Stochastic Capacitance Extraction Considering Process Variation with Spatial Correlation Tuck Chan EE Department UCLA, CA 90095 tuckie@ucla.edu Fang Gong EE Department UCLA, CA 90095 gongfang@ucla.edu ABSTRACT The increasing process variation and spatial correlation between process parameters makes capacitance extraction of interconnects an important yet challenging problem in modern VLSI designs. In this presentation, we will first introduce basic capacitance extraction flow follow by advance model considering process variation (E.g., orthogonal polynomial method). After that, approaches that solve the resulted augmented system will be presented, such as spectral methods based on polynomial chaos, including Galerkin and collocation methods. Experiment results will also be presented to compare the performance between these methods as well as Monte Carlo method. Finally, some potential research topics within this area will be discussed. Categories and Subject Descriptors B.7.2 [Integrated Circuits]: Design Aids simulation, verification General Terms Design, Algorithms Keywords Process variations, capacitance, parasitic extraction, random variable reduction, principle factor analysis, orthogonal polynomial 1. INTRODUCTION Along with the scaling of CMOS technology, process-induced variability is increasing and has huge impacts on the circuit performance in sub-100nm VLSI technologies. Such process variation has to be considered throughout VLSI design flows to ensure robust circuit design as well as good yield rate. In order to consider the impacts of process variations on interconnects, an efficient stochastic RLC extraction method is required. In this report, we surveyed several approaches on statistical extraction of capacitance considering process variations proposed in [4, 5, 7, 6, 2, 1] under different variational models. Method in [4], uses analytical formulae to consider the variations in capacitance extraction and it has only first-order accuracy. The FastSies program considers the rough surface effects of the interconnect conductors [7]. It assumes only Gaussian distributions and has high computation costs. In [5], a method combining hierarchical extraction and principle factor analysis is proposed. The capacitance extraction is done based on second-order perturbation, which can generate quadratic variational capacitance for better accuracy. Recently, a spectral stochastic collocation based capacitance extraction method was proposed [2]. This approach is based on the Hermite orthogonal polynomial representation of the variational capacitance. It applies the collocation idea, where the capacitance extraction processes (by solving the potential coefficient matrices)are performed many times by sampling so that the coefficients of orthogonal polynomials of variational capacitance can be computed using the weighted least square method. The number of samplings is O(m2), where m is the number of variables. So if m is large, the approach will lose it efficiency compared to the Monte Carlo method. Instead of using collocation method, [1] use a different spectral stochastic method, where the Galerkin scheme is used. Galerkin-based Spectral stochastic method has been applied for statisitical interconenct modeling and on-chip power grid analysis consider process variations. The new method, called statcap, first transforms the original stochastic potential coefficient equations into a deterministic and larger one and then solves it using iterative method. It avoids the sampling process in the existing collocation-based extraction approach. As a result, the potential coefficient equations and the corresponding augmented system only need to be set up once versus many times in the collocation based sampling method. This can lead to a significant saving in CPU time. Also the augmented potential coefficient system is sparse, symmetric and low-rank, which is further exploited by an iterative solver to gain further speedup. The rest of this report is organized as follows: Section 2 reviews the capacitance extraction method in [3]. Section 3 and 4 present the details of stochastic capacitance extraction method proposed in [5] and [1], respectively. Section 5 shows the summary of experimental results and Section 6 concludes this report.
2. PRELIMINARY The capacitances among m conductors can be summarized by an m m capacitance matrix C, Cṽ = q, (1) where q,ṽ R m 1 are conductor charge distribution and potential vectors, respectively. The diagonal entries C ii of C are positive, representing the self-capacitance of conductor i. The non-diagonal entries C ij are negative, representing the coupling capacitance between conductors i and j. The j th column of C can be calculated by solving for the total charges on each of the conductors when the j th conductor is at unit potential and all the other conductors are at zero potential. Then the charge on conductor i, q i, is equal to C ij. This procedure is repeated m times to compute all columns of C. 2.1 BEM Capacitance Extraction Boundary element methods (BEM), also referred to as panel methods or the method of moments, have been adopted as the main approach for 3D capacitance calculation. Due to the fact that the charge is restricted to the surface of the conductors, the surfaces of m conductors with non-uniform charge distribution need to be discretized into a total of n twodimensional panels and the charge distribution on each panel is assumed to be even. Then for each panel k, an equation is written that relates the potential at the center of the k th panel to the sum of contributions to that potential from the charge distribution on all n panels and the contribution from the l th panel is determined by the potential coefficient, P kl = 1 a l Z panel 1 1 x k x da, (2) x k x l where x l and x k are the centers of the lth and kth panels. Then a system of equations can be constructed to solve for the discretized conductor surface charges Pq = v, (3) where P R n n is the potential coefficient matrix and q, v R nx1 are panel charge distribution and potential vectors. To compute the j th column of the capacitance matrix, Eq. 3 must be solved for q, given a v vector where v k = 1 if panel k is on the j th conductor, and v k = 0. Then C ij of the capacitance matrix is computed by summing all the panel charges on the ith conductor, X C ij = q k, (4) k conductor i 2.2 Hierarchical Capacitance Algorithms The main obstacle of solving q is that the coefficient matrix in Eq. 3 is very dense and direct linear system solvers, such as Gaussian elimination or Cholesky decomposition, become computationally intractable if the number of panels exceeds several hundred. Therefore, multipole accelerated and hierarchical algorithms have been proposed to address this problem. Conductor surfaces can be hierarchically, instead of uniformly, divided into smaller panels. The hierarchical panel refinement can be fully described by a multipletree, in which the root panel of each tree corresponds to a conductor surface. If the estimated potential coefficient between two panels is larger than a threshold value P, they are further divided into smaller panels. Otherwise, a link recording the potential coefficient is created between these two panels. All recorded potential coefficients compose a link matrix H R NxN, where N is the number of all panels. For any two panels i and j with no links in between, the corresponding entry in H is zero, otherwise, the recorded potential coefficient evaluated by using Eq. 2 is filled into H ij. For each panel in the hierarchical data structure, its charge is equal to the summation of charges on its two child panels. Therefore, one can choose a specific set of panels, called basis panels, such that all panel charges can be uniquely represented as linear combinations of charges on those panels. The coefficient matrix of those linear combinations is called the structure matrix J R Nxn, where n is the number of leaf panels. For a particular multiple-tree structure, there are many possible bases and each of them has its own structure matrix J and potential coefficient matrix P, which can be expressed in terms of H and J as charges on the ith conductor, P = J T HJ. (5) Then in the new linear system, the q vector in Eq. 3 will represent charges on those basis panels instead of leaf panels. It has been discovered in [3] that the potential coefficient matrix in Eq. 5 is dense when leaf panels are chosen as the basis. On the contrary, if all root panels and left hand side panels are chosen as the basis, it is provable that the P matrix related to this basis is sparse and contains O(n) non-zeros. Therefore, equations in Eq. 3 can be efficiently solved by preconditioned Krylov subspace solvers in linear time. 3. STOCHASTIC CAPACITANCE EXTRAC- TION USING PFA In this section, the details of solving statistical system equation, modeling of surface fluctuation due to process variation and PFA method are described. 3.1 Variational Capacitance Approximation Assume for now that process variations induce some perturbations in the nominal potential coefficient P kl between panel k and panel l in Eq. 2, and the variational potential coefficient P can be represented in terms of the nominal value P and normal random variables δ = [δ 1 δ 2... δ k ] T as P = P + X J T H i Jδ i + X J T H ij Jδ iδ j, i i,j {z } P where P i = J T H i J and P ij = J T H ij J. P is the potential coefficient matrix without considering the process variations, and P, which is the summation of the second and third terms in Eq. 6, represents the variational part of (6)
In [5], the spatial variation between these panels are modeled by Gaussian correlation function: Γ ij = e x i x j 2 /η2, (11) Figure 1: Conductor with position perturbations on leaf panels where e is Euler constant and η is user specified correlation length. x i and x j are the centers of leaf panels i and j, respectively. Then the correlation matrix can be written as P. Let q denote the variational charge distribution vector, q is expressed in a quadratic form, such that Γ( ñ) = (Γ ij) n n (12) q = q + X q i δ i + X q ij δ iδ j, i i,j {z } q where q, q i, q ijr n 1. From Eq. 7, it is clear that the quadratic expressions of self and coupling capacitances can be easily obtained by using Eq. 4. From Eq. 6 and Eq. 7, the variational linear system can then represented as (7) (P + P)(q + q) = v. (8) Substituting the normal equation in Eq. 3 into Eq. 8 and applying T aylor expansion (up to second order terms), q can be expressed as q = Aq + A 2 q, A = P 1 P. Let the quadratic form representation of the first term on the right hand side of Eq. 9, q 1, to be q 1 = X i q i 1δ i + X ij q ij 1 δ iδ j. By using Eq. 6 and P q 1 = Pq, we can get P q i 1 = P i q, P q ij 1 = P ij q (9) (10) Therefore, the quadratic expression of q 1 can be calculated by solving (k + k 2 ) linear systems. Since P is sparse, each linear system in Eq. 10 can be efficiently solved by preconditioned iterative methods with O(n) complexity. So the total complexity of solving q 1 is O((k 2 + k)n). 3.2 Process Variation Modeling After the hierarchical panel discretization process, the positions of those most delicate panels, leaf panels, may be varying due to process variations. The surface fluctuation of a conductor can be described as a statistical perturbation on each nominal leaf panel smooth surface along its normal direction as shown in Fig. 1. The leaf panel position variation is modeled as a random variable vector ñ, where the random perturbation on the i th leaf is denote as ñ i and the variance-covariance matrix Σ of ñ can be written as Σ( ñ) = (Γ ijσ iσ j) n n (13) where σ is the variance for a particular leaf panel. 3.3 Random Variable Reduction Since only a portion of the random variables have significant effects on the conductor surface fluction, principle factor analysis is used to reduce number of random variables in [5]. The orthogonal principle factor analysis (OPFA), also referred to as principle component model, assumes that µ(d) = 0, Σ(d) = I. After applying PFA on to the covariance matrix, Σ, it can be represents as Σ( ñ) = L δ. (14) where L is a loading matrix L R n k and is given by L = [λ 1e 1λ 2e 2...λ k e k ], (15) where λ and e are the eigen pair of Σ( ñ). It should be noted that the number of random variable has been reduced from n to k using this method. In [5], it is shown that about 90% of the total random variables can be compressed/discarded with minor error in accuracy. 4. ORTHOGONAL POLYNOMIAL METHOD In this section, we will briefly review orthogonal polynomial based stochastic analysis methods that introduced in [1]. 4.1 Hermite Polynomial Basis First, we can consider a random variable ξ(θ) which is expressed as a function of θ that is the random event. Hermite Polynomial utilizes a series of orthogonal polynomials (with respect to the Gaussian distribution) to facilitate stochastic analysis. So, these polynomials are used as orthogonal basis to decompose a random process. With the Hermite Polynomials, if given a random variable v(t, ξ) with variation, where ξ = [ξ 1,..., ξ n] denotes a vector of orthonormal Gaussian random variables with zero mean,
the random variable can be approximated by a truncated Hermite PC expansion as follows: v(t,ξ) = PX a k Hk n (ξ) (16) k=0 where n is the number of independent random variables, H n k (ξ) are n-dimensional Hermite polynomials, and a k are the deterministic coefficients. The Hermite polynomials are orthogonal with respect to Gaussian weighted expectation, which can be expressed with following: < H i(ξ),h j(ξ) >=< H 2 i (ξ) > δ ij (17) where δ ij is the Kronecker delta and <, > denotes an inner product. Thus, in order to determine the orthogonal expansion coefficients a k, the orthogonality can be used to evaluate by the projection operation onto the Hermite Polynomial basis: a k (t) = < v(t, ξ), H k(ξ) > H 2 k (ξ) (18) 4.2 Extraction With Hermite Polynomials With Hermite Polynomials, we can present the capacitance extraction with them considering the process variation in this section. or CG. As such, the capacitance extraction under process variation is to fix the orthogonal expansion of q as: q(ξ) = q 0 + KX q ih i(ξ) (23) Once the Hermite Polynomial Expansion of q(ξ) is known, the mean and variance of q(ξ) can be evaluated trivially. Given an example, for one random variable, the mean and variance are calculated as: i=1 E(q(ξ)) = q 0 (24) V ar(q(ξ)) = q 2 1 + 2q 2 2 (25) 5. EXPERIMENT In this section, we compare the results of both PFA and OP based methods against the Monte Carlo method. The two approaches are tested on 2x2 bus test case with 352 panels, which is shown in Fig.2. Also, the standard deviation is set as 10% of the wire width and the η, the correlation length, as 200of the wire width. 4.2.1 Process Variation Modeling For each panel, we assume there is a perturbation n i along its normal direction, which can describe the geometric variation on conductor surface. Also, the potential coefficient matrix P can be approximated with its integral kernel, when panels are far away (their distance is much larger than the panel area). We can have the following approximation: P ij G(x i, x j), i j (19) G(x i, x j) = 1 x i x j (20) Therefore, the approximation for P ij can be obtained: nx P ij(ξ) = p k ξ k (21) k=1 As we have mentioned, the coefficients p k will be evaluated with orthogonal property in Equa.(18). 4.2.2 OP based Stochastic Capacitance Extraction The nominal system Pq = v can be converted to P(ξ)q(ξ) = v, and potential coefficient matrix: P(ξ) = P 0 + P 1 = P 0 + PX p 1iH i(ξ). (22) In this way, the perturbated system P(ξ)q(ξ) = v will be solved with Iterative Linear System Solver, such as GMRES i=1 Figure 2: 2x2 Bus Test Example. 5.1 PFA based method First, for the 2x2 bus crossing problem, probability density functions (PDF) obtained from the canonical linear model and the quadratic model are shown in Fig.3 and compared with that from Monte Carlo simulation. It is illustrated that there is a significant accuracy improvement by using the second order quadratic model instead of the canonical model. In Table 1, the run times of Monte Carlo method and the quadratic model with 10 dominant factors for 2x2 bus crossing benchmarks are compared. It is clear that the quadratic model exhibits over 100X speedup compared with Monte Carlo simulation. Statistical distribution-related parameters, such as mean value, standard deviation, and skewness are normally within 3% errors. Combined with the results from previous experiments, We can safely conclude that, currently, the second order approximation is accurate enough for variational parasitic capacitance modeling. In the second experiment, the PDFs of the second order quadratic models with different number of dominant factors
Figure 3: First and second order capacitance models and their comparisons with Monte Carlo method for the bus 2x2 benchmark (δ = 20%). Table 1: Simulation Comparison between MC and PFA method. Method Time Mean µ std variation δ skewness η MC 1826-78.56 106.01 1.868 QualMod 9.78-81.43 103.64 1.927 Speedup/Err 186.7X 3.7% 2.2% 3.2 are compared in Fig. 4. In this test, PFA with only ten factors is very close to the result PDF from Monte Carlo simulation, so that ninety percent random variable reduction has been achieved by PFA. And in this case, the error compared with Monte Carlo is less than 3%. Furthermore, as the number of factors increases, the PDFs from the quadratic models quickly converge to those from Monte Carlo simulation. 5.2 OP based method First, we compare the CPU times of the two methods. The results are summarized in Table 1, where MC(1000) means that 1000 runs used in MC method.it can be seen that OP method (268 second) is much faster than Monte Carlo method (286620 second), which can deliver about three orders of magnitude speedup over Monte Carlo method. Next, we perform the accuracy comparison. For 2x2 bus case, we carry out 6000 times runs for Monte Carlo simulation. The results are summarized in Table 2. 6. CONCLUSION In this paper, we survey some papers about stochastic capacitance extraction considering process variation. Especially, we explain the main flows of Principle Factor Analysis and Table 2: Capacitance mean values for the 2x2 bus. 2x2 Bus, Panel num=352, δ = 0.1, η = 2 MC OP C11 242.54 242.41 C12-82.46-82.48 C13-47.48-47.34 C14-47.42-47.33 C22 242.34 242.07 C33 242.52 241.99 C44 242.47 242.02 Figure 4: Second order parasitic capacitance modeling with different number of factors and the comparison with Monte Carlo method for bus 2x2 benchmark. orthogonal polynomial methods, which involve variational geometric modeling, perturbated system building and solution. Other related methods base on these approaches to improve the accuracy or speedup. Experiments show that both methods are many orders of magnitude faster than the Monte Carlo method. 7. REFERENCES [1] J. Cui, G. Chen, R. Shen, S. Tan, W. Yu, and J. Tong. Variational capacitance modeling using orthogonal polynomial method. In in Proc. 18th ACM Great lake s symposium on VLSI., pages 23 28. ACM, 2008. [2] H.Zhu, X.Zeng, W.Cai, J.Xue, and D.Zhou. A sparse grid based spectral stochastic collocation method for variations-aware capacitance extraction of interconnects under nanometer process technology. In in Proc. European Design and Test Conf. (DATE), pages 1514 1519. IEEE, 2007. [3] R. Jiang, Y.-H. Chang, and C.-P. Chen. Iccap: A linear time sparse transformation and reordering algorithm for 3d ben capacitance extraction. In in Proc. Design Automation Conf. (DAC), pages 163 166. IEEE, 2005. [4] A. Labun. Rapid method to account for process variation in full-chip capacitance extraction. IEEE Trans. on Computer-Aided Design of Inregrated Circuits and Systems, 23:941 951, June 2004. [5] R.jiang, W.Fu, J. Wang, V.lin, and C.-P. Chen. Efficient statistical capacitance variability modeling with orthogonal principle factor analysis. In in Proc. Int. Conf. on Computer Aided Design (ICCAD), pages 683 690. IEEE, 2005. [6] Y.Zhou, Z.Li, Y.Tian, W.Shi, and F.Liu. A new methodology for interconnect parasitics extraction considering photo-lithography effects. In in Proc. Asia South pacific design Automation Conf. (ASPDAC), pages 450 455. IEEE, January 2007. [7] Z.Zhu and J.White. Fastsies: a fast stochastic integral equation solver for modeling the rough surface effect. In in Proc. Int. Conf. on Computer Aided Design (ICCAD), pages 675 682. IEEE, 2005.