Optical measurement of the bulk plasmon dispersion in silver

Solid State Communications, Vol. 51, No. 11, pp. 905-908, 1984. Printed in Great Britain.

0038-1098/84 $3.00 + .00 Pergamon Press Ltd.

OPTICAL MEASUREMENT OF THE BULK PLASMON DISPERSION IN SILVER G. Piazza* and D.M. Kolb ¢ Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4 - 6 , D-1000 Berlin 33, West Germany and K. Kempa and F. Forstmann Institut fiir Theorie der Kondensierten Materie, Freie Universitiit, Arnimallee 14, D-1000 Berlin 33, West Germany

(Received 4 April 1984 by M. Cardona) By using electroreflectance spectroscopy, we detected an oscillatory behavior in the optical response of a thin Ag overlayer during its continuous growth on a flat Au (1 1 1) surface, when p-polarized light around the plasmon frequency of Ag was used. These oscillations are due to standing plasma waves in the Ag overlayer, and they allow a reliable measurement of the bulkplasmon dispersion for silver in the small-k regime (0.12 < k < 0.16 A-1).

THE BULK PLASMON dispersion is usually obtained by electron loss spectroscopy on metal foils, measuring the relation between energy loss and scattering angle [1 ]. This technique, however, cannot be extended into the region of small wave vectors, where the theory of plasmon dispersion is most trustworthy. We therefore present results of an optical technique, which complements the energy loss measurements in the small-k regime (k = 0.12-0.16 A-l). The optical measurement is based on our previous finding [2] that an electroreflectance (ER) signal from a thin silver overlayer on a metal substrate is strongly dependent on the thickness of the silver layer, if ppolarized light around the plasma frequency of silver (h(.op = 3.8 eV) is employed. Pursuing this thickness dependence further we discovered an oscillatory behaviour of the ER-signal with growing layer thickness due to standing plasma waves in the silver layer. The relation between photon energy and layer thickness leads directly to the bulk plasmon dispersion. ER measures the relative change AR/R o f the reflectivity of a surface due to an applied electric field [3-5]. In order to achieve large enough fields, the investigation was done in an electrochemical cell where a potential drop is applied across the very narrow double layer region of the electrode-electrolyte interface which for metals constitutes a plate condenser of extremely high capacitance [6]. The second virtue of

* Permanent address: Istituto Chimico Ciamician, University of Bologna, Italy. i" To whom all correspondence should be addressed.

this arrangement is the possibility to grow a silver film continuously by electrodeposition under controlled conditions and to measure with high accuracy the layer thickness by coulometry [6]. The silver f'rims were deposited on Au (1 1 1) from aqueous solutions of 0.5 M H2SO4 + 0.1 mM AgNO3 under diffusion controlled conditions. Hence, the sinusoidal potential modulation of 180 Hz and 100 mV peak-to-peak, which was superimposed on the bias potential of 0.1 V (vs a saturated calomel reference electrode) for detecting the ER, did not infuence the deposition rate. A flat substrate was mandatory for obtaining Ag layers of a thickness uniform enough not to average out the oscillations in AR/R. With lock-in amplification we achieved a sensitivity in AR/R of better than 10 -4 . The sign of AR/R is taken positive when a positive change of the electrode potential increased the reflectance. The ER signal was recorded at constant wavelength as a function of the deposited amount of silver. This amount was converted into layer thickness by assuming bulk silver density. Further experimental details will be given elsewhere [7]. The ER signal for an Ag overlayer on Au(1 1 1) as a function of Ag film thickness D is shown in Fig. 1 (left-hand side) for various photon energies. As we know from previous studies, the optical properties of the first one or two monolayers of a metal deposit are different from those of the bulk [8]. Up to a thickness of about 1 nm the reflectivity change AR/R shows marked variations with D for both polarizations, indicating that the film optical properties are still a function of thickness. For D > 1 nm, the ER effect for spolarization changes monotonically and smoothly, if at

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MEASUREMENT OF THE BULK PLASMON DISPERSION IN SILVER

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Fig. 1. Electroreflectance AR/R for Ag overlayers on Au(1 1 1) as a function of layer thickness D. Angle of incidence is 45 °. s ( - - - ) and p ( ) polarization. Experimental curves are on the left-hand side, calculated curves on right-hand side. all, which signals that the film optical properties have now reached bulk values and are therefore no longer depending on D. In this region, however, where/XR/R for s-polarized light shows hardly any variation with D, we notice pronounced oscillations for p-polarized light, especially in the photon energy region of plasmon excitation in Ag. This oscillatory behaviour we ascribe to standing plasma waves in the Ag film. When calculating the reflectivity R of the electrode surface, the Ag overlayer thickness D enters the equation for R in the form R = f{exp

(ikzD)},

(1)

where kz is the wave vector component normal to the surface for the longitudinal plasma wave. Equation (1) is correct for thicknesses D much smaller than the wavelength of the light and much greater than the transition region at the interfaces, no matter how complicated the phase relations in the surface transition region may be. Hence, the reflectivity is a periodic function of D, and the same is true for reflectance differences as measured in ER. This periodicity is the reason for the oscillatory behaviour shown in Fig. 1. The distance between two adjacent maxima or minima in AR/R can be taken directly as the plasmon wave length, because the tangential component of the plasmon wave vector is less than 1%o of the normal component. Although the relation between plasmon wavelength and distance of maxima is only strict for undamped waves, we have checked by a hydrodynamical calculation, that this correlation holds also for the strong-damping conditions in silver.

The key point for this evaluation of the plasmon wave length at optically enforced frequencies is that we do not have to deal with problems of surface and interface transition regions, boundary conditions, phase jumps etc., which in general complicate metal optics [9-11 ]. We measure the plasmon wave length directly by the thickness difference corresponding to two adjacent maxima in --/XR/R. Lindau and Nilsson [12] reported the only previous optical measurement of the silver plasmon dispersion where transmission minima for Ag films were determined. They evaluated the plasmon wave length from the total layer thickness and therefore could not avoid boundary problems. In Fig. 2 we show our co2(k2) relation together with previous evaluations, taken from a compilation by Raether [1 ]. The differences between our results and those of previous measurements seemed surprising, until we realized that optical measurements require an analysis which differs from that for the energy loss experiments. We evaluate for a real frequency the real part of a complex wave vector at which eL(co, k) = 0, while the electron energy loss experiments yield for a real k (fixed by the scattering angle) the real co, at which Im {eL(co,k) -1 } has its maximum. Essential differences between these two aspects of the eigenmode dispersion can be easily demonstrated for a free-electron dielectric function: %(~o, k) =

eL(CO,k)

1 - co~/[co(co + i v ) - - ~k2l.

= 0 yields the eigenmode dispersion for the optical experiment:

(2)

MEASUREMENT OF THE BULK PLASMON DISPERSION IN SILVER

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This approximation for the spatial dispersion in silver has been successfully used in a number of cases [13, 15-17]. The imaginary part of e L is taken independent of k and is included in eb(co). e L ( c o , k) = 0 then yields the plasmon dispersion relation: k 2 = /3-1 [co= _ c o ~ l e b ( c o ) ] .

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Fig. 2. The bulk plasmon dispersion curves for Ag, as evaluated from this work (solid line, full squares and dotted line) and from a compilation by Raether [1]. ( R e k ) 2 = /3-1 [(co2 _ co~)2 + 9'2co211,2 cos 2 ~ ;

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(3)

The energy loss experiment yields Max {Ira eL1} for real k and real co: co 2 = ~(cop= + / 3 k ~ - - 9'2/2)

+ t [ ( 4 + ~k2 -- 9'2/8)2 + 39'*/64] 1/2.

(4)

This dispersion [equation (4)] can hardly be distinguished from that for Re (co) and real k, derived from eL(CO , k) = 0: 9'2 (Re co)2 = co~ +/3k 2 4 ' (5) where co2 is exactly linear in k 2 as seen in the energy loss experiments (at higher k values). The characteristics of the optical dispersion curve [equation (3)] are: (Re k) 2 goes to zero for co + 0 and over a certain region the slope is approximately/3/cos2 ~b, i.e., larger than/3 in equation (5). We have included the complications due to the non-free-electron nature of silver in a manner first suggested by Zacharias and Kliewer [13]. The measured dielectric function of bulk silver [14] is separated into a bound and a free electron contribution: e(co) = 1 + Xb(co) + X f ( c o ) = eb(O3) - - (.02/(-.O2 •

(6)

The numerator, con2 = 4 1 r n e 2 / m , in the Drude part is derived from measurements in the infrared regime [ 14], where eb(co) = 1. For Ag, l~co,, = 9 eV. Near the plasma frequency the metal becomes spatially dispersive, i.e., the longitudinal dielectric function shows a dependence on the wave vector k [9, 10, 15]. This k-dependence is assumed only for the free-electron part of the longitudinal dielectric constant and is introduced according to the random.phase approximation (RPA) for a freeelectron gas:

(8)

We have determined the parameter/3 from the least squares fit of equation (8) to the experimental data, which is shown by the heavy line in Fig. 2. In order to achieve agreement in the absolute positions of measured and calculated dispersion curves, we adjusted the frequency scale for the data of [14] by a small downward shift of eAg of 73 meV. The smallness of this correction which in essence removes minute discrepancies in the plasmon frequencies of different Ag samples, again confirms the high quality of the electrodeposited Ag layers. The solid curve in Fig. 2 shows the characteristics discussed for the free-electron model [equation (3)]. The upwards bent for (Re k) 2 > 0.02 A-2 is due to peculiarities of the silver dielectric function. The energy loss curve is insensitive to these details and remains linear in k 2 . We prove this by showing in Fig. 2 as dotted line also Max (Im eLt) for the same parameters /3, 7, cop and e b which we evaluated from the optical dispersion curve. Our result for the dispersion parameter is 13= (2.57 +- 0.1) 1012 m 2 s e c - 2 . This is exactly the same value derived by Zacharias and Kliewer [ 13] from the energy loss measurements at higher k, which proves the validity of their extrapolation into the smaU-k regime. The/3 value is larger by a factor of 2.2 compared to/3RPA = 3V2F/5 for an electron gas of the density, which fits the infrared dielectric function of Ag. Freeelectron-gas calculations predict/3-values even lower than/3RPA [18]. Band structure calculations for Ag indicate a very small deviation of the effective mass from the free electron mass [19]. We therefore conclude, that the d-electrons in silver must contribute to the plasmon dispersion in a way, which is not yet understood. The other two previously reported evaluations for Ag by Otto and Petri [20] and Raether [1] did not take the bound electrons into account. They used equation (2) for the dielectric function of Ag and found/3 values three times smaller than ours. It seems, however, rather implausible to approximate the dielectric function of Ag by equation (2) because the low value of the plasma frequency cannot be explained by any reasonable assumption about the electron gas. Since the analyses of optical as well as electron loss experiments lead to

908

MEASUREMENT OF THE BULK PLASMON DISPERSION IN SILVER

identical dispersion parameters, it supports equation (7) as being the correct expression for the dielectric function eL(co, k). In order to prove that the distance between the maxima in (-- AR/R) (see Fig. 1) really is the plasmon wave length, we have done a model calculation for the ER signal which included the plasmon waves in the metal optics. This was done according to the method outlined in [2, 15, 16]. The result is shown on the right-hand side of Fig. 1. It demonstrates that the characteristic oscillations are well reproduced by this model calculation. As found in the experiment, the position of the first peak in -- AR/R hardly moves with photon energy while the second peak shifts to lower D-values with increasing hw. The agreement is not perfect, however, there are other effects in the ER of silver, e.g., due to bound electrons, which are not included in the model. But these effects do not disturb the periodicity due to the standing plasma waves. We have shown, that optical measurements should yield a dispersion relation for the bulk plasma waves in metals, which is distinctly different from the dispersion derived from energy loss experiments. Our measurements are the first ones which really show this difference, and they are done in the small-k regime, which is theoretically most relevant but was previously only reached by extrapolation from the large-k region. We derive from our measurements exactly the same dispersion parameter ~ (see equation 7) as previously obtained from the energy loss measurements, which justifies the extrapolation to small k in the latter case. The value of/3 = (2.57 -+ 0.1) 1012 m 2 sec -2 cannot be understood within the free-electron model and calls for a calculation of the spatial dispersion including the d-electrons of silver.

Acknowledgements - One of us (G.P.) wants to acknowledge the financial support by the Consiglio Nazionale

Vol. 51, No. 11

della R6cerche and the Max-Planck-GesellschafL Part of this work was supported by the Deutsche Forschungsmeinschaft.

REFERENCES H. Raether, Springer Tracts in Modern Physics, Vol. 88, p. 83 (1980). 2. F. Forstmann, K. Kempa & D.M. Kolb, J. Electroanal. Chem. 150,241 (1983). 3. J.D.E. Mclntyre, Surf. ScL 37,658 (1973). 4. D.M. Kolb,J. Phys. (Paris) 38, C5-167 (1977). 5. T.E. Furtak & D.W. Lynch, J. Electroanal. Chem. 79, 1 (1977). 6. H. Gerischer, D.M. Kolb & J.K. Sass, Adv. Phys. 27,437 (1978); R. K6tz & D.M. Kolb, Surf. ScL 97,575 (1980). 7. K. Kempa, F. Forstmann, G. Piazza & D.M. Kolb (in preparation). 8. D.M. Kolb, in Adv. Electrochem. Electrochem. Engin. Vol. 11 (Edited by H. Gerischer & C.W. Tobias), p. 125. Wiley, New York (1978). 9. K.L. Kliewer, Surf. ScL 101,57 (1980). 10. P.J. Feibelman, Progr. Surf. Sci. 12,287 (1982). 11. K. Kempa & F. Forstmann, Surf. Sci. 129, 516 (1983). 12. I. Lindau & P.O. Nilsson, Phys. Scripta 3, 87 (1971). 13. P. Zacharias & K.L. Kliewer, Solid State Commun. 18, 23 (1976). 14. P.B. Johnson & R.W. Christy, Phys. Rev. B6, 4370 (1972). 15. F. Forstmann & R. Gerhardts, in Festk6rperprobleme/Advances in Solid State Phys., Vol. XXII, p. 291. Vieweg, Braunschweig (1982). 16. R. K6tz, D.M. Kolb & F. Forstmann, Surf. ScL 91,489 (1980). 17. T. Lopez-Rios, F. Abel6s & G. Vuye, J. Phys. (Paris) 40, L343 (1979). 18. K. Sturm,Adv. Phys. 31, 1 (1982). 19. N.E. Christensen, Phys. Status Solidi 31,635 (1969). 20. A. Otto & E. Petri, Solid State Commun. 20, 823 (1976). 1.