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Variability-aware circuit design
by Fang Gong
With the semiconductor industry adopting 45nm and smaller technology nodes, process variation has become an increasingly challenging issue for both integrate circuit (IC) design and manufacturing, which results from the uncertainties from chemical mechanical polishing (CMP), etching, lithography, etc. These process variations significantly alter the device parameters of fabricated circuits (such as effective channel length, channel width, oxide thickness, etc.) and can change circuit behaviors. As such, the potential for circuit failure is high, which leads to decreased yields and possibly even expensive respins of the circuit.
The most effective approach for IC designers is to predict the stochastic behaviors of IC designs due to process variations before real manufacturing, which is called “variability-aware circuit design”. Using this technique, IC designers can know how the fabricated circuits would behave under typical process variations and, thereby, tune their designs to improve the statistical performance merits and yield rate. A critical issue is to accurately model the stochastic behavior of circuits.
Many statistical methods have been developed in the past few years. For example, asymptotic waveform evaluation (AWE)  approximates the “arbitrary” circuit behavioral distribution with the impulse response of a linear time-invariant (LTI) system by matching a few high-order moments. This approach requires no prior knowledge of the circuit behavioral distribution but needs expensive computational efforts to evaluate the high-order moments.
To resolve the performance issues associated with AWE, response-surface-method (RSM)-based methods have been proposed which model the circuit behavior as a polynomial function of the variable process parameters and further evaluate the high-order moments. For example, asymptotic probability extraction (APEX)  evaluates the RSM model using an ordinary least-square (OLS) regression method so that the number of needed SPICE simulations is equal to the total number of unknown coefficients in the polynomial function of RSM model. However, fully nonlinear AMS circuits tend to need higher-order RSM models (e.g. strongly nonlinear functions of random process variables) where the number of unknown coefficients and required SPICE simulations in OLS-based RSM method can increase exponentially. Therefore, an efficient and accurate method to evaluate high-order moments and further extract the stochastic behavior of AMS circuits is, still, urgently sought.
Solution of UCLA researcher
One UCLA graduate student researcher, Fang Gong, has been engaged in this research field since 2008 focusing on stochastic modeling and simulation for analog/mixed-signal (AMS) circuits. Recently he has successfully developed an approach to accurately model the stochastic behaviors of AMS circuits under process variations. It is worthwhile to point out the benefits that his solution can offer as below:
1. The proposed approach can recover “arbitrary” probabilistic distribution of circuit behavior without any assumption of the distribution types.
2. A large number of variable parameters (e.g., tens or hundreds of variable parameters) can be handled with linear complexity.
3. The stochastic behaviors of strongly nonlinear circuits can be extracted without any accuracy compromise.
These advantages enable IC designers to predict the stochastic circuit behavior accurately without resorting to expensive Monte Carlo simulations. In fact, the proposed method has been used by Samsung Company in Korea.
The proposed solution uses the moment-matching method in AWE  to recover the “arbitrary” PDF/CDF of stochastic circuit behavior, which can avoid potential errors by assuming distribution type beforehand. The key problem is to estimate the high-order moments of circuit performance merits. For example, the p-th order moment of performance y can be estimated as:
To compute this integral, APEX  first fits a polynomial function of performance merit y using a regression method and then calculates the high-order moments. As we mentioned before, the polynomial function in APEX leads to exponential complexity for strongly nonlinear circuit and large number of variable parameters, thereby, making it infeasible for practical applications.
The proposed solution evaluates the high order moments using sampling-like method, which picks a few “representative” samples of y and weights them with Pj to approximate the integral value as:
For example, Figure 1 shows a probability density map consisting of two normal distributed variables. Only part of the probability density map is plotted in order to show the interior “representative” sampling points.
Fig 1. The probability density map and “representative” sampling points.
To better approximate the entire sampling space, these “representative” samples should be chosen according to these conditions:
1. The samples for each variable parameter should follow its known marginal distribution;
2. Various correlations and other relationships between the variable parameters should remain intact;
3. The chosen samples should fully cover the entire sampling space to provide closer approximation;
4. These samples should be incoherent so that the minimum number of samples are needed.
As such, only a few of the samples (e.g., sample-size is linear with number of variable parameters) can be used to accurately estimate high-order moments of circuit performance merits and further recover their PDF/CDF.
The proposed method has been validated on a number of different circuits, and one case study is shown here: an operational amplifier (OPAMP) with a total of 90 random variables, where the AC performance merit (e.g., bandwidth) is considered as circuit behavior.
Fig 2. The extracted PDF/CDF from proposed methods for OPAMP example.
Some important observations can be found in the results: first, the proposed solution can accurately predict non-Gaussian distribution and other arbitrary distributions for strongly nonlinear circuit behavior; second, the proposed solution can achieve 1% error, on average, for CDF approximation using only 90 samples, while Monte Carlo method needs 1.7e+5 samples for the same accuracy. It implies that the proposed method achieves 1888X speedup over Monte Carlo and nearly linear complexity in this high-dimensional problem.
A novel solution has been developed to accurately predict the stochastic circuit behavior for analog/mixed-signal circuits, in particular when strongly nonlinear circuits are studied. More importantly, the proposed method can achieve linear complexity in terms of SPICE simulation cost which can be of benefit to many high-dimensional problems found in the industry.
More details can be checked out from the website: http://www.ee.ucla.edu/~fang08/ or email
 Lawrence T. Pillage, Ronald A. Rohrer: "Asymptotic waveform evaluation for timing analysis". IEEE Trans. on CAD of Integrated Circuits and Systems 9(4): 352-366 (1990)
 Xin Li, Jiayong Le, Padmini Gopalakrishnan and Lawrence Pileggi, "Asymptotic probability extraction for nonnormal performance distributions," IEEE Trans. on Computer-Aided Design of Integrated Circuits and Systems (TCAD), vol. 26, no. 1, pp. 16-37, Jan. 2007.
Fang Gong is a Ph.D. candidate and graduate student researcher in the Department of Electrical Engineering at University of California, Los Angeles (UCLA) since 2008. His research interests include stochastic circuit modeling and simulation, parallel and distributed computing, and healthcare. He has around 20 referred publications on top-ranked international conferences and top-tier journals. In addition, he has served as a peer reviewer for many conference and journal papers including DAC, ICCAD, ASPDAC, TVLSI, TCAD, TCAS, TODAES.
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