Pulsed Current and Potential Response of Acid Copper System with Additives and the Double Layer Effect

Transcription

1 Journal of The Electrochemical Society, C229-C /2002/149 5 /C229/8/$7.00 The Electrochemical Society, Inc. Pulsed Current and Potential Response of Acid Copper System with Additives and the Double Layer Effect Wen-Ching Tsai, Chi-Chao Wan,*,z and Yung-Yun Wang Department of Chemical Engineering, National Tsing-Hwa University, Hsinchu, Taiwan 300 C229 A mathematical model for analyzing the double layer dl effect on pulse plating in the presence of additives is presented. Numerical simulations are employed to predict the influence of additives with various inclusion rates and capacities on current and potential response and mass transfer. It is shown that capacitive current density will not only increase in the presence of additives due to the increase in overpotential but also will increase with increasing dl capacity. Then, the variation of surface concentrations of additives and metal ions will even out as a result of the charging and discharging of the dl. The predicted results are compared with the experimental overpotential response on a copper rotating disk electrode in a bath containing M CuSO 4 and 2.06 M H 2 SO 4 with and without 100 ppm polyethylene glycol. Good agreement between them is found The Electrochemical Society. DOI: / All rights reserved. Manuscript submitted June 26, 2001; revised manuscript received November 15, Available electronically March 20, * Electrochemical Society Active Member. z ccwan@mx.nthu.edu.tw Choosing copper as the material for on-chip interconnects and vias can alleviate reliability limitations and enable faster processing. As the width of interconnects decreases and aspect ratio increases, it is difficult to plate void-free copper in submicrometer trenches or vias because of depletion of active ions. 1,2 In plating, cupric ions are reduced to copper in the trenches or vias. Because the depletion rate of cupric ions is faster than the rate of cupric ions diffusing from the bulk solution into the bottom of trenches or vias, the concentration of cupric ions at the bottom is lower than that at the top. This concentration gradient leads to different deposition rates. So, the deposition rate at the top is faster than that at the bottom. Eventually, voids are formed inside. The concentration depletion is similar to the process of platedthrough-holes PTHs in manufacturing printed circuit board PCB. The trend in the manufacture of multilayer PCBs is toward more layers with smaller interconnection holes, which implies higher ratio of through-holes, and more difficulty encountered in achieving uniform copper deposition. 3 Pulse plating is commonly employed in such cases. Pulse plating has a periodic relaxation time during which reactant can be supplied from the bulk solution to reduce the concentration difference. The enhancement of mass transport by high frequency pulses renders more uniform deposition and void-free deposits can thus be obtained. To minimize voids size in the interconnect, West 4 suggested that the deposition time should be smaller than the diffusion time constant and the off-time should be on the order of the diffusion time constant. The diffusion time constant, d, is to be defined as d h 0 2 /D M, where h 0 is the depth of the trench and D M is the diffusion coefficient of cupric ions. h cm for a high aspect ratio trenches or vias and D M cm 2 s 1 for an acid-copper bath. 5 Thus, d 2.36 ms, implying that the pulse period is on the order of ms. In the plating industry, various additives are applied for individual purposes. For instance, copper deposited in the absence of additives will be columnar in structure and contain large grains, which is disadvantageous to void-free fill. To achieve superfilling, additives are added in the copper electroplating. The additives are generally organic compounds that act as inhibitors, grain refiners, ductilizers and may be codeposited with copper. 6,7 Extensive theoretical work was done on the mass transport and the electrode kinetics in pulse plating Since capacitance is sensitive to the adsorptive organic additives, the double layer dl effect could become significant especially in systems with very short pulses. Yin 15 studied theoretically the effect of inert blocking inhibitors on the galvanostatic pulse plating, but he neglected the dl effect. Ibl 16,17 proposed the concept of dl effect on the faradic current density for pulse plating with the absence of additives and estimated the magnitude of capacitance effect. The influence of the dl effect on mass transport and electrode kinetics of pulse plating in the presence of additives is less well known. A theoretical analysis of the dl effect on pulse plating in the presence of additives is presented in this study. Mathematical Model Faradic and capacitive current densities. Owing to the existence of a significant dl at the interface, the total current density applied, i p, is divided into two parts. One is the capacitive current density. i C, which charges the dl, and the other is the faradic current density, i F, which contributes to the charge transfer reaction. 12,16 Their relation is as follows i p i C i F The capacitive current density is related to the charge of potential by Eq. 2 i C C dl d s dt where s is the surface overpotential and C dl is the capacity of double layer which is independent of s. The electrode kinetics in the presence of additives. Jordan and Tobias 18 proposed estimating the surface coverage of additive, A, by the following relation A 1 i F i F, A 0 where i F and i F, A 0 are the faradic current density for metal deposition with and without additives, respectively. They also mentioned that the surface fraction of unblocked area (1 A ) is equal to the ratio of the faradic current density for metal deposition, i F, to the faradic current density without additives, i F, A 0. For an electrochemical reaction consisting of a metal electrode and an ion of the metal M n ne M the faradic current density without additives, i F, A 0, at the electrode is related to the electrode surface overpotential, s, by the Butler-Volmer equation

2 C230 Journal of The Electrochemical Society, C229-C N d k d A 9 where k a and k d are adsorption and desorption rate constants, respectively. Since the additives are assumed to be incorporated into the deposit, the inclusion rate is related to the growth velocity, v, of the deposit, which can be computed as follows v M w i F nf 1 A 10 where M w and are the metal s molecular weight and density, respectively. The lifetime for an adsorbed additive of diameter d A to be covered up is d A /v. The inclusion rate can then be expressed by N incl Av d A 11 Figure 1. Schematic diagram of the metal-additive deposition on the rotating disk electrode. exp anf i F, A 0 i 0 c Ms c Mb s exp cnf s 5 where i 0 is the exchange current density of the reaction according to the bulk concentration of M n, c Ms is the concentration of M n at the electrode surface, a and c are the anodic and cathodic transfer coefficients, respectively, n is the number of electrons transferred, F is the Faraday constant, is a kinetic parameter related to the order of the electrochemical reaction, and s is the surface overpotential. So combining Eq. 5 and Eq. 3 gives the kinetics relation with the effect of additives included exp anf i F i 0 1 A c Ms c Mb s exp cnf s Many studies have reported that additives used in electrodeposition of metals increase the polarization of the electrode. Although additives may decrease i 0 and change a and c, we assume that i 0,, a, and c are not much affected by the presence of additives. This assumption is commonly used in modeling the effect of additives on the kinetics Following Roha and Landau s 29 and Yin s 15 models, we assume that additives diffuse from the bulk solution to the deposit, and are adsorbed onto the deposit, then desorbed back to the diffusion layer, and incorporated into the deposit. These processes are illustrated schematically in Fig. 1. A mass balance on the surface leads to Eq. 7 A t N a N d N incl where approximated by 1/ (d A /2) 2 N av is the surface concentration of one complete additive monolayer, N av is Avogadro s number, and d A is the diameter of additive, N a is the adsorption flux of additives, N d is the desorption flux of additives, and N incl is the inclusion rate of additives. The adsorption flux, N a, is proportional to the number of vacant sites (1 A ) and the surface concentration of additives, c As Na kacas 1 A 8 The desorption flux, N d, is proportional to the number of adsorption sites occupied A 6 7 Substituting Eq. 10 into Eq. 11 produces Eq. 12 A N incl k i i F 1 A 12 where the inclusion rate constant, k i is defined by k i M w / ( nfd A ). Hence, the inclusion rate is proportional to the surface coverage of additive and the faradic current density. Finally, combining Eq. 8, 9, 12, and 7 leads to Eq. 13 A t A k d c As 1 A k d A k i i F 1 A 13 For pulse plating, the surface coverage is a function of time. It is related to the adsorption, desorption, and inclusion rate of additives. If i F 0, the surface concentration of additives should be equal to its bulk concentration, c As c Ab. By setting A / t 0, the surface coverage in dynamic equilibrium with no faradic current is then obtained A 0 Kc Ab 1 Kc Ab 14 Equation 14 is the classical Langmuir isotherm, where K k a /k d is the adsorption equilibrium constant. Mass transfer of additives and metal ions. A material balance within the diffusion layer illustrated in Fig. 1 leads to the differential conservation law c i t N i R i 15 Since both metal ion species M n and additives are reacted on the surface of electrode, the reaction term of species i, R i, in the diffusion layer is zero. With excess supporting electrolyte, the contribution of ionic migration to the flux, N i, becomes negligible. The flux of species i near the electrode surface on the rotating disk electrode expressed in one dimension is as follows N i D i c i y v yc i 16 Equation 16 implies that the flux of species i is contributed only by diffusion and convection. The normal velocity, v y, of the solution can be expressed as 19 v y /2 y 2 17

3 Journal of The Electrochemical Society, C229-C C231 Table I. Parameters for the simulation of a copper RDE in M CuSO 4 and 2.06 M H 2 SO 4 at 300 K. T 300 K 52.4 rad s rpm re 0.01 cm i Acm 25 a c n cm 2 s gcm 1 s 15 R 0.02 cm 5 l 0.2 cm c Mb mol cm 3 D M cm 2 s 15 M w g mol g cm 3 c Ab mol cm 3 k a cm 3 mol 1 s 1 k d s 1 k i 5, 50, 250 cm 2 s 1 A 1 d A , , cm D A , , cm 2 s 1 C dl , , Fcm Fcm 25 C dl where is the rotation speed of the rotating disk electrode RDE, and is the kinematic viscosity of the solution. Substituting of Eq. 16 into Eq. 15 yields c i c i v t y y D 2 c i i y 2 18 The initial condition is c M c Mb c A c Ab at t 0 18a the boundary condition is c M c Mb c A c Ab at y re 18b D M c M y i F t nf D A c A y A k i i F t at y 0 1 A 18c For pulse plating, the current density is a function of time. According to Eq. 1, the faradic current density during the on period is i F i p i C i p C dl d s dt and the faradic current density during the off period is 19 d s i F i C C dl 20 dt Substituting Eq. 6 and Eq. 2 into Eq. 20 yields d s C dl i dt 0 1 A c Ms c Mb exp anf s exp cnf s 21 Response of overpotential. The total overpotential of an electrode consists of three components. They are i the surface overpotential, s, ii concentration overpotential, conc, and iii ohmic 19 overpotential, ohm E t s t conc t ohm t 22 The surface overpotential is related to the faradic current density at the electrode surface, as expressed in Eq. 6. The concentration overpotential is caused by the variation in concentration between the electrode surface and the bulk solution. For the metal/metal ion system, it can be expressed by the Nernst equation as conc t nf ln a Ms 23 a Mb where a Ms and a Mb are the activity of the surface and bulk solution, respectively. If the applied current is low, the concentration difference between the electrode surface and the bulk solution is small, and the activity coefficient of the surface and bulk solution, Ms and Mb, can be equal. Hence, Eq. 23 can be simplified to Eq. 24 conc t nf ln c Ms 24 c Mb The ohmic overpotential comes from the resistance of the electrolyte. It can be expressed in terms of one-dimensional Ohm s law as ohm t R li t 25 where R is the resistivity of electrolyte and l is the distance between the working and the reference electrode. Simulation procedure. The numerical calculations for the potential responses as well as charge and discharge phenomena of the dl in pulse plating were made using a copper RDE in a bath containing M CuSO 4 and 2.06 M H 2 SO 4. The reduction of cupric ions is assumed to be complete with no side reaction. The adsorption and desorption rate constants, k a and k d, are assumed to be and , respectively. These values of k a and k d would correspond to 90.9% surface coverage with no applied current, 0 A, as calculated from Eq. 14. The diffusion coefficient of additives, D A, is estimated from the Stokes-Einstein equation 30 assuming that additives are spherical and there are no fluid slips at the surface of the additives D A 1 T 3 d A 26 where is the viscosity of the solution and is the Boltzmann s constant. The capacity of the dl is sensitive to the adsorptive organic additives. The capacity decreases in the presence of poly ethylene glycol PEG and Cl or the commercial additive, LP-1. 28,5 However, the capacity increases in the presence of thiourea. 31 Three capacity values were therefore chosen for system with additives, which are smaller than C 0 dl, equal to C 0 dl and larger than C 0 dl, respectively. The effects of inclusion rate and capacity on the potential response and mass transfer were simulated. The parameters for the simulation are listed in Table I. The set of Eq. 18 is discretized to the finite-difference form by the forward-time and centered-space scheme. Equation 13 and Eq are discretized by the forward-time finite difference scheme. 32 The simulation procedures are described below For the on period 1. The initial values of c M, c A, and A are known. 2. A of the next time is calculated from Eq Guess i F to calculate c Ms, c As, and s from Eq. 18 and Eq. 6.

4 C232 Journal of The Electrochemical Society, C229-C Figure 2. dl effect on overpotential response in millisecond range pulse plating with additives. The duty cycle is 0.5, and the pulse period is 4 ms. k a , k d , k i 50, C dl 50 F/cm 2, and i p Figure 4. Faradic current density response for additives with different inclusion rate constants. The duty cycle is 0.5, and the pulse period is 4 ms. k a , k d , C 0 dl 25 F/cm 2, C dl 25 F/cm 2, and i p 4. Compute i F from Eq. 19. Compare this value with the guessed one. If these two values are different, then go back to step 3. Otherwise, go back to step 2 to calculate the next time step. For the off period 1. A and s of the next time are calculated from Eq. 13, and Eq. 21, respectively. 2. Compute i F from Eq. 20 and calculate c Ms and c As from Eq Repeat steps 1 and 2 until the end of off time. Results and Discussion Figure 2 shows the importance of the dl effect on overpotential response in millisecond range pulse plating with additives. The charge and discharge time are comparable to the millisecond pulse period in the presence of additives, and the dl effect cannot be neglected. In the following, we discuss the effects of different inclusion rates and capacities of additives on the current and potential response and mass transfer with the dl effect. Current and potential response. Figure 3 shows the effect of the additives inclusion rate constant on the surface coverage for pulse plating at 0.05 A/cm 2. The dl capacities of systems with and without additives are both 25 F/cm 2. The pulse period is 4 ms and 0 the duty cycle is 0.5. In this model, k a and k d are estimated from A and c Ab according to the classical Langmuir isotherm. The small value of k d ( ) indicates that the desorption of additives is not important and that the adsorption of additives is irreversible. Since c Ab contained in the plating bath is usually very small ( mol cm 3 ), the adsorption flux, N a is small. Hence, the surface coverage is only related to inclusion rate of additives. During the on period, the surface coverage diminishes with time since it is consumed gradually in the inclusion of additives. During the off period, the surface coverage is independent of time with no applied current. The surface coverage decreases with increasing inclusion rate constant. For small k i, e.g., k i 5, the fluctuation of surface coverage reaches a steady state after a few pulses. For large k i, e.g., k i 250, the fluctuation of surface coverage decays with plating time. Yin 15 chose a higher order of adsorption rate constant ( ) and adsorption equilibrium constant ( ) than those used in this study. In his study, the surface coverage of additives increases with time during the off period because of the high flux of adsorption. However, large adsorption equilibrium constant leads to almost complete surface coverage before current is applied, which is unreasonable in practice. Hence, we assume that k a and k d , and a more reasonable surface coverage, 0 A 0.909, is obtained. In the dl the amount of electricity being charged during the charge period is equal to the amount being discharged during the discharge period, and the following conservation relation must be satisfied 0 t cic chg dt ton t dic dis dt 27 Figure 3. Variation in surface coverage of additives during plating with different inclusion rate constants. The duty cycle is 0.5, and the pulse period is 4 ms. k a , k d , C dl 25 F/cm 2, and i p where i C chg is the capacitive current density during the charge period, i C dis is the capacitive current density during the discharge period, and t c and t d are the charge and discharge times, respectively. In pulse plating, i C chg is larger than i C dis because i C chg is charged by the pulse generator and i C dis is discharged naturally. Hence, the charge time is much shorter than the discharge time according to Eq. 27. The dl effect on the faradic current density with different inclusion rate constants is shown in Fig. 4. During the charge period, the charge time is shorter than the pulse period. The difference in faradic cur-

5 Journal of The Electrochemical Society, C229-C C233 Figure 5. Overpotential response for additives with different inclusion rate constants. The duty cycle is 0.5, and the pulse period is 4 ms. k a , k d , C 0 dl 25 F/cm 2, C dl 25 F/cm 2, and i p Figure 6. Variation in surface coverage of additives during plating with different capacity. The duty cycle is 0.5, and the pulse period is 4 ms. k a , k d , k i 50, and i p The fifth pulse cycle is magnified in the upper right. rent density among different inclusion rate constants is not apparent. During the discharge period, the discharge time is comparable to the pulse period. Although the faradic current density is quite small, the variation in faradic current density with various k i is quite significant. The faradic current density takes much longer to decrease to zero with low inclusion rate constant. More details about the effect of inclusion rate constant on the charging and discharging of the dl are discussed in the overpotential response. In the absence of additives, charging and discharging the dl can be neglected because the capacitive current density with small overpotential variation is very small. The overpotential response for additives with different inclusion rate constants is shown in Fig. 5. Although the difference in faradic current density is not obvious during the on period, the overpotential varies with different inclusion rate constants. In the presence of additives, the overpotential increases faster than that without additives since the electrode is covered with additives in the former case. For high inclusion rate constant, e.g., k i 250, the surface coverage of additives diminishes quickly, as shown in Fig. 3, during the on period and the overpotential drops sharply. For low inclusion rate constant, e.g., k i 50 or k i 5, the surface coverage of additives remains high and decreases more smoothly, so the overpotential is also larger than that with high inclusion rate constant and decreases slightly. For instance, at t s, if k i 250, the E is 0.28; in the case of k i 5, the E is During the on period, the overpotential increases with decreasing k i. The capacitive current density is in proportion to the overpotential. As the overpotential increases with decreasing k i, the capacitive current density increases with increasing overpotential. Hence, the overpotential with small k i takes more time to decrease to zero owing to a larger capacitive current density during the off period. In the absence of additives, the overpotential increases with time because of depletion of metal ions near the electrode during the on period. During the off period, the overpotential decreases quickly to zero due to the small capacitive current density. The effect of dl capacity on surface coverage of additives is shown in Fig. 6. The capacitive current density increases with increasing C dl with the same overpotential. As the capacitive current density increases, the faradic current density decreases. During the on period, surface coverage with large C dl diminishes more slowly than that with small C dl owing to the smaller i F. During the off period, surface coverage still diminishes due to the discharging of the dl. The A decays more slowly during off period to its steady state as C dl increases, which again represents the effect of dl. Figure 7 and 8, show the faradic current density and overpotential response for additives with different dl capacities. As the capacitive current density increases with increasing C dl, the faradic current density and overpotential with large C dl take more time to reach i p and the related overpotential during the on period, and take more time to decrease to zero during the off period. Mass transfer of additives and metal ions. Figure 9 shows the variation in surface concentrations of additives with respect to time for pulse plating. For large k i, e.g., k i 250, it is seen that during the on period, the surface concentrations of additives decrease with time and reach a minimum value at the end of each on period. During the off period, the surface concentration rises again owing to the diffusion of additives from the bulk solution. For lower k i, e.g., k i 5 or 50, the enhancement of mass transfer by the pulse current causes c As to approach c Ab. The variation in surface concentration of metal ions during plating with different inclusion rate constants is shown in Fig. 10. The fluctuations of c Ms are similar to those of c As, as shown in Fig. 9. To show clearly the dl effect on the mass transfer of c Ms, we only Figure 7. Faradic current density response for additives with different capacities. The duty cycle is 0.5, and the pulse period is 4 ms. k a , k d , k i 50, C 0 dl 25 F/cm 2, and i p

6 C234 Journal of The Electrochemical Society, C229-C Figure 8. Overpotential response for additives with different capacities. The duty cycle is 0.5, and the pulse period is 4 ms. k a , k d , k i 50, C 0 dl 25 F/cm 2, and i p Figure 10. Variation in surface concentration of metal ions during plating with different inclusion rate constants. The duty cycle is 0.5, and the pulse period is 4 ms. k a , k d , C 0 dl 25 F/cm 2, C dl 25 F/cm 2, and i p display c Ms of the fifth pulse. Because the applied current is far lower than the dc-limiting current density, the surface concentration is still fairly high in pulse plating. During the on period, c Ms with smaller k i diminishes slowly due to the smaller i F. During the off period, metal ions still reduce to metal by discharging the dl, and the surface concentration rises slowly. However, the fluctuation of c Ms is slightly affected by the dl because i C is small compared to i F. Figures 11 and 12 show the variation in surface concentration of additives and metal ions with different capacities. The effect of different dl capacities on mass transfer of additives and metal ions is similar to that displayed in Fig. 10. During the on period, c As and c Ms with larger C dl diminish slowly because of the larger i C. During the off period, c As and c Ms increase slowly with larger C dl because more additives and metal ions are consumed by the larger i F discharged by the dl. Note that, if the dl affects significantly the mass transfer, the variation in c As and c Ms will even out due to the charging and discharging of the dl. This result is the same as that obtained by Puippe and Ibl. 17 However, their estimation of the magnitude of charge and discharge time is in microseconds in the absence of additives. In this study, the magnitude of charge and discharge time increases to milliseconds in the presence of additives because additives block the electrode and increase the overpotential, thus increasing i C. Experiments of overpotential response. To verify the model, we measured the overpotential response of pulse plating. The plating bath of 250 ml contained M CuSO 4 and 2.06 M H 2 SO 4. The additive chosen was 100 ppm PEG with 4000 molecular weight. The working electrode WE was a copper rotating disk with 1 cm diam. The WE was polished with SAMPL-KWICK powder no , subsequently immersed into 1MNaOHfor1min,andfinally rinsed with deionized water 18 m cm, Millipore by 3 min ultrasonic oscillation before each experiment. The counter electrode was platinum and saturated calomel electrode SCE served as the reference electrode; both were placed directly in the cell far from the WE. The pulse current was generated by a potentiostat/galvanostat EG&G PARC model 362 which was controlled by a function generator BNC Corp. model 625. The overpotential responses 5000 data/s were stored through an analog output unit of interface card UEI PD2-MF-16-50/16H in a personal computer. The duty cycle is 0.5, the pulse period is 4 ms, 500 rpm, and i p The experimental overpotential responses of pulse plating with and Figure 9. Variation in surface concentration of additives during plating with different inclusion rate constants. The duty cycle is 0.5, and the pulse period is 4 ms. k a , k d , C dl 25 F/cm 2, and i p Figure 11. Variation in surface concentration of additives during plating with different capacities. The duty cycle is 0.5, and the pulse period is 4 ms. k a , k d , k i 50, and i p The fifth pulse cycle is magnified in the upper right.

7 Journal of The Electrochemical Society, C229-C C235 charging and discharging of the dl. The model predications follow the same trend as the experimental overpotential responses. National Tsing-Hwa University assisted in meeting the publication costs of this article. List of Symbols Figure 12. Variation in surface concentration of metal ions during plating with different capacities. The duty cycle is 0.5, and the pulse period is 4 ms. k a , k d , k i 50, C dl 25 F/cm 2, and i p without 100 ppm PEG were shown in Fig. 13. Same as that predicted by the model, the overpotential with additives is higher than that without additives because the additives block the electrode, and i C with additives is larger than that without additives because of the higher s. The potential with PEG is about 1.5 times larger than that without PEG, and k i of PEG is probably small according to this model. Conclusions Numerical simulations had been presented for the analysis of the dl effect on current and potential response and mass transfer in pulse plating with additives. It is shown that overpotential increases with decreasing inclusion rate constant of additives due to the increase in the blocking area of the electrode. As the overpotential increases, capacitive current density will increase, and the dl effect is significant. On the other hand, capacitive current density also increases with increasing capacity of the dl. In the absence of additives, capacitive current density can be neglected owing to the smaller overpotential. If the dl effect is significant, the variation in surface concentrations of additives and metal ions will even out because of the c Ab bulk concentration of additives, mol cm 3 c As surface concentration of additives, mol cm 3 C dl capacity of double layer, F cm 2 0 C dl capacity of double layer without additives, F cm 2 c Mb bulk concentration of metal ions, mol cm 3 c Ms surface concentration of metal ions, mol cm 3 d A additive diameter, cm D A diffusion coefficient of additives, cm 2 s 1 D M diffusion coefficient of metal ions, cm 2 s 1 E total overpotential, V i 0 exchange current density, A cm 2 i C capacitive current density, A cm 2 i F faradic current density, A cm 2 i F, A 0 faradic current density without additives, A cm 2 i p pulse current density, A cm 2 K adsorption equilibrium constant, cm 3 mol 1 k a additive adsorption rate constant, cm 3 mol 1 s 1 k d additive desorption rate constant, s 1 k i additive inclusion rate constant, cm 2 s 1 A 1 l distance between the working and the reference electrode, cm M w molecular weight of metal, g mol 1 n number of electrons transferred in the electrode reaction, kg equivalent/kg mol N a additive adsorption flux, mol cm 2 s 1 N av Avogadro s number, mol 1 N d additive desorption flux, mol cm 2 s 1 N incl additive inclusion rate, mol cm 2 s 1 T absolute temperature, K t time, s t c charge time, s t d discharge time, s v growth velocity of deposit, cm s 1 v y normal fluid velocity, cm s 1 Greek a anodic charge transfer coefficient, dimensionless c cathodic charge transfer coefficient, dimensionless re reference position where no concentration gradient of species exists, cm con concentration overpotential, V ohm ohmic overpotential, V s surface overpotential, V surface concentration of one complete additive monolayer, mol cm 2 kinetic parameter in Eq. 5, dimensionless Boltzmann s constant, JK 1 viscosity of solution, g cm 1 s 1 metal density, g cm 3 R resistivity of electrolyte, cm A surface coverage of additives, dimensionless 0 A surface coverage of additives with no applied current, dimensionless kinematic viscosity of solution, cm 2 s 1 disk rotation speed, rad s 1 References Figure 13. Experimental overpotential responses of pulse plating with and without additives 100 ppm PEG. The plating bath contains M CuSO 4, 2.06 M H 2 SO 4. The duty cycle is 0.5, the pulse period is 4 ms, and i p 1. V. Dubin, C. Ting, and R. W. Cheung, U.S. Pat. 5,972, T. N. Theis, IBM J. Res. Dev., 44, E. J. Taylor, J. J. Sun, and M. E. Inman, Plat. Surf. Finish., 87, A. C. West, C. C. Cheng, and B. C. Baker, J. Electrochem. Soc., 145, S. Goldbach, W. Messing, T. Daenen, and F. Lapicque, Electrochim. Acta, 44, R. D. Mikkola, Q.-T. Jiang, and B. Carpenter, Plat. Surf. Finish., 87, J. Reid, S. Mayer, E. Broadbent, E. Klawuhn, and K. Ashtiani, Solid State Technol., 43, H. Y. Cheh, J. Electrochem. Soc., 118, K. Viswanathan, M. A. F. Epstein, and H. Y. Cheh, J. Electrochem. Soc., 125, K. Viswanathan and H. Y. Cheh, J. Electrochem. Soc., 126, D.-T. Chin, J. Electrochem. Soc., 130, S. Venkatesh, M. Meyyappan, and D. T. Chin, J. Colloid Interface Sci., 85, K.-M. Yin and R. E. White, AIChE J., 36, H. Schultz and M. Pritzker, J. Electrochem. Soc., 145, K.-M. Yin, J. Electrochem. Soc., 145, N. Ibl, Surf. Technol., 10, J. Cl. Puippe and N. Ibl, J. Appl. Electrochem., 10, K. G. Jordan and C. W. Tobias, J. Electrochem. Soc., 138,

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