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Relation between Morphology and Alternating Current Electrical Properties of Granular Metallic Films Close to Percolation Threshold† S. Blacher,*,‡ F. Brouers,§ A. Sarychev,§ A. Ramsamugh,| and P. Gadenne⊥ Ge´ nie Chimique, Institut de Chimie, Lie` ge University, B4000 Lı`ege, Belgium, Physique de Mate´ riaux, Institut de Physique, Lie` ge University, B4000 Belgium, Department of Physics, University of the West Indies, Mona, Jamaica, West India, and Laboratoire d’Optique des Solides, Unite´ Associe´ e au CNRS, Universite´ P. et M. Curie, 4 Place Jussieu, 75252 Paris Cedex 05, France Received September 1, 1994. In Final Form: December 5, 1994X The purpose of this paper is to show how the low frequency electrical and dielectric properties may be related to the morphology of metallic thin films. To achieve this goal, we first summarize a recent comparative morphological study of granular gold films deposited on amorphous and polycrystalline substrates. Then we present unpublished measurement of the frequency-dependent conductivity of the same thin films samples in the range 102-106 Hz in the immediate vicinity of the percolation. These measurements indicate that the dispersion of the conductivity does not obey the classical percolation scaling laws. Taking advantage of our morphological study of these films, we show that it is possible to discuss the nonuniversality of the frequency variation of the electrical conductivity observed in these films in the framework of a new tunneling scaling law which depends on a morphology related tunneling factor and new scaling exponents.
Introduction Two-dimensional discontinuous metal films are obtained during early stages of film growth by evaporation or sputtering a metal onto a nonconducting substrate. The deposited metal first forms isolated islands. These islands undergo coalescence growth. Then as the film thickness increases they eventually form a continuous film via direct impingement followed by surface diffusion. The electrical properties of such a system strongly depend on the metal content. When a concentration of metal is low, the electrical conductivity is close to zero and highly activated. As the proportion of metal is increased, the activation energy falls and, eventually for a given coverage (percolation threshold), continuous paths extending through the material are established. At this stage, the system undergoes a “insulator-metal” transition to a conducting state. For higher metal concentrations, the morphology is inverted: insulating inclusions in a metallic matrix. As the proportion of insulating inclusions is reduced to zero, the conductivity continues to improve toward its bulk value. In fact, the growth of these films is a complex nonlinear process complicated by surface tension and substrate wetting effects giving rise to a highly correlated geometry which determines the concentration pc at which the insulator-metal transition occurs. This transition is complicated by the presence of conductivity mechanisms such as intergrain tunneling and phonon-assisted hopping which can smoothen the sharp threshold of the conductivity at pc. Despite these complications, metallic thin films have been used by many authors as a convenient system to test the rich variety of universal scaling behaviors predicted by the classical theory of percolation near the percolation threshold. Indeed it is now well†Presented at the symposium on Advances in the Measurement and Modeling of Surface Phenomena, San Luis, Argentina, August 24-30, 1994. ‡ Institute de Chimie, Lie ` ge University. § Institute de Physique, Lie ` ge University. | Department of Physics, University of the West Indies. ⊥ Universite ´ P. et M. Curie. X Abstract published in Advance ACS Abstracts, January 1, 1996.
0743-7463/96/2412-0183$12.00/0
known that the morphology of these films may be different from the simple Boolean model assumed in the classical theory of percolation and that critical and scaling exponents for continuous percolation may in some conditions become nonuniversal and morphology dependent. It is also known that close to the percolation threshold tunneling induced conductivity plays an important role and can lead to a scaling law different from those of the classical random percolation theory. This new scaling law, which depends on a morphologic parameter, is relevant to analyze the low frequency dispersion of the frequency dependence of the conductivity and dielectric constant of granular thin films. The purpose of this paper is to show how the low frequency electrical and dielectric properties may be related to the morphology of these films. To achieve this goal, we will first summarize a recent comparative morphological study of granular gold films deposited on amorphous and polycrystalline substrates.1 Then we will present unpublished measurement of the frequencydependent conductivity of the same thin films samples in the range 102-106 Hz in the immediate vicinity of the percolation. These measurements indicate that the dispersion of the conductivity does not obey the classical percolation scaling laws. This has been noticed by several other authors.2-7 Taking advantage of our morphological study of these films, we will show that it is possible to discuss the nonuniversality of the frequency variation of the electrical conductivity observed in these films in the framework of a new tunneling scaling law which depends on a morphology related tunneling factor and new scaling exponents. (1) Blacher, S.; Brouers, F.; Gadenne, P.; Lafait, J. J. Appl. Phys. 1993, 74, 207. (2) Laibowitz, R. B.; Gefen, Y. Phys. Rev. Lett. 1984, 53, 380. (3) Song, Y. Phys. Rev. B 1986, 33, 904. (4) Hundley, M. F.; Zettel, A. Phys. Rev. B 1988, 38, 10290. (5) Yoon, C. S.; Lee, S. Phys. Rev. B 1990, 42, 4594. (6) In-Gann Chen.; Johnson, W. B. J. Mater. Sci. 1991, 26, 1567. (7) McLachlan, D. S.; Oblakova, I. I.; Pakhomov,A. B.; Brouers, F.; Sarychev, A. To be submitted for publication.
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Figure 1. Sample TEM micrograph at two stages of the gold deposition on two different substrates: intermediate stage on AMS (a, top left) and PCS (b, top right); metallic coverage before percolation on AMS (c, bottom left) and PCS (d, bottom right).
Sample Preparation A full description of the experimental setup can be found in ref 8. In order to realize different growth conditions and to get different morphologies, two kinds of substrates were used: amorphous substrates (AMS), borosilicate glass highly polished (rugosity smaller than the optical wavelength); polycrystalline substrates (PCS), float glass covered with a very thin (5-25 nm) microcrystalline layer of Sn and Cd oxides). Image Analysis A full description of the image treatment can be found in refs 1 and 8. In this work we have considered two stages of deposition on each substrate (amorphous and polycrystalline): an intermediate stage where gold clusters are large (10-30 nm) but still have Euclidean shapes (Figure 1a,b); a metallic coverage close to, but below, percolation (Figure 1 c,d). The image statistical analyses were performed using the followings tools of the mathematical morphology: the standard granulometry, the opening size granulometry distribution, and the covariance measure. The classical granulometry yields the radius distribution of gold clusters in the low-concentration regime when the clusters are well separated. As for higher concentrations, the films have a co-continuous structure and we are interested in the Au cluster size evolution with concentration, we calculate the opening size distribution. This type of granulometry can be used for both semicontinuous and individual particles on the gold and the void phase. Near the percolation threshold the size distribution of the gold phase evolves differently on the two types of (8) Gadenne, P.; Beghdadi, A.; Lafait, J. Opt. Commun. 1987, 52, 373, and references herewith.
substrate (Figure 2a,b). For the amorphous substrate, the curve obtained from the micrograph near percolation is strongly shifted with respect to the curve obtained for low concentration. The great dispersion in sizes is a consequence of the richness of the geometrical forms and sizes obtained after successive opening transformations. By contrast, the size distribution on the polycrystalline substrate keeps roughly the same shape and is slightly shifted toward larger sizes. This difference in the mean cluster sizes of films close to percolation on amorphous and polycrystalline substrates can be related to the difference in the growth process on these substrates. We have performed the same granulometric techniques on the complementary images (Figure 2c,d). It allows us to determine a measure of the “pores”. Here too the differences in the distribution can be related to the film growing mechanisms. The films and the “pore” covariance curves (see ref 1) exhibit an oscillatory behavior which is a manifestation of the existence of a characteristic length, in the cluster morphology. The value of the characteristic length which is concentration and substrate dependent, can be determined by the power spectra of the covariance curves. These characteristic lengths reflect the history of the film formation and the competition between the nucleation and growth processes which are obviously substrate dependent. ac Conductivity Measurements We present here data (Figure 3a,b) showing the lowfrequency dispersion of the electrical conductivity for the two series of films whose morphology has been discussed in the previous section. These results are in agreement with the same type of measurements done by other authors2-7 on similar physical systems. These results are in disagreement with the predictions of the dynamic
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Figure 2. Granulometry density curves for Au clusters for two different concentrations on AMS (a) and PSC (b) substrates and for its complementary images on AMS (c) and PCS (d) substrates. The error bars, corresponding to the length of one pixel on the image, are shown in the upper right part of the figure.
scaling theory of the percolation theory.9 The conductivity is nonzero for Au concentration below the percolation threshold and precise measurements would reveal a exponential behavior of the conductivity with respect to the concentration (or coverage, or film width) for p < pc. The dispersion occurs in a frequency region where this effect should not be observable and the slope exponent in the log-log plot is very different from the one predicted by the percolation theory. Recently two of us10 have proposed a new scaling model which seems able to account for results such as those presented here. This new general scaling law which incorporates the previous percolation scaling emphasizes the role and the importance of tunneling effect in insulating-metallic composites. It has been applied successfully to NbC-KCl composites.7 We discuss here the possibility of using ac measurements and this theory to extract physical parameters simply related to the film morphology.
σeff(p,0) ∝ σm ∆pt
(
σeff(p,ω) ) σ(p,0) φ
)
-i ω eff(p,0) σeff(p,0)
(1)
where the static effective conductivity and dielectric constant are given by (9) Clerc, J. P.; Giraud, G.; Luck, J. M. Adv. Phys. 1990, 39, 191 and references herein. (10) Sarychev, A.; Brouers, F. Phys. Rev. Lett. 1994, 73, 2895.
p g pc
and
eff(p,0) ∝ |∆p|-s where σm and i are the static metal conductivity and insulator dielectric constant, p is the metal concentration, pc is the critical metal insulator concentration, and ∆p ) p - pc. For any type of lattice the critical exponents t and s should be universal and depend only on the dimensionality d of the system (e.g., t ≈ 2.0, s ≈ 0.7 for d ) 3 and t ) s ) 1.3 for d ) 2). From this law it follows that when both |∆p| and the ratio iω/σm are small in the critical region around pc defined by |∆p| < (iω/σm) t/(t + s), the ac conductivity and dielectric constant exhibit a dispersive behavior
σeff(ω) ∝ ωx
The Tunneling Scaling Law The classical theory for the behavior of the complex conductivity of metal-dielectric mixtures is believed to be well understood in terms of the percolation theory.9 For finite frequencies in a critical region near the percolation threshold pc the complex conductivity is assumed to obey a scaling law of the form
for
and
eff(ω) ∝ ω-y with x ) t/(t + s) and x + y ) 1. As a typical metal conductivity σm is of the order of 1017 s-1 if the measurement frequency ω < 108 Hz, the concentration range where the dispersion can be observed shrinks to |∆p| < 10-4. In that region, however, dispersion of the conductivity and conductivity has been observed and reported for thin Au films of varying thickness or concentration p near pc from 100 Hz to 10 MHz 2. It was found that x ≈ 0.95 and y ≈ 0.13, which are different from the predictions for either d ) 2 (x ) y ) 0.5) or for d ) 3 (x ) 0.74 and y ) 0.26). Similar results have been reported for carbon-Teflon mixtures, computer modeled aluminum percolation thin films, and NbC-KCl composites.7
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Figure 3. Measurement of the frequency-dependent conductivity of thin films for different Au concentrations on AMS (a) and PSC (b) substrates.
Recently two of us have presented a new percolation theory incorporating the effect of quantum tunneling and introduced a new “tunneling” scaling equation to describe the AC properties of such composite materials10
(
eff(p,ω) ) σeff(p,0)φ
)
Σω (λ(∆p))a Σt
(2)
where λ is a renormalized tunneling constant which will be defined later.
As one might expect the universal scaling function φ depends on the ratio of the capacitive conductance Σω to the tunneling conductance Σt, both being between the large metallic clusters in the system. It also depends on a renormalized tunneling constant λ to some power a, where a is a new critical exponent. The analytical function φ(x) must have the following asymptotic behavior : φ(x) f constant when x f 0 and φ(x) ∝ xr when x f ∞ where r is yet another critical exponent, which will not necessarily have the values expected from classical percolation theory.
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Figure 4. Theoretical collapse curves of the frequency-dependent conductivity obtained from the Sarychev-Brouers scaling law for Au thin films on AMS (a) and PSC (b) substrates.
To estimate values of Σω and Σt, when the concentration approaches pc , one proceeds as follows. The typical size of a metallic cluster is of the order of the correlation length ξ ≈ a0 ∆|p|-ν where a0 is the size of a metal grain and the critical exponent ν ) 4/3 for d ) 2 and ν ≈ 0.9 for d ) 3. These clusters are extremely large just below pc and have a very ramified shape. The typical intercluster capacitive conductance Σω between two adjacent conducting clusters, proportional to the number of points of close approach N(∆p) is
Σω ∝ -i ω a0iN(∆p) ∝ -i ω|∆p)|-s
(3)
The number of points of closest approach N(∆p) goes to infinity when p f pc , leading to a divergence of the static dielectric constant. To determine the typical tunneling conductance, one has to consider two large metal clusters connected through N(∆p) tunneling junctions. The tunneling conductance of a single junction is given by the well-known exponential
expression
Σt ≈ a0 σm exp(-l/lt) The intergranular separation is l with 0 < l < a0 and the quantum tunneling distance lt is of the order of a few Å. The conductance of the ith junction is then written as
Σit ≈ a0 σm exp(-λ0xi) where λ0 ) a0/lt and the dimensionless distances xi ) lt /a0 are distributed in the interval 0 < xi < 1. The contact conductances are spread out over an exponentially large range and only the lowest of these makes a significant contribution to the tunneling current. If one now considers all the adjacent cluster pairs (i.e., all jk), it is easy to show10,11 that the distribution of the intercluster conductances can be written as (11) Ambegaokar, V.; Halperin, B. I.; Langer, J. S. Phys. Rev. B 1971, 4, 2612.
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Σjk t ≈ a0 σm exp(-λ(∆p)y)
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