Multi Physics Modelling of the Electrodeposition Process
Michael Hughes, Christopher Bailey, Kevin McManus
University of Greenwich
Park Row, Greenwich, London SE10 9LS
[email protected], [email protected], [email protected]
Abstract
This paper describes some of the initial work in the
development of a high fidelity multi-physics model of
electrodeposition undertaken as part of the MEMSA
project. This is a collaborative research project between
the universities of Heriot-Watt and Greenwich which
aims to investigate advanced electrodeposition. processes.
A key component of this research is the development of a
numerical model that can be verified against experimental
work which will be undertaken at HW. Model
development focusses on; (i) the represention of the
moving interface through a level-set technique, (ii) the
implementation of the associated moving boundary
conditions and source terms together with considerations
regarding the electrode kinetics boundary condition.
Accurate modelling of the electrode kinetics is crucial to
any electrodeposition model as it drives the deposition
process and influences the distribution of the solved
variables of which it is itself a non linear function. The
unstructured Control-Volume based multi-physics CFD
code Physica provides the framework in which the
electrodeposition models will be built and this paper
should be of particular interest to applied modellers
wanting to modify CFD codes to simulate the
electrodeposition process.
1. Introduction
Electrodeposition is a process that is truly multiphysics in its nature and of considerable importance to the
microsystems and semiconductor industries. The
reduction in length scales and the replacement of
aluminium interconnects and trenches with copper has
increased the operational speed of CMOS devices, largely
due to copper’s higher conductivity and reduced
metallization capacitance. Much important modeling
work has been applied to the simulation of these types of
trenches and investigating the feature filling with respect
to void formation. In particular Wheeler et al. [1,2], have
published interesting results from modeling at the submicron scale where chemical additives have been used to
produce a superconformal, ‘bottom-up’ filling of highaspect ratio features for the deposition of Damescene
copper. This process is known as CEAC (Curvature
Enhanced Accelerator Coverage), However at larger
length scales, (mm), this process does not necessarily
scale up and the problem of ion depletion within the highaspect ratio features can cause problems of void
formation, through effects such as ‘current-crowding’.
This situation may be ameliorated by using forced
convection to replenish the supply of reacting ions to the
electrode surface. However because of the possibility of
flow dead-zones in these trenches forced convection may
not be sufficient to improve matters. The application of
pulse reversed waveforms may diminish void formation
[3] by improving the distribution of the time averaged
deposition rate along the trench side walls, possibly
because the concentration of reacting ions has sufficient
time to recover during the plating off-time
Attempts to numerically model the electrodeposition
process are challenging as they must solve a system of
coupled non-linear equations with the added complication
that the governing equation set changes under different
physical situations; for example as the deposition current
varies from primary to secondary, tertiary or diffusion
limited regimes [4,5] Additionally the representation of
electrode kinetics, the driving force for deposition, is of
key importance and is complicated by its influence from
the electrode surface overpotential and the concentration
of reacting ions in the immediate vicinity of the
depositing interface. These factors can in turn be
influenced by effects such as forced convection of the
electrolyte replenishing the ion supply to the deposition
interface, the electrode potential difference or total
current applied to the electrolytic cell. The governing
equations may therefore include all or a combination of
the momentum, heat, concentration and electric potential
equations with various degrees of intercoupling by
electromigration, convection and importantly through the
reaction rate boundary condition at the electrode surface.
Much of the modeling work to date has focused on
deposition within particular current distribution regimes
where assumptions can be made about the deposition
process and some simplification of the equations may
therefore be possible, for example electromigration if an
excessive supporting electrolyte is used [4,6], or electric
field if the deposition rate is diffusion controlled and the
surface overpotential can be provided from experimental
voltammetry [1]. It is clear that developing a model to
solve the full equation set with the electrode overpotential
being implicitly calculated is a challenging task which
must consider the underlying physics carefully, ideally
supported by experimental results.
A suitable technique must be chosen to represent the
moving interface, here the Level Set Method [1,2,7,10]
has been chosen. This paper considers these issues from
the viewpoint of developing a model from scratch within
a CFD framework. This framework aims to provide a
good starting point for model development as momentum
and heat equations are then implicitly handled. Further to
this a sensible first step is to build a model for the
simplest electrodeposition scenario, namely, that of the
primary current distribution regime with a single ionic
species. In this scenario a DC current drives the process
and the deposition rate is governed by Ohm’s law this is
discussed later in Section 5. Further to this, progression of
the model into secondary and tertiary current distributions
is considered in Section 6, these deposition regimes
introduce more numerical complexity because of the nonlinear reaction rate and boundary constraints at the
deposition surface. In Section 7 future work towards the
goals of the MEMSA project are considered together with
some thoughts on how to address them. Implementation
of the model framework at this stage of the project aims
to establish numerical stability and reflect qualitative
experimental behaviour.
We begin with an overview of the governing
equations and a brief discussion of the current deposition
regimes and subsequent boundary conditions.
2. Governing equations and deposition current
deposition regimes.
The governing equations for the electrodeposition
process are:
The Navier-Stokes equation if the electrolyte is under
the influence of forced convection:
du
 uu  P  2u  Su 1
dt
where S u represents momentum source for forced

convection such as electroyte stirring. Together with the
continuity equation:
 (.u )  0
2
and the temperature equation with external heating ST
C p
dT
 C p uT  k2T  ST
dt
3
The flux of ionic species is given by Paunovic and
Schlesinger [8]:
Ni   zi ei ci   Di ci  uci
where
 , ci , Di , e,i , zi are
4
respectively; electrolyte
electric potential, concentration and diffusion coefficent
of the ith ionic species, elementary charge, ion
mobility.and ion species valency. The first term on the
RHS represents ion drift due to the electric field, the
second diffusion of ions and the last term movement by
convection. Ionic mobility is given by
i 
Di
; k is the Boltzman constant 5
kT
Migration is essentially an electrostatic effect that
arises due the application of a voltage on the electrodes if
there is a large quantity of the electrolyte (relative to the
reactants) it is possible to ensure that the electrolysis
reaction is shielded and not significantly affected by
migration, in such circumstances the first term on the
RHS can be neglected [4].
Concentration of ionic species can be represented by
taking the divergence of the above term and expressing
this in the total derivative for concentration of species to
give the equation:
c i
   (uc i )    ( Di c i )  ez i   ( i c i  )
t



6
convection diffusion migration
The equation set is closed with the electric potential
equation togther with suitable boundary conditions for the
equation set. The time scale for establishing a DC field is
much faster than for establishing concentration gradients
so under DC conditions the electric field can be expressed
through electric potential as a Poisson equation without
time influence:
2  
4

 ez c
i i
;
7
where ε is the dielectric constant. An alternative to
solving equation 7 given by Griffiths [9] is to enforce
electroneutrality in the bulk electrolyte, in which case the
electric field becomes an unknown constant which is
determined as part of the overall solution from the
governing condition:
 zc 0
n
1 i i
8
As with equation 7 this condition applies at every
point in the solution domain, except at the thin layers
adjacent to the electrode boundaries, the electrical double
layer [9] which is of the order of <~1000 Angstrons in
width. In these thin layers the deposition current is
accounted for by an electrode kinetic function, typically
the Butler-Volmer equation [1,4,5,9]. In this electrical
double layer the electroneutrality condition breaks down
and a spacial charge exists [8]. This charge is referred to
as the surface overpotential and its value is one of the
parameters that drive the reaction rate through the ButlerVolmer equation. (Figure 1) In line with this authors
present knowledge and literature read to date, the double
layer is not explicitly taken account of with DC
conditions. Instead the overpotential is either specified [1]
or details of its explicit calculation are not given special
attention [4,5]. However this region will effectively
present a discontinuity to the electric potential distribution
and therefore some thought is necessary towards the
application of the boundary conditions for equation 7, this
is discussed later in Section 6.
For AC conditions at low frequency it is likely that the
above equations (6,7) can still be utilised. However at
higher frequencies and if the numerical model is to
implicitly calculate the overpotential it may be necessary
to introduce a sub-model to calculate the overpotential
which approximates the layer as a plate capacitor. This
complication may be bypassed if sufficient overpotential
vs applied voltage or current data is available.
3. Boundary Conditions
Ritter et al. [4] and Drese [5] give concise descriptions
of four deposition regimes, the relevant equations and
boundary conditions are listed here, the heavy line in
Figure 1 below being the cathode-electrolyte interface
To advect the deposition interface the level set method
of Osher and Sethian [10] was chosen.. This is a
numerical method for tracking interfaces and shapes that
has been sucessfully applied to the electrodeposition
process [1, 7]. It has the advantage of sucessfully
handling surfaces that have sharp cusps or corners,
without smearing, through a fixed mesh and has the
advantages of a Eularian approach. Within the CFD code
Physica the existing Level Set algorithm can be readily
modified by decoupling its propagation from the
momentum velocity and replacing this with a deposition
velocity, vdep which is calculated as below:
vdep 
Figure 1: Boundary Condition schematic
 Tertiary current distribution
The deposition current, ibv, at the cathode is given by the
Butler-Volmer equation and is a function of the local
interface concentration to bulk concentration of reacting
ions Cint fce / Cbulk and electrode overpotential, η. At the
electrolyte-cathode interface condition b in Figure 1 needs
to be enforced.
 Secondary current distribution
If concentration gradients can be ignored because the
concentration of ions is very high then the electric
potential equation is solved with condition c in Figure 1.
 Primary current distribution
If the resistance of the electrolyte is much higher then that
of the interface then the Current density passing through
the electrode is given by Ohms law, condition a in Figure
1 is applied.
 Diffusion limited current distribution
At sufficiently high overpotentials, a limiting current
is reached as the ionic concentration at the interface
approaches zero and electric potential equation can be
ignored. At the interface, c = 0, and the deposition current
is calculated as IDL = nFD dc
dn
4. Moving the interface
iBV 
; where Ώ is molecular volume,
nF
n is charge number and F, Faradays constant, the units of
vdep are metres/sec.
In the level set method a variable φ is used to keep
track of the moving interface. This variable is initialised
to zero along the interface at the start of a simulation and
at all other places in the computational domain stores a
value representing the shortest distance to the interface
with positive values in front and negative values behind.
The calculated deposition velocity, vdep, is then used to
advect the interface and the new interfacial distances are
updated as φ is reinitialised. Because the distance
function, φ, is updated at computational nodes diffusional
smearing from the propagation of the front can be kept to
a minimum. The procedure is as follows:
 Initialise the level set function, φ(x,y,z)=0
 Update material properties



Solve

 vdep , this updates the interface
dt
position only.
At the end of the timestep reinitialise φ in
locations other then the interface to update
distance from the interface by iterating:
i1  i   S (o )(1   ) where φo is the
value of the variable at the start of the reinitialisation, Δτ
is a pseudo time step that is set to be 1 th of the
10
minimum distance between the current computational
cell centre and the centre of the adjacent cell
Max
,
d ap
which is closest to the zero level set.
S (o )
S ( ) 
is
a
sign
0
max 2
( 0) 2  (d ap
)
function
calculated
by
. Futher details of the
scheme can be found in [1,7].
5. Simulating the Primary current Regime
The primary current distribution provides a good first
target for model development, because of the simpler
governing equation set. Under these conditions the
deposition current can be modeled using Ohm’s law. If
we make the assumption that the concentration of reacting
ions is sufficiently high then we can ignore the influence
of the ion concentration in equation 6, which has the
advantages of reducing the equation set to that of solving
a Laplace equation for electric potential using equation 7
with the RHS reduced to zero. At the deposition interface
the current normal to the surface is given by boundary
condition a in Figure 1.
The computational grid is shown in Figure 2 below,
together with the boundary conditions for the electric
potential equation. In this instance potentials are fixed at
either ends of the domain. An alternative is to replace the
fixed potential boundary condition,   1.0 , by
specifying a current boundary condition, i.e.
k

 I anode .
n
average. If this is applied then the electric potential across
the interface is rendered numerically continuous and the
interface condition becomes similar to conjugate heat
transfer in standard CFD simulations in that the condition
k electrolye


 K metal
is automatically satisfied.
n
n
Additionally it is convenient to note that under the
circumstances where Kmetal >> Kelectrolyte, and Kmetal is large
as is the case with metals and given that the currents
imvolved are small, the cathode boundary condition
(   0 ) will permeate through the metallic deposited
layer and anchor the interface electric potential on the
metal side to the applied boundary condition.
Calculating the electric potential gradients and hence
current from Ohm’s law can be achieved in an
unstructured discretisation scheme by utilising Gauss’s
divergence formula as shown below where k is the
electric potential and norx, nory, norz are the Cartesian
components of the face normal vectors. So for example
when calculating the x-direction gradient, then only norx
contributions are used.

k
faces
1
 face  Area face  (norx face | nory face | norz face )
Cell _ volume
The gradients are taken from both sides of the interface
separately, with the value at the interfacial cell face  face ,
calculated by extrapolation along the gradient of
shown in Figure 4 below.
 as
Figure 2: Computational grid and wall boundary conditions
for DC conditions
At trench interfaces ‘current crowding’ effects occur
because of a pinching effect on the electric field from
sharp corners. In these instances voids may be formed as
the current and hence deposition rate is higher at these
edges. The results of this phenomenon on deposition can
be seen in Figure 3 below as time increases a void in
enclosed in the trench as seen in the RHS picture.
Figure 4: Calculating gradients across an interface
This complication is necessary for taking gradients
across a region that encloses materials with different
electrical conductivities. It may not be necessary for the
calculation of the deposition driving current as the electric
current is only required in the electrolyte region of the
computational domain and in particular in the cells
adjacent to the interface on the electrolyte side
Figure 3: deposition through time
Under the primary regime, no special attention needs to
be made at the interface boundary in terms of the
conditions except to ensure that the electric conductivity
across the interface is calculated using a harmonic
A schematic of the solution domain is shown in Figure
6 above where BV stands for the deposition current as
given by a Butler-Volmer equation
A complication with these boundary conditions is that
the current passing through the interface is goverened by
the surface kinetics and is defined by the Butler-Volmer
equation. If we now consider
the solution of equation 7.
over the entire solution
domain M   E , as was
the case in section 5, then the
current, Idep, passing across
the interface  INT is governed
Figure 5: Current distribution through interface
6. Simulating the Tertiary Regime
Advancing the model to introduce ionic concentration
involves greater restrictions at the deposition interface.
This is now considered under DC conditions with a
single ionic reacting species and an assumed constant
overpotential.
In this scenario, equation 6 is solved for bulk
concentration c 
c
cbulk
together with the equation for
electric potential (7) with the RHS again equated to zero
as only one species is considered. At the boundary
between the metal-electrolyte interface, condition b in
Figure 1 needs to be satisfied and hence the position of
electrolyte side interface cells must be tracked throughout
the simulation so that the boundary source terms for
equations 6 and 7 can be applied.
by conduction alone and of
course influenced by the
relative values of the applied
boundary conditions.
In the tertiary and
secondary current regimes the
current crossing the interface
is governed by the surface
kinetic function and therefore an appropriate ‘sink’
boundary condition must be applied to the electrolyte-side
computational cells that are adjacent to the interface.
Assuming that an appropriate boundary condition is
applied here and equation 5 is computed over the entire
domain,  M   E , then the current passing across the
interface will be incorrect as in addition to the applied
sink it will contain a conduction contribution. This can be
avoided by splitting the computation domain into two
sides,  M and  E and linking these regions by
appropriate sink/source type boundary conditions; the
current leaving the electrolyte should be equal to the
current entering the deposited metal. To recover current
from this type of calculation the technique discussed
section 5 is used. Figure 8 below explains this idea
showing the results from a 1D test case in which the
unequal spacing of the grid cells around the interface area
is a way of testing the current calculation within the
model (otherwise not a sensible grid arrangement).
Current passing across the interface is continuous and is
of equal magnitude to the computed deposition current
from the Butler-Volmer equation.
Figure 6: The solution domain
Figure 8: Current across an interface; splitting the domain.
This situation requires careful application of the
current source terms in the  M side when handling
corners. Figure 9 below shows the recovered current from
such a situation, here the required source for the electric
potential equation in  M at the corner cell, A, is
calculated as the sum of the sinks at B and C.
being more likely to give a smoother deposition profile at
the surface as it is a solved variable and will consequently
smooth the deposition current as calculated from the
Butler-Volmer equation.
The Boundary condition for the solution of reacting
ion concentration may also be applied in a similar manner
to the current, as a flux loss at the interface,
D
c
  I BV or by fixing the interface concentration to
dn
zero if the concentration drops below a tolerance and the
deposition current enters the diffusion limited regime.
The application of the latter of these conditions is similar
to equation 9:
S  A face  Coeff  (0  cell ) ;
Figure 9: Current across an interface
These sources and sinks are calculated using the ‘upto- date’ iteration values of deposition current as returned
by the Butler-Volmer equation and are applied as flux
type boundary conditions either side of the interface as
shown below
S  A face  Coeff  (
 I BV
 cell ) ;
Coeff
electric potential at the cell center and a small value is
used for the Coeff (i.e. 1E-10) and the cell face area, Aface
is estimated in the computational cell as:

i 1
Start of time step
Store interface position in variable
9
where IBV is the deposition current, cell is the value of
all _ cell _ faces
Coeff is calculated from D  Area / dn where dn is
the distance between the computational cell centre and the
interface. The larger the value of Coeff then the stronger
is the tie of the computational cell centre to the applied
surface zero value. If the magnitude of the diffusional
coefficient, D, is extremely small (i.e. 10-9-10-10) , then in
practice it may be necessary to increase the value of Coeff
by a possible order of magnitude so that the influence of
this source term is felt.
The solution procedure for the simulation is as
follows:

Area face 
 n face  d face
|  |
where   is the gradient of the level set function and
Area, n are the cell face areas and normals repectively and
d has the value 1 if the cell face is on the interface and
zero elsewhere. It is updated at the end of a time step in
line with the level set distance variable  and gradient
 .
For simulations in which surface overpotential data is
known a priori the distribution of electric potential is only
required in  E because there is no need to calculate the
overpotential from the distribution of electric potential
across the interface. Under these circumstances the
solution of the electric potential equation may be limited
to the electrolyte region,  E , together with appropriate
current sink at the interface and the deposition current
estimated from the current that is recovered from the
electric potential in the cells adjacent to the interface or
from Butler-Volmer equation itself. The former of these
1  int erface
d
0  otherwise
Start of iterations

Calculate Idep from Butler-Volmer function *

Solve for
 in E ;
applying  k   I
at
BV
n
d=1

Solve for
c in E applying  D c  I BV at
n
d=1
End of iterations

Update surface level set variable and associated
parameters
End of time step
To improve numerical stability the calulation of Idep can be moved from
the iteration loop to the start of timestep.
Figure 10 (a-d) shows the deposition profile through
time, showing a tendency for the deposition to be more
concentrated at the top corners of the trench. This is
highlighted in Figure 11 which shows the profile of Idep, it
has higher magnitude at the top trench edges in line with
the deposition rate. Figure 12 shows vectors of the current
profile in the electrolyte side of the trench region and also
highlights a higher concentration of electric current at the
trench corners.
7. Consideration of the overpotential calculation
The overpotential which acts over a tiny ‘doublelayer’ effectively presenting a discontinuity to the electric
potential euqtion. Its influence also feeds back into the
electric field and ionic concentration equations by means
of the electrode kinetics and hence errors in its calculation
can result in numerical instability.
Many publications concerning the numerical simulation
of the electrodeposition process do not give special
attention the calculation of the overpotential and details
seem to be hidden in the discretisation scheme or implied
to be calculated as the potential difference
metal  electrolyte across the interface. Consideration of

Figures 10 (a-d): deposition profile through time.

the boundary conditions gives some insight into the
behaviour of these equations at the interface. The
boundary conditions b in Figure 1 form a closed set of
equations that can be iterated to a steady state solution by
assuming that the gradients are to be taken over a layer
that spans the interface boundary into the bulk
region, n
for electric potential and nc for
concentration. Known constant values are applied for the
cathode transfer coefficient,  c , Temperature, T and
exchange current density, Io. The suffix i denotes interface
values.
ci
  F 
I dep  I o
exp  c 
Cbulk
 RT 
I
 i 1  dep n  bulk
k
I dep
c i 1 
nc  cbulk
zDF
10
If we couple into the above equation an estimation of the
overpotential as the difference in potential across the
boundary n , i.e.
Figure 11: Surface Current from Butler-Volmer equation
Figure 12 Current vectors in electrolyte region
   i  bulk we find that the final
converged values are srongly dependent on the distance
n as shown in Figure 13 below. An obvious high level
concluson might be drawn: It seems likely that any
numerical model which attempts to implicitly calculate
 will encounter difficulty in resolving the grid spacing
around the interface region to a sufficient level and that
the calculated deposition current will therefore be mesh
dependent. Future work in this area will be essential in
moving the model forwards into AC waveforms. Two
lines of investigation are presently envisaged to introduce
a sub-model from which the overpotential can be
calculated namely:
i) Solve the equation set (10) in computational cells
immediately adjacent to the interface in the electrolyte
region. This method would use the cell values of
concentration and electric potential within these interface
cells as the bulk values and attempt to provide a more
accurate estimation of the interface concentration,
potential and deposition current by iterating the equation
set.
ii) To estimate capacitance and charge based on
parallel plate capacitor approaches such as the Stern
model [8].
Both approaches will be investigated during the course of
this project ideally with aid of accompanying
experimental verification.
R –Universal gas constant – 8.314 J/mole k
T –temperature – K
 c –Cathode transfer coefficient.
 – Boltzman constant – 1.38-23m2kg/s2k.
 – electric potential – volts.
 – Level Set distance variable – m.
 – metal electric conductivity – volts.
Acknowledgments
Thanks to Dr Nick Croft of Swansea University and
Dan Wheeler of Nist for comments and conversations.
Figure 13: Deposition current vs log deltan
8
Conclusions and future work
The
code
framework
for
modelling
the
electrodeposition process has been implemented within
the control-volume CFD code, Physica. Issues concerning
the application of surface source terms and movement
have been considered. The model shows numerical
stability and exhibits qualitative behaviour for primary
and tertiary current regimes with known overpotential.
Future work will address:
i) Testing the model against experimental data and
subsequent tuning to give quantitatively correct
behaviour.
ii) Investigating the calculation of surface overpotential
through sub-models or relationships that may link the
overpotential to concentration distribution.
iii) Advancing the model towards the simulation of pulse
reverse plating and AC waveforms.
iv) Application of megasonic vibration within the model
to encourage mixing and the replenishment of ions within
trenches of high aspect ratio.
Glossary
C – concentartion of reacting ions – moles/m3.
Ci – interface concentration of reacting ions. – moles/m3.
Cbulk – interface ionic concentration – moles/m3
c –Dimensionless concentration ratio [0,1]
D –Diffusion coefficient – m2/s.
e – Elementary charge – 1.602.10-19.
F – Faradays constant – C/mole.
IBV – Deposition current from But-Vol equation – A/m2.
Idep –Deposition current – A/m2.
Io –Exchange current density – A/m2.
K –electrical conductivity – m  .
1
n|Z –ion valency
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