A graph-theoretic approach for quantification of surface morphology variation and its application to chemical mechanical planarization process

Abstract

We present an algebraic graph-theoretic approach for quantification of surface morphology. Using this approach, heterogeneous, multi-scaled aspects of surfaces; e.g., semiconductor wafers, are tracked from optical micrographs as opposed to reticent profile mapping techniques. Therefore, this approach can facilitate in situ real-time assessment of surface quality. We report two complementary methods for realizing graph-theoretic representation and subsequent quantification of surface morphology variations from optical micrograph images. Experimental investigations with specular finished copper wafers (surface roughness (Sa) ∼ 6 nm) obtained using a semiconductor chemical mechanical planarization process suggest that the graph-based topological invariant Fiedler number (λ₂) was able to quantify and track variations in surface morphology more effectively compared to other quantifiers reported in literature.

Keywords:

• Surface quality
• surface morphology quantification
• optical metrology
• chemical mechanical planarization (CMP)
• algebraic graph theory
• Fiedler number
• graph-based image processing

Appendices

Appendix A

Key Properties of the Normalized Laplacian Matrix ():

1. Every graph has a unique Laplacian (). The converse is not true. is a Hermitian matrix (); is a real symmetric matrix ). is positive semi-definite ().

2. The eigenvectors of are orthonormal, i.e., v₁⊥v₂⋅⋅⋅⊥⋅⋅⋅v_n; v_i, v_j = 0; v_i, v_i = 1. This implies, the Gramian of is an identity matrix .

3. The first eigenvector of is an identity vector v₁ = e = 1, the first eigenvalue is zero (λ₁ = 0). The multiplicity of λ₁( = 0) as an eigenvalue is equal to the number of connected components in the graph.

4. All eigenvalues of are non-negative (λ_i ⩾ 0). All eigenvalues of are less than or equal to two; the equality holds if and only if the graph is bipartite (λ* ⩽ 2).

5. The second eigenvalue of is greater than zero if the graph is connected (λ₂ > 0) and zero (λ₂ = 0) if and only if the graph is disconnected. The second eigenvalue of is greater than one (λ₂ > 1) if an only if the graph is a complete graph (i.e., every node is connected to all other nodes). The second eigenvalue is 1 (λ₂ = 1) if and only if the graph is bipartite.

Appendix BI

Procedure for simulation of different surface morphologies described in Section 4

• • Simulation procedure for Type 1 surface morphology Given a 1000 × 1000 pixel surface with no features (black-colored surface), essentially a defect-free surface, in practice a 1000 × 1000 matrix of zeros, a random coordinate sampled from a discrete uniform distribution with bounds [1 1000]— i.e., —is selected. At this coordinate we create a point feature by changing the matrix element at the identified location to a one. We continue in this manner, taking care that sampled locations do not repeat (sampling without replacement), until 3% of the total surface—i.e., 30 000 points out of the total (1000 × 1000 =) 10⁶ possible locations—have been converted into features (white spots).

• • Simulation procedure for Type 2 surface morphology We simulate a surface with uniformly distributed scratches between 20 and 200 pixels in length (discrete uniform distribution; ) at random locations inclined at ± 45° to the horizontal. The starting locations for the scratches are sampled from a discrete uniform distribution in a manner identical to the procedure described for the previous case. We note that three parameters are simultaneously manipulated for this type of surface, viz.

  • i. the location of the scratch, which is randomly sampled (without replacement) from the distribution .

  • ii. the slope of the scratch, alternated as 1 or −1; i.e., if a scratch has a positive slope, the subsequent scratch will have a negative slope; and

  • iii. the length of the scratch sampled from a uniform random distribution . Once the starting location has been identified, a scratch with the appropriate slope (± 1) and length is generated. This procedure is recursively implemented until ρ = 3%; i.e., 30 000 points out of the total (1000 × 1000 =) 10⁶ possible locations of the total surface is covered in scratches.

• • Simulation procedure for Type 3 surface morphology The procedure for simulating Type 3 surfaces is largely identical to the Type 1 surface. However, instead of a point feature as in Type 1, for Type 3 surfaces the neighboring 20 to 40 pixels; i.e., the area of a feature is 20 pixel² to 40 pixel² sq, sampled from a discrete uniform distribution in a square loci around a random location are also converted into ones (white spots). We note that there is a chance of overlap between features. The total area occupied by features (ρ) is restricted to 3%.

Additional information

Notes on contributors

Prahalad K. Rao

Prahalad K. Rao is currently an Assistant Professor in the System Science and Industrial Engineering (SSIE) department at State University of New York at Binghamton (Binghamton University). Prior to joining Binghamton University he was a Post-Doctoral Research Associate at Virginia Polytechnic Institute and State University, Blacksburg, Virginia. He received a B.Engg. degree (First Class) from Victoria Jubilee Technical Institute (VJTI), Bombay University, India, in 2003, and M.S. and Ph.D. degrees in Industrial Engineering from Oklahoma State University (OSU), Stillwater, Oklahoma in 2006 and 2013, respectively. His research focuses on sensor-based monitoring of manufacturing processes (ultraprecision diamond turning, chemical mechanical planarization, and additive manufacturing processes). He is a member of the Institute of Industrial Engineers, American Society for Quality, and Institute for Operations Research and the Management Sciences. He was awarded the Alpha Pi Mu Outstanding Research Assistant Award by OSU in 2008. He is an amateur radio ARRL General Class licensee with the call sign K5RAO.

Omer F. Beyca

Omer F. Beyca is with the Department of Industrial Engineering at Faith University, Istanbul, Turkey, as an Assistant Professor. He received a B.S. degree in Industrial Engineering from Fatih University, Istanbul, Turkey in 2007 and a Ph.D. degree in 2013 from the School of Industrial Engineering and Management, Oklahoma State University, Stillwater, Oklahoma. His current research interests are modeling nonlinear dynamic systems and quality improvement in micromachining.

Zhenyu (James) Kong

Zhenyu (James) Kong he is currently an Associate Professor with the Grado Department of Industrial and Systems Engineering, Virginia Polytechnic institute and State University, Virginia. Prior to that, he was a Faculty member with the School of Industrial Engineering and Management, Oklahoma State University, Stillwater, Oklahoma. He received B.S. and M.S. degrees in Mechanical Engineering from the Harbin Institute of Technology, China, in 1993 and 1995, respectively, and a Ph.D. degree from the Department of Industrial and System Engineering, University of Wisconsin–Madison, Madison, Wisconsin, in 2004. He has authored or co-authored a number of articles in various journals. His research is sponsored by the National Science Foundation, the Oklahoma Transportation Center, and Dimensional Control Systems Inc. His research focuses on automatic quality control for large and complex manufacturing processes/systems. He is a member of the Institute of Industrial Engineers, Institute for Operations Research and the Management Sciences, American Society of Mechanical Engineers, and the Society of Mechanical Engineers.

Satish T. S. Bukkapatnam

Satish Bukkapatnam serves as a Rockwell International Professor of Industrial and Systems Engineering at Texas A&M University with a joint (courtesy) appointment in the Department of Biomedical Engineering and the director of Texas Engineering Experimentation Station (TEES) Institute for Manufacturing Systems. He has previously served as an AT&T Professor at the Oklahoma State University and as an Assistant Professor at the University of Southern California. His research addresses the harnessing of high-resolution nonlinear dynamic information, particularly from wireless MEMS sensors, to improve the monitoring and prognostics, mainly of ultraprecision and nanomanufacturing processes and machines and cardiorespiratory processes. His research has led to 125 peer-reviewed publications (73 published/accepted in journals and 52 in conference proceedings) and five pending patents and has been the basis for 10 Ph.D. dissertations. His research has received support from federal agencies, including, the National Science Foundation, Department of Energy, and Department of Defense, and the private sector, including General Motors, Ford, National Instruments, and the Central Rural Electric Cooperative. His work with his students has received over a dozen best paper/poster/innovation awards, and his contributions have been recognized with Oklahoma State University regents distinguished researcher (2011), Halliburton outstanding college of engineering faculty (2011 and 2012), the Institute of Industrial Engineers (IIE) Eldin outstanding young industrial engineer (2012), IIE Boeing Technical Innovation (2014), and the Society of Manufacturing Engineers (SME) Dougherty outstanding young manufacturing engineer (2005) awards. He received his master's degree and Ph.D. from the Pennsylvania State University and undergraduate degree from S.V. University, Tirupati, India.

Kenneth E. Case

Kenneth E. Case is Regents Professor Emeritus of Industrial Engineering and Management at Oklahoma State University (OSU). He is a graduate of OSU (BSEE, MSIE, and Ph.D.) and taught at both Virginia Tech and OSU for 35 years. His primary interests are in quality and reliability engineering and economic analysis. He has been Project Director on 25 sponsored research projects, published over 100 articles, and co-authored four books. As a consultant, he has assisted 60 organizations over 35 years. He was elected in 1990 to the National Academy of Engineering and to the International Academy for Quality. He is a licensed professional engineer and was the Outstanding Engineer in Oklahoma in 1987. He holds five ASQ certifications and is a Six Sigma Black Belt. He was a Senior Examiner on the Malcolm Baldrige National Quality Award from 1988 to 1990 and on the Panel of Judges from 1991 to 1993. He received the Quality Oklahoman Award from Governor Brad Henry in 2003 and the Oklahoma Medal for Excellence in College/University Teaching from the Oklahoma Foundation for Excellence in 2005. He is a Fellow of the American Society for Quality, for which he served as President in 2003–2004. He is also a recipient of the Eugene L. Grant Medal. He is a Fellow of the Institute of Industrial Engineers (IIE), for which he served as President in 1986–1987. In 2002 he became the 29th person since 1962 to receive the Frank and Lillian Gilbreth Industrial Engineering Award, the highest honor presented by the IIE. He is an amateur radio ARRL Extra Class licensee with the call sign K5KC.

Ranga Komanduri

Ranga Komanduri (1942–2011) received a B. Engg. degree in Mechanical Engineering and an M.E. degree in Heat Power from Osmania University, Hyderabad, India, and Ph.D. and D.Eng. degrees from Monash University, Melbourne, Australia. He was a Regents Professor and the A.H. Nelson, Jr., Chair in Engineering with the School of Mechanical and Aerospace Engineering, Oklahoma State University (OSU), Stillwater, Oklahoma. He authored or co-authored approximately 230 publications and holds 22 U.S. Patents. He was a Fellow of the American Society of Mechanical Engineers (ASME), the SME, and the CIRP. He was the recipient of several prominent awards in the areas of manufacturing and materials, including the SME Albert M. Sargent Progress Award, the ASME International William T. Ennor Manufacturing Technology Award, the ASME Charles Russ Richards Memorial Award, the ASME Blackall Machine Tool and Gage Award, the ASME/SME M. Eugene Merchant Manufacturing Medal, the F. W. Taylor Medal of CIRP, OSU's Eminent Faculty Award, the OSU Regents Distinguished Research Award, the OSU President's Distinguished Service Award, the OSU Halliburton Outstanding Faculty Award, and the OSU Phoenix Outstanding Graduate Faculty Award. He was also a Distinguished Honorary Professor at the Indian Institute of Technology Kanpur, Kanpur, India. Dr. Komanduri passed away on September 6, 2011.