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The kinetics of thin film island growth at initial stages
Vacuum 53 (1999) 377 — 380
The kinetics of thin film island growth at initial stages E. Urbonavic\ ius *, E. Pe tnyc\ yte , A. Galdikas Vytautas Magnus University, S. Daukanto 28, 3000 Kaunas, Lithuania Kaunas University of Technology, Studentu 50, 3028 Kaunas, Lithuania * Received 13 January 1998; accepted 14 October 1998
Abstract The simulation of the early stages of thin film formation is performed using a phenomenological approach considering the surface diffusion of the adatoms and the migration of islands. The model describes the kinetics of low surface coverage as well as of higher surface coverage. The effects of atom evaporation and island dissociation are not included in this model. The variation of the phenomenological coefficients gives the possibility to obtain the time dependence of the main characteristics such as surface coverage, island density and average island size describing initial stages of the film growth. The surface roughening during deposition can also be considered by the proposed model. The calculated curves are in good qualitative and quantitative agreement with the experimental results. 1999 Elsevier Science Ltd. All rights reserved.
Introduction The process of nucleation and kinetics of thin metallic film growth have been studied for many years and many of the features are well understood [1]. It is known that thin film growth can occur in three modes [2]: (1) the island or Volmer—Weber mode; (2) the layer or Frank— van der Merwe mode; (3) layer plus island or Stranski— Krastanov growth mode which is an intermediate case. The growth of islands in vapor deposition experiments is governed by two distinct mechanisms. The first process is direct adsorption from the vapor and the second is coalescence [3]. Initially, surface coverage is low and the growth process is dominant by deposition of the atoms on the substrate and their diffusion until they form aggregates with existing islands. The island dissociation and the evaporation of the single atoms from the surface takes place at this stage. If evaporation from the surface is negligible, all atoms deposited diffuse and combine with existing islands. This is called an aggregation regime [4]. When the surface coverage becomes sufficiently large, the separation of islands decreases and upon contact they coalesce to form larger islands [3, 5]. However, small islands up to 50—100 As can migrate on the substrate. Island mobility results from the gradual translation of an island center of mass as a result of atomic
* Corresponding author. E-mail: [email protected]
motions in the island and is expected to be characterized by activation energies on the order of those for single atom surface diffusion [6]. Small islands may be charged that may have influence on the mobility of islands [7]. The motion of islands on the substrate results in the so-called mobility coalescence [8] which may have a significant effect on the growth of thin films. For most metals on insulating surfaces, as material is continuously deposited, the islands not fully coalesce, but at some critical island size they form elongated structures [4]. As the growth proceeds, these elongated structures grow longer and interconnect to form a percolating structure and, finally, the channels between the structures fill in to form a continuous hole-free film. The morphology of a thin film is strongly dependent on the nature of the substrate [9]. In this work the kinetics of the thin island film growth is considered. The time dependencies of the main characteristic quantities such as surface coverage u(t), island density N(t) and average island size (area) 1S(t)2 are calculated using proposed phenomenological model. The model describes continuous film growth and is valid for low surface coverage as well as for higher surface coverage u)1. The processes of atom evaporation and dissociation of islands are not included. Phenomenological coefficients in the proposed model describe the different film growth mechanisms discussed above. The variation of these coefficients allows to calculate the time dependencies of the main characteristic quantities and determine
0042-207X/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 2 - 2 0 7 X ( 9 8 ) 0 0 3 4 4 - 3
378
E. Urbonavic\ ius et al. / Vacuum 53 (1999) 377—380
the qualitative differences in the different film growth mechanisms.
2. Phenomenological model The fraction of the atoms arriving at the surface and fall on the bare substrate is proportional to the uncovered area and is equal to i (1!u), where u" S /S is the coverage of the first monolayer (0)u )1), S and S are the covered and total areas of surface, respectively, and i "I /C is the relative flux of arriving atoms (s\), where I is the flux of atoms arriving to the surface (cm\ s\) and C is the surface atomic concentration (cm\). The sticking coefficient a defines the prob ability for an atom to stick and stay on the bare substrate in the adsorbed state. Then the time variation of the coverage of the first monolayer can be expressed as du "i a (1!u). dt
(1)
In order to consider layer-plus-island morphology of the growing film, the surface coverage by single atoms u and islands u (of two or more atoms) are intro1 ! duced (u#u"u). The surface coverage by is! 1 lands increases when (1) the arriving atom sticks at the already existing single adatom with probability (or sticking coefficient) a and/or (2) it sticks at the edge of already existing island with probability a . In the latter ! case, the atom must arrive at the edge of the island, i.e. in an area ¸a around the island, where ¸ is the perimeter of the island and a the distance which is approximately equal to lattice parameter. Considering the circle-like islands, the radius of one island is equal to r"(S /n, ! where S is the area of one island. Island growth is ! governed by the diffusion radius jI within which the adatom can move. The island could capture an adatom if it falls at a distance from the edge of the island less than jI . The area around the N islands with the distance jI from the edge is expressed as S*"N(n(r#jI )!nr)"NnjI #2njI (NS /n. (2) After dividing this equation by the total surface area S"N na, the dimensionless area u*"S* /S is ob tained. N is the number of the atoms on the surface of the bare substrate, a is approximately equal to the lattice parameter. Eq. (2) becomes 1 u*"(NnjI #2njI (NS /n) N na "nj#2j(nu , (3) where n"N/N is the relative island density and j"jI /a is the diffusion radius in units of a. The same characteristic for area around the singles may be obtained in
a similar way and is expressed as u*"n j#2j(n u , (4) 1 1 1 1 where n is the relative density of singles. The atoms arriving at u and u* nucleate with prob ability a . The nucleation term may be expressed as nucl"a i (u #u* ). The increase in the number of 1 1 islands per unit time is dN /dt"nucl/2u , where u is the area of one atom (dimensionless) multiplied by a factor of 2 because two stuck atoms become an island. From u "S /S and N "S/S where S is the area of one atom, it follows that N "1/u . Then dN /dt"(nucl/2) N and (dn/dt) "0.5a i (u #u* ). L 1 1 In the last equation the coalescence effect is not included. The term of coalescence may be obtained from the assumption that coalescence occurs when u #u ! @ *1, where u is the dimensionless area around the island @ within the range of the migration radius of the island b. u may be obtained in the same way as u* or u* and is @ expressed as u "nb#2b(nu . (5) @ The coalescence term is (dn/dt) "a C (1!u !u ), ! @ where C is the normalization factor which may take two values:
u #u (1, @ (6) u #u *1. @ The coefficient a is the frequency probability of coales cence (s\). As (u #u )P1 the areas u* and u* decrease. As suming that u* and u* decrease proportionally, we need to find some coefficient to normalize u "u # * 1 a(t)u*#u #a(t)u* , which must fulfill the condition 1 ! ! u "1, and after simple arithmetic a(t) is given by * 0, C " 1,
1,
u (1, *
a(t)" 1!u (t)!u (t) , u *1. * u* (t)#u* (t)
(7)
Thus, the final equation for the kinetics of island density is dn 1 " a i (u #a(t)u*)#a C (1!u !u ), 1 ! @ dt 2 1
(8)
and the final equation for the kinetics of coverage by singles is the following: du "C a i (1!u )!a i (u #a(t)u* ). * dt
(9)
Considering the growth of higher monolayers K'1, the migration of adatoms on these monolayers must be taken into account. Assuming that j) is the diffusion 2
E. Urbonavic\ ius et al. / Vacuum 53 (1999) 377—380
radius of the atom on the Kth monolayer (the same for all monolayers K*2) the parameter u) is introduced havH ing a meaning close to u* and may be obtained in the same way: u)"2j)\ (nu)\#n (j)\). 2 H 2
379
3. Results and discussions The calculated results obtained after numerical integration of Eq. (14) are presented in Fig. 1a—c and are
(10)
There may be two different cases: (1) if u))u)\! H u) then the atoms stick on the (K!1)th monolayer and form the Kth monolayer; and (2) if u)'u)\!u) H then atoms move to (K!2)th monolayer and form (K!1)th monolayer. Thus, the final equation for the coverage of Kth monolayer is du) "!A)a i (u)\!u)\) (u)\!u)) 2 dt #B)a i (u)!u)>) (u)\!u)), 2
(11)
where for K"2, u)\"1. The coefficients A) and B) are introduced for normalizing of u) to 1 and may be obtained by
u)\!u)(u) H , K"2, 3, . . . , N 1, u)\!u)*u) H 0, u)!u)>*u) H , K"1, 2, . . . , N. B)" 1, u)!u)>(u) H (12) A)"
0,
Then the coverage by islands of the first monolayer K"1 is expressed as du !"2a i (u #a(t)u * )#a(t)a i u * 1 1 ! ! dt #Ba i (1!u) (u!u). !
(13)
The final set of equations consists of Eqs. (8), (9), (11) and (13). du 1"C a i (1!u )!a i (u #a(t)u *), * 1 1 dt du !"2 a i (u #a(t)u *)#a(t)a i u * 1 1 ! ! dt #Ba i (1!u)(u!u), ! du) "!A)a i (u)\!u)\)(u)\!u)) 2 dt #B)a
i (u)!u)>)(u)\!u)), 2
K"2, 3, 2 , N, dn 1 " a i (u #a(t)u *)#a C (1!u !u ). (14) 1 ! @ dt 2 1
Fig. 1. (a) The calculated kinetics (line) of island density. (b) The calculated kinetics (lines) of first monolayer coverage. (c) The calculated dependence of coverage of monolayers on monolayer number for different moments of deposition time. Experimental results (squares) are for sputter deposition of gold on amorphous carbon. Calculation parameters are the same for the all presented figures: I "10\, b"33, j"35, j)"10, a "0.4, a "0.5, a "0.55, a "0.8, a "1. ! 2 2
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E. Urbonavic\ ius et al. / Vacuum 53 (1999) 377—380
compared with the experimental results for sputter deposition of gold on the amorphous carbon substrate with the deposition rate of 1.2;10 cm\ s\ at room temperature and at 5;10\ Pa pressure [10]. The experimental time dependence of island density (Fig. 1a) shows that the number of islands decreases even at low values of surface coverage (Fig. 1b) and differs from both: power law t and logarithmic law ln t usually observed for three and two dimensional islands, respectively [4], during the first stage (aggregation) growth. This dependence is in agreement with the second (coalescence) stage in which the number of islands has a negative exponent larger than one [3]. It means that the process of island coalescence cannot be ignored even at the initial stages of deposition. The mechanism of coalescence which can be expected at such a low value of surface coverage is the migration of islands on the surface, prior to the situation of immobile islands in which the coalescence occurs at high values of surface coverage when the islands meet each other [11]. The process of island migration in the presented model is described by the parameter b, which is defined as the radius within which the center of island mass can move. At higher values of b coalescence occurs at lower values of surface coverage. The island density strongly depends on other parameters as well, first of all, on the parameter j which represents the radius within which the adatom can move. With the increase in parameter j, the number of islands decreases and the size of islands increases. The set of parameters used for the calculations in which the calculated curves are in best quantitative agreement with the experimentally measured time dependencies of island density (Fig. 1a) and surface coverage (Fig. 1b) is the following: b"33, j"35, j)"10, a "0.4, a "0.5, 2 a "0.55, a "0.8. ! 2 The parameter b was taken as a constant for all deposition time. In reality, it depends on time as the size of islands increases with time and the mobility of islands decreases. At some critical value of island size, the islands become immobile [12], i.e. b must be taken as zero (b"0). In the presented calculations b"const., because final coverage is not high (u)0.4) and the decrease of b is not sufficient. The surface coverage does not depend on the parameters b and j, but it strongly depends on the parameters a . From this dependence and ratios of parameters G H a the information about the possible mechanism of film G H growth can be obtained, i.e. if a /a (1, the growth by ! islands takes place; and if a /a '1 the islands are 2 ! three-dimensional [13]. From the set of parameters used for the presented calculations it can be seen that in this case the film growth corresponds with the three-
dimensional island growth mechanism. This conclusion is in agreement with the many experimental observations for gold deposition on amorphous carbon [4, 10, 14]. In Fig. 1c the coverage of different monolayers at t"80 s is presented. As the total deposition time is not high and coverage of first monolayers u(0.5 the three-dimensional island growth is not sufficient. It is possible that at the initial stages two-dimensional island growth takes place. When the islands reach some critical size the transition from the two-dimensional to the three-dimensional islands may occur [12].
4. Conclusions (1) The proposed phenomenological model includes the aggregation and coalescence regimes of sputter deposition of thin film growth, and, depending on the ratios of sticking coefficients, it may be applied for three different film growth mechanisms: epitaxial, two- and three-dimensional island growth. (2) The calculated results are in good qualitative and quantitative agreement with the experimental ones for the sputter deposition of gold on the amorphous carbon substrate and show for this system the threedimensional island growth mechanism. (3) The coalescence cannot be ignored at the initial stages of deposition and occurs as a result of migration of islands on the surface.
References [1] Praneviv\ ius L. Coating technology: ion beam deposition. Werwick, RI: Satas and Associates, 1993:453. [2] Venables JA, Spiller GDT, Hanbucken M. Rep Prog Phys 1984;47:399. [3] Family F, Meakin P. Phys Rev A 1989;40(7):3836. [4] Blackman JA, Wilding A. Europhys Lett 1991;16(1):115. [5] Bartelt MC, Evans JW. Surf Sci 1993;298:421. [6] Geguzin YaE, Kaganovskij YuS. Diffusion processes on surfaces of crystals. Moscow, 1984:123 (in Russian). [7] Levine JR, Cohen JB, Chung YW. Surf Sci 1991;248:215. [8] Stoyanov S, Kashchiev D. In: Kaldis E, editor. Current topics in materials science, vol. 7. Amsterdam: North-Holland, 1981:69. [9] Biham O, Barkema GT, Breaman M. Surf Sci 1995;324:47. [10] Muzard S, Templier C, Delafond J, Girard JC, Thiaudiere D, Pranevic\ ius L, Galdikas A. Poster in E-MRS’97, Strasbourg, France, 16—20 June 1997. Surface Coating Technol, submitted. [11] Bartelt MC, Evans JW. Surf Sci 1994;314:L229. [12] Rakoc\ evic Z, S[ trbac S, Bibic N, Nenadovic T. Surf Sci 1995;343: 247. [13] Galdikas A, Pranevic\ ius A, Templier C, Delafond J, Jaouen M, Muzard S. Medz\ iagotyra 1996;2(1):11. [14] Xu S, Evans BL, Flynn DI, En C. Thin Solid Films 1994;238:54.
The kinetics of thin film island growth at initial stages E. Urbonavic\ ius *, E. Pe tnyc\ yte , A. Galdikas Vytautas Magnus University, S. Daukanto 28, 3000 Kaunas, Lithuania Kaunas University of Technology, Studentu 50, 3028 Kaunas, Lithuania * Received 13 January 1998; accepted 14 October 1998
Abstract The simulation of the early stages of thin film formation is performed using a phenomenological approach considering the surface diffusion of the adatoms and the migration of islands. The model describes the kinetics of low surface coverage as well as of higher surface coverage. The effects of atom evaporation and island dissociation are not included in this model. The variation of the phenomenological coefficients gives the possibility to obtain the time dependence of the main characteristics such as surface coverage, island density and average island size describing initial stages of the film growth. The surface roughening during deposition can also be considered by the proposed model. The calculated curves are in good qualitative and quantitative agreement with the experimental results. 1999 Elsevier Science Ltd. All rights reserved.
Introduction The process of nucleation and kinetics of thin metallic film growth have been studied for many years and many of the features are well understood [1]. It is known that thin film growth can occur in three modes [2]: (1) the island or Volmer—Weber mode; (2) the layer or Frank— van der Merwe mode; (3) layer plus island or Stranski— Krastanov growth mode which is an intermediate case. The growth of islands in vapor deposition experiments is governed by two distinct mechanisms. The first process is direct adsorption from the vapor and the second is coalescence [3]. Initially, surface coverage is low and the growth process is dominant by deposition of the atoms on the substrate and their diffusion until they form aggregates with existing islands. The island dissociation and the evaporation of the single atoms from the surface takes place at this stage. If evaporation from the surface is negligible, all atoms deposited diffuse and combine with existing islands. This is called an aggregation regime [4]. When the surface coverage becomes sufficiently large, the separation of islands decreases and upon contact they coalesce to form larger islands [3, 5]. However, small islands up to 50—100 As can migrate on the substrate. Island mobility results from the gradual translation of an island center of mass as a result of atomic
* Corresponding author. E-mail: [email protected]
motions in the island and is expected to be characterized by activation energies on the order of those for single atom surface diffusion [6]. Small islands may be charged that may have influence on the mobility of islands [7]. The motion of islands on the substrate results in the so-called mobility coalescence [8] which may have a significant effect on the growth of thin films. For most metals on insulating surfaces, as material is continuously deposited, the islands not fully coalesce, but at some critical island size they form elongated structures [4]. As the growth proceeds, these elongated structures grow longer and interconnect to form a percolating structure and, finally, the channels between the structures fill in to form a continuous hole-free film. The morphology of a thin film is strongly dependent on the nature of the substrate [9]. In this work the kinetics of the thin island film growth is considered. The time dependencies of the main characteristic quantities such as surface coverage u(t), island density N(t) and average island size (area) 1S(t)2 are calculated using proposed phenomenological model. The model describes continuous film growth and is valid for low surface coverage as well as for higher surface coverage u)1. The processes of atom evaporation and dissociation of islands are not included. Phenomenological coefficients in the proposed model describe the different film growth mechanisms discussed above. The variation of these coefficients allows to calculate the time dependencies of the main characteristic quantities and determine
0042-207X/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 2 - 2 0 7 X ( 9 8 ) 0 0 3 4 4 - 3
378
E. Urbonavic\ ius et al. / Vacuum 53 (1999) 377—380
the qualitative differences in the different film growth mechanisms.
2. Phenomenological model The fraction of the atoms arriving at the surface and fall on the bare substrate is proportional to the uncovered area and is equal to i (1!u), where u" S /S is the coverage of the first monolayer (0)u )1), S and S are the covered and total areas of surface, respectively, and i "I /C is the relative flux of arriving atoms (s\), where I is the flux of atoms arriving to the surface (cm\ s\) and C is the surface atomic concentration (cm\). The sticking coefficient a defines the prob ability for an atom to stick and stay on the bare substrate in the adsorbed state. Then the time variation of the coverage of the first monolayer can be expressed as du "i a (1!u). dt
(1)
In order to consider layer-plus-island morphology of the growing film, the surface coverage by single atoms u and islands u (of two or more atoms) are intro1 ! duced (u#u"u). The surface coverage by is! 1 lands increases when (1) the arriving atom sticks at the already existing single adatom with probability (or sticking coefficient) a and/or (2) it sticks at the edge of already existing island with probability a . In the latter ! case, the atom must arrive at the edge of the island, i.e. in an area ¸a around the island, where ¸ is the perimeter of the island and a the distance which is approximately equal to lattice parameter. Considering the circle-like islands, the radius of one island is equal to r"(S /n, ! where S is the area of one island. Island growth is ! governed by the diffusion radius jI within which the adatom can move. The island could capture an adatom if it falls at a distance from the edge of the island less than jI . The area around the N islands with the distance jI from the edge is expressed as S*"N(n(r#jI )!nr)"NnjI #2njI (NS /n. (2) After dividing this equation by the total surface area S"N na, the dimensionless area u*"S* /S is ob tained. N is the number of the atoms on the surface of the bare substrate, a is approximately equal to the lattice parameter. Eq. (2) becomes 1 u*"(NnjI #2njI (NS /n) N na "nj#2j(nu , (3) where n"N/N is the relative island density and j"jI /a is the diffusion radius in units of a. The same characteristic for area around the singles may be obtained in
a similar way and is expressed as u*"n j#2j(n u , (4) 1 1 1 1 where n is the relative density of singles. The atoms arriving at u and u* nucleate with prob ability a . The nucleation term may be expressed as nucl"a i (u #u* ). The increase in the number of 1 1 islands per unit time is dN /dt"nucl/2u , where u is the area of one atom (dimensionless) multiplied by a factor of 2 because two stuck atoms become an island. From u "S /S and N "S/S where S is the area of one atom, it follows that N "1/u . Then dN /dt"(nucl/2) N and (dn/dt) "0.5a i (u #u* ). L 1 1 In the last equation the coalescence effect is not included. The term of coalescence may be obtained from the assumption that coalescence occurs when u #u ! @ *1, where u is the dimensionless area around the island @ within the range of the migration radius of the island b. u may be obtained in the same way as u* or u* and is @ expressed as u "nb#2b(nu . (5) @ The coalescence term is (dn/dt) "a C (1!u !u ), ! @ where C is the normalization factor which may take two values:
u #u (1, @ (6) u #u *1. @ The coefficient a is the frequency probability of coales cence (s\). As (u #u )P1 the areas u* and u* decrease. As suming that u* and u* decrease proportionally, we need to find some coefficient to normalize u "u # * 1 a(t)u*#u #a(t)u* , which must fulfill the condition 1 ! ! u "1, and after simple arithmetic a(t) is given by * 0, C " 1,
1,
u (1, *
a(t)" 1!u (t)!u (t) , u *1. * u* (t)#u* (t)
(7)
Thus, the final equation for the kinetics of island density is dn 1 " a i (u #a(t)u*)#a C (1!u !u ), 1 ! @ dt 2 1
(8)
and the final equation for the kinetics of coverage by singles is the following: du "C a i (1!u )!a i (u #a(t)u* ). * dt
(9)
Considering the growth of higher monolayers K'1, the migration of adatoms on these monolayers must be taken into account. Assuming that j) is the diffusion 2
E. Urbonavic\ ius et al. / Vacuum 53 (1999) 377—380
radius of the atom on the Kth monolayer (the same for all monolayers K*2) the parameter u) is introduced havH ing a meaning close to u* and may be obtained in the same way: u)"2j)\ (nu)\#n (j)\). 2 H 2
379
3. Results and discussions The calculated results obtained after numerical integration of Eq. (14) are presented in Fig. 1a—c and are
(10)
There may be two different cases: (1) if u))u)\! H u) then the atoms stick on the (K!1)th monolayer and form the Kth monolayer; and (2) if u)'u)\!u) H then atoms move to (K!2)th monolayer and form (K!1)th monolayer. Thus, the final equation for the coverage of Kth monolayer is du) "!A)a i (u)\!u)\) (u)\!u)) 2 dt #B)a i (u)!u)>) (u)\!u)), 2
(11)
where for K"2, u)\"1. The coefficients A) and B) are introduced for normalizing of u) to 1 and may be obtained by
u)\!u)(u) H , K"2, 3, . . . , N 1, u)\!u)*u) H 0, u)!u)>*u) H , K"1, 2, . . . , N. B)" 1, u)!u)>(u) H (12) A)"
0,
Then the coverage by islands of the first monolayer K"1 is expressed as du !"2a i (u #a(t)u * )#a(t)a i u * 1 1 ! ! dt #Ba i (1!u) (u!u). !
(13)
The final set of equations consists of Eqs. (8), (9), (11) and (13). du 1"C a i (1!u )!a i (u #a(t)u *), * 1 1 dt du !"2 a i (u #a(t)u *)#a(t)a i u * 1 1 ! ! dt #Ba i (1!u)(u!u), ! du) "!A)a i (u)\!u)\)(u)\!u)) 2 dt #B)a
i (u)!u)>)(u)\!u)), 2
K"2, 3, 2 , N, dn 1 " a i (u #a(t)u *)#a C (1!u !u ). (14) 1 ! @ dt 2 1
Fig. 1. (a) The calculated kinetics (line) of island density. (b) The calculated kinetics (lines) of first monolayer coverage. (c) The calculated dependence of coverage of monolayers on monolayer number for different moments of deposition time. Experimental results (squares) are for sputter deposition of gold on amorphous carbon. Calculation parameters are the same for the all presented figures: I "10\, b"33, j"35, j)"10, a "0.4, a "0.5, a "0.55, a "0.8, a "1. ! 2 2
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E. Urbonavic\ ius et al. / Vacuum 53 (1999) 377—380
compared with the experimental results for sputter deposition of gold on the amorphous carbon substrate with the deposition rate of 1.2;10 cm\ s\ at room temperature and at 5;10\ Pa pressure [10]. The experimental time dependence of island density (Fig. 1a) shows that the number of islands decreases even at low values of surface coverage (Fig. 1b) and differs from both: power law t and logarithmic law ln t usually observed for three and two dimensional islands, respectively [4], during the first stage (aggregation) growth. This dependence is in agreement with the second (coalescence) stage in which the number of islands has a negative exponent larger than one [3]. It means that the process of island coalescence cannot be ignored even at the initial stages of deposition. The mechanism of coalescence which can be expected at such a low value of surface coverage is the migration of islands on the surface, prior to the situation of immobile islands in which the coalescence occurs at high values of surface coverage when the islands meet each other [11]. The process of island migration in the presented model is described by the parameter b, which is defined as the radius within which the center of island mass can move. At higher values of b coalescence occurs at lower values of surface coverage. The island density strongly depends on other parameters as well, first of all, on the parameter j which represents the radius within which the adatom can move. With the increase in parameter j, the number of islands decreases and the size of islands increases. The set of parameters used for the calculations in which the calculated curves are in best quantitative agreement with the experimentally measured time dependencies of island density (Fig. 1a) and surface coverage (Fig. 1b) is the following: b"33, j"35, j)"10, a "0.4, a "0.5, 2 a "0.55, a "0.8. ! 2 The parameter b was taken as a constant for all deposition time. In reality, it depends on time as the size of islands increases with time and the mobility of islands decreases. At some critical value of island size, the islands become immobile [12], i.e. b must be taken as zero (b"0). In the presented calculations b"const., because final coverage is not high (u)0.4) and the decrease of b is not sufficient. The surface coverage does not depend on the parameters b and j, but it strongly depends on the parameters a . From this dependence and ratios of parameters G H a the information about the possible mechanism of film G H growth can be obtained, i.e. if a /a (1, the growth by ! islands takes place; and if a /a '1 the islands are 2 ! three-dimensional [13]. From the set of parameters used for the presented calculations it can be seen that in this case the film growth corresponds with the three-
dimensional island growth mechanism. This conclusion is in agreement with the many experimental observations for gold deposition on amorphous carbon [4, 10, 14]. In Fig. 1c the coverage of different monolayers at t"80 s is presented. As the total deposition time is not high and coverage of first monolayers u(0.5 the three-dimensional island growth is not sufficient. It is possible that at the initial stages two-dimensional island growth takes place. When the islands reach some critical size the transition from the two-dimensional to the three-dimensional islands may occur [12].
4. Conclusions (1) The proposed phenomenological model includes the aggregation and coalescence regimes of sputter deposition of thin film growth, and, depending on the ratios of sticking coefficients, it may be applied for three different film growth mechanisms: epitaxial, two- and three-dimensional island growth. (2) The calculated results are in good qualitative and quantitative agreement with the experimental ones for the sputter deposition of gold on the amorphous carbon substrate and show for this system the threedimensional island growth mechanism. (3) The coalescence cannot be ignored at the initial stages of deposition and occurs as a result of migration of islands on the surface.
References [1] Praneviv\ ius L. Coating technology: ion beam deposition. Werwick, RI: Satas and Associates, 1993:453. [2] Venables JA, Spiller GDT, Hanbucken M. Rep Prog Phys 1984;47:399. [3] Family F, Meakin P. Phys Rev A 1989;40(7):3836. [4] Blackman JA, Wilding A. Europhys Lett 1991;16(1):115. [5] Bartelt MC, Evans JW. Surf Sci 1993;298:421. [6] Geguzin YaE, Kaganovskij YuS. Diffusion processes on surfaces of crystals. Moscow, 1984:123 (in Russian). [7] Levine JR, Cohen JB, Chung YW. Surf Sci 1991;248:215. [8] Stoyanov S, Kashchiev D. In: Kaldis E, editor. Current topics in materials science, vol. 7. Amsterdam: North-Holland, 1981:69. [9] Biham O, Barkema GT, Breaman M. Surf Sci 1995;324:47. [10] Muzard S, Templier C, Delafond J, Girard JC, Thiaudiere D, Pranevic\ ius L, Galdikas A. Poster in E-MRS’97, Strasbourg, France, 16—20 June 1997. Surface Coating Technol, submitted. [11] Bartelt MC, Evans JW. Surf Sci 1994;314:L229. [12] Rakoc\ evic Z, S[ trbac S, Bibic N, Nenadovic T. Surf Sci 1995;343: 247. [13] Galdikas A, Pranevic\ ius A, Templier C, Delafond J, Jaouen M, Muzard S. Medz\ iagotyra 1996;2(1):11. [14] Xu S, Evans BL, Flynn DI, En C. Thin Solid Films 1994;238:54.