PHYSICAL REVIEW B, VOLUME 65, 245113
Optical constants of gold blacks: Fractal network models and experimental data Juan A. Sotelo,1,2 Vitaly N. Pustovit,1,3,* and Gunnar A. Niklasson1 1
Department of Materials Science, Division of Solid State Physics, University of Uppsala, The Angstrom Laboratory, SE-75121 Uppsala, Sweden 2 Departamento de Fisica, Informatica y Matematicas, Universidad Peruana Cayetano Heredia, Apartado Postal 4314 Lima, Peru 3 Institute of Surface Chemistry, NAS of Ukraine, 17 Gen. Naumova Street, Kiev 03164, Ukraine 共Received 7 November 2001; revised manuscript received 8 February 2002; published 14 June 2002兲 We demonstrate that the optical properties of gas-evaporated gold blacks can be qualitatively explained using fractal admittance network models. A consistent set of optical constants of gold black is obtained by performing a Kramers-Kronig analysis on literature data in the wavelength range from the near ultraviolet to the far infrared. In order to describe the optical properties we put forward two three-dimensional fractal network models. In one of them we consider a percolating backbone structure; in the other we also include effects of dangling ends and finite clusters. The shape of the absorption coefficient and refractive index spectra predicted by the models are in qualitative agreement with the experimental data. DOI: 10.1103/PhysRevB.65.245113
PACS number共s兲: 78.67.⫺n
INTRODUCTION
Optical properties of metal network films close to percolation have attracted a lot of interest.1,2 In particular, the limits of validity of scaling3 and effective-medium theories4 have been discussed.5 The works mentioned above mainly concern two-dimensional structures, although generalizations of scaling theory to three dimensions have been put forward.6 On the other hand, experimental studies on threedimensional metal network films have often been carried out. Such films can easily be prepared by the gas-evaporation technique;7 they are often called ‘‘metal blacks.’’ The optical properties of gold blacks were thoroughly studied by Harris and co-workers as far back as the 1940’s;8 –10 a comprehensive report on these studies was later published as a monograph.11 The optical properties of the coatings were studied in an unusually large frequency range, from the near infrared ( ⫽4.1⫻1012 s⫺1 ) to the near ultraviolet ( ⫽7.4 ⫻1015 s⫺1 ). Later studies12,13 have been restricted to narrower frequency ranges; hence Harris’s data are still the best for obtaining the optical constants of gold black. Previously used theoretical models based on equivalent circuit modeling14 or effective-medium theories with general shape distributions,12 included many fitting parameters and were not suitable for a critical comparison with experiments. The purpose of this paper is twofold. First, we obtain a consistent set of optical constants of gold black by using the tabulated data of Harris.11 The availability of optical constants in a wide frequency range is ideal for critical comparisons with theoretical models. Second, we compare these optical constants to computations by admittance deterministic fractal lattice 共YDFL兲 models. We use fractal geometry to model the structure of the gold blacks; apart from this no fitting parameters enter in the comparison of theory with experimental data. Fractal network models were first used by Zabel and Stroud,15 who showed that fractal clusters have a much larger absorption than clusters with the same amount of randomly distributed particles. The results obtained by Zabel and Stroud15 were clearly confirmed by later results of Brouers and Clerc,16 who studied the impedance two0163-1829/2002/65共24兲/245113共7兲/$20.00
dimensional deterministic fractal lattice 共ZDFL兲 model. Our previous work on the ZDFL model using the experimental bulk dielectric function of some noble metals 共including gold兲 highlighted the damping effects expected in reality.17 Many numerical techniques can be used for the calculation of the optical response of percolating gold blacks clusters. Here we briefly mention some basic models and approaches traditionally used to describe electrical and optical properties of random percolating media. The random resistor 共R兲 and resistor-inductor-capacitor 共RLC兲 network models are widely used to study the dc and ac conductivity, respectively.18 The most effective methods used for numerical simulations of RLC networks are the transfer-matrix 共TM兲 suggested by Derrida and co-workers19 and the Y-⌬ transformation developed by Frank and Lobb.20 The Frank and Lobb algorithm consists of repeated applications of a sequence of series, parallel and star-triangle Y-⌬ transformations to the bonds of the lattice, finally reducing the lattice to a single bond having the same conductance as the entire lattice. Although, the Y-⌬ algorithm is faster than the TM approach, it can be applied only to two-dimensional systems. The TM approach, on the other hand, treats a d-dimensional network as being built up by successive additions of (d⫺1) dimensional layers in a specific direction;21 in this way a recursive updating of the network impedance matrix is achieved. Indeed, the possibility to explore optical properties of fractal aggregates in three-dimensional space looks promising since experimental aggregates very often have this kind of structure. In the following we present a different type of deterministic fractal lattice 共DFL兲 model for threedimensional structures and compare it to experimental data for gold blacks. THEORY
Our three-dimensional YDFL models find their roots in the work of Clerc and co-workers,22,23 who mapped into a deterministic hierarchic impedance network, the DFL introduced by Kirkpatrick18 and Mandelbrot24 to study the properties of percolating clusters. We use an equivalent formula-
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FIG. 3. The cutting procedure of an insulating cube, used recursively to build the DFL. Here we show the procedure leading to model 共II兲.
FIG. 1. The snowflake fractal at iterations S⫽0,1,2. The fractal dimension is D f ⫽ln(7)/ln(3)⬇1.771.
tion in terms of admittances, hence the name YDFL, because it gives us directly the renormalization transform for the dielectric function and, hence, for the conductivity of the YDFL. To model a percolating cluster just above the percolation threshold we use a DFL that is inspired by the fractal shown in Fig. 1. The DFL is obtained recursively as follows: Suppose a starting cube of metal is divided in cubes of side a third of the original’s, as a Rubik’s cube; replace then the boundary cubes in the upper and bottom layers, and the corner cubes in the middle layer, with insulator cubes of the same size, see Fig. 2. Proceed likewise with a starting cube of insulator material; from any two of its three layers, choose at random a total of at most seven of the smaller insulator cubes and replace them with metal cubes; in Fig. 3, for example, one insulator cube of the middle layer and six of the upper layer have been replaced by metal cubes. This process allows the introduction of a population of dangling ends and finite clusters, like in the real percolating systems.23 Setting to zero and seven the number of metal cubes introduced in the insulator’s transformation rule yields, respectively, two limiting models: model 共I兲, a model of the percolation backbone, and model 共II兲, a model of the percolation backbone augmented with a population of dangling ends and finite clusters. There are several ways to implement model 共II兲, we choose the one suggested by Fig. 3, whereby the seven metal cubes are distributed between any two of the three layers 共middle and top layers in Fig. 3兲 in a 1:6 ratio;
FIG. 2. The cutting procedure of a conducting cube, used recursively to build the DFL.
both the layers and the location of the metal cubes in the layers are chosen at random. We shall concentrate on these two limiting models. Following Clerc and co-workers,23 we map each of the models into corresponding YDFLs. Let us denote by Y c,s ( ) the frequency-dependent admittance of a metallic component at the sth level of recursion, and by Y i,s ( ) that of an insulator component; their ratio will be denoted by h s ( ). By definition, Y c,s ( ) is the admittance of the YDFL at the sth generation. We assume that the starting values of the admittances, Y c,0( ) and Y i,0( ), and hence that of their ratio h 0 ( ) are known quantities and the same for both models. Furthermore, whenever necessary, we will stress the model a particular variable refers to, by adding to the variable the model’s name as a superindex. Figures 4共a兲 and 4共b兲 show the recursive procedure for building the YDFL of model 共I兲. It follows from this that the metal volume fraction in the YDFL at the sth level of recurs sion is f (I) c ⫽(7/27) . In a similar way, we show in Figs. 4共a兲 and 4共c兲 the recursive procedure for building YDFL of model 共II兲; unlike in the previous case, the metal volume fraction for this YDFL is independent of the recursive level and equals f (II) c ⫽7/27.
FIG. 4. The metal and insulator transformation rules from one generation to the next for both models. 共a兲 The common metal transformation rule for models 共I兲 and 共II兲. 共b兲 The insulator transformation rule for model 共I兲. 共c兲 The insulator transformation rule for model 共II兲.
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c,s 共 兲 ⫽⫺i 0 c,s 共 兲 ,
As shown in Fig. 4共a兲, the two models have a common transformation rule for the metallic admittances,
i,s 共 兲 ⫽⫺i 0 i,s ,
Y c,s 共 兲 ⫽ 关 Y c,s⫺1 共 兲 ⫹8Y i,s⫺1 共 兲兴 5Y c,s⫺1 共 兲 ⫹4Y i,s⫺1 共 兲 ⫻ , s⭓1, 11Y c,s⫺1 共 兲 ⫹16Y i,s⫺1 共 兲
共1兲
which can be expressed in terms of the Y c,s ( ) and h s ( ) as Y c,s 共 兲 ⫽Y c,s⫺1 共 兲
关 h s⫺1 共 兲 ⫹8 兴关 5h s⫺1 共 兲 ⫹4 兴 , h s⫺1 共 兲关 11h s⫺1 共 兲 ⫹16兴
Y i,s 共 兲 ⫽3Y i,s⫺1 共 兲 ,
共3兲
Taking the ratio of Eqs. 共2兲 and 共3兲 yields a recursive relation for h s ( ) h 共0I兲 共 兲 ⫽h 0 共 兲 , I兲 h 共sI兲 共 兲 ⫽F 关 h 共s⫺1 共 兲兴 ,
共5兲
s⭓1,
Y c,s 共 兲 ⫽3 s c,s 共 兲 ,
s⭓0,
共7兲
Y i,s 共 兲 ⫽3 s i,s 共 兲 ,
s⭓0,
共8兲
s
where 3 is the length of the YDFL at the sth generation. Next, we write the metal and insulator conductivities in terms of the dielectric functions for metal, c,s ( ), and insulator, i,s ( ), respectively,
Y i,s 共 兲 ⫽9Y i,s⫺1 共 兲
c,s 共 兲 , i,s
共i,sI兲 ⫽ i,0 ,
h 共sI兲 共 兲 ⫽
2 2h s⫺1 共 兲 ⫹38h s⫺1 共 兲 ⫹41
s⭓0,
共11兲 共12兲
共13兲
s⭓0,
I兲 共c,s 共兲
i,0
,
s⭓0.
共14兲
Insertion of this equation into Eq. 共5兲 yields the desired expression for the dielectric function of the YDFL of model 共I兲 at the sth generation, I兲 共c,0 共 兲 ⫽ c,0共 兲 , I兲 共c,s 共兲
i,0
⫽F
冉
I兲 共c,s⫺1 共兲
i,0
冊
,
s⭓1.
共15兲
We can now use this expression to compute the refractive (I) index of the material in the usual way, N( )⫽ 冑 c,s ( ). We proceed likewise for model 共II兲. In this case, the transformation rule for the insulator admittances, shown in Fig. 4共c兲, has the form
2 2 2Y c,s⫺1 共 兲 ⫹38Y c,s⫺1 共 兲 Y i,s⫺1 共 兲 ⫹41Y i,s⫺1 共兲
关 h s⫺1 共 兲 ⫹8 兴关 2h s⫺1 共 兲 ⫹1 兴
s⭓0,
consequently, Eq. 共12兲 gives
关 Y c,s⫺1 共 兲 ⫹8Y i,s⫺1 共 兲兴关 2Y c,s⫺1 共 兲 ⫹Y i,s⫺1 共 兲兴
which, as for the metallic component, can be expressed in terms of the Y i,s ( ) and h s ( ) as
共10兲
which are valid for both models. For model 共I兲, Eq. 共12兲 can be further simplified. Downward iteration of Eq. 共3兲 to the zero level; substitution of Eq. 共8兲 into the resulting expression; and subsequent application of Eq. 共10兲 leads to
共6兲
where the rational function F is of degree 2 and defines the renormalization-group transform of the YDFL. In both models, the dielectric function, c,s ( ), of the YDFL at the sth generation is found as follows. First, we write the metal and insulator admittances in terms of their corresponding conductivities c,s ( ) and i,s ( ),
s⭓1.
h s共 兲 ⫽
共4兲
1 共 h⫹8 兲共 5h⫹4 兲 F共 h 兲⫽ , 3 11h⫹16
Y i,s 共 兲 ⫽9Y i,s⫺1 共 兲
Y c,s 共 兲 ⫽⫺3 s i 0 c,s 共 兲 ,
共2兲
s⭓1.
s⭓0,
共9兲
where 0 is the permittivity of the vacuum, and both c,0( ) and i,0 are known quantities and the same for both models. Inserting Eq. 共9兲 into Eq. 共7兲, Eq. 共10兲 into Eq. 共8兲, and substituting the resulting expressions into the definition of the h s ( ) we obtain the relations,
s⭓1.
The transformation rule for the insulator admittances, on the other hand, is model dependent. For model 共I兲 the rule, see Fig. 4共b兲, leads to the recursive relation
s⭓0,
,
s⭓1,
共16兲
Taking the quotient of Eq. 共2兲 by Eq. 共17兲 leads to a functional recursive relation for h s ( ) h 共0II兲 共 兲 ⫽
,
共17兲 245113-3
c,0共 兲 , i,0
II兲 h 共sII兲 共 兲 ⫽T„h 共s⫺1 共 兲 …,
s⭓1,
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FIG. 5. The real part of the refractive index, n( ), as computed from the experimental data for gold blacks 共curve exp兲; and as computed using model 共I兲 共curve I兲 and model 共II兲 共curve II兲 at S ⫽5 iterations.
FIG. 6. The imaginary part of the refractive index, k( ), as computed from the experimental data for gold blacks 共curve exp兲; and as computed using model 共I兲 共curve I兲 and model 共II兲 共curve II兲 at S⫽5 iterations.
1 共 5h⫹4 兲共 2h 2 ⫹38h⫹41兲 T共 h 兲⫽ , 9 共 11h⫹16兲共 2h⫹1 兲
micrographs6 and indirect evidence from light scattering,27 it seems that aggregate sizes could be larger than a micron. It has been shown in later studies28 –30 that gas-evaporated aggregates of various nonmagnetic metals have a fractal structure with fractal dimension D around 1.9. It is believed that the individual particles travel in ballistic trajectories in the low-pressure gas and stick to one another upon collisions. The mean free path of a particle in a few Torr of gas is usually larger that the distance from evaporation source to substrate position. Simulations of the ballistic cluster-cluster aggregation process give fractal structures with D⬇1.9, in good agreement with the experiments.31 We next describe how the complex refractive index, N( )⫽n( )⫹ik( ), of gold black was obtained from the published data. Harris11 found that the optical density was proportional to the weight per unit area of the deposits. This shows the good reproducibility of his results. The weight per unit area W/A is related to the volume fraction f and thickness d by the relation:
共18兲
where the renormalization-group transform T(h) of the YDFL is now a rational function of degree 3. Substituting Eq. 共11兲 into Eq. 共2兲, and in view of Eqs. 共18兲, we obtain a double recursive relation for the dielectric function of the YDFL of model 共II兲 at the sth generation, II兲 共c,0 共 兲 ⫽ c,0共 兲 ,
II兲 II兲 共c,s 共 兲 ⫽ 共c,s⫺1 共兲
II兲 F„h 共s⫺1 共 兲… II兲 h 共s⫺1 共兲
,
s⭓1.
共19兲
As we have already stated, the volume fraction of metal components in this YDFL is independent of the recursion level, namely, f (II兲 c ⫽7/27. To deal with structures with a general value of f, we make use of an effective-medium concept. Assume that percolating aggregates with dielectric function (II) ( ) are embedded in additional insulator with dielectric c,s function i,0 . Considering that the percolating aggregates and the insulator components are all connected in parallel we get for the refractive index of the material25 II兲 ˜ 共 兲 ⫽ 冑 f 共c,s N 共 兲 ⫹ 共 1⫺ f 兲 i,0,
共20兲
W ⫽ f d, A
共21兲
where is the density of bulk gold. Approximately, the transmittance is related to thickness by the relation32
where f ⫽ f exp /f (II) and f exp is the volume fraction of the c metallic component in the real material.
T共 兲 ⫽exp关 ⫺ ␣ 共 兲 d 兴 1⫺R 共 兲
ANALYSIS OF EXPERIMENTAL DATA
where ␣ ( )⫽2 k( )/c is the absorption coefficient of gold black and c is the speed of light. The good accuracy of using a single absorption coefficient for different samples, in the frequency range above 1014 s⫺1 , is evident from Fig. 4 in Ref. 11. At lower frequencies, the optical properties of the samples depend on their dc conductivity, as discussed below. Having obtained the imaginary part of the refractive index from Eq. 共22兲, the real part can be obtained from KramersKronig analysis by
The gold blacks considered in this paper were produced by evaporation of gold in the presence of 3 Torr of pure nitrogen gas.11 Subsequently, the coatings were stabilized by annealing for 24 h at 342 K. Very porous coatings, with volume fractions of gold as low as 0.002 to 0.003, were obtained.9 Transmission electron microscopy showed rounded Au particles aggregated into chains and irregular clusters.9,26 The particle sizes were highly dependent on evaporation conditions; particle diameters ranging from 1.5 to 20 nm have been reported.11 A structural analysis of the aggregation was not carried out, but from published
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n 共 0 兲 ⫽1⫹ 共 2/ 兲 P
冕
u
0
k共 兲 d ⫹B, 共 2 ⫺ 20 兲
共22兲
共23兲
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FIG. 7. The frequency dependence 共log-log scale兲 of the optical constants n( ) and k( ) for different number of iterations S ⫽2, 4, 6 in model 共I兲 for gold blacks.
where P denotes the principal value of the integral, the integration is over angular frequency, and u is the upper limit of the frequency range where values of k( ) are available. The available data for k( ) values were extended to zero frequency by using published dc conductivity values; this procedure introduces substantial uncertainties in n( ) at the lowest frequencies. The constant B accounts for the contribution to n( ) from processes at frequencies higher than those used in the measurements. This high-frequency extrapolation was fixed by assigning the values of n( ) at frequencies lower than 1014 s⫺1 in the infrared to be consistent with reported4 specular reflectance data. The errors introduced by this form of high-frequency extrapolation were studied by Nilsson and Munkby.33 There are various sources of uncertainty involved in the above procedures. First, the samples showed some diffuse scattering of light. We neglected this because values of diffuse transmittance and reflectance were available only in part of the frequency range. The errors due to this are largest in the visible and near UV. From reported diffuse transmittances values, we find that k( ) is overestimated by about 10% in the visible frequency range and the error rises to 20% at the highest frequency, namely, 7.4⫻1015 s⫺1 . Second, since the reflectance in Eq. 共22兲 is very low, its accuracy is
much less than that of the transmittance. In order to minimize errors from this we used in our analysis Harris’s tabulated data for a rather thick layer of gold black with W/A ⫽1 g/m2 . Third, the Kramers-Kronig analysis introduces additional errors in the absolute value of n( ), because an approximate value of B was determined from reflectance measurements of unknown accuracy. However, we estimate this error to be less than 10%. The experimental optical constants of gold black are displayed as a function of frequency in Figs. 5 and 6. It is seen that both n( ) and k( ) increase towards lower frequencies, with a small hump in the region of 1014 s⫺1 . The refractive index is close to unity at high frequencies, as expected from the very small measured reflectances.11 We note that the lowest frequency points are uncertain, since they are influenced by the low-frequency extrapolation used. RESULTS AND DISCUSSION
We have carried out calculations by both fractal network models for gold in the frequency region below 2 ⫻1015 s⫺1 . We do not consider here the interband transition region discussed in our previous paper.17 Experimental bulk optical constants for gold were used; in the low-frequency
FIG. 8. The frequency dependence 共log-log scale兲 of the optical constants n( ) and k( ) for different number of iterations S ⫽2, 4, 6 in model 共II兲 for gold blacks.
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region they are well described by the Drude model with parameters p ⫽1.365⫻1016 s⫺1 and p ⫽1.73⫻10⫺14 s. 34 As in our previous paper, the dielectric function of small gold particles was corrected to include the influence of electron scattering from the boundaries of the particles,17 we used V F ⫽1.372⫻1015 nm/s for the Fermi velocity of the free electrons in gold, and set the average radius of the particles to R⫽3 nm. As shown in Fig. 7, the application of model 共I兲 produces a peak in the optical spectra of n( ) and k( ). As the number S of iterations increases, the peak becomes more damped and also lower in magnitude. We have observed that for a fixed iteration S as the particle size increases, the hump region becomes less pronounced. The hump at ⬃1014 s⫺1 in Fig. 7 appears only after taking into account the effect of electron scattering from the boundaries of particles and is completely absent in the bulk data. This size effect could be due to electron correlations in the particles at the percolating limit. For model 共II兲, on the other hand, as the number of iterations increases, at first, the peak becomes more damped and lower in magnitude as for model 共I兲; after four iterations, however, the magnitude and shape of the peak do not change with increasing number of iterations, see Fig. 8. This happens because for this model the metal volume fraction is fixed, f (II) c ⫽7/27. Following the effective-medium concept, we get an effective dielectric function of the metal-insulator material at the percolation limit, Eq. 共20兲, by modeling the material as percolating aggregates corresponding to model 共II兲, connected in parallel and embedded in an insulating host. Here, the volume fraction of percolating aggregates in the material is f ⫽ f exp /f c , and f exp is the metal volume fraction given by the experimental data. Comparison of the two models with experimental data requires that the number of iterations as well as the volume fraction be specified. In order to perform calculations with a metal volume fraction close to the experimental one for gold blacks ( f exp⬃0.003), we use in model 共I兲 S⫽5 iterations, leading to a volume fraction f (I) c ⬇0.001. In model 共II兲 we also use S⫽5 in order to ensure convergence of the results. The volume fraction f exp was put to 0.003; hence f ⫽0.012 in Eq. 共20兲. In Figs. 5 and 6, the n( ) and k( ) for gold black, obtained from experimental data, are compared to the computations. A rough qualitative agreement is observed. The experimental data shows less pronounced structure in the region around 1014 s⫺1 . This might be due to variations in particle size and shape in the experimental samples, as well as to their random fractal structure. The quasideterministic fractal models we use are expected to give rise to sharper features than those that can be observed experimentally. In order to compare the theoretical models, we stress that model 共I兲 presents in a direct way the construction of the YDFL network, producing as a result a percolating backbone cluster. It also has a fractal dimension close to that of gasevaporated metal blacks. In model 共II兲, on the other hand, the fractal percolation backbone is augmented with a population of dangling ends and finite clusters. It is seen, that model 共I兲 gives a fair description of the magnitude of n( ) and k( ), taking into account the uncertainty in the actual values of
n( ). On the other hand model 共II兲 gives results with a shape of the spectra more similar to the experimental one, in the region above 1014 s⫺1 . However, it appears that model 共II兲 substantially underestimates n( ) and k( ) in the whole frequency range. This discrepancy is especially severe at low frequencies. Earlier models of the optical properties of gold black12,14 employed several fitting parameters. We think that our simple deterministic models represent a substantial improvement, since we do not employ fitting parameters. It should also be noted that metal blacks are random fractals and a model that includes the randomness would probably improve the agreement with experimental data. Another possible reason for the discrepancies between our theoretical models and the experiments is suggested by recent work by Stockman, Faleev, and Bergman35 on the solution of the electrostatic problem for random planar composites 共RPC兲 films. The application of the real-space renormalization model, as introduced by Kadanoff et al.36 to percolating systems, assumes an insensitivity of the results to the details of the large- and small-scale structure of the system. In contrast, Stockman, Faleev, and Bergman35 found that the surface plasmon excitations on their structures are simultaneously singular on the minimum scale and correlated over a length that may be of the same order as the length of the total system. Because of the singularities, they found that the distribution of the surface plasmons were very sensitive to small changes in the structure of the system at the minimum scale. These findings, Stockman et al.35 argue, suggest that the electrostatic theory as applied to RPC films may not be renormalizable and that it may be impossible to give a universal description of the optical properties of such structures based on a percolation model. Such inhomogeneous localization effects are probably present in the gold black samples and may, at least partly, account for the discrepancies between theory and experiment. In the far-infrared ( ⬍1014 s⫺1 ) region the data of Harris depend on the dc conduction of the samples. In a previous work37 a scaling region was found for the conductivity ⬃ 0.7 at ⬍1014 s⫺1 , indicating anomalous diffusion on a fractal structure. This behavior is not seen in the present models. Probably, our quasideterministic models do not reproduce correctly the scaling behavior of random fractals. In the far infrared, model 共I兲 gets closer to the experimental scaling behavior, provided that the particle radius R⬎5 nm; we found that approximately ⬃ in this case. CONCLUSIONS
We have applied two different analytical threedimensional fractal admittance network models for the qualitative description of gas-evaporated gold blacks. A consistent set of optical constants of gold blacks was obtained by performing a Kramers-Kronig analysis on literature data in the wavelength range from the near ultraviolet to the far infrared. It appears that model 共I兲 developed for a percolating backbone structure of particles fits better with experiment regarding the magnitudes of n( ) and k( ), while model 共II兲,
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which, besides the backbone also includes dangling ends and finite clusters, gives a better curve shape. Remaining discrepancies may be due to inhomogeneous localization of surface plasmons, an effect that is not accounted for in renormalization theories.
ACKNOWLEDGMENT
V.P. acknowledges support from the Wenner-Gren Foundation. J.S. and G.N. acknowledge support from the Swedish Natural Science Research Council.
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