Abstract

An in-plane light scattering setup that is capable of measuring large azimuthal scattering angles is presented. This type of measurement makes it easier to probe large k at a fixed k value (k and k are momentum transfer vectors parallel and perpendicular to the surface, respectively). Therefore the system allows us to explore small lateral scale and large vertical roughness (∼λ, the wavelength of the probe beam) of a rough surface. In-plane intensity measurements from a rough backside Si wafer and a Cu thin-film surface are reported. The structure factor that is related to surface roughness parameters is obtained from the measured in-plane intensity profiles. Both scalar (Beckmann–Kirchhoff) and vector (Rayleigh–Rice) theories have been applied to interpret the experimental data. The roughness parameters obtained from the scattering measurements are compared with those measured by atomic-force microscopy.

© 2000 Optical Society of America

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  1. Z.-H. Gu, A. A. Maradudin, eds., Scattering and Surface Roughness, Proc. SPIE3141 (1997).
  2. J. C. Stover, ed., Optical Scattering in the Optics, Semiconductor, and Computer Disk Industries, Proc. SPIE2541 (1995).
  3. J. C. Stover, ed., Optical Scattering: Applications, Measurement, and Theory II, Proc. SPIE1995 (1993).
  4. H.-N. Yang, K. Fang, T.-M. Lu, G.-C. Wang, “Diffuse-light-scattering study of Pb(110) surface-roughening,” Phys. Rev. B 47, 15,842–15,847 (1993).
    [CrossRef]
  5. Y.-P. Zhao, Y.-J. Wu, H.-N. Yang, G.-C. Wang, T.-M. Lu, “In situ real-time study of chemical etching process of Si(100) using light scattering,” Appl. Phys. Lett. 69, 221–223 (1996).
    [CrossRef]
  6. T. Pinnington, C. Lavoie, T. Tiedje, B. Haveman, E. Nodwell, “Surface morphology dynamics in strained epitaxial InGaAs,” Phys. Rev. Lett. 79, 1698–1701 (1997).
    [CrossRef]
  7. E. Chason, M. B. Sinclair, J. A. Floro, J. A. Hunter, R. Q. Hwang, “Spectroscopic light scattering for real-time measurements of thin film and surface evolution,” Appl. Phys. Lett. 72, 3276–3278 (1998).
    [CrossRef]
  8. J. C. Stover, Optical Scattering: Measurement and Analysis, 2nd ed. (SPIE Press, Bellingham, Wash., 1995).
    [CrossRef]
  9. Y.-P. Zhao, C.-F. Cheng, G.-C. Wang, T.-M. Lu, “Power law behavior in diffraction from fractal surfaces,” Surf. Sci. 409, L703–L708 (1998).
    [CrossRef]
  10. Y.-P. Zhao, I. Wu, C.-F. Cheng, U. Block, G.-C. Wang, T.-M. Lu, “Characterization of random rough surfaces by in-plane light scattering,” J. Appl. Phys. 84, 2571–2582 (1998).
    [CrossRef]
  11. Y.-P. Zhao, C.-F. Cheng, G.-C. Wang, T.-M. Lu, “Characterization of pitting corrosion in aluminum films by light scattering,” Appl. Phys. Lett. 73, 2432–2434 (1998).
    [CrossRef]
  12. T. A. Germer, C. C. Asmail, “A goniometric optical scatter instrument for bidirectional reflectance distribution function measurements with out-of-plane and polarimetry capabilities,” in Scattering and Surface Roughness, Z.-H. Gu, A. A. Maradudin, eds., Proc. SPIE3141, 220–231 (1997); T. A. Germer, C. C. Asmail, B. W. Scheer, “Polarization of out-of-plane scattering from microrough silicon,” Opt. Lett. 22, 1284–1286 (1997); T. A. Germer, “Angular dependence and polarization of out-of-plane optical scattering from particulate contamination, subsurface defects, and surface microroughness,” Appl. Opt. 36, 8798–8805 (1997); T. A. Germer, “Application of bidirectional ellipsometry to the characterization of roughness and defects in dielectric layers,” in Flatness, Roughness, and Discrete Defect Characterization for Computer Disks, Wafers, and Flat Panel Displays II, J. C. Stover, ed., Proc. SPIE3275, 121–131 (1998); T. A. Germer, C. C. Asmail, “Polarization of light scattered by microrough surfaces and subsurface defects,” J. Opt. Soc. Am. A 16, 1326–1332 (1999).
    [CrossRef]
  13. H.-N. Yang, G.-C. Wang, T.-M. Lu, Diffraction from Rough Surfaces and Dynamic Growth Fronts (World Scientific, Singapore, 1993).
  14. A.-L. Barabasi, H. E. Stanley, Fractal Concepts in Surface Growth (Cambridge U. Press, New York, 1995).
    [CrossRef]
  15. F. Family, T. Vicsek, Dynamics of Fractal Surfaces (World Scientific, Singapore, 1990).
  16. S. K. Sinha, E. B. Sirota, S. Garoff, “X-ray and neutron scattering from rough surfaces,” Phys. Rev. B 38, 2297–2311 (1988).
    [CrossRef]
  17. E. L. Church, “Fractal surface finish,” Appl. Opt. 27, 1518–1526 (1988).
    [CrossRef] [PubMed]
  18. P.-Z. Wong, A. J. Bray, “Scattering by rough surfaces,” Phys. Rev. B 37, 7751–7758 (1988).
    [CrossRef]
  19. H.-N. Yang, G.-C. Wang, T.-M. Lu, “Diffraction from surface growth fronts,” Phys. Rev. B 47, 3911–3922 (1993).
    [CrossRef]
  20. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).
  21. J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Adam Hilger, New York, 1991).
  22. P. Beckmann, “Shadowing of random rough surfaces,” IEEE Trans. Antennas Propag. AP-13, 384–388 (1965).
    [CrossRef]
  23. R. J. Wagner, “Shadowing of randomly rough surfaces,” J. Acoust. Soc. Am. 41, 138–147 (1966).
    [CrossRef]
  24. One can calculate w from a single in-plane structure factor profile by using the relation Rδ = ∫ d2k‖Sδ(k‖, k⊥)/∫ d2k‖S(k) = exp(-Ω). But see Ref. 10 and Chap. 3 of Ref. 13 for the detailed methods of w and ξ calculation from a structure factor.
  25. E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, Orlando, Fla., 1985), p. 565.
  26. D. E. Barrick, Radar Cross Section Handbook (Plenum, New York, 1970), Chap. 9.

1998

E. Chason, M. B. Sinclair, J. A. Floro, J. A. Hunter, R. Q. Hwang, “Spectroscopic light scattering for real-time measurements of thin film and surface evolution,” Appl. Phys. Lett. 72, 3276–3278 (1998).
[CrossRef]

Y.-P. Zhao, C.-F. Cheng, G.-C. Wang, T.-M. Lu, “Power law behavior in diffraction from fractal surfaces,” Surf. Sci. 409, L703–L708 (1998).
[CrossRef]

Y.-P. Zhao, I. Wu, C.-F. Cheng, U. Block, G.-C. Wang, T.-M. Lu, “Characterization of random rough surfaces by in-plane light scattering,” J. Appl. Phys. 84, 2571–2582 (1998).
[CrossRef]

Y.-P. Zhao, C.-F. Cheng, G.-C. Wang, T.-M. Lu, “Characterization of pitting corrosion in aluminum films by light scattering,” Appl. Phys. Lett. 73, 2432–2434 (1998).
[CrossRef]

1997

T. Pinnington, C. Lavoie, T. Tiedje, B. Haveman, E. Nodwell, “Surface morphology dynamics in strained epitaxial InGaAs,” Phys. Rev. Lett. 79, 1698–1701 (1997).
[CrossRef]

1996

Y.-P. Zhao, Y.-J. Wu, H.-N. Yang, G.-C. Wang, T.-M. Lu, “In situ real-time study of chemical etching process of Si(100) using light scattering,” Appl. Phys. Lett. 69, 221–223 (1996).
[CrossRef]

1993

H.-N. Yang, K. Fang, T.-M. Lu, G.-C. Wang, “Diffuse-light-scattering study of Pb(110) surface-roughening,” Phys. Rev. B 47, 15,842–15,847 (1993).
[CrossRef]

H.-N. Yang, G.-C. Wang, T.-M. Lu, “Diffraction from surface growth fronts,” Phys. Rev. B 47, 3911–3922 (1993).
[CrossRef]

1988

S. K. Sinha, E. B. Sirota, S. Garoff, “X-ray and neutron scattering from rough surfaces,” Phys. Rev. B 38, 2297–2311 (1988).
[CrossRef]

E. L. Church, “Fractal surface finish,” Appl. Opt. 27, 1518–1526 (1988).
[CrossRef] [PubMed]

P.-Z. Wong, A. J. Bray, “Scattering by rough surfaces,” Phys. Rev. B 37, 7751–7758 (1988).
[CrossRef]

1966

R. J. Wagner, “Shadowing of randomly rough surfaces,” J. Acoust. Soc. Am. 41, 138–147 (1966).
[CrossRef]

1965

P. Beckmann, “Shadowing of random rough surfaces,” IEEE Trans. Antennas Propag. AP-13, 384–388 (1965).
[CrossRef]

Asmail, C. C.

T. A. Germer, C. C. Asmail, “A goniometric optical scatter instrument for bidirectional reflectance distribution function measurements with out-of-plane and polarimetry capabilities,” in Scattering and Surface Roughness, Z.-H. Gu, A. A. Maradudin, eds., Proc. SPIE3141, 220–231 (1997); T. A. Germer, C. C. Asmail, B. W. Scheer, “Polarization of out-of-plane scattering from microrough silicon,” Opt. Lett. 22, 1284–1286 (1997); T. A. Germer, “Angular dependence and polarization of out-of-plane optical scattering from particulate contamination, subsurface defects, and surface microroughness,” Appl. Opt. 36, 8798–8805 (1997); T. A. Germer, “Application of bidirectional ellipsometry to the characterization of roughness and defects in dielectric layers,” in Flatness, Roughness, and Discrete Defect Characterization for Computer Disks, Wafers, and Flat Panel Displays II, J. C. Stover, ed., Proc. SPIE3275, 121–131 (1998); T. A. Germer, C. C. Asmail, “Polarization of light scattered by microrough surfaces and subsurface defects,” J. Opt. Soc. Am. A 16, 1326–1332 (1999).
[CrossRef]

Barabasi, A.-L.

A.-L. Barabasi, H. E. Stanley, Fractal Concepts in Surface Growth (Cambridge U. Press, New York, 1995).
[CrossRef]

Barrick, D. E.

D. E. Barrick, Radar Cross Section Handbook (Plenum, New York, 1970), Chap. 9.

Beckmann, P.

P. Beckmann, “Shadowing of random rough surfaces,” IEEE Trans. Antennas Propag. AP-13, 384–388 (1965).
[CrossRef]

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).

Block, U.

Y.-P. Zhao, I. Wu, C.-F. Cheng, U. Block, G.-C. Wang, T.-M. Lu, “Characterization of random rough surfaces by in-plane light scattering,” J. Appl. Phys. 84, 2571–2582 (1998).
[CrossRef]

Bray, A. J.

P.-Z. Wong, A. J. Bray, “Scattering by rough surfaces,” Phys. Rev. B 37, 7751–7758 (1988).
[CrossRef]

Chason, E.

E. Chason, M. B. Sinclair, J. A. Floro, J. A. Hunter, R. Q. Hwang, “Spectroscopic light scattering for real-time measurements of thin film and surface evolution,” Appl. Phys. Lett. 72, 3276–3278 (1998).
[CrossRef]

Cheng, C.-F.

Y.-P. Zhao, C.-F. Cheng, G.-C. Wang, T.-M. Lu, “Characterization of pitting corrosion in aluminum films by light scattering,” Appl. Phys. Lett. 73, 2432–2434 (1998).
[CrossRef]

Y.-P. Zhao, I. Wu, C.-F. Cheng, U. Block, G.-C. Wang, T.-M. Lu, “Characterization of random rough surfaces by in-plane light scattering,” J. Appl. Phys. 84, 2571–2582 (1998).
[CrossRef]

Y.-P. Zhao, C.-F. Cheng, G.-C. Wang, T.-M. Lu, “Power law behavior in diffraction from fractal surfaces,” Surf. Sci. 409, L703–L708 (1998).
[CrossRef]

Church, E. L.

Family, F.

F. Family, T. Vicsek, Dynamics of Fractal Surfaces (World Scientific, Singapore, 1990).

Fang, K.

H.-N. Yang, K. Fang, T.-M. Lu, G.-C. Wang, “Diffuse-light-scattering study of Pb(110) surface-roughening,” Phys. Rev. B 47, 15,842–15,847 (1993).
[CrossRef]

Floro, J. A.

E. Chason, M. B. Sinclair, J. A. Floro, J. A. Hunter, R. Q. Hwang, “Spectroscopic light scattering for real-time measurements of thin film and surface evolution,” Appl. Phys. Lett. 72, 3276–3278 (1998).
[CrossRef]

Garoff, S.

S. K. Sinha, E. B. Sirota, S. Garoff, “X-ray and neutron scattering from rough surfaces,” Phys. Rev. B 38, 2297–2311 (1988).
[CrossRef]

Germer, T. A.

T. A. Germer, C. C. Asmail, “A goniometric optical scatter instrument for bidirectional reflectance distribution function measurements with out-of-plane and polarimetry capabilities,” in Scattering and Surface Roughness, Z.-H. Gu, A. A. Maradudin, eds., Proc. SPIE3141, 220–231 (1997); T. A. Germer, C. C. Asmail, B. W. Scheer, “Polarization of out-of-plane scattering from microrough silicon,” Opt. Lett. 22, 1284–1286 (1997); T. A. Germer, “Angular dependence and polarization of out-of-plane optical scattering from particulate contamination, subsurface defects, and surface microroughness,” Appl. Opt. 36, 8798–8805 (1997); T. A. Germer, “Application of bidirectional ellipsometry to the characterization of roughness and defects in dielectric layers,” in Flatness, Roughness, and Discrete Defect Characterization for Computer Disks, Wafers, and Flat Panel Displays II, J. C. Stover, ed., Proc. SPIE3275, 121–131 (1998); T. A. Germer, C. C. Asmail, “Polarization of light scattered by microrough surfaces and subsurface defects,” J. Opt. Soc. Am. A 16, 1326–1332 (1999).
[CrossRef]

Haveman, B.

T. Pinnington, C. Lavoie, T. Tiedje, B. Haveman, E. Nodwell, “Surface morphology dynamics in strained epitaxial InGaAs,” Phys. Rev. Lett. 79, 1698–1701 (1997).
[CrossRef]

Hunter, J. A.

E. Chason, M. B. Sinclair, J. A. Floro, J. A. Hunter, R. Q. Hwang, “Spectroscopic light scattering for real-time measurements of thin film and surface evolution,” Appl. Phys. Lett. 72, 3276–3278 (1998).
[CrossRef]

Hwang, R. Q.

E. Chason, M. B. Sinclair, J. A. Floro, J. A. Hunter, R. Q. Hwang, “Spectroscopic light scattering for real-time measurements of thin film and surface evolution,” Appl. Phys. Lett. 72, 3276–3278 (1998).
[CrossRef]

Lavoie, C.

T. Pinnington, C. Lavoie, T. Tiedje, B. Haveman, E. Nodwell, “Surface morphology dynamics in strained epitaxial InGaAs,” Phys. Rev. Lett. 79, 1698–1701 (1997).
[CrossRef]

Lu, T.-M.

Y.-P. Zhao, I. Wu, C.-F. Cheng, U. Block, G.-C. Wang, T.-M. Lu, “Characterization of random rough surfaces by in-plane light scattering,” J. Appl. Phys. 84, 2571–2582 (1998).
[CrossRef]

Y.-P. Zhao, C.-F. Cheng, G.-C. Wang, T.-M. Lu, “Power law behavior in diffraction from fractal surfaces,” Surf. Sci. 409, L703–L708 (1998).
[CrossRef]

Y.-P. Zhao, C.-F. Cheng, G.-C. Wang, T.-M. Lu, “Characterization of pitting corrosion in aluminum films by light scattering,” Appl. Phys. Lett. 73, 2432–2434 (1998).
[CrossRef]

Y.-P. Zhao, Y.-J. Wu, H.-N. Yang, G.-C. Wang, T.-M. Lu, “In situ real-time study of chemical etching process of Si(100) using light scattering,” Appl. Phys. Lett. 69, 221–223 (1996).
[CrossRef]

H.-N. Yang, G.-C. Wang, T.-M. Lu, “Diffraction from surface growth fronts,” Phys. Rev. B 47, 3911–3922 (1993).
[CrossRef]

H.-N. Yang, K. Fang, T.-M. Lu, G.-C. Wang, “Diffuse-light-scattering study of Pb(110) surface-roughening,” Phys. Rev. B 47, 15,842–15,847 (1993).
[CrossRef]

H.-N. Yang, G.-C. Wang, T.-M. Lu, Diffraction from Rough Surfaces and Dynamic Growth Fronts (World Scientific, Singapore, 1993).

Nodwell, E.

T. Pinnington, C. Lavoie, T. Tiedje, B. Haveman, E. Nodwell, “Surface morphology dynamics in strained epitaxial InGaAs,” Phys. Rev. Lett. 79, 1698–1701 (1997).
[CrossRef]

Ogilvy, J. A.

J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Adam Hilger, New York, 1991).

Pinnington, T.

T. Pinnington, C. Lavoie, T. Tiedje, B. Haveman, E. Nodwell, “Surface morphology dynamics in strained epitaxial InGaAs,” Phys. Rev. Lett. 79, 1698–1701 (1997).
[CrossRef]

Sinclair, M. B.

E. Chason, M. B. Sinclair, J. A. Floro, J. A. Hunter, R. Q. Hwang, “Spectroscopic light scattering for real-time measurements of thin film and surface evolution,” Appl. Phys. Lett. 72, 3276–3278 (1998).
[CrossRef]

Sinha, S. K.

S. K. Sinha, E. B. Sirota, S. Garoff, “X-ray and neutron scattering from rough surfaces,” Phys. Rev. B 38, 2297–2311 (1988).
[CrossRef]

Sirota, E. B.

S. K. Sinha, E. B. Sirota, S. Garoff, “X-ray and neutron scattering from rough surfaces,” Phys. Rev. B 38, 2297–2311 (1988).
[CrossRef]

Spizzichino, A.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).

Stanley, H. E.

A.-L. Barabasi, H. E. Stanley, Fractal Concepts in Surface Growth (Cambridge U. Press, New York, 1995).
[CrossRef]

Stover, J. C.

J. C. Stover, Optical Scattering: Measurement and Analysis, 2nd ed. (SPIE Press, Bellingham, Wash., 1995).
[CrossRef]

Tiedje, T.

T. Pinnington, C. Lavoie, T. Tiedje, B. Haveman, E. Nodwell, “Surface morphology dynamics in strained epitaxial InGaAs,” Phys. Rev. Lett. 79, 1698–1701 (1997).
[CrossRef]

Vicsek, T.

F. Family, T. Vicsek, Dynamics of Fractal Surfaces (World Scientific, Singapore, 1990).

Wagner, R. J.

R. J. Wagner, “Shadowing of randomly rough surfaces,” J. Acoust. Soc. Am. 41, 138–147 (1966).
[CrossRef]

Wang, G.-C.

Y.-P. Zhao, C.-F. Cheng, G.-C. Wang, T.-M. Lu, “Power law behavior in diffraction from fractal surfaces,” Surf. Sci. 409, L703–L708 (1998).
[CrossRef]

Y.-P. Zhao, I. Wu, C.-F. Cheng, U. Block, G.-C. Wang, T.-M. Lu, “Characterization of random rough surfaces by in-plane light scattering,” J. Appl. Phys. 84, 2571–2582 (1998).
[CrossRef]

Y.-P. Zhao, C.-F. Cheng, G.-C. Wang, T.-M. Lu, “Characterization of pitting corrosion in aluminum films by light scattering,” Appl. Phys. Lett. 73, 2432–2434 (1998).
[CrossRef]

Y.-P. Zhao, Y.-J. Wu, H.-N. Yang, G.-C. Wang, T.-M. Lu, “In situ real-time study of chemical etching process of Si(100) using light scattering,” Appl. Phys. Lett. 69, 221–223 (1996).
[CrossRef]

H.-N. Yang, G.-C. Wang, T.-M. Lu, “Diffraction from surface growth fronts,” Phys. Rev. B 47, 3911–3922 (1993).
[CrossRef]

H.-N. Yang, K. Fang, T.-M. Lu, G.-C. Wang, “Diffuse-light-scattering study of Pb(110) surface-roughening,” Phys. Rev. B 47, 15,842–15,847 (1993).
[CrossRef]

H.-N. Yang, G.-C. Wang, T.-M. Lu, Diffraction from Rough Surfaces and Dynamic Growth Fronts (World Scientific, Singapore, 1993).

Wong, P.-Z.

P.-Z. Wong, A. J. Bray, “Scattering by rough surfaces,” Phys. Rev. B 37, 7751–7758 (1988).
[CrossRef]

Wu, I.

Y.-P. Zhao, I. Wu, C.-F. Cheng, U. Block, G.-C. Wang, T.-M. Lu, “Characterization of random rough surfaces by in-plane light scattering,” J. Appl. Phys. 84, 2571–2582 (1998).
[CrossRef]

Wu, Y.-J.

Y.-P. Zhao, Y.-J. Wu, H.-N. Yang, G.-C. Wang, T.-M. Lu, “In situ real-time study of chemical etching process of Si(100) using light scattering,” Appl. Phys. Lett. 69, 221–223 (1996).
[CrossRef]

Yang, H.-N.

Y.-P. Zhao, Y.-J. Wu, H.-N. Yang, G.-C. Wang, T.-M. Lu, “In situ real-time study of chemical etching process of Si(100) using light scattering,” Appl. Phys. Lett. 69, 221–223 (1996).
[CrossRef]

H.-N. Yang, G.-C. Wang, T.-M. Lu, “Diffraction from surface growth fronts,” Phys. Rev. B 47, 3911–3922 (1993).
[CrossRef]

H.-N. Yang, K. Fang, T.-M. Lu, G.-C. Wang, “Diffuse-light-scattering study of Pb(110) surface-roughening,” Phys. Rev. B 47, 15,842–15,847 (1993).
[CrossRef]

H.-N. Yang, G.-C. Wang, T.-M. Lu, Diffraction from Rough Surfaces and Dynamic Growth Fronts (World Scientific, Singapore, 1993).

Zhao, Y.-P.

Y.-P. Zhao, C.-F. Cheng, G.-C. Wang, T.-M. Lu, “Characterization of pitting corrosion in aluminum films by light scattering,” Appl. Phys. Lett. 73, 2432–2434 (1998).
[CrossRef]

Y.-P. Zhao, C.-F. Cheng, G.-C. Wang, T.-M. Lu, “Power law behavior in diffraction from fractal surfaces,” Surf. Sci. 409, L703–L708 (1998).
[CrossRef]

Y.-P. Zhao, I. Wu, C.-F. Cheng, U. Block, G.-C. Wang, T.-M. Lu, “Characterization of random rough surfaces by in-plane light scattering,” J. Appl. Phys. 84, 2571–2582 (1998).
[CrossRef]

Y.-P. Zhao, Y.-J. Wu, H.-N. Yang, G.-C. Wang, T.-M. Lu, “In situ real-time study of chemical etching process of Si(100) using light scattering,” Appl. Phys. Lett. 69, 221–223 (1996).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

Y.-P. Zhao, Y.-J. Wu, H.-N. Yang, G.-C. Wang, T.-M. Lu, “In situ real-time study of chemical etching process of Si(100) using light scattering,” Appl. Phys. Lett. 69, 221–223 (1996).
[CrossRef]

E. Chason, M. B. Sinclair, J. A. Floro, J. A. Hunter, R. Q. Hwang, “Spectroscopic light scattering for real-time measurements of thin film and surface evolution,” Appl. Phys. Lett. 72, 3276–3278 (1998).
[CrossRef]

Y.-P. Zhao, C.-F. Cheng, G.-C. Wang, T.-M. Lu, “Characterization of pitting corrosion in aluminum films by light scattering,” Appl. Phys. Lett. 73, 2432–2434 (1998).
[CrossRef]

IEEE Trans. Antennas Propag.

P. Beckmann, “Shadowing of random rough surfaces,” IEEE Trans. Antennas Propag. AP-13, 384–388 (1965).
[CrossRef]

J. Acoust. Soc. Am.

R. J. Wagner, “Shadowing of randomly rough surfaces,” J. Acoust. Soc. Am. 41, 138–147 (1966).
[CrossRef]

J. Appl. Phys.

Y.-P. Zhao, I. Wu, C.-F. Cheng, U. Block, G.-C. Wang, T.-M. Lu, “Characterization of random rough surfaces by in-plane light scattering,” J. Appl. Phys. 84, 2571–2582 (1998).
[CrossRef]

Phys. Rev. B

S. K. Sinha, E. B. Sirota, S. Garoff, “X-ray and neutron scattering from rough surfaces,” Phys. Rev. B 38, 2297–2311 (1988).
[CrossRef]

P.-Z. Wong, A. J. Bray, “Scattering by rough surfaces,” Phys. Rev. B 37, 7751–7758 (1988).
[CrossRef]

H.-N. Yang, G.-C. Wang, T.-M. Lu, “Diffraction from surface growth fronts,” Phys. Rev. B 47, 3911–3922 (1993).
[CrossRef]

H.-N. Yang, K. Fang, T.-M. Lu, G.-C. Wang, “Diffuse-light-scattering study of Pb(110) surface-roughening,” Phys. Rev. B 47, 15,842–15,847 (1993).
[CrossRef]

Phys. Rev. Lett.

T. Pinnington, C. Lavoie, T. Tiedje, B. Haveman, E. Nodwell, “Surface morphology dynamics in strained epitaxial InGaAs,” Phys. Rev. Lett. 79, 1698–1701 (1997).
[CrossRef]

Surf. Sci.

Y.-P. Zhao, C.-F. Cheng, G.-C. Wang, T.-M. Lu, “Power law behavior in diffraction from fractal surfaces,” Surf. Sci. 409, L703–L708 (1998).
[CrossRef]

Other

J. C. Stover, Optical Scattering: Measurement and Analysis, 2nd ed. (SPIE Press, Bellingham, Wash., 1995).
[CrossRef]

Z.-H. Gu, A. A. Maradudin, eds., Scattering and Surface Roughness, Proc. SPIE3141 (1997).

J. C. Stover, ed., Optical Scattering in the Optics, Semiconductor, and Computer Disk Industries, Proc. SPIE2541 (1995).

J. C. Stover, ed., Optical Scattering: Applications, Measurement, and Theory II, Proc. SPIE1995 (1993).

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).

J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Adam Hilger, New York, 1991).

T. A. Germer, C. C. Asmail, “A goniometric optical scatter instrument for bidirectional reflectance distribution function measurements with out-of-plane and polarimetry capabilities,” in Scattering and Surface Roughness, Z.-H. Gu, A. A. Maradudin, eds., Proc. SPIE3141, 220–231 (1997); T. A. Germer, C. C. Asmail, B. W. Scheer, “Polarization of out-of-plane scattering from microrough silicon,” Opt. Lett. 22, 1284–1286 (1997); T. A. Germer, “Angular dependence and polarization of out-of-plane optical scattering from particulate contamination, subsurface defects, and surface microroughness,” Appl. Opt. 36, 8798–8805 (1997); T. A. Germer, “Application of bidirectional ellipsometry to the characterization of roughness and defects in dielectric layers,” in Flatness, Roughness, and Discrete Defect Characterization for Computer Disks, Wafers, and Flat Panel Displays II, J. C. Stover, ed., Proc. SPIE3275, 121–131 (1998); T. A. Germer, C. C. Asmail, “Polarization of light scattered by microrough surfaces and subsurface defects,” J. Opt. Soc. Am. A 16, 1326–1332 (1999).
[CrossRef]

H.-N. Yang, G.-C. Wang, T.-M. Lu, Diffraction from Rough Surfaces and Dynamic Growth Fronts (World Scientific, Singapore, 1993).

A.-L. Barabasi, H. E. Stanley, Fractal Concepts in Surface Growth (Cambridge U. Press, New York, 1995).
[CrossRef]

F. Family, T. Vicsek, Dynamics of Fractal Surfaces (World Scientific, Singapore, 1990).

One can calculate w from a single in-plane structure factor profile by using the relation Rδ = ∫ d2k‖Sδ(k‖, k⊥)/∫ d2k‖S(k) = exp(-Ω). But see Ref. 10 and Chap. 3 of Ref. 13 for the detailed methods of w and ξ calculation from a structure factor.

E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, Orlando, Fla., 1985), p. 565.

D. E. Barrick, Radar Cross Section Handbook (Plenum, New York, 1970), Chap. 9.

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Figures (10)

Fig. 1
Fig. 1

(a) Light-scattering geometry for reflection diffraction and (b) reciprocal space geometry for reflection diffraction.

Fig. 2
Fig. 2

Two types of light-scattering geometry: (a) in-plane scattering and (b) out-of-plane scattering.

Fig. 3
Fig. 3

(a) k /k and k /k as functions of the rotation angle at θ = 45° in the range |Δϕ| ≤ 10° for the in-plane geometry. (b) k /k and k x /k as a function of the rotation angle at θ = 45° in the range |Δθ| ≤ 10° for the out-of-plane geometry.

Fig. 4
Fig. 4

In-plane scattering experimental setup (top view).

Fig. 5
Fig. 5

(a) In-plane scattering experimental geometry (side view) and (b) out-of-plane scattering experimental geometry.

Fig. 6
Fig. 6

AFM images of (a) a backside Si wafer for a 90 µm × 90 µm sampling size and (b) a Cu film surface for a 20 µm × 20 µm sampling size.

Fig. 7
Fig. 7

In-plane intensity profiles of a Si backside for incident angles 83°, 75°, and 48°.

Fig. 8
Fig. 8

Diffraction profile obtained from the out-of-plane diffraction geometry for the same backside Si wafer as for Fig. 7 at a 75° incident angle.

Fig. 9
Fig. 9

(a) Normalized and corrected diffraction profiles from both scalar and vector theories for the in-plane diffraction intensity of the backside Si wafer at an 83° incident angle. Curve a, the normalized intensity. Structure factors labeled S Q (curve b) and S FB (curve c) are obtained from the Rayleigh vector and the Beckmann scalar theories, respectively. (b) In-plane light-scattering normalized intensity profile and normalized corrected structure factors for the scattering from an electroplated Cu film with a 52° incident angle.

Fig. 10
Fig. 10

Normalized intensity and structure factors versus k profiles of the backside Si wafer at an 83° incident angle on the log–log scale.

Tables (2)

Tables Icon

Table 1 Calculated α, w, and ξ Values from AFM Measurements for a Backside Si Wafer and for an Electroplated Cu Film

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Table 2 Calculated α, w, and ξ Values from the In-Plane Light-Scattering Measurements for a Backside Si Wafer and for an Electroplated Cu Film

Equations (44)

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k=kcos θs+cos θi,
kx=ksin θs cos ϕs-sin θi,
ky=k sin θs sin ϕs,
k=2k cos θ,
kx=k sin θcos Δϕ-1,
ky=k sin θ sin Δϕ.
k=kcosθ+Δθ+cos θ,
kx=ksinθ+Δθ-sin θ,
ky=0.
Ik=FPk * Sk,
Sk= Ck, rexpik · rd2r,
Ck, r=exp-k2Hr2,
Hr=Hr=2w2fr/ξ,
fr/ξr/ξ2αr  ξ1r  ξ,
Sk=Sδk, k+Sdiffk, k,
Sdiffk, k=exp-Ω  d2r×expΩ1-fr/ξexpik · r.
Sdiff=2πk2 exp-ΩPSDk,
PSDkx, ky=1A12π  zx, yexpikxx+kyydxdy2,
Sk=2π2 exp-Ωδk+2πk2 exp-ΩPSDk.
Sdiffk, k2πξ-2αΩBk-2-2α,
Sdiffk  k2 exp-Ωλ4FB2cos θi+cos θs2 Ik,
FB=b+akx+cky/k2 cos θi,
a=1-Rsin θi+1+Rsin θs cos ϕs,  b=1+Rcos θs-1-Rcos θi,  c=1+Rsin θs sin ϕs.
Sdiffk  k2 exp-Ωλ4Q cos2 θs cos θi Ik,
Q=|qss|2+|qpp|2+|qsp|2+|qps|2,
Sk  FIk,
F1/FB2from scalar theory with FB=FBϕs, R1/Qfrom vector theory with Q=Qϕs, .
FB2=1+cos2 θ-sin2 θ cos ϕs2 cos2 θ
qss=-cos ϕs,  qsp=sin ϕs/cos θ,  qps=-sin ϕs/cos θ,  qpp=sin2 θ-cos ϕs/cos2 θ.
E1sE2s=SE1iE2i,
S=a11a12a21a22,
E=Es+Ep=Essˆ+Ep expiδpˆ,
EssEps=assaspapsappEsiEpi.
E=ER+EL=ERRˆ+EL expiΨLˆ,
ERsESs=aRRaRLaLRaLLERiESi.
ERES=121-i1+iEsEp,  EsEp=1211+i-iEREL.
aRR=-app+ass-iasp+aps2,  aRL=+app+ass+iasp-aps2,  aLR=+app+ass-iasp-aps2,  aLL=-app+ass+iasp+aps2.
σkl=4πr2Eks · Eks*Eli · Eli*,
σkl=4πr2|akl|2.
γkldiff=σkldiffA/cos θi=4π 2πλ4 cos2 θi cos2 θs|qkl|2 PSDk,
qss=1-cos ϕscos θi+ -sin2 θicos θs+-sin2 θs,  qsp=-1-sin2 θs sin ϕs cos θi+-sin2 θicos θs+-sin2 θs,  qps=1--sin2 θs sinϕscos θi+-sin2 θi cos θs+-sin2 θs,  qpp=-1 sin θi sin θs--sin2 θs-sin2 θi cos ϕs cos θi+-sin2 θi cos θs+-sin2 θs,
qRR=-qpp+qss+iqps+qsp2,  qRL=qpp+qss-iqsp-qps2,  qLR=qpp+qss+iqsp-qps2,  qLL=-qpp+qss-iqps+qsp2.
Isk  |ERRs|2+|ELRs|2  IiQR cos2 θs cos θiλ4 PSDk,
Isk  |ERLs|2+|ELLs|2  IiQL cos2 θs cos θiλ4 PSDk,

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