Abstract

We present what are to our knowledge the first theoretical and experimental results for dispersion in the anisotropy (σ) of χ(3) associated with optical third-harmonic generation in Si. The theory is based on a full empirical tight-binding band-structure calculation, and the results agree well with our measurements for 0.72 μm < λ < 1.9 μm. The calculated low-frequency limit of σ agrees with measurements from the literature better than earlier calculations do. In addition, the dispersion in the relative phase of χ1212(3)/χ1111(3) also shows good agreement with experiment.

© 1989 Optical Society of America

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  1. M. S. Hybertsen, S. G. Louie, Phys. Rev. B 34, 5390 (1986).
  2. D. J. Moss, J. E. Sipe, H. M. van Driel, Phys. Rev. B 36, 9708 (1987).
  3. C. Y. Fong, Y. R. Shen, Phys. Rev. B 12, 2325 (1975).
  4. K. Arya, S. S. Jha, Phys. Rev. B 20, 1611 (1979).
  5. D. J. Moss, J. E. Sipe, H. M. van Driel, “Band-structure calculation of dispersion and anisotropy in χ↔(3) for third-harmonic generation in Si, Ge, and GaAs,” Phys. Rev. B (to be published).
  6. S. S. Jha, N. Bloembergen, Phys. Rev. 171, 891 (1968).
  7. J. C. Phillips, J. A. van Vechten, Phys. Rev. 183, 709 (1969).
  8. C. Flytzanis, Phys. Lett. 31A, 273 (1969).
  9. J. A. van Vechten, D. E. Aspnes, Phys. Lett. 30A, 346 (1968); J. A. van Vechten, M. Cardona, D. E. Aspnes, R. M. Martin, in Proceedings of the Tenth International Conference on the Physics of Semiconductors, S. P. Keller, J. C. Hensel, F. Stern, eds. (USAEC Division of Technical Information Extension, Oak Ridge, Tenn., 1970), p. 82.
  10. D. J. Moss, E. Ghahramani, J. E. Sipe, H. M. van Driel, Phys. Rev. B 34, 8758 (1986).
  11. J. E. Sipe, D. J. Moss, H. M. van Driel, Phys. Rev. B 35, 1129 (1987).
  12. D. J. Moss, H. M. van Driel, J. E. Sipe, Appl. Phys. Lett. 48, 1150 (1986).
  13. C. C. Wang, J. Bomback, W. T. Donlon, C. R. Huo, J. V. James, Phys. Rev. Lett. 57, 1647 (1986).
  14. C. C. Wang, N. W. Ressler, Phys. Rev. B 2, 1827 (1970).
  15. J. J. Wynne, G. D. Boyd, Appl. Phys. Lett. 12, 191 (1968); J. J. Wynne, Phys. Rev. 178, 1295 (1969).
  16. E. Yablonovitch, C. Flytzanis, N. Bloembergen, Phys. Rev. Lett. 29, 865 (1972).
  17. W. K. Burns, N. Bloembergen, Phys. Rev. B 4, 3437 (1971).
  18. J. Lee, C. C. Wang, C. Y. Tung, W. L. Smith, S. Hahn, M. Arst, Appl. Phys. Lett. 51, 54 (1987).

1987 (3)

D. J. Moss, J. E. Sipe, H. M. van Driel, Phys. Rev. B 36, 9708 (1987).

J. E. Sipe, D. J. Moss, H. M. van Driel, Phys. Rev. B 35, 1129 (1987).

J. Lee, C. C. Wang, C. Y. Tung, W. L. Smith, S. Hahn, M. Arst, Appl. Phys. Lett. 51, 54 (1987).

1986 (4)

D. J. Moss, H. M. van Driel, J. E. Sipe, Appl. Phys. Lett. 48, 1150 (1986).

C. C. Wang, J. Bomback, W. T. Donlon, C. R. Huo, J. V. James, Phys. Rev. Lett. 57, 1647 (1986).

M. S. Hybertsen, S. G. Louie, Phys. Rev. B 34, 5390 (1986).

D. J. Moss, E. Ghahramani, J. E. Sipe, H. M. van Driel, Phys. Rev. B 34, 8758 (1986).

1979 (1)

K. Arya, S. S. Jha, Phys. Rev. B 20, 1611 (1979).

1975 (1)

C. Y. Fong, Y. R. Shen, Phys. Rev. B 12, 2325 (1975).

1972 (1)

E. Yablonovitch, C. Flytzanis, N. Bloembergen, Phys. Rev. Lett. 29, 865 (1972).

1971 (1)

W. K. Burns, N. Bloembergen, Phys. Rev. B 4, 3437 (1971).

1970 (1)

C. C. Wang, N. W. Ressler, Phys. Rev. B 2, 1827 (1970).

1969 (2)

J. C. Phillips, J. A. van Vechten, Phys. Rev. 183, 709 (1969).

C. Flytzanis, Phys. Lett. 31A, 273 (1969).

1968 (3)

J. A. van Vechten, D. E. Aspnes, Phys. Lett. 30A, 346 (1968); J. A. van Vechten, M. Cardona, D. E. Aspnes, R. M. Martin, in Proceedings of the Tenth International Conference on the Physics of Semiconductors, S. P. Keller, J. C. Hensel, F. Stern, eds. (USAEC Division of Technical Information Extension, Oak Ridge, Tenn., 1970), p. 82.

S. S. Jha, N. Bloembergen, Phys. Rev. 171, 891 (1968).

J. J. Wynne, G. D. Boyd, Appl. Phys. Lett. 12, 191 (1968); J. J. Wynne, Phys. Rev. 178, 1295 (1969).

Arst, M.

J. Lee, C. C. Wang, C. Y. Tung, W. L. Smith, S. Hahn, M. Arst, Appl. Phys. Lett. 51, 54 (1987).

Arya, K.

K. Arya, S. S. Jha, Phys. Rev. B 20, 1611 (1979).

Aspnes, D. E.

J. A. van Vechten, D. E. Aspnes, Phys. Lett. 30A, 346 (1968); J. A. van Vechten, M. Cardona, D. E. Aspnes, R. M. Martin, in Proceedings of the Tenth International Conference on the Physics of Semiconductors, S. P. Keller, J. C. Hensel, F. Stern, eds. (USAEC Division of Technical Information Extension, Oak Ridge, Tenn., 1970), p. 82.

Bloembergen, N.

E. Yablonovitch, C. Flytzanis, N. Bloembergen, Phys. Rev. Lett. 29, 865 (1972).

W. K. Burns, N. Bloembergen, Phys. Rev. B 4, 3437 (1971).

S. S. Jha, N. Bloembergen, Phys. Rev. 171, 891 (1968).

Bomback, J.

C. C. Wang, J. Bomback, W. T. Donlon, C. R. Huo, J. V. James, Phys. Rev. Lett. 57, 1647 (1986).

Boyd, G. D.

J. J. Wynne, G. D. Boyd, Appl. Phys. Lett. 12, 191 (1968); J. J. Wynne, Phys. Rev. 178, 1295 (1969).

Burns, W. K.

W. K. Burns, N. Bloembergen, Phys. Rev. B 4, 3437 (1971).

Donlon, W. T.

C. C. Wang, J. Bomback, W. T. Donlon, C. R. Huo, J. V. James, Phys. Rev. Lett. 57, 1647 (1986).

Flytzanis, C.

E. Yablonovitch, C. Flytzanis, N. Bloembergen, Phys. Rev. Lett. 29, 865 (1972).

C. Flytzanis, Phys. Lett. 31A, 273 (1969).

Fong, C. Y.

C. Y. Fong, Y. R. Shen, Phys. Rev. B 12, 2325 (1975).

Ghahramani, E.

D. J. Moss, E. Ghahramani, J. E. Sipe, H. M. van Driel, Phys. Rev. B 34, 8758 (1986).

Hahn, S.

J. Lee, C. C. Wang, C. Y. Tung, W. L. Smith, S. Hahn, M. Arst, Appl. Phys. Lett. 51, 54 (1987).

Huo, C. R.

C. C. Wang, J. Bomback, W. T. Donlon, C. R. Huo, J. V. James, Phys. Rev. Lett. 57, 1647 (1986).

Hybertsen, M. S.

M. S. Hybertsen, S. G. Louie, Phys. Rev. B 34, 5390 (1986).

James, J. V.

C. C. Wang, J. Bomback, W. T. Donlon, C. R. Huo, J. V. James, Phys. Rev. Lett. 57, 1647 (1986).

Jha, S. S.

K. Arya, S. S. Jha, Phys. Rev. B 20, 1611 (1979).

S. S. Jha, N. Bloembergen, Phys. Rev. 171, 891 (1968).

Lee, J.

J. Lee, C. C. Wang, C. Y. Tung, W. L. Smith, S. Hahn, M. Arst, Appl. Phys. Lett. 51, 54 (1987).

Louie, S. G.

M. S. Hybertsen, S. G. Louie, Phys. Rev. B 34, 5390 (1986).

Moss, D. J.

D. J. Moss, J. E. Sipe, H. M. van Driel, Phys. Rev. B 36, 9708 (1987).

J. E. Sipe, D. J. Moss, H. M. van Driel, Phys. Rev. B 35, 1129 (1987).

D. J. Moss, H. M. van Driel, J. E. Sipe, Appl. Phys. Lett. 48, 1150 (1986).

D. J. Moss, E. Ghahramani, J. E. Sipe, H. M. van Driel, Phys. Rev. B 34, 8758 (1986).

D. J. Moss, J. E. Sipe, H. M. van Driel, “Band-structure calculation of dispersion and anisotropy in χ↔(3) for third-harmonic generation in Si, Ge, and GaAs,” Phys. Rev. B (to be published).

Phillips, J. C.

J. C. Phillips, J. A. van Vechten, Phys. Rev. 183, 709 (1969).

Ressler, N. W.

C. C. Wang, N. W. Ressler, Phys. Rev. B 2, 1827 (1970).

Shen, Y. R.

C. Y. Fong, Y. R. Shen, Phys. Rev. B 12, 2325 (1975).

Sipe, J. E.

D. J. Moss, J. E. Sipe, H. M. van Driel, Phys. Rev. B 36, 9708 (1987).

J. E. Sipe, D. J. Moss, H. M. van Driel, Phys. Rev. B 35, 1129 (1987).

D. J. Moss, E. Ghahramani, J. E. Sipe, H. M. van Driel, Phys. Rev. B 34, 8758 (1986).

D. J. Moss, H. M. van Driel, J. E. Sipe, Appl. Phys. Lett. 48, 1150 (1986).

D. J. Moss, J. E. Sipe, H. M. van Driel, “Band-structure calculation of dispersion and anisotropy in χ↔(3) for third-harmonic generation in Si, Ge, and GaAs,” Phys. Rev. B (to be published).

Smith, W. L.

J. Lee, C. C. Wang, C. Y. Tung, W. L. Smith, S. Hahn, M. Arst, Appl. Phys. Lett. 51, 54 (1987).

Tung, C. Y.

J. Lee, C. C. Wang, C. Y. Tung, W. L. Smith, S. Hahn, M. Arst, Appl. Phys. Lett. 51, 54 (1987).

van Driel, H. M.

D. J. Moss, J. E. Sipe, H. M. van Driel, Phys. Rev. B 36, 9708 (1987).

J. E. Sipe, D. J. Moss, H. M. van Driel, Phys. Rev. B 35, 1129 (1987).

D. J. Moss, H. M. van Driel, J. E. Sipe, Appl. Phys. Lett. 48, 1150 (1986).

D. J. Moss, E. Ghahramani, J. E. Sipe, H. M. van Driel, Phys. Rev. B 34, 8758 (1986).

D. J. Moss, J. E. Sipe, H. M. van Driel, “Band-structure calculation of dispersion and anisotropy in χ↔(3) for third-harmonic generation in Si, Ge, and GaAs,” Phys. Rev. B (to be published).

van Vechten, J. A.

J. C. Phillips, J. A. van Vechten, Phys. Rev. 183, 709 (1969).

J. A. van Vechten, D. E. Aspnes, Phys. Lett. 30A, 346 (1968); J. A. van Vechten, M. Cardona, D. E. Aspnes, R. M. Martin, in Proceedings of the Tenth International Conference on the Physics of Semiconductors, S. P. Keller, J. C. Hensel, F. Stern, eds. (USAEC Division of Technical Information Extension, Oak Ridge, Tenn., 1970), p. 82.

Wang, C. C.

J. Lee, C. C. Wang, C. Y. Tung, W. L. Smith, S. Hahn, M. Arst, Appl. Phys. Lett. 51, 54 (1987).

C. C. Wang, J. Bomback, W. T. Donlon, C. R. Huo, J. V. James, Phys. Rev. Lett. 57, 1647 (1986).

C. C. Wang, N. W. Ressler, Phys. Rev. B 2, 1827 (1970).

Wynne, J. J.

J. J. Wynne, G. D. Boyd, Appl. Phys. Lett. 12, 191 (1968); J. J. Wynne, Phys. Rev. 178, 1295 (1969).

Yablonovitch, E.

E. Yablonovitch, C. Flytzanis, N. Bloembergen, Phys. Rev. Lett. 29, 865 (1972).

Appl. Phys. Lett. (3)

D. J. Moss, H. M. van Driel, J. E. Sipe, Appl. Phys. Lett. 48, 1150 (1986).

J. J. Wynne, G. D. Boyd, Appl. Phys. Lett. 12, 191 (1968); J. J. Wynne, Phys. Rev. 178, 1295 (1969).

J. Lee, C. C. Wang, C. Y. Tung, W. L. Smith, S. Hahn, M. Arst, Appl. Phys. Lett. 51, 54 (1987).

Phys. Lett. (2)

C. Flytzanis, Phys. Lett. 31A, 273 (1969).

J. A. van Vechten, D. E. Aspnes, Phys. Lett. 30A, 346 (1968); J. A. van Vechten, M. Cardona, D. E. Aspnes, R. M. Martin, in Proceedings of the Tenth International Conference on the Physics of Semiconductors, S. P. Keller, J. C. Hensel, F. Stern, eds. (USAEC Division of Technical Information Extension, Oak Ridge, Tenn., 1970), p. 82.

Phys. Rev. (2)

S. S. Jha, N. Bloembergen, Phys. Rev. 171, 891 (1968).

J. C. Phillips, J. A. van Vechten, Phys. Rev. 183, 709 (1969).

Phys. Rev. B (8)

C. C. Wang, N. W. Ressler, Phys. Rev. B 2, 1827 (1970).

W. K. Burns, N. Bloembergen, Phys. Rev. B 4, 3437 (1971).

D. J. Moss, E. Ghahramani, J. E. Sipe, H. M. van Driel, Phys. Rev. B 34, 8758 (1986).

J. E. Sipe, D. J. Moss, H. M. van Driel, Phys. Rev. B 35, 1129 (1987).

M. S. Hybertsen, S. G. Louie, Phys. Rev. B 34, 5390 (1986).

D. J. Moss, J. E. Sipe, H. M. van Driel, Phys. Rev. B 36, 9708 (1987).

C. Y. Fong, Y. R. Shen, Phys. Rev. B 12, 2325 (1975).

K. Arya, S. S. Jha, Phys. Rev. B 20, 1611 (1979).

Phys. Rev. Lett. (2)

E. Yablonovitch, C. Flytzanis, N. Bloembergen, Phys. Rev. Lett. 29, 865 (1972).

C. C. Wang, J. Bomback, W. T. Donlon, C. R. Huo, J. V. James, Phys. Rev. Lett. 57, 1647 (1986).

Other (1)

D. J. Moss, J. E. Sipe, H. M. van Driel, “Band-structure calculation of dispersion and anisotropy in χ↔(3) for third-harmonic generation in Si, Ge, and GaAs,” Phys. Rev. B (to be published).

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Figures (2)

Fig. 1
Fig. 1

Theoretical and experimental results for the dispersion in |σ(ω)|. The results at λ = 10.6 μm are from Refs. 1416, and the results at all other wavelengths are listed in Table 1.

Fig. 2
Fig. 2

Results for the phase (ϕ) of B/A (B/A = |B/A|eiϕ). Experimentally there is an uncertainty in the sign of ϕ, so the magnitude of the angle is plotted.

Tables (1)

Tables Icon

Table 1 Experimental Results

Equations (4)

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P i ( 3 ω ) = 3 χ 1212 ( 3 ) E i ( E E ) + [ χ 1111 ( 3 ) 3 χ 1212 ( 3 ) ] E i E i E i B E i ( E E ) + ( A B ) E i E i E i ,
χ ˜ ( 3 ) ( ω ) = + π 6 ( e ћ m ) 4 i i j j B Z d k 4 π 3 R e { p i j vc p j i cv p i j vc p j i cv } × 1 E j i δ [ 3 6 δ ( E j i 3 ћ ω ) ( 3 E j i E j i ) + δ ( E j i ћ ω ) ( E j i 3 E j i ) ] ,
THG ϕ = 45 THG ϕ = 0 = 1 4 | 1 + B A | 2 R 1 ,
THG circ THG ϕ = 0 = 1 4 | 1 B A | 2 = 1 4 | σ | 2 R 2 .

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