Electroplating Simulation
Theory and Physics
What is Electroplating Simulation?
What does electroplating simulation predict?
In short, it's a technology to predict the uniformity of plating film thickness. You want to apply plating of the same thickness over the entire surface of the workpiece (the object to be plated), but in reality, some areas become thick and others thin. Electroplating simulation is about determining this in advance through calculation.
Why does the thickness vary by location?
Because current wants to take the "electrically shortest path." For example, when chrome plating an automobile bumper, current concentrates on convex areas near the anode, making the plating thick, while in concave areas and the depths of holes, the current density becomes low and the film becomes thin. This "throwing power (uniform deposition property)" problem is the biggest motivation behind the development of electroplating simulation.
In what specific situations is it used?
Typical applications include——
- PCB via hole inner plating: Uniformly applying Cu to the depths of holes with diameters of 0.1–0.3 mm. The higher the aspect ratio of the through-hole, the more difficult it becomes.
- Semiconductor damascene wiring: Complete filling (superfilling) of trenches with widths below 10 nm with Cu.
- Automotive decorative plating: Uniform chrome plating on complex 3D shapes like bumpers and grilles.
- Hard chrome plating for aircraft parts: Film thickness management for parts requiring wear resistance.
- Plating jig/rack design: Optimizing the placement of auxiliary cathodes and shielding plates to improve uniformity.
It's used in quite familiar products! So, how is it calculated theoretically?
Electroplating simulation has three levels of models depending on accuracy: primary current distribution, secondary current distribution, and tertiary current distribution. As the level increases, physics is incorporated more precisely, but at the cost of increased computational expense. Let's look at them in order.
Primary Current Distribution (Laplace Equation)
First, what is primary current distribution?
Primary current distribution is a model that completely ignores the resistance of electrode reactions and determines current solely from the potential distribution in the electrolyte. Assuming the electrolyte conductivity $\kappa$ is uniform and there is no influence from chemical reactions, the potential $\phi$ obeys the Laplace equation:
The current density is obtained from the gradient of the potential. This is the differential form of Ohm's law:
Huh, that's it? That's pretty simple.
Yes, the equation itself is simple. But the primary distribution gives the most non-uniform distribution, the "worst-case scenario." Because the electrode surface is treated as an equipotential surface, current tries to take the geometrically shortest path. Current concentrates extremely on the edges of convex areas. In actual plating, the resistance of the electrode reaction works to make it more uniform than this. Therefore, the primary distribution can be used as a "lower limit for uniformity."
Boundary Conditions for Primary Current Distribution
- Cathode surface: $\phi = \phi_{\text{cathode}}$ (equipotential)
- Anode surface: $\phi = \phi_{\text{anode}}$ (equipotential)
- Insulating wall surface: $\frac{\partial \phi}{\partial n} = 0$ (normal component of current is zero)
The essential assumption of the primary distribution is ignoring the overpotential $\eta$ at the electrode/electrolyte interface.
Secondary Current Distribution (Butler-Volmer Equation)
What changes in the secondary distribution?
The secondary current distribution considers charge transfer reactions at the electrode surface (electrode reaction kinetics). In other words, the reaction where metal ions receive electrons to become metal atoms has a certain "resistance." This relationship is described by the Butler-Volmer equation:
Lots of symbols appeared... Could you explain them one by one?
| Symbol | Meaning | Typical Value / Note |
|---|---|---|
| $i$ | Current density at electrode surface | [A/m²] |
| $i_0$ | Exchange current density | Depends on reaction system (Cu: ~1–10 A/m², Ni: ~0.1–1 A/m²) |
| $\alpha_a, \alpha_c$ | Anodic and cathodic transfer coefficients | Usually $\alpha_a + \alpha_c = 1$ (for single-electron reactions) |
| $F$ | Faraday constant | 96,485 C/mol |
| $\eta$ | Overpotential (activation overpotential) | $\eta = \phi_m - \phi_s - E^{\text{eq}}$ (metal potential - solution potential - equilibrium potential) |
| $R$ | Gas constant | 8.314 J/(mol·K) |
| $T$ | Absolute temperature | [K] |
The overpotential $\eta$ is the key parameter, right? The larger it is, the faster the reaction?
Exactly. The overpotential $\eta$ is the driving force from equilibrium, and the larger $|\eta|$ is, the more current increases. But the important point is that this equation functions as a nonlinear boundary condition. The Laplace equation inside the electrolyte is the same as in the primary distribution, but instead of setting $\phi$ as equipotential at the electrode surface, the Butler-Volmer equation connects $i$ and $\eta$ (i.e., $\phi$). Thanks to this nonlinearity, the current trying to concentrate on convex areas "hits the wall of reaction resistance," resulting in a more uniform distribution than the primary distribution.
I see! So plating baths with slower reactions (smaller $i_0$) actually deposit more uniformly?
Sharp observation. That's correct. The smaller $i_0$ is, the more the electrode reaction becomes rate-limiting, and the current distribution tends to become more uniform. The excellent uniformity of cyanide-based copper plating is largely due to this reaction kinetics effect.
Tafel Approximation (High Overpotential Region)
When $|\eta| \gg RT/F$ (approx. 25 mV at 25°C), the cathodic side of the Butler-Volmer equation can be simplified to a single exponential function:
$$ i \approx -i_0 \exp\left(-\frac{\alpha_c F \eta}{RT}\right) $$Taking the logarithm yields a linear relationship (Tafel plot), a standard method to determine $i_0$ and $\alpha_c$ from experimental data:
$$ \eta = a + b \log|i| $$Here $a = -\frac{RT}{\alpha_c F}\ln i_0$, $b = -\frac{2.303RT}{\alpha_c F}$ (Tafel slope).
Tertiary Current Distribution (Nernst-Planck Equation)
What else is considered in the tertiary current distribution?
The tertiary current distribution considers mass transport. As plating progresses, the metal ion concentration near the electrode decreases (concentration polarization), and supply cannot keep up. The movement of ions is governed by the Nernst-Planck equation:
The three terms on the right side are:
- $-D_i \nabla c_i$: Diffusion (Fick's law due to concentration gradient)
- $-\frac{z_i F D_i c_i}{RT} \nabla \phi$: Migration (movement of charged ions due to electric field)
- $c_i \mathbf{v}$: Convection (transport due to fluid flow)
Here $D_i$ is the diffusion coefficient, $c_i$ is the concentration of species $i$, $z_i$ is the charge number, and $\mathbf{v}$ is the flow velocity vector.
Since convection is included, does that mean fluid calculation is also necessary?
Yes. To do tertiary distribution rigorously, you need to solve the flow field with the Navier-Stokes equations and input it into the convection term of the Nernst-Planck equation. That becomes quite a heavy calculation. However, in practice, a simplified method is often used that approximates the diffusion boundary layer near the electrode with the Nernst diffusion layer model, giving only the layer thickness $\delta_N$ (typical value 10–500 $\mu$m) as a parameter.
For example, in PCB via holes, the bottom of the hole has little fluid movement, so concentration tends to drop easily, right?
Exactly right. The bottom of a via hole is almost stagnant with no convection. Since ions are supplied only by diffusion, concentration polarization becomes more severe as the aspect ratio (hole depth/diameter) increases, leading to thinner film thickness. Accurately predicting this effect is where the tertiary current distribution comes into play.
Limiting Current Density
When the ion concentration at the electrode surface becomes zero, no more current can flow. This upper limit is the limiting current density $i_L$:
$$ i_L = \frac{nFD c_{\infty}}{\delta_N} $$$c_{\infty}$ is the bulk concentration, $\delta_N$ is the Nernst diffusion layer thickness. If $i > i_L$, deposition quality deteriorates rapidly, causing burning and dendritic deposition.
Electroneutrality Condition
In the bulk region of the electrolyte, local charge is assumed neutral:
$$ \sum_i z_i c_i = 0 $$The complete formulation of the tertiary current distribution involves solving this condition together with various Nernst-Planck equations and the Butler-Volmer equation at the electrode surface.
Wagner Number and Uniform Deposition Property
Is there a criterion for choosing between primary, secondary, and tertiary? Is it safe to just do tertiary for everything?
The computational cost is completely different, so the judgment to select a model with sufficient and necessary accuracy is important. The Wagner number Wa can be used for that judgment:
$\kappa$ is the electrolyte conductivity, $L$ is the characteristic dimension of the workpiece, $\partial\eta/\partial i$ is the slope of the polarization curve (differential resistance of the electrode reaction). Physically, it's the ratio of "electrode reaction resistance / electrolyte resistance".
- Wa ≪ 1: Electrolyte resistance dominates → close to primary distribution (non-uniform)
- Wa ≫ 1: Electrode reaction resistance dominates → uniform current distribution
- Wa ≈ 1: Both have comparable influence → secondary distribution calculation is essential
So by looking at the Wa number, you can judge whether primary is enough or secondary is needed. What are typical values in the field?
| Plating Bath | Typical Wa | Uniformity Tendency |
|---|---|---|
| Copper sulfate plating (acidic) | 0.1–1 | Low–Medium |
| Cyanide copper plating | 5–50 | High (excellent uniformity) |
| Watts nickel plating | 0.5–5 | Medium |
| Hard chrome plating | 0.01–0.1 | Very low |
| Tin plating | 1–10 | Medium–High |
Hard chrome plating has a Wa of 0.01... That seems like it would be very non-uniform.
Exactly. That's why in chrome plating, auxiliary cathodes (robbers) and shielding plates are used to control the current distribution. Optimizing the placement of these jigs is precisely where simulation excels.
Film Growth Model
Once the current distribution is known, how is the film thickness calculated?
Use Faraday's law of electrolysis to determine the film growth rate from the local current density:
- $\delta$: Film thickness [m]
- $M$: Molar mass of deposited metal [kg/mol] (Cu: 0.0636, Ni: 0.0587, Cr: 0.0520)
- $i_n$: Normal current density at electrode surface [A/m²]
- $n$: Number of electrons transferred (Cu²⁺→Cu: $n=2$, Cr⁶⁺→Cr: $n=6$)
- $\rho$: Density of deposited metal [kg/m³]
- $\eta_{\text{CE}}$: Current efficiency (subtracting side reactions. Chrome plating is extremely low, ~10–25%)
For example, in copper sulfate plating with $i = 300$ A/m² and 100% current efficiency, the growth rate is about 3.7 $\mu$m/min.
Electroplating Simulation Inside Your Smartphone
Your smartphone's processor has billions of copper wiring lines with widths below 10 nm crisscrossing it. All of these are formed by electroplating (damascene process). Particularly challenging is "superfilling," growing copper from the bottom of high-aspect-ratio (narrow and deep) trenches. Inhibitors like PEG (polyethylene glycol), accelerators like SPS (bis(3-sulfopropyl) disulfide), levelers like JGB (Janus Green B)——the competition of adsorption, diffusion, and consumption of these three types of organic additives is calculated using mass transport models to explore process conditions that achieve bottom-up filling. The computational load of plating simulation required to manufacture one semiconductor chip is said to be thousands of times that for one automobile bumper.
Numerical Methods and Implementation
Choosing Between FEM and BEM
What numerical methods are used in electroplating simulation? Finite element method?
There are mainly two methods.
- Finite Element Method (FEM): Divides the entire electrolyte region into elements. Strong for secondary/tertiary current distributions and nonlinear materials (like concentration-dependent conductivity). Used by COMSOL, Ansys.
- Boundary Element Method (BEM): Meshes only the surfaces of electrodes and insulating walls. For primary distribution (Laplace equation), no internal volume mesh is needed, making mesh generation easier. Advantageous for 3D complex shape plating jig design. ElSyCA PlatingMaster is a representative example.
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