Abstract

A rigorous theory of optical rectification in zinc-blende semiconductors is developed. This theory combines the bonding-orbitals representation of the electrons in the semiconductor with the band-structure representation. It is shown that when the semiconductor is excited above the absorption edge there is a strong resonant enhancement of the optical rectification signal and of the terahertz-radiation emission related to it. Both the magnitude and the temporal characteristics of this signal are closely related to the intraband relaxation processes in the valence band.

© 1994 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. Bass, P. A. Franken, J. F. Ward, and G. Weinreich, Phys. Rev. Lett. 9, 446 (1962).
    [CrossRef]
  2. J. F. Ward, Phys. Rev. 143, 569 (1966).
    [CrossRef]
  3. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), pp. 57–60.
  4. S. L. Chuang, S. Schmitt-Rink, B. J. Greene, P. N. Saeta, and A. F. J. Levi, Phys. Rev. Lett. 68, 102 (1992).
    [CrossRef] [PubMed]
  5. X.-C. Zhang, Y. Jin, K. Yang, and L.-J. Scholwalter, Phys. Rev. Lett. 69, 2303 (1992).
    [CrossRef] [PubMed]
  6. T. D. Hewitt, Y. Jin, W. Ellis, and X.-C. Zhang, in Conference on Lasers and Electro-Optics, Vol. 11 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C.1993), paper CWJ61.
  7. P. C. M. Planken, M. C. Nuss, I. Brener, K. W. Goosen, M. S. C. Luo, S. L. Chuang, and L. Pfeifer, Phys. Rev. Lett. 69, 3800 (1992).
    [CrossRef] [PubMed]
  8. X.-C. Zhang, B. B. Hu, J. T. Darrow, and D. H. Auston, Appl. Phys. Lett. 56, 1011 (1990).
    [CrossRef]
  9. B. B. Hu, X.-C. Zhang, and D. H. Austin, Phys. Rev. Lett. 67, 2709 (1991).
    [CrossRef] [PubMed]
  10. M. Planken, M. C. Nuss, W. H. Knox, D. A. B. Miller, and K. W. Goosen, Appl. Phys. Lett. 61, 2009 (1992).
    [CrossRef]
  11. P. N. Saeta, B. I. Greene, and S. L. Chuang, Appl. Phys. Lett. 63, 3482 (1993).
    [CrossRef]
  12. B. F. Levine, Phys. Rev. B 7, 2591, 2600 (1973).
    [CrossRef]
  13. C. L. Tang and C. Flytzanis, Phys. Rev. B 4, 2520 (1971).
    [CrossRef]
  14. C. Y. Fond and Y. R. Shen, Phys. Rev. B 12, 2325 (1975).
    [CrossRef]
  15. E. O. Kane, J. Phys. Chem. Solids 1, 249 (1957).
    [CrossRef]
  16. M. Lindberg and S. W. Koch, Phys. Rev. B 38, 3342 (1988).
    [CrossRef]
  17. C. Flytzanis and J. Ducuing, Phys. Rev. 178, 1218 (1969).
    [CrossRef]
  18. M. Hass and B. W. Henvis, J. Phys. Chem. Solids 23, 1099 (1962).
    [CrossRef]
  19. J. C. Phillips, Bonds and Bands in Semiconductors (Academic, New York, 1973).
  20. A. Yariv, Optical Electronics (Saunders, Philadelphia, Pa., 1991), p. 500.
  21. J. M. Ziman, Principles of the Theory of Solids (Cambridge U. Press, London, 1979), p. 222.
  22. J. Singh, Physics of Semiconductors and Their Heterostructures (McGraw-Hill, New York, 1993), Chaps. 10–12, and references therein.
  23. B. K. Ridley, Quantum Processes in Semiconductors (Clarendon, Oxford, 1982).

1993 (1)

P. N. Saeta, B. I. Greene, and S. L. Chuang, Appl. Phys. Lett. 63, 3482 (1993).
[CrossRef]

1992 (4)

S. L. Chuang, S. Schmitt-Rink, B. J. Greene, P. N. Saeta, and A. F. J. Levi, Phys. Rev. Lett. 68, 102 (1992).
[CrossRef] [PubMed]

X.-C. Zhang, Y. Jin, K. Yang, and L.-J. Scholwalter, Phys. Rev. Lett. 69, 2303 (1992).
[CrossRef] [PubMed]

P. C. M. Planken, M. C. Nuss, I. Brener, K. W. Goosen, M. S. C. Luo, S. L. Chuang, and L. Pfeifer, Phys. Rev. Lett. 69, 3800 (1992).
[CrossRef] [PubMed]

M. Planken, M. C. Nuss, W. H. Knox, D. A. B. Miller, and K. W. Goosen, Appl. Phys. Lett. 61, 2009 (1992).
[CrossRef]

1991 (1)

B. B. Hu, X.-C. Zhang, and D. H. Austin, Phys. Rev. Lett. 67, 2709 (1991).
[CrossRef] [PubMed]

1990 (1)

X.-C. Zhang, B. B. Hu, J. T. Darrow, and D. H. Auston, Appl. Phys. Lett. 56, 1011 (1990).
[CrossRef]

1988 (1)

M. Lindberg and S. W. Koch, Phys. Rev. B 38, 3342 (1988).
[CrossRef]

1975 (1)

C. Y. Fond and Y. R. Shen, Phys. Rev. B 12, 2325 (1975).
[CrossRef]

1973 (1)

B. F. Levine, Phys. Rev. B 7, 2591, 2600 (1973).
[CrossRef]

1971 (1)

C. L. Tang and C. Flytzanis, Phys. Rev. B 4, 2520 (1971).
[CrossRef]

1969 (1)

C. Flytzanis and J. Ducuing, Phys. Rev. 178, 1218 (1969).
[CrossRef]

1966 (1)

J. F. Ward, Phys. Rev. 143, 569 (1966).
[CrossRef]

1962 (2)

M. Bass, P. A. Franken, J. F. Ward, and G. Weinreich, Phys. Rev. Lett. 9, 446 (1962).
[CrossRef]

M. Hass and B. W. Henvis, J. Phys. Chem. Solids 23, 1099 (1962).
[CrossRef]

1957 (1)

E. O. Kane, J. Phys. Chem. Solids 1, 249 (1957).
[CrossRef]

Austin, D. H.

B. B. Hu, X.-C. Zhang, and D. H. Austin, Phys. Rev. Lett. 67, 2709 (1991).
[CrossRef] [PubMed]

Auston, D. H.

X.-C. Zhang, B. B. Hu, J. T. Darrow, and D. H. Auston, Appl. Phys. Lett. 56, 1011 (1990).
[CrossRef]

Bass, M.

M. Bass, P. A. Franken, J. F. Ward, and G. Weinreich, Phys. Rev. Lett. 9, 446 (1962).
[CrossRef]

Brener, I.

P. C. M. Planken, M. C. Nuss, I. Brener, K. W. Goosen, M. S. C. Luo, S. L. Chuang, and L. Pfeifer, Phys. Rev. Lett. 69, 3800 (1992).
[CrossRef] [PubMed]

Chuang, S. L.

P. N. Saeta, B. I. Greene, and S. L. Chuang, Appl. Phys. Lett. 63, 3482 (1993).
[CrossRef]

P. C. M. Planken, M. C. Nuss, I. Brener, K. W. Goosen, M. S. C. Luo, S. L. Chuang, and L. Pfeifer, Phys. Rev. Lett. 69, 3800 (1992).
[CrossRef] [PubMed]

S. L. Chuang, S. Schmitt-Rink, B. J. Greene, P. N. Saeta, and A. F. J. Levi, Phys. Rev. Lett. 68, 102 (1992).
[CrossRef] [PubMed]

Darrow, J. T.

X.-C. Zhang, B. B. Hu, J. T. Darrow, and D. H. Auston, Appl. Phys. Lett. 56, 1011 (1990).
[CrossRef]

Ducuing, J.

C. Flytzanis and J. Ducuing, Phys. Rev. 178, 1218 (1969).
[CrossRef]

Ellis, W.

T. D. Hewitt, Y. Jin, W. Ellis, and X.-C. Zhang, in Conference on Lasers and Electro-Optics, Vol. 11 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C.1993), paper CWJ61.

Flytzanis, C.

C. L. Tang and C. Flytzanis, Phys. Rev. B 4, 2520 (1971).
[CrossRef]

C. Flytzanis and J. Ducuing, Phys. Rev. 178, 1218 (1969).
[CrossRef]

Fond, C. Y.

C. Y. Fond and Y. R. Shen, Phys. Rev. B 12, 2325 (1975).
[CrossRef]

Franken, P. A.

M. Bass, P. A. Franken, J. F. Ward, and G. Weinreich, Phys. Rev. Lett. 9, 446 (1962).
[CrossRef]

Goosen, K. W.

P. C. M. Planken, M. C. Nuss, I. Brener, K. W. Goosen, M. S. C. Luo, S. L. Chuang, and L. Pfeifer, Phys. Rev. Lett. 69, 3800 (1992).
[CrossRef] [PubMed]

M. Planken, M. C. Nuss, W. H. Knox, D. A. B. Miller, and K. W. Goosen, Appl. Phys. Lett. 61, 2009 (1992).
[CrossRef]

Greene, B. I.

P. N. Saeta, B. I. Greene, and S. L. Chuang, Appl. Phys. Lett. 63, 3482 (1993).
[CrossRef]

Greene, B. J.

S. L. Chuang, S. Schmitt-Rink, B. J. Greene, P. N. Saeta, and A. F. J. Levi, Phys. Rev. Lett. 68, 102 (1992).
[CrossRef] [PubMed]

Hass, M.

M. Hass and B. W. Henvis, J. Phys. Chem. Solids 23, 1099 (1962).
[CrossRef]

Henvis, B. W.

M. Hass and B. W. Henvis, J. Phys. Chem. Solids 23, 1099 (1962).
[CrossRef]

Hewitt, T. D.

T. D. Hewitt, Y. Jin, W. Ellis, and X.-C. Zhang, in Conference on Lasers and Electro-Optics, Vol. 11 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C.1993), paper CWJ61.

Hu, B. B.

B. B. Hu, X.-C. Zhang, and D. H. Austin, Phys. Rev. Lett. 67, 2709 (1991).
[CrossRef] [PubMed]

X.-C. Zhang, B. B. Hu, J. T. Darrow, and D. H. Auston, Appl. Phys. Lett. 56, 1011 (1990).
[CrossRef]

Jin, Y.

X.-C. Zhang, Y. Jin, K. Yang, and L.-J. Scholwalter, Phys. Rev. Lett. 69, 2303 (1992).
[CrossRef] [PubMed]

T. D. Hewitt, Y. Jin, W. Ellis, and X.-C. Zhang, in Conference on Lasers and Electro-Optics, Vol. 11 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C.1993), paper CWJ61.

Kane, E. O.

E. O. Kane, J. Phys. Chem. Solids 1, 249 (1957).
[CrossRef]

Knox, W. H.

M. Planken, M. C. Nuss, W. H. Knox, D. A. B. Miller, and K. W. Goosen, Appl. Phys. Lett. 61, 2009 (1992).
[CrossRef]

Koch, S. W.

M. Lindberg and S. W. Koch, Phys. Rev. B 38, 3342 (1988).
[CrossRef]

Levi, A. F. J.

S. L. Chuang, S. Schmitt-Rink, B. J. Greene, P. N. Saeta, and A. F. J. Levi, Phys. Rev. Lett. 68, 102 (1992).
[CrossRef] [PubMed]

Levine, B. F.

B. F. Levine, Phys. Rev. B 7, 2591, 2600 (1973).
[CrossRef]

Lindberg, M.

M. Lindberg and S. W. Koch, Phys. Rev. B 38, 3342 (1988).
[CrossRef]

Luo, M. S. C.

P. C. M. Planken, M. C. Nuss, I. Brener, K. W. Goosen, M. S. C. Luo, S. L. Chuang, and L. Pfeifer, Phys. Rev. Lett. 69, 3800 (1992).
[CrossRef] [PubMed]

Miller, D. A. B.

M. Planken, M. C. Nuss, W. H. Knox, D. A. B. Miller, and K. W. Goosen, Appl. Phys. Lett. 61, 2009 (1992).
[CrossRef]

Nuss, M. C.

P. C. M. Planken, M. C. Nuss, I. Brener, K. W. Goosen, M. S. C. Luo, S. L. Chuang, and L. Pfeifer, Phys. Rev. Lett. 69, 3800 (1992).
[CrossRef] [PubMed]

M. Planken, M. C. Nuss, W. H. Knox, D. A. B. Miller, and K. W. Goosen, Appl. Phys. Lett. 61, 2009 (1992).
[CrossRef]

Pfeifer, L.

P. C. M. Planken, M. C. Nuss, I. Brener, K. W. Goosen, M. S. C. Luo, S. L. Chuang, and L. Pfeifer, Phys. Rev. Lett. 69, 3800 (1992).
[CrossRef] [PubMed]

Phillips, J. C.

J. C. Phillips, Bonds and Bands in Semiconductors (Academic, New York, 1973).

Planken, M.

M. Planken, M. C. Nuss, W. H. Knox, D. A. B. Miller, and K. W. Goosen, Appl. Phys. Lett. 61, 2009 (1992).
[CrossRef]

Planken, P. C. M.

P. C. M. Planken, M. C. Nuss, I. Brener, K. W. Goosen, M. S. C. Luo, S. L. Chuang, and L. Pfeifer, Phys. Rev. Lett. 69, 3800 (1992).
[CrossRef] [PubMed]

Ridley, B. K.

B. K. Ridley, Quantum Processes in Semiconductors (Clarendon, Oxford, 1982).

Saeta, P. N.

P. N. Saeta, B. I. Greene, and S. L. Chuang, Appl. Phys. Lett. 63, 3482 (1993).
[CrossRef]

S. L. Chuang, S. Schmitt-Rink, B. J. Greene, P. N. Saeta, and A. F. J. Levi, Phys. Rev. Lett. 68, 102 (1992).
[CrossRef] [PubMed]

Schmitt-Rink, S.

S. L. Chuang, S. Schmitt-Rink, B. J. Greene, P. N. Saeta, and A. F. J. Levi, Phys. Rev. Lett. 68, 102 (1992).
[CrossRef] [PubMed]

Scholwalter, L.-J.

X.-C. Zhang, Y. Jin, K. Yang, and L.-J. Scholwalter, Phys. Rev. Lett. 69, 2303 (1992).
[CrossRef] [PubMed]

Shen, Y. R.

C. Y. Fond and Y. R. Shen, Phys. Rev. B 12, 2325 (1975).
[CrossRef]

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), pp. 57–60.

Singh, J.

J. Singh, Physics of Semiconductors and Their Heterostructures (McGraw-Hill, New York, 1993), Chaps. 10–12, and references therein.

Tang, C. L.

C. L. Tang and C. Flytzanis, Phys. Rev. B 4, 2520 (1971).
[CrossRef]

Ward, J. F.

J. F. Ward, Phys. Rev. 143, 569 (1966).
[CrossRef]

M. Bass, P. A. Franken, J. F. Ward, and G. Weinreich, Phys. Rev. Lett. 9, 446 (1962).
[CrossRef]

Weinreich, G.

M. Bass, P. A. Franken, J. F. Ward, and G. Weinreich, Phys. Rev. Lett. 9, 446 (1962).
[CrossRef]

Yang, K.

X.-C. Zhang, Y. Jin, K. Yang, and L.-J. Scholwalter, Phys. Rev. Lett. 69, 2303 (1992).
[CrossRef] [PubMed]

Yariv, A.

A. Yariv, Optical Electronics (Saunders, Philadelphia, Pa., 1991), p. 500.

Zhang, X.-C.

X.-C. Zhang, Y. Jin, K. Yang, and L.-J. Scholwalter, Phys. Rev. Lett. 69, 2303 (1992).
[CrossRef] [PubMed]

B. B. Hu, X.-C. Zhang, and D. H. Austin, Phys. Rev. Lett. 67, 2709 (1991).
[CrossRef] [PubMed]

X.-C. Zhang, B. B. Hu, J. T. Darrow, and D. H. Auston, Appl. Phys. Lett. 56, 1011 (1990).
[CrossRef]

T. D. Hewitt, Y. Jin, W. Ellis, and X.-C. Zhang, in Conference on Lasers and Electro-Optics, Vol. 11 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C.1993), paper CWJ61.

Ziman, J. M.

J. M. Ziman, Principles of the Theory of Solids (Cambridge U. Press, London, 1979), p. 222.

Appl. Phys. Lett. (3)

X.-C. Zhang, B. B. Hu, J. T. Darrow, and D. H. Auston, Appl. Phys. Lett. 56, 1011 (1990).
[CrossRef]

M. Planken, M. C. Nuss, W. H. Knox, D. A. B. Miller, and K. W. Goosen, Appl. Phys. Lett. 61, 2009 (1992).
[CrossRef]

P. N. Saeta, B. I. Greene, and S. L. Chuang, Appl. Phys. Lett. 63, 3482 (1993).
[CrossRef]

J. Phys. Chem. Solids (2)

M. Hass and B. W. Henvis, J. Phys. Chem. Solids 23, 1099 (1962).
[CrossRef]

E. O. Kane, J. Phys. Chem. Solids 1, 249 (1957).
[CrossRef]

Phys. Rev. (2)

C. Flytzanis and J. Ducuing, Phys. Rev. 178, 1218 (1969).
[CrossRef]

J. F. Ward, Phys. Rev. 143, 569 (1966).
[CrossRef]

Phys. Rev. B (4)

B. F. Levine, Phys. Rev. B 7, 2591, 2600 (1973).
[CrossRef]

C. L. Tang and C. Flytzanis, Phys. Rev. B 4, 2520 (1971).
[CrossRef]

C. Y. Fond and Y. R. Shen, Phys. Rev. B 12, 2325 (1975).
[CrossRef]

M. Lindberg and S. W. Koch, Phys. Rev. B 38, 3342 (1988).
[CrossRef]

Phys. Rev. Lett. (5)

S. L. Chuang, S. Schmitt-Rink, B. J. Greene, P. N. Saeta, and A. F. J. Levi, Phys. Rev. Lett. 68, 102 (1992).
[CrossRef] [PubMed]

X.-C. Zhang, Y. Jin, K. Yang, and L.-J. Scholwalter, Phys. Rev. Lett. 69, 2303 (1992).
[CrossRef] [PubMed]

B. B. Hu, X.-C. Zhang, and D. H. Austin, Phys. Rev. Lett. 67, 2709 (1991).
[CrossRef] [PubMed]

M. Bass, P. A. Franken, J. F. Ward, and G. Weinreich, Phys. Rev. Lett. 9, 446 (1962).
[CrossRef]

P. C. M. Planken, M. C. Nuss, I. Brener, K. W. Goosen, M. S. C. Luo, S. L. Chuang, and L. Pfeifer, Phys. Rev. Lett. 69, 3800 (1992).
[CrossRef] [PubMed]

Other (7)

T. D. Hewitt, Y. Jin, W. Ellis, and X.-C. Zhang, in Conference on Lasers and Electro-Optics, Vol. 11 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C.1993), paper CWJ61.

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), pp. 57–60.

J. C. Phillips, Bonds and Bands in Semiconductors (Academic, New York, 1973).

A. Yariv, Optical Electronics (Saunders, Philadelphia, Pa., 1991), p. 500.

J. M. Ziman, Principles of the Theory of Solids (Cambridge U. Press, London, 1979), p. 222.

J. Singh, Physics of Semiconductors and Their Heterostructures (McGraw-Hill, New York, 1993), Chaps. 10–12, and references therein.

B. K. Ridley, Quantum Processes in Semiconductors (Clarendon, Oxford, 1982).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

States in the VB and the CB involved in this discussion.

Fig. 2
Fig. 2

Feynman diagram of the OR components: (a) fast component, (b) slow component.

Fig. 3
Fig. 3

Bonding orbitals in the VB of the zinc-blende semiconductor.

Fig. 4
Fig. 4

Magnitude of the PDM of the heavy-hole state as a function of the direction of the wave vector k.

Fig. 5
Fig. 5

Magnitude of the projection of the PDM of the heavy-hole state onto the axis [100] as a function of the direction of the wave vector k.

Fig. 6
Fig. 6

Dispersion of the OR coefficient near the absorption edge of GaAs: (a) fast component, (b) slow component, (c) combined.

Equations (95)

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

χ i j k ( 2 ) ( 0 = ω - ω ) = χ j k i ( 2 ) ( ω = ω + 0 ) = χ k i j ( 2 ) ( ω = 0 + ω ) - n 2 r i j k ( ω ) ,
E = E 1 cos ( ω 1 t ) + E 2 cos ( ω 2 t ) .
W ( t ) = 1 2 η [ E 1 2 + E 2 2 + E 1 2 cos 2 ω 1 t + E 2 2 cos 2 ω 2 t + 2 E 1 E 2 cos ( ω 1 - ω 2 ) t + 2 E 1 E 2 cos ( ω 1 + ω 2 ) t ] ,
Ψ c ( k ) = 1 V 1 / 2 u c , k exp ( i k · r ) ,
Ψ v ( k ) = 1 V 1 / 2 u v , k exp ( i k · r ) ,
ρ k = [ ρ v v , k ρ v c , k ρ c v , k ρ c c , k ] .
ρ k = i [ ρ , H ] ,
H ( k ) = H 0 + μ ( k ) · e ^ E ( t )
H 0 = [ E v , k 0 0 E c , k ] .
μ ( k ) = [ μ v v , k μ v c , k μ c v , k μ c c , k ] .
μ c c , k = - e u c , k r u c , k ,
μ v v , k = - e u v , k r u v , k
ω 0 , k = 1 ( E c , k - E v , k ) ,
μ ¯ k = ½ ( μ v v , k + μ c c , k ) ,
Δ μ k = μ v v , k - μ c c , k ,
Δ ρ k = ρ v v , k - ρ c c , k .
Δ ρ ˙ k = 2 i E ( t ) e ^ · ( ρ c v , k μ v c , k - ρ c v , k * μ v c , k * ) - Δ ρ k - Δ ρ ¯ k T 1 , s - Δ ρ k - 1 T 1 , r ,
ρ ˙ c v , k = - i [ ω 0 , k + - 1 Δ μ k E ( t ) ] × ρ c v , k + i μ v c , k · e ^ E ( t ) - ρ c v , k T 2 ,
Δ ρ ¯ k = ρ ¯ v v ( k ) - ρ ¯ c c ( k ) = f ( E v , k ) - f ( E c , k ) .
f c ( v ) [ E c ( v ) ] = { e [ E c ( v ) - F c ( v ) k T ] + 1 } - 1 ,
n = 0 f c ( E ) g c ( E ) d E = 1 4 π 3 0 ρ c c , k d 3 k
p = n = - 0 [ 1 - f v ( E ) ] g v ( E ) d E = 1 4 π 3 0 ( 1 - ρ v v , k ) d 3 k
n = 1 8 π 3 0 ( 1 - Δ ρ k ) d 3 k .
n ˙ = i 4 π 3 0 e ^ E ( t ) · ( ρ c v , k μ v c , k - ρ v c , k * μ v c , k * ) d 3 k - n T 1 , r .
μ v c , k = T r ( ρ μ v c , k ) = μ v c , k ( ρ c v , k + ρ c v , k * ) + μ ¯ k + ½ Δ μ k Δ ρ k ,
P = 1 V k μ v c , k = 1 V k ( μ v c , k ρ c v , k + μ v c , k * ρ c v , k * ) + 1 2 V k Δ μ k Δ ρ k + P ¯ ,
P ¯ = 1 V k μ ¯ k ,
ρ c v , k = σ c v , k exp ( - i ω t ) + σ v c , k exp ( i ω t ) + σ 0 , k .
σ c v , k = ½ - 1 μ v c , k * · e ^ E 0 ω 0 , k - ω - i / T 2 ,
σ v c , k = ½ - 1 μ v c , k * · e ^ E 0 ω 0 , k + ω - i / T 2 ,
σ 0 , k = - ½ - 1 Δ μ k · e ^ E 0 ( σ c v , k + σ v c , k ) ω 0 , k + i / T 2 ,
Δ ρ k - Δ ρ ¯ k = i - 1 e ^ E 0 · T 1 , s × [ ( σ c v , k μ v c , k - σ c v , k * μ v c , k * ) + ( σ c v , k μ v c , k - σ c v , k * μ v c , k * ) ] ,
P dc = 1 V k [ ( μ v c , k σ 0 , k + μ v c , k * ) + 1 2 Δ μ k Δ ρ k ] = 1 2 V k [ - μ v c , k - 2 Δ μ k · e ^ E 0 ω 0 , k - 1 μ v c , k * · e ^ E 0 × Re ( 1 ω 0 , k - ω - i / T 2 + 1 ω 0 , k + ω - i / T 2 ) - Δ μ k - 2 μ v c , k · e ^ E 0 T 1 , s μ v c , k * · e ^ E 0 × Im ( 1 ω 0 , k - ω - i / T 2 + 1 ω 0 , k + ω - i / T 2 ) ] + 1 2 V k Δ ρ ¯ k Δ μ k .
P dc , V = - 1 2 V ( E 0 ) 2 k μ v c , k ( Δ μ k · e ^ ) ( μ v c , k * · e ^ ) ω 0 , k - 1 × [ ω 0 , k - ω ( ω 0 , k - ω ) 2 + T 2 - 2 + ω 0 , k + ω ( ω 0 , k + ω ) 2 + T 2 - 2 ] .
P dc , R = - 1 2 V ( E 0 ) 2 k Δ μ k μ v c , k · e ^ 2 T 1 , s T 2 × [ 1 ( ω 0 , k - ω ) 2 + T 2 - 2 + 1 ( ω 0 , k + ω ) 2 + T 2 - 2 ] .
μ b , i = - e b i * r b i = μ b b ^ i
μ b a , i = - e a i * r b i = μ b a b ^ i .
A abs , i = μ b a , i b ^ i · e ^ E 0 2 2 2 T 2 1 ( ω 0 , k - ω ) 2 + T 2 - 2 = μ b a E 0 2 2 2 T 2 1 ( ω 0 , k - ω ) 2 + T 2 - 2 b ^ i · b ^ 1 2 .
μ a , i = - e a i * r a i = μ a b ^ i ,
d d t Δ μ i ( E ) = A abs , i ( μ a - μ b ) b ^ i .
d d t Δ μ ( E ) = i = 1 4 Δ μ i ( E ) = ( μ a - μ b ) × μ b a E 0 2 2 2 T 2 1 ( ω 0 , k - ω ) 2 + T 2 - 2 × i = 1 4 ( b ^ i · b ^ 1 ) 2 b ^ i .
d d t Δ μ ( E ) = 8 9 ( μ a - μ b ) μ b a E 0 2 2 2 T 2 1 ( ω 0 , k - ω ) 2 + T 2 - 2 b ^ 1 .
P dc , [ 111 ] = N Δ μ ( E ) T 1 , s = 8 9 ( μ a - μ b ) × μ b a E 0 2 2 2 T 1 , s T 2 1 ( ω 0 , k - ω ) 2 + T 2 - 2 b ^ 1 ,
h c ( a ) ( 1 ) = ½ [ S c ( a ) + X c ( a ) + Y c ( a ) + Z c ( a ) ] directed along 111 ,
h c ( a ) ( 2 ) = ½ [ S c ( a ) + X c ( a ) - Y c ( a ) - Z c ( a ) ] directed along 1 11 ¯ ,
h c ( a ) ( 3 ) = ½ [ S c ( a ) - X c ( a ) + Y c ( a ) - Z c ( a ) ] directed along 1 ¯ 1 1 ¯ ,
h c ( a ) ( 4 ) = ½ [ S c ( a ) - X c ( a ) - Y c ( a ) + Z c ( a ) ] directed along 11 ¯ 1 .
b i = ( 1 + α p 2 ) 1 / 2 [ h a ( i ) + α p h c ( i ) ] ,
a i = ( 1 + α p 2 ) 1 / 2 [ h c ( i ) - α p h a ( i ) ] ,
p = ( 1 - α p 2 ) / ( 1 + α p 2 ) .
μ b , i = μ b b ^ i = q b l b b ^ i ,
q b = 2 e p = 2 e 1 - α p 2 1 + α p 2
μ a , i = - a i * e r a i = - μ b , i = - μ b b ^ i .
- b i * e r b j μ i δ i j .
μ c , k = S A = ( 1 + α p 2 ) 1 / 2 ( S c - α p S a ) ( ) .
u c , k = ½ ( a 1 + a 2 + a 3 + a 4 ) ( ) .
μ c c , k = - u c , k * - e r u c , k = i = 1 4 μ a ( i ) = - μ b b ^ i = 0.
Δ μ k = μ v v , k .
u h , k = 2 1 / 2 [ ( X b cos ϕ + Y b sin ϕ ) + i ( - X b cos θ sin ϕ + Y b cos θ sin ϕ - Z b sin θ ] ( ) ,
u l , k = ( 2 / 3 ) 1 / 2 ( - X b ) sin θ sin ϕ + Y b sin θ cos ϕ + Z b cos θ ) ( ) + ( 1 / 6 ) 1 / 2 [ ( X b cos ϕ + Y b sin ϕ ) + i ( - X b cos θ sin ϕ + Y b cos θ cos ϕ - Z b sin θ ) ] ( ) ,
X b = ( 1 + α 2 ) 1 / 2 ( X a + X c )
u h , k = 2 - 3 / 2 { b 1 [ ( cos ϕ + sin ϕ ) + i ( - cos θ sin ϕ + cos θ cos ϕ - sin θ ) ] + b 2 [ ( cos ϕ - sin ϕ ) + i ( - cos θ sin ϕ - cos θ cos ϕ + sin θ ) ] + b 3 [ ( - cos ϕ + sin ϕ ) + i ( cos θ sin ϕ + cos θ cos ϕ + sin θ ) ] + b 4 [ - ( cos ϕ + sin ϕ ) + i ( cos θ sin ϕ - cos θ cos ϕ - sin θ ) ] } ( ) ,
u l , k = 6 - 1 / 2 [ b 1 ( - sin θ sin ϕ + sin θ cos ϕ + cos θ ) + b 2 ( - sin θ sin ϕ - sin θ cos ϕ - cos θ ) + b 3 ( sin θ sin ϕ + sin θ cos ϕ - cos θ ) + b 4 ( sin θ sin ϕ - sin θ cos ϕ + cos θ ) ] ( ) × 12 - 1 / 2 { b 1 [ ( cos ϕ + sin ϕ ) + i ( - cos θ sin ϕ + cos θ cos ϕ - sin θ ) ] + b 2 [ ( cos ϕ - sin ϕ ) + i ( - cos θ sin ϕ - cos θ cos ϕ + sin θ ) ] + b 3 [ ( - cos ϕ + sin ϕ ) + i ( cos θ sin ϕ + cos θ cos ϕ + sin θ ) ] + b 4 [ - ( cos ϕ + sin ϕ ) + i ( cos θ sin ϕ - cos θ cos ϕ - sin θ ) ] } ( ) .
μ h h , k = μ h , k * - e r u h , k = ¼ [ ( μ b , 1 + μ b , 2 + μ b , 3 + μ b , 4 ) + sin 2 θ cos ϕ sin ϕ ( μ b , 1 - μ b , 2 - μ b , 3 + μ b , 4 ) + cos θ sin θ sin ϕ ( μ b , 1 - μ b , 2 + μ b , 3 - μ b , 4 ) + cos θ sin θ cos ϕ ( - μ b , 1 - μ b , 2 + μ b , 3 - μ b , 4 ) ]
μ l l , k = u l , k * - e r u h , k = ¼ [ ( μ b , 1 + μ b , 2 + μ b , 3 + μ b , 4 ) - sin 2 θ cos ϕ sin ϕ ( μ b , 1 - μ b , 2 - μ b , 3 + μ b , 4 ) - cos θ sin θ sin ϕ ( μ b , 1 - μ b , 2 + μ b , 3 - μ b , 4 ) + cos θ sin θ cos ϕ ( μ b , 1 + μ b , 2 - μ b , 3 - μ b , 4 ) ]
μ h h , k = μ b 2 3 ( - sin 2 θ · cos ϕ x ^ + sin 2 θ · sin ϕ y ^ + sin 2 θ · sin 2 ϕ z ^ ) ,
μ l l , k = μ b 2 3 ( sin 2 θ · cos ϕ x ^ - sin 2 θ · sin ϕ y ^ - sin 2 θ · sin 2 ϕ z ^ ) .
u h , [ 111 ] = 1 3 [ b 2 + b 3 exp ( i 2 π / 3 ) + b 4 exp ( - i 2 π / 3 ) ( ) ,
u l , [ 111 ] = 1 2 b 1 ( ) - 1 3 ( 2 ) ( b 2 + b 3 + b 4 ) ( ) + 1 3 [ b 2 + b 3 exp ( i 2 π / 3 ) + b 4 exp ( i 2 π / 3 ) ] ( ) .
μ h h , k = μ l l , k = μ b 2 3 ( sin 2 2 θ + sin 4 θ sin 2 2 ϕ ) ,
μ c h , k = P ω 0 m 0 2 [ ( cos ϕ - i cos θ · sin ϕ ) x ^ + ( sin ϕ + i cos θ · cos ϕ ) y ^ - i sin θ z ^ ] ,
μ c l , k = P ω 0 m 0 2 [ ( - 2 sin θ · sin ϕ + cos ϕ - i cos θ · sin ϕ ) x ^ + ( 2 sin θ · cos ϕ + sin ϕ + i cos θ · cos ϕ ) y ^ + ( 2 cos θ - j sin θ ) z ^ ] ,
P d c = r ( ω ) : e ^ e ^ ɛ 0 E 0 2 .
r = r h , R + r h , r + r l , R + r l , V ,
( r ν , R ) l m n ( ω ) = 1 2 V k ( μ v v , k ) l ( μ c v , k ) m ( μ c v , k ) n × F R ( ω , ω 0 , k ) ,
( r ν , V ) l m n ( ω ) = 1 2 V k ( μ c v , k ) l ( μ v v , k ) m ( μ c v , k ) n × F V ( ω , ω 0 , k ) ,
F R ( ω , ω 0 , k ) = 1 ɛ 0 2 T 1 , s T 2 [ 1 ( ω 0 , k - ω ) 2 + T 2 - 2 + 1 ( ω 0 , k - ω ) 2 + T 2 2 ] ,
F V ( ω , ω 0 , k ) = 1 ɛ 0 2 ω 0 , k [ ω 0 , k - ω ( ω 0 , k + ω ) 2 - T 2 - 2 + ω 0 , k + ω ( ω 0 , k + ω ) 2 + T 2 2 ] .
( r ν , R ) l m n ( ω ) = 1 2 V 0 π 0 2 π ( μ v v , k ) l ( μ c v , k ) m ( μ c v , k ) n d θ d ϕ × 0 k 2 F R ( ω , ω 0 , k ) d k = 1 8 π 0 π 0 2 π ( μ v v , k ) l ( μ c v , k ) m ( μ c v , k ) n d θ d ϕ × 0 g c v ( ω 0 ) F R ( ω , ω 0 , k ) d ( ω 0 ) ,
( r ν , V ) l m n ( ω ) = 1 8 π 0 π 0 2 π ( μ c v , k ) l ( μ v v , k ) m ( μ c v , k ) n d θ d ϕ × 0 g c v ( ω 0 ) F V ( ω , ω 0 , k ) d ( ω 0 ) ,
g c v ( ω 0 ) = 1 2 π 2 [ 2 m c m v 2 ( m c + m v ) ] 3 / 2 ( ω 0 - E gap ) 1 / 2 .
1 4 π 0 π 0 2 π ( μ c v , k ) l ( μ v v , k ) m ( μ c v , k ) n d θ d ϕ = 1 4 π 0 π 0 2 π ( μ v v , k ) l ( μ c v , k ) m ( μ c v , k ) n d θ d ϕ = 1 30 μ b 3 e 2 p 2 ω 0 2 m 0 2 [ ( 1 - δ l m ) ( 1 - δ m n ) ( 1 - δ l n ) ] .
( r ν , R ) x y z ( ω ) = 1 60 μ b 3 e 2 p 2 ω 0 2 m 0 2 T 1 , s [ 2 m c m v 2 ( m c + m v ) ] 3 / 2 1 2 π ɛ 0 × ( ω - E gap ) 1 / 2 ,
( r u , V ) x y z ( ω ) = 1 60 μ b 3 e 2 p 2 ω 0 2 m 0 2 E gap - 1 / 2 [ 2 m c m v 2 ( m c + m y ) ] 3 / 2 × 1 2 π ɛ 0 S V ( ω ) ,
S v ( ω ) = { E gap E gap - ω + E gap ω < E gap E gap ω ω > E gap .
r v , R r v , V T 1 , s [ ( ω - ω gap ) ω gap ] 1 / 2 ,
ω - E gap 2 T 1 , s 2 E gap - 1
α = ω c n e 2 p 2 3 ω 0 2 m 0 2 [ 2 m c m v 2 ( m c + m v ) ] 3 / 2 1 2 π ɛ 0 ( ω - E gap ) 1 / 2 .
( r R ) x y z = 1 20 μ b 3 T 1 , s n α λ 2 π .
P dc = r : e ^ e ^ ɛ 0 E 2 = μ b 10 3 ( T 1 , s + T V ) α I ω × 2 ( sin 2 θ sin 2 2 ϕ + 4 cos 2 θ ) 1 / 2 ,
W ( k , k ) = W ( Θ k ) ,
1 / T 2 = W ( Θ k ) d Ω
1 / τ v = ( 1 - cos Θ k ) W ( Θ k ) d Ω .
1 / T 1 , s 3 / 2 sin 2 Θ k W ( Θ k ) d Ω .
τ v > T 1 , s > T 2 .

Metrics