Optimization of Cathode Shielding in Cell Design Using Computer Simulation

Optimization of Cathode Shielding in Cell Design Using Computer Simulation | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Optimization of Cathode Shielding in Cell Design Using Computer Simulation Allan Reed, Aleksander Jaworski This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6173934/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract This paper describes a straightforward numerical method for solving the Laplace equation to optimize the design of an electroplating cell by placement of the anode, cathode and cathode shields to achieve a uniform deposit thickness on a flat cathode in a cell tailored for a specific purpose. The method uses an iterative finite-difference algorithm to calculate the deposit thickness distributions for selected sets of cell geometry parameters so that the effect of changes in these parameters can be seen, thus allowing the optimum set of geometric parameters to be selected. The algorithm shows how sub-optimum geometry leads to either over-shielding or under-shielding and where the optimum cell geometry lies between these two conditions. The Tafel slope for the reaction, the exchange current density and the electrolyte conductivity are also considered. However, these factors have considerably less of an effect on the deposit distribution than do the geometric cell design parameters. This paper demonstrates the universality of Prentice’s numerical method by allowing the parametrization as illustrated with an exemplary symmetrical electrochemical cylindrical cell. Such a geometry is of utmost relevance in semiconductor manufacturing. Achieving a uniform deposit by optimization of cell design can lead to a significant reduction of subsequent corrective procedures such as chemical mechanical planarization commonly employed in the semiconductor industry. Current distribution cell design optimization Laplace equation finite difference simulation copper electrodeposition Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 Figure 11 Figure 12 Figure 13 Figure 14 1. Introduction Cathode shielding in electroplating cells is often used to reduce edge buildup, also known as dog-boning. Edge buildup occurs on cathodes in electroplating cells because the current distribution in a cell is greater at the edges of an electrode than in its central regions because of the shape of the electrical field in the cell. The shape of the electrical field and hence the current distribution is affected by the size and shape of the non-conducting portion of the cell. Strategically-placed non-conducting shapes (usually called cathode shields) can be used to distort the natural current distribution in a cell to make it more uniform over the surface of the cathode. The mathematical calculations for modeling the current and potential distribution in an electrochemical cell have been reviewed by Schlesinger [ 1 ] and Zamani [ 2 ]. Landau [ 3 ] also reviewed analytical and numerical methods for solving the Laplace equation for modeling the current distribution in a cell. Kasper [ 4 ] published analytical solutions for several simple cell geometries considering only linear polarization of the electrodes and neglecting mass transfer. However, analytical solutions for more practical cell geometries and more realistic electrode polarization functions are limited and require considerable mathematical sophistication [ 5 ]. While these methods provide a rigorous basis for understanding the current and potential distribution in a cell, there seems to be no discussion of the design and optimization of an appropriate cell for any particular purpose. Whatever has been done is apparently empirical and proprietary since there are no reports in the open literature of any methods for cell optimization, including cathode shielding. The lack of published information on shield design would seem to indicate the difficulty (or impossibility) of applying analytical methods to this problem since a plating cell has a shape in which electrical fields and currents cannot be consistently well defined. However, the problem of trying to understand and calculate the current density distribution is important in both a theoretical and practical sense since it is usually desirable to produce a deposit whose thickness is as uniform as possible. Calculation of the current density distribution in an electrochemical cell requires solving the Laplace equation for the particular cell geometry coupled with the kinetic parameters of the electrode reactions and the prevailing mass transfer conditions. Optimization of cathode shielding particularly in the design of cylindrical cells used in semiconductor wafer manufacturing may also reduce the amount of chemical mechanical planarization (CMP), also known as chemical mechanical polishing. One of the functions of CMP is to remove excess plated copper to produce a wafer surface having the uniform planarity required for subsequent steps in the manufacturing process. CMP requires the use of a number of different types of abrasives as well as chemicals including oxidizers, chelating agents, surfactants, polyelectrolytes and inhibitors [ 6 ]. Reducing the amount of CMP required should also reduce the use of these chemicals leading to a reduction in chemical consumption and cost in the overall manufacturing process. Numerical solutions can be used to calculate the current density distribution in idealized design. Klingert et al. [ 7 ] were the first to implement an iterative computer procedure to solve the Laplace equation for an electrochemical cell using the finite difference method. They considered a two-dimensional cell having one L-shaped electrode and one flat electrode and logarithmic (Tafel) polarization. Prentice and Tobias [ 8 ] extended this method to consider shape changes at the corner of the L-shaped electrode and later published a Fortran program to calculate the current density distribution in a cell with an L-shaped electrode [ 9 ]. This paper extends earlier work done to calculate the potential and current distribution in a cylindrical cell that was used for plating semiconductor wafers [ 10 , 11 ] with a circular cathode and anode at the bottom and top the cell, respectively. The finite-difference method presented by Prentice [ 9 ] is particularly suited to solving the Laplace equation for geometries that can be divided into square elements as shown in Fig. 1 . This figure shows a cross sectional plane of a cell having a diameter (CW) of 10 inches (25 cm). The intersections of the dotted lines that form the square elements in Fig. 1 are the nodes where the electric field potentials are calculated using the finite-difference method. The nodes in Fig. 1 are shown as a coarse grid to illustrate the method. However, the actual finite-difference calculations use nodes that are more closely spaced to form a finer mesh that approximates a continuum. 2. The Model 2.1. Cell Design The idealized cell design that was used is shown in Figure 1. It is a cylindrical cell with both the anode and cathode being circular to avoid any corner effects on the current density (CD) distribution. Since a cylindrical cell has radial symmetry, the CD distribution over the surface of the cathode can be calculated by considering a two-dimensional cell in a perpendicular plane through the diameter of the cylinder. The algorithm presented by Prentice [9] was modified for a two-dimensional plane of the cylindrical cell and rewritten in the C programming language [12]. The CD distribution curves presented in this paper were calculated using the same algorithm implemented in MATLAB R2021b (The MathWorks, Inc., Natick, MA, USA). The MATLAB source code used to calculate the cell geometry for the curves in Figures 2 through 14 is given in the Appendix. All dimensions that are shown can be varied except the cell diameter which is fixed at 10.0 inches (25 cm) as shown. The shield thickness (SHLDTH) was held constant at 0.4 inches. Cao et al. [13] described a cell having a similar design with a cathode shield between the anode and cathode for plating on cathodes having a thin, resistive seed layer. Matlosz et al. [14] showed that the cathode CD distribution becomes uniform as the deposit resistivity decreases due to its increasing thickness. The work presented in this paper describes how the shield diameter and position affects the CD distribution on a highly conductive cathode so that the initial surface resistance of the cathode can be neglected. 2.2. Digitization of Cell Design Geometry Cathode CD distributions were calculated using the above-mentioned computer program for cells having cathodes with diameters (CAW) ranging from 2.0 to 8.0 inches and anode diameters (ANW) ranging from 2.0 to 8.0 inches. The open diameters in the shield (SHLDDM) ranged from 0.4 to 2.0 inches less than the cathode diameter and the shield heights (SHLDHT) ranging from 0.5 to 3.0 inches above the cathode. The cathode to anode (CH) spacing was held constant at 6.0 inches as this had been found in all early calculations to be an optimum spacing to achieve uniform cathode CD distributions. Even though these calculations are illustrated using an acid copper plating bath, the same calculations appear to be useful for trying to understand how variations in shield placement could be applied to produce more uniform deposits of any metal in an appropriately designed cell. The Laplace equation in Cartesian coordinates has the form: for every point in the domain shown in Figure 1. 2.3. Assumptions in the Model The following assumptions and simplifications are implicit in the model: 1. The electrodeposition reaction at the cathode is 100 percent efficient so that the deposit thickness distribution is essentially identical to the current density distribution on the cathode. 2. Both anode and cathode are shape invariant with respect to time and all conditions are steady state. 3. Only the cathode is polarizable; the anode remains unpolarized. 4. The bulk electrolyte is well-stirred so that it contains no concentration gradients. 5. The Butler-Volmer equation can be used to describe the relationship between the CD and surface potential (overpotential) at the cathode and the overpotential is sufficiently large that the cathode is operating in the Tafel regime. 6. The cathodic CD is low with respect to the mass-transport limiting CD so that the effects of concentration polarization can be neglected. 2.4. Calculation Algorithm A summary of the algorithm used to calculate the distributions in these simulations is as follows: 1. Specify the electrolyte conductivity, exchange current density, Tafel slope, anode and cathode potentials, grid spacing, and convergence criteria. 2. Specify the cell geometry (anode and cathode size, shape, and space). 3. Set the potentials of the anode and cathode to their initial values. 4. Initialize the grid by setting the potential at all nodes to the average of the potentials applied to the anode and cathode. 5. Calculate new values of all interior potential nodes using equation (2). Convergence can be hastened by using an overrelaxation method with a relaxation constant of 1.85 as used by Prentice and Tobias [15] and Prentice [9]. 6. Calculate the potential at the nodes on the insulating surfaces using the image point method. Since equipotential lines are always perpendicular to an insulating surface, the potential at the image point inside the insulator is the same as the potential at the corresponding mirror-image position in the electrolyte. The initializations of steps 2 through 6 is shown in the code in the Appendix, 7. Calculate the CD at each cathode node as the product of the two-point normal derivative of the surface potential and the electrolyte conductivity. 8. Calculate the overpotential at each cathode node from the CD at that node and the Tafel expression. 9. Repeat steps 5 through 8 until the convergence criteria are met or until the preset maximum of iterations is reached. Then display the results of the calculation. The iterative procedure of steps 7 through 9 is based on the algorithm of Prentice [9]. 3. Results and Discussion The algorithm outlined above follows that presented by Prentice [ 9 ]. However, the Prentice’s [ 9 ] algorithm was just for a specific L-shaped cell (defined in step 2 of calculation algorithm). In our approach the cell shape is parameterized in step 2 as demonstrated on the example of the cylindrical shape with shielded cathode. By doing so it was demonstrated that Prentice’s algorithm could be universally applied to various cell geometries (as long as they are defined, parameterized and discretized). The applied value of this algorithm is much broader than just a particular, L-shaped cell for calculation of which this model was originally presented. We suspect that this universal algorithm did not receive the attention it deserves because it was rigidly assigned to the specific cell geometry for which it was originally introduced. The objective of this paper is to demonstrate the universality and potential for exploitation of the general algorithm by parameterizing cell design (step 2). This algorithm was used to calculate the CD across a cathode surface. In order to compare the effects of shield diameter and height above the cathode, the calculations were normalized to the maximum value for any set of calculations with the maximum value for each curve always equal to 1. Plots of such calculations are shown in Figs. 2 , 3 and 4 for an 8-inch diameter cathode and a 6-inch diameter anode. Figure 2 shows the CD distribution for a shield diameter of 7.2 inches, Fig. 3 shows the distribution for a shield diameter of 7.4 inches and Fig. 4 for a shield diameter of 7.6 inches. Similar calculations are shown in Figs. 5 , 6 and 7 for a 6-inch diameter cathode and a 6-inch diameter anode. Figure 5 shows the CD distribution for a shield diameter of 5.2 inches, Fig. 6 shows the distribution for a shield diameter of 5.4 inches and Fig. 7 for a shield diameter of 5.6 inches. The CD distribution curves in Figs. 2 through 7 were calculated using the parameters shown in Table 1. The conductivity is that of a high-acid copper sulfate plating solution (1.8 M sulfuric acid, 0.3 M copper sulfate) calculated to be 0.563 Ω −1 cm −1 (1.43 Ω −1 in −1 ) using the correlation of Hsueh [ 16 ] as reported by Cabán and Chapman [ 17 ]. The exchange CD was reported by Pesco and Cheh [ 18 ] as 1.00 x 10 − 3 A/cm 2 (6.45 x 10 − 3 A/in 2 ). Since it is commonly thought that the CD distribution is affected by the conductivity of a plating bath, current density distribution curves were calculated and plotted using an electrolyte conductivity of 0.135 Ω −1 in −1 (that of 0.1 M sulfuric acid) and 0.039 Ω −1 in −1 (that of 0.01 M sulfuric acid). These curves are shown in Figs. 8 through 10 and 11 through 13 , respectively. These conductivities were calculated using the correlation of Hsueh [ 16 ]. The curves in Figs. 2 through 13 show that the CD distribution varies with shield height above the cathode for all shield diameters. Three different shield heights are shown in each figure. In all figures, when the shield is closest to the cathode the CD is lower at the edges of the cathode and greater near its center. This is called “over-shielding.” Likewise, when the shield is farthest from the cathode, the CD density is higher at the edges of the cathode and lower at its center. This is called “under-shielding.” The most uniform current density distribution is between these two positions. This is shown in Fig. 14 for a 6-inch diameter cathode, a 6-inch anode and a 5.4-in diameter shield opening at heights of 0.6, 0.9 and 1.2 inches above the cathode in an electrolyte having a conductivity of 1.43 Ω −1 in −1 . When the shield is 0.6 inches above the cathode, the deposit thickness is low at the edges of the cathode, indicating over-shielding; when the shield is 1.2 inches above the cathode, the deposit is high at the edges and lower at the center, indicating under-shielding. The optimum shield position to produce a deposit with the most uniform thickness is somewhere around 0.9 inches above the cathode. This same behavior is found with electrolytes of 0.135 Ω −1 in −1 and 0.039 Ω −1 in −1 as seen in Figs. 9 and 12 , respectively. Calculations over a wide range of cathode and anode diameters always show the same characteristic of over-shielding and under shielding as the height of the shield above the cathode is changed. The diameter of the shield opening has a somewhat similar effect. If its diameter is too large, there is little or no shielding and if it is too small, over-shielding occurs. Table 1. Parameters used in Calculations 4. Conclusions This paper describes how the current density distribution in a symmetrical electrochemical cylindrical cell can be modeled by solving the Laplace equation using iterative finite difference calculations. It has been shown that cathode shielding has a significant effect on the current density distribution in the cell and can be used to improve the deposit thickness uniformity on a circular cathode such as commonly used for semiconductor manufacturing. A shield for this cell configuration consists of a circular ring attached to the wall of the cell. The optimum diameter of the ring opening and its distance from the cathode surface can be determined by observing how the current density distribution changes with the shield opening diameter and position above the cathode. When the shield is too close to the cathode, the current density distribution is lower at the edge of the cathode and greater near its center. This is called “over-shielding.” Conversely, when the shield is too far from the cathode, the current density distribution is higher at the edge of the cathode and lower near its center. This is called “under-shielding.” The diameter of the shield opening has a similar effect. If the opening diameter is too large, “under-shielding” occurs and if it is too small, “over-shielding” occurs. Although the work presented in this paper is focused on a symmetrical cell of immediate practical importance, it should be possible to digitize other cell geometries to model and optimize the current distribution in them using the same algorithm as used in this work. The numerical method presented here offers the possibility of process-oriented optimization of plating cell design to significantly reducing the need for CMP post-processing. Numerical simulation is a more effective way of plating cell design optimization than iterative cell redesign based on empirical experience. This paper demonstrates the universal value of Prentice’s method by offering the possibility of the parameterization of cell design. Therefore, the method presented is of critical utility for prompt and efficient (single-step rather than iterative) custom-designing of plating cells optimized for an electrodeposition process in order to minimize post-processing of the metal deposit. Declarations Author Contribution Allan Reed: Conceptualization, Software, Writing: Original Draft Preparation, Reviewing, EditingAleksander Jaworski: Software, Data Curation, Writing: Reviewing, Editing References Schlesinger M, “Mathematical Modeling in Electrochemistry,” in Modern Aspects of Electrochemistry , Vol. 43, Schlesinger M, Ed., Springer, New York, 2008. Zamani NG, “Numerical Modeling of Certain Electrochemical Processes,” in Modern Aspects of Electrochemistry , Vol. 44, Schlesinger M, Ed., Springer, New York, 2009. Landau U, “Current Distribution in Electrochemical Cells: Analytical and Numerical Modeling,” in Modern Aspects of Electrochemistry , Vol. 44, Schlesinger M, Ed., Springer, New York, 2009. Kasper C, Trans. Electrochem. Soc ., 77 , 353 (1940); 77 , 365 (1940); 78 ,131 (1940); 78, 147 (1940); 82 ,153 (1942). Prentice GA and Tobias CW , J. Electrochem. Soc , 129 , 72 (1982). Krishnan M, Nalaskowski JW and Cook LM, Chem. Rev. 110 , 178 (2010). Klingert JA, Lynn S and Tobias CW, Electrochemica Acta , 9 , 297 (1964). Prentice GA and Tobias CW , J. Electrochem. Soc , 129 , 78 (1982). Prentice G, Electrochemical Engineering Principles , Prentice-Hall, Englewood Cliffs, NJ (1991). Reed AH, “Optimization of Wafer Plating Cell Design Using Finite Difference Calculations,” AESF Sur/Fin ’99 Proceedings (1999). Reed AH, “The Effect of Cell Geometry on Deposit Thickness Uniformity in a Wafer Plating Cell,” AESF Sur/Fin ’99 Proceedings (2000). Kernighan BW and Ritchie DM, The C Programming Language, Second Edition, Prentice-Hall, Englewood Cliffs, NJ (1988). Cao Y, Lee J.-M and West AC, Plating & Surface Finishing, 40 (November 2003). Matlosz M. Vallotton P-H, West AC and Landolt D, J. Electrochem. Soc , 139 , 752 (1992). Prentice GA and Tobias CW, AIChE. J. , 28 , 486 (1982). Hsueh L, Dissertation UCRL – 18597, University of California, Berkeley (1968). Cabán R and Chapman TW, J. Electrochem. Soc., 124 , 1371 (1977). Pesco AM and Cheh HY, J. Electrochem. Soc., 136 , 399 (1989). Additional Declarations No competing interests reported. Supplementary Files GraphicalabstractAReedAJaworski03062025.jpg Appendix.docx Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-6173934","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":425475676,"identity":"20c86ade-80bd-4e55-9db2-801552f25739","order_by":0,"name":"Allan Reed","email":"","orcid":"","institution":"Technic, Inc","correspondingAuthor":false,"prefix":"","firstName":"Allan","middleName":"","lastName":"Reed","suffix":""},{"id":425475677,"identity":"31671695-07dc-4c0b-a7c4-1d21b81995a2","order_by":1,"name":"Aleksander Jaworski","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA3klEQVRIie3PsQrCMBCA4UjgXIJdUyr6CimBTgUfpqu7W6kIdXJ38yFcHByuBHTpAwg6WIWu6iIUF2vdHNK6CeYbbgj3E44Qw/hJWE0B7YiQVkTYFwlTjZM3QXhQJfWsscr5bR3Kjn1JjsXq0LUimp12moTjxrPnufLACag7S3PGEaQcak9Bb88Q/TIBpxUrJggDR5f0cXsvk9AHW7UfjRKB6esX6gGnQBslLqajYo5KAgukPSsTrmpu6e22S3HF0F1Mk+xaxGpgTSfZWXs+x88Xqlt/saK6DcMwjL/3BFSiS9BRJMTOAAAAAElFTkSuQmCC","orcid":"","institution":"Technic, Inc","correspondingAuthor":true,"prefix":"","firstName":"Aleksander","middleName":"","lastName":"Jaworski","suffix":""}],"badges":[],"createdAt":"2025-03-07 00:53:07","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-6173934/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-6173934/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":78142472,"identity":"d6f24873-72e1-43e1-bed3-cdf88952d645","added_by":"auto","created_at":"2025-03-10 10:24:06","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":74424,"visible":true,"origin":"","legend":"\u003cp\u003eCell design showing finite-difference grid.\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-6173934/v1/7c9b12eb79d961b22f571111.png"},{"id":78142451,"identity":"a7dc3852-a465-4978-ac1a-bcb1838f2d82","added_by":"auto","created_at":"2025-03-10 10:24:02","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":39156,"visible":true,"origin":"","legend":"\u003cp\u003eDistribution of current density (CD) normalized to maximum (CD\u003csub\u003emax\u003c/sub\u003e) across an 8-inch cathode using a 6-inch diameter anode and a shield diameter of 7.2 inches.\u0026nbsp; Electrolyte conductivity 1.43 Ω\u003csup\u003e-1\u003c/sup\u003ein\u003csup\u003e-1\u003c/sup\u003e.\u0026nbsp;\u0026nbsp; Other parameters as in Table 1.\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-6173934/v1/095ef0f3d50d0c9929ffb15a.png"},{"id":78142770,"identity":"46d345ca-37d8-42bd-a2f7-2460e89d887e","added_by":"auto","created_at":"2025-03-10 10:32:06","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":42866,"visible":true,"origin":"","legend":"\u003cp\u003eDistribution of current density (CD) normalized to maximum (CD\u003csub\u003emax\u003c/sub\u003e) across an 8-inch cathode using a 6-inch diameter anode and a shield diameter of 7.4 inches.\u0026nbsp; Electrolyte conductivity 1.43 Ω\u003csup\u003e-1\u003c/sup\u003ein\u003csup\u003e-1\u003c/sup\u003e.\u0026nbsp;\u0026nbsp; Other parameters as in Table 1.\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-6173934/v1/ef9602e77fe55fe0f75eab63.png"},{"id":78142765,"identity":"e7169063-c945-44be-ac9a-d7511c65c0c9","added_by":"auto","created_at":"2025-03-10 10:32:04","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":42908,"visible":true,"origin":"","legend":"\u003cp\u003eDistribution of current density (CD) normalized to maximum (CD\u003csub\u003emax\u003c/sub\u003e) across an 8-inch cathode using a 6-inch diameter anode and a shield diameter of 7.6 inches.\u0026nbsp; Electrolyte conductivity 1.43 Ω\u003csup\u003e-1\u003c/sup\u003ein\u003csup\u003e-1\u003c/sup\u003e.\u0026nbsp;\u0026nbsp; Other parameters as in Table 1.\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-6173934/v1/f13cec40c8de7f77d38bf484.png"},{"id":78142503,"identity":"430deb39-7fe2-444d-9969-76067246a5c6","added_by":"auto","created_at":"2025-03-10 10:24:08","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":42217,"visible":true,"origin":"","legend":"\u003cp\u003eDistribution of current density (CD) normalized to maximum (CD\u003csub\u003emax\u003c/sub\u003e) across a 6-inch cathode using a 6-inch diameter anode and a shield diameter of 5.2 inches.\u0026nbsp; Electrolyte conductivity 1.43 Ω\u003csup\u003e-1\u003c/sup\u003ein\u003csup\u003e-1\u003c/sup\u003e.\u0026nbsp;\u0026nbsp; Other parameters as in Table 1.\u003c/p\u003e","description":"","filename":"5.png","url":"https://assets-eu.researchsquare.com/files/rs-6173934/v1/5b74dfa559542a8e38e8c3f3.png"},{"id":78142773,"identity":"783f8dc2-520a-449f-80fe-b764e2992c6b","added_by":"auto","created_at":"2025-03-10 10:32:08","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":41504,"visible":true,"origin":"","legend":"\u003cp\u003eDistribution of current density (CD) normalized to maximum (CD\u003csub\u003emax\u003c/sub\u003e) across a 6-inch cathode using a 6-inch diameter anode and a shield diameter of 5.4 inches.\u0026nbsp; Electrolyte conductivity 1.43 Ω\u003csup\u003e-1\u003c/sup\u003ein\u003csup\u003e-1\u003c/sup\u003e.\u0026nbsp;\u0026nbsp; Other parameters as in Table 1.\u003c/p\u003e","description":"","filename":"6.png","url":"https://assets-eu.researchsquare.com/files/rs-6173934/v1/0669b2bfb078c1688a00885c.png"},{"id":78142453,"identity":"bc47fbba-ac36-4c1d-b81b-79e5e22355eb","added_by":"auto","created_at":"2025-03-10 10:24:04","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":43013,"visible":true,"origin":"","legend":"\u003cp\u003eDistribution of current density (CD) normalized to maximum (CD\u003csub\u003emax\u003c/sub\u003e) across a 6-inch cathode using a 6-inch diameter anode and a shield diameter of 5.6 inches.\u0026nbsp; Electrolyte conductivity 1.43 Ω\u003csup\u003e-1\u003c/sup\u003ein\u003csup\u003e-1\u003c/sup\u003e.\u0026nbsp;\u0026nbsp; Other parameters as in Table 1.\u003c/p\u003e","description":"","filename":"7.png","url":"https://assets-eu.researchsquare.com/files/rs-6173934/v1/8e8419754fb2a6c842b9a3af.png"},{"id":78142463,"identity":"908793df-bd29-4230-b063-b52185183332","added_by":"auto","created_at":"2025-03-10 10:24:05","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":42685,"visible":true,"origin":"","legend":"\u003cp\u003eDistribution of current density (CD) normalized to maximum (CD\u003csub\u003emax\u003c/sub\u003e) across a 6-inch cathode using a 6-inch diameter anode and a shield diameter of 5.2 inches.\u0026nbsp; Electrolyte conductivity 0.135 Ω\u003csup\u003e-1\u003c/sup\u003ein\u003csup\u003e-1\u003c/sup\u003e.\u0026nbsp;\u0026nbsp; Other parameters as in Table 1.\u003c/p\u003e","description":"","filename":"8.png","url":"https://assets-eu.researchsquare.com/files/rs-6173934/v1/76f7a0fba65614f8adb17775.png"},{"id":78142471,"identity":"50eaa49a-7565-4477-a673-5a3035f4f156","added_by":"auto","created_at":"2025-03-10 10:24:06","extension":"png","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":42405,"visible":true,"origin":"","legend":"\u003cp\u003eDistribution of current density (CD) normalized to maximum (CD\u003csub\u003emax\u003c/sub\u003e) across a 6-inch cathode using a 6-inch diameter anode and a shield diameter of 5.4 inches.\u0026nbsp; Electrolyte conductivity 0.135 Ω\u003csup\u003e-1\u003c/sup\u003ein\u003csup\u003e-1\u003c/sup\u003e.\u0026nbsp;\u0026nbsp; Other parameters as in Table 1.\u003c/p\u003e","description":"","filename":"9.png","url":"https://assets-eu.researchsquare.com/files/rs-6173934/v1/fd6a59b352505d3b8f673288.png"},{"id":78142466,"identity":"935e26c1-84bf-4be4-b750-9a8f132fc82a","added_by":"auto","created_at":"2025-03-10 10:24:05","extension":"png","order_by":10,"title":"Figure 10","display":"","copyAsset":false,"role":"figure","size":43932,"visible":true,"origin":"","legend":"\u003cp\u003eDistribution of current density (CD) normalized to maximum (CD\u003csub\u003emax\u003c/sub\u003e) across a 6-inch cathode using a 6-inch diameter anode and a shield diameter of 5.6 inches.\u0026nbsp; Electrolyte conductivity 0.135 Ω\u003csup\u003e-1\u003c/sup\u003ein\u003csup\u003e-1\u003c/sup\u003e.\u0026nbsp;\u0026nbsp; Other parameters as in Table 1.\u003c/p\u003e","description":"","filename":"10.png","url":"https://assets-eu.researchsquare.com/files/rs-6173934/v1/f789f72f1f2dba5bc43543fa.png"},{"id":78142491,"identity":"d162be0c-6e6c-4eea-b241-b5afd2d6db54","added_by":"auto","created_at":"2025-03-10 10:24:08","extension":"png","order_by":11,"title":"Figure 11","display":"","copyAsset":false,"role":"figure","size":45446,"visible":true,"origin":"","legend":"\u003cp\u003eDistribution of current density (CD) normalized to maximum (CD\u003csub\u003emax\u003c/sub\u003e) across a 6-inch cathode using a 6-inch diameter anode and a shield diameter of 5.2 inches.\u0026nbsp; Electrolyte conductivity 0.039 Ω\u003csup\u003e-1\u003c/sup\u003ein\u003csup\u003e-1\u003c/sup\u003e.\u0026nbsp;\u0026nbsp; Other parameters as in Table 1.\u003c/p\u003e","description":"","filename":"11.png","url":"https://assets-eu.researchsquare.com/files/rs-6173934/v1/6f8942af92e24bcf6b01e18c.png"},{"id":78142767,"identity":"3508d16c-f7d6-44fa-8a53-d2c99664ae5b","added_by":"auto","created_at":"2025-03-10 10:32:06","extension":"png","order_by":12,"title":"Figure 12","display":"","copyAsset":false,"role":"figure","size":43548,"visible":true,"origin":"","legend":"\u003cp\u003eDistribution of current density (CD) normalized to maximum (CD\u003csub\u003emax\u003c/sub\u003e) across a 6-inch cathode using a 6-inch diameter anode and a shield diameter of 5.4 inches.\u0026nbsp; Electrolyte conductivity 0.039 Ω\u003csup\u003e-1\u003c/sup\u003ein\u003csup\u003e-1\u003c/sup\u003e.\u0026nbsp;\u0026nbsp; Other parameters as in Table 1.\u003c/p\u003e","description":"","filename":"12.png","url":"https://assets-eu.researchsquare.com/files/rs-6173934/v1/76c0783185fb9b527affa01b.png"},{"id":78142454,"identity":"9eace7a0-acaf-40f1-baba-eeb4884c25ff","added_by":"auto","created_at":"2025-03-10 10:24:04","extension":"png","order_by":13,"title":"Figure 13","display":"","copyAsset":false,"role":"figure","size":44333,"visible":true,"origin":"","legend":"\u003cp\u003eDistribution of current density (CD) normalized to maximum (CD\u003csub\u003emax\u003c/sub\u003e) across a 6-inch cathode using a 6-inch diameter anode and a shield diameter of 5.6 inches.\u0026nbsp; Electrolyte conductivity 0.039 Ω\u003csup\u003e-1\u003c/sup\u003ein\u003csup\u003e-1\u003c/sup\u003e.\u0026nbsp;\u0026nbsp; Other parameters as in Table 1.\u003c/p\u003e","description":"","filename":"13.png","url":"https://assets-eu.researchsquare.com/files/rs-6173934/v1/27dc62e796f28d37782e7844.png"},{"id":78142501,"identity":"9e03611d-a1bb-411d-8998-e791e0ca065d","added_by":"auto","created_at":"2025-03-10 10:24:08","extension":"png","order_by":14,"title":"Figure 14","display":"","copyAsset":false,"role":"figure","size":42210,"visible":true,"origin":"","legend":"\u003cp\u003eDistribution of current density (CD) normalized to maximum (CD\u003csub\u003emax\u003c/sub\u003e) across a 6-inch cathode using a 6-inch diameter anode and a shield diameter of 5.4 inches.\u0026nbsp; Electrolyte conductivity 1.43 Ω\u003csup\u003e-1\u003c/sup\u003ein\u003csup\u003e-1\u003c/sup\u003e.\u0026nbsp;\u0026nbsp; Other parameters as in Table 1.\u003c/p\u003e","description":"","filename":"14.png","url":"https://assets-eu.researchsquare.com/files/rs-6173934/v1/67e14cd1b4cacf667af9b505.png"},{"id":79001646,"identity":"51fc3bea-1e7c-4b99-9569-f1a399921af5","added_by":"auto","created_at":"2025-03-22 05:32:43","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":968700,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-6173934/v1/de34f065-9400-4b21-a926-cae27355be19.pdf"},{"id":78142456,"identity":"bcd3bf72-8726-4a6e-90d9-c018316f90de","added_by":"auto","created_at":"2025-03-10 10:24:04","extension":"jpg","order_by":1,"title":"","display":"","copyAsset":false,"role":"supplement","size":114368,"visible":true,"origin":"","legend":"","description":"","filename":"GraphicalabstractAReedAJaworski03062025.jpg","url":"https://assets-eu.researchsquare.com/files/rs-6173934/v1/b4459c5a526f0899d3f9e4a6.jpg"},{"id":78142470,"identity":"7f96ba3c-f6cf-4bf9-920f-c4cee0181080","added_by":"auto","created_at":"2025-03-10 10:24:06","extension":"docx","order_by":2,"title":"","display":"","copyAsset":false,"role":"supplement","size":17462,"visible":true,"origin":"","legend":"","description":"","filename":"Appendix.docx","url":"https://assets-eu.researchsquare.com/files/rs-6173934/v1/8a91a6ec859f1fdcfb12fa0b.docx"}],"financialInterests":"No competing interests reported.","formattedTitle":"Optimization of Cathode Shielding in Cell Design Using Computer Simulation","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eCathode shielding in electroplating cells is often used to reduce edge buildup, also known as dog-boning. Edge buildup occurs on cathodes in electroplating cells because the current distribution in a cell is greater at the edges of an electrode than in its central regions because of the shape of the electrical field in the cell. The shape of the electrical field and hence the current distribution is affected by the size and shape of the non-conducting portion of the cell. Strategically-placed non-conducting shapes (usually called cathode shields) can be used to distort the natural current distribution in a cell to make it more uniform over the surface of the cathode. The mathematical calculations for modeling the current and potential distribution in an electrochemical cell have been reviewed by Schlesinger [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e] and Zamani [\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e]. Landau [\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e] also reviewed analytical and numerical methods for solving the Laplace equation for modeling the current distribution in a cell. Kasper [\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e] published analytical solutions for several simple cell geometries considering only linear polarization of the electrodes and neglecting mass transfer. However, analytical solutions for more practical cell geometries and more realistic electrode polarization functions are limited and require considerable mathematical sophistication [\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eWhile these methods provide a rigorous basis for understanding the current and potential distribution in a cell, there seems to be no discussion of the design and optimization of an appropriate cell for any particular purpose. Whatever has been done is apparently empirical and proprietary since there are no reports in the open literature of any methods for cell optimization, including cathode shielding. The lack of published information on shield design would seem to indicate the difficulty (or impossibility) of applying analytical methods to this problem since a plating cell has a shape in which electrical fields and currents cannot be consistently well defined. However, the problem of trying to understand and calculate the current density distribution is important in both a theoretical and practical sense since it is usually desirable to produce a deposit whose thickness is as uniform as possible. Calculation of the current density distribution in an electrochemical cell requires solving the Laplace equation for the particular cell geometry coupled with the kinetic parameters of the electrode reactions and the prevailing mass transfer conditions.\u003c/p\u003e \u003cp\u003eOptimization of cathode shielding particularly in the design of cylindrical cells used in semiconductor wafer manufacturing may also reduce the amount of chemical mechanical planarization (CMP), also known as chemical mechanical polishing. One of the functions of CMP is to remove excess plated copper to produce a wafer surface having the uniform planarity required for subsequent steps in the manufacturing process. CMP requires the use of a number of different types of abrasives as well as chemicals including oxidizers, chelating agents, surfactants, polyelectrolytes and inhibitors [\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e]. Reducing the amount of CMP required should also reduce the use of these chemicals leading to a reduction in chemical consumption and cost in the overall manufacturing process.\u003c/p\u003e \u003cp\u003eNumerical solutions can be used to calculate the current density distribution in idealized design. Klingert \u003cem\u003eet al.\u003c/em\u003e [\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e] were the first to implement an iterative computer procedure to solve the Laplace equation for an electrochemical cell using the finite difference method. They considered a two-dimensional cell having one L-shaped electrode and one flat electrode and logarithmic (Tafel) polarization. Prentice and Tobias [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e] extended this method to consider shape changes at the corner of the L-shaped electrode and later published a Fortran program to calculate the current density distribution in a cell with an L-shaped electrode [\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e]. This paper extends earlier work done to calculate the potential and current distribution in a cylindrical cell that was used for plating semiconductor wafers [\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e, \u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e] with a circular cathode and anode at the bottom and top the cell, respectively.\u003c/p\u003e \u003cp\u003eThe finite-difference method presented by Prentice [\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e] is particularly suited to solving the Laplace equation for geometries that can be divided into square elements as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e. This figure shows a cross sectional plane of a cell having a diameter (CW) of 10 inches (25 cm). The intersections of the dotted lines that form the square elements in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e are the nodes where the electric field potentials are calculated using the finite-difference method. The nodes in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e are shown as a coarse grid to illustrate the method. However, the actual finite-difference calculations use nodes that are more closely spaced to form a finer mesh that approximates a continuum.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e"},{"header":"2. The Model","content":"\u003cp\u003e\u003cstrong\u003e2.1. Cell Design\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe idealized cell design that was used is shown in Figure 1. \u0026nbsp;It is a cylindrical cell with both the anode and cathode being circular to avoid any corner effects on the current density (CD) distribution. Since a cylindrical cell has radial symmetry, the CD distribution over the surface of the cathode can be calculated by considering a two-dimensional cell in a perpendicular plane through the diameter of the cylinder. \u0026nbsp;The algorithm presented by Prentice [9] was modified for a two-dimensional plane of the cylindrical cell and rewritten in the C programming language [12]. \u0026nbsp;The CD distribution curves presented in this paper were calculated using the same algorithm implemented in MATLAB R2021b (The MathWorks, Inc., Natick, MA, USA). \u0026nbsp; The MATLAB source code used to calculate the cell geometry for the curves in Figures 2 through 14 is given in the Appendix.\u003c/p\u003e\n\u003cp\u003eAll dimensions that are shown can be varied except the cell diameter which is fixed at 10.0 inches (25 cm) as shown. \u0026nbsp;The shield thickness (SHLDTH) was held constant at 0.4 inches. \u0026nbsp;Cao \u003cem\u003eet al.\u003c/em\u003e [13] described a cell having a similar design with a cathode shield between the anode and cathode for plating on cathodes having a thin, resistive seed layer. \u0026nbsp;Matlosz \u003cem\u003eet al.\u0026nbsp;\u003c/em\u003e[14] showed that the cathode CD distribution becomes uniform as the deposit resistivity decreases due to its increasing thickness. \u0026nbsp;The work presented in this paper describes how the shield diameter and position affects the CD distribution on a highly conductive cathode so that the initial surface resistance of the cathode can be neglected.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e2.2. Digitization of Cell Design Geometry\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eCathode CD distributions were calculated using the above-mentioned computer program for cells having cathodes with diameters (CAW) \u0026nbsp;ranging from 2.0 to 8.0 inches and anode diameters (ANW) ranging from 2.0 to 8.0 inches. The open diameters in the shield (SHLDDM) ranged from 0.4 to 2.0 inches less than the cathode diameter and the shield heights (SHLDHT) ranging from 0.5 to 3.0 inches above the cathode. \u0026nbsp;The cathode to anode (CH) spacing was held constant at 6.0 inches as this had been found in all early calculations to be an optimum spacing to achieve uniform cathode CD distributions. Even though these calculations are illustrated using an acid copper plating bath, the same calculations appear to be useful for trying to understand how variations in shield placement could be applied to produce more uniform deposits of any metal in an appropriately designed cell.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eThe Laplace equation in Cartesian coordinates has the form:\u003c/p\u003e\n\u003cp\u003e\u003cimg 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\"\u003e\u003c/p\u003e\n\u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n\u003cp\u003efor every point in the domain shown in Figure 1.\u003c/p\u003e\n\u003ch1\u003e2.3. Assumptions in the Model\u003c/h1\u003e\n\u003cp\u003eThe following assumptions and simplifications are implicit in the model:\u003c/p\u003e\n\u003cp\u003e1. \u0026nbsp;The electrodeposition reaction at the cathode is 100 percent efficient so that the deposit thickness distribution is essentially identical to the current density distribution on the cathode.\u003c/p\u003e\n\u003cp\u003e2. \u0026nbsp;Both anode and cathode are shape invariant with respect to time and all conditions are steady state.\u003c/p\u003e\n\u003cp\u003e3. \u0026nbsp;Only the cathode is polarizable; the anode remains unpolarized.\u003c/p\u003e\n\u003cp\u003e4. \u0026nbsp;The bulk electrolyte is well-stirred so that it contains no concentration gradients.\u003c/p\u003e\n\u003cp\u003e5. \u0026nbsp;The Butler-Volmer equation can be used to describe the relationship between the CD and surface potential (overpotential) at the cathode and the overpotential is sufficiently large that the cathode is operating in the Tafel regime.\u003c/p\u003e\n\u003cp\u003e6. \u0026nbsp;The cathodic CD is low with respect to the mass-transport limiting CD so that the effects of concentration polarization can be neglected.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e2.4. Calculation Algorithm\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eA summary of the algorithm used to calculate the distributions in these simulations is as follows:\u003c/p\u003e\n\u003cp\u003e1. Specify the electrolyte conductivity, exchange current density, Tafel slope, anode and cathode potentials, grid spacing, and convergence criteria.\u003c/p\u003e\n\u003cp\u003e2. Specify the cell geometry (anode and cathode size, shape, and space).\u003c/p\u003e\n\u003cp\u003e3. Set the potentials of the anode and cathode to their initial values.\u003c/p\u003e\n\u003cp\u003e4. Initialize the grid by setting the potential at all nodes to the average of the potentials applied to the anode and cathode.\u003c/p\u003e\n\u003cp\u003e5. Calculate new values of all interior potential nodes using equation (2). \u0026nbsp;Convergence can be hastened by using an overrelaxation method with a\u003csup\u003e\u0026nbsp;\u003c/sup\u003erelaxation constant of 1.85 as used by Prentice and Tobias [15] and Prentice [9].\u003c/p\u003e\n\u003cp\u003e6. Calculate the potential at the nodes on the insulating surfaces using the image point method. \u0026nbsp;Since equipotential lines are always perpendicular to an insulating surface, the potential at the image point inside the insulator is the same as the potential at the corresponding mirror-image position in the electrolyte.\u003c/p\u003e\n\u003cp\u003eThe initializations of steps 2 through 6 is shown in the code in the Appendix,\u003c/p\u003e\n\u003cp\u003e7. Calculate the CD at each cathode node as the product of the two-point normal derivative of the surface potential and the electrolyte conductivity.\u003c/p\u003e\n\u003cp\u003e8. Calculate the overpotential at each cathode node from the CD at that node and the Tafel expression.\u003c/p\u003e\n\u003cp\u003e9. Repeat steps 5 through 8 until the convergence criteria are met or until the preset maximum of iterations is reached. \u0026nbsp;Then display the results of the calculation.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eThe iterative procedure of steps 7 through 9 is based on the algorithm of Prentice [9].\u003c/p\u003e"},{"header":"3. Results and Discussion","content":"\u003cp\u003eThe algorithm outlined above follows that presented by Prentice [\u003cspan class=\"CitationRef\"\u003e9\u003c/span\u003e]. However, the Prentice\u0026rsquo;s [\u003cspan class=\"CitationRef\"\u003e9\u003c/span\u003e] algorithm was just for a specific L-shaped cell (defined in step 2 of calculation algorithm). In our approach the cell shape is parameterized in step 2 as demonstrated on the example of the cylindrical shape with shielded cathode. By doing so it was demonstrated that Prentice\u0026rsquo;s algorithm could be universally applied to various cell geometries (as long as they are defined, parameterized and discretized). The applied value of this algorithm is much broader than just a particular, L-shaped cell for calculation of which this model was originally presented. We suspect that this universal algorithm did not receive the attention it deserves because it was rigidly assigned to the specific cell geometry for which it was originally introduced. The objective of this paper is to demonstrate the universality and potential for exploitation of the general algorithm by parameterizing cell design (step 2).\u003c/p\u003e\n\u003cp\u003eThis algorithm was used to calculate the CD across a cathode surface. In order to compare the effects of shield diameter and height above the cathode, the calculations were normalized to the maximum value for any set of calculations with the maximum value for each curve always equal to 1. Plots of such calculations are shown in Figs. \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e, \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e and \u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003e for an 8-inch diameter cathode and a 6-inch diameter anode. Figure \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e shows the CD distribution for a shield diameter of 7.2 inches, Fig. \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e shows the distribution for a shield diameter of 7.4 inches and Fig. \u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003e for a shield diameter of 7.6 inches. Similar calculations are shown in Figs. \u003cspan class=\"InternalRef\"\u003e5\u003c/span\u003e, \u003cspan class=\"InternalRef\"\u003e6\u003c/span\u003e and \u003cspan class=\"InternalRef\"\u003e7\u003c/span\u003e for a 6-inch diameter cathode and a 6-inch diameter anode. Figure \u003cspan class=\"InternalRef\"\u003e5\u003c/span\u003e shows the CD distribution for a shield diameter of 5.2 inches, Fig. \u003cspan class=\"InternalRef\"\u003e6\u003c/span\u003e shows the distribution for a shield diameter of 5.4 inches and Fig. \u003cspan class=\"InternalRef\"\u003e7\u003c/span\u003e for a shield diameter of 5.6 inches.\u003c/p\u003e\n\u003cp\u003eThe CD distribution curves in Figs. \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e through \u003cspan class=\"InternalRef\"\u003e7\u003c/span\u003e were calculated using the parameters shown in Table 1. The conductivity is that of a high-acid copper sulfate plating solution (1.8 M sulfuric acid, 0.3 M copper sulfate) calculated to be 0.563\u003c/p\u003e\n\u003cp\u003eΩ\u003csup\u003e\u0026minus;1\u003c/sup\u003ecm\u003csup\u003e\u0026minus;1\u003c/sup\u003e (1.43 Ω\u003csup\u003e\u0026minus;1\u003c/sup\u003ein\u003csup\u003e\u0026minus;1\u003c/sup\u003e) using the correlation of Hsueh [\u003cspan class=\"CitationRef\"\u003e16\u003c/span\u003e] as reported by Cab\u0026aacute;n and Chapman [\u003cspan class=\"CitationRef\"\u003e17\u003c/span\u003e]. The exchange CD was reported by Pesco and Cheh [\u003cspan class=\"CitationRef\"\u003e18\u003c/span\u003e] as 1.00 x 10\u003csup\u003e\u0026minus;\u0026thinsp;3\u003c/sup\u003e A/cm\u003csup\u003e2\u003c/sup\u003e (6.45 x 10\u003csup\u003e\u0026minus;\u0026thinsp;3\u003c/sup\u003eA/in\u003csup\u003e2\u003c/sup\u003e). Since it is commonly thought that the CD distribution is affected by the conductivity of a plating bath, current density distribution curves were calculated and plotted using an electrolyte conductivity of 0.135 Ω\u003csup\u003e\u0026minus;1\u003c/sup\u003ein\u003csup\u003e\u0026minus;1\u003c/sup\u003e (that of 0.1 M sulfuric acid) and 0.039 Ω\u003csup\u003e\u0026minus;1\u003c/sup\u003ein\u003csup\u003e\u0026minus;1\u003c/sup\u003e (that of 0.01 M sulfuric acid). These curves are shown in Figs. \u003cspan class=\"InternalRef\"\u003e8\u003c/span\u003e through \u003cspan class=\"InternalRef\"\u003e10\u003c/span\u003e and \u003cspan class=\"InternalRef\"\u003e11\u003c/span\u003e through \u003cspan class=\"InternalRef\"\u003e13\u003c/span\u003e, respectively. These conductivities were calculated using the correlation of Hsueh [\u003cspan class=\"CitationRef\"\u003e16\u003c/span\u003e].\u003c/p\u003e\n\u003cp\u003eThe curves in Figs. \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e through \u003cspan class=\"InternalRef\"\u003e13\u003c/span\u003e show that the CD distribution varies with shield height above the cathode for all shield diameters. Three different shield heights are shown in each figure. In all figures, when the shield is closest to the cathode the CD is lower at the edges of the cathode and greater near its center. This is called \u0026ldquo;over-shielding.\u0026rdquo; Likewise, when the shield is farthest from the cathode, the CD density is higher at the edges of the cathode and lower at its center. This is called \u0026ldquo;under-shielding.\u0026rdquo; The most uniform current density distribution is between these two positions. This is shown in Fig. \u003cspan class=\"InternalRef\"\u003e14\u003c/span\u003e for a 6-inch diameter cathode, a 6-inch anode and a 5.4-in diameter shield opening at heights of 0.6, 0.9 and 1.2 inches above the cathode in an electrolyte having a conductivity of 1.43 Ω\u003csup\u003e\u0026minus;1\u003c/sup\u003ein\u003csup\u003e\u0026minus;1\u003c/sup\u003e. When the shield is 0.6 inches above the cathode, the deposit thickness is low at the edges of the cathode, indicating over-shielding; when the shield is 1.2 inches above the cathode, the deposit is high at the edges and lower at the center, indicating under-shielding. The optimum shield position to produce a deposit with the most uniform thickness is somewhere around 0.9 inches above the cathode. This same behavior is found with electrolytes of 0.135 Ω\u003csup\u003e\u0026minus;1\u003c/sup\u003ein\u003csup\u003e\u0026minus;1\u003c/sup\u003e and 0.039 Ω\u003csup\u003e\u0026minus;1\u003c/sup\u003ein\u003csup\u003e\u0026minus;1\u003c/sup\u003e as seen in Figs. \u003cspan class=\"InternalRef\"\u003e9\u003c/span\u003e and \u003cspan class=\"InternalRef\"\u003e12\u003c/span\u003e, respectively. Calculations over a wide range of cathode and anode diameters always show the same characteristic of over-shielding and under shielding as the height of the shield above the cathode is changed. The diameter of the shield opening has a somewhat similar effect. If its diameter is too large, there is little or no shielding and if it is too small, over-shielding occurs.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eTable 1. \u0026nbsp;Parameters used in Calculations\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e\u003cimg 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\"\u003e\u003c/strong\u003e\u003cbr\u003e\u003c/p\u003e"},{"header":"4. Conclusions","content":"\u003cp\u003eThis paper describes how the current density distribution in a symmetrical electrochemical cylindrical cell can be modeled by solving the Laplace equation using iterative finite difference calculations. It has been shown that cathode shielding has a significant effect on the current density distribution in the cell and can be used to improve the deposit thickness uniformity on a circular cathode such as commonly used for semiconductor manufacturing.\u003c/p\u003e \u003cp\u003eA shield for this cell configuration consists of a circular ring attached to the wall of the cell. The optimum diameter of the ring opening and its distance from the cathode surface can be determined by observing how the current density distribution changes with the shield opening diameter and position above the cathode. When the shield is too close to the cathode, the current density distribution is lower at the edge of the cathode and greater near its center. This is called \u0026ldquo;over-shielding.\u0026rdquo; Conversely, when the shield is too far from the cathode, the current density distribution is higher at the edge of the cathode and lower near its center. This is called \u0026ldquo;under-shielding.\u0026rdquo; The diameter of the shield opening has a similar effect. If the opening diameter is too large, \u0026ldquo;under-shielding\u0026rdquo; occurs and if it is too small, \u0026ldquo;over-shielding\u0026rdquo; occurs.\u003c/p\u003e \u003cp\u003eAlthough the work presented in this paper is focused on a symmetrical cell of immediate practical importance, it should be possible to digitize other cell geometries to model and optimize the current distribution in them using the same algorithm as used in this work.\u003c/p\u003e \u003cp\u003eThe numerical method presented here offers the possibility of process-oriented optimization of plating cell design to significantly reducing the need for CMP post-processing. Numerical simulation is a more effective way of plating cell design optimization than iterative cell redesign based on empirical experience. This paper demonstrates the universal value of Prentice\u0026rsquo;s method by offering the possibility of the parameterization of cell design. Therefore, the method presented is of critical utility for prompt and efficient (single-step rather than iterative) custom-designing of plating cells optimized for an electrodeposition process in order to minimize post-processing of the metal deposit.\u003c/p\u003e"},{"header":"Declarations","content":"\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003eAllan Reed: Conceptualization, Software, Writing: Original Draft Preparation, Reviewing, EditingAleksander Jaworski: Software, Data Curation, Writing: Reviewing, Editing\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n \u003cli\u003eSchlesinger M, \u0026ldquo;Mathematical Modeling in Electrochemistry,\u0026rdquo; in \u003cem\u003eModern Aspects of\u003c/em\u003e \u003cem\u003eElectrochemistry\u003c/em\u003e, Vol. 43, Schlesinger M, Ed., Springer, New York, 2008.\u003c/li\u003e\n \u003cli\u003eZamani NG, \u0026ldquo;Numerical Modeling of Certain Electrochemical Processes,\u0026rdquo; in \u003cem\u003eModern Aspects of\u003c/em\u003e \u003cem\u003eElectrochemistry\u003c/em\u003e, Vol. 44, Schlesinger M, Ed., Springer, New York, 2009.\u003c/li\u003e\n \u003cli\u003eLandau U, \u0026ldquo;Current Distribution in Electrochemical Cells: Analytical and Numerical Modeling,\u0026rdquo; in \u003cem\u003eModern Aspects of\u003c/em\u003e \u003cem\u003eElectrochemistry\u003c/em\u003e, Vol. 44, Schlesinger M, Ed., Springer, New York, 2009.\u003c/li\u003e\n \u003cli\u003eKasper C, \u003cem\u003eTrans. Electrochem. Soc\u003c/em\u003e., \u003cstrong\u003e77\u003c/strong\u003e, 353 (1940); \u003cstrong\u003e77\u003c/strong\u003e, 365 (1940); \u003cstrong\u003e78\u003c/strong\u003e,131 (1940); \u003cstrong\u003e78,\u003c/strong\u003e 147 (1940); \u003cstrong\u003e82\u003c/strong\u003e,153 (1942).\u003c/li\u003e\n \u003cli\u003ePrentice GA and Tobias CW\u003cem\u003e, J. Electrochem. Soc\u003c/em\u003e, \u003cstrong\u003e129\u003c/strong\u003e, 72 (1982).\u003c/li\u003e\n \u003cli\u003eKrishnan M, Nalaskowski JW and Cook LM, \u003cem\u003eChem. Rev.\u003c/em\u003e \u003cstrong\u003e110\u003c/strong\u003e, 178 (2010).\u003c/li\u003e\n \u003cli\u003eKlingert JA, Lynn S and Tobias CW, \u003cem\u003eElectrochemica Acta\u003c/em\u003e, \u003cstrong\u003e9\u003c/strong\u003e, 297 (1964).\u003c/li\u003e\n \u003cli\u003ePrentice GA and Tobias CW\u003cem\u003e, J. Electrochem. Soc\u003c/em\u003e, \u003cstrong\u003e129\u003c/strong\u003e, 78 (1982).\u003c/li\u003e\n \u003cli\u003ePrentice G, \u003cem\u003eElectrochemical Engineering Principles\u003c/em\u003e, Prentice-Hall, Englewood Cliffs, NJ (1991).\u003c/li\u003e\n \u003cli\u003eReed AH, \u0026ldquo;Optimization of Wafer Plating Cell Design Using Finite Difference Calculations,\u0026rdquo; AESF Sur/Fin \u0026rsquo;99 Proceedings (1999).\u003c/li\u003e\n \u003cli\u003eReed AH, \u0026ldquo;The Effect of Cell Geometry on Deposit Thickness Uniformity in a Wafer Plating Cell,\u0026rdquo; AESF Sur/Fin \u0026rsquo;99 Proceedings (2000).\u003c/li\u003e\n \u003cli\u003eKernighan BW and Ritchie DM, \u003cem\u003eThe C Programming Language,\u0026nbsp;\u003c/em\u003eSecond Edition, Prentice-Hall, Englewood Cliffs, NJ (1988).\u003c/li\u003e\n \u003cli\u003eCao Y, Lee J.-M and West AC, \u003cem\u003ePlating \u0026amp; Surface Finishing,\u0026nbsp;\u003c/em\u003e40 (November 2003).\u003c/li\u003e\n \u003cli\u003eMatlosz M. Vallotton P-H, West AC and Landolt D, \u003cem\u003eJ. Electrochem. Soc\u003c/em\u003e, \u003cstrong\u003e139\u003c/strong\u003e, 752 (1992).\u003c/li\u003e\n \u003cli\u003ePrentice GA and Tobias CW, \u003cem\u003eAIChE. J.\u003c/em\u003e,\u003cem\u003e\u0026nbsp;\u003c/em\u003e\u003cstrong\u003e28\u003c/strong\u003e, 486 (1982).\u003c/li\u003e\n \u003cli\u003eHsueh L, Dissertation UCRL \u0026ndash; 18597, University of California, Berkeley (1968).\u003c/li\u003e\n \u003cli\u003eCab\u0026aacute;n R and Chapman TW, \u003cem\u003eJ. Electrochem. Soc.,\u003c/em\u003e\u003cstrong\u003e\u0026nbsp;124\u003c/strong\u003e, 1371 (1977).\u003c/li\u003e\n \u003cli\u003ePesco AM and Cheh HY, \u003cem\u003eJ. Electrochem. Soc.,\u003c/em\u003e\u003cstrong\u003e\u0026nbsp;136\u003c/strong\u003e, 399 (1989).\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Current distribution, cell design optimization, Laplace equation, finite difference simulation, copper electrodeposition","lastPublishedDoi":"10.21203/rs.3.rs-6173934/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-6173934/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThis paper describes a straightforward numerical method for solving the Laplace equation to optimize the design of an electroplating cell by placement of the anode, cathode and cathode shields to achieve a uniform deposit thickness on a flat cathode in a cell tailored for a specific purpose. The method uses an iterative finite-difference algorithm to calculate the deposit thickness distributions for selected sets of cell geometry parameters so that the effect of changes in these parameters can be seen, thus allowing the optimum set of geometric parameters to be selected. The algorithm shows how sub-optimum geometry leads to either over-shielding or under-shielding and where the optimum cell geometry lies between these two conditions. The Tafel slope for the reaction, the exchange current density and the electrolyte conductivity are also considered. However, these factors have considerably less of an effect on the deposit distribution than do the geometric cell design parameters.\u003c/p\u003e \u003cp\u003eThis paper demonstrates the universality of Prentice\u0026rsquo;s numerical method by allowing the parametrization as illustrated with an exemplary symmetrical electrochemical cylindrical cell. Such a geometry is of utmost relevance in semiconductor manufacturing. Achieving a uniform deposit by optimization of cell design can lead to a significant reduction of subsequent corrective procedures such as chemical mechanical planarization commonly employed in the semiconductor industry.\u003c/p\u003e","manuscriptTitle":"Optimization of Cathode Shielding in Cell Design Using Computer Simulation","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-03-10 10:23:49","doi":"10.21203/rs.3.rs-6173934/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"cfe302ee-aafd-455e-9d55-fd3dd2d0743e","owner":[],"postedDate":"March 10th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2025-03-22T05:08:33+00:00","versionOfRecord":[],"versionCreatedAt":"2025-03-10 10:23:49","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-6173934","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-6173934","identity":"rs-6173934","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}