Ensuring Uniform Material Removal with Magnetic Abrasive Finishing of Semiconductor Wafers by Mode Control of Machining Parameters

Abstract

The paper describes a solution technique for direct and inverse problems of modeling the process of magnetic abrasive finishing (MAF) of semiconductor wafers. Solution of the direct problem enables calculation of allowance removal function for the prescribed machining parameters, and solution of the inverse problem enables determination of machining parameters required for realization of the prescribed allowance removal function.  The direct problem is solved by means  of Preston equation, which is usually used for description of material removal rate when polishing optical parts. The inverse problem is considered in a matrix formulation, and its least squares solution is determined by means of generalized inverse Moore-Penrose matrix. Based on the solution of the direct problem with constant values of kinematic and magnetic machining  parameters shows that MAF with constant values of machining parameters does not ensure uniformity of material removal.  On the basis of numerical examples it is shown that close to uniform material removal can be ensured by control of magnetic machining parameters with the law of variation determined from solution of the inverse problem. It is demonstrated that the smoothness of solution of the initial ill-conditioned inverse problem can be improved by means of Tikhonov’s regularization, which in turn technically simplifies control of machining parameters