Ei = 16.80 eV, Os = 81.0°; the data denoted by ( 0 ) , ( x ) and ( + ) were recorded on a slightly damaged crystal; their frequencies are systematically higher than for the other sets o f data. The effect is, however, within experimental error and was not observed along (1 TO); ( b ) (1 i 0 ) ( x ) Ei = 5.94 eV, Os = 74.9°; ( D ) Ei = 8.71 eV, Os = 58.6°; ( 0 ) Ei = 18.7 eV, 0s = 78.0°; ( + ) El = 8.13 eV, 0s = 76.4°; ( o ) Ei = 8.85 eV, 0s = 58.6 °. The bars indicate the integration w i ndow s in reciprocal space for the different experimental conditions. T he best-fit parameters o f the curves are reported in Table 6.

Ag(ll0) along (001) for which a linear fit is appropriate. The deviation from linear dispersion is on the other hand appreciable for A g ( l l l ) and Ag(ll0) along (li0). The data for Ag(001) were recorded at room temperature. When cooling the sample to 100 K a minor shift of tOsptoward higher frequencies was observed, while the dispersion coefficients remain substantially unaffected. Temperature effects will be addressed in more detail in Section 5.2.3. In case of Ag(ll0) the dispersion curve is clearly anisotropic with respect to crystal azimuth. To be quantitative the data were fitted either with the quadratic form tOsp(qll) = tOsp(0) + Aql I + B q~,

(21)

or with the linear form (Osp(qll)

(22)

= O.)sp(0 ) "-]- A qll"

4.2 '1'o'

Iqll'O'

'

'

I

I

qll<0 El= IO.7eV

iI ?

II

qll>O

4.2

l'j

....

Q

I

4.1

'

/

EI=IS.0eV

_

4.1

x

> ~)

4.0

.... ,'~',

I .... I .... ~:11<> ~ Z,=to.vev

/ ~ q

qll>O qll <0

4,0

I ' ' ' /

Di= 15.0eV x

3.9

3.9

3

a) 100K

a.a

Ag (111

3.7

( ' 1~ 1 ' 1~ "Ag K

3.6

....

3.a

~+

3.7

3.6

.... 0

i ....

I ....

0.1

0,2

q,

(~-')

i .... 0.3

0.4

i .... 0.1

I ....

0.2

qll (~-')

J,,, 0.3

) I

0.4

Fig. 50. Surface plasmon dispersion on A g ( 111 ) (a) at 100 K and (b) at room temperature. The bars indicate the integration w i n d o w s in reciprocal space for the different experimental conditions.

49

M. Rocca/Surface Science Reports 22 (1995) 1-71

Table 6 Best-fit values romp(0),A and B for the differentAg surfacesaccordingto Eqs. (21) and (22) (the referencesare: Ag(001) [47,87]; Ag(ll0) [48]; Ag(lll) [50]) Face

Azimuth

T = 100 K

T = 300 K

tOsp(0) (eV)

A (eVA)

B (eV/~2)

(001)

(100) 8.8°off (110)

3.71 (1)

1.4 (1)

(110)

(001) (li0)

3.699 (5) 3.705 (5)

1.13 (2) 0.42 (2)

1.1 (1) 2.06 (7)

3.708 (6)

0.60 (9)

2.2 (4)

(111)

tosp(0)(eV)

A (eV/~)

B (eV/~2)

3.69 (1) 3.68 (2) 3.67 (1)

1.4 (1) 1.6 (1) 1.5 (1)

0.06(44)

3.692 (5)

0.64 (9)

2.0 (3)

The number in parenthesesindicates the error on the last digit. The best-fit of the parameters and the uncertainties, computed by minimizing the X 2 by using the MINUIT computing routine, are reported in Fig. 51 and Table 6 for all Ag surfaces. The following conclusions may be drawn: (i) Within experimental error tOsp(qll = 0) is the same for the three surfaces in accord with classical dielectric theory; the surface plasmon energy at 3.71 eV at 100 K and 3.69 eV at 300 K is slightly larger than expected from optical data (at 300 K 3.65 eV according to Ref. [63] and 3.68 eV according to Ref. [60] ). This difference is small but beyond experimental error. The maximum of the loss frequency in the EEL spectra might be shifted by the contribution of interband transitions to the high-energy-loss side. I doubt, however, that such effect is important as, if it would be the case, the measured loss frequency at qll = 0 should be different for the different crystal surfaces as different interband transitions contribute to the spectra. When comparing EELS data with optical data one should keep in mind that the latter were measured on polycrystalline films with rough surfaces. (ii) The dominant coefficient of surface plasmon dispersion for small qll is always linear in qll but the slope is positive contrary to the results for simple metals [43]; quadratic terms become ....

4.0

I ....

- _ _

I ....

I ....

(110) < 1 1 0 >

....... (11o) ,

1-~ >

,

....

,"

;

"

///

(111) ...-/

o

3.8

I :."' ' , ; " I ' ) " .."i/// /

..-"i

,,,I

0

....

0.05

] ....

I,,,,I

0.1 0 . 1 5 0.2 qll (;~-')

....

I,,,

0.25

0.3

Fig. 5 ]. Comparison of the surface plasmon dispersion curves measured on the different low-Miller-index surfaces of A g at T = 100 K.

50

M. Rocca/SurfaceScienceReports22 (1995) 1-71

important for large qfl only for A g ( l l 0 ) along (li0) and A g ( l l l ) . (iii) The surface plasmon dispersion of A g ( l l 0 ) depends on the crystal direction while no dependence is present on the isotropic Ag(001) [87]. (iv) Along the close-packed Ag(110) (1 i0) rows the linear dispersion coefficient is smallest and the quadratic terms are most important; on Ag(111) the situation is similar but the linear term is 50% larger than along Ag(110) (1 i0). (v) Across the rows, along Ag(110) (001), the linear dispersion coefficient is dominant and is by 20% smaller than on Ag(001). These results were confirmed recently by Plummer's group for Ag(110) [49] and Ag( 111 ). The fit to their data yields, however, systematically smaller values for the linear coefficient which are compensated by larger quadratic terms, e.g. for Ag(110) along (li0) they report: tOsp(qll) = 3.70-4-0.01 + (0.484-0.13) qPI + (2.234-0.63) q~, and agree therefore with our results, while along (001) they obtain: ~°m(qll) = 3.70 4- 0.01 + (0.65 4- 0.10) qll q- (3.4 4- 0.80) q~, and question whether the anisotropy is really connected to the linear term, as suggested by Table 6, or by the quadratic term. In the latter case the anisotropy could be ascribed to some bulk property. In my opinion these objections are not too serious as the error bars given by Lee et al. are so large that, contrary to evidence, the anisotropy itself would not be beyond experimental uncertainty! On the other hand, the raw data look nearly identical on visual inspection. It is at the moment not clear whether the small difference in the data, which produces the different weight of linear and quadratic terms, is real, e.g. connected with a different quality of the surface, or an artifact due to better or worse focusing of the EEL spectrometer.

5.2.1.2. Ag clusters. The results for surface plasmon dispersion should be compared with those for the Mie resonance shift on Ag clusters. Different data sets exist: Ganirre et al. [ 138] measured the electronic excitation for mass-selected silver particles embedded in a glass matrix, Charl6 et al. [ 139] for Ag clusters embedded in rare-gas matrices, and Tiggesb~iumker and coworkers [ 136,137] for free ionic Ag clusters. The data of Ref. [ 137] are shown in Fig. 52; as one can see, a well defined peak is present at the Mie frequency which shifts to higher frequency (i.e. to the blue) with 1/R. The data of Tiggesb~iurnker et al. are collected in Fig. 53 with those of Charl6 et al., after correction of the dielectric constant of the matrix. This result is at variance with the red shift observed for alkali metal clusters [ 113 ] and confirms nicely the anomaly of Ag inferred from surface plasmon dispersion. 5.2.1.3. Interpretation of the positive surface plasmon dispersion. The positive dispersion of Wsp(qll) would be indicative of a subsurface position of the centroid of the induced charge if one could apply the theory developed for jellium. While such an assumption is surely not justified in the presence of d-electrons, an inward shift of the screening charge at oJsp appears reasonable if one considers that tom is only 2% lower than ~Op, which corresponds to the threshold for the penetration of electric fields. Considering the ¢o dependence of d± reported in Fig. 7 for jellium one can see that the centroid of the screening charge has to move subsurface as Wp is approached. To be more quantitative one can try to correct Eq. (8) and Eq. (9) starting from the formula for the optical reflectivity [27] :

M. Rocca/Surface Science Reports 22 (1995) 1-71

r

r

105

51

7

A

Agg÷

20

~

Ag]-~+

10~

I

4O~o

~

# 5 t~ 3O

A~

.

10

2

3

4

5 6 energy(eV)

Fig. 52. Optical absorption spectra for mass-selected free Ag clusters (from Ref. [ 137], used with permission). I

I

I

i

i

i

i {~

~ 4.0 0

3 4:: 3.5 I

0.1 I

I

2

Io

0

I

0.3 I

I

11R (J, -~ )

Fig. 53. I IR dispersion of the Mie frequency for Ag: ( + ) A I particles in an Ar matrix [ 139 ], ( o ) free Ag + clusters, ( . ) free A g - clusters [ 137] (from Ref. [ 137], used with permission).

52

M. Rocca/ Surface Science Reports 22 (1995) 1-71

Table 7 dx(~Osp) as derived from the data in Table 6 and from Ref. [136] (the distance of the first lattice plane a± from the geometrical surface is given for comparison) Face

Direction

(111) (001) (110) (110) Ag cluster

dl(tOsp) (/~)

a± (/~)

-1.0

-1.18 - 1.02 -0.72 -0.72

-2.6 (001) (1 i0)

- 1.9 -0.7 -2.1

• + 1 - (• - 1) [d± (o.O - dll(~O)] qll = O. Substituting • = •f -q- •b w i t h •f at toM) one obtains: A~o

1

tOsp

1 -~- •b ((-Osp)

=

(23)

--(.O~/tO 2 and the numeric value for •b (equal to 5.3 at tO~pand 4

Re [d±(wsp)] qll,

Ato 3 Re [dr(OM) ] = tOM 2 + E b ( ~ i ) R

(24)

(25)

Eq. (25) applied to the data of Ref. [137] yields dr = -2.1 /~. Applying Eq. (24) to the data for surface plasmon dispersion on Ag one obtains values ranging from - 1 . 0 A for Ag(111) to - 2 . 6 / ~ for Ag(001). These values are compared in Table 7 with the distance of the outermost lattice plane from the geometric surface which reads 1.18 A, for (111), 1.02 A for (001) and 0.72 A for (110). Giving credit to this analysis the centroid would therefore be very close to the nuclei of the first crystal layer for the close-packed direction of Ag(110) and for the close-packed Ag( 111 ) surface, and close to the second crystal layer for the more open Ag(001) and for the open direction on A g ( l l 0 ) . It is not clear whether the coincidence between d± and the interplanar distance and between close-packed planes and directions is accidental or not. These values are in any case surprisingly large; analyzing the data with Eqs. (8) and (11) one would on the other hand obtain the more reasonable values of - 0 . 3 4 / k for A g ( l l l ) , - 0 . 8 / ~ for Ag (001) and dr = -0.85 A for Ag clusters. An analysis in terms of only d± is inappropriate for Ag(110) as for a corrugated surface dll may depend on the crystallographic direction.

5.2.1.4. Recent theoretical efforts. Quantitative calculations are difficult for Ag as tOsp is too near to O~p and surface integrals tend to diverge. Two efforts to compute d± appeared so far: by Liebsch and by Feibelman. Liebsch [52] started from the following considerations schematically illustrated in Fig. 54: (i) the oscillating charge is composed of s-electrons; (ii) the effect of the presence of d-electrons on the charge influenced by the electric potential associated with the surface plasmon depends on qll because of the wavelength-dependent penetration of the surface plasmon field; (iii) the d-electrons may be described by a polarizable medium.

M. Rocca/Surface Science Reports 22 (1995) 1-71

. . . .

I

J

1.0 . . . . . . . . . . .

i

I

.

.

.

.

i

'

'

,

53

'

no xx

0.0 -10

"~ -5

0

Zd

5

10

z (a.u.)

Fig. 54. Schematic model for the dynamical response o f a two-component 5s--4d electron system: no denotes the ground-state density profile, 8n the induced density, 8~b the qll-dependent potential associated with the surface plasmon. T he positive background is located in the half-space with za < 0 (from Ref. [ 5 2 ] , used with permission).

According to Liebsch the surface plasmon thus feels the presence of the polarizable medium more at vanishing qll than at large qll, where it is more localized in the spill-out tail of the electron density, As the surface plasmon of a more free electron-like gas with the density of Ag lies at ~ 6 . 4 eV, tom would shift upward with qll as then the surface plasmon field would feel a more free-like electron gas. The resulting loss function is shown in Fig. 55 for different qll, while the forecasted dispersion of tOsp(qll) is reported in Fig. 56. As one can see, the correct sign for the dispersion is recovered. This model was applied with success also to the qll dependence of dispersion and damping for Hg [ 106] where, however, the influence of the d-electrons is weaker.

I ' ' ' ' I ' ' ' ' I ....

. . . .

I ' ' ' ' I ' ' ' '

A ,'/ I p~

E

s J"

u

, ...

~

. ..-

s ~

0.20

Li

. "

I

C

It

--

,y ~

t 3.4

,

,

,

I ~.5

,

,

,

,

I 3.6

,

l

l

I

,

~.7

,

,

t

I 3.8

J

,

,

,

l 3.9

t

~

~

i 4.0

,~ (,v) Fig. 55. Frequency dependence o f logarithm o f the surface loss function for A g at several qll for Zd = 0 (from Ref. [ 5 2 ] , used with permission).

54

M. Rocca/Surface Science Reports 22 (1995) 1-71

6.8

'

J

'

'

i

'

'

'

'

I

'

'

'

'

.

6.6 //. 6.4 > 0

0-

3

6.2 6.0 3.8 3.6 , , , , I

3.4

. . . .

0

....

I

.1

.2

.3

q (~-') Fig. 56. Dispersion of surface plasmon of Ag for Zd = 0 and for a semi-infinite electron gas with rs = 3 within RPA response treatment (solid curves) and TDLDA (dashed curves) (from Ref. [52], used with permission).

4.1

4.0 3.9 3.8 3.7 3.6 3.5

. . . . 0

I

,

.I

,

,

,

I .2

q

,

~ ,

, .3

(~,-')

Fig. 57. Dispersion of the surface plasmon of Ag for different za values calculated within RPA. Dotted and dashed lines denote the experimental data for the (001) [47] and (111) [41] faces of Ag. The triangles indicate the dispersion of the bulk plasmon [66] (from Ref. [52], used with permission).

M. Rocca/SurfaceScienceReports22 (1995)1-71 I

'

'

'

I

.

.

.

.

55

~

II

10

(b)

(o)

.2

V

.1 v

N e" j

,

t,O

iI

..1

-40

l

,

~

1

t

t

-20 z (o.-.)

,~, I

0

t

t

I

-40

-20 z

0

(o...)

Fig. 58. Induced charge density 8n(z, qll, ¢o) for qll = 0.15 /~-i calculated within TDLDA at (a) to = 1.0 eV and (b) co = 3.7 eV (from Ref. [52], used with permission). The magnitude of the effect depends on the distance, z a, between the polarizable medium and the jellium edge as shown in Fig. 57. Naively one could assume that Za should be larger for Ag(111) than for Ag(001) giving rise to a face dependence of surface plasmon dispersion. Such dependence is, however, beyond the scope of the model as z a is not a ground-state quantity since it refers to the optical response. Liebsch draws no conclusion about the position of the centroid of the screening charge which could be anywhere outside the polarizable medium to give the positive dispersion. The calculations reported for qll = 0.15 t~ -1 (shown in Fig. 58) do in fact indicate a subsurface position of the centroid of 8n(z, qll, to) at tosp as expected in view of the proximity of tosp to top. The picture is, however, complicated by the dependence of 8n(z, qlr, to) on qll" Unfortunately this quantity cannot be computed for qll = 0 as the surface integrals diverge. Feibelman gave a completely different explanation of the positive dispersion sign. His approach starts from the consideration that due to the vicinity of the onset of interband transitions, the surface plasmon on Ag must be considered as a frustrated, collective d --, sp excitation. Such transition can be induced by matrix elements which couple d-states to sp-states and the shift of the centroid, 6d±, is consequently given by:

8d± oc f dz~(z)**~(z)~a,(z) 2,

(26)

where • ( z ) is the electric potential associated to the surface plasmon and the subscripts ,~ and ,V refer to one-electron states with wavefunctions ff'~(z) and ~ a , ( z ) . a corresponds to an "extra" electron, i.e. to an occupied d-orbital while A' refers to an unoccupied state, e.g. a 5s level above EF. Since d-states and s-states differ by 2 in angular momentum the transition will be taking place only when ~ has a strong second derivative with respect to z, i.e. near the surface. Deeper lying 4d

56

M. Rocca/ Surface Science Reports 22 (1995) 1-71

electrons will thus not be excited to 5s states because of the smooth dependence of • on z. 5p states should then be considered but the author calls to mind that the function ~ ( z ) does not couple 4d states with Px and py orbitals and that the density of Pz states near EF at the surface is for Ag only half that of s-states according to electronic structure calculations [ 142]. While this theory does not pretend to be quantitative several qualitative conclusions may be drawn: (i) the position of the centroid is naturally placed inside the crystal as it is related to the position of the localized d-electrons; (ii) as the distance of the nuclei from the truncation plane of the surface depends on the vertical lattice parameter the effect on the global position of the centroid of the induced charge will be smaller for close-packed surfaces than for more open ones; (iii) an anisotropy of the dispersion is expected for heavily corrugated systems as the component of the electric field parallel to the surface can then penetrate into the crystal and contribute to the d ~ s transitions, too. More recently Li et al. [143] showed in a theoretical work that also the mass of the excited electrons plays a role in determining the slope of surface plasmon dispersion. Such mass would differ on the different faces of Ag as different surface states are present.

5.2.2. Surface plasmon damping To extract the line width of the surface plasmon one should correct the experimental loss width according to:

AhWsp = V/(AEioss) 2 - (AEexp) 2,

(27)

whereby AEexp is given by the transfer width of the spectrometer and depends on the uncertainty on the impact energy AEi and on the angular acceptance. The two terms are independent and add therefore quadratically. The latter term is connected with the slope of the dispersion of the energy loss with qll and dominates for the experimental conditions used in the experiment: e.g. for Ei = 16 eV and 0s = 60 ° one has AE~xp = 52 meV, of which 48 meV are due to integration over Aql I and 20 meV to the monochromatization of the incident electron beam. As AElos~ ,-~ 110 meV, Ahwsp ~, 95 meV at room temperature. The correction is thus important for vanishing qll. As we will show later in detail the loss width depends moreover on crystal temperature. The qll dependence of AO~p is shown in Figs. 59-61 for the different Ag surfaces, while similar plots v e r s u s h(.Osp are given in Figs. 62-64. As one can see, initially A~sp grows linearly with qll for Ag(001) and for Ag(110)(001) while it decreases for A g ( l l l ) and Ag(110)(110). For the latter surfaces Atosp grows then again at larger qll. The effect is small but beyond experimental error as shown in Fig. 65 for Ag(110) (1 TO) in an extended zone scheme. An initial decrease of ACosp(qll) was discussed already for the films of Mg [97] and Hg [ 106] and suggested to be associated to a non-negligible crystal potential. The importance of such effect would be doubtful for Mg but the explanation is reasonable for Hg and Ag. In view of the anisotropy of the effect with respect to the different Ag surfaces the relevant property must be connected to the band structure at the surface. Surface plasmon damping is thus anisotropic with respect to crystal face and, for Ag(110), azimuthal direction as was the case for surface plasmon dispersion, too. The existence of a link between these two quantities is appealing and was indeed suggested by Li et al. in a theoretical paper [ 143]. It can be rationalized thinking that both are influenced by the band structure of the surface.

M. Rocca/Surface Science Reports 22 (1995) 1-71

....

I ....

I ....

I

-,~-~,

t ,°,,,

I ....

1,

57

1.0 >,a.,)

0.8

0.6

[.~ 0.4 0.2 0.0

0

0.1

0.2

0.3

qll (~ ') Fig. 59. (a) Collection of plasmon width versus qll for Ag(001 ). The best-fit parameters are summarized in Table 8. Above htosp = 3.84 eV and qll = 0 . 1 0 / ~ - ~ an extra decay channel due to interband transitions between surface electronic states opens up. 300

....

I ....

I ....

I ....

I ....

Ag(ll0)

~" zoo

I ....

300

<001>

'''1

i

53-

-o-

~"

o

M

~.

100

<1

I ....

] ....

I ....

I ....

Ag (110)

<110>

aoo

E

....

[

~

-oxo

1oo

b)

<

....

I ....

I ....

o.os

o.1

I

....

I ....

o. s

qll

(i

0.2

I ....

0

0.25

0.3

''

t . . . . . .

o

L ....

o.os

I ....

I ....

0.2

*)

qlr

0.3

(~ 1)

Fig. 60. Same as Fig. 59 for A g ( l l 0 ) at T = 100 K: (a) {001}, (b) (110}.

. . . .

400

o

[]

> @

300

X

+

v

:=

I

qll> 0 qll<0 qll>0 qll<0

. . . .

I

'

.....

'

/ E,= 15.0 eV

I .... 0.1

'

I .... o E3

Ei= 10.7 eV <1)

/

I ....

Ei= IO.7 eV

qlt>O qll<0

E~

I ....

"× +

300

15.0

--

200

a) 100Kt g ( 1 1 1) 0.2

I ....

qil>O qrl
>

/

oO }--+*~e-×-a-- + ....

I

400

aoo

I00

. . . .

t .... 0.3

[]+ [] 0

0.4

qll ( 1 - ' ) Fig. 61. Same as Fig. 59 for A g ( l l l ) :

°°×

100

.... 0

[]

OK

+

Ag(111)

+

I ....

I ....

0.1

0.2

qll (a) T = 100 K, T = 300 K.

t .... 0.3

(1-')

0.4

M. Rocca/Surface Science Reports 22 (1995) 1-71

58

. . . .

I . . . .

I . . . .

I '

'

'

1.0 q? 0.8

;> 09

~9

0.6 X 0.4 0.2

Ag(001)

0.0

I .... 3.7

I .... 3.8

I .... 3.9

I .... 4

4.

hCOsp ( e V ) Fig. 62. Dependence of the surface plasmon loss line width versus loss energy for Ag(001). . . . .

I

. . . .

I

. . . .

I

I . . . .

. . . .

I . . . .

I . . . .

I

....

400

400

Ag(110)

;>

~>

<001>

a)

(1)

E

300

[] 200

200

+

i

b)

Ag _(110) <110>

[] []

<:1

<3

IO0

0

.... 3.7

I .... 3.8

I .... 3.9

~C0sp

I .... 4

0

I . . . .

I . . . .

3.7

4.1

3.8

I . . . .

I

3.9 i¢-1C0 s p

(eV)

....

4

4.

(eV)

Fig. 63. Same as Fig. 62 for the two high-symmetry directions of Ag(110): (a) (001), (b) (li0). . . . .

400 ;> 0

I

. . . .

I

. . . .

I

....

. . . .

400

2 q,>o lO.7 ev u qll
x qll>0 + qll
300

1 5 . 0 eV

>o

x-

u~

~b

0

~$

o ×

D a) lOOK_

+ 0

I

3.7

. . . .

Fig.

15.0 eV

qll>0 qll
x +

200

I

. . . .

I

100

3.9

64.

Same

....

4

[]

as

Fig.

62

....

0

3.7

4.

for

Ag(lll):

x ×

0

x []

+

b)

+

300K

Ag(lll)

A g ( l l 1)

I . . . .

3.8

300

I ....

10.7 eV

0

X

200 i00

qll > 0 t2 qll
I ....

v

v

¢.r..

I ....

0

I ....

3.8

I ....

3.9

( a ) T = 100 K, ( b ) T = 3 0 0 K.

I ....

4

4.

M. Rocca/Surface Science Reports 22 (1995) 1-71

500 - ' ' " 1

.... I .... I ....

59

I .... I ....

Ag (ii0)

400



> 300

200

-0-

<1

0 -0.3

,,,L .... I .... I .... ] .... -0.2

-0.1

0

q, (A-1)

0.1

I .... 0.2

0.3

Fig. 65. (a) Dependence of surface-plasmon damping versus qll for Ag(110) (1 i0} on an extended zone scheme to highlight the negative initial dispersion.

An anisotropy for the excitation spectrum of interband transitions was found for Ag(110) also in optical reflectance measurements [ 144]. These data were analyzed on the basis of theoretical calculations by Tarriba and Mochan [145] which predict a sharp peak in the real part of the surface conductivity along (001} at 3.82 eV. Such peak should, however, show up also in the EEL spectra and should be resolvable from the surface plasmon contribution at least for vanishing qll, contrary to our experimental results. We notice that no extra broadening of the losses is present at the onset of the bulk interband transitions involving either d-states or sp-bands. According to optical data [63,64] transitions involving d-states are distributed over a large portion of reciprocal space with an onset at 4.03 eV at T = 0 K, which shifts to 3.94 eV at room temperature, while EF ~ L4+ (see Fig. 66) transitions occur at 4.01 eV at T = 0 and shift to 3.86 eV at room temperature. As one can see in Fig. 59 for Ag(001), the loss width grows abruptly beyond qll = 0.10 4- 0.02 ~-1 (or htOsp = 3.84 4- 0.03 eV) indicating the opening of a new efficient decay channel. Such Table 8 Values of parameters of surface plasmon damping on Ag surfaces: Atosp(0), Bl, which describes the dependence of the FWHM on qll (Eq. ( 2 6 ) ) , and B2, idem on o)sp (Eq. (27)) (the number in parentheses indicates the error on the last digit) Face (001) (001) (110) (110) (110) (111) (111) (111) (111)

Azimuth qll < 0.1 A qll > 0.1 A (001) (110} (110}

qll qtl qll qll qll qll

<0.1A > 0.16 < 0.16 > 0.16 < 0.16 > 0.16

A A A A A

T (K)

Atosp(0) (eV)

B1 (eVA)

B2

300 300 100 100 100 100 100 300 300

0.092 (5)

1.0 (1) 4.4 (3) 0.60 (5) <0 0.66 (7) <0 1.8 (1) 0.15 (5) 1.8 (1)

0.66 (3) 3.0 (2) 0.40 (4) <0 0.68 (7) N0 0.95 (10) 0.6 (6) 0.6 (6)

References: Ag(001) [87]; A g ( l l 0 ) [135]; A g ( i l l )

0.095 (5) 0.11 (1) 0.090 (5) 0.090 (5)

[50].

M. Rocca/Surface Science Reports 22 (1995) 1-71

60

s band mSll= 0.126

+ L4( L 1)

ms j_= 5.160

3.87 ,~Wg= 3.85

p band

E~

7[I/4 a

~/4a,

rap.L= 0.320

i_4( L2 ) mP11=0.172 ~.

4.03

,.lh~O= 3.68

+

d band

L6+5 (L3) rod j_= 2 . 5 8 0

mdl I-- 2.075

kj. W

k II ~

L

~

r'

Fig. 66. Bulk band structure of Ag near L with the indication of the interband gaps which determine the onset of sp --* sp and d --~ sp transitions (from Ref. [ 146], used with permission).

phenomenon is observed also for A g ( l l l ) , although at a different critical qll and with a smaller efficiency, and is absent for Ag(110). Interestingly, the critical frequency turns out to be the same for Ag(001 ) and Ag(111). The extra channel is probably connected to decay into an electron-hole pair involving a filled Shockley state, as indeed present near }( on Ag(001) and near ~ on A g ( l l l ) , and an empty image state. Alternatively, transitions between Shockley states are possible near X on Ag(001). Such mechanisms would be absent for Ag(110) as (a) the image states are absent near F as no gap in the projection of the bulk states is present, (b) empty and filled Shockley states are present at X and Y but do not match with surface plasmon energy and momentum. To extract the dispersion coefficients for the FWHM the data were fitted in each relevant qll range by the linear forms:

M. Rocca/Surface Science Reports 22 (1995) 1-71

0.14

....

I ....

I ....

I

. . . .

61

I'

/O ~-~

//////

0.12

3

<1 0.10

0.08

0

100

200

300

400

T (K) Fig. 67. Dependence of AtOsp on T for the different Ag surfaces. The data for Ag(110) are reported explicitly ( o ) . They were recorded along ( l i 0 ) at Ei = 8.41 eV and Os = 77.7 °, which correspond to qll = 0.01 /~-t; in such conditions the contribution of the interband transition to the inelastic intensity is smallest. The data for Ag(001 ) (solid line) and Ag( 111 ) (dashed line) are taken from Fig. 69 and Figs. 61 and 64, respectively. Note that the extrapolation to T ~ 0 of AW~p(T) converges to the same value for Ag(001 ) and Ag( 111 ). A h t o s p ( q l l ) = Ahtospo + nlqll

(28)

AtOsp(tOsp ) = AtOsp0 + B2tosp,

(29)

and

where Ahtosp 0 is the line width at vanishing momentum transfer. The best-fit values are reported in Table 8.

5.2.3. Effect of crystal temperature tOsp and Atosp depend on crystal temperature, T, for all Ag surfaces. Data for A g ( l l l ) were already presented in Figs. 50, 61 and 64. In general tOsp decreases with T, while Atosp grows. The effect on Atosp is small for Ag(001) and larger for A g ( l l l ) and A g ( l l 0 ) as shown in Fig. 67. For Ag(110) T affects strongly the relative contribution of the interband transitions (which is particularly strong for this face) to the inelastic intensity; the reported data were thus recorded for the lowest possible Ei where the latter contribution is minimal. The extrapolation of Atosp to 0 K converges to the same value for Ag(001) and Ag( 111 ), as shown in Fig. 67. This result increases our confidence in the correctness of our analysis of the data which excluded further contributions to the inelastic intensity for these surfaces. The data for A g ( l l 0 ) ( l i 0 ) are slightly higher indicating either that the contribution of the interband transition to the loss intensity is small but not negligible or that the presence of interband transitions at the surface reduces the lifetime of the surface plasmon. The effect of T on surface plasmon dispersion is shown in Fig. 48a for Ag(001) (compare the data recorded under identical experimental conditions: (o) at 295 K and ( × ) at 125 K) and in Fig. 50 for A g ( l l l ) . As one can see, the dispersion coefficient is affected little, the major effect of T being to shift rigidly the dispersion curve. Effects connected to T were studied more in detail for Ag(001) where the losses are more symmetric and the damping is minimal for vanishing qll- A series of EEL spectra obtained for this surface at different crystal temperatures is shown in Fig. 68 [134]. The T dependence of tOsp is

62

M. Rocca/ Surface Science Reports 22 (1995) 1-71

,qg{O01) , "X q , , = 0 . 0 0 7 A-~ .// ~T =138K

E i = 1 6 . 0 0 eV

e~=6o.5° e'=el "3°

H

I0

3

o X.

Z o

m F~ ~E

>-I-----t

..........

Z W Z

I

I

3.4

I

I

3.B 3.6 ENERGY

I

3.7 3.8 3.9 LOSS ( e V }

Fig. 68. EEL spectra recorded at different temperatures for Ag(001). reported in Fig. 69. In the limited temperature range under consideration it can be fitted with a linear form A + B T and one obtains: htosp(T) = (3.724 + 0.003) - (9 + 1 ) 1 0 - S T eV with T expressed in K. This dependence should be compared with the shift o f - 1 . 3 x 10 -4 eV K -1 o f the onset o f the d ---* EF interband transition at 3.98 eV and with the much larger shift o f - 6 . 5 x 10 -4 eV K - ! o f the sp to sp L4- ~ L4+ transition at 3.86 eV [63,64]. As shown in Fig. 69 the shift in tosp(T) is proportional to the thermal expansion o f the crystal. Such proportionality can be understood as a 1.005 O O

....

I ....

I ....

I ....

I'

1.000 0.995

Z: 0.990

0.985

....

I .... i00

AT

I .... 200

I .... 300

I, 400

(K)

Fig. 69. Dependence of the surface plasmon frequency on Ag(001) with crystal temperature. The data are compared with the thermal expansion of the crystal, denoted by (0) and by the solid line.

M. Rocca/Surface Science Reports 22 (1995) 1-71

0.12

. . . .

]

.

.

.

.

.

.

.

.

.

.

.

.

63

.

0.11 3

0.10

<1 0.09 0.08 0

100

200

300

400

T (K) Fig. 70. Surface plasmon damping on Ag(001 ) versus T.

consequence of the linearity of the shift of the onset of the d ---+EF interband transitions with lattice expansion. Alternatively, it can be rationalized thinking that the electron density, n, enters into the determination of tOp and tOsp and that n decreases with thermal expansion. The T dependence of Atosp is reported in Fig. 70. Assuming again a linear dependence in first approximation one obtains: Ahtosp(T) = (81 4- 4) × 10 -3 q- (7 q- 2) × 10-ST eV. Over the same T range Ahtom(T) is thus affected much more than to~p(T). The effect amounts in fact to some 30% over 100 K, while the frequency shift is less than 1%. It could be related either to the shift of the interband transitions (Landau mechanism) or to a phonon-assisted decay (Drude mechanism). The gap between htosp and the onset of interband transitions is reduced indeed from T = 0 K to 295 K by 50% for sp ---+ sp and by 15% for d --+ EF. Hasegawa [75] proposed on the other hand that up to one third of the broadening of bulk plasmons is due to a phonon-assisted Drude mechanism. Such contribution might be expected to be even larger at the surface because of the lower Debye temperature.

6. Surface electronic excitations of transition metals: Pd(llO) Surface plasmons on d-metals are usually strongly damped by interband transitions. Pd is a remarkable exception to this rule: optical data [ 147] indicate that the collective electronic excitation has a width of ~ 2 eV but is still quite well defined in the energy-loss spectra (see Fig. 71 ). The assignment of this peak in experimental work was, however, controversial as bulk and surface plasmons differ by only 0.3 eV and cannot be resolved as separated peaks. Several authors concluded in favor of a bulk excitation as they were unable to observe any appreciable dispersion. The reported frequencies are, however, scattered over a remarkable range and a downshifi of the loss frequency was observed upon oxygen exposure [ 148]. The latter effect is clearly in contradiction with the assignment to a bulk loss, as this should be independent of the presence of adsorbates. Moreover, if the peak would have been composed of both bulk and surface contributions a shift to higher frequencies would be expected when quenching the surface plasmon contribution! The probable reason for the poor experimental reproducibility is that strong interband transitions are present both at lower and at higher loss energies and their contribution to the inelastic intensity is not proportional to the one of the

M. Rocca /Surface Science Reports 22 (1995) 1-71

64

100

O.3 oS

O

2

-

d v

0.2

60 £.

{3_

t~

40 E = tn

0.1 (3

~C

20

0,0

~--

0

0 20

40

60 E n e r g y (eV)

80

100

Fig. 71. Surface electronic excitation spectrum of Pd at qll = 0 as calculated by Lovd6 and Gumhalter using a semiempidcal approach. Solid and dashed lines are the result for the sum rule describing the condition of overall charge neutrality and particle conservation ( f - s u m rule) of the system (from Ref. [ 130], used with permission).

E, Os

]

= 2301 eV

1.28

72.8o~~.~

0.00~ I

.6 Ei =

/ ' ~

eV

599.92

o,

683 --

Ei = 78.8 eV 0s = 72.9° / ..__.._.-.'t"

0 "~

i

~ .

0. 069

1

I

I

6.36 0.179

I

16

I

8 loss

energy

I

1

0 (eV)

Fig. 72. Energy-loss spectra for Pd(110) along (li0). 7.25

r

'

'

'

'

[

I

'

Pd(110)

7.00

Ox

>

q) 6.75 X X

6.50

3

,

6.25

6.00

0

x

<001> i o <110> ,

,

1 ,

,

I

0.1

,

,

I

02

qlf (1-1) Fig. 73. Surface plasmon dispersion of Pd(110). A negative dispersion is present in spite of the presence of d-electrons.

M. Rocca/Surface Science Reports 22 (1995) 1-71

65

collective excitation. Similar problems were already encountered in the study of Ag(110) where we learned that the relative contributions are mediated by the scattering mechanism and change with Ei and 0. A more accurate study of Pd(110) was performed recently in collaboration with the group of Paolucci in Trieste [ 132]. We found in agreement with previous work that at low Ei the contribution of interband transitions dominates the spectra, while for larger Ei, as shown in Fig. 72, the collective excitation dominates. At high energy the dipole lobe shrinks and the data were accordingly recorded in-specular. Grazing incidence was used to enhance surface sensitivity. The assignment to a surface excitation follows from the consideration that: (a) the extrapolation of the loss frequency to qll = 0 gives the value of 7.3 e V / h and a loss width of ~2 eV as expected from optical data; (b) the loss disperses with qH, while no dispersion was observed for the bulk loss [ 131]. The data show a negative initial dispersion with qll while the losses become broader. Interband transitions show up in the spectra as minor peaks, as e.g. for Ei = 2301 eV. An artifact due to the contribution of interband transitions to the EEL spectra mimicking the negative dispersion is excluded as interband transitions contribute differently to the two azimuths, while the observed dispersion is isotropic. A quantitative evaluation of surface plasmon damping is difficult in view of the short lifetime present already at vanishing qll. The data for surface plasmon dispersion are collected in Fig. 73; as one can see, the initial dispersion is strongly negative in spite of the presence of d-electrons. The situation for Pd is, however, very different than for Ag as in this case the d-bands are cut by the Fermi level and d-electrons contribute to the electrical conductivity. On Ag on the contrary they did not contribute and built a polarizable medium. Electron mobility seems therefore to play a role in determining the sign of surface plasmon dispersion, s- and d-electrons thus play an equally important role for surface plasmon dispersion for Pd. Fitting the initial dispersion with a linear form one would obtain: htosp = (7.36 4-0.05) - (7.52 40.73)ql I eV. The linear dispersion term is thus much larger than the values reported so far whose magnitudes were typically of the order of 1 eV/~. Applying Eq. (8) one would obtain d± = 2 / ~ outside the geometric surface, i.e. 2 to 4 times larger than for the alkali metals. Such value has hence no physical meaning and indicates the need for a more refined theory.

7. Synopsis and conclusions In conclusion I hope to have convinced the readers that the study of surface electronic excitations disclosed an unexpectedly rich field. While free-electron-like metals can be considered to be reasonably understood, the presence of d-electrons causes completely different behavior whose microscopic origin has only been marginally grasped. While waiting for the development of new more refined theories, a widening of the experimental data basis is still possible. Although no other material presents such favorable conditions for surface plasmon measurements as simple metals and Ag, reasonably sharp surface collective excitations should be present for Nb [61 ] and W [ 130] ; a revisiting of AI and In would also be instructive. Well defined surface collective excitations are known to be present also on Si, although strongly damped. In spite of the fact that all low-Miller-index surfaces of Ag were investigated in detail the special properties of this material with respect to electronic excitations are not exhausted and can be exploited further: e.g. interesting effects are expected for stepped surfaces, which are strongly azimuthally anisotropic and for thin Ag films grown on other materials as in this

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case lattice spacing and electron density could be varied at will. Particularly important would be the study of the surface plasmon dispersion of a monoatomic Ag film deposited on a free-electron-like substrate as this case could be treated theoretically starting from a jellium model for the substrate [ 51 ]. The understanding of electronic excitations can thus keep people active for several more years, in spite of having been intensively studied for several decades.

8. Symbols and acronyms Charge of the electron Electron Energy-Loss Spectroscopy Ei Impact energy of the electron beam Energy transfer E|oss Energy of the scattered electrons Es EXAFS Extended X-ray Absorption Fine Structure FWHM Full-Width at Half-Maximum hqll Transferred momentum of the scattered electrons and quasi-momentum of the surface excitation HREELS High-Resolution Electron Energy-Loss Spectroscopy ILEED Inelastic LEED Wavevector of the impinging electron ki Wavevector of the scattered electron ks LDA Local Density Approximation LEED Low-Energy Electron Diffraction m Mass of the electron m* Effective mass n Concentration of free carriers NEXAFS Near-Edge X-ray Appearance Fine Structure (n,,,,) Ensemble of phonon occupation numbers of the modes t with momentum hql I PES Photoemission spectroscopy P(qll, ~o) Energy-loss function rlt Coordinate parallel to the surface (R} Ensemble of atomic positions Ri and Rs Amplitude of elastic reflectivity corresponding to processes of loss after and before reflection RPA Random Phase Approximation Equivalent radius denoting jellium density, expressed in Bohr units a0 = 0.53/~. rs T Crystal temperature TDLDA Time-Dependent LDA vii and v± Velocity components of the impinging electrons Coordinate vertical to the surface Z e

EELS

~'b Es

Bulk dielectric function Dielectric function in the surface layer

M. Rocca/Surface Science Reports 22 (1995) 1-71

Angle of incidence Angle of scattering Angular frequency Solid angle

~s (1)

[l

Acknowledgement The author acknowledges the critical reading of the manuscript by A. Liebsch.

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