Application of Importance Sampling using Contaminated Normal Distribution to Multidimensional Variation Analysis

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1 1, 2 1 3, Monte Carlo g(x) g(x) g(x) g(x) g(x) / 6-24 SRAM Monte Carlo 2 5 Application of Importance Sampling using Contaminated Normal Distribution to Multidimensional Variation Analysis Shiho Hagiwara, 1, 2 Takanori Date, 1 Takumi Uezono, 3, 4 Kazuya Masu 1 and Takashi Sato 3 Process variation by miniaturization has been inducing yield degradation. Design-time yield estimation is required. Monte Carlo method, which is one of the effective yield estimation methods, has a problem that its convergence becomes slower when analyzing a low probability event. Although importance sampling can overcome this problem, it is valid only when the alternative probability density function, g(x), is appropriate. This paper proposes a procedure to determine appropriate g(x) when g(x) is a mixture gaussian distribution. In the proposed procedure, clustering result determines the number of normal distributions constructing g(x) and mean of each gaussian is determined through bisection method. g(x) determined by the proposed procedure can sample near the boundary of failure region and this accelerates yield estimation by importance sampling. SRAM yield estimations of 6 to 24 dimensions are also conducted as examples. The number of Monte Carlo trials has been reduced by 2-5 orders compared to a crude Monte Carlo simulation. 1. SRAM 10 Mbit SRAM 0.1 % SRAM [4] Monte Carlo Monte Carlo [5] SRAM Monte Carlo [5] 0 [1, 3, 4, 6, 8] [2] Monte Carlo 6 1 Integrated Research Institute, Tokyo Institute of Technology 2 DC Research Fellow of the Japan Society for the Promotion of Science 3 Graduate School of Informatics Kyoto University 4 PD Research Fellow of the Japan Society for the Promotion of Science 1

2 [9] [9] ( ) SRAM [2, 9] 6 24 Monte Carlo SRAM 5 2. Monte Carlo Monte Carlo 2.1 Monte Carlo x p(x) P P = J (x) p(x)dx (1) J (x) (0) (1) f(x) f th 1 f(x) > f th J (x) = (2) 0 f(x) f th (1) Monte Carlo P MC P MC = 1 N J (x i ) (3) N N x i p(x) P MC Var(P MC ) [3] Var(P MC ) = 1 N (P MC P MC 2 ). (4) Var(P MC) Var(PMC ) P MC ρ(p MC ) P MC P MC 100(1 ɛ)% ±100δ% ρ 0 δ ρ 0 = Φ 1 (ɛ/2). (5) Φ 1 (p) 95% (ɛ = 0.05) ±20% ρ 0.10 (4) N MC = 1 PMC 1 (6) ρ 2 P MC ρ 2 P MC Monte Carlo N 2.2 Monte Carlo 1 [5] (1) P = J (x) p(x) g(x) g(x)dx = J (x) w(x) g(x)dx (7) g(x) P P P IS P IS = 1 N N J (x i )w(x i ) (8) P IS Var(P IS) ( ) Var(P IS ) = 1 1 N J (x i ) w(x i ) 2 2 P IS. (9) N N 2

3 g(x) = cj (x)p(x) ( (c ) ) Var(P IS) 1 N Var(P IS ) = Var J (x i )w(x i ) = Var(1/c) = 0. (10) N J (x)p(x) g (x) 0 g (x) g (x) g(x) g(x) g(x) 3. d x p(x) [ ] 1 p(x) = exp xt x. (11) (2π) d/2 2 p(x) g (x) g(x) (1) (2) (3) r S(r) r 1 n s r n f n f r 1 n f n s 1/n s n f /n s e 1/n s < e n f /n s n f > 1/e (12) Fig. 1 1 Incremental hypersphere sampling. θ c k g k 2 T k Fig. 2 Definition of solid of revolution T k. n F n f r 1 S(r 1) F F 0 1 ( 1 ) k(k > 1) g k g k c k g k c k θ k ( ) gk c k θ k = arccos (13) g k c k c k g k T k ( 2) r T k D k (r) D k (r) / 1: r max = r 1 2: r min = 0 3: repeat 4: r = (r max + r min )/2 5: S fail := a set of failure samples found in D k (r) 3

4 6: if the number of S fail > 0 then 7: r max := r 8: F k := S fail 9: else 10: r min := r 11: end if 12: until r max r min > r th 13: m k := argmax f(s) s F k s 5 n s (θ k /π) r max r min r th (12 ) r th D k (r max ) f(x) f th m k m k m(x) m(x) p(m 1) m(x) = nc p(m k=1 k) p(x m1) + p(m 2) nc p(m p(x m2) +... k=1 k) + p(m n c ) nc p(m k=1 k) p(x m n c ). (14) n c P IS ρ(p IS ) ρ 0 4. SRAM 3 6 SRAM PTM 65nm [7] 1 1 nmos 2 access tr., driver tr. SNM 0 Monte Carlo 6 ( Word Line Bit Line load tr. access tr. VL driver tr. VDD = 1V GND load tr. V R driver tr. access tr. 3 SRAM Fig. 3 Schematic of SRAM cell. #Bit Line 1 SRAM Table 1 Parameters of transistors in SRAM. nmos pmos (mv) , (nm) (mm 2 /Vs) (nm) (nm) 110, ) 12 (+ ) 18 (+) 24 (+ ) Monte Carlo ρ ( ) ρ < % (ɛ = 0.05) ±20% 3 n s Monte Carlo 4 20 Monte Carlo 1 Monte Carlo (6) N MC Monte Carlo (6,12 ) 95% 20% (5) ( n c ) 6,12,18 20 n c = n c = 2 2 n c = 3 n s ( ms 3.1e

5 2 Monte Carlo SRAM Table 2 Comparison of estimation results and number of samples. Monte Carlo p p IHS IS 6 - (1.8e+10) 5.6e k 11.8k k 0.8k - 2.0k 38.0k k 12 - (3.1e+08) 3.3e k 29.0k k 0.6k - 2.9k 80.1k k e e e k 31.2k k 1.4k k 122.1k k e e e k 75.1k k 1.3k k 197.9k k IHS IS 1/2 ) n c = 3 [10] ( ) ( ) 4.2 m(x) Monte Carlo d n s 2 d n s d Monte Carlo m(x)( m k ) m(x) J (x)p(x) x m(x) {x f } J (x f )p(x f ) m(x f ) ( ) 4 d ( ) 1.08 d SRAM 3 SRAM (6 ) [3,4,9] 3 [9] 20 3 [9] 1 2 m [9] [4] 3 [3] [ 10σ, 10σ] 2 Monte Carlo 5 [4] 5

6 250k 200k 150k 100k 50k Fig. 4 Breakdown of number of samples. Table 3 50k 40k 30k 20k 10k m(x) Fig. 5 The relation between standard deviation and the number of samples used for importance sampling. 3 Speed comparison with other importance sampling techniques. p IS () 5.6e k k [9] 5.6e k k [4] 5.7e k 296k [3] 5.0e k 220k 230k [3] 5. SRAM Monte Carlo Monte Carlo Monte Carlo NEDO 1) Chen, G., Blaauw, D., Mudge, T., Sylvester, D. and Kim, N.S.: Yield-driven nearthreshold SRAM design, Proceedings of IEEE/ACM International Conference on Computer-Aided Design, pp (2007). 2) Date, T., Hagiwara, S., Masu, K. and Sato, T.: Robust importance sampling for efficient SRAM yield analysis, Proceedings of IEEE International Symposium on Quality Electronic Design, pp (2010). 3) Dolecek, L., Qazi, M., Shah, D. and Chandrakasan, A.: Breaking the simulation barrier: SRAM evaluation through norm minimization, Proceedings of IEEE/ACM International Conference on Computer-Aided Design, pp (2008). 4) Doorn, T., ter Maten, E., Croon, J., DiBucchianico, A. and Wittich, O.: Importance sampling Monte Carlo simulations for accurate estimation of SRAM yield, Proceedings of European Solid-State Circuits Conference, pp (2008). 5) Hesterberg, T.C.: Advances in Importance Sampling, PhD Thesis, Statistics Department, Stanford Univ. (1988). 6) Kanj, R., Joshi, R. and Nassif, S.: Mixture importance sampling and its application to the analysis of SRAM designs in the presence of rare failure events, Proceedings of IEEE/ACM Design Automation Conference, pp (2006). 7) Predictive Technology Model (PTM): 8) Wang, J., Yaldiz, S., Li, X. and Pileggi, L.: SRAM parametric failure analysis, Proceedings of IEEE/ACM Design Automation Conference, pp (2009). 9),,, VLSI No.462, pp (2010). 10) (2)Vol.18, No.1 (2003). 6