Numerical modelling of the electrodeposition process

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDSInt. J. Numer. Meth. Fluids 2009; 00:1–33 Prepared using fldauth.cls [Version: 2002/09/18 v1.01]

Numerical modelling of the electrodeposition process

Michael Hughes1,∗ , Nadia Strussevitch1, Christopher Bailey1, Kevin McManus1,Jens Kaufmann2 and Marc P.Y. Desmulliez2

1 Centre for Numerical Modelling and Process Analysis (CNMPA), University Of Greenwich, Park Row,Greenwich, London, SE10 9LS, United Kingdom

2 MicroSystems Engineering Centre (MISEC), School of Engineering & Physical Science Earl MountbattenBuilding, Heriot-Watt University, Edinburgh, EH14 4AS, United Kingdom

SUMMARY

Electrodeposition is a widely used technique for the fabrication of high aspect ratio microstructurecomponents. In recent years much research has been focused within this area aiming to understandthe physics behind the filling of high-aspect ratio vias and trenches on substrates and in particularhow they can be made without the formation of voids in the deposited material. This paperreports on fundamental work towards the advancement of numerical algorithms that can predictthe electrodeposition process. It is split into three sections:Part I, Compares and contrasts two different numerical approaches developed to calculate themovement of the deposition interface.In Part II, 2-D simulations are presented for two deposition regimes; those where surface kinetics isgoverned by Ohm’s law and the Butler-Volmer equation respectively.Part III, examines modelling of acoustic forces and their subsequent impact on the deposition profilethrough convection. Copyright c© 2009 John Wiley & Sons, Ltd.

key words: CFD, Electrodeposition,Level Set Method, Megasonic Agitation

1. Introduction

Electrodeposition is a complex process and truly multi-physics in nature. It couples togetherseveral physical phenomena such as fluid flow, heat transfer, ionic concentration, electriccurrent and chemical reaction rates. In recent years, the necessity of manufacturing tiny, yetintricate, electronic components of high quality has focused the study of eletrodeposition onmicro scale high aspect ratio features. Key to the deposition quality in recessed features is theability to replenish the ionic species. Deposition rates are otherwise limited by the speed ofdiffusion processes in the electrolyte, as these features are consequently flow dead zones there isa tendency for void formation in the bottom of the trenches because the concentration of ionic

∗Correspondence to: [email protected]

Contract/grant sponsor: EPSRC ; contract/grant number: EP/C513061/1

ReceivedCopyright c© 2009 John Wiley & Sons, Ltd. Revised 21 Jan 2009

2 HUGHES ET AL

species becomes depleted. Due to these reasons, there has been much interest in using eitherchemical additives that can produce a ‘superconformal’ acceleration of the deposition from thebottom of the trench, see [1, 3]; or more recently, high frequency sound waves that stimulateelectrolye flow into these recessed features, replenishing the supply of ionic species [2]. For anymodel of electrodeposition, the presence of an accurate method for representing the interfacemovement is of key importance. This paper discusses two such methods; one relatively simpleapproach that tracks the motion of the interface through a stored variable and depositionrate and the other more complex, driving the interface motion through the advection of afree surface using the Level Set method. These methods are discussed in detail and compared.Simulation results from both techniques are presented for two deposition regimes; one drivenby electric current as calculated by Ohm’s law and the other by surface kinetics describedby the Butler-Volmer equation. In the third part of this paper the methodology used toincorporate acoustic forces and the effect of their implementation on the electrodepositionprocess is discussed.

2. Part I

Key to the development of numerical models of deposition is the accurate representation ofthe moving metallic/electrolyte interface. Previous work by Hughes et al [4], has utilized aLevel Set technique to represent this interface motion, however in this paper an alternativeand computationally cheap approach has been developed that is based on a novel variationof the donor-acceptor technique. Results are presented for two types of interface depositionkinetics, driven respectively by Ohm’s law and Butler-Volmer equation.

2.1. Numerical Algorithms

In this section the two methods will be examined. A 1-D test case is presented were thedeposition is driven by a fixed and constant potential difference to help illustrate the algorithmsand also to validate that both techniques exhibit time-step independent behaviour. This issuehad caused minor problems during the development of the Level Set approach. This issuewas resolved by careful calculation of current density across the deposition interface. This is ofprimary importance as it is the physical driver for deposit motion. In fact the calculation of thecurrent density across a moving interface requires careful programming to ensure continuityof the normal components. There are a number of possible routes to accomplish this goal andfinding a less complex route was a major motivation for the development of an alternativemethod to the Level Set approach.

2.1.1. Explicit Interface Tracking Method - EITM Using this technique a 1-D test case forthe deposition of copper has been simulated under the primary current regime (PCR), wherethe current density is calculated from Ohm’s law j = −σ (∇φ) and a scalar variable is storedto represent the cell filling. This stored variable has a value of 0 for a fluid cell and 1 for afilled cell. The filling of the cells is governed by the deposition velocity that is a function of thecurrent vdep = jΩ

2F where Ω and F are the atomic volume and Faradays constant, respectively,and 2 is the valiancy of cupric ions Cu2+. Multiplying vdep by the cell face area and ∂t ,thetime step length, a volumetric growth is found for the time step in m3. This volumetric growth

Copyright c© 2009 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 2009; 00:1–33Prepared using fldauth.cls

COMPUTATIONAL ELECTRODEPOSITION 3

can then be divided by the cell volume to give a dimensionless proportion of the cell that isfilled during a time step. The stored variable for the required cell is then updated with theaddition of the new proportion of growth.

Figure 1: excess redistribution

If during a time step any growth within a computational cell exceeds unity then the excessgrowth is spread to the neighbour cells that have spare capacity. This excess is spread accordingto the current distribution in the surrounding cells. In Figure 1 the current that enters cellA from cell B is greater than that entering from cell C. The excess is therefore redistributedto the neighbour cells B and C according to the relative current magnitudes; cell B receives

currentBcurrentB+currentC and cell C receives currentC

currentB+currentC . The cell growth is updated at the endof each time step as illustrated in Figure 2, Step (iv).

In this method, the current and hence the deposition rate are calculated only in the fluiddomain and in the vicinity of the interface. This calculation does not take into account theelectric potential in the deposition layer, it assumes that the potential in this solid region isfixed to the applied cathode voltage; because of the huge value of electrical conductivity in thedeposited solid region, 5.8E + 7Ωm−1 in comparison to electrolyte. This assumption removesthe complexity of calculating the current across the solid/fluid interface where the values for theelectrical conductivity differ by orders of magnitude, for example 51.1 : 5.8E + 7. Resolving thecurrent accurately across the interface is however essential for the Level Set method where theevolving interface is the result of the solution of an advection equation driven by the interfacecurrent. It is also important to note that with the EITM the advantage of the deposit growthbeing directly applied to the cells numerically in a controlled manner, is countered by therequirement that the computational domain consists of regular cuboid cells. This is becausethe growth is added as a dimensionless proportion of cell volume. If, for example, two adjacentcells have the same filling rate but are not the same size, then the interface shape could berepresented incorrectly unless proportional allocations of cell growth were applied. Pictures ofdeposition height and velocity using this technique are presented against predicted values inFigures 3a and 3b. The ’step-like’ deviation from the expected results in Figure 3b is becauseupdating of the deposition velocity is made at the end of each discrete timestep, remainingconstant throughout. This situation can be ameliorated by using smaller simulation timestepsand the rate of growth should be ideally limited to approximately 1

3 cell per timestep to achievea smooth transient filling.


4 HUGHES ET AL

Figure 2: Explicit Interface Tracking Method algorithm

2.1.2. Level Set Approach The essence of this technique is that the motion of the depositioninterface is not explicitly defined, but comes from the solution of an advection equation inwhich the driving velocity is the current distribution. An accurate current distribution isrequired in the deposit and interface region to give a time independent motion of the interfaceand therefore the electric current must be calculated accurately across the moving interface.This is not a trivial task partly because of the large differences in electrical conductivityacross these regions, but also because the calculation of current should be grid independentas there is a practical necessity for the current to be calculated across an unstructured grid.



(a) deposition height (b) deposition vecolity

Figure 3: Explicit Interface Tracking Method vs expected results:series 1 - expected values, series 2 - simulated values

Several methods where explored; the most successful of these will be discussed in detail inPart II. The algorithm for this approach is illustrated in Figure 4. At the end of each time stepthe current is calculated as with the previous method, it is then converted into a depositionvelocity measured in m3 positioned at the cell faces through the following process. First, thecurrent is normalised into jx, jy and jz components. Current is then calculated on cell facesusing arithmetic averaging (using cell centre to face distance weightings). The current valuesat cell face are converted into deposition velocities vdep, by multiplication of Ω

2F , where Ω, F

and 2 represent the atomic volume, Faraday’s constant and the valency of Cu2+ cupric ionsrespectively. The deposition rate that drives the free surface is stored at each cell face andis a volumetric rate in units m3

s . It is calculated by multiplication of the deposition velocityby cell face areas and then taking the scalar product of this result by the normalised currentcomponents

vrate at face = vdep at face ×Af · (jx, jy, jz) (1)

where Af denotes the area of a face f .Once the vrate at face values are calculated, the free surface can be advected at the end of a

simulation time step by solving the following advection equation (see Step (vi) of Figure 4 ):

∂φ

∂t+ u · φ = 0. (2)

In equation 2 u is a cell face velocity recovered from the vrate at face. The variant of theLevel Set algorithm used in this research is described in [6], however some modifications wererequired as detailed in Section 2.1.3 below. Figures 5a and 5b below show comparisons of thecurrent distribution for Level Set and Explicit Interface Tracking methods with simulationtime steps of dt = 0.1 sec and 0.01 sec respectively.

Figures 6a and 6b show deposition height plotted against time for the two methods.

2.1.3. Level Set Algorithm The process is as follows: at the start of the simulation an initial‘seed’ solid region is required, in which the value of the free surface variable φ is initialisedto be a positive constant, conversely in the remaining fluid (electrolyte) region it is set to a


6 HUGHES ET AL

Figure 4: Level Set Deposition Algorithm

negative constant value. The zero Level Set represents the deposit/liquid interface. This isdriven by the solution of an advection equation, see step (vi) of Figure 4. At the end of a timestep, the values of the free surface variable φ are passed to the Level Set algorithm where theyare updated by solving the reinitialisation equation 3 below.

∂φ

∂τ= S(φ0)(1− |∇φ|). (3)

Here φ0 represents the advected values from the free surface variable and τ represents apseudo time step that is set to be 0.1 times the minimum distance between any two adjacent



(a) current dt=0.1 (b) current dt=0.01

Figure 5: Deposition Velocities: Explicit Interface Tracking Method vs Level Set Method

(a) current dt=0.1 (b) current dt=0.01

Figure 6: Deposition Heights: Explicit Interface Tracking Method vs Level Set Method

cell centres. The sign function S(φ) in (3) ensures that the values either side of the interface areeither all positive or negative in accordance with how the free surface variable was initialised.

It is important to note that an initial seed layer for the solid region is required in thisformulation of the Level Set algorithm and that this seed layer should consist of at least twocells. This is because the immediate cells either side of the interface are not updated by thereinitialisation equation and therefore without additional code modification the zero Level Setwill advance at a slower and incorrect rate. Ensuring a minimum of two cells in the solid regionat the start of the simulation avoids this issue.

The values of φ in all cells except those containing the interface converge towards thenumerical value of their normal distance from the interface.

Equation 3 is discretised in the following way

φi+1P = φi

P + ∆τ S(φ0)P (1− |∇φP |), (4)

where i + 1 and i denote the present and previous iteration, respectively.In equation 4, the gradient ∇φP and the sign function S(φ0)P , with respect to a cell P are

defined in the following manner. The gradient of a variable on an unstructured mesh is derived


8 HUGHES ET AL

from Gauss’s divergence theorem as

∇φ =∑

f

Af

VPnf φf , (5)

where the summation is taken over all faces f of cell P , VP is the cell volume, nf is anoutward face normal.

However, the Level Set discretisation uses the following numerical modifications to thegradient that allow information to be passed from the interface while maintaining possibleinterface extremities, [6]

Upwinding applied to equation 5 gives

φf = φP if w.n ≥ 0;φf = φA if w.n < 0.

and the gradients are calculated from

∇φ =∆ΦVP

,

where

∆Φ = max mag δ+i , δ−i , δi

and

δ+i =

∑

f+

(AfniφP ) +∑

f−(Afniφf );

δ−i =∑

f+

(Afniφf ) +∑

f−(AfniφP );

δi =∑

f

(Afniφf ).

The magnitude of the gradient function is defined with respect to the L2 norm as

|∇φ| =√√√√∑

i

(∆iΦVP

)2

and the sign function is calculated as

S(φ) =φ0√

(φ0)2 + (d2AP )

,

where dAP is the minimum distance between cell centres for any adjacent cells within theentire computational grid, [6]



3. Part II

In this section we begin with a description of the governing equations and show resultscomparing the Level Set approach against the EITM for 2-D simulations of the PrimaryCurrent Regime (PCR) and the Tertiary Current Regime. Additionally as the calculatedcurrent distribution is paramount in interface development for the PCR some different methodsof calculating its distribution across the interface are discussed.

3.1. Governing Equations

The governing equations for the electrodeposition process are namely, the Navier-Stokesequation if the electrolyte is under the influence of forced convection:

ρ∂u

∂t+ ρu∇u = −∇P + µ∇2u + Su,

where Su represents momentum source for forced convection such as electrolye stirring.Together with the continuity equation

ρ∇ · u = 0,

and the temperature equation with external heating ST if present.

ρCp∂T

∂t+ ρCpu∇T = k∇2T + ST .

The flux of ionic species is given in Paunovic and Schlesinger [7] as

Ni = −zieνici∇φ−Di∇ci + uci, (6)

where φ, ci, Di, e, νi, zi represent electrolyte electric potential, concentration and diffusioncoefficient of the i−th ionic species, elementary charge, ion mobility and ion species valency,respectively. The first term in the right-hand side of (6) represents ion drift due to the electricfield, the second corresponds to diffusion of ions and the last term is movement by convection.Ionic mobility is given by

νi =Di

kT,

where k is the Boltzmann constant.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) itis possible to ensure that the electrolysis reaction is shielded and not significantly affected bymigration. In such circumstances the first term in the right-hand side of (6) can be neglected.

Concentration of ionic species can be represented by taking the divergence of (6) andexpressing this in the total derivative for concentration of species, this gives the ionicconcentration equation

∂ci

∂t+∇ · (uci) = ∇ · (Dici) + ezi∇ · (νici∇φ) , (7)


10 HUGHES ET AL

where the second term on the left-hand side is convection of ionic species and the two termsin the right-hand side are diffusion and migration respectively.

The equation set is closed with the electric potential equation together with suitableboundary conditions for the entire equation set. The time scale for establishing a DC fieldis much faster than for concentration gradients so under DC conditions the electric field canbe expressed through electric potential as a Poisson equation without time influence

∇2φ =4π

ε

n∑

i=1

ezici, (8)

where ε is the dielectric constant. An alternative to solving equation 8 is given in [8] andis to enforce electroneutrality in the bulk electrolyte. In this case the electric field becomesan unknown constant which is determined as part of the overall solution from the governingcondition

n∑

i=1

zici = 0.

As with equation 8, this condition applies at every point in the solution domain, except atthe thin layers adjacent to the electrode boundaries; the electrical double layer (see [8]) whichis less than 1000 A in width. In these thin layers, the deposition current is accounted for by anelectrode kinetic function, typically the Butler-Volmer equation; see [5, 7, 8]. In this electricaldouble layer the electroneutrality condition breaks down and a spatial charge exists; see [7].This charge is referred to as the surface overpotential and its value is one of the parametersthat drive the reaction rate through the Butler-Volmer equation.

In line with the literature to date, the double layer is not explicitly taken into account withDC conditions. Instead, the overpotential either is specified, as in [1, 3, 5], or details of itsexplicit calculation are not given special attention, as in [9, 10].

For AC conditions at low frequency it is likely that the above equations 7 and 8 can stillbe used. However, at higher frequencies and if the numerical model is to implicitly calculatethe overpotential, it may be necessary to introduce a numerical sub-model to calculate theoverpotential, which approximates the layer as a plate capacitor. This complication may bebypassed if sufficient overpotential vs applied voltage or current data is available.

3.2. Boundary Conditions

Ritter et al. [9] and Drese [10] give concise descriptions of four deposition regimes. The relevantequations and boundary conditions are listed here, the heavy line in Figure 7 below representsthe cathode-electrolyte interface.

• Tertiary current distributionThe deposition current, ibv, at the cathode is given by the Butler-Volmer equation andis a function of the ratio of the local interface concentration over bulk concentrationof reacting ions c

c∞ and electrode overpotential, ν. At the electrolyte-cathode interface,condition b in Figure 7 needs to be enforced.

• Secondary current distributionIf concentration gradients can be ignored because the concentration of ions is very highthen the electric potential equation is solved with condition c in Figure 7.



Figure 7: Boundary Conditions

• Primary current distributionIf the resistance of the electrolyte is much higher than that of the interface, the currentdensity passing through the electrode is given by Ohm’s law, condition a in Figure 7 isapplied.

• Diffusion limited current distributionAt sufficiently high overpotentials, a limiting current is reached as the ionic concentrationat the interface approaches zero and the electric potential equation can be ignored. Atthe interface, c = 0, and the deposition current is calculated as IDL = nFD ∂c

∂n .

3.3. Simulating the Primary Current Regime

The primary current distribution provides a good first target for model development, becauseof the simpler governing equation set. Under these conditions the deposition current is modelledusing Ohm’s law. If we make the assumption that the concentration of reacting ions issufficiently high, then we can ignore the influence of the ion concentration in equation 7.This has the advantage of reducing the equation set to that of solving a Laplace equation forelectric potential using equation 8 with the right-hand side reduced to zero. At the depositioninterface the current normal to the surface is given by boundary condition a in Figure 7.

The computational grid is shown in Figure 8 below, together with the boundary conditionsfor the electric potential equation. In this instance potentials are fixed at either ends of thedomain. An alternative is to replace the fixed potential boundary condition at the anode, byspecifying an incoming current boundary condition, i.e. k ∂φ

∂n = Ianode.Both solution methods produce the expected current crowding effects whereby at trench


12 HUGHES ET AL

Figure 8: Computational grid and electric potential boundary conditions

interfaces there is a larger deposition rate because of a pinching effect on the electric field fromsharp corners. In these instances voids may be formed as current and hence deposition rate ishigher at these edges. For comparative reasons both methods have been run on a regular gridbecause of the previously mentioned limitation of this requirement for the EITM; the LevelSet method however has no such limitation.

What is important to the Level Set method is accurate calculation of the background drivingvelocity that is the electric current. This is complicated by the existence of materials that havemassively different electrical conductivities and a moving interface and is discussed in detailin Section 3.4. It can be seen from comparing Figures 19 and 20 that a point is generatedwith the Level Set method. This point attracts current and therefore continues to grow as anextremity. This growth may be physically unrealistic, however given the model parameters;that current here is the sole generator of movement, it seems to be a plausible solution. Inreality the deposition regime would most likely change during the deposition process, limitingthe growth of this extremity as reliance on Ohm’s Law becomes less important . The ExplicitInterface Tracking Method does not exhibit a similar manner of growth. This is because ofthe way that the filling is calculated; these matters are discussed in Section 2.1.1. The lattermethod spreads any excess cell filling to the adjacent neighbour cells (see Figure 1) and hencethe extremity is smoothed away.



3.4. Current Calculation

Before presenting the simulation results, calculation of the current is now discussed. Theelectric current across an interface of different materials must conform to the condition thatthe normal flux across the interface is continuous as given by

kelectrolytedφ

dn= kmetal

dφ

dn.

In the instance of metallic deposition, kmetal À kelectrolyte. Since kmetal is generally largeand given that the currents involved are small, the cathode boundary condition φ = const willpermeate through the metallic deposited layer, anchoring the interface electric potential to theapplied cathode voltage. Using this knowledge to fix φ’s value in the deposit can significantlysimplify the calculation of current across the interface. However, the error in this approximationwill increase as current flow becomes larger. Therefore, it is important to be able to fullycalculate the electric potential in the deposit and use these potential values to accuratelycalculate the interface flux. When either an unstructured CFD code is used, or a method suchas the Level Set method is applied, several routes are possible for the calculation of current.Four possible options have been examined and are now discussed.

Calculating the electric potential gradients and hence current from Ohm’s law can beachieved in an unstructured discretisation scheme by utilising Gauss’s divergence formulabelow.

∇φ = k ×No faces∑

f=1

φf ·Af · (nx, ny, nz)VP

, (9)

Here k is the electric potential and nx, ny and nz are the Cartesian components of thecomputational cell face normal vectors. For example, when calculating the x−direction gradientonly nx components are required. Across an interface where k varies, its value can beapproximated by taking a weighted harmonic average given either by

kharm =kmetal × kelectrolyte

w1 × kmetal + w2 × kelectrolyte; w1 =

d1

d1 + d2; w2 =

d1

d1 + d2

or by

kharm =2kmetalkelectrolyte

kmetal + kelectrolyte,

if the computational grid has regular square cells.

The flux across an interface can be reasonably approximated by using kharm∂φ∂n , akin to

calculating the heat flux through composite materials. However, as can be seen in Figure 9a,this approximation is only exact if the cells spanning the interface have the same length, sothat ∂φ

∂n as well as kharm sit at the same location on the interface.Otherwise, as shown in Figure 9a the value of ∂φ

∂n is not collocated with kharm, it lies midwaybetween the two cell centers. This gives an approximation to the interfacial flux, but not anexact value. This discrepancy can be improved either if the cells surrounding the interface arefiner or if the values of φ used in the approximation are close to the interface. This is illustrated


14 HUGHES ET AL

(a) approximating current (b) improving the approximation

Figure 9: calculating the interface flux

in Figure 9b, where the dotted line of circles represents the electric potential on both sides ofthe interface and the value of ∂φ

∂n as calculated from the difference of cell center values is shownat position 1 in Figure 9b. The value of ∂φ

∂n is much better approximated at the interface inposition 2. This can be achieved without using a fine grid since the distribution of φ either sideof the interface is linear. It is therefore possible to calculate φ at equidistant intervals acrossand close to the interface by either extrapolation or employing a one-dimensional sub-modelthat solves the equation k∇2φ = 0 across the interface using the boundary conditions providedby the values of φ at the cell centres of the interfacial fluid and solid cells (see Section 3.4.3).

Four techniques for calculating the current are now discussed together with their merits.

3.4.1. Finite Difference Approximation This approximation is the most obvious choice touse when working within a structured CFD code. The current is calculated taking consistentlyeither forward or backward differences in each cell and using the harmonic conductivity acrossinterface cells. Current vectors can be seen for a 2-D test case as shown in Figure 10a andthe numerical values for the i−th current component across a section of the interface areshown in Figure 10b. These values illustrate that current components in the x−directionare not continuous across the interface. Similarly this is also the case for the y−directioncomponents and although the current distribution is a reasonable approximation it can bebetter approximated using other methods. Current is not conserved because of the reasonsdiscussed previously and illustrated in Figure 9a.

3.4.2. Interface Flux Balancing Using this method, the current in fluid and solid domainsare calculated at cell centres using equation 9. The values at the interface get special attention,the value of φ at this location is obtained from an estimated interface flux that is obtainedfrom a finite-difference calculation of kharm

∂φ∂n . The value of φoppface (Figure 11) at the far face

of the cell and directly opposite to the interface is known (by weighted average) and the facevalue of φintface at the nearest interface cell face is obtained by a reverse calculation from theflux. This calculation is made in both solid and fluid cells in the same manner. For example,



(a) Finite difference currentcalculation

(b) i-component

Figure 10: Finite difference current calculation

in the fluid side, we use the formula

φint =flux

kfluid+ φf ; flux = kharm

∂φ

∂n.

For the solid cell, we replace kfluid with ksolid and use φs . This gives an unstructuredmethod to calculate kfluid

∂φ∂n = flux either side of the interface based on the smae flux. When

using this procedure on both sides of the interface we get an estimation of the current similarto that in Figure 10, but with the added guarantee that the normal current components arecontinuous as they are both reverse calculated from the same flux estimate.

It should be noted that this method treats the solid/fluid interface as being positioned at theshared computational cell face between adjacent fluid and solid cells that span the interface.The weights w1 , w2, that are used in the calculation to find the harmonic mean are the cellcenter to shared face distances, divided by the distance between the solid/fluid cell centres.

This technique will give exact values for current if the cells spanning the interface have thesame length or if the computational grid is regular, otherwise it gives a good estimation thatguarantees conservation of the normal component.

Figure 12 shows the current distribution from a 2-D test, the distribution looks similarto that given in Figure 10, however the normal current components across the interface arecontinuous as shown in Figure 12b and 12c. Notice the difference in current distribution at thecorner between fluid and solid region in Figure 12a compared to Figure 10. Such differencescan result in significantly different deposition growth with the Level Set method.

3.4.3. Interface Flux Balancing Using a Sub-Model An extension of the above method inSection 3.4.2, can be made for a more accurate estimation of the interface flux by using valuesof φ that are equidistant and much closer to the interface. This ensures that the value of ∂φ

∂nis positioned extremely close to the interface and almost collocated with the position of theharmonic conductivity as illustrated in Figure 9. The diagram in Figure 13 shows how this isdone. The flux at the interface is calculated using kharm

φSI−φF I

dxf +dxswhere dxf = dxs and the size

of these dx’s can be variable in what is a pseudo grid or sub-model between the cell centres.A one dimensional equation is solved using the TDMA algorithm with the values of φ at the


16 HUGHES ET AL

Figure 11: Unstructured gradient calculation

(a) Flux current calculation (b) i-component (c) j -component

Figure 12: Interface flux current calculation

cell centres providing the boundary conditions for this equation. The results of this calculationcan be seen in the dotted line of circles in Figure 9. There is a different gradient either sideof the interface because of the unequal electrical conductivities. Using the values of φSI andφFI either side of the interface guarantees that the value of the flux lies in close proximityat position 2 (see Figure 13). This avoids the situation where kharm lies at the interface andthe value of ∂φ

∂n does not, (i.e. at position 1). This approximation is only necessary if the cellseither size of the interface are irregularly spaced and would be unnecessary otherwise.

The distribution of current illustrated in Figure 14a, is vertually identical to that shownin Figure 12a. Figures 14b and 14c show piecewise parts of the grid and illustrate that thenormal current components are continuous across the interface. The numerical values of thesecomponents are identical up to 3 significant digits to those shown in 12b and 12c.



Figure 13: Using a submodel to accurately approximate current flux

(a) Interface flux current calcu-lation

(b) i-component (c) j -component

Figure 14: Interface flux current calculation

3.4.4. Calculating Current with Electrical Conductivity as a Function of the Level Set positionThis technique is generally applied when calculating material properties using the Level Set

method. The electrical conductivity in computational cells at the interface can be calculatedeither as a proportion of the cells solid/fluid (electrolyte) mixture, or more generally by usinga predetermined function. Typically for the Level Set, function 10 below is used.

ν =

ν1, if φ > αν2, if φ < −αν1 + ν2

2+

ν1 − ν2

2sin

(πφ

2α

), if |φ| ≤ α,

(10)

The interface width is typically given by the function below. This by default spans 1.5 cellseither side of the interface, but it is a variable and can be easily changed.

α =3dAP

2.

If we use the functions above to define the interface, we find that a domain containing two


18 HUGHES ET AL

materials effectively becomes a domain with a composite number of materials depending onhow many cells are defined for the interface width. Figure 15 below shows the distributionof electrical conductivities around the interface using equation 10. Here the interface widthwas defined as 2 cells, one cell spanning either side of the interface. This effectively turns thedomain into a domain of 4 ’pseudo’ materials with the result that there are now four regionswith different conductivity. This is illustrated in Figure 15. This is a significant complication tocurrent calculation and together with the reasons discussed previously is why electric currenthas been calculated using the methods discussed in Sections 3.4.1-3.4.3. Note that the defaultinterface width of 3 cells (1.5 cells either side of the interface) would result in more than fourdifferent values of electrical conductivity within the computational domain.

Figure 15: Electrical conductivities calculated from equation 10 with interface width definedas 2 cells

Current calculation in Sections 3.4.1- 3.4.3 have been based on the assumption that the freesurface lies on the shared computational cell face of the two cells spanning the interface. Thereis good reason for this assumption. It can best be appreciated by examining the alternativeapproach, that is to calculate current based on the exact position of the interface and on theLevel Set values that define normal (shortest) distance to the interface. The Level Set distancesare illustrated in Figure 16 as d1 and d2.

To calculate the current from the electric potential, the gradient is of course required and ifusing equation 9, values are needed at the face locations, n, e, w and s, see Figure 16. If cell Phas a composite electrical conductivity given by, say, equation 10, then cells N, P and E willeach have different electrical conductivities. The values of φ at cell face locations n, e, w and sneed to be calculated from the stored cell center locations and so must take into account thatthese face locations bridge computational cells with different conductivities. Therefore thereis an inconsistency with equation 9 that assumes the value of electrical conductivity at cellcenter P is used in conjunction with the face (n, e, w, s) values of φ to calculate the current.

Additionally, there is a problem if d1 is very small, d1 ¿ d2. In this instance, numericalrounding can cause significant errors in the continuity of the current and a sub-model such asthat discussed in Section 3.4.3 would be required to accurately estimate the interface value ofφ.

A more straightforward and computationally consistent way to calculate the current is to



Figure 16: considerations of the interface when calculating current

assume only two electrical conductivities, those of the metal and electrolyte. When solvingequation 8 for the electric potential, the conductivity of cell P is then always positioned abovethe interface and so has conductivity kfluid, while the conductivity at cell S is located in thedeposit and has the value ksolid. The solution of the potential equation then assumes that theentire cell P is in the fluid region and conversely cell S is immersed entirely in the solid. Thenumerical solution is then consistent with the interface being at the cell face s of cell P . Thevalues of φ at all cell faces except s can then be readily calculated from weighted averagesfrom neighbour cells if they are on the same side of the interface, i.e. have the same electricalconductivity. The value of at the cell face s can then be obtained either from an estimationof the current flux that passes through the interface (3.4.2) or from using a submodel to solvefor φ across the interface as described previously in Section 3.4.3. Using these techniques thecalculation of current can be accurately recovered from the solution of the electric potentialequation without the added complication of considering multiple cells of different electricalconductivity or from rounding error caused by situations where either d1 ¿ d2 or d1 À d2.

3.5. Simulating the Tertiary Current Regime

Advancing the model to introduce ionic concentration involves greater numerical restrictionsat the deposition interface. This is now considered under DC conditions with a single ionicreacting species and an assumed constant overpotential. In this scenario, equation 7 is solvedfor bulk concentration

(c = c

c∞

)together with the equation (8) for electric potential . If only

one species is considered, the right-hand side of equation 8 is set equal to zero. At the boundarybetween the metal-electrolytic interface, condition b of Figure 7 needs to be satisfied, and hencethe positions of the electrolytic side of the interfacial cells must be tracked throughout thesimulation, so that the boundary conditions for equations 7 and 8 can be applied. A diagramof the solution domain is shown in Figure 17, where BV stands for the deposition currentdescribed by the Butler-Volmer equation. At the interface, the deposition current is governedby the surface kinetic function and therefore an appropriate ‘sink’ boundary condition mustbe applied to the electrolyte-side computational cells. To avoid any unphysical loss of ionicconcentration from conduction the diffusion coefficient across the interface should be set to asmall value approaching zero so that the only loss of concentration comes from the Butler-


20 HUGHES ET AL

Volmer equation. This situation can be avoided by splitting the computation domain into twosections and linking these regions by appropriate sink/source type boundary conditions; thecurrent leaving the electrolyte side should be equal to the current entering the deposit region.To recover current from this type of calculation, the techniques discussed in section 3.3 canbe applied. Figure 18 below illustrates this idea showing the results from a 1-D test case, inwhich the unequal spacing of the grid cells around the interface is a means to test currentcalculation within the model, otherwise this grid arrangement is not sensible. As required, thecurrent calculated across the interface is continuous and is of equal magnitude to the computeddeposition current from the Butler-Volmer equation.

Figure 17: The Tertiary Current Solution Domain

(a) (b)

Figure 18: Interface considerations of the Tertiary Current Regime

3.6. Results - Primary Current Distribution

The results for both methods are briefly presented for the PCR. It should be noted that thisscenario may be somewhat physically unrealistic throughout an entire simulation. It is onlyvalid if the resistance of the electrolyte is greater than that of the kinetics at the interface.During deposition, the regime may change at some stage and the deposition current may no



longer be governed by Ohm’s law. The results presented in this section are from a model thatassumes the interface motion is governed by Ohm’s law throughout the entire simulation.

Figures 19a and 19b show the deposit after 30 seconds, the level of growth is similar, howeverFigure 19a shows a more rounded growth at the deposit corner. This is due to the spreadingof the excess cell filling to neighbour cells as discussed in Section 2.1.1 and illustrated in 1.Current vectors for the Level Set method are shown in Figure 19c.

(a) Explicit Interface Tracking (b) Level Set (c) Current

Figure 19: Explicit Interface method VS Level Set at 30 secs

Figures 20a and 20b show the deposition further in time. The profile from the Level Set hasdeveloped a ’point’ profile that has grown from the corner of the trench. This point attracts thecurrent producing further growth in a perpetual cycle. This has not happened with the EITRmethod because of the smoothing in the algorithm and highlights one of the strengths of theLevel Set method in that it does not smooth interface extremities. The current vectors shownin Figure 20c illustrate the attraction of current to the point and explain physically why thesharp extremity grows. With both methods the ’current crowding’ effect produced by the PCRcan be clearly seen, the differences in profile at 150 seconds is explained by differences in thealgorithms; the Level Set method profile follows that of the current and the EITM smoothespossible extremities as illustrated in 1 and discussed earlier.

3.7. Results - Tertiary Current Distribution

Results for the Tertiary current regime are now presented. In this regime, the depositionvelocity is governed by interface kinetics that is described by the Butler-Volmer equation; seecondition b in Figure 7. The deposition rate is a function of the concentration of cupric ionsand is a scalar rather than the vector current density of the PCR. Deposition growth is normalto the interface surface, and it is differences in the magnitude of this deposition rate that driveany non-conformal deposit growth . In this regime the extremities of ‘current crowding’ arelikely to be less extreme than with the PCR, but will increase as the diffusion coefficient ofequation 7 decreases in magnitude.


22 HUGHES ET AL

(a) Explicit Interface Tracking (b) Level Set (c) Current


Figure 21 below shows a similar rate and shape of growth between the two methods at 400seconds.

(a) Explicit Interface Tracking (b) Level Set

Figure 21: TCR: Explicit Interface method VS Level Set at 400 secs

Figure 22 shows the deposition at 900 seconds with the growth looking very similar. Plots ofconcentration are shown in Figures 23; these are in close agreement and explain the similarity ofthe deposition profiles, since concentration of ionic species is the driving force of the depositionrate. Figures 24 illustrate the rate, in moles m−2sec−1, at which concentration of Cu2+ isabsorbed at the interface.

Plots of the deposition at 1400 seconds show the formation of a void in the trench. In




Figure 22: TCR: Explicit Interface method VS Level Set at 900 secs

(a) Explicit InterfaceTracking

(b) Level Set

Figure 23: Concentration: Explicit Interface method VS Level Set at 900 secs

Figures 25 the growth can be seen to be non-conformal with the creation of a void renderingthe deposition ill-equipped for any function that involves electrical circuitry. In this case thetrench has an aspect ratio of 3 : 1. In Part III of this paper, simulations of acoustic forces areincluded as a means to generate a conformal growth without creating such voids.


24 HUGHES ET AL


Figure 24: Interface sink: Explicit Interface method VS Level Set at 900 secs



4. Part III

Megasonic agitation provides a promising means to solve the problems concerning voidformation in high aspect ratio trenches and vias by enhancing and replenishing ionic transportthat would otherwise not penetrate deep into these features. The physical effect of this processis to decrease the Nernst diffusion layer and therefore increase the kinetic rate at the interface.Experiments at Heriot-Watt by Kaufmann et al. [11], have shown an improvement in boththe deposition rates and deposit quality for microvias with features of aspect-ratio up to 2:1being successfully completed. Presently a limitation for trenches with ratios larger than 2:1 is



reported, due to seeding problems; the physical application of conductive seed layer throughoutthe microcavity.

The Acoustic Streaming process is generated from the high frequency agitation (1 MHz) ofa Piezoelectric plate generating an acoustic intensity of 60 kW/m2 with corresponding highfrequency pressure variations within the feature. These pressure variations generate Reynoldsstresses that drive a steady fluid recirculation within the cavity that replenishes the supply ofthe reacting ionic species from the bulk of the containing bath.

4.0.1. Solution Procedure The numerical solution process closely follows that presented byNilson and Griffiths [2], and is in two stages; the first concerns the analytical solution of theLinearised Euler equations as developed by Rayleigh and Nyborg for channels between infiniteparallel plates and is detailed concisely by Nilson and Griffiths in[2]. The second stage involvescalculating the time averaged Reynolds stresses that are produced from these Linearised Eulerequations and their numerical integration into the Navier-Stokes equations. For validationsimulations have been made to examine the effects of acoustic streaming in extremely narrowfeatures with widths in the order of tens of microns so that the resulting flow profiles can becompared against those presented in [2]. For completeness the solution process is now brieflydiscussed. The linearised Euler equations 11-12, are solved analytically.

∂ρa

∂t+ ρ0∇ · ua = 0, (11)

ρ0∂ua

∂t= −∇Pa + µ∇2ua +

(µb +

13µ

)∇ (∇ · ua) (12)

µ, µb are shear and bulk viscosities, c is the sound speed, ρa, Pa are the acoustic density andpressure variations and ρ0 the background electrolyte density. The analytical solutions for theacoustic velocities are given in equations 9-10 of [2], the real components can be expressed inthe following form, 13 - 14

va ≈ ua0e−αy

[cos

(ωt− 2πy

λ

)− e

xd cos

(ωt− 2πy

λ− x

d

)], (13)

ua ≈ πd

λ[Re (va)− Im (va)] , (14)

ua0 =

√I

ρc; d =

√2µ

ωρ; λ =

c

f. (15)

In the simulation presented in this paper the electrolyte was assumed to have the propertiesof water and a sound intensity of 60 kW/m2 was estimated based on a megasonic frequencyof 1 MHz. Equations 13-14 were computed within the trench region across the domain fromone side of the deposit to the other, but only within cells containing the electrolyte. Theseacoustic velocities ua, va are functions of time and time harmonic so their values repeat overwavelengths of period 1Mhz. The wavelength was discretised into twelve pseudo time stepsthat were calculated at the beginning of each time step and the subsequent values of ua (t)and va (t) were stored as a requirement for calculating the time averaged acoustic forces.


26 HUGHES ET AL

The acoustic forces are then calculated from (16) where the brackets 〈〉 denote timeaveraging, they are subsequently integrated into the Navier-Stokes equations as volumetricsources.

F = ρ0 〈(ua · ∇) ua + ua (∇ · ua)〉 (16)

To accurately represent the acoustic forces the computational grid must be fine enough tocapture the large Reynolds stresses that occur next to the feature walls. A grid length of 1µmusing regularly spaced cells was used within the trench region as shown in Figure 26a. Inthis case the acoustic forces permeate throughout the feature from top to bottom and have acalculated magnitude of the order 3.5×104 N/m3. The distribution of these forces is illustratedin Figure 26b.

(a) Acoustic Forces ComputationalGrid

(b) Acoustic Force

Figure 26: Acoustic Computational Grid - Acoustic Force Distribution

When integrated into the Navier-Stokes equations the forces produce a flow profile thatis driven downwards at the feature walls and recirculates back through the feature centerdue to continuity. This is illustrated in Figure 27. In this simulation there was a backgroundelectrolytic flow from left to right as indicated by the ’red vectors’, the deep recirculatingwithin the trench is caused by the acoustic agitation. It is this effect that makes the possibilityof conformal growth within the feature possible.

Normalised flow profiles through the channels are affected by the channel width and arefastest in the region of the largest Reynolds stresses adjacent to the feature walls. The velocitiesin this region are asymptotic with increasing channel width. In Figure 28, the value of w

drepresents a dimensionless channel width; the smaller it’s value the narrower the channel. Theplotted velocities have been normalised by a nominal streaming speed of 2.6 · 10−6µm/sec,Ua0 of equation 15, and are therefore tiny. The profiles shown in Figure 28 are however ingood agreement with the results presented by Nilson and Griffiths [8].

It is interesting to note that as the channel width increases the velocity profile becomessimilar to one that has been generated through wall movement in a static fluid through africtional non-slip wall condition and as such the complications of the streaming physics mightbe replaced in numerical simulations that have larger width features with the application of asuitable fixed wall velocity.



Figure 27: Acoustic Flow Profile

Figure 28: Acoustic Velocity Profiles

4.0.2. Results: Acoustic agitation Results are presented for simulations with and withoutacoustic forces for the EITM method. This method was chosen because it has significantlyless computational expense compared to the Level Set method, especially with the addedburden of acoustic agitation and the fine grid required. The development of the routines thatcalculate the acoustic forces are independent to the deposition algorithms and the resultingfluid motion couples into both methods only through the convection term of equation 7.The Figures 29 and 30 show profiles of the deposit growth with and without megasonicagitation. With agitation significantly faster and more conformal growth occurs, as can beseen by comparing the figures. This effect is produced because the concentration of Cu2+ ionsin the trench has been maintained by the convection plume that permeates throughout thetrench. The concentration of Cu2+ is much higher in the feature with agitation as illustrated in


28 HUGHES ET AL

Figure 31. The corresponding rate of growth could be numerically mimicked (without agitation)by dramatically increasing the diffusion coefficient by several orders of magnitude over the valueused in this simulation(order E − 10). Physically the resulting forced motion of concentrationCu2+, reduces the Nernst Diffusion layer, that in the presence of agitation is altered from avelocity dependence to the inverse of frequency as shown below in (17) and (18), where νis kinematic viscosity and ω is frequency. In this continuum model, an increase in frequencycorresponds to larger acoustic forces, more convection and higher interface kinetics.

∂hydrodynamic = 0.16( ν

Ux

) 12

, (17)

∂acoustic =(

2ν

ω

) 12

. (18)

(a) 100 secs (b) 400 secs

(c) 900 secs (d) 1300 secs

Figure 29: deposition growth with megasonic agitation

Figures 31 below compare the concentration profile at 400 seconds, a higher concentrationof Cu2+ is maintained throughout the simulation. In the run without megasonic agitation theconcentration profile quickly depletes at the interface and becomes limited to the speed atwhich the Cu2+ ions can reach the vicinity of the interface by the process of diffusion.

The boundary condition of equation 7 for both simulations can be seen at 400 secondsin Figures 32 below. This is a sink condition that represents the loss of Cu2+ in units ofmoles/(m2sec) at the interface. It is proportional to the deposition current and therefore the



(a) 100 secs (b) 400 secs

(c) 900 secs (d) 1300 secs

Figure 30: deposition growth without megasonic agitation

(a) with forces (b) without forces

Figure 31: Concentration profiles at 200 seconds with and without agitation

growth rate. Figure 32 illustrates that this sink is an order of magnitude larger within thetrench for the simulation with acoustic agitation.

The velocity profile of the electrolyte can be seen within the trench at 400 seconds. Thevelocity profile in the solid deposit region has been fixed to zero by inclusion of Darcy forces in


30 HUGHES ET AL

(a) With forces (b) Without forces

Figure 32: Cu2+ sink at the Interface

the momentum equations; the following source term was applied to the momentum equation.

Sφ = frac(1− f)2(max( f, ε) )3µV

e

where f is the electrolyte fraction content of a cell that is zero in cells containing solid, ε isthe solidification modulus, its value is 1E-10, V is the cell volume, µ is the dynamic viscosityand finally e is the permeability coefficient that has a value close to zero.

(a) Velocity Vectors (b) Acoustic Force Dist’n

Figure 33: Velocity vectors and Force distribution at 400 seconds

The acoustic forces are prevalent along the sides of the deposit, these are calculated at thestart of every time step to keep them in track with the deposit growth. Figure 33 illustratesthese forces that have a value of approximately 2E + 4 N/m2 400 seconds into the simulation.

In this simulation, megasonic agitation has roughly doubled the growth rate of the deposit.without acoustic agitation the growth rate is approximately 0.5 micron per minute. However,it should be stressed that numerically this is dependent on the parameters chosen for thesimulation such as overpotential.



5. Conclusions

Two numerical models of electrodeposition have been presented in this paper. Both methodshave been incorporated into the unstructured CFD code Physica. The CFD framework providesthe perfect host to electrodeposition simulations as the solution of momentum, concentrationand enthalpy equations are implicit to CFD codes. The Explicit Interface Tracking Methodrepresents a fast, relatively simple-to-implement procedure, while the Level Set method issignificantly more complex in terms of development and application. Both methods give goodapproximations to the predicted filling of a 1-D test case driven by current governed by Ohm’slaw, but in 2-D there are significant differences in the deposit profile under the Primary CurrentDistribution. These differences are explained in the way that the current is calculated in themethods. Using the Level Set method the current is calculated throughout the domain asvalues are required in the solid deposit to drive the deposit growth. In the EITM the currentis only calculated in the electrolytic side and the growth is smoothed at the interface by thepotential spreading of any excess growth in a computational cell to its neighbours to avoidoverfilling within a time step.

Both methods give similar and physically realistic deposit profiles when the growth isgoverned by surface kinetics as defined by the Butler-Volmer equation. The Level Set methodgenerally seems to give sharper and cleaner interface growth between time steps, this is becausethe motion is controlled by the solution of an advection equation rather than directly applied.However in this authors opinion the extra increase in effort of coding. labour and computingruntimes over the EITM method make this latter algorithm a more attractive proposition forthe modeller who wishes to quickly modify his CFD code to handle ’one-off’ application specificelectrodeposition simulations. The physical effect of Acoustic agitation have been examined inPart III of this paper. The Acoustic forces have been calculated from an analytical formulaas given in [2] and integrated into the Navier-Stokes equations as volumetric sources. Thegenerated velocity profiles show good agreement with those presented in [2] and the effect ofconformal faster deposit growth can be clearly seen in the simulations. The computational gridmust be sufficiently fine in the near interface region to capture the acoustic forces (≤ 1µm),adding significant computational run cost to the models. It is hoped that this paper will providevaluable information to anyone wishing to develop continuum models of electrodeposition.

REFERENCES

1. Wheeler, D. et al“Numerical Simulation of Super-conformal Electrodeposition using the Level-Set Method”,Proceedings of the International Conference on Modeling and Simulation of Microsystems,(2002).

2. Nilson, R.H., Griffiths, S.K.,”Enhanced Transport by Acoustic Streaming in Deep Trench-Like Cavities”,Journal of the Electrochemical Society,149 (4) G268-G296,(2002).

3. Wheeler D, Guyer J.E, Warren J.A,“A Finite Volume PDE Solver Using Phython”,URL http://www.ctcms.nist.gov/fipy/download/fipy.pdf,(2007).

4. Michael Hughes, Christopher Bailey, Kevin McManus“Multi-Physics Modelling of the Electrodepostion Process”,


32 HUGHES ET AL

Proceedings of EuroSimE London, IEEE,(2007).

5. Wheeler D, Josell D, Moffat T.P,“Modeling Superconformal Electrodeposition Using the Level Set Method.”J.Electrochem. Soc. Vol 150, No 5,(2003), ppC302-C310.

6. The physica reference manual-TheoryURL : http :www.physica.co.uk

7. Paunovic M, Schlesinger M,“Fundamentals of Electrochemical Deposition”,Wiley-Interscience,ISMN 0-471-16820-3.

8. Griffiths S.K et al, “ Modeling Electrodeposition for LIGA Microdevice fabrication”,SAND98-8231,(1998),Distribution Category UC-411.

9. Ritter G, McHugh P, Wilson G, Ritzdorf T,“Three dimensional numerical modelling of copper electroplating for advanced ULSI metallisation”,Solid-State Electronics, 44, (2000), pp 797-807.

10. Drese K.S,”Design Rules for Electroforming in LIGA Process”,J.Electrochem. Soc,Vol 151, No 6, (2004), D39-D45.

11. Kaufmann, J. et al“Megasonic Enhanced Electro-deposition”,Proceedings of DTIP of MEMS & MOEMS,9-11 April (2008).


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Numerical modelling of the electrodeposition process