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The simple analytical model of Boning and co-workers is extended by taking variable removal rate ͑RR͒ into account. The advantage of rapid process characterization is retained and model parameters can be obtained from experimental data, and more importantly, analytical solutions are available to provide necessary analysis. The result shows that tradeoff between electrochemically plated copper thickness and throughput exists when the conventional constant RR strategy is applied. At the low RR extreme, minimal Cu thickness is obtained while low throughput is also experienced as a result of low polish rate. At the high RR end, however, highest throughput can be achieved but extremely thick Cu layer should be plated in order to meet planarization specifications. This results in additional operating cost for post-CMP waste water treatment. Therefore, an optimal control problem is formulated for CMP planarization and an analytical solution for optimal RR trajectory is derived. The optimal trajectory achieves the highest throughput with minimal amount of copper thickness ͑47% less than the conventional practice͒. In addition to the variable RR strategy to carry out optimal process control, stepwise implementation is also proposed and it can readily be practiced on present CMP stations with little extra engineering effort.The complexity of current microelectronic devices demands global planarity at different metallization levels. Copper-oxide damascene provides a promising way to pattern Cu lines and it typically involves the following steps: trench patterning, electrochemical plating ͑ECP͒ of Cu, and chemical mechanical polishing ͑CMP͒. CMP has the capability of achieving such stringent requirements over a step height of several micrometers. 1 However, nonplanarity may be created as a result of underlying pattern. The within-die nonplanarity may subsequently create focus problems in the lithography process. 2-8 Moreover, dishing of the soft Cu and erosion of oxide reduce the Cu height in interconnects and result in unnecessarily high electrical resistance, possibly resulting in potential problems in manufacturing. Therefore, it is necessary to devise a CMP polishing strategy to maintain stable CMP operation. [9][10][11][12] The last decade has seen substantial progress in CMP modeling. Nanz and Camilletti 13 provide an earlier review and Ouma and Boning 14 give an updated summary on modeling up to the year 2000. Detailed models have been proposed by considering kinematic, fluid flow, pressure distribution, contact wear effects, etc., and they provide necessary insight into polishing phenomena. [15][16][17][18][19] Most of the research focuses on the effects of pattern geometry on the removal of materials on the interlevel dielectric ͑ILD͒ surface and copper/oxide surface 2,3 with detailed modeling.Much less work has been done on the operational aspects of Cu CMP, especially model-based operating and control strategies for Cu CMP where better accuracy is required. Here we are more interested in the control-relevant model...

This work compliments and extends the capability of the run-to-run (R2R) control by sequencing the incomings such that improved control performance can be achieved. Unlike chemical or mechanical systems, this is important for semiconductor manufacturing processes because some prior information about the incoming wafers is generally available. First, the limitation of a R2R control type of feedback system is explained. The frequency domain explanation is as follows: a negative feedback system is effective rejecting a low-frequency type of disturbance. From the feedback property, then the answer to the feed sequencing problem becomes clear: arrange the feed (which we are capable of doing) in such a way that it gives a low-frequency characteristic. Furthermore, a scalar load strength indicator is derived, and an engineering approach is taken to sort the incomings effectively. Two policies, from thin to thick (in terms of prethickness, policy L2R) and from thick to thin (policy R2L), are proposed, and they are shown to provide a better control performance over the conventional random feed policy (R). The feed sequencing problems are tested for systems with different dimensions, e.g., single input single output, single input multiple output, and multiple input multiple output systems, which include the model of an experimental chemical mechanical polishing process, and the results show that an improved control performance can be obtained simply by arranging the feeds.

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