Abstract

A fully microscopical theory for the photoluminescence of a quantum-well in an arbitrary one-dimensional stack structure is presented. For strong-coupling configurations, the full semiconductor luminescence equations are solved. For the weak-coupling regime, a frequency-dependent filter function is directly derived from the semiconductor luminescence equations with the knowledge of the dielectric structure. Via that filter function, the detected luminescence can be related to the pure quantum-well emission in vacuum. The approach is generalized to include corrections to the emitted peak width due to the photonic-environment-dependent radiative decay, and the corrections are shown to be obtainable from the mode functions alone. The applicability of the method is thoroughly tested up to the onset of normal-mode coupling.

© 2008 Optical Society of America

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  1. E. F. Schubert, Y. H. Wang, A. Y. Cho, L. W. Tu, and G. J. Zydyik, "Resonant cavity light-emitting diode," Appl. Phys. Lett. 60, 921-924 (1992).
    [CrossRef]
  2. E. F. Schubert, Light-Emitting Diodes (Cambride U. Press, 2006).
    [CrossRef]
  3. T. H. Maiman, "Stimulated optical radiation in ruby," Nature 187, 493-494 (1960).
    [CrossRef]
  4. S. L. Chuang, Physics of Optoelectronic Devices (Wiley, 1995).
  5. W. W. Chow and S. W. Koch, Semiconductor Laser Fundamentals, 1st ed. (Springer Verlag, 1999).
  6. E. M. Purcell, "Spontaneous emission probabilities at radio frequencies," Phys. Rev. 69, 681 (1946).
    [CrossRef]
  7. F. D. Martini, G. Innocenti, G. R. Jacobovitz, and P. Mataloni, "Spontaneous emission probabilities at radio frequencies," Phys. Rev. Lett. 59, 2955-2958 (1987).
    [CrossRef] [PubMed]
  8. J. Martorell and N. M. Lawandy, "Observation of inhibited spontaneous emission in a periodic dielectric structure," Phys. Rev. Lett. 65, 1877-1880 (1990).
    [CrossRef] [PubMed]
  9. J. P. Dowling and C. M. Bowden, "Atomic emission rates in inhomogeneous media with applications to photonic band structures," Phys. Rev. A 46, 612-622 (1992).
    [CrossRef] [PubMed]
  10. A. G. Kofman, G. Kurizki, and B. Sherman, "Spontaneous and induced atomic decay in photonic band structures," J. Mod. Opt. 41, 353-384 (1994).
    [CrossRef]
  11. M. D. Tocci, M. Scalora, M. J. Bloemer, J. P. Dowling, and C. M. Bowden, "Measurement of spontaneous-emission enhancement near the one-dimensional photonic band edge of semiconductor heterostructures," Phys. Rev. A 53, 2799-2803 (1996).
    [CrossRef] [PubMed]
  12. P. Lambropoulos, G. M. Nikolopoulos, T. R. Nielsen, and S. Bay, "Fundamental quantum optics in structured reservoirs," Rep. Prog. Phys. 63, 455-503 (2000).
    [CrossRef]
  13. E. Yablanovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059-2062 (1987).
    [CrossRef]
  14. R. E. Slusher and C. Weisbuch, "Optical microcavities in condensed matter systems," Solid State Commun. 92, 149-157 (1994).
    [CrossRef]
  15. C. Weisbuch, M. Nishioka, A. Ishikawa, and Y. Arakawa, "Observation of the coupled exciton-photon mode splitting in a semiconductor quantum microcavity," Phys. Rev. Lett. 69, 3314-3317 (1992).
    [CrossRef] [PubMed]
  16. F. Jahnke, M. Kira, S. W. Koch, G. Khitrova, E. K. Lindmark, T. R. Nelson, Jr., D. V. Wick, J. D. Berger, O. Lyngnes, H. M. Gibbs, and K. Tai, "Excitonic nonlinearities of semiconductor microcavities in the nonperturbative regime," Phys. Rev. Lett. 77, 5257-5260 (1996).
    [CrossRef] [PubMed]
  17. G. Khitrova, H. M. Gibbs, M. Kira, S. W. Koch, and A. Scherer, "Vacuum Rabi splitting in semiconductors," Nat. Phys. 2, 81-90 (2006)
    [CrossRef]
  18. G. Khitrova, H. M. Gibbs, F. Jahnke, M. Kira, and S. W. Koch, "Nonlinear optics of normal-mode-coupling semiconductor microcavities," Rev. Mod. Phys. 71, 1591-1639 (1999).
    [CrossRef]
  19. G. Hernandez, Fabry-Perot Interferometers (Cambridge U. Press, 1986).
  20. V. M. Agranovich and O. A. Dubowskii, "Effect of retarded interaction of exciton spectrum in 1-dimensional and 2-dimensional crystals," JETP Lett. 3, 223-226 (1966).
  21. J. Feldmann, G. Peter, E. O. Göbel, P. Dawson, K. Moore, C. Foxon, and R. J. Elliott, "Linewidth dependence of radiative exciton lifetimes in quantum wells," Phys. Rev. Lett. 59, 2337-2340 (1987).
    [CrossRef] [PubMed]
  22. E. Hanamura, "Rapid radiative decay and enhanced optical nonlinearity of excitons in a quantum well," Phys. Rev. B 38, 1228-1234 (1988).
    [CrossRef]
  23. B. Deveaud, F. Clérot, N. Roy, K. Satzke, B. Sermage, and D. S. Katzer, "Enhanced radiative recombination of free excitons in GaAs quantum wells," Phys. Rev. Lett. 67, 2355-2358 (1991).
    [CrossRef] [PubMed]
  24. M. Kira, F. Jahnke, W. Hoyer, and S. W. Koch, "Quantum theory of secondary emission in optically excited semiconductor quantum wells," Prog. Quantum Electron. 23, 189-279 (1999).
    [CrossRef]
  25. M. Kira and S. W. Koch, "Many-body correlations and exciton effects in semiconductor spectroscopy," Prog. Quantum Electron. 30, 155-296 (2006).
    [CrossRef]
  26. E. Mertzbacher, Quantum Mechanics, 3rd ed. (Wiley, 1998).
  27. M. Born and E. Wolf, Principles of Optics, Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge U. Press, 1999).
    [PubMed]
  28. M. Kira, F. Jahnke, and S. W. Koch, "Microscopic theory of excitonic signatures in semiconductor photoluminescence," Phys. Rev. Lett. 81, 3263-3266 (1998).
    [CrossRef]
  29. S. Chatterjee, C. Ell, S. Mosor, G. Khitrova, H. M. Gibbs, W. Hoyer, M. Kira, S. W. Koch, J. P. Prineas, and H. Stolz, "Excitonic photoluminescence in semiconductor quantum wells: plasma versus excitons," Phys. Rev. Lett. 92, 067402 (2004).
    [CrossRef] [PubMed]
  30. W. Hoyer, M. Kira, S. W. Koch, J. Hader, and J. V. Moloney, "Coulomb effects on quantum-well luminescence spectra and radiative recombination times," J. Opt. Soc. Am. 24, 1344-1353 (2007).
    [CrossRef]
  31. H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors, 4th ed. (World Scientific, 2004).
  32. M. Schafer, M. Werchner, W. Hoyer, M. Kira, and S. W. Koch, "Quantum theory of luminescence in multiple-quantum-well Bragg structures," Phys. Rev. B 74, 155315 (2006).
    [CrossRef]

2007

W. Hoyer, M. Kira, S. W. Koch, J. Hader, and J. V. Moloney, "Coulomb effects on quantum-well luminescence spectra and radiative recombination times," J. Opt. Soc. Am. 24, 1344-1353 (2007).
[CrossRef]

2006

M. Schafer, M. Werchner, W. Hoyer, M. Kira, and S. W. Koch, "Quantum theory of luminescence in multiple-quantum-well Bragg structures," Phys. Rev. B 74, 155315 (2006).
[CrossRef]

M. Kira and S. W. Koch, "Many-body correlations and exciton effects in semiconductor spectroscopy," Prog. Quantum Electron. 30, 155-296 (2006).
[CrossRef]

G. Khitrova, H. M. Gibbs, M. Kira, S. W. Koch, and A. Scherer, "Vacuum Rabi splitting in semiconductors," Nat. Phys. 2, 81-90 (2006)
[CrossRef]

2004

S. Chatterjee, C. Ell, S. Mosor, G. Khitrova, H. M. Gibbs, W. Hoyer, M. Kira, S. W. Koch, J. P. Prineas, and H. Stolz, "Excitonic photoluminescence in semiconductor quantum wells: plasma versus excitons," Phys. Rev. Lett. 92, 067402 (2004).
[CrossRef] [PubMed]

2000

P. Lambropoulos, G. M. Nikolopoulos, T. R. Nielsen, and S. Bay, "Fundamental quantum optics in structured reservoirs," Rep. Prog. Phys. 63, 455-503 (2000).
[CrossRef]

1999

G. Khitrova, H. M. Gibbs, F. Jahnke, M. Kira, and S. W. Koch, "Nonlinear optics of normal-mode-coupling semiconductor microcavities," Rev. Mod. Phys. 71, 1591-1639 (1999).
[CrossRef]

M. Kira, F. Jahnke, W. Hoyer, and S. W. Koch, "Quantum theory of secondary emission in optically excited semiconductor quantum wells," Prog. Quantum Electron. 23, 189-279 (1999).
[CrossRef]

1998

M. Kira, F. Jahnke, and S. W. Koch, "Microscopic theory of excitonic signatures in semiconductor photoluminescence," Phys. Rev. Lett. 81, 3263-3266 (1998).
[CrossRef]

1996

M. D. Tocci, M. Scalora, M. J. Bloemer, J. P. Dowling, and C. M. Bowden, "Measurement of spontaneous-emission enhancement near the one-dimensional photonic band edge of semiconductor heterostructures," Phys. Rev. A 53, 2799-2803 (1996).
[CrossRef] [PubMed]

F. Jahnke, M. Kira, S. W. Koch, G. Khitrova, E. K. Lindmark, T. R. Nelson, Jr., D. V. Wick, J. D. Berger, O. Lyngnes, H. M. Gibbs, and K. Tai, "Excitonic nonlinearities of semiconductor microcavities in the nonperturbative regime," Phys. Rev. Lett. 77, 5257-5260 (1996).
[CrossRef] [PubMed]

1994

R. E. Slusher and C. Weisbuch, "Optical microcavities in condensed matter systems," Solid State Commun. 92, 149-157 (1994).
[CrossRef]

A. G. Kofman, G. Kurizki, and B. Sherman, "Spontaneous and induced atomic decay in photonic band structures," J. Mod. Opt. 41, 353-384 (1994).
[CrossRef]

1992

J. P. Dowling and C. M. Bowden, "Atomic emission rates in inhomogeneous media with applications to photonic band structures," Phys. Rev. A 46, 612-622 (1992).
[CrossRef] [PubMed]

E. F. Schubert, Y. H. Wang, A. Y. Cho, L. W. Tu, and G. J. Zydyik, "Resonant cavity light-emitting diode," Appl. Phys. Lett. 60, 921-924 (1992).
[CrossRef]

C. Weisbuch, M. Nishioka, A. Ishikawa, and Y. Arakawa, "Observation of the coupled exciton-photon mode splitting in a semiconductor quantum microcavity," Phys. Rev. Lett. 69, 3314-3317 (1992).
[CrossRef] [PubMed]

1991

B. Deveaud, F. Clérot, N. Roy, K. Satzke, B. Sermage, and D. S. Katzer, "Enhanced radiative recombination of free excitons in GaAs quantum wells," Phys. Rev. Lett. 67, 2355-2358 (1991).
[CrossRef] [PubMed]

1990

J. Martorell and N. M. Lawandy, "Observation of inhibited spontaneous emission in a periodic dielectric structure," Phys. Rev. Lett. 65, 1877-1880 (1990).
[CrossRef] [PubMed]

1988

E. Hanamura, "Rapid radiative decay and enhanced optical nonlinearity of excitons in a quantum well," Phys. Rev. B 38, 1228-1234 (1988).
[CrossRef]

1987

J. Feldmann, G. Peter, E. O. Göbel, P. Dawson, K. Moore, C. Foxon, and R. J. Elliott, "Linewidth dependence of radiative exciton lifetimes in quantum wells," Phys. Rev. Lett. 59, 2337-2340 (1987).
[CrossRef] [PubMed]

F. D. Martini, G. Innocenti, G. R. Jacobovitz, and P. Mataloni, "Spontaneous emission probabilities at radio frequencies," Phys. Rev. Lett. 59, 2955-2958 (1987).
[CrossRef] [PubMed]

E. Yablanovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059-2062 (1987).
[CrossRef]

1966

V. M. Agranovich and O. A. Dubowskii, "Effect of retarded interaction of exciton spectrum in 1-dimensional and 2-dimensional crystals," JETP Lett. 3, 223-226 (1966).

1960

T. H. Maiman, "Stimulated optical radiation in ruby," Nature 187, 493-494 (1960).
[CrossRef]

1946

E. M. Purcell, "Spontaneous emission probabilities at radio frequencies," Phys. Rev. 69, 681 (1946).
[CrossRef]

Appl. Phys. Lett.

E. F. Schubert, Y. H. Wang, A. Y. Cho, L. W. Tu, and G. J. Zydyik, "Resonant cavity light-emitting diode," Appl. Phys. Lett. 60, 921-924 (1992).
[CrossRef]

J. Mod. Opt.

A. G. Kofman, G. Kurizki, and B. Sherman, "Spontaneous and induced atomic decay in photonic band structures," J. Mod. Opt. 41, 353-384 (1994).
[CrossRef]

J. Opt. Soc. Am.

W. Hoyer, M. Kira, S. W. Koch, J. Hader, and J. V. Moloney, "Coulomb effects on quantum-well luminescence spectra and radiative recombination times," J. Opt. Soc. Am. 24, 1344-1353 (2007).
[CrossRef]

JETP Lett.

V. M. Agranovich and O. A. Dubowskii, "Effect of retarded interaction of exciton spectrum in 1-dimensional and 2-dimensional crystals," JETP Lett. 3, 223-226 (1966).

Nat. Phys.

G. Khitrova, H. M. Gibbs, M. Kira, S. W. Koch, and A. Scherer, "Vacuum Rabi splitting in semiconductors," Nat. Phys. 2, 81-90 (2006)
[CrossRef]

Nature

T. H. Maiman, "Stimulated optical radiation in ruby," Nature 187, 493-494 (1960).
[CrossRef]

Phys. Rev.

E. M. Purcell, "Spontaneous emission probabilities at radio frequencies," Phys. Rev. 69, 681 (1946).
[CrossRef]

Phys. Rev. A

M. D. Tocci, M. Scalora, M. J. Bloemer, J. P. Dowling, and C. M. Bowden, "Measurement of spontaneous-emission enhancement near the one-dimensional photonic band edge of semiconductor heterostructures," Phys. Rev. A 53, 2799-2803 (1996).
[CrossRef] [PubMed]

J. P. Dowling and C. M. Bowden, "Atomic emission rates in inhomogeneous media with applications to photonic band structures," Phys. Rev. A 46, 612-622 (1992).
[CrossRef] [PubMed]

Phys. Rev. B

M. Schafer, M. Werchner, W. Hoyer, M. Kira, and S. W. Koch, "Quantum theory of luminescence in multiple-quantum-well Bragg structures," Phys. Rev. B 74, 155315 (2006).
[CrossRef]

E. Hanamura, "Rapid radiative decay and enhanced optical nonlinearity of excitons in a quantum well," Phys. Rev. B 38, 1228-1234 (1988).
[CrossRef]

Phys. Rev. Lett.

B. Deveaud, F. Clérot, N. Roy, K. Satzke, B. Sermage, and D. S. Katzer, "Enhanced radiative recombination of free excitons in GaAs quantum wells," Phys. Rev. Lett. 67, 2355-2358 (1991).
[CrossRef] [PubMed]

M. Kira, F. Jahnke, and S. W. Koch, "Microscopic theory of excitonic signatures in semiconductor photoluminescence," Phys. Rev. Lett. 81, 3263-3266 (1998).
[CrossRef]

S. Chatterjee, C. Ell, S. Mosor, G. Khitrova, H. M. Gibbs, W. Hoyer, M. Kira, S. W. Koch, J. P. Prineas, and H. Stolz, "Excitonic photoluminescence in semiconductor quantum wells: plasma versus excitons," Phys. Rev. Lett. 92, 067402 (2004).
[CrossRef] [PubMed]

F. D. Martini, G. Innocenti, G. R. Jacobovitz, and P. Mataloni, "Spontaneous emission probabilities at radio frequencies," Phys. Rev. Lett. 59, 2955-2958 (1987).
[CrossRef] [PubMed]

J. Martorell and N. M. Lawandy, "Observation of inhibited spontaneous emission in a periodic dielectric structure," Phys. Rev. Lett. 65, 1877-1880 (1990).
[CrossRef] [PubMed]

J. Feldmann, G. Peter, E. O. Göbel, P. Dawson, K. Moore, C. Foxon, and R. J. Elliott, "Linewidth dependence of radiative exciton lifetimes in quantum wells," Phys. Rev. Lett. 59, 2337-2340 (1987).
[CrossRef] [PubMed]

E. Yablanovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059-2062 (1987).
[CrossRef]

C. Weisbuch, M. Nishioka, A. Ishikawa, and Y. Arakawa, "Observation of the coupled exciton-photon mode splitting in a semiconductor quantum microcavity," Phys. Rev. Lett. 69, 3314-3317 (1992).
[CrossRef] [PubMed]

F. Jahnke, M. Kira, S. W. Koch, G. Khitrova, E. K. Lindmark, T. R. Nelson, Jr., D. V. Wick, J. D. Berger, O. Lyngnes, H. M. Gibbs, and K. Tai, "Excitonic nonlinearities of semiconductor microcavities in the nonperturbative regime," Phys. Rev. Lett. 77, 5257-5260 (1996).
[CrossRef] [PubMed]

Prog. Quantum Electron.

M. Kira, F. Jahnke, W. Hoyer, and S. W. Koch, "Quantum theory of secondary emission in optically excited semiconductor quantum wells," Prog. Quantum Electron. 23, 189-279 (1999).
[CrossRef]

M. Kira and S. W. Koch, "Many-body correlations and exciton effects in semiconductor spectroscopy," Prog. Quantum Electron. 30, 155-296 (2006).
[CrossRef]

Rep. Prog. Phys.

P. Lambropoulos, G. M. Nikolopoulos, T. R. Nielsen, and S. Bay, "Fundamental quantum optics in structured reservoirs," Rep. Prog. Phys. 63, 455-503 (2000).
[CrossRef]

Rev. Mod. Phys.

G. Khitrova, H. M. Gibbs, F. Jahnke, M. Kira, and S. W. Koch, "Nonlinear optics of normal-mode-coupling semiconductor microcavities," Rev. Mod. Phys. 71, 1591-1639 (1999).
[CrossRef]

Solid State Commun.

R. E. Slusher and C. Weisbuch, "Optical microcavities in condensed matter systems," Solid State Commun. 92, 149-157 (1994).
[CrossRef]

Other

G. Hernandez, Fabry-Perot Interferometers (Cambridge U. Press, 1986).

S. L. Chuang, Physics of Optoelectronic Devices (Wiley, 1995).

W. W. Chow and S. W. Koch, Semiconductor Laser Fundamentals, 1st ed. (Springer Verlag, 1999).

E. F. Schubert, Light-Emitting Diodes (Cambride U. Press, 2006).
[CrossRef]

E. Mertzbacher, Quantum Mechanics, 3rd ed. (Wiley, 1998).

M. Born and E. Wolf, Principles of Optics, Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge U. Press, 1999).
[PubMed]

H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors, 4th ed. (World Scientific, 2004).

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Figures (6)

Fig. 1
Fig. 1

Refractive index profile is shown as a gray shaded area. The dashed and solid curves depict the propagating and counterpropagating modes that are resonant to the exciton- 1 s energy. The arrows indicate that the propagating mode consists of the incident plane wave with amplitude 1, which is partially reflected ( R + ) and transmitted ( T + ) .

Fig. 2
Fig. 2

PL spectra of a QW positioned either in the mode minimum (left column) or mode maximum (right column). Frames a and b show the refractive index profile as a shaded area while the (counter-)propagating mode is shown as dashed (solid) curve. The QW position is indicated by the black arrow. Frames c and d show the PL spectra corresponding to the QW positions depicted in a and b for a homogeneous broadening of γ = 0.21 meV and a radiative broadening of γ = 0.084 meV . The shaded area shows the result of the full SLE calculation. The dashed curve is the product of the filter function and the free-space PL. The solid black curve is gained if one multiplies the filter function and the free-space PL with adjusted homogeneous dephasing constant. Frames e and f show the same comparison when the homogeneous broadening is γ = 2.1 meV and therefore much larger than the radiative broadening.

Fig. 3
Fig. 3

PL spectra for the structure depicted in Fig. 4a. Frame a shows the PL perpendicular to the QW plane while frame b shows the spectra at an angle of 80 ° . The shaded areas indicate the full SLE PL while the dashed curves indicate the free-space PL perpendicular to the QW plane without the radiative broadening. The solid curves can be obtained by multiplying the filter functions with the free-space PL. In b, the shaded area and the solid line are multiplied by a factor of four for better visibility of the spectral shape. The three peak maxima are labeled via the letters A, B, and C for the subsequent discussion. The inset in both frames indicate the computed filter functions.

Fig. 4
Fig. 4

Frames a and b show the propagating and counterpropagating modes for the frequencies corresponding to the peak maximum A (B) in Fig. 3b as a dashed (solid) curve. The mode function corresponding to peak C is qualitatively similar to case A. The QW position is indicated via a black arrow. The shaded area shows the refractive index profile. The frequency-dependent reflection and transmission coefficients are indicated as solid and dashed curves in frame c. The filter function for a detector positioned right was already shown as an inset in Fig. 3b and for better comparison is shown in frame d again.

Fig. 5
Fig. 5

Comparison of QW microcavity PL full versus extended filter-function approach. a, Radiative decay constant (squares) evaluated from Eq. (16) and half-width of the cavity mode (triangles) are presented as function of number of Bragg layers with the DBR mirrors. b, The position of PL peak evaluated using the extended filter-function approach (squares) versus the full PL (triangles). c, Comparison of the maximum PL computed by the full SLE (triangles) and the maximum PL using the extended filter-function approach (squares).

Fig. 6
Fig. 6

Cavity with 15 Bragg periods of n = 2.97 and 3.63 to each side. The size of the cavity is 3 λ 0 2 n , where λ 0 is the light-wave length in vacuum corresponding to the 1 s energy and n is the refractive index. The shaded area shows the refractive index profile. The black solid line curve depicts the resonant mode. The QW position at the mode maximum is indicated by the black arrow.

Equations (17)

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[ 2 + n 2 ( z ) ω q 2 c 2 ] U q ( r ) = 0 .
i t Δ B q , q B q , q = ( ω q ω q ) Δ B q , q B q , q + i k F q , q Π q , q k + i k F q , q ( Π q , q k ) ,
i t Π q , q k = ( ϵ ̃ k , q ω q ) Π q , q k ( 1 f k + q e f k h ) k V k k Π q , q k + i F q , q Ω S E ( k , q ) ( 1 f k + q e f k h ) q i F q , q Δ B q , q B q , q + i t Π q , q k scatt ,
Π q , q k Δ B q , q a v , k a c , k + q ,
Ω S E ( k , q ) = f k + q e f k h + l c X q , k , l .
B q , q = F q , q b q , q .
Δ B q , q B q , q = F q , q ( F q , q ) Δ b q , q b q , q ,
Π q , q k = F q , q π q , q k F q , q Δ b q , q a v , k a c , k + q .
i t Δ b q , q b q , q = ( ω q ω q ) Δ b q , q b q , q + i k π q , q k + i k ( π q , q k ) ,
i t π q , q k = ( ϵ ̃ k , q ω q ) π q , q k ( 1 f k + q e f k h ) k V k k π q , q k + i Ω S E ( k , q ) ( 1 f k + q e f k h ) q i F q , q 2 Δ b q , q b q , q + i t π q , q k scatt .
d q L = R q + B + q + q z L q z R T q B q = R q + ( F + q ) b + q + q z L q z R T q ( F q ) b q ,
d q d q L = R q + 2 F + q 2 Δ b + q b + q + q z L q z R T q 2 F q 2 Δ b q b q + q z L q z R [ ( R q + ) T q F + q ( F q ) Δ b + q b q ] + [ R q + ( T q ) F q ( F + q ) Δ b q b + q ] .
d q d q L F q Δ b q b q ,
F q = R q + 2 F + q 2 + q z L q z R T q 2 F q 2 + q z L q z R [ ( R q + ) T q F + q ( F q ) ]
[ + R q + ( T q ) F q ( F + q ) ] .
Γ rad = d c v 2 ϕ 1 s ( 0 ) 2 U q 1 s 2 2 ε 0 q 1 s ,
Γ rad actual = Γ rad free ( n = 1 ) U + q 2 + U q 2 2 .

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