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Applied Optics

Applied Optics

APPLICATIONS-CENTERED RESEARCH IN OPTICS

  • Vol. 22, Iss. 7 — Apr. 1, 1983
  • pp: 1099–1119
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Optical properties of the metals Al, Co, Cu, Au, Fe, Pb, Ni, Pd, Pt, Ag, Ti, and W in the infrared and far infrared

M. A. Ordal, L. L. Long, R. J. Bell, S. E. Bell, R. R. Bell, R. W. Alexander, Jr., and C. A. Ward  »View Author Affiliations


Applied Optics, Vol. 22, Issue 7, pp. 1099-1119 (1983)
http://dx.doi.org/10.1364/AO.22.001099


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Abstract

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 1 ( ω ) = 2 ( ω ) ω p 2 / ( 2 ω τ 2 ) at the damping frequency ω = ωτ. Also −1(ωτ) ≃ −(½) 1(0), where the plasma frequency is ωp.

© 1983 Optical Society of America

I. Introduction

Many measurements of the optical constants of metals have been made, primarily at near IR, visible, and UV wavelengths. Brandli and Sievers [1]

1. G. Brandli and A. J. Sievers, Phys. Rev. B 5, 3550 (1972). [CrossRef]

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 [2]

2. P. Drude, Theory of Optics (Longmans, Green, New York, 1922;

2. Dover, New York, 1968).

2. A more modern reference is F. Wooten, Optical Properties of Solids (Academic, New York, 1972), p. 52.

2. For the Drude model and surface impedance see B. Donovan, Elementary Theory of Metals (Pergamon, New York, 1967), p. 220.

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. [3]

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).

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. [4]

4. G. Haas and L. Hadley, in American Institute of Physics Handbook, D. E. Gray, Ed. (McGraw-Hill, New York, 1972), p. 6–118.

However, this includes data only up to 1967. Except for a few cases, the data presented here are more recent.

Bennett and Bennett [5]

5. H. E. Bennett and J. M. Bennett, in Optical Properties and Electronic Structure of Metals and Alloys, F. Abeles, Ed., (North-Holland, Amsterdam;

5. 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.

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

c1+i2nc2(n+ik)2.
(1)

The Drude model dielectric function is

c=ωp2ω2+iωωτ,
(2)

where ω, ωp, and ωτ have units of cm−1. Separating the real and imaginary parts yields

1=ωp2ω2+ωτ2,
(3)
2=ωp2ωτω3+ωωτ2.
(4)

In these equations, the plasma frequency [6]

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]

is

ωp(cm1)=12πc(4πNe2m*)1/2,
(5)

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

ωτ(cm1)=12πcτ,
(6)

where τ is the electron lifetime in seconds and c is the velocity of light. Note that for low frequencies

1(0)(ωpωτ)2.
(7)

σ0=ωp2/(4πωτ)
(8)

with σ0 having units of cm−1. This can be expressed in terms of the dc resistivity ρ0:

σ0(cm1)=1/[2πcρo(s)]=(9×1011)/[2πcρ0(Ωcm)].
(9)

To analyze the data of Brandli, and Sievers [1]

1. G. Brandli and A. J. Sievers, Phys. Rev. B 5, 3550 (1972). [CrossRef]

it is convenient to write the surface impedance Z(ω) for the Drude model [2]

2. P. Drude, Theory of Optics (Longmans, Green, New York, 1922;

2. Dover, New York, 1968).

2. A more modern reference is F. Wooten, Optical Properties of Solids (Academic, New York, 1972), p. 52.

2. For the Drude model and surface impedance see B. Donovan, Elementary Theory of Metals (Pergamon, New York, 1967), p. 220.

:

Z(ω)R(ω)+iX(ω)=4πc(1+i)(ωωτ2ωp2)1/2(1+iωωτ)1/2.
(10)

We shall need only R(ω):

R(ω)=4πc(ωωτ2ωp2)1/2[ωωτ+(1+ω2ωτ2)1/2]1/2.
(11)

III. Determination of Drude Model Parameters

All data in the form of n and k were changed to 1 and 2. Equations (3) and (4) were solved for ωτ, eliminating ωp:

ωτ=ω2(11).
(12)

This equation was solved to determine ωτ using 1 and 2 at some frequency ω. Then ωp was obtained from

ωp2=(11)(ω2+ωτ2).
(13)

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.

The one exception to this process was the measurements of Brandli and Sievers [1]

1. G. Brandli and A. J. Sievers, Phys. Rev. B 5, 3550 (1972). [CrossRef]

for Au and Pb. They reported values of R(ω)/Z0 where Z0 = (4π)/c. For the far IR, Eq. (11) reduces to

R(ω)Z0=(ωωτ2ωp2)1/2.
(14)

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 ωp2/(2ωτ2). 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 1(0)ωp2/ωτ2; so −½1(0) ≃ −1(ωτ).

IV. Data

Figures 112 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 112 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 [19]

19. J. Babiskin and J. R. Anderson, in American Institute of Physics Handbook, (McGraw-Hill, New York, 1972), p. 9–39.

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. [20]

20. G. R. Parkins, W. E. Lawrence, and R. W. Christy, Phys. Rev. B 23, 6408 (1981). [CrossRef]

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

Fig. 1 Aluminum: −1(ω) and 2(ω) vs frequency. The solid line is the Drude model. The data from Ref. [7] are: Shiles et al., □ for both −1 and 2; Bennett and Bennett * for −1 and 2; Schulz, ⋄ for −1 and 2.
Fig. 2 Copper: −1(ω) and 2(ω) vs frequency. The solid line is the Drude model. The data from Ref. [8] are: Schulz, ⋄ for both −1 and 2; Lenham and Treherne, * for −1 and 2; Robusto and Braunstein, □ for both; Hageman et al., × for both; and Dold and Mecke, Δ for both.
Fig. 3 Gold: −1(ω) and 2(ω) vs frequency. The solid line is the Drude model. The data from Ref. [9] are: Bennett and Bennett, * for both −1 and 2; Schulz, ⋄ for both; Motulevich and Shubin, □ for both; Padalka and Shklyarevskii, ○ for both; Bolotin et al., × for both; Brandli and Sievers, + for both; Weaver et al., Δ for both.
Fig. 4 Lead: −1(ω) and 2(ω) vs frequency. The solid line represents the Drude model. The data from Ref. [10] are: Brandli and Sievers, × for −1 and + for 2; and Golovashkin and Motulevich, Δ for −1 and □ for 2.
Fig. 5 Silver: −1(ω) and 2(ω) vs frequency. The solid line is the Drude model. The data from Ref. [11] are: Bennett and Bennett, * for both −1 and 2; Schulz, ⋄ for both; and Hagemann et al., × for both.
Fig. 6 Colbalt: −1(ω) and 2(ω) vs frequency. The data from Ref. [12] are: Kirillova and Charikov, + for −1 and □ for 2; Johnson and Christy, ⋄ for −1 and ○ for 2; and Weaver et al, × for −1 and Δ for 2.
Fig. 7 Iron: −1(ω) and 2(ω) vs frequency. The data from Ref. [13] are: Weaver et al., × for −1 and Δ for 2; Bolotin et al., ⋄ for −1 and ○ for 2.
Fig. 8 Nickel: −1(ω) and 2(ω) vs frequency. The data from Ref. [14] are: Lynch et al., × for −1 and Δ for 2; Johnson and Christy, ⋄ for −1 and ○ for 2.
Fig. 9 Palladium: −1(ω) and 2(ω) vs frequency. The data from Ref. [15] are: Weaver and Benbow, ⋄ for −1 and ○ for 2; Bolotin et al., + for −1 and □ for 2; Johnson and Christy, × for −1 and Δ for 2.
Fig. 10 Platinum: −1(ω) and 2(ω) vs frequency. The data from Ref. [16] are Weaver et al., Δ for −1 and □ for 2.
Fig. 11 Titanium: −1(ω) and 2(ω) vs frequency. The data from Ref. [17] are: Kirillova and Charikov, □ for both −1 and 2; Lynch et al., Δ for both; Johnson and Christy, ○ for both; Kirillova and Charikov, + for both; Bolotin et al., × for both.
Fig. 12 Tungsten: −1(ω) and 2(ω) vs frequency. The solid line is the Drude model. The data from Ref. [18] are: Nomerovannaya et al., □ for both −1 and 2; Weaver et al., Δ for both.

TABLE 1. Al, ALUMINUME. Shiles, T. Sasaki, M. Inokuti, and D. Y. Smith, Phys. Rev. B 22, 1612 (1980)

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H. E. Bennett and J. M. Bennett, Optical Properties and Electronics Structure of Metals and Alloys, ed. F. Abeles (North–Holland, 1966), p. 175.

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L. G. Schulz, J. Opt. Soc. Am. 44, 357 (1954) and 362 (1954).

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TABLE 2. Cu, COPPERL. G. Schulz, J. Opt. Am. 44, 357 and 362 (1954).

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A. P. Lenham and D. M. Treherne, J. Opt. Soc. Am. 56, 683 (1966).

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P. F. Robusto and Braunstein, Phys. Stat. Sol. (b) 107, 443 (1981).

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H. J. Hagemann, W. Gudat, and C. Kunz, J. Opt. Soc. Am. 65, 742 (1975).

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B. Dold and R. Mecke, Optik 22, 435 (1965).

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TABLE 3. Au, GOLDH. E. Bennett and J. M. Bennett, Optical Properties and Electronic Structure of Metals and Alloys edited by F. Abeles (North–Holland, Amsterdam, 1966), p. 175.

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L. G. Schulz, J. Opt. Soc. Am. 44, 357 and 362 (1954).

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G. P. Motulevich and A. A. Shubin, Soviet Phys. JETP 20, 560 (1965).

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V. G. Padalka and I. N. Shklyarevskii, Opt. Spectr. U.S.S.R. 11, 285 (1961).

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G. A. Bolotin, A. N. Voloshinskii, M. M. Neskov, A. V. Sokolov, and B. A. Charikov, Phys. Met. and Met. 13, 823 (1962).

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G. Brandli and A. J. Sievers, Phy. Rev. B 5, 3550 (1972).

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J. H. Weaver, C. Krafka, D. W. Lynch, and E. E. Koch (with C. G. Olson), Physics Data, Optical Properties of Metals, (Fach–Information Zentrum, Kalsrube, FOR, 1981).

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TABLE 4. Pb, LEADG. Brandli and A. J. Sievers, Phys. Rev. B 5, 3550 (1972).

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A. I. Golovashkin and G. P. Motulevich, Soviet Physics JETP 26, 881 (1968)

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TABLE 5. Ag, SILVERH. E. Bennett and J. M. Bennett in Optical Properties and Electronic Structure of Metals and Alloys, edited by F. Abeles (North–Holland, Amsterdam, 1966), p. 175.

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L. G. Schulz, J. Opt. Soc. Am. 44, p. 357 and 362 (1954).

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H. J. Hageman, W. Gudat, and C. Kunz, J. Opt. Soc. Am. 65, 742 (1975).

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TABLE 6. Co, COBALTM. M. Kirillova and B. A. Charikov, Opt. Spectry. 17, 134 (1964).

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P. B. Johnson and R. W. Christy, Phys. B 9, 5056 (1974).

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J. H. Weaver, E. Colavita, D. W. Lynch and R. Rosei, Phys. Rev. B 19, 3850 (1979).

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TABLE 7. Fe, IronJ. H. Weaver, E. Colavita, D. W. Lynch, and R. Rosei, Phys. Rev. B 19, 3850 (1979).

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G. A. Bolotin, M. M. Kirillova, and V. M. Mayevskiy, Phys. Met. Metall, 27(2) 31 (1969).

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TABLE 8. Ni, NICKELD. W. Lynch, R. Rosei and J. H. Weaver, Solid State Commun. 9, 2195 (1971).

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B. Johnson and R. W. Christy, Phys. Rev. B 9, 5056 (1974).

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TABLE 9. Pd, PalladiumJ. H. Weaver and R. L. Bendow, Phys. Rev. B 12, 3509 (1975).

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G. A. Bolotin, M. M. Kirilova, L. V. Nomerovannaya, and M. M. Noskov, Fiz. Metal. Metalloved 23, 463 (1967).

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P. B. Johnson and R. W. Christy, Phys. Rev. B 9, 5056 (1974).

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TABLE 10. Pt, PlatinumJ. H. Weaver, Phys. Rev. B 11, 1416 (1975).

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J. H. Weaver, D. W. Lynch, and C. G. Olson, Phys. Rev. B 10, 501 (1974).

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TABLE 11. Ti, TITANIUMM. M. Kirillova and B. A. Charikov, Opt. Spectry 17, 134 (1964).

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D. W. Lynch, C. G. Olson, and J. H. Weaver, Phys. Rev. B 11, 3617 (1975).

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P. B. Johnson and R. W. Christy, Phys. Rev. B 9, 5056 (1974).

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M. M. Kirillova and B. A. Charikov, Phys. Met. 15, 138 (1963).

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G. A. Bolotin, A. N. Voloshinskii, M. M. Neskov, A. V. Sokolov, and B. A. Charikov, Phys. Met. and Met. 13, 823 (1962).

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