现代电子元器件工艺水平评价模型与算法研究
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摘要
在当今微电子制造领域,随着工艺过程和设备的日益复杂,电子元器件产品自身功能的不断完善,以及客户对工艺水平要求的不断提高,使得评价、控制工艺水平的技术日益显得重要。同时,目前军用产品实施工艺评价标准,明确要求军方电子元器件生产厂家全面实施工艺评价技术;且越来越多的国内企业为了向国际型企业标准迈进,开始重视质量管理与工艺水平评价技术。因此,全面实施工艺水平评价技术具有重要意义。
本论文选取工序能力指数为研究对象,对工艺水平评价中遇到的几个主要问题建立相应的模型和算法,以及提出解决方案。在工序能力指数与成品率关系,非正态工序能力指数模型与算法,多变量工序能力指数模型与算法,高精度正态分布函数计算以及样本容量对工序能力影响等问题做了一定的研究工作,主要工作和成果总结如下:
1,在分析工序能力指数C_p与成品率一一对应的关系基础上,指出工序能力指数C_(pk)与成品率没有一一对应关系的原因,然后引入中间变量,取数据的均值和规范限中值的差与规范长度的一半的比值作为系数,推导出工序能力指数C_(pk)与成品率之间的关系表达式。然后针对实施6σ设计技术情况下C_(pk)与成品率关系的选取,即采用偏离1.5σ作为参考,得出C_(pk)与成品率关系。
2,建立了非正态分布工序能力指数模型与算法。该方法直接分析数据的均值、标准偏差、偏度和峰度,引用这四个变量,使用切比雪夫—埃尔米特多项式展开积分函数,有效地解决了已有非正态模型(如分位点)法无法计算有效工艺区域的问题。结果显示通过该模型能反映工序的能力,能有效的避免其他方法在数据分布与正态分布偏差大时计算不准确的情况,有效完成工艺评价的要求,实现半导体生产线的质量控制。此外,在三类典型皮尔逊(Pearson)分布的中心矩与原点矩进行分析研究的基础上,推导了针对该分布的拟合表达式,得到了相应分布类型的参数,并给出分布的拟合步骤。
3,建立基于成品率的多变量工序能力指数模型与算法。该模型将工序能力指数与成品率直接联系起来,能让使用者从成品率角度理解工序能力指数的含义。在变量独立的情况下,利用综合成品率等于单个变量成品率乘积关系,然后通过单变量工序能力指数与成品率的关系得出多变量工序能力指数计算式。此算法不会因为变量个数的增多而难以计算,并讨论其应用时根据工艺水平等级得出单变量工序能力指数的范围。在变量不独立的情况下,因为不能利用不相关变量综合成品率与单个变量成品率的关系,则应用函数积分计算成品率,直接对函数进行多重积分。
4,建立基于权重系数的多变量工序能力指数模型与算法。使用主成分分析法计算因子载荷,提取因子分析中全部因子,保证数据所有信息不丢失,建立计算模型。该方法不用考虑变量相互之间是否相关,不用考虑变量个数的多少。结合基于成品率的多变量工序能力指数模型以及精度考虑,为多变量工序能力指数的计算提出了现实可行的解决方案。
5,PPM水平工艺评价需要高精度的分布函数值。针对一维正态分布,本论文根据正态分布函数和误差函数的关式,采用有限连分式展开此关系式,并用泰勒级数表达对应指数项,建立一维正态分布函数高积分限和高精度的算法,然后得出结果。由已有资料表明该算法正确,同时列出了积分变量大于5的部分数值。
针对二维正态分布,分析了相关系数对二维正态分布的重要影响,提出相关系数的拟合算法。给出了实现高精度二维正态分布函数值的算法。以数组作为大数存储结构,解决受到计算机字节长度制约无法实现高精度运算的问题;并引入复化辛普森公式、复化柯特斯公式和龙贝格公式加速收敛,最终得到所需要的结果。
该算法得到的数据不仅能解决当前工序能力评价中一维和二维正态分布函数的精度要求,而且对于其他行业的高精度要求也提供有价值的参考。
6实际使用工序能力指数时,往往要面对样本量到底要取多大的问题。获取大的样本量会带来经济上浪费,或者很难得到大样本量;获取少量的样本量是否可以得到能正确表征工艺的工序能力指数值等这些问题直接面对工序能力指数的使用者。本论文区分完整样本容量和非完整样本容量两种情况,讨论了样本量的变化对工序能力分析的影响,并以图形表示两者之间的关系曲线,为使用者根据要求取样量的选择提供了参考。
7,此外,在本课题的研究基础上,开发了工序能力评价软件系统。该软件不仅包含常规工序能力指数的计算,通常遇到的分布函数的计算与拟合,而且具有非正态和多变量工序能力指数计算两个重要模块。成为目前针对工序能力评价的专用分析工具。
In the field of modern microcircuit manufacturing, because process and equipments are becoming more and more complicated, the function of electronic devices is becoming more perfect and users bring more requirements to process, it is necessary that we should pay more attention to evaluating and controlling technologies of process. At the same time, all factories which produce electronic device for army equipment are required to adopt systemically process evaluating technologies; and in order to keep abreast of time and compete with international companies, domestic companies begin to implement technologies of quality management and process evaluation. So, their implement are of important significant.
In the dissertation, process capability index is chosen as research object, many main problems which come from using process evaluating technologies are solved though setting up relevant model and algorithm. Such research works include studying relationship between process capability index and yield, building model and algorithm of process capability index for non-normal distribution, building model and algorithm of process capability index for multivariate, getting more precise value of normal distribution and relationship between sample numbers with process capability index analysis. The major achievements are listed as followed:
1. After analyzing relationship between process capability index Cp and yield, the reason that Cpk value alone is not sufficient to determine the yield are pointed out, then using intermediate variable and setting the ratio of the difference of mean and median to half of specification width, the equation about Cpk and yield is developed successfully. Last, relationship about process capability index and yield when concerning 6σdesign technology is analyzed.
2. Model and algorithm of process capability index for non-normal distribution are built successfully. It is common to compute process capability index for non-normal data when concerning the level of semiconductor process. Firstly, analyzing several main process capability indexs which have been already presented, their advantages and disadvantages are presented. Then, based on Chebyshev-Hermite polynomials, a model for computing process capability index for non-normal is given when regarding the fact that these four moments, i.e. mean, standard deviation, skewness, and kurtosis, are suitable to approximately characterize the data distribution properties , which also work effectively even data deviation to normal distribution is large.
In order to evaluate modern process, a probability distribution family named Pearson distribution is introduced and its three types of probability density function are deeply analyzed. Based on the study of distribution's central moments and origin moments, the parameters of the distribution are obtained and fitting method is presented.
3. Model and algorithm of process capability index for multivariate based on yield are built successfully. The model connects process capability index with yield, which makes user understand process capability index meaning easier. When variables are independent, the model is built according to relationship between single variable yield and total yield. The model is still effective when variable number is much large, and can deduce single process capability index interval according to process level class. When variable is not independent, the model is built successfully after using multi-dimension function integral.
4. Model and algorithm of process capability index for multivariate based on weighting Coefficient are built successfully. After analyzing factor analysis and using principal component analysis to compute load matrix, then considering all public factors and contributing ratio, the model is presented, which has not requirement of variable number and of variable correlation. An example of calculating process capability index for multivariate is given. Real application shows that the method presented is effective and actual. A system scheme computing process capability index for multivariate is given when also considering process capability index for multivariate based on yield.
5. At PPM(Parts Per Million) level, using process evaluation needs high accuracy distribution function value. As for one-dimension normal distribution, after using relationship between normal distribution function and error function, adopting continued-fraction expansion and Taylor series expansion for exponential term, algorithm of high range of integration and accuracy of normal distribution is built successfully.
As for two-dimension normal distribution, algorithm for fitting relative coefficient and getting high precise two-dimension normal distribution value are presented. The dissertation puts forward a way of testing two-dimension normality using testing marginal distribution and data figure, and analyzes important significance of relative coefficient on two-dimension normal distribution value. It needs high precise function value of normal distribution. After building big data storage structure and constructing decimal code array, an algorithm for high-accuracy value of two-dimensional normal distribution is built through using Laguerre Polynomial expansion, and accelerating operation with Compound Simpson formula, Compound Cotes formula and Romberg formula. At last result is gotten.
6. Sample number has much influence on computing process capability index. When sample number is large, the cost will be also large and it will waste more time and cost; when sample number is small, it is possible the precise process capability index value can not be gotten. The dissertation discusses the influence of sample number to process capability evaluation in two cases which are full sample number and non-full sample number. Last curve about sample number and process capability index is gotten, from which users can determine sample number according to their need.
7. At last, based on the models and algorithms discussed above, the computer-aid process capability evaluation software is developed. The software system not only performs common process capability index computation, common distribution function computation and curve fitting, but also has the function especially for computing process capability index for non-normal distribution and multivariate. The software offers an effective analysis tool for electronic element evaluation.
本论文选取工序能力指数为研究对象,对工艺水平评价中遇到的几个主要问题建立相应的模型和算法,以及提出解决方案。在工序能力指数与成品率关系,非正态工序能力指数模型与算法,多变量工序能力指数模型与算法,高精度正态分布函数计算以及样本容量对工序能力影响等问题做了一定的研究工作,主要工作和成果总结如下:
1,在分析工序能力指数C_p与成品率一一对应的关系基础上,指出工序能力指数C_(pk)与成品率没有一一对应关系的原因,然后引入中间变量,取数据的均值和规范限中值的差与规范长度的一半的比值作为系数,推导出工序能力指数C_(pk)与成品率之间的关系表达式。然后针对实施6σ设计技术情况下C_(pk)与成品率关系的选取,即采用偏离1.5σ作为参考,得出C_(pk)与成品率关系。
2,建立了非正态分布工序能力指数模型与算法。该方法直接分析数据的均值、标准偏差、偏度和峰度,引用这四个变量,使用切比雪夫—埃尔米特多项式展开积分函数,有效地解决了已有非正态模型(如分位点)法无法计算有效工艺区域的问题。结果显示通过该模型能反映工序的能力,能有效的避免其他方法在数据分布与正态分布偏差大时计算不准确的情况,有效完成工艺评价的要求,实现半导体生产线的质量控制。此外,在三类典型皮尔逊(Pearson)分布的中心矩与原点矩进行分析研究的基础上,推导了针对该分布的拟合表达式,得到了相应分布类型的参数,并给出分布的拟合步骤。
3,建立基于成品率的多变量工序能力指数模型与算法。该模型将工序能力指数与成品率直接联系起来,能让使用者从成品率角度理解工序能力指数的含义。在变量独立的情况下,利用综合成品率等于单个变量成品率乘积关系,然后通过单变量工序能力指数与成品率的关系得出多变量工序能力指数计算式。此算法不会因为变量个数的增多而难以计算,并讨论其应用时根据工艺水平等级得出单变量工序能力指数的范围。在变量不独立的情况下,因为不能利用不相关变量综合成品率与单个变量成品率的关系,则应用函数积分计算成品率,直接对函数进行多重积分。
4,建立基于权重系数的多变量工序能力指数模型与算法。使用主成分分析法计算因子载荷,提取因子分析中全部因子,保证数据所有信息不丢失,建立计算模型。该方法不用考虑变量相互之间是否相关,不用考虑变量个数的多少。结合基于成品率的多变量工序能力指数模型以及精度考虑,为多变量工序能力指数的计算提出了现实可行的解决方案。
5,PPM水平工艺评价需要高精度的分布函数值。针对一维正态分布,本论文根据正态分布函数和误差函数的关式,采用有限连分式展开此关系式,并用泰勒级数表达对应指数项,建立一维正态分布函数高积分限和高精度的算法,然后得出结果。由已有资料表明该算法正确,同时列出了积分变量大于5的部分数值。
针对二维正态分布,分析了相关系数对二维正态分布的重要影响,提出相关系数的拟合算法。给出了实现高精度二维正态分布函数值的算法。以数组作为大数存储结构,解决受到计算机字节长度制约无法实现高精度运算的问题;并引入复化辛普森公式、复化柯特斯公式和龙贝格公式加速收敛,最终得到所需要的结果。
该算法得到的数据不仅能解决当前工序能力评价中一维和二维正态分布函数的精度要求,而且对于其他行业的高精度要求也提供有价值的参考。
6实际使用工序能力指数时,往往要面对样本量到底要取多大的问题。获取大的样本量会带来经济上浪费,或者很难得到大样本量;获取少量的样本量是否可以得到能正确表征工艺的工序能力指数值等这些问题直接面对工序能力指数的使用者。本论文区分完整样本容量和非完整样本容量两种情况,讨论了样本量的变化对工序能力分析的影响,并以图形表示两者之间的关系曲线,为使用者根据要求取样量的选择提供了参考。
7,此外,在本课题的研究基础上,开发了工序能力评价软件系统。该软件不仅包含常规工序能力指数的计算,通常遇到的分布函数的计算与拟合,而且具有非正态和多变量工序能力指数计算两个重要模块。成为目前针对工序能力评价的专用分析工具。
In the field of modern microcircuit manufacturing, because process and equipments are becoming more and more complicated, the function of electronic devices is becoming more perfect and users bring more requirements to process, it is necessary that we should pay more attention to evaluating and controlling technologies of process. At the same time, all factories which produce electronic device for army equipment are required to adopt systemically process evaluating technologies; and in order to keep abreast of time and compete with international companies, domestic companies begin to implement technologies of quality management and process evaluation. So, their implement are of important significant.
In the dissertation, process capability index is chosen as research object, many main problems which come from using process evaluating technologies are solved though setting up relevant model and algorithm. Such research works include studying relationship between process capability index and yield, building model and algorithm of process capability index for non-normal distribution, building model and algorithm of process capability index for multivariate, getting more precise value of normal distribution and relationship between sample numbers with process capability index analysis. The major achievements are listed as followed:
1. After analyzing relationship between process capability index Cp and yield, the reason that Cpk value alone is not sufficient to determine the yield are pointed out, then using intermediate variable and setting the ratio of the difference of mean and median to half of specification width, the equation about Cpk and yield is developed successfully. Last, relationship about process capability index and yield when concerning 6σdesign technology is analyzed.
2. Model and algorithm of process capability index for non-normal distribution are built successfully. It is common to compute process capability index for non-normal data when concerning the level of semiconductor process. Firstly, analyzing several main process capability indexs which have been already presented, their advantages and disadvantages are presented. Then, based on Chebyshev-Hermite polynomials, a model for computing process capability index for non-normal is given when regarding the fact that these four moments, i.e. mean, standard deviation, skewness, and kurtosis, are suitable to approximately characterize the data distribution properties , which also work effectively even data deviation to normal distribution is large.
In order to evaluate modern process, a probability distribution family named Pearson distribution is introduced and its three types of probability density function are deeply analyzed. Based on the study of distribution's central moments and origin moments, the parameters of the distribution are obtained and fitting method is presented.
3. Model and algorithm of process capability index for multivariate based on yield are built successfully. The model connects process capability index with yield, which makes user understand process capability index meaning easier. When variables are independent, the model is built according to relationship between single variable yield and total yield. The model is still effective when variable number is much large, and can deduce single process capability index interval according to process level class. When variable is not independent, the model is built successfully after using multi-dimension function integral.
4. Model and algorithm of process capability index for multivariate based on weighting Coefficient are built successfully. After analyzing factor analysis and using principal component analysis to compute load matrix, then considering all public factors and contributing ratio, the model is presented, which has not requirement of variable number and of variable correlation. An example of calculating process capability index for multivariate is given. Real application shows that the method presented is effective and actual. A system scheme computing process capability index for multivariate is given when also considering process capability index for multivariate based on yield.
5. At PPM(Parts Per Million) level, using process evaluation needs high accuracy distribution function value. As for one-dimension normal distribution, after using relationship between normal distribution function and error function, adopting continued-fraction expansion and Taylor series expansion for exponential term, algorithm of high range of integration and accuracy of normal distribution is built successfully.
As for two-dimension normal distribution, algorithm for fitting relative coefficient and getting high precise two-dimension normal distribution value are presented. The dissertation puts forward a way of testing two-dimension normality using testing marginal distribution and data figure, and analyzes important significance of relative coefficient on two-dimension normal distribution value. It needs high precise function value of normal distribution. After building big data storage structure and constructing decimal code array, an algorithm for high-accuracy value of two-dimensional normal distribution is built through using Laguerre Polynomial expansion, and accelerating operation with Compound Simpson formula, Compound Cotes formula and Romberg formula. At last result is gotten.
6. Sample number has much influence on computing process capability index. When sample number is large, the cost will be also large and it will waste more time and cost; when sample number is small, it is possible the precise process capability index value can not be gotten. The dissertation discusses the influence of sample number to process capability evaluation in two cases which are full sample number and non-full sample number. Last curve about sample number and process capability index is gotten, from which users can determine sample number according to their need.
7. At last, based on the models and algorithms discussed above, the computer-aid process capability evaluation software is developed. The software system not only performs common process capability index computation, common distribution function computation and curve fitting, but also has the function especially for computing process capability index for non-normal distribution and multivariate. The software offers an effective analysis tool for electronic element evaluation.
引文
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