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The simulation of thin film growth
Thin Solid Films, 85 ( 1981 ) 285 292
285
PREPARATION AND CHARACTERIZATION
T H E S I M U L A T I O N OF T H I N F I L M G R O W T H * R. HRACH AND V. STAR¢/"
Department oJ Electronics and Vacuum Physics, Faculty of Mathematics and Physics, Charles University, Povltavska I, 180 Of) Prague 8 ( C:echoslovakia)
The simulation of the intermediate stage of thin film growth is described. The change in the condensation coefficient on the free surface and the randomness of the initial distribution of nuclei (shown by the effects on the capture area) were studied. The time dependences of the coverage and of the integral condensation coefficient, the histograms of the island radii and of the distances to nearest neighbours etc. were obtained. The results derived show that the Monte Carlo method could be successfully used for this type of physical problem.
1. I N T R O D U C T I O N
Computer simulation of thin film growth is often used for the study of various problems in thin film physics. This approach has been used recently to obtain new results 1-4. In a previous paper 5 the intermediate stage of film growth or, more exactly, the growth and coalescence of three-dimensional islands before their coalescence into a semicontinuous film was dealt with. We introduced some very simple assumptions for the computer calculation of the following thin film parameters: the coverage O, the integral condensation coefficient K~, the island density N, the mean value and the distribution of the grain radius r, and the mean value and distribution of the separation ! between nearest neighbours. These assumptions are as follows. (1) The nuclei, i.e. centres of condensation with zero volume, arc distributed randomly over the surface. Their density has a saturated value N.~ at the onset of the process being simulated (i.e. the primary nucleation process is complete). (2) The real condensation coefficients K and K~ of the atoms deposited onto the island and onto the substrate respectively are both equal to unity. (3) The islands are cup shaped and are in thermodynamic equilibrium. The condition of minimum surface energy implies that the ratio of the height h of an island to its radius r is constant. (4) Island coalescence takes place by the physical contact of pairs ofislands in a * Paper presented at the International Summer School on Processes of Thin Film Formation, Fonyod, Lake Balaton, Hungary, September28 to October 4, 1980. 0040-6090/8 I;0000-0000/$02.50
((~)ElsevierSequoia'Printed in The Netherlands
286
R. HRACH, V. STARY
liquid-like manner, and the centre of the new island is situated at the centre of gravity of the pair. (5) Secondary nucleation and the migration of nuclei and islands over the surface are not taken into account. 2.
METHOD OF CALCULATION
The fundamental concept of our modt~l is described in ref. 5. The chosen initial n u m b e r n s of nuclei were distributed on a square of 500 x 500 spots by means of a r a n d o m n u m b e r generator. For n~ = 128 and a spot separation of 0.5 nm, the density N s of the nuclei was 2 x 109 m m - 2 which is in accordance with typical experimental values. The intercept area S was constructed as a set of spots with a shorter separation distance from the periphery of the island (or nucleus) in question than from the periphery of another island (or nucleus). If the condensation coefficients K s and K for the free surface and for the island respectively are equal to unity, all the a t o m s reaching the intercept area of a given island (or nucleus) will join this island. The M o n t e Carlo method was used for the simulation. In the present calculation we make our assumptions less simple and therefore simulate the growth in a more realistic manner. Firstly, we assume that K s varies from 10- 3 to 1 whereas K is constant and equal to unity which is in accordance with experiment 6. Secondly, we assume that a diffusion area exists in which nucleation is reduced to zero 7. Therefore we introduce a further parameter, the minimum separation Xml . of the nuclei, which varies between 1 and 50 units. All the calculations reported here were performed for n s = 128. 3.
RESULTS AND DISCUSSION
The effects of changes in the film parameters due to changes in K s and in Xm~. are studied separately. Therefore we first investigate the effect of K s for Xmi . = 0. The results of calculations of the dependence on time of the n u m b e r n of islands, of the surface coverage O and of the integral condensation coefficient K~ which is defined as K~ = O + ( 1 - - O ) K
s
are shown in Figs. 1-3. The relative units of the time axis are given by the n u m b e r of evaporated (not deposited) atoms. The time variable tre j is related to the mean film thickness a measured in monolayers. The value tre I = l corresponds to l06 atoms evaporated o n t o the observed area. At K s = I and a spot separation of 0.5 nm we obtain N s ~ 1.5 x 10 ~3 atoms m m -2 which corresponds to a = 3 ~ , monolayers. The experimental values of K i obtained by Elliot 8 for the system Au/mica at = 365 : C and an evaporation rate of 8 x 101° atoms m m - 2 s- ~ are very similar to our results for K~ = 0.01. The distribution of the island radii is shown in Fig. 4. Firstly, the distribution is narrower at the lower values of K~ owing to the smaller effect of the magnitude S of the intercept area on the growth velocity of the islands. Secondly, the distribution is skewed towards higher values of the radii. We believe that this behaviour is due to r a n d o m fluctuations in the island radii which increase during the growth process.
287
S I M U L A T I O N OF T H I N FILM G R O W T H
140
t
100
80
60 .
"~ \°
.
40
20
5
El
15
20
trel
25
Fig. 1. N u m b e r n of nuclei t:.s. the time t,~ of e v a p o r a t i o n for v a r i o u s c o n d e n s a t i o n coefficients K~ o n the free surfacc: O, K , = I ; + , K , = 0 . 1 : O , K , = 0 . 0 1 ; x , K , = 0.001.
o
1,10
///
//
1,10"
/
1 .I0-'
1 alO "s
1
0
.5
~
I
|
I
10
15
20
trel
25
Fig. 2. S u r f a c e c o v e r a g e O t,s. time tr,i o f e v a p o r a t i o n : O, K , = 1; + , K s = 0.1; O , K~ = 0.01; x , K , = 0.001.
The islands with large diameters (due to the random growth process) grow more quickly than the others. This effect is more pronounced after a longer period of evaporation as can be seen by a comparison of Figs. 4(a) and 4(b). The effect of changing the minimum separation Xmi. of the nuclei for K~ = 1 is
288
R. HRACH, V. STAR~"
1.10'
;~__.__==,----~.
,,0v//" 1.16'/ 1,10"3
i
i
i
5
I0
20
25
Fig. 3. Integral condensation coefficient K~ t,s. time i',,~ of. evaporalion: x , K , = 1; O , K,=0.1; +, K,=0.01; e, K~ = 0.001.
Ks= I
t
30
lret
1I I
t
o
r':l~
1
,
Ks =0.'f
3O
oA 0
__
,
A
i
Ks=O.01
i K s : 0.001
o
(a)
r
i
10
20
r
30
h
0
(b)
10
~ 20
n r
30
Fig.4. H i s t o g r a m s of the island radii (expressed as percentages) obtained for different values of K, after the evaporation of (a} 0.25× l0 s atoms (tr,~ = 0.25) and (b) 106 atoms (t,a = 1).
289
SIMULATION OF THIN FILM G R O W T H
. • i.'
•
;
'.
i
.
.
•" i i
-.
•
.
.
.j
•
•
.
•
.
':::"
- :
" " "~ "'"
.'"
. . . . .
•
.
•
•
.
i
I
.
.
.j ."
".
.
•
.
•
i
'
(a)
(b)
(c)
Fig. 5. D i s t r i b u t i o n of the nuclei over the surface: (a) X m l n = 1, n, = 128: (b) X,m,, = 30, n~ = 128: (c) Xm+. = 50, n, = 65 (a larger n u m b e r of nuclei were not g e n e r a t e d even after long exposures)•
140
r
100
r
•
40
20
0
4 I
,
+
1
8 I
~
2
Fig. 6. I s l a n d n u m b e r n c.s. t i m e t,c ~o f e v a p o r a t i o n O,
Xmi
n =
30;
x+
gmi
n =
50.
12 I
3
,
16 t
a
4
fret
f o r v a r i o u s v a l u e s o f Xm+ . : O • Xmm = 1 : + , Xml . = 20:
aVis the a p p r o x i m a t e average thickness of the film in monolayers.
shown in Fig. 5. X,,+, is assumed to be given by 6 Xmi. = (r,D) t'2 where z. is the residence time and D is the diffusion coefficient of the atoms on the substrate surface. The value of Xml, is chosen before starting the simulation• If the value of Xm+, exceeds some limit, e.g. in Fig. 5 if Xmm >. 40, the initial value of ns can be reduced in the same way as at higher substrate temperatures. Furthermore, as can be seen clearly in Fig. 5, the randomness of the distribution of the nuclei on the surface is decreased• The time dependence of the number of nuclei is shown in Fig. 6 for several
290
R. H R A C H ,
V. STAR~(
Xml n = 1 20
0
_
71 Xmrn = 20
20
3_
Xml n = 30 20
_1-
Xmi n = 50 20
0
Jl 10
20
(a)
J
r
i
30
10
r
310
i
20
r
(b)
Fig. 7. H i s t o g r a m s of the d i s t r i b u t i o n of island radii vs. X,,,~. (expressed as percentages) after the e v a p o r a t i o n o f l a ) 0.25 x 106 a t o m s (t,~ I = 0.25, J = I ) and (b) 106 a t o m s (t,~j = 1, d = 4).
values of Xml.. The distributions of the island radii and of the distances to nearest neighbours are shown in Figs. 7 and 8 respectively. 4. CONCI.USIONS Our results show that the Monte Carlo method can be applied successfully to the simulation of the intermediate stage of thin film growth. However, a higher initial number of nuclei and a larger process area than those employed here should be used for statistical evaluation. The disadvantage of this method is the progressive increase in computation time, but this difficulty could probably be overcome by a more economic program. The major advantage of this method is that the two processes taking place during evaporation at elevated substrate temperatures can be separated. These processes can only be studied simultaneously in experimental investigations, and therefore the stepwise simulation of the growth process allows a deeper understand-
291
SIMULATION OF THIN FILM G R O W T H
1
×rain: I ]
Xmm: 20 10
,
Xm, n = 30
10
,
i i . , ~
~,,~,
xml X mrn : 50
P]
10
0
30
(a~
60
t
,! .... 1 0
30
60
I
90
(bl
Fig. 8. Histograms ofthe distance to the nearest neighbour rs. Xm~.lexpressed aspercentages)after the evaporation of(a)0.25 x 10 ~ atoms(t,~j = 0.25, d = I ) a n d (b)10" atoms(t,,1 = I, d = 4j.
ing to be obtained of temperature phenomena. However, the simultaneous simulation of both processes is possible if required. The results of the simulation can be compared with experimental rcsuhs (mainly with in situ transmission electron microscopy observations) and in this way the relations between the real experimental conditions and the model parameters can be found. We are now making improvements in the program which will allow the simulation of secondary nucleation, island migration and the creation of a continuous film. REFERENCES I 2 3 4
F . F . Abraham and G. M. White, J. Appl. Phys., 41 (1970) 1841. R. Vincent, Proc. Phys. Soc.. London, Sect. A, 321 (1971) 53. T . J . Coutts and B. Hopewcll, Thin Solid Films, 9 ( 1971) 37. ,~. Barna, P. B. Barna, G. Radn6czi, H. Sugawara and P. Thomas, Thin Solid Films. 48 (1978) 163.
5
R. HrachandV. Star~',Czech. J. Phys. B, 28(1978) 1382.
292 6 7 8
K.L. Chopra, Thin Film Phenomena, McGraw-Hill, New York, 1969. B. Lewis and D. S. Campbell, J. Vat'. Sci. Technol., 4 (1967) 209. A.G. Elliot, Surf Sci., 44 (1974) 337.
R. HRACH, V. STARY
285
PREPARATION AND CHARACTERIZATION
T H E S I M U L A T I O N OF T H I N F I L M G R O W T H * R. HRACH AND V. STAR¢/"
Department oJ Electronics and Vacuum Physics, Faculty of Mathematics and Physics, Charles University, Povltavska I, 180 Of) Prague 8 ( C:echoslovakia)
The simulation of the intermediate stage of thin film growth is described. The change in the condensation coefficient on the free surface and the randomness of the initial distribution of nuclei (shown by the effects on the capture area) were studied. The time dependences of the coverage and of the integral condensation coefficient, the histograms of the island radii and of the distances to nearest neighbours etc. were obtained. The results derived show that the Monte Carlo method could be successfully used for this type of physical problem.
1. I N T R O D U C T I O N
Computer simulation of thin film growth is often used for the study of various problems in thin film physics. This approach has been used recently to obtain new results 1-4. In a previous paper 5 the intermediate stage of film growth or, more exactly, the growth and coalescence of three-dimensional islands before their coalescence into a semicontinuous film was dealt with. We introduced some very simple assumptions for the computer calculation of the following thin film parameters: the coverage O, the integral condensation coefficient K~, the island density N, the mean value and the distribution of the grain radius r, and the mean value and distribution of the separation ! between nearest neighbours. These assumptions are as follows. (1) The nuclei, i.e. centres of condensation with zero volume, arc distributed randomly over the surface. Their density has a saturated value N.~ at the onset of the process being simulated (i.e. the primary nucleation process is complete). (2) The real condensation coefficients K and K~ of the atoms deposited onto the island and onto the substrate respectively are both equal to unity. (3) The islands are cup shaped and are in thermodynamic equilibrium. The condition of minimum surface energy implies that the ratio of the height h of an island to its radius r is constant. (4) Island coalescence takes place by the physical contact of pairs ofislands in a * Paper presented at the International Summer School on Processes of Thin Film Formation, Fonyod, Lake Balaton, Hungary, September28 to October 4, 1980. 0040-6090/8 I;0000-0000/$02.50
((~)ElsevierSequoia'Printed in The Netherlands
286
R. HRACH, V. STARY
liquid-like manner, and the centre of the new island is situated at the centre of gravity of the pair. (5) Secondary nucleation and the migration of nuclei and islands over the surface are not taken into account. 2.
METHOD OF CALCULATION
The fundamental concept of our modt~l is described in ref. 5. The chosen initial n u m b e r n s of nuclei were distributed on a square of 500 x 500 spots by means of a r a n d o m n u m b e r generator. For n~ = 128 and a spot separation of 0.5 nm, the density N s of the nuclei was 2 x 109 m m - 2 which is in accordance with typical experimental values. The intercept area S was constructed as a set of spots with a shorter separation distance from the periphery of the island (or nucleus) in question than from the periphery of another island (or nucleus). If the condensation coefficients K s and K for the free surface and for the island respectively are equal to unity, all the a t o m s reaching the intercept area of a given island (or nucleus) will join this island. The M o n t e Carlo method was used for the simulation. In the present calculation we make our assumptions less simple and therefore simulate the growth in a more realistic manner. Firstly, we assume that K s varies from 10- 3 to 1 whereas K is constant and equal to unity which is in accordance with experiment 6. Secondly, we assume that a diffusion area exists in which nucleation is reduced to zero 7. Therefore we introduce a further parameter, the minimum separation Xml . of the nuclei, which varies between 1 and 50 units. All the calculations reported here were performed for n s = 128. 3.
RESULTS AND DISCUSSION
The effects of changes in the film parameters due to changes in K s and in Xm~. are studied separately. Therefore we first investigate the effect of K s for Xmi . = 0. The results of calculations of the dependence on time of the n u m b e r n of islands, of the surface coverage O and of the integral condensation coefficient K~ which is defined as K~ = O + ( 1 - - O ) K
s
are shown in Figs. 1-3. The relative units of the time axis are given by the n u m b e r of evaporated (not deposited) atoms. The time variable tre j is related to the mean film thickness a measured in monolayers. The value tre I = l corresponds to l06 atoms evaporated o n t o the observed area. At K s = I and a spot separation of 0.5 nm we obtain N s ~ 1.5 x 10 ~3 atoms m m -2 which corresponds to a = 3 ~ , monolayers. The experimental values of K i obtained by Elliot 8 for the system Au/mica at = 365 : C and an evaporation rate of 8 x 101° atoms m m - 2 s- ~ are very similar to our results for K~ = 0.01. The distribution of the island radii is shown in Fig. 4. Firstly, the distribution is narrower at the lower values of K~ owing to the smaller effect of the magnitude S of the intercept area on the growth velocity of the islands. Secondly, the distribution is skewed towards higher values of the radii. We believe that this behaviour is due to r a n d o m fluctuations in the island radii which increase during the growth process.
287
S I M U L A T I O N OF T H I N FILM G R O W T H
140
t
100
80
60 .
"~ \°
.
40
20
5
El
15
20
trel
25
Fig. 1. N u m b e r n of nuclei t:.s. the time t,~ of e v a p o r a t i o n for v a r i o u s c o n d e n s a t i o n coefficients K~ o n the free surfacc: O, K , = I ; + , K , = 0 . 1 : O , K , = 0 . 0 1 ; x , K , = 0.001.
o
1,10
///
//
1,10"
/
1 .I0-'
1 alO "s
1
0
.5
~
I
|
I
10
15
20
trel
25
Fig. 2. S u r f a c e c o v e r a g e O t,s. time tr,i o f e v a p o r a t i o n : O, K , = 1; + , K s = 0.1; O , K~ = 0.01; x , K , = 0.001.
The islands with large diameters (due to the random growth process) grow more quickly than the others. This effect is more pronounced after a longer period of evaporation as can be seen by a comparison of Figs. 4(a) and 4(b). The effect of changing the minimum separation Xmi. of the nuclei for K~ = 1 is
288
R. HRACH, V. STAR~"
1.10'
;~__.__==,----~.
,,0v//" 1.16'/ 1,10"3
i
i
i
5
I0
20
25
Fig. 3. Integral condensation coefficient K~ t,s. time i',,~ of. evaporalion: x , K , = 1; O , K,=0.1; +, K,=0.01; e, K~ = 0.001.
Ks= I
t
30
lret
1I I
t
o
r':l~
1
,
Ks =0.'f
3O
oA 0
__
,
A
i
Ks=O.01
i K s : 0.001
o
(a)
r
i
10
20
r
30
h
0
(b)
10
~ 20
n r
30
Fig.4. H i s t o g r a m s of the island radii (expressed as percentages) obtained for different values of K, after the evaporation of (a} 0.25× l0 s atoms (tr,~ = 0.25) and (b) 106 atoms (t,a = 1).
289
SIMULATION OF THIN FILM G R O W T H
. • i.'
•
;
'.
i
.
.
•" i i
-.
•
.
.
.j
•
•
.
•
.
':::"
- :
" " "~ "'"
.'"
. . . . .
•
.
•
•
.
i
I
.
.
.j ."
".
.
•
.
•
i
'
(a)
(b)
(c)
Fig. 5. D i s t r i b u t i o n of the nuclei over the surface: (a) X m l n = 1, n, = 128: (b) X,m,, = 30, n~ = 128: (c) Xm+. = 50, n, = 65 (a larger n u m b e r of nuclei were not g e n e r a t e d even after long exposures)•
140
r
100
r
•
40
20
0
4 I
,
+
1
8 I
~
2
Fig. 6. I s l a n d n u m b e r n c.s. t i m e t,c ~o f e v a p o r a t i o n O,
Xmi
n =
30;
x+
gmi
n =
50.
12 I
3
,
16 t
a
4
fret
f o r v a r i o u s v a l u e s o f Xm+ . : O • Xmm = 1 : + , Xml . = 20:
aVis the a p p r o x i m a t e average thickness of the film in monolayers.
shown in Fig. 5. X,,+, is assumed to be given by 6 Xmi. = (r,D) t'2 where z. is the residence time and D is the diffusion coefficient of the atoms on the substrate surface. The value of Xml, is chosen before starting the simulation• If the value of Xm+, exceeds some limit, e.g. in Fig. 5 if Xmm >. 40, the initial value of ns can be reduced in the same way as at higher substrate temperatures. Furthermore, as can be seen clearly in Fig. 5, the randomness of the distribution of the nuclei on the surface is decreased• The time dependence of the number of nuclei is shown in Fig. 6 for several
290
R. H R A C H ,
V. STAR~(
Xml n = 1 20
0
_
71 Xmrn = 20
20
3_
Xml n = 30 20
_1-
Xmi n = 50 20
0
Jl 10
20
(a)
J
r
i
30
10
r
310
i
20
r
(b)
Fig. 7. H i s t o g r a m s of the d i s t r i b u t i o n of island radii vs. X,,,~. (expressed as percentages) after the e v a p o r a t i o n o f l a ) 0.25 x 106 a t o m s (t,~ I = 0.25, J = I ) and (b) 106 a t o m s (t,~j = 1, d = 4).
values of Xml.. The distributions of the island radii and of the distances to nearest neighbours are shown in Figs. 7 and 8 respectively. 4. CONCI.USIONS Our results show that the Monte Carlo method can be applied successfully to the simulation of the intermediate stage of thin film growth. However, a higher initial number of nuclei and a larger process area than those employed here should be used for statistical evaluation. The disadvantage of this method is the progressive increase in computation time, but this difficulty could probably be overcome by a more economic program. The major advantage of this method is that the two processes taking place during evaporation at elevated substrate temperatures can be separated. These processes can only be studied simultaneously in experimental investigations, and therefore the stepwise simulation of the growth process allows a deeper understand-
291
SIMULATION OF THIN FILM G R O W T H
1
×rain: I ]
Xmm: 20 10
,
Xm, n = 30
10
,
i i . , ~
~,,~,
xml X mrn : 50
P]
10
0
30
(a~
60
t
,! .... 1 0
30
60
I
90
(bl
Fig. 8. Histograms ofthe distance to the nearest neighbour rs. Xm~.lexpressed aspercentages)after the evaporation of(a)0.25 x 10 ~ atoms(t,~j = 0.25, d = I ) a n d (b)10" atoms(t,,1 = I, d = 4j.
ing to be obtained of temperature phenomena. However, the simultaneous simulation of both processes is possible if required. The results of the simulation can be compared with experimental rcsuhs (mainly with in situ transmission electron microscopy observations) and in this way the relations between the real experimental conditions and the model parameters can be found. We are now making improvements in the program which will allow the simulation of secondary nucleation, island migration and the creation of a continuous film. REFERENCES I 2 3 4
F . F . Abraham and G. M. White, J. Appl. Phys., 41 (1970) 1841. R. Vincent, Proc. Phys. Soc.. London, Sect. A, 321 (1971) 53. T . J . Coutts and B. Hopewcll, Thin Solid Films, 9 ( 1971) 37. ,~. Barna, P. B. Barna, G. Radn6czi, H. Sugawara and P. Thomas, Thin Solid Films. 48 (1978) 163.
5
R. HrachandV. Star~',Czech. J. Phys. B, 28(1978) 1382.
292 6 7 8
K.L. Chopra, Thin Film Phenomena, McGraw-Hill, New York, 1969. B. Lewis and D. S. Campbell, J. Vat'. Sci. Technol., 4 (1967) 209. A.G. Elliot, Surf Sci., 44 (1974) 337.
R. HRACH, V. STARY