Generalized Stochastic Collocation Method for Variation-Aware

IEICE TRANS. ELECTRON., VOL.E92–C, NO.4 APRIL 2009

508

PAPER

Special Section on Low-Leakage, Low-Voltage, Low-Power and High-Speed Technologies for System LSIs in Deep-Submicron Era

Generalized Stochastic Collocation Method for Variation-Aware Capacitance Extraction of Interconnects Considering Arbitrary Random Probability Hengliang ZHU† , Student Member, Xuan ZENG†∗a) , Member, Xu LUO† , and Wei CAI†† , Nonmembers

SUMMARY For variation-aware capacitance extraction, stochastic collocation method (SCM) based on Homogeneous Chaos expansion has the exponential convergence rate for Gaussian geometric variations, and is considered as the optimal solution using a quadratic model to model the parasitic capacitances. However, when geometric variations are measured from the real test chip, they are not necessarily Gaussian, which will significantly compromise the exponential convergence property of SCM. In order to pursue the exponential convergence, in this paper, a generalized stochastic collocation method (gSCM) based on generalized Polynomial Chaos (gPC) expansion and generalized Sparse Grid quadrature is proposed for variation-aware capacitance extraction that further considers the arbitrary random probability of real geometric variations. Additionally, a recycling technique based on Minimum Spanning Tree (MST) structure is proposed to reduce the computation cost at each collocation point, for not only “recycling” the initial value, but also “recycling” the preconditioning matrix. The exponential convergence of the proposed gSCM is clearly shown in the numerical results for the geometric variations with arbitrary random probability. key words: variation-aware capacitance extraction, geometric variations, generalized polynomial chaos

1.

Introduction

Parasitic extraction technique used to build the equivalent RLC circuit model for interconnects is based on numerical methods solving Maxwell equations [1], and in most case, assuming the shape of interconnects is ideally cuboid. When it comes to 45 nm technology node and beyond, the geometric variations of interconnects, both systematic and random, can no longer be ignored [2]. The systematic variations, of which the variability is deterministic, are mainly layout dependent [2]. This kind of variations can be handled by combining the parasitic extraction method with the lithography or CMP (Chemical Mechanical Polishing) simulation tools [3]. Apart from the systematic variations, a great portion of geometric variations are random in nature. These random variations are mostly due to the uncertainty of processes [2]. Unlike the systematic variations, interconnect parasitic model with random variations is difficult to capture with the traditional parasitic extraction tools. So far, two Manuscript received August 4, 2008. Manuscript revised November 14, 2008. † The authors are with the State Key Lab. of ASIC & System, Microelectronics Dept., Fudan Univ., Shanghai, P.R. China. †† The author is with the Department of Mathematics, University of North Carolina at Charlotte, USA. ∗ Corresponding Author is Xuan Zeng. a) E-mail: [email protected] DOI: 10.1587/transele.E92.C.508

Fig. 1

Histogram of line width distribution.

methods, namely perturbation method in [4] and stochastic collocation method (SCM) in [5], have been developed for parasitic extraction to model random geometric variations of on-chip interconnects. In general, the geometric variations are modeled as Gaussian random fields. [6] proposed an efficient nonsampling method named FastSies for capacitance extraction with the random field model representing the off-chip rough surface. For capacitance extraction of on-chip interconnects, a high-order polynomial model is generally required representing the nonlinear dependency of parasitic capacitances on multiple variation sources [4], [5]. Generally, Principal Factor Analysis (PFA) [4] or K-L expansion [5] can be used to transform the geometric variations into a set of independent random variables. These methods in [4] and [5] develop a polynomial model with respect to these random variables for capacitance parameters to capture the effect of geometric variations. Perturbation method [4] is based on the Taylor expansion, which is limited to small variation applications. On the other hand, SCM [5] is based on the Homogeneous Chaos expansion. Taking advantage of the exponential convergence rate of stochastic spectral method, SCM is very promising for the variation-aware capacitance extraction of interconnects. Both of the above two methods are based on the Gaussian assumption of the geometric variations, mostly due to the ubiquitousness of Gaussian distributions in practice. However, this is not always true in IC fabrication. Some of the process variations do show severe deviation from the Gaussian distribution [7]. Figure 1 is an example of the statistical distribution of the line width, simulated by Synopsys Progen [8] and ICWB [9] lithography simulation tool, using a 128-bit bus interconnect structure as the test case. The

c 2009 The Institute of Electronics, Information and Communication Engineers Copyright

ZHU et al.: GSCM FOR VARIATION-AWARE CAPACITANCE EXTRACTION

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data are gathered from the Monte Carlo sampling of process parameters including defocus, NA (Numerical Aperture), dose, etc. It is shown that the probability density function (pdf) of line width is asymmetric and is quite different from the standard Gaussian random distribution with the same mean value and variance. In this case, the random distribution of line width is better modeled by a Gaussianlike random distribution as formulated in (1), which has two different variances σ1 and σ2 at two sides of the mean value m0 . ⎧ (x−m0 )2 ⎪ − ⎪ ⎪ 2 2σ2 ⎪ 1 , √ x < m0 ; ⎪ ⎨ 2π(σ1 +σ2 ) e pd f (x) = ⎪ (1) (x−m0 )2 ⎪ ⎪ − ⎪ ⎪ 2σ2 ⎩ √ 2 2 , e x ≥ m . 0 2π(σ +σ ) 1

2

Gaussian assumption for the geometric variations like (1) could lead to large errors of the resulting capacitance parameters, as will be illustrated in the numerical results. We should note that the distribution of a real geometric variation could be arbitrary, which is neither Gaussian nor a closed form as in (1). Consequently, it is very desirable for a robust variation-aware parasitic extraction of interconnects to tackle the real geometric variations, whose random probability is arbitrary. It has long been recognized that, in order to achieve the exponential convergence rate of stochastic spectral method, different random processes should be represented by different orthogonal polynomials according to the property of the processes. In 1938, Weiner first proposed the Homogenous Chaos [5], i.e. Hermit Polynomial, for Gaussian process. Later on Charlie Polynomial was developed by Ogura to represent the Poisson process. Wiener-Askey Polynomial Chaos† was proposed in [10], which uses Askey scheme to define the orthogonal polynomials for most of the typical random processes including Gaussian, gamma, beta, Poisson, etc.. More recently, generalized polynomial chaos (gPC) [11] was constructed to represent arbitrary random probability which neither has explicit formula nor belongs to the typical random processes listed above. It has been proved that each Wiener-Askey Polynomial Chaos or gPC can achieve optimal (exponential) convergence rate for its corresponding random process [10], [11]. In this paper, the stochastic collocation method (SCM) in [5] is extended to the generalized stochastic collocation method (gSCM), in order to accommodate the arbitrary random probability of geometric variations. The algorithm can use the real test data of geometric variations as the inputs, applies K-L expansion and Second Order Decomposition (SOD) to reduce the number of random variables for geometric variations and finds the most “independent” random variables. It is pointed out in this paper that the geometric variation decomposition could affect the accuracy of latter gSCM. Therefore, SOD is proposed to replace the Independent Component Analysis (ICA) method [7], [12] in order to improve the accuracy of gSCM. The generalized Polynomial Chaos (gPC), as well as the generalized Sparse Grid, is then constructed “on-the-fly” according to the samples of these “independent” random variables, and is used

instead of Homogeneous Chaos or Wiener-Askey Polynomial Chaos in order to achieve the exponential convergence rate for arbitrary random probability. Moreover, the beauty of SCM is inherited by gSCM that it only requires solving deterministic problems at some collocation points to build the gPC expansion model of capacitances. Furthermore, a recycling technique based on the Minimum Spanning Tree (MST) in [5] is developed for the “recycling” of initial values and preconditioning matrices, further reducing the computation cost. The rest of this paper is organized as follows. Background for reviewing SCM is presented in Sect. 2. The gSCM is proposed to handle the arbitrary random probability by introducing K-L expansion and SOD approach in Sect. 3, and by using gPC and generalized Sparse Grids in Sect. 4. Recycling technique for SCM is further proposed for the “recycling” of preconditioning matrices in Sect. 5. Numerical results are provided in Sect. 6. Finally, the conclusions are drawn in Sect. 7. 2.

Background

In this section, we first formulate the problem of capacitance extraction of on-chip interconnects with geometric variations, and then briefly review SCM for variation-aware capacitance extraction with Gaussian geometric variations. 2.1 Problem Definition Capacitance extraction of interconnects is generally based on the Boundary Element Method [1], solving the following integral equation, 1 (2) q(r )da = v(r ), r − r | sur f aces 4πε0 | where q(r ) is the charge distributed on the interconnect surfaces, v(r ) is the potential and 4πε01|r−r | is the free space Green’s function. When geometric variations are considered, the geometric parameters of interconnects including CD (Critical Dimension), metal layer thickness, ILD (Inter-Layer Dielectric) and etc. are modeled as random fields [4]–[6], and (2) becomes a stochastic integral equation (SIE), which can be written in the form of [4]–[6] P(Δr1 , · · · , Δrm ) · q(Δr1 , · · · , Δrm ) = v,

(3)

where the potential coefficient matrix P and the charge density q are both functions of the m random variables {Δri }m i=1 representing the fluctuation of panels due to the variations † In some papers, Wiener-Askey Polynomial Chaos is also known as generalized Polynomial Chaos. In this paper, we would like to distinguish the generalized Polynomial Chaos, which uses three-term recurrent formula to construct the orthogonal polynomials for arbitrary random probabilities, from the Wiener-Askey Polynomial Chaos, which uses Askey scheme only for some typical random processes.


510

∂R = ∂ci1 ,··· ,id = 0, j1 +···+ jd ≤2.

of geometric parameters. 2.2 Stochastic Collocation Method (SCM) SCM [5] solves the variation-aware capacitance extraction problem with two steps. K-L expansion or Principal Component Analysis (PCA) is first applied to find the major variations of geometric variations, denoted by ξ as (4) Δr1 , · · · , Δrm T = L ξ1 , · · · , ξd T . The equation in (3) can then be reformulated using ξ as the random input, P(ξ1 , · · · , ξd ) · q(ξ1 , · · · , ξd ) = v.

(5)

After that, a quadratic Homogeneous Chaos expansion w.r.t. the principal components ξ is employed to model the capacitance parameters, ˆ ξ ) = ci1 ,··· ,id Hdi1 ,··· ,id (ξ ), (6) C(ξ ) ≈ C( i1 +···+id ≤2

where Hdi1 ,··· ,id (ξ ) denotes the d-dimensional Hermite polynomials. The coefficients of the expansion is computed by Sparse Grid quadrature [5] ci1 ,··· ,id =

M

C(ξˆ i )Hdi1 ,··· ,id (ξˆ i )wi ,

i1 +···+id ≤2,

(7)

i=1

based on which, the stochastic problem in (5) is transformed to the deterministic problems that compute the capacitance ˜ ξˆ i ) at some quadrature points [5]. C( Compared with perturbation method, SCM has the exponential convergence rate for solving the variation-aware capacitance extraction problem [5]. Moreover, the solution of SCM in (6) and (7) is actually the optimal polynomial model that the residual is minimal in the Mean Square Error sense. Theorem 1 The Homogeneous Chaos expansion in (6) with its expansion coefficients computed in (7) is the optimal polynomial model that minimizes the Mean Square Error (MSE) ˆ ξ ), C(ξ ) − C( ˆ ξ ) >, R =< C(ξ ) − C( where < ·, · > is the inner product defined as follows, < f (ξ ), g(ξ ) >= f (ξ )g(ξ )ρ(ξ )dξ.

(8)

(9)

ρ(ξ ) is the probability density function (pdf) of ξ. Theorem 1 can be proved by applying the Least Square Method. Actually, all second-order polynomial models can be written in the form of (6). To determine the best one that minimizes the MSE in (8), the coefficients of Hermite polynomials have to be calculated using Least Square Method, which leads to

(10)

Based on the fact that Hermite polynomial is the orthogonal polynomial for the Gaussian pdf ρ(ξ ), Eq. (10) can actually be reduced to ci1 ,··· ,id =< C(ξ ), Hdi1 ,··· ,id (ξ ) >,

j1 +···+ jd ≤2.

(11)

By applying the Sparse Grid quadrature, an optimal solution in (11) can be computed by (7) [5]. Therefore, it is clear that the property of optimal approximation for SCM as well as its exponential convergence rate depends on the orthogonalization of Hermite polynomial and the accuracy of Sparse Grid quadrature. However, when geometric variations are measured from real test chips, they are not necessarily Gaussian. Hermite polynomial is not the orthogonal polynomial in the non-Gaussian random space. Furthermore, the Sparse Grid quadrature based on Hermite polynomial [5] will be significantly inaccurate to calculate the integration in (11). SCM can no longer produce the optimal solution that achieve the exponential convergence rate for capacitance extraction problem with the real geometric variations. Unsatisfied with this limitation, in this paper, we propose a generalized stochastic collocation method (gSCM) for variation-aware capacitance extraction, which achieves exponential convergence rate when the arbitrary random probability of geometric variations is considered. Three important issues are addressed in the following that constitute the major parts of gSCM algorithm. 1) How to calculate the principal components ξ from the real test data of geometric variations that are generally independent; 2) how to define the generalized Polynomial Chaos (gPC) instead of Homogeneous Chaos as the orthogonal polynomial of the arbitrary random space; 3) and how to define the generalized Sparse Grid quadrature to calculate the inner product < ·, · > in (9) with arbitrary random probability. 3.

Geometric Variation Modeling

In general, the geometric variations of interconnects include CD variation, metal layer thickness variation and ILD variation. These geometric variations are generally modeled as random fields in variation-aware capacitance extraction [4], [5], as well as Statistical Static Timing Analysis (SSTA) [7], [13]. In order to characterize the random field, test data of geometric variations are measured from different locations on the test chips [14], which can be denoted as h1 , · · · , h M for M measurements of the geometric variations h = [h(r1 ), · · · , h(r p )]T that are located at p different positions of the same test chip. These raw data cannot be directly used in capacitance extraction. Instead, principal components ξ are calculated from measured data by K-L expansion or PCA [4], [5], for the reasons that 1) the number of principal components ξ


511

is generally much smaller than that of random variables in h, which improve the computation efficiency of the algorithm; and 2) the principal components ξ are independent for Gaussian random geometric variations, which are very important to define the orthogonal polynomials and the Sparse Grid quadrature and achieves exponential convergence rate in SCM. The idea of K-L expansion is based on the truncated eigen decomposition of the correlation matrix < h h T >† to find the major variation source of random field h. It can be formulated as

h ≈ E · diag λ1/2 · ζ, (12) where λ ∈ Rd is the d largest eigenvalues of the correlation matrix < h h T >, and E ∈ R p×d is the corresponding eigenvectors (d p). And conversely,

ζ = diag λ−1/2 · ET · h. (13) It has been proved that K-L expansion in (12) is the best root mean square approximation of the random field h [15]. Furthermore, the new random variables ζ are proved to be uncorrelated, which are independent when only considering Gaussian random geometric variations. However, when the geometric variations are measured from real test data, which are generally non-Gaussian, additional factorization is needed to transform the random variables ζ to independent components ξ. To the best of authors’ knowledge, ICA [12] is the best option that has been well studied for many years to find the “most independent” components ξ based on the linear assumption ζ = AT · ξ,

(14)

and can be formulated as the following optimization problem [12], minimize: I(ξ; A) = di=1 H(ξi ) − H(ξ ), (15) s.t.: AAT = I. There are several definitions of the objection function I(ξ; A) in ICA. The most popular one is the mutual information of random vector ξ, defined by the differential entropy H of random variables as in (15) [12]. FastICA [12] uses Newton method to solve the above optimization problem provided the samples of ζ, say ζ1 , · · · , ζ M , which are calculated from the measured data h1 , · · · , h M using (13). Generally, the Gaussian components in ξ are forbidden in ICA for the reason that it might compromise the convergence [12]. Nevertheless, this is not a problem in practice, since we can set up a maximum iteration number for ICA, which is already implemented in the new version of FastICA [12]. Due to the limitation of the space, the details of ICA method are not discussed in the paper. One important observation is that the linear assumption in (14) is generally unsatisfied. As a result, the “independent” components obtained from ICA are not “exactly

independent.” In this case, ICA seems a little bit heuristic. A more proper definition for the objection function I(ξ; A) is needed to find the “independent” components that may not be exactly independent but still can guarantee the exponential convergence rate of gSCM. Generally, the independence of random variables ξ can be formulated as that, for any function f and g, f (ξi ) and g(ξ j ) are uncorrelated, i.e. < f (ξi ), g(ξ j ) >=< f (ξi ) >< g(ξ j ) >, i j.

(16)

Noting that only a second order polynomial form of f (ξi ) and g(ξ j ) is used in a quadratic model for capacitance extraction, we proposed to apply the following objection function F to find the most independent components ξ. 2 minimize: F = k,l=1,2 i j Jkl (ξi , ξ j ) , (17) s.t.: AAT = I, where Jkl (ξi , ξ j ) =< pk (ξi )pl (ξ j ) > − < pk (ξi ) >< pl (ξ j ) >, (18) and p1 (x) = x and p2 (x) = x2 . Since the solution of (17) minimizes the correlation between second order polynomials of ξ, we call this the Second Order Decomposition (SOD). When higher order polynomials are employed in gPC, the SOD can be simply extended to include higher order terms in (17). The accuracy of SCM largely depends on the orthogonality of gPC as defined in (21), which is actually based on the uncorrelatedness between gPC of ξ. On the other hand, for correlated gPC, large errors will also be introduced in the numerical quadrature, and consequently compromise the exponential convergence rate of SCM. Therefore, the above proposed post-decomposition method, minimizing the correlation between second order polynomials, is designed directly to pursue the uncorrelatedness of gPC for a quadratic gPC expansion. Compared with the ICA problem in (15), SOD can find a more proper matrix A that the definition of multi-dimensional gPC in (21) is more close to delta function, and consequently, improve the accuracy of gSCM. 4.

Generalized Stochastic Collocation Method

Based on the above geometric variation modeling, we can rewrite (3) using the independent components ξ as the random input as in (5), which is very similar to the original SCM [5]. Since the geometric variations within each grid cell on the test chip are assumed to be uniform [14], we can calculate the geometric variations of panels {Δri }m i=1 from geometric variations h using linear interpolation, Δr1 , · · · , Δrm T = Lh. †

The mean of h is assumed to be zero.

(19)


512

Substituting (12) and (14) into (19), {Δri }m i=1 can be represented as a linear combination of ξ,

(20) Δr1 , · · · , Δrm T = LE · diag λ−1/2 · AT ξ. Therefore, the variance-aware capacitance extraction problem can now be reformulated using the independent com ponents ξ instead of Δr1 , · · · , Δrm T as the random input in (5). With the samples of “independent” components ξ, we are now able to construct the multi-dimensional gPC “on the fly” using the well-known three-term recurrent formula, build the corresponding generalized Sparse Grids, and extend SCM to gSCM for variation-aware capacitance extraction problem with the arbitrary random probability of real geometric variations. 4.1 Generalized Polynomial Chaos (gPC) The gPC is a set of orthogonal polynomial bases of the random space [10]. < Hdi1 ,··· ,id (ξ ), Hdj1 ,··· , jd (ξ ) >= δi1 ,··· ,id ; j1 ,··· , jd .

(21)

Here the subscript i1 , · · · , id denotes the maximum order of Hdi1 ,··· ,id (ξ ) for each dimension, and the inner product < ·, · > in (9) should be redefined based on the test data in order to take the probability of geometric variations into consideration. M j j j=1 f (ξ )g(ξ ) < f, g > , (22) M where {ξ j } M j=1 is the samples of ξ calculated from (12) and (14) using the samples of h. Instead of using Askey scheme for some specific random probability, we employ the well known three-term recurrent formula to construct the orthogonal polynomial for each dimension [11]. H1k+1 (ξi ) = (ξi − αk )H1k (ξi ) − βk H1k−1 (ξi ),

(23)

with the initial value H1−1 = 0 and H10 = 1. The coefficients αk and βk are determined by αk =

< ξi H1k , H1k > < H1k , H1k >

, βk =

< H1k , H1k > < H1k−1 , H1k−1 >

,

(24)

where the inner product is calculated using the samples {ξ j } M j=1 , as denoted in (22). Based on the uncorrelatedness of gPC of ξ, we can construct the orthogonal polynomials for each dimension separately. The gPC can then be obtained by tensor product of these one-dimensional orthogonal polynomials, in which the orthogonality of gPC is guaranteed. Hdi1 ,··· ,id (ξ )

=

H1i1 (ξ1 ) · · · H1id (ξd ).

(25)

Note that the orthogonal polynomials used in (25) for each

dimension are not necessarily the same. We can build different types of orthogonal polynomials according to the distribution of random variables ξi . For arbitrary random probability, which may not fall into the Askey scheme, gPC can achieve the exponential convergence rate, while Wiener-Askey Polynomial Chaos (including Hermite polynomial) fail to. Therefore, we use gPC to discretize the unknown charges as in (6) where the Hermite polynomials Hdi1 ,··· ,id (ξ ) are now replaced by the gPC in (25). The coefficients ci1 ,··· ,id are calculated by gSCM using generalized Sparse Grid, as will be discussed in the following. 4.2 Generalized Sparse Grids Sparse Gird constructs the multi-dimensional collocation points based on the linear combination of tensor product [5] for computing the multi-dimensional integration in (9) or (22) (Θl11 × · · · × Θl1d ), (26) Θkd = k+1≤l1 +···+ld ≤d+k

where Θl1i denotes the set of collocation points for the ith dimension, with the li level of accuracy. The formula is actually so general that any selection of one-dimensional collocation points can be applied. [5] employs the Gauss-Hermite quadrature rule for Θl1i , i = 1, · · · , d that achieves the highest polynomial exactness. This can be generalized by using the roots of orthogonal polynomials in (23) as the collocation points for each dimension. Θl1i = {ξî1 , · · · , ξîli } := roots of H1li (ξi ),

(27)

where ξîj ( j=1,··· ,li ) denotes the jth collocation point for one dimensional li -level Gaussian quadrature rule using the roots of H1li (ξi ) [16]. The generalized Sparse Grid based on (27) inherits the merit from the original Sparse Grid, i.e. the minimum number of collocation points and the highest polynomial exactness. It is readily proved that, the generalized Sparse Grid has the same number of collocation points as the one used in [5], which is several orders less than that of the Monte Carlo method and the direct tensor product method [5]. Moreover, the generalized Sparse Grid based on Θkd has 2k + 1 polynomial exactness as guaranteed by Theorem 2. Theorem 2 One-dimensional Gaussian quadrature rule based on collocation points Θl1i has the 2li − 1 polynomial exactness for integration with probability density function ρ(ξi ) [16]. Thereby, the d-dimensional k-level Sparse Grid Θkd constructed by (26) can have the 2k + 1 polynomial exactness [17]. The proposed variation-aware capacitance extraction method for arbitrary random probability is summarized as follows. Based on the samples of independent components ξ, the one dimensional orthogonal polynomial H k (ξi ) can 1


513

be generated “on-the-fly” by three-term recurrent formula in (23) for each dimension. The gPC is a tensor product of H1k (ξi ) as in (25), which is used to replace the Homogeneous Chaos in [5]. The generalized Sparse Grid is constructed using the roots of H1k (ξi ) as in (26) and (27), which has the same number of collocation points and polynomial exactness as the original one in [5], but is designed for arbitrary random probability. The unknown charges can be approximated by a gPC expansion as in (6) with the Hermite polynomials replacing by the gPC. In order to calculate the coefficients of the expansion, capacitances are calculated by solving the deterministic problems (5) at generalized Sparse Grids. Finally, the coefficients are computed using (7). 5.

Recycling Technique

In collocation method, Eq. (5) has to be solved at N collocation points of Sparse Grids. The results of deterministic problems in (5) at the previous nearest collocation point can be used to further reduce the computation cost at the latter collocation points. We call this the “recycling technique.” A Minimum Spanning Tree (MST) structure is proposed in [5] as a schematics of the similarity of Sparse Grids, and is used to “recycle” the solution of the parent node as the initial value of GMRES for (5) at the child node. It is observed that the pre-setup time of GMRES at each collocation point, including the computation time for the preconditioning matrix, actually takes the most part of the total computation time. As a consequence, it would be more helpful to extend the idea in [5] to include the “recycling” of the preconditioning matrix to further reduce the total number of the preconditioning matrix. Figure 2 is a simple example of MST structure for 8point Sparse Grids. The idea of preconditioning matrix recycling is to share the preconditioning matrix at the ancestor node with some of its descendants. For example, in Fig. 2, we calculate the preconditioning matrix every L levels. Say L = 2, that is to calculate the preconditioning matrices for ξˆ 0 at the 0th level, and ξˆ 6 , ξˆ5 , ξˆ 2 at the 2nd level. Then at node ξˆ 1 and ξˆ7 , the preconditioning matrix of node ξˆ 0 is used. Similarly, ξˆ 3 and ξˆ 4 share the preconditioning matrix of node ξˆ 5 . Furthermore, we can set a lower bound LB for the number of descendants. For those nodes with the number of descendants less than LB, we further reduce the computation cost by using the preconditioning matrix of their ancestors. For example, since node ξˆ 6 and ξˆ 2 in Fig. 2 have no descendants, preconditioning matrix of node ξˆ0 is used for these two nodes. Therefore, totally, only two precondi-

Fig. 2

MST for Sparse Grid: an example.

tioning matrices at ξˆ0 and ξˆ5 are computed for these eight collocation points. 6.

Numerical Results

In this section, a 3D 3-bit bus structure in Fig. 3 is used to validate the accuracy and efficiency of methods for variation-aware capacitance extraction. The conductors of 3-bit bus are of 20 μm long, with 0.3 μm line width and 0.53 μm metal height. Monte Carlo method with 10,000 samplings is used as the benchmark of the experiments. The accuracy and efficiency of gSCM is evaluated by the mean values and variances of capacitances C11 , C12 and C22 of the 3-bit bus, where C11 and C22 are the self-capacitance of the first and second conductor respectively, and C12 is the mutual-capacitance of these two conductors. For solving the deterministic problem in (5) at each collocation point, MIT pfft++ package [1] is applied as a fast solver. Line-width variation and metal-thickness variation are considered in the numerical experiment, and are modeled by homogeneous isotropic random fields [14]. The asymmetric random probability based on the lithography simulation, as defined in (1), is used to model the variations of the line width caused by the lithography process. For simplicity, we further assume that the variation of the metal thickness caused by CMP process follows the similar random distribution but has the different mean value and variance. In a real application, the variation of the metal thickness should be measured from test chips. Considering that the variance σ of geometric variations in Fig. 1 is about 10% of the nominal value, the 3σ range of geometric variations is chosen as 30% of the dimension of conductors in the test case. Specifically, the mean value m0 , σ1 and σ2 of line width are set as 0.3 μm, 0.021 μm and 0.039 μm, respectively; while the mean value m0 , σ1 and σ2 of metal height are set as 0.53 μm, 0.030 μm and 0.068 μm, respectively. The Gaussian correlation function [4], [5] is used. Considering the 30 μm planarization length in CMP [18], the correlation lengths are set as 15 μm and 25 μm, respectively, for the line-width variation and the metal-height variation. In this simple example, since line-width variation and metal-thickness variation are caused by different processes, we further assume the random fields for modeling these geometric variations are independent, and apply K-L expansion to each of these random fields. By preserving the eigen components that have the eigenvalues no less than 1% of the largest eigenvalue in (12), the random fields are reduced

Fig. 3

3-bit bus test case (μm).


514 Table 1 KL ICA SOD

F 0.041 0.037 0.008

Correlation of the polynomials of ξ. maxi, j J12 (ξi , ξ j ) 0.316 0.370 0.121

maxi, j J22 (ξi , ξ j ) 0.119 0.089 0.085

C11 C22 C12 C11 C22 C12

Fig. 4

Variance errors of C11 , C12 and C22 in 3-bit bus case.

to 8 random variables. SOD in (17) is further applied to find the “independent” components from these random variables. The gSCM uses the gPC model w.r.t. these “independent” components to approximate the capacitances, and computes the coefficients of the gPC model by solving the deterministic problems at generalized Sparse Grids of these “independent” components. In the following, the accuracy and efficiency of SOD and gSCM are demonstrated. 6.1 Efficiency of Post-Decomposition The efficiency of SOD is validated in Table 1, where F is the object function in (17), and Jkl (ξi , ξ j ) is defined in (18). It is shown that SOD proposed in this paper efficiently reduces the correlation between second-order polynomials of random variables ξ. Benefited from this, gSCM using SOD has much better convergence rate compared with the ones using ICA or K-L expansion. The convergence rate of gSCM based on different decomposition approaches is illustrated in Fig. 4(a), by examining the variance errors when using different polynomial orders of gPC expansion. It shows that the error of gSCM using SOD reduces much faster than those ones using ICA or K-L expansion when the polynomial order of gPC expansion increases. The proposed SOD can improve the accuracy of gSCM very efficiently. 6.2 Exponential Convergence Rate of gPC Based on the results of SOD, the accuracy of the linear and

Table 2 Relative error of 3-bit bus test case. Monte Carlo Relative Errors (10−18 F) Pert(%) SCM(%) gSCM(%) Arb. Gau. lin. qua. lin. qua. lin. qua. mean values 794 791 2.80 1.20 0.56 0.49 0.39 0.21 1197 1192 3.76 1.69 0.78 0.52 0.55 0.20 −564 −561 4.00 1.83 0.78 0.71 0.54 0.31 variances 147.2 133.3 24.5 18.7 20.2 7.2 16.3 4.2 280.8 252.5 25.8 19.8 21.7 8.1 17.6 4.7 141.3 127.1 25.6 19.7 21.5 7.6 17.3 4.4

quadratic capacitance model for different methods is verified by the relative errors of their mean values and variances in Table 2. The exponential convergence rate of gSCM is illustrated in Fig. 4(b). Here “exponential convergence rate” means that the variance errors reduce exponentially when the polynomial order of gPC increases. 1) The abbreviation “Arb.” denotes the Monte Carlo results considering the random probability of geometric variations, while the “Gau.” stands for using Gaussian distribution with the same mean value and variance to approximate the geometric variations. Although in this case the resulting mean values in “Gau.” are very close to those results in “Arb.,” the variances still have large errors of about 10%. It suggests that the random probability of geometric variations should be considered for an accurate model of interconnects. 2) Stochastic spectral methods, including SCM and gSCM, have higher accuracy and better convergence rate compared with the perturbation method. 3) Compared with SCM, gSCM has the optimal (exponential) convergence rate. As shown in Fig. 4(b), SCM based on either K-L expansion or SOD has the worst convergence rate. Taking advantage of gPC, the results of gSCM based on K-L expansion are more accurate than the results of SCM based on either K-L expansion or SOD. Among all, gSCM based on SOD has the best (exponential) convergence rate. The quadratic model of gSCM based on SOD has nearly the same accuracy in the approximation of variances compared with the cubic model of SCM based on either K-L expansion or SOD. They all have the error around 4%, as shown by the dash line in Fig. 4(b). Figure 4(b) clearly shows that in order to achieve the optimal (exponential) convergence rate, both SOD and gPC should be used in gSCM. 6.3 Efficiency of Recycling Technique The efficiency of recycling technique is verified by comparing the computation time in Table 3. It shows that the “recycling” of the initial value and the “recycling” of the preconditioning matrix can save about 5% and 50% computation time, respectively, compared with those ones without recycling technique. Overall, SCM and gSCM using recycling technique can achieve the same computation time compared with the perturbation method for the same model order. In conclusion, with the same accuracy, linear model


515 Table 3

Computation time. (seconds)

linear model MC Pert SCM / gSCM

quadratic model cubic model 119632 119 950 no recycling 278/282 2537/2550 15968/16147 initial value recycling 269/270 2273/2423 13275/15239 initial value & preconditioning matrix recycling 114/115 899/941 5197/6095

of SCM and gSCM saves nearly one order of the computation time than that of the quadratic model of perturbation method. Similarly, to achieve the same accuracy, quadratic model of gSCM spends less than one fifth of the time cost by the cubic model of SCM. With the same computation cost, gSCM has much higher accuracy compared with the existing methods. Therefore, the above numerical results have well demonstrated that gSCM has higher accuracy and faster convergence rate than SCM and perturbation method when geometric variations with arbitrary random probability are considered. Noting that the cubic model is only calculated to validate the exponential convergence rate of gSCM, the quadratic model of gSCM is accurate enough and is very efficient for variation-aware capacitance extraction of interconnects. 7.

Conclusion

In this paper, a generalized stochastic collocation method (gSCM) is proposed for variation-aware capacitance extraction to handle the geometric variations with arbitrary random probability. It has been pointed out and is well demonstrated in this paper that 1) using the proposed Second Order Decomposition (SOD) to find the major variation sources ξ that are most “independent” can improve both the efficiency and accuracy of gSCM; 2) for real geometric variations with arbitrary random probability, the proposed gSCM can automatically construct the corresponding generalized Polynomial Chaos (gPC) and generalized Sparse Grids that achieve the exponential convergence rate of stochastic spectral methods; 3) furthermore, the proposed recycling technique for the “recycling” of both initial values and preconditioning matrices can efficiently reduce the computation cost at each collocation point. It is well demonstrated that, compared with the existing methods, gSCM has the exponential convergence rate, and is more promising to tackle the real geometric variations with arbitrary random probability. In some applications with large number of random variables, Sparse Grids could still be in the order of hundreds. In the future, we will further study more efficient dimension reduction techniques to reduce the number of random variables, and also the parallel computing techniques for gSCM with recycling scheme to improve the computational efficiency.

Acknowledgements This research is supported partly by NSFC research project 90307017, 60676018 and 60806013, partly by the National Basic Research Program of China under the grant 2005CB321701, partly by the doctoral program foundation of Ministry of Education of China 20050246082, partly by the International Science and Technology Cooperation program foundation of Shanghai 08510700100, partly by the program for Outstanding Academic Leader of Shanghai. W. Cai is supported partly by the US National Science Foundation (grant number: CCF0727791) and the US Department of Energy (grant number: DEFG0205ER25678). References [1] J.R. Phillips and J.K. White, “A precorrected-FFT method for electrostatic analysis of complicated 3-D structures,” IEEE Trans. Comput.-Aided Des. Integrated Circuits Syst., vol.16, no.10, pp.1059–1072, Oct. 1997. [2] C. Chiang and J. Kawa, Design for Manufacturability and Yield for Nano-Scale CMOS, Springer, 2007. [3] Y. Zhou, Z. Li, Y. Tian, W. Shi, and F. Liu, “A new methodology for interconnect parasitic extraction considering photo-lithography effects,” IEEE/ACM Proc. ASPDAC, pp.450–455, Jan. 2007. [4] R. Jiang, W. Fu, J.M. Wang, V. Lin, and C.C.P. Chen, “Efficient statistical capacitance variability modeling with orthogonal principle factor analysis,” IEEE/ACM Proc. ICCAD, pp.683–690, Nov. 2005. [5] 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,” IEEE/ACM Proc. DATE, pp.1–6, April 2007. [6] Z. Zhu and J. White, “FastSies: A fast stochastic integral equation solver for modeling the rough surface effect,” IEEE/ACM Proc. ICCAD, pp.675–682, Nov. 2005. [7] J. Singh and S. Sapatnekar, “Statistical timing analysis with correlated non-Gaussian parameters using independent component analysis,” IEEE/ACM Proc. DAC, pp.155–160, July 2006. [8] “Proteus, progen, prospector full-chip optical proximity correction,” http://www.synopsys.com/products/avmrg/pdfs/proteus ds.pdf [9] “IC workbench high speed layout visualization and lithography analysis,” http://www.synopsys.com/products/icwkbch/icworkbch ds. pdf [10] D. Xiu and G.E. Karniadakis, “The wiener-asky polynomial chaos for stochastic differential equations,” SIAM J. Sci. Comput., vol.24, no.2, pp.619–644, Oct. 2002. [11] X. Wan and G.E. Karniadakis, “Multi-element generalized polynomial chaos for arbitrary probability measures,” SIAM J. Sci. Comput., vol.28, no.3, pp.901–928, 2006. [12] A. Hyvarinen, J. Karhunen, and E. Oja, Independent Component Analysis, John Wiley & Sons, 2001. [13] S. Bhardwaj, P. Ghanta, and S. Vrudhula, “A framework for statistical timing analysis using non-linear delay and slew models,” IEEE/ACM Proc. ICCAD, pp.225–230, Nov. 2006. [14] J. Xiong, V. Zolotov, and L. He, “Robust extraction of spatial correlation,” IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst., vol.26, no.4, pp.619–631, April 2007. [15] R.G. Ghanem and P.D. Spanos, Stochastic Finite Elements: A Spectral Approach, Springer-Verlag, New York, 1991. [16] G.H. Golub and J.H. Welsch, “Calculation of Gauss quadrature rules,” Mathematics of Computation, vol.23, no.106, pp.221–230, April 1969. [17] E. Novak and K. Ritter, “Simple cubature formulas with high


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Hengliang Zhu received the B.E. in electronic engineering from University of Science and Technology of China, Hefei, China, in 2004. He is currently working toward the Ph.D. degree at the State Key Lab of ASIC and System of Fudan University, Shanghai, China. His research interests include variational-aware parameter extraction algorithms for VLSI circuits.

Xuan Zeng received the B.Sc. and Ph.D. degrees in electrical engineering from Fudan University, Shanghai, China, in 1991 and 1997, respectively. She joined the Electrical Engineering Department, Fudan University in 1997 and became a full professor in Microelectronics Department in 2001. Now she serves as the Director of the State key Lab of ASIC and System of Fudan University. She was a visiting professor in the Electrical Engineering Department, Texas A&M University, U.S.A. and Microelectronics Department of TU Delft, Netherland in 2002 and 2003 respectively. Her research interests include DFM, analog and mixed signal design automation (behavioral modeling, circuit simulation and analog layout generation), high speed interconnect analysis and design and ASIC design. Dr. Zeng received the Cross-Century Outstanding Scholar Award from the Ministry of Education of China in 2002. She was selected into IT Top 10 in Shanghai China in 2003. She served in the technical program committee of IEEE/ACM ASP-DAC in 2000 and 2005.

Xu Luo received the B.S. in Electronic Science and Technology from Xidian University, Xi’an, China, in 2004. He is currently working toward the Ph.D. degree at the State Key Lab of ASIC and System, Fudan University, Shanghai, China. His research interests include parameter extraction and variation-aware stochastic methods for the modeling and simulation of VLSI circuits.

Wei Cai received the Ph.D. degree in applied mathematics from Brown University, Providence, RI in 1989. He joined the Department of Mathematics, the University of North Carolina at Charlotte in 1989 as an assistant professor, later became an associate professor in 1995, and a full Professor in 1999. His research interest includes numerical methods for model reduction, computational electromagnetics for parameter extraction for computer packaging and VLSI designs and photonic devices.