Investigation On Several Issues In VLSI Interconnect Process Variations

With the great development of very large scale integrated circuits manufacturing,interconnect process variations have been a very important factor impacting the design andmanufacture of integrated circuits. At present, the analysis approaches for processvariations use the equivalent lumped model of interconnects generally, which reduces thecomplexity of the problem and simplies the analysis. However, the equivalent lumpedmodel of interconnects is just the approximation of interconnects, and it leads to the loss ofthe accuracy.This dissertation concentrates on the problem of interconnect process variations inVLSI. The researches are based on the interconnect model with distributed parameters. Weemphatically study the impact of interconnect process variations on the parasiticparameters of interconnects, and then research the impact of interconnect processvariations on the transmission performance of interconnects in several ways such astransient response, sinusoidal steady state response, delay matric and ABCD transmissionmatrix. Generally, the main research points in this paper are as follows:We provide the detailed derivative of finite element method for capacitance extractionof interconnects, and make a comparison on computational complexity, computed speedand calculation accuracy of three kinds of field methods (Finite Difference Method, FiniteElement Method and Measured Equation of Invariance) for capacitance extraction. Tocalculate the capacitance variation due to the slight variation of conductor dimensions, theLagrange expansion is applied to the voltage expression, and a novel method withnon-uniform grids is proposed to complete the calculation of coefficients in the case of theaberration of conductor sizes. Our revision method can get accurate results fast and stably,without any change of the scale of system of equations for the computation of electricfield.Aimed at the singular problem of reentrant points during the process of capacitanceextraction, the mathematical model of electric field near the reentrant is given and itsanalytical solution format is derived. Finite series expansion is used to approximate theanalytical solution, and a novel method is proposed to calculate the charge of conductor.The error analysis of asymptotic approximation of finite series expansion is also given. Ourmethod avoids choosing the integral length of the reentrant corner, and can be used tocalculate the charge of the conductor conveniently and efficiently. Based on the telegraph equation, the stochastic model for transmission lines isproposed, and the precise integration algorithm is combined with Monte Carlo method toanalyze the transient response of the stochastic model. Jarque-Bera test is made for thenormality of the model's output and the worst-case estimation is given. In the case ofsinusoidal excitation, lossless transmission lines are considered. The analytic form of firstmoment of the corresponding stochastic differential equation's solution is derived by usingIto's formula, and the numerical computation method for second moment is given, finallythe upper and lower bound of the output signal's amplitude and phase shift is estimated.We propose the interconnect delay formula affected by process variations, which isbased on Elmore delay. In the case of that there is no analytical form between processvariations and parasitic parameters, we apply numerical simulations and curve fitting toobtain the approximation expression between them. First order variation and second ordervariation are analyzed separately, and the method to calculate the statistical characteristicsof interconnect delay is proposed. As for the case of that the analytical form exists (suchas wire width variation), the formulas to calculate the mean value and variance ofinterconnect delay are derived.Considering the spatial correlations of interconnect process variations, we modelprocess variations as continual stochastic processes with correlation. For decoupleing thespatial correlation, we express them as an expanded form of independent random variablesby Karhunen-Loeve expansion. Then we use stochastic Galerkin method to solve thedecoupled telegraph equation in the presence of process variations. In this way, greatnumber of samples needed by Monte-Karlo method is avoided, and the semi analyticalexpansion of transmission signal can be obtained.We derive the integrated equation that the ABCD matrix of interconnects shouldsatisfy, and apply stochastic Galerkin method to solve it. During the solving, mathematicaltransformation is proposed to make the process simpler. Considering that ABCD matrix issuited to the analysis about the cascade problem of two-port networks, we propose that wecan separate the line into segments and then cascade them to obtain the ABCD matrix ofthe line.