Abstract

Recently, there has been significant research investigating new optical technologies for dimensional metrology of features 22 nm in critical dimension and smaller. When modeling optical measurements, a library of curves is assembled through the simulation of a multidimensional parameter space. A nonlinear regression routine described in this paper is then used to identify an optimum set of parameters that yields the closest experiment-to-theory agreement. However, parametric correlation, measurement noise, and model inaccuracy all lead to measurement uncertainty in the fitting process for optical critical dimension measurements. To improve the optical measurements, other techniques such as atomic force microscopy and scanning electronic microscopy can also be used to provide supplemental a priori information. In this paper, a Bayesian statistical approach is proposed to allow the combination of different measurement techniques that are based on different physical measurements. The effect of this hybrid metrology approach will be shown to reduce the uncertainties of the parameter estimators.

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  1. R. M. Silver, B. M. Barnes, R. Attota, R. Jun, M. Stocker, E. Marx, and H. J. Patrick, “Scatterfield microscopy for extending the limits of image-based optical metrology,” Appl. Opt. 46, 4248–4257 (2007).
    [CrossRef]
  2. H. J. Patrick, R. Attota, B. M. Barnes, T. A. Germer, R. G. Dixson, M. T. Stocker, R. M. Silver, and M. R. Bishop, “Optical critical dimension measurement of silicon grating targets using back focal plane scatterfield microscopy,” J. Micro/Nanolith. MEMS MOEMS 7, 013012 (2008).
    [CrossRef]
  3. B. M. Barnes, R. Attota, L. P. Howard, P. Lipscomb, M. T. Stocker, and R. M. Silver, “Zero-order and super-resolved imaging of arrayed nanoscale lines using scatterfield microscopy,” in 2007 International Conference on Frontiers of Characterization and Metrology , Vol. 931 of AIP Conference Proceedings (2007), pp. 397–401.
  4. R. M. Silver, B. M. Barnes, H. Zhou, N. F. Zhang, and R. Dixson, “Angle-resolved optical metrology using multi-technique nested uncertainties,” Proc. SPIE 7390, 73900P (2009).
    [CrossRef]
  5. M. G. Moharam, M. G. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1076 (1995).
    [CrossRef]
  6. M. G. Moharam, D. A. Pommet, E. B. Grann, and T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. A 12, 1077–1086 (1995).
    [CrossRef]
  7. P. Lalanne and G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996).
    [CrossRef]
  8. A. Taflove and M. E. Brodwin, “Numerical solution of steady-state electromagnetic scattering problems using time-dependent Maxwell’s Equations,” IEEE Trans. Microwave Theory Tech. 23, 623–630 (1975).
    [CrossRef]
  9. R. M. Silver, N. F. Zhang, B. M. Barnes, H. Zhou, A. Heckert, R. Dixson, T. A. Germer, and B. Bunday, “Improving optical measurement accuracy using multi-technique nested uncertainties,” Proc. SPIE 7272, 727202 (2009).
    [CrossRef]
  10. R. M. Silver, B. M. Barnes, A. Heckert, R. Attota, R. Dixson, and J. Jun, “Angle resolved optical metrology,” Proc. SPIE 6922, 69221M (2008).
    [CrossRef]
  11. J. Neter, W. Wasserman, and M. H. Kutner, Applied Linear Regression Models (Richard D. Irwin, 1983).
  12. C. R. Rao and H. Toutenburg, Linear Models: Least Squares and Alternatives (Springer, 1995).
  13. D. M. Bates and D. G. Watts, Nonlinear Regression Analysis and Its Applications (Wiley, 1988).
  14. R. M. Silver, T. A. Germer, R. Attota, B. M. Barnes, B. Bunday, J. Allgair, E. Marx, and J. Jun, “Fundamental limits of optical critical dimension metrology: a simulation study,” Proc. SPIE 6518, 65180U (2007).
    [CrossRef]
  15. A. Vaid, B. B. Yan, Y. T. Jiang, M. Kelling, C. Hartig, J. Allgair, P. Ebersbach, M. Sendelbach, N. Rana, A. Katnani, E. Mclellan, C. Archie, C. Bozdog, H. Kim, M. Sendler, S. Ng, B. Sherman, B. Brill, I. Turovets, and R. Urensky, “Holistic metrology approach: hybrid metrology utilizing scatterometry, critical dimension-atomic force microscope and critical dimension-scanning electron microscope,” J. Micro/Nanolith. MEMS MOEMS 10, 043016 (2011).
  16. S. Press, Bayesian Statistics: Principles, Models, and Applications (Wiley, 1989).
  17. A. Gelman, J. B. Carlin, H. S. Stern, and D. B. Rubin, Bayesian Data Analysis (Chapman & Hall/CRC, 2004).
  18. D. J. Lunn, A. Thomas, N. Best, and D. Spiegelhalter, “WinBUGS—a Bayesian modeling framework: concepts, structure, and extensibility,” Stat. Comput. 10, 325–337 (2000).
    [CrossRef]
  19. D. V. Lindley and A. F. M. Smith, “Bayes estimates for the linear model,” J. R. Stat. Soc. Ser. B. Methodol.1–41 (1972).
  20. S. Kocherlakota and K. E. Kocherlakota, Encyclopedia of Statistical Sciences (Wiley, 1983), Vol. 3, pp. 354–355.
  21. H. K. Iyer, C. M. Wang, and D. F. Vecchia, “Consistency tests for key comparison data,” Metrologia 41, 223–230 (2004).
    [CrossRef]
  22. M. G. Cox, “The evaluation of key comparison data,” Metrologia 39, 589–595 (2002).
    [CrossRef]
  23. Certain commercial equipment, instruments, or materials are identified in this paper in order to specify the experimental procedure adequately. Such identification is not intended to imply recommendation or endorsement by the National Institute of Standards and Technology, nor is it intended to imply that the materials or equipment identified are necessarily the best available for the purpose.
  24. G. H. Golub and C. H. Van Loan, Matrix Computations (Johns Hopkins University, 1989).

2011

A. Vaid, B. B. Yan, Y. T. Jiang, M. Kelling, C. Hartig, J. Allgair, P. Ebersbach, M. Sendelbach, N. Rana, A. Katnani, E. Mclellan, C. Archie, C. Bozdog, H. Kim, M. Sendler, S. Ng, B. Sherman, B. Brill, I. Turovets, and R. Urensky, “Holistic metrology approach: hybrid metrology utilizing scatterometry, critical dimension-atomic force microscope and critical dimension-scanning electron microscope,” J. Micro/Nanolith. MEMS MOEMS 10, 043016 (2011).

2009

R. M. Silver, B. M. Barnes, H. Zhou, N. F. Zhang, and R. Dixson, “Angle-resolved optical metrology using multi-technique nested uncertainties,” Proc. SPIE 7390, 73900P (2009).
[CrossRef]

R. M. Silver, N. F. Zhang, B. M. Barnes, H. Zhou, A. Heckert, R. Dixson, T. A. Germer, and B. Bunday, “Improving optical measurement accuracy using multi-technique nested uncertainties,” Proc. SPIE 7272, 727202 (2009).
[CrossRef]

2008

R. M. Silver, B. M. Barnes, A. Heckert, R. Attota, R. Dixson, and J. Jun, “Angle resolved optical metrology,” Proc. SPIE 6922, 69221M (2008).
[CrossRef]

H. J. Patrick, R. Attota, B. M. Barnes, T. A. Germer, R. G. Dixson, M. T. Stocker, R. M. Silver, and M. R. Bishop, “Optical critical dimension measurement of silicon grating targets using back focal plane scatterfield microscopy,” J. Micro/Nanolith. MEMS MOEMS 7, 013012 (2008).
[CrossRef]

2007

R. M. Silver, B. M. Barnes, R. Attota, R. Jun, M. Stocker, E. Marx, and H. J. Patrick, “Scatterfield microscopy for extending the limits of image-based optical metrology,” Appl. Opt. 46, 4248–4257 (2007).
[CrossRef]

R. M. Silver, T. A. Germer, R. Attota, B. M. Barnes, B. Bunday, J. Allgair, E. Marx, and J. Jun, “Fundamental limits of optical critical dimension metrology: a simulation study,” Proc. SPIE 6518, 65180U (2007).
[CrossRef]

2004

H. K. Iyer, C. M. Wang, and D. F. Vecchia, “Consistency tests for key comparison data,” Metrologia 41, 223–230 (2004).
[CrossRef]

2002

M. G. Cox, “The evaluation of key comparison data,” Metrologia 39, 589–595 (2002).
[CrossRef]

2000

D. J. Lunn, A. Thomas, N. Best, and D. Spiegelhalter, “WinBUGS—a Bayesian modeling framework: concepts, structure, and extensibility,” Stat. Comput. 10, 325–337 (2000).
[CrossRef]

1996

1995

1975

A. Taflove and M. E. Brodwin, “Numerical solution of steady-state electromagnetic scattering problems using time-dependent Maxwell’s Equations,” IEEE Trans. Microwave Theory Tech. 23, 623–630 (1975).
[CrossRef]

1972

D. V. Lindley and A. F. M. Smith, “Bayes estimates for the linear model,” J. R. Stat. Soc. Ser. B. Methodol.1–41 (1972).

Allgair, J.

A. Vaid, B. B. Yan, Y. T. Jiang, M. Kelling, C. Hartig, J. Allgair, P. Ebersbach, M. Sendelbach, N. Rana, A. Katnani, E. Mclellan, C. Archie, C. Bozdog, H. Kim, M. Sendler, S. Ng, B. Sherman, B. Brill, I. Turovets, and R. Urensky, “Holistic metrology approach: hybrid metrology utilizing scatterometry, critical dimension-atomic force microscope and critical dimension-scanning electron microscope,” J. Micro/Nanolith. MEMS MOEMS 10, 043016 (2011).

R. M. Silver, T. A. Germer, R. Attota, B. M. Barnes, B. Bunday, J. Allgair, E. Marx, and J. Jun, “Fundamental limits of optical critical dimension metrology: a simulation study,” Proc. SPIE 6518, 65180U (2007).
[CrossRef]

Archie, C.

A. Vaid, B. B. Yan, Y. T. Jiang, M. Kelling, C. Hartig, J. Allgair, P. Ebersbach, M. Sendelbach, N. Rana, A. Katnani, E. Mclellan, C. Archie, C. Bozdog, H. Kim, M. Sendler, S. Ng, B. Sherman, B. Brill, I. Turovets, and R. Urensky, “Holistic metrology approach: hybrid metrology utilizing scatterometry, critical dimension-atomic force microscope and critical dimension-scanning electron microscope,” J. Micro/Nanolith. MEMS MOEMS 10, 043016 (2011).

Attota, R.

R. M. Silver, B. M. Barnes, A. Heckert, R. Attota, R. Dixson, and J. Jun, “Angle resolved optical metrology,” Proc. SPIE 6922, 69221M (2008).
[CrossRef]

H. J. Patrick, R. Attota, B. M. Barnes, T. A. Germer, R. G. Dixson, M. T. Stocker, R. M. Silver, and M. R. Bishop, “Optical critical dimension measurement of silicon grating targets using back focal plane scatterfield microscopy,” J. Micro/Nanolith. MEMS MOEMS 7, 013012 (2008).
[CrossRef]

R. M. Silver, B. M. Barnes, R. Attota, R. Jun, M. Stocker, E. Marx, and H. J. Patrick, “Scatterfield microscopy for extending the limits of image-based optical metrology,” Appl. Opt. 46, 4248–4257 (2007).
[CrossRef]

R. M. Silver, T. A. Germer, R. Attota, B. M. Barnes, B. Bunday, J. Allgair, E. Marx, and J. Jun, “Fundamental limits of optical critical dimension metrology: a simulation study,” Proc. SPIE 6518, 65180U (2007).
[CrossRef]

B. M. Barnes, R. Attota, L. P. Howard, P. Lipscomb, M. T. Stocker, and R. M. Silver, “Zero-order and super-resolved imaging of arrayed nanoscale lines using scatterfield microscopy,” in 2007 International Conference on Frontiers of Characterization and Metrology , Vol. 931 of AIP Conference Proceedings (2007), pp. 397–401.

Barnes, B. M.

R. M. Silver, B. M. Barnes, H. Zhou, N. F. Zhang, and R. Dixson, “Angle-resolved optical metrology using multi-technique nested uncertainties,” Proc. SPIE 7390, 73900P (2009).
[CrossRef]

R. M. Silver, N. F. Zhang, B. M. Barnes, H. Zhou, A. Heckert, R. Dixson, T. A. Germer, and B. Bunday, “Improving optical measurement accuracy using multi-technique nested uncertainties,” Proc. SPIE 7272, 727202 (2009).
[CrossRef]

R. M. Silver, B. M. Barnes, A. Heckert, R. Attota, R. Dixson, and J. Jun, “Angle resolved optical metrology,” Proc. SPIE 6922, 69221M (2008).
[CrossRef]

H. J. Patrick, R. Attota, B. M. Barnes, T. A. Germer, R. G. Dixson, M. T. Stocker, R. M. Silver, and M. R. Bishop, “Optical critical dimension measurement of silicon grating targets using back focal plane scatterfield microscopy,” J. Micro/Nanolith. MEMS MOEMS 7, 013012 (2008).
[CrossRef]

R. M. Silver, B. M. Barnes, R. Attota, R. Jun, M. Stocker, E. Marx, and H. J. Patrick, “Scatterfield microscopy for extending the limits of image-based optical metrology,” Appl. Opt. 46, 4248–4257 (2007).
[CrossRef]

R. M. Silver, T. A. Germer, R. Attota, B. M. Barnes, B. Bunday, J. Allgair, E. Marx, and J. Jun, “Fundamental limits of optical critical dimension metrology: a simulation study,” Proc. SPIE 6518, 65180U (2007).
[CrossRef]

B. M. Barnes, R. Attota, L. P. Howard, P. Lipscomb, M. T. Stocker, and R. M. Silver, “Zero-order and super-resolved imaging of arrayed nanoscale lines using scatterfield microscopy,” in 2007 International Conference on Frontiers of Characterization and Metrology , Vol. 931 of AIP Conference Proceedings (2007), pp. 397–401.

Bates, D. M.

D. M. Bates and D. G. Watts, Nonlinear Regression Analysis and Its Applications (Wiley, 1988).

Best, N.

D. J. Lunn, A. Thomas, N. Best, and D. Spiegelhalter, “WinBUGS—a Bayesian modeling framework: concepts, structure, and extensibility,” Stat. Comput. 10, 325–337 (2000).
[CrossRef]

Bishop, M. R.

H. J. Patrick, R. Attota, B. M. Barnes, T. A. Germer, R. G. Dixson, M. T. Stocker, R. M. Silver, and M. R. Bishop, “Optical critical dimension measurement of silicon grating targets using back focal plane scatterfield microscopy,” J. Micro/Nanolith. MEMS MOEMS 7, 013012 (2008).
[CrossRef]

Bozdog, C.

A. Vaid, B. B. Yan, Y. T. Jiang, M. Kelling, C. Hartig, J. Allgair, P. Ebersbach, M. Sendelbach, N. Rana, A. Katnani, E. Mclellan, C. Archie, C. Bozdog, H. Kim, M. Sendler, S. Ng, B. Sherman, B. Brill, I. Turovets, and R. Urensky, “Holistic metrology approach: hybrid metrology utilizing scatterometry, critical dimension-atomic force microscope and critical dimension-scanning electron microscope,” J. Micro/Nanolith. MEMS MOEMS 10, 043016 (2011).

Brill, B.

A. Vaid, B. B. Yan, Y. T. Jiang, M. Kelling, C. Hartig, J. Allgair, P. Ebersbach, M. Sendelbach, N. Rana, A. Katnani, E. Mclellan, C. Archie, C. Bozdog, H. Kim, M. Sendler, S. Ng, B. Sherman, B. Brill, I. Turovets, and R. Urensky, “Holistic metrology approach: hybrid metrology utilizing scatterometry, critical dimension-atomic force microscope and critical dimension-scanning electron microscope,” J. Micro/Nanolith. MEMS MOEMS 10, 043016 (2011).

Brodwin, M. E.

A. Taflove and M. E. Brodwin, “Numerical solution of steady-state electromagnetic scattering problems using time-dependent Maxwell’s Equations,” IEEE Trans. Microwave Theory Tech. 23, 623–630 (1975).
[CrossRef]

Bunday, B.

R. M. Silver, N. F. Zhang, B. M. Barnes, H. Zhou, A. Heckert, R. Dixson, T. A. Germer, and B. Bunday, “Improving optical measurement accuracy using multi-technique nested uncertainties,” Proc. SPIE 7272, 727202 (2009).
[CrossRef]

R. M. Silver, T. A. Germer, R. Attota, B. M. Barnes, B. Bunday, J. Allgair, E. Marx, and J. Jun, “Fundamental limits of optical critical dimension metrology: a simulation study,” Proc. SPIE 6518, 65180U (2007).
[CrossRef]

Carlin, J. B.

A. Gelman, J. B. Carlin, H. S. Stern, and D. B. Rubin, Bayesian Data Analysis (Chapman & Hall/CRC, 2004).

Cox, M. G.

M. G. Cox, “The evaluation of key comparison data,” Metrologia 39, 589–595 (2002).
[CrossRef]

Dixson, R.

R. M. Silver, N. F. Zhang, B. M. Barnes, H. Zhou, A. Heckert, R. Dixson, T. A. Germer, and B. Bunday, “Improving optical measurement accuracy using multi-technique nested uncertainties,” Proc. SPIE 7272, 727202 (2009).
[CrossRef]

R. M. Silver, B. M. Barnes, H. Zhou, N. F. Zhang, and R. Dixson, “Angle-resolved optical metrology using multi-technique nested uncertainties,” Proc. SPIE 7390, 73900P (2009).
[CrossRef]

R. M. Silver, B. M. Barnes, A. Heckert, R. Attota, R. Dixson, and J. Jun, “Angle resolved optical metrology,” Proc. SPIE 6922, 69221M (2008).
[CrossRef]

Dixson, R. G.

H. J. Patrick, R. Attota, B. M. Barnes, T. A. Germer, R. G. Dixson, M. T. Stocker, R. M. Silver, and M. R. Bishop, “Optical critical dimension measurement of silicon grating targets using back focal plane scatterfield microscopy,” J. Micro/Nanolith. MEMS MOEMS 7, 013012 (2008).
[CrossRef]

Ebersbach, P.

A. Vaid, B. B. Yan, Y. T. Jiang, M. Kelling, C. Hartig, J. Allgair, P. Ebersbach, M. Sendelbach, N. Rana, A. Katnani, E. Mclellan, C. Archie, C. Bozdog, H. Kim, M. Sendler, S. Ng, B. Sherman, B. Brill, I. Turovets, and R. Urensky, “Holistic metrology approach: hybrid metrology utilizing scatterometry, critical dimension-atomic force microscope and critical dimension-scanning electron microscope,” J. Micro/Nanolith. MEMS MOEMS 10, 043016 (2011).

Gaylord, T. K.

Gelman, A.

A. Gelman, J. B. Carlin, H. S. Stern, and D. B. Rubin, Bayesian Data Analysis (Chapman & Hall/CRC, 2004).

Germer, T. A.

R. M. Silver, N. F. Zhang, B. M. Barnes, H. Zhou, A. Heckert, R. Dixson, T. A. Germer, and B. Bunday, “Improving optical measurement accuracy using multi-technique nested uncertainties,” Proc. SPIE 7272, 727202 (2009).
[CrossRef]

H. J. Patrick, R. Attota, B. M. Barnes, T. A. Germer, R. G. Dixson, M. T. Stocker, R. M. Silver, and M. R. Bishop, “Optical critical dimension measurement of silicon grating targets using back focal plane scatterfield microscopy,” J. Micro/Nanolith. MEMS MOEMS 7, 013012 (2008).
[CrossRef]

R. M. Silver, T. A. Germer, R. Attota, B. M. Barnes, B. Bunday, J. Allgair, E. Marx, and J. Jun, “Fundamental limits of optical critical dimension metrology: a simulation study,” Proc. SPIE 6518, 65180U (2007).
[CrossRef]

Golub, G. H.

G. H. Golub and C. H. Van Loan, Matrix Computations (Johns Hopkins University, 1989).

Grann, E. B.

Grann, M. G.

Hartig, C.

A. Vaid, B. B. Yan, Y. T. Jiang, M. Kelling, C. Hartig, J. Allgair, P. Ebersbach, M. Sendelbach, N. Rana, A. Katnani, E. Mclellan, C. Archie, C. Bozdog, H. Kim, M. Sendler, S. Ng, B. Sherman, B. Brill, I. Turovets, and R. Urensky, “Holistic metrology approach: hybrid metrology utilizing scatterometry, critical dimension-atomic force microscope and critical dimension-scanning electron microscope,” J. Micro/Nanolith. MEMS MOEMS 10, 043016 (2011).

Heckert, A.

R. M. Silver, N. F. Zhang, B. M. Barnes, H. Zhou, A. Heckert, R. Dixson, T. A. Germer, and B. Bunday, “Improving optical measurement accuracy using multi-technique nested uncertainties,” Proc. SPIE 7272, 727202 (2009).
[CrossRef]

R. M. Silver, B. M. Barnes, A. Heckert, R. Attota, R. Dixson, and J. Jun, “Angle resolved optical metrology,” Proc. SPIE 6922, 69221M (2008).
[CrossRef]

Howard, L. P.

B. M. Barnes, R. Attota, L. P. Howard, P. Lipscomb, M. T. Stocker, and R. M. Silver, “Zero-order and super-resolved imaging of arrayed nanoscale lines using scatterfield microscopy,” in 2007 International Conference on Frontiers of Characterization and Metrology , Vol. 931 of AIP Conference Proceedings (2007), pp. 397–401.

Iyer, H. K.

H. K. Iyer, C. M. Wang, and D. F. Vecchia, “Consistency tests for key comparison data,” Metrologia 41, 223–230 (2004).
[CrossRef]

Jiang, Y. T.

A. Vaid, B. B. Yan, Y. T. Jiang, M. Kelling, C. Hartig, J. Allgair, P. Ebersbach, M. Sendelbach, N. Rana, A. Katnani, E. Mclellan, C. Archie, C. Bozdog, H. Kim, M. Sendler, S. Ng, B. Sherman, B. Brill, I. Turovets, and R. Urensky, “Holistic metrology approach: hybrid metrology utilizing scatterometry, critical dimension-atomic force microscope and critical dimension-scanning electron microscope,” J. Micro/Nanolith. MEMS MOEMS 10, 043016 (2011).

Jun, J.

R. M. Silver, B. M. Barnes, A. Heckert, R. Attota, R. Dixson, and J. Jun, “Angle resolved optical metrology,” Proc. SPIE 6922, 69221M (2008).
[CrossRef]

R. M. Silver, T. A. Germer, R. Attota, B. M. Barnes, B. Bunday, J. Allgair, E. Marx, and J. Jun, “Fundamental limits of optical critical dimension metrology: a simulation study,” Proc. SPIE 6518, 65180U (2007).
[CrossRef]

Jun, R.

Katnani, A.

A. Vaid, B. B. Yan, Y. T. Jiang, M. Kelling, C. Hartig, J. Allgair, P. Ebersbach, M. Sendelbach, N. Rana, A. Katnani, E. Mclellan, C. Archie, C. Bozdog, H. Kim, M. Sendler, S. Ng, B. Sherman, B. Brill, I. Turovets, and R. Urensky, “Holistic metrology approach: hybrid metrology utilizing scatterometry, critical dimension-atomic force microscope and critical dimension-scanning electron microscope,” J. Micro/Nanolith. MEMS MOEMS 10, 043016 (2011).

Kelling, M.

A. Vaid, B. B. Yan, Y. T. Jiang, M. Kelling, C. Hartig, J. Allgair, P. Ebersbach, M. Sendelbach, N. Rana, A. Katnani, E. Mclellan, C. Archie, C. Bozdog, H. Kim, M. Sendler, S. Ng, B. Sherman, B. Brill, I. Turovets, and R. Urensky, “Holistic metrology approach: hybrid metrology utilizing scatterometry, critical dimension-atomic force microscope and critical dimension-scanning electron microscope,” J. Micro/Nanolith. MEMS MOEMS 10, 043016 (2011).

Kim, H.

A. Vaid, B. B. Yan, Y. T. Jiang, M. Kelling, C. Hartig, J. Allgair, P. Ebersbach, M. Sendelbach, N. Rana, A. Katnani, E. Mclellan, C. Archie, C. Bozdog, H. Kim, M. Sendler, S. Ng, B. Sherman, B. Brill, I. Turovets, and R. Urensky, “Holistic metrology approach: hybrid metrology utilizing scatterometry, critical dimension-atomic force microscope and critical dimension-scanning electron microscope,” J. Micro/Nanolith. MEMS MOEMS 10, 043016 (2011).

Kocherlakota, K. E.

S. Kocherlakota and K. E. Kocherlakota, Encyclopedia of Statistical Sciences (Wiley, 1983), Vol. 3, pp. 354–355.

Kocherlakota, S.

S. Kocherlakota and K. E. Kocherlakota, Encyclopedia of Statistical Sciences (Wiley, 1983), Vol. 3, pp. 354–355.

Kutner, M. H.

J. Neter, W. Wasserman, and M. H. Kutner, Applied Linear Regression Models (Richard D. Irwin, 1983).

Lalanne, P.

Lindley, D. V.

D. V. Lindley and A. F. M. Smith, “Bayes estimates for the linear model,” J. R. Stat. Soc. Ser. B. Methodol.1–41 (1972).

Lipscomb, P.

B. M. Barnes, R. Attota, L. P. Howard, P. Lipscomb, M. T. Stocker, and R. M. Silver, “Zero-order and super-resolved imaging of arrayed nanoscale lines using scatterfield microscopy,” in 2007 International Conference on Frontiers of Characterization and Metrology , Vol. 931 of AIP Conference Proceedings (2007), pp. 397–401.

Lunn, D. J.

D. J. Lunn, A. Thomas, N. Best, and D. Spiegelhalter, “WinBUGS—a Bayesian modeling framework: concepts, structure, and extensibility,” Stat. Comput. 10, 325–337 (2000).
[CrossRef]

Marx, E.

R. M. Silver, B. M. Barnes, R. Attota, R. Jun, M. Stocker, E. Marx, and H. J. Patrick, “Scatterfield microscopy for extending the limits of image-based optical metrology,” Appl. Opt. 46, 4248–4257 (2007).
[CrossRef]

R. M. Silver, T. A. Germer, R. Attota, B. M. Barnes, B. Bunday, J. Allgair, E. Marx, and J. Jun, “Fundamental limits of optical critical dimension metrology: a simulation study,” Proc. SPIE 6518, 65180U (2007).
[CrossRef]

Mclellan, E.

A. Vaid, B. B. Yan, Y. T. Jiang, M. Kelling, C. Hartig, J. Allgair, P. Ebersbach, M. Sendelbach, N. Rana, A. Katnani, E. Mclellan, C. Archie, C. Bozdog, H. Kim, M. Sendler, S. Ng, B. Sherman, B. Brill, I. Turovets, and R. Urensky, “Holistic metrology approach: hybrid metrology utilizing scatterometry, critical dimension-atomic force microscope and critical dimension-scanning electron microscope,” J. Micro/Nanolith. MEMS MOEMS 10, 043016 (2011).

Moharam, M. G.

Morris, G. M.

Neter, J.

J. Neter, W. Wasserman, and M. H. Kutner, Applied Linear Regression Models (Richard D. Irwin, 1983).

Ng, S.

A. Vaid, B. B. Yan, Y. T. Jiang, M. Kelling, C. Hartig, J. Allgair, P. Ebersbach, M. Sendelbach, N. Rana, A. Katnani, E. Mclellan, C. Archie, C. Bozdog, H. Kim, M. Sendler, S. Ng, B. Sherman, B. Brill, I. Turovets, and R. Urensky, “Holistic metrology approach: hybrid metrology utilizing scatterometry, critical dimension-atomic force microscope and critical dimension-scanning electron microscope,” J. Micro/Nanolith. MEMS MOEMS 10, 043016 (2011).

Patrick, H. J.

H. J. Patrick, R. Attota, B. M. Barnes, T. A. Germer, R. G. Dixson, M. T. Stocker, R. M. Silver, and M. R. Bishop, “Optical critical dimension measurement of silicon grating targets using back focal plane scatterfield microscopy,” J. Micro/Nanolith. MEMS MOEMS 7, 013012 (2008).
[CrossRef]

R. M. Silver, B. M. Barnes, R. Attota, R. Jun, M. Stocker, E. Marx, and H. J. Patrick, “Scatterfield microscopy for extending the limits of image-based optical metrology,” Appl. Opt. 46, 4248–4257 (2007).
[CrossRef]

Pommet, D. A.

Press, S.

S. Press, Bayesian Statistics: Principles, Models, and Applications (Wiley, 1989).

Rana, N.

A. Vaid, B. B. Yan, Y. T. Jiang, M. Kelling, C. Hartig, J. Allgair, P. Ebersbach, M. Sendelbach, N. Rana, A. Katnani, E. Mclellan, C. Archie, C. Bozdog, H. Kim, M. Sendler, S. Ng, B. Sherman, B. Brill, I. Turovets, and R. Urensky, “Holistic metrology approach: hybrid metrology utilizing scatterometry, critical dimension-atomic force microscope and critical dimension-scanning electron microscope,” J. Micro/Nanolith. MEMS MOEMS 10, 043016 (2011).

Rao, C. R.

C. R. Rao and H. Toutenburg, Linear Models: Least Squares and Alternatives (Springer, 1995).

Rubin, D. B.

A. Gelman, J. B. Carlin, H. S. Stern, and D. B. Rubin, Bayesian Data Analysis (Chapman & Hall/CRC, 2004).

Sendelbach, M.

A. Vaid, B. B. Yan, Y. T. Jiang, M. Kelling, C. Hartig, J. Allgair, P. Ebersbach, M. Sendelbach, N. Rana, A. Katnani, E. Mclellan, C. Archie, C. Bozdog, H. Kim, M. Sendler, S. Ng, B. Sherman, B. Brill, I. Turovets, and R. Urensky, “Holistic metrology approach: hybrid metrology utilizing scatterometry, critical dimension-atomic force microscope and critical dimension-scanning electron microscope,” J. Micro/Nanolith. MEMS MOEMS 10, 043016 (2011).

Sendler, M.

A. Vaid, B. B. Yan, Y. T. Jiang, M. Kelling, C. Hartig, J. Allgair, P. Ebersbach, M. Sendelbach, N. Rana, A. Katnani, E. Mclellan, C. Archie, C. Bozdog, H. Kim, M. Sendler, S. Ng, B. Sherman, B. Brill, I. Turovets, and R. Urensky, “Holistic metrology approach: hybrid metrology utilizing scatterometry, critical dimension-atomic force microscope and critical dimension-scanning electron microscope,” J. Micro/Nanolith. MEMS MOEMS 10, 043016 (2011).

Sherman, B.

A. Vaid, B. B. Yan, Y. T. Jiang, M. Kelling, C. Hartig, J. Allgair, P. Ebersbach, M. Sendelbach, N. Rana, A. Katnani, E. Mclellan, C. Archie, C. Bozdog, H. Kim, M. Sendler, S. Ng, B. Sherman, B. Brill, I. Turovets, and R. Urensky, “Holistic metrology approach: hybrid metrology utilizing scatterometry, critical dimension-atomic force microscope and critical dimension-scanning electron microscope,” J. Micro/Nanolith. MEMS MOEMS 10, 043016 (2011).

Silver, R. M.

R. M. Silver, N. F. Zhang, B. M. Barnes, H. Zhou, A. Heckert, R. Dixson, T. A. Germer, and B. Bunday, “Improving optical measurement accuracy using multi-technique nested uncertainties,” Proc. SPIE 7272, 727202 (2009).
[CrossRef]

R. M. Silver, B. M. Barnes, H. Zhou, N. F. Zhang, and R. Dixson, “Angle-resolved optical metrology using multi-technique nested uncertainties,” Proc. SPIE 7390, 73900P (2009).
[CrossRef]

H. J. Patrick, R. Attota, B. M. Barnes, T. A. Germer, R. G. Dixson, M. T. Stocker, R. M. Silver, and M. R. Bishop, “Optical critical dimension measurement of silicon grating targets using back focal plane scatterfield microscopy,” J. Micro/Nanolith. MEMS MOEMS 7, 013012 (2008).
[CrossRef]

R. M. Silver, B. M. Barnes, A. Heckert, R. Attota, R. Dixson, and J. Jun, “Angle resolved optical metrology,” Proc. SPIE 6922, 69221M (2008).
[CrossRef]

R. M. Silver, T. A. Germer, R. Attota, B. M. Barnes, B. Bunday, J. Allgair, E. Marx, and J. Jun, “Fundamental limits of optical critical dimension metrology: a simulation study,” Proc. SPIE 6518, 65180U (2007).
[CrossRef]

R. M. Silver, B. M. Barnes, R. Attota, R. Jun, M. Stocker, E. Marx, and H. J. Patrick, “Scatterfield microscopy for extending the limits of image-based optical metrology,” Appl. Opt. 46, 4248–4257 (2007).
[CrossRef]

B. M. Barnes, R. Attota, L. P. Howard, P. Lipscomb, M. T. Stocker, and R. M. Silver, “Zero-order and super-resolved imaging of arrayed nanoscale lines using scatterfield microscopy,” in 2007 International Conference on Frontiers of Characterization and Metrology , Vol. 931 of AIP Conference Proceedings (2007), pp. 397–401.

Smith, A. F. M.

D. V. Lindley and A. F. M. Smith, “Bayes estimates for the linear model,” J. R. Stat. Soc. Ser. B. Methodol.1–41 (1972).

Spiegelhalter, D.

D. J. Lunn, A. Thomas, N. Best, and D. Spiegelhalter, “WinBUGS—a Bayesian modeling framework: concepts, structure, and extensibility,” Stat. Comput. 10, 325–337 (2000).
[CrossRef]

Stern, H. S.

A. Gelman, J. B. Carlin, H. S. Stern, and D. B. Rubin, Bayesian Data Analysis (Chapman & Hall/CRC, 2004).

Stocker, M.

Stocker, M. T.

H. J. Patrick, R. Attota, B. M. Barnes, T. A. Germer, R. G. Dixson, M. T. Stocker, R. M. Silver, and M. R. Bishop, “Optical critical dimension measurement of silicon grating targets using back focal plane scatterfield microscopy,” J. Micro/Nanolith. MEMS MOEMS 7, 013012 (2008).
[CrossRef]

B. M. Barnes, R. Attota, L. P. Howard, P. Lipscomb, M. T. Stocker, and R. M. Silver, “Zero-order and super-resolved imaging of arrayed nanoscale lines using scatterfield microscopy,” in 2007 International Conference on Frontiers of Characterization and Metrology , Vol. 931 of AIP Conference Proceedings (2007), pp. 397–401.

Taflove, A.

A. Taflove and M. E. Brodwin, “Numerical solution of steady-state electromagnetic scattering problems using time-dependent Maxwell’s Equations,” IEEE Trans. Microwave Theory Tech. 23, 623–630 (1975).
[CrossRef]

Thomas, A.

D. J. Lunn, A. Thomas, N. Best, and D. Spiegelhalter, “WinBUGS—a Bayesian modeling framework: concepts, structure, and extensibility,” Stat. Comput. 10, 325–337 (2000).
[CrossRef]

Toutenburg, H.

C. R. Rao and H. Toutenburg, Linear Models: Least Squares and Alternatives (Springer, 1995).

Turovets, I.

A. Vaid, B. B. Yan, Y. T. Jiang, M. Kelling, C. Hartig, J. Allgair, P. Ebersbach, M. Sendelbach, N. Rana, A. Katnani, E. Mclellan, C. Archie, C. Bozdog, H. Kim, M. Sendler, S. Ng, B. Sherman, B. Brill, I. Turovets, and R. Urensky, “Holistic metrology approach: hybrid metrology utilizing scatterometry, critical dimension-atomic force microscope and critical dimension-scanning electron microscope,” J. Micro/Nanolith. MEMS MOEMS 10, 043016 (2011).

Urensky, R.

A. Vaid, B. B. Yan, Y. T. Jiang, M. Kelling, C. Hartig, J. Allgair, P. Ebersbach, M. Sendelbach, N. Rana, A. Katnani, E. Mclellan, C. Archie, C. Bozdog, H. Kim, M. Sendler, S. Ng, B. Sherman, B. Brill, I. Turovets, and R. Urensky, “Holistic metrology approach: hybrid metrology utilizing scatterometry, critical dimension-atomic force microscope and critical dimension-scanning electron microscope,” J. Micro/Nanolith. MEMS MOEMS 10, 043016 (2011).

Vaid, A.

A. Vaid, B. B. Yan, Y. T. Jiang, M. Kelling, C. Hartig, J. Allgair, P. Ebersbach, M. Sendelbach, N. Rana, A. Katnani, E. Mclellan, C. Archie, C. Bozdog, H. Kim, M. Sendler, S. Ng, B. Sherman, B. Brill, I. Turovets, and R. Urensky, “Holistic metrology approach: hybrid metrology utilizing scatterometry, critical dimension-atomic force microscope and critical dimension-scanning electron microscope,” J. Micro/Nanolith. MEMS MOEMS 10, 043016 (2011).

Van Loan, C. H.

G. H. Golub and C. H. Van Loan, Matrix Computations (Johns Hopkins University, 1989).

Vecchia, D. F.

H. K. Iyer, C. M. Wang, and D. F. Vecchia, “Consistency tests for key comparison data,” Metrologia 41, 223–230 (2004).
[CrossRef]

Wang, C. M.

H. K. Iyer, C. M. Wang, and D. F. Vecchia, “Consistency tests for key comparison data,” Metrologia 41, 223–230 (2004).
[CrossRef]

Wasserman, W.

J. Neter, W. Wasserman, and M. H. Kutner, Applied Linear Regression Models (Richard D. Irwin, 1983).

Watts, D. G.

D. M. Bates and D. G. Watts, Nonlinear Regression Analysis and Its Applications (Wiley, 1988).

Yan, B. B.

A. Vaid, B. B. Yan, Y. T. Jiang, M. Kelling, C. Hartig, J. Allgair, P. Ebersbach, M. Sendelbach, N. Rana, A. Katnani, E. Mclellan, C. Archie, C. Bozdog, H. Kim, M. Sendler, S. Ng, B. Sherman, B. Brill, I. Turovets, and R. Urensky, “Holistic metrology approach: hybrid metrology utilizing scatterometry, critical dimension-atomic force microscope and critical dimension-scanning electron microscope,” J. Micro/Nanolith. MEMS MOEMS 10, 043016 (2011).

Zhang, N. F.

R. M. Silver, N. F. Zhang, B. M. Barnes, H. Zhou, A. Heckert, R. Dixson, T. A. Germer, and B. Bunday, “Improving optical measurement accuracy using multi-technique nested uncertainties,” Proc. SPIE 7272, 727202 (2009).
[CrossRef]

R. M. Silver, B. M. Barnes, H. Zhou, N. F. Zhang, and R. Dixson, “Angle-resolved optical metrology using multi-technique nested uncertainties,” Proc. SPIE 7390, 73900P (2009).
[CrossRef]

Zhou, H.

R. M. Silver, N. F. Zhang, B. M. Barnes, H. Zhou, A. Heckert, R. Dixson, T. A. Germer, and B. Bunday, “Improving optical measurement accuracy using multi-technique nested uncertainties,” Proc. SPIE 7272, 727202 (2009).
[CrossRef]

R. M. Silver, B. M. Barnes, H. Zhou, N. F. Zhang, and R. Dixson, “Angle-resolved optical metrology using multi-technique nested uncertainties,” Proc. SPIE 7390, 73900P (2009).
[CrossRef]

Appl. Opt.

IEEE Trans. Microwave Theory Tech.

A. Taflove and M. E. Brodwin, “Numerical solution of steady-state electromagnetic scattering problems using time-dependent Maxwell’s Equations,” IEEE Trans. Microwave Theory Tech. 23, 623–630 (1975).
[CrossRef]

J. Micro/Nanolith. MEMS MOEMS

H. J. Patrick, R. Attota, B. M. Barnes, T. A. Germer, R. G. Dixson, M. T. Stocker, R. M. Silver, and M. R. Bishop, “Optical critical dimension measurement of silicon grating targets using back focal plane scatterfield microscopy,” J. Micro/Nanolith. MEMS MOEMS 7, 013012 (2008).
[CrossRef]

A. Vaid, B. B. Yan, Y. T. Jiang, M. Kelling, C. Hartig, J. Allgair, P. Ebersbach, M. Sendelbach, N. Rana, A. Katnani, E. Mclellan, C. Archie, C. Bozdog, H. Kim, M. Sendler, S. Ng, B. Sherman, B. Brill, I. Turovets, and R. Urensky, “Holistic metrology approach: hybrid metrology utilizing scatterometry, critical dimension-atomic force microscope and critical dimension-scanning electron microscope,” J. Micro/Nanolith. MEMS MOEMS 10, 043016 (2011).

J. Opt. Soc. Am. A

J. R. Stat. Soc. Ser. B. Methodol.

D. V. Lindley and A. F. M. Smith, “Bayes estimates for the linear model,” J. R. Stat. Soc. Ser. B. Methodol.1–41 (1972).

Metrologia

H. K. Iyer, C. M. Wang, and D. F. Vecchia, “Consistency tests for key comparison data,” Metrologia 41, 223–230 (2004).
[CrossRef]

M. G. Cox, “The evaluation of key comparison data,” Metrologia 39, 589–595 (2002).
[CrossRef]

Proc. SPIE

R. M. Silver, B. M. Barnes, H. Zhou, N. F. Zhang, and R. Dixson, “Angle-resolved optical metrology using multi-technique nested uncertainties,” Proc. SPIE 7390, 73900P (2009).
[CrossRef]

R. M. Silver, T. A. Germer, R. Attota, B. M. Barnes, B. Bunday, J. Allgair, E. Marx, and J. Jun, “Fundamental limits of optical critical dimension metrology: a simulation study,” Proc. SPIE 6518, 65180U (2007).
[CrossRef]

R. M. Silver, N. F. Zhang, B. M. Barnes, H. Zhou, A. Heckert, R. Dixson, T. A. Germer, and B. Bunday, “Improving optical measurement accuracy using multi-technique nested uncertainties,” Proc. SPIE 7272, 727202 (2009).
[CrossRef]

R. M. Silver, B. M. Barnes, A. Heckert, R. Attota, R. Dixson, and J. Jun, “Angle resolved optical metrology,” Proc. SPIE 6922, 69221M (2008).
[CrossRef]

Stat. Comput.

D. J. Lunn, A. Thomas, N. Best, and D. Spiegelhalter, “WinBUGS—a Bayesian modeling framework: concepts, structure, and extensibility,” Stat. Comput. 10, 325–337 (2000).
[CrossRef]

Other

S. Kocherlakota and K. E. Kocherlakota, Encyclopedia of Statistical Sciences (Wiley, 1983), Vol. 3, pp. 354–355.

J. Neter, W. Wasserman, and M. H. Kutner, Applied Linear Regression Models (Richard D. Irwin, 1983).

C. R. Rao and H. Toutenburg, Linear Models: Least Squares and Alternatives (Springer, 1995).

D. M. Bates and D. G. Watts, Nonlinear Regression Analysis and Its Applications (Wiley, 1988).

B. M. Barnes, R. Attota, L. P. Howard, P. Lipscomb, M. T. Stocker, and R. M. Silver, “Zero-order and super-resolved imaging of arrayed nanoscale lines using scatterfield microscopy,” in 2007 International Conference on Frontiers of Characterization and Metrology , Vol. 931 of AIP Conference Proceedings (2007), pp. 397–401.

S. Press, Bayesian Statistics: Principles, Models, and Applications (Wiley, 1989).

A. Gelman, J. B. Carlin, H. S. Stern, and D. B. Rubin, Bayesian Data Analysis (Chapman & Hall/CRC, 2004).

Certain commercial equipment, instruments, or materials are identified in this paper in order to specify the experimental procedure adequately. Such identification is not intended to imply recommendation or endorsement by the National Institute of Standards and Technology, nor is it intended to imply that the materials or equipment identified are necessarily the best available for the purpose.

G. H. Golub and C. H. Van Loan, Matrix Computations (Johns Hopkins University, 1989).

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Figures (3)

Fig. 1.
Fig. 1.

Parametric variables defined for electromagnetic scattering simulations. (a) Cross section for a nitride line on polysilicon. The height (h), top width (wtop), middle width (wmiddle), and bottom width (wbottom) are varied. In addition, the optical index of refraction (n) was also scaled as a floating parameter. (b) Cross section for an amorphous silicon line on oxidized silicon. The three floating parameters are total height (h), sidewall angle (SWA), and total middle width (wmiddle). A 2 nm fixed-thickness layer of conformal native oxide is assumed in the model to coat these lines.

Fig. 2.
Fig. 2.

Example set of experimental data (markers) and library data fits (curves) for the reflectivity from a patterned nitride line array on polysilicon illustrated in Fig. 1(a). The four curves in each plot correspond to the four combinations of scan direction and orthogonal linear polarizations shown in the schematic.

Fig. 3.
Fig. 3.

Examples of experimental reflectivity data (markers) and library data fits (curves) for oxide-coated amorphous silicon lines on oxidized silicon. The scan parameters and legend are defined in Fig. 1 for the structure shown in Fig. 2(b).

Tables (2)

Tables Icon

Table 1. Parametric OCD Fits to the Data in Fig. 2 Before and After the Inclusion of Data from AFM a

Tables Icon

Table 2. Parametric OCD Fits to the Data in Fig. 3 Before and After the Inclusion of Data from an AFM a

Equations (41)

Equations on this page are rendered with MathJax. Learn more.

yi=y(xi,a)+εifori=1,,N,
y(xi;a)y(xi;a(0))+k=1K([y(xi;a)ak]a=a(0)(akak(0))),
yi=y(xi;a(0))+k=1K([y(xi;a)ak]a=a(0)(akak(0)))+εi,i=1,,N.
yi(0)=k=1KDik(0)βk(0)+εi,i=1,,N,
βk(0)=akak(0),
Dik(0)=[y(xi;a)ak]a=a(0),
yi(0)=yiy(xi;a(0)).
Y(0)=D(0)·β(0)+ε,
β^(0)=(D(0)TV1D(0))1D(0)TV1Y(0).
a^k=β^k(0)+ak(0),
Cov[a^]=Cov[β^(0)]=(D(0)T·V1·D(0))1.
a*=(a1*,,ap*)T,
βk*(0)=E[βk(0)]=ak*ak(0),
Σβp(0)=diag[σa12,,σap2].
β1*(0)=β1(0)+εN+1,
βk*(0)=βk(0)+εN+k.
Y*(0)=D*(0)·β(0)+ε*,
D*(0)=(D(0)1)=(D11(0),,D1K(0)DN1(0),,DNK(0)1,0,0,,00,1,0,,00,0,0.0,1,0,0),
Y*(0)=(y1(0),,yN(0),β1*(0),,βp*(0))T,
ε*=(ε1,,εN,εN+1,,εN+p)T
V*=Cov[ε*]=diag[σ12,,σN2,σa12,,σap2].
β^#(0)=(D*(0)TV*1D*(0))1D*(0)TV*1Y*(0)
Cov[a^#]=Cov[β^#(0)]=(D*(0)T·V*1·D*(0))1,
β^#(0)=1|Q|[|Q1|β^(0)+|Q1|β1*(0)σa12(Var[β^1(0)]Cov[β^1(0),β^2(0)])+(0g2σa12)],
Cov[β^#(0)]Cov[β^(0)].
|Cov[β^#(0)]||Cov[β^(0)]|.
Var[β^k#(0)]Var[β^k(0)]fork=1,,K.
Var[β^i#(0)]σβi2,
Z=a^OCDaAFM*ua^2+ua*2,
Q1=D(0)TV1D(0)=(q11,q12q12,q22).
Q=D*(0)TV*1D*(0)=D(0)TV1D(0)+1TΣβ2(0)11=(q11+1/σa12,q12q12,q22+1/σa22).
D*(0)T=(D11(0),,DN1(0),1,0D12(0),,DN2(0),0,1)=(D(0)T,(1,00,1)),
V*1=(V1,00,(σa12,00,σa22)).
Q=D*(0)TV*1D*(0)=D(0)TV1D(0)+(σa12,00,σa22)=(q11+σa12,q12q12,q22+σa22).
Q11=(D(0)TV1D(0))1=(q22,q12q12,q11)|Q1|
(D*(0)TV*1D*(0))1=Q1=1|Q|[(q22,q12q12,q11)+(σa22,00,σa12)]=1|Q|[|Q1|(D(0)TV1D(0))1+(σa22,00,σa12)].
D*(0)TV*1=(D(0)T,(1,00,1))(V1,00,(σa12,00,σa22))=(D(0)TV1,(σa12,00,σa22))
D*(0)TV*1Y*(0)=(D(0)TV1,(σa12,00,σa22))(y1(0)yN(0)β1*(0)β2*(0))=D(0)TV1Y(0)+Σβp(0)1β*(0).
D(0)TV1Y(0)=(g1g2).
β^#(0)=(D*(0)TV*1D*(0))1D*(0)TV*1Y*(0)=1|Q|[|Q1|(D(0)TV1D(0))1+(σa22,00,σa12)][D(0)TV1Y(0)+Σβp(0)1β*(0)]=1|Q|[|Q1|(D(0)TV1D(0))1D(0)TV1Y(0)+|Q1|(D(0)TV1D(0))1Σβp(0)1β*(0)+(σa22,00,σa12)D(0)TV1Y(0)+β*(0)σa12σa22]=1|Q|[|Q1|β^(0)+|Q1|Cov[β^(0)]Σβp(0)1β*(0)+(g1σa22g2σa12)+β*(0)σa12σa22]=1|Q|[|Q1|β^(0)+[|Q1|Cov[β^(0)]Σβp(0)1+1σa12σa22]β*(0)+(g1σa22g2σa12)].
β^#(0)=1|Q|[|Q1|β^(0)+|Q1|β1*(0)σa12(Var[β^1(0)]Cov[β^1(0),β^2(0)])+(0g2σa12)].

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