Abstract

The electronic band structures and optical properties of type-II superlattice (T2SL) photodetectors in the mid-infrared (IR) range are investigated. We formulate a rigorous band structure model using the 8-band k · p method to include the conduction and valence band mixing. After solving the 8 × 8 Hamiltonian and deriving explicitly the new momentum matrix elements in terms of envelope functions, optical transition rates are obtained through the Fermi’s golden rule under various doping and injection conditions. Optical measurements on T2SL photodetectors are compared with our model and show good agreement. Our modeling results of quantum structures connect directly to the device-level design and simulation. The predicted doping effect is readily applicable to the optimization of photodetectors. We further include interfacial (IF) layers to study the significance of their effect. Optical properties of T2SLs are expected to have a large tunable range by controlling the thickness and material composition of the IF layers. Our model provides an efficient tool for the designs of novel photodetectors.

© 2012 OSA

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  1. G. A. Sai-Halasz, R. Tsu, and L. Esaki, “A new semiconductor superlattice,” Appl. Phys. Lett. 30, 651–653 (1977).
    [CrossRef]
  2. D. L. Smith and C. Mailhiot, “Proposal for strained type II superlattice infrared detectors,” J. Appl. Phys. 62, 2545–2548 (1987).
    [CrossRef]
  3. J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264, 553–556 (1994).
    [CrossRef] [PubMed]
  4. C. S. Kim, M. Kim, W. W. Bewley, J. R. Lindle, C. L. Canedy, J. Abell, I. Vurgaftman, and J. R. Meyer, “Corrugated-sidewall interband cascade lasers with single-mode midwave-infrared emission at room temperature,” Appl. Phys. Lett. 95, 231103 (2009).
    [CrossRef]
  5. J. V. Li, R. Q. Yang, C. J. Hill, and S. L. Chuang, “Interband cascade detectors with room temperature photovoltaic operation,” Appl. Phys. Lett. 86, 101102 (2005).
    [CrossRef]
  6. H. Mohseni, V. I. Litvinov, and M. Razeghi, “Interface-induced suppression of the Auger recombination in type-II InAs/GaSb superlattices,” Phys. Rev. B 58, 15378–15380 (1998).
    [CrossRef]
  7. E. Kane, “The k · p method,” in Physics of III–V Cmopounds, Vol. 1 of Semiconductors and Semimetals,R. Willardson and A. Beer, eds. (Academic Press, New York, 1966), pp. 75–100.
    [PubMed]
  8. J. M. Luttinger and W. Kohn, “Motion of electrons and holes in perturbed periodic fields,” Phys. Rev. 97, 869–883 (1955).
    [CrossRef]
  9. G. Liu and S. L. Chuang, “Modeling of Sb-based type-II quantum cascade lasers,” Phys. Rev. B 65, 165220 (2002).
    [CrossRef]
  10. S. L. Chuang, Physics of Photonic Devices, 2nd ed. (Wiley, New York, 2009), Chap. 4 and 9.
  11. L.-W. Wang, S.-H. Wei, T. Mattila, A. Zunger, I. Vurgaftman, and J. R. Meyer, “Multiband coupling and electronic structure of (InAs)n/(GaSb)n superlattices,” Phys. Rev. B 60, 5590–5596 (1999).
    [CrossRef]
  12. Y.-M. Mu and S. S. Pei, “Effects of anisotropic k.p interactions on energy bands and optical properties of type-II interband cascade lasers,” J. Appl. Phys. 96, 1866–1879 (2004).
    [CrossRef]
  13. S. Mou, J. V. Li, and S. L. Chuang, “Quantum efficiency analysis of InAs-GaSb type-II superlattice photodiodes,” IEEE J. Quantum Electron. 45, 737–743 (2009).
    [CrossRef]
  14. F. Szmulowicz, “Derivation of a general expression for the momentum matrix elements within the envelope-function approximation,” Phys. Rev. B 51, 1613–1623 (1995).
    [CrossRef]
  15. Y.-C. Chang and R. B. James, “Saturation of intersubband transitions in p-type semiconductor quantum wells,” Phys. Rev. B 39, 12672–12681 (1989).
    [CrossRef]
  16. E. O. Kane, “Band structure of indium antimonide,” J. Phys. Chem. Solids 1, 249–261 (1957).
    [CrossRef]
  17. P. O. Löwdin, “A note on the quantum-mechanical perturbation theory,” J. Chem. Phys. 19, 1396–1401 (1951).
    [CrossRef]
  18. C. S. Chang and S. L. Chuang, “Modeling of strained quantum-well lasers with spin-orbit coupling,” IEEE J. Sel. Top. Quantum Electron. 1, 218–229 (1995).
    [CrossRef]
  19. J. V. Li, C. J. Hill, J. Mumolo, S. Gunapala, S. Mou, and S. L. Chuang, “Midinfrared type-II InAs/GaSb super-lattice photodiodes toward room temperature operation,” Appl. Phys. Lett. 93, 163505 (2008).
    [CrossRef]
  20. C. J. Hill, J. V. Li, J. M. Mumolo, and S. D. Gunapala, “MBE grown type-II MWIR and LWIR superlattice photodiodes,” Infrared Phys. Techn. 50, 187–190 (2007).
    [CrossRef]
  21. A. Khoshakhlagh, E. Plis, S. Myers, Y. D. Sharma, L. R. Dawson, and S. Krishna, “Optimization of InAs/GaSb type-II superlattice interfaces for long-wave (∼8μm) infrared detection,” J. Cryst. Growth 311, 1901–1904 (2009).
    [CrossRef]
  22. H. J. Haugan, G. J. Brown, L. Grazulis, K. Mahalingam, and D. H. Tomich, “Optimization of InAs/GaSb type-II superlattices for high performance of photodetectors,” Physica E 20, 527–530 (2004).
    [CrossRef]
  23. Y. Huang, J.-H. Ryou, R. D. Dupuis, A. Petschke, M. Mandl, and S. L. Chuang, “InAs/GaSb type-II superlattice structures and photodiodes grown by metalorganic chemical vapor deposition,” Appl. Phys. Lett. 96, 251107 (2010).
    [CrossRef]
  24. Y. Wei and M. Razeghi, “Modeling of type-II InAs/GaSb superlattices using an empirical tight-binding method and interface engineering,” Phys. Rev. B 69, 085316 (2004).
    [CrossRef]
  25. G. L. Bir and G. E. Pikus, Symmetry and Strain-Induced Effects in Semiconductors (Wiley, New York, 1974).
  26. T. B. Bahder, “Analytic dispersion relations near the γ point in strained zinc-blende crystals,” Phys. Rev. B 45, 1629–1637 (1992).
    [CrossRef]
  27. T. E. Ostromek, “Evaluation of matrix elements of the 8 × 8 k · p Hamiltonian with k-dependent spin-orbit contributions for the zinc-blende structure of GaAs,” Phys. Rev. B 54, 14467–14479 (1996).
    [CrossRef]
  28. G. Dresselhaus, “Spin-orbit coupling effects in zinc blende structures,” Phys. Rev. 100, 580–586 (1955).
    [CrossRef]
  29. I. Vurgaftman, J. R. Meyer, and L. R. Ram-Mohan, “Band parameters for IIIV compound semiconductors and their alloys,” J. Appl. Phys. 89, 5815–5875 (2001).
    [CrossRef]
  30. C. G. Van de Walle, “Band lineups and deformation potentials in the model-solid theory,” Phys. Rev. B 39, 1871–1883 (1989).
    [CrossRef]

2010

Y. Huang, J.-H. Ryou, R. D. Dupuis, A. Petschke, M. Mandl, and S. L. Chuang, “InAs/GaSb type-II superlattice structures and photodiodes grown by metalorganic chemical vapor deposition,” Appl. Phys. Lett. 96, 251107 (2010).
[CrossRef]

2009

A. Khoshakhlagh, E. Plis, S. Myers, Y. D. Sharma, L. R. Dawson, and S. Krishna, “Optimization of InAs/GaSb type-II superlattice interfaces for long-wave (∼8μm) infrared detection,” J. Cryst. Growth 311, 1901–1904 (2009).
[CrossRef]

C. S. Kim, M. Kim, W. W. Bewley, J. R. Lindle, C. L. Canedy, J. Abell, I. Vurgaftman, and J. R. Meyer, “Corrugated-sidewall interband cascade lasers with single-mode midwave-infrared emission at room temperature,” Appl. Phys. Lett. 95, 231103 (2009).
[CrossRef]

S. Mou, J. V. Li, and S. L. Chuang, “Quantum efficiency analysis of InAs-GaSb type-II superlattice photodiodes,” IEEE J. Quantum Electron. 45, 737–743 (2009).
[CrossRef]

2008

J. V. Li, C. J. Hill, J. Mumolo, S. Gunapala, S. Mou, and S. L. Chuang, “Midinfrared type-II InAs/GaSb super-lattice photodiodes toward room temperature operation,” Appl. Phys. Lett. 93, 163505 (2008).
[CrossRef]

2007

C. J. Hill, J. V. Li, J. M. Mumolo, and S. D. Gunapala, “MBE grown type-II MWIR and LWIR superlattice photodiodes,” Infrared Phys. Techn. 50, 187–190 (2007).
[CrossRef]

2005

J. V. Li, R. Q. Yang, C. J. Hill, and S. L. Chuang, “Interband cascade detectors with room temperature photovoltaic operation,” Appl. Phys. Lett. 86, 101102 (2005).
[CrossRef]

2004

Y.-M. Mu and S. S. Pei, “Effects of anisotropic k.p interactions on energy bands and optical properties of type-II interband cascade lasers,” J. Appl. Phys. 96, 1866–1879 (2004).
[CrossRef]

H. J. Haugan, G. J. Brown, L. Grazulis, K. Mahalingam, and D. H. Tomich, “Optimization of InAs/GaSb type-II superlattices for high performance of photodetectors,” Physica E 20, 527–530 (2004).
[CrossRef]

Y. Wei and M. Razeghi, “Modeling of type-II InAs/GaSb superlattices using an empirical tight-binding method and interface engineering,” Phys. Rev. B 69, 085316 (2004).
[CrossRef]

2002

G. Liu and S. L. Chuang, “Modeling of Sb-based type-II quantum cascade lasers,” Phys. Rev. B 65, 165220 (2002).
[CrossRef]

2001

I. Vurgaftman, J. R. Meyer, and L. R. Ram-Mohan, “Band parameters for IIIV compound semiconductors and their alloys,” J. Appl. Phys. 89, 5815–5875 (2001).
[CrossRef]

1999

L.-W. Wang, S.-H. Wei, T. Mattila, A. Zunger, I. Vurgaftman, and J. R. Meyer, “Multiband coupling and electronic structure of (InAs)n/(GaSb)n superlattices,” Phys. Rev. B 60, 5590–5596 (1999).
[CrossRef]

1998

H. Mohseni, V. I. Litvinov, and M. Razeghi, “Interface-induced suppression of the Auger recombination in type-II InAs/GaSb superlattices,” Phys. Rev. B 58, 15378–15380 (1998).
[CrossRef]

1996

T. E. Ostromek, “Evaluation of matrix elements of the 8 × 8 k · p Hamiltonian with k-dependent spin-orbit contributions for the zinc-blende structure of GaAs,” Phys. Rev. B 54, 14467–14479 (1996).
[CrossRef]

1995

C. S. Chang and S. L. Chuang, “Modeling of strained quantum-well lasers with spin-orbit coupling,” IEEE J. Sel. Top. Quantum Electron. 1, 218–229 (1995).
[CrossRef]

F. Szmulowicz, “Derivation of a general expression for the momentum matrix elements within the envelope-function approximation,” Phys. Rev. B 51, 1613–1623 (1995).
[CrossRef]

1994

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264, 553–556 (1994).
[CrossRef] [PubMed]

1992

T. B. Bahder, “Analytic dispersion relations near the γ point in strained zinc-blende crystals,” Phys. Rev. B 45, 1629–1637 (1992).
[CrossRef]

1989

C. G. Van de Walle, “Band lineups and deformation potentials in the model-solid theory,” Phys. Rev. B 39, 1871–1883 (1989).
[CrossRef]

Y.-C. Chang and R. B. James, “Saturation of intersubband transitions in p-type semiconductor quantum wells,” Phys. Rev. B 39, 12672–12681 (1989).
[CrossRef]

1987

D. L. Smith and C. Mailhiot, “Proposal for strained type II superlattice infrared detectors,” J. Appl. Phys. 62, 2545–2548 (1987).
[CrossRef]

1977

G. A. Sai-Halasz, R. Tsu, and L. Esaki, “A new semiconductor superlattice,” Appl. Phys. Lett. 30, 651–653 (1977).
[CrossRef]

1957

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

1955

J. M. Luttinger and W. Kohn, “Motion of electrons and holes in perturbed periodic fields,” Phys. Rev. 97, 869–883 (1955).
[CrossRef]

G. Dresselhaus, “Spin-orbit coupling effects in zinc blende structures,” Phys. Rev. 100, 580–586 (1955).
[CrossRef]

1951

P. O. Löwdin, “A note on the quantum-mechanical perturbation theory,” J. Chem. Phys. 19, 1396–1401 (1951).
[CrossRef]

Abell, J.

C. S. Kim, M. Kim, W. W. Bewley, J. R. Lindle, C. L. Canedy, J. Abell, I. Vurgaftman, and J. R. Meyer, “Corrugated-sidewall interband cascade lasers with single-mode midwave-infrared emission at room temperature,” Appl. Phys. Lett. 95, 231103 (2009).
[CrossRef]

Bahder, T. B.

T. B. Bahder, “Analytic dispersion relations near the γ point in strained zinc-blende crystals,” Phys. Rev. B 45, 1629–1637 (1992).
[CrossRef]

Bewley, W. W.

C. S. Kim, M. Kim, W. W. Bewley, J. R. Lindle, C. L. Canedy, J. Abell, I. Vurgaftman, and J. R. Meyer, “Corrugated-sidewall interband cascade lasers with single-mode midwave-infrared emission at room temperature,” Appl. Phys. Lett. 95, 231103 (2009).
[CrossRef]

Bir, G. L.

G. L. Bir and G. E. Pikus, Symmetry and Strain-Induced Effects in Semiconductors (Wiley, New York, 1974).

Brown, G. J.

H. J. Haugan, G. J. Brown, L. Grazulis, K. Mahalingam, and D. H. Tomich, “Optimization of InAs/GaSb type-II superlattices for high performance of photodetectors,” Physica E 20, 527–530 (2004).
[CrossRef]

Canedy, C. L.

C. S. Kim, M. Kim, W. W. Bewley, J. R. Lindle, C. L. Canedy, J. Abell, I. Vurgaftman, and J. R. Meyer, “Corrugated-sidewall interband cascade lasers with single-mode midwave-infrared emission at room temperature,” Appl. Phys. Lett. 95, 231103 (2009).
[CrossRef]

Capasso, F.

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264, 553–556 (1994).
[CrossRef] [PubMed]

Chang, C. S.

C. S. Chang and S. L. Chuang, “Modeling of strained quantum-well lasers with spin-orbit coupling,” IEEE J. Sel. Top. Quantum Electron. 1, 218–229 (1995).
[CrossRef]

Chang, Y.-C.

Y.-C. Chang and R. B. James, “Saturation of intersubband transitions in p-type semiconductor quantum wells,” Phys. Rev. B 39, 12672–12681 (1989).
[CrossRef]

Cho, A. Y.

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264, 553–556 (1994).
[CrossRef] [PubMed]

Chuang, S. L.

Y. Huang, J.-H. Ryou, R. D. Dupuis, A. Petschke, M. Mandl, and S. L. Chuang, “InAs/GaSb type-II superlattice structures and photodiodes grown by metalorganic chemical vapor deposition,” Appl. Phys. Lett. 96, 251107 (2010).
[CrossRef]

S. Mou, J. V. Li, and S. L. Chuang, “Quantum efficiency analysis of InAs-GaSb type-II superlattice photodiodes,” IEEE J. Quantum Electron. 45, 737–743 (2009).
[CrossRef]

J. V. Li, C. J. Hill, J. Mumolo, S. Gunapala, S. Mou, and S. L. Chuang, “Midinfrared type-II InAs/GaSb super-lattice photodiodes toward room temperature operation,” Appl. Phys. Lett. 93, 163505 (2008).
[CrossRef]

J. V. Li, R. Q. Yang, C. J. Hill, and S. L. Chuang, “Interband cascade detectors with room temperature photovoltaic operation,” Appl. Phys. Lett. 86, 101102 (2005).
[CrossRef]

G. Liu and S. L. Chuang, “Modeling of Sb-based type-II quantum cascade lasers,” Phys. Rev. B 65, 165220 (2002).
[CrossRef]

C. S. Chang and S. L. Chuang, “Modeling of strained quantum-well lasers with spin-orbit coupling,” IEEE J. Sel. Top. Quantum Electron. 1, 218–229 (1995).
[CrossRef]

S. L. Chuang, Physics of Photonic Devices, 2nd ed. (Wiley, New York, 2009), Chap. 4 and 9.

Dawson, L. R.

A. Khoshakhlagh, E. Plis, S. Myers, Y. D. Sharma, L. R. Dawson, and S. Krishna, “Optimization of InAs/GaSb type-II superlattice interfaces for long-wave (∼8μm) infrared detection,” J. Cryst. Growth 311, 1901–1904 (2009).
[CrossRef]

Dresselhaus, G.

G. Dresselhaus, “Spin-orbit coupling effects in zinc blende structures,” Phys. Rev. 100, 580–586 (1955).
[CrossRef]

Dupuis, R. D.

Y. Huang, J.-H. Ryou, R. D. Dupuis, A. Petschke, M. Mandl, and S. L. Chuang, “InAs/GaSb type-II superlattice structures and photodiodes grown by metalorganic chemical vapor deposition,” Appl. Phys. Lett. 96, 251107 (2010).
[CrossRef]

Esaki, L.

G. A. Sai-Halasz, R. Tsu, and L. Esaki, “A new semiconductor superlattice,” Appl. Phys. Lett. 30, 651–653 (1977).
[CrossRef]

Faist, J.

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264, 553–556 (1994).
[CrossRef] [PubMed]

Grazulis, L.

H. J. Haugan, G. J. Brown, L. Grazulis, K. Mahalingam, and D. H. Tomich, “Optimization of InAs/GaSb type-II superlattices for high performance of photodetectors,” Physica E 20, 527–530 (2004).
[CrossRef]

Gunapala, S.

J. V. Li, C. J. Hill, J. Mumolo, S. Gunapala, S. Mou, and S. L. Chuang, “Midinfrared type-II InAs/GaSb super-lattice photodiodes toward room temperature operation,” Appl. Phys. Lett. 93, 163505 (2008).
[CrossRef]

Gunapala, S. D.

C. J. Hill, J. V. Li, J. M. Mumolo, and S. D. Gunapala, “MBE grown type-II MWIR and LWIR superlattice photodiodes,” Infrared Phys. Techn. 50, 187–190 (2007).
[CrossRef]

Haugan, H. J.

H. J. Haugan, G. J. Brown, L. Grazulis, K. Mahalingam, and D. H. Tomich, “Optimization of InAs/GaSb type-II superlattices for high performance of photodetectors,” Physica E 20, 527–530 (2004).
[CrossRef]

Hill, C. J.

J. V. Li, C. J. Hill, J. Mumolo, S. Gunapala, S. Mou, and S. L. Chuang, “Midinfrared type-II InAs/GaSb super-lattice photodiodes toward room temperature operation,” Appl. Phys. Lett. 93, 163505 (2008).
[CrossRef]

C. J. Hill, J. V. Li, J. M. Mumolo, and S. D. Gunapala, “MBE grown type-II MWIR and LWIR superlattice photodiodes,” Infrared Phys. Techn. 50, 187–190 (2007).
[CrossRef]

J. V. Li, R. Q. Yang, C. J. Hill, and S. L. Chuang, “Interband cascade detectors with room temperature photovoltaic operation,” Appl. Phys. Lett. 86, 101102 (2005).
[CrossRef]

Huang, Y.

Y. Huang, J.-H. Ryou, R. D. Dupuis, A. Petschke, M. Mandl, and S. L. Chuang, “InAs/GaSb type-II superlattice structures and photodiodes grown by metalorganic chemical vapor deposition,” Appl. Phys. Lett. 96, 251107 (2010).
[CrossRef]

Hutchinson, A. L.

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264, 553–556 (1994).
[CrossRef] [PubMed]

James, R. B.

Y.-C. Chang and R. B. James, “Saturation of intersubband transitions in p-type semiconductor quantum wells,” Phys. Rev. B 39, 12672–12681 (1989).
[CrossRef]

Kane, E.

E. Kane, “The k · p method,” in Physics of III–V Cmopounds, Vol. 1 of Semiconductors and Semimetals,R. Willardson and A. Beer, eds. (Academic Press, New York, 1966), pp. 75–100.
[PubMed]

Kane, E. O.

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

Khoshakhlagh, A.

A. Khoshakhlagh, E. Plis, S. Myers, Y. D. Sharma, L. R. Dawson, and S. Krishna, “Optimization of InAs/GaSb type-II superlattice interfaces for long-wave (∼8μm) infrared detection,” J. Cryst. Growth 311, 1901–1904 (2009).
[CrossRef]

Kim, C. S.

C. S. Kim, M. Kim, W. W. Bewley, J. R. Lindle, C. L. Canedy, J. Abell, I. Vurgaftman, and J. R. Meyer, “Corrugated-sidewall interband cascade lasers with single-mode midwave-infrared emission at room temperature,” Appl. Phys. Lett. 95, 231103 (2009).
[CrossRef]

Kim, M.

C. S. Kim, M. Kim, W. W. Bewley, J. R. Lindle, C. L. Canedy, J. Abell, I. Vurgaftman, and J. R. Meyer, “Corrugated-sidewall interband cascade lasers with single-mode midwave-infrared emission at room temperature,” Appl. Phys. Lett. 95, 231103 (2009).
[CrossRef]

Kohn, W.

J. M. Luttinger and W. Kohn, “Motion of electrons and holes in perturbed periodic fields,” Phys. Rev. 97, 869–883 (1955).
[CrossRef]

Krishna, S.

A. Khoshakhlagh, E. Plis, S. Myers, Y. D. Sharma, L. R. Dawson, and S. Krishna, “Optimization of InAs/GaSb type-II superlattice interfaces for long-wave (∼8μm) infrared detection,” J. Cryst. Growth 311, 1901–1904 (2009).
[CrossRef]

Li, J. V.

S. Mou, J. V. Li, and S. L. Chuang, “Quantum efficiency analysis of InAs-GaSb type-II superlattice photodiodes,” IEEE J. Quantum Electron. 45, 737–743 (2009).
[CrossRef]

J. V. Li, C. J. Hill, J. Mumolo, S. Gunapala, S. Mou, and S. L. Chuang, “Midinfrared type-II InAs/GaSb super-lattice photodiodes toward room temperature operation,” Appl. Phys. Lett. 93, 163505 (2008).
[CrossRef]

C. J. Hill, J. V. Li, J. M. Mumolo, and S. D. Gunapala, “MBE grown type-II MWIR and LWIR superlattice photodiodes,” Infrared Phys. Techn. 50, 187–190 (2007).
[CrossRef]

J. V. Li, R. Q. Yang, C. J. Hill, and S. L. Chuang, “Interband cascade detectors with room temperature photovoltaic operation,” Appl. Phys. Lett. 86, 101102 (2005).
[CrossRef]

Lindle, J. R.

C. S. Kim, M. Kim, W. W. Bewley, J. R. Lindle, C. L. Canedy, J. Abell, I. Vurgaftman, and J. R. Meyer, “Corrugated-sidewall interband cascade lasers with single-mode midwave-infrared emission at room temperature,” Appl. Phys. Lett. 95, 231103 (2009).
[CrossRef]

Litvinov, V. I.

H. Mohseni, V. I. Litvinov, and M. Razeghi, “Interface-induced suppression of the Auger recombination in type-II InAs/GaSb superlattices,” Phys. Rev. B 58, 15378–15380 (1998).
[CrossRef]

Liu, G.

G. Liu and S. L. Chuang, “Modeling of Sb-based type-II quantum cascade lasers,” Phys. Rev. B 65, 165220 (2002).
[CrossRef]

Löwdin, P. O.

P. O. Löwdin, “A note on the quantum-mechanical perturbation theory,” J. Chem. Phys. 19, 1396–1401 (1951).
[CrossRef]

Luttinger, J. M.

J. M. Luttinger and W. Kohn, “Motion of electrons and holes in perturbed periodic fields,” Phys. Rev. 97, 869–883 (1955).
[CrossRef]

Mahalingam, K.

H. J. Haugan, G. J. Brown, L. Grazulis, K. Mahalingam, and D. H. Tomich, “Optimization of InAs/GaSb type-II superlattices for high performance of photodetectors,” Physica E 20, 527–530 (2004).
[CrossRef]

Mailhiot, C.

D. L. Smith and C. Mailhiot, “Proposal for strained type II superlattice infrared detectors,” J. Appl. Phys. 62, 2545–2548 (1987).
[CrossRef]

Mandl, M.

Y. Huang, J.-H. Ryou, R. D. Dupuis, A. Petschke, M. Mandl, and S. L. Chuang, “InAs/GaSb type-II superlattice structures and photodiodes grown by metalorganic chemical vapor deposition,” Appl. Phys. Lett. 96, 251107 (2010).
[CrossRef]

Mattila, T.

L.-W. Wang, S.-H. Wei, T. Mattila, A. Zunger, I. Vurgaftman, and J. R. Meyer, “Multiband coupling and electronic structure of (InAs)n/(GaSb)n superlattices,” Phys. Rev. B 60, 5590–5596 (1999).
[CrossRef]

Meyer, J. R.

C. S. Kim, M. Kim, W. W. Bewley, J. R. Lindle, C. L. Canedy, J. Abell, I. Vurgaftman, and J. R. Meyer, “Corrugated-sidewall interband cascade lasers with single-mode midwave-infrared emission at room temperature,” Appl. Phys. Lett. 95, 231103 (2009).
[CrossRef]

I. Vurgaftman, J. R. Meyer, and L. R. Ram-Mohan, “Band parameters for IIIV compound semiconductors and their alloys,” J. Appl. Phys. 89, 5815–5875 (2001).
[CrossRef]

L.-W. Wang, S.-H. Wei, T. Mattila, A. Zunger, I. Vurgaftman, and J. R. Meyer, “Multiband coupling and electronic structure of (InAs)n/(GaSb)n superlattices,” Phys. Rev. B 60, 5590–5596 (1999).
[CrossRef]

Mohseni, H.

H. Mohseni, V. I. Litvinov, and M. Razeghi, “Interface-induced suppression of the Auger recombination in type-II InAs/GaSb superlattices,” Phys. Rev. B 58, 15378–15380 (1998).
[CrossRef]

Mou, S.

S. Mou, J. V. Li, and S. L. Chuang, “Quantum efficiency analysis of InAs-GaSb type-II superlattice photodiodes,” IEEE J. Quantum Electron. 45, 737–743 (2009).
[CrossRef]

J. V. Li, C. J. Hill, J. Mumolo, S. Gunapala, S. Mou, and S. L. Chuang, “Midinfrared type-II InAs/GaSb super-lattice photodiodes toward room temperature operation,” Appl. Phys. Lett. 93, 163505 (2008).
[CrossRef]

Mu, Y.-M.

Y.-M. Mu and S. S. Pei, “Effects of anisotropic k.p interactions on energy bands and optical properties of type-II interband cascade lasers,” J. Appl. Phys. 96, 1866–1879 (2004).
[CrossRef]

Mumolo, J.

J. V. Li, C. J. Hill, J. Mumolo, S. Gunapala, S. Mou, and S. L. Chuang, “Midinfrared type-II InAs/GaSb super-lattice photodiodes toward room temperature operation,” Appl. Phys. Lett. 93, 163505 (2008).
[CrossRef]

Mumolo, J. M.

C. J. Hill, J. V. Li, J. M. Mumolo, and S. D. Gunapala, “MBE grown type-II MWIR and LWIR superlattice photodiodes,” Infrared Phys. Techn. 50, 187–190 (2007).
[CrossRef]

Myers, S.

A. Khoshakhlagh, E. Plis, S. Myers, Y. D. Sharma, L. R. Dawson, and S. Krishna, “Optimization of InAs/GaSb type-II superlattice interfaces for long-wave (∼8μm) infrared detection,” J. Cryst. Growth 311, 1901–1904 (2009).
[CrossRef]

Ostromek, T. E.

T. E. Ostromek, “Evaluation of matrix elements of the 8 × 8 k · p Hamiltonian with k-dependent spin-orbit contributions for the zinc-blende structure of GaAs,” Phys. Rev. B 54, 14467–14479 (1996).
[CrossRef]

Pei, S. S.

Y.-M. Mu and S. S. Pei, “Effects of anisotropic k.p interactions on energy bands and optical properties of type-II interband cascade lasers,” J. Appl. Phys. 96, 1866–1879 (2004).
[CrossRef]

Petschke, A.

Y. Huang, J.-H. Ryou, R. D. Dupuis, A. Petschke, M. Mandl, and S. L. Chuang, “InAs/GaSb type-II superlattice structures and photodiodes grown by metalorganic chemical vapor deposition,” Appl. Phys. Lett. 96, 251107 (2010).
[CrossRef]

Pikus, G. E.

G. L. Bir and G. E. Pikus, Symmetry and Strain-Induced Effects in Semiconductors (Wiley, New York, 1974).

Plis, E.

A. Khoshakhlagh, E. Plis, S. Myers, Y. D. Sharma, L. R. Dawson, and S. Krishna, “Optimization of InAs/GaSb type-II superlattice interfaces for long-wave (∼8μm) infrared detection,” J. Cryst. Growth 311, 1901–1904 (2009).
[CrossRef]

Ram-Mohan, L. R.

I. Vurgaftman, J. R. Meyer, and L. R. Ram-Mohan, “Band parameters for IIIV compound semiconductors and their alloys,” J. Appl. Phys. 89, 5815–5875 (2001).
[CrossRef]

Razeghi, M.

Y. Wei and M. Razeghi, “Modeling of type-II InAs/GaSb superlattices using an empirical tight-binding method and interface engineering,” Phys. Rev. B 69, 085316 (2004).
[CrossRef]

H. Mohseni, V. I. Litvinov, and M. Razeghi, “Interface-induced suppression of the Auger recombination in type-II InAs/GaSb superlattices,” Phys. Rev. B 58, 15378–15380 (1998).
[CrossRef]

Ryou, J.-H.

Y. Huang, J.-H. Ryou, R. D. Dupuis, A. Petschke, M. Mandl, and S. L. Chuang, “InAs/GaSb type-II superlattice structures and photodiodes grown by metalorganic chemical vapor deposition,” Appl. Phys. Lett. 96, 251107 (2010).
[CrossRef]

Sai-Halasz, G. A.

G. A. Sai-Halasz, R. Tsu, and L. Esaki, “A new semiconductor superlattice,” Appl. Phys. Lett. 30, 651–653 (1977).
[CrossRef]

Sharma, Y. D.

A. Khoshakhlagh, E. Plis, S. Myers, Y. D. Sharma, L. R. Dawson, and S. Krishna, “Optimization of InAs/GaSb type-II superlattice interfaces for long-wave (∼8μm) infrared detection,” J. Cryst. Growth 311, 1901–1904 (2009).
[CrossRef]

Sirtori, C.

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264, 553–556 (1994).
[CrossRef] [PubMed]

Sivco, D. L.

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264, 553–556 (1994).
[CrossRef] [PubMed]

Smith, D. L.

D. L. Smith and C. Mailhiot, “Proposal for strained type II superlattice infrared detectors,” J. Appl. Phys. 62, 2545–2548 (1987).
[CrossRef]

Szmulowicz, F.

F. Szmulowicz, “Derivation of a general expression for the momentum matrix elements within the envelope-function approximation,” Phys. Rev. B 51, 1613–1623 (1995).
[CrossRef]

Tomich, D. H.

H. J. Haugan, G. J. Brown, L. Grazulis, K. Mahalingam, and D. H. Tomich, “Optimization of InAs/GaSb type-II superlattices for high performance of photodetectors,” Physica E 20, 527–530 (2004).
[CrossRef]

Tsu, R.

G. A. Sai-Halasz, R. Tsu, and L. Esaki, “A new semiconductor superlattice,” Appl. Phys. Lett. 30, 651–653 (1977).
[CrossRef]

Van de Walle, C. G.

C. G. Van de Walle, “Band lineups and deformation potentials in the model-solid theory,” Phys. Rev. B 39, 1871–1883 (1989).
[CrossRef]

Vurgaftman, I.

C. S. Kim, M. Kim, W. W. Bewley, J. R. Lindle, C. L. Canedy, J. Abell, I. Vurgaftman, and J. R. Meyer, “Corrugated-sidewall interband cascade lasers with single-mode midwave-infrared emission at room temperature,” Appl. Phys. Lett. 95, 231103 (2009).
[CrossRef]

I. Vurgaftman, J. R. Meyer, and L. R. Ram-Mohan, “Band parameters for IIIV compound semiconductors and their alloys,” J. Appl. Phys. 89, 5815–5875 (2001).
[CrossRef]

L.-W. Wang, S.-H. Wei, T. Mattila, A. Zunger, I. Vurgaftman, and J. R. Meyer, “Multiband coupling and electronic structure of (InAs)n/(GaSb)n superlattices,” Phys. Rev. B 60, 5590–5596 (1999).
[CrossRef]

Wang, L.-W.

L.-W. Wang, S.-H. Wei, T. Mattila, A. Zunger, I. Vurgaftman, and J. R. Meyer, “Multiband coupling and electronic structure of (InAs)n/(GaSb)n superlattices,” Phys. Rev. B 60, 5590–5596 (1999).
[CrossRef]

Wei, S.-H.

L.-W. Wang, S.-H. Wei, T. Mattila, A. Zunger, I. Vurgaftman, and J. R. Meyer, “Multiband coupling and electronic structure of (InAs)n/(GaSb)n superlattices,” Phys. Rev. B 60, 5590–5596 (1999).
[CrossRef]

Wei, Y.

Y. Wei and M. Razeghi, “Modeling of type-II InAs/GaSb superlattices using an empirical tight-binding method and interface engineering,” Phys. Rev. B 69, 085316 (2004).
[CrossRef]

Yang, R. Q.

J. V. Li, R. Q. Yang, C. J. Hill, and S. L. Chuang, “Interband cascade detectors with room temperature photovoltaic operation,” Appl. Phys. Lett. 86, 101102 (2005).
[CrossRef]

Zunger, A.

L.-W. Wang, S.-H. Wei, T. Mattila, A. Zunger, I. Vurgaftman, and J. R. Meyer, “Multiband coupling and electronic structure of (InAs)n/(GaSb)n superlattices,” Phys. Rev. B 60, 5590–5596 (1999).
[CrossRef]

Appl. Phys. Lett.

C. S. Kim, M. Kim, W. W. Bewley, J. R. Lindle, C. L. Canedy, J. Abell, I. Vurgaftman, and J. R. Meyer, “Corrugated-sidewall interband cascade lasers with single-mode midwave-infrared emission at room temperature,” Appl. Phys. Lett. 95, 231103 (2009).
[CrossRef]

J. V. Li, R. Q. Yang, C. J. Hill, and S. L. Chuang, “Interband cascade detectors with room temperature photovoltaic operation,” Appl. Phys. Lett. 86, 101102 (2005).
[CrossRef]

G. A. Sai-Halasz, R. Tsu, and L. Esaki, “A new semiconductor superlattice,” Appl. Phys. Lett. 30, 651–653 (1977).
[CrossRef]

J. V. Li, C. J. Hill, J. Mumolo, S. Gunapala, S. Mou, and S. L. Chuang, “Midinfrared type-II InAs/GaSb super-lattice photodiodes toward room temperature operation,” Appl. Phys. Lett. 93, 163505 (2008).
[CrossRef]

Y. Huang, J.-H. Ryou, R. D. Dupuis, A. Petschke, M. Mandl, and S. L. Chuang, “InAs/GaSb type-II superlattice structures and photodiodes grown by metalorganic chemical vapor deposition,” Appl. Phys. Lett. 96, 251107 (2010).
[CrossRef]

IEEE J. Quantum Electron.

S. Mou, J. V. Li, and S. L. Chuang, “Quantum efficiency analysis of InAs-GaSb type-II superlattice photodiodes,” IEEE J. Quantum Electron. 45, 737–743 (2009).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

C. S. Chang and S. L. Chuang, “Modeling of strained quantum-well lasers with spin-orbit coupling,” IEEE J. Sel. Top. Quantum Electron. 1, 218–229 (1995).
[CrossRef]

Infrared Phys. Techn.

C. J. Hill, J. V. Li, J. M. Mumolo, and S. D. Gunapala, “MBE grown type-II MWIR and LWIR superlattice photodiodes,” Infrared Phys. Techn. 50, 187–190 (2007).
[CrossRef]

J. Appl. Phys.

Y.-M. Mu and S. S. Pei, “Effects of anisotropic k.p interactions on energy bands and optical properties of type-II interband cascade lasers,” J. Appl. Phys. 96, 1866–1879 (2004).
[CrossRef]

D. L. Smith and C. Mailhiot, “Proposal for strained type II superlattice infrared detectors,” J. Appl. Phys. 62, 2545–2548 (1987).
[CrossRef]

I. Vurgaftman, J. R. Meyer, and L. R. Ram-Mohan, “Band parameters for IIIV compound semiconductors and their alloys,” J. Appl. Phys. 89, 5815–5875 (2001).
[CrossRef]

J. Chem. Phys.

P. O. Löwdin, “A note on the quantum-mechanical perturbation theory,” J. Chem. Phys. 19, 1396–1401 (1951).
[CrossRef]

J. Cryst. Growth

A. Khoshakhlagh, E. Plis, S. Myers, Y. D. Sharma, L. R. Dawson, and S. Krishna, “Optimization of InAs/GaSb type-II superlattice interfaces for long-wave (∼8μm) infrared detection,” J. Cryst. Growth 311, 1901–1904 (2009).
[CrossRef]

J. Phys. Chem. Solids

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

Phys. Rev.

G. Dresselhaus, “Spin-orbit coupling effects in zinc blende structures,” Phys. Rev. 100, 580–586 (1955).
[CrossRef]

J. M. Luttinger and W. Kohn, “Motion of electrons and holes in perturbed periodic fields,” Phys. Rev. 97, 869–883 (1955).
[CrossRef]

Phys. Rev. B

G. Liu and S. L. Chuang, “Modeling of Sb-based type-II quantum cascade lasers,” Phys. Rev. B 65, 165220 (2002).
[CrossRef]

H. Mohseni, V. I. Litvinov, and M. Razeghi, “Interface-induced suppression of the Auger recombination in type-II InAs/GaSb superlattices,” Phys. Rev. B 58, 15378–15380 (1998).
[CrossRef]

L.-W. Wang, S.-H. Wei, T. Mattila, A. Zunger, I. Vurgaftman, and J. R. Meyer, “Multiband coupling and electronic structure of (InAs)n/(GaSb)n superlattices,” Phys. Rev. B 60, 5590–5596 (1999).
[CrossRef]

F. Szmulowicz, “Derivation of a general expression for the momentum matrix elements within the envelope-function approximation,” Phys. Rev. B 51, 1613–1623 (1995).
[CrossRef]

Y.-C. Chang and R. B. James, “Saturation of intersubband transitions in p-type semiconductor quantum wells,” Phys. Rev. B 39, 12672–12681 (1989).
[CrossRef]

T. B. Bahder, “Analytic dispersion relations near the γ point in strained zinc-blende crystals,” Phys. Rev. B 45, 1629–1637 (1992).
[CrossRef]

T. E. Ostromek, “Evaluation of matrix elements of the 8 × 8 k · p Hamiltonian with k-dependent spin-orbit contributions for the zinc-blende structure of GaAs,” Phys. Rev. B 54, 14467–14479 (1996).
[CrossRef]

Y. Wei and M. Razeghi, “Modeling of type-II InAs/GaSb superlattices using an empirical tight-binding method and interface engineering,” Phys. Rev. B 69, 085316 (2004).
[CrossRef]

C. G. Van de Walle, “Band lineups and deformation potentials in the model-solid theory,” Phys. Rev. B 39, 1871–1883 (1989).
[CrossRef]

Physica E

H. J. Haugan, G. J. Brown, L. Grazulis, K. Mahalingam, and D. H. Tomich, “Optimization of InAs/GaSb type-II superlattices for high performance of photodetectors,” Physica E 20, 527–530 (2004).
[CrossRef]

Science

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264, 553–556 (1994).
[CrossRef] [PubMed]

Other

S. L. Chuang, Physics of Photonic Devices, 2nd ed. (Wiley, New York, 2009), Chap. 4 and 9.

E. Kane, “The k · p method,” in Physics of III–V Cmopounds, Vol. 1 of Semiconductors and Semimetals,R. Willardson and A. Beer, eds. (Academic Press, New York, 1966), pp. 75–100.
[PubMed]

G. L. Bir and G. E. Pikus, Symmetry and Strain-Induced Effects in Semiconductors (Wiley, New York, 1974).

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Figures (7)

Fig. 1.
Fig. 1.

Wavefunctions at kt = 0 solved using the Dirichlet boundary condition (DBC) and the periodic boundary condition (PBC). |g(1)| of the lowest C1 band (green) and |g(2)| of the highest HH1 band (red) are plotted with C1 and HH1 band edges (blue dashed). The zero energy reference is chosen as the unstrained conduction band edge of InAs.

Fig. 2.
Fig. 2.

Energy dispersion relation to transverse wave number. Blue dashed lines indicate the band edge energies. The zero energy reference is chosen as the unstrained conduction band edge of InAs.

Fig. 3.
Fig. 3.

Theoretical TE absorption spectra of an InAs/GaSb 44 Å/21 Å T2SL on GaSb substrate using Dirichlet boundary condition (DBC) with 7 (blue solid), 9 (green solid), and 11 (red solid) periods, and using the periodic boundary condition (PBC) with 11 (red dashed), 15 (green dashed), and 19 (blue dashed) periods. Inset: Total absorption (blue), first heavy-hole (HH1) to all conduction subbands (CBs) absorption (black), first light-hole (LH1) to all CBs absorption (green), and second heavy-hole (HH2) to all CBs absorption (red).

Fig. 4.
Fig. 4.

Modeled external quantum efficiency and photoluminescence (PL) spectra of a 300-period lightly p-doped InAs/GaSb 45 Å/24 Å T2SL absorber with NA = 5 × 1016 cm−3 (cross), NA = 2 × 1016 cm−3 (dashed), NA = 5 × 1015 cm−3 (solid).

Fig. 5.
Fig. 5.

Comparison between theoretical model and experimental data [19] for an InAs/GaSb 18 Å/22 Å MWIR T2SL on GaSb substrate. Both the periodic boundary condition (PBC) and Dirichlet boundary condition (DBC) (inset) are used, along with different linewidth parameter γ.

Fig. 6.
Fig. 6.

Comparison between our theoretical model and experimental data [20] for an InAs/GaSb 48 Å/22 Å LWIR T2SL on GaSb substrate. Both the periodic boundary condition (PBC) and Dirichlet boundary condition (DBC) are used and different valence band offset (VBO) values take into account the interfacial effect if actual interfacial layers are not included in the model.

Fig. 7.
Fig. 7.

Modeled TE absorption (cross) and PL (solid) spectra for InAs/GaSb 44 Å/21 Å T2SL on GaSb substrate when including InSb interfacial layers of different thicknesses: 0 Å (purple), 1 Å (red), 1.5 Å (green), 2 Å (blue), 2.5 Å (black). The inset shows the band edge profile at InAs/GaSb interface with forced InSb layer in between. The blue and black lines are unstrained conduction band and valence band edges respectively. The green and red dashed lines are conduction and heavy-hole band edges respectively including strain effect.

Tables (1)

Tables Icon

Table 1. Input material parameters for the 8-band k · p method

Equations (29)

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

H 8 × 8 LK = [ H 4 × 4 U ( k t ) 0 0 H 4 × 4 L ( k t ) ] ,
Ψ m U ( k t , r ) = e i k t r t A i = 1 4 g m ( i ) ( k t , z ) ) | u i , Ψ n L ( k t , r ) = e i k t r t A i = 5 8 g n ( i ) ( k t , z ) ) | u i .
H 4 × 4 U ( k t ) [ g ( 1 ) ( k t , z ) g ( 2 ) ( k t , z ) g ( 3 ) ( k t , z ) g ( 4 ) ( k t , z ) ] = E U ( k t ) [ g ( 1 ) ( k t , z ) g ( 2 ) ( k t , z ) g ( 3 ) ( k t , z ) g ( 4 ) ( k t , z ) ] ,
u 1 , 5 | H 8 × 8 LK | u 1 , 5 = E c + A ɛ + ( h ¯ 2 2 m 0 + A ) ( k t 2 + k z 2 ) = E c + A ɛ + h ¯ 2 2 m c ( k t 2 + k z 2 ) , u 2 , 6 | H 8 × 8 LK | u 2 , 6 = E v P ɛ Q ɛ h ¯ 2 2 m 0 [ ( γ 1 + γ 2 ) k t 2 + ( γ 1 2 γ 2 ) k z 2 ] , u 3 , 7 | H 8 × 8 LK | u 3 , 7 = E v P ɛ + Q ɛ h ¯ 2 2 m 0 [ ( γ 1 γ 2 ) k t 2 + ( γ 1 + 2 γ 2 ) k z 2 ] , u 4 , 8 | H 8 × 8 LK | u 4 , 8 = E v P ɛ Δ h ¯ 2 2 m 0 γ 1 ( k t 2 + k z 2 ) .
A = h ¯ 2 m 0 2 j Γ 5 | S | p x | u j | 2 E c E j
m 0 m c = 1 + 2 m 0 h ¯ 2 A + E p E g + 2 Δ / 3 E g ( E g + Δ ) .
m 0 m c = m 0 m c E p E g + 2 Δ / 3 E g ( E g + Δ ) .
γ 1 = γ 1 L E p 3 E g + Δ , γ 2 = γ 2 L 1 2 ( E p 3 E g + Δ ) , γ 3 = γ 3 L 1 2 ( E p 3 E g + Δ ) .
g m ( i ) ( k t , L T / 2 ) = g m ( i ) ( k t , L T / 2 ) = 0
{ g m ( i ) ( k t , z + L p ) = e iqLp g m ( i ) ( k t , z ) , g m ( i ) ( k t , z + L T ) = g m ( i ) ( k t , z + N L p ) = g m ( i ) ( k t , z ) .
e iNq L p = 1 , and q = 2 π n NL p .
n = { 0 , ± 1 , ± 2 , , ± N 2 N is even , 0 , ± 1 , ± 2 , , ± N 1 2 N is odd .
α ( h ¯ ω ) = π e 2 n r c ɛ 0 m 0 2 ω σ 1 , σ 2 U , L n , m 1 L T 0 2 π d ϕ 2 π 0 k t dk t 2 π | Ψ c σ 1 , n | e ^ p | Ψ v σ 2 , m | 2 × [ f v σ 2 , m ( k t ) f c σ 1 , n ( k t ) ] L ( k t , h ¯ ω ) ,
f c n ( k t ) = 1 1 + exp ( E c , n ( k t ) F c k T ) , f v m ( k t ) = 1 1 + exp ( E v , m ( k t ) F v kT ) .
p 8 × 8 = m 0 h ¯ k H 8 × 8 ,
Ψ c σ 1 , n ( k t , z ) | e ^ p | Ψ v σ 2 , m ( k t , z ) = e ^ i , j dz ( g n ( i ) ( k t , z ) ) * [ m 0 h ¯ k H ij ] g m ( j ) ( k t , z ) ,
L ( k t , h ¯ ω ) = 1 γ 2 π exp ( ( E c , n ( k t ) E v , m ( k t ) h ¯ ω ) 2 2 γ 2 ) .
r spon ( h ¯ ω ) = n r e 2 ω c 3 π h ¯ ɛ 0 m 0 2 σ 1 , σ 2 U , L n , m 1 L T 0 2 π d ϕ 2 π 0 k t dk t 2 π | Ψ c σ 1 , n | e ^ p | Ψ v σ 2 , m | 2 f c σ 1 , n ( k t ) [ 1 f v σ 2 , m ( k t ) ] L ( k t , h ¯ ω ) .
g ( h ¯ ω ) = α ( h ¯ ω ) .
n = σ 1 U , L n 1 2 π L T 0 k t dk t f c σ 1 , n ( k t ) , p = σ 2 U , L m 1 2 π L T 0 k t dk t [ 1 f v σ 2 , m ( k t ) ] .
n + N A = p + N D + ,
n 0 = n i 2 p 0 δ n = δ p p 0 = N A N D + , for p-type , p 0 = n i 2 n 0 δ n = δ p n 0 = N D + N A , for n-type , n 0 = p 0 = n i δ n = δ p , for intrinsic .
H 4 × 4 U = [ E c + A 3 V ρ V ρ + i 2 U 2 V ρ i U 3 V ρ E v P Q R ρ + iS ρ 2 R ρ i 1 2 S ρ V ρ i 2 U R ρ iS ρ E v P + Q 2 Q i 3 2 S ρ 2 V ρ + iU 2 R ρ + i 1 2 S ρ 2 Q + i 3 2 S ρ E v P Δ ] , H 4 × 4 L = [ E c + A 3 V ρ V ρ i 2 U 2 V ρ + iU 3 V ρ E v P Q R ρ iS ρ 2 R ρ + i 1 2 S ρ V ρ + i 2 U R ρ + iS ρ E v P + Q 2 Q + i 3 2 S ρ 2 V ρ iU 2 R ρ + i 1 2 S ρ 2 Q i 3 2 S ρ E v P Δ ] ,
A = h ¯ 2 2 m c ( k t 2 + k z 2 ) + A ɛ , A ɛ = a c ( ɛ xx + ɛ yy + ɛ zz ) , P = h ¯ 2 2 m 0 γ 1 ( k t 2 + k z 2 ) + P ɛ , P ɛ = a v ( ɛ xx + ɛ yy + ɛ zz ) , Q = h ¯ 2 2 m 0 γ 2 ( k t 2 2 k z 2 ) + Q ɛ , Q ɛ = b 2 ( ɛ xx + ɛ yy 2 ɛ zz ) , R ρ = h ¯ 2 2 m 0 3 ( γ 2 + γ 3 2 ) k t 2 = h ¯ 2 2 m 0 3 γ ¯ k t 2 , S ρ = h ¯ 2 2 m 0 2 3 γ 3 k t 2 , V ρ = 1 6 P cv k t , U = 1 3 P cv k z , P cv = h ¯ m 0 iS | h ¯ i x | X = ( h ¯ 2 / 2 m 0 ) E p , ɛ xx = ɛ yy = a 0 a a , ɛ zz = 2 C 12 C 11 ɛ xx ,
| u 1 = 1 2 [ | 1 e i ϕ / 2 + i | 5 e i ϕ / 2 ] , | u 5 = 1 2 [ | 1 e i ϕ / 2 i | 5 e i ϕ / 2 ] , | u 2 = 1 2 [ | 2 e i 3 ϕ / 2 i | 6 e i 3 ϕ / 2 ] , | u 6 = 1 2 [ | 2 e i 3 ϕ / 2 + i | 6 e i 3 ϕ / 2 ] , | u 3 = 1 3 [ i | 3 e i ϕ / 2 | 7 e i ϕ / 2 ] , | u 7 = 1 2 [ i | 3 e i ϕ / 2 | 7 e i ϕ / 2 ] , | u 4 = 1 2 [ i | 4 e i ϕ / 2 | 8 e i ϕ / 2 ] , | u 8 = 1 2 [ i | 4 e i ϕ / 2 | 8 e i ϕ / 2 ] .
| 1 = | i S , | 5 = | i S , | 2 = 1 2 | ( X + i Y ) , | 6 = 1 2 | ( X i Y ) , | 3 = 1 6 | ( X + i Y ) + 2 Z , | 7 = 1 6 | ( X i Y ) + 2 Z , | 4 = 1 3 | ( X + i Y ) + Z , | 8 = 1 3 | ( X i Y ) z .
Ψ c n , σ 1 = U | x ^ p | Ψ v m , σ 2 = U = cos ϕ d z { h ¯ k t [ m 0 m c g 1 , n * g 1 , m ( γ 1 + γ 2 ) g 2 , n * g 2 , m ( γ 1 γ 2 ) g 3 , n * g 3 , m γ 1 g 4 , n * g 4 , m 3 γ ¯ g 2 , n * g 3 , m 6 γ ¯ g 2 , n * g 4 , m 3 γ ¯ g 3 , n * g 2 , m 6 γ ¯ g 4 , n * g 2 , m 2 γ 2 g 3 , n * g 4 , m 2 γ 2 g 4 , n * g 3 , m ] P c v m 0 6 h ¯ [ 3 g 1 , n * g 2 , m + g 1 , n * g 3 , m + 2 g 1 , n * g 4 , m + 3 g 2 , n * g 1 , m + g 3 , n * g 1 , m + 2 g 4 , n * g 1 , m ] + 3 γ 3 h ¯ 2 [ g 2 , n * d d z g 3 , m g 3 , m d d z g 2 , n * g 3 , n * d d z g 2 , m + g 2 , m d d z g 3 , n * ] + 3 γ 3 h ¯ 2 2 [ g 4 , m d d z g 2 , n * g 2 , n * d d z g 4 , m g 2 , m d d z g 4 , n * + g 4 , n * d d z g 2 , m ] + 3 γ 3 h ¯ 2 2 [ g 4 , m d d z g 3 , n * g 3 , n * d d z g 4 , m g 3 , m d d z g 4 , n * + g 4 , n * d d z g 3 , m ] } .
Ψ c n , σ 1 = U | x ^ p | Ψ v m , σ 2 = L = i sin ϕ dz { h ¯ k t [ 3 γ ¯ g 2 , n * g 7 , m 6 γ ¯ g 2 , n * g 8 , m + 3 γ ¯ g 3 , n * g 6 , m + 6 γ ¯ g 4 , n * g 6 , m ] + P c v m 0 6 h ¯ [ 3 g 1 , n * g 6 , m g 1 , n * g 7 , m 2 g 1 , n * g 8 , m 3 g 2 , n * g 5 , m + g 3 , n * g 5 , m + 2 g 4 , n * g 5 , m ] + 3 γ 3 h ¯ 2 [ g 2 , n * d d z g 7 , m g 7 , m d d z g 2 , n * + g 3 , n * d d z g 6 , m g 6 , m d d z g 3 , n * ] + 3 γ 3 h ¯ 2 2 [ g 6 , m d d z g 4 , n * g 4 , n * d d z g 6 , m g 8 , m d d z g 2 , n * + g 2 , n * d d z g 8 , m ] + 3 γ 3 h ¯ 2 2 [ g 8 , m d d z g 3 , n * g 3 , n * d d z g 8 , m g 7 , m d d z g 4 , n * + g 4 , n * d d z g 7 , m ] } .
Ψ c n , σ 1 = U | z ^ p | Ψ v m , σ 2 = U = i d z { P c v m 0 3 h ¯ [ 2 g 1 , n * g 3 , m g 1 , n * g 4 , m 2 g 3 , n * g 1 , m + g 4 , n * g 1 , m ] + 3 2 γ 3 k t [ 2 g 2 , n * g 3 , m g 2 , n * g 4 , m 2 g 3 , n * g 2 , m ] + g 4 , n * g 2 , m 3 g 3 , n * g 4 , m + 3 g 4 , n * g 3 , m ] m 0 2 m c [ g 1 , n * d d z g 1 , m g 1 , m d d z g 1 , n * ] + γ 1 2 γ 2 2 [ g 2 , n * d d z g 2 , m g 2 , m d d z g 2 , n * ] + γ 1 + 2 γ 2 2 [ g 3 , n * d d z g 3 , m g 3 , m d d z g 3 , n * ] + γ 1 2 [ g 4 , n * d d z g 4 , m g 4 , m d d z g 4 , n * ] } , Ψ c n , σ 1 = U | z ^ p | Ψ v m , σ 2 = L = 0 , Ψ c n , σ 1 = L | z ^ p | Ψ v m , σ 2 = U = 0 .

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