Bruno Scibilia, Altis Semiconductor

Statistical process control (SPC) is used extensively to identify the causes of defect excursions as early as possible. Because of the increasing complexity of semiconductor devices and the steady decrease in feature size, it is necessary to eliminate defect sources and reduce the density of systematic and random defects per wafer in all semiconductor processes to maintain high device yields. Traditionally, IC development has focused on performance and technological improvements. Today, however, more attention is being paid to techniques such as SPC to improve manufacturing efficiency.

Defect data are particularly difficult to monitor using SPC charts, partly because they exhibit erratic behavior and noisiness. While measurement data such as critical dimensions and depths follow a Gaussian, or normal, distribution (the well-known bell-shaped curve), defect data have an asymmetrical distribution. Most defect data are positive and relatively close to zero. This violation of the Gaussian assumption must be taken into account in process control operations.

Defects arise from two types of causes: common causes and assignable causes. Common causes include routine or chronic sources of variations in the fab, such as the failure of environmental cleanroom controls, a lack of repeatability, and measurement errors. By definition, fab personnel cannot eliminate common causes from processes. Deviations from the norm can also have assignable causes. Defects with assignable causes can lead to processes that are out of statistical control. Once a control chart has triggered a true alarm, assignable causes must be discovered and corrected.

Figure 1: Theoretical Poisson distributions with different means.

Traditionally, four types of charts have been used in SPC applications: np and p charts for binomial random distributions, and c or u charts for Poisson random distributions. Figure 1 illustrates theoretical Poisson distributions with different means.

The shape of the random distribution (whether it is a normal, Poisson, or other type of distribution) has an important influence on where control limits are set. SPC limits on defect data are usually set according to the Poisson distribution.¹ Control limits are helpful to identify systematic departures from random distributions and are therefore set at the very limits of these distributions. However, statistician engineers have observed that in the semiconductor industry, such limits are often too tight, leading to many false alarms. To minimize the incidence of false alarms, many practitioners resort to limits that are not based on statistical principles, a pragmatic approach that is valid only when the behavior of defect data is well understood.

The objective of this article is to present a simple and effective SPC approach to monitoring defect data. The article begins with a summary of the techniques and concepts used in the IC industry. Then it presents examples of defect distributions and proposes an approach based on exponential distributions to account for the overdispersion of defect data.

Results from a Benchmark Survey

Engineers at Altis Semiconductor (Corbeil-Essonnes, France) conducted an informal benchmark survey covering six firms in the semiconductor industry. The survey concluded that the Poisson model was the most widely used SPC method throughout the industry, although other, alternative approaches are also implemented.

In two firms, control limits were based purely on "engineering judgment." In one of these firms, control limits were set so that a certain percentage of lots (say 10%) would be rechecked. Sometimes, defect-yield correlations and yield-impact calculations were used to identify defect levels at which the impact on yield became noticeable. Actually a specification limit, that level was not considered a control limit.

An expert from an equipment supplier proposed that a logarithmic transformation be used to normalize distributions. Transformed values, however, have little physical significance and are therefore not popular in the industry.

At some companies, exponentially weighted moving average (EWMA) charts were used in addition to standard charts to track gradually increasing particle counts. However, when Altis engineers tested the efficiency of EWMA charts using the fab's own data, they observed that whenever defect levels were out of control, they were so much above the control limits that the statistical results from EWMA charts and standard charts were the same. Hence, EWMA charts did not provide useful information in that context.

Overdispersion of Defect Data

While within-wafer processing conditions are quite homogeneous, wafer-to-wafer conditions can vary extensively. Most of the time, small groups of defects, or even isolated defects, appear on wafers. However, when a common cause (routine run-to-run variations) exposes a wafer to particle contamination, clusters of defects rather than isolated particles appear on the surface. In such cases, the number of defects increases exponentially according to the number and size of the clusters.

A well-known consequence of this clustering phenomenon is the overdispersion of defect data, which results in high false-alarm rates when the effect is not properly taken into account. Because of within-wafer and within-lot correlations, the Poisson distribution assumption is inaccurate in this situation.

Some researchers advocate the use of a square-root transformation of particle counts to account for overdispersion of defect data.² According to them, particle counts often follow a χ² distribution. Quantile-quantile plots indicate that after transformation, the data structure is much closer to a normal distribution. However, the tails of the transformed defect data distribution still deviate somewhat from the Gaussian model.

Other researchers recommend that a mixture distribution be used to account for overdispersion and the correlation of within-wafer data.³ In this approach, a Poisson distribution is combined with a gamma distribution to achieve a negative binomial distribution. The shape of the gamma distribution can be adjusted, providing a very high degree of flexibility. Although the flexibility of the mixture distribution model enables it to fit defect data well, the technique is complex. It requires that maximum-likelihood estimations be calculated using quasi-Newtonian methods.

Experiments and Results

Defect Monitoring in the Low-Pressure Chemical Vapor Deposition (LPCVD) Area. The LPCVD poly gate deposition furnace that was selected for this study is used to process batches of 150 wafers at a time. At the time of the experiment, defects were often caused by particles and dust released from the quartz racks that supported the wafers. Because the LPCVD process involves several cooling and heating cycles, the racks gradually deteriorated. (Since the experiment was conducted, the quartz racks have been replaced by more-expensive silicon racks, resulting in a significant reduction in defect levels.)

Figure 2: Histogram of the raw sum of defects gathered over a five-month period.

Control charts were used to determine when the racks needed to be replaced. When measurements indicated that defect levels were out of control, the tool was cleaned. Production resumed if postcleaning measurements were considered acceptable. Otherwise, the tool was shut down and serviced by the maintenance team.

Tool defect data were collected 200 times over a five-month period and divided according to their value into nine bins of uniform width. The results of the data acquisition are presented in the histograms in Figures 2 and 3, in which the smallest sum of defects is represented by the leftmost bar and the largest by the rightmost bar.

Figure 3: Histogram based on the same data as in Figure 2 without outliers.

The data shown in Figure 2 were clearly distorted by outliers. After the outliers were eliminated, the data, presented in Figure 3, seemed to follow an exponential distribution, a theoretical example of which is presented in Figure 4 for comparison. The similarity between the data curve from the LPCVD tool and the theoretical exponential distribution is striking. More importantly, the estimates of the mean and standard deviation values in Figure 3—13.2 and 11.8σ, respectively—are very close to each other, which is typical of exponential distributions.

Figure 4: Theoretical exponential distribution.

A positive random variable is exponentially distributed when f(x) = ß exp(–ßx) for x > 0. The random variable, x, is more likely to have small rather than large positive values. The single parameter, ß, determines the exponential distribution and all of its moments. For example,

where E(X) stands for mean, σ(X) stands for standard deviation, and V(X) stands for variance. The mean and standard deviation are therefore equal theoretically and very close to each other in practice.

In an exponential distribution, density function (f(x)), distribution function (F(x)), mean (µ), variance (σ²), and standard deviation (σ) are determined by the following equations: