Abstract

A theory of electromagnetic surface-wave propagation on a plane superconductor–vacuum interface is presented. On the basis of a nonlocal description taking into account the finite size of the pair-bound state, the transverse and longitudinal parts of the conductivity response tensor are analyzed. Special emphasis is given to a study of the quasi-particle scattering and the two-quasi-particle emission and absorption processes in the near-local regime, i.e., the regime where nonlocal effects are incorporated in lowest order. Within the framework of the well-known semiclassical infinite-barrier model, the hydrodynamic model, extended to include nonlocal effects in the divergence-free part of the response, is used to establish a simple analytical expression for the dispersion relation of surface waves on a Cooper-paired superconductor. Attention is paid to the frequency dispersion of the waves near the superconducting gap.

© 1990 Optical Society of America

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References

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  1. V. M. Agranovich and D. L. Mills, ed., Surface Polaritons, Vol. 1 of Modern Problems in Condensed Matter Sciences (North-Holland, Amsterdam, 1982).
  2. A. D. Boardman, ed., Electromagnetic Surface Modes (Wiley, New York, 1982).
  3. R. F. Wallis and G. I. Stegeman, eds., Electromagnetic Surface Excitations, Vol. 3 of Springer Series on Wave Phenomena (Springer-Verlag, Heidelberg, 1986).
    [Crossref]
  4. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings, Vol. III of Springer Tracts in Modern Physics (Springer-Verlag, Heidelberg, 1988).
  5. J. Bardeen, L. N. Cooper, and J. R. Schrieffer, “Theory of superconductivity,” Phys. Rev. 108, 1175–1204 (1957).
    [Crossref]
  6. J. R. Schrieffer, Theory of Superconductivity (Benjamin, New York, 1964).
  7. P. J. Feibelman, “Surface electromagnetic fields,” Prog. Surf. Sci. 12, 287–408 (1982).
    [Crossref]
  8. O. Keller and J. H. Pedersen, “A nonlocal description of the dispersion relation and the energy flow associated with surface electromagnetic waves on metals,” in Scattering and Diffraction, H. A. Ferwada, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1029, 18–26 (1989).
    [Crossref]
  9. G. Rickayzen, Theory of Superconductivity (Interscience, New York, 1964).
  10. F. Forstmann and R. R. Gerhardts, Metal Optics near the Plasma Frequency, Vol. 109 of Springer Tracts in Modern Physics (Springer-Verlag, Heidelberg, 1986).
    [Crossref]
  11. K. Sturm, “Elektromagnetische Oberflächenwellen am Metallhalbraum (III),” Z. Phys. 209, 329–347 (1968).
    [Crossref]
  12. R. Fuchs and K. L. Kliewer, “Surface plasmon in a semi-infinite free-electron gas,” Phys. Rev. B 3, 2270–2278 (1971).
    [Crossref]
  13. T. Timusk and D. B. Tanner, “Infrared properties of high TC superconductors,” in Physical Properties of High Temperature Superconductors I, D. M. Ginsberg, ed. (World Scientific, London, 1989), Chap. 7, pp. 339–407.
  14. A. Liu and O. Keller, “Surface polaritons on a BCS-paired jellium,” Opt. Commun. (to be published).
  15. I. Bozovic, K. Char, S. J. B. Yoo, A. Kapitulnik, M. R. Beasley, T. H. Geballe, Z. Z. Wang, S. Hagen, N. P. Ong, D. E. Aspnes, and M. K. Kelly, “Optical anisotropy of YBa2Cu3O7−x,” Phys. Rev. B 38, 5077–5080 (1988).
    [Crossref]

1988 (1)

I. Bozovic, K. Char, S. J. B. Yoo, A. Kapitulnik, M. R. Beasley, T. H. Geballe, Z. Z. Wang, S. Hagen, N. P. Ong, D. E. Aspnes, and M. K. Kelly, “Optical anisotropy of YBa2Cu3O7−x,” Phys. Rev. B 38, 5077–5080 (1988).
[Crossref]

1982 (1)

P. J. Feibelman, “Surface electromagnetic fields,” Prog. Surf. Sci. 12, 287–408 (1982).
[Crossref]

1971 (1)

R. Fuchs and K. L. Kliewer, “Surface plasmon in a semi-infinite free-electron gas,” Phys. Rev. B 3, 2270–2278 (1971).
[Crossref]

1968 (1)

K. Sturm, “Elektromagnetische Oberflächenwellen am Metallhalbraum (III),” Z. Phys. 209, 329–347 (1968).
[Crossref]

1957 (1)

J. Bardeen, L. N. Cooper, and J. R. Schrieffer, “Theory of superconductivity,” Phys. Rev. 108, 1175–1204 (1957).
[Crossref]

Aspnes, D. E.

I. Bozovic, K. Char, S. J. B. Yoo, A. Kapitulnik, M. R. Beasley, T. H. Geballe, Z. Z. Wang, S. Hagen, N. P. Ong, D. E. Aspnes, and M. K. Kelly, “Optical anisotropy of YBa2Cu3O7−x,” Phys. Rev. B 38, 5077–5080 (1988).
[Crossref]

Bardeen, J.

J. Bardeen, L. N. Cooper, and J. R. Schrieffer, “Theory of superconductivity,” Phys. Rev. 108, 1175–1204 (1957).
[Crossref]

Beasley, M. R.

I. Bozovic, K. Char, S. J. B. Yoo, A. Kapitulnik, M. R. Beasley, T. H. Geballe, Z. Z. Wang, S. Hagen, N. P. Ong, D. E. Aspnes, and M. K. Kelly, “Optical anisotropy of YBa2Cu3O7−x,” Phys. Rev. B 38, 5077–5080 (1988).
[Crossref]

Bozovic, I.

I. Bozovic, K. Char, S. J. B. Yoo, A. Kapitulnik, M. R. Beasley, T. H. Geballe, Z. Z. Wang, S. Hagen, N. P. Ong, D. E. Aspnes, and M. K. Kelly, “Optical anisotropy of YBa2Cu3O7−x,” Phys. Rev. B 38, 5077–5080 (1988).
[Crossref]

Char, K.

I. Bozovic, K. Char, S. J. B. Yoo, A. Kapitulnik, M. R. Beasley, T. H. Geballe, Z. Z. Wang, S. Hagen, N. P. Ong, D. E. Aspnes, and M. K. Kelly, “Optical anisotropy of YBa2Cu3O7−x,” Phys. Rev. B 38, 5077–5080 (1988).
[Crossref]

Cooper, L. N.

J. Bardeen, L. N. Cooper, and J. R. Schrieffer, “Theory of superconductivity,” Phys. Rev. 108, 1175–1204 (1957).
[Crossref]

Feibelman, P. J.

P. J. Feibelman, “Surface electromagnetic fields,” Prog. Surf. Sci. 12, 287–408 (1982).
[Crossref]

Forstmann, F.

F. Forstmann and R. R. Gerhardts, Metal Optics near the Plasma Frequency, Vol. 109 of Springer Tracts in Modern Physics (Springer-Verlag, Heidelberg, 1986).
[Crossref]

Fuchs, R.

R. Fuchs and K. L. Kliewer, “Surface plasmon in a semi-infinite free-electron gas,” Phys. Rev. B 3, 2270–2278 (1971).
[Crossref]

Geballe, T. H.

I. Bozovic, K. Char, S. J. B. Yoo, A. Kapitulnik, M. R. Beasley, T. H. Geballe, Z. Z. Wang, S. Hagen, N. P. Ong, D. E. Aspnes, and M. K. Kelly, “Optical anisotropy of YBa2Cu3O7−x,” Phys. Rev. B 38, 5077–5080 (1988).
[Crossref]

Gerhardts, R. R.

F. Forstmann and R. R. Gerhardts, Metal Optics near the Plasma Frequency, Vol. 109 of Springer Tracts in Modern Physics (Springer-Verlag, Heidelberg, 1986).
[Crossref]

Hagen, S.

I. Bozovic, K. Char, S. J. B. Yoo, A. Kapitulnik, M. R. Beasley, T. H. Geballe, Z. Z. Wang, S. Hagen, N. P. Ong, D. E. Aspnes, and M. K. Kelly, “Optical anisotropy of YBa2Cu3O7−x,” Phys. Rev. B 38, 5077–5080 (1988).
[Crossref]

Kapitulnik, A.

I. Bozovic, K. Char, S. J. B. Yoo, A. Kapitulnik, M. R. Beasley, T. H. Geballe, Z. Z. Wang, S. Hagen, N. P. Ong, D. E. Aspnes, and M. K. Kelly, “Optical anisotropy of YBa2Cu3O7−x,” Phys. Rev. B 38, 5077–5080 (1988).
[Crossref]

Keller, O.

A. Liu and O. Keller, “Surface polaritons on a BCS-paired jellium,” Opt. Commun. (to be published).

O. Keller and J. H. Pedersen, “A nonlocal description of the dispersion relation and the energy flow associated with surface electromagnetic waves on metals,” in Scattering and Diffraction, H. A. Ferwada, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1029, 18–26 (1989).
[Crossref]

Kelly, M. K.

I. Bozovic, K. Char, S. J. B. Yoo, A. Kapitulnik, M. R. Beasley, T. H. Geballe, Z. Z. Wang, S. Hagen, N. P. Ong, D. E. Aspnes, and M. K. Kelly, “Optical anisotropy of YBa2Cu3O7−x,” Phys. Rev. B 38, 5077–5080 (1988).
[Crossref]

Kliewer, K. L.

R. Fuchs and K. L. Kliewer, “Surface plasmon in a semi-infinite free-electron gas,” Phys. Rev. B 3, 2270–2278 (1971).
[Crossref]

Liu, A.

A. Liu and O. Keller, “Surface polaritons on a BCS-paired jellium,” Opt. Commun. (to be published).

Ong, N. P.

I. Bozovic, K. Char, S. J. B. Yoo, A. Kapitulnik, M. R. Beasley, T. H. Geballe, Z. Z. Wang, S. Hagen, N. P. Ong, D. E. Aspnes, and M. K. Kelly, “Optical anisotropy of YBa2Cu3O7−x,” Phys. Rev. B 38, 5077–5080 (1988).
[Crossref]

Pedersen, J. H.

O. Keller and J. H. Pedersen, “A nonlocal description of the dispersion relation and the energy flow associated with surface electromagnetic waves on metals,” in Scattering and Diffraction, H. A. Ferwada, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1029, 18–26 (1989).
[Crossref]

Raether, H.

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings, Vol. III of Springer Tracts in Modern Physics (Springer-Verlag, Heidelberg, 1988).

Rickayzen, G.

G. Rickayzen, Theory of Superconductivity (Interscience, New York, 1964).

Schrieffer, J. R.

J. Bardeen, L. N. Cooper, and J. R. Schrieffer, “Theory of superconductivity,” Phys. Rev. 108, 1175–1204 (1957).
[Crossref]

J. R. Schrieffer, Theory of Superconductivity (Benjamin, New York, 1964).

Sturm, K.

K. Sturm, “Elektromagnetische Oberflächenwellen am Metallhalbraum (III),” Z. Phys. 209, 329–347 (1968).
[Crossref]

Tanner, D. B.

T. Timusk and D. B. Tanner, “Infrared properties of high TC superconductors,” in Physical Properties of High Temperature Superconductors I, D. M. Ginsberg, ed. (World Scientific, London, 1989), Chap. 7, pp. 339–407.

Timusk, T.

T. Timusk and D. B. Tanner, “Infrared properties of high TC superconductors,” in Physical Properties of High Temperature Superconductors I, D. M. Ginsberg, ed. (World Scientific, London, 1989), Chap. 7, pp. 339–407.

Wang, Z. Z.

I. Bozovic, K. Char, S. J. B. Yoo, A. Kapitulnik, M. R. Beasley, T. H. Geballe, Z. Z. Wang, S. Hagen, N. P. Ong, D. E. Aspnes, and M. K. Kelly, “Optical anisotropy of YBa2Cu3O7−x,” Phys. Rev. B 38, 5077–5080 (1988).
[Crossref]

Yoo, S. J. B.

I. Bozovic, K. Char, S. J. B. Yoo, A. Kapitulnik, M. R. Beasley, T. H. Geballe, Z. Z. Wang, S. Hagen, N. P. Ong, D. E. Aspnes, and M. K. Kelly, “Optical anisotropy of YBa2Cu3O7−x,” Phys. Rev. B 38, 5077–5080 (1988).
[Crossref]

Phys. Rev. (1)

J. Bardeen, L. N. Cooper, and J. R. Schrieffer, “Theory of superconductivity,” Phys. Rev. 108, 1175–1204 (1957).
[Crossref]

Phys. Rev. B (2)

R. Fuchs and K. L. Kliewer, “Surface plasmon in a semi-infinite free-electron gas,” Phys. Rev. B 3, 2270–2278 (1971).
[Crossref]

I. Bozovic, K. Char, S. J. B. Yoo, A. Kapitulnik, M. R. Beasley, T. H. Geballe, Z. Z. Wang, S. Hagen, N. P. Ong, D. E. Aspnes, and M. K. Kelly, “Optical anisotropy of YBa2Cu3O7−x,” Phys. Rev. B 38, 5077–5080 (1988).
[Crossref]

Prog. Surf. Sci. (1)

P. J. Feibelman, “Surface electromagnetic fields,” Prog. Surf. Sci. 12, 287–408 (1982).
[Crossref]

Z. Phys. (1)

K. Sturm, “Elektromagnetische Oberflächenwellen am Metallhalbraum (III),” Z. Phys. 209, 329–347 (1968).
[Crossref]

Other (10)

T. Timusk and D. B. Tanner, “Infrared properties of high TC superconductors,” in Physical Properties of High Temperature Superconductors I, D. M. Ginsberg, ed. (World Scientific, London, 1989), Chap. 7, pp. 339–407.

A. Liu and O. Keller, “Surface polaritons on a BCS-paired jellium,” Opt. Commun. (to be published).

J. R. Schrieffer, Theory of Superconductivity (Benjamin, New York, 1964).

O. Keller and J. H. Pedersen, “A nonlocal description of the dispersion relation and the energy flow associated with surface electromagnetic waves on metals,” in Scattering and Diffraction, H. A. Ferwada, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1029, 18–26 (1989).
[Crossref]

G. Rickayzen, Theory of Superconductivity (Interscience, New York, 1964).

F. Forstmann and R. R. Gerhardts, Metal Optics near the Plasma Frequency, Vol. 109 of Springer Tracts in Modern Physics (Springer-Verlag, Heidelberg, 1986).
[Crossref]

V. M. Agranovich and D. L. Mills, ed., Surface Polaritons, Vol. 1 of Modern Problems in Condensed Matter Sciences (North-Holland, Amsterdam, 1982).

A. D. Boardman, ed., Electromagnetic Surface Modes (Wiley, New York, 1982).

R. F. Wallis and G. I. Stegeman, eds., Electromagnetic Surface Excitations, Vol. 3 of Springer Series on Wave Phenomena (Springer-Verlag, Heidelberg, 1986).
[Crossref]

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings, Vol. III of Springer Tracts in Modern Physics (Springer-Verlag, Heidelberg, 1988).

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Figures (2)

Fig. 1
Fig. 1

The four basic photon absorption processes of a Cooper-paired jellium. In the two upper diagrams, a photon absorption causes a quasi-particle scattering from state k to state k + q and from state k + q to k, respectively (●: filled state, ○: empty state). In the two lower diagrams, the photon absorption gives rise to, respectively, creation and destruction of two quasi-particles. The large circle indicates the Fermi surface. In general quasi-particles exist both inside and outside the Fermi surface. Four equivalent diagrams can be drawn for the photon emission.

Fig. 2
Fig. 2

Imaginary part of the wave number for electromagnetic surface waves on a superconductor (YBa2Cu3O7−y) as a function of the frequency in the clean limit. Curves are plotted for three temperatures. The resonance peaks occur at the gap frequencies 2Δ0(T)/ħ. For comparison we also show the imaginary part of the wave number in the normal, almost lossless phase.

Equations (54)

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σ ( q , ω ) = i n e 2 m ω U i ω ( e ћ 2 m ) 2 1 V k ( ( 2 k + q ) ( 2 k + q ) × { ( u k u k + q + υ k υ k + q ) 2 [ f ( E k ) f ( E k + q ) ] × ( 1 E k + q E k + ћ ω + 1 E k + q E k ћ ω ) + ( u k + q υ k υ k + q u k ) 2 [ 1 f ( E k ) f ( E k + q ) ] × ( 1 E k + q + E k ћ ω + 1 E k + q + E k + ћ ω ) } ) .
| G = [ k ( u k + υ k a k + a k + ) ] | 0 ,
E k = + [ ( k μ ) 2 + Δ k 2 ( T ) ] 1 / 2
f ( E k ) = [ exp ( E k / k B T ) + 1 ] 1 .
u k = ( 1 / 2 ) ( 1 + ˜ k / E k ) 1 / 2 ,
υ k = ( 1 / 2 ) ( 1 ˜ k / E k ) 1 / 2 ,
σ ( q , ω ) = [ σ T ( q , ω ) 0 0 0 σ T ( q , ω ) 0 0 0 σ L ( q , ω ) ] ,
σ ( q , ω ) = σ ( 0 , ω ) + 1 2 q 2 2 σ ( 0 , ω ) q 2 + .
σ ( q , ω ) = σ ( q , ω ) ,
σ nl ( q , ω ) = ( i n e 2 / m ω ) [ U + α ( ω , T ) ( q / ω ) 2 ( U + 2 e q e q ) ] ,
α ( ω , T ) = ћ 4 ω 2 3 n m 3 1 V k ( k · e q ) 4 × { 2 f ( E k ) [ 1 f ( E k ) ] k B T ω 2 [ 1 Δ k 2 ( T ) E k 2 ] + ћ 2 Δ k 2 ( T ) E k 3 1 2 f ( E k ) ( ћ ω ) 2 ( 2 E k ) 2 } .
α ( ω , T ) = ћ 4 ω 2 30 π 2 n m 3 × 0 k 6 { 2 f ( E k ) [ 1 f ( E k ) ] k B T ω 2 [ 1 Δ 0 2 ( T ) E k 2 ] + ћ 2 Δ 0 2 ( T ) E k 3 1 2 f ( E k ) ( ћ ω ) 2 ( 2 E k ) 2 } d k .
1 = g ( 0 ) W 0 ћ ω D tanh { [ ˜ 2 + Δ 0 2 ( T ) ] 1 / 2 2 k B T } d ˜ [ ˜ 2 + Δ 0 2 ( T ) ] 1 / 2 ,
1 = g ( 0 ) W 0 ћ ω D tanh ( ˜ 2 k B T c ) d ˜ ˜ ,
Δ 0 ( 0 ) = ћ ω D sinh [ 1 / g ( 0 ) W ] .
α ( ω , 0 ) = ћ 6 ω 2 Δ 0 2 ( 0 ) 30 π 2 n m 3 0 k 6 E k 3 d k ( ћ ω ) 2 ( 2 E k ) 2 .
α ( ω , T > T C ) = ћ 4 15 π 2 n m 3 k B T 0 k 6 f ( ˜ k ) [ 1 f ( ˜ k ) ] d k ,
i π 1 q 2 [ q 2 N L ( q , ω ) + q 2 N T ( q , ω ) ] exp ( i q 0 + ) d q = q 0 ( c 0 ω ) 2 .
N T ( q , ω ) = ( ω c 0 ) 2 [ 1 + i σ T ( q , ω ) 0 ω ] q 2 ,
N L ( q , ω ) = ( ω c 0 ) 2 [ 1 + i σ L ( q , ω ) 0 ω ] ,
q 2 L κ L 2 + ( κ T ) 2 T κ T 2 = q 0 2 ( c 0 ω ) 2 ,
T , L = κ T , L ( N T , L / q ) | q = κ T , L κ T , L .
κ T = [ ω 2 ω p 2 c 0 2 + α ( ω , T ) ( ω p / ω ) 2 ] 1 / 2 ,
σ L nl ( q , ω ) = i n e 2 m ω 1 1 3 α ( q / ω ) 2 .
κ L = [ ω 2 ω p 2 3 α ( ω , T ) ] 1 / 2 ,
T = 1 2 κ T [ 1 + α c 0 2 ( ω p ω ) 2 ] 1
L = 1 6 κ L c 0 2 α ( ω p ω ) 2 ,
q 2 [ 1 T ( q 0 , ω ) ] + κ L [ κ T q 0 T ( q 0 , ω ) ] = 0 ,
T ( q 0 , ω ) = 1 ( ω p / ω ) 2
q = ω c 0 T 1 / 2 ( q 0 , ω ) × { 1 T ( q 0 , ω ) [ 1 + α ( ω , T ) c 0 2 ( ω p ω ) 2 ] [ 1 T 2 ( q 0 , ω ) ] [ 1 + α ( ω , T ) c 0 2 ( ω p ω ) 2 ] } 1 / 2 .
q = ω c 0 1 / 2 ( ω ) × { 1 ( ω ) [ 1 + α ( ω + i / τ , T ) c 0 2 ω p 2 ω ( ω + i / τ ) ] [ 1 2 ( ω ) ] [ 1 + α ( ω + i / τ , T ) c 0 2 ω p 2 ω ( ω + i / τ ) ] } 1 / 2 ,
( ω ) = [ ω p 2 / ω ( ω + i / τ ) ] .
1 + α ( ω , T ) c 0 2 ( ω p ω ) 2 0 ,
A 1 = ( u k u k + q + υ k υ k + q ) 2 ,
B 1 = f ( E k ) f ( E k + q ) ,
C 1 = ( E k + q E k + ћ ω ) 1 + ( E k + q E k ћ ω ) 1 .
2 ( A 1 B 1 C 1 ) q 2 | 0 = 2 B 1 q C 1 q | 0 .
A 2 = ( u k + q υ k + υ k + q u k ) 2 ,
B 2 = 1 f ( E k ) f ( E k + q ) ,
C 2 = ( E k + q + E k ћ ω ) 1 + ( E k + q + E k + ћ ω ) 1 .
2 ( A 2 B 2 C 2 ) q 2 | 0 = B 2 C 2 2 A 2 q 2 | 0 .
B 1 q = ( k B T ) 1 f ( E k + q ) [ 1 f ( E k + q ) ] E k + q q
E k + q q = ћ 2 m ( q + k · e q ) ˜ k + q E k + q ,
B 1 q | 0 = ћ 2 m k B T ˜ k E k k · e q f ( E k ) [ 1 f ( E k ) ] ,
C 1 q | 0 = 2 m ω 2 ˜ k E k k · e q .
2 ( A 1 B 1 C 1 ) q 2 | 0 = 4 ћ 2 ( k · e q ) 2 m 2 k B T ω 2 ( Δ k 2 E k 2 1 ) f ( E k ) [ 1 f ( E k ) ] ,
2 A 2 q 2 | 0 = 2 [ q ( u k + q υ k υ k + q u k ) ] 2 | 0 ,
q ( u k + q υ k υ k + q u k ) = ћ 2 4 m ( q + k · e q ) Δ k 2 E k + q 3 × ( υ k u k + q + u k υ k + q ) ,
2 A 2 q 2 | 0 = ћ 4 Δ k 4 ( k · e q ) 2 8 m 2 u k 2 υ k 2 E k 6 = ћ 4 Δ k 4 ( k · e q ) 2 2 m 2 E k 4
2 ( A 2 B 2 C 2 ) q 2 | 0 = 2 ћ 4 Δ k 2 ( k · e q ) 2 m 2 E k 3 1 2 f ( E k ) ( 2 E k ) 2 ( ћ ω ) 2 ,
T = 1 V k 4 k x 2 S k · e q
2 ( D 0 A i B i C i ) q 2 | 0 = 4 ( k · e q ) 2 2 ( A i B i C i ) q 2 | 0 ,
L = 1 V k 4 k z 2 S k · e q .
L = 3 T .

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