“New trends in statistical
methods applied
in a semiconductor company”
Luigi Radaelli
Statistical 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
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􀂄 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
©2009 Micron Technologies, Inc. All rights reserved. Products are warranted only to meet Micron’s production data sheet specifications. Information, products, and/or specifications
are subject to change without notice. All information is provided on an “AS IS” basis without warranties of any kind. Dates are estimates only. Drawings are not to scale. Micron and
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©2009 Micron Technologies, Inc. All rights reserved. Products are warranted only to meet Micron’s production data sheet specifications. Information, products, and/or specifications
are subject to change without notice. All information is provided on an “AS IS” basis without warranties of any kind. Dates are estimates only. Drawings are not to scale. Micron and
the Micron logo are trademarks of Micron Technology, Inc. All other trademarks are the property of their respective owners.
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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.
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are subject to change without notice. All information is provided on an “AS IS” basis without warranties of any kind. Dates are estimates only. Drawings are not to scale. Micron and
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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.
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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
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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.
Cluster
Cluster
•
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?
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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.
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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.
envpp
No Clusters
0.2
0.2
0.4
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F(r)
F(r)
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envpp
Clusters
0
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theo
hi
lo
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theo
hi
lo
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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
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©2009 Micron Technologies, Inc. All rights reserved. Products are warranted only to meet Micron’s production data sheet specifications. Information, products, and/or specifications
are subject to change without notice. All information is provided on an “AS IS” basis without warranties of any kind. Dates are estimates only. Drawings are not to scale. Micron and
the Micron logo are trademarks of Micron Technology, Inc. All other trademarks are the property of their respective owners.
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.
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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.
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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.
campione ottimo
49 sites starting map
100
Optimal fitness=
submap
499.911- 20sites
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-100
•
-100
-50
50
0
100
X
©2009 Micron Technologies, Inc. All rights reserved. Products are warranted only to meet Micron’s production data sheet specifications. Information, products, and/or specifications
are subject to change without notice. All information is provided on an “AS IS” basis without warranties of any kind. Dates are estimates only. Drawings are not to scale. Micron and
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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.
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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.
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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.
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Region expected to be
not on the target
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Cpk
contour plot
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0 2.18 2.5 2.78 3.02 3.22 3.36 3.44 3.46 3.41 3.3 3.13 2.91 2.65 2.37 2.07
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2.6 2.79 2.93 3.02 3.04 3.01 2.82
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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.
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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.
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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
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Thank you for your kind attention
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