Abstract

A quasi-dimensional analysis is presented for the N-photon interband absorption coefficients (KN) of direct-gap semiconductors. Regardless of the value of N, we show that

KN(e2c)NN-1P2N-3nNEg4N-5FN(NωEg),

where FN varies little over the relevant range of radiation frequencies. The dominant material dependence of KN is through the energy-gap factor; the N-photon coefficient decreases as Eg4N−5.

© 1984 Optical Society of America

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  1. A. Miller, A. Johnston, J. Dempsey, J. Smith, C. R. Pidgeon, and G. D. Holah, "Two-photon absorption in InSb and Hg1−xCdxTe," J. Phys. C 12, 4839–4849 (1978).
    [CrossRef]
  2. E. W. Van Stryland, M. A. Woodall, W. E. Williams, and M. H. Soileau, "Two- and three-photon aborption in semiconductors with subsequent absorption by photogenerated carriers," in Proceedings of 1981 Symposium on Laser Induced Damage in Optical Materials, Boulder, Nov. 1981, NBS Spec. Pub. (to be published).
  3. Compare C. R. Pidgeon, B. S. Wherrett, A. M. Johnston, J. Dempsey, and A. Miller, "Two-photon absorption in zinc-blende semiconductors," Phys. Rev. Lett. 42, 1785–1788 (1979), and references therein.
    [CrossRef]
  4. R. Braunstein and N. Ockman, "Optical double-photon absorption in CdS," Phys. Rev. A 134, 499–507 (1964).
  5. S. S. Mitra, N. H. K. Judell, A. Vaidyanathan, and A. H. Guenther, "Three-photon absorption in direct-gap crystals," Opt. Lett. 7, 307–309 (1982).
    [CrossRef] [PubMed]
  6. J. H. Yee, "Three-photon absorption in semiconductors," Phys. Rev. B 5, 449–458 (1972).
    [CrossRef]
  7. E. O. Kane, "Band structure of indium antimonide," J. Phys. Chem. Solids 1, 249–261 (1957).
    [CrossRef]
  8. M. Weiler, "Nonparabolicity and exciton effects in two-photon absorption in zincblende semiconductors," Solid State Commun. 39, 937–940 (1981).
    [CrossRef]
  9. C. C. Lee and H. Y. Fan, "Two-photon absorption with exciton effect for degenerate levels," Phys. Rev. B 9, 3502–3516 (1974).
    [CrossRef]
  10. M. G. Basov, A. Z. Grasiuk, I. G. Zubarev, V. A. Katulin, and O. N. Krokhin, "Semiconductor quantum generation with two-photon optical excitation," Sov. Phys. JETP 23, 366–371 (1966).
  11. H. J. Fossum and D. B. Chang, "Two-photon excitation rate in indium antimonide," Phys. Rev. B 8, 2842–2848 (1973).
    [CrossRef]
  12. A. Vaidynanthan, T. Walker, A. H. Guenther, S. S. Mitra, and L. M. Narducci, "Two-photon absorption in several direct-gap crystals," Phys. Rev. B 21, 743–748 (1980).
    [CrossRef]
  13. Compare C. R. Pidgeon, B. S. Wherrett, A. M. Johnston, J. Dempsey, and A. Miller, "Two-photon absorption in zinc-blende semiconductors—erratum," Phys. Rev. Lett. 43, 1843 (1979).
    [CrossRef]
  14. A. Vaidyanathan, A. H. Guenther, and S. S. Mitra, "Two-photon absorption in direct-gap crystals—an addendum," Phys. Rev. B 22, 6480–6483 (1980).
    [CrossRef]
  15. J. H. Bechtel and W. L. Smith, "Two-photon absorption in semiconductors with picosecond laser pulses," Phys. Rev. B 13, 3515–3522 (1976).
    [CrossRef]
  16. S. S. Mitra, L. M. Narducci, R. A. Shatas, Y. F. Tsay, and A. Vaidyanathan, "Nonlinear absorption in direct-gap semiconductors," Appl. Opt. 14, 3038–3042 (1975).
    [CrossRef] [PubMed]
  17. I. M. Catalano and A. Gingolani, "Non-parabolic band effect on two-photon absorption in ZnSe and CdTe," Solid State Commun. 43, 213–215 (1982).
    [CrossRef]

1982 (2)

I. M. Catalano and A. Gingolani, "Non-parabolic band effect on two-photon absorption in ZnSe and CdTe," Solid State Commun. 43, 213–215 (1982).
[CrossRef]

S. S. Mitra, N. H. K. Judell, A. Vaidyanathan, and A. H. Guenther, "Three-photon absorption in direct-gap crystals," Opt. Lett. 7, 307–309 (1982).
[CrossRef] [PubMed]

1981 (1)

M. Weiler, "Nonparabolicity and exciton effects in two-photon absorption in zincblende semiconductors," Solid State Commun. 39, 937–940 (1981).
[CrossRef]

1980 (2)

A. Vaidynanthan, T. Walker, A. H. Guenther, S. S. Mitra, and L. M. Narducci, "Two-photon absorption in several direct-gap crystals," Phys. Rev. B 21, 743–748 (1980).
[CrossRef]

A. Vaidyanathan, A. H. Guenther, and S. S. Mitra, "Two-photon absorption in direct-gap crystals—an addendum," Phys. Rev. B 22, 6480–6483 (1980).
[CrossRef]

1979 (2)

Compare C. R. Pidgeon, B. S. Wherrett, A. M. Johnston, J. Dempsey, and A. Miller, "Two-photon absorption in zinc-blende semiconductors—erratum," Phys. Rev. Lett. 43, 1843 (1979).
[CrossRef]

Compare C. R. Pidgeon, B. S. Wherrett, A. M. Johnston, J. Dempsey, and A. Miller, "Two-photon absorption in zinc-blende semiconductors," Phys. Rev. Lett. 42, 1785–1788 (1979), and references therein.
[CrossRef]

1978 (1)

A. Miller, A. Johnston, J. Dempsey, J. Smith, C. R. Pidgeon, and G. D. Holah, "Two-photon absorption in InSb and Hg1−xCdxTe," J. Phys. C 12, 4839–4849 (1978).
[CrossRef]

1976 (1)

J. H. Bechtel and W. L. Smith, "Two-photon absorption in semiconductors with picosecond laser pulses," Phys. Rev. B 13, 3515–3522 (1976).
[CrossRef]

1975 (1)

1974 (1)

C. C. Lee and H. Y. Fan, "Two-photon absorption with exciton effect for degenerate levels," Phys. Rev. B 9, 3502–3516 (1974).
[CrossRef]

1973 (1)

H. J. Fossum and D. B. Chang, "Two-photon excitation rate in indium antimonide," Phys. Rev. B 8, 2842–2848 (1973).
[CrossRef]

1972 (1)

J. H. Yee, "Three-photon absorption in semiconductors," Phys. Rev. B 5, 449–458 (1972).
[CrossRef]

1966 (1)

M. G. Basov, A. Z. Grasiuk, I. G. Zubarev, V. A. Katulin, and O. N. Krokhin, "Semiconductor quantum generation with two-photon optical excitation," Sov. Phys. JETP 23, 366–371 (1966).

1964 (1)

R. Braunstein and N. Ockman, "Optical double-photon absorption in CdS," Phys. Rev. A 134, 499–507 (1964).

1957 (1)

E. O. Kane, "Band structure of indium antimonide," J. Phys. Chem. Solids 1, 249–261 (1957).
[CrossRef]

Basov, M. G.

M. G. Basov, A. Z. Grasiuk, I. G. Zubarev, V. A. Katulin, and O. N. Krokhin, "Semiconductor quantum generation with two-photon optical excitation," Sov. Phys. JETP 23, 366–371 (1966).

Bechtel, J. H.

J. H. Bechtel and W. L. Smith, "Two-photon absorption in semiconductors with picosecond laser pulses," Phys. Rev. B 13, 3515–3522 (1976).
[CrossRef]

Braunstein, R.

R. Braunstein and N. Ockman, "Optical double-photon absorption in CdS," Phys. Rev. A 134, 499–507 (1964).

Catalano, I. M.

I. M. Catalano and A. Gingolani, "Non-parabolic band effect on two-photon absorption in ZnSe and CdTe," Solid State Commun. 43, 213–215 (1982).
[CrossRef]

Chang, D. B.

H. J. Fossum and D. B. Chang, "Two-photon excitation rate in indium antimonide," Phys. Rev. B 8, 2842–2848 (1973).
[CrossRef]

Dempsey, J.

Compare C. R. Pidgeon, B. S. Wherrett, A. M. Johnston, J. Dempsey, and A. Miller, "Two-photon absorption in zinc-blende semiconductors," Phys. Rev. Lett. 42, 1785–1788 (1979), and references therein.
[CrossRef]

Compare C. R. Pidgeon, B. S. Wherrett, A. M. Johnston, J. Dempsey, and A. Miller, "Two-photon absorption in zinc-blende semiconductors—erratum," Phys. Rev. Lett. 43, 1843 (1979).
[CrossRef]

A. Miller, A. Johnston, J. Dempsey, J. Smith, C. R. Pidgeon, and G. D. Holah, "Two-photon absorption in InSb and Hg1−xCdxTe," J. Phys. C 12, 4839–4849 (1978).
[CrossRef]

Fan, H. Y.

C. C. Lee and H. Y. Fan, "Two-photon absorption with exciton effect for degenerate levels," Phys. Rev. B 9, 3502–3516 (1974).
[CrossRef]

Fossum, H. J.

H. J. Fossum and D. B. Chang, "Two-photon excitation rate in indium antimonide," Phys. Rev. B 8, 2842–2848 (1973).
[CrossRef]

Gingolani, A.

I. M. Catalano and A. Gingolani, "Non-parabolic band effect on two-photon absorption in ZnSe and CdTe," Solid State Commun. 43, 213–215 (1982).
[CrossRef]

Grasiuk, A. Z.

M. G. Basov, A. Z. Grasiuk, I. G. Zubarev, V. A. Katulin, and O. N. Krokhin, "Semiconductor quantum generation with two-photon optical excitation," Sov. Phys. JETP 23, 366–371 (1966).

Guenther, A. H.

S. S. Mitra, N. H. K. Judell, A. Vaidyanathan, and A. H. Guenther, "Three-photon absorption in direct-gap crystals," Opt. Lett. 7, 307–309 (1982).
[CrossRef] [PubMed]

A. Vaidynanthan, T. Walker, A. H. Guenther, S. S. Mitra, and L. M. Narducci, "Two-photon absorption in several direct-gap crystals," Phys. Rev. B 21, 743–748 (1980).
[CrossRef]

A. Vaidyanathan, A. H. Guenther, and S. S. Mitra, "Two-photon absorption in direct-gap crystals—an addendum," Phys. Rev. B 22, 6480–6483 (1980).
[CrossRef]

Holah, G. D.

A. Miller, A. Johnston, J. Dempsey, J. Smith, C. R. Pidgeon, and G. D. Holah, "Two-photon absorption in InSb and Hg1−xCdxTe," J. Phys. C 12, 4839–4849 (1978).
[CrossRef]

Johnston, A.

A. Miller, A. Johnston, J. Dempsey, J. Smith, C. R. Pidgeon, and G. D. Holah, "Two-photon absorption in InSb and Hg1−xCdxTe," J. Phys. C 12, 4839–4849 (1978).
[CrossRef]

Johnston, A. M.

Compare C. R. Pidgeon, B. S. Wherrett, A. M. Johnston, J. Dempsey, and A. Miller, "Two-photon absorption in zinc-blende semiconductors," Phys. Rev. Lett. 42, 1785–1788 (1979), and references therein.
[CrossRef]

Compare C. R. Pidgeon, B. S. Wherrett, A. M. Johnston, J. Dempsey, and A. Miller, "Two-photon absorption in zinc-blende semiconductors—erratum," Phys. Rev. Lett. 43, 1843 (1979).
[CrossRef]

Judell, N. H. K.

Kane, E. O.

E. O. Kane, "Band structure of indium antimonide," J. Phys. Chem. Solids 1, 249–261 (1957).
[CrossRef]

Katulin, V. A.

M. G. Basov, A. Z. Grasiuk, I. G. Zubarev, V. A. Katulin, and O. N. Krokhin, "Semiconductor quantum generation with two-photon optical excitation," Sov. Phys. JETP 23, 366–371 (1966).

Krokhin, O. N.

M. G. Basov, A. Z. Grasiuk, I. G. Zubarev, V. A. Katulin, and O. N. Krokhin, "Semiconductor quantum generation with two-photon optical excitation," Sov. Phys. JETP 23, 366–371 (1966).

Lee, C. C.

C. C. Lee and H. Y. Fan, "Two-photon absorption with exciton effect for degenerate levels," Phys. Rev. B 9, 3502–3516 (1974).
[CrossRef]

Miller, A.

Compare C. R. Pidgeon, B. S. Wherrett, A. M. Johnston, J. Dempsey, and A. Miller, "Two-photon absorption in zinc-blende semiconductors—erratum," Phys. Rev. Lett. 43, 1843 (1979).
[CrossRef]

Compare C. R. Pidgeon, B. S. Wherrett, A. M. Johnston, J. Dempsey, and A. Miller, "Two-photon absorption in zinc-blende semiconductors," Phys. Rev. Lett. 42, 1785–1788 (1979), and references therein.
[CrossRef]

A. Miller, A. Johnston, J. Dempsey, J. Smith, C. R. Pidgeon, and G. D. Holah, "Two-photon absorption in InSb and Hg1−xCdxTe," J. Phys. C 12, 4839–4849 (1978).
[CrossRef]

Mitra, S. S.

S. S. Mitra, N. H. K. Judell, A. Vaidyanathan, and A. H. Guenther, "Three-photon absorption in direct-gap crystals," Opt. Lett. 7, 307–309 (1982).
[CrossRef] [PubMed]

A. Vaidynanthan, T. Walker, A. H. Guenther, S. S. Mitra, and L. M. Narducci, "Two-photon absorption in several direct-gap crystals," Phys. Rev. B 21, 743–748 (1980).
[CrossRef]

A. Vaidyanathan, A. H. Guenther, and S. S. Mitra, "Two-photon absorption in direct-gap crystals—an addendum," Phys. Rev. B 22, 6480–6483 (1980).
[CrossRef]

S. S. Mitra, L. M. Narducci, R. A. Shatas, Y. F. Tsay, and A. Vaidyanathan, "Nonlinear absorption in direct-gap semiconductors," Appl. Opt. 14, 3038–3042 (1975).
[CrossRef] [PubMed]

Narducci, L. M.

A. Vaidynanthan, T. Walker, A. H. Guenther, S. S. Mitra, and L. M. Narducci, "Two-photon absorption in several direct-gap crystals," Phys. Rev. B 21, 743–748 (1980).
[CrossRef]

S. S. Mitra, L. M. Narducci, R. A. Shatas, Y. F. Tsay, and A. Vaidyanathan, "Nonlinear absorption in direct-gap semiconductors," Appl. Opt. 14, 3038–3042 (1975).
[CrossRef] [PubMed]

Ockman, N.

R. Braunstein and N. Ockman, "Optical double-photon absorption in CdS," Phys. Rev. A 134, 499–507 (1964).

Pidgeon, C. R.

Compare C. R. Pidgeon, B. S. Wherrett, A. M. Johnston, J. Dempsey, and A. Miller, "Two-photon absorption in zinc-blende semiconductors—erratum," Phys. Rev. Lett. 43, 1843 (1979).
[CrossRef]

Compare C. R. Pidgeon, B. S. Wherrett, A. M. Johnston, J. Dempsey, and A. Miller, "Two-photon absorption in zinc-blende semiconductors," Phys. Rev. Lett. 42, 1785–1788 (1979), and references therein.
[CrossRef]

A. Miller, A. Johnston, J. Dempsey, J. Smith, C. R. Pidgeon, and G. D. Holah, "Two-photon absorption in InSb and Hg1−xCdxTe," J. Phys. C 12, 4839–4849 (1978).
[CrossRef]

Shatas, R. A.

Smith, J.

A. Miller, A. Johnston, J. Dempsey, J. Smith, C. R. Pidgeon, and G. D. Holah, "Two-photon absorption in InSb and Hg1−xCdxTe," J. Phys. C 12, 4839–4849 (1978).
[CrossRef]

Smith, W. L.

J. H. Bechtel and W. L. Smith, "Two-photon absorption in semiconductors with picosecond laser pulses," Phys. Rev. B 13, 3515–3522 (1976).
[CrossRef]

Soileau, M. H.

E. W. Van Stryland, M. A. Woodall, W. E. Williams, and M. H. Soileau, "Two- and three-photon aborption in semiconductors with subsequent absorption by photogenerated carriers," in Proceedings of 1981 Symposium on Laser Induced Damage in Optical Materials, Boulder, Nov. 1981, NBS Spec. Pub. (to be published).

Stryland, E. W. Van

E. W. Van Stryland, M. A. Woodall, W. E. Williams, and M. H. Soileau, "Two- and three-photon aborption in semiconductors with subsequent absorption by photogenerated carriers," in Proceedings of 1981 Symposium on Laser Induced Damage in Optical Materials, Boulder, Nov. 1981, NBS Spec. Pub. (to be published).

Tsay, Y. F.

Vaidyanathan, A.

Vaidynanthan, A.

A. Vaidynanthan, T. Walker, A. H. Guenther, S. S. Mitra, and L. M. Narducci, "Two-photon absorption in several direct-gap crystals," Phys. Rev. B 21, 743–748 (1980).
[CrossRef]

Walker, T.

A. Vaidynanthan, T. Walker, A. H. Guenther, S. S. Mitra, and L. M. Narducci, "Two-photon absorption in several direct-gap crystals," Phys. Rev. B 21, 743–748 (1980).
[CrossRef]

Weiler, M.

M. Weiler, "Nonparabolicity and exciton effects in two-photon absorption in zincblende semiconductors," Solid State Commun. 39, 937–940 (1981).
[CrossRef]

Wherrett, B. S.

Compare C. R. Pidgeon, B. S. Wherrett, A. M. Johnston, J. Dempsey, and A. Miller, "Two-photon absorption in zinc-blende semiconductors—erratum," Phys. Rev. Lett. 43, 1843 (1979).
[CrossRef]

Compare C. R. Pidgeon, B. S. Wherrett, A. M. Johnston, J. Dempsey, and A. Miller, "Two-photon absorption in zinc-blende semiconductors," Phys. Rev. Lett. 42, 1785–1788 (1979), and references therein.
[CrossRef]

Williams, W. E.

E. W. Van Stryland, M. A. Woodall, W. E. Williams, and M. H. Soileau, "Two- and three-photon aborption in semiconductors with subsequent absorption by photogenerated carriers," in Proceedings of 1981 Symposium on Laser Induced Damage in Optical Materials, Boulder, Nov. 1981, NBS Spec. Pub. (to be published).

Woodall, M. A.

E. W. Van Stryland, M. A. Woodall, W. E. Williams, and M. H. Soileau, "Two- and three-photon aborption in semiconductors with subsequent absorption by photogenerated carriers," in Proceedings of 1981 Symposium on Laser Induced Damage in Optical Materials, Boulder, Nov. 1981, NBS Spec. Pub. (to be published).

Yee, J. H.

J. H. Yee, "Three-photon absorption in semiconductors," Phys. Rev. B 5, 449–458 (1972).
[CrossRef]

Zubarev, I. G.

M. G. Basov, A. Z. Grasiuk, I. G. Zubarev, V. A. Katulin, and O. N. Krokhin, "Semiconductor quantum generation with two-photon optical excitation," Sov. Phys. JETP 23, 366–371 (1966).

Appl. Opt. (1)

J. Phys. C (1)

A. Miller, A. Johnston, J. Dempsey, J. Smith, C. R. Pidgeon, and G. D. Holah, "Two-photon absorption in InSb and Hg1−xCdxTe," J. Phys. C 12, 4839–4849 (1978).
[CrossRef]

J. Phys. Chem. Solids (1)

E. O. Kane, "Band structure of indium antimonide," J. Phys. Chem. Solids 1, 249–261 (1957).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (1)

R. Braunstein and N. Ockman, "Optical double-photon absorption in CdS," Phys. Rev. A 134, 499–507 (1964).

Phys. Rev. B (6)

J. H. Yee, "Three-photon absorption in semiconductors," Phys. Rev. B 5, 449–458 (1972).
[CrossRef]

C. C. Lee and H. Y. Fan, "Two-photon absorption with exciton effect for degenerate levels," Phys. Rev. B 9, 3502–3516 (1974).
[CrossRef]

H. J. Fossum and D. B. Chang, "Two-photon excitation rate in indium antimonide," Phys. Rev. B 8, 2842–2848 (1973).
[CrossRef]

A. Vaidynanthan, T. Walker, A. H. Guenther, S. S. Mitra, and L. M. Narducci, "Two-photon absorption in several direct-gap crystals," Phys. Rev. B 21, 743–748 (1980).
[CrossRef]

A. Vaidyanathan, A. H. Guenther, and S. S. Mitra, "Two-photon absorption in direct-gap crystals—an addendum," Phys. Rev. B 22, 6480–6483 (1980).
[CrossRef]

J. H. Bechtel and W. L. Smith, "Two-photon absorption in semiconductors with picosecond laser pulses," Phys. Rev. B 13, 3515–3522 (1976).
[CrossRef]

Solid State Commun. (2)

M. Weiler, "Nonparabolicity and exciton effects in two-photon absorption in zincblende semiconductors," Solid State Commun. 39, 937–940 (1981).
[CrossRef]

I. M. Catalano and A. Gingolani, "Non-parabolic band effect on two-photon absorption in ZnSe and CdTe," Solid State Commun. 43, 213–215 (1982).
[CrossRef]

Sov. Phys. JETP (1)

M. G. Basov, A. Z. Grasiuk, I. G. Zubarev, V. A. Katulin, and O. N. Krokhin, "Semiconductor quantum generation with two-photon optical excitation," Sov. Phys. JETP 23, 366–371 (1966).

Other (3)

Compare C. R. Pidgeon, B. S. Wherrett, A. M. Johnston, J. Dempsey, and A. Miller, "Two-photon absorption in zinc-blende semiconductors—erratum," Phys. Rev. Lett. 43, 1843 (1979).
[CrossRef]

E. W. Van Stryland, M. A. Woodall, W. E. Williams, and M. H. Soileau, "Two- and three-photon aborption in semiconductors with subsequent absorption by photogenerated carriers," in Proceedings of 1981 Symposium on Laser Induced Damage in Optical Materials, Boulder, Nov. 1981, NBS Spec. Pub. (to be published).

Compare C. R. Pidgeon, B. S. Wherrett, A. M. Johnston, J. Dempsey, and A. Miller, "Two-photon absorption in zinc-blende semiconductors," Phys. Rev. Lett. 42, 1785–1788 (1979), and references therein.
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Schematic of the simplest N-step transitions leading to a net interband excitation (a) for three-photon absorption and (b) for two-photon absorption.

Fig. 2
Fig. 2

The function FN(Nħω/Eg) plotted over the relevant frequency ranges for N = 2 to N = 7.

Fig. 3
Fig. 3

Schematic of transition schemes leading to modification of the essential forms of KN (see Section 3).

Tables (3)

Tables Icon

Table 1 Examples of Estimated K2 and K3 values from the Present Analysis Compared with Previous Experimental and Theoretical Quotes

Tables Icon

Table 2 Material Parameters for N-Photon Absorption Calculations and Values for Two-Photon Photon and Three-Photon Coefficients Normalized to Those of InSb at Low Temperaturea

Tables Icon

Table 3 KN Values for InSb for Use with Table 2 and Fig. 2a

Equations (35)

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K N ( e 2 c ) N N - 1 P 2 N - 3 n N E g 4 N - 5 F N ( N ω E g ) ,
K N = W N N ω I - N .
W N = 2 π - 1 M c v ( N ) ( k ) 2 δ ( E c v ( k ) - N ω ) .
M c v ( 1 ) = H c v ,
M c v ( 2 ) = a H c a H a v [ E a v ( k ) - ω ] - 1 ,
M c v ( 3 ) = a , b H c b H b a H a v [ E b v ( k ) - 2 ω ] - 1 [ E a v ( k ) - ω ] - 1 .
H f i = e i m ω ( 2 π I n c ) 1 / 2 · p f i .
W N = 1 π [ ( 2 m c v 2 ) 3 / 2 ( N ω - E g ) 1 / 2 ] × [ ( e m ω ) 2 N ( 2 π I n c ) N S c v ( N ) ( k r ) 2 ] .
k r 2 = 2 m c v ( N ω - E g ) - 2 .
S c v ( 1 ) = p c v ,
S c v ( 2 ) = a p c a p a v [ E a v ( k r ) - ω ] - 1 ,
S c v ( 3 ) = a , b p c b p b a p a v [ E b v ( k r ) - 2 ω ] - 1 [ E a v ( k r ) - ω ] - 1 .
p i i = m k / m i .
m m c = - m m v = 2 p c v 2 m E g .
p c v 2 N = f 2 N ( m P / ) 2 N .
K N = [ N 2 N 2 ( 2 π ) N - 1 ( e 2 c ) N f - 3 ] 3 N - 1 m 2 N [ 1 n N P 3 E g 2 N - 3 ] × ( N ω / E g - 1 ) 1 / 2 ( N ω / E g ) 2 N - 1 S c v ( N ) ( k r ) 2 .
S c v ( 3 ) = p c v p v c p c v / { ( - 2 ω ) ( 2 ω ) } ,
S c v ( N ) = p c v p c v N - 1 × { ( - 1 ) ( N - 1 ) / 2 2 N - 1 [ ( N - 1 2 ) ! ] 2 ( ω ) N - 1 } - 1 .
K N = C N F N ( N ω / E g ) N - 1 P 2 N - 3 n N E g 4 N - 5 .
C N = ( e 2 c ) N N 4 N - 2 π N - 1 2 - N 2 [ ( N - 1 2 ) ! ] - 4 f 2 N f - 3 ,
F N ( N ω / E g ) = ( N ω / E g - 1 ) 1 / 2 / ( N ω / E g ) 4 N - 3 .
S c v ( 2 ) = [ p c v p v v / ( - ω ) ] + [ p c c p c v / ( ω ) ] ,
S c v ( 2 ) = p c v m k ˙ r / [ m c v ω ] .
S c v ( N ) N 2 ( N - 1 ) ! p c v p c v N - 2 m k r / { m c v ( ω ) N - 1 } .
K N = C N F N ( N ω / E g ) N - 1 P 2 N - 3 n N E g 4 N - 5 .
C N ( e 2 c ) N N 4 N π N - 1 2 N - 1 2 [ ( N - 1 ) ! ] 2 f 2 N - 2 f - 1 ,
F N = ( N ω / E g - 1 ) 3 / 2 / ( N ω / E g ) 4 N - 3 .
H ( band ) = k · ( p / m ) ,
H ( interaction ) = [ - i ( 2 π I n e 2 c ) 1 / 2 E g ω ] 3 / 2 E g · ( p m ) .
χ ( N ) = e N + 1 P N - 2 E g 2 N - 1 F N ( ω E g ; E g E g ; P P ) .
K N = 2 ω ( 2 π / n c ) N Im [ χ ( 2 N - 1 ) ] ,
K N N - 1 P 2 N - 3 n N E g 4 N - 5 ( P c c 2 P c v 2 P 4 × E g 2 E g 2 ) ( N ω / E g - 1 ) 1 / 2 ( N ω / E g ) 4 N - 5 .
K 2 = 2 9 2 π ( e 2 c ) 2 f 2 f P n 2 E g 3 ( 2 ω / E g - 1 ) 3 / 2 ( 2 ω / E g ) .
K 3 = 3 10 2 8 π 2 ( e 2 c ) 3 2 P 3 n 3 E g 7 ( 3 ω / E g - 1 ) 1 / 2 ( 3 ω / E g ) 9 .
I c = ( n E g 4 / P 2 ) ( c / e 2 ) .

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