Choosing the Best Yield Model for Your Product

Today's complete yield model must meet a tall order of requirements. It must differentiate between random and systematic yield loss. The modeled yield must agree well with actual yield. It should give insight into causes of yield loss. Finally, the model must be able to partition and quantify yield losses resulting from design, process, test and random defects. Ron Ross and Nick Atchison of TI's Silicon Systems (Santa Cruz, Calif.) find that depending on the distribution of die sizes of a given product and the distribution pattern of the defects, different yield models will best fit the data. For instance, the negative binomial model best matches data for mixed-signal memory products because it contains three fitting parameters. Here we present pros and cons to using the Poisson, Murphy, Seeds, Bose-Einstein and negative binomial models.

Each model assumes a particular defect density distribution: random for the Poisson model, triangular in the Murphy model, exponential in the Seeds model and gamma for the negative binomial model (Fig. 1).

The Poisson yield model is the simplest to use. The probability of finding a given number, k, of defects on any single die is given by the distribution shown (Fig. 2), where l is the product of the chip area and defect density, D₀. D₀ is the total number of defects divided by total wafer area. Yield is then the probability of a die having zero defects (P(0)), giving the yield equation shown. The Poisson model tends to predict yields that are lower than the actual because random defect density usually varies across the wafer and from wafer to wafer. Adding a defect distribution function, F(D), gives a modified Poisson equation that takes such variation into account.

With the Murphy model, the yield of small chips typically falls below the actual yield curve. The Seeds model can provide a better forecast for small chips, but it usually fits poorly with technologies containing a wide range of chip sizes.

The Bose-Einstein model takes into account the number of mask levels, n. Unfortunately, it assumes the same defect density at each layer, which typically is not the case.

The negative binomial model uses a cluster factor, a, to estimate the tendency of killer defects to depart from total randomness. A smaller value of a means a higher degree of clustering and greater variation in D_o across the wafer. Alternatively, as a approaches infinity one gets the modified Poisson yield model.

If systematic defects make up a significant portion of the total number of killer defects, use of random defect yield models to estimate total yield can be misleading. Instead, the systematic component can be removed from the equation using a cluster analysis (see March 1999 SI, p.44 for details). For instance, using the negative binomial model, parameters a, D and Y_s are simultaneously optimized (where Y_t=Y_sY_r). Then random yield loss is calculated using the equation shown. •