COMSOL 6.4 - Fountain Flow Effects on Electrodeposition on a Rotating Wafer

Fountain Flow Effects on Electrodeposition on a Rotating Wafer

This example extends the analysis made in the model Electrodeposition on a Resistive Patterned Wafer by including the diffusion and convection of copper ions in the electrolyte.

The coupled mass transport convection–diffusion effects are of interest in this type of reactor since they are accentuated toward the rim of the wafer, limiting the current density. This counterbalances the activation overpotential, which is highest at the rim due to electric current conduction effects in the wafer. By the design of the reactor, the relation between mass transport and activation potential effects can be optimized to make the current distribution over the wafer more uniform.

This application requires the CFD Module.

Model Definition

Due to symmetry the model geometry is made axisymmetric in two dimensions. The model geometry is shown in Figure 1. A wafer is mounted in a holder of isolating material at the top of the cell and rotates at a given rotational speed of 30 rpm. Electrolyte enters the reactor (which does not rotate) from the bottom at a given flow rate and exits through the narrow gap at the top right in the figure. Copper is deposited on the rotating cathode wafer at a known average current density.

Figure 1: Model geometry, axisymmetric around r = 0. Two domains are used, with the copper concentration being solved for in the top domain only to save computational time.

Choice of physics

Use the Electrode, Shell physics for solving for the potential on the wafer boundary, ϕs,wafer (V).

In the cell, use a supporting electrolyte so that the electrolyte conductivity is not affected by local copper concentration changes. This implies that the Secondary Current Distribution physics can be used to model the electrolyte potential ϕl (V), and the Transport of Diluted Species physics to model the mass transfer, solving for the electrolyte concentration of copper ions, c (mol/m3).

The relatively low rotational speed, in combination with the large wafer size, implies that the flow is not turbulent. This means that a Laminar Flow interface is suitable for solving for the velocity, v (m/s), and pressure p (Pa).

Electrode, Shell

Use a resistivity of 1.7·10−6 (Ω·cm) and an electrode layer thickness of either 75 nm or 150 nm. The latter parameter will be varied by the Continuation settings in the solver. Ground the right endpoint of the wafer. Use a Normal Current Density feature to create a current source, which becomes coupled to the Electrode Reaction current density set up by the Secondary Current Distribution physics.

Secondary Current Distribution

Use an electrolyte conductivity of 0.5 (S/m). Set the total current at the inlet boundary to correspond to an average wafer current density (0.5 or 2 A/dm2), which is varied by the Continuation settings in the solver.

On the wafer boundary, use an Electrode Surface node to set up the electrode current density according to

(1)

where i0 (10 A/m2) is the exchange current density, F is Faraday’s constant (96,485 C/mol), R is the molar gas constant (8.3145 J/(mol·K)), T is the temperature (298.15 K), and cb (0.3 mol/l) is the bulk concentration of copper ions.

The overpotential, η (V), is defined as

Laminar Flow

Enable Swirl Flow to solve also for the velocity in the φ direction. Specify the inflow velocity to correspond to a total flow of 30 l/min, and set the outlet pressure to 0 Pa. Use a Wall boundary condition, with the Sliding Wall feature enabled, to define a rotational velocity for the rotating wafer and the holder boundaries.

Transport of Diluted Species

Since concentration gradients are expected only in the vicinity of the electrode surface, the computational time can be reduced by modeling the copper ion transport in upper domain only.

Use a diffusion coefficient of 5·10−6 (cm2/s) and couple the convective flow to the velocity solved for by the Laminar Flow physics. Enable Migration and couple it to the electrolyte potential solved for by Secondary Current Distribution physics.

Set the concentration at the inlet boundary to cb and set zero diffusional flux at the outlet using an Outlet condition.

Couple the flux of copper ions at the wafer to the electrode current density in the Secondary Current Distribution physics boundary using an Electrode-Electrolyte Interface Coupling. This sets up a flux condition using Faraday’s law.

Study

Since the flow profile is not affected by any other physics, solve the model in two steps with Laminar Flow only in the first step, and the other physics in the second step. Use a Study Extension with Continuation in the second study step to solve for two different average wafer current densities (0.5 and 2 A/dm2) and two different wafer seed layer thicknesses (75 and 150 nm).

Results and Discussion

Figure 2 shows the concentration in copper ions in the electrolyte. The concentration is lowest toward the rim of the rotating wafer.

Figure 2: Copper concentration for an average wafer current of 200 A/m2 and a seed layer thickness of 150 nm.

Figure 3 shows the velocity steam lines in the reactor. The velocity magnitude is highest close to the rotating surfaces, as indicated by the higher density of streamlines.

Figure 3: Velocity streamlines.

Figure 4 shows the electrode current density as described by Equation 1. The cathodic current density increases toward the rim of the wafer until concentration effects start to dominate, at the end of the wafer the current density drops significantly.

Figure 4: Cathode current density versus radial coordinate.

Notes About the COMSOL Implementation

Use second-order elements in all physics interfaces.

Electrodeposition_Module/Tutorials/fountain_flow

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

Add the four different physics that are used in this model.

In the tree, select > .

Click .

In the tree, select > > .

Click .

In the tree, select > .

Click .

In the text field, type phis_wafer.

In the tree, select > > .

Click .

Click

In the tree, select > .

Click

Global Definitions

Parameters 1

Load the model parameters from a text file.

In the window, under click .

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file fountain_flow_parameters.txt.

Geometry 1

Draw the geometry as the union of two rectangles, split one boundary by adding a point. The lower part of the split boundary will become the outlet.

Rectangle 1 (r1)

In the toolbar, click

In the window for , locate the section.

In the text field, type d_cell/2.

In the text field, type h_cell.

Locate the section. In the text field, type -h_cell.

Rectangle 2 (r2)

In the toolbar, click

In the window for , locate the section.

In the text field, type d_wafer/2.

In the text field, type h_gap.

Point 1 (pt1)

In the toolbar, click

In the window for , locate the section.

In the text field, type d_wafer/2.

In the text field, type h_outlet.

Click

Click the

button in the toolbar.

The finalized geometry should now look like the figure below:

Definitions

Add some explicit selections to facilitate boundary selection when setting up the physics.

Cathode

In the toolbar, click

In the window for , locate the section.

From the list, choose .

Select Boundary 5 only.

In the text field, type Cathode.

Inlet

In the toolbar, click

In the window for , locate the section.

From the list, choose .

Select Boundary 2 only.

In the text field, type Inlet.

Outlet

In the toolbar, click

In the window for , locate the section.

From the list, choose .

Select Boundary 8 only.

In the text field, type Outlet.

Rotating wall

In the toolbar, click

In the window for , locate the section.

From the list, choose .

Select Boundaries 5 and 9 only.

In the text field, type Rotating wall.

Laminar Flow (spf)

Now start defining the Laminar flow settings of the physics. Enable Swirl Flow to also solve for the velocity in the φ direction.

In the window, under click .

In the window for , locate the section.

Select the checkbox.

Fluid Properties 1

In the window, under > click .

In the window for , locate the section.

From the ρ list, choose . In the associated text field, type rho.

From the μ list, choose . In the associated text field, type mu.

Inlet 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. In the U0 text field, type v_in.

Outlet 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Wall 2

Use a sliding wall to define the rotating surfaces. The velocity is zero in the rz-plane and ωr in the φ direction.

In the toolbar, click

and choose .

Select Boundaries 5 and 9 only.

In the window for , click to expand the section.

Select the checkbox.

In the vw text field, type omega*r.

Initial Values 1

In the window, click .

In the window for , locate the section.

Specify the u vector as


0	r
0	phi
v_in	z

In the p text field, type 1.

Secondary Current Distribution (cd)

Electrolyte 1

Now set up the model for the electrolyte currents. Start with the electrolyte conductivity.

In the window, under > click .

In the window for , locate the section.

From the σl list, choose . In the associated text field, type sigma.

Electrode Surface 1

In the toolbar, click

and choose .

Set the electric potential of the electrode surface to the potential solved for by the Electrode, Shell physics.

In the window for , locate the section.

From the list, choose .

Locate the section. In the ϕs,ext text field, type phis_wafer.

Electrode Reaction 1

Set the electrode kinetics to be dependent on the local concentration of copper ions.

In the window, click .

In the window for , locate the section.

From the Eeq list, choose .

In the CO text field, type c/c_bulk.

Locate the section. In the n text field, type 2.

Locate the section. From the list, choose .

From the list, choose .

In the i0,ref(T) text field, type i0.

In the αa text field, type 1.5.

Electrolyte Current 1

Define the total current of the cell on the bottom electrolyte boundary.

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. In the Il,total text field, type I_tot.

Transport of Diluted Species (tds)

Now define the physics for the transport of copper ions. To save computational time, model only the domain in direct vicinity to the electrode surface.

In the window, under click .

In the window for , locate the section.

Click

Select Domain 2 only.

Locate the section. Select the checkbox.

Species Charges

In the window, under > click .

In the window for , locate the section.

In the zc text field, type 2.

Fluid 1

In the window, click .

In the window for , locate the section.

In the V text field, type phil.

Locate the section. In the Dc text field, type D_Cu.

The velocity field will be coupled to Laminar Flow later using the Reacting Flow multiphysics feature.

Initial Values 1

In the window, click .

In the window for , locate the section.

In the c text field, type c_bulk.

Inflow 1

In the toolbar, click

and choose .

Select Boundary 4 only.

In the window for , locate the section.

In the c0,c text field, type c_bulk.

Outflow 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Electrode Surface Coupling 1

Define the flux of copper ions on the electrode surface by coupling it to the electrode currents.

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Reaction Coefficients 1

In the window, expand the node, then click .

In the window for , locate the section.

From the iloc list, choose .

Locate the section. In the n text field, type 2.

In the νc text field, type -1.

Electrode, Shell (els)

Now set up the model for the electric potential in the wafer.

In the window, under click .

In the window for , locate the section.

From the list, choose .

Electrode 1

In the window, under > click .

In the window for , locate the section.

In the s text field, type t_seed.

From the σ list, choose . In the associated text field, type 1/r_seed.

Ground 1

In the toolbar, click

and choose .

Select Point 8 only.

Normal Current Density 1

Couple the wafer potential model to the electrode reaction by adding a current source corresponding to the electrode reaction.

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the in list, choose .

Multiphysics

Finally, set up the Reactive Flow multiphysics feature.

Reacting Flow, Diluted Species 1 (rfd1)

In the toolbar, click

and choose > .

Mesh 1

Create a triangular mesh with higher resolution close to the electrode surface. Add boundary layers to further improve the accuracy where needed.

Size 1

In the window, under right-click and choose .

Size

In the window for , locate the section.

From the list, choose .

Size 1

In the window, click .

In the window for , locate the section.

From the list, choose .

Locate the section. Click the button.

Locate the section.

Select the checkbox. In the associated text field, type 1.5e-4.

Free Triangular 1

In the toolbar, click

Boundary Layers 1

In the toolbar, click

Boundary Layer Properties

In the window, click .

Select Boundaries 5–7 and 9 only.

In the window for , locate the section.

In the text field, type 5.

In the window, right-click and choose .

The finalized mesh should look like this:

Transport of Diluted Species (tds)

Use 2nd-order elements in all interfaces (Electrode, Shell uses 2nd-order elements by default).

In the window, under click .

In the window for , click to expand the section.

From the list, choose .

Secondary Current Distribution (cd)

In the window, under click .

In the window for , click to expand the section.

From the list, choose .

Laminar Flow (spf)

In the window, under click .

In the window for , click to expand the section.

From the list, choose .

Study 1

Step 1: Stationary

In the window, under click .

In the window for , locate the section.

In the column of the table, under , clear the checkboxes for , , and .

Step 2: Stationary 2

In the toolbar, click

In the window for , locate the section.

In the column of the table, under , clear the checkbox for .

Click to expand the section. Select the checkbox.

From the list, choose .

Click

In the table, enter the following settings:


Parameter name	Parameter value list	Parameter unit
j_avg_wafer (Average current density on wafer)	0.5[A/dm^2] 2[A/dm^2]	A/m^2
t_seed (Seed layer thickness)	75[nm] 150[nm]	m