Interconnect crosstalk noise evaluation in deep-submicron technologies

Microelectronics Reliability 49 (2009) 170–177

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Interconnect crosstalk noise evaluation in deep-submicron technologies Xiaoxiao Liu *, Guangsheng Ma, Jingbo Shao, Zhi Yang, Guanjun Wang College of Computer Science and Technology, Harbin Engineering University, No.145 Nantong Street, Harbin 150001, China

a r t i c l e

i n f o

Article history: Received 14 May 2008 Received in revised form 21 November 2008 Available online 9 January 2009

a b s t r a c t To solve the crosstalk noise problem in deep-submicron technologies, a statistical method for analyzing crosstalk noise with reduced distributed RC-p model is proposed in this paper. First, quiet aggressor net and tree branch reduction techniques are introduced into the distributed RC-p model, and a new spatial correlation model for both Gaussian and non-Gaussian process variations among segments is created. Then, principal components analysis (PCA) and independent component analysis (ICA) techniques are applied to reduce correlations of process variations. Finally, the moment matching scheme is used to obtain the probability density function (PDF) of crosstalk noise in victim coupled with multiple aggressors. Experimental results show that our method maintains the efficiency of previous approaches, and significantly improves on their accuracy. Ó 2008 Elsevier Ltd. All rights reserved.

1. Introduction As the package density and clock frequency of VLSI circuits increase, interconnect parasitic effects have become increasingly significant. Crosstalk noise has been one of the most challenging problems in the design and verification of modern VLSI circuits. Another unwanted side effect of CMOS process technology scaling is the increase in process variations. Differences between identical features in a certain lithographic process are referred to as process variations [1]. Lithography steps generate more process variations in smaller geometric feature sizes. This phenomenon can create a rather large deviation of key circuit parameters from their designed values, which may in turn increase crosstalk, produce timing uncertainty, interfere circuit performance [2], therefore highly sophisticated and robust crosstalk-aware analysis and optimization tools are needed. Statistical analysis is viewed as an essential methodology for nanometer process technologies, and it enables the application of actual statistics of the process technology parameters to the accurate calculation of design characteristics such as delay and noise [3]. Monte Carlo simulation can obtain the most accurate result for crosstalk noise estimation, but its run time is too long. The statistical model of [4] uses a lumped RC model to explore crosstalkinduced pulse effect, where a single resistance is extracted to capture the effect of total self resistance of interconnect, regardless of its length. The case for self and coupling capacitances is similar. The statistical model proposed in [5] is more sophisticated and uses a circuit model with higher number of nodes; however, still

* Corresponding author. E-mail address: [email protected] (X. Liu). 0026-2714/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.microrel.2008.11.013

a single capacitance is extracted to model the total coupling effect, which makes it inappropriate for long interconnect lines. The authors of [6] apply more accurate distributed RC-p model to study the statistics of crosstalk effect, but the complexity of model reduction becomes huge because of considering process variations. Additionally, all the above approaches assume that parameter variables are statistically Gaussian independent. The assumption about the Gaussian nature of process parameters makes it very convenient for statistical analysis. However, it is well known that some process parameters deviate significantly from a Gaussian distribution [7]. The normality assumption is not always valid, which may sometimes lead to significant sources of errors in crosstalk noise analysis. In this paper, a new methodology is proposed to derive the statistical distribution of on-chip crosstalk noise, which presents a solution in the presence of both correlated Gaussian and nonGaussian process variations. Further the paper is organized as follows: In Section 2, the quiet aggressor and tree reduction techniques are introduced into the distributed RC-p model. In Section 3, both Gaussian and non-Gaussian process variations in the physical dimensions of each interconnect segments are accurately extracted from the reduced model, and then the crosstalk noise is represented as a linear canonical function of this process variations set. In Section 4, the grid-based model is used to generate the spatial correlation among each neighboring segment of the reduced distributed RC-p model. In Section 5, PCA [8] and ICA [9] technologies are employed to transform correlated sets of Gaussian and non-Gaussian variables into sets of statistically independent variables correspondingly, and then the PDF of crosstalk noise in a victim line coupled with multiple aggressors is obtained using the moment matching scheme. Section 6 shows experimental results while Section 7 concludes this paper.

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2. Reduced distributed RC-p model Consider three coupled interconnect lines in some metal layer, which lies in between two dielectric plates (cf. Fig. 1a). The three interconnect lines run in parallel and are capacitive coupled. A distributed RC-p model (cf. Fig. 1b) is used to accurately model the abovementioned interconnect line configuration. In this circuit, each RC-p stage represents an interconnect segment of a predefined length, Lseg , which is an important factor when considering spatial correlation among physical parameters, as will be explained in Section 4. The couplings between each two interconnect lines along segment i are captured by the coupling capacitances C i;1c2 , C i;1c3 , and C i;2c3 , respectively. The self capacitance and resistance

se g

L

se g

a

L

se g

L

g-i Se

I

II

III

b Ri ,2 Ci ,1c 2

Ci ,2

Ci ,1c 3

Ci ,2 c 3

Fig. 1. Distributed capacitive modeling of coupled interconnects.

Ria

of line II in segment i are denoted by C i;2 and Ri;2 , respectively. Note that although lengths of all wire segments are identical, due to process variations, parameter values of each segment are different from those of the other segments. In general, a net has a tree structure instead of being a simple wire. Millions of interconnect lines in VLSI circuit limit the application of distributed RC-p model in real designs. The authors of [10] proposed the quiet aggressor and tree reduction techniques and used a 6-node template circuit for fast crosstalk estimation. They also pointed out that the 6-node template circuit is only suitable for short to medium interconnects because it uses only one lumped coupling capacitor, and more complex template circuits with larger number of coupling capacitors should be employed for long interconnect, but the aggressor and tree reduction methods are generally applicable and not restricted to the specific circuit topology. Consequently, in this paper we use distributed RC-p model to consider all sorts of interconnects under deep-submicron process techniques, and inject the quiet aggressor and tree reduction techniques to the distributed RC-p model which help well eliminate quiet aggressors that do not affect the state of victim in fact. Fig. 2 shows the application of reduction techniques to the distributed RC-p model. Fig. 2a shows the quiet aggressor reduction process to reduce aggressors non-switching and not switching simultaneously with victim V in segment i, while Fig. 2b shows the tree branches reduction process of the victim or aggressor in distributed RC-p model. Effective capacitances Ceffia and Cefft can be easily derived by matching the lower order Taylor series expansion coefficients of the admittances at the coupling point and branching point correspondingly. More details of the quiet aggressor and tree reduction techniques please refer to [10]. Therefore, the crosstalk noise estimation problem for a general RC tree structure circuit is transformed into a much simpler one of solving the reduced distributed RC-p circuit, which makes our approach be applicable to real designs. Using the proposed reduced distributed RC-p model, we can obtain more accurate crosstalk noise estimation results while maintaining the analysis efficiency, compared with the simple distributed RC-p model where the quiet aggressor nets are grounded during superposition and the resistive shielding effects in the branches are not considered. Experimental results in Section 6.1 prove the validity of our reduced distributed RC-p model.

Ria

A

Cia

Cic

Ceffia

V (a) Quiet aggressor net reduction

A(V ) A(V )

R

V

A(V )

C1 C2

(b) Tree branches reduction Fig. 2. Reduction processes of the distributed RC-p model.

Cefft

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X. Liu et al. / Microelectronics Reliability 49 (2009) 170–177

3. Variables in reduced distributed RC-p model A complete statistical crosstalk noise analysis method based on the reduced distributed RC-p model (RD-SCNA) is proposed in this paper. Different from the former approaches, our RD-SCNA method only requires the moments of process parameters distributions as inputs because it is more realistic to obtain the moments of the variations from a process engineer than to extract precise distributions from process data. The kth moments of process parameter xi in segment can be evaluated from the process files directly as Eq. (1), which also enables us to simply use skewness and kurtosis to describe the non-Gaussianity character of random variables.

mk ðxi Þ ¼ E½xki  ¼

X

xk Probalilityðxi ¼ xÞ

ð1Þ

x

Skewness is a measure of the degree that a parameter distribution deviates from the symmetric distribution. The parameter distribution whose skewness is not equal to zero must be asymmetrical. Kurtosis characterizes the general form of a statistical frequency curve near the center of the distribution. Both skewness and kurtosis of Gaussian random variable are permanently equal to zero, but for non-Gaussian random variable either skewness or kurtosis is not permanently equal to zero. Skewness S and kurtosis K are easy to compute, and have a good linear nature. For a random variable X, S and K are defined as follows:

S¼

E½X  EðXÞ3

K¼

2 1:5

fE½X  EðXÞ g

E½X  EðXÞ4 fE½X  EðXÞ2 g2

3

However, the sources of variation may exhibit non-Gaussian distributions [11,12] and therefore result in non-Gaussian crosstalk noise distributions. To overcome the fatal defect of not considering non-Gaussian parameters in previous statistical crosstalk analysis methods, we incorporate the effects of both Gaussian and non-Gaussian parameters into our RD-SCNA framework. All crosstalk noises are represented in a linear form as

V ¼lþ

n X i¼1

b i  xi þ

m X

cj  yj þ e  z ¼ l þ BT X þ C T Y þ e  z

ð2Þ

j¼1

where V is the random variable corresponding to crosstalk noise on victim line caused by a coupled aggressor line, xi [ yj ] is a nonGaussian [Gaussian] random variable corresponding to a physical parameter variation, bi [ cj ] is the first order sensitivity of the crosstalk noise with respect to the ith non-Gaussian [jth Gaussian] parameter, z is the uncorrelated parameter which could be a Gaussian or a non-Gaussian random variable, e is the sensitivity with respect to the uncorrelated variable, and n [m] is the number of correlated non-Gaussian [Gaussian] variables. In the vector form, B and C are the sensitivity vectors for X, the random vector of non-Gaussian parameter variations, and Y, the random vector of Gaussian random variables, respectively. The mean of each random variable is adjusted to zero. We use the analytical crosstalk noise model proposed in [13] and the first order Taylor series terms to obtain the linear sensitivities of crosstalk noise with respect to different Gaussian and nonGaussian parameters in Eq. (2). However, clearly using a linear approximation to model crosstalk noise is not a restriction, as our canonical form is similar in form to those in [14,15]. What is more, for real circuits many parameters are correlated due to presence of the inherent spatial and non-spatial correlations. In this paper, we consider the spatial correlation of systematic interconnect variation (e.g. line height, line width, inter-layer dielectric (ILD) thickness, recall that we consider line width and spacing to be perfectly inversely correlated). According to [6,16], we know that the spatial correlation exists only among the same type of interconnect process parameter while different types of

interconnect process parameters are not spatially correlated to one another, thus Gaussian parameters and non-Gaussian parameters in Eq. (2) are statistically independent, as in [15,17]. 4. Spatial correlations The spatial correlation among the variation of neighboring segments is quite important for the crosstalk noise on victim line. For example, most of the variations resulting from chemical-mechanical polishing (CMP) of the inter-layer dielectric are based on systematic spatial effects and vary substantially within-die [18]. CMP effects show several millimeter long ranges, typically in the 2–10 mm range. Over that large range most of interconnect lines are coupled with each other [19], which warrants the spatial correlation analysis. To generate the exacter spatial correlations of process variables in segments of the reduced distributed RC-p model, the chip is partitioned into grids and each grid cell is assigned to an individual random variation for the considered parameter, as shown in Fig. 3. All segments in the grid cell share the same variation as assigned if the considered parameter affects the crosstalk noise, so each grid will be associated with (n + m) variations. Since the segments close to each other are more likely to have more similar characteristics than those placed far away, the correlation function Corrðxi ; xj ; yi ; yj Þ between any two points (xi ; yi ) and (xj ; yj ) in the grid depends on the distance v ij between them, as follows:

Corrðxi ; xj ; yi ; yj Þ ¼ Corr

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðxi  xj Þ2 þ ðyi  yj Þ2 ¼ Corrðv ij Þ

It has been observed that the spatial correlation function Corrðv ij Þ is a monotonically decreasing function of distance v ij . However, not all monotonically decreasing functions qualify for the spatial correlation function. A necessary and sufficient condition for function Corrðv ij Þ to be a valid spatial correlation function of a homogeneous and isotropic random field is that it can be represented in the form of [20]:

Corrðv ij Þ ¼

Z

1

J 0 ðwv ij Þdð/ðwÞÞ

ð3Þ

0

where J 0 ðtÞ is the Bessel function of order zero and /ðwÞ is a real non-decreasing function on ½0; 1 such that for some non-negative p:

Z 0

1

d/ðwÞ <1 ð1 þ w2 Þp

It has been found through a series of characterization and extraction experiments by the manufacturers that the spatial correlation between the parameter values corresponding to two points along the interconnect length is an exponential decay function of distance, as Eq. (2.2) in [6]. Thus on the premise of satisfying

Fig. 3. The grid-model of spatial correlations.

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qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi conditions proposed in [20], we fix /ðwÞ ¼ 1  1= 1 þ w2 L2seg to obtain the spatial correlation function of all segments in the grid. By plugging it into Eq. (3), we obtain the corresponding correlation function as:

Corrðv ij Þ ¼ expðv ij =Lseg Þ where Lseg is a predefined segment length that regulates the decaying rate of the correlation function with respect to distance v ij . For each type of parameter in the reduced distributed RC-p model, a sparse correlation matrix M of size g  g represents the spatial correlations of such a structure.

2

1

Corrðv 12 Þ Corrðv 13 Þ . . . Corrðv 1g Þ

3

Fig. 4. Two RC-p stages circuit of coupled interconnect lines.

7 6 6 Corrðv 12 Þ 1 Corrðv 23 Þ . . . Corrðv 2g Þ 7 7 6 7 6 7 6 Corrðv 13 Þ Corrðv 23 Þ 1 . . . Corrð v Þ 3g 7 M¼6 7 6 7 6 .. 7 6 . 5 4 Corrðv 1g Þ

Corrðv 2g Þ

Corrðv 3g Þ

ing-based PDF evaluation. For each interconnect we assume that the width W to be Gaussian random variables and the height H to be uniformly distributed random variables. Using a first order approximation, the crosstalk noise in this circuit can be written as

1

...

V ¼ l þ a1a W 1a þ a2a W 2a þ a1v W 1v þ a2v W 2v þ b1a H1a þ b2a H2a þ b1v H1v þ b2v H2v

5. PDF evaluation of crosstalk noise When correlations among process parameters are taken into consideration, the statistical crosstalk noise analysis becomes even more complicated. To make the problem tractable, we employ the mathematical techniques PCA and ICA to accomplish the desired goal of transforming Gaussian and non-Gaussian correlated random variables sets into random variables sets that are statistically as independent as possible, respectively. Hence, crosstalk noise in the reduced distributed RC-p circuit model can be represented in a linear form of new sets of independent Gaussian and non-Gaussian random variables. In Eq. (2), ICA is applied to the non-Gaussian parameters X and PCA to the Gaussian variables Y, to obtain a set of statistically independent non-Gaussian variables S and a set of independent Gaussian variables R. We then arrive at the following canonical noise model:

V ¼ l þ B0T S þ C 0T R þ e  z ¼ l þ

n X i¼1

0

bi  si þ

m X

c0j  r j þ e  z

ð4Þ

j¼1

where B0T ¼ BT A½C 0T ¼ C T P1 y  is the new sensitivity vector with respect to the statistically independent non-Gaussian components s1 ; s2 ; . . . ; sn [Gaussian principal components r 1 ; r 2 ; . . . ; r m ], A [P y ] is the n  n [m  m] transformation matrix. Given the underlying process variables and their moments, after performing ICA [PCA] the moments of the independent components s1 ; s2 ; . . . ; sn [r1 ; r2 ; . . . ; r m ] can be easily computed from the moments of the correlated non-Gaussian parameters x1 ; x2 ; . . . ; xn [correlated Gaussian parameters y1 ; y2 ; . . . ; ym ] successively. These moments of the process variables are inputs to our RD-SCNA algorithm. Finally, to compute the PDF/CDF of the crosstalk noise random variable we adapt the probability extraction scheme APEX [21]. Given 2M moments of a random variable as inputs to the APEX algorithm, the scheme employs an AWE (asymptotic waveform evaluate) technique to match the 2M moments in order to generate an Mth order LTI (linear time invariant) system. The scheme then approximates the PDF [CDF] of a crosstalk noise random variable in the reduced distributed RC-p model by an impulse response h(t) [step response s(t)] of the Mth order LTI system. It is clear that we borrow some techniques from existing algorithm; however, the overall algorithm is distinctly different from any existing crosstalk noise analysis method. We use a simple 2 RC-p stages circuit of a pair of coupled interconnect lines, as shown in Fig. 4, to illustrate the moment match-

where a1a , a1v , b1a , b1v [ a2a , a2v , b2a , b2v ] are sensitivities of the crosstalk noise with respect to the zero-mean randomly varying parameters W 1a , W 1v , H1a , H1v [ W 2a , W 2v , H2a , H2v ] of seg-1[seg2], respectively, and l is the nominal crosstalk noise of the circuit. ^ ¼ V  l, then we have: Let V

^ ¼ a1a W 1a þ a2a W 2a þ a1v W 1v þ a2v W 2v þ b1a H1a þ b2a H2a V þ b1v H1v þ b2v H2v

ð5Þ

We consider three possible cases: Case I: If all of the parameters are considered to be statistically independent, we take the expectation of both sides of Eq. (5) to obtain:

^ ¼ E½ða1a W 1a þ a2a W 2a þ a1v W 1v þ a2v W 2v þ b1a H1a þ b2a H2a mk ½V þ b1v H1v þ b2v H2v Þk  In the above equation, all of the k moments of W and H are known from the underlying normal and uniform distribution, thus the kth ^ can be simplified by performing an efficient multinomoment of V mial expansion using the binomial moment evaluation technique. The moments are computed successively, starting from the first to the second to the third, and so on. Case II: If both (W 1a , W 2a , W 1v , W 2v ) and (H1a , H2a , H1v , H2v ) are considered to be perfectly correlated identical random variables, we have:

^ ¼aW þbH V where a ¼ a1a þ a2a þ a1v þ a2v ; b ¼ b1a þ b2a þ b1v þ b2v . Since variables W and H are statistically independent, the kth moment ^ can be computed by using the binomial expansion formula of V as:

^ ¼ mk ½V

k   X k i¼0

i

ai b

ki

mi ðWÞmki ðHÞ

where all of the k moments of W and H are known from the underlying normal and uniform distribution. Case III: If at least one of (W 1a , W 2a , W 1v , W 2v ) and (H1a , H2a , H1v , H2v ) is considered to be spatial correlated, then the PCA or ICA technique is applied to transform the Gaussian or non-Gaussian correlated random variables sets into statistically independent random variables sets, respectively. After this processing, the kth ^ can be computed according to Cases I and II. moment of V

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X. Liu et al. / Microelectronics Reliability 49 (2009) 170–177

^ we can now employ the Having computed 2M moments of V, ^ AWE-based PDF evaluation scheme to approximate the PDF of V by an impulse response as:

8 M > < P ^r  ep^i v^ i fV^ ðv^ Þ ¼ > i¼1 : 0

v^ P 0 v^ < 0

^ is the residue [pole] of the LTI approximation. where ^r½p Then, the evaluated crosstalk noise PDF is given as: fV ðv Þ ¼ fV^ ðv þ lÞ. We can generalize the PDF evaluation idea, illustrated in the above example, to compute the PDF [CDF] of any crosstalk noise random variable expressed in the canonical form of Eq. (4). It is well known that a victim usually has multiple coupled aggressors switching simultaneously. The crosstalk noise on a victim is the superposition of that generated by each aggressor. When the crosstalk glitches induced by different aggressors switch in the same [opposite] direction[s], the generated crosstalk noise effect on the victim is added [counteracted] with each other. Thus, our proposed crosstalk noise computation procedure involves two atomic operations of ‘‘sum” and ‘‘subtraction”. Consider two random variables, V 1 and V 2 expressed as:

V 1 ¼ l1 þ

n X

0

bi1  si þ

i¼1

V 2 ¼ l2 þ

n X

m X

c0j1  rj þ e1  z1

j¼1

0

bi2  si þ

i¼1

m X

c0j2  rj þ e2  z2

j¼1

The sum V 3 of V 1 and V 2 can be expressed in canonical form as

V 3 ¼ l3 þ

n X i¼1

0

bi3  si þ

m X

c0j3  rj þ e3  z3

j¼1 0

0

0

where l3 ¼ l1 þ l2 , bi3 ¼ bi1 þ bi2 , c0j3 ¼ c0j1 þ c0j2 , and e3 z3 ¼ e1 z1 þ e2 z2 . The uncorrelated non-Gaussian random variable e3 z3 serves as a place holder to store the moments of (e1 z1 þ e2 z2 ). In other words, rather than propagating an uncorrelated component z in the canonical form, we propagate its 2M moments. The ‘‘subtraction” operation of crosstalk noise random variable is similar to the ‘‘sum” operation. So, the proposed superposition operation of crosstalk noise random variables expressed in the linear canonical form of Eq. (4) is largely straightforward. 6. Experimental results 6.1. Comparison of circuit models To verify the correctness of our reduced distributed RC-p model, firstly we have applied the proposed circuit model to seven noiseprone nets from a high-performance processor designed in a 0.15 lm process technology. In these nets, the drivers of the circuits are replaced by linear resistors using the technique described

in [22]. Each piece of distributed circuitry consisting of 25 RC-p stages is used. Some information of the circuits and comparison of our model results with Spice simulation, 6-node template circuit based estimation [10], and simple distributed RC-p model based estimation are shown in Table 1. The second column shows the number of total aggressor nets and that of switching aggressor nets. The third column is the total number of RC elements in a given circuit. The average number of all the nets is 154. Table 1 shows that the average error of the peak noise voltage values calculated using the proposed model is 3.7%, while the six-node template circuit and the simple approach have an average error of 4.4% and 9.1%, respectively. We can see that in short to medium circuits, our reduced distributed RC-p model can obtain more accurate noise results than the six-node template circuit based estimation in [10], while simple distributed RC-p model based estimation obtains the worst peak noise voltages because of not well modeling the reduced quiet aggressors and tree branches in circuits. Secondly, we repeat the experiments corresponding to Table 1 on some larger circuits to further verify the effect of our reduced distributed RC-p model. The models are tested on seven MCNC benchmark circuits synthesized in 0.13 lm technology. The synthesized benchmarks are placed-and-routed by using a commercial APR tool. A summary of the experimental results obtained for MCNC benchmark circuits is listed in Table 2. The details of all circuits are given in the first three columns, and the average errors of different circuit models compared with Spice simulation are given in the last four columns. We see that for large circuits our reduced distributed RC-p model still gives the most accurate results, where the maximum average error of circuit i6 is 6.2% and the minimum average error of circuit i4 is 4.4%. Results in Table 2 testify to the accuracy of our reduced distributed RC-p model on large circuits. Compared Table 2 with Table 1, we can see that the average errors of both lumped 6-node circuit model and the simple approach in large circuits (Table 2) are far higher than those in small and medium circuits (Table 1). However, our reduced distributed RCp model can obtain crosstalk noise results within a small error on both small and large circuits, which proves that it is well suitable for crosstalk noise analysis of integrated circuits under deep-submicron technologies. 6.2. Statistical comparison of crosstalk models To show the accuracy and efficiency of our RD-SCNA approach, two case studies are carried on: (I) a simple two coupled interconnect structure (cf. Fig. 5 (a)) is applied to simulate the crosstalk noise between interconnect lines; (II) a typical five coupled interconnect structure (cf. Fig. 5 (b)) is applied to analyze the statistical superposition of crosstalk noise on the victim with multiple aggressors. In both cases, we all treat the interconnect with input V in and output V out as the victim line and the other(s) as the aggressor line(s). The cells are taken from a standard 0.13 lm, 1.2V

Table 1 Experimental results on peak noise voltages. Circuit

Na/Ns

#RC

Spice (V)

[10] (Err%)

Simple (Err%)

Proposed (Err%)

1 2 3 4 5 6 7 AVE

2/2 9/9 4/4 5/5 7/6 3/3 10/9 5.7/5.4

97 215 132 128 144 116 247 154

0.841 0.807 0.799 0.796 0.731 0.664 0.651 –

0.902(6.1) 0.786(2.6) 0.821(2.8) 0.815(2.3) 0.783(7.1) 0.641(3.4) 0.607(6.8) 4.4%

0.789(6.2) 0.713(11.6) 0.709(11.3) 0.692(13.1) 0.715(2.2) 0.627(5.6) 0.564(13.4) 9.1%

0.889(5.7) 0.792(1.8) 0.816(2.1) 0.812(2.0) 0.771(5.5) 0.643(3.2) 0.613(5.8) 3.7%

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X. Liu et al. / Microelectronics Reliability 49 (2009) 170–177 Table 2 Experimental results for MCNC benchmark circuits. Circuit

#of Nets

Na/Ns

i1 i2 i3 i4 i5 i6 i7

46 221 126 230 138 668 870

232/103 706/324 551/281 1181/610 1853/794 7298/3066 9605/4925

Table 4 The statistic results of skewness and kurtosis.

Average error (%) [10]

Simple

Proposed

7.6 11.9 9.3 13.5 11.4 14.7 13.8

8.2 12.4 9.5 12.7 10.9 13.3 14.1

4.7 5.3 4.9 4.4 5.5 6.2 5.1

Fig. 5. The coupled interconnect test cases.

production library. Distributed circuit consisting of 25 identical RC-p segments is used to model 2.5mm Cu interconnect with segment length Lseg ¼ 100lm. All experiments are performed on the Linux machine with a clock speed of 3.2GHZ and 1GB memory. We consider interconnect parameters the width, W, the height, H, and the dielectric thickness, T, as the sources of variation in each interconnect segment. The proposed framework can be easily extended to include other parameters of variations. The process parameters and their max percentages of deviations from the nominal values (Table 3) used here are based on predictions from [23]. To realistically model the parameter distributions, for each process parameter we sample 100 times by 25 lm gap on the tested circuit. Carrying on statistics to 100 samples, we compute the results

Table 3 Parameters used in the experiments.

l r

Width (nm)

Height (nm)

Dielectric thickness (nm)

130 25%

270 20%

70 35%

S K

Width

Height

Dielectric thickness

0.8 1.6

0.1 0.3

0.6 1.7

of skewness and kurtosis for each interconnect process parameter, as shown in Table 4. It is obvious that H can be approximated as Gaussian distribution, while W and T have significantly non-Gaussian probability distributions where W follows either a uniform distribution or a symmetric triangular distribution, and T follows a Poisson distribution. Besides, since Cu CMP effects are considered here the parameters W, H are spatial correlated, and the parameter T is uncorrelated according to [18]. We use the grid-based model described in Section 4 to generate the spatial correlations for parameters W and H of interconnect segments, and adopt the grid cell with size of 100  100 lm. Fig. 6a and b show comparisons of the results of our RD-SCNA method with the Gauss D-SCNA method [6] and the Monte Carlo (MC) simulation using 2000 samples for Case (I) and Case (II), respectively. When modeling the non-Gaussian parameters W, T as normally distributed random variables, the performed MC simulations for test circuits is termed as MCGauss . As seen in Fig. 6, the results of our RD-SCNA approach for both Case (I) and Case (II) are able to match well with those of MC simulations within small errors. These errors are reasonably small as compared with the accuracy penalty paid of the Gauss D-SCNA method and MCGauss . Thus, modeling the non-Gaussian parameters as normally distributed ones leads to significant inaccuracy. Moreover, to show the importance of considering spatial correlations, we consider the difference between performing statistical analysis while considering spatial correlation and ignoring it. Therefore, we run another set of Monte Carlo simulations, termed as MCNoCorr , on the same test circuits, this time assuming zero correlations among parameters. It can be observed in Fig. 6 that the PDF and CDF curves of MCNoCorr deviate significantly from those of MC simulations. Thus, statistical crosstalk noise analysis without considering spatial correlation may incorrectly predict the real performance of the circuit and could even overestimate the performance of the circuit. This underlines the meaning of developing efficient statistical crosstalk noise analysis methods that can incorporate spatial correlations. Run time in Case (I) and Case (II) is correspondingly about 5.3 s and 6.2 s of our RD-SCNA method, 58.6 s and 69.7 s of the MC simulation, and 4.9 s and 5.7 s of the Gauss D-SCNA approach. It can be seen that our RD-SCNA method is considerably faster than the MC simulation, but has a higher runtime cost as compared with the Gauss D-SCNA approach due to handling additional non-Gaussian variables (no more than 0.5 s). In addition, we further choose different Lseg values to study how the accuracy of our RD-SCNA algorithm changes. Table 5 shows the errors of l and r of our obtained crosstalk noise distributions corresponding to different Lseg values, compared with the MC simulation results for both Case (I) and Case (II). From Table 5, we can see that when the segment length changes from 50 to 125 lm, there is only slight increase of errors for all noise results and our algorithm still gives quite accurate results. This convincingly shows that the proposed RD-SCNA approach can warrant the validity of the unavoidable measurement noise. When the Lseg value change to 250 lm, we see a larger error (but less than 1% for l, 6% for r) in the obtained crosstalk noise distributions. It indicates that there is an upper bound on the value of segment length in order to achieve reasonably accurate crosstalk noise estimation results.

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Fig. 6. Comparisons of crosstalk noise models and MC simulations.

Table 5 The errors of our RD-SCNA results with different Lseg values. Error (%)

l r

50 lm

100 lm

125 lm

250 lm

Case (I)

Case (II)

Case (I)

Case (II)

Case (I)

Case (II)

Case (I)

Case (II)

0.21 0.56

0.24 0.68

0.36 0.79

0.39 0.91

0.45 1.07

0.51 1.22

0.91 4.25

0.98 5.71

7. Conclusions and future works As the VLSI technology scales down, the role of interconnect parasitic effects in the signal integrity becomes increasingly significant, which may result in the aggravation of crosstalk noise amplitude and duration, and the circuit faults. In this paper a new statistical crosstalk noise analysis method in deep-micron technologies is presented. The contributions of our approach include: (1) Quiet aggressor and tree reduction technologies are introduced into the distributed RC-p model to improve the accuracy of circuit model. (2) Both Gaussian and non-Gaussian local process variations are considered, which significantly improve the accuracy of crosstalk noise analysis. (3) A new spatial correlation model is proposed for interconnect process variations in the reduced distrib-

uted RC-p model, which enables our RD-SCNA method to consider effects of all the correlated and independent process parameters. (4) The proposed method is able to obtain a very precise crosstalk noise result of the victim coupled with multiple aggressors, and general enough to work even for the cases when the PDF of the sources of variation is not available by using the moment matching scheme. Experimental results show that compared with previous methods, the proposed RD-SCNA approach significantly improves the accuracy of crosstalk noise analysis while maintaining the efficiency. What is more, our method will play a more important role in crosstalk noise analysis in future 90 nm or even less IC manufacturing process. As a meaningful future work, we can continue to find a statistical crosstalk noise analysis method of coupled inter-

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connects that must handle the process parameter correlations due to all possible sources simultaneously, including the spatial and non-spatial (structural) correlations. Acknowledgements This work is supported by the National Natural Science Foundation of China under Grant No. 60273081 and Harbin Engineering University Foundation under Grant No. F04088. References [1] Hao Yue, Ming-e Jing, Peijun Ma. State of the art on study of parametric yield and its optimization for VLSI. Acta Electron Sin 2003;31(12A):1971–4. [2] Feng Gang, Guangsheng Ma, Du Zhenjun. Signal-related crosstalk optimization for detailed routing. J Comput-Aid Des Comput Graph 2005;17(5):1074–8. [3] Agarwal Kanak, Nassif Sani. Characterizing process variation in nanometer CMOS. In: Proceedings of the 44th annual conference on design automation, San Diego, 2007. p. 396–9. [4] Maurizio Martina, Guido Masera. A statistical model for estimating the effect of process variations on crosstalk noise. In: Proceedings of the international workshop on system level interconnect prediction, Paris, 2004. p. 115–20. [5] Agarwal K, Agarwal M, Sylvester D, Blaauw D. Statistical interconnect metrics for physical-design optimization. IEEE Trans Comput-Aid Des Integr Circ Syst 2006;25(7):1273–88. [6] Nazarian Shahin, Iranli Ali, Pedram Massoud. Crosstalk analysis in nanometer technologies. In: Proceedings of the 16th ACM Great Lakes symposium on VLSI, Philadelphia, 2006. p. 253–8. [7] Nagaraj NS, Bonifield Tom, Singh Abha, Bittlestone Clive, Narasimha Usha, Le Viet, et al. BEOL variability and impact on RC extraction. In: Proceedings of the 42nd annual conference on Design automation, California; 2005. p. 758–9. [8] Dunteman George H. Principal components analysis. California: Sage Publications; 1989. [9] Hyvarinen A, Oja E. Independent component analysis: algorithms and applications. Neural Networks 2000;13(4–5):411–30. [10] Li Ding, David Blaauw, Pinaki Mazumder. Accurate crosstalk noise modeling for early signal integrity analysis. IEEE Trans Comput-Aid Des Integr Circ Syst 2003;22(5):595–600.

177

[11] Chang Hongliang, Zolotov Vladimir, Narayan Sambasivan, Visweswariah Chandu. Parameterized block-based statistical timing analysis with nonGaussian parameters, nonlinear delay functions. In: Proceedings of the 42nd annual conference on design automation, California, 2005. p. 71–6. [12] Damerdji H, Dasdan A, Kolay S. On the assumption of normality in statistical static timing analysis. In: Proceedings of ACM/IEEE international workshop on timing issues, Texas, 2005. p. 2–7. [13] Heydari P, Pedram M. Capacitive coupling noise in high-speed VLSI circuits. IEEE Trans Comput-Aid Des Integr Circ Syst 2005;24(3):478–88. [14] Agarwal Mridul, Agarwal Kanak, Sylvester Dennis, Blaauw David. Statistical modeling of cross-coupling effects in VLSI interconnects. In: Proceedings of the conference on Asia South Pacific design automation, Shanghai, 2005. p. 503–6. [15] Fatemi Hanif, Abbaspour Soroush, Pedram Massoud, Ajami Amir H, Tuncer Emre. SACI: statistical static timing analysis of coupled interconnects. In: Proceedings of the 16th ACM Great Lakes symposium on VLSI, Philadelphia, 2006. p. 241–6. [16] Nassif SR. Modeling and analysis of manufacturing variations. In: Proceedings of the IEEE custom integrated circuits conference, San Diego, 2001. p. 223–8. [17] Li Tao, Yu Zhiping. Statistical analysis of full-chip leakage power considering junction tunneling leakage. In: Proceedings of the 44th annual conference on design automation, California, 2007. p. 99–102. [18] Zhang X, He L, Gerousis V, Song L, Teng C-C. Case study and efficient modelling for variational chemical–mechanical planarisation. IET Circ Dev Syst 2008;2(1):30–6. [19] Semiconductor Industry Association. National technology roadmap for semiconductors, 2003. [20] Xiong Jinjun, Zolotov Vladimir, He Lei. Robust extraction of spatial correlation. In: Proceedings of the 2006 international symposium on physical design, California, 2006. p. 2–9. [21] Li Xin, Le Jiayong, Gopalakrishnan Padmini, Pileggi Lawrence T. Asymptotic probability extraction for non-normal performance distributions. IEEE Trans Comput-Aid Des Integr Circ Syst 2007;26(1):16–37. [22] Levy Rafi, Blaauw David, Braca Gabi, Dasgupta Aurobindo, Grinshpon Ami, Oh Chanlee, et al. ClariNet: a noise analysis tool for deep submicron design. In: Proceedings of the 37th conference on design automation, Los Angeles, 2000. p. 233–8. [23] Cong J. Challenges and opportunities for design innovations in nanometer technologies. SRC design science concept paper, 1997.