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Electrodeposition of a Microconnector Bump in 2D
Introduction
This model demonstrates the impact of convection and diffusion on the transport-limited electrodeposition of a copper microconnector bump (metal post). Microconnector bumps are used in various types of electronic applications for interconnecting components, for instance liquid crystal displays (LCDs) and driver chips.
The location of the bumps on the electrode surface is controlled by the use of a photoresist mask. Control of the current distribution in terms of uniformity and shape is important for ensuring the shape and resulting reliability of the interconnector bumps.
The cell is running at a high overpotential so the deposition rate is governed by the transport rate of the depositing ion in the electrolyte. A result of this operating condition is that the electric potentials in the electrolyte and electrode need not be modeled to determine the current distribution on the bump.
The model is based on a paper by Kondo and others (Ref. 1).
For an extension of this model to 3D, including the microconnector shape evolution over time, see Electrodeposition of a Microconnector Bump with Deforming Geometry in 3D.
Model Definition
The model geometry is shown in Figure 1. The rectangular hole in the photoresist film is significantly longer in one direction so the model can be represented in 2D.
Figure 1: Model geometry.
Flow Model
Assume laminar incompressible flow conditions (described by the Navier-Stokes equations), and use the Laminar Flow interface to model the electrolyte flow.
Apply a linear velocity profile (Couette flow) on the left vertical inlet boundary, ranging from zero at the photoresist surface up to a specified bulk flow velocity, depending on the Peclet number (see Equation 1). Set the velocity at the top boundary equal to the bulk velocity using a moving wall condition. Apply a pressure conditions on the right vertical outlet boundary.
The default no slip condition applies to all other boundaries.
Mass Transport Model
Assume the electrolyte to be diluted so that the transport of copper ions can be described using Fick’s law. Use the Transport of Diluted Species interface coupled to the flow velocity in the Laminar Flow interface to model the mass transport.
Apply a fixed bulk concentration on the left inlet and top boundaries. Use outflow conditions on the right outlet boundary.
The cell is operating at high overpotential and is limited by transport of copper ions to the electrode surface. This is described by applying a zero concentration of ions at the electrode surface.
The default no-flux condition applies to all other boundaries.
The Peclet Number
The current density distribution on the electrode surface for this cell is governed by the Peclet number, which is a dimensionless quantity that relates the convective transport to the diffusion. For this cell, use the following definition of the Peclet number:
(1)
where h is the thickness of the photoresist film, uPe is the velocity in the x direction at a height of 10 μm at the inlet boundary, and D is the diffusion coefficient of the copper ions.
Study Settings
Use a Parametric Sweep of a Stationary Study to solve the problem for two different Peclet numbers (1.31 and 41.6).
Results and Discussion
Figure 2 and Figure 3 show the streamlines and velocity magnitudes for the two Peclet numbers. The shape of the streamlines are quite similar, although the velocity magnitudes differ. Vortices that are formed close the corners both on the upstream and downstream sides.
Figure 2: Velocity streamlines for Pe = 1.31
.
Figure 3: Velocity stream lines for Pe = 41.6.
Figure 4 and Figure 5 show the concentrations for the two different Peclet number cases. The concentration profiles are quite different, with the depletion zone of copper ions extending farther into the electrolyte for the lower Peclet number.
Figure 4: Copper ion concentration for Pe = 1.31.
Figure 5: Copper ion concentration for Pe = 41.6.
The fluxes of the copper ions over the electrode surface in combination with Faraday’s law can be used to calculate the local current density on the electrode surface:
where n is the number of electrons in the electrode reactions, F is Faraday’s constant, and NCu- is the normal flux of copper ions over the electrode surface.
Figure 6 shows the local current density for the two Peclet numbers. The higher flow velocity for the higher Peclet number increases the local current density significantly due to the increased transport velocity of copper ions.
Figure 6: Local current densities at the electrode surface
In Figure 7 the local current densities have been normalized to their maximum values. The lower Peclet number results in a more uniform current density distribution on the electrode surface.
Figure 7: Local current density at the electrode surface normalized by the maximum value.
Reference
1. K. Kondo, K. Fukui, K. Uno, and K. Shonohara, “Shape Evolution of Electrodeposited Copper Bumps,” J. Electrochemical Society, vol. 143, pp 1880–1886, 1996.
Application Library path: Electrodeposition_Module/Verification_Examples/microconnector_bump_2d
Modeling Instructions
From the File menu, choose New.
New
In the New window, click  Model Wizard.
Model Wizard
1
In the Model Wizard window, click  2D.
2
In the Select Physics tree, select Chemical Species Transport>Transport of Diluted Species (tds).
3
Click Add.
4
In the Select Physics tree, select Fluid Flow>Single-Phase Flow>Laminar Flow (spf).
5
Click Add.
6
Click  Study.
7
In the Select Study tree, select General Studies>Stationary.
8
Click  Done.
Global Definitions
Load the model parameters from a text file.
Parameters 1
1
In the Model Builder window, under Global Definitions click Parameters 1.
2
In the Settings window for Parameters, locate the Parameters section.
3
Click  Load from File.
4
Browse to the model’s Application Libraries folder and double-click the file microconnector_bump_parameters.txt.
Geometry 1
Create the geometry by drawing two rectangles. The union of the two defines the final geometry.
Rectangle 1 (r1)
1
In the Geometry toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type L1+L2+L3.
4
In the Height text field, type h2.
5
Locate the Position section. In the x text field, type -L3.
6
In the y text field, type h1.
Rectangle 2 (r2)
1
In the Geometry toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type L1.
4
In the Height text field, type h1.
Union 1 (uni1)
1
In the Geometry toolbar, click  Booleans and Partitions and choose Union.
2
Click in the Graphics window and then press Ctrl+A to select both objects.
3
In the Settings window for Union, locate the Union section.
4
Clear the Keep interior boundaries check box.
5
In the Geometry toolbar, click  Build All.
Definitions
Add some variable expressions from a text file.
Variables 1
1
In the Home toolbar, click  Variables and choose Local Variables.
2
In the Settings window for Variables, locate the Variables section.
3
Click  Load from File.
4
Browse to the model’s Application Libraries folder and double-click the file microconnector_bump_2d_variables.txt.
Note that the i_loc_norm variable (that you will use for results processing) is colored orange. It needs a maximum operator that you have yet to define.
Maximum 1 (maxop1)
1
In the Definitions toolbar, click  Nonlocal Couplings and choose Maximum.
2
In the Settings window for Maximum, locate the Source Selection section.
3
From the Geometric entity level list, choose Boundary.
4
Select Boundary 5 only.
Variables 1
If you return to the variables again, the i_loc_norm variable should now have turned black.
Laminar Flow (spf)
Now start setting up the physics, beginning with the fluid flow model.
Fluid Properties 1
1
In the Model Builder window, under Component 1 (comp1)>Laminar Flow (spf) click Fluid Properties 1.
2
In the Settings window for Fluid Properties, locate the Fluid Properties section.
3
From the ρ list, choose User defined. In the associated text field, type rho.
4
From the μ list, choose User defined. In the associated text field, type mu.
Inlet 1
1
In the Physics toolbar, click  Boundaries and choose Inlet.
2
Select Boundary 1 only.
3
In the Settings window for Inlet, locate the Velocity section.
4
Click the Velocity field button.
5
Specify the u0 vector as
u_profile
x
0
y
Wall 2
1
In the Physics toolbar, click  Boundaries and choose Wall.
2
Select Boundary 3 only.
3
In the Settings window for Wall, click to expand the Wall Movement section.
4
From the Translational velocity list, choose Manual.
5
Specify the utr vector as
u_bulk
x
0
y
Outlet 1
1
In the Physics toolbar, click  Boundaries and choose Outlet.
2
Select Boundary 8 only.
3
In the Settings window for Outlet, locate the Pressure Conditions section.
4
Select the Normal flow check box.
Initial Values 1
1
In the Model Builder window, click Initial Values 1.
2
In the Settings window for Initial Values, locate the Initial Values section.
3
Specify the u vector as
u_bulk
x
0
y
Transport of Diluted Species (tds)
Transport Properties 1
Now set up the mass transport model of the ions. Start by coupling the mass transport to the fluid velocity in the flow model, and specify the diffusion coefficient.
1
In the Model Builder window, under Component 1 (comp1)>Transport of Diluted Species (tds) click Transport Properties 1.
2
In the Settings window for Transport Properties, locate the Convection section.
3
From the u list, choose Velocity field (spf).
4
Locate the Diffusion section. In the Dc text field, type D.
Concentration 1
1
In the Physics toolbar, click  Boundaries and choose Concentration.
2
Select Boundaries 1 and 3 only.
3
In the Settings window for Concentration, locate the Concentration section.
4
Select the Species c check box.
5
In the c0,c text field, type c_bulk.
Outflow 1
1
In the Physics toolbar, click  Boundaries and choose Outflow.
2
Select Boundary 8 only.
Concentration 2
1
In the Physics toolbar, click  Boundaries and choose Concentration.
2
Select Boundary 5 only.
3
In the Settings window for Concentration, locate the Concentration section.
4
Select the Species c check box.
Initial Values 1
1
In the Model Builder window, click Initial Values 1.
2
In the Settings window for Initial Values, locate the Initial Values section.
3
In the c text field, type c_bulk.
Study 1
The model is now ready for solving. Create a parametric sweep to solve for two different Peclet numbers.
Parametric Sweep
1
In the Study toolbar, click  Parametric Sweep.
2
In the Settings window for Parametric Sweep, locate the Study Settings section.
3
Click  Add.
4
In the table, enter the following settings:
Parameter name
Parameter value list
Parameter unit
Pe (Peclet number)
1.31 41.6
 
5
In the Study toolbar, click  Compute.
Results
Concentration (tds)
1
Click the  Zoom Extents button in the Graphics toolbar.
One concentration plot is created by default (Figure 5). To reproduce the concentration plot for the lower Peclet number (Figure 4), do the following.
2
In the Settings window for 2D Plot Group, locate the Data section.
3
From the Parameter value (Pe) list, choose 1.31.
4
In the Concentration (tds) toolbar, click  Plot.
2D Plot Group 4
Create a new plot group with a streamline plot to reproduce Figure 2 and Figure 3.
In the Home toolbar, click  Add Plot Group and choose 2D Plot Group.
Streamline 1
1
Right-click 2D Plot Group 4 and choose Streamline.
2
In the Settings window for Streamline, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1 (comp1)>Laminar Flow>Velocity and pressure>u,v - Velocity field.
3
Locate the Streamline Positioning section. From the Positioning list, choose Uniform density.
4
In the Separating distance text field, type 0.01.
Color Expression 1
1
Right-click Streamline 1 and choose Color Expression.
2
In the Settings window for Color Expression, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1 (comp1)>Laminar Flow>Velocity and pressure>spf.U - Velocity magnitude - m/s.
3
In the 2D Plot Group 4 toolbar, click  Plot.
2D Plot Group 4
1
In the Model Builder window, click 2D Plot Group 4.
2
In the Settings window for 2D Plot Group, locate the Data section.
3
From the Parameter value (Pe) list, choose 1.31.
4
In the 2D Plot Group 4 toolbar, click  Plot.
1D Plot Group 5
Now plot the current density on the electrode surface using the defined i_loc variable (Figure 6).
1
In the Home toolbar, click  Add Plot Group and choose 1D Plot Group.
2
In the Settings window for 1D Plot Group, click to expand the Title section.
3
From the Title type list, choose Manual.
4
In the Title text area, type Local current density for different Peclet numbers.
Line Graph 1
1
Right-click 1D Plot Group 5 and choose Line Graph.
2
Select Boundary 5 only.
3
In the Settings window for Line Graph, click Replace Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Component 1 (comp1)>Definitions>Variables>i_loc - Local current density - A/m².
4
Locate the x-Axis Data section. From the Parameter list, choose Expression.
5
In the Expression text field, type x.
6
Click to expand the Legends section. Select the Show legends check box.
7
In the 1D Plot Group 5 toolbar, click  Plot.
1D Plot Group 6
Proceed in a similar way as above to reproduce the normalized current density plot (Figure 7).
1
In the Home toolbar, click  Add Plot Group and choose 1D Plot Group.
2
In the Settings window for 1D Plot Group, locate the Title section.
3
From the Title type list, choose Manual.
4
In the Title text area, type Normalized local current density for different Peclet numbers.
Line Graph 1
1
Right-click 1D Plot Group 6 and choose Line Graph.
2
Select Boundary 5 only.
3
In the Settings window for Line Graph, click Replace Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Component 1 (comp1)>Definitions>Variables>i_loc_norm - Normalized local current density.
4
Locate the x-Axis Data section. From the Parameter list, choose Expression.
5
In the Expression text field, type x.
6
Locate the Legends section. Select the Show legends check box.
7
In the 1D Plot Group 6 toolbar, click  Plot.