The effect of rate of surface growth on roughness scaling

Article abstract of DOI:10.1149/1.1921767

The effect of rate of deposition on the scaling properties of roughness during surface growth is studied by silver electrodeposition in a flow cell and atomic force microscopy measurements. The rate of growth is controlled by the applied current density. The Ag-electrodeposited surface becomes rougher as the current density increases, approaching the diffusion limiting current. Furthermore, as the current density increases, the number of growth sites decreases and the lateral surface features become wider. Scaling analysis of the self-affine surface indicates that the saturated root-mean-square height increases with increasing current density, over a range of 2 decades; however, below saturation, the dependence is more complicated: Initially, there is no local effect; however, as the rate increases, negative local effect is observed, indicating that the local roughness grows more slowly at higher rates, as the current approaches the diffusion limiting current in the electrolyte. The discrepancy between the obtained average roughness exponent α = 0.52 and the growth rate exponent β_r = 0.3, similar to experiments under low deposition rate and for various deposition times (α = 0.62, β_t, = 0.71), indicates that the rate plays a decisive role in the selection of the growth kinetics by affecting the initial number of nucleation sites and their subsequent merge. Only at low rates, the rate and the time of growth behave interchangeably, and therefore the roughness scales with the average height, the product of the rate, and the time of growth.

Full text of DOI:10.1149/1.1921767

C462
Journal of The Electrochemical Society, 152 ͑7͒ C462-C465 ͑2005͒
0
013-4651/2005/152͑7͒/C462/4/$7.00 © The Electrochemical Society, Inc.
The Effect of Rate of Surface Growth on Roughness Scaling
a,b
a,c
a,
David G. Foster, Yonathan Shapir, and Jacob Jorne *
a
c
Department of Chemical Engineering, and Department of Physics and Astronomy, University of
Rochester, Rochester, New York 14627, USA
b
Eastman Kodak Company, Rochester, New York 14650, USA
The effect of rate of deposition on the scaling properties of roughness during surface growth is studied by silver electrodeposition
in a flow cell and atomic force microscopy measurements. The rate of growth is controlled by the applied current density. The
Ag-electrodeposited surface becomes rougher as the current density increases, approaching the diffusion limiting current. Further-
more, as the current density increases, the number of growth sites decreases and the lateral surface features become wider. Scaling
analysis of the self-affine surface indicates that the saturated root-mean-square height increases with increasing current density,
over a range of 2 decades; however, below saturation, the dependence is more complicated: Initially, there is no local effect;
however, as the rate increases, negative local effect is observed, indicating that the local roughness grows more slowly at higher
rates, as the current approaches the diffusion limiting current in the electrolyte. The discrepancy between the obtained average
roughness exponent ␣ = 0.52 and the growth rate exponent ␤ = 0.3, similar to experiments under low deposition rate and for
r
various deposition times ͑␣ = 0.62, ␤ = 0.71͒, indicates that the rate plays a decisive role in the selection of the growth kinetics
t
by affecting the initial number of nucleation sites and their subsequent merge. Only at low rates, the rate and the time of growth
behave interchangeably, and therefore the roughness scales with the average height, the product of the rate, and the time of growth.
©
2005 The Electrochemical Society. ͓DOI: 10.1149/1.1921767͔ All rights reserved.
Manuscript submitted August 24, 2004; revised manuscript received January 6, 2005. Available electronically June 3, 2005.
␣
Kinetic roughening of self-affine surfaces has been the subject of
W ϳ L
͓3͔
͓4͔
intense investigation during the last decade. Surface growth repre-
sents an example of scaling of nonequilibrium systems, where the
main issue is the establishment of scale invariance and universality,
as has been observed in equilibrium critical phenomena and nonlin-
ear dynamics. Surface roughness has been studied primarily from
while for t Ӷ Lz
␤
W ϳ t
where ␤ = ␣/z is the growth exponent. It is expected that all pro-
cesses with the same universality class share the same critical expo-
nent.
In the present paper, we focus on the effect of growth rate on the
scaling exponents of electrochemically deposited growing surfaces.
The growth rate can be easily maintained and manipulated by the
applied current density. For electrodeposition, it is proposed here
that the roughness scales as before with the length and with the
average thickness of the deposited layer, rather than simply with the
time
1
-5
6
the point of view of scaling behavior,
surface width, height
4
7
8-10
velocity, maximal height, and cyclical surface growth,
among
11
others. The effect of growth rate has been recently addressed for
the case of kinetic roughening in polymer film growth by vapor
deposition. The growth rate was controlled by the pressure of the
deposition chamber. It was found that the scaling behavior did not
11
change for growth rate spanning almost 1 order of magnitude,
despite the fact that the rate at which material arrives at the surface
is expected to affect the kinetic roughening. The effect of the growth
rate of the surface on its scaling properties has not been extensively
studied, mainly because it is usually difficult to control the rate of
growth. Electrodeposition offers a convenient way to investigate this
effect because the rate of growth can be easily controlled and ma-
nipulated by the applied current. The rate of growth is directly pro-
portional to the current density through Faraday’s law. Thus, the
scaling exponents obtained from experiments conducted under vari-
ous current densities ͑under constant deposition time͒ can be com-
pared with scaling exponents obtained from experiments conducted
under various growth times ͑under constant current density͒. Agree-
ment or disagreement between the obtained scaling exponents could
reveal the unique role of the rate in determining the growth mecha-
nism. The effect of the current density on roughening scaling was
␣
1/z
W͑L,h͒ = L f ͓L/͑h͒
͔
͓5͔
The average thickness under constant growth rate ͑or constant
current density͒ is given by
h ϰ it
͓6͔
where i is the current density ͑rate of growth͒ and t is the elec-
trodeposition time. Then, the width scales with the lateral length and
current density according to
␣
1/z
W͑L,it͒ = L f ͓L/͑it͒
͔
͓7͔
If the time of electrodeposition is maintained constant, then
␣
1/z
12
W͑L,i͒ = L f ͓L/i
Thus, for current density i ӷ Lz
W ϳ L
͔
͓8͔
attributed to local growth exponent, which becomes significant as
the surface growth rate approaches the diffusion-limiting current
from the bulk.
␣
͓9͔
Self-affine growing surfaces can be described by scaling analysis
of the surface roughness. The width of the surface W͑L,t͒, where L
is the spatial size and t is the time, is defined by
while for i Ӷ Lz
␤
W ϳ i
͓10͔
2
1/2
where ␤ = ␣/z is the growth exponent. This is expected under con-
stant time, and as long as the current is significantly lower than the
diffusion limiting current ͑no diffusion effect͒.
W͑L,t͒ = ͓͗h͑r,t͒ − ͗h͑r,t͔͒͘ ͘
͓1͔
and it scales as
Thus, by logarithmically plotting the root-mean-square ͑rms͒ of
the height as a function of the length, one can obtain the roughness
exponent ␣ ͑Eq. 9͒, and by logarithmically plotting the saturated
rms height as a function of the current density, under constant depo-
␣
1/z
W͑L,t͒ ϳ L f͑L/t
͒
͓2͔
where f is the scaling function. For large time t ӷ Lz
sition time, one can obtain the growth exponent ␤ ͑Eq. 10͒. These
r
exponents then can be compared with the time exponent ␤ obtained
t
under constant current density and for various deposition times ͑Eq.
4͒. An agreement between the growth exponents will indicate that
*
Electrochemical Society Active Member.
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Journal of The Electrochemical Society, 152 ͑7͒ C462-C465 ͑2005͒
C463
the surface roughness scales with the current density, the growth rate
of the surface, and the roughness scales with the average height, Eq.
. A disagreement will indicate that the roughness scales with the
5
rate in a more complicated way, as the rate determines the various
growth kinetics regimes.
Silver electrodeposition from its thiosulfate solution was selected
to test the effect of growth rate on roughness scaling because silver
reduction is a relatively simple one-electron reaction and well-
studied system. The subject of silver electrodeposition has been pre-
13
viously reviewed.
Experimental
Experiments of surface growth were conducted in a flow channel
by silver electrodeposition from silver thiosulfate solution
3
−
0
2−
3
͓
Ag͑S O ͒ ͔ + e = Ag + 2S O
2 3 2 2
The plating solution was prepared by dissolving 0.1 M AgBr in
solution containing 0.20 M ammonium sulfite and 0.25 M ammo-
nium thiosulfate or sodium thiosulfate in Millipore-Q water. The
ratio of 2.5 mol of thiosulfate to every mole of silver was chosen to
ensure that all the silver ions were complexed by the excess thiosul-
fate ions. The natural pH of this solution is around 7, and was not
adjusted. The electrodeposition was performed in a glass flow cell to
ensure adequate supply of silver ions and to minimize any diffusion
limitations. The current efficiency for silver deposition was close to
Figure 1. Limiting current density for silver electrodeposition under flow
rate of 1.08 cm /s ͑Reynolds number is 300͒.
3
100%, eliminating the possibility of any side reaction. Silver sub-
strates served as the cathodes for the electrodeposition experiments.
The 1 ϫ 1 cm substrates, prepared from pure silver ͑99.999% Alfa
Aesar, Inc.͒, were polished down to 50 nm with alumina, rinsed in
an ultrasound water bath, and fitted into the bottom of the flow cell.
Platinum foil served as the anode facing the silver cathode.
corresponds to a Reynolds number of 300. The limiting current is
measured at 19 mA/cm ; thus, all experiments were conducted sub-
stantially below the limiting current density and consequently diffu-
sion in the fluid plays little effect.
Figure 2 shows the AFM images of four surfaces under the vari-
ous current densities. As the current density increases, the number of
deposition sites decreases and the surface becomes rougher, as indi-
cated by the higher measured rms of the surface roughness. The
decrease in the number of bumps can be accounted for by the merg-
ing of roughness features, because in high current experiments the
amount of deposited silver was the highest.
2
The AFM flow cell ͑Digital Instruments͒, 10 cm long, 0.5 cm
wide, and 0.25 cm high and horizontally placed, served as the elec-
trodeposition cell. The electrodeposition occurred at the cathode fac-
ing up. The counter anode was made of silver sheet. Constant cur-
2
rents of 0.2-8 mA/cm for a period of 1250 s were applied by a
PAR 273A potentiostat using EG&G 270/259 software. Experiments
2
were also conducted under constant current density of 0.8 mA/cm
Figure 3 shows the logarithmic plots of the rms roughness versus
length scale for silver electrodeposition under various current den-
sities. The initial slopes are quite similar, resulting in an average
roughness exponent of ␣ = 0.52 ± 0.11, although the slope at the
for various deposition times of 800-2000 s ͑which correspond to
2
charge passed of 0.2-0.45 C/cm .͒ Before use, the glass flow cell
was rinsed with 97% sulfuric acid and Millipore-Q filtered water.
The plating solution was continuously fed into the flow cell by grav-
2
highest current density of 8 mA/cm is significantly steeper, prob-
ity from an elevated reservoir, and the flow rate was maintained at
ably because the amount deposited was higher and merging of fea-
tures occurred. It is not clear why at small correlation lengths the
3
65 cm /min, which corresponds to a linear velocity of 11 cm/s, par-
2
2
allel to the surface. The corresponding Reynolds number was 300,
clearly within the fully developed laminar flow regime. All experi-
ments were conducted under constant flow velocity, and the current
roughness for 8 mA/cm is lower than that for 4 mA/cm , although
the corresponding saturation roughness increases monotonically
11,12
with current density, as expected,
indicating that the surface be-
2
density varied between 1 and 10 mA/cm . The electrodeposition
comes rougher at higher growth rates. It is quite possible that be-
cause the magnitude of the roughness measured here is relatively
high ͑Ͼ100 nm͒, hence the aspect ratios at small correlation lengths
is relatively high. This might prevent the AFM tip from accurately
measuring the roughness at very small correlation lengths.
time was held constant at 1250 s, which corresponds to average
deposit thickness of 1.3-13 ␮m, while for the constant current ex-
2
periments ͑0.8 mA/cm ͒, the electrodeposition time varied from
800 to 1200 s.
The roughness of the electrodeposited silver surface was mea-
Figure 4 shows a plot of the saturated roughness as a function of
the current density. The increase in saturated roughness with rate is
sured after each experiment by imaging the surface on a Digital
Nanoscope III atomic force microscope ͑AFM͒, operated in a con-
tact mode. Gold-coated pyramidal NanoProbe Si N tips mounted
11,12
monotonic, as expected.
The slope, according to Eq. 10, gives a
growth exponent ␤ = 0.30.
3
4
r
on a gold-coated v-shaped cantilever were used to image the sur-
face. The correction for the tilt distortion was applied and the image
files were exported as text files to a spreadsheet program for numeri-
For comparison, Fig. 5 and 6 show similar plots for silver elec-
2
trodeposition at constant current density of 0.8 mA/cm and under
1
0,13
various deposition times: 800-2000 s.
Again, monotonic in-
1
4
cal calculations of surface roughness at each pixel.
crease of roughness with time is observed only for the saturation
region. The roughness exponent for these experiments was obtained
as ␣ = 0.62 ± 0.05 and the average growth exponent was found to
Results and Discussion
1
0,13
The purpose of this work was to determine the effect, if any, of
rate of growth on the roughness of electrodeposited surface, and to
compare the obtained scaling exponents with those obtained under
constant growth rate. The maximum rate of deposition can be found
by measuring the limiting current density, which is the rate con-
trolled by diffusion in the electrolyte. Figure 1 shows the limiting
current density for the cell under a flow velocity of 11 cm/s, which
be ␤ = 0.71 ± 0.10.
The slightly different slopes in Fig. 6 are
t
probably due to the complexing ability of ammonium ion in com-
parison to the sodium ion. It appears that the growth exponent de-
12
pends on the rate in a complicated way. Huo and Schwarzacher
attempted to explain anomalous scaling of the surface width during
copper electrodeposition by introducing a local growth exponent
␤loc, which was found to become significant when the rate ap-
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C464
Journal of The Electrochemical Society, 152 ͑7͒ C462-C465 ͑2005͒
Figure 3. Logarithmic plot of surface roughness vs. current densities: 1, 2, 4,
2
8
mA/cm .
by the fraction of the limiting current. The disagreement between the
present rate growth exponent ␤ = 0.3 and the time growth exponent
r
␤t = 0.71 indicates that the roughness scales with the growth rate
11
and with the growth time in a dissimilar way. Zhao et al. con-
cluded that the scaling behaviors of the growth front, in the case of
polymer deposition, do not change for growth rate spanning almost
1
order of magnitude. This apparent discrepancy can be resolved if
one looks into the rates involved. The polymer growth rates from the
vapor phase in Ref. 11 ranged from 5.5 to 13.9 nm/min, while the
present current densities for silver deposition correspond to much
higher rates of 100-1000 nm/min. In addition, silver deposition
from a liquid phase is expected to be more diffusion limited than in
the case of vapor deposition; thus, the local effect is expected to be
1
2
more pronounced. Using SEM cross-sectional studies of copper
1
5
electrodeposition, Kang and Gewirth showed that identical scaling
parameters do not necessarily ensure identical growth mechanism.
This might be true in the present case, especially at the highest
2
current density of 8 mA/cm , where the slope below saturation is
Figure 2. AFM images of silver electrodeposition under various current
2
densities: 1, 2, 4, 8 mA/cm .
proached the diffusion-limiting current in the electrolyte. Similar
analysis of the current data resulted in ␤ = 0 at low rates ͑i
loc
2
Ͻ 2 mA/cm ͒, because in this range, the employed current densities
were well below the diffusion limiting current of 19 mA/cm , as
shown in Fig. 1. For higher rates ͑i = 4-8 mA/cm ͒, a negative local
2
2
growth exponent is observed, ␤loc = −0.61, indicating that the local
roughness grows more slowly as the rate of deposition starts to
approach the diffusion-limiting current. The growth exponent for the
higher current regime is calculated as the difference between the
experimental and the local growth exponent: ␤ = 0.30 − ͑−0.61͒
=
0.91. Both experiments, at constant time and at constant rate, es-
tablish the lack of local growth effect at low rates of growth, in
12
agreement with Huo and Schwarzacher, who related the local ef-
fect to the effect of diffusion in the electrolyte, which is expressed
Figure 4. Logarithmic plot of saturation rms roughness vs. various current
densities: 1, 2, 4, 8 mA/cm .
2
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Journal of The Electrochemical Society, 152 ͑7͒ C462-C465 ͑2005͒
C465
nent based on the average height, of ␤ = 0.36. The growth expo-
h
nent predicted from the ballistic and the KPZ models is 0.23 and
2
0.25, respectively.
The effect of the growth rate might have been more highlighted
by comparing the time growth exponents under various growth
rates, rather than by the present approach of comparing the time and
the rate growth exponents; however, this comparison requires a large
number of experiments which the present approach is attempting to
avoid. It should be pointed out that the range of average height in
2
the constant time experiments ͑1.25-10 C/cm ͒ is higher than the
2
range in the constant current experiments ͑0.64-1.6 C/cm ͒; how-
ever, this difference has little effect on the calculated exponents
because the saturation roughness was achieved much earlier.
Conclusions
The effect of growth rate on the roughness scaling was studied
using silver electrodeposition and AFM roughness measurements.
As the growth rate is increased, the surface becomes rougher with
larger lateral feature size. The obtained roughness exponent ␣
=
0.52 is in good agreement with a value of 0.62, obtained from
10
silver electrodeposition under constant current. It slightly increases
with increasing the current density. The rate growth exponent ␤r
Figure 5. rms roughness vs. length scale for various deposition times
=
0.3, obtained over 2 decades from the dependence of the rough-
2
͑
800-2000 s͒ under constant current density 0.8 mA/cm .
ness on the current density, is in disagreement with ␤ = 0.71, ob-
t
tained under constant rate and various times, indicating that the rate
of growth affects the kinetics of roughness growth. Even after cor-
rection for local effects, it is found that the rate determines the
scaling exponents by affecting the initial number of nucleation sites
and their subsequent merge. The scaling analysis of the self-affine
surface indicates that the roughness scales with the average height of
the growing surface, in accordance with Eq. 10 and the growth
exponent is ␤ = 0.36.
significantly steeper ͑see Fig. 3͒, resulting in a higher ␣ value at
higher rate and indicating different mechanism of merging of rough-
1
4
ness features, as has been observed in copper electrodeposition.
The scaling of the saturation roughness with the average height ͑i
times t͒, observed for silver electrodeposition in the presence of
either sodium or ammonium thiosulfate, resulted in a growth expo-
Acknowledgments
The authors gratefully acknowledge Eastman Kodak Company
and the Office of Naval Research ONR grant no. N00014-00-1-
0057.
References
1
. Dynamics of Fractal Surfaces, F. Family and T. Vicsek, Editors, Cambridge Uni-
versity Press, Cambridge, U.K. ͑1995͒.
2
3
. A.-L. Barabasi and H. E. Stanley, Fractal Concepts in Surface Growth, Cambridge
University Press, Cambridge, U.K. ͑1995͒.
. J. Villain and A. Pimpinelli, Physique de la Croissance Crystalline, Editions Ey-
rolles, Paris ͑1995͒.
4
5
. T. Halpin-Healy and Y.-C. Zhang, Phys. Rep., 254, 215 ͑1995͒.
. P. Meakin, Fractals, Scaling and Growth Far from Equilibrium, Cambridge Uni-
versity Press, Cambridge, U.K. ͑1998͒.
6
7
. Z. Racz and M. Plischke, Phys. Rev. E, 50, 3530 ͑1994͒.
. S. Raychaudhuri, M. Cranston, C. Przybyla, and Y. Shapir, Phys. Rev. Lett., 87,
1
36101 ͑2001͒.
. Y. Shapir, S. Raychaudhuri, D. G. Foster, and J. Jorne, Phys. Rev. Lett., 84, 3029
2000͒.
. S. Raychaudhuri, Y. Shapir, D. G. Foster, and J. Jorne, Phys. Rev. E, 64, 051604
2001͒.
0. D. G. Foster, Ph.D. Thesis, University of Rochester, Rochester, NY ͑1999͒.
8
9
͑
͑
1
11. Y.-P Zhao, J. B. Fortin, G. Bonvallet, G.-C. Wand, and T.-M. Lu, Phys. Rev. Lett.,
8
5, 3229 ͑2000͒.
1
1
2. S. Hua and W. Schwarzacher, Phys. Rev. Lett., 86, 256 ͑2001͒.
3. D. G. Foster, J. Jorne, and Y. Shapir, J. Electrochem. Soc., 150, C375 ͑2003͒.
Figure 6. Logarithmic plot of saturation rms roughness vs. deposition time
under constant current density of 0.8 mA/cm .
14. A. Osafo-Acquaah, Ph.D. Thesis, University of Rochester, Rochester, NY ͑2004͒.
15. M. Kang and A. A. Gewirth, J. Electrochem. Soc., 150, C426 ͑2003͒.
2
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