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Letter Voids in thin metallic films RICHARD ZITO Department oJ Physics, University oJ A rizona, Tucson, Ari=. 85721 (U.S.A.) (Received June 23, 1978 : accepted July 27, 1978)

In this letter the size and the number of large voids in thin metal films are predicted. This number is important in determining the structural strength of metal coatings. The predicted numbers are compared with observations on gold films. Voids appear to be an intrinsic characteristic of thin films due to statistical fluctuations on deposition. During the deposition of thin reflective metallic films many large voids {called pores or holes in some literature) are formed. These voids place limits on the structural strength of reflective coatings at high temperatures 1. They also have a large effect on the electrical resistance of metal films 2. A few measurements on voids in gold films have been carried out by Nakahara et al. 2' 3 In this paper we attempt to predict the number of voids from first principles. The starting point for the calculation presented here is inhomogeneous nucleation theory. During rapid deposition, material may be non-uniformly deposited on the substrate and may result in small voids being left behind. If these holes have a certain critical radius r*, they will be "stable" in the sense that the surface tension of the metal will not cause them to "evaporate" into monovacancies. Indeed, monovacancies (whose number density is4 about 101° cm-3) may condense onto these small voids and may allow them to grow to their observed size. The situation is not unlike that encountered in the study of raindrops. Nucleation theory predicts that the smallest possible critical radius is approximately given by 5' 6 P~rnin ~

2o-v E

where a is the surface tension of the condensed phase, v is the average volume per atom or molecule in the condensed phase and E is the heat of vaporization per particle. For the present calculation, a for crystalline gold is approximated by a for the liquid metal ( ~ 1 N m-1). An atomic diameter typical .of the reflective metals (gold, silver, aluminum, copper, chromium, platinum, rhodium, molybdenum, tungsten) is 3 ,~. Hence the volume per vacancy in the condensed (void) phase is approximately (3 A) 3. The quantity E is taken as the formation energy of a monovacancy (i.e. E will be released when a monovacancy is removed by becoming part" of a void). The formation energy is about 1 eV for gold v. Hence, r*ml" ,,~ 3.4 x 10 -a° m, which is of the same order of magnitude as but smaller than the large 20 A voids 2. The volume V*ml. corresponding to r*mi. is approximately 165 × 10- 3o m 3. This volume may hold about 12 atomic volumes, but packing is not

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completely efficient. Hence after multiplying by the packing factor t0,74 for gold) we find that nine atoms must be absent in order to create V*,,~n. A void where only nine atoms are missing is probably on the limit of acceptability for our continuum mechanics treatment. Next we assume that an e m p D 1*,,,~., is created by the inhomogeneous placement of material during deposition due to statistical fluctuations. The substrate is considered "cold" in the sense that impinging atoms ~'stick'" to their point of impact. This approximation is acceptable since in this letter we are primarily interested in describing voids in gold films deposited at r o o m temperature. This temperature is below the 400 C needed to give gold atoms a high mobility ~. It should also be noted that, via h o m o g e n e o u s nucleation theory, the probability of producing a void of critical size without deposition fluctuations from an initially uniform distribution of monovacancies is extremely small ~. Let us consider a metal film and let us carve it up into cubic boxes of size l'*m~,, each holding nine atoms. The probability that one of these boxes will be empty after a r a n d o m filling process is given by the Poisson distribution ~ P=

(,,)"e N!

w h e r e ( n ) is the average n u m b e r of atoms normally present in one of our cubic boxes

(i.e. ( n ) = 9) and N is the actual n u m b e r we are interested in li.e. N = 01. Hence P = 1.2 x 10 4. The n u m b e r of voids iper cubic centimeter) of volume 1/*m~. is then given by the product of P with the n u m b e r of cubic boxes per cubic centimeter. Hence we have 7.3 x 10 ~; voids cm 3 of size V*m~.. The measured n u m b e r 3 of voids in electrodeposited gold films is 3 × I01 ~ voids cm ~ 3. The probability of larger voids being formed on deposition falls off rapidly and therefore these larger voids have been ignored in our rough calculation. However, approximately 102 monovacancies may condense on each void of size V*m~n. Hence our voids may grow to more typical sizes, as reported by Lloyd and N a k a h a r a 2. Thus both the size and the n u m b e r of voids in gold films are predicted. The calculation is admittedly rough. The nature and temperature of the substrate as well as the method of deposition m a y change 2' 3 the n u m b e r of voids per cubic centimeter by a factor of 106. However, it is now clear that a certain nonnegligible n u m b e r of voids are created on deposition owing to statistical inhomogeneities during the placement of atoms. In this sense, growth of films during v a c u u m deposition is very different from the growth of metal crystars in a furnace. A similar conclusion has been reached experimentally by N a k a h a r a 3. The a u t h o r would like to thank R. Y o u n g and R. Emrick, both of the Department of Physics at the University of Arizona, for m a n y valuable discussions. 1 2 3 4 5

A . E . B . Presland, G. L. Price and D. L. Trimm, Su(L Sci.~ 29 (1972) 435. J.R. Lloyd and S. Nakahara, Thin Solid Films, 45 (1977) 41 I. S. Nakahara, Thin Solid Films, 45 (1977) 421. L . H . Van Vlack, Elements o! Materials Science. Addison-Wesley. Reading, Massachusetts. 1964, p. 90. L. Dufour and R. Defay, Thermadvnamics ~?/ Clouds, Academic Press, New York. 1963, pp. 106 108.

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B.J. Mason, The Physics of Clouds, Oxford University Press, London, 1971, p. 51. 7 C.P. Flynn, J, BassandD. Lazarus, Philos. Mag.,11(1965) 521. 8 E.N.C. Andrade, Trans. Faraday Soc., 31(1935) 1137. 9 R.H. Doremus, Glass Science, Wiley, New York, 1973, p. 47; see also Thermodynamics oJ Clouds, 6

Academic Press, New York, 1963, p. 5. 10 L.D. Landau and E. M. Lifshitz, Statistical Physics, Addison-Wesley, Reading, Massachusetts, 1969, p. 357.