Millimeter Wave Absorption in Superconducting Aluminum. II. Calculation of the Skin Depth

Manfred A. Biondi; M. P. Garfunkel

doi:10.1103/PhysRev.116.862

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Millimeter Wave Absorption in Superconducting Aluminum. II. Calculation of the Skin Depth

Manfred A. Biondi

M. P. Garfunkel

Westinghouse Research Laboratories, Pittsburgh, Pennsylvania

University of Pittsburgh, Pittsburgh, Pennsylvania

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Phys. Rev. 116, 862 – Published 15 November, 1959

DOI: https://doi.org/10.1103/PhysRev.116.862

Abstract

The skin depth in superconducting aluminum is calculated from the measured frequency dependence of the surface resistance through the Kronig-Kramers integral transforms. At absolute zero, it is found that the skin depth $δ$ is independent of frequency at low frequencies but begins to increase at higher frequencies. The maximum rate of increase of $δ$ occurs when the photon energy equals the gap energy, $h ν = 3.2 k T_{c} = E_{g}$ ; at this point $\frac{δ (h ν = E_{g})}{δ (h ν = 0)} \approx 1.12$ . The maximum value of $δ$ occurs at $h ν \approx 4 k T_{c}$ . The superconducting penetration depth $λ$ [i.e., $δ (h ν = 0)$ ] is found to vary approximately as $λ (t) = λ (0) {(1 - t^{4})}^{- \frac{1}{2}}$ , with $λ (0) = 5.15 \times 10^{- 6}$ cm and $t \equiv \frac{T}{T_{c}}$ . The effects of changes in the skin depth have been eliminated from the determination of the energy gap by calculation of the real part of the complex conductivity, $σ_{r}$ . The energy gap values deduced from the behavior of $σ_{r}$ differ only slightly from the results obtained directly from the surface resistance measurements.

References (21)

M. A. Biondi and M. P. Garfunkel, preceding paper [Phys. Rev. 116, 853 (1959)]
A. B. Pippard, Advances in Electronics and Electron Physics, edited by L. Marton (Academic Press, Inc., New York, 1954), Vol. 6, pp. 4, 5
([18])
P. L. Richards and M. Tinkham, Phys. Rev. Letters 1, 318 (1958)
Omitted endnote
T. E. Faber and A. B. Pippard, Proc. Roy. Soc. (London) A231, 336 (1955)
G. E. H. Reuter and E. H. Sondheimer, Proc. Roy. Soc. (London) A195, 336 (1948)
R. B. Dingle, Physica 19, 311 (1953)
Omitted endnote
N. E. Phillips, Proceedings of the Fifth International Conference on Low-Temperature Physics, Madison, Wisconsin, 1957 (University of Wisconsin Press, Madison, Wisconsin, 1958), p. 414
Omitted endnote
Bardeen, Cooper, and Schrieffer [Phys. Rev. 108, 1175 (1957)] L. C. Hebel and C. P. Slichter [ 107, 901 (1957)]
Khalatnikov and Abrikosov, Advances in Phys. 8, No. 29, 45 (1959)
A. L. Schawlow and G. E. Devlin, Phys. Rev. 113, 120 (1959)
P. Miller, Phys. Rev. 113, 1209 (1959)
F. London, Superfluids (John Wiley & Sons, Inc., New York, 1950), Vol. 1, Chap. B
R. E. Glover, III, and M. Tinkham, Phys. Rev. 108, 243 (1957)
D. C. Mattis and J. Bardeen, Phys. Rev. 111, 412 (1958)
Omitted endnote
R. A. Ferrell and M. Tinkham, Phys. Rev. Letters 2, 331 (1959)
R. A. Ferrell, Bull. Am. Phys. Soc. Ser. II, 4, 225 (1959)

[c1] M. A. Biondi and M. P. Garfunkel, preceding paper [Phys. Rev. 116, 853 (1959)]

[c2] A. B. Pippard, Advances in Electronics and Electron Physics, edited by L. Marton (Academic Press, Inc., New York, 1954), Vol. 6, pp. 4, 5

[c3] ([18])

[c4] P. L. Richards and M. Tinkham, Phys. Rev. Letters 1, 318 (1958)

[c5] Omitted endnote

[c6] T. E. Faber and A. B. Pippard, Proc. Roy. Soc. (London) A231, 336 (1955)

[c7] G. E. H. Reuter and E. H. Sondheimer, Proc. Roy. Soc. (London) A195, 336 (1948)

[c8] R. B. Dingle, Physica 19, 311 (1953)

[c9] Omitted endnote

[c10] N. E. Phillips, Proceedings of the Fifth International Conference on Low-Temperature Physics, Madison, Wisconsin, 1957 (University of Wisconsin Press, Madison, Wisconsin, 1958), p. 414

[c11] Omitted endnote

[c12] Bardeen, Cooper, and Schrieffer [Phys. Rev. 108, 1175 (1957)] L. C. Hebel and C. P. Slichter [ 107, 901 (1957)]

[c13] Khalatnikov and Abrikosov, Advances in Phys. 8, No. 29, 45 (1959)

[c14] A. L. Schawlow and G. E. Devlin, Phys. Rev. 113, 120 (1959)

[c15] P. Miller, Phys. Rev. 113, 1209 (1959)

[c16] F. London, Superfluids (John Wiley & Sons, Inc., New York, 1950), Vol. 1, Chap. B

[c17] R. E. Glover, III, and M. Tinkham, Phys. Rev. 108, 243 (1957)

[c18] D. C. Mattis and J. Bardeen, Phys. Rev. 111, 412 (1958)

[c19] Omitted endnote

[c20] R. A. Ferrell and M. Tinkham, Phys. Rev. Letters 2, 331 (1959)

[c21] R. A. Ferrell, Bull. Am. Phys. Soc. Ser. II, 4, 225 (1959)

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Millimeter Wave Absorption in Superconducting Aluminum. II. Calculation of the Skin Depth

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Millimeter Wave Absorption in Superconducting Aluminum. I. Temperature Dependence of the Energy Gap

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