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Optical properties of nanostructured thin films containing noble metal clusters: AuN, (Au0.5Ag0.5)N and AgN
Scripta mater. 44 (2001) 1235–1238 www.elsevier.com/locate/scriptamat
OPTICAL PROPERTIES OF NANOSTRUCTURED THIN FILMS CONTAINING NOBLE METAL CLUSTERS: AuN, (Au0.5Ag0.5)N AND AgN B. Pre´vel1, J. Lerme´2, M. Gaudry2, E. Cottancin2, M. Pellarin2, M. Treilleux1, P. Me´linon1, A. Perez1, J.L. Vialle2 and M. Broyer2 1
De´partement de Physique des Mate´riaux and 2Laboratoire de Spectrome´trie Ionique et Mole´culaire, Universite´ Claude Bernard Lyon-1, 43, bd du 11 novembre 1918, 69622 Villeurbanne Cedex, France (Received August 21, 2000) (Accepted in revised form December 27, 2000) Abstract—The optical properties of nanocomposite thin films of gold, silver and bimetallic silvergold clusters embedded in a porous alumina matrix have been investigated in the size range 2– 6.7 nm. The metallic particles are produced by laser vaporization of either an Au0.5Ag0.5 alloy or a pure metal target whereas the dielectric matrix is evaporated by an electron gun. Samples involving a low metal concentration have been characterized by several complementary techniques in order to determine their composition, morphology and cluster size distribution. The mixed particles have the same stoichiometry as the target rod. Optical absorption spectra exhibit a surface plasmon resonance whose position is shifting with cluster mean size, giving evidence of finite size effects. Theoretical calculations in the framework of Time-Dependent-Local-Density-Approximation (TDLDA), taking into account an inner skin of ineffective screening and the porosity of the matrix, are consistent with observed size evolutions of the Mie frequency in each type of sample. © 2001 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved.
Keywords: Optical spectroscopy; Optical properties; Metals; Surface plasmon resonance Introduction As compared to the bulk, noble metal particles exhibit specific optical properties mainly due to the multipolar collective excitations arising from the dielectric confinement (1). In the nanometer range the photoabsorption spectra are dominated by an absorption band whose position and width are ruled by the intrinsic properties of the clusters and the surrounding medium (surface plasmon -or Mie -resonance). These features appear both to be size-dependent, although no size effect is predicted by the classical Mie’s theory in the quasistatic limit, except for a mere volume scaling factor. The measured size effects which reflect those of the electronic structure have indeed a quantum origin. In contrast with alkali clusters which exhibit a noticeable red-shift trend with decreasing cluster size due to the spillout phenomenon (2), this trend is reversed in the case of free AgN⫹ and AuN⫹ clusters (3,4), namely a slight blue shift trend is observed, reflecting the influence of the d-electrons. The understanding of the optical properties of pure metal particles makes the extension of this study to bimetallic clusters in the small size domain very interesting. Bimetallic particles can form either alloys or segregated (referred to -as coated- or -core-shell- particles) systems (see for example refs. 5 to 7). Up to now, most of the experiments on alloyed and coated Ag/Au systems are realized for particles larger than 5 nm in diameter (6,8). In this communication, we report on experimental and theoretical investigations on the size 1359-6462/01/$–see front matter. © 2001 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. PII: S1359-6462(01)00852-1
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Figure 1. Absorption coefficient versus energy of alumina embedded particles: (a) AuN, (b) (Au0.5Ag0.5)N, (c) AgN, The optical diameters ⬍Dopt⬎ are given in the figures. The vertical lines, drawn for guiding the eyes, are the classical Mie frequencies and the arrows are located at the peak plasmon maximum.
evolution of the optical properties of alumina-embedded silver, gold and mixed silver-gold clusters with mean optical diameter lower than 7 nm, prepared by Low Energy Cluster Beam Deposition technique (LECBD). For these three systems the size effects can be consistently explained within a common model. Sample Production and Characterization Composite films are produced using the LECBD technique described elsewhere (9). The alumina matrix is evaporated from a crucible heated by an electron gun. Clusters and alumina matrix are co-deposited at room temperature under a vacuum of 10⫺7 Torr, on a substrate whose nature depends on the experiments to be performed. In order to avoid any cluster coalescence and to deduce the optical properties of non-interacting isolated particles from the experiment, the metal volumic fraction is chosen to be less than 5% by controlling both deposition rates. The alumina matrix has an amorphous structure with a high porosity (⬇45%) and is slightly overstoichiometric in oxygen according to the raw formula Al2O3.2. The relative atomic fraction 50% Ag and 50% Au is measured in the deposited bimetallic clusters by Energy Dispersive X-Ray (EDX) and Rutherford Back Scattering (RBS) analyses. Those techniques confirm also the low volumic metal concentration in each sample. Transmission Electron Microscopy (TEM) micrographs reveal that the particles are roughly spherical and randomly distributed in the matrix and allow the determination of the size distributions and the corresponding mean diameter ⬍D⬎ over a population of 700 to 3000 particles. For comparison with the theoretical absorption spectra, calculated for a restricted set of selected sizes, each size distribution has been characterized by the mean optical diameter ⬍Dopt⬎. ⬍Dopt⬎3, proportional to the mean cluster volume, is defined as ⬍D3⬎. ⬍Dopt⬎ varies from 2.2 to 4.5 nm, from 2 to 4 nm and from 3.6 to 6.7 nm for AuN, (Au0.5Ag0.5)N, and AgN clusters, respectively. Optical Properties Transmission measurements performed on the films deposited on silica substrates (suprasil), in the spectral range 200 – 800 nm (1.55– 6.2 eV), lead to the determination of the absorption coefficient versus photon energy. Results are displayed in Figure 1(a-c) for AuN, (Au0.5Ag0.5)N and AgN clusters embedded in alumina matrix for various size distributions.
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The spectra exhibit a more or less pronounced absorption band due to the surface plasmon resonance and an increasing absorption in the UV region due to interband transitions. The absorption band lies respectively around 2.4 eV for AuN samples, 2.6 eV for (Au0.5Ag0.5)N and 2.9 eV for AgN samples. In the case of (Au0.5Ag0.5)N clusters, the resonance occurs almost halfway between those of pure gold and pure silver embedded clusters. Comparison of the spectra recorded in the case of the mixed clusters with the absorption spectra of alloyed and segregated Au-Ag clusters reported in the literature (10 –12) highly suggests an alloying of the two metals inside the clusters. Let us focus now on the size effects in the optical response. The most conspicuous features are the broadening and the damping of the surface plasmon as the cluster size decreases, especially for AuN clusters, and, to a lesser extent, for mixed Au/Ag clusters. Moreover a noticeable blue-shift of the resonance with decreasing cluster size is observed in the case of AuN clusters (from 2.33 eV to 2.52 eV), as well as (Au0.5Ag0.5)N clusters (from 2.56 eV to 2.68 eV), whereas for AgN clusters the size effect is almost completely quenched (9,13). The observed blue-shift trend in the case of (Au0.5Ag0.5)N is intermediate between the size evolution obtained in pure AuN and AgN nanoparticles. As a matter of fact, the blue-shift of the surface plasmon frequency is the main factor ruling the broadening and the damping of the Mie band, due to the increasing coupling with (and the increasing magnitude of) the interband excitations in the plasmon band spectral range.
Discussion and Conclusion The quantum size effects in the optical absorption of alumina-embedded nanoscaled Ag and Au clusters are now well-understood (13,14). They can be reproduced in the frame of a mixed quantum/classical theoretical approach where the conduction electrons are quantum-mechanically treated (TDLDA), whereas the optical properties of the matrix and the ionic core background (interband transitions from the d band towards the s-p conduction band, and polarization effects) are classically described through bulk-like dielectric functions. The model includes the inner surface skin of ineffective screening by the ionic cores (15) and the local porosity at the particle-matrix interface (13,14). The frequencydependence of the real component of ⑀d() (where ⑀d() is the interband contribution in the dielectric function of the metal) was found to be the key feature ruling the competition between the blue-shift trend induced by the skin regions of reduced polarizability and the red-shift trend induced by the spill-out phenomenon, the net result of which leading to the quenching of the size effects in silver clusters and the blue-shift trend in gold clusters. Owing to the propensity of the bimetallic system Ag/Au for alloying, at any relative concentration, we have assumed that both materials are homogeneously and similarly distributed inside the clusters, allowing the optical properties of the mixture to be mimicked by an effective dielectric function. As a first approximation the volumic compositionweighted average of the dielectric functions of silver and gold has been taken. Although such a convenient assumption was often used in the literature whatever the relative composition is, it is worthwhile pointing out that this simple ansatz can be supported in the present case. A general formula, suitable for any kind of heterogeneities, was derived by Brown (16). The effective dielectric function is expressed as a series expansion of the dielectric contrast parameter ␦⑀ ⫽ ⑀Ag-⑀Au, whose coefficients depend on the volumic fraction x ⫽ VAg/(VAu⫹VAg) and on the statistical spatial distributions of both metals (corrective terms which depend explicitly on the detailed geometry of the mixture). The zero-order term is nothing else but the composition average of both dielectric functions. For the volumic ratio x ⫽ 0.5 (silver and gold have almost identical volume per atom) the leading term in the geometry-dependent expansion vanishes if both materials in the mixture are similarly distributed. Moreover we have checked that the remaining geometry-independent term can be numerically neglected, supporting therefore our simple ansatz. Since silver and gold are characterized by almost
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Figure 2. The location of the Mie resonance frequency for AgN (triangles), AuN (rhombs), and (Au0.5Ag0.5)N] (circles) clusters. R is defined as ⬍Dopt⬎/2. The open symbols are the experimental data. The black symbols, joined by a full line, are the theoretical TDLDA results. The dashed line shows the results obtained within the extended Mie’s theory (three interfaces) in the case of mixed Ag/Au clusters.
identical electronic density and effective mass, the Drude components in the dielectric function of both metals are similar, and the volumic average assumption is thus also appropriate for defining the interband contribution ⑀d() of the Ag/Au mixture. The hybrid classical/quantum theoretical model has been applied in order to study the quantum finite-size effects of alloyed Ag/Au clusters. The results obtained with respect to the location of the Mie resonance are summarized in Fig. 2. The quite good consistent agreement with the experimental data, for the three systems and with the same model parameters (thicknesses of the inner and outer skins of reduced polarizability at the cluster-matrix interface) emphasizes the suitability of the theoretical approach. More conclusive information will be provided by varying the relative metal concentration and by studying other alloyed systems from noble metals. References 1. U. Kreibig and M. Vollmer, Optical Properties of Metal Clusters, Springer, Berlin (1995). 2. V. V. Kresin, Phys. Rep. 220, 1 (1992). 3. J. Tiggesbau¨mker, L. Ko¨ller, K. H. Meiwes-Broer, and A. Liebsch, Phys. Rev. A. 48, R1749 (1993). 4. M. Lindinger, K. Dasgupta, G. Dietrich, S. Kru¨ckeberg, S. Kuznetsov, K. Lu¨tzenkirchen, L. Schweikhard, C. Walther, and J. Ziegler, Z. Phys. D. 40, 347 (1997). 5. A. Henglein and M. Giersig, J. Phys. Chem. 98, 6931 (1994). 6. L. M. Liz-Marzan and A. P. Philipse, J. Phys. Chem. 99, 15120 (1995). 7. J. L. Rousset, A. Renouprez, and A. M. Cadrot, Phys. Rev. B. 58, 2150 (1998). 8. S. Link, Z. L. Wang, and M. A. El-Sayed, J. Phys. Chem. 103, 3529 (1999). 9. B. Palpant, B. Pre´vel, J. Lerme´, E. Cottancin, M. Pellarin, M. Treilleux, A. Perez, J. L. Vialle, and M. Broyer, Phys. Rev. B. 57, 1963 (1998). 10. G. C. Papavassiliou, J. Phys. F. 6, L103 (1976). 11. J. Sinzig, U. Radtke, M. Quinten, and U. Kreibig, Z. Phys. D. 26, 242 (1993). 12. B. K. Teo, K. Keating, and Y. H. Kao, J. Am. Chem. Soc. 109, 3494 (1987). 13. J. Lerme´, B. Palpant, B. Pre´vel, M. Pellarin, M. Treilleux, J. L. Vialle, A. Perez, and M. Broyer, Phys. Rev. Lett. 80, 5105 (1998). 14. J. Lerme´, B. Palpant, B. Pre´vel, E. Cottancin, M. Pellarin, M. Treilleux, J. L. Vialle, A. Perez, and M. Broyer, Eur. Phys. J. D. 4, 95 (1998). 15. A. Liebsch, Phys. Rev. B. 48, 11317 (1993). 16. W. F. J. Brown, J. Chem. Phys. 23, 1514 (1955).
OPTICAL PROPERTIES OF NANOSTRUCTURED THIN FILMS CONTAINING NOBLE METAL CLUSTERS: AuN, (Au0.5Ag0.5)N AND AgN B. Pre´vel1, J. Lerme´2, M. Gaudry2, E. Cottancin2, M. Pellarin2, M. Treilleux1, P. Me´linon1, A. Perez1, J.L. Vialle2 and M. Broyer2 1
De´partement de Physique des Mate´riaux and 2Laboratoire de Spectrome´trie Ionique et Mole´culaire, Universite´ Claude Bernard Lyon-1, 43, bd du 11 novembre 1918, 69622 Villeurbanne Cedex, France (Received August 21, 2000) (Accepted in revised form December 27, 2000) Abstract—The optical properties of nanocomposite thin films of gold, silver and bimetallic silvergold clusters embedded in a porous alumina matrix have been investigated in the size range 2– 6.7 nm. The metallic particles are produced by laser vaporization of either an Au0.5Ag0.5 alloy or a pure metal target whereas the dielectric matrix is evaporated by an electron gun. Samples involving a low metal concentration have been characterized by several complementary techniques in order to determine their composition, morphology and cluster size distribution. The mixed particles have the same stoichiometry as the target rod. Optical absorption spectra exhibit a surface plasmon resonance whose position is shifting with cluster mean size, giving evidence of finite size effects. Theoretical calculations in the framework of Time-Dependent-Local-Density-Approximation (TDLDA), taking into account an inner skin of ineffective screening and the porosity of the matrix, are consistent with observed size evolutions of the Mie frequency in each type of sample. © 2001 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved.
Keywords: Optical spectroscopy; Optical properties; Metals; Surface plasmon resonance Introduction As compared to the bulk, noble metal particles exhibit specific optical properties mainly due to the multipolar collective excitations arising from the dielectric confinement (1). In the nanometer range the photoabsorption spectra are dominated by an absorption band whose position and width are ruled by the intrinsic properties of the clusters and the surrounding medium (surface plasmon -or Mie -resonance). These features appear both to be size-dependent, although no size effect is predicted by the classical Mie’s theory in the quasistatic limit, except for a mere volume scaling factor. The measured size effects which reflect those of the electronic structure have indeed a quantum origin. In contrast with alkali clusters which exhibit a noticeable red-shift trend with decreasing cluster size due to the spillout phenomenon (2), this trend is reversed in the case of free AgN⫹ and AuN⫹ clusters (3,4), namely a slight blue shift trend is observed, reflecting the influence of the d-electrons. The understanding of the optical properties of pure metal particles makes the extension of this study to bimetallic clusters in the small size domain very interesting. Bimetallic particles can form either alloys or segregated (referred to -as coated- or -core-shell- particles) systems (see for example refs. 5 to 7). Up to now, most of the experiments on alloyed and coated Ag/Au systems are realized for particles larger than 5 nm in diameter (6,8). In this communication, we report on experimental and theoretical investigations on the size 1359-6462/01/$–see front matter. © 2001 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. PII: S1359-6462(01)00852-1
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Figure 1. Absorption coefficient versus energy of alumina embedded particles: (a) AuN, (b) (Au0.5Ag0.5)N, (c) AgN, The optical diameters ⬍Dopt⬎ are given in the figures. The vertical lines, drawn for guiding the eyes, are the classical Mie frequencies and the arrows are located at the peak plasmon maximum.
evolution of the optical properties of alumina-embedded silver, gold and mixed silver-gold clusters with mean optical diameter lower than 7 nm, prepared by Low Energy Cluster Beam Deposition technique (LECBD). For these three systems the size effects can be consistently explained within a common model. Sample Production and Characterization Composite films are produced using the LECBD technique described elsewhere (9). The alumina matrix is evaporated from a crucible heated by an electron gun. Clusters and alumina matrix are co-deposited at room temperature under a vacuum of 10⫺7 Torr, on a substrate whose nature depends on the experiments to be performed. In order to avoid any cluster coalescence and to deduce the optical properties of non-interacting isolated particles from the experiment, the metal volumic fraction is chosen to be less than 5% by controlling both deposition rates. The alumina matrix has an amorphous structure with a high porosity (⬇45%) and is slightly overstoichiometric in oxygen according to the raw formula Al2O3.2. The relative atomic fraction 50% Ag and 50% Au is measured in the deposited bimetallic clusters by Energy Dispersive X-Ray (EDX) and Rutherford Back Scattering (RBS) analyses. Those techniques confirm also the low volumic metal concentration in each sample. Transmission Electron Microscopy (TEM) micrographs reveal that the particles are roughly spherical and randomly distributed in the matrix and allow the determination of the size distributions and the corresponding mean diameter ⬍D⬎ over a population of 700 to 3000 particles. For comparison with the theoretical absorption spectra, calculated for a restricted set of selected sizes, each size distribution has been characterized by the mean optical diameter ⬍Dopt⬎. ⬍Dopt⬎3, proportional to the mean cluster volume, is defined as ⬍D3⬎. ⬍Dopt⬎ varies from 2.2 to 4.5 nm, from 2 to 4 nm and from 3.6 to 6.7 nm for AuN, (Au0.5Ag0.5)N, and AgN clusters, respectively. Optical Properties Transmission measurements performed on the films deposited on silica substrates (suprasil), in the spectral range 200 – 800 nm (1.55– 6.2 eV), lead to the determination of the absorption coefficient versus photon energy. Results are displayed in Figure 1(a-c) for AuN, (Au0.5Ag0.5)N and AgN clusters embedded in alumina matrix for various size distributions.
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The spectra exhibit a more or less pronounced absorption band due to the surface plasmon resonance and an increasing absorption in the UV region due to interband transitions. The absorption band lies respectively around 2.4 eV for AuN samples, 2.6 eV for (Au0.5Ag0.5)N and 2.9 eV for AgN samples. In the case of (Au0.5Ag0.5)N clusters, the resonance occurs almost halfway between those of pure gold and pure silver embedded clusters. Comparison of the spectra recorded in the case of the mixed clusters with the absorption spectra of alloyed and segregated Au-Ag clusters reported in the literature (10 –12) highly suggests an alloying of the two metals inside the clusters. Let us focus now on the size effects in the optical response. The most conspicuous features are the broadening and the damping of the surface plasmon as the cluster size decreases, especially for AuN clusters, and, to a lesser extent, for mixed Au/Ag clusters. Moreover a noticeable blue-shift of the resonance with decreasing cluster size is observed in the case of AuN clusters (from 2.33 eV to 2.52 eV), as well as (Au0.5Ag0.5)N clusters (from 2.56 eV to 2.68 eV), whereas for AgN clusters the size effect is almost completely quenched (9,13). The observed blue-shift trend in the case of (Au0.5Ag0.5)N is intermediate between the size evolution obtained in pure AuN and AgN nanoparticles. As a matter of fact, the blue-shift of the surface plasmon frequency is the main factor ruling the broadening and the damping of the Mie band, due to the increasing coupling with (and the increasing magnitude of) the interband excitations in the plasmon band spectral range.
Discussion and Conclusion The quantum size effects in the optical absorption of alumina-embedded nanoscaled Ag and Au clusters are now well-understood (13,14). They can be reproduced in the frame of a mixed quantum/classical theoretical approach where the conduction electrons are quantum-mechanically treated (TDLDA), whereas the optical properties of the matrix and the ionic core background (interband transitions from the d band towards the s-p conduction band, and polarization effects) are classically described through bulk-like dielectric functions. The model includes the inner surface skin of ineffective screening by the ionic cores (15) and the local porosity at the particle-matrix interface (13,14). The frequencydependence of the real component of ⑀d() (where ⑀d() is the interband contribution in the dielectric function of the metal) was found to be the key feature ruling the competition between the blue-shift trend induced by the skin regions of reduced polarizability and the red-shift trend induced by the spill-out phenomenon, the net result of which leading to the quenching of the size effects in silver clusters and the blue-shift trend in gold clusters. Owing to the propensity of the bimetallic system Ag/Au for alloying, at any relative concentration, we have assumed that both materials are homogeneously and similarly distributed inside the clusters, allowing the optical properties of the mixture to be mimicked by an effective dielectric function. As a first approximation the volumic compositionweighted average of the dielectric functions of silver and gold has been taken. Although such a convenient assumption was often used in the literature whatever the relative composition is, it is worthwhile pointing out that this simple ansatz can be supported in the present case. A general formula, suitable for any kind of heterogeneities, was derived by Brown (16). The effective dielectric function is expressed as a series expansion of the dielectric contrast parameter ␦⑀ ⫽ ⑀Ag-⑀Au, whose coefficients depend on the volumic fraction x ⫽ VAg/(VAu⫹VAg) and on the statistical spatial distributions of both metals (corrective terms which depend explicitly on the detailed geometry of the mixture). The zero-order term is nothing else but the composition average of both dielectric functions. For the volumic ratio x ⫽ 0.5 (silver and gold have almost identical volume per atom) the leading term in the geometry-dependent expansion vanishes if both materials in the mixture are similarly distributed. Moreover we have checked that the remaining geometry-independent term can be numerically neglected, supporting therefore our simple ansatz. Since silver and gold are characterized by almost
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Figure 2. The location of the Mie resonance frequency for AgN (triangles), AuN (rhombs), and (Au0.5Ag0.5)N] (circles) clusters. R is defined as ⬍Dopt⬎/2. The open symbols are the experimental data. The black symbols, joined by a full line, are the theoretical TDLDA results. The dashed line shows the results obtained within the extended Mie’s theory (three interfaces) in the case of mixed Ag/Au clusters.
identical electronic density and effective mass, the Drude components in the dielectric function of both metals are similar, and the volumic average assumption is thus also appropriate for defining the interband contribution ⑀d() of the Ag/Au mixture. The hybrid classical/quantum theoretical model has been applied in order to study the quantum finite-size effects of alloyed Ag/Au clusters. The results obtained with respect to the location of the Mie resonance are summarized in Fig. 2. The quite good consistent agreement with the experimental data, for the three systems and with the same model parameters (thicknesses of the inner and outer skins of reduced polarizability at the cluster-matrix interface) emphasizes the suitability of the theoretical approach. More conclusive information will be provided by varying the relative metal concentration and by studying other alloyed systems from noble metals. References 1. U. Kreibig and M. Vollmer, Optical Properties of Metal Clusters, Springer, Berlin (1995). 2. V. V. Kresin, Phys. Rep. 220, 1 (1992). 3. J. Tiggesbau¨mker, L. Ko¨ller, K. H. Meiwes-Broer, and A. Liebsch, Phys. Rev. A. 48, R1749 (1993). 4. M. Lindinger, K. Dasgupta, G. Dietrich, S. Kru¨ckeberg, S. Kuznetsov, K. Lu¨tzenkirchen, L. Schweikhard, C. Walther, and J. Ziegler, Z. Phys. D. 40, 347 (1997). 5. A. Henglein and M. Giersig, J. Phys. Chem. 98, 6931 (1994). 6. L. M. Liz-Marzan and A. P. Philipse, J. Phys. Chem. 99, 15120 (1995). 7. J. L. Rousset, A. Renouprez, and A. M. Cadrot, Phys. Rev. B. 58, 2150 (1998). 8. S. Link, Z. L. Wang, and M. A. El-Sayed, J. Phys. Chem. 103, 3529 (1999). 9. B. Palpant, B. Pre´vel, J. Lerme´, E. Cottancin, M. Pellarin, M. Treilleux, A. Perez, J. L. Vialle, and M. Broyer, Phys. Rev. B. 57, 1963 (1998). 10. G. C. Papavassiliou, J. Phys. F. 6, L103 (1976). 11. J. Sinzig, U. Radtke, M. Quinten, and U. Kreibig, Z. Phys. D. 26, 242 (1993). 12. B. K. Teo, K. Keating, and Y. H. Kao, J. Am. Chem. Soc. 109, 3494 (1987). 13. J. Lerme´, B. Palpant, B. Pre´vel, M. Pellarin, M. Treilleux, J. L. Vialle, A. Perez, and M. Broyer, Phys. Rev. Lett. 80, 5105 (1998). 14. J. Lerme´, B. Palpant, B. Pre´vel, E. Cottancin, M. Pellarin, M. Treilleux, J. L. Vialle, A. Perez, and M. Broyer, Eur. Phys. J. D. 4, 95 (1998). 15. A. Liebsch, Phys. Rev. B. 48, 11317 (1993). 16. W. F. J. Brown, J. Chem. Phys. 23, 1514 (1955).