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Capillary instabilities in thin, continuous films
Thin Solid Films, 208 ( 1992) 23 28
Capillary instabilities in thin, continuous films E. Jiran*
and C. V. Thompson
Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139 (U.S.A.) (Received
June 3. 1991; accepted
August
23, 1991)
Abstract The agglomeration of continuous films was studied in 60 nm thick, polycrystalline, gold films deposited on polished fused silica substrates, Nucleation of holes in any particular sample showed nucleation frequencies characteristic of site saturation. The growth of the holes resulted in roughly circular agglomerated areas. The area fraction transformed was measured as a function of time for several temperatures in oxygen and argon. Using a JohnsonMehl-Avrami analysis and assuming a constant hole growth rate, the time dependence of the agglomeration can be described. The growth rates, which contain important kinetic information, could thus be determined. Growth rates for anneals in argon and oxygen were found to be similar.
1. Introduction Because thin films have a high surface-to-volume ratio, on heating, a thin, solid, continuous film may develop holes and may eventually agglomerate into shapes with a lower surface-to-volume ratio. This process can occur well below the melting point of the material, and can thus interfere with device fabrication when processing involves heating subsequent to the deposition of thin films, or when a reaction between two thin films must occur to produce, for example, a silicide used for interconnects [ 11. In an earlier study of the agglomeration of thin gold films patterned into wide lines [2], we showed that, once a hole forms (as simulated by the spaces between the lines) a complex morphology of void cells develops (Fig. 1(a)). This void morphology propagates by means of the growth of void cells at an average rate which is independent of time and varies as the inverse of the cube of the film thickness. The result of this growth is an agglomerated area consisting of islands of material and exposed substrate. In contrast with patterned films, agglomeration of continuous thin films involves the nucleation of voids as well as their growth. Preliminary experiments on continuous gold films showed that, although the growth of the void is complex, the agglomerated areas formed are roughly circular and nucleate at approximately the same time (Fig. l(b)). In this paper we report a study of the kinetics of agglomeration of initially continuous gold films on fused silica substrates. By studying agglomeration of
*Present address: Institut National de la Recherche Energie, Varennes, Quebec J3X lS2, Canada.
0040-6090/92/$5.00
Scientifique
-
continuous films after studying the agglomeration of patterned films, we have been able to isolate the nucleation component of the agglomeration process, and to obtain kinetic characterizations of the nucleation rate from measurements of the overall agglomeration rate.
2. Experimental
details
In order to determine the time dependence of the process of agglomeration in continuous films, agglomerated area vs. time curves were obtained, in situ, at different temperatures for films annealed in either oxygen or argon. Fused silica substrates were cleaned using a schedule designed for the cleaning of oxide substrates in the manufacture of electronic devices, involving, in succession, cleaning in solvents with ultrasonic agitation, a hot detergent bath with ultrasonic agitation and a bath of 5: 1:l deionized water:hydrogen peroxide:ammonium hydroxide at 80 “C. Hot deionized water was used for rinsing the substrates and water, with a resistivity of 18 MR cm, was used as a final cleaning step, and to store the substrates in until they were loaded into the evaporator. Just before mounting and loading, the substrates were dried and UV-ozone cleaned. The substrates were immediately loaded into an electron beam evaporator. 600 A thick gold (99.999% pure) was deposited in a base vacuum of about 10e6 Torr at a rate of 50 nm s-‘. To minimize contamination, samples were stored in a vacuum desiccator until used and the temperature of the experiment was chosen so that the anneals could be performed in relatively quick succession. An overpressure of argon or oxygen was maintained during annealing.
c 1992 -
Elsevier
Sequoia.
All rights
reserved
24
E. J/ran, ('. V. Thompson / C~q~i/ho3' instabilities m nmthmous /ihm
ation frequency for each sample, the anneal was terminated while individual agglomerated areas were still distinguishable, i.e. before about 20% of the transformation had taken place. Size distributions of holes were obtained from micrographs. 2. 1. L a s e r - a s s i s t e d
~ m
50 t.trn
m
substrate 6
(a)
measurement
of growth
rates
The extent of the transformation from a continuous to an agglomerated film was monitored in s i t u by measuring the intensity of laser light transmitted through the substrate as a function of time. The equipment and techniques used arc described in more detail in a previous publication [2]. The continuous films were nearly opaque to the frequency used, so that the intensity of transmitted laser light was directly related to the area of the substrate uncovered by agglomeration of the film in the path of the beam. An actual sample trace t\~r a continuous film, with the corresponding temperature profile superimposed, is shown in Fig. 2. Also shown are micrographs illustrating the state of agglomeration associated with a particular value of the laser signal. The initial signal at the detector decreases slightly as the sample and the chamber are heated up to the annealing temperature. This is due to a decrease in transmittance with increase in temperature of the fused silica substrate and chamber. The signal at the detector increases as the film develops voids and as the voids grow. The transmitted intensity levels off towards the value of a fully agglomerated film as the agglomerated areas impinge.
3. Modelingof agglomeration
I
50 gm
|
Agglomeration in a continuous film occurs through the nucleation and growth of circular agglomerated areas, consisting of beads and exposed substrate. If these areas nucleate at approximately the same time, in an experiment started at time t = 0, each of the round areas will have an area at time t of
(b) Fig. 1. (a) Agglomeration of a gold film patterned into lines, on a fused silica substrate. An optical micrograph, made using transmitted light, ( 80 n m thick film) with a scanning electron micrograph close-up ( 50 nm thick film) illustrating the complex, cell-like morphology of the receding edge of the film. The complex morphology results in beads of gold being left behind as the edge propagates across the gold lines. (b) An optical micrograph, made using transmitted light, of void formation and subsequent agglomeration in a 50 nm thick continuous gold film on a fused silica substrate. The black areas are gold and the white areas are exposed silica. Agglomeration initiates at only a few locations and proceeds through the growth of roughly circular agglomerated areas.
Data were obtained for films annealed in oxygen at 750 'C, 775 C , 800 C , and 825 ~'C, and films annealed in argon at 775 C , 800 C , and 825 ~'C for comparison with the oxygen results. In order to analyze the nucle-
a, = ~zu2(t - r) 2
(1)
where u is the velocity of the interface between agglomerated and continuous film (i.e. the growth rate of the void) and r is the time at which nucleation occurs (i.e. an incubation time). If all the nuclei form at once and there is no impingement, the total area A ~ ~ of the transformed region at time t is A aggl = NhA~z[u(t - r)] 2,
N h ¢.[(T)
(2)
where Nh is the number of nuclei per unit area and A is the total area of unagglomerated film. If impingement of the voids takes place, eqn. (2) overestimates the area transformed. This overestimated area is called the extended area and is indicated by the subscript "ex". The ratio of the change dA ~ggl in actual area agglomerated
E. Jiran, C. V. Thompson / Capillary instabilities in continuous films
3000
• ,
temperature profile
25
- 825°C
.I..~
2000
B
C~
m 20°C
1000
0 0.0
60.0 TIME
120.0
180.0
(min)
100 ~ m
(a)
(b)
100 ~ m (c)
(d)
Fig. 2. (a) Sample trace of the intensity of the laser light transmitted through a continuous gold film during an isothermal anneal. The temperature profile is superimposed. (b)-(d) Micrographs illustrating the state of agglomeration associated with three particular values of the laser intensity indicated by A - C respectively in (a).
to the change dA ~x ggj in extended area agglomerated is given by the ratio of untransformed area A - A aggl remaining, to A [3]: dm aggl=
1
dAagglex
(3)
Equation (3) can be integrated to solve for the extended area, so that Aaggl= --A In 1 -"~
ex
(4)
Substituting for the extended a r e a A axgg| of eqn. (2) with the expression for extended area of eqn. (4) gives - l n ( 1 - X) = N h ~ u 2 ( t -- r) 2
(5)
where X is the area fraction A aggl/A agglomerated. Equation (5) defines a transformation curve, X vs. t, with
an incubation time + and which levels off towards X = 1 as the continuous film is consumed by agglomeration. An agglomeration curve obeying eqn. (5) is shown in Fig. 3. As long as the holes are large enough to develop the steady state morphology described in ref. 2, u is given by the following expression for the void growth rate A x / A t , derived in ref. 2: Ax u = A~ =
fl e x p ( - Q s / k T ) kTh 3
(6)
where /3 = 2DoTsV122/Tt, Do is the surface diffusion preexponential, 7s is the surface energy, 12 is the atomic volume, Qs is the activation energy for surface diffusion, T is the temperature and h is the film thickness. If N h is not a function of either time or temperature, eqns. (5) and (6) will represent the complete time, temperature and film thickness dependence.
26
E. Jiran, C. 1/. Thompson / Capillary instahilities m continuous/i/ms
;4
T A B L E I. A g g l o m e r a t i o n f r o n t velocities u a n d i n c u b a t i o n times r c a l c u l a t e d f r o m eqn. (5)
lm
0
+,,,,a
0
E ©
Tern p e r a t u re (C) 750 (11 750 (2) 775
,,,,,,,-I
800
825
rcu 2 ~" h
~';h
It
T
(rain 2)
(Cgl] 2)
(l'In]S ~)
(mini
1.07 × I0 1.00 x 10 0.95 × 10 18.40 × 10 3.50 x l0
5.0 7.7 4.9 14.2 5.3
4.5 3.5 4.2 11 7.S
127 100 75 44 49
x I0 ~
x x x ,~
I0 ~ 10 ~ 10 ~ I0 ~
© ( ( I I a n d (2) i n d i c a l c s a m p l e s in Figs. 4 ( a ) a n d 4(b))
0
0
"C time, t
Fig. 3. T h e o r e t i c a l p r e d i c t i o n for the a r e a f r a c t i o n X a g g l o m e r a l c d as a f u n c t i o n o f time, a c c o r d i n g to eqn. (5).
4. Results
From eqn. (5), I n ( l - X ) should have a parabolic dependence on time t. We have fitted our data to eqn. (5) using the method of least squares. The calculation is repeated with different values of T until the best fit is found as indicated by a minimum in the sum of the squares of the deviations. Both experimental and fitted transformation curves X vs. t for the oxygen samples are plotted in Fig. 4. The fraction transformed sometimes appears to fall below zero. This is due to slight fluctuations about a mean value with time in the laser intensity. The standard deviations of the fitted curves are given for each curve and indicate, overall, an excellent fit to the experimental results. The appropriateness of the fitting procedure described above depends on the validity of the assumption that all the nuclei formed simultaneously. Figure 5, the size distributions of agglomerated areas, shows that this assumption is not precisely valid. The distributions do, however, show, in most cases, clear, relatively narrow peaks while in the case of constant nucleation no peak would occur. Even though they would have been easily detected with an optical microscope, no very small agglomerated areas (compared with the median area) were found. Although the distributions exhibit peaks of finite width, the good agreement between the model and the experiments supports the approximation of simultaneous nucleation. The growth rate u can be calculated from eqn. (5) given measurements of X vs. t and Nh, the number of agglomerated areas per unit area. For example, in the sample at 800 ~"C, the value of Nh is approximately 1.42 x 104 nuclei cm 2, So that the growth rate is 11 nm s ~. The particularly rapid transformation in the sample at 800 '~C (Fig. 4(d)) is due to the large nucle-
ation density in the film (Table 1). The growth rate determined from experiments oil patterned fihns of the same thickness was 7.8 nm s ~ [2]. The rates calculated l\~r samples annealed in oxygen are listed ill Table 1 and plotted against l / k T in Fig. 6. There is scatter m the data and, therefore, not a well-defined linearity, but forcing a fit yields an activation energy of 1.2 _+ 0.4 eV and Do value of 0.04 cm 2 s '. The calculated activation energy is consistent with the value of 1.8 + 0.4 eV obtained from patterned films and in particular with the 1.6 eV obtained for patterned films 60 nm thick. Growth rates were also calculated for the anneals in argon. Figure 6 shows that growth rates measured in oxygen and argon using continuous films are similar and agree with growth rates measured from patterned films annealed in oxygen.
5. Discussion
The purpose of studying continuous films was to include the effect of nucleation on agglomeration. The transformation curves (Fig. 4) and the size distributions (Fig. 5) show that there is an incubation time r before the transformation occurs. This can be explained as follows. One of the ways in which holes form in thin films is by grain boundary grooving. Triple points are even more susceptible to penetration than grain boundaries [4]. Because a grain boundary groove deepens as the fourth root of time [5], it is expected that a finite amount of time is needed for penetration of the film, leading to an incubation time. The thinner the film, the shorter the expected incubation time would be. Although the edges of the voids are uneven because of the complex morphology of the agglomeration process, the agglomerated areas exhibit an approximately circular morphology until impingement. This indicates that the rate is the same in all directions in the plane of the film and would not be expected to vary from void to void. Therefore, the range in the sizes of the nuclei in a particular sample is due to a time-dependent nucleation frequency. Since grain boundary grooves evolve at different rates, depending on the orientations of the
E. Jiran, C. V. Thompson / Capillary instabilities in continuous.films
.30
.30 Oxygen, 750°C (1) s.d. = 0.004 .
.24
27
X
m
Oxygen, 800°C s.d. = 0.006 .24-
.18
-
.12
.12
-
.06
.06
.00
.00
.18
E o
60
120 180 t (min)
(a)
240
60
300
I
I
I
120
180
240
300
t (min) (d)
.30
.30
X
Oxygen, 825°C s.d. = 0.004
Oxygen, 750°C (2) s.d. = 0.005 .24
o
~
.18
o
~'~ .12
~
.06
.24 .18 .12 .06
.00
--"-:;_~,-." 60
.00 120
180
240
300
60
t (min)
120
I
I
180
240
300
t (min)
(b)
(e)
.30 Oxygen, 775°C s.d. = 0.004 X ~
_o
.24 .18
-
.12
-
e~0
'
fl
.06 .00 " ~ -
.-,,k_ 60
120 180 t (min)
I 240
300
(c) Fig. 4. Experimental and fitted traces of X vs. t for originally continuous films annealed in oxygen: , experimental curves; of eqn. (5) to the experimental data. The standard deviation (s.d.) for the fit for each curve is also given.
, best fits
28
E. Jiran, ('. V. Thompson / ('apilhtry instabilities in continuous Blm,~
60
grains that meet, nucleation would occur over a finite time interval as different grain boundaries would intersect the substrate at different times. This would result in a distribution of the sizes of the nuclei, such as seen in Fig. 5.
Oxygen 800°C
Oxygen 750°C (2)
40 s.d. = 5 lain
s.d. = 2.5 tam
t~
~- 20
6. Summary and conclusions U"l
0
5
5 101520253035404550
J"-
10 15 20 25 30 35
I
60 Oxygen 775°C
Oxygen 825°C
40 s'd'= 71am
I.
¢1,
s.d. = 5 ~m
1
20
,g = I
10 15 20 25 30 35 40
10 15 20 25 30 35 40 45 50
radius (gm)
radius (gm)
Fig. 5. Distribution of the sizes of unimpinged agglomerated areas for four of the samples in Fig. 4. The range for the horizontal axis was chosen so that it would include areas formed over the duration of the experiment.
T(°C) 825
800
775
750
1.0
0.1 10.5
The agglomeration of continuous films was modeled as a two-dimensional phase transformation: from a continous film to a beaded film. A constant rate of growth of agglomerated areas and a constant number of nuclei (agglomerated areas) was assumed. Under these conditions, taking impingement of agglomerated areas into account, the fraction X transformed has the following dependence on time (eqn. (5)):
[]
oxygen data
•
argon data I
ll.O
"
i
t
11.5
l/kT (1/eV) Fig. 6. Arrhenius plot of the growth rate of agglomerated areas in continuous tilms annealed in oxygen and argon. The rates were calculated using eqn. (5) , best fit to the oxygen data points: • best fit found in previous work using patterned films.
exp[ - Nh 7ru'(t - r)-~]
(7)
Although there was some time dependence of the nucleation frequency, the fraction transformed t's. time, as observed in experiments, is described very well by this equation. From the measured number Nh of voids and the fit of eqn. (7) to the experimental data from 600,~ thick films, the value of the agglomeration rate u was determined. In these experiments, continuous films annealed in oxygen and argon were found to agglomerate at similar rates. The rates reported here for continuous films are in agreement with data reported earlier for patterned gold films of the same thickness and annealed in oxygen. Void nucleation in continuous films exhibited incubation times with small but definite variations. The incubation time can be explained as the time needed lk~r surface grooves at grain boundaries or grain boundary triple junctions to penetrate the film. The spread in this time would then be due to the presence of various types of grain boundaries and triple junctions deepening at different rates and penetrating the film at different times.
References 1 K.-N. Tu. Annu. Ret,. Mater. Sci., 15(1985) 147. 2 E. Jiran and C, V. Thompson, J. Eh'ctron. Mater., 19(1990) 1153. 3 J. W. Christian, The Theop3' ~1" Transjormation Rates in Meta/s and Alloys, Part I, Pergamon. Oxford, 1975. p. 18. 4 D. J. Srolovitz and S. A. Safran, J. Appl. Phys., 60 (1986) 247. 5 W. Mullins, J. Appl. Phys., 28 ( 1957} 333.
Capillary instabilities in thin, continuous films E. Jiran*
and C. V. Thompson
Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139 (U.S.A.) (Received
June 3. 1991; accepted
August
23, 1991)
Abstract The agglomeration of continuous films was studied in 60 nm thick, polycrystalline, gold films deposited on polished fused silica substrates, Nucleation of holes in any particular sample showed nucleation frequencies characteristic of site saturation. The growth of the holes resulted in roughly circular agglomerated areas. The area fraction transformed was measured as a function of time for several temperatures in oxygen and argon. Using a JohnsonMehl-Avrami analysis and assuming a constant hole growth rate, the time dependence of the agglomeration can be described. The growth rates, which contain important kinetic information, could thus be determined. Growth rates for anneals in argon and oxygen were found to be similar.
1. Introduction Because thin films have a high surface-to-volume ratio, on heating, a thin, solid, continuous film may develop holes and may eventually agglomerate into shapes with a lower surface-to-volume ratio. This process can occur well below the melting point of the material, and can thus interfere with device fabrication when processing involves heating subsequent to the deposition of thin films, or when a reaction between two thin films must occur to produce, for example, a silicide used for interconnects [ 11. In an earlier study of the agglomeration of thin gold films patterned into wide lines [2], we showed that, once a hole forms (as simulated by the spaces between the lines) a complex morphology of void cells develops (Fig. 1(a)). This void morphology propagates by means of the growth of void cells at an average rate which is independent of time and varies as the inverse of the cube of the film thickness. The result of this growth is an agglomerated area consisting of islands of material and exposed substrate. In contrast with patterned films, agglomeration of continuous thin films involves the nucleation of voids as well as their growth. Preliminary experiments on continuous gold films showed that, although the growth of the void is complex, the agglomerated areas formed are roughly circular and nucleate at approximately the same time (Fig. l(b)). In this paper we report a study of the kinetics of agglomeration of initially continuous gold films on fused silica substrates. By studying agglomeration of
*Present address: Institut National de la Recherche Energie, Varennes, Quebec J3X lS2, Canada.
0040-6090/92/$5.00
Scientifique
-
continuous films after studying the agglomeration of patterned films, we have been able to isolate the nucleation component of the agglomeration process, and to obtain kinetic characterizations of the nucleation rate from measurements of the overall agglomeration rate.
2. Experimental
details
In order to determine the time dependence of the process of agglomeration in continuous films, agglomerated area vs. time curves were obtained, in situ, at different temperatures for films annealed in either oxygen or argon. Fused silica substrates were cleaned using a schedule designed for the cleaning of oxide substrates in the manufacture of electronic devices, involving, in succession, cleaning in solvents with ultrasonic agitation, a hot detergent bath with ultrasonic agitation and a bath of 5: 1:l deionized water:hydrogen peroxide:ammonium hydroxide at 80 “C. Hot deionized water was used for rinsing the substrates and water, with a resistivity of 18 MR cm, was used as a final cleaning step, and to store the substrates in until they were loaded into the evaporator. Just before mounting and loading, the substrates were dried and UV-ozone cleaned. The substrates were immediately loaded into an electron beam evaporator. 600 A thick gold (99.999% pure) was deposited in a base vacuum of about 10e6 Torr at a rate of 50 nm s-‘. To minimize contamination, samples were stored in a vacuum desiccator until used and the temperature of the experiment was chosen so that the anneals could be performed in relatively quick succession. An overpressure of argon or oxygen was maintained during annealing.
c 1992 -
Elsevier
Sequoia.
All rights
reserved
24
E. J/ran, ('. V. Thompson / C~q~i/ho3' instabilities m nmthmous /ihm
ation frequency for each sample, the anneal was terminated while individual agglomerated areas were still distinguishable, i.e. before about 20% of the transformation had taken place. Size distributions of holes were obtained from micrographs. 2. 1. L a s e r - a s s i s t e d
~ m
50 t.trn
m
substrate 6
(a)
measurement
of growth
rates
The extent of the transformation from a continuous to an agglomerated film was monitored in s i t u by measuring the intensity of laser light transmitted through the substrate as a function of time. The equipment and techniques used arc described in more detail in a previous publication [2]. The continuous films were nearly opaque to the frequency used, so that the intensity of transmitted laser light was directly related to the area of the substrate uncovered by agglomeration of the film in the path of the beam. An actual sample trace t\~r a continuous film, with the corresponding temperature profile superimposed, is shown in Fig. 2. Also shown are micrographs illustrating the state of agglomeration associated with a particular value of the laser signal. The initial signal at the detector decreases slightly as the sample and the chamber are heated up to the annealing temperature. This is due to a decrease in transmittance with increase in temperature of the fused silica substrate and chamber. The signal at the detector increases as the film develops voids and as the voids grow. The transmitted intensity levels off towards the value of a fully agglomerated film as the agglomerated areas impinge.
3. Modelingof agglomeration
I
50 gm
|
Agglomeration in a continuous film occurs through the nucleation and growth of circular agglomerated areas, consisting of beads and exposed substrate. If these areas nucleate at approximately the same time, in an experiment started at time t = 0, each of the round areas will have an area at time t of
(b) Fig. 1. (a) Agglomeration of a gold film patterned into lines, on a fused silica substrate. An optical micrograph, made using transmitted light, ( 80 n m thick film) with a scanning electron micrograph close-up ( 50 nm thick film) illustrating the complex, cell-like morphology of the receding edge of the film. The complex morphology results in beads of gold being left behind as the edge propagates across the gold lines. (b) An optical micrograph, made using transmitted light, of void formation and subsequent agglomeration in a 50 nm thick continuous gold film on a fused silica substrate. The black areas are gold and the white areas are exposed silica. Agglomeration initiates at only a few locations and proceeds through the growth of roughly circular agglomerated areas.
Data were obtained for films annealed in oxygen at 750 'C, 775 C , 800 C , and 825 ~'C, and films annealed in argon at 775 C , 800 C , and 825 ~'C for comparison with the oxygen results. In order to analyze the nucle-
a, = ~zu2(t - r) 2
(1)
where u is the velocity of the interface between agglomerated and continuous film (i.e. the growth rate of the void) and r is the time at which nucleation occurs (i.e. an incubation time). If all the nuclei form at once and there is no impingement, the total area A ~ ~ of the transformed region at time t is A aggl = NhA~z[u(t - r)] 2,
N h ¢.[(T)
(2)
where Nh is the number of nuclei per unit area and A is the total area of unagglomerated film. If impingement of the voids takes place, eqn. (2) overestimates the area transformed. This overestimated area is called the extended area and is indicated by the subscript "ex". The ratio of the change dA ~ggl in actual area agglomerated
E. Jiran, C. V. Thompson / Capillary instabilities in continuous films
3000
• ,
temperature profile
25
- 825°C
.I..~
2000
B
C~
m 20°C
1000
0 0.0
60.0 TIME
120.0
180.0
(min)
100 ~ m
(a)
(b)
100 ~ m (c)
(d)
Fig. 2. (a) Sample trace of the intensity of the laser light transmitted through a continuous gold film during an isothermal anneal. The temperature profile is superimposed. (b)-(d) Micrographs illustrating the state of agglomeration associated with three particular values of the laser intensity indicated by A - C respectively in (a).
to the change dA ~x ggj in extended area agglomerated is given by the ratio of untransformed area A - A aggl remaining, to A [3]: dm aggl=
1
dAagglex
(3)
Equation (3) can be integrated to solve for the extended area, so that Aaggl= --A In 1 -"~
ex
(4)
Substituting for the extended a r e a A axgg| of eqn. (2) with the expression for extended area of eqn. (4) gives - l n ( 1 - X) = N h ~ u 2 ( t -- r) 2
(5)
where X is the area fraction A aggl/A agglomerated. Equation (5) defines a transformation curve, X vs. t, with
an incubation time + and which levels off towards X = 1 as the continuous film is consumed by agglomeration. An agglomeration curve obeying eqn. (5) is shown in Fig. 3. As long as the holes are large enough to develop the steady state morphology described in ref. 2, u is given by the following expression for the void growth rate A x / A t , derived in ref. 2: Ax u = A~ =
fl e x p ( - Q s / k T ) kTh 3
(6)
where /3 = 2DoTsV122/Tt, Do is the surface diffusion preexponential, 7s is the surface energy, 12 is the atomic volume, Qs is the activation energy for surface diffusion, T is the temperature and h is the film thickness. If N h is not a function of either time or temperature, eqns. (5) and (6) will represent the complete time, temperature and film thickness dependence.
26
E. Jiran, C. 1/. Thompson / Capillary instahilities m continuous/i/ms
;4
T A B L E I. A g g l o m e r a t i o n f r o n t velocities u a n d i n c u b a t i o n times r c a l c u l a t e d f r o m eqn. (5)
lm
0
+,,,,a
0
E ©
Tern p e r a t u re (C) 750 (11 750 (2) 775
,,,,,,,-I
800
825
rcu 2 ~" h
~';h
It
T
(rain 2)
(Cgl] 2)
(l'In]S ~)
(mini
1.07 × I0 1.00 x 10 0.95 × 10 18.40 × 10 3.50 x l0
5.0 7.7 4.9 14.2 5.3
4.5 3.5 4.2 11 7.S
127 100 75 44 49
x I0 ~
x x x ,~
I0 ~ 10 ~ 10 ~ I0 ~
© ( ( I I a n d (2) i n d i c a l c s a m p l e s in Figs. 4 ( a ) a n d 4(b))
0
0
"C time, t
Fig. 3. T h e o r e t i c a l p r e d i c t i o n for the a r e a f r a c t i o n X a g g l o m e r a l c d as a f u n c t i o n o f time, a c c o r d i n g to eqn. (5).
4. Results
From eqn. (5), I n ( l - X ) should have a parabolic dependence on time t. We have fitted our data to eqn. (5) using the method of least squares. The calculation is repeated with different values of T until the best fit is found as indicated by a minimum in the sum of the squares of the deviations. Both experimental and fitted transformation curves X vs. t for the oxygen samples are plotted in Fig. 4. The fraction transformed sometimes appears to fall below zero. This is due to slight fluctuations about a mean value with time in the laser intensity. The standard deviations of the fitted curves are given for each curve and indicate, overall, an excellent fit to the experimental results. The appropriateness of the fitting procedure described above depends on the validity of the assumption that all the nuclei formed simultaneously. Figure 5, the size distributions of agglomerated areas, shows that this assumption is not precisely valid. The distributions do, however, show, in most cases, clear, relatively narrow peaks while in the case of constant nucleation no peak would occur. Even though they would have been easily detected with an optical microscope, no very small agglomerated areas (compared with the median area) were found. Although the distributions exhibit peaks of finite width, the good agreement between the model and the experiments supports the approximation of simultaneous nucleation. The growth rate u can be calculated from eqn. (5) given measurements of X vs. t and Nh, the number of agglomerated areas per unit area. For example, in the sample at 800 ~"C, the value of Nh is approximately 1.42 x 104 nuclei cm 2, So that the growth rate is 11 nm s ~. The particularly rapid transformation in the sample at 800 '~C (Fig. 4(d)) is due to the large nucle-
ation density in the film (Table 1). The growth rate determined from experiments oil patterned fihns of the same thickness was 7.8 nm s ~ [2]. The rates calculated l\~r samples annealed in oxygen are listed ill Table 1 and plotted against l / k T in Fig. 6. There is scatter m the data and, therefore, not a well-defined linearity, but forcing a fit yields an activation energy of 1.2 _+ 0.4 eV and Do value of 0.04 cm 2 s '. The calculated activation energy is consistent with the value of 1.8 + 0.4 eV obtained from patterned films and in particular with the 1.6 eV obtained for patterned films 60 nm thick. Growth rates were also calculated for the anneals in argon. Figure 6 shows that growth rates measured in oxygen and argon using continuous films are similar and agree with growth rates measured from patterned films annealed in oxygen.
5. Discussion
The purpose of studying continuous films was to include the effect of nucleation on agglomeration. The transformation curves (Fig. 4) and the size distributions (Fig. 5) show that there is an incubation time r before the transformation occurs. This can be explained as follows. One of the ways in which holes form in thin films is by grain boundary grooving. Triple points are even more susceptible to penetration than grain boundaries [4]. Because a grain boundary groove deepens as the fourth root of time [5], it is expected that a finite amount of time is needed for penetration of the film, leading to an incubation time. The thinner the film, the shorter the expected incubation time would be. Although the edges of the voids are uneven because of the complex morphology of the agglomeration process, the agglomerated areas exhibit an approximately circular morphology until impingement. This indicates that the rate is the same in all directions in the plane of the film and would not be expected to vary from void to void. Therefore, the range in the sizes of the nuclei in a particular sample is due to a time-dependent nucleation frequency. Since grain boundary grooves evolve at different rates, depending on the orientations of the
E. Jiran, C. V. Thompson / Capillary instabilities in continuous.films
.30
.30 Oxygen, 750°C (1) s.d. = 0.004 .
.24
27
X
m
Oxygen, 800°C s.d. = 0.006 .24-
.18
-
.12
.12
-
.06
.06
.00
.00
.18
E o
60
120 180 t (min)
(a)
240
60
300
I
I
I
120
180
240
300
t (min) (d)
.30
.30
X
Oxygen, 825°C s.d. = 0.004
Oxygen, 750°C (2) s.d. = 0.005 .24
o
~
.18
o
~'~ .12
~
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.00
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.00 120
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I
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.30 Oxygen, 775°C s.d. = 0.004 X ~
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.24 .18
-
.12
-
e~0
'
fl
.06 .00 " ~ -
.-,,k_ 60
120 180 t (min)
I 240
300
(c) Fig. 4. Experimental and fitted traces of X vs. t for originally continuous films annealed in oxygen: , experimental curves; of eqn. (5) to the experimental data. The standard deviation (s.d.) for the fit for each curve is also given.
, best fits
28
E. Jiran, ('. V. Thompson / ('apilhtry instabilities in continuous Blm,~
60
grains that meet, nucleation would occur over a finite time interval as different grain boundaries would intersect the substrate at different times. This would result in a distribution of the sizes of the nuclei, such as seen in Fig. 5.
Oxygen 800°C
Oxygen 750°C (2)
40 s.d. = 5 lain
s.d. = 2.5 tam
t~
~- 20
6. Summary and conclusions U"l
0
5
5 101520253035404550
J"-
10 15 20 25 30 35
I
60 Oxygen 775°C
Oxygen 825°C
40 s'd'= 71am
I.
¢1,
s.d. = 5 ~m
1
20
,g = I
10 15 20 25 30 35 40
10 15 20 25 30 35 40 45 50
radius (gm)
radius (gm)
Fig. 5. Distribution of the sizes of unimpinged agglomerated areas for four of the samples in Fig. 4. The range for the horizontal axis was chosen so that it would include areas formed over the duration of the experiment.
T(°C) 825
800
775
750
1.0
0.1 10.5
The agglomeration of continuous films was modeled as a two-dimensional phase transformation: from a continous film to a beaded film. A constant rate of growth of agglomerated areas and a constant number of nuclei (agglomerated areas) was assumed. Under these conditions, taking impingement of agglomerated areas into account, the fraction X transformed has the following dependence on time (eqn. (5)):
[]
oxygen data
•
argon data I
ll.O
"
i
t
11.5
l/kT (1/eV) Fig. 6. Arrhenius plot of the growth rate of agglomerated areas in continuous tilms annealed in oxygen and argon. The rates were calculated using eqn. (5) , best fit to the oxygen data points: • best fit found in previous work using patterned films.
exp[ - Nh 7ru'(t - r)-~]
(7)
Although there was some time dependence of the nucleation frequency, the fraction transformed t's. time, as observed in experiments, is described very well by this equation. From the measured number Nh of voids and the fit of eqn. (7) to the experimental data from 600,~ thick films, the value of the agglomeration rate u was determined. In these experiments, continuous films annealed in oxygen and argon were found to agglomerate at similar rates. The rates reported here for continuous films are in agreement with data reported earlier for patterned gold films of the same thickness and annealed in oxygen. Void nucleation in continuous films exhibited incubation times with small but definite variations. The incubation time can be explained as the time needed lk~r surface grooves at grain boundaries or grain boundary triple junctions to penetrate the film. The spread in this time would then be due to the presence of various types of grain boundaries and triple junctions deepening at different rates and penetrating the film at different times.
References 1 K.-N. Tu. Annu. Ret,. Mater. Sci., 15(1985) 147. 2 E. Jiran and C, V. Thompson, J. Eh'ctron. Mater., 19(1990) 1153. 3 J. W. Christian, The Theop3' ~1" Transjormation Rates in Meta/s and Alloys, Part I, Pergamon. Oxford, 1975. p. 18. 4 D. J. Srolovitz and S. A. Safran, J. Appl. Phys., 60 (1986) 247. 5 W. Mullins, J. Appl. Phys., 28 ( 1957} 333.