Statistical methods need to be part of every lithographer's toolbox, because lithographic processes contain intrinsic levels of variation. This variation is a consequence of the nature of the world. For example, petroleum is typically the starting material from which photoresists are synthesized, and the composition of crude oil varies from well to well. Lithographic processes and tools are affected by environmental parameters such as barometric pressure and relative humidity, and these factors vary with the weather. Lithography is a manufacturing science implemented and ultimately exercised by human beings, each of whom is a unique individual, different from all others. When people are involved, there is a special element of variation interjected into the process. The analytical methods used by lithography engineers and managers must be capable of dealing with variation in equipment, materials, and people.
The objective of any process control methodology is the reduction of variation, in order to maintain conformance to standards or to meet a higher standard. Variations in gate lengths can lead to degraded yield or slower parts, which usually sell for a lower price than faster devices. Higher manufacturing costs result from variation, in the form of scrap, reduced yield, rework, and low equipment utilization. Often there are costs associated with attempts to reduce variation. Since the objective of process control is to maximize profitability, the most effective methods are those which accomplish control in the most cost-effective manner. Indeed, Dr. Walter Shewhart, the inventor of statistical process control, titled his book, Economic Control of Quality of Manufactured Product, with the first word of the title identifying the monetary considerations motivating his methods. Generally, the most economical approaches require that particular levels of variation be tolerated, and the purpose of statistical process control is the identification of variation in excess of the norms of a controlled process.
Statistics is the mathematical science for making inferences about quantities which are probabilistic (in contrast to deterministic) in nature. Many statistical methods are therefore applicable only to situations that are random, in which events are independent of each other. A large fraction of the first three chapters of this Tutorial Text will involve the examination of commonly occurring situations in microlithography in which measurements are not independent. The discussion will lead to methods for applying statistical control techniques correctly in such situations.
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