“New trends in statistical methods applied in a semiconductor company”

1. “New trends in statistical methods applied in a semiconductor company” Luigi RadaelliStatistical Methods eng. PC & Robustness group - Micron Workshop on “Statistical methods applied in microelectronics” Catholic University of Milan and University of Milan-Bicocca June 13th, 2011

2. 􀂄 Incorporated 1978, Boise, Idaho 􀂄 20,794 worldwide 􀂄 Stock Information • Traded on the NYSE - Symbol, MU 􀂄 Developer of DRAM, NAND and Image Sensors • Only US producer of DRAM 􀂄 Company Divisions • Crucial Technology – Meridian, ID • SpecTek – Nampa, ID • Lexar Media – San Jose, CA • Aptina Imaging – San Jose, CA 􀂄 Related Joint Ventures • TECH Semiconductor – DRAM • IM Flash Technologies – Flash • IM Flash Singapore – Flash • Inotera Memories – DRAM • MP Mask Technology Center

4. Introduction The continuous technology development requires complex manufacturing processes with consequent increasing of the difficulties in monitoring their evolution over time. Most process monitoring involve several quality characteristics. A large number of variables, often strongly correlated, must be kept under control to guarantee the effectiveness of the manufacturing process. For these reasons, although the usual univariate techniques are adequate for the single variable, the use of multivariate approaches, jointly considering the variables, avoid inefficient and erroneous conclusions.

5. Introduction Because of the complexities of the process the standard methodologies do not always give a satisfying answer; so new approaches, conform to more strict requirements, must be developed. A further frequent drawback, looking at the data describing manufacturing processes, is the evidence that they show distributions different from the Gaussian. Seldom the characteristics are independent and heteroschedasticity is often observed. As a consequence, most of the classical statistical techniques must be integrated with modern non parametric statistics inferential procedures.

6. Some new trends… Among the non standard methodologies, geostatistic was implemented in the development of new and interesting applications for process control. Also about the LogVariance methodology, interesting application can be performed In this presentation some case studies will be showed: the study of the spatial distribution of the defectivity over the wafer surface the optimization of the size of the maps used to measure parameters on wafer joint modelling of mean and variance surface to monitor Critical Dimension parameter

7. The spatial distribution defectivity over the wafer surface The aim is to investigate if the defects on the surface are dislocated in clusters after a washing step process. • Are any defectivity clusters present on the surface of the wafer? And if yes where are they dislocated? • How can we decide if any clusters are present?

8. The spatial distribution defectivity over the wafer surface A situation where no clusters are present is desiderable, because it means the process is in control (e.g. no “special” source of particles is present) This situation is statistically modellized by a Homogeneous Poisson Process (HPP). The process is defined Complete Spatial Randomness (CSR). The defectivity pattern is evaluated by the estimation of the distribution function of the euclidean distances between pairs of defects.

9. The spatial distribution defectivity over the wafer surface A comparison between the empirical cumulative distribution function (EDF) and the CSR cumulative distribution function is performed by a graphical test. No Clusters Clusters • The CSR hypothesis should be rejected if the EDF (the continuous black colored line), lies outside the envelope identified by the dotted lines. The EDF is far below the lower bound of the envelope: we can assume that there is a presence of clusters

10. The optimization of the size of the measurement maps Starting from an in-use monitoring map, the goal is to consider all the possible configurations of a reduced map, to evaluate for each of them the fitness function and to select the optimum by some criteria. • This means to consider a very high • number of possible configurations that is • impossible to evaluate by enumeration. • The simulating annealing, a combinatorial • optimization algorithm, allows to draw the • optimum reduced map. • Because technical constraints, only a • sub-grid of the n starting point grid can • be selected.

11. The optimization of the size of the measurement maps Universal kriging was used to predict the deposition surface assuming a complete polynomial of second order for the mean function of the process. To select the optimum reduced map, the fitness function should be related to the kriging variance, evaluated at the current sample configuration. The algorithm starts from a random configuration of sample points and sequentially updates it. At each step i, the current configuration is modified by replacing one point by one point. The candidate point for replacement is selected randomly and it is accepted if this determines an improvement in the value of the fitness function.

12. The optimization of the size of the measurement maps The procedure is iterated until the value of the fitness function gets stable and cannot be further reduced. 49 sites starting map Optimal submap - 20sites

13. The LogVariance methods The early development phase for a technological step is the study of a working point able to both center the specification limits and minimize the unavoidable differences among positions on wafer area. In this phase the measurement time and data analysis are only possible for a small number of samples. If the system response match the specification limits, the technological step is implemented on an experimental WIP, in order to monitor its behavior. Data produced are very important to estimate whether the process will be stable enough for production volumes or not. The aim of the method is to give a quantitative evaluation of the degree of stability of a technological step.

14. The LogVariance methods The wafers come from different lots processed in different times and they are randomly selected and supposed to be stochastically independent each other and by time. The aim is to evaluate the homogeneity in mean and variance on the whole wafer. Standard deviation RS Mean RS

15. The LogVariance methods Using the mean and the standard deviation estimated over the whole wafer surface by the L-RS model, it is possible both to point out whether a region is far from the target and to find the distribution of the Cpk over the surface. Region expected to be not on the target Cpk contour plot

16. The LogVariance methods Possible other applications The algorithm proposed may be used in other applications: to check the alignment between two equipments that produce wafers (are they aligned regarding average and variance?) in a DOE experiment to find the area with the smallest variance and the best average value or to understand what are the most relevant factors.

17. Conclusions The application of the approaches here presented allows to save measurement time and to reduce equipment access time. The use of non appropriate methods can yield misleading results with consequent erroneous decisions and waste of time and materials. The study and the development of adequate statistical approaches are fundamentals also for a better understanding of the process.

18. References Diggle PJ. 2003, Statistical Analysis of Spatial Point Patterns. 2-nd edition, Arnold London Illian J, Penttinen A, Stoyan D. 2008 Statistical Analysis and Modeling of Spatial Point Patterns. Wiley New York Aarts, E., Korst, J.: Simulated Annealing and Boltzmann Machines - A Stochastic Approach to Combinatorial Optimization and Neural Computing. Wiley, New York (1990) Chiles, J.P., Delfiner, P.: Geostatistics: Modeling Spatial Uncertainty, John Wiley & Sons, New York (1999) Aitkin, M. (1987), Modelling Variance Heterogeneity in Normal Regression Using GLIM. Applied Statistics, 36, 332-339. Faraway, J. J. (2006), Extending the Linear Model with R. Chapmanand Hall

19. Thank you for your kind attention