On a generalized framework for modeling the effects of process variations on circuit delay performance using response surface methodology
A generalized methodology for modeling the effects of process variations on circuit delay performance is proposed by directly relating the variations in process parameters to variations in delay metric of a digital circuit. The 2-input NAND gate is used as a library element for 65 nm gate length technology, whose delay is extensively characterized by mixed-mode simulations. This information is then used in a general-purpose circuit simulator SEQUEL, by incorporating appropriate templates for the,NAND gate library. A 4-bit x 4-bit Wallace tree multiplier circuit, consisting of about 300 2-input NAND gates, is used as a representative combinational circuit to demonstrate the proposed methodology. The variation in the multiplier delay is characterized by an extensive Monte Carlo analysis. To extend this methodology for a generic technology library with a variety of library elements, modeling of NAND gate delays by response surface methodology (RSM), in terms of process parameters, is carried out using design of experiments (DOE). A simple piecewise quadratic model, based on the least squares method (LSM), is proposed for one-parameter variation to address significant cubic effects observed in the delay response function. Then, a hybrid model for gate delays is generated by superimposing the interaction terms of DOE-RSM model upon the quadratic model of one-parameter variation to address the generalized case of simultaneous variations in multiple process parameters. The proposed methodology has been demonstrated for NAND gate library with 266 gates, and the simplicity and generality of the approach make it equally applicable to a large library of cells for both statistical timing analysis and statistical circuit simulation at the gate level.
mixed-mode simulations monte carlo analysis response surface methodology (rsm) design of experiments (doe) least squares method (lsm) hybrid model delay distribution process sensitivity