Anisotropic Slope Distribution and Bidirectional Reflectance of a Rough Silicon Surface

[+] Author and Article Information
Q. Z. Zhu, Z. M. Zhang

The George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332

J. Heat Transfer 126(6), 985-993 (Jan 26, 2005) (9 pages) doi:10.1115/1.1795244 History: Received January 27, 2004; Revised May 04, 2004; Online January 26, 2005
Copyright © 2004 by ASME
Your Session has timed out. Please sign back in to continue.


Zhang,  Z. M., 2000, “Surface Temperature Measurement Using Optical Techniques,” Annu. Rev. Heat Transfer, 11, pp. 351–411.
Zhou,  Y. H., Shen,  Y. J., Zhang,  Z. M., Tsai,  B. K., and DeWitt,  D. P., 2002, “A Monte Carlo Model for Predicting the Effective Emissivity of the Silicon Wafer in Rapid Thermal Processing Furnaces,” Int. J. Heat Mass Transfer, 45, pp. 1945–1949.
Modest, M. F., 1993, Radiative Heat Transfer, McGraw-Hill, New York.
Siegel, R., and Howell, J. R., 2002, Thermal Radiation Heat Transfer, Fourth Edition, Taylor & Francis, New York.
Barnes, P. Y., Early, E. A., and Parr, A. C., 1998, Spectral Reflectance, NIST Special Publication 250-48.
Zaworski,  J. R., Welty,  J. R., Palmer,  B. J., and Drost,  J. R., 1996, “Comparison of Experiment With Monte Carlo Simulations on a Reflective Gap Using a Detailed Surface Properties Model,” ASME J. Heat Transfer, 118, pp. 388–393.
Maradudin,  A. A., Michel,  T., McGurn,  A. R., and Méndez,  E. R., 1990, “Enhanced Backscattering of Light from a Random Grating,” Ann. Phys. (San Diego), 203, pp. 255–307.
Tang,  K., Yang,  Y., and Buckius,  R. O., 1999, “Theory and Experiments on Scattering from Rough Interfaces,” Annu. Rev. Heat Transfer, 10, pp. 100–140.
Saillard,  M., and Sentenac,  A., 2001, “Rigorous Solutions for Electromagnetic Scattering from Rough Surfaces,” Waves Random Media, 11, pp. R103–R137.
Beckmann, P., and Spizzichino, A., 1987, The Scattering of Electromagnetic Waves From Rough Surfaces, Artech House, Norwood, MA.
Tang,  K., Dimenna,  R. A., and Buckius,  R. O., 1997, “Regions of Validity of the Geometric Optics Approximation for Angular Scattering From Very Rough Surfaces,” Int. J. Heat Mass Transfer, 40, pp. 49–59.
Zhou,  Y. H., and Zhang,  Z. M., 2003, “Radiative Properties of Semitransparent Silicon Wafers With Rough Surfaces,” ASME J. Heat Transfer, 125, pp. 462–470.
Tang,  K., and Buckius,  R. O., 2001, “A Statistical Model of Wave Scattering From Random Rough Surfaces,” Int. J. Heat Mass Transfer, 44, pp. 4059–4073.
Torrance,  K., and Sparrow,  E., 1967, “Theory for Off-Specular Reflection From Roughed Surfaces,” J. Opt. Soc. Am., 57, pp. 1105–1114.
Caron,  J., Lafait,  J., and Andraud,  C., 2003, “Catastrophe Theory Interpretation of Multiple Peaks Produced by Light Scattering From Very Rough Dielectric Surfaces,” Physica B, 325, pp. 76–85.
Phong,  B. T., 1975, “Illumination for Computer Generated Images,” Commun. ACM, 18, pp. 311–317.
Blinn,  J. F., 1977, “Models of Light Reflection for Computer Synthesized Pictures,” Comput. Graph., 11, pp. 192–198.
Schleef,  D., Schaefer,  D. M., Andres,  R. P., and Reifenberger,  R., 1997, “Radial-Histogram Transform of Scanning-Probe-Microscope Images,” Phys. Rev. B, 55, pp. 2535–2542.
Hegeman,  J. B. J. W., Kooi,  B. J., Groen,  H. B., and de Hosson,  J. Th. M., 1999, “Analyses of Small Facets Imaged With Scanning-Probe Microscopy,” J. Appl. Phys., 86, pp. 3661–3669.
Bennett, J. M., and Mattsson, L., 1999, Introduction to Surface Roughness and Scattering, Second Edition, Optical Society of America, Washington, DC.
Thomas, T. R., 1999, Rough Surface, Second Edition, Imperial College Press, London.
Wiesendanger, R., 1994, Scanning Probe Microscopy and Spectroscopy: Methods and Applications, Cambridge University Press, Cambridge, England.
Stover, J. C., Ivakhnenko, V. I., and Scheer, C. A., 1998, “Comparison of Surface PSD’s Calculated From Both AFM Profiles and Scatter Data,” Proc. of SPIE, 3275 , pp. 37–46.
Bawolek,  E. J., Mohr,  J. B., Hirleman,  E. D., and Majumda,  A., 1993, “Light Scattering From Polysilicon and Aluminum Surfaces and Comparison With Surface-Roughness Statistics by Atomic Force Microscopy,” Appl. Opt., 32, pp. 3377–3400.
Cao,  L. X., Vorburger,  T. V., Lieberman,  A. G., and Lettieri,  T. R., 1991, “Light-Scattering Measurement of the RMS Slopes of Rough Surfaces,” Appl. Opt., 30, pp. 3221–3226.
Nee,  S.-M. F., Dewees,  R. V., Nee,  T.-W., Johnson,  L. F., and Moran,  M. B., 2000, “Slope Distribution of a Rough Surface Measured by Transmission Scattering and Polarization,” Appl. Opt., 39, pp. 1561–1569.
Guérin,  C.-A., 2002, “Scattering on Rough Surfaces With Alpha-Stable Non-Gaussian Height Distributions,” Waves Random Media, 12, pp. 293–306.
Ward,  G. J., 1992, “Measuring and Modeling Anisotropic Reflection,” Comput. Graph., 26, pp. 265–272.
Smith,  B., 1967, “Geometrical Shadowing of a Random Rough Surface,” IEEE Trans. Antennas Propag., 15, pp. 668–671.
Shen,  Y. J., Zhang,  Z. M., Tsai,  B. K., and DeWitt,  D. P., 2001, “Bidirectional Reflectance Distribution Function of Rough Silicon Wafers,” Int. J. Thermophys., 22, pp. 1311–1326.
Tsang,  L., and Kong,  J. A., 1980, “Energy Conservation for Reflectivity and Transmissivity at a Very Rough Surface,” J. Appl. Phys., 51, pp. 673–380.
Priest, R. G., and Germer, T. A., 2000, “Polarimetric BRDF in the Micro-Facet Model: Theory and Measurements,” Proc. of 2000 Meeting of the Military Sensing Symposia Specialty Group on Passive Sensors, Infrared Information Analysis Center, Ann Arbor, MI, Vol. 1, pp. 169–181.
Zhu, Q. Z., Sin, S., and Zhang, Z. M., 2002, “Surface Characterization of the Rough Side of Silicon Wafers,” Developments in Theoretical and Applied Mechanics, A. J. Kassab et al., eds., Rivercross Publishing, Orlando, pp. 49–55.
Flury, B., 1997, A First Course in Multivariate Statistics, Springer, New York.
Zhang,  Z. M., Fu,  C. J., and Zhu,  Q. Z., 2003, “Optical and Thermal Radiative Properties of Semiconductors Related to Micro/Nanotechnology,” Adv. Heat Transfer, 37, pp. 179–296.
Drolen,  B. L., 1992, “Bidirectional Reflectance and Specularity of Twelve Control Materials,” J. Thermophys. Heat Transfer, 6, pp. 672–679.
Ford,  J. N., Tang,  K., and Buckius,  R. O., 1995, “Fourier Transform Infrared System Measurement of the Bidirectional Reflectivity of Diffuse and Grooved Surfaces,” ASME J. Heat Transfer, 117, pp. 955–962.
Roy,  S., Bang,  S. Y., Modest,  M. F., and Stubican,  V. S., 1993, “Measurement of Spectral Directional Reflectivities of Solids at High Temperatures Between 9 and 11 μm,” Appl. Opt., 32, pp. 3550–3558.
Shen,  Y. J., Zhu,  Q. Z., and Zhang,  Z. M., 2003, “A Scatterometer for Measuring the Bidirectional Reflectance and Transmittance of Semiconductor Wafers With Rough Surfaces,” Rev. Sci. Instrum., 74, pp. 4885–4892.
Edwards, D. F., 1985, “Silicon(Si),” Handbook of Optical Constants of Solids, E. D. Palik, ed., Academic Press, Orlando, FL, pp. 547–569.
Resnik,  D., Vrtacnik,  D., and Amon,  S., 2000, “Morphological Study of {311} Crystal Planes Anisotropically Etched in (100) Silicon: Role of Etchants and Etching Parameters,” J. Micromech. Microeng., 10, pp. 430–439.


Grahic Jump Location
Illustration of the BRDF definition and the specular reflection from a microfacet. The z-axis is normal to the mean surface, n is the normal of the microfacet m, α is the angle between the z-axis and n , and ψ is local incidence angle. For specular reflection, n bisects the directions of incidence and reflection.
Grahic Jump Location
Topographic characteristics of the rough side of the silicon wafer: (a) 3D surface image; (b) height distribution
Grahic Jump Location
Nodal network and the slope distributions: (a) schematic of the nodal network; (b) 1D slope distributions, p1x); (c) 2D slope distribution, p(ζxy)
Grahic Jump Location
The experimental setup of the TAAS
Grahic Jump Location
Comparisons of the predicted and measured BRDFs at λ=635 nm: (a) p polarization; (b) s polarization
Grahic Jump Location
Effect of wavelength on BRDF: (a) p polarization, (b) s polarization
Grahic Jump Location
Comparison of BRDFs at azimuthal angle ϕi=0 deg
Grahic Jump Location
BRDFs at different azimuthal angles for p polarization: (a) predicted; (b) measured
Grahic Jump Location
BRDFs at different azimuthal angles for s polarization: (a) predicted; (b) measured



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In