Abstract

Semiconductors are known to show in the band-gap region relative large changes of their optical properties with increasing light intensity. These changes are due to the creation of electron–hole pairs, which modify by band filling the intraband and interband contributions to the complex optical dielectric function. Most important, with increasing concentration of electron–hole pairs, the band gap shrinks, and the Coulomb forces are strongly reduced so that excitonic effects disappear. The theoretical description of these phenomena in three- and quasi-two-dimensional semiconductors is reviewed.

© 1985 Optical Society of America

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References

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  1. H. Haug and S. Schmitt-Rink, Prog. Quantum Electron. 9, 3 (1984).
    [CrossRef]
  2. H. Haug, J. Lumin. 30, 171 (1985).
    [CrossRef]
  3. S. Schmitt-Rink and C. Ell, J. Lumin. 30, 385 (1985).
    [CrossRef]
  4. Here, we do not discuss the so-called high-field limit that is characterized by the opening of a field-induced gap in the quasi-particle spectrum [see, e.g., V. M. Galitskii, S. P. Goreslavskii, and V. F. Elesin, Sov. Phys. JETP 30, 117 (1970) or Ref. 2]. This effect has not yet been observed because of the strong electron and hole intraband and interband relaxation processes.
  5. H. Haug and S. Schmitt-Rink, Phil. Trans. R. Soc. London Ser. A 313, 221 (1984).
    [CrossRef]
  6. M. Rösier, R. Zimmermann, and W. Richert, Phys. Status Solidi b121, 609 (1984).
  7. H. Haug and F. F. Abraham, Phys. Rev. B 23, 2960 (1981).
    [CrossRef]
  8. K. Bohnert, M. Anselment, G. Kobbe, C. Klingshirn, H. Haug, S. W. Koch, S. Schmitt-Rink, and F. F. Abraham, Z. Physik B 42, 1 (1981).
    [CrossRef]
  9. H. Schweizer, A. Forchel, A. Hangleiter, S. Schmitt-Rink, J. P. Löwenau, and H. Haug, Phys. Rev. Lett. 51, 698 (1983).
    [CrossRef]
  10. J. P. Löwenau, S. Schmitt-Rink, and H. Haug, Phys. Rev. Lett. 49, 1511 (1982).
    [CrossRef]
  11. H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, A. C. Gossard, A. Passner, and W. Wiegmann, Appl. Phys. Lett. 35, 451 (1979).
    [CrossRef]
  12. G. W. Gobeli and H. Y. Fan, Phys. Rev. 119, 613 (1960).
    [CrossRef]
  13. D. A. B. Miller, S. D. Smith, and A. Johnston, Appl. Phys. Lett. 35, 658 (1979).
    [CrossRef]
  14. D. S. Chemla, Helv. Phys. Acta 56, 607 (1983).
  15. D. S. Chemla, D. A. B. Miller, P. W. Smith, A. C. Gossard, and W. Wiegmann, IEEE J. Quantum Electron. 20, 265 (1984).
    [CrossRef]
  16. S. Schmitt-Rink, C. Ell, S. W. Koch, H. E. Schmidt, and H. Haug, Solid State Commun. 52, 123 (1984).
    [CrossRef]
  17. H. Totsuji, J. Phys. Soc. Jpn. 40, 857 (1976).
    [CrossRef]
  18. H. M. Gibbs, S. S. Tarng, J. L. Jewell, D. A. Weinberger, K. Tai, A. C. Gossard, S. L. McCall, A. Passner, and W. Wiegmann, Appl. Phys. Lett. 41, 221 (1982).
    [CrossRef]
  19. H. Stolz, Einführung in die Vielelektronentheorie der Kristalle (Akademie-Verlag, Berlin, 1974).
  20. M. Reed and B. Simon, Methods of Modern Mathematical Physics (Academic, New York, 1978), Vol. IV.

1985 (2)

H. Haug, J. Lumin. 30, 171 (1985).
[CrossRef]

S. Schmitt-Rink and C. Ell, J. Lumin. 30, 385 (1985).
[CrossRef]

1984 (5)

H. Haug and S. Schmitt-Rink, Phil. Trans. R. Soc. London Ser. A 313, 221 (1984).
[CrossRef]

M. Rösier, R. Zimmermann, and W. Richert, Phys. Status Solidi b121, 609 (1984).

H. Haug and S. Schmitt-Rink, Prog. Quantum Electron. 9, 3 (1984).
[CrossRef]

D. S. Chemla, D. A. B. Miller, P. W. Smith, A. C. Gossard, and W. Wiegmann, IEEE J. Quantum Electron. 20, 265 (1984).
[CrossRef]

S. Schmitt-Rink, C. Ell, S. W. Koch, H. E. Schmidt, and H. Haug, Solid State Commun. 52, 123 (1984).
[CrossRef]

1983 (2)

D. S. Chemla, Helv. Phys. Acta 56, 607 (1983).

H. Schweizer, A. Forchel, A. Hangleiter, S. Schmitt-Rink, J. P. Löwenau, and H. Haug, Phys. Rev. Lett. 51, 698 (1983).
[CrossRef]

1982 (2)

J. P. Löwenau, S. Schmitt-Rink, and H. Haug, Phys. Rev. Lett. 49, 1511 (1982).
[CrossRef]

H. M. Gibbs, S. S. Tarng, J. L. Jewell, D. A. Weinberger, K. Tai, A. C. Gossard, S. L. McCall, A. Passner, and W. Wiegmann, Appl. Phys. Lett. 41, 221 (1982).
[CrossRef]

1981 (2)

H. Haug and F. F. Abraham, Phys. Rev. B 23, 2960 (1981).
[CrossRef]

K. Bohnert, M. Anselment, G. Kobbe, C. Klingshirn, H. Haug, S. W. Koch, S. Schmitt-Rink, and F. F. Abraham, Z. Physik B 42, 1 (1981).
[CrossRef]

1979 (2)

H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, A. C. Gossard, A. Passner, and W. Wiegmann, Appl. Phys. Lett. 35, 451 (1979).
[CrossRef]

D. A. B. Miller, S. D. Smith, and A. Johnston, Appl. Phys. Lett. 35, 658 (1979).
[CrossRef]

1976 (1)

H. Totsuji, J. Phys. Soc. Jpn. 40, 857 (1976).
[CrossRef]

1960 (1)

G. W. Gobeli and H. Y. Fan, Phys. Rev. 119, 613 (1960).
[CrossRef]

Abraham, F. F.

H. Haug and F. F. Abraham, Phys. Rev. B 23, 2960 (1981).
[CrossRef]

K. Bohnert, M. Anselment, G. Kobbe, C. Klingshirn, H. Haug, S. W. Koch, S. Schmitt-Rink, and F. F. Abraham, Z. Physik B 42, 1 (1981).
[CrossRef]

Anselment, M.

K. Bohnert, M. Anselment, G. Kobbe, C. Klingshirn, H. Haug, S. W. Koch, S. Schmitt-Rink, and F. F. Abraham, Z. Physik B 42, 1 (1981).
[CrossRef]

Bohnert, K.

K. Bohnert, M. Anselment, G. Kobbe, C. Klingshirn, H. Haug, S. W. Koch, S. Schmitt-Rink, and F. F. Abraham, Z. Physik B 42, 1 (1981).
[CrossRef]

Chemla, D. S.

D. S. Chemla, D. A. B. Miller, P. W. Smith, A. C. Gossard, and W. Wiegmann, IEEE J. Quantum Electron. 20, 265 (1984).
[CrossRef]

D. S. Chemla, Helv. Phys. Acta 56, 607 (1983).

Ell, C.

S. Schmitt-Rink and C. Ell, J. Lumin. 30, 385 (1985).
[CrossRef]

S. Schmitt-Rink, C. Ell, S. W. Koch, H. E. Schmidt, and H. Haug, Solid State Commun. 52, 123 (1984).
[CrossRef]

Fan, H. Y.

G. W. Gobeli and H. Y. Fan, Phys. Rev. 119, 613 (1960).
[CrossRef]

Forchel, A.

H. Schweizer, A. Forchel, A. Hangleiter, S. Schmitt-Rink, J. P. Löwenau, and H. Haug, Phys. Rev. Lett. 51, 698 (1983).
[CrossRef]

Gibbs, H. M.

H. M. Gibbs, S. S. Tarng, J. L. Jewell, D. A. Weinberger, K. Tai, A. C. Gossard, S. L. McCall, A. Passner, and W. Wiegmann, Appl. Phys. Lett. 41, 221 (1982).
[CrossRef]

H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, A. C. Gossard, A. Passner, and W. Wiegmann, Appl. Phys. Lett. 35, 451 (1979).
[CrossRef]

Gobeli, G. W.

G. W. Gobeli and H. Y. Fan, Phys. Rev. 119, 613 (1960).
[CrossRef]

Gossard, A. C.

D. S. Chemla, D. A. B. Miller, P. W. Smith, A. C. Gossard, and W. Wiegmann, IEEE J. Quantum Electron. 20, 265 (1984).
[CrossRef]

H. M. Gibbs, S. S. Tarng, J. L. Jewell, D. A. Weinberger, K. Tai, A. C. Gossard, S. L. McCall, A. Passner, and W. Wiegmann, Appl. Phys. Lett. 41, 221 (1982).
[CrossRef]

H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, A. C. Gossard, A. Passner, and W. Wiegmann, Appl. Phys. Lett. 35, 451 (1979).
[CrossRef]

Hangleiter, A.

H. Schweizer, A. Forchel, A. Hangleiter, S. Schmitt-Rink, J. P. Löwenau, and H. Haug, Phys. Rev. Lett. 51, 698 (1983).
[CrossRef]

Haug, H.

H. Haug, J. Lumin. 30, 171 (1985).
[CrossRef]

H. Haug and S. Schmitt-Rink, Prog. Quantum Electron. 9, 3 (1984).
[CrossRef]

H. Haug and S. Schmitt-Rink, Phil. Trans. R. Soc. London Ser. A 313, 221 (1984).
[CrossRef]

S. Schmitt-Rink, C. Ell, S. W. Koch, H. E. Schmidt, and H. Haug, Solid State Commun. 52, 123 (1984).
[CrossRef]

H. Schweizer, A. Forchel, A. Hangleiter, S. Schmitt-Rink, J. P. Löwenau, and H. Haug, Phys. Rev. Lett. 51, 698 (1983).
[CrossRef]

J. P. Löwenau, S. Schmitt-Rink, and H. Haug, Phys. Rev. Lett. 49, 1511 (1982).
[CrossRef]

H. Haug and F. F. Abraham, Phys. Rev. B 23, 2960 (1981).
[CrossRef]

K. Bohnert, M. Anselment, G. Kobbe, C. Klingshirn, H. Haug, S. W. Koch, S. Schmitt-Rink, and F. F. Abraham, Z. Physik B 42, 1 (1981).
[CrossRef]

Jewell, J. L.

H. M. Gibbs, S. S. Tarng, J. L. Jewell, D. A. Weinberger, K. Tai, A. C. Gossard, S. L. McCall, A. Passner, and W. Wiegmann, Appl. Phys. Lett. 41, 221 (1982).
[CrossRef]

Johnston, A.

D. A. B. Miller, S. D. Smith, and A. Johnston, Appl. Phys. Lett. 35, 658 (1979).
[CrossRef]

Klingshirn, C.

K. Bohnert, M. Anselment, G. Kobbe, C. Klingshirn, H. Haug, S. W. Koch, S. Schmitt-Rink, and F. F. Abraham, Z. Physik B 42, 1 (1981).
[CrossRef]

Kobbe, G.

K. Bohnert, M. Anselment, G. Kobbe, C. Klingshirn, H. Haug, S. W. Koch, S. Schmitt-Rink, and F. F. Abraham, Z. Physik B 42, 1 (1981).
[CrossRef]

Koch, S. W.

S. Schmitt-Rink, C. Ell, S. W. Koch, H. E. Schmidt, and H. Haug, Solid State Commun. 52, 123 (1984).
[CrossRef]

K. Bohnert, M. Anselment, G. Kobbe, C. Klingshirn, H. Haug, S. W. Koch, S. Schmitt-Rink, and F. F. Abraham, Z. Physik B 42, 1 (1981).
[CrossRef]

Löwenau, J. P.

H. Schweizer, A. Forchel, A. Hangleiter, S. Schmitt-Rink, J. P. Löwenau, and H. Haug, Phys. Rev. Lett. 51, 698 (1983).
[CrossRef]

J. P. Löwenau, S. Schmitt-Rink, and H. Haug, Phys. Rev. Lett. 49, 1511 (1982).
[CrossRef]

McCall, S. L.

H. M. Gibbs, S. S. Tarng, J. L. Jewell, D. A. Weinberger, K. Tai, A. C. Gossard, S. L. McCall, A. Passner, and W. Wiegmann, Appl. Phys. Lett. 41, 221 (1982).
[CrossRef]

H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, A. C. Gossard, A. Passner, and W. Wiegmann, Appl. Phys. Lett. 35, 451 (1979).
[CrossRef]

Miller, D. A. B.

D. S. Chemla, D. A. B. Miller, P. W. Smith, A. C. Gossard, and W. Wiegmann, IEEE J. Quantum Electron. 20, 265 (1984).
[CrossRef]

D. A. B. Miller, S. D. Smith, and A. Johnston, Appl. Phys. Lett. 35, 658 (1979).
[CrossRef]

Passner, A.

H. M. Gibbs, S. S. Tarng, J. L. Jewell, D. A. Weinberger, K. Tai, A. C. Gossard, S. L. McCall, A. Passner, and W. Wiegmann, Appl. Phys. Lett. 41, 221 (1982).
[CrossRef]

H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, A. C. Gossard, A. Passner, and W. Wiegmann, Appl. Phys. Lett. 35, 451 (1979).
[CrossRef]

Reed, M.

M. Reed and B. Simon, Methods of Modern Mathematical Physics (Academic, New York, 1978), Vol. IV.

Richert, W.

M. Rösier, R. Zimmermann, and W. Richert, Phys. Status Solidi b121, 609 (1984).

Rösier, M.

M. Rösier, R. Zimmermann, and W. Richert, Phys. Status Solidi b121, 609 (1984).

Schmidt, H. E.

S. Schmitt-Rink, C. Ell, S. W. Koch, H. E. Schmidt, and H. Haug, Solid State Commun. 52, 123 (1984).
[CrossRef]

Schmitt-Rink, S.

S. Schmitt-Rink and C. Ell, J. Lumin. 30, 385 (1985).
[CrossRef]

H. Haug and S. Schmitt-Rink, Prog. Quantum Electron. 9, 3 (1984).
[CrossRef]

H. Haug and S. Schmitt-Rink, Phil. Trans. R. Soc. London Ser. A 313, 221 (1984).
[CrossRef]

S. Schmitt-Rink, C. Ell, S. W. Koch, H. E. Schmidt, and H. Haug, Solid State Commun. 52, 123 (1984).
[CrossRef]

H. Schweizer, A. Forchel, A. Hangleiter, S. Schmitt-Rink, J. P. Löwenau, and H. Haug, Phys. Rev. Lett. 51, 698 (1983).
[CrossRef]

J. P. Löwenau, S. Schmitt-Rink, and H. Haug, Phys. Rev. Lett. 49, 1511 (1982).
[CrossRef]

K. Bohnert, M. Anselment, G. Kobbe, C. Klingshirn, H. Haug, S. W. Koch, S. Schmitt-Rink, and F. F. Abraham, Z. Physik B 42, 1 (1981).
[CrossRef]

Schweizer, H.

H. Schweizer, A. Forchel, A. Hangleiter, S. Schmitt-Rink, J. P. Löwenau, and H. Haug, Phys. Rev. Lett. 51, 698 (1983).
[CrossRef]

Simon, B.

M. Reed and B. Simon, Methods of Modern Mathematical Physics (Academic, New York, 1978), Vol. IV.

Smith, P. W.

D. S. Chemla, D. A. B. Miller, P. W. Smith, A. C. Gossard, and W. Wiegmann, IEEE J. Quantum Electron. 20, 265 (1984).
[CrossRef]

Smith, S. D.

D. A. B. Miller, S. D. Smith, and A. Johnston, Appl. Phys. Lett. 35, 658 (1979).
[CrossRef]

Stolz, H.

H. Stolz, Einführung in die Vielelektronentheorie der Kristalle (Akademie-Verlag, Berlin, 1974).

Tai, K.

H. M. Gibbs, S. S. Tarng, J. L. Jewell, D. A. Weinberger, K. Tai, A. C. Gossard, S. L. McCall, A. Passner, and W. Wiegmann, Appl. Phys. Lett. 41, 221 (1982).
[CrossRef]

Tarng, S. S.

H. M. Gibbs, S. S. Tarng, J. L. Jewell, D. A. Weinberger, K. Tai, A. C. Gossard, S. L. McCall, A. Passner, and W. Wiegmann, Appl. Phys. Lett. 41, 221 (1982).
[CrossRef]

Totsuji, H.

H. Totsuji, J. Phys. Soc. Jpn. 40, 857 (1976).
[CrossRef]

Venkatesan, T. N. C.

H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, A. C. Gossard, A. Passner, and W. Wiegmann, Appl. Phys. Lett. 35, 451 (1979).
[CrossRef]

Weinberger, D. A.

H. M. Gibbs, S. S. Tarng, J. L. Jewell, D. A. Weinberger, K. Tai, A. C. Gossard, S. L. McCall, A. Passner, and W. Wiegmann, Appl. Phys. Lett. 41, 221 (1982).
[CrossRef]

Wiegmann, W.

D. S. Chemla, D. A. B. Miller, P. W. Smith, A. C. Gossard, and W. Wiegmann, IEEE J. Quantum Electron. 20, 265 (1984).
[CrossRef]

H. M. Gibbs, S. S. Tarng, J. L. Jewell, D. A. Weinberger, K. Tai, A. C. Gossard, S. L. McCall, A. Passner, and W. Wiegmann, Appl. Phys. Lett. 41, 221 (1982).
[CrossRef]

H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, A. C. Gossard, A. Passner, and W. Wiegmann, Appl. Phys. Lett. 35, 451 (1979).
[CrossRef]

Zimmermann, R.

M. Rösier, R. Zimmermann, and W. Richert, Phys. Status Solidi b121, 609 (1984).

Appl. Phys. Lett. (3)

D. A. B. Miller, S. D. Smith, and A. Johnston, Appl. Phys. Lett. 35, 658 (1979).
[CrossRef]

H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, A. C. Gossard, A. Passner, and W. Wiegmann, Appl. Phys. Lett. 35, 451 (1979).
[CrossRef]

H. M. Gibbs, S. S. Tarng, J. L. Jewell, D. A. Weinberger, K. Tai, A. C. Gossard, S. L. McCall, A. Passner, and W. Wiegmann, Appl. Phys. Lett. 41, 221 (1982).
[CrossRef]

Helv. Phys. Acta (1)

D. S. Chemla, Helv. Phys. Acta 56, 607 (1983).

IEEE J. Quantum Electron. (1)

D. S. Chemla, D. A. B. Miller, P. W. Smith, A. C. Gossard, and W. Wiegmann, IEEE J. Quantum Electron. 20, 265 (1984).
[CrossRef]

J. Lumin. (2)

H. Haug, J. Lumin. 30, 171 (1985).
[CrossRef]

S. Schmitt-Rink and C. Ell, J. Lumin. 30, 385 (1985).
[CrossRef]

J. Phys. Soc. Jpn. (1)

H. Totsuji, J. Phys. Soc. Jpn. 40, 857 (1976).
[CrossRef]

Phil. Trans. R. Soc. London Ser. A (1)

H. Haug and S. Schmitt-Rink, Phil. Trans. R. Soc. London Ser. A 313, 221 (1984).
[CrossRef]

Phys. Rev. (1)

G. W. Gobeli and H. Y. Fan, Phys. Rev. 119, 613 (1960).
[CrossRef]

Phys. Rev. B (1)

H. Haug and F. F. Abraham, Phys. Rev. B 23, 2960 (1981).
[CrossRef]

Phys. Rev. Lett. (2)

H. Schweizer, A. Forchel, A. Hangleiter, S. Schmitt-Rink, J. P. Löwenau, and H. Haug, Phys. Rev. Lett. 51, 698 (1983).
[CrossRef]

J. P. Löwenau, S. Schmitt-Rink, and H. Haug, Phys. Rev. Lett. 49, 1511 (1982).
[CrossRef]

Phys. Status Solidi (1)

M. Rösier, R. Zimmermann, and W. Richert, Phys. Status Solidi b121, 609 (1984).

Prog. Quantum Electron. (1)

H. Haug and S. Schmitt-Rink, Prog. Quantum Electron. 9, 3 (1984).
[CrossRef]

Solid State Commun. (1)

S. Schmitt-Rink, C. Ell, S. W. Koch, H. E. Schmidt, and H. Haug, Solid State Commun. 52, 123 (1984).
[CrossRef]

Z. Physik B (1)

K. Bohnert, M. Anselment, G. Kobbe, C. Klingshirn, H. Haug, S. W. Koch, S. Schmitt-Rink, and F. F. Abraham, Z. Physik B 42, 1 (1981).
[CrossRef]

Other (3)

Here, we do not discuss the so-called high-field limit that is characterized by the opening of a field-induced gap in the quasi-particle spectrum [see, e.g., V. M. Galitskii, S. P. Goreslavskii, and V. F. Elesin, Sov. Phys. JETP 30, 117 (1970) or Ref. 2]. This effect has not yet been observed because of the strong electron and hole intraband and interband relaxation processes.

H. Stolz, Einführung in die Vielelektronentheorie der Kristalle (Akademie-Verlag, Berlin, 1974).

M. Reed and B. Simon, Methods of Modern Mathematical Physics (Academic, New York, 1978), Vol. IV.

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Figures (5)

Fig. 1
Fig. 1

Renormalized band gap of bulk GaAs as a function of the free-carrier density n for various temperatures T (according to Ref. 6).

Fig. 2
Fig. 2

Measured and calculated absorption spectra at the direct gap of Ge for various excitation intensities I and a lattice temperature TL = 30 K (according to Ref. 9).

Fig. 3
Fig. 3

Calculated absorption and refraction spectra of bulk GaAs for various free-carrier densities n and a temperature T = 10 K (according to Ref. 10).

Fig. 4
Fig. 4

Calculated absorption and refraction spectra of InSb for various excitation intensities I and a temperature T = 77 K; the excitation frequency is ℏωexc = 225 meV (according to Ref. 10). The experimental points of the low-intensity absorption spectrum are taken from Ref. 12.

Fig. 5
Fig. 5

Renormalized band gap of a quasi-two-dimensional semiconductor as a function of the free-carrier density n for various temperatures T (according to Ref. 16).

Equations (40)

Equations on this page are rendered with MathJax. Learn more.

n ( ω ) = ( ½ { ( ω ) + [ 2 ( ω ) + 2 ( ω ) ] 1 / 2 } ) 1 / 2
α ( ω ) = ω ( ω ) n ( ω ) c .
( ω ) = 1 ω 0 2 ( ω + i δ ) 2 8 π e 2 V × k , k [ r vc ( k ) r vc * ( k ) G eh 0 ( k , k , ω ) + c . c . ( ω ) ] ,
ω 0 2 = 8 π e 2 V k [ f e ( k ) m e ( k ) + f h ( k ) m h ( k ) ] ,
G eh 0 ( k , k , ω ) = 1 f e ( k ) f h ( k ) ω + i δ e ( k ) h ( k ) δ k , k
f i ( k ) = 1 exp { β [ i ( k ) μ i ] } + 1 ,
| r vc ( k ) | = ( E g / 4 m ) 1 / 2 E g + 2 k 2 2 m ,
n = 2 V k f i ( k ) .
= α ( ω ) I ω R spon ,
R spon = 0 d ω [ ω n ( ω ) π c ] 2 α ( ω ) 1 exp [ β ( ω μ ) ] 1 ,
V s ( q , ω ) = V ( q ) / ( q , ω ) ,
( q , ω ) = 0 8 π e 2 V q 2 i , k f i ( k ) f i ( k + q ) ω + i δ i ( k + q ) + i ( k ) ,
1 ( q , ω ) = 1 0 [ 1 + ω pl 2 ( ω + i δ ) 2 ω 2 ( q ) ] ,
ω 2 ( q ) = ω pl 2 ( 1 + q 2 κ 2 ) + η 2 ( q ) .
κ 2 = 4 π e 2 0 i n μ i ,
( q , ω ) = 0 4 π n e 2 q 2 n | 1 s | exp ( i α qr ) exp ( i β qr ) | n | 2 × 2 [ ω 1 s ( 0 ) ω n ( q ) ] ( ω + i δ ) 2 [ ω 1 s ( 0 ) ω n ( q ) ] 2 .
Σ i s x ( k ) = 1 V k V s ( k k ) f i ( k )
Σ i Ch = 1 2 V k [ V ( k ) / 0 V s ( k ) ] ,
e i ( k ) = i ( k ) + i ( k )
E g = E g + i i ( 0 ) .
E g = E g + i , k f i ( k ) i ( k ) / k f i ( k ) .
( ω ) = 8 π e 2 V k , k r vc ( k ) r vc * ( k ) G eh ( k , k , ω ) ,
G eh ( k , k , ω ) = G eh 0 ( k , k , ω ) 1 V k , k G eh 0 ( k , k , ω ) × V s ( k k ) G eh ( k , k , ω ) ,
G eh 0 ( k , k , ω ) = 1 f e ( k ) f h ( k ) ω + i δ e e ( k ) e h ( k ) δ k , k
( q , ω ) = 0 4 π e 2 A q i , k f i ( k ) f i ( k + q ) ω + i δ i ( k + q ) + i ( k ) ,
ω 2 ( q ) = ω pl 2 ( q ) ( 1 + g κ ) + η 2 ( q ) ,
ω pl 2 ( q ) = 2 π n e 2 q 0 m ,
κ = 2 π e 2 0 i n μ i
χ ( ω ) χ 0 ( ω ) = 1 1 + | E | 2 / | E | sat 2 ,
G eh ( k , k , ω ) = n | F ( k ) | 1 / 2 ϕ n ( k ) | F ( k ) | 1 / 2 ϕ n * ( k ) ω + i δ ω n × sgn ( ω n μ ) ,
F ( k ) = 1 f e ( k ) f h ( k ) .
ω n ϕ n ( k ) = k H ( k , k ) ϕ n ( k ) ,
k ϕ n * ( k ) sgn [ F ( k ) ] ϕ m ( k ) = δ n , m sgn ( ω n μ ) .
H ( k , k ) = [ e e ( k ) + e h ( k ) ] δ k , k sgn [ F ( k ) ] | F ( k ) | 1 / 2 V s ( k k ) | F ( k ) | 1 / 2 .
G eh ( k , k , ω ) = n ϕ n ( k ) ϕ n * ( k ) ω + i δ ω n ,
ω n ϕ n ( k ) = [ e ( k ) + h ( k ) ] ϕ n ( k ) k V ( k k ) ϕ n ( k ) / 0
k ϕ n * ( k ) ϕ m ( k ) = δ n , m
χ ( ω ) = χ 0 ( ω ) ( 1 n n sat ) ,
= χ ( ω ) d | E | 2 2 n τ ,
| E | sat 2 = 2 n sat d τ χ 0 ( ω ) ,

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