Abstract

This article presents a topology optimization approach for micro- and nano-devices fabricated by optical projection lithography. Incorporating the photolithography process and the manufacturing uncertainties into the topology optimization process results in a binary mask that can be sent directly to manufacturing without additional optical proximity correction (OPC). The performance of the optimized device is robust toward the considered process variations. With the proposed unified approach, the design for photolithography is achieved by considering the optimal device performance and manufacturability at the same time. Only one optimization problem is solved instead of two as in the conventional separate procedures by (1) blueprint design and (2) OPC. A micro-gripper design example is presented to demonstrate the potential of this approach.

© 2014 Optical Society of America

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  1. C. Mack, Fundamental Principles of Optical Lithography: The Science of Micro-Fabrication (Wiley, 2007).
  2. Z. Cui, Nanofabrication: Principles, Capabilities and Limits (Springer, 2008).
  3. Y. Granik, “Fast pixel-based mask optimization for inverse lithography,” J. Micro/Nanolith. MEMS MOEMS 5, 043002 (2006).
  4. A. Poonawala and P. Milanfar, “Mask design for optical microlithography an inverse imaging problem,” IEEE Trans. Image Process. 16, 774–788 (2007).
    [CrossRef]
  5. M. Levenson, N. Viswanathan, and R. Simpson, “Improving resolution in photolithography with a phaseshifting mask,” IEEE Trans. Electron Devices 29, 1828–1836 (1982).
    [CrossRef]
  6. X. Ma and G. R. Arce, “Generalized inverse lithography methods for phase-shifting mask design,” Opt. Express 15, 15066–15079 (2007).
    [CrossRef]
  7. X. Ma, Z. Song, Y. Li, and G. R. Arce, “Block-based mask optimization for optical lithography,” Appl. Opt. 52, 3351–3363 (2013).
    [CrossRef]
  8. X. Ma and G. R. Arce, “Pixel-based simultaneous source and mask optimization for resolution enhancement in optical lithography,” Opt. Express 17, 5783–5793 (2009).
    [CrossRef]
  9. N. Jia and E. Y. Lam, “Pixelated source mask optimization for process robustness in optical lithography,” Opt. Express 19, 19384–19398 (2011).
    [CrossRef]
  10. J. Li, Y. Shen, and E. Y. Lam, “Hotspot-aware fast source and mask optimization,” Opt. Express 20, 21792–21804 (2012).
    [CrossRef]
  11. J. Li, S. Liu, and E. Y. Lam, “Efficient source and mask optimization with augmented Lagrangian methods in optical lithography,” Opt. Express 21, 8076–8090 (2013).
    [CrossRef]
  12. S. Li, X. Wang, and Y. Bu, “Robust pixel-based source and mask optimization for inverse lithography,” Opt. Laser Technol. 45, 285–293 (2013).
    [CrossRef]
  13. Y. Shen, N. Jia, N. Wong, and E. Y. Lam, “Robust levelset-based inverse lithography,” Opt. Express 19, 5511–5521 (2011).
    [CrossRef]
  14. N. Jia and E. Y. Lam, “Machine learning for inverse lithography: using stochastic gradient descent for robust photomask synthesis,” J. Opt. 12, 045601 (2010).
    [CrossRef]
  15. J. L. Sturtevant, J. A. Torres, J. Word, and P. L. Y. Granik, “Consideration for the use of defocus models for OPC,” Proc. SPIE 5756, 427–436 (2005).
    [CrossRef]
  16. S. Choy, N. Jia, C. Tong, M. Tang, and E. Lam, “A robust computational algorithm for inverse photomask synthesis in optical projection lithography,” SIAM J. Imag. Sci. 5, 625–651 (2012).
  17. P. Yu, D. Z. Pan, and C. A. Mack, “True process variation aware optical proximity correction with variational lithography modeling and model calibration,” J. Micro/Nanolith. MEMS MOEMS 6, 031004 (2007).
  18. X. Ma and G. R. Arce, “Pixel-based OPC optimization based on conjugate gradients,” Opt. Express 19, 2165–2180 (2011).
    [CrossRef]
  19. M. Bendsøe and N. Kikuchi, “Generating optimal topologies in structural design using a homogenization method,” Comp. Meth. App. Mech. Eng. 71, 197–224 (1988).
  20. J. Jensen and O. Sigmund, “Topology optimization for nano-photonics,” Laser Photon. Rev. 5, 308–321 (2011).
  21. J. Jonsmann, O. Sigmund, and S. Bouwstra, “Compliant thermal microactuators,” Sens. Act. 76, 463–469 (1999).
  22. O. Sardan, V. Eichhorn, D. Petersen, S. Fatikow, O. Sigmund, and P. Bøggild, “Rapid prototyping of nanotube-based devices using topology-optimized microgrippers,” Nanotechnology 19, 495503 (2008).
    [CrossRef]
  23. M. A. Philippine, O. Sigmund, G. M. Rebeiz, and T. W. Kenny, “Topology optimization of stressed capacitive RF MEMS switches,” J. Microelectromech. Syst. 22, 206–215 (2013).
  24. M. Bendsøe and O. Sigmund, Topology Optimization: Theory, Methods and Applications (Springer, 2003).
  25. M. Jansen, B. S. Lazarov, M. Schevenels, and O. Sigmund, “On the similarities between micro/nano lithography and topology optimization projection methods,” Struct. Multidiscip. Optim. 48, 717–730 (2013).
  26. J. Guest, J. Prevost, and T. Belytschko, “Achieving minimum length scale in topology optimization using nodal design variables and projection functions,” Int. J. Numer. Methods Eng. 61, 238–254 (2004).
  27. F. Wang, B. Lazarov, and O. Sigmund, “On projection methods, convergence and robust formulations in topology optimization,” Struct. Multidiscip. Optim. 43, 767–784 (2011).
  28. O. Sigmund, “Morphology-based black and white filters for topology optimization,” Struct. Multidiscip. Optim. 33, 401–424 (2007).
  29. B. S. Lazarov, M. Schevenels, and O. Sigmund, “Topology optimization considering material and geometric uncertainties using stochastic collocation methods,” Struct. Multidiscip. Optim. 46, 597–612 (2012).
  30. O. Sigmund, “On the design of compliant mechanisms using topology optimization,” Mech. Struct. Machines 25, 493–524 (1997).
  31. K. Svanberg, “The method of moving asymptotes—a new method for structural optimization,” Int. J. Numer. Methods Eng. 24, 359–397 (1987).
  32. Y. C. Pati and T. Kailath, “Phase-shifting masks for microlithography: automated design and mask requirements,” J. Opt. Soc. Am. A 11, 2438–2452 (1994).
    [CrossRef]

2013

M. A. Philippine, O. Sigmund, G. M. Rebeiz, and T. W. Kenny, “Topology optimization of stressed capacitive RF MEMS switches,” J. Microelectromech. Syst. 22, 206–215 (2013).

M. Jansen, B. S. Lazarov, M. Schevenels, and O. Sigmund, “On the similarities between micro/nano lithography and topology optimization projection methods,” Struct. Multidiscip. Optim. 48, 717–730 (2013).

S. Li, X. Wang, and Y. Bu, “Robust pixel-based source and mask optimization for inverse lithography,” Opt. Laser Technol. 45, 285–293 (2013).
[CrossRef]

J. Li, S. Liu, and E. Y. Lam, “Efficient source and mask optimization with augmented Lagrangian methods in optical lithography,” Opt. Express 21, 8076–8090 (2013).
[CrossRef]

X. Ma, Z. Song, Y. Li, and G. R. Arce, “Block-based mask optimization for optical lithography,” Appl. Opt. 52, 3351–3363 (2013).
[CrossRef]

2012

J. Li, Y. Shen, and E. Y. Lam, “Hotspot-aware fast source and mask optimization,” Opt. Express 20, 21792–21804 (2012).
[CrossRef]

B. S. Lazarov, M. Schevenels, and O. Sigmund, “Topology optimization considering material and geometric uncertainties using stochastic collocation methods,” Struct. Multidiscip. Optim. 46, 597–612 (2012).

S. Choy, N. Jia, C. Tong, M. Tang, and E. Lam, “A robust computational algorithm for inverse photomask synthesis in optical projection lithography,” SIAM J. Imag. Sci. 5, 625–651 (2012).

2011

F. Wang, B. Lazarov, and O. Sigmund, “On projection methods, convergence and robust formulations in topology optimization,” Struct. Multidiscip. Optim. 43, 767–784 (2011).

J. Jensen and O. Sigmund, “Topology optimization for nano-photonics,” Laser Photon. Rev. 5, 308–321 (2011).

X. Ma and G. R. Arce, “Pixel-based OPC optimization based on conjugate gradients,” Opt. Express 19, 2165–2180 (2011).
[CrossRef]

Y. Shen, N. Jia, N. Wong, and E. Y. Lam, “Robust levelset-based inverse lithography,” Opt. Express 19, 5511–5521 (2011).
[CrossRef]

N. Jia and E. Y. Lam, “Pixelated source mask optimization for process robustness in optical lithography,” Opt. Express 19, 19384–19398 (2011).
[CrossRef]

2010

N. Jia and E. Y. Lam, “Machine learning for inverse lithography: using stochastic gradient descent for robust photomask synthesis,” J. Opt. 12, 045601 (2010).
[CrossRef]

2009

2008

O. Sardan, V. Eichhorn, D. Petersen, S. Fatikow, O. Sigmund, and P. Bøggild, “Rapid prototyping of nanotube-based devices using topology-optimized microgrippers,” Nanotechnology 19, 495503 (2008).
[CrossRef]

2007

X. Ma and G. R. Arce, “Generalized inverse lithography methods for phase-shifting mask design,” Opt. Express 15, 15066–15079 (2007).
[CrossRef]

P. Yu, D. Z. Pan, and C. A. Mack, “True process variation aware optical proximity correction with variational lithography modeling and model calibration,” J. Micro/Nanolith. MEMS MOEMS 6, 031004 (2007).

O. Sigmund, “Morphology-based black and white filters for topology optimization,” Struct. Multidiscip. Optim. 33, 401–424 (2007).

A. Poonawala and P. Milanfar, “Mask design for optical microlithography an inverse imaging problem,” IEEE Trans. Image Process. 16, 774–788 (2007).
[CrossRef]

2006

Y. Granik, “Fast pixel-based mask optimization for inverse lithography,” J. Micro/Nanolith. MEMS MOEMS 5, 043002 (2006).

2005

J. L. Sturtevant, J. A. Torres, J. Word, and P. L. Y. Granik, “Consideration for the use of defocus models for OPC,” Proc. SPIE 5756, 427–436 (2005).
[CrossRef]

2004

J. Guest, J. Prevost, and T. Belytschko, “Achieving minimum length scale in topology optimization using nodal design variables and projection functions,” Int. J. Numer. Methods Eng. 61, 238–254 (2004).

1999

J. Jonsmann, O. Sigmund, and S. Bouwstra, “Compliant thermal microactuators,” Sens. Act. 76, 463–469 (1999).

1997

O. Sigmund, “On the design of compliant mechanisms using topology optimization,” Mech. Struct. Machines 25, 493–524 (1997).

1994

1988

M. Bendsøe and N. Kikuchi, “Generating optimal topologies in structural design using a homogenization method,” Comp. Meth. App. Mech. Eng. 71, 197–224 (1988).

1987

K. Svanberg, “The method of moving asymptotes—a new method for structural optimization,” Int. J. Numer. Methods Eng. 24, 359–397 (1987).

1982

M. Levenson, N. Viswanathan, and R. Simpson, “Improving resolution in photolithography with a phaseshifting mask,” IEEE Trans. Electron Devices 29, 1828–1836 (1982).
[CrossRef]

Arce, G. R.

Belytschko, T.

J. Guest, J. Prevost, and T. Belytschko, “Achieving minimum length scale in topology optimization using nodal design variables and projection functions,” Int. J. Numer. Methods Eng. 61, 238–254 (2004).

Bendsøe, M.

M. Bendsøe and N. Kikuchi, “Generating optimal topologies in structural design using a homogenization method,” Comp. Meth. App. Mech. Eng. 71, 197–224 (1988).

M. Bendsøe and O. Sigmund, Topology Optimization: Theory, Methods and Applications (Springer, 2003).

Bøggild, P.

O. Sardan, V. Eichhorn, D. Petersen, S. Fatikow, O. Sigmund, and P. Bøggild, “Rapid prototyping of nanotube-based devices using topology-optimized microgrippers,” Nanotechnology 19, 495503 (2008).
[CrossRef]

Bouwstra, S.

J. Jonsmann, O. Sigmund, and S. Bouwstra, “Compliant thermal microactuators,” Sens. Act. 76, 463–469 (1999).

Bu, Y.

S. Li, X. Wang, and Y. Bu, “Robust pixel-based source and mask optimization for inverse lithography,” Opt. Laser Technol. 45, 285–293 (2013).
[CrossRef]

Choy, S.

S. Choy, N. Jia, C. Tong, M. Tang, and E. Lam, “A robust computational algorithm for inverse photomask synthesis in optical projection lithography,” SIAM J. Imag. Sci. 5, 625–651 (2012).

Cui, Z.

Z. Cui, Nanofabrication: Principles, Capabilities and Limits (Springer, 2008).

Eichhorn, V.

O. Sardan, V. Eichhorn, D. Petersen, S. Fatikow, O. Sigmund, and P. Bøggild, “Rapid prototyping of nanotube-based devices using topology-optimized microgrippers,” Nanotechnology 19, 495503 (2008).
[CrossRef]

Fatikow, S.

O. Sardan, V. Eichhorn, D. Petersen, S. Fatikow, O. Sigmund, and P. Bøggild, “Rapid prototyping of nanotube-based devices using topology-optimized microgrippers,” Nanotechnology 19, 495503 (2008).
[CrossRef]

Granik, P. L. Y.

J. L. Sturtevant, J. A. Torres, J. Word, and P. L. Y. Granik, “Consideration for the use of defocus models for OPC,” Proc. SPIE 5756, 427–436 (2005).
[CrossRef]

Granik, Y.

Y. Granik, “Fast pixel-based mask optimization for inverse lithography,” J. Micro/Nanolith. MEMS MOEMS 5, 043002 (2006).

Guest, J.

J. Guest, J. Prevost, and T. Belytschko, “Achieving minimum length scale in topology optimization using nodal design variables and projection functions,” Int. J. Numer. Methods Eng. 61, 238–254 (2004).

Jansen, M.

M. Jansen, B. S. Lazarov, M. Schevenels, and O. Sigmund, “On the similarities between micro/nano lithography and topology optimization projection methods,” Struct. Multidiscip. Optim. 48, 717–730 (2013).

Jensen, J.

J. Jensen and O. Sigmund, “Topology optimization for nano-photonics,” Laser Photon. Rev. 5, 308–321 (2011).

Jia, N.

S. Choy, N. Jia, C. Tong, M. Tang, and E. Lam, “A robust computational algorithm for inverse photomask synthesis in optical projection lithography,” SIAM J. Imag. Sci. 5, 625–651 (2012).

N. Jia and E. Y. Lam, “Pixelated source mask optimization for process robustness in optical lithography,” Opt. Express 19, 19384–19398 (2011).
[CrossRef]

Y. Shen, N. Jia, N. Wong, and E. Y. Lam, “Robust levelset-based inverse lithography,” Opt. Express 19, 5511–5521 (2011).
[CrossRef]

N. Jia and E. Y. Lam, “Machine learning for inverse lithography: using stochastic gradient descent for robust photomask synthesis,” J. Opt. 12, 045601 (2010).
[CrossRef]

Jonsmann, J.

J. Jonsmann, O. Sigmund, and S. Bouwstra, “Compliant thermal microactuators,” Sens. Act. 76, 463–469 (1999).

Kailath, T.

Kenny, T. W.

M. A. Philippine, O. Sigmund, G. M. Rebeiz, and T. W. Kenny, “Topology optimization of stressed capacitive RF MEMS switches,” J. Microelectromech. Syst. 22, 206–215 (2013).

Kikuchi, N.

M. Bendsøe and N. Kikuchi, “Generating optimal topologies in structural design using a homogenization method,” Comp. Meth. App. Mech. Eng. 71, 197–224 (1988).

Lam, E.

S. Choy, N. Jia, C. Tong, M. Tang, and E. Lam, “A robust computational algorithm for inverse photomask synthesis in optical projection lithography,” SIAM J. Imag. Sci. 5, 625–651 (2012).

Lam, E. Y.

Lazarov, B.

F. Wang, B. Lazarov, and O. Sigmund, “On projection methods, convergence and robust formulations in topology optimization,” Struct. Multidiscip. Optim. 43, 767–784 (2011).

Lazarov, B. S.

M. Jansen, B. S. Lazarov, M. Schevenels, and O. Sigmund, “On the similarities between micro/nano lithography and topology optimization projection methods,” Struct. Multidiscip. Optim. 48, 717–730 (2013).

B. S. Lazarov, M. Schevenels, and O. Sigmund, “Topology optimization considering material and geometric uncertainties using stochastic collocation methods,” Struct. Multidiscip. Optim. 46, 597–612 (2012).

Levenson, M.

M. Levenson, N. Viswanathan, and R. Simpson, “Improving resolution in photolithography with a phaseshifting mask,” IEEE Trans. Electron Devices 29, 1828–1836 (1982).
[CrossRef]

Li, J.

Li, S.

S. Li, X. Wang, and Y. Bu, “Robust pixel-based source and mask optimization for inverse lithography,” Opt. Laser Technol. 45, 285–293 (2013).
[CrossRef]

Li, Y.

Liu, S.

Ma, X.

Mack, C.

C. Mack, Fundamental Principles of Optical Lithography: The Science of Micro-Fabrication (Wiley, 2007).

Mack, C. A.

P. Yu, D. Z. Pan, and C. A. Mack, “True process variation aware optical proximity correction with variational lithography modeling and model calibration,” J. Micro/Nanolith. MEMS MOEMS 6, 031004 (2007).

Milanfar, P.

A. Poonawala and P. Milanfar, “Mask design for optical microlithography an inverse imaging problem,” IEEE Trans. Image Process. 16, 774–788 (2007).
[CrossRef]

Pan, D. Z.

P. Yu, D. Z. Pan, and C. A. Mack, “True process variation aware optical proximity correction with variational lithography modeling and model calibration,” J. Micro/Nanolith. MEMS MOEMS 6, 031004 (2007).

Pati, Y. C.

Petersen, D.

O. Sardan, V. Eichhorn, D. Petersen, S. Fatikow, O. Sigmund, and P. Bøggild, “Rapid prototyping of nanotube-based devices using topology-optimized microgrippers,” Nanotechnology 19, 495503 (2008).
[CrossRef]

Philippine, M. A.

M. A. Philippine, O. Sigmund, G. M. Rebeiz, and T. W. Kenny, “Topology optimization of stressed capacitive RF MEMS switches,” J. Microelectromech. Syst. 22, 206–215 (2013).

Poonawala, A.

A. Poonawala and P. Milanfar, “Mask design for optical microlithography an inverse imaging problem,” IEEE Trans. Image Process. 16, 774–788 (2007).
[CrossRef]

Prevost, J.

J. Guest, J. Prevost, and T. Belytschko, “Achieving minimum length scale in topology optimization using nodal design variables and projection functions,” Int. J. Numer. Methods Eng. 61, 238–254 (2004).

Rebeiz, G. M.

M. A. Philippine, O. Sigmund, G. M. Rebeiz, and T. W. Kenny, “Topology optimization of stressed capacitive RF MEMS switches,” J. Microelectromech. Syst. 22, 206–215 (2013).

Sardan, O.

O. Sardan, V. Eichhorn, D. Petersen, S. Fatikow, O. Sigmund, and P. Bøggild, “Rapid prototyping of nanotube-based devices using topology-optimized microgrippers,” Nanotechnology 19, 495503 (2008).
[CrossRef]

Schevenels, M.

M. Jansen, B. S. Lazarov, M. Schevenels, and O. Sigmund, “On the similarities between micro/nano lithography and topology optimization projection methods,” Struct. Multidiscip. Optim. 48, 717–730 (2013).

B. S. Lazarov, M. Schevenels, and O. Sigmund, “Topology optimization considering material and geometric uncertainties using stochastic collocation methods,” Struct. Multidiscip. Optim. 46, 597–612 (2012).

Shen, Y.

Sigmund, O.

M. Jansen, B. S. Lazarov, M. Schevenels, and O. Sigmund, “On the similarities between micro/nano lithography and topology optimization projection methods,” Struct. Multidiscip. Optim. 48, 717–730 (2013).

M. A. Philippine, O. Sigmund, G. M. Rebeiz, and T. W. Kenny, “Topology optimization of stressed capacitive RF MEMS switches,” J. Microelectromech. Syst. 22, 206–215 (2013).

B. S. Lazarov, M. Schevenels, and O. Sigmund, “Topology optimization considering material and geometric uncertainties using stochastic collocation methods,” Struct. Multidiscip. Optim. 46, 597–612 (2012).

F. Wang, B. Lazarov, and O. Sigmund, “On projection methods, convergence and robust formulations in topology optimization,” Struct. Multidiscip. Optim. 43, 767–784 (2011).

J. Jensen and O. Sigmund, “Topology optimization for nano-photonics,” Laser Photon. Rev. 5, 308–321 (2011).

O. Sardan, V. Eichhorn, D. Petersen, S. Fatikow, O. Sigmund, and P. Bøggild, “Rapid prototyping of nanotube-based devices using topology-optimized microgrippers,” Nanotechnology 19, 495503 (2008).
[CrossRef]

O. Sigmund, “Morphology-based black and white filters for topology optimization,” Struct. Multidiscip. Optim. 33, 401–424 (2007).

J. Jonsmann, O. Sigmund, and S. Bouwstra, “Compliant thermal microactuators,” Sens. Act. 76, 463–469 (1999).

O. Sigmund, “On the design of compliant mechanisms using topology optimization,” Mech. Struct. Machines 25, 493–524 (1997).

M. Bendsøe and O. Sigmund, Topology Optimization: Theory, Methods and Applications (Springer, 2003).

Simpson, R.

M. Levenson, N. Viswanathan, and R. Simpson, “Improving resolution in photolithography with a phaseshifting mask,” IEEE Trans. Electron Devices 29, 1828–1836 (1982).
[CrossRef]

Song, Z.

Sturtevant, J. L.

J. L. Sturtevant, J. A. Torres, J. Word, and P. L. Y. Granik, “Consideration for the use of defocus models for OPC,” Proc. SPIE 5756, 427–436 (2005).
[CrossRef]

Svanberg, K.

K. Svanberg, “The method of moving asymptotes—a new method for structural optimization,” Int. J. Numer. Methods Eng. 24, 359–397 (1987).

Tang, M.

S. Choy, N. Jia, C. Tong, M. Tang, and E. Lam, “A robust computational algorithm for inverse photomask synthesis in optical projection lithography,” SIAM J. Imag. Sci. 5, 625–651 (2012).

Tong, C.

S. Choy, N. Jia, C. Tong, M. Tang, and E. Lam, “A robust computational algorithm for inverse photomask synthesis in optical projection lithography,” SIAM J. Imag. Sci. 5, 625–651 (2012).

Torres, J. A.

J. L. Sturtevant, J. A. Torres, J. Word, and P. L. Y. Granik, “Consideration for the use of defocus models for OPC,” Proc. SPIE 5756, 427–436 (2005).
[CrossRef]

Viswanathan, N.

M. Levenson, N. Viswanathan, and R. Simpson, “Improving resolution in photolithography with a phaseshifting mask,” IEEE Trans. Electron Devices 29, 1828–1836 (1982).
[CrossRef]

Wang, F.

F. Wang, B. Lazarov, and O. Sigmund, “On projection methods, convergence and robust formulations in topology optimization,” Struct. Multidiscip. Optim. 43, 767–784 (2011).

Wang, X.

S. Li, X. Wang, and Y. Bu, “Robust pixel-based source and mask optimization for inverse lithography,” Opt. Laser Technol. 45, 285–293 (2013).
[CrossRef]

Wong, N.

Word, J.

J. L. Sturtevant, J. A. Torres, J. Word, and P. L. Y. Granik, “Consideration for the use of defocus models for OPC,” Proc. SPIE 5756, 427–436 (2005).
[CrossRef]

Yu, P.

P. Yu, D. Z. Pan, and C. A. Mack, “True process variation aware optical proximity correction with variational lithography modeling and model calibration,” J. Micro/Nanolith. MEMS MOEMS 6, 031004 (2007).

Appl. Opt.

Comp. Meth. App. Mech. Eng.

M. Bendsøe and N. Kikuchi, “Generating optimal topologies in structural design using a homogenization method,” Comp. Meth. App. Mech. Eng. 71, 197–224 (1988).

IEEE Trans. Electron Devices

M. Levenson, N. Viswanathan, and R. Simpson, “Improving resolution in photolithography with a phaseshifting mask,” IEEE Trans. Electron Devices 29, 1828–1836 (1982).
[CrossRef]

IEEE Trans. Image Process.

A. Poonawala and P. Milanfar, “Mask design for optical microlithography an inverse imaging problem,” IEEE Trans. Image Process. 16, 774–788 (2007).
[CrossRef]

Int. J. Numer. Methods Eng.

J. Guest, J. Prevost, and T. Belytschko, “Achieving minimum length scale in topology optimization using nodal design variables and projection functions,” Int. J. Numer. Methods Eng. 61, 238–254 (2004).

K. Svanberg, “The method of moving asymptotes—a new method for structural optimization,” Int. J. Numer. Methods Eng. 24, 359–397 (1987).

J. Micro/Nanolith. MEMS MOEMS

P. Yu, D. Z. Pan, and C. A. Mack, “True process variation aware optical proximity correction with variational lithography modeling and model calibration,” J. Micro/Nanolith. MEMS MOEMS 6, 031004 (2007).

Y. Granik, “Fast pixel-based mask optimization for inverse lithography,” J. Micro/Nanolith. MEMS MOEMS 5, 043002 (2006).

J. Microelectromech. Syst.

M. A. Philippine, O. Sigmund, G. M. Rebeiz, and T. W. Kenny, “Topology optimization of stressed capacitive RF MEMS switches,” J. Microelectromech. Syst. 22, 206–215 (2013).

J. Opt.

N. Jia and E. Y. Lam, “Machine learning for inverse lithography: using stochastic gradient descent for robust photomask synthesis,” J. Opt. 12, 045601 (2010).
[CrossRef]

J. Opt. Soc. Am. A

Laser Photon. Rev.

J. Jensen and O. Sigmund, “Topology optimization for nano-photonics,” Laser Photon. Rev. 5, 308–321 (2011).

Mech. Struct. Machines

O. Sigmund, “On the design of compliant mechanisms using topology optimization,” Mech. Struct. Machines 25, 493–524 (1997).

Nanotechnology

O. Sardan, V. Eichhorn, D. Petersen, S. Fatikow, O. Sigmund, and P. Bøggild, “Rapid prototyping of nanotube-based devices using topology-optimized microgrippers,” Nanotechnology 19, 495503 (2008).
[CrossRef]

Opt. Express

Opt. Laser Technol.

S. Li, X. Wang, and Y. Bu, “Robust pixel-based source and mask optimization for inverse lithography,” Opt. Laser Technol. 45, 285–293 (2013).
[CrossRef]

Proc. SPIE

J. L. Sturtevant, J. A. Torres, J. Word, and P. L. Y. Granik, “Consideration for the use of defocus models for OPC,” Proc. SPIE 5756, 427–436 (2005).
[CrossRef]

Sens. Act.

J. Jonsmann, O. Sigmund, and S. Bouwstra, “Compliant thermal microactuators,” Sens. Act. 76, 463–469 (1999).

SIAM J. Imag. Sci.

S. Choy, N. Jia, C. Tong, M. Tang, and E. Lam, “A robust computational algorithm for inverse photomask synthesis in optical projection lithography,” SIAM J. Imag. Sci. 5, 625–651 (2012).

Struct. Multidiscip. Optim.

M. Jansen, B. S. Lazarov, M. Schevenels, and O. Sigmund, “On the similarities between micro/nano lithography and topology optimization projection methods,” Struct. Multidiscip. Optim. 48, 717–730 (2013).

F. Wang, B. Lazarov, and O. Sigmund, “On projection methods, convergence and robust formulations in topology optimization,” Struct. Multidiscip. Optim. 43, 767–784 (2011).

O. Sigmund, “Morphology-based black and white filters for topology optimization,” Struct. Multidiscip. Optim. 33, 401–424 (2007).

B. S. Lazarov, M. Schevenels, and O. Sigmund, “Topology optimization considering material and geometric uncertainties using stochastic collocation methods,” Struct. Multidiscip. Optim. 46, 597–612 (2012).

Other

M. Bendsøe and O. Sigmund, Topology Optimization: Theory, Methods and Applications (Springer, 2003).

C. Mack, Fundamental Principles of Optical Lithography: The Science of Micro-Fabrication (Wiley, 2007).

Z. Cui, Nanofabrication: Principles, Capabilities and Limits (Springer, 2008).

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Figures (14)

Fig. 1.
Fig. 1.

Failure example using the conventional two-step design scheme. (a) Step 1: a microgripper blueprint. (b) Step 2: a corrected mask from a OPC method for robust patterns (Section 3). (c) Developed pattern containing freely hanging structural members. (d) Another pattern with disconnected structural joints. (both patterns are developed at the considered lithography conditions in OPC.)

Fig. 2.
Fig. 2.

Optical projection lithography system.

Fig. 3.
Fig. 3.

(a) PSF at nominal focus β=0; (b) normalized projection Eq. (6) with different steepness; (c) sigmoid function.

Fig. 4.
Fig. 4.

(a) Input mask of dimension 2000nm(width)×1950nm(height); (b) the aerial image; (c) the output pattern, PE=18.67%.

Fig. 5.
Fig. 5.

Effect of focus variation, patterns developed at η=0.2 with different β: (a) β=40nm, PE=21.09%; (b) β=50nm, PE=29.15%; (c) β=60nm, PE=43.96%.

Fig. 6.
Fig. 6.

Effect of photoresist variation, patterns developed at 50 nm defocus with different η: (a) η=0.15, PE=50.23%; (b) η=0.2, PE=29.15%; (c) η=0.25, PE=16.47%.

Fig. 7.
Fig. 7.

Inverse lithography for different sized blueprints (left column: corrected mask; middle: aerial image; right: developed pattern): (a)–(c) 2000nm×1950nm, PE=0.75%; (d)–(f) 1600nm×1580nm, PE=5.10%; (g)–(i) 1200nm×1184nm, PE=16.31%.

Fig. 8.
Fig. 8.

(a) Optimized binary mask (2000nm×1950nm); (b) PE=1.250% at nominal focus and η=0.2; (c) PE=20.92% at β=50nm and η=0.175; (d) PE=14.83% at β=50nm and η=0.225.

Fig. 9.
Fig. 9.

(a) Robust binary mask (2000nm×1950nm); (b) PE=6.92% at nominal focus and η=0.2; (c) PE=9.35% at β=50nm and η=0.175; (d) PE=10.06% at β=50nm and η=0.225.

Fig. 10.
Fig. 10.

Pattern error of using the standard (sold line) and robust (dash line) binary mask at η=0.175 (red), 0.2 (green), and 0.225 (blue). The solid (purple) markers represent three design points in the robust formulation.

Fig. 11.
Fig. 11.

Micro-gripper design. (a) Design domain (blue) and boundary condition. (b) Mask embedding design domain (blue) nondesign domain (gray) and opaque supportive features (red).

Fig. 12.
Fig. 12.

Micro-gripper design using the unified approach. (a) Optimized binary mask with supportive features. (b) Aerial image. (c) Pattern (inside the design domain) developed at η=0.5 without defocus, output displacement (magnitude) 33.3 nm.

Fig. 13.
Fig. 13.

Comparison of gripper geometry using the robust binary mask under process variations. (a) Output displacement (magnitude) 32.1 nm, developed at β=50nm and η=0.3. (b) Output displacement (magnitude) 33.6 nm, developed at β=50nm and η=0.5. (c) Output displacement (magnitude) 32.3 nm, developed at β=50nm and η=0.7.

Fig. 14.
Fig. 14.

Output displacement of the micro-gripper developed at different lithography conditions: η=0.3 (red); (b) η=0.5 (green); (c) η=0.7 (blue). The solid (purple) markers represent three design points in the robust formulation.

Equations (18)

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IP(x)=Γ(IβA(ρ(x))),
IβA(x)=|(ρHβ)(x)|2,
Hβ(x)=IFT1{hβ(v)},
hβ(v)=h(v)·exp{jπβ(λ|v|2NA22λ)},
h(v)={1,ifvNA/λ0,otherwise,
Γ(IβA(x))=tanh(α·η)+tanh(α·(IβA(x)η))tanh(α·η)+tanh(α·(1.0η)),
PE=IP(x)I0(x)2I0(x)2,
min:F=1neIP(x)I0(x))2,s.t.:0ρ(x)1,
F¯=F+λ1nee=1neρ(xe)(1ρ(xe))+λ2nee=1ne|ρ(xe)|2,
IbinaryP(xe)={1,ifIP(xe)0.50,otherwise.
min:L=max{Fηβ,β={βmin,βmax},η={ηmin,ηmid,ηmax}}+λ1nee=1neρ(xe)(1ρ(xe))+λ2nee=1ne|ρ(xe)|2,s.t.:0ρ(x)1,
min:L¯=max{Fηβ}+λ1nee=1neρ(xe)(1ρ(xe))+λ2nee=1ne|ρ(xe)|2,whereFηβ=lTuηβ,β={βmin,βmax},η={ηmin,ηmid,ηmax},s.t.:Kηβuηβ=f,[e=1neρ˜¯(xe)veVD]ηmidβminV*,0ρ(x)1,
ρ˜(x)=|(ρHβ)(x)|2,
ρ˜¯(x)=tanh(α·η)+tanh(α·(ρ˜(x)η))tanh(α·η)+tanh(α·(1.0η)).
Ke=(Emin+ρ˜¯p(xe)(E0Emin))K0,
Fρ˜¯=λTKρ˜¯u,
Fρ=Fρ˜¯·ρ˜¯ρ˜·ρ˜ρ=[Hβ(Fρ˜¯·ρ˜¯ρ˜·(Hβ*ρ))+Hβ*(Fρ˜¯·ρ˜¯ρ˜·(Hβρ))],
ρ˜¯ρ˜=α·(1tanh2(α·(ρ˜(x)η))tanh(α·η)+tanh(α·(1.0η)),

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