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Volume 2011 |Article ID 471903 | https://doi.org/10.1155/2011/471903

Rouwaida Kanj, Rajiv Joshi, Sani Nassif, "The Impact of Statistical Leakage Models on Design Yield Estimation", VLSI Design, vol. 2011, Article ID 471903, 12 pages, 2011. https://doi.org/10.1155/2011/471903

The Impact of Statistical Leakage Models on Design Yield Estimation

Academic Editor: Shiyan Hu
Received19 Oct 2010
Accepted11 Dec 2010
Published22 Feb 2011


Device mismatch and process variation models play a key role in determining the functionality and yield of sub-100 nm design. Average characteristics are often of interest, such as the average leakage current or the average read delay. However, detecting rare functional fails is critical for memory design and designers often seek techniques that enable accurately modeling such events. Extremely leaky devices can inflict functionality fails. The plurality of leaky devices on a bitline increase the dimensionality of the yield estimation problem. Simplified models are possible by adopting approximations to the underlying sum of lognormals. The implications of such approximations on tail probabilities may in turn bias the yield estimate. We review different closed form approximations and compare against the CDF matching method, which is shown to be most effective method for accurate statistical leakage modeling.

1. Introduction

With technology scaling, memory designs are the first to suffer from process variation. Density requirements in memory cells, latches, and register files aggravate variability effects and often undergo performance and yield degradation. More recently, the trend is starting to become more visible in peripheral logic [1], and random variations can lead to reduced noise margins. From a leakage perspective, general interest has been to model the statistical leakage distribution of the full-chip [2]. However, under extreme device conditions, leakages may inflict delays or faulty behavior: example false reads. In memory designs, bitline leakages can further aggravate the variability effects and impact readability of the design. Often many devices are stacked on the bitlines, and it has become necessary to statistically account for and model those many independent and, in most cases, identical, leaky sources. A typical array column can have around 16 to 64 cells per bitline. With the aim of capturing rare fail events accurately and modeling the variability space [3], it becomes important to emulate those independent variability sources by a single statistically equivalent device, the main goal here being to reduce the dimensionality space. This is especially important for response surface modeling-based integration methods or variance reduction techniques.

The threshold voltage variation due to random dopant fluctuation [4] for the single device, however, is different from that of the equivalent device. In a typical linear model, the equivalent statistical model would be derived from linear relationships with respect to the individual devices, and what the designers often refer to as the square root law; as the device area (𝐴) increases, the threshold voltage variation scales ~√1/𝐴. This would let us model “𝑛” small devices with one large device whose variation is ~√1/𝑛 times the original variation. However, as was indicated in [5, 6] this model lacks accuracy when modeling the highly nonlinear leakage currents where the dependency is exponential on the threshold voltage (the source of variation). In fact, as we will explain in the following sections, the leakage current is distributed lognormally, and the problem of the equivalent device is known mathematically as the sum of lognormals. Mathematically, a true and exact closed form does not exist for the method, and historically there have been many approximations. There is a lot of literature on the sum of lognormals, and applications span a wide range of domains including economics, finance, actuarial sciences, and others, with engineering being one of the oldest. The most famous approximations to the sum of lognormals are Fenton-Wilkinson’s [7] and Schwartz-Yeh (SY) [8]. For circuit design, those models have been studied in [5, 6] to enable modeling the threshold voltage distribution of the equivalent device. The authors in [5] relied on the Fenton-Wilkinson’s (FW) method to maintain a reasonable 3-sigma estimate (99th percentile). It is often indicated that when ğœŽâ€‰dB > 4, FW method may not result in proper approximation of the lognormal distribution. On the other hand, the authors in [6] proposed to use the Schwartz-Yeh (SY) methodology to model the equivalent threshold voltage distribution of a 3-fin device. Again, the approximation is qualified, in this case for a small number of summands, based on a thousand samples thereby ignoring the accuracy of the model in the tail regions which is critical to estimation of low fail probabilities. In the following sections, we review the modeling trends in the critical regions, focusing on the cumulative density function (CDF) matching method [9] as a viable solution for reliable statistical modeling when the tail region modeling is key to the accuracy of the yield estimate.

The paper is organized as follows. Section 2 introduces a basic circuit example to describe the need for the single equivalent device modeling. Section 3 reviews the mathematical background and assumptions of the most common sum of lognormals approximation methods. It also provides a summary chart of the different approaches under study. Section 4 evaluates the impact on fail probability estimations. Finally conclusions are presented in Section 5.

2. Equivalent Device Modeling

Figure 1 illustrates an 8T cell/register file. The circuit operates like a 6T SRAM cell during cell Write. During Read, the read wordline turns ON. If the cell is storing a “1” on the right-side-node, in this example, the Read stack will turn ON, thereby discharging the read bitline (RBL).

Driven by density requirements, many cells share the same bitline and sense amplifier (or read logic). Only one cell per column is accessed at a given time; other cells remain unaccessed. For the example of Figure 1, the RWL of the unaccessed cells is turned OFF. However, based on the data storage, it is possible that the data stack device turns on, thereby rendering the stack leaky. Figure 2 illustrates a scenario where the accessed cell is OFF. RBL is expected to remain high. However, leakages through the accessed stack and unaccessed stacks may falsely discharge the RBL node. This together with noise, mismatch in the read logic, and other environmental effects can degrade the yield of the system and increase the probability of a false read.

Note that this phenomenon is also manifested in other memory and domino logic designs. For example in the case of 6T SRAM, the devices sharing the WBL (see Figure 1) can be storing 1’s thereby leaking onto the WBL and degrading the Read “0” performance in a 6T fashion. Hence there is a need to account for the statistical effects of the leaky devices in any design yield optimization and study. This can dramatically increase the dimensionality of the problem thereby reducing the efficiency of statistical integration, response surface methods, or any methods that try to model the failure region and that suffer from sparsity and the curse of dimensionality. Consider for example a study that involves variability in 64 devices on the bitline along with variability sense amp devices. Thus, a model for a statistically equivalent device to emulate the multiple leaky devices can significantly simplify the complexity. Figure 3 summarizes the problem. Given:𝑛 devices 𝑇1,…,𝑇𝑛. The threshold voltage variation 𝛿𝑉𝑇 is normally distributed with zero mean and standard deviation ğœŽğ‘‰Ti=ğœŽ0. That is, 𝛿𝑉Ti∼𝑁(0,ğœŽ0)Find:𝛿𝑉Teqv distribution for the device 𝑇eqv such that“ 𝑛∗𝐼Teqv” is distributed according to “∑𝐼Ti”.

Note that we assume here independent and identical devices; the problem can be generalized to nonidentical/correlated devices.

Leakage Current Model
Leakage current as a function of the threshold voltage variation can be modeled according to (1). Hence it portrays an exponential dependency on the random variable 𝛿𝑉𝑇𝐼leak=ğ‘’ğ‘Žğ›¿ğ‘‰ğ‘‡+𝑏,ğ‘Žâˆğ¾ğ‘‡ğ‘ž(1)ğ‘ž is the electron charge, 𝐾 is boltzmann constant, and 𝑇 is the temperature. For simplicity, 𝑇eqv has the same device width as the other 𝑇𝑖, but it is followed by a current multiplier to model the equivalent leaky device and solve for the new 𝛿𝑉Teqv distribution: the set of (2) represents how the problem can be reduced. Parameters 𝑍𝑖 are normal variables, 𝑒Zi are lognormal variables (see next section for definitions), and 𝑒𝑍eqv (and 𝑌) are distributed according to a sum of lognormal. A sum of lognormals is not distributed lognormally, and hence 𝑍eqv is not normal. In the following section we review the sum of lognormals characteristics and possible methods of approximation that can be used to generate the statistical characteristics of 𝑍eqv and hence 𝛿𝑉Teqv𝐼sum=𝐼leak1+⋯+𝐼leak𝑛=𝑛∗𝐼leak-eqv,𝐼sum=𝑛𝑖=1ğ‘’ğ‘Žğ›¿ğ‘‰Ti+𝑏=ğ‘›âˆ—ğ‘’ğ‘Žğ›¿ğ‘‰Teqv+𝑏,𝑍eqv=ğ‘Žğ›¿ğ‘‰Teqv;𝑍𝑖=ğ‘Žğ›¿ğ‘‰Ti,𝑒𝑍eqv=1𝑛𝑛𝑖=1𝑒Zi=1𝑛𝑌.(2)

3. Sum of Lognormals

In this section we will focus on methods to approximate models for 𝑌 (2), and compare their accuracy to the CDF matching approach. First we present some background information.

Lognormal Distribution
Let 𝑍=ln(𝑋). If 𝑍 is normally distributed ~𝑁(𝜇𝑧,ğœŽğ‘§), then 𝑋 is lognormally distributed and its probability density function is [10] ğ‘“ğ‘‹âŽ§âŽªâŽ¨âŽªâŽ©1(𝑥)=√2ğœ‹ğœŽğ‘§ğ‘’âˆ’(𝑧−𝜇𝑧)2/2ğœŽğ‘§2𝑥>00𝑥≤0.(3) Often the Gaussian Variable 𝑉=10log10(𝑋) is defined. 𝑉 is in decibels (dB), and 𝑉≅4.34𝑍; if 𝑍∼𝑁(0,1), 𝑉∼𝑁(0,4.34 dB).

Sum of Lognormals
A common way to compute the probability density function (pdf) of a sum of independent random variables, would be from the product of the characteristic functions, CF, of the summands [11]; a characteristic function of random variable 𝑍 is the expected value (𝐸[𝑒𝑗𝑤𝑍]). However, for the case of the sum of lognormals, ∑𝑒𝑌=Zi, a closed form of the CF does not exist, and approximations are used instead to represent the density function of the sum of lognormals. Some approximations include modeling the sum of lognormals as an “approximately” lognormal variable (see (4)); where 𝑍𝑚. is normally distributed (𝑚 refers to the method of approximation) 𝑌=𝑛𝑖=1𝑒Zi≈𝑒𝑍𝑚.(4) Common techniques are SY and FW methods mentioned earlier. In the following section, we will revisit those methods and compare them to CDF matching. Figure 4 summarizes the four techniques that will be visited: (1)Fenton-Wilkinson method,(2)Schwartz-Yeh method, (3)Log Moment Matching approximation, (4)CDF matching.All these methods except the CDF matching one assume that 𝑌 is a lognormal variable; that is, that 𝑍𝑚 (and hence 𝛿𝑉𝑡) is a normally distributed Gaussian variable whose mean and standard deviation are to be found.

3.1. Fenton-Wilkinson

Fenton and Wilkinson method estimates the standard deviation and mean of the Gaussian variable 𝑍fw by matching the first two moments of 𝑌, 𝐸[𝑌] and 𝐸[𝑌2], to those of the sum of lognormals (the two sides of (4)) according to (5) 𝑌=𝑒Zfw,𝐸𝑒𝑍fw𝑒=𝐸𝑍1+⋯+𝑒𝑍𝑛,𝐸𝑒2𝑍fw𝑒=𝐸𝑍1+⋯+𝑒𝑍𝑛2.(5) It is a well-known fact [10] that given a Gaussian variable, 𝑥, with mean and standard deviation 𝜇𝑥 and ğœŽğ‘¥ then (6) holds𝐸𝑒𝑛𝑥=𝑒𝑛𝜇𝑥+(1/2)𝑛2ğœŽ2𝑥(6) by relying on (5) and (6) we can obtain a closed form for 𝜇Zfw and ğœŽZfw according to (7) for the case of independent variables 𝑧𝑖∼𝑁(𝜇𝑖,ğœŽğ‘§)ğœŽZfw=î„¶î„µî„µâŽ·îƒ¬î€·ğ‘’lnğœŽğ‘§2∑𝑒−12𝜇𝑖∑𝑒𝜇𝑖2,𝜇+1Zfw𝑒=ln𝜇𝑖+ğœŽğ‘§22âˆ’ğœŽ2Zfw2.(7)

3.2. Schwartz and Yeh

Again the methodology starts with the assumption that the sum of lognormals is approximately lognormally distributed 𝑌=𝑒Zsy,(8) where 𝑍sy is a Gaussian variable ~𝑁(𝜇Zsy,ğœŽZsy). It approximates the mean and standard deviation of the 𝑍SY=ln(𝑌), with numerical recursion; whereas FW estimates the first two moments of 𝑌. Schwartz and Yeh method relies on the ability to exactly compute the mean, 𝜇Zsy, and standard deviation, ğœŽZsy, of 𝑍SY=ln(𝑌) for the case when the number of lognormal variables in the sum 𝑛=2. For 𝑛>2, the method then relies on a recursive approach, adding one factor at a time. Hence, the mean and standard deviation of 𝑍sy𝑒=ln𝑍1+⋯+𝑒𝑍𝑛(9) can be derived from the following set of generalized equations (10); 𝑍sy(𝑘−1) is assumed to be normally distributed; 𝑘=2,…,𝑛, and the 𝑍𝑖 are assumed to be uncorrelated in (10) 𝑍sy(𝑘)𝑒=ln𝑍sy(𝑘−1)+𝑒𝑍𝑘,𝜇Zsy(𝑘)=𝜇Zsy(𝑘−1)+𝐺1,ğœŽ2Zsy(𝑘)=ğœŽ2Zsy(𝑘−1)+𝐺2−2âˆ—ğœŽZsy(𝑘−1)𝐺3ğœŽ2𝑤(𝑘),(10) where 𝑤𝑘 and 𝐺𝑖(𝑖=1,2,3) are defined in (11). Finally set of (12)-(13) illustrates how the 𝐺𝑖’s can be computed according to [12]; this slightly modified implementation was intended to circumvent the round off error of integration of the original Schwartz and Yeh implementation [13]𝑤𝑘=𝑍𝑘−𝑍sy(𝑘−1),𝐺𝑖=𝐺𝑖𝜇𝑤𝑘,ğœŽğ‘¤ğ‘˜î€¸ğœ‡ğ‘–=1,2,3,𝑤𝑘=𝜇𝑍𝑘−𝜇Zsy(𝑘−1),ğœŽ2𝑤𝑘=ğœŽ2𝑍𝑘+ğœŽğ‘2sy(𝑘−1),(11)𝐺1=𝐴0+𝐼1,𝐺2=𝐼3+2𝐼4+ğœŽ2𝑤𝑘𝐼0+𝜇𝑤𝑘𝐴0,𝐺3=ğœŽ2𝑤𝑘𝐼0+𝐼2,𝐴0=ğœŽğ‘¤ğ‘˜âˆšğ‘’2𝜋−𝜂2/2+𝜇𝑤𝑘𝐼0,𝜇𝜂=âˆ’ğ‘¤ğ‘˜ğœŽğ‘¤ğ‘˜,𝐼𝑖=10â„Žğ‘–(𝑣)𝑣−1𝑑𝑣=∞−∞2ℎ(𝑣)𝑣𝑢cosh𝑥𝑑𝑥=âˆžâˆ’âˆžğ»(𝑥)𝑑𝑥,𝑢=exp(−2sinh𝑥);𝑣=(1+𝑢)−1.(12) Thus, at each step of the recursion, we compute the mean and standard deviation of 𝑤𝑘. The integrals 𝐼𝑖 are then numerically computed using the functions ℎ defined in (13). Their values are used to solve for the 𝐺𝑖’s in order to evaluate (10). The final estimate for the 𝑍SY mean and standard deviation is reached at 𝑘=ğ‘›â„Žğ‘–âŽ§âŽªâŽªâŽªâŽ¨âŽªâŽªâŽªâŽ©1(𝑣)=√𝑒2𝜋−(ln𝑣−𝜂)2/2𝑓,𝑖=0,𝑤(ln𝑣)−𝑓𝑤(𝑓−ln𝑣)ln(1+𝑣),𝑖=1,𝑤(ln𝑣)−𝑓𝑤(−ln𝑣)1+𝑣−1−1𝑓,𝑖=2,𝑤(ln𝑣)−𝑓𝑤(−ln𝑣)ln2(1+𝑣),𝑖=3,−𝑓𝑤1(−ln𝑣)ln(𝑣)ln(1+𝑣),𝑖=4,fw=√2ğœ‹ğœŽ2𝑤𝑘𝑒−(𝑤𝑘−𝜇𝑤𝑘)2/2ğœŽ2𝑤𝑘.(13)

3.3. Log Moment Matching Approximation

While the previous two approaches find the moment matching using analytical or recursive analytical solutions, this approach relies on sampling instead to compute the moments according to (10) [9]. Similar to the SY method, it estimates the 1st two moments of ln(𝑌). It does not pertain to a closed-form solution, but it maintains the lognormal assumption. Thus, the recursive analytical approach in SY and this approximation do not always converge as we will see in Section 3.5, especially that the SY assumptions is held exact when 𝑛=2𝑌≈𝑒𝑍lmm,𝜇𝑍lmm[],ğœŽ=𝐸ln(𝑌)𝑍lmm=𝐸(ln𝑌)2[]−𝐸ln(𝑌)2.(14)

3.4. CDF Matching

This is the closest possible to the true sum. Unlike the previous approximations, it does not rely on the lognormal approximation. We build a piece-wise linear probability density function from the Monte Carlo samples. Our goal is to demonstrate the difference between this approach and other lognormal approximations, when it comes to tail regions modeling and the resultant probability of fail estimations. Key features of the problem are(i)the piece-wise linear fit for the density function of ln(𝑌) is non-Gaussian.(a)𝑔pwl(⋅) is defined by pairs (𝑃𝑗,𝑌0𝑗), such that Prob(𝑌>𝑌0𝑗)=𝑃𝑗, hence Prob(ln(𝑌)>ln(𝑌0𝑗))=𝑃𝑗.(b)The pwl function can be sparse in the center of the distribution and more dense in the tails to adequately model the low fail probability region. In an extreme fashion, the tail probabilities can be recomputed from tail samples to avoid interpolation errors. (ii)generating the 𝑌 samples is cheap, so is the 𝐼sum sample once the function or even tables of (𝐼,𝛿𝑉𝑇) are available. Interpolation, bootstrapping, and other techniques can reduce the number of real simulations needed and still enable good confidence in the density function. After all, the previous approaches do rely on the availability of a closed form function for 𝐼(𝛿𝑉𝑇). The number of samples is inversely proportional to the tail probability of interest; for example, if we are looking for accurate probabilities in the range of 1𝑒−4, then we need to have replications of samples that are larger than 1𝑒4. Replications add to the confidence in the expected tail probability. After all the interest is in the CDF tails mainly,(iii)finally, this model can accommodate any complex nonlinear function (𝐼=𝑓(𝛿𝑉𝑇)); even if it is different from the exponential approximation above, (iv)most importantly, once the distribution of 𝐼sum is available, 𝑉𝑇 distribution is derived by reverse fitting (𝑓−1(𝐼sum)) samples.

Note that for purposes of comparison, in this study all methods share the same exponential current model.

3.5. Theoretical Experiments

In this section, we compare the different methods ability to model 𝑌 (Figure 4). We study both the moment matching abilities as well as the upper and lower CDF tails. The study is performed over different combinations of 𝑛 and ğœŽğ‘§ (the standard deviation of 𝑍𝑖):î€·ğœŽğ‘§î€¸Ã—=0.5,1,2,3,4(𝑛=2,8,16,64)(15)ğœŽğ‘§ range corresponds to the range 2 dB–16 dB. Recall that often 4 dB is considered as a critical threshold for how accurate FW methods can model the distribution. Also as demonstrated in [6] standard deviation of the threshold voltage of scaled devices can exceed 4–8 dB (this is based on the leakage coefficient in (1)).

For each (𝑛,ğœŽ) combination, multiple replications of 1 million samples are studied; this enables good estimates for the low fail probabilities. Figure 5 plots the mean and standard deviation of ln(𝑌) obtained from the different approximations. FW method falls behind for larger ğœŽğ‘§ (>1 or 4 dB). Recall that this method was intended to match the moments of 𝑌 and not ln(𝑌). We note that the FW estimates do underestimate the mean and overestimate the variance of ln(𝑌) for larger dB values. Finally, the SW and LMM methods do match the CDF well, yet they do differ from each other a bit for larger sigma values, given that the former is a recursive approximation.

Figure 6 illustrates histogram (pdf plot) of ln(𝑌); the FW method does not model the body of the (𝑍 or 𝛿𝑉𝑡) distribution well compared to other methods; this was also indicated in [6] for small 𝑛. The trend is even more obvious as the number of summands (𝑛) increases. However, to obtain a complete picture, there is a need to study the lognorm matching and the tail regions which we will cover next.

Figure 7 plots the mean and standard deviation of 𝑌 obtained from the different approximations. SY and LMM methods that match moments of ln(𝑌) falls behind for larger ğœŽğ‘§ (>1 or 4 dB). This is true for large 𝑛 (>2), and we note that SY tends to particularly underestimate the variance of the sum of lognorms; this effect is most visible when the variables are identical and uncorrelated as is the case in this study. To study the tail regions, we rely on a “tail log plot” as illustrated in Figure 8. Without loss of generality we set the 𝑥-axis in these plots to be in dB; a small shift can mean large change in 𝑌 values. The plot is derived from the density function (𝑃(𝑦>𝑦0)) and its complement (1−𝑃). It is such tail probabilities that are linked to the fail probabilities of a design. Note that for the case of leakages, the right-side tail is critical for the fails (larger Y values correlate with larger leakage values in real applications).

Figure 9 illustrates the tail probability plot as function of ğœŽğ‘§ for 𝑛=64. We get good match for all the methods for small ğœŽ. For critical ğœŽ values (4–8 dB), we note that SY and FW do miss the right tail modeling. As ğœŽ increases, FW tries to catch towards modeling the right tail by missing on the left tail model. Figure 10 illustrates the tail probability plots as function of 𝑛 for ğœŽğ‘§=2 (8 dB). SY and LMM methods have larger errors in modeling the right tail. FW error increases with increasing 𝑛.

4. Case Study: Leak-Down Time Comparisons

In this section, we extend the analysis to study the impact of the different modeling schemes, compared to the CDF matching method, on the probability of fail estimations. Figure 11 illustrates a summary of 𝛿𝑉Teqv distribution approximations based on the relation between 𝛿𝑉Teqv and 𝑌 in (2); except for the CDF matching method, 𝛿𝑉Teqv is modeled as Gaussian random variable ~𝑁(𝜇eqv,ğœŽeqv). The distributions are then used to analyze the leak-down of RBL in the circuit of Figure 12. The time for the bitline to discharge to 50% of the rail under extreme noisy corner voltage is then estimated.

Figures 13, 14, and 15 illustrate the normalized time-to-leak distributions for the case of 16 cells/bitline. 𝑠0 represents a lower limit on the threshold voltage standard deviation of a 45 nm cell device; its value is set to the equivalent of ğœŽğ‘§=0.6 (or 2.5 dB). This is conservative especially that as technology scales we expect more variability and additional sources of variability like the random telegraphic noise can add to the 𝑉𝑇 variation; cases for standard deviation of 1.3s0 and 1.6s0 are also plotted. SY, FW, and LMM methods overestimate the time-to-leak from 10% for s0 to close to 100% for 1.6s0 (see horizontal arrow in the figures; this corresponds to system yield around 4.5 sigma). More importantly, this leads to underestimating the probability of the number of elements failing at a given leak-time. Note that time-to-leak values can be critical relative to operating frequencies and accurate prediction is needed for robust designs. Thus, we are interested in computing the ratio of the probability of fails (vertical arrows in figures) for predicted time-to-leak values. Figure 16 summarizes the ratio of the true (cdf) probability of fail compared to that of the other methods (SY, FW, and LMM) at their 4.5 sigma yield leak-time. Each experiment is based on the average of 25 × 1 million replications. This is done for the case of 16 and 64 cells/bitline and at increments of 0.1∗s0. We note that the SY, FW, and LMM methods underestimate the probability of fail 10×–147×.

5. Conclusions

We study the ability of different sums of lognormal approximation to emulate the leakage of multiple leaky devices by a single equivalent device. With the goal of rare event estimation, tail distributions are examined closely. Modeling the tail probability by CDF matching is found to be critical compared to the Fenton-Wilkinson and Schwartz-Yeh methods that are found to underestimate the sum of lognorms and hence overestimate the fail probability by 10×–147×; this trend is expected to increase as the variability increases with scaling technology.


  1. R. Joshi, A. Pelella, A. Tuminaro, Y. Chan, and R. Kanj, “The dawn of predictive chip yield design: along and beyond the memory lane,” IEEE Design and Test of Computers, vol. 27, no. 6, pp. 36–45, 2010. View at: Publisher Site | Google Scholar
  2. E. Acar, K. Agarwal, and S. R. Nassif, “Characterization of total chip leakage using inverse (reciprocal) gamma distribution,” in Proceedings of the IEEE International Symposium on Circuits and Systems (ISCAS '06), pp. 3029–3032, May 2006. View at: Google Scholar
  3. R. Kanj, R. Joshi, and S. Nassif, “Mixture importance sampling and its application to the analysis of SRAM designs in the presence of rare failure events,” in Proceedings of the 43rd Annual Design Automation Conference (DAC '06), pp. 69–72, 2006. View at: Publisher Site | Google Scholar
  4. A. Asenov, “Random dopant induced threshold voltage lowering and fluctuations in sub-0.1 fjim MOSFET's: a 3-D "atomistic" simulation study,” IEEE Transactions on Electron Devices, vol. 45, no. 12, pp. 2505–2513, 1998. View at: Google Scholar
  5. J. Gu, S. S. Sapatnekar, and C. Kim, “Width-dependent statistical leakage modeling for random dopant induced threshold voltage shift,” in Proceedings of the 44th ACM/IEEE Design Automation Conference (DAC '07), pp. 87–92, June 2007. View at: Publisher Site | Google Scholar
  6. S. H. Rasouli, K. Endo, and K. Banerjee, “Variability analysis of FinFET-based devices and circuits considering electrical confinement and width quantization,” in Proceedings of the IEEE/ACM International Conference on Computer-Aided Design (ICCAD '09), pp. 505–512, November 2009. View at: Google Scholar
  7. L. F. Fenton, “The sum of log-normal probability distibutions in scattered transmission systems,” IRE Transactions on Communications Systems, vol. 8, pp. 57–67, 1960. View at: Google Scholar
  8. S. C. Schwartz and Y. S. Yeh, “On the distribution function and moments of power sums with log-normal components,” The Bell System Technical Journal, vol. 61, no. 7, pp. 1441–1462, 1982. View at: Google Scholar
  9. R. Kanj, R. Joshi, and S. Nassif, “Statistical leakage modeling for accurate yield analysis: the CDF matching method and its alternatives,” in Proceedings of the International Symposium on Low Power Electronics and Design (ISLPED '10), pp. 337–342, Austin, Tex, USA, 2010. View at: Publisher Site | Google Scholar
  10. N. C. Beaulieu, A. A. Abu-Dayya, and P. J. McLane, “Estimating the distribution of a sum of independent lognormal random variables,” IEEE Transactions on Communications, vol. 43, no. 12, pp. 2869–2873, 1995. View at: Publisher Site | Google Scholar
  11. A. Papoulis, Probability, Random Variables and Stochastic Processes, McGraw Hill, New York, NY, USA, 1991.
  12. P. Cardieri and T. S. Rappaport, “Statistics of the sum of lognormal variables in wireless communications,” in Proceedings of the 51st Vehicular Technology Conference (VTC '00), pp. 1823–1827, May 2000. View at: Google Scholar
  13. C. L. Ho, “Calculating the mean and variance of power sums with two log-normal components,” IEEE Transactions on Vehicular Technology, vol. 44, no. 4, pp. 756–762, 1995. View at: Publisher Site | Google Scholar

Copyright © 2011 Rouwaida Kanj et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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