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fter many weeks or months of laboring over your design with the best available
models and a favorite simulator, you and your management commit it to production.
You breathe a sigh of relief as the first manufactured units work their
way to final test. Then, just as you are transitioning to the next project,
the bad news comes: yields are unacceptably low.
A likely cause is that you overlooked the effects of the stochastic nature of the manufacturing process. Or perhaps you assumed that random fluctuations in the myriad of process steps would more or less cancel each other out--and this time they seem to be canceling rather less than more. Whatever the reason, too many of the units failed to measure up and became scrap.
Now management is on edge. Customers are unsettled and looking around for options. The part can be produced, but the low yield means that manufacturing costs will be too high. Yields can be improved, but that will require more engineering time. In retrospect, it seems likely that the application of statistical circuit design techniques could have largely prevented these problems.
NBA captures the statistics of the process from data vectors--collections of parameter values for all of a design's statistically variant components. These parameters can be just about anything so long as a given set of values uniquely represents each component sample. Herein lies NBA's flexibility. It allows the input data vectors to be formed from measurements or from model parameters that represent each of the design's components. Each component's model can have any number of parameters, and the number of components in the design can be quite large.
The technique is a straightforward means of estimating a lower bound on yield, and requires much less computational effort than the well-known technique of Monte Carlo yield estimation. The Monte Carlo approach, on the other hand, provides an actual yield estimate, not just a lower bound. It randomly samples the distribution of combinations, performs a complete circuit simulation, and counts how many give an acceptable result. For typical distributions, most of the computation effort is expended on simulations that have near-nominal responses.
NBA requires less computational effort because it first examines the overall distribution of model parameter values or measurements of components and then selects only those combinations in the tails of the distribution--the combinations most likely to cause yield problems--for a complete circuit simulation. The combinations selected enclose a user-specified fraction of the distribution. NBA's lower bound on yield is obtained by simulating all the selected combinations and determining whether they all result in circuits with acceptable responses. If they do, then the yield will be at least as large as the user-specified enclosure fraction. If not, then the enclosure fraction is reduced and the process is repeated until a fraction is found for which all of the circuits work properly.
In addition to analyzing yield, the technique identifies a nominal model for each of the statistically variant components in a design. Nominal models are useful for initiating the design process because they emulate the various components' ideal behaviors.
We can also see why NBA provides only a lower bound on the yield and why this bound can be pessimistic. Compare the 85 percent enclosure in the I-V space with the hypothetical acceptability region outlined in white. NBA provides only a lower bound because the 85 percent enclosure is round and the acceptability region has corners that are not enclosed. Grossly pessimistic yield estimates can be obtained by applying NBA only once and noting that all the outcomes are acceptable. This situation would pertain if, for example, the hypothetical acceptability region were expanded. This difficulty can be corrected simply by iterating the NBA enclosure percentage upward until a failure is found.
NBA has a number of other constraints and limitations on its practical application. An important constraint is that the functions that map the input space to the response space must be approximately monotonic if the property of equal enclosures is to hold. Without equal enclosures in the two spaces, the algorithm does not provide a lower bound on yield. Approximately monotonic means that the mapping functions can have some ripple. However, the ripple must be small relative to the overall change in the response functions across the acceptable response region.
Although approximate monotonicity is difficult to guarantee in all cases, the author conjectures that the vast majority of practical circuit designs are approximately monotonic. The monotonicity of a design can be explored by artificially adjusting the component parameters to trace paths along rays emanating from the nominal design while verifying that the circuit responses are monotonically increasing or decreasing. Another method consists of iteratively increasing the NBA enclosure percentage while verifying for all pair-wise combinations of the response variables that the area (or volume) of enclosure in the response space is constantly increasing.
Ultimately, even if the circuit is not approximately monotonic with respect to its parameters, NBA can be used to explore for worst-case designs in the performance space.
Another point to watch regarding dimensionality is that the method's density estimate degrades as dimensionality increases. The author has shown proper functionality with dimensionalities as high as 320. Proper performance into low- to mid-thousands is viable.
Beyond the logistics of getting the data and algorithm performance, the user should be aware of an interesting phenomena of very high-dimensional spaces: as the number of dimensions goes to infinity, the volume contained in a very thin shell on the surface of a hypersphere containing one unit of volume also goes to unity. That is, all the volume of a high-dimensional hypersphere is on its surface. This phenomena ultimately limits NBA's ability to return a meaningful set of boundary points for a given enclosure percentage.
Several tools are available to help the user of NBA limit dimensionality. Direct knowledge and engineering judgment assist in selecting appropriate model parameters or measurements to represent each component in the overall design. Obviously, parameters to which the component's responses are not sensitive should be avoided. If using measurements, do not choose data that shows little variation across the samples. Avoid using parameters or measurements that are highly correlated with one another. Beyond engineering judgment, employ correlation and sensitivity analyses to determine the important parameters or measurements.
NBA has different data requirements from other statistical circuit design methodologies. This is both good and bad. To apply other methods, it is normal to use a compact set of statistics derived from the raw data--mean vectors and covariance matrices for Gaussian distributions or more exotic distributions along with their pertinent statistics. NBA, however, needs the raw data. This is not as onerous a demand as it may at first appear, however. To get the compact statistics in the first place, the raw data must be gathered--no extra work here. Although the volume of NBA's data is much larger, it is not impractically so, and today's Internet makes user access simple and quick.
A big advantage of NBA is that it never requires that a probability density function be fitted to the data. It thereby avoids a host of problems, not the least being the often unjustified assumption that the data fits a Gaussian model. Even if the probability density model is appropriate, fitting any model to any data inherently leads to an imperfect emulation.
The last major concern for NBA is how large a circuit can be analyzed. To answer that question, it is helpful first to consider the various statistical circuit design flows that use NBA. As is turns out, there are several answers depending on the flow that one employs.
In common with all the others, the MCYE flow assumes that the design under analysis has a total of five transistors of different sizes [light blue block]. Two hundred samples of each of the transistors are measured to produce 1000 measurement datasets [light purple block]. One thousand models are then extracted [orange block]. Next, form as many combinations of five transistors as it is practical to simulate (200-1000) by randomly selecting a model from each pool of 200 transistor models [green block]. Then run full-circuit simulations with the combinations and get a yield estimate [gray block]. If the yield estimate is low, repeat as shown until the yield becomes satisfactory [black block].
The NBA-MD flow introduces an extra step into the MCYE flow--perform NBA after choosing a minimum desired yield [tan block]. In addition, it transposes the steps for forming the random combinations and extracting models. This transposition matters both from the standpoint of how many combinations are considered (coverage of a high-dimensional space) and because of the amount of effort expended on extracting the models and running the full-circuit simulations.
The NBA-MP flow is essentially that of the MCYE flow. The NBA-MP flow has a "perform NBA" block interjected between the "form random combinations" and "full-circuit simulation" blocks of the MYCE analysis. This flow diminishes the effort in the iteration loop but still requires extraction of a model for all the measurement datasets.
The NBA-RMP flow begins like the NBA-MP flow but relies on a priori knowledge of how to translate the model parameter values of the measured reference transistor into the parameter values for all the other transistors. If this translation can be done effectively, less effort by far is needed in gathering the measured datasets.
The main difference between MCYE and the NBA methods is the number of full-circuit simulations required to interrogate the nominal circuit design. For the present example, which assumes typical counts, MCYE requires a factor of 10-50 more full-circuit simulations per iteration. Given that simulation time can be large and that the full-circuit simulations are in an iteration loop, significant savings in engineering time will result from using NBA.
Another key point in comparing MCYE to NBA-MD and NBA-MP is the extent to which the methods explore the input data space. Note the number of combinations formed for the MCYE, NBA-MD and NBA-MP methods. The NBA methods accommodate far greater numbers of combinations because they select only a tiny subset of the input points, which then are run through full-circuit simulation. MCYE's weakness is that it runs all the combinations through full-circuit simulation.
Comparing the NBA methods indicates that NBA-RMP requires the least effort and NBA-MP the most. NBA-RMP enjoys its position because fairly few transistors are measured and extracted. The weak link in NBA-RMP is the transistor scaling operation [light green block]. For example, it implicitly assumes that all of a design's transistors are deterministically related--that there are no intra-die variations.
NBA-MP takes the most effort, and may not offer any advantage in accuracy over NBA-MD, which requires the mid-range effort. Low data dimensionality and data fidelity drive accuracy in any NBA method. If the underlying models are sufficiently compact, then NBA-MP offers lower dimensionality. All the same, this lower dimensionality from models as compared to measurements is not a given. NBA-MD always offers higher fidelity than NBA-MP because extracting a model from measurements always introduces noise into the NBA input data.
Far larger circuits can be accommodated if the NBA-RMP method is employed. With this method, each of 100 statistically variant reference devices can represent tens to even thousands of individual devices in the circuit designs. The number of devices in an amenable circuit then grows into the 1 000 to 100 000 range.
The result of most interest is how well nonparametric boundary analysis helps determine a lower bound for the yield. Since acceptable circuit responses are usually specified in terms of independent upper and lower limits for each response, acceptability regions are hyperboxes. (A hyperbox is the generalization of a rectangle in two dimensions and a box in three dimensions to higher-dimensional spaces.) Three enclosure percentages are compared with Monte Carlo yield estimates. The Monte Carlo estimates are then 34, 89, and 97 percent for the hyperboxes enclosing the 25-, 50-, and 100-percent boundary point sets. The expected result is that the analysis enclosure percentages represent lower bounds on the actual yields.
Calculating a lower bound on yield is also possible with parametric worst-case methods so long as the underlying data distributions are Gaussian. Inspection of Fig. 4 illustrates that the compression versus large-signal (LS) gain response data are clearly non-Gaussian. Also, we see overlays of the nominal point and the boundary points for 25, 50, and 100 percent enclosures. Visually, the nominal and boundary points appear consistent with the response scatter although the analysis was done on the input variables.
The data presented is a projection of the responses, which are eight-dimensional, onto the compression versus LS-gain plane. Interesting problems arise when a human being tries to examine multidimensional boundaries on a two-dimensional sheet of paper--to wit, they can't be seen. This does not mean that the boundary is absent, but rather that the boundary points do not form an enclosed region in this two-dimensional projection.
Michael D. Meehan and John Purviance present a lively review of statistical circuit design in their book Yield and Reliability in Microwave Circuit and System Design (Artech House, Boston, 1993). The authors discuss not only the many algorithms in the field, but also when and how to apply these methods.
David W. Scott's book, Multivariate Density Estimation Theory, Practice, and Visualization (John Wiley & Sons, New York, 1992), provides the reader with an introduction to density estimation in multivariate (multidimensional) spaces.
Further information about the theory and application of nonparametric boundary analysis is available in the form of U.S. Patent No. 5,835,891, "Device Modeling Using Non-Parametric Statistical Determination of Boundary Data Vectors," which was issued to the author, Dan Stoneking, on 10 November 1998. Dan also wrote "Statistical Circuit Design and IC-CAP's Non-Parametric Boundary Analysis," which is in HP EEsof's Characterization Solutions (Order No. 5965-8931EUS).