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Journal of the Electrochemical Society, Vol.150, No.6, C401-C412, 2003
Natural convection in trenches of high aspect ratio
Filling deep trench-like features by electrodeposition is often limited by ion transport from the electrolyte bath to the plating surface at the feature bottom. This transport may be substantially enhanced by buoyancy-driven convection induced by metal-ion depletion adjacent to the plating surface. Numerical solutions of the Navier-Stokes and species transport equations are used to determine the magnitude of transport enhancement, expressed as a Sherwood number, for Rayleigh numbers ranging from 103 to 108 and for feature aspect ratios of depth to width ranging from 1 to 16 both for open trenches and for fully enclosed rectangular domains. To facilitate extrapolation of these numerical results, an exact analytical solution is derived for aspect ratios much greater than unity. This is used in conjunction with the known asymptotic behavior for large Rayleigh numbers to construct a composite formula relating the Sherwood number to the Rayleigh number and aspect ratio. The results indicate that buoyancy-driven convection may provide significant transport enhancement during electrodeposition into features having depths greater than about 100 mum and that enhancement exceeding a factor of ten may occur in LIGA features having depths of 1 mm or more. It is also shown that a moderate inclination of the substrate helps to suppress the formation of multiple vertically stacked convective cells that would otherwise reduce the overall transport. (C) 2003 The Electrochemical Society.