Abstract

Two-photon absorption (TPA) in a semiconducting quantum well (QW) is investigated theoretically in the presence of the electric field. Numerical results show that the TPA depends strongly on the relative orientations of the electric field, the direction of optical polarization, and the direction of carrier confinement in the QW. The TPA is finite at photon energy below the threshold energy where the TPA would vanish in the absence of the electric field for any case. The TPA is large when the electric field is along the direction of carrier confinement in the QW. Moreover, when the electric field is in the plane of the QW, the difference Δα between the TPAs in a finite electric field and in zero field at the threshold energy increases almost linearly with increasing the electric field. In addition, it is also found that, in all cases, the TPA is less in the QW than that in the bulk.

© 2010 Optical Society of America

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    [Crossref]
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    [Crossref]
  4. Y. Liu and H. K. Tsang, “Time dependent density of free carriers generated by two photon absorption in silicon waveguides,” Appl. Phys. Lett. 90, 211105 (2007).
    [Crossref]
  5. M. Sheik-Bahae, A. A. Said, T. H. Wei, D. Hagan, and E. W. V. Stryland, “Sensitive measurements of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
    [Crossref]
  6. B. Gu, J. He, W. Ji, and H. T. Wang, “Three-photon absorption saturation in ZnO and ZnS crystals,” J. Appl. Phys. 103, 073105 (2008).
    [Crossref]
  7. X. Li, J. Embden, J. W. M. Chon, and M. Gu, “Enhanced two-photon absorption of CdS nanocrystal rods,” Appl. Phys. Lett. 94, 103117 (2009).
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2009 (4)

J. Bravo-Abad, E. P. Ippen, and M. Soljačić, “Ultrafast photodetection in an all-silicon chip enabled by two-photon,” Appl. Phys. Lett. 94, 241103 (2009).
[Crossref]

X. Li, J. Embden, J. W. M. Chon, and M. Gu, “Enhanced two-photon absorption of CdS nanocrystal rods,” Appl. Phys. Lett. 94, 103117 (2009).
[Crossref]

C. Xia and H. N. Spector, “Nonlinear Franz–Keldysh effect: Two photon absorption in semiconducting quantum wires and quantum boxes,” J. Appl. Phys. 106, 124302 (2009).
[Crossref]

C. Xia and H. N. Spector, “Franz–Keldysh effect in the interband optical absorption of semiconducting nanostructures,” J. Appl. Phys. 105, 084313 (2009).
[Crossref]

2008 (3)

B. Gu, J. He, W. Ji, and H. T. Wang, “Three-photon absorption saturation in ZnO and ZnS crystals,” J. Appl. Phys. 103, 073105 (2008).
[Crossref]

P. Aivaliotis, E. A. Zibik, L. R. Wilson, J. W. Cockburn, M. Hopkinson, and N. Q. Vinh, “Two photon absorption in quantum dot-in-a-well infrared photodetectors,” Appl. Phys. Lett. 92, 023501 (2008).
[Crossref]

H. Y. Cui, Z. F. Li, Z. L. Liu, C. Wang, X. S. Chen, X. N. Hu, Z. H. Ye, and W. Lub, “Modulation of the two-photon absorption by electric fields in HgCdTe photodiode,” Appl. Phys. Lett. 92, 021128 (2008).
[Crossref]

2007 (3)

Y. Liu and H. K. Tsang, “Time dependent density of free carriers generated by two photon absorption in silicon waveguides,” Appl. Phys. Lett. 90, 211105 (2007).
[Crossref]

A. D. Bristow, N. Rotenberg, and H. M. van Driel, “Two-photon absorption and Kerr coefficients of silicon for 850–2200 nm,” Appl. Phys. Lett. 90, 191104 (2007).
[Crossref]

L. A. Padilha, J. Fu, D. Hagan, E. Van Stryland, C. Cesar, L. Barbosa, D. Buso, and A. Martucci, “Frequency degenerate and nondegenerate two-photon absorption spectra of semiconductor quantum dots,” Phys. Rev. B 75, 075325 (2007).
[Crossref]

2006 (1)

H. Garcia, “Tunneling assisted two-photon absorption: The nonlinear Franz–Keldysh effect,” Phys. Rev. B 74, 035212 (2006).
[Crossref]

1996 (1)

A. V. Fedorov, A. V. Baranov, and K. Inoue, “Two-photon transitions in systems with semiconductor quantum dots,” Phys. Rev. B 54, 8627–8632 (1996).
[Crossref]

1995 (1)

1994 (1)

1992 (1)

1990 (3)

A. Pasquarello and A. Quattropani, “Two-photon transitions to excitons in quantum wells,” Phys. Rev. B 42, 9073–9079 (1990).
[Crossref]

A. Pasquarello and A. Quattropani, “Effect of continuum states on two-photon absorption in quantum wells,” Phys. Rev. B 41, 12728–12734 (1990).
[Crossref]

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. Hagan, and E. W. V. Stryland, “Sensitive measurements of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[Crossref]

1989 (1)

A. Shimizu, “Two-photon absorption in quantum-well structures near half the direct band gap,” Phys. Rev. B 40, 1403–1406 (1989).
[Crossref]

1988 (1)

A. Pasquarello and A. Quattropani, “Gauge-invariant two-photon transitions in quantum wells,” Phys. Rev. B 38, 6206–6210 (1988).
[Crossref]

1987 (1)

H. Spector, “Two-photon absorption in semiconducting quantum-well structures,” Phys. Rev. B 35, 5876–5879 (1987).
[Crossref]

1964 (1)

R. Braunstein and N. Ockman, “Optical double-photon absorption in CdS,” Phys. Rev. 134, A499–A507 (1964).
[Crossref]

1962 (1)

R. Braunstein, “Nonlinear optical effects,” Phys. Rev. 125, 475–477 (1962).
[Crossref]

Aivaliotis, P.

P. Aivaliotis, E. A. Zibik, L. R. Wilson, J. W. Cockburn, M. Hopkinson, and N. Q. Vinh, “Two photon absorption in quantum dot-in-a-well infrared photodetectors,” Appl. Phys. Lett. 92, 023501 (2008).
[Crossref]

Baranov, A. V.

A. V. Fedorov, A. V. Baranov, and K. Inoue, “Two-photon transitions in systems with semiconductor quantum dots,” Phys. Rev. B 54, 8627–8632 (1996).
[Crossref]

Barbosa, L.

L. A. Padilha, J. Fu, D. Hagan, E. Van Stryland, C. Cesar, L. Barbosa, D. Buso, and A. Martucci, “Frequency degenerate and nondegenerate two-photon absorption spectra of semiconductor quantum dots,” Phys. Rev. B 75, 075325 (2007).
[Crossref]

Braunstein, R.

R. Braunstein and N. Ockman, “Optical double-photon absorption in CdS,” Phys. Rev. 134, A499–A507 (1964).
[Crossref]

R. Braunstein, “Nonlinear optical effects,” Phys. Rev. 125, 475–477 (1962).
[Crossref]

Bravo-Abad, J.

J. Bravo-Abad, E. P. Ippen, and M. Soljačić, “Ultrafast photodetection in an all-silicon chip enabled by two-photon,” Appl. Phys. Lett. 94, 241103 (2009).
[Crossref]

Bristow, A. D.

A. D. Bristow, N. Rotenberg, and H. M. van Driel, “Two-photon absorption and Kerr coefficients of silicon for 850–2200 nm,” Appl. Phys. Lett. 90, 191104 (2007).
[Crossref]

Buso, D.

L. A. Padilha, J. Fu, D. Hagan, E. Van Stryland, C. Cesar, L. Barbosa, D. Buso, and A. Martucci, “Frequency degenerate and nondegenerate two-photon absorption spectra of semiconductor quantum dots,” Phys. Rev. B 75, 075325 (2007).
[Crossref]

Cesar, C.

L. A. Padilha, J. Fu, D. Hagan, E. Van Stryland, C. Cesar, L. Barbosa, D. Buso, and A. Martucci, “Frequency degenerate and nondegenerate two-photon absorption spectra of semiconductor quantum dots,” Phys. Rev. B 75, 075325 (2007).
[Crossref]

Chen, X. S.

H. Y. Cui, Z. F. Li, Z. L. Liu, C. Wang, X. S. Chen, X. N. Hu, Z. H. Ye, and W. Lub, “Modulation of the two-photon absorption by electric fields in HgCdTe photodiode,” Appl. Phys. Lett. 92, 021128 (2008).
[Crossref]

Chon, J. W. M.

X. Li, J. Embden, J. W. M. Chon, and M. Gu, “Enhanced two-photon absorption of CdS nanocrystal rods,” Appl. Phys. Lett. 94, 103117 (2009).
[Crossref]

Cockburn, J. W.

P. Aivaliotis, E. A. Zibik, L. R. Wilson, J. W. Cockburn, M. Hopkinson, and N. Q. Vinh, “Two photon absorption in quantum dot-in-a-well infrared photodetectors,” Appl. Phys. Lett. 92, 023501 (2008).
[Crossref]

Cui, H. Y.

H. Y. Cui, Z. F. Li, Z. L. Liu, C. Wang, X. S. Chen, X. N. Hu, Z. H. Ye, and W. Lub, “Modulation of the two-photon absorption by electric fields in HgCdTe photodiode,” Appl. Phys. Lett. 92, 021128 (2008).
[Crossref]

Embden, J.

X. Li, J. Embden, J. W. M. Chon, and M. Gu, “Enhanced two-photon absorption of CdS nanocrystal rods,” Appl. Phys. Lett. 94, 103117 (2009).
[Crossref]

Fedorov, A. V.

A. V. Fedorov, A. V. Baranov, and K. Inoue, “Two-photon transitions in systems with semiconductor quantum dots,” Phys. Rev. B 54, 8627–8632 (1996).
[Crossref]

Fu, J.

L. A. Padilha, J. Fu, D. Hagan, E. Van Stryland, C. Cesar, L. Barbosa, D. Buso, and A. Martucci, “Frequency degenerate and nondegenerate two-photon absorption spectra of semiconductor quantum dots,” Phys. Rev. B 75, 075325 (2007).
[Crossref]

Garcia, H.

H. Garcia, “Tunneling assisted two-photon absorption: The nonlinear Franz–Keldysh effect,” Phys. Rev. B 74, 035212 (2006).
[Crossref]

Gu, B.

B. Gu, J. He, W. Ji, and H. T. Wang, “Three-photon absorption saturation in ZnO and ZnS crystals,” J. Appl. Phys. 103, 073105 (2008).
[Crossref]

Gu, M.

X. Li, J. Embden, J. W. M. Chon, and M. Gu, “Enhanced two-photon absorption of CdS nanocrystal rods,” Appl. Phys. Lett. 94, 103117 (2009).
[Crossref]

Hagan, D.

L. A. Padilha, J. Fu, D. Hagan, E. Van Stryland, C. Cesar, L. Barbosa, D. Buso, and A. Martucci, “Frequency degenerate and nondegenerate two-photon absorption spectra of semiconductor quantum dots,” Phys. Rev. B 75, 075325 (2007).
[Crossref]

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. Hagan, and E. W. V. Stryland, “Sensitive measurements of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[Crossref]

He, J.

B. Gu, J. He, W. Ji, and H. T. Wang, “Three-photon absorption saturation in ZnO and ZnS crystals,” J. Appl. Phys. 103, 073105 (2008).
[Crossref]

Hopkinson, M.

P. Aivaliotis, E. A. Zibik, L. R. Wilson, J. W. Cockburn, M. Hopkinson, and N. Q. Vinh, “Two photon absorption in quantum dot-in-a-well infrared photodetectors,” Appl. Phys. Lett. 92, 023501 (2008).
[Crossref]

Hu, X. N.

H. Y. Cui, Z. F. Li, Z. L. Liu, C. Wang, X. S. Chen, X. N. Hu, Z. H. Ye, and W. Lub, “Modulation of the two-photon absorption by electric fields in HgCdTe photodiode,” Appl. Phys. Lett. 92, 021128 (2008).
[Crossref]

Hutchings, D. C.

Inoue, K.

A. V. Fedorov, A. V. Baranov, and K. Inoue, “Two-photon transitions in systems with semiconductor quantum dots,” Phys. Rev. B 54, 8627–8632 (1996).
[Crossref]

Ippen, E. P.

J. Bravo-Abad, E. P. Ippen, and M. Soljačić, “Ultrafast photodetection in an all-silicon chip enabled by two-photon,” Appl. Phys. Lett. 94, 241103 (2009).
[Crossref]

Ji, W.

B. Gu, J. He, W. Ji, and H. T. Wang, “Three-photon absorption saturation in ZnO and ZnS crystals,” J. Appl. Phys. 103, 073105 (2008).
[Crossref]

Khurgin, J.

Li, X.

X. Li, J. Embden, J. W. M. Chon, and M. Gu, “Enhanced two-photon absorption of CdS nanocrystal rods,” Appl. Phys. Lett. 94, 103117 (2009).
[Crossref]

Li, Z. F.

H. Y. Cui, Z. F. Li, Z. L. Liu, C. Wang, X. S. Chen, X. N. Hu, Z. H. Ye, and W. Lub, “Modulation of the two-photon absorption by electric fields in HgCdTe photodiode,” Appl. Phys. Lett. 92, 021128 (2008).
[Crossref]

Liu, Y.

Y. Liu and H. K. Tsang, “Time dependent density of free carriers generated by two photon absorption in silicon waveguides,” Appl. Phys. Lett. 90, 211105 (2007).
[Crossref]

Liu, Z. L.

H. Y. Cui, Z. F. Li, Z. L. Liu, C. Wang, X. S. Chen, X. N. Hu, Z. H. Ye, and W. Lub, “Modulation of the two-photon absorption by electric fields in HgCdTe photodiode,” Appl. Phys. Lett. 92, 021128 (2008).
[Crossref]

Lub, W.

H. Y. Cui, Z. F. Li, Z. L. Liu, C. Wang, X. S. Chen, X. N. Hu, Z. H. Ye, and W. Lub, “Modulation of the two-photon absorption by electric fields in HgCdTe photodiode,” Appl. Phys. Lett. 92, 021128 (2008).
[Crossref]

Martucci, A.

L. A. Padilha, J. Fu, D. Hagan, E. Van Stryland, C. Cesar, L. Barbosa, D. Buso, and A. Martucci, “Frequency degenerate and nondegenerate two-photon absorption spectra of semiconductor quantum dots,” Phys. Rev. B 75, 075325 (2007).
[Crossref]

Obeidat, A.

Ockman, N.

R. Braunstein and N. Ockman, “Optical double-photon absorption in CdS,” Phys. Rev. 134, A499–A507 (1964).
[Crossref]

Padilha, L. A.

L. A. Padilha, J. Fu, D. Hagan, E. Van Stryland, C. Cesar, L. Barbosa, D. Buso, and A. Martucci, “Frequency degenerate and nondegenerate two-photon absorption spectra of semiconductor quantum dots,” Phys. Rev. B 75, 075325 (2007).
[Crossref]

Pasquarello, A.

A. Pasquarello and A. Quattropani, “Effect of continuum states on two-photon absorption in quantum wells,” Phys. Rev. B 41, 12728–12734 (1990).
[Crossref]

A. Pasquarello and A. Quattropani, “Two-photon transitions to excitons in quantum wells,” Phys. Rev. B 42, 9073–9079 (1990).
[Crossref]

A. Pasquarello and A. Quattropani, “Gauge-invariant two-photon transitions in quantum wells,” Phys. Rev. B 38, 6206–6210 (1988).
[Crossref]

Quattropani, A.

A. Pasquarello and A. Quattropani, “Two-photon transitions to excitons in quantum wells,” Phys. Rev. B 42, 9073–9079 (1990).
[Crossref]

A. Pasquarello and A. Quattropani, “Effect of continuum states on two-photon absorption in quantum wells,” Phys. Rev. B 41, 12728–12734 (1990).
[Crossref]

A. Pasquarello and A. Quattropani, “Gauge-invariant two-photon transitions in quantum wells,” Phys. Rev. B 38, 6206–6210 (1988).
[Crossref]

Rotenberg, N.

A. D. Bristow, N. Rotenberg, and H. M. van Driel, “Two-photon absorption and Kerr coefficients of silicon for 850–2200 nm,” Appl. Phys. Lett. 90, 191104 (2007).
[Crossref]

Said, A. A.

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. Hagan, and E. W. V. Stryland, “Sensitive measurements of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[Crossref]

Sheik-Bahae, M.

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. Hagan, and E. W. V. Stryland, “Sensitive measurements of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[Crossref]

Shimizu, A.

A. Shimizu, “Two-photon absorption in quantum-well structures near half the direct band gap,” Phys. Rev. B 40, 1403–1406 (1989).
[Crossref]

Soljacic, M.

J. Bravo-Abad, E. P. Ippen, and M. Soljačić, “Ultrafast photodetection in an all-silicon chip enabled by two-photon,” Appl. Phys. Lett. 94, 241103 (2009).
[Crossref]

Spector, H.

H. Spector, “Two-photon absorption in semiconducting quantum-well structures,” Phys. Rev. B 35, 5876–5879 (1987).
[Crossref]

Spector, H. N.

C. Xia and H. N. Spector, “Nonlinear Franz–Keldysh effect: Two photon absorption in semiconducting quantum wires and quantum boxes,” J. Appl. Phys. 106, 124302 (2009).
[Crossref]

C. Xia and H. N. Spector, “Franz–Keldysh effect in the interband optical absorption of semiconducting nanostructures,” J. Appl. Phys. 105, 084313 (2009).
[Crossref]

Stryland, E. W. V.

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. Hagan, and E. W. V. Stryland, “Sensitive measurements of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[Crossref]

Tsang, H. K.

Y. Liu and H. K. Tsang, “Time dependent density of free carriers generated by two photon absorption in silicon waveguides,” Appl. Phys. Lett. 90, 211105 (2007).
[Crossref]

van Driel, H. M.

A. D. Bristow, N. Rotenberg, and H. M. van Driel, “Two-photon absorption and Kerr coefficients of silicon for 850–2200 nm,” Appl. Phys. Lett. 90, 191104 (2007).
[Crossref]

Van Stryland, E.

L. A. Padilha, J. Fu, D. Hagan, E. Van Stryland, C. Cesar, L. Barbosa, D. Buso, and A. Martucci, “Frequency degenerate and nondegenerate two-photon absorption spectra of semiconductor quantum dots,” Phys. Rev. B 75, 075325 (2007).
[Crossref]

Van Stryland, E. W.

Vinh, N. Q.

P. Aivaliotis, E. A. Zibik, L. R. Wilson, J. W. Cockburn, M. Hopkinson, and N. Q. Vinh, “Two photon absorption in quantum dot-in-a-well infrared photodetectors,” Appl. Phys. Lett. 92, 023501 (2008).
[Crossref]

Wang, C.

H. Y. Cui, Z. F. Li, Z. L. Liu, C. Wang, X. S. Chen, X. N. Hu, Z. H. Ye, and W. Lub, “Modulation of the two-photon absorption by electric fields in HgCdTe photodiode,” Appl. Phys. Lett. 92, 021128 (2008).
[Crossref]

Wang, H. T.

B. Gu, J. He, W. Ji, and H. T. Wang, “Three-photon absorption saturation in ZnO and ZnS crystals,” J. Appl. Phys. 103, 073105 (2008).
[Crossref]

Wei, T. H.

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. Hagan, and E. W. V. Stryland, “Sensitive measurements of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[Crossref]

Wilson, L. R.

P. Aivaliotis, E. A. Zibik, L. R. Wilson, J. W. Cockburn, M. Hopkinson, and N. Q. Vinh, “Two photon absorption in quantum dot-in-a-well infrared photodetectors,” Appl. Phys. Lett. 92, 023501 (2008).
[Crossref]

Xia, C.

C. Xia and H. N. Spector, “Franz–Keldysh effect in the interband optical absorption of semiconducting nanostructures,” J. Appl. Phys. 105, 084313 (2009).
[Crossref]

C. Xia and H. N. Spector, “Nonlinear Franz–Keldysh effect: Two photon absorption in semiconducting quantum wires and quantum boxes,” J. Appl. Phys. 106, 124302 (2009).
[Crossref]

Ye, Z. H.

H. Y. Cui, Z. F. Li, Z. L. Liu, C. Wang, X. S. Chen, X. N. Hu, Z. H. Ye, and W. Lub, “Modulation of the two-photon absorption by electric fields in HgCdTe photodiode,” Appl. Phys. Lett. 92, 021128 (2008).
[Crossref]

Zibik, E. A.

P. Aivaliotis, E. A. Zibik, L. R. Wilson, J. W. Cockburn, M. Hopkinson, and N. Q. Vinh, “Two photon absorption in quantum dot-in-a-well infrared photodetectors,” Appl. Phys. Lett. 92, 023501 (2008).
[Crossref]

Appl. Phys. Lett. (6)

X. Li, J. Embden, J. W. M. Chon, and M. Gu, “Enhanced two-photon absorption of CdS nanocrystal rods,” Appl. Phys. Lett. 94, 103117 (2009).
[Crossref]

A. D. Bristow, N. Rotenberg, and H. M. van Driel, “Two-photon absorption and Kerr coefficients of silicon for 850–2200 nm,” Appl. Phys. Lett. 90, 191104 (2007).
[Crossref]

J. Bravo-Abad, E. P. Ippen, and M. Soljačić, “Ultrafast photodetection in an all-silicon chip enabled by two-photon,” Appl. Phys. Lett. 94, 241103 (2009).
[Crossref]

P. Aivaliotis, E. A. Zibik, L. R. Wilson, J. W. Cockburn, M. Hopkinson, and N. Q. Vinh, “Two photon absorption in quantum dot-in-a-well infrared photodetectors,” Appl. Phys. Lett. 92, 023501 (2008).
[Crossref]

H. Y. Cui, Z. F. Li, Z. L. Liu, C. Wang, X. S. Chen, X. N. Hu, Z. H. Ye, and W. Lub, “Modulation of the two-photon absorption by electric fields in HgCdTe photodiode,” Appl. Phys. Lett. 92, 021128 (2008).
[Crossref]

Y. Liu and H. K. Tsang, “Time dependent density of free carriers generated by two photon absorption in silicon waveguides,” Appl. Phys. Lett. 90, 211105 (2007).
[Crossref]

IEEE J. Quantum Electron. (1)

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. Hagan, and E. W. V. Stryland, “Sensitive measurements of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[Crossref]

J. Appl. Phys. (3)

B. Gu, J. He, W. Ji, and H. T. Wang, “Three-photon absorption saturation in ZnO and ZnS crystals,” J. Appl. Phys. 103, 073105 (2008).
[Crossref]

C. Xia and H. N. Spector, “Nonlinear Franz–Keldysh effect: Two photon absorption in semiconducting quantum wires and quantum boxes,” J. Appl. Phys. 106, 124302 (2009).
[Crossref]

C. Xia and H. N. Spector, “Franz–Keldysh effect in the interband optical absorption of semiconducting nanostructures,” J. Appl. Phys. 105, 084313 (2009).
[Crossref]

J. Opt. Soc. Am. B (3)

Phys. Rev. (2)

R. Braunstein, “Nonlinear optical effects,” Phys. Rev. 125, 475–477 (1962).
[Crossref]

R. Braunstein and N. Ockman, “Optical double-photon absorption in CdS,” Phys. Rev. 134, A499–A507 (1964).
[Crossref]

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Figures (7)

Fig. 1
Fig. 1

The TPA as a function of the photon energy in the QW with well 20 nm wide for zero field and the electric field of 300 kV/cm (curve a); the electric field is along the direction of carrier confinement and the polarization is in the plane of the QWs (curve b); the electric field and the polarization are in the plane of the well but perpendicular to each other (curve c). Here E g = E g + π 2 2 / 2 m μ L 2 .

Fig. 2
Fig. 2

The TPA as a function of the photon energy when the electric field is in the plane of the well while the polarization is along the direction of carrier confinement is shown for zero electric field and for an electric field of 300 kV/cm and a well 20 nm wide.

Fig. 3
Fig. 3

The difference in the TPA, Δ α = α ( w , F ) α ( w , 0 ) , is shown as a function of the electric field at the threshold photon energy w = E g / 2 in the QW with a well width of 20 nm. Curves b and c are of the same meaning as in Fig. 1.

Fig. 4
Fig. 4

The difference in the TPA, Δ α = α ( w , F ) α ( w , 0 ) , is shown as a function of the electric field at the threshold photon energy w = E g / 2 , where E g = E g + π 2 2 / 2 m e L 2 + 4 π 2 2 / 2 m h L 2 for TPA in the absence of the field for the case where the electric field is in the plane of the well and the polarization is along the direction of carrier confinement.

Fig. 5
Fig. 5

The ratio of the TPA in the QW with a well width of 20 nm to that in the bulk as a function of the photon energy, where the electric field and the optical polarization are in the plane of the well but perpendicular to each other.

Fig. 6
Fig. 6

The ratio of the TPA in the QW with a well width of 20 nm to that in the bulk as a function of the photon energy, where the electric field is along the direction of carrier confinement and the polarization is in the plane of the well.

Fig. 7
Fig. 7

The ratio of the TPA in the QW with a well width of 20 nm to that in the bulk as a function of the photon energy, where the electric field is in the plane of the well while the polarization is along the direction of confinement.

Equations (16)

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W tot = 1 V 2 π e 4 A 0 4 m 0 4 c 4 c , v i | c | ε P | i i | ε P | v E i E v w | 2 δ ( E c E v 2 w ) ,
α = ( 4 w I 2 ) W tot ,
W tot = e 4 A 0 2 m 0 4 c 4 ( m μ 2 2 2 e F ) 1 / 3 ( ε P c v ) 2 L z ( w ) 2 ( 2 m μ E 0 2 ) 3 / 2 n ( ε ε n ) 1 / 2 | A i ( ε ) | 2 d ε ,
E 0 = ( e 2 F 2 2 2 m μ ) 1 / 3 ,
ε n = ( n 2 π 2 2 2 m μ L z 2 + E g 2 w ) / E 0 .
α 2 D = 4 ( ε P c v ) 2 2 L z w ( e 2 c m 0 2 ) 2 ( 8 π η w 2 ) 2 ( m μ 2 2 2 e F ) 1 / 3 ( 2 m μ E 0 2 ) 3 / 2 n ( ε ε n ) 1 / 2 | A i ( ε ) | 2 d ε .
α 2 D 0 = 8 ( ε P c v ) 2 w L ( m μ 2 ) 2 e 4 m 0 4 c 2 ( 8 π η w 2 ) 2 n [ 2 w n 2 π 2 2 2 m μ L z 2 E g ] S ( 2 w n 2 π 2 2 2 m μ L z 2 E g ) ,
W tot = ( e A 0 m 0 c ) 4 2 ( ε P c v ) 2 L z 3 ( m μ 2 2 2 e F ) 1 / 3 ( 2 m μ E 0 2 ) 1 / 2 n , n I 2 ( n , n ) [ [ w + ( n 2 n 2 ) π 2 2 2 m e L z 2 ] 2 + [ w ( n 2 n 2 ) π 2 2 2 m h L z 2 ] 2 ] ε n n ( ε ε n n ) 1 / 2 | A i ( ε ) | 2 d ε ,
I ( n , n ) = [ 1 ( 1 ) n n ] n n n 2 n 2 ,
ε n n = n 2 π 2 2 2 m e L z 2 + n 2 π 2 2 2 m h L z 2 + E g 2 w E 0 .
α 2 D = e 4 m 0 4 c 2 ( 8 π η w 2 ) 2 8 w ( ε P c v ) 2 L z 3 ( m μ 2 2 2 e F ) 1 / 3 ( 2 m μ E 0 2 ) 1 / 2 n , n I 2 ( n , n ) [ [ w + ( n 2 n 2 ) π 2 2 2 m e L z 2 ] 2 + [ w ( n 2 n 2 ) π 2 2 2 m h L z 2 ] 2 ] ε n n ( ε ε n n ) 1 / 2 | A i ( ε ) | 2 d ε .
α 2 D 0 = [ 16 2 w ( ε P c v ) 2 L 3 ] ( e 4 m 0 4 c 2 ) ( 8 π η w 2 ) 2 ( m μ 2 ) n , n I 2 ( n , n ) [ [ w + ( n 2 n 2 ) π 2 2 2 m e L 2 ] 2 + [ w ( n 2 n 2 ) π 2 2 2 m h L 2 ] 2 ] S [ 2 w n 2 π 2 2 2 m e L 2 n 2 π 2 2 2 m h L 2 E g ] .
W tot = ( ε P c v ) 2 2 L z w 2 ( e A 0 m 0 c ) 4 ( 2 m μ 2 ) 2 n , m I n m 2 [ 2 w E n E m E g ] S ( 2 w E n E m E g ) ,
I n m 2 = | L z / 2 L z / 2 ϕ m ( z ) ϕ n ( z ) d z | 2 L z / 2 L z / 2 | ϕ m ( z ) | 2 d z L z / 2 L z / 2 | ϕ n ( z ) | 2 d z .
α 2 D = 2 ( ε P c v ) 2 L z w e 4 m 0 4 c 2 ( 8 π η w 2 ) 2 ( 2 m μ 2 ) 2 n , m I n m 2 [ 2 w E n E m E g ] S ( 2 w E n E m E g ) .
Δ α = α ( w , F ) α ( w , 0 ) .

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