Numerical Modeling of Electrodeposition Phenomena
Nadia Strusevich, Michael Hughes, Christopher Bailey, Georgi Djambazov
University of Greenwich
Park Row, Greenwich, London SE10 9LS
{n.stroussevitch,m.s.hughes,c.bailey, g.djambazov}@gre.ac.uk
Abstract
Electrodeposition is a widely used technique for the
fabrication of high aspect ratio microstructure
components. In recent years much research has been
focused within this area with an aim to understanding the
physics behind the filling of high-aspect ratio vias and
trenches on PCB’s and in particular how they can be
made without the formation of voids in the deposited
material. This paper describes some of the fundamental
work towards the advancement of numerical models that
can predict the electrodeposition process and addresses:
i)
ii)
iii)
A novel technique for interface motion based on
a variation of a donor-acceptor technique
A methodology for the investigation of stress
profiles in deposits
The implementation of acoustic forces to
generate replenishing electrolytic flow
circulation in recessed features
Introduction
Electrodeposition is a complex process and truly
multi-physics in its nature, coupling together the solutions
of several physical phenomena such as fluid-flow,
temperature, ionic concentrations, electric current and
chemical reaction rates. In recent years the drive to
produce high quality, tiny yet intricate components has
lead research effort in the deposition of micro scale high
aspect ratio features.
Key to the deposition quality in these recessed
features is the ability to replenish the ionic species.
Deposition rates are otherwise limited to the speed of
diffusion processes because they are flow dead zones and
subsequently there is a tendency for void formation in the
bottom of the trenches as the concentration of ionic
species depletes. Towards this aim there has been much
interest in using either chemical additives that can
produce a ‘superconformal’ acceleration of the deposition
from the bottom of the trench, known as CEAC, see [1];
or more recently high frequency sound waves that
generate electrolytic flow deep into these recessed
features and replenish the supply of ionic species to
produce a faster, symmetrical void-free filling of the
trench, see [2].
This paper describes some of the fundamental work
concerning the development of numerical models for the
electrodeposition process of copper. This work has been
undertaken in collaboration with Herriot-Watt University,
as part of the MEMSA project in order to explore such
scenarios as the deposition of high-aspect ratio features
and the use of megasonic agitation to improve the
deposition quality.
Key to this research is the development of the
underlying numerical models of deposition; this in
particular involves the accurate representation of the
moving metallic/electrolyte interface. Previous work by
Hughes et al. [3], has utilized a level-set technique to
represent this interface motion, however in this paper an
alternative and more accurate approach has been
developed that is based on a novel variation of the donoracceptor technique. Results are presented for two types of
interface deposition kinetics; driven respectively by
Ohm's and Butler-Volmer laws.
Additionally a finite element based approach to
examine the intrinsic and residual stresses in the
electrodeposits is described; as the strength and durability
of these deposits is of crucial importance to the
operational expectations of the derived devices. The
method described is generic and can be adapted to
examine such affects as variation of grain size on stress
profiles.
Finally, the effect of acoustic agitation on the
electrolytic flow within recessed, high aspect trenches has
been considered; as a pre-cursor to their implementation
in the full electrodeposition model. CFD simulations have
been computed over trenches of varying aspect ratio to
examine the influence of the acoustic streaming on the
electrolytic transport.
All software development and numerical analysis has
been implemented within the unstructured, control
volume CFD code PHYSICA [4] and results from the
modeling simulations are discussed in the subsequent
sections.
Modelling of Electrodeposition: Primary Current
Distribution
The formula for the deposition rate that evaluates how
much metal is deposited during each time step is
described by the equation below, where v has units (m/s)
in
v
.
zF
Here  is the molar volume of copper (equal to
7.1·10-6m3/mol), i is the current density (A/m2), z is the
charge number (equal to 2 for copper), F is Faraday’s
constant (equal to 96485.33 Coul/mol), and n is the
normal to the electrolyte/metal interface pointing into the
electrolyte.
Dependent on the electrodeposition regime to be
simulated, we distinguish the primary current distribution,
governed by Ohm’s law, and the tertiary current
distribution, governed by the Butler-Volmer law.
The following assumptions are made: (i) a single
cathodic reaction takes place at the cathode; (ii) the
physical properties of the electrolyte remain constant;
(iii) the activation overpotential at the cathode is constant;
(iv) the anode does not change shape during
electrodeposition and the cathode is of primary interest,
(v) only basic electrodeposition is considered with no use
of electrolytic additives or accelerators.
This section discusses the numerical modelling of the
electrodeposition process of copper, provided that the
current distribution follows Ohm’s Law, also known as
the primary current distribution. This distribution is
normally applicable if the resistance of the electrolyte is
much higher than the resistance of the interface, and takes
into account the migratory properties of ions in electrolyte
and does not consider diffusion mass transport. The
current density in this case can be written as
i   el  ,
where σel is the electrolyte’s electric conductivity
(A2s3/kg/m3 ) and  is the electric potential (V).
We start with the simplest situation, in which copper
is deposited on the horizontal surface. This geometry
restricts the growth of the deposited metal to 1-D. The
governing equation for the potential field in the
electrolyte obeys the Laplace equation  2  0 . In this
case the potential  depends linearly on the vertical
coordinate y of the domain, y[h, L], where h is the
height of the electrolyte/metal interface that changes as
the deposition process develops, and L is the overall
height of the domain. We assume that (L) is kept equal
to 200 V, while (h) = 0. Thus, as the interface moves
higher, the gradient of the potential increases.
The numerical solution is obtained by PHYSICA by
applying a variation of the donor-acceptor method. The
domain is covered of by a mesh of rectangular elements.
In each time step, we determine the layer of elements that
contain the interface, and update the domain to become
the electrolyte above the new location of the interface.
For the interface layer PHYSICA finds the potential, and
based on that, the current density i and the deposition rate
v are computed to determine a new position of the
interface.
The values of the analytically computed and
numerically obtained deposition rates are plotted in
Figure 1. The graph for the numerical results has a
staircase-like shape, different from the linear shape of the
analytical solution. This is due to the fact that our
numerical method does not update the deposition rate
while the interface remains inside the same layer of
elements. When such a transition to a new layer takes
place, both graphs coincide. Notice that the deposition
rate grows as the gradient of the potential grows over
time. On the other hand, the graphs for the interface
position (analytically and numerically found) are virtually
indistinguishable, and therefore are not presented here.
Now we turn to a two-dimensional situation, in
which copper is deposited on a surface with a trench. The
2-D geometry is presented in Figure 2; due to symmetry,
only the left half is shown.
Figure 2: Two-dimensional geometry (the left half) with
the governing equation and the boundary conditions for
the primary current distribution. The interface is marked
with a thick line
The governing equation for the potential field in the
electrolyte obeys the Laplace equation. The aspect ratio r
of the trench is defined as b/(2a).
4.1000E-05
Deposition Velocity (m/sec.)
4.0500E-05
4.0000E-05
3.9500E-05
3.9000E-05
3.8500E-05
Analytical
Numerical
3.8000E-05
3.7500E-05
3.7000E-05
0
5
10
15
20
Time (sec.)
Figure 1: Analytically and numerically computed
deposition rate for the primary current distribution
Figure 3: (a) the interface position after 3.5 min,
(b) deposition thickness in the middle part of the side
wall (the top line) and at the bottom of the trench (the
lower line)
The implementation of the numerical solution by
PHYSICA in the 2-D case is more complicated than in
the 1-D case. A special code is written for determining the
position of the interface in each time step. For each
element that contains the interface in the beginning of a
time step, the volume of deposited metal by the end of
this step is computed. If that volume exceeds the
remaining capacity of the element, the excess is
recursively redistributed to its neighbouring elements in
proportion to the difference of potentials at the elements,
until the whole excess is redistributed. Numerical results
of electrodeposition for the cell with aspect ratio 1:1,
a = 75μm, b = 150μm are shown in Figure 3.
As can be in Figure 3, the deposited metal layer grows
faster on the side wall than at the bottom of the trench,
thereby leaving a void. A similar behaviour is observed
for other aspect ratios, e.g., 2:1 and 3:1.
Modelling of Electrodeposition: Butler-Volmer
Current Distribution
The tertiary current distribution model takes into
consideration ion transport due to diffusion described by
Fick’s second law. Concentration is the variable factor in
the diffusion layer and the kinetic expression at the
electrode has to take into account the concentration of
ions at the surface. For the tertiary current distribution,
the current density i is given by the Butler-Volmer
equation
i  i0
a = 75μm, L-b = 500μm. The initial concentration was
taken to be 350 mol/m3, α = 0.5, η = -0.3V, i0 = 26 A/m2
and the temperature T = 293ºK. The simulation was run
until either the trench is filled or the copper ions
concentration is depleted.
Figure 5: Results for r=1:1
For each aspect ratio, the numerical results after the
simulation was completed are shown in Figures 5-7,
respectively, where part (a) shows the deposition level
and part (b) shows the concentration distribution. The
simulation time for r=1:1 was 22 min and, as shown in
Figure 5, the trench is completely filled, while the
minimum concentration in the neighbourhood of the
interface was still fairly large.
cint
 F
exp(
),

c
RT
where i0 is the exchange current density (A/m2), cint and
c∞ are the molar concentration of copper at the interface
and in the far field (mol/m3), α the transfer coefficient, R
is the gas constant (equal to 8.314 J/mol/ºK), T is the
temperature (ºK) and η is the overpotential (V). Unlike
the primary current distribution, the current density i is
depends on the metal concentration and the overpotential.
The governing equation and the boundary conditions are
shown in Figure 4.
Figure 6: Results for r=2:1
Figure 4: 2-D geometry with the governing equation and
the boundary conditions for the tertiary current
distribution. The interface is marked with a thick line.
We
performed
numerical
modelling
of
electrodeposition for three values of the aspect ratio of the
trench equal to 1:1, 2:1 and 3:1. Recall that the aspect
ratio r is defined by a/(2b); see Figure 2. In all cases
Figure 7: Results for r=3:1
The simulation time for r=2:1 was 25 min and, as
shown in Figure 6, a void is left in the trench; the
minimum concentration in the void is depleted, i.e., too
small to get rid of the void. The simulation time for r=3:1
was 23 min and, as shown in Figure 7, the metal is
deposited as a layer along the trench, and the minimum
concentration is depleted, i.e., too small to get rid of the
void.
Prediction of Intrinsic Stress Distribution by Finite
Element Analysis
The presence of residual or intrinsic stress in the
electrodeposited films is the important factor to consider
in devices because such a stress causes the deposited layer
depending on its thickness and elastic properties to
expand or contract either to a concave or to a convex
shape. In some cases intrinsic stress exceeds the strength
of the deposited layer, resulting in film cracking,
deformation of devices, and interfacial failure. Even
moderate tensile or compressive stresses in films may
lead to geometric distortions, warping or shrinkage. Thus,
the intrinsic stress in films deposited on substrates should
be made as small as possible by controlling the deposition
conditions of a film.
Many factors affect internal stress in electrodeposited
films, including film composition, nature of the substrate
surface and the deposit, current density, the deposit
thickness, among others. It is essential to determine
effects that process variables may have on stress. This
will allow us to find such values of the critical process
variables that the deposition process is optimized with
respect to a chosen objective, e.g., a desirable stress
profile or a high deposition rate.
We are not aware of any established mathematical
models that describe the influence of process variables on
intrinsic stress. Most of the studies in this area are based
on experimental measurements. For example, in [5] Kim
et al. study the behaviour of the intrinsic stress of copper
film electrodeposited on the Au seed. The authors
measure the substrate’s radius of curvature before and
after film deposition and compute the stress by Stoney’s
equation. They demonstrate how the value of stress
depends on the film thickness and on the applied current
density.
Engelstad et al. [6] point out that Stoney’s equation
can only be reliably used under the assumption on the
uniform film stress. To overcome this drawback, they
make stress predictions by the Finite Element analysis,
taking interpolated measured curvatures as input and
deliver intrinsic stress as a function of location and
direction.
In this work, we rely on the data presented by Kim et
al. [5] for the average values for tensile intrinsic stress
obtained for 50-, 100-, and 150-μm thick copper films
deposited on an Au seed and with the current density
varying within the interval 25mA/cm2-65mA/cm2. For the
purpose of predicting the intrinsic stress values we use the
analysis software package PHYSICA. The nature of the
intrinsic stress does not allow us to compute it directly.
Instead, we use the finite element approach to the
numerical modelling of another stress-generating process
and then convert the obtained solution into the desired
values on the intrinsic stress. In particular, we select heat
transfer as this alternative process and take either the
thermal expansion coefficient or the temperature change
as the process parameter. We establish such values of the
chosen parameter that generate the values of tensile stress
close to the measured intrinsic stress values reported in
[5]. Our approach is close to the one used by Chen and
Ou [7], who express intrinsic strains in film and substrate
in terms of the equivalent thermal expansion coefficients
and the equivalent temperature difference.
Having conducted numerical experiments with
PHYSICA, we have observed that the average values of
the process parameter  ( h, i ) can be accurately
approximated by a function of two variables, the film
thickness h and current density i, given by
 (h, i)  f1 (h)ln(i)  f 2 (h) ,
where f1(h)= 0.0279h2 - 2.3202h + 128.45 and f2(h)=
0.0279h2 - 2.3202h + 128.4; see Figure 8.
Figure 8: Process parameter  ( h, i ) as
a function of two variables
In order to obtain the values of intrinsic stress reported
in [5] for a given pair of values of h and i, we fix the
process parameter to be equal to  ( h, i ), run the heat
transfer simulation and output the computed average
stress value.
The average value of the process parameter can be
expressed as
h
 (h, i ) 
  (h, i)dh
0
h
,
where  (h, i ) is an appropriate distribution function.
Differentiating, we find this process parameter
distribution function for a fixed current density as
 (h, i)   (h, i)  h
 (h, i) .
h
Substituting the value of the parameter  ( h, i ) into
this formula we derive the expression for the distribution
function
 (h, i)  ln(i )(0.0279h 2  4.64h  128.45)
 0.2703h 2  15.7464h  417.
This function is then embedded into the PHYSICA
code, which allows us to view the process parameter
distribution; e.g., see Figure 9. The figure shows one half
of the geometry symmetric with respect to the vertical
axis.
Figure 9. Process parameter distribution for the 100-μm
thick copper film deposited at i = 35 mA/cm2
In turn, based on this distribution function we are able
to simulate numerically the values of intrinsic stress in the
elements of the deposited copper; see Figures 10 and 11.
Figure 10. Intrinsic stress distribution in copper film
deposited on Au seed for h = 100 μm:
(a) i = 35 mA/cm2; (b) i = 55 mA/cm2
Figure 11. Intrinsic stress distribution in copper film
deposited on Au seed for h = 150μm:
(a) i = 35 mA/cm2; (b) i = 55 mA/cm2
Acoustic Streaming
Megasonic agitation provides a promising means to
solve the problems concerning void formation in high
aspect ratio trenches and vias by enhancing a replenishing
ionic transport that would otherwise not penetrate deep
into these features. The effect of this is to decrease the
Nernst diffusion layer and therefore increase the kinetic
rate at the interface. Experiments at Heriot-Watt by
Kaufmann et al. [8], have shown an improvement in both
the deposition rates and deposit quality for microvias with
features of aspect-ratio up to 2:1. being successfully
filled; the limitation of trenches with ratios larger than 2:1
being reported as due to the lack of conductive seed layer
deep into the microcavity.
The Acoustic Streaming process is generated from the
high frequency agitation (~ 1MHz) of a Piezoelectric
plate generating an acoustic intensity of 60kW/m2 with
corresponding high frequency pressure variations within
the feature. These pressure variations generate Reynolds
stresses that drive a steady fluid recirculation within the
cavity that replenishes the supply of the reacting ionic
species from the bulk of the containing bath.
The numerical solution process is in two stages; the
first concerns the analytical solution of the time
dependent wave equations as developed by Rayleigh and
Nyborg for channels between infinite parallel plates and
is detailed concisely by Nilson and Griffiths [2]. The
second stage involves calculating the time averaged
Reynolds stresses that are produced from these harmonic
wave solutions and their numerical implementation into
the Navier-Stokes equations.
Simulations have been made to examine the effects of
acoustic streaming in extremely narrow features with
widths in the order of tens of microns so that the resulting
flow profiles can be compared with those produced by
Nilson and Griffiths [2].
In order to accurately represent the forces that drive
the acoustic flow the computational grid must be fine
enough to capture the large Reynolds stresses that occur
next to the feature walls. A grid size of 49x99 regularly
spaced cells was sufficient to achieve this within the
trench region as shown in Figure 12 below. The acoustic
forces permeate throughout the feature from top to bottom
and have a magnitude in the order of 3.5×10-4 N/m3, the
distribution of these forces is illustrated in Figure 13.
As can be seen in Figures 10 and 11, the increase in
current density during electrodeposition induces larger
stress and also that stress of thick copper films (over
50μm) increases with further growth of the deposited
copper. The average values of internal stress in all
examples shown in Figures 10 and 11 are fairly closed to
the data reported by Kim et al. [5].
Figure 12: Computational grid
Figure 13: Acoustic Force distribution
These forces produce a flow profile that is driven
downwards at the feature walls and recirculates through
the middle of the feature due to continuity, as shown in
Figure 14.
Conclusions
The deposition simulations presented in the first
section of this paper illustrate that the deposition models
exhibit physically realistic behaviour and as such the
interface tracking method appears to work well.
Regarding intrinsic stress, our approach can deliver the
relevant predictions, provided that appropriate
experimental measurements under various deposition
conditions are supplied. It is an interesting research goal
to adapt our approach for predicting the variation of the
grain size of the deposited metal.
The Acoustic Streaming modelling highlights the
effect of the fluid profile within the features and shows
reasonable agreement with profiles given in
Acknowledgments
This research was supported by the EPSRC. The
authors are grateful to the industrial and academic
partners Merlin Circuit Technology Limited and HeriotWatt University. Also thanks to Drs H. Lou and S. Ridout
of the University of Greenwich for useful discussions.
References
Figure 14: Contours of velocity within the feature
Normalised flow profiles through the channels can be
seen to be affected by the channel width and are fastest in
the region of the maximum Reyolds stresses at the feature
walls and are asymptotic with increasing width. In
Figure 15, the generated velocities have been normalised
by a nominal streaming speed of 2.6×10-6 µm and are
therefore tiny, the profiles shown in Figure 15 are
however in good agreement with the results presented by
Nilson and Griffiths [2].
1.
2.
3.
4.
5.
6.
7.
Figure 15: Acoustic Streaming Velocities
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becomes similar to one that has been generated through
wall movement in a static fluid (ignoring the non-slip wall
condition) and as such the complications of the streaming
physics might be replaced in numerical simulations for
larger width features with the application of a fixed wall
velocity.
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