Abstract

Transmission measurements for the optical constants δ, β of the complex refractive index n = 1 - δ + iβ of molybdenum are performed in the energy range 60–930 eV. Free-standing C/Mo/C foils of five different thicknesses are used, and the results are normalized for the presence of the carbon layers in the samples. These absorption results are combined with previous experimental data in the lower energy range and values from the atomic tables to obtain the imaginary (absorptive) part of the refractive index for molybdenum in the range 1–30,000 eV. The real (dispersive) part of n was calculated from Kramers–Kronig analysis with the above absorption data. An evaluation with the partial sum rules demonstrates that this new compilation provides an improved set of values for n covering a wider energy range compared with the current tabulated values. The new results are applied so as to calculate the normal-incidence reflectivities of Mo/Si and Mo/Be multilayer mirrors.

© 1998 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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  13. D. L. Windt, Lucent Technologies-Bell Laboratories, Murray Hill, N.J. 07974-0636 (personal communication, 1992).
  14. E. M. Gullikson, “X-ray interactions with matter,” http://www-cxro.lbl.gov/optical_constants , currently available on the World Wide Web.

1993

B. L. Henke, E. M. Gullikson, J. C. Davis, “X-ray interactions: photoabsorption, scattering, transmission and reflection at E = 50–30,000 eV, Z = 1–92,” At. Data Nucl. Data Tables 54, 181–343 (1993).
[CrossRef]

D. G. Stearns, R. S. Rosen, S. P. Vernon, “Multilayer mirror technology for soft-x-ray lithography,” Appl. Opt. 32, 6952–6960 (1993).
[CrossRef] [PubMed]

1991

1988

1987

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

1978

D. Y. Smith, E. Shiles, “Finite-energy f-sum rules for valence electrons,” Phys. Rev. B 17, 4689–4694 (1978).
[CrossRef]

1975

1974

J. H. Weaver, D. W. Lynch, C. G. Olson, “Optical properties of V, Ta, and Mo from 0.1 to 35 eV,” Phys. Rev. B 10, 501–516 (1974).
[CrossRef]

1968

Arendt, P.

Batson, P. J.

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, Rev. Sci. Instrum.67(9), available in CD ROM only (1996).

Cash, J. W. C.

Davis, J. C.

B. L. Henke, E. M. Gullikson, J. C. Davis, “X-ray interactions: photoabsorption, scattering, transmission and reflection at E = 50–30,000 eV, Z = 1–92,” At. Data Nucl. Data Tables 54, 181–343 (1993).
[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, Rev. Sci. Instrum.67(9), available in CD ROM only (1996).

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. F.

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, Rev. Sci. Instrum.67(9), available in CD ROM only (1996).

Gudat, W.

Gullikson, E. M.

B. L. Henke, E. M. Gullikson, J. C. Davis, “X-ray interactions: photoabsorption, scattering, transmission and reflection at E = 50–30,000 eV, Z = 1–92,” At. Data Nucl. Data Tables 54, 181–343 (1993).
[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, Rev. Sci. Instrum.67(9), available in CD ROM only (1996).

Hagemann, H.-J.

Henke, B. L.

B. L. Henke, E. M. Gullikson, J. C. Davis, “X-ray interactions: photoabsorption, scattering, transmission and reflection at E = 50–30,000 eV, Z = 1–92,” At. Data Nucl. Data Tables 54, 181–343 (1993).
[CrossRef]

Hunter, W. R.

D. W. Lynch, W. R. Hunter, “Optical constants of metals,” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, San Diego, Calif., 1985), pp. 303–313.

Juenker, D. W.

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, Rev. Sci. Instrum.67(9), available in CD ROM only (1996).

Kunz, C.

LeBlanc, L. J.

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]

Lynch, D. W.

J. H. Weaver, D. W. Lynch, C. G. Olson, “Optical properties of V, Ta, and Mo from 0.1 to 35 eV,” Phys. Rev. B 10, 501–516 (1974).
[CrossRef]

D. W. Lynch, W. R. Hunter, “Optical constants of metals,” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, San Diego, Calif., 1985), pp. 303–313.

Martin, C. R.

Newnam, B.

Olson, C. G.

J. H. Weaver, D. W. Lynch, C. G. Olson, “Optical properties of V, Ta, and Mo from 0.1 to 35 eV,” Phys. Rev. B 10, 501–516 (1974).
[CrossRef]

Rosen, R. S.

Scott, M.

Shiles, E.

D. Y. Smith, E. Shiles, “Finite-energy f-sum rules for valence electrons,” Phys. Rev. B 17, 4689–4694 (1978).
[CrossRef]

Smith, D. Y.

D. Y. Smith, E. Shiles, “Finite-energy f-sum rules for valence electrons,” Phys. Rev. B 17, 4689–4694 (1978).
[CrossRef]

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. R. Samson, D. Y. Smith, eds., Proc. SPIE911, 86–99 (1988).
[CrossRef]

Stearns, D. G.

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, Rev. Sci. Instrum.67(9), available in CD ROM only (1996).

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, Rev. Sci. Instrum.67(9), available in CD ROM only (1996).

Underwood, J. H.

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, Rev. Sci. Instrum.67(9), available in CD ROM only (1996).

Vernon, S. P.

Weaver, J. H.

J. H. Weaver, D. W. Lynch, C. G. Olson, “Optical properties of V, Ta, and Mo from 0.1 to 35 eV,” Phys. Rev. B 10, 501–516 (1974).
[CrossRef]

Windt, D. L.

Appl. Opt.

At. Data Nucl. Data Tables

B. L. Henke, E. M. Gullikson, J. C. Davis, “X-ray interactions: photoabsorption, scattering, transmission and reflection at E = 50–30,000 eV, Z = 1–92,” At. Data Nucl. Data Tables 54, 181–343 (1993).
[CrossRef]

J. Opt. Soc. Am.

Phys. Rev. B

J. H. Weaver, D. W. Lynch, C. G. Olson, “Optical properties of V, Ta, and Mo from 0.1 to 35 eV,” Phys. Rev. B 10, 501–516 (1974).
[CrossRef]

D. Y. Smith, E. Shiles, “Finite-energy f-sum rules for valence electrons,” Phys. Rev. B 17, 4689–4694 (1978).
[CrossRef]

Phys. Scr.

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

Other

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. R. Samson, D. Y. Smith, eds., Proc. SPIE911, 86–99 (1988).
[CrossRef]

D. W. Lynch, W. R. Hunter, “Optical constants of metals,” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, San Diego, Calif., 1985), pp. 303–313.

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, Rev. Sci. Instrum.67(9), available in CD ROM only (1996).

D. L. Windt, Lucent Technologies-Bell Laboratories, Murray Hill, N.J. 07974-0636 (personal communication, 1992).

E. M. Gullikson, “X-ray interactions with matter,” http://www-cxro.lbl.gov/optical_constants , currently available on the World Wide Web.

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Figures (8)

Fig. 1
Fig. 1

Transmission data from the five C/Mo/C films are shown, corresponding to molybdenum layers of 305, 460, 925, 1510, and 1900 Å each, deposited between two 145-Å-thick carbon layers.

Fig. 2
Fig. 2

Fitting procedure for μ at four energies. The data points represent the transmission of the C/Mo/C foils plotted versus Mo thickness. At 400 eV, high absorption prevented reliable measurements from the thick samples, so only data from the foils with 305-, 460-, and 925-Å Mo thickness were used. The dashed lines represent the fit to the experimental results at each energy which yields the values of μ (square centimeters per gram) and T 0.

Fig. 3
Fig. 3

Results for T0 are plotted versus photon energy. The most prominent feature is the carbon K edge at 284.2 eV. The small edge at the Ar 2p 3/2 energy (248.4 eV) indicates the presence of argon in the samples. The oxygen K edge feature at 543.1 eV is attributed to a layer of photoresist left on the samples. The experimental results are fitted to a calculated transmission of 290 Å of C98Ar2 with 400 Å of photoresist, shown with the dashed curve. The optical constants from the 1993 atomic tables were used for the fitting. The disagreement between the experimental results and the fit in the region above the carbon K edge could be due to poor knowledge of the optical constants of carbon in this region.

Fig. 4
Fig. 4

New set of data for the Mo absorption coefficient from 1 eV to 30 keV is shown with the solid curve. The values of μ in the range 60–930 eV are obtained from the present measurements. Data from Refs. 1-3 are used in the low-energy region (1–35 eV). In the rest of the spectrum, the values of the 1993 atomic tables are used with small corrections around the energy region of the present research. In the inset, the molybdenum M 2,3 structure and its deviation from the smoothed tabulated values are shown in detail.

Fig. 5
Fig. 5

Optical constants δ, β of the refractive index of Mo n = 1 - δ + iβ are plotted versus photon energy with the new absorption data in the complete spectrum. (a) δ is calculated from dispersion analysis and (b) β is derived directly from μ. The values in the 1993 atomic tables6 are also shown (dashed curve) for comparison.

Fig. 6
Fig. 6

(a) β-sum rule is shown for the new Mo absorption data (solid curve) and for the data from the 1993 atomic tables (dashed curve). As the photon energy approaches 0, there is an oscillator strength deficiency of 2.3 electrons in the tabulated values. The new data give the correct result of zero electrons. (b) The sum rules for ∊2, β, and Im(-∊-1) are shown, obtained from the values of the new set of data. In the low-energy region, they behave exactly as predicted by the theoretical model discussed in Section 2. The agreement of all three sum rules at the energies above the first Mo core absorption level demonstrates self-consistency among all the calculations performed.

Fig. 7
Fig. 7

New experimental values for μ (solid curve) are shown in comparison with previous reflectance data from Refs. 4, 5, and 13 in the wavelength range of interest for Mo/Si and Mo/Be multilayer mirror applications. The values in the 1993 atomic tables (dashed curve) in this range were based on Refs. 4 and 5 combined with theoretical calculations.7

Fig. 8
Fig. 8

Peak (Bragg) reflectivity at normal incidence for an infinite multilayer stack of Mo/Si and Mo/Be is plotted versus wavelength. The ratio of the Mo thickness to the multilayer period was 0.4, and the period was optimized for each wavelength point. The Bragg peak is shown for a Mo/Si mirror with a 6.87-nm period and for a Mo/Be mirror with a 5.75-nm period. These theoretical calculations predict the reflectance to be 2% higher for Mo/Si and 1.8% lower for Mo/Be compared with calculations from the values in the 1993 atomic tables.6

Equations (10)

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n = 1 - δ + i β 1 - 2 π r 0 n a c 2 ω 2   f ,
f 1 ω - Z * = - 2 π 0 uf 2 u u 2 - ω 2 d u ,
Z * = 2 π m 0 n a e 2 0   u 2 u d u ,
Z * = 4 π m 0 n a e 2 0   u β u d u ,
Z * = 2 π m 0 n a e 2 0   u   Im - - 1 u d u ,
N eff , 2 ω = Z * - 2 π m 0 n a e 2 ω   u 2 u d u ,
N eff , β ω = Z * - 4 π m 0 n a e 2 ω   u β u d u ,
N eff , - 1 ω = Z * - 2 π m 0 n a e 2 ω   u   Im - - 1 u d u ,
T = T 0 exp - μ ρ x ,
β = μ ( λ ρ / 4 π ) ,

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