Abstract

A method is presented for performing the inversion of ellipsometric data using a hybrid approach involving a particle swarm optimization algorithm and a Levenberg–Marquardt algorithm. A sample may be composed of any number of layers of transparent or absorbing materials on a substrate. The method described is applicable to single- or multiple-angle, single-wavelength ellipsometry. The results of the particle swarm optimization algorithm agree well with previously published data calculated using different ellipsometric inversion algorithms, and converges for wide ranges of initial parameter estimates.

© 2010 Optical Society of America

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References

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  1. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).
  2. F. K. Urban III, “Ellipsometry algorithm for absorbing films,” Appl. Opt. 32, 2339-2344 (1993).
    [Crossref]
  3. S. Bosch, F. Monzonís, and E. Masetti, “Ellipsometric methods for absorbing layers: a modified downhill simplex algorithm,” Thin Solid Films 289, 54-58 (1996).
    [Crossref]
  4. Y. Zhaoxian and M. Dang, “Generalized simulated annealing algorithm applied in the ellipsometric inversion problem,” Thin Solid Films 425, 108-112 (2003).
    [Crossref]
  5. G. Cormier and R. Boudreau, “Genetic algorithm for ellipsometric data inversion of absorbing layers,” J. Opt. Soc. Am. A 17, 129-134 (2000).
    [Crossref]
  6. Z. Peng, Z. Li, Y. Hu, L. Tang, and X. Yang, “Thickness and refractivity computation in ellipsometry measurement by genetic algorithm,” Proc. SPIE 4077, 492-495 (2000).
    [Crossref]
  7. Y. A. Zaghloul and A. R. M. Zaghloul, “Single-angle-of-incidence ellipsometry,” Appl. Opt. 47, 4579-4588 (2008).
    [Crossref] [PubMed]
  8. S. Bosch and F. Monzonís, “General inversion method for single-wavelength ellipsometry of samples with an arbitrary number of layers,” J. Opt. Soc. Am. A 12, 1375-1379 (1995).
    [Crossref]
  9. A. Kudla, “Application of the genetic algorithms in spectroscopic ellipsometry,” Thin Solid Films 455-456, 804-808 (2004).
    [Crossref]
  10. J. Kennedy and R. Eberhart, “Particle swarm optimization,” in IEEE International Conference on Neural Networks (IEEE, 1995), pp. 1942-1948.
    [Crossref]
  11. G. E. Jellison Jr., “Use of the biased estimator in the interpretation of spectroscopic ellipsometry data,” Appl. Opt. 30, 3354-3360 (1991).
    [Crossref] [PubMed]
  12. Y. Shi and R. Eberhart, “A modified particle swarm optimizer,” in Proceedings of IEEE International Conference on Evolutionary Computation (IEEE, 1998), pp. 69-73.
  13. J. Kennedy, R. Eberhart, and Y. Shi, Swarm Intelligence (Morgan Kaufmann, 2001).

2008 (1)

2004 (1)

A. Kudla, “Application of the genetic algorithms in spectroscopic ellipsometry,” Thin Solid Films 455-456, 804-808 (2004).
[Crossref]

2003 (1)

Y. Zhaoxian and M. Dang, “Generalized simulated annealing algorithm applied in the ellipsometric inversion problem,” Thin Solid Films 425, 108-112 (2003).
[Crossref]

2000 (2)

G. Cormier and R. Boudreau, “Genetic algorithm for ellipsometric data inversion of absorbing layers,” J. Opt. Soc. Am. A 17, 129-134 (2000).
[Crossref]

Z. Peng, Z. Li, Y. Hu, L. Tang, and X. Yang, “Thickness and refractivity computation in ellipsometry measurement by genetic algorithm,” Proc. SPIE 4077, 492-495 (2000).
[Crossref]

1996 (1)

S. Bosch, F. Monzonís, and E. Masetti, “Ellipsometric methods for absorbing layers: a modified downhill simplex algorithm,” Thin Solid Films 289, 54-58 (1996).
[Crossref]

1995 (1)

1993 (1)

1991 (1)

Azzam, R. M. A.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).

Bashara, N. M.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).

Bosch, S.

S. Bosch, F. Monzonís, and E. Masetti, “Ellipsometric methods for absorbing layers: a modified downhill simplex algorithm,” Thin Solid Films 289, 54-58 (1996).
[Crossref]

S. Bosch and F. Monzonís, “General inversion method for single-wavelength ellipsometry of samples with an arbitrary number of layers,” J. Opt. Soc. Am. A 12, 1375-1379 (1995).
[Crossref]

Boudreau, R.

Cormier, G.

Dang, M.

Y. Zhaoxian and M. Dang, “Generalized simulated annealing algorithm applied in the ellipsometric inversion problem,” Thin Solid Films 425, 108-112 (2003).
[Crossref]

Eberhart, R.

J. Kennedy and R. Eberhart, “Particle swarm optimization,” in IEEE International Conference on Neural Networks (IEEE, 1995), pp. 1942-1948.
[Crossref]

Y. Shi and R. Eberhart, “A modified particle swarm optimizer,” in Proceedings of IEEE International Conference on Evolutionary Computation (IEEE, 1998), pp. 69-73.

J. Kennedy, R. Eberhart, and Y. Shi, Swarm Intelligence (Morgan Kaufmann, 2001).

Hu, Y.

Z. Peng, Z. Li, Y. Hu, L. Tang, and X. Yang, “Thickness and refractivity computation in ellipsometry measurement by genetic algorithm,” Proc. SPIE 4077, 492-495 (2000).
[Crossref]

Jellison, G. E.

Kennedy, J.

J. Kennedy, R. Eberhart, and Y. Shi, Swarm Intelligence (Morgan Kaufmann, 2001).

J. Kennedy and R. Eberhart, “Particle swarm optimization,” in IEEE International Conference on Neural Networks (IEEE, 1995), pp. 1942-1948.
[Crossref]

Kudla, A.

A. Kudla, “Application of the genetic algorithms in spectroscopic ellipsometry,” Thin Solid Films 455-456, 804-808 (2004).
[Crossref]

Li, Z.

Z. Peng, Z. Li, Y. Hu, L. Tang, and X. Yang, “Thickness and refractivity computation in ellipsometry measurement by genetic algorithm,” Proc. SPIE 4077, 492-495 (2000).
[Crossref]

Masetti, E.

S. Bosch, F. Monzonís, and E. Masetti, “Ellipsometric methods for absorbing layers: a modified downhill simplex algorithm,” Thin Solid Films 289, 54-58 (1996).
[Crossref]

Monzonís, F.

S. Bosch, F. Monzonís, and E. Masetti, “Ellipsometric methods for absorbing layers: a modified downhill simplex algorithm,” Thin Solid Films 289, 54-58 (1996).
[Crossref]

S. Bosch and F. Monzonís, “General inversion method for single-wavelength ellipsometry of samples with an arbitrary number of layers,” J. Opt. Soc. Am. A 12, 1375-1379 (1995).
[Crossref]

Peng, Z.

Z. Peng, Z. Li, Y. Hu, L. Tang, and X. Yang, “Thickness and refractivity computation in ellipsometry measurement by genetic algorithm,” Proc. SPIE 4077, 492-495 (2000).
[Crossref]

Shi, Y.

Y. Shi and R. Eberhart, “A modified particle swarm optimizer,” in Proceedings of IEEE International Conference on Evolutionary Computation (IEEE, 1998), pp. 69-73.

J. Kennedy, R. Eberhart, and Y. Shi, Swarm Intelligence (Morgan Kaufmann, 2001).

Tang, L.

Z. Peng, Z. Li, Y. Hu, L. Tang, and X. Yang, “Thickness and refractivity computation in ellipsometry measurement by genetic algorithm,” Proc. SPIE 4077, 492-495 (2000).
[Crossref]

Urban, F. K.

Yang, X.

Z. Peng, Z. Li, Y. Hu, L. Tang, and X. Yang, “Thickness and refractivity computation in ellipsometry measurement by genetic algorithm,” Proc. SPIE 4077, 492-495 (2000).
[Crossref]

Zaghloul, A. R. M.

Zaghloul, Y. A.

Zhaoxian, Y.

Y. Zhaoxian and M. Dang, “Generalized simulated annealing algorithm applied in the ellipsometric inversion problem,” Thin Solid Films 425, 108-112 (2003).
[Crossref]

Appl. Opt. (3)

J. Opt. Soc. Am. A (2)

Proc. SPIE (1)

Z. Peng, Z. Li, Y. Hu, L. Tang, and X. Yang, “Thickness and refractivity computation in ellipsometry measurement by genetic algorithm,” Proc. SPIE 4077, 492-495 (2000).
[Crossref]

Thin Solid Films (3)

S. Bosch, F. Monzonís, and E. Masetti, “Ellipsometric methods for absorbing layers: a modified downhill simplex algorithm,” Thin Solid Films 289, 54-58 (1996).
[Crossref]

Y. Zhaoxian and M. Dang, “Generalized simulated annealing algorithm applied in the ellipsometric inversion problem,” Thin Solid Films 425, 108-112 (2003).
[Crossref]

A. Kudla, “Application of the genetic algorithms in spectroscopic ellipsometry,” Thin Solid Films 455-456, 804-808 (2004).
[Crossref]

Other (4)

J. Kennedy and R. Eberhart, “Particle swarm optimization,” in IEEE International Conference on Neural Networks (IEEE, 1995), pp. 1942-1948.
[Crossref]

Y. Shi and R. Eberhart, “A modified particle swarm optimizer,” in Proceedings of IEEE International Conference on Evolutionary Computation (IEEE, 1998), pp. 69-73.

J. Kennedy, R. Eberhart, and Y. Shi, Swarm Intelligence (Morgan Kaufmann, 2001).

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).

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Figures (8)

Fig. 1
Fig. 1

Evolution of the best particles in three different swarms for the case of a layer of Ti O 2 on a glass substrate.

Fig. 2
Fig. 2

Evolution of the best particle in a swarm compared with the average performance of the swarm for the case of a Ti O 2 layer on glass.

Fig. 3
Fig. 3

Cross-section of a sample having three layers stacked on a silicon substrate.

Fig. 4
Fig. 4

Evolution of the best particles in three different swarms for the case of a multiple-layer sample on a silicon substrate.

Fig. 5
Fig. 5

Evolution of the best particle in a swarm compared with the average performance of the swarm for the multiple-layer sample.

Fig. 6
Fig. 6

Convergence regions for the proposed particle swarm approach (where the algorithm can be expected to converge consistently) and the Levenberg–Marquardt algorithm for the multi-layer problem with noisy measurements.

Fig. 7
Fig. 7

Value of the objective function for the best particles in three different swarms for the case of a sample with a rough surface using a Bruggeman model.

Fig. 8
Fig. 8

Evolution of the best particle in a swarm compared with the average performance of the swarm for a sample having a rough surface.

Tables (6)

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Table 1 Parameters Used in the Particle Swarm Optimization Algorithm

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Table 2 Calculated Values of Δ and Ψ for Ti O 2 on a Glass Substrate

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Table 3 Values of Δ and Ψ for Ti O 2 on a Glass Substrate with Random Noise Added to Measurements (Standard Deviation 0.02° on Δ and 0.01° on Ψ) a

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Table 4 Values of Δ and Ψ for Ti O 2 on a Glass Substrate with Random Noise Added to Measurements (Standard Deviation 0.5° for Δ and 0.25° for Ψ) a

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Table 5 Values of Δ and Ψ for Tungsten on a Silicon Substrate

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Table 6 Values of Δ and Ψ for the Sample Shown in Fig. 3 with Noise Added (Standard Deviation 0.02° for Δ and 0.01° for Ψ )

Equations (4)

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

ρ = r p r s = tan ( Ψ ) exp ( j Δ ) .
χ 2 = 1 N k = 1 N [ ( Δ m k Δ c ε Δ ) 2 + ( Ψ m k Ψ c ε Ψ ) 2 ] .
v k [ t + 1 ] = w v k [ t ] + c 1 r 1 ( x ̂ g x k [ t ] ) + c 2 r 2 ( x ̂ k x k [ t ] ) ,
x k [ t + 1 ] = x k [ t ] + v k [ t + 1 ] .

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