Authors: S. Rao, S. Saxena, P. Apte, P.K. Mozumder, J. Davis, R. Burch and K. Vasanth
Affilation: Texas Instruments, Inc., United States
Pages: 122 - 126
Keywords: process modeling, yield
The trend towards smaller feature sizes has increased the need to accurately characterize the distribution of process and device responses to predict and improve yield . The usual approach to characterization assumes that the response distributions have a parametric form (e.g., normal distribution) and that the parameters of the distribution are estimated from experimental data. For instance, using a designed experiment it is possible to build response surface models (RSM) for the mean and variance as a function of process settings. These models, together with the assumption of a parametric distribution completely and compactly characterizes the predicted response distribution at any interior point in the model space and may be used for yield prediction. One problem with the traditional approach is that the response data may not have a parametric or a closed form distribution possibly because of within-wafer spatial dependency. As an example, we looked at the gate sheet and contact resistance of a salicide process (0.25 micron technology) as a function of four process settings: implant dose, Titanium thickness, anneal temperature, and anneal time. As shown in Figure 1, the within-wafer measurements at each of the design point do not follow any standard parametric distribution. In this situation, assuming a parametric distribution for yield calculation increases the prediction error. We propose an approach to compactly model non-standard distributions from limited design points to predict the distribution at any interpolated point in the design space. The distribution is modeled in two steps. First, using the data from the designed experiments we build the usual RSM models for predicting the mean and variance at any interior point in the design space. Next, we generate an empirical cumulative distribution function (CDF) that represents the overall shape of the data by accumulating shape evidence from measurements at each of the available design point. Figure 2 shows the shape of this standardized distribution function. This cumulative distribution function, together with the predicted values of mean and variance, allows us to compactly represent the predicted distribution at any interpolated point in the design space. We have found that, on the average, the empirical CDF approach has substantially less prediction error compared to the normal approximation approach. As an illustration, suppose we are interested in predicting the probability that the scaled Rc is less than 2.5 Ohm-micrometer at the process conditions corresponding to those for Wafer B. From the observed Rc data from Wafer B, the fraction falling below 2.5 Ohm-micrometer is 11/23= 0.4783. Table 1 shows that the prediction error from normal approximation is about 4 times that from the empirical CDF approach. There are several advantages to the empirical CDF approach. First, one can represent arbitrary shaped distributions which may even incorporate within wafer spatial dependency information. Second, the class of parametric distributions assumed in the traditional method is a special case of our general approach. Third, the complete predictive distribution at any interior point in the design space can be characterized compactly by three quantities, an empirical CDF function, a mean estimate and a variance estimate. Using this approach, we also demonstrate the optimization of the salicide process for the 0.25 micron CMOS technology for minimum contact resistance with minimum variance.