Infrared optical constants collected from the literature are tabulated. The data for the noble metals and Al, Pb, and W can be reasonably fit using the Drude model. It is shown that at the damping frequency ω = ωτ. Also −∊1(ωτ) ≃ −(½) ∊1(0), where the plasma frequency is ωp.
© 1983 Optical Society of America
Many measurements of the optical constants of metals have been made, primarily at near IR, visible, and UV wavelengths. Brandli and Sievers have measured Au and Pb in the far IR. For the near and far IR we have compiled these data and have tabulated the real and imaginary parts of the dielectric function, ∊1 and ∊2, respectively, the index of refraction n and the extinction index k for each metal. Drude model parameters giving a reasonable fit to the data are given for Au, Ag, Cu, Al, Pb, and W. In general, the Drude model is not expected to be appropriate for transition metals in the near and middle IR, but a good fit can be obtained for W with a Drude model dielectric function.
Weaver et al. have compiled extensive tables or optical properties of metals which have been recently published. Most of their tables do not extend beyond 12-μm wavelength, while our compilation extends to the longest wavelength for which data are available. Another standard compilation is that of Haas and Hadley in the AMERICAN INSTITUTE OF PHYSICS HANDBOOK. However, this includes data only up to 1967. Except for a few cases, the data presented here are more recent.
Bennett and Bennett have shown that the Drude model fits the measured reflectance of gold, silver, and aluminum in the 3–30-μm wavelength range with one adjustable parameter; i.e., the Drude model parameters were obtained from the dc resistivity and fitted with one free electron per atom for gold and silver and 2.6 free electrons per atom for aluminum. Brandli and Sievers have shown that the Drude model is an excellent fit to their far IR measurements on lead and provides a good fit for gold with no adjustable parameters.
II. Definitions and Equations
In keeping with IR spectroscopic notation, all frequencies will be expressed in cm−1. The complex dielectric function ∊c and the complex index of refraction nc are defined as
The Drude model dielectric function is
where ω, ωp, and ωτ have units of cm−1. Separating the real and imaginary parts yields
In these equations, the plasma frequency is
where N is the free electron density, e is the electron charge, m* is the effective mass of the electrons, and ∊∞ is the high frequency dielectric constant. The damping frequency ωτ expressed in cm−1 is
where τ is the electron lifetime in seconds and c is the velocity of light. Note that for low frequencies
The dc conductivity σ0 is related to ωp and ωτ by
with σ0 having units of cm−1. This can be expressed in terms of the dc resistivity ρ0:
We shall need only R(ω):
III. Determination of Drude Model Parameters
This equation was solved to determine ωτ using ∊1 and ∊2 at some frequency ω. Then ωp was obtained from
This was done for several values of ω to obtain several pairs of ωτ and ωp, which produce the curve with the best eyeball fit to the data.
ωτ was obtained from this data using ωn from the near IR fit. This value of ωτ was used for gold and lead rather than the ωτ obtained from the near IR fit.
We note from Eq. (12) the frequency for which −∊1(ω) = ∊2(ω) is very nearly ω = ωτ since −∊1 ≫ 1. With ω = ωτ both components (−∊1 and ∊2) of the dielectric function are . Thus the Drude parameters, ωτ and ωp, can be determined at the crossover from ω = ωτ and the value of the dielectric function. Note that ; so −½∊1(0) ≃ −∊1(ωτ).
Figures 1–12 are plots of −∊1(ω) and ∊2(ω) for the twelve metals. The high frequency termination occurs where the Drude model becomes invalid. The solid lines are calculated from the Drude model with the parameters listed in Table 13. Tables 1–12 present the collected values of ∊1, ∊2, n and k. Table 13 summarizes the Drude model parameters from our fit (for Ag, Au, Cu, Al, Pb, and W) as well as ωτ calculated from ωp and the AIP Handbook values of the dc resistivity. Dielectric functions for all metals considered in this article except Pb have been tabulated by Weaver et al. for the UV, visible, and near IR.
Finally, we disclaim any physical signficance for the Drude model. The intent is only to parametrize the optical constants for these metals even when there is some question as to the physical meaning of the parameters. The transition metals show interband transitions and cannot be fit with a Drude model in the IR (with the exception of W). Even the noble metals in the IR can have small interband contributions to the dielectric constants.
This work was partially supported by the U.S. Army, DAAK-11-82-C-0052. We gratefully acknowledge the valuable advice of Jean M. Bennett, David Begley, David Bryan, Kul Bhasin, and W. F. Parks.
Figures and Tables
1. G. Brandli and A. J. Sievers, Phys. Rev. B 5, 3550 (1972). [CrossRef]
2. P. Drude, Theory of Optics (Longmans, Green, New York, 1922; Dover, New York, 1968). A more modern reference is F. Wooten, Optical Properties of Solids (Academic, New York, 1972), p. 52. For the Drude model and surface impedance see B. Donovan, Elementary Theory of Metals (Pergamon, New York, 1967), p. 220.
3. J. H. Weaver, C. Krafka, D. W. Lynch, and E. E. Koch, “Part 1: The Transition Metals,” “Part 2, Noble Metals, Aluminum, Scandium, Yttrium, the Lanthanides, and the Actinides,” in Physics Data, Optical Properties of Metals (Fachinformationszentrum 7514 Eggenstein-Leopoldshafen 2, Karlsruhe, Federal Republic of Germany, 1981).
4. G. Haas and L. Hadley, in American Institute of Physics Handbook, D. E. Gray, Ed. (McGraw-Hill, New York, 1972), p. 6–118.
5. H. E. Bennett and J. M. Bennett, in Optical Properties and Electronic Structure of Metals and Alloys, F. Abeles, Ed., (North-Holland, Amsterdam; Wiley, New York, 1966), Sec. II.6, p. 175. For Ag, Au, and AL for ω, they estimated 145, 216, and 663 cm−1, respectively.
6. For a single carrier type (electrons) the plasma frequency ωp is as given in Eq. (5) where the dielectric constant is ∊∞ (the contribution from the core electrons at high frequencies). Often m* = m and ∊∞ = 1 are assumed. For discussion see H. Ehrenreich and M. H. Cohen, Phys. Rev. 115, 786 (1959); the last paragraph on p. 790 is most relevant. [CrossRef]
7. Al: E. Shiles, T. Sasaki, M. Inokuti, and D. Y. Smith, Phys. Rev. B 22, 1612 (1980); [CrossRef] H. E. Bennett and J. M. Bennett, Optical Properties and Electronic Structure of Metals and Alloys, F. Abeles, Ed. (North Holland, Amsterdam, 1966), p. 175; L. G. Schulz, J. Opt. Soc. Am. 44, 357, 362 (1954). [CrossRef]
8. Cu: L. G. Schulz, J. Opt. Soc. Am. 44, 357, 362 (1954); [CrossRef] A. P. Lenham and D. M. Treherne, J. Opt. Soc. Am. 56, 683 (1966); [CrossRef] P. F. Robusto and R. Braunstein, Phys. Status Solidi B 107, 443 (1981); [CrossRef] H. J. Hageman, W. Gudat, and C. Kunz, J. Opt. Soc. Am. 65, 742 (1975); [CrossRef] B. Dold and R. Mecke, Optik 22, 435 (1965).
9. Au: H. E. Bennett and J. M. Bennett, Optical Properties and Electronic Structure of Metals and Alloys, F. Abeles, Ed. (North-Holland, Amsterdam, 1966), p. 75; L. G. Schulz, J. Opt. Soc. Am. 44, 357, 362 (1954); [CrossRef] G. P. Motulevich and A. A. Shubin, Sov. Phys. JETP 20, 560 (1965); V. G. Padalka and I. N. Shklyarevskii, Opt. Spectrosc. 11, 285 (1961); G. A. Bolotin, A. N. Voloshinskii, M. M. Kirilbra, M. M. Neskov, A. V. Sokolov, and B. A. Charikov, Fiz. Met. Metalloved. 13, 823 (1962); G. Brändli and A. J. Sievers, Phys. Rev. B 5, 3550 (1972). [CrossRef]
10. Pb: G. Brandli and A. J. Sievers, Phys. Rev. B 5, 3550 (1972); [CrossRef] A. I. Golovashkin and G. P. Motulevich, Sov. Phys. JETP 26, 881 (1968).
11. Ag: H. E. Bennett and J. M. Bennett, in Optical Properties and Electronic Structure of Metals and AlloysF. Abeles, Ed. (North-Holland, Amsterdam, 1966), p. 175; L. G. Schulz, J. Opt. Soc. Am. 44, 357, and 362 (1954); [CrossRef] H. J. Hagemann, W. Endat, and C. Kunz, J. Opt. Soc. Am. 65, 742 (1975). [CrossRef]
12. Co: M. M. Kirillova and B. A. Charikov, Opt. Spectrosc. 17, 134 (1964); P. B. Johnson and R. W. Christy, Phys. Rev. B 9, 5056 (1974); [CrossRef] J. H. Weaver, E. Colavita, D. W. Lynch, and R. Rosei, Phys. Rev. B 19, 3850 (1979). [CrossRef]
13. Fe: J. H. Weaver, E. Colavita, D. W. Lynch, and R. Rosei, Phys. Rev. B 19, 3850 (1979); [CrossRef] G. A. Bolotin, M. M. Krillova, and V. M. Mayevskiy, Phys. Met. Mettalogr. USSR 27, No. 2, 31 (1969).
15. Pd: J. H. Weaver and R. L. Benbow, Phys. Rev. B 12, 3509 (1975); [CrossRef] G. A. Bolotin, M. M. Kirillova, L. V. Nomerovannaya, and M. M. Noskov, Fiz. Met. Mettalloved. 23, 463 (1967); P. B. Johnson and R. W. Christy, Phys. Rev. B 9, 5056 (1974). [CrossRef]
17. Ti: M. M. Kirillova and B. A. Charikov, Opt. Spectrosc. 17, 134 (1964); D. W. Lynch, C. G. Olson, and J. H. Weaver, Phys. Rev. B 11, 3617 (1975); [CrossRef] P. B. Johnson and R. W. Christy, Phys. Rev. B 9, 5056 (1974); [CrossRef] M. M. Kirillova and B. A. Charikov, Phys. Met. 15, 138 (1963); G. A. Bolotin, A. N. Voloshinskii, M. M. Noskov, A. V. Sokolov, and B. A. Charikov, Phys. Met. Metallogr. USSR 13, 823 (1962).
18. W: L. V. Nomerovannaya, M. M. Kirillova, and M. M. Noskov, Opt. Spectrosc. 17, 134 (1964); J. H. Weaver, D. W. Lynch, and C. G. Olson, Phys. Rev. B 12, 1293 (1975). [CrossRef]
19. J. Babiskin and J. R. Anderson, in American Institute of Physics Handbook, (McGraw-Hill, New York, 1972), p. 9–39.
20. G. R. Parkins, W. E. Lawrence, and R. W. Christy, Phys. Rev. B 23, 6408 (1981). [CrossRef]