Optical absorption in discontinuous gold films

Solid State Communications, Vol. 23, PP. 261—265, 1977.

Pergamon Press.

Printed in Great Britain

OFFICAL ABSORPTION IN DISCONTINUOUS GOLD FILMS S. Norrman, 1. Andersson and C.G. Granqvist Physics Dept., Chalmers University of Technology, Fack,S.402 20 Gothenburg, Sweden and 0. Hunderi Dept. of Physical Metallurgy,Norwegian Institute of Technology, 7034 Trondheini

—

NTH, Norway

(Received 28 February 1977; in revised form 25 March 1977 by L. Hedin) Discontinuous gold films with mass thicknesses 1.1—4.0 nm were prepared by UHV-deposition onto glass substrates. The islands were accurately represented by randomly oriented prolate spheroids (symmetry axis parallel with substrate plane) with varying sizes and eccentricities. Optical transmittance was recorded for wavelengths 0.3—2.5 aim. Extensive computations, based on the Maxwell—Garnett formalism, were carried out for islands surrounded by air. The calculated transmittances were much larger than the experimental ones. The most likely explanation is that local field effects are more prominent than estimated from recent theories. ANOMALOUS optical absorption in discontinuous noble metal films has been known for over a century, and it is widely accepted that the effect stems from the island-like nature of the deposits. The anomalous absorption has been described theoretically in terms of Mie scattering [11,which reduces to the Maxwell—Garnett theory [2] for sufficiently small islands. This latter theory has been compared with experimental data for discontinuous gold films in several publications [3—8]. A qualitative agreement has been found thus verifying that the theoretical approach is appropriate while quantitative discrepancies have led to discussions of the roles of (i) the shape and size of the islands, (ii) their dipole—dipole coupling, (iii) interactions with the substrate, and (iv) dielectric coatings. The relative importance of (i)—(iv) has remained conjectural owing to the general lack of detailed morphological treatments of the investigated films. With the purpose of improving on this situation we report here on measurements of the optical absorption caused by gold islands which can be accurately represented by prolate spheroids having varying sizes and eccentricities. We then show from detailed computations (with no fitting parameters) that the absorption predicted from the Maxwell—Garnett formalism, applied to islands surrounded by air, is much too low to be reconciled with the experimental data. l’his result can be explained if we assume that local field effects are stronger than expected from recent theories, or that interaction from the substrate plays an important role in determining the optical properties of discontinuous films, Two of the present authors in a recent article [9] —

—

analyzed the morphology of islands in discontinuous gold films. The samples of this work were similarly prepared and therefore the discussion of the experimental technique is brief. Gold with purity 99.99% was deposited from a resistively’heated alumina crucible onto Coming 7059 glass substrates at room temperature with a pressure <4 x I O~torr during evaporation. An electric field of 20 V cm~was maintained in the sub. strate plane. The mass thickness was measured on a quartz crystal microbalance calibrated to ±5% by interference microscopy. After deposition the films were aged in ultrahigh vacuum for> 20 hr; they were then transferred to a spectrophotometer for optical measurements. When these were completed we evaporated a stabilizing carbon layer over the gold films, dissolved the glass substrates in 5% HF acid and studied the free films by high resolution electron microscopy. Films with mass thicknesses of 1.1-4.0 nm were produced by evaporation at 0.02—0.04 nm sec’. Representative electron micrographs for two thicknesses are shown in Fig. I, where it is seen that the mean size of the islands as well as their irregularity is enhanced when the mass thickness, t, is increased. It is obvious from Fig. 1 that the gold islands look far from circular and that a much better approximation is achieved if we describe the two dimensional images instead as being ellipses with areas (ir/4~kbk, where ak(bk) denotes the major (minor) axis of the kth island. The three dimensional form of the islands cannot be judged from aa and bk alone, but we also have to employ determinations [9] of the area fraction of the substrates covered with metal or, alternatively, the

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Fig. 1. Electron micrographs of discontinuous gold films with two different mass thicknesses. A Philips EM300 electron microscope was used. density of islands. From such analyses we argued earlier [9] that a model of whole prolate spheroids with symmetry axis parallel with the substrate plane Ls a good approximation for the three dimensional shape of the islands at t ~ 3 rim. The island volumes, m,~,are hence 1ir’6~”b2 k ‘~ ~ ~‘ “ ‘ v =

Optical transmittance at normal incidence was recorded in the wavelength interval 033
Vol. 23, No.4

OPTICAL ABSORPTION IN DISCONTINUOUS GOLD FILMS isfoundedontheMaxwell—Garnett[2]model,which

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where Xk is the size of the kth particle (sphere diameter) and ~k is its dielectric permeability. The above criterion is fulfilled for our discontinuous gold films with t ~ 3 nm. The model also presumes that the Lorentz local field correction applies and that the induced polarisation of one particle from its surrounding neighbours is nonretarded [11]. The Maxwell’--Garnett theory yields an effective dielectric permeability, ~, that can be written

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dielectric permeability for the medium surrounding the islands, and cr~is proportional to the polarisability of the kth island. In our morphological model for the islands we regarded ellipsoids with one of their axes perpendicular to a plane but with otherwise random orientations; for these we obtain [12] 2

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Fig. 2. Transmittance vs wavelength for discontinuous gol&fllms with two mass thicknesses. Solid curves show experimental data as measured with a UNICAM SP700 double beam spectrophotometer with a film on its substrate in one light beam and an uncoated substrate in the other. Dashed curves represent computed results, obtained by averaging over an ensemble of islands with varying sizes and eccentricities, as discussed in the main text. Note that the transmittances are larger than 55% forailcurves! decrease at A ~ 0.45 sin is due to interband transitions, The location of the transmittance minimum shifts towards larger wavelengths when t is increased, which agrees with the commonly observed [3—8] behaviour of discontinuous gold films. The islands are orders of magnitude smaller than the wavelength of the incident light, implying that the optical properties can be characterized by an effective medium. Our theoretical description of the transmittance

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where f signifies the “filling factor”, i.e. the volume fraction of metal in the films. Local field effects can be incorporated in the above effective medium formalism for two dimensional arrays of islands by defining the “optical” thickness (t 0~~) of the film and its filling factor appropriately, as discussed recently by Bedaux and Vlieger [14]. They carried out extensive calculations on the optical properties of islands with “amorphous” structures and prove that

264

OPTICAL ABSORPTION IN DISCONTINUOUS GOLD FILMS

f

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Vol. 23, No.4

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(9) these values were used throughout our computations, and furthermore we set where D is equal to the average distance between the centers of neighbouring islands and ~ 4/3 was shown Em = 1 [14] to be at least a rough estimate. For equal islands as we do not want to incorporate any interactions from forming a square lattice we obtain the empirically docu- the substrate. mented relation 112(~,/2), (10) The above equations constitute a complete scheme for computing ë, from which we derived the optical D = (ir/A) where 2~is the mean sphere diameter of the islands and transmittance by the standard equations [18] for a thin A is the area fraction of the substrate covered with film on a substrate. The dashed curves in Fig. 2 refer to metal, which is easily determined from the electron averaging at each thickness over a randomly selected micrographs. Using experimental values we obtained area containing 150 islands, i.e. the effective medium consistently 0.11
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randomly oriented prolate spheroids. Starting from this structure characterization developed Maxweil—Gamett formalism towe provide whatthe we believe to be the most detailed transmittance computations yet performed for islands lacking substrate interaction. The reliability was achieved by averaging over a large set of

(14) islands using their measured sizes and eccentricities. No adjustable parameters were invoked. The main issue is where VF is the Fermi velocity and ‘k is the mean free the very large differences between theoretical and path for the kth island. For a spherical island the mean experimental data, which was illustrated in Fig. 2. In free path is equal to its radius [171, but, as far as we order for the two kinds of data to coincide our pre. know, no quantitative results are known for prolate ~ studies indicate that either we need Em> 1, spheroids. It is obvious that bk <7~1k
= bk!2. From Winsemius [16] the bulk parameters are

h/Tb

=

8.55 eV

=

0.108eV

(15) estimation [14]. The latter explanation is easiest to justify theoretically and should therefore be preferred. Our work on the detailed interpretation of the anomalous absorption in discontinuous films is continuing and we hope to present a more detailed report in the near future. REFERENCES

1.

MIE G.,Ann. Phys: (Leipzig) 25, 377 (1908).

2. 3.

MAXWELL—GARNETT J.C.,PhiL Trans. R. Soc. (London) 203, 385 (1904); 205,237 (1906). YAMAGUCHI S.,J. Phys. Soc. Japan 15, 1577 (1960).

Vol. 23, No.4

OPTICAL ABSORPTION IN DISCONTINUOUS GOLD FILMS

4.

RASIGNI G. & ROUARD P.,J. Opt. Soc. Am. 53,604(1963).

5.

DOREMUS R.H.,J. App!. Phys. 37, 2775 (1966).

6.

CARLAN A.,Ann. Phys. (Paris) 4, 5 (1969).

7.

TRUONG V.V. & SCOTF G.D.,J. Opt. Soc. Am. 66, 124 (1976).

8.

JARRETF D.N. & WARD L.,J.Phys. D: App!. Phys. 9,1515(1976).

9;

ANDERSSON T. & GRANQVIST C.G.,J. App!. Phys. 48, 1673 (1977).

10.

GRANQVIST C.G. & HUNDERI 0., Phys. Rev. B (in press).

11.

GRANQVIST C.G. & HIJNDERI 0. (to be published).

12.

HUNDERI O.,Phys. Rev. B7, 3419 (1973).

13.

LANDAU L. & LIFSHITZ E.M.,Electrodynamics of Continuous Media, Section 4. Pergamon, New York (1960).

14. 15.

BEDAUX D. & VLIEGER J.,Physica (Utrecht) 73, 287 (1974). GRANQVIST C.G. & HUNDERI O.,Solid State Commun. 19,939(1976).

16.

WINSEMIUS P., Unpublished Ph.D. Thesis, Rijksuniversiteit te Leiden, The Netherlands (1973).

17.

EULERJ.,Z.Phys. 137, ~l8 (1954).

18.

See, for example, HEAVENS O.S., Optical Properties of Thin Solid Films. Dover, New York (1965).

265