Abstract

The refractive index n = 1 - δ + iβ of Si in the energy range 50–180 eV is investigated with angle-dependent reflectance measurements. The optical constants δ and β are both determined by fitting to the Fresnel equations. The results of this method are compared with the values in the atomic tables derived from experimental data for β and implementation of the Kramers–Kronig relations for δ. The samples were prepared by UV irradiation and HF:ethanol dipping to H passivate the surface. It is found that the values of δ in the atomic tables are 8–15% too high in the region 50–90 eV. This is attributed to missing oscillator strength in the tabulated absorption coefficient for Si. The measured values of β for crystalline Si exhibit structure below the L 2,3 edge (99.8 eV), as was previously observed in transmission measurements of Si(111). It is also found that the method of least-squares fitting reflectance data to obtain optical constants is most effective for energies well below the edge, where δ > β, while for a range of energies around and above the edge, where δ < β, the optical constants are determined with large uncertainties. This behavior is not unique to the Si L 2,3 edge.

© 1997 Optical Society of America

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References

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  1. B. L. Henke, E. M. Gullikson, J. C. Davis, At. Data Nucl. Data Tables 54 (1993).
  2. D. L. Windt, W. C. Cash, M. Scott, P. Arendt, B. Newnam, R. Fisher, A. B. Swartzlander, P. Z. Takacs, J. M. Pinneo, “Optical constants for thin films of C, diamond, Al, Si, and CVD SiC from 24 Å to 1216 Å,” Appl. Opt. 27, 279–295 (1988).
    [CrossRef] [PubMed]
  3. F. R. Bartsch, H. G. Birken, C. Kunz, R. Wolf, “Reflectance and total photoelectric yield measurements of silicon wafers in the XUV spectral range,” Semicond. Sci. Technol. 5, 974–979 (1990).
    [CrossRef]
  4. E. M. Gullikson, P. Denham, S. Mrowka, J. H. Underwood, “Absolute photoabsorption measurements of Mg, Al, and Si in the soft-x-ray region below the L2,3 edges,” Phys. Rev. B 49, 16283–16288 (1994).
    [CrossRef]
  5. D. Y. Smith, “X-ray optical properties: a review of the constraints and the data base,” in X-Ray and Vacuum Ultraviolet Interaction Data Bases, Calculations, and Measurements, N. K. Del Grande, P. Lee, J. A. Samson, D. Y. Smith, eds., Proc. SPIE911, 86–99 (1988).
    [CrossRef]
  6. R. W. C. Hansen, J. Wolske, D. Wallace, M. Bissen, “Cleaning of optical surfaces with photogenerated reactants,” Nucl. Instrum. Methods A 347, 249–253 (1994).
    [CrossRef]
  7. J. M. C. Thornton, R. H. Williams, “S/XPS Study of hydrogen terminated, ordered silicon (100) and (111) surfaces prepared by chemical etching,” Phys. Scr. 41, 1047–1052 (1990).
    [CrossRef]
  8. T. Takahagi, I. Nagai, A. Ishitani, H. Kuroda, “Formation of hydrogen passivated silicon single-crystal surfaces using ultraviolet cleaning and HF etching,” J. Appl. Phys. 64, 3516–3521 (1988).
    [CrossRef]
  9. A. F. Carley, M. W. Roberts, “X-ray photoelectron spectroscopic study of the interaction of oxygen and nitric oxide with aluminum,” Proc. R. Soc. London Ser. A 363, 403–424 (1978).
    [CrossRef]
  10. E. M. Gullikson, J. H. Underwood, P. J. Batson, V. Nikitin, “Soft x-ray/EUV reflectometer based on a laser produced plasma source,” J. X-Ray Sci. Technol. 3, 283–299 (1992).
    [CrossRef]
  11. J. H. Underwood, E. M. Gullikson, M. Koike, P. J. Batson, P. E. Denham, K. D. Franck, R. E. Tackaberry, W. F. Steele, “Calibration and standards beamline 6.3.2 at the advanced light source,” in Conference on Synchrotron Radiation Instrumentation ’95, in Rev. Sci. Instrum.67 (1996), available only on CD-ROM.
  12. M. Lampton, B. Margon, S. Bowyer, “Parameter estimation in x-ray astronomy,” Astrophys. J. 208, 177–190 (1976).
    [CrossRef]
  13. W. R. Hunter, “Measurement of optical properties of materials in the vacuum ultraviolet spectral region,” Appl. Opt. 21, 2103–2114 (1982).
    [CrossRef] [PubMed]
  14. C. Gähwiller, F. C. Brown, “Photoabsorption near the LII,III edge of silicon and aluminum,” Phys. Rev. B 2, 1918–1925 (1970).
    [CrossRef]
  15. G. Doolen, D. A. Liberman, “Calculations of photoabsorption by atoms using a linear response method,” Phys. Scr. 36, 77–79 (1987).
    [CrossRef]

1994 (2)

E. M. Gullikson, P. Denham, S. Mrowka, J. H. Underwood, “Absolute photoabsorption measurements of Mg, Al, and Si in the soft-x-ray region below the L2,3 edges,” Phys. Rev. B 49, 16283–16288 (1994).
[CrossRef]

R. W. C. Hansen, J. Wolske, D. Wallace, M. Bissen, “Cleaning of optical surfaces with photogenerated reactants,” Nucl. Instrum. Methods A 347, 249–253 (1994).
[CrossRef]

1993 (1)

B. L. Henke, E. M. Gullikson, J. C. Davis, At. Data Nucl. Data Tables 54 (1993).

1992 (1)

E. M. Gullikson, J. H. Underwood, P. J. Batson, V. Nikitin, “Soft x-ray/EUV reflectometer based on a laser produced plasma source,” J. X-Ray Sci. Technol. 3, 283–299 (1992).
[CrossRef]

1990 (2)

J. M. C. Thornton, R. H. Williams, “S/XPS Study of hydrogen terminated, ordered silicon (100) and (111) surfaces prepared by chemical etching,” Phys. Scr. 41, 1047–1052 (1990).
[CrossRef]

F. R. Bartsch, H. G. Birken, C. Kunz, R. Wolf, “Reflectance and total photoelectric yield measurements of silicon wafers in the XUV spectral range,” Semicond. Sci. Technol. 5, 974–979 (1990).
[CrossRef]

1988 (2)

D. L. Windt, W. C. Cash, M. Scott, P. Arendt, B. Newnam, R. Fisher, A. B. Swartzlander, P. Z. Takacs, J. M. Pinneo, “Optical constants for thin films of C, diamond, Al, Si, and CVD SiC from 24 Å to 1216 Å,” Appl. Opt. 27, 279–295 (1988).
[CrossRef] [PubMed]

T. Takahagi, I. Nagai, A. Ishitani, H. Kuroda, “Formation of hydrogen passivated silicon single-crystal surfaces using ultraviolet cleaning and HF etching,” J. Appl. Phys. 64, 3516–3521 (1988).
[CrossRef]

1987 (1)

G. Doolen, D. A. Liberman, “Calculations of photoabsorption by atoms using a linear response method,” Phys. Scr. 36, 77–79 (1987).
[CrossRef]

1982 (1)

1978 (1)

A. F. Carley, M. W. Roberts, “X-ray photoelectron spectroscopic study of the interaction of oxygen and nitric oxide with aluminum,” Proc. R. Soc. London Ser. A 363, 403–424 (1978).
[CrossRef]

1976 (1)

M. Lampton, B. Margon, S. Bowyer, “Parameter estimation in x-ray astronomy,” Astrophys. J. 208, 177–190 (1976).
[CrossRef]

1970 (1)

C. Gähwiller, F. C. Brown, “Photoabsorption near the LII,III edge of silicon and aluminum,” Phys. Rev. B 2, 1918–1925 (1970).
[CrossRef]

Arendt, P.

Bartsch, F. R.

F. R. Bartsch, H. G. Birken, C. Kunz, R. Wolf, “Reflectance and total photoelectric yield measurements of silicon wafers in the XUV spectral range,” Semicond. Sci. Technol. 5, 974–979 (1990).
[CrossRef]

Batson, P. J.

E. M. Gullikson, J. H. Underwood, P. J. Batson, V. Nikitin, “Soft x-ray/EUV reflectometer based on a laser produced plasma source,” J. X-Ray Sci. Technol. 3, 283–299 (1992).
[CrossRef]

J. H. Underwood, E. M. Gullikson, M. Koike, P. J. Batson, P. E. Denham, K. D. Franck, R. E. Tackaberry, W. F. Steele, “Calibration and standards beamline 6.3.2 at the advanced light source,” in Conference on Synchrotron Radiation Instrumentation ’95, in Rev. Sci. Instrum.67 (1996), available only on CD-ROM.

Birken, H. G.

F. R. Bartsch, H. G. Birken, C. Kunz, R. Wolf, “Reflectance and total photoelectric yield measurements of silicon wafers in the XUV spectral range,” Semicond. Sci. Technol. 5, 974–979 (1990).
[CrossRef]

Bissen, M.

R. W. C. Hansen, J. Wolske, D. Wallace, M. Bissen, “Cleaning of optical surfaces with photogenerated reactants,” Nucl. Instrum. Methods A 347, 249–253 (1994).
[CrossRef]

Bowyer, S.

M. Lampton, B. Margon, S. Bowyer, “Parameter estimation in x-ray astronomy,” Astrophys. J. 208, 177–190 (1976).
[CrossRef]

Brown, F. C.

C. Gähwiller, F. C. Brown, “Photoabsorption near the LII,III edge of silicon and aluminum,” Phys. Rev. B 2, 1918–1925 (1970).
[CrossRef]

Carley, A. F.

A. F. Carley, M. W. Roberts, “X-ray photoelectron spectroscopic study of the interaction of oxygen and nitric oxide with aluminum,” Proc. R. Soc. London Ser. A 363, 403–424 (1978).
[CrossRef]

Cash, W. C.

Davis, J. C.

B. L. Henke, E. M. Gullikson, J. C. Davis, At. Data Nucl. Data Tables 54 (1993).

Denham, P.

E. M. Gullikson, P. Denham, S. Mrowka, J. H. Underwood, “Absolute photoabsorption measurements of Mg, Al, and Si in the soft-x-ray region below the L2,3 edges,” Phys. Rev. B 49, 16283–16288 (1994).
[CrossRef]

Denham, P. E.

J. H. Underwood, E. M. Gullikson, M. Koike, P. J. Batson, P. E. Denham, K. D. Franck, R. E. Tackaberry, W. F. Steele, “Calibration and standards beamline 6.3.2 at the advanced light source,” in Conference on Synchrotron Radiation Instrumentation ’95, in Rev. Sci. Instrum.67 (1996), available only on CD-ROM.

Doolen, G.

G. Doolen, D. A. Liberman, “Calculations of photoabsorption by atoms using a linear response method,” Phys. Scr. 36, 77–79 (1987).
[CrossRef]

Fisher, R.

Franck, K. D.

J. H. Underwood, E. M. Gullikson, M. Koike, P. J. Batson, P. E. Denham, K. D. Franck, R. E. Tackaberry, W. F. Steele, “Calibration and standards beamline 6.3.2 at the advanced light source,” in Conference on Synchrotron Radiation Instrumentation ’95, in Rev. Sci. Instrum.67 (1996), available only on CD-ROM.

Gähwiller, C.

C. Gähwiller, F. C. Brown, “Photoabsorption near the LII,III edge of silicon and aluminum,” Phys. Rev. B 2, 1918–1925 (1970).
[CrossRef]

Gullikson, E. M.

E. M. Gullikson, P. Denham, S. Mrowka, J. H. Underwood, “Absolute photoabsorption measurements of Mg, Al, and Si in the soft-x-ray region below the L2,3 edges,” Phys. Rev. B 49, 16283–16288 (1994).
[CrossRef]

B. L. Henke, E. M. Gullikson, J. C. Davis, At. Data Nucl. Data Tables 54 (1993).

E. M. Gullikson, J. H. Underwood, P. J. Batson, V. Nikitin, “Soft x-ray/EUV reflectometer based on a laser produced plasma source,” J. X-Ray Sci. Technol. 3, 283–299 (1992).
[CrossRef]

J. H. Underwood, E. M. Gullikson, M. Koike, P. J. Batson, P. E. Denham, K. D. Franck, R. E. Tackaberry, W. F. Steele, “Calibration and standards beamline 6.3.2 at the advanced light source,” in Conference on Synchrotron Radiation Instrumentation ’95, in Rev. Sci. Instrum.67 (1996), available only on CD-ROM.

Hansen, R. W. C.

R. W. C. Hansen, J. Wolske, D. Wallace, M. Bissen, “Cleaning of optical surfaces with photogenerated reactants,” Nucl. Instrum. Methods A 347, 249–253 (1994).
[CrossRef]

Henke, B. L.

B. L. Henke, E. M. Gullikson, J. C. Davis, At. Data Nucl. Data Tables 54 (1993).

Hunter, W. R.

Ishitani, A.

T. Takahagi, I. Nagai, A. Ishitani, H. Kuroda, “Formation of hydrogen passivated silicon single-crystal surfaces using ultraviolet cleaning and HF etching,” J. Appl. Phys. 64, 3516–3521 (1988).
[CrossRef]

Koike, M.

J. H. Underwood, E. M. Gullikson, M. Koike, P. J. Batson, P. E. Denham, K. D. Franck, R. E. Tackaberry, W. F. Steele, “Calibration and standards beamline 6.3.2 at the advanced light source,” in Conference on Synchrotron Radiation Instrumentation ’95, in Rev. Sci. Instrum.67 (1996), available only on CD-ROM.

Kunz, C.

F. R. Bartsch, H. G. Birken, C. Kunz, R. Wolf, “Reflectance and total photoelectric yield measurements of silicon wafers in the XUV spectral range,” Semicond. Sci. Technol. 5, 974–979 (1990).
[CrossRef]

Kuroda, H.

T. Takahagi, I. Nagai, A. Ishitani, H. Kuroda, “Formation of hydrogen passivated silicon single-crystal surfaces using ultraviolet cleaning and HF etching,” J. Appl. Phys. 64, 3516–3521 (1988).
[CrossRef]

Lampton, M.

M. Lampton, B. Margon, S. Bowyer, “Parameter estimation in x-ray astronomy,” Astrophys. J. 208, 177–190 (1976).
[CrossRef]

Liberman, D. A.

G. Doolen, D. A. Liberman, “Calculations of photoabsorption by atoms using a linear response method,” Phys. Scr. 36, 77–79 (1987).
[CrossRef]

Margon, B.

M. Lampton, B. Margon, S. Bowyer, “Parameter estimation in x-ray astronomy,” Astrophys. J. 208, 177–190 (1976).
[CrossRef]

Mrowka, S.

E. M. Gullikson, P. Denham, S. Mrowka, J. H. Underwood, “Absolute photoabsorption measurements of Mg, Al, and Si in the soft-x-ray region below the L2,3 edges,” Phys. Rev. B 49, 16283–16288 (1994).
[CrossRef]

Nagai, I.

T. Takahagi, I. Nagai, A. Ishitani, H. Kuroda, “Formation of hydrogen passivated silicon single-crystal surfaces using ultraviolet cleaning and HF etching,” J. Appl. Phys. 64, 3516–3521 (1988).
[CrossRef]

Newnam, B.

Nikitin, V.

E. M. Gullikson, J. H. Underwood, P. J. Batson, V. Nikitin, “Soft x-ray/EUV reflectometer based on a laser produced plasma source,” J. X-Ray Sci. Technol. 3, 283–299 (1992).
[CrossRef]

Pinneo, J. M.

Roberts, M. W.

A. F. Carley, M. W. Roberts, “X-ray photoelectron spectroscopic study of the interaction of oxygen and nitric oxide with aluminum,” Proc. R. Soc. London Ser. A 363, 403–424 (1978).
[CrossRef]

Scott, M.

Smith, D. Y.

D. Y. Smith, “X-ray optical properties: a review of the constraints and the data base,” in X-Ray and Vacuum Ultraviolet Interaction Data Bases, Calculations, and Measurements, N. K. Del Grande, P. Lee, J. A. Samson, D. Y. Smith, eds., Proc. SPIE911, 86–99 (1988).
[CrossRef]

Steele, W. F.

J. H. Underwood, E. M. Gullikson, M. Koike, P. J. Batson, P. E. Denham, K. D. Franck, R. E. Tackaberry, W. F. Steele, “Calibration and standards beamline 6.3.2 at the advanced light source,” in Conference on Synchrotron Radiation Instrumentation ’95, in Rev. Sci. Instrum.67 (1996), available only on CD-ROM.

Swartzlander, A. B.

Tackaberry, R. E.

J. H. Underwood, E. M. Gullikson, M. Koike, P. J. Batson, P. E. Denham, K. D. Franck, R. E. Tackaberry, W. F. Steele, “Calibration and standards beamline 6.3.2 at the advanced light source,” in Conference on Synchrotron Radiation Instrumentation ’95, in Rev. Sci. Instrum.67 (1996), available only on CD-ROM.

Takacs, P. Z.

Takahagi, T.

T. Takahagi, I. Nagai, A. Ishitani, H. Kuroda, “Formation of hydrogen passivated silicon single-crystal surfaces using ultraviolet cleaning and HF etching,” J. Appl. Phys. 64, 3516–3521 (1988).
[CrossRef]

Thornton, J. M. C.

J. M. C. Thornton, R. H. Williams, “S/XPS Study of hydrogen terminated, ordered silicon (100) and (111) surfaces prepared by chemical etching,” Phys. Scr. 41, 1047–1052 (1990).
[CrossRef]

Underwood, J. H.

E. M. Gullikson, P. Denham, S. Mrowka, J. H. Underwood, “Absolute photoabsorption measurements of Mg, Al, and Si in the soft-x-ray region below the L2,3 edges,” Phys. Rev. B 49, 16283–16288 (1994).
[CrossRef]

E. M. Gullikson, J. H. Underwood, P. J. Batson, V. Nikitin, “Soft x-ray/EUV reflectometer based on a laser produced plasma source,” J. X-Ray Sci. Technol. 3, 283–299 (1992).
[CrossRef]

J. H. Underwood, E. M. Gullikson, M. Koike, P. J. Batson, P. E. Denham, K. D. Franck, R. E. Tackaberry, W. F. Steele, “Calibration and standards beamline 6.3.2 at the advanced light source,” in Conference on Synchrotron Radiation Instrumentation ’95, in Rev. Sci. Instrum.67 (1996), available only on CD-ROM.

Wallace, D.

R. W. C. Hansen, J. Wolske, D. Wallace, M. Bissen, “Cleaning of optical surfaces with photogenerated reactants,” Nucl. Instrum. Methods A 347, 249–253 (1994).
[CrossRef]

Williams, R. H.

J. M. C. Thornton, R. H. Williams, “S/XPS Study of hydrogen terminated, ordered silicon (100) and (111) surfaces prepared by chemical etching,” Phys. Scr. 41, 1047–1052 (1990).
[CrossRef]

Windt, D. L.

Wolf, R.

F. R. Bartsch, H. G. Birken, C. Kunz, R. Wolf, “Reflectance and total photoelectric yield measurements of silicon wafers in the XUV spectral range,” Semicond. Sci. Technol. 5, 974–979 (1990).
[CrossRef]

Wolske, J.

R. W. C. Hansen, J. Wolske, D. Wallace, M. Bissen, “Cleaning of optical surfaces with photogenerated reactants,” Nucl. Instrum. Methods A 347, 249–253 (1994).
[CrossRef]

Appl. Opt. (2)

Astrophys. J. (1)

M. Lampton, B. Margon, S. Bowyer, “Parameter estimation in x-ray astronomy,” Astrophys. J. 208, 177–190 (1976).
[CrossRef]

At. Data Nucl. Data Tables (1)

B. L. Henke, E. M. Gullikson, J. C. Davis, At. Data Nucl. Data Tables 54 (1993).

J. Appl. Phys. (1)

T. Takahagi, I. Nagai, A. Ishitani, H. Kuroda, “Formation of hydrogen passivated silicon single-crystal surfaces using ultraviolet cleaning and HF etching,” J. Appl. Phys. 64, 3516–3521 (1988).
[CrossRef]

J. X-Ray Sci. Technol. (1)

E. M. Gullikson, J. H. Underwood, P. J. Batson, V. Nikitin, “Soft x-ray/EUV reflectometer based on a laser produced plasma source,” J. X-Ray Sci. Technol. 3, 283–299 (1992).
[CrossRef]

Nucl. Instrum. Methods A (1)

R. W. C. Hansen, J. Wolske, D. Wallace, M. Bissen, “Cleaning of optical surfaces with photogenerated reactants,” Nucl. Instrum. Methods A 347, 249–253 (1994).
[CrossRef]

Phys. Rev. B (2)

E. M. Gullikson, P. Denham, S. Mrowka, J. H. Underwood, “Absolute photoabsorption measurements of Mg, Al, and Si in the soft-x-ray region below the L2,3 edges,” Phys. Rev. B 49, 16283–16288 (1994).
[CrossRef]

C. Gähwiller, F. C. Brown, “Photoabsorption near the LII,III edge of silicon and aluminum,” Phys. Rev. B 2, 1918–1925 (1970).
[CrossRef]

Phys. Scr. (2)

G. Doolen, D. A. Liberman, “Calculations of photoabsorption by atoms using a linear response method,” Phys. Scr. 36, 77–79 (1987).
[CrossRef]

J. M. C. Thornton, R. H. Williams, “S/XPS Study of hydrogen terminated, ordered silicon (100) and (111) surfaces prepared by chemical etching,” Phys. Scr. 41, 1047–1052 (1990).
[CrossRef]

Proc. R. Soc. London Ser. A (1)

A. F. Carley, M. W. Roberts, “X-ray photoelectron spectroscopic study of the interaction of oxygen and nitric oxide with aluminum,” Proc. R. Soc. London Ser. A 363, 403–424 (1978).
[CrossRef]

Semicond. Sci. Technol. (1)

F. R. Bartsch, H. G. Birken, C. Kunz, R. Wolf, “Reflectance and total photoelectric yield measurements of silicon wafers in the XUV spectral range,” Semicond. Sci. Technol. 5, 974–979 (1990).
[CrossRef]

Other (2)

J. H. Underwood, E. M. Gullikson, M. Koike, P. J. Batson, P. E. Denham, K. D. Franck, R. E. Tackaberry, W. F. Steele, “Calibration and standards beamline 6.3.2 at the advanced light source,” in Conference on Synchrotron Radiation Instrumentation ’95, in Rev. Sci. Instrum.67 (1996), available only on CD-ROM.

D. Y. Smith, “X-ray optical properties: a review of the constraints and the data base,” in X-Ray and Vacuum Ultraviolet Interaction Data Bases, Calculations, and Measurements, N. K. Del Grande, P. Lee, J. A. Samson, D. Y. Smith, eds., Proc. SPIE911, 86–99 (1988).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Calculation of the left-hand side of the partial sum rule, Eq. (6), with the data for f 2 from Ref. 1. Although Eq. (6) requires that N eff(ℏω → 0) = 0, the calculation with experimental data gives N eff(ℏω → 0) = 0.2. This missing oscillator strength should cause overestimation of δ at low energies.

Fig. 2
Fig. 2

XPS survey scan results for a Si(111) wafer (a) as received and (b) after H passivation.

Fig. 3
Fig. 3

Electron yield measurements (arbitrary units) on a Si(100) wafer at θ = 3° (grazing angle), performed at the ALS beamline 6.3.2; before cleaning (dotted curve) there is a pronounced peak at the O K edge (543.1 eV) that is due to the presence of the SiO2 on the surface; the peak becomes barely visible after surface cleaning (solid curve).

Fig. 4
Fig. 4

Optical constants δ (○) and β (□) for a Si(100) sample measured at the ALS beamline 6.3.2. The values for δ and β (dashed curve) from the atomic tables1 are also shown.

Fig. 5
Fig. 5

Contour plots of S generated with data on a Si(100) sample measured at the ALS. The energies shown are 60, 70, 80, 95, 105, 130, 150, and 180 eV in a δ–β axis system. Each contour area encloses the optical constant pairs fitted by the least-squares algorithm within ±1σ of their true values. This is equivalent to drawing the contours containing (δ, β) fitted with SS min + 2.3, where S min is the least-squared error, at each energy.2,12 The line β/δ = 0.5 indicates the boundary between regions of reliable and uncertain fitting.

Fig. 6
Fig. 6

Reflectivity curves calculated for s-polarized light and with δ = -0.00725, β = 0.0105, shown as point 105H in Fig. 5 and with δ = -0.015, β = 0.001, shown as point 105L in Fig. 5. The reflectance data at 105 eV for a Si(100) sample measured at the ALS beamline 6.3.2 are also shown.

Fig. 7
Fig. 7

Calculated reflectivity curves for s-polarized radiation and a range of β/δ values. On the horizontal axis the grazing angle (θ) is normalized to the critical angle θ c = 2δ.

Fig. 8
Fig. 8

Fitted values of β determined from reflectance measurements at the laser plasma source for (a) a Si(111) sample, (b) a Si(100) sample, and (c) an amorphous Si sample. Small corrections (≤10%) for the presence of residual contamination were applied for each of the sets of data. The dotted curves represent transmission data for Si(111) presented by Gullikson et al. in Ref. 4.

Fig. 9
Fig. 9

Fitted values of δ determined from reflectance measurements on a Si(111) sample measured at the laser plasma source in the region 55–90 eV are shown to be in agreement with the Kramers–Kronig calculations for δ (solid curve), by use of absorption data that satisfy the sum rule. Consistent results for δ in this energy range were obtained from measurements of several Si(100) and Si(111) samples by use of both the laser-plasma source and the ALS. The Kramers–Kronig calculations from the tables in Ref. 1 (dotted curve) appear to overestimate the values for δ by 8–15%. Theoretical calculations with a linear response approximation model15 (dashed–dotted curve) are also shown.

Tables (1)

Tables Icon

Table 1 Overlayer (SiO2 and C) Thickness Calculations Corresponding to the XPS Results Shown in Fig. 2 for a Si(111) Wafera

Equations (9)

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

n=1-δ+iβ1-r02π λ2naf,
δr02πλ2naf1,
βr02πλ2naf2.
f1ω-Z=-2π0uf2uu2-ω2du,
Z=2π0f2uudu,
Z-2πωf2uudu=Neffω,
Rsθ=sinθ-1-δ+iβ2-cos2θ1/2sinθ+1-δ+iβ2-cos2θ1/22,
Rpθ=1-δ+iβ2 sinθ-1-δ+iβ2-cos2θ1/21-δ+iβ2 sinθ+1-δ+iβ2-cos2θ1/22,
S=i=1NRthθi-Rexpθi2σi2,

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