Resistive Substrates

Up to this point we have assumed that the material being plated is highly conductive and essentially at a single potential; in other words, we have assumed that the conductivity of the material is sufficiently high that the potential gradient accompanying current flow in the material is very small and can be neglected. There are important practical situations where this assumption is not true. Resistive materials and even insulators can be electroplated. In the case of insulators, a thin conductive layer is first deposited by techniques such as chemical vapor deposition, physical vapor deposition, or electroless deposition. Electrodeposition, which is driven by an applied potential or current, is then used to provide a layer with the desired thickness and properties. A couple of important examples that involve resistive substrates are copper electrodeposition to form interconnects on semiconductor chips and chrome plating on plastics. The key challenge with resistive substrates is that the current distribution is influenced by the potential drop in the solid substrate as well as by the potential drop in the electrolyte solution.

Figure 13.15 illustrates a simple electrochemical cell with a resistive substrate. It looks similar to the electrochemical cell shown in Figure 13.1, except that the electrical resistance of the cathode is now important. The scale of the thin conductive layer in the figure is not necessarily accurate in proportion to the rest of the dimensions. For example, a semiconductor wafer 300 mm in diameter may have an initial layer on the order of 10 nm thick, upon which electrodeposition would be performed. While intentionally simple, Figure 13.15 also illustrates an important aspect of behavior with resistive substrates, namely, that the general flow of current in the substrate is orthogonal to the flow of current in solution. Hence, the problem is inherently multidimensional and does not, in general, lend itself to simple numerical analysis, even when concentration effects are unimportant.

Figure 13.15 Schematic diagram of deposition cell for deposition on a resistive substrate where the potential drop in the substrate is important.

However, if we assume that the kinetic resistance is large relative to the ohmic resistance in the solution (Wa > 10), then the potential in the electrolyte solution at the surface of the working electrode is essentially constant along the length of the electrode, and the current distribution will be controlled by the potential drop in the resistive substrate. The situation is illustrated in Figure 13.16. This assumption is essentially the opposite of that which we have made in previous chapters of this text. Initially, we assumed that the metal substrate was much more conductive than the solution, so the potential drop in the metal was not significant and the metal was at a constant potential. Now we assume that the solution is much more conductive than the resistive substrate, so the solution is at a constant potential and the behavior is controlled by the potential drop in the resistive substrate. This assumption will allow us to develop a dimensionless number analogous to the Wagner number to describe the ratio of the kinetic resistance to the substrate resistance in order to determine whether the substrate resistance is important.

Figure 13.16 Schematic diagram of the physical situation described by the model below. ϕmetal decreases from x = L to x = 0 (not shown). Current flow shown qualitatively by filled arrows.

Consistent with the physical situation shown in Figure 13.16, we model one-dimensional current flow in a thin layer that is connected to the current collector at x = 0, and continues for a length L. We assume a rectangular electrode of width W with an area L × W exposed to the electrolyte. We begin by writing a charge balance for current flow in the resistive substrate (shown in gray in the diagram)

(13.33)
where i1 is the current density in the thin metal layer, a is the area of the metal electrode exposed to the electrolyte per volume of metal, and in is the current flow between the solution and metal. Note that for deposition, which is a cathodic process, in is negative and the term on the right-hand side of Equation 13.33 is positive since current flows from the solution to the metal. The boundary conditions are as follows:

where I is a positive number and represents the current in amperes, and is cross-sectional area of the conductive layer through which that current flows. The specified boundary condition is negative at x = 0 since the current flow is out of the electrode (negative x direction) for a cathodic reaction. The current flow through the metal layer is related to the potential drop:

(13.34)
where is the potential in the metal and is the solution overpotential. The two derivatives are equal since the solution potential is constant. Assuming Tafel kinetics, Equation 13.33 becomes

(13.35)
Differentiating this equation with respect to x yields

(13.36)
This second-order differential equation can be solved analytically for as a function of x. Before doing so, it is instructive to make the equation dimensionless. To do so, we define the following:

The resulting dimensionless equation is

(13.37)
The dimensionless group represents the ratio of the substrate resistance to the kinetic resistance. It is expressed in terms of I, the current density in the metal at the current collector. At small values of this ratio, equation approaches

(13.38)
This implies a uniform current distribution (you should understand and be able to explain why!).

It is common to address these types of deposition problems in terms of the average current density in the cell, which is based on the area WL. The average current density is related to the current I by

(13.39)
Using this relationship, Equation 13.37 becomes

(13.40)
where

(13.41)
The dimensionless number is analogous to Wa and represents the ratio of the charge-transfer resistance to the electrical resistance of the substrate. Large values of are desired for a uniform current density. It is interesting to note the role of L, which is squared in this ratio. This reflects its dual role as both the distance that the current must travel in the substrate and the magnitude of the overall current for a given value of iavg scale with L. It is necessary for both Wa ≫ 1 (unless the primary current density is uniform) and for the current density to be uniform when electrodepositing on a resistive substrate. Since the ohmic resistance in solution and in the substrate are essentially in series, one must also consider the ratio of the two resistances in situations where Wa and indicate that both of these resistances are important relative to the charge transfer resistance, as it is possible that one or the other of the resistances dominates. The relevant ratio is

(13.42a)
Subscripts have been added to distinguish the characteristic lengths for the solution () and the substrate (). In cases where the characteristic lengths are similar, this ratio becomes

(13.42b)
To summarize the use of these dimensionless numbers:

Wa ≫ 1 and : current density is uniform, charge-transfer resistance dominates.
Wa ≤ 1 and : resistive substrate is not important, current distribution determined by Wa.
Wa ≫ 1 and : solution resistance is not important, current distribution determined by .
Wa ≤ 1 and : both substrate and solution resistances potentially important; need Wa, , and ratio (Equation 13.42).
When we introduced the current distribution in Chapter 4, we assumed that , which is common for highly conductive substrates. Hence, we considered only the first two possibilities in the above list. Items 3 and 4 allow us to consider the impact of the substrate. We now return to the solution of Equation 13.40. The analytical solution is

(13.43)
where and are integration constants. Application of the boundary conditions yields two equations that can be solved for the two constants:

(13.44a)
(13.44b)
We can take advantage of the fact that , which comes from the boundary conditions, to simplify Equation 13.43:

(13.45)
Thus, only the second relationship of 13.44 is needed. Note that is bound between 0 and . Equation 13.45 can be differentiated to yield the local current density across the electrode–solution interface (see Equation 13.33):

(13.46)
Remember that implicit to this solution is the condition that Wa ≫ 1, which was used as an assumption in the derivation. Solution of a problem with these equations will typically involve the use of either Equation 13.45 or 13.46 with Equation 13.44b, and may require iterative solution of 13.44b to find . Figure 13.17 shows the local current distribution as a function of dimensionless distance for several different values of .

ILLUSTRATION 13.7
How thick of a copper layer must be deposited on a rectangular electrode 15 cm long and 5 cm wide connected to the current collector on one side in order that the current density across the electrode does not vary by more than 15%? Assume that the conductivity of the copper is equal to its bulk value, and . Also, assume that the initial copper layer on the surface is of uniform thickness. The average current density is 100 mA·cm−2.

SOLUTION:
Application of Equation 13.46 at the two ends of the resistive substrate (x = 0 and x = L) must yield current densities that differ by no more than 15%. Therefore,

Solving this expression for yields . We can now use this value of in Equation 13.44b to find χ.

From the definition of χ,

Figure 13.17 Influence of the dimensionless ratio χ on the uniformity of the current distribution for a resistive substrate.

Closure
In this chapter, we examined several aspects of electrodeposition that range from the fundamental processes that contribute to the formation of new phases to macroscopic issues such as the role of side reactions and of the current distribution. Importantly, the connection between the fundamentals and the characteristics of electrodeposited materials was discussed. Finally, the influence of the substrate resistance was investigated and quantified in terms of a dimensionless number similar to the Wagner number.

Further Reading
Bockris, J.O’M. and Reddy, A.K.N. (1970) Modern Electrochemistry, vol. 2, Plenum Press, New York.
Budevski, E.B., Staikov, G.T., and Lorenz, W.F. (1996) Electrochemical Phase Formation and Growth, Wiley-VCH Verlag GmbH, Weinheim, Germany.
Paunovic, M. and Schlesinger, M. (2006) Fundamentals of Electrochemical Deposition, 2nd edition, John Wiley & Sons, Inc., Hoboken, NJ.
Pletcher, D. and Walsh, F.C. (1993) Industrial Electrochemistry, 2nd edition, Springer Science+Business Media, LLC.
Schlesinger, M. and Paunovic, M. (2010) Modern Electroplating, 5th edition, John Wiley & Sons, Inc., Hoboken, NJ.
West, A.C. (2012) Electrochemistry and Electrochemical Engineering: An Introduction, Independent Publishing Platform.
Problems
13.1. Corrosion protection is frequently provided by coating a steel structure with a thin coating of zinc. Please determine the time required to electroplate a 25 μm layer. The density of zinc is 7.14 g·cm−3. The current density is 250 A·m−2, and the faradaic efficiency is 80%.

13.2. Nickel is plated from a sulfate bath at 95% faradaic efficiency onto a surface with a total area of 0.6 m2. The density of nickel is 8910 kg·m−3, and plating is performed at a current of 300 A. What is the thickness of the plated nickel after 30 minutes? What is the average rate of deposition?

13.3. Electrorefining of copper involves the plating of copper onto large electrodes in tank-type cells. In one design, electrodes are plated with 5 mm of copper before they are mechanically stripped. If the electrode area is 2 m2 and plating takes place at 200 A·m−2 at a faradaic efficiency of 96%, how long does it take to deposit a layer of the desired thickness? What is the mass of the plated copper? What is the total current for a production cell that contains 50 cathodes? The density of copper is 8960 kg·m−3.

13.4. Chromium is plated from a hexavalent bath at a faradaic efficiency of about 20%. A metal piece is to be plated to a thickness of 0.5 μm. The current density is 500 A·m−2.

How long will it take to deposit the desired amount of Cr?
How does the low faradaic efficiency impact the time and energy required to plate Cr?
There is a movement away from the use of hexavalent Cr. Why?
13.5. Please determine the B value for a 3D cubic cluster of atoms.

13.6. Show that the perimeter for a 2D cluster. What is in this expression? Why do we need P as a function of N? How does the variation of P with N affect the size of the critical size of the nucleus? The critical number of atoms in a nucleus changes with overpotential. Does that mean that the relationship between P and N changes? Please explain.

13.7. Please determine the critical cluster size for the system shown in Figure 13.6 at an overpotential of −40 mV. Also, plot the Gibbs energy of cluster formation as a function of cluster size for this value of the overpotential, similar to Figure 13.6. Why does the critical cluster size change with overpotential? How does this impact nucleation?

13.8. The following data were taken for the number of nuclei as a function of time at the overpotentials indicated for the same system considered in Illustration 13.4, but over a different range of overpotentials. Please use these data to determine the nucleation rate at each of the two overpotentials given. Then, fit the nucleation rate data from these two points together with the data from the illustration to the expression for the 3D nucleation rate. Finally, estimate the critical value of the overpotential. How do your results compare to those from the illustration?

94 mV 98 mV
t [ms] Znuc [cm−2] t [ms] Znuc [cm−2]
0.112 6.0 0.057 4.6
0.145 10.5 0.072 13.6
0.189 20.5 0.089 24.6
0.231 33.2 0.110 38.4
0.289 49.7 0.129 54.9
0.346 65.6 0.156 77.6
13.9. Please derive the expression for r as a function of t for a growing hemispherical nucleus beginning with a mass balance similar to that given in Equation 13.21. Compare the resulting relationship to that given for a 2D cylindrical nucleus in Equation 13.23. Comment on similarities and differences between the two equations. What does the equation represent physically and what was a key assumption in its derivation?

13.10. Figure 13.10 shows current as a function of time for 2D layer growth under the assumptions of instantaneous and progressive nucleation.

In both cases, the current drops to zero at “long” times. Why?
Please sketch analogous curves for 3D nucleation and growth. How are they different? Why?
13.11. In this problem, we examine instantaneous nucleation both with and without overlap.

Calculate the current as a function of time for 2D instantaneous nucleation without overlap and plot the results (t ∼ 10 seconds). Assume a nucleation density of . Assuming equally spaced nuclei, estimate the time at which you would expect overlap to occur.
Calculate the current as a function of time assuming overlap. Plot the results on the same figure as the data from part (a). Assume the same nucleation density. Based on the results, comment on the accuracy of the estimate made in part (a).
The following parameters are known (Ag): n = 1; MAg = 107.87 g·mol−1; ρ = 10.49 g·cm−3; isurf = 0.005 A·cm−2; h = 0.288 nm.

13.12. In the section on deposit morphology, we discussed the morphological development of deposits in terms of the fundamental processes that occur as discussed in previous sections.

Please reread that section and comment briefly on the role of the overpotential in at least one aspect of deposit growth.
Assume that an additive is added to the solution that preferentially adsorbs and inhibits growth at step sites and kink sites. How might this affect deposit growth?
13.13. Suppose that you are the engineer put in charge of implementing an electroplating plating process for your company. After considerable effort, you get the process running only to discover that the uniformity of the plating layer is not acceptable. Using Wa as a guide, and assuming that the solid phase is very conductive, describe three or four changes that you might try to improve plating uniformity. Please justify each recommended change.

13.14. Suppose that you are conducting tests on a plating bath with use of a Hull cell.

Assume that the bath does not plate uniformly. Where and under what conditions would you expect to see the highest deposition rate?
In measuring the local deposition rate, you find that the deposition rate on the surface closest to the anode is less than observed in the middle of the cathode. Please provide a possible explanation.
13.15. A Haring–Blum cell is used to measure the throwing power of a plating bath. The distance ratio, x1/x2 is 5, and the conductivity of the solution is 20 S·m−1. The measured deposit loading at x1 is 0.8 kg·m−2, and that at x2 is 1.4 kg·m−2. Both electrodes have the same surface area. What is the throwing power?

13.16. Iron is electrodeposited from a bath that is 0.5 M in Fe2+ at a pH of 5. The anode is also iron, and the potential applied across the cell is −3.0 V. Due to the larger size and surface area of the anode, the anodic surface overpotential can be neglected for this problem, as it was in the illustration (not true in general); concentration gradients can also be neglected. However, iR losses in solution are important. Because of this, the current density is not uniform. We are interested in the relative deposition rate at two specific points on the surface, and the impact of the side reaction (hydrogen evolution) on that relative rate, as well as on the current efficiency. The solution resistance from the anode to the cathode is 0.005 Ω·m2 at the first point of interest, and 0.006 Ω·m2 at the second. Please determine the value of the current density at each of the two points, as well as the relative rate of deposition. Next, include the hydrogen reaction and repeat the calculation. In this example, how did H2 evolution impact the absolute and relative rates of deposition? The following parameters are known:

Iron reaction: Tafel slope ≈ −0.1 V; i0 = 0.8 A·m−2.
Hydrogen evolution reaction: Tafel slope: −0.11 V; i0 = 10−2.3 A·m−2.
13.17. Electrodeposition is performed on a thin chrome layer that has been deposited on an insulating substrate. The layer is 0.5 μm thick and of uniform thickness. About how far from the point where electrical connection is made can electrodeposition be performed (distance L) if the current density across the surface must not vary by more than 20%? The solution phase is not limiting. Assume that the conductivity of the chrome is equal to its bulk value and . The average current density is 200 A·m−2.

13.18. Assuming Wa ≫ 1, please use the appropriate equations from the chapter to generate Figure 13.17. Plot for values of 10, 1, and 0.05 (in other words, replace 0.1 with 0.05). Hint: You will need the value that corresponds to each value of .

13.19. Copper is to be electrodeposited onto a thin copper seed layer, 20 nm thick, which has been deposited onto a silicon oxide insulator. If the solution-phase conductivity is high, how much would you expect the deposition current to vary over a 150 mm distance if the average current is 10 A·m−2? Assume that the conductivity of the copper is the same as its bulk conductivity. Thin layers often have conductivities that are significantly lower than the bulk value. Comment on the impact of the conductivity on the results reported above.

13.20. Electrodeposition is being performed onto a thin metal layer (σ = 6 × 106 S·m−1) that has been deposited onto an insulating substrate. The length scale associated with solid phase (distance from connection point) is 0.5 m, and the thickness of the solid layer is 100 nm. The conductivity of the solution is 20 S·m−1, and the length scale associated with the solution transport is 0.1 m. The average current density is 500 A·m−2. Would you expect the current distribution to be uniform? If not, which phase (solution, solid, or both) would determine the nonuniform current distribution? In which direction would the thickness of the metal layer need to change in order to change the limiting process?

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