Copper Deposition in a Trench

This model simulates the electroplating of copper in a microcavity typically found in the plating of copper onto circuit boards. The plating cell is a laboratory cell under potentiostatic control where the anode is placed in close vicinity of the cathode. The example is based on a scientific paper (Ref. 1).

The purpose of the model is to demonstrate the use of deforming geometries for plating processes and to investigate the influence of the cavity on the plating result. The deforming geometry makes it possible to simulate the growth of the cathode boundary as the process proceeds.

For a detailed description of how to build this model, including screenshots, see the Introduction to the Electrodeposition Module book.

Model Definition

The model simulates the deposition process at pH 4, which implies that the proton concentration is very low compared to the copper and sulfate ion concentrations. For this reason, the material balance of the protons does not need to be modeled. Sulfate is also treated as a fully dissociated ion. The deposition at the cathode and the dissolution at the anode are assumed to take place with 100% current yield, which means that the model does not include possible side reactions. During the process, differences in electrolyte density arise in the enclosed cell, giving higher density at the anode compared to the cathode. This can induce free convection in the cell. Under the modeled conditions, however, the variations in composition are small and it is therefore possible to neglect free convection.

The process is inherently time-dependent because the cathode boundary moves as the deposition process takes place. The model is defined by the material balances for the involved ions (copper, Cu2+, and sulfate, SO42-) and the electroneutrality condition. This gives three unknowns and three model equations. The dependent variables are the copper ion concentration, sulfate ion concentration, and ionic potentials. Additional variables keep track of the deformation of the mesh.

The model geometry is shown in Figure 1. The upper horizontal boundary represents the anode, while the cathode is placed at the bottom. The vertical walls correspond to the pattern on the master electrode and are assumed to be insulating.

Figure 1: Model domain with boundaries corresponding to the anode, cathode, and vertical symmetry walls.

The flux for each of the ions in the electrolyte is given by the Nernst-Planck equation

where Ni denotes the transport vector (mol/(m2·s)), ci the concentration in the electrolyte (mol/m3), zi the charge for the ionic species, ui the mobility of the charged species (m2/(s·J·mole)), F Faraday’s constant (As/mole), and

the potential in the electrolyte (V). The material balances are expressed through

one for each species, that is i = 1, 2. The electroneutrality condition is given by the following expression:

The boundary conditions for the anode and cathode are given by the Butler-Volmer equation for copper deposition. The deposition process is assumed to take place through the following simplified mechanism:

where the first step is rate determining step, RDS, and the second step is assumed to be at equilibrium (Ref. 1). This gives the following relation for the local current density as a function of potential and copper concentration:

where η denotes the overpotential defined as

where

denotes the electronic potential of the respective electrode. This gives the following condition for the cathode:

where n denotes the normal vector to the boundary. The condition at the anode is

All other boundaries are insulating:

For the sulfate ions, insulating conditions apply everywhere:

The initial conditions set the composition of the electrolyte according to

You set up Equation 1 through Equation 11 using the Tertiary Current Distribution, Nernst-Planck Equations interface. The Deformed Geometry node keeps track of the deformation of the mesh.

Using the Electrode Surface boundary node, with an added Dissolving-Depositing species, the ion fluxes and the boundary mesh velocity are based on the reaction currents, the number of electrons, and the specified stoichiometric coefficients of the electrode reactions. The sign of the stoichiometric coefficient for a species depends on whether the species is getting oxidized (positive) or reduced (negative) in the reaction. In the case for the total reaction in this model

the stoichiometric coefficient is

for the copper ions in the electrolyte, and

for the copper atoms in the electrodes.

Results and Discussion

Figure 2 shows the concentration distribution of copper ions, the isopotential lines, the current density lines, and the displacement of the cathode and anode surfaces after 14 seconds of operation. The figure clearly shows that the mouth of the cavity is starting to narrow due to the nonuniform thickness of the deposition. This effect can be detrimental to the quality of the deposition because a trapped electrolyte can later cause corrosion of components in the circuit board. In addition, the simulation shows substantial variations in copper ion concentration in the cell. Such variations eventually cause free convection in the cell. The model is symmetric along a vertical line in the middle of the cell. An asymmetric result would be a sign of poor mesh resolution.

Figure 2: Copper ion concentration (mol/m3), isopotential lines, current density streamlines, and electrode displacement in the cell after 14 seconds of operation.

Figure 3 shows the thickness of the deposition along one of the vertical cathodic surfaces. The lines reveal the development of the nonuniform deposition due to nonuniform current density distribution. This effect is accentuated by the depletion of copper ions along the depth of the cavity.