Cyclic Tableaux and Symmetric Functions

We introduce the notion of cyclic tableaux and develop involutions for Waring's formulas expressing the power sum symmetric function p_n in terms of the elementary symmetric function e_n and the homogeneous symmetric function h_n. The coefficients appearing in Waring's formulas are shown to be a cyclic analog of the multinomial coefficients, a fact that seems to have been neglected before. Our involutions also spell out the duality between these two forms of Waring's formulas, which turns out to be exactly the “duality between sets and multisets.” We also present an involution for permutations in cycle notation, leading to probably the simplest combinatorial interpretation of the Möbius function of the partition lattice and a purely combinatorial treatment of the fundamental theorem on symmetric functions. This paper is motivated by Chebyshev polynomials in connection with Waring's formula in two variables.     

Process characterization and statistical analysis of oxide CMP on a silicon wafer with sparse data

Continuous advancements in chemical mechanical planarization (CMP) process, such as new polishing pads, slurry materials, and abrasive particles necessitate optimization of the key process input parameters for maximum material removal rate (MRR) and/or minimum within wafer non-uniformity (WIWNU) using sparse experimental results. In this investigation a methodology is proposed for developing process models and optimization of input parameters (both main and interaction parameters) for maximum MRR and minimum WIWNU. This approach will be equally applicable for polishing other materials, such as copper, dielectrics and low-k materials. Complex relationships exist between several machine-specific and material-specific input parameters and the output performance variables, chiefly MRR and WIWNU. However, only a few of the input parameters are changed on a regular basis. Hence, only those subsets of relationships need to be considered for optimizing the CMP process. In this investigation, CMP process was characterized for polishing a thin layer of silicon dioxide on top of a silicon wafer. Statistical analysis of the experimental data was performed to obtain the order of significance of the input variables (machine and material parameters and their interactions). Both linear and logarithmic regression models were developed and used to determine optimum process conditions for maximizing MRR and minimizing WIWNU. While the main input parameters were responsible for maximum MRR, interaction parameters were found to be responsible for minimizing WIWNU. This may vary for different materials and polishing environments. PACS 81.00.00; 81.05.Gc; 81.65.Ps