Abstract

We improve the methods used to interpolate the responsivity of unbiased silicon photodetectors in the near-ultraviolet region. This improvement is achieved by the derivation of an interpolation function for the quantum yield of silicon and by consideration of this function in the interpolation of the internal quantum efficiency of photodiodes. The calculated quantum-yield and spectral-responsivity values are compared with measurement results obtained by the study of a silicon trap detector and with values reported by other research groups. The comparisons show agreement with a standard deviation of 0.4% between our measured and modeled values for both the quantum yield and the spectral responsivity within the wavelength region from 260 to 400 nm. The proposed methods thus extend the predictability of the spectral responsivity of silicon photodetectors to the wavelength region from 260 to 950 nm. Furthermore, an explanation is proposed for the change in the spectral responsivity of silicon photodiodes that is due to UV radiation. In our improved quantum efficiency model the spectral change can be accounted for completely by the adjustment of just one parameter, i.e., the collection efficiency near the SiO2/Si interface.

© 2000 Optical Society of America

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References

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  1. T. R. Gentile, J. M. Houston, C. L. Cromer, “Realization of a scale of absolute spectral response using the National Institute of Standards and Technology high-accuracy cryogenic radiometer,” Appl. Opt. 35, 4392–4403 (1996).
    [CrossRef] [PubMed]
  2. F. J. Wilkinson, A. J. D. Farmer, J. Geist, “The near UV quantum yield of silicon,” J. Appl. Phys. 54, 1172–1174 (1983).
    [CrossRef]
  3. A. Bittar, “Extension of a silicon-based detector spectral-responsivity scale into the ultraviolet,” Metrologia 32, 497–500 (1995/96).
    [CrossRef]
  4. N. M. Durant, N. P. Fox, “A physical basis for the extrapolation of silicon photodiode quantum efficiency into the ultraviolet,” Metrologia 30, 345–350 (1993).
    [CrossRef]
  5. J. Geist, C. S. Wang, “New calculations of the quantum yield of silicon in the near ultraviolet,” Phys. Rev. B 27, 4841–4847 (1983).
    [CrossRef]
  6. M. V. Fischetti, S. E. Laux, “Monte Carlo analysis of electron transport in small semiconductor devices including band-structure and space-charge effects,” Phys. Rev. B 38, 9721–9745 (1988).
    [CrossRef]
  7. R. C. Alig, S. Bloom, C. W. Struck, “Scattering by ionization and phonon emission in semiconductors,” Phys. Rev. B 22, 5565–5581 (1980).
    [CrossRef]
  8. R. Goebel, R. Köhler, R. Pello, “Some effects of low-power ultraviolet radiation on silicon photodiodes,” Metrologia 32, 515–518 (1995/96).
    [CrossRef]
  9. R. Korde, J. Geist, “Quantum efficiency stability of silicon photodiodes,” Appl. Opt. 26, 5284–5290 (1987).
    [CrossRef] [PubMed]
  10. M. Born, E. Wolf, Principles of Optics, 3rd ed. (Pergamon, Oxford, 1965), pp. 40, 632–633.
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    [CrossRef]
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    [CrossRef]
  13. G. E. Jellison, “Optical functions of silicon determined by two-channel polarization modulation ellipsometry,” Opt. Mater. 1, 41–47 (1992).
    [CrossRef]
  14. P. Kärhä, P. Toivanen, F. Manoochehri, E. Ikonen, “Development of a detector-based absolute spectral irradiance scale in the 380–900-nm spectral range,” Appl. Opt. 36, 8909–8918 (1997).
    [CrossRef]
  15. T. Kübarsepp, P. Kärhä, E. Ikonen, “Characterization of a polarization-independent transmission trap detector,” Appl. Opt. 36, 2807–2812 (1997).
    [CrossRef] [PubMed]
  16. H. J. Hovel, Semiconductors and Semimetals, Vol. 11 of Solar Cells Series, R. K. Willardson, ed. (Academic Press, New York, 1975), pp. 17–20.
  17. J. Singh, Physics of Semiconductors and Their Heterostructures (McGraw-Hill, New York, 1993), pp. 572–575, 595–599.
  18. N. P. Fox, “Trap detectors and their properties,” Metrologia 28, 197–202 (1991).
    [CrossRef]
  19. J. Geist, D. Chandler-Horowitz, A. M. Robinson, C. R. James, “Numerical modeling of silicon photodiodes for high-accuracy applications. Part I. Simulation programs,” J. Res. Natl. Inst. Stand. Technol. 96, 463–469 (1991).
    [CrossRef]
  20. F. Lei, J. Fischer, “Characterization of photodiodes in the UV and visible spectral region based on cryogenic radiometry,” Metrologia 30, 297–303 (1993).
    [CrossRef]
  21. P. J. Caplan, E. H. Poindexter, R. Morrison, “Ultraviolet bleaching and regeneration of ·Si ≡ Si3 centers at the Si/SiO2 interface of thinly oxidized silicon wafers,” J. Appl. Phys. 53, 541–545 (1982).
    [CrossRef]
  22. K. Vanheusden, W. L. Warren, J. R. Schwank, D. M. Fleetwood, M. R. Shaneyfelt, P. S. Winokur, R. A. B. Devine, “Nonuniform oxide charge and paramagnetic interface traps in high-temperature annealed Si/SiO2/Si structures,” Appl. Phys. Lett. 68, 2117–2119 (1996).
    [CrossRef]
  23. L. Werner, “Ultraviolet stability of silicon photodiodes,” Metrologia 35, 407–411 (1998).
    [CrossRef]

1998 (2)

L. Werner, “Ultraviolet stability of silicon photodiodes,” Metrologia 35, 407–411 (1998).
[CrossRef]

A. Haapalinna, P. Kärhä, E. Ikonen, “Spectral reflectance of silicon photodiodes,” Appl. Opt. 37, 729–732 (1998).
[CrossRef]

1997 (2)

1996 (2)

T. R. Gentile, J. M. Houston, C. L. Cromer, “Realization of a scale of absolute spectral response using the National Institute of Standards and Technology high-accuracy cryogenic radiometer,” Appl. Opt. 35, 4392–4403 (1996).
[CrossRef] [PubMed]

K. Vanheusden, W. L. Warren, J. R. Schwank, D. M. Fleetwood, M. R. Shaneyfelt, P. S. Winokur, R. A. B. Devine, “Nonuniform oxide charge and paramagnetic interface traps in high-temperature annealed Si/SiO2/Si structures,” Appl. Phys. Lett. 68, 2117–2119 (1996).
[CrossRef]

1993 (2)

N. M. Durant, N. P. Fox, “A physical basis for the extrapolation of silicon photodiode quantum efficiency into the ultraviolet,” Metrologia 30, 345–350 (1993).
[CrossRef]

F. Lei, J. Fischer, “Characterization of photodiodes in the UV and visible spectral region based on cryogenic radiometry,” Metrologia 30, 297–303 (1993).
[CrossRef]

1992 (1)

G. E. Jellison, “Optical functions of silicon determined by two-channel polarization modulation ellipsometry,” Opt. Mater. 1, 41–47 (1992).
[CrossRef]

1991 (2)

N. P. Fox, “Trap detectors and their properties,” Metrologia 28, 197–202 (1991).
[CrossRef]

J. Geist, D. Chandler-Horowitz, A. M. Robinson, C. R. James, “Numerical modeling of silicon photodiodes for high-accuracy applications. Part I. Simulation programs,” J. Res. Natl. Inst. Stand. Technol. 96, 463–469 (1991).
[CrossRef]

1988 (1)

M. V. Fischetti, S. E. Laux, “Monte Carlo analysis of electron transport in small semiconductor devices including band-structure and space-charge effects,” Phys. Rev. B 38, 9721–9745 (1988).
[CrossRef]

1987 (1)

1983 (2)

F. J. Wilkinson, A. J. D. Farmer, J. Geist, “The near UV quantum yield of silicon,” J. Appl. Phys. 54, 1172–1174 (1983).
[CrossRef]

J. Geist, C. S. Wang, “New calculations of the quantum yield of silicon in the near ultraviolet,” Phys. Rev. B 27, 4841–4847 (1983).
[CrossRef]

1982 (1)

P. J. Caplan, E. H. Poindexter, R. Morrison, “Ultraviolet bleaching and regeneration of ·Si ≡ Si3 centers at the Si/SiO2 interface of thinly oxidized silicon wafers,” J. Appl. Phys. 53, 541–545 (1982).
[CrossRef]

1980 (1)

R. C. Alig, S. Bloom, C. W. Struck, “Scattering by ionization and phonon emission in semiconductors,” Phys. Rev. B 22, 5565–5581 (1980).
[CrossRef]

1965 (1)

Alig, R. C.

R. C. Alig, S. Bloom, C. W. Struck, “Scattering by ionization and phonon emission in semiconductors,” Phys. Rev. B 22, 5565–5581 (1980).
[CrossRef]

Bittar, A.

A. Bittar, “Extension of a silicon-based detector spectral-responsivity scale into the ultraviolet,” Metrologia 32, 497–500 (1995/96).
[CrossRef]

Bloom, S.

R. C. Alig, S. Bloom, C. W. Struck, “Scattering by ionization and phonon emission in semiconductors,” Phys. Rev. B 22, 5565–5581 (1980).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 3rd ed. (Pergamon, Oxford, 1965), pp. 40, 632–633.

Caplan, P. J.

P. J. Caplan, E. H. Poindexter, R. Morrison, “Ultraviolet bleaching and regeneration of ·Si ≡ Si3 centers at the Si/SiO2 interface of thinly oxidized silicon wafers,” J. Appl. Phys. 53, 541–545 (1982).
[CrossRef]

Chandler-Horowitz, D.

J. Geist, D. Chandler-Horowitz, A. M. Robinson, C. R. James, “Numerical modeling of silicon photodiodes for high-accuracy applications. Part I. Simulation programs,” J. Res. Natl. Inst. Stand. Technol. 96, 463–469 (1991).
[CrossRef]

Cromer, C. L.

Devine, R. A. B.

K. Vanheusden, W. L. Warren, J. R. Schwank, D. M. Fleetwood, M. R. Shaneyfelt, P. S. Winokur, R. A. B. Devine, “Nonuniform oxide charge and paramagnetic interface traps in high-temperature annealed Si/SiO2/Si structures,” Appl. Phys. Lett. 68, 2117–2119 (1996).
[CrossRef]

Durant, N. M.

N. M. Durant, N. P. Fox, “A physical basis for the extrapolation of silicon photodiode quantum efficiency into the ultraviolet,” Metrologia 30, 345–350 (1993).
[CrossRef]

Farmer, A. J. D.

F. J. Wilkinson, A. J. D. Farmer, J. Geist, “The near UV quantum yield of silicon,” J. Appl. Phys. 54, 1172–1174 (1983).
[CrossRef]

Fischer, J.

F. Lei, J. Fischer, “Characterization of photodiodes in the UV and visible spectral region based on cryogenic radiometry,” Metrologia 30, 297–303 (1993).
[CrossRef]

Fischetti, M. V.

M. V. Fischetti, S. E. Laux, “Monte Carlo analysis of electron transport in small semiconductor devices including band-structure and space-charge effects,” Phys. Rev. B 38, 9721–9745 (1988).
[CrossRef]

Fleetwood, D. M.

K. Vanheusden, W. L. Warren, J. R. Schwank, D. M. Fleetwood, M. R. Shaneyfelt, P. S. Winokur, R. A. B. Devine, “Nonuniform oxide charge and paramagnetic interface traps in high-temperature annealed Si/SiO2/Si structures,” Appl. Phys. Lett. 68, 2117–2119 (1996).
[CrossRef]

Fox, N. P.

N. M. Durant, N. P. Fox, “A physical basis for the extrapolation of silicon photodiode quantum efficiency into the ultraviolet,” Metrologia 30, 345–350 (1993).
[CrossRef]

N. P. Fox, “Trap detectors and their properties,” Metrologia 28, 197–202 (1991).
[CrossRef]

Geist, J.

J. Geist, D. Chandler-Horowitz, A. M. Robinson, C. R. James, “Numerical modeling of silicon photodiodes for high-accuracy applications. Part I. Simulation programs,” J. Res. Natl. Inst. Stand. Technol. 96, 463–469 (1991).
[CrossRef]

R. Korde, J. Geist, “Quantum efficiency stability of silicon photodiodes,” Appl. Opt. 26, 5284–5290 (1987).
[CrossRef] [PubMed]

J. Geist, C. S. Wang, “New calculations of the quantum yield of silicon in the near ultraviolet,” Phys. Rev. B 27, 4841–4847 (1983).
[CrossRef]

F. J. Wilkinson, A. J. D. Farmer, J. Geist, “The near UV quantum yield of silicon,” J. Appl. Phys. 54, 1172–1174 (1983).
[CrossRef]

Gentile, T. R.

Goebel, R.

R. Goebel, R. Köhler, R. Pello, “Some effects of low-power ultraviolet radiation on silicon photodiodes,” Metrologia 32, 515–518 (1995/96).
[CrossRef]

Haapalinna, A.

Houston, J. M.

Hovel, H. J.

H. J. Hovel, Semiconductors and Semimetals, Vol. 11 of Solar Cells Series, R. K. Willardson, ed. (Academic Press, New York, 1975), pp. 17–20.

Ikonen, E.

James, C. R.

J. Geist, D. Chandler-Horowitz, A. M. Robinson, C. R. James, “Numerical modeling of silicon photodiodes for high-accuracy applications. Part I. Simulation programs,” J. Res. Natl. Inst. Stand. Technol. 96, 463–469 (1991).
[CrossRef]

Jellison, G. E.

G. E. Jellison, “Optical functions of silicon determined by two-channel polarization modulation ellipsometry,” Opt. Mater. 1, 41–47 (1992).
[CrossRef]

Kärhä, P.

Köhler, R.

R. Goebel, R. Köhler, R. Pello, “Some effects of low-power ultraviolet radiation on silicon photodiodes,” Metrologia 32, 515–518 (1995/96).
[CrossRef]

Korde, R.

Kübarsepp, T.

Laux, S. E.

M. V. Fischetti, S. E. Laux, “Monte Carlo analysis of electron transport in small semiconductor devices including band-structure and space-charge effects,” Phys. Rev. B 38, 9721–9745 (1988).
[CrossRef]

Lei, F.

F. Lei, J. Fischer, “Characterization of photodiodes in the UV and visible spectral region based on cryogenic radiometry,” Metrologia 30, 297–303 (1993).
[CrossRef]

Malitson, I. H.

Manoochehri, F.

Morrison, R.

P. J. Caplan, E. H. Poindexter, R. Morrison, “Ultraviolet bleaching and regeneration of ·Si ≡ Si3 centers at the Si/SiO2 interface of thinly oxidized silicon wafers,” J. Appl. Phys. 53, 541–545 (1982).
[CrossRef]

Pello, R.

R. Goebel, R. Köhler, R. Pello, “Some effects of low-power ultraviolet radiation on silicon photodiodes,” Metrologia 32, 515–518 (1995/96).
[CrossRef]

Poindexter, E. H.

P. J. Caplan, E. H. Poindexter, R. Morrison, “Ultraviolet bleaching and regeneration of ·Si ≡ Si3 centers at the Si/SiO2 interface of thinly oxidized silicon wafers,” J. Appl. Phys. 53, 541–545 (1982).
[CrossRef]

Robinson, A. M.

J. Geist, D. Chandler-Horowitz, A. M. Robinson, C. R. James, “Numerical modeling of silicon photodiodes for high-accuracy applications. Part I. Simulation programs,” J. Res. Natl. Inst. Stand. Technol. 96, 463–469 (1991).
[CrossRef]

Schwank, J. R.

K. Vanheusden, W. L. Warren, J. R. Schwank, D. M. Fleetwood, M. R. Shaneyfelt, P. S. Winokur, R. A. B. Devine, “Nonuniform oxide charge and paramagnetic interface traps in high-temperature annealed Si/SiO2/Si structures,” Appl. Phys. Lett. 68, 2117–2119 (1996).
[CrossRef]

Shaneyfelt, M. R.

K. Vanheusden, W. L. Warren, J. R. Schwank, D. M. Fleetwood, M. R. Shaneyfelt, P. S. Winokur, R. A. B. Devine, “Nonuniform oxide charge and paramagnetic interface traps in high-temperature annealed Si/SiO2/Si structures,” Appl. Phys. Lett. 68, 2117–2119 (1996).
[CrossRef]

Singh, J.

J. Singh, Physics of Semiconductors and Their Heterostructures (McGraw-Hill, New York, 1993), pp. 572–575, 595–599.

Struck, C. W.

R. C. Alig, S. Bloom, C. W. Struck, “Scattering by ionization and phonon emission in semiconductors,” Phys. Rev. B 22, 5565–5581 (1980).
[CrossRef]

Toivanen, P.

Vanheusden, K.

K. Vanheusden, W. L. Warren, J. R. Schwank, D. M. Fleetwood, M. R. Shaneyfelt, P. S. Winokur, R. A. B. Devine, “Nonuniform oxide charge and paramagnetic interface traps in high-temperature annealed Si/SiO2/Si structures,” Appl. Phys. Lett. 68, 2117–2119 (1996).
[CrossRef]

Wang, C. S.

J. Geist, C. S. Wang, “New calculations of the quantum yield of silicon in the near ultraviolet,” Phys. Rev. B 27, 4841–4847 (1983).
[CrossRef]

Warren, W. L.

K. Vanheusden, W. L. Warren, J. R. Schwank, D. M. Fleetwood, M. R. Shaneyfelt, P. S. Winokur, R. A. B. Devine, “Nonuniform oxide charge and paramagnetic interface traps in high-temperature annealed Si/SiO2/Si structures,” Appl. Phys. Lett. 68, 2117–2119 (1996).
[CrossRef]

Werner, L.

L. Werner, “Ultraviolet stability of silicon photodiodes,” Metrologia 35, 407–411 (1998).
[CrossRef]

Wilkinson, F. J.

F. J. Wilkinson, A. J. D. Farmer, J. Geist, “The near UV quantum yield of silicon,” J. Appl. Phys. 54, 1172–1174 (1983).
[CrossRef]

Winokur, P. S.

K. Vanheusden, W. L. Warren, J. R. Schwank, D. M. Fleetwood, M. R. Shaneyfelt, P. S. Winokur, R. A. B. Devine, “Nonuniform oxide charge and paramagnetic interface traps in high-temperature annealed Si/SiO2/Si structures,” Appl. Phys. Lett. 68, 2117–2119 (1996).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 3rd ed. (Pergamon, Oxford, 1965), pp. 40, 632–633.

Appl. Opt. (5)

Appl. Phys. Lett. (1)

K. Vanheusden, W. L. Warren, J. R. Schwank, D. M. Fleetwood, M. R. Shaneyfelt, P. S. Winokur, R. A. B. Devine, “Nonuniform oxide charge and paramagnetic interface traps in high-temperature annealed Si/SiO2/Si structures,” Appl. Phys. Lett. 68, 2117–2119 (1996).
[CrossRef]

J. Appl. Phys. (2)

P. J. Caplan, E. H. Poindexter, R. Morrison, “Ultraviolet bleaching and regeneration of ·Si ≡ Si3 centers at the Si/SiO2 interface of thinly oxidized silicon wafers,” J. Appl. Phys. 53, 541–545 (1982).
[CrossRef]

F. J. Wilkinson, A. J. D. Farmer, J. Geist, “The near UV quantum yield of silicon,” J. Appl. Phys. 54, 1172–1174 (1983).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Res. Natl. Inst. Stand. Technol. (1)

J. Geist, D. Chandler-Horowitz, A. M. Robinson, C. R. James, “Numerical modeling of silicon photodiodes for high-accuracy applications. Part I. Simulation programs,” J. Res. Natl. Inst. Stand. Technol. 96, 463–469 (1991).
[CrossRef]

Metrologia (6)

F. Lei, J. Fischer, “Characterization of photodiodes in the UV and visible spectral region based on cryogenic radiometry,” Metrologia 30, 297–303 (1993).
[CrossRef]

R. Goebel, R. Köhler, R. Pello, “Some effects of low-power ultraviolet radiation on silicon photodiodes,” Metrologia 32, 515–518 (1995/96).
[CrossRef]

A. Bittar, “Extension of a silicon-based detector spectral-responsivity scale into the ultraviolet,” Metrologia 32, 497–500 (1995/96).
[CrossRef]

N. M. Durant, N. P. Fox, “A physical basis for the extrapolation of silicon photodiode quantum efficiency into the ultraviolet,” Metrologia 30, 345–350 (1993).
[CrossRef]

N. P. Fox, “Trap detectors and their properties,” Metrologia 28, 197–202 (1991).
[CrossRef]

L. Werner, “Ultraviolet stability of silicon photodiodes,” Metrologia 35, 407–411 (1998).
[CrossRef]

Opt. Mater. (1)

G. E. Jellison, “Optical functions of silicon determined by two-channel polarization modulation ellipsometry,” Opt. Mater. 1, 41–47 (1992).
[CrossRef]

Phys. Rev. B (3)

J. Geist, C. S. Wang, “New calculations of the quantum yield of silicon in the near ultraviolet,” Phys. Rev. B 27, 4841–4847 (1983).
[CrossRef]

M. V. Fischetti, S. E. Laux, “Monte Carlo analysis of electron transport in small semiconductor devices including band-structure and space-charge effects,” Phys. Rev. B 38, 9721–9745 (1988).
[CrossRef]

R. C. Alig, S. Bloom, C. W. Struck, “Scattering by ionization and phonon emission in semiconductors,” Phys. Rev. B 22, 5565–5581 (1980).
[CrossRef]

Other (3)

H. J. Hovel, Semiconductors and Semimetals, Vol. 11 of Solar Cells Series, R. K. Willardson, ed. (Academic Press, New York, 1975), pp. 17–20.

J. Singh, Physics of Semiconductors and Their Heterostructures (McGraw-Hill, New York, 1993), pp. 572–575, 595–599.

M. Born, E. Wolf, Principles of Optics, 3rd ed. (Pergamon, Oxford, 1965), pp. 40, 632–633.

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Figures (4)

Fig. 1
Fig. 1

Schematic diagram of the silicon band structure depicting a simplified energy budget of direct transitions in silicon (modified from Ref. 6). All energy bands are not shown. Possible transitions after the absorption of a photon with an energy of h ν > E g d: Direct transition of an electron (dashed line). Impact-ionization scattering of the created electron [at Γ (boldface solid curve)]. Generation of a secondary electron [at X (boldface solid curve)]. Electron–phonon interaction of the primary electron (dotted line).

Fig. 2
Fig. 2

Comparison of the measured quantum-yield values: Results from this study are represented by the filled circles, those of Wilkinson et al.2 by the open squares, and those of Durant and Fox4 by the crosses. In this study the interpolated quantum yield (solid curve) was obtained with the parameters listed in Table 1.

Fig. 3
Fig. 3

Comparison between the calculated and the measured spectral-responsivity values obtained with a silicon trap detector: (a) the measured (filled circles) and the calculated (solid curve) responsivities and (b) the relative difference (crosses) between the modeled and the measured responsivity values. The average of the differences is zero (solid line). In this study the modeled responsivity was obtained with the parameters listed in Table 2.

Fig. 4
Fig. 4

Comparison of the change in the spectral responsivity of a silicon photodiode as measured by Goebel et al.8 (open squares) and the modeled changes in the responsivity (solid curve). The relative change reported in Ref. 8 was obtained with a low radiation dose at a wavelength of 248 nm. We obtained the modeled change by increasing the value of the collection efficiency P f by 0.6%.

Tables (2)

Tables Icon

Table 1 Parameters Used to Fit the Interpolation Function of the Quantum Yield [Eq. (10)] to the Measured Values of the Quantum Yield

Tables Icon

Table 2 Parameters Used to Fit the Interpolation Function of the Internal Quantum Efficiency [Eq. (2)] to the Measured Values of the Spectral Responsivity

Equations (11)

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

Rλ=1-ρλiλλK,
iλ=Pf+1-PfαλT1-exp-αλT-1-PrαλD-T×exp-αλT-exp-αλD-Pr exp-αλd,
iλ=ηλiλ,
ηhν=1+ Phν, ENEdE,
Ekin=hν-Egin,
Ekd=hν-Egd,
Pehν, Ekin  αhν-1Ba3ρEkin,
NEkin  1+105A2πhν-Egin-ω01/2hν-2Egin7/2-1,
Ek,mind=0, Ek,maxd=hν-Egd,  hνEgd.
ηhν=1+a3Bαhν ρhν-Egin1+105A2π×hν-Egin-ω01/2hν-2Egin7/2-1hν-Egd.
hν=hν-ΔE,

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