numerical algorithms for modelling electrodeposition: tracking the deposition front under forced...

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDSInt. J. Numer. Meth. Fluids 2010; 64:237–268Published online 30 September 2009 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/fld.2140

Numerical algorithms for modelling electrodeposition: Trackingthe deposition front under forced convection

from megasonic agitation

Michael Hughes1,∗,†, Nadia Strussevitch1, Christopher Bailey1, Kevin McManus1,Jens Kaufmann2, David Flynn2 and Marc P. Y. Desmulliez2

1Centre for Numerical Modelling and Process Analysis (CNMPA), University of Greenwich,Park Row, Greenwich, London SE10 9LS, U.K.

2MicroSystems Engineering Centre (MISEC), School of Engineering and Physical Science, Heriot-Watt University,Earl Mountbatten Building, Edinburgh EH14 4AS, U.K.

SUMMARY

Electrodeposition is a widely used technique for the fabrication of high aspect ratio microstructures. Inrecent years, much research has been focused within this area aiming to understand the physics behindthe filling of high aspect ratio vias and trenches on substrates and in particular how they can be madewithout the formation of voids in the deposited material. This paper reports on the fundamental worktowards the advancement of numerical algorithms that can predict the electrodeposition process in micronscaled features. Two different numerical approaches have been developed, which capture the motion of thedeposition interface and 2-D simulations are presented for both methods under two deposition regimes:those where surface kinetics is governed by Ohm’s law and the Butler–Volmer equation, respectively.

In the last part of this paper the modelling of acoustic forces and their subsequent impact on thedeposition profile through convection is examined. Copyright q 2009 John Wiley & Sons, Ltd.

Received 28 January 2009; Revised 11 June 2009; Accepted 18 June 2009

KEY WORDS: CFD; electrodeposition; level set method; megasonic agitation; microsystems; LIGA

1. INTRODUCTION

Electrodeposition is a complex process and truly multiphysics in nature. It couples together withseveral physical phenomena such as fluid flow, heat transfer, ionic concentration, electric current

∗Correspondence to: Michael Hughes, Centre for Numerical Modelling and Process Analysis (CNMPA), Universityof Greenwich, Park Row, Greenwich, London SE10 9LS, U.K.

†E-mail: [email protected]

Contract/grant sponsor: Engineering and Physical Sciences Research Council (EPSRC); contract/grant numbers:EP/C513061/1, GR/S12395/1

Copyright q 2009 John Wiley & Sons, Ltd.

238 M. HUGHES ET AL.

and chemical reaction rates. In recent years, the manufacturing of intricate-integrated electronicand electromechanical components of high quality has focused the study of electrodeposition onmicro scale high aspect ratio structures. Key to the deposition quality in recessed features is theability to replenish ionic species in the electrolyte. Deposition rates are otherwise limited by thespeed of diffusion processes in the electrolyte. Recessed features are typically flow dead zones andthere is a tendency for void formation in features as the concentration of ionic species becomesdepleted over time. For this reason, there has been much interest in using either chemical additivesthat can produce a ‘superconformal’ acceleration of the deposition in sub micron geometries [1, 2],or more recently, high-frequency sound waves that stimulate electrolyte flow into micron scaledrecesses [3, 4]. Accurate calculation of the interface movement is of key importance in any modelof electrodeposition. This paper addresses two methods of interface motion: one relatively simpleapproach that tracks the motion of the interface through a stored marker variable and depositionrate, and the other a more complex method drives the interface motion through the advection of afree surface using the level set method [5–7]. Both algorithms have been implemented on blockstructured orthogonal meshes within the cell-centred finite volume unstructured CFD environmentdescribed in Croft et al. [8] and Bailey et al. [9]. These algorithms are described in detail andcompared. Simulation results from both techniques are presented for two deposition regimes: onedriven by electric current as calculated by Ohm’s law and the other by surface kinetics describedby the Butler–Volmer equation. In the final part of this paper, the methodology used to incorporatemegasonic agitation and its subsequent effect on the electrodeposition process through acousticstreaming is discussed.

We begin with a description of the governing equations and boundary conditions.

2. GOVERNING EQUATIONS AND NUMERICAL ALGORITHMS

2.1. Governing equations

The governing equations for the electrodeposition process are namely, the Navier–Stokes equationif the electrolyte is under the influence of forced convection

��u�t

+�u∇u=−∇P+�∇2u+Su (1)

where Su represents momentum sources for forced convection such as electrolyte stirring. Togetherwith the continuity equation

�∇ ·u=0 (2)

and the temperature equation with external heating, ST , if present

�Cp�T�t

+�Cpu∇T =k∇2T +ST (3)

The flux of ionic species is given in Paunovic and Schlesinger [10] asNi =−zi e�i ci∇�−Di∇ci +uci (4)

where �, ci , Di , e, �i , zi represent electrolyte electric potential, concentration and diffusioncoefficient of the i th ionic species, elementary charge, ion mobility and ion species valency,

Copyright q 2009 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 2010; 64:237–268DOI: 10.1002/fld

COMPUTATIONAL ELECTRODEPOSITION 239

respectively. The first term in the right-hand side of Equation (4) represents ion drift due to theelectric field, the second corresponds to the diffusion of ions and the last term is movement byconvection. Ionic mobility is given by

�i = Di

kT(5)

where k is the Boltzmann constant. Migration is essentially an electrostatic effect that arises dueto the application of a potential difference between the electrodes. If there is a large quantity ofthe electrolyte (relative to the reactants), it is possible to ensure that the electrolysis reaction isshielded and not significantly affected by migration [10–12]. In such circumstances the first termin the right-hand side of Equation (4) can be neglected.

Concentration of ionic species can be represented by taking the divergence of (4). Expressingthis divergence in the total derivative for concentration of species yields the ionic concentrationequation

�ci�t

+∇ ·(uci )=∇ ·(Dici )+ezi∇ ·(�i ci∇�) (6)

where the second term in the left-hand side is the convection of the ionic species and the twoterms in the right-hand side are diffusion and migration, respectively.

The equation set is closed with the electric potential equation together with suitable boundaryconditions for the entire equation set. The time scale for establishing a DC field is much fasterthan for concentration gradients, so that under DC conditions the electric field can be expressedthrough electric potential as a Poisson equation without time influence

∇2�= 4�

ε

n∑i=1

ezi ci (7)

where ε is the dielectric constant. An alternative to solving Equation (7) is given in Griffithset al. [12] and is to enforce electroneutrality in the bulk electrolyte. In this case the electric fieldbecomes an unknown constant which is determined as a part of the overall solution from thegoverning condition

n∑i=1

zi ci =0 (8)

As with Equation (7), this condition applies at every point in the solution domain, except atthe thin layers adjacent to the electrode boundaries, the electrical double layer [12], which is lessthan 1000 A in width. In these thin layers, a potential difference exists and the electroneutralitycondition breaks down [10]. The deposition current in this region is accounted for by an electrodekinetic function that is typically the Butler–Volmer equation [10, 12, 13]. This potential differenceis referred to as the surface overpotential and its value is one of the parameters that drive thereaction rate.

In line with the literature to date, the double layer is not explicitly taken into account with DCconditions. Instead, the overpotential is either specified, as in Wheeler et al. [1, 2, 13], or detailsof its explicit calculation are not given special attention, Drese [14], Ritter et al. [11].

For AC conditions at low frequency, it is likely that the above equations, (6) and (7) can stillbe used. However, at higher frequencies and if the numerical model is to implicitly calculate the



overpotential, it may be necessary to introduce a numerical sub-model to calculate the overpotentialthat approximates the layer as a plate capacitor. This complication may be bypassed if sufficientoverpotential versus applied voltage or current data is available.

2.2. Boundary conditions

Ritter et al. [11] and Drese [14] give concise descriptions of four deposition regimes. The relevantequations and boundary conditions are listed here, the heavy line in Figure 1 represents thecathode–electrolyte interface.

• Tertiary current distribution: The deposition current, ibv, at the cathode is given by the Butler–Volmer equation and is a function of the ratio of the local interface concentration over bulkconcentration of reacting ions c/c∞ and electrode overpotential, �. At the electrolyte–cathodeinterface, condition b in Figure 1 needs to be enforced.

• Secondary current distribution: If concentration gradients can be ignored, because the concen-tration of ions is very high, then the electric potential equation is solved with condition c inFigure 1.

• Primary current distribution: If the resistance of the electrolyte is much higher than that ofthe interface, the current density passing through the electrode is given by Ohm’s law andthe condition a in Figure 1 is applied.

• Diffusion limited current distribution: At sufficiently high overpotential, a limiting currentis reached as the ionic concentration at the interface is exhausted and approaches zero. Inthis instance, the electric potential equation can be ignored. At the interface, c=0, and thedeposition current is calculated as IDL=nFD�c/�n.

The key to the development of numerical models in electrodeposition is the accurate repre-sentation of the moving metallic/electrolyte interface. Previous work by Hughes et al. [15]and Wheeler et al. [2, 13] has utilized the level set method for interface motion. The level setpresented in this paper is an extension of that presented in [15] and originally based on theformulation presented by Williams et al. [5]. It distinguishes itself from the formulation presentedin Wheeler et al. [2, 13], through its application on larger scale features, without the applica-tion of leveling agents and therefore through the mechanism of interface propagation. In thispaper an alternative and computationally cheap approach to the level set have been developedwhich is based on a novel variation of the donor–acceptor technique. Results are presentedfor the two types of interface kinetics, driven, respectively, by Ohm’s law and Butler–Volmerequation.

2.3. Numerical algorithms

In this section the two methods are examined. A 1-D test case is presented where the depositionis driven by a constant potential difference in order to test the algorithms and validate that bothtechniques exhibit time-step independent behaviour. This issue, which caused problems during thedevelopment of the level set approach, is resolved by the careful calculation of current densityacross the deposition interface. Calculation of the current density across a moving interface isof primary importance as it drives the deposition. As will be shown, care is required to ensurecontinuity of the normal components. There are a number of possible routes to accomplish this goaland finding a less complex method was a major motivation for the development of an alternativemethod to the level set.



Figure 1. Boundary conditions.

2.3.1. Explicit interface tracking method (EITM). Using this technique a 1-D test case for thedeposition of copper has been simulated under the primary current regime (PCR), where the currentdensity is calculated from Ohm’s law j =−�(∇�) and a scalar variable is stored to represent thecell filling. This stored variable has a value of 0 for a fluid cell and 1 for a filled cell. The fillingof the cells is governed by the deposition velocity that is a function of the current vdep= j�/2F ,where � and F are the atomic volume and Faraday’s constant, respectively, and 2 is the valency ofcupric ions Cu2+. Multiplying vdep by the cell face area and �t , the time-step length, a volumetricgrowth is found for the time step in m3. This volumetric growth can then be divided by the cellvolume to give a dimensionless proportion of the cell that is filled during a time step. The storedvariable for the required cell is then updated with the addition of the new proportion of growth.

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 according tothe current distribution in the surrounding cells. In Figure 2, the current that enters cell A from cell Bis greater than that entering from cell C. The excess is therefore redistributed to the neighbour cells Band C according to the relative current magnitudes; cell B receives current B/(current B+currentC)

and cell C receives currentC/(current B+currentC) at the end of each computational time step.The current and hence the deposition rate are calculated only in the fluid domain and in the vicinityof the interface. This calculation does not take into account the electric potential in the depositionlayer. It assumes that the potential in this solid region is fixed to the applied cathode voltage becauseof the high magnitude of electrical conductivity for the metal; 5.8E+7�m−1 in comparison withthe electrolyte value 5.1�m−1. This assumption removes the complexity of calculating the currentacross the solid/fluid interface where the values for the electrical conductivity differ by orders ofmagnitude. Resolving the current accurately across the interface is however essential for the level



Figure 2. Excess redistribution.

Figure 3. Explicit interface tracking method versus expected results: series 1—expected values,series 2—simulated values: (a) deposition height and (b) deposition velocity.

set method where the evolving interface is the result of solving an advection equation driven by theinterface current. With the EITM, the advantage of the deposit growth being directly applied to thecells, numerically in a controlled manner, is countered by the requirement that the computationaldomain consists of cuboid cells, since the growth is added as a dimensionless proportion of cellvolume. If, for example, two adjacent cells have the same filling rate but are not the same size,then the interface shape could be represented incorrectly unless proportional allocations of cellgrowth are applied. Pictures of deposition height and velocity using this technique are presentedagainst predicted values in Figures 3(a) and (b).

The ‘step-like’ deviation from the expected results in Figure 3(b) stems from the updating ofthe deposition velocity being made at the end of each discrete time step only. This situation can beimproved by using smaller simulation time steps and the rate of growth should be ideally limitedto approximately 1

3 cell per time step to achieve a smooth transient filling.

2.3.2. Level set approach. The essence of this technique is that the motion of the depositioninterface is not defined explicitly, but comes from the solution of an advection equation in whichthe driving velocity is the current distribution. An accurate current distribution is required in thedeposit and across the interface to give a time step independent motion of the front. The electriccurrent must therefore be calculated accurately across the interface. This is not a trivial task, partly



because of the large differences in electrical conductivity, but also because the calculation of currentshould be independent with respect to grid spacing. There is a practical necessity on more complexgeometries for the current to be calculated on block structured grids with irregular cell spacings.Several methods for current calculation were explored and are discussed in Section 2.4. The levelset algorithm is described in Section 2.3.3. At the end of each time step, the current is calculatedas for the EITM; it is then converted into a deposition velocity measured in m3 and positionedat the cell faces through the following process. First, the current is normalized into jx , jy and jzcomponents. Current is then calculated on cell faces using arithmetic averaging (cell centre to facedistance weightings) and the values at cell faces are converted into deposition velocities vdep, bymultiplication of �/2F . Here, �, F and 2 represent the atomic volume, Faraday’s constant andthe valency of Cu2+ cupric ions, respectively. The deposition rate is stored at each cell face and isa volumetric rate in units m3/s. It is calculated by multiplying the deposition velocity by cell faceareas and then taking the scalar product of this result with the normalized current components

vrate at face=vdep at face×A f ·( jx , jy, jz) (9)

where A f denotes the area of cell face f . Once the vrate at face values are calculated, the freesurface can be advected at the end of a simulation time step by solving the advection equation

��

�t+u ·�=0 (10)

In Equation (10), 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 [5, 16]; however, some modificationswere required as detailed in Section 2.3.3. Figures 4(a) and (b) show comparisons of the currentdistribution for level set and explicit interface tracking methods with simulation time steps ofdt=0.1 and 0.01 s, respectively. Figures 5(a) and (b) show deposition height plotted against timefor the two methods.

2.3.3. 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, �, is initialized tobe a positive constant. Conversely, in the remaining fluid (electrolyte) region it is set to a negativeconstant value. The zero level set represents the deposit/liquid interface. This is driven by thesolution of Equation (10). At the end of a time step, the values of the free surface variable � arepassed to the level set algorithm where they are updated by solving a reinitialization equation (11)

��

��= S(�0)(1−|∇�|) (11)

Here, �0 represents the advected values from the free surface variable and � represents a pseudo-time step that is set to be 0.1 times the minimum distance between any two adjacent cell centres.The sign function, S(�), in (11), ensures that the values on either side of the interface are eitherall positive or all negative in accordance with how the free surface variable was initialized.

The initial seed layer for the solid region is necessary in this formulation for the level setalgorithm and this seed layer should consist of at least two cells. This is because the immediatecells on either side of the interface are not updated by the reinitialization equation and thereforewithout additional code modification the ‘zero level’ will advance at a slower and incorrect rate.Ensuring a minimum of two cells in the solid region at the start of the simulation avoids this issue.



Figure 4. Deposition velocities: explicit interface tracking method versus level set method.(a) current dt=0.1 and (b) current dt=0.01.

Figure 5. Deposition heights: explicit interface tracking method versus level set method.(a) current dt=0.1 and (b) current dt=0.01.

The values of � in all cells except those containing the interface converge towards the numericalvalue of their normal distance from the interface.

Equation (11) is discretized in the following way:

�i+1P =�i

P +��S(�0)P(1−|∇�P |) (12)

where i+1 and i denote the present and previous iteration, respectively.In Equation (12), the gradient ∇�P and the sign function S(�0)P with respect to a cell P are

defined in the following manner. The gradient of a variable on an unstructured mesh is derivedfrom Gauss’s divergence theorem as

∇�=∑f

A f

VPn f � f (13)



where the summation is taken over all faces f of cell P , A f represents cell face area, VP is thecell volume and n f is an outward face normal.

However, the level set discretization uses the following numerical modifications to the gradientthat allows information to be passed from the interface while maintaining possible interfaceextremities [5, 16].

Upwinding applied to Equation (13) gives

� f =�P if w ·n�0

� f =�A if w ·n<0

and the gradients are calculated from

∇�= ��

VP

where

��=max mag{+i ,−

i ,i }and

+i =∑

f +(A f ni�P)+∑

f −(A f ni� f )

−i =∑

f +(A f ni� f )+

∑f −

(A f ni�P)

i =∑f(A f ni� f )

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

|∇�|=√∑

i

(�i�

VP

)2

and the sign function is calculated as

S(�)= �0√(�0)

2+(dAP)2

where dAP is the minimum distance between cell centres for any adjacent cells within the entirecomputational grid [5, 16].2.4. Current calculation across the interface

As the calculated current distribution is paramount in the interface development for the PCR,some different methods of calculating its distribution are discussed before presenting the resultsin Section 3. The electric current across an interface of different materials must conform to thecondition that the normal flux is continuous as given by

kelectrolyted�

dn=kmetal

d�

dn



In the instance of metallic deposition, kmetal�kelectrolyte. Since kmetal is generally large andgiven that the currents involved are not large, the cathode boundary condition �=const willpermeate through the metallic deposited layer, anchoring the interface electric potential to theapplied cathode voltage. Using this knowledge to fix �’s value in the deposit can significantlysimplify the calculation of current across the interface. However, the error in this approximationwill increase as the 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 accurately calculatethe interface flux. Four possible options have been examined, which can be applied on blockstructured orthogonal meshes and are now discussed.

Calculating the electric potential gradients and hence current from Ohm’s law can be achievedin an unstructured discretization scheme by

∇�=k×No faces∑

f =1

� f ·A f ·(nx ,ny,nz)VP

(14)

Here, � is the electric potential and nx , ny and nz are the Cartesian components of the compu-tational cell face normal vectors. For example, when calculating the x-direction gradient only nxcomponents are required. Across an interface where k varies, its value can be approximated bytaking a weighted harmonic average given either by

kharm= kmetal×kelectrolytew1×kmetal+w2×kelectrolyte

, w1= d1d1+d2

, w2= d2d1+d2

if the computational grid is block structured, orthogonal and irregular, or by

kharm= 2kmetalkelectrolytekmetal+kelectrolyte

if the computational grid has regular square cells.On block structured orthogonal grids the flux across an interface can be reasonably approx-

imated by using kharm(��/�n), akin to calculating the heat flux through composite materials.However, as can be seen in Figure 6(a), this approximation is only exact if the cells spanning theinterface have the same length, so that ��/�n as well as kharm sit at the same location on theinterface.

If this is not the case, the value of ��/�n is not co-located with kharm and lies midway betweenthe two cell centres. This gives an approximation to the interface flux, but not an exact value.This discrepancy can be improved if the cells surrounding the interface are finer or if the values of� used in the approximation are closer to the interface. This is illustrated in Figure 6(b), where thedotted lines represent the electric potential on both sides of the interface and the value of ��/�n ascalculated from the difference of cell centre values is shown at position 1 in Figure 6(b). The valueof ��/�n is much better approximated at the interface in position 2. This can be achieved withoutusing a fine grid since the distribution of � on either side of the interface is linear. It is thereforepossible to calculate � at equidistant intervals close to the interface by either extrapolation, or asshown in later results by employing a 1-D sub-model that solves the equation k∇2�=0 acrossthe interface, using the boundary conditions provided by the values of � at the cell centres of thecells that span the interface, see Section 2.4.3.

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



Figure 6. Calculating the interface flux: (a) approximating current and (b) improving the approximation.

Figure 7. Finite difference current calculation: (a) finite difference current calculation and (b) i-component.

2.4.1. Finite difference approximation. This approximation is the most obvious choice whenworking within a structured CFD code. The current is calculated by taking consistently forward orbackward differences in each cell and using the harmonic conductivity across the interface cells.Current vectors can be seen for a 2-D test case as shown in Figure 7(a) and the numerical valuesfor the i th current component across a section of the interface are shown in Figure 7(b). Thesevalues illustrate that current components in the x-direction are not continuous across the interface.Similarly, this is also the case for the y-direction components and, although the current distributionis a reasonable approximation, it can be better approximated using other methods. Current is notconserved because of the reasons discussed previously and illustrated in Figure 6(a).



Figure 8. Unstructured gradient calculation.

2.4.2. Interface flux balancing (IFB). Using this method, the current in fluid and solid domainsis calculated at cell centres using Equation (14). The values at the interface get special attention;the value of � at this location is obtained from an estimated interface flux that is obtained froma finite difference calculation of kharm(��/�n). The value of �oppface (Figure 8) at the far face ofthe cell and directly opposite to the interface is known (by weighted average) and the face valueof �intface at the nearest interface cell face is obtained by a reverse calculation from this flux. Thiscalculation is made in the same manner in both the solid and the fluid cells. For example, in thefluid side, the following formula is used:

�int=flux

kfluid+� f , flux=kharm

��

�n

For the solid cell, we replace kfluid with ksolid and use �s . This gives a method to calculatekfluid(��/�n)=flux that will ensure continuity on a structured grid as the current on either side ofthe interface is based on the same flux. When using this procedure on both sides of the interface,an estimation of the current is obtained which is similar to that in Figure 7, but with the addedguarantee that the normal current components are continuous.

This method treats the solid/fluid interface as being positioned at the shared computational cellface between adjacent fluid and solid cells that span the interface. The weights w1, w2, which areused in the calculation to find the harmonic mean, are the cell centre 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 the samelength or if the computational grid is regular; otherwise, it gives a good estimation that guaranteesconservation of the normal component.

Figure 9 shows the current distribution from a 2-D test, the distribution looks similar to thatgiven in Figure 7; however, the normal current components across the interface are continuousas shown in Figures 9(b) and (c). Notice that the difference in current distribution at the cornerbetween the fluid and solid region in Figure 9(a) compared with Figure 7(a). Such differences can



Figure 9. Interface flux current calculation with IFB: (a) flux current calculation;(b) i-component; and (c) j-component.

Figure 10. Using a sub-model to accurately approximate current flux.

result in significantly different deposition growth calculation with the level set method betweensimulations.

2.4.3. IFB using a sub-model. An extension of the above method in Section 2.4.2 can be madefor a more accurate estimation of the interface flux, by using values of � which are equidistant andmuch closer to the interface. This ensures that the value of ��/�n is positioned extremely close tothe interface and almost co-located with the position of the harmonic conductivity as illustrated inFigure 6. The diagram in Figure 10 shows how this is done. The flux at the interface is calculatedusing kharm((�SI−�FI)/(dx f +dxs)), where dx f =dxs and the size of these dx’s can be variablein what is a pseudo-grid or sub-model between the cell centres. A 1-D equation is solved using theTDMA algorithm with the values of � at the cell centres providing the boundary conditions forthis equation. The results of this calculation can be seen in the dotted lines in Figure 6. There is adifferent gradient on either side of the interface because of the different electrical conductivities.Using the values of �SI and �FI on either side of the interface guarantees that the value of theflux lies in close proximity at position 2 (see Figure 10). This avoids the situation where kharm liesat the interface and the value of ��/�n does not, (i.e. at position 1). This approximation is onlynecessary if the cells on either side of the interface are irregularly spaced and would otherwise beunnecessary.

The distribution of current, illustrated in Figure 11(a), is virtually identical to that shown inFigure 9(a). Figures 11(b) and (c) show piecewise parts of the grid and illustrate that the normal



Figure 11. Interface flux current calculation: (a) interface flux current calculation;(b) i-component; and (c) j-component.

current components are continuous across the interface. The numerical values of these componentsare identical up to three significant digits to those shown in Figures 9(b) and (c).

2.4.4. Calculating current with electrical conductivity as a function of the level set position.When calculating material properties using the level set method the electrical conductivity incomputational cells at the interface can be calculated either as a proportion of the cells’ solid/fluid(electrolyte) mixture, or more generally by using a predetermined function. Typically for the levelset, function (15) is used

�=

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

�1 if �>

�2 if �<−

�1+�22

+ �1−�22

sin

(��

2

)if |�|�

(15)

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 readily changed.

= 3dAP2

Using the functions above to define the interface, we find that a domain containing two materialseffectively becomes a domain with a composite number of materials depending on how manycells are defined for the interface width. Figure 12 shows the distribution of electrical conductiv-ities around the interface using Equation (15). Here, the interface width was defined as 2 cells,one cell spanning either side of the interface. This effectively turns the domain into one of 4‘pseudo’ materials with the result that there are now four regions with different conductivity, asillustrated in Figure 12. This is a significant complication to current calculation and, together withthe reasons discussed previously, is why electric current has been calculated using the methodsdiscussed in Sections 2.4.1–2.4.3. The default interface width of 3 cells (1.5 cells either side ofthe interface) would result in more than four different values of electrical conductivity within thecomputational domain.



Figure 12. Electrical conductivities calculated from level set function.

Figure 13. Considerations at the interface when calculating current.

Current calculation in Sections 2.4.1–2.4.3 has been based on the assumption that the freesurface lies on the shared computational cell face of the two cells spanning the interface. There isgood reason for this assumption. It can best be appreciated by examining the alternative approach,that is to calculate current based on the exact position of the interface and on the level set valuesthat define normal (shortest) distance to the interface. The level set distances are illustrated inFigure 13 as d1 and d2.

To calculate the current from the electric potential, the gradient is of course required and ifusing Equation (14), values are needed at the face locations, n, e, w and s, see Figure 13. If cell Phas a composite electrical conductivity given by, for example, Equation (15), then the neighbourcells may also have different conductivities. The values of � at cell face centre locations n, e, w

and s need to be estimated from these neighbour cells that have different electrical conductivities.Therefore, there is an inconsistency when using Equation (14), as it assumes that the value ofelectrical conductivity at cell centre P is used in conjunction with the face (n,e,w,s) values of �.



Additionally, there is a problem if d1 is very small, d1�d2. In this instance, numerical roundingcan cause significant errors in the continuity of the current and a sub-model such as that discussedin Section 2.4.3 would be required to achieve better accuracy.

A more straightforward and computationally consistent way to calculate the current is to assumeonly two electrical conductivities, those of the metal and electrolyte. When solving Equation (7)for the electric potential, the conductivity of cell P is then always positioned above the interfaceand so has conductivity kfluid, while the conductivity at cell S is located in the deposit and hasthe value ksolid. The solution of the potential equation then assumes that the entire cell P is in thefluid region and conversely cell S is immersed entirely in the solid. The numerical solution is thenconsistent with the interface being at the cell face s of cell P . The values of � at all cell faces,except s, can then be readily calculated from the weighted averages from neighbour cells if theyare on the same side of the interface, i.e. have the same electrical conductivity. The value at thecell face s can then be obtained either from an estimation of the current flux that passes throughthe interface (2.4.2), or from using a sub-model to solve for � across the interface as described inSection 2.4.3.

2.4.5. Current limitations and non-conjunctionality considerations. The calculation of currentdiscussed in Sections 2.4.1–2.4.3 assumes that the grid cells are block structured and orthogonal.In this arrangement the grid nodes are co-located in the cartesian x , y, z directions and the directionof the estimated interface current flux lies parallel to the face normal vector and close to the cellface centroid.

The block structured grid arrangement can effectively represent many practical geometrical struc-tures, however, for complex geometries, it may be necessary to relax the orthogonality constraintand allow the grid cells to skew. The discretization of the governing equations and the currentcalculation methods presented earlier will require modification to reduce errors that will otherwisegrow in the simulation with increasing grid skew.

Figure 14 illustrates two distinct classes of mesh skew. In Figure 14(a), the line connectingthe cell centres intersects the cell face at its centre, but is not parallel to the face normal. This isreferred to here as non-orthogonal. Conversely in Figure 14(b), the line connecting cell centresis parallel to the cell face normal but does not intersect the face at its centre. This is referred tohere as non-conjunctional. Using the above distinctions for skewness the errors created by non-orthogonality are restricted to the diffusion term in the governing equations and finite differencegradient calculations of the electric potential as both require the estimation of a derivative withrespect to the face normal. The correction of the diffusion term is well reported in the literatureand is described in detail in Croft et al. [8], Physica [16] and McBride et al. [17].

Non-conjunctionality affects both convection and diffusion terms in the governing equationsas both require estimates of the values of quantities at cell face centres. Corrections will also berequired for Equation (14) as the face value of � is assumed to be co-located at the face centre.The corrections for the convection and diffusion terms are again well reported and discussed indetail in the literature [8, 16, 17].

Because a finite difference calculation over non-orthogonal cells will require redirection toalign the resulting vector with respect to the face normal, it is convenient to replace it by usingEquation (14). This is possible in the regions away from the interface (of equal electrical conduc-tivity) as the electric potential, �, can be easily interpolated from cell centre to cell face centresbecause of its linear distribution. However, across the interface, it is not easy to take a weightedaverage of � because the difference in electrical conductivites produces a piecewise linear



Figure 14. Non-orthogonal cells: (a) non-orthogonal cell face; (b) non-conjunctionalcell face; and (c) finite difference flux correction.

distribution, as shown in Figure 6. Similar to the orthogonal grid, there are two obvious options:the first is to extrapolate the value of � towards the face centre separately from both cells spanningthe interface and then use these values in Equation (14) and the second is to use a finite differencegradient calculation at the interface cells. The gradient from this calculation could be renderedmore accurate by using the values of � much closer to interface, but it should be realigned in thedirection of the face normal vector because it points along the line connecting cell centres.

For cells that are not conjunct the values of electric potential used in Equation (14) will requirethe following modification, as knowledge of � at face centroid locations is necessary:

��

�xi= 1

V

∑fA f nxi

[�I +

(��

�x jdIf

)](16)

Here, dIf is the distance vector from the intersection point, I , to the face center, f (Figure 14(c)).The term ��/�x j represents the previous iterations value of the cell gradients.

At the interface it will still be necessary to take a finite difference calculation to estimate thecurrent flux if using the interface flux balancing scheme of Section 2.4.2.

The calculation of this flux should be modified by calculating the harmonic conductivity usingthe distances dP and dA from the cell centres to the intersection point as shown in Figure 14(c).Using these distances will ensure that the harmonic conductivity lies at the intersection point I .Similarly to when the grid is orthogonal the value of the gradient ��/�n will not be co-located atposition I , unless the distances dA and dP are equal and will be closer to I as the grid becomesfiner. The formula to relocate a variable � from the intersection point, I , to the cell face centre,f , is given by

� f =�I −dIf∇�xi (17)

Substituting for � with the flux kharm∇� in (17), it should be noted that the required flux at thecell face centre is approximated by the value at I , kharm f ∇� f =kharmI ∇�I , because the correctiontends to zero. This is because the flux as a gradient is linear and so that the gradient of a gradienttends to zero as it is a second-order differential. The flux should now be redirected with respectto the cell face normal. This is achieved by multiplying the gradient with v ·n, where v is the unitvector along the line joining cell centres and n is the unit outward cell face normal, as shown inFigure 14(c).



Errors from this approximation will increase in correlation with mesh skewness and thereforecare should be taken to keep the grid as orthogonal as possible.

3. SIMULATING THE PRIMARY AND TERTIARY CURRENT REGIMES (TCR) IN 2-D

In this section we show that the results comparing the level set approach against the EITM for2-D simulations of the PCR and the tertiary current regime (TCR).

3.1. Simulating the PCR

The primary current distribution provides a good first target for model development because ofthe simpler governing equation set. Under these conditions, the deposition current is modelledusing Ohm’s law. If the concentration of reacting ions is assumed sufficiently high, we can ignorethe influence of the ion concentration in Equation (6). This has the advantage of reducing theequation set to that of solving a Laplace equation for electric potential using Equation (7) withthe right-hand side reduced to zero. At the deposition interface the current normal to the surfaceis given by the boundary condition a in Figure 15(a).

Figure 15. Computation grid and boundary conditions: (a) boundaryconditions and (b) computational grid.



The computational grid is shown in Figure 15(b), together with the boundary conditions for theelectric potential equation. In this instance the electric potentials have been fixed at either end ofthe domain. An alternative is to replace the fixed potential boundary condition at the anode, byspecifying an incoming current boundary condition, i.e. k��/�n= Ianode.

Both solution methods produce the expected current crowding effects where, at trench interfaces,there is a larger deposition rate because of a pinching effect on the electric field from sharp corners.In these instances, voids may be formed as current and hence the deposition rate is higher atthese edges. For comparative reasons both methods have been run on a regular grid because ofthe previously mentioned mesh limitation for the EITM.

What is important to the level set method is the accurate calculation of the driving velocitythat in this case is the electric current. This is complicated by the existence of materials that havemassively different electrical conductivities and a moving interface. This is discussed in detail inSection 2.4. It can be seen from comparing Figures 17 and 18 that a point is generated with thelevel set method. This point attracts current and therefore continues to grow as an extremity. Thisgrowth may be physically unrealistic, however given the model parameters, that current here isthe sole generator of movement, it seems to be a plausible solution. Actually, levelling agentscan be applied to the electrolyte, which are specifically targeted at suppressing the surface growthof unwanted perturbations and such extremities are practically unrealistic. The explicit interfacetracking method does not exhibit a similar manner of growth because of the way that the fillingis calculated; these matters are discussed in Section 2.3.1. The latter method spreads any excesscell filling to the adjacent neighbour cells and hence the extremity is smoothed away.

3.2. Simulating the TCR

The introduction of ionic concentration such as to advance the model involves greater numericalrestrictions at the deposition interface. This is now considered under DC conditions with a singleionic reacting species and an assumed constant overpotential. In this scenario, Equation (6) issolved for bulk concentration (c=c/c∞) together with Equation (7) for electric potential. If onlyone species is considered, the right-hand side of Equation (7) equates to zero. At the boundarybetween the metal–electrolyte interface, condition b of Figure 15(a) needs to be enforced. Thepositions of the electrolyte side of the interface cells must be tracked throughout the simulation sothat the boundary conditions of Equations (6) and (7) can be applied. A diagram of the solutiondomain is shown in Figure 16(a), where BV stands for the deposition current calculated from theButler–Volmer equation. At the interface, the deposition current is governed by this surface kineticfunction and therefore an appropriate ‘sink’ boundary condition must be applied to the electrolyte-side computational cells. To avoid any physically unrealistic loss of ionic concentration fromconduction the diffusion coefficient across the interface should be set to a small value approachingzero so that the only loss of concentration comes from the Butler–Volmer equation. This situationcan be avoided by splitting the computation domain into two sections and linking these regions byappropriate sink/source-type boundary conditions: the current leaving the electrolyte side shouldbe equal to the current entering the deposit region. To recover current from this type of calculation,the techniques discussed in Sections 2.4.1–2.4.3 can be applied. Figure 16 illustrates this ideashowing the results from a 1-D test case, in which the unequal spacing of the grid cells around theinterface is a means to test current calculation within the model, otherwise this grid arrangement isnot sensible. As required, the current calculated across the interface is continuous and is of equalmagnitude to the computed deposition current from the Butler–Volmer equation.



Figure 16. Tertiary current regime considerations: (a) tertiary current domainand (b) TCR-interface-considerations.

Figure 17. PCR: explicit interface method versus level set at 30 s. (a) Explicit interfacetracking; (b) level set; and (c) current.

3.3. Results—primary current distribution

The results for both methods are now presented for the PCR. This scenario may be somewhatphysically unrealistic throughout an entire simulation. It is only valid if the resistance of theelectrolyte is greater than that of the kinetics at the interface. During deposition, the regime maychange at some stage and the deposition current may no longer be governed by Ohm’s law. Theresults presented in this section are from a model that assumes the interface motion is governedby Ohm’s law throughout the entire simulation.

Figures 17(a) and (b) show the deposit after 30 s, the level of growth is similar,; however,Figure 17(a) shows a more rounded growth at the deposit corner. This is due to the spreading ofthe excess cell filling to neighbour cells as discussed in Section 2.3.1 and illustrated in Figure 2.Current vectors for the level set method are shown in Figure 17(c).



Figure 18. PCR: explicit interface method versus level set at 150 s. (a) Explicitinterface tracking; (b) level set; and (c) current.

Figures 18(a) and (b) 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 EITMmethod 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 18(c) 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 s are explained by the differences in thealgorithms, the level set method profile follows that of the current and the EITM smoothes possibleextremities.

3.4. Results—tertiary current distribution

In the TCR the deposition velocity is governed by interface kinetics that is described by the Butler–Volmer equation, condition b in Figure 1. The deposition rate is a function of the concentration ofcupric ions and is a scalar rather than the vector current density of the PCR. Deposition growth isnormal to the interface surface, and it is the differences in the magnitude of this deposition rate thatdrives any 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 of Equation(6) decreases in magnitude. Figure 19 shows a similar rate and shape of growth between the twomethods at 400 s. Figure 20 shows the deposition at 900 s with the growth looking very similar. Plotsof concentration are shown in Figure 21, these are in close agreement and explain the similarityof the deposition profiles, since the concentration of ionic species is the driving force of thedeposition rate. Figure 22 illustrates the rates in moles/m−2 s−1 at which the concentration ofCu2+ is absorbed at the interface.

Plots of the deposition at 1400 s show the formation of a void in the trench. In Figure 23 thegrowth can be seen to be non-conformal with the creation of a void rendering the deposition



Figure 19. TCR: explicit interface method versus level set at 400 s. (a) Explicitinterface tracking and (b) level set.

Figure 20. TCR: explicit interface method versus level set at 900 s. (a) Explicitinterface tracking and (b) level set.

ill-equipped for any function that involves electrical circuitry. In this case, the trench has an aspectratio of 3 :1. In the next section of this paper, simulations of acoustic forces are included as ameans to generate a conformal growth without creating these voids.



Figure 21. Concentration: explicit interface method versus level set at 900 s.(a) Explicit interface tracking and (b) level set.

Figure 22. Interface sink: explicit interface method versus level set at 900 s.(a) Explicit interface tracking and (b) level set.

3.5. Computational runtimes

The simulations were run on an Intel 2.8GHz PC Xeon (TM) running Windows XP. The gridconfiguration consisted of 3627 cells with a total simulation time for the PCR of 200 s discretizedinto 200 time steps of duration 1 s. For the PCR simulation runtimes were 235 CPU seconds for theEITM and 2208 s for the level set. The TCR simulations had a total runtime of 2000 s discretized



Figure 23. Explicit interface method versus level set at 1400 s.(a) Explicit interface tracking and (b) level set.

into 200 time steps of 10 s. The EITM simulation took 309 CPU seconds compared with 3317 s forthe level set. Although these are hardly benchmark results, it is apparent that the EITM is about10× faster than the level set method.

4. ENHANCED DEPOSITION THROUGH MEGASONIC AGITATION

Megasonic agitation provides a promising means to solve the problems concerning void formationin high aspect ratio trenches and vias. It works by enhancing and replenishing ionic transport thatwould otherwise not penetrate deep into these features. The physical effect of this process is todecrease the Nernst diffusion layer [10] and therefore increase the kinetic rate at the interface.Experiments at Heriot–Watt by Kaufmann et al. [3] have shown an improvement in both thedeposition rates and deposit quality for microvias with features of aspect ratio up to 2 :1 beingsuccessfully completed. Presently a limitation for trenches with ratios larger than 2 :1 is reported dueto seeding problems; the physical application of conductive seed layer throughout the microcavity.The acoustic streaming is generated from a piezoelectric transducer placed in the electrolyte thatstimulates high-frequency agitation (1MHz) at an acoustic intensity of 60KW/m2. The streamingprocess arises from the resulting pressure fluctuations within the feature. They generate Reynoldsstresses at the feature walls that drive a steady fluid recirculation replenishing the supply of reactingionic species from the bulk of the containing bath deep into the feature.

4.1. Solution procedure

The numerical process closely follows that presented by Nilson and Griffiths [4] and is in twostages. The first concerns the analytical solution of the linearized Euler equations as developedby Rayleigh and Nyborg for channels between infinite parallel plates and is detailed concisely by



Nilson and Griffiths in [4]. The second stage involves the calculation of the time-averaged Reynoldsstresses that are produced from these linearized Euler equations and their numerical integrationinto the Navier–Stokes equations. For validation purposes, simulations have been made to examinethe effects of acoustic streaming in narrow features with widths in the order of tens of microns sothat the resulting flow profiles can be compared against those presented in [4]. For completenessthe solution process is now briefly discussed. The linearized Euler Equations (18)–(19) are solvedanalytically

��a�t

+�0∇ ·ua =0 (18)

�0�ua�t

=−∇Pa+�∇2ua+(

�b+ 1

3�

)∇(∇ ·ua) (19)

Here, �, �b are shear and bulk viscosities, c is the sound speed, �a , Pa are the acoustic density andpressure variations, �0 the background electrolyte density and ua0 the nominal streaming velocity.The analytical solutions for the acoustic velocities (ua,va) are given in Equations (9)–(10) ofNilson and Griffiths [4] and the real components can be expressed in the following form (20)–(21):

va ≈ua0e−y

[cos

(�t− 2�y

)−ex/d cos

(�t− 2�y

− x

d

)](20)

ua ≈ �d

[Re(va)−Im(va)] (21)

ua0≈√

I

�c, d≈

√2�

��, ≈ c

f(22)

For the simulation presented here, the electrolyte was assumed to have the properties of waterand a sound intensity of 60kW/m2 at 1MHz was estimated. Equations (20)–(21) were computedwithin the trench region across the domain from one side of the deposit to the other, but onlywithin cells containing the electrolyte. These acoustic velocities ua , va are harmonic functions oftime and hence their values repeat over wavelengths at the frequency of 1MHz. The wavelengthwas discretized into 12 pseudo-time steps that were calculated at the beginning of each time stepand the subsequent values of ua(t) and va(t) were stored as a requirement for calculating thetime-averaged acoustic forces. The symbol in Equation (20) represents an attenuation coefficientfor the wave and has been set to zero in this instance as the height of the feature is less than thewavelength (1.5mm).

The acoustic forces are then calculated from (23) where the brackets 〈〉 denote timeaveraging. They are subsequently integrated into the Navier–Stokes equations as volumetricsources

F=�0〈(ua ·∇)ua+ua(∇ ·ua)〉 (23)

To accurately represent the acoustic forces the computational grid must be fine enough to capturethe large Reynolds stresses that occur next to the feature walls. A grid pitch of 1�m using regularlyspaced cells was used within the trench region as shown in Figure 24(a). In this case the acousticforces permeate throughout the feature from top to bottom and have a calculated magnitude of



Figure 24. Acoustic computational grid—Acoustic force distribution: (a) acousticforces computational grid and (b) acoustic force.

Figure 25. Acoustic flow profile.

the order 3.5×104N/m3. The distribution of these forces is illustrated in Figure 24(b) and showstheir prominence near to the walls.

When integrated into the Navier–Stokes equations these forces produce a flow profile thatis driven downwards at the feature walls and recirculates back through the feature centre fromcontinuity. This is illustrated in Figure 25. In this simulation there was a forced backgroundelectrolytic flow from left to right. The deep recirculating within the trench makes conformalgrowth possible within the feature.

Normalized flow profiles through the channels are affected by the channel width and are fastestin the region of the largest Reynolds stresses adjacent to the feature walls. The velocities in thisregion are asymptotic with increasing channel width. In Figure 26, the value of w/ represents adimensionless channel width, the smaller its value the narrower the channel. The plotted velocitieshave been normalized by a nominal streaming speed of 2.6×10−6m/s, ua0 of Equation (22) andare therefore tiny. The profiles shown in Figure 26 are however in good agreement with the resultspresented by Nilson and Griffiths [12].



Figure 26. Acoustic velocity profiles.

As the channel width increases, the velocity profile becomes similar to one that has beengenerated through wall movement in a static fluid by a non-slip wall boundary condition. As suchthe complications of the streaming physics might be replaced in numerical simulations with largerwidth features by the application of a suitable fixed wall velocity.

4.2. Results: acoustic agitation

Results are presented on the simulations with and without acoustic forces for the EITM method.This method was chosen because it has significantly less computational expense compared with thelevel set method, especially with the added burden of acoustic agitation and the required fine grid.The routines that calculate the acoustic forces are independent to the deposition algorithms and theresulting fluid motion couples into both methods only through the convection term of Equation (6).Figures 27 and 28 show profiles of the deposit growth with and without megasonic agitation.With agitation significantly faster and more conformal growth occurs. This effect is producedbecause the concentration of Cu2+ ions in the trench has been maintained by the convectionplume that permeates throughout the trench. The concentration of Cu2+ is much higher in thefeature with agitation as illustrated in Figure 29. The corresponding rate of growth could benumerically mimicked (without agitation) by dramatically increasing the diffusion coefficient byseveral orders of magnitude over the value used in this simulation (order E−10). Physically, theresulting forced motion of Cu2+ concentration reduces the Nernst diffusion layer [3, 10]. In thepresence of agitation this is altered from a velocity dependence to the inverse of frequency asshown, respectively, in Equations (24) and (25), where � is the kinematic viscosity and � is thefrequency [3]. In this continuum model, an increase in frequency corresponds to a larger acousticforces, more convection and higher interface kinetics

�hydrodynamic=0.16( �

Ux

)1/2(24)



Figure 27. EITM: deposition growth with megasonic agitation. (a) 100 s;(b) 400 s; (c) 900 s; and (d) 1300 s.

�acoustic=(2�

�

)1/2

(25)

Figure 29 compares the concentration profile at 400 s, a higher concentration of Cu2+ is main-tained throughout the simulation. In the run without megasonic agitation the concentration profilequickly depletes at the interface and becomes limited to the speed at which the Cu2+ ions canreach the vicinity of the interface by diffusion.

The boundary condition of Equation (6) for both simulations can be seen at 400 s in Figure 30.This is a sink condition that represents the loss of Cu2+ in units of moles/(m2 s) at the interface.It is proportional to the deposition current and therefore the growth rate. Figure 30 illustrates thatthis sink is an order of magnitude higher with acoustic agitation.

The velocity profile of the electrolyte can be seen within the trench at 400 s. The velocity profilein the solid deposit region has been fixed to zero by inclusion of Darcy forces in the momentumequations. The following source term was applied to the momentum equation as:

S� = frac(1− f )2(max( f,�))3�V

e

where f is the electrolyte fraction content of a cell that is zero in cells containing solid, � is thesolidification modulus, its value is 1E−10, V is the cell volume, � is the dynamic viscosity andfinally e is the permeability coefficient that has a value close to zero.



Figure 28. EITM: deposition growth without megasonic agitation. (a) 100 s;(b) 400 s; (c) 900 s; and (d) 1300 s.

Figure 29. Concentration profiles at 400 s with and without agitation:(a) with agitation and (b) without agitation.

The acoustic forces are prevalent along the sides of the deposit; these are calculated at the startof every time step to keep them in track with the deposit growth. Figure 31 illustrates that theseforces have a value of approximately 2e+4N/m2, 400 s into the simulation. In this simulation,megasonic agitation has roughly doubled the growth rate of the deposit. Without acoustic agitationthe growth rate is approximately 0.5�m/min. It should be stressed that numerically this rateis dependent on the model parameters such as overpotential. The computational grid for these



Figure 30. Cu2+ sink at the interface: (a) with forces and (b) without forces.

Figure 31. Velocity vectors and force distribution at 400 s: (a) velocity vectorsand (b) acoustic force Dist’n.

simulations consisted of 34 394 cells. The simulated runtime was 1300 s split into 130 time steps oflength 10 s. Simulation runtimes were 57 414 CPU seconds without megasonic agitation, slowingby a factor of 20% to 70 882 CPU seconds with the inclusion of the forces.

5. CONCLUSIONS

Two numerical models of deposition motion have been presented, which have been incorporatedinto an unstructured CFD environment and developed to simulate filling in regular shaped micronscaled features. The CFD framework provides the perfect host to electrodeposition simulations asthe solutions of momentum, concentration and enthalpy equations are implicit to these codes. Themodels presented in this research have been developed within an unstructured CFD multiphysicsframework, but necessitate the use of block structured grids for accurate interface electric currentcalculation. The EITM formulation makes limited use of meshing methodology in that it is restrictedto regular distributed computational cells. The level set formulation is more flexible and can beapplied to block structured grids with irregular cell spacing and can be extended to correct forgrid skew. The numerical issues of calculating electric current across a moving interface have



been examined and techniques have been presented which can be successfully applied with thelevel set formulation. The explicit interface tracking method represents a fast, relatively simple-to-implement procedure, while the level set method is significantly more complex in terms ofdevelopment and application. Both methods give good approximations to the predicted filling of a1-D test case driven by current governed by Ohm’s law. But in 2-D there are significant differencesin the deposit profile under the primary current distribution. These differences are explained in theway that the current is calculated. Using the level set method the current is calculated throughoutthe domain as values are required across the interface in both deposit and electrolyte to accuratelydrive the deposit growth. For the EITM, the current is only calculated in the electrolyte regionand the electric potential in the deposit is anchored to the cathode boundary condition becauseof the magnitude of electrical conductivity in the deposit. The deposit growth is then smoothedat the interface with possible spreading of excess cell growth in a time step transferred to itsneighbour cells. This avoids overfilling within a time step, but because current is not explicitlycalculated across the interface, beyond 1-D, its results must be viewed with suspicion as interfaceperturbations and extremities will be removed. Both methods give similar and physically realisticdeposit profiles when the growth is governed by surface kinetics as defined by the Butler–Volmerequation. The level set method generally seems to give sharper and cleaner interface growth betweentime steps, this is because the motion is controlled by the solution of an advection equation ratherthan directly applied. The ability of the level set method to capture surface perturbations and itsflexibility regarding mesh distribution make it favourable over the EITM. With the EITM, a noteof caution must be stressed regarding the numerical smoothing applied to the interface. This canlead to results that may not accurately reflect the surface propagation when the rate of depositionis fast, as under the PCR, where extremities can develop that lead to excessive current crowding.

A model of acoustic agitation has been implemented and its physical effects have been examinedin Section 4. The acoustic forces have been calculated from an analytical formula described byNilson and Griffiths [4] and integrated into the Navier–Stokes equations as volumetric sources.The generated velocity profiles show good agreement with those presented in [4] and the effect ofconformal faster deposit growth can be clearly seen in the simulations. The computational grid mustbe sufficiently fine in the feature interface region to capture the acoustic forces (�1�m), addingsignificant computational cost to the models. Future work should address the possibility of applyingan adaptive grid in this region. It is hoped by this author that this paper will provide pragmaticand useful information to anyone wishing to develop continuum models of electrodeposition.

ACKNOWLEDGEMENTS

The author team wishes to acknowledge the support from the Engineering and Physical Sciences ResearchCouncil (EPSRC) who funded this work and also to our industrial partner Merlin Circuits. The authorthanks his former colleagues Dr Nick Croft and Dr Alison Williams of Swansea University for theirconversations and advice regarding the use of the Physica simulation software.

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