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Optical properties and microstructure of thin silver films
Optics Communications North-Holland
85
( 199 I ) 70-82
OPTICS COMMUNICATIONS
FUN length article
Optical properties and microstructure C.A. Davis,
D.R. McKenzie
of thin silver films
and R.C. McPhedran
School qf Physics.Universily qfS.vdnev.S.vdney.Auslralia2006
Received
I7 December
I990
Several previous studies have shown that the optical transmittance of thin gold films at the metal-insulator transition is constant over a wide wavelength band. We show that this holds true for silver films in the spectral range from I .5 urn to at least 43 urn. The optical properties of films near the transition are shown to be sensitive to variations in the microstructure of films grown on different amorphous substrates. The optical absorption near the transition is found to be 40% and essentially independent of wavelength. Narrow necks in the conducting pathways are argued to be an essential feature of the microstructure. In particular a correlation length cannot be defined for either silver or void clusters for films near the critical concentration. The optical absorption is evidently due to losses at the narrow necks in the conducting pathways. The scattering properties of films as a function of thickness are discussed, and the absence of any enhanced scattering at the transition is explained in terms of the particle size distribution.
1. Introduction Very thin metal films deposited by vacuum evaporation consist of isolated islands of metal, and so are electrically insulating. In thicker films the isolated islands join up to form a continuous metal pathway through the film, which is then conducting. Thin metal films therefore undergo a metal-insulator phase transition as p, the fraction of surface area covered, increases with thickness. The surface area coverage (or metal ‘concentration’) at which a conducting metal path across the sample is first established is called the critical concentration pc. Thick cermet (inhomogeneous metal-insulator composite) films show a similar transition as the metal volume fraction is increased. It has recently been noted that the transmittance, reflectance, and optical absorption of metal films with concentration near pc are independent of wavelength [ 1,2]. The resistance of such films is strongly dependent on the concentration and weakly dependent on temperature near the critical concentration [ 31. Thus metal films or cermets near the critical concentration are of considerable interest due to their favourable properties for a variety of technological applications. 70
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The wavelength independence of the optical properties can not be explained by the existing formalism of Effective Medium Theories (EMTs) [ 41 and several phenomenological explanations have been proposed [2,5,6]. At present, however, a satisfactory level of understanding is yet to be reached and more experimental data is required to differentiate between competing theoretical approaches. The optical properties of silver are the closest of all metals to those of an ‘ideal’ free electron metal. Most of the published work in this area has been on gold thin films and gold cermets. Silver films are, however, of even greater interest because of silver’s unique properties and are the subject of this work. In this paper the most important results of the various theoretical approaches are summarised. We then present the results of optical and microstructural measurements made on thin silver films close to the critical concentration. These results are interpreted in terms of the dynamical theory proposed by Josifovska et al. [7].
2. Theory Effective 0
I99
Medium
Theories
I Elsevier Science Publishers
(EMTs)
have
had
B.V. All rights reserved.
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some success in describing the optical properties of cermets over a range of concentrations [4]. In the region of the metal-insulator transition, however, they are of limited applicability. This is because EMTs rely on the quasistatic assumption. The “particles” (usually defined as connected regions of homogeneous material) which comprise the system are assumed to “see” only a region of constant applied electric field. This only applies when the particles are small compared with the wavelength. Near the threshold concentration very large particles are observed (see fig. 1) and the quasistatic assumption is therefore not valid.
showing
the labyrinthine
1991
In 1987 Yagil and Deutscher [ 1 ] used a percolation theory approach to explain the wavelength independent transmittance of films at the threshold. An important parameter in their theory is the correlation length <, defined as the separation above which the probability of two sites belonging to the same cluster decays exponentially. It is essentially a measure of the average diameter of all clusters. Near the threshold concentration pC the correlation length is assumed to be described by the critical exponent V, which is defined by the equation
(1)
Fig. 1. Transmission electron micrograph tion. The scale bar is 400 nm.
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structure
-
--
of’ a silver film in the region of the metal-insulator
transi-
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where a is a constant. The value of v has been calculated as approximately 4/3 for a two-dimensional system [ 8 1. Eq. ( 1) shows that (diverges to infinity as p approaches pc, so the particle ‘diameter’ is larger than 2 for all wavelengths at p=p=. Close to the critical concentration the correlation length will be large but finite. For wavelengths small compared to 5 a realspace renormalisation argument [ 91 gives the transmittance as T=T,+(A/2na)““(p,-p)(T,-T,).
(2)
Here T,, T,,,, and T, are constants argued to be only weakly wavelength dependent on the basis of the scale size invariance of the particle structure predicted by percolation theory. The constant a is identified as the “grain size” of the film structure. The definition of grain size in terms of measurable parameters, however, is not clear. Eq. (2) predicts that the transmittance is constant for p=pC and has a power law wavelength dependence for concentrations near pC for which the correlation length is large compared to the wavelength. Recently, Robin and Souillard [ 10,l 11, and Yagil et al. [ 12 ] have proposed two different arguments to give the transmittance, reflectance and optical absorption. Both of these detailed derivations predict that the optical properties will be constant over a Iinite range of wavelengths. A completely different approach has been put forward by Josifovska et al. [ 71 who note that a simple model is available which has been exactly solved. The model consists of a thin chequer-board in which the black squares are perfectly conducting and the white squares are void. Compton et al. [ 13 ] show that the reflectance and transmittance of such a chequerboard are independent of wavelength for wavelengths greater than the chequer-board period. This approach could be labelled as a “dynamical” theory since the quasistatic assumption is avoided by working directly from Maxwell’s equations. Thus there exists a simple model which has been exactly solved and which exhibits wavelength independent optical properties. It should be noted that this result is for ideal chequer-boards, in which the metallic squares are connected only by a geometric point. If there is a finite connection between the metal squares then currents can flow over distances comparable with the wave72
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length and the reflectance is high - this is known as inductive behavior [ 141 (by analogy with the longwavelength behavior of inductive circuits). On the other hand, if the metal squares are clearly disconnected then the reflectance is low at long wavelengths, as if the squares were capacitively coupled, and hence this is known as capacitive behavior. Thus the wavelength independent optical properties are only seen when both the silver and the dielectric phases are “connected” over distances larger than the wavelength. This metal insulator structural symmetry is possible because the point contacts between the squares look identical to both phases. In fact the squares are neither connected nor disconnected in the ideal chequer-board. Thus the correlation length is not infinite (as in the percolation theory approach) but is actually undefined. We shall argue later that the correlation length in silver films at the critical concentration is also undefined.
3. Thin film preparation The silver films were deposited by vacuum evaporation at a base pressure of 4 x 10p4 Pa. Deposition rates were typically between 0.02 rim/s and 0.05 rim/s,, as measured by a quartz crystal oscillator microbalance. The films were simultaneously deposited on several substrates which were located in close proximity in the vacuum chamber. Scanning electron microscope studies of the effect of the substrate surface on film growth have indicated that the microstructure of silver films is qualitatively similar for various amorphous substrates [ 4,151. Therefore all substrates used were chosen to have an amorphous surface. It will be shown below, however, that the assumption of identical optical properties for films on different amorphous substrates is not necessarily valid, because of the sensitivity of these properties to quite fine structural details. Suprasil fused silica optical glass was used for the visible to near infrared optical measurements. For transmittance measurements at longer wavelengths KBr discs coated with a thin ( lo-20 nm) layer of amorphous carbon were used. Films to be studied by Transmission Electron Microscopy (TEM) were deposited on standard copper TEM grids which were
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coated first with parlodion (a polymer film) and then with amorphous carbon. The resistance of the films undergoes a sharp transition in the region of the critical concentration [ 7,161. The resistance of the films was therefore monitored during deposition so that film growth could be consistently stopped in the region of interest. The resistance was measured during deposition by a two-electrode method and after deposition using a four-electrode method. The measured resistance was converted to resistivity (p) using the dimensions of the pre-deposited nickel-chromium electrodes and the film thickness (given by the quartz crystal oscillator, assuming bulk density). A typical set of results from the in situ resistance monitoring, shown in fig. 2, shows no sign of the expected sharp transition. Below the transition thickness the in situ resistance measured was lower than expected. It was observed, however, that if film growth was interrupted the resistance value immediately began to increase. For thicker films the resistance decreased during momentary interruptions of the deposition. It was found that the change in resistance during an interruption of the deposition became small when the Iilm was close to the critical concentration. Increases in the post deposition resistance of island-like thin metal films have previously been stud-
Thickness
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ied [ 171. Possibly in this case the film is not in equilibrium during growth and the changes in resistance are due to thermal settling of the film after deposition ceases. At low metal concentrations, the tendency to form larger islands is favoured - leading to an increase in resistance. At higher concentrations the tendency of the film to become continuous dominates and the resistance decreases.
4. Optical properties Measurements of transmittance and reflectance were made for wavelengths between 0.2 pm and 3 urn using a Varian Cary 2300 spectrophotometer. The transmittance of silver films on amorphous-carbon coated potassium bromide discs was measured in the infrared using a Bio-Rad Fourier Transform Infrared spectrophotometer (FTIR). The FTIR consists of two spectrophotometers with effective wavelength ranges of 2.5 urn to 20 urn (the FTS 20-80) and 20 urn to 100 urn (the FTS 15-80). Fig. 3 shows typical results for films above the percolation threshold (inductive, T approaches zero for long wavelengths) and below the threshold (capacitive, T approaches 100% in the infrared). The main region of interest is wavelengths longer than 1.5 pm for which the optical properties are dominated by
(nm)
Fig. 2. Resistivity as a function of thickness (given by the quartz crystal oscillator, assuming bulk density) measured tion ofa film at a rate of 0.02 rim/s.. Note the discontinuity at 10.7 nm which corresponds to a two minute interruption
during the deposiofthe film growth.
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aF :
60.0% -
J
2
z E
40.0% -
: ? l20.0% 1
0.0%
1 Wavelength
Wavelength
Fig. 3. The spectral transmittance of a film (AG29, p= 5 x 10m4 Rem) above the threshold concentration which shows typical inductive behaviour (low transmittance) at long wavelengths. The film AC3 I (p= 7 X IO5 Rem) is below the threshold concentration and shows typical capacitive behaviour (high transparency) in the infrared.
the free electron behaviour [ 61. Fig. 4 shows the transmittance and reflectance of a sample near the critical concentration on a Suprasil substrate (AG30S). The wavelength independence of the transmittance and reflectance above about 1.5 pm is clear. The optical absorption of the same film versus wavelength is shown in fig. 5. The absorption at long wavelengths is constant, and has a value of close to 40%. Gadenne, Yagil and Deutscher [ 181 also found an absorption of 40% for two gold films
at the percolation threshold which were grown at different rates and had different thicknesses. Thus very similar results for the absorption are seen, not only for different film thicknesses, but also for both silver and gold thin films. The results of Gadenne, Beghdadi and Lafait [ 21, however, show two different gold films with constant absorptions of about 32% and 22%, so this effect is apparently not universal. The transmittance of AG30C (deposited simultaneously with AG30S, but on a carbon coated KBr
60.0% -
60.0%
-
40.0%
-
20.0%
-
Wavelength Fig. 4. The spectral
74
transmittance
and reflectance
of the film AG30S @=3x
IO-* fkm)
are essentially
constant
in
the infrared.
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60.0%
0.0%
0
1
I
1000
2000
I
3000
Wavelength Fig. 5. The optical absorption cr= 1-R - Tof the film AG30S is anomalously high (40%) and independent of wavelength in the infrared.
The magnitude of the transmittance of AG30C near 3 urn is around 3096, whilst the transmittance of AG30S at the same wavelength is 16% and so the results are discontinuous. The two spectrophotometers were calibrated by measuring the transmittan& of AG30S independently, and they agreed to within 2% at the region of wavelength overlap near
substrate) in the infrared is shown in fig. 6. The transmittance is only weakly dependent on the wavelength to the limit of the measurement (on the FTS 20430) at I= 18 urn and is slowly decreasing with wavelength, so the behaviour is inductive in nature and the film is therefore slightly above the critical concentration. 100
80
1
0’
I
I
I
I
I
I
4
6
8
10
12
14
Wavelength
I
16
I
18
(pm)
Fig. 6. The transmittance of AG3OC at wavelengths up to 18 pm is very slowly decreasing. We conclude that this film is slightly above the critical concentration.
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1
60-
: :
60-
._ E VI
: 40,. c
0-20
I 22
I 24
I 26
Wavelength
Fig. 7. The transmittance dependence can be seen.
I 28
I 30
I 36
I 38
I 40
I 42
(pm)
of AG30C in the far infrared. The noise is due to water absorbed by the KBr substrate. No strong wavelength
3 urn. The transmittance of the film on Suprasil is very constant whereas the transmittance of the film on amorphous carbon is slowly decreasing. Thus the films grown on Suprasil and amorphous carbon give qualitatively different transmittance spectra. Since films grown on Suprasil and amorphous carbon have been previously shown to be similar in structure [ 151, we conclude that the transmittance spectrum is very sensitive to fine details of the film structure which are caused by variations in the substrate surface-film interaction. The transmittance of AG30C at even longer wavelengths, measured on the FTS 15-80, is shown in fig. 7. The high noise level is due to the vibrational modes of water present even after the spectrophotometer had been flushed with dry nitrogen gas for over 30 minutes (possibly due to water adsorbed by the KBr substrate). Nevertheless, it is clear that the transmittance is at most weakly dependent on the wavelength for wavelengths as large as 43 urn (the limit of the transparency of the KBr substrate). Preliminary measurements at a wavelength of 433 urn gave an approximate value for the transmittance of AG3OS in the range 10 f 5%. These results lend support to the predictions of Yagil and Deutscher [ 5 1, and Josifovska et al. [ 71 that the transmittance of films exactly at the threshold would be constant for arbitrarily long wavelengths. 76
I 34
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The non-specular component of the reflected light was also investigated in detail. Gold was used as a test case in order to compare with the results of Gadenne, Yagil and Deutscher [ 19 1, who observed nonspecular scattering to form less than 1% of the total, over the whole range of surface coverage. The configuration is as shown in fig. 8. A green HeNe laser Top View
I
1
I
Fig. 8. Experimental set-up used to measure the scattered component of the light reflected by a gold film during deposition.
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beam of wavelength 543 nm was used to illuminate the substrate surface and the scattered light intensity was measured as a function of angle in real time using a photodiode array (EG&G PARC OMAIII). Gold was sputtered from a gold planar magnetron target using argon as the sputtering gas. The substrate was a glass slide. The distribution of scattered intensity at various times during deposition is shown in fig. 9. The intensity of the scattered light increased as the gold coverage of the substrate (and therefore the specular reflectance of the film) increased. The angular dependence of the scattered light did not depend on the mass thickness coverage of the substrate. We conclude that the scattering observed is primarily due to light scattered by surface roughness and other localized defects and not by the connected regions of gold on the substrate. A pronounced enhancement of scattered light at a particular mass thickness might be expected if the size distribution of connected regions had a well defined maximum at a value comparable with the illuminating wavelength. The measured distribution of particle sizes, shown in fig. 10, shows no such maximum except at sizes very much smaller than the wavelength of visible light and
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Particle
size (microns)
Fig. 10. Typical distribution of particle sizes seen in a film just below the critical concentration. The field of view is a square 2 pm across, note the broad tail at large sizes.
therefore enhanced scattering at visible wavelengths is not expected, in agreement with experiment. The particle size distribution is further discussed in section 5 below.
10000
5. Microstructure
8000
0 0
I
scattering
2
angle
(degrees)
Fig. 9. Angular distribution of the intensity of the scattered component of light reflected from a gold film at various stages of the deposition. Curves from bottom to top: show data taken at 0,30, 60, I20 and 180 seconds after commencing deposition. There is no evidence of enhanced scattering at any particular thickness.
Thin silver films deposited on a-C coated copper TEM grids were studied using the Philips EM400 and the Jeol 1OOCXtransmission electron microscopes. Standard bright-field imaging techniques were used to produce micrographs similar to the one shown in fig. 1. Selected TEM micrographs were studied using the Tracer Northern TN-8502 Image Analysis system. An image of the micrograph from a black and white TV camera was converted into a greyscale image of 5 12 x 5 12 pixels. The greyscale image was then converted into a binary image by selecting the critical greyscale value G. Pixels with greyscale value less than G are assigned a value of one in the binary image, all other pixels are set to zero. The value of G is varied until the binary image is the most accurate possible representation of the original image. 77
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It is important to note that, despite careful attempts to carry out this procedure in a reproducible way, the decision of exactly what value of G gives the best representation of the original remains a subjective choice. Also, information is certainly lost during the process. The main losses are due to optical imperfections in the camera and the finite number of pixels [20]. Computerised image analysis has previously been used to calculate percolation theory critical exponents for this type of film [ 2 11. Here we study the microstructure in a more qualitative manner. Fig. 11 shows a typical result for a film at the critical concentration. The binary image has a concentration of 63% (63% of the pixels are silver) and none of the clusters span the field of view, so the correlation length is less than 2 pm. Thus the sample is below the critical concentration. A very small difference in the binary image creation process, however, leads to the result shown in fig. 12. It is easily seen that the grey coloured cluster spans the field of view, so the correlation length is more than 2 pm if the film has a concentration of 64%. In fact, a spanning cluster is seen across other regions of the film when the binary image has a concentration of 64%. Thus if the concentration is 64% the correlation length must be much larger than 2 urn. Following percolation theory we will call any connected region much larger than 2 urn a percolating network. The question now arises as to which of the above images is the correct representation of the film. Both binary images seem at first sight to be good representations of the micrograph. The problem can be resolved by noting that the only difference between the two images is a small number of points which appear in the higher concentration image. These points form narrow necks which connect the separate clusters to form one large cluster. Thus the sharp qualitative change between 63% and 64% is due to the fact that between these two concentrations the silver particles are joined into a percolating network. The fact that the observed transition in the equilibrium film conductivity has rounded edges [ 7 1, however, suggests that the transition in the connectedness of the particles is not as sharp as it would initially appear. Gadenne, Yagil, and Deutscher [ 191 have noted the importance of narrow necks they observed in the 78
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microstructure. Close inspection of our TEM micrographs shows that it is often diff’icult to ascertain whether or not two particles are connected at a narrow neck without obtaining a micrograph of higher magnification. When two particles are this close we propose that the two particles are neither connected nor disconnected but they in fact have some intermediate connectivity. According to this interpretation, the correlation length for silver particles is undefined in this region of concentration.
The narrow necks observed in the microstructure are very much like the point contacts in the chequer board model in that they exhibit intermediate conductivity through both phases. For example two metal clusters which are very close but do not touch will be capacitively’coupled and therefore exhibit tinite AC conductivity. The void phase is a narrow neck at this point, so it is also in some sense connected. Fig. 13 shows a comparison of the inverses of the two previous images. The inverse binary images were obtained by a NOT binary operation, and so show the structure of the void component of the film. It is seen that the 63% image contains a spanning void cluster whilst the 64W image does not. So for concentrations at and above 64% we have a spanning cluster in the silver phase but not in the void phase whereas for 63W and below we have a spanning cluster in the void phase but not in the silver phase. Therefore both the silver and void phases of this film form percolating networks for concentrations near 63.5%. If the narrow necks between the clusters behave like point contacts as we propose then the silver and void phases can simultaneously exist as percolating networks in this region of concentration. Thus there is some range of concentrations for which the correlation lengths of both the silver and void phases are not defined. Without making a detailed study of the narrow necks we can only guess the size of this concentration region. The relative ease with which a film was produced with transmittance constant to more than 43 pm, however, suggests that this range is appreciable. A large absorption is seen in a range of concentrations around the critical concentration [ 18,191. We argue that this absorption is strongly related to the presence of narrow necks in the microstructure
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Fig. I 1. Digit&d image of the film AG25, showing separate clusters in different colours. The area concentration of metal in this image is 63%, and there is no connected metal region which spans the field of view. Note that the clusters contain many tortuously wandering paths of roughly constant width.
which occur in this range. Clearly the conductivity through a narrow neck is much smaller than the conductivity through the bulk of the conducting net-
work, and it is also insensitive to the thickness at which the critical concentration is reached. The relative insensitivity of the magnitude of the absorp79
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Fig. 12. An almost identical image of AG25, with metal concentration 64%. The large grey region forms a connected metal pathway across the field of view (approximately 2 lrn across). Note that the only significant difference between this image and the one with 63% concentration is the degree of connectedness between the silver clusters. tion to the film thickness
which is seen in the literature [ 18,191 is therefore consistent with the suggestion that the optical absorption is primarily due to losses at weak links between particles. (Yagil et al. 80
[ 121 emphasise the importance of intercluster capacitance in understanding the optical properties of films. Clearly, actual films will contain a variety of close-connection morphologies, ranging from good
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and void networks consist of many tortuous paths which seem to be of roughly constant width. An estimate of the average width was made by randomly selecting points on the image and measuring the width of the path at these points. The values were then averaged to give rough estimates of the widths. For AG30C the average width of the silver paths was found to be 60? 5 nm and the average width of the void paths was found to be 20 ? 2 nm. The difference between the average widths of the silver and void particles is due to the fact that silver arriving at the substrate during deposition is more likely to adhere to a silver cluster than to the bare substrate. The silver width seems to be the only scale size that can be associated with the small-scale structure of the silver clusters. It is therefore equated with the ‘grain size’ parameter in the percolation theory approach. It should be noted that varying the relative widths of the silver and void clusters changes the critical concentration, so the value of between 63% and 64Oh is not universal. Films grown at different rates or under slightly different conditions were found to have varying threshold concentrations. The film AG30C, for example, had a threshold concentration of about 72%. The distribution of sizes of the silver particles was also studied. The maximum lateral extent of each particle was calculated in order to assign a representative “size” to each connected region. Fig. 10 shows the distribution of particle sizes for a film of concentration 63%, just below the critical concentration. The peak at small sizes is due to small silver
Fig. 13. The images in figs. 11 and 12 have been inverted to give a comparison of the connectedness of the void component of the films. The pink area in the lower image, with 63% metal concentration, spans the field of view. The upper image (metal concentration 64%) does not. Thus the void component shows a transition in connectedness which is complementary to the transition in the metal connectivity.
inductive
contacts,
through
point
contacts
to highly
capacitive linkages. The influence of each contact type would need to be taken into account in a complete model of optical behaviour.) It can be seen in figs. 12 and 13 that both the silver
particles which have grown by the accumulation of silver atoms as they are deposited. This is the predominant process at low area concentrations. In the region of the critical concentration, however, the small particles begin to overlap and very large particles are formed by aggregation. These large particles cause the very broad tail in the size distribution seen in the region of the critical concentration. Note that the broad tail (which accounts for all particles of size comparable to the wavelength of light) contains no well defined maximum, in agreement with the scattering results of section 4.
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6. Conclusions The transmittance of thin silver films at the metal insulator transition has been found to be constant for wavelengths from 1.5 urn to more than 43 urn. The optical properties have been shown to be very sensitive to the nature of the substrate surface. Even similar amorphous surfaces, which were previously thought to produce identical films, can result in films near transition having significantly different optical properties. Microstructural results are presented which are consistent with the ‘dynamic scattering’ model of Josifovska et al. [ 71. An important feature of both. the model and the actual microstructure is the presence of tenuous contacts between clusters. These cause the correlation lengths in both the metal and the void phases to be undefined. We suggest that the high optical absorption can be explained as being primarily due to losses at the narrow necks in the microstructure. The elaboration of the dynamic scattering model requires a quite complicated electromagnetic calculation. Work on this is in progress.
Acknowledgements Craig Davis acknowledges the support of an Australian Postgraduate Research Award. The authors thank Mr. Zhen Hua Wang for providing the measurements of scattered light on growing gold films, and Mr. Peter Ring for help in the measurement of film transmittance at 433 urn. The authors acknowledge the Science Foundation for Physics and His Royal Highness Prince Nawaf Bin Abdul Aziz of the Kingdom of Saudi Arabia for the provision of facilities.
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References [ 11 Y. Yagil and G. Deutscher, Thin Solid Films 152 ( 1987) 465. [2] P. Gadenne, A. Beghdadi and J. Lafait, Optics Comm. 65 (1988) 17. [3] B. Abeles, H.L. Pinch and J.I. Gittleman, Phys. Rev. Lett. 35 (1975) 247. [4] M. Gajdardziska-Josifovska, MSc Thesis, University of Sydney (1986). [ 51 Y. Yagil and G. Deutscher, Appl. Phys. Lett. 52 (1988) 373. [ 61 M. Gadenne, J. Iafait and P. Gadenne, Optics Comm. 7 1 (1989) 273. [ 71 M. Gajdardziska-Josifovska, R.C. McPhedran, D.R. McKenzie and R.E. Collins, Appl. Optics 28 ( 1989) 2744. [ 81 D. Stauffer, Phys. Rep. 54 ( 1979) 1. [9] D. Stauffer, Introduction to percolation theory (Taylor and Francis, London, 1985). [IO] T. Robin and B. Souillard, Physica A 157 (1989) 285. [ 111T. Robin and B. Souillard, Optics Comm. 71 (1989) 15. [ 121 Y. Yagil, M. Yosetin, D.J. Bergman, G. Deutscher and P. Gadenne, Paper presented at ICMC, San Diego, USA, April 1990. [ 131 R.C. Compton, J.C. Macfarlane, L.B. Whitboum, M.M. Blanc0 and R.C. McPhedran, Optica Acta 3 1 ( 1984) 5 15. 141 R. Uhich, Infrared Phys. 7 (1967) 37. 151 M. Gajdardziska-Josifovska, R.C. McPhedran, D.J.H. Cockayne, D.R. McKenzie and R.E. Collins, Appl. Optics 28 (1989) 2736. 161 S. Berthier, J. Lafait, C. Sella and Thran-Khanh-Vien, Thin Solid Films 125 (1985) 171. 171 M. Nishiura and A. Kinbara, Thin Solid Films 24 ( 1974) 79. [ 181 P. Gadenne, Y. Yagil and G. Deutscher, Physica A 157 (1989) 279. [ 191 P. Gadenne, Y. Yagil and G. Deutscher, J. Appl. Phys. 66 (1989) 3019. [ 201 A. Beghdadi, M. Gadenne, J. Lafait, A. Le Negrate and A. Constans, Physica A 157 ( 1989) 64. [ 2 1] R.F. Voss, R.B. Laibowitz and E.I. AUessandrini, Phys. Rev. Lett.49 (1982) 1441.
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FUN length article
Optical properties and microstructure C.A. Davis,
D.R. McKenzie
of thin silver films
and R.C. McPhedran
School qf Physics.Universily qfS.vdnev.S.vdney.Auslralia2006
Received
I7 December
I990
Several previous studies have shown that the optical transmittance of thin gold films at the metal-insulator transition is constant over a wide wavelength band. We show that this holds true for silver films in the spectral range from I .5 urn to at least 43 urn. The optical properties of films near the transition are shown to be sensitive to variations in the microstructure of films grown on different amorphous substrates. The optical absorption near the transition is found to be 40% and essentially independent of wavelength. Narrow necks in the conducting pathways are argued to be an essential feature of the microstructure. In particular a correlation length cannot be defined for either silver or void clusters for films near the critical concentration. The optical absorption is evidently due to losses at the narrow necks in the conducting pathways. The scattering properties of films as a function of thickness are discussed, and the absence of any enhanced scattering at the transition is explained in terms of the particle size distribution.
1. Introduction Very thin metal films deposited by vacuum evaporation consist of isolated islands of metal, and so are electrically insulating. In thicker films the isolated islands join up to form a continuous metal pathway through the film, which is then conducting. Thin metal films therefore undergo a metal-insulator phase transition as p, the fraction of surface area covered, increases with thickness. The surface area coverage (or metal ‘concentration’) at which a conducting metal path across the sample is first established is called the critical concentration pc. Thick cermet (inhomogeneous metal-insulator composite) films show a similar transition as the metal volume fraction is increased. It has recently been noted that the transmittance, reflectance, and optical absorption of metal films with concentration near pc are independent of wavelength [ 1,2]. The resistance of such films is strongly dependent on the concentration and weakly dependent on temperature near the critical concentration [ 31. Thus metal films or cermets near the critical concentration are of considerable interest due to their favourable properties for a variety of technological applications. 70
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The wavelength independence of the optical properties can not be explained by the existing formalism of Effective Medium Theories (EMTs) [ 41 and several phenomenological explanations have been proposed [2,5,6]. At present, however, a satisfactory level of understanding is yet to be reached and more experimental data is required to differentiate between competing theoretical approaches. The optical properties of silver are the closest of all metals to those of an ‘ideal’ free electron metal. Most of the published work in this area has been on gold thin films and gold cermets. Silver films are, however, of even greater interest because of silver’s unique properties and are the subject of this work. In this paper the most important results of the various theoretical approaches are summarised. We then present the results of optical and microstructural measurements made on thin silver films close to the critical concentration. These results are interpreted in terms of the dynamical theory proposed by Josifovska et al. [7].
2. Theory Effective 0
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Medium
Theories
I Elsevier Science Publishers
(EMTs)
have
had
B.V. All rights reserved.
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some success in describing the optical properties of cermets over a range of concentrations [4]. In the region of the metal-insulator transition, however, they are of limited applicability. This is because EMTs rely on the quasistatic assumption. The “particles” (usually defined as connected regions of homogeneous material) which comprise the system are assumed to “see” only a region of constant applied electric field. This only applies when the particles are small compared with the wavelength. Near the threshold concentration very large particles are observed (see fig. 1) and the quasistatic assumption is therefore not valid.
showing
the labyrinthine
1991
In 1987 Yagil and Deutscher [ 1 ] used a percolation theory approach to explain the wavelength independent transmittance of films at the threshold. An important parameter in their theory is the correlation length <, defined as the separation above which the probability of two sites belonging to the same cluster decays exponentially. It is essentially a measure of the average diameter of all clusters. Near the threshold concentration pC the correlation length is assumed to be described by the critical exponent V, which is defined by the equation
(1)
Fig. 1. Transmission electron micrograph tion. The scale bar is 400 nm.
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structure
-
--
of’ a silver film in the region of the metal-insulator
transi-
71
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where a is a constant. The value of v has been calculated as approximately 4/3 for a two-dimensional system [ 8 1. Eq. ( 1) shows that (diverges to infinity as p approaches pc, so the particle ‘diameter’ is larger than 2 for all wavelengths at p=p=. Close to the critical concentration the correlation length will be large but finite. For wavelengths small compared to 5 a realspace renormalisation argument [ 91 gives the transmittance as T=T,+(A/2na)““(p,-p)(T,-T,).
(2)
Here T,, T,,,, and T, are constants argued to be only weakly wavelength dependent on the basis of the scale size invariance of the particle structure predicted by percolation theory. The constant a is identified as the “grain size” of the film structure. The definition of grain size in terms of measurable parameters, however, is not clear. Eq. (2) predicts that the transmittance is constant for p=pC and has a power law wavelength dependence for concentrations near pC for which the correlation length is large compared to the wavelength. Recently, Robin and Souillard [ 10,l 11, and Yagil et al. [ 12 ] have proposed two different arguments to give the transmittance, reflectance and optical absorption. Both of these detailed derivations predict that the optical properties will be constant over a Iinite range of wavelengths. A completely different approach has been put forward by Josifovska et al. [ 71 who note that a simple model is available which has been exactly solved. The model consists of a thin chequer-board in which the black squares are perfectly conducting and the white squares are void. Compton et al. [ 13 ] show that the reflectance and transmittance of such a chequerboard are independent of wavelength for wavelengths greater than the chequer-board period. This approach could be labelled as a “dynamical” theory since the quasistatic assumption is avoided by working directly from Maxwell’s equations. Thus there exists a simple model which has been exactly solved and which exhibits wavelength independent optical properties. It should be noted that this result is for ideal chequer-boards, in which the metallic squares are connected only by a geometric point. If there is a finite connection between the metal squares then currents can flow over distances comparable with the wave72
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length and the reflectance is high - this is known as inductive behavior [ 141 (by analogy with the longwavelength behavior of inductive circuits). On the other hand, if the metal squares are clearly disconnected then the reflectance is low at long wavelengths, as if the squares were capacitively coupled, and hence this is known as capacitive behavior. Thus the wavelength independent optical properties are only seen when both the silver and the dielectric phases are “connected” over distances larger than the wavelength. This metal insulator structural symmetry is possible because the point contacts between the squares look identical to both phases. In fact the squares are neither connected nor disconnected in the ideal chequer-board. Thus the correlation length is not infinite (as in the percolation theory approach) but is actually undefined. We shall argue later that the correlation length in silver films at the critical concentration is also undefined.
3. Thin film preparation The silver films were deposited by vacuum evaporation at a base pressure of 4 x 10p4 Pa. Deposition rates were typically between 0.02 rim/s and 0.05 rim/s,, as measured by a quartz crystal oscillator microbalance. The films were simultaneously deposited on several substrates which were located in close proximity in the vacuum chamber. Scanning electron microscope studies of the effect of the substrate surface on film growth have indicated that the microstructure of silver films is qualitatively similar for various amorphous substrates [ 4,151. Therefore all substrates used were chosen to have an amorphous surface. It will be shown below, however, that the assumption of identical optical properties for films on different amorphous substrates is not necessarily valid, because of the sensitivity of these properties to quite fine structural details. Suprasil fused silica optical glass was used for the visible to near infrared optical measurements. For transmittance measurements at longer wavelengths KBr discs coated with a thin ( lo-20 nm) layer of amorphous carbon were used. Films to be studied by Transmission Electron Microscopy (TEM) were deposited on standard copper TEM grids which were
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coated first with parlodion (a polymer film) and then with amorphous carbon. The resistance of the films undergoes a sharp transition in the region of the critical concentration [ 7,161. The resistance of the films was therefore monitored during deposition so that film growth could be consistently stopped in the region of interest. The resistance was measured during deposition by a two-electrode method and after deposition using a four-electrode method. The measured resistance was converted to resistivity (p) using the dimensions of the pre-deposited nickel-chromium electrodes and the film thickness (given by the quartz crystal oscillator, assuming bulk density). A typical set of results from the in situ resistance monitoring, shown in fig. 2, shows no sign of the expected sharp transition. Below the transition thickness the in situ resistance measured was lower than expected. It was observed, however, that if film growth was interrupted the resistance value immediately began to increase. For thicker films the resistance decreased during momentary interruptions of the deposition. It was found that the change in resistance during an interruption of the deposition became small when the Iilm was close to the critical concentration. Increases in the post deposition resistance of island-like thin metal films have previously been stud-
Thickness
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ied [ 171. Possibly in this case the film is not in equilibrium during growth and the changes in resistance are due to thermal settling of the film after deposition ceases. At low metal concentrations, the tendency to form larger islands is favoured - leading to an increase in resistance. At higher concentrations the tendency of the film to become continuous dominates and the resistance decreases.
4. Optical properties Measurements of transmittance and reflectance were made for wavelengths between 0.2 pm and 3 urn using a Varian Cary 2300 spectrophotometer. The transmittance of silver films on amorphous-carbon coated potassium bromide discs was measured in the infrared using a Bio-Rad Fourier Transform Infrared spectrophotometer (FTIR). The FTIR consists of two spectrophotometers with effective wavelength ranges of 2.5 urn to 20 urn (the FTS 20-80) and 20 urn to 100 urn (the FTS 15-80). Fig. 3 shows typical results for films above the percolation threshold (inductive, T approaches zero for long wavelengths) and below the threshold (capacitive, T approaches 100% in the infrared). The main region of interest is wavelengths longer than 1.5 pm for which the optical properties are dominated by
(nm)
Fig. 2. Resistivity as a function of thickness (given by the quartz crystal oscillator, assuming bulk density) measured tion ofa film at a rate of 0.02 rim/s.. Note the discontinuity at 10.7 nm which corresponds to a two minute interruption
during the deposiofthe film growth.
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I
aF :
60.0% -
J
2
z E
40.0% -
: ? l20.0% 1
0.0%
1 Wavelength
Wavelength
Fig. 3. The spectral transmittance of a film (AG29, p= 5 x 10m4 Rem) above the threshold concentration which shows typical inductive behaviour (low transmittance) at long wavelengths. The film AC3 I (p= 7 X IO5 Rem) is below the threshold concentration and shows typical capacitive behaviour (high transparency) in the infrared.
the free electron behaviour [ 61. Fig. 4 shows the transmittance and reflectance of a sample near the critical concentration on a Suprasil substrate (AG30S). The wavelength independence of the transmittance and reflectance above about 1.5 pm is clear. The optical absorption of the same film versus wavelength is shown in fig. 5. The absorption at long wavelengths is constant, and has a value of close to 40%. Gadenne, Yagil and Deutscher [ 181 also found an absorption of 40% for two gold films
at the percolation threshold which were grown at different rates and had different thicknesses. Thus very similar results for the absorption are seen, not only for different film thicknesses, but also for both silver and gold thin films. The results of Gadenne, Beghdadi and Lafait [ 21, however, show two different gold films with constant absorptions of about 32% and 22%, so this effect is apparently not universal. The transmittance of AG30C (deposited simultaneously with AG30S, but on a carbon coated KBr
60.0% -
60.0%
-
40.0%
-
20.0%
-
Wavelength Fig. 4. The spectral
74
transmittance
and reflectance
of the film AG30S @=3x
IO-* fkm)
are essentially
constant
in
the infrared.
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60.0%
0.0%
0
1
I
1000
2000
I
3000
Wavelength Fig. 5. The optical absorption cr= 1-R - Tof the film AG30S is anomalously high (40%) and independent of wavelength in the infrared.
The magnitude of the transmittance of AG30C near 3 urn is around 3096, whilst the transmittance of AG30S at the same wavelength is 16% and so the results are discontinuous. The two spectrophotometers were calibrated by measuring the transmittan& of AG30S independently, and they agreed to within 2% at the region of wavelength overlap near
substrate) in the infrared is shown in fig. 6. The transmittance is only weakly dependent on the wavelength to the limit of the measurement (on the FTS 20430) at I= 18 urn and is slowly decreasing with wavelength, so the behaviour is inductive in nature and the film is therefore slightly above the critical concentration. 100
80
1
0’
I
I
I
I
I
I
4
6
8
10
12
14
Wavelength
I
16
I
18
(pm)
Fig. 6. The transmittance of AG3OC at wavelengths up to 18 pm is very slowly decreasing. We conclude that this film is slightly above the critical concentration.
75
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1
60-
: :
60-
._ E VI
: 40,. c
0-20
I 22
I 24
I 26
Wavelength
Fig. 7. The transmittance dependence can be seen.
I 28
I 30
I 36
I 38
I 40
I 42
(pm)
of AG30C in the far infrared. The noise is due to water absorbed by the KBr substrate. No strong wavelength
3 urn. The transmittance of the film on Suprasil is very constant whereas the transmittance of the film on amorphous carbon is slowly decreasing. Thus the films grown on Suprasil and amorphous carbon give qualitatively different transmittance spectra. Since films grown on Suprasil and amorphous carbon have been previously shown to be similar in structure [ 151, we conclude that the transmittance spectrum is very sensitive to fine details of the film structure which are caused by variations in the substrate surface-film interaction. The transmittance of AG30C at even longer wavelengths, measured on the FTS 15-80, is shown in fig. 7. The high noise level is due to the vibrational modes of water present even after the spectrophotometer had been flushed with dry nitrogen gas for over 30 minutes (possibly due to water adsorbed by the KBr substrate). Nevertheless, it is clear that the transmittance is at most weakly dependent on the wavelength for wavelengths as large as 43 urn (the limit of the transparency of the KBr substrate). Preliminary measurements at a wavelength of 433 urn gave an approximate value for the transmittance of AG3OS in the range 10 f 5%. These results lend support to the predictions of Yagil and Deutscher [ 5 1, and Josifovska et al. [ 71 that the transmittance of films exactly at the threshold would be constant for arbitrarily long wavelengths. 76
I 34
I 32
The non-specular component of the reflected light was also investigated in detail. Gold was used as a test case in order to compare with the results of Gadenne, Yagil and Deutscher [ 19 1, who observed nonspecular scattering to form less than 1% of the total, over the whole range of surface coverage. The configuration is as shown in fig. 8. A green HeNe laser Top View
I
1
I
Fig. 8. Experimental set-up used to measure the scattered component of the light reflected by a gold film during deposition.
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beam of wavelength 543 nm was used to illuminate the substrate surface and the scattered light intensity was measured as a function of angle in real time using a photodiode array (EG&G PARC OMAIII). Gold was sputtered from a gold planar magnetron target using argon as the sputtering gas. The substrate was a glass slide. The distribution of scattered intensity at various times during deposition is shown in fig. 9. The intensity of the scattered light increased as the gold coverage of the substrate (and therefore the specular reflectance of the film) increased. The angular dependence of the scattered light did not depend on the mass thickness coverage of the substrate. We conclude that the scattering observed is primarily due to light scattered by surface roughness and other localized defects and not by the connected regions of gold on the substrate. A pronounced enhancement of scattered light at a particular mass thickness might be expected if the size distribution of connected regions had a well defined maximum at a value comparable with the illuminating wavelength. The measured distribution of particle sizes, shown in fig. 10, shows no such maximum except at sizes very much smaller than the wavelength of visible light and
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Particle
size (microns)
Fig. 10. Typical distribution of particle sizes seen in a film just below the critical concentration. The field of view is a square 2 pm across, note the broad tail at large sizes.
therefore enhanced scattering at visible wavelengths is not expected, in agreement with experiment. The particle size distribution is further discussed in section 5 below.
10000
5. Microstructure
8000
0 0
I
scattering
2
angle
(degrees)
Fig. 9. Angular distribution of the intensity of the scattered component of light reflected from a gold film at various stages of the deposition. Curves from bottom to top: show data taken at 0,30, 60, I20 and 180 seconds after commencing deposition. There is no evidence of enhanced scattering at any particular thickness.
Thin silver films deposited on a-C coated copper TEM grids were studied using the Philips EM400 and the Jeol 1OOCXtransmission electron microscopes. Standard bright-field imaging techniques were used to produce micrographs similar to the one shown in fig. 1. Selected TEM micrographs were studied using the Tracer Northern TN-8502 Image Analysis system. An image of the micrograph from a black and white TV camera was converted into a greyscale image of 5 12 x 5 12 pixels. The greyscale image was then converted into a binary image by selecting the critical greyscale value G. Pixels with greyscale value less than G are assigned a value of one in the binary image, all other pixels are set to zero. The value of G is varied until the binary image is the most accurate possible representation of the original image. 77
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It is important to note that, despite careful attempts to carry out this procedure in a reproducible way, the decision of exactly what value of G gives the best representation of the original remains a subjective choice. Also, information is certainly lost during the process. The main losses are due to optical imperfections in the camera and the finite number of pixels [20]. Computerised image analysis has previously been used to calculate percolation theory critical exponents for this type of film [ 2 11. Here we study the microstructure in a more qualitative manner. Fig. 11 shows a typical result for a film at the critical concentration. The binary image has a concentration of 63% (63% of the pixels are silver) and none of the clusters span the field of view, so the correlation length is less than 2 pm. Thus the sample is below the critical concentration. A very small difference in the binary image creation process, however, leads to the result shown in fig. 12. It is easily seen that the grey coloured cluster spans the field of view, so the correlation length is more than 2 pm if the film has a concentration of 64%. In fact, a spanning cluster is seen across other regions of the film when the binary image has a concentration of 64%. Thus if the concentration is 64% the correlation length must be much larger than 2 urn. Following percolation theory we will call any connected region much larger than 2 urn a percolating network. The question now arises as to which of the above images is the correct representation of the film. Both binary images seem at first sight to be good representations of the micrograph. The problem can be resolved by noting that the only difference between the two images is a small number of points which appear in the higher concentration image. These points form narrow necks which connect the separate clusters to form one large cluster. Thus the sharp qualitative change between 63% and 64% is due to the fact that between these two concentrations the silver particles are joined into a percolating network. The fact that the observed transition in the equilibrium film conductivity has rounded edges [ 7 1, however, suggests that the transition in the connectedness of the particles is not as sharp as it would initially appear. Gadenne, Yagil, and Deutscher [ 191 have noted the importance of narrow necks they observed in the 78
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microstructure. Close inspection of our TEM micrographs shows that it is often diff’icult to ascertain whether or not two particles are connected at a narrow neck without obtaining a micrograph of higher magnification. When two particles are this close we propose that the two particles are neither connected nor disconnected but they in fact have some intermediate connectivity. According to this interpretation, the correlation length for silver particles is undefined in this region of concentration.
The narrow necks observed in the microstructure are very much like the point contacts in the chequer board model in that they exhibit intermediate conductivity through both phases. For example two metal clusters which are very close but do not touch will be capacitively’coupled and therefore exhibit tinite AC conductivity. The void phase is a narrow neck at this point, so it is also in some sense connected. Fig. 13 shows a comparison of the inverses of the two previous images. The inverse binary images were obtained by a NOT binary operation, and so show the structure of the void component of the film. It is seen that the 63% image contains a spanning void cluster whilst the 64W image does not. So for concentrations at and above 64% we have a spanning cluster in the silver phase but not in the void phase whereas for 63W and below we have a spanning cluster in the void phase but not in the silver phase. Therefore both the silver and void phases of this film form percolating networks for concentrations near 63.5%. If the narrow necks between the clusters behave like point contacts as we propose then the silver and void phases can simultaneously exist as percolating networks in this region of concentration. Thus there is some range of concentrations for which the correlation lengths of both the silver and void phases are not defined. Without making a detailed study of the narrow necks we can only guess the size of this concentration region. The relative ease with which a film was produced with transmittance constant to more than 43 pm, however, suggests that this range is appreciable. A large absorption is seen in a range of concentrations around the critical concentration [ 18,191. We argue that this absorption is strongly related to the presence of narrow necks in the microstructure
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Fig. I 1. Digit&d image of the film AG25, showing separate clusters in different colours. The area concentration of metal in this image is 63%, and there is no connected metal region which spans the field of view. Note that the clusters contain many tortuously wandering paths of roughly constant width.
which occur in this range. Clearly the conductivity through a narrow neck is much smaller than the conductivity through the bulk of the conducting net-
work, and it is also insensitive to the thickness at which the critical concentration is reached. The relative insensitivity of the magnitude of the absorp79
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Fig. 12. An almost identical image of AG25, with metal concentration 64%. The large grey region forms a connected metal pathway across the field of view (approximately 2 lrn across). Note that the only significant difference between this image and the one with 63% concentration is the degree of connectedness between the silver clusters. tion to the film thickness
which is seen in the literature [ 18,191 is therefore consistent with the suggestion that the optical absorption is primarily due to losses at weak links between particles. (Yagil et al. 80
[ 121 emphasise the importance of intercluster capacitance in understanding the optical properties of films. Clearly, actual films will contain a variety of close-connection morphologies, ranging from good
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and void networks consist of many tortuous paths which seem to be of roughly constant width. An estimate of the average width was made by randomly selecting points on the image and measuring the width of the path at these points. The values were then averaged to give rough estimates of the widths. For AG30C the average width of the silver paths was found to be 60? 5 nm and the average width of the void paths was found to be 20 ? 2 nm. The difference between the average widths of the silver and void particles is due to the fact that silver arriving at the substrate during deposition is more likely to adhere to a silver cluster than to the bare substrate. The silver width seems to be the only scale size that can be associated with the small-scale structure of the silver clusters. It is therefore equated with the ‘grain size’ parameter in the percolation theory approach. It should be noted that varying the relative widths of the silver and void clusters changes the critical concentration, so the value of between 63% and 64Oh is not universal. Films grown at different rates or under slightly different conditions were found to have varying threshold concentrations. The film AG30C, for example, had a threshold concentration of about 72%. The distribution of sizes of the silver particles was also studied. The maximum lateral extent of each particle was calculated in order to assign a representative “size” to each connected region. Fig. 10 shows the distribution of particle sizes for a film of concentration 63%, just below the critical concentration. The peak at small sizes is due to small silver
Fig. 13. The images in figs. 11 and 12 have been inverted to give a comparison of the connectedness of the void component of the films. The pink area in the lower image, with 63% metal concentration, spans the field of view. The upper image (metal concentration 64%) does not. Thus the void component shows a transition in connectedness which is complementary to the transition in the metal connectivity.
inductive
contacts,
through
point
contacts
to highly
capacitive linkages. The influence of each contact type would need to be taken into account in a complete model of optical behaviour.) It can be seen in figs. 12 and 13 that both the silver
particles which have grown by the accumulation of silver atoms as they are deposited. This is the predominant process at low area concentrations. In the region of the critical concentration, however, the small particles begin to overlap and very large particles are formed by aggregation. These large particles cause the very broad tail in the size distribution seen in the region of the critical concentration. Note that the broad tail (which accounts for all particles of size comparable to the wavelength of light) contains no well defined maximum, in agreement with the scattering results of section 4.
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6. Conclusions The transmittance of thin silver films at the metal insulator transition has been found to be constant for wavelengths from 1.5 urn to more than 43 urn. The optical properties have been shown to be very sensitive to the nature of the substrate surface. Even similar amorphous surfaces, which were previously thought to produce identical films, can result in films near transition having significantly different optical properties. Microstructural results are presented which are consistent with the ‘dynamic scattering’ model of Josifovska et al. [ 71. An important feature of both. the model and the actual microstructure is the presence of tenuous contacts between clusters. These cause the correlation lengths in both the metal and the void phases to be undefined. We suggest that the high optical absorption can be explained as being primarily due to losses at the narrow necks in the microstructure. The elaboration of the dynamic scattering model requires a quite complicated electromagnetic calculation. Work on this is in progress.
Acknowledgements Craig Davis acknowledges the support of an Australian Postgraduate Research Award. The authors thank Mr. Zhen Hua Wang for providing the measurements of scattered light on growing gold films, and Mr. Peter Ring for help in the measurement of film transmittance at 433 urn. The authors acknowledge the Science Foundation for Physics and His Royal Highness Prince Nawaf Bin Abdul Aziz of the Kingdom of Saudi Arabia for the provision of facilities.
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References [ 11 Y. Yagil and G. Deutscher, Thin Solid Films 152 ( 1987) 465. [2] P. Gadenne, A. Beghdadi and J. Lafait, Optics Comm. 65 (1988) 17. [3] B. Abeles, H.L. Pinch and J.I. Gittleman, Phys. Rev. Lett. 35 (1975) 247. [4] M. Gajdardziska-Josifovska, MSc Thesis, University of Sydney (1986). [ 51 Y. Yagil and G. Deutscher, Appl. Phys. Lett. 52 (1988) 373. [ 61 M. Gadenne, J. Iafait and P. Gadenne, Optics Comm. 7 1 (1989) 273. [ 71 M. Gajdardziska-Josifovska, R.C. McPhedran, D.R. McKenzie and R.E. Collins, Appl. Optics 28 ( 1989) 2744. [ 81 D. Stauffer, Phys. Rep. 54 ( 1979) 1. [9] D. Stauffer, Introduction to percolation theory (Taylor and Francis, London, 1985). [IO] T. Robin and B. Souillard, Physica A 157 (1989) 285. [ 111T. Robin and B. Souillard, Optics Comm. 71 (1989) 15. [ 121 Y. Yagil, M. Yosetin, D.J. Bergman, G. Deutscher and P. Gadenne, Paper presented at ICMC, San Diego, USA, April 1990. [ 131 R.C. Compton, J.C. Macfarlane, L.B. Whitboum, M.M. Blanc0 and R.C. McPhedran, Optica Acta 3 1 ( 1984) 5 15. 141 R. Uhich, Infrared Phys. 7 (1967) 37. 151 M. Gajdardziska-Josifovska, R.C. McPhedran, D.J.H. Cockayne, D.R. McKenzie and R.E. Collins, Appl. Optics 28 (1989) 2736. 161 S. Berthier, J. Lafait, C. Sella and Thran-Khanh-Vien, Thin Solid Films 125 (1985) 171. 171 M. Nishiura and A. Kinbara, Thin Solid Films 24 ( 1974) 79. [ 181 P. Gadenne, Y. Yagil and G. Deutscher, Physica A 157 (1989) 279. [ 191 P. Gadenne, Y. Yagil and G. Deutscher, J. Appl. Phys. 66 (1989) 3019. [ 201 A. Beghdadi, M. Gadenne, J. Lafait, A. Le Negrate and A. Constans, Physica A 157 ( 1989) 64. [ 2 1] R.F. Voss, R.B. Laibowitz and E.I. AUessandrini, Phys. Rev. Lett.49 (1982) 1441.