Abstract

We convert calculations of the bound-to-continuum absorption in type-II semiconductor quantum wells into an equivalent source-radiation problem under the effective-mass approximation with band mixing. Perfectly matched layers corresponding to the eight-band Luttinger-Kohn Hamiltonian are introduced to incorporate the effect of quasi-bound states in open regions. In this way, the interplay between quantum tunneling and optical transitions is fully taken into account. From resulted lineshapes of the Fano resonance, we can evaluate tunneling rates of these metastable states and related absorption strengths relative to those of the continuum. The approach here is useful in estimations of carrier extraction rates from type-II nanostructures for photovoltaic applications.

© 2013 Optical Society of America

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  1. J. R. Meyer, C. L. Felix, W. W. Bewley, I. Vurgaftman, E. H. Aifer, L. J. Olafsen, J. R. Lindle, C. A. Hoffman, M.-J. Yang, B. R. Bennett, B. V. Shanabrook, H. Lee, C.-H. Lin, S. S. Pei, and R. H. Miles, “Auger coefficients in type-II InAs/Ga1−x Inx Sb quantum wells,” Appl. Phys. Lett.73, 2857–2859 (1998).
    [CrossRef]
  2. S. Kim, B. Fisher, H.-J. Eisler, and M. Bawendi, “Type-II quantum dots: CdTe/CdSe(core/shell) and CdSe/ZnTe(core/shell) heterostructures,” J. Am. Chem. Soc.125, 11466–11467 (2003).
    [CrossRef] [PubMed]
  3. A. Ahland, D. Schulz, and E. Voges, “Accurate mesh truncation for Schrodinger equations by a perfectly matched layer absorber: Application to the calculation of optical spectra,” Phys. Rev. B60, R5109–R5112 (1999).
    [CrossRef]
  4. J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys.114, 185–200 (1994).
    [CrossRef]
  5. W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates,” Microw. Opt. Technol. Lett.7, 599–604 (1994).
    [CrossRef]
  6. F. L. Teixeira and W. C. Chew, “General closed-form PML constitutive tensors to match arbitrary bianisotropic and dispersive linear media,” IEEE Microw. Guided Wave Lett.8, 223–225 (1998).
    [CrossRef]
  7. S. Odermatt, M. Luisier, and B. Witzigmann, “Bandstructure calculation using the k· p method for arbitrary potentials with open boundary conditions,” J. Appl. Phys.97, 046104 (2005).
    [CrossRef]
  8. M. Karner, A. Gehring, and H. Kosina, “Efficient calculation of lifetime based direct tunneling through stacked dielectrics,” J. Comput. Electron.5, 161–165 (2006).
    [CrossRef]
  9. A. Ahland, M. Wiedenhaus, D. Schulz, and E. Voges, “Calculation of exciton absorption in arbitrary layered semiconductor nanostructures with exact treatment of the coulomb singularity,” IEEE J. Quantum Electron.36, 842–848 (2000).
    [CrossRef]
  10. T.-Y. Zhang and W. Zhao, “Magnetoexcitonic optical absorption in semiconductors under strong magnetic fields and intense terahertz radiation in the Voigt configuration,” EPL82, 67001 (2008).
    [CrossRef]
  11. U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev.124, 1866–1878 (1961).
    [CrossRef]
  12. S.-L. Chuang, S. Schmitt-Rink, D. A. B. Miller, and D. S. Chemla, “Exciton Green’s-function approach to optical absorption in a quantum well with an applied electric field,” Phys. Rev. B43, 1500–1509 (1991).
    [CrossRef]
  13. S. Glutsch, D. S. Chemla, and F. Bechstedt, “Numerical calculation of the optical absorption in semiconductor quantum structures,” Phys. Rev. B54, 11592–11601 (1996).
    [CrossRef]
  14. J. M. Luttinger and W. Kohn, “Motion of electrons and holes in perturbed periodic fields,” Phys. Rev.97, 869–883 (1955).
    [CrossRef]
  15. C. Y.-P. Chao and S. L. Chuang, “Spin-orbit-coupling effects on the valence-band structure of strained semiconductor quantum wells,” Phys. Rev. B46, 4110–4122 (1992).
    [CrossRef]
  16. G. Liu and S.-L. Chuang, “Modeling of Sb-based type-II quantum cascade lasers,” Phys. Rev. B65, 165220 (2002).
    [CrossRef]
  17. P.-F. Qiao, S. Mou, and S. L. Chuang, “Electronic band structures and optical properties of type-II superlattice photodetectors with interfacial effect,” Opt. Express20, 2319–2334 (2012).
    [CrossRef] [PubMed]
  18. S. L. Chuang, Physics of Photonic Devices, 2 (John Wiley & Sons, New Jersey, 2009).
  19. G. L. Bir and G. E. Pikus, Symmetry and Strain-Induced Effects in Semiconductors (John Wiley & Sons, New York, 1974).
  20. G. D. Mahan, Many-particle Physics, 3 (Kluwer Academic/Plenum Publishers, New York, 2000).
  21. S. L. Chuang, Physics of Optoelectronic Devices (John Wiley & Sons, New York, 1995).
  22. C.-S. Chang and S. L. Chuang, “Modeling of strained quantum-well lasers with spin-orbit coupling,” IEEE J. Sel. Topics Quantum Electron.1, 218–229 (1995).
    [CrossRef]
  23. S. L. Chuang and C. S. Chang, “A band-structure model of strained quantum-well wurtzite semiconductors,” Semicond. Sci. Technol.12, 252–263 (1997).
    [CrossRef]
  24. D. Ahn and S.-L. Chuang, “Optical gain in a strained-layer quantum-well laser,” IEEE J. Quantum Electron.24, 2400–2406 (1988).
    [CrossRef]
  25. F. Szmulowicz, “Derivation of a general expression for the momentum matrix elements within the envelope-function approximation,” Phys. Rev. B51, 1613–1623 (1995).
    [CrossRef]
  26. Y.-C. Chang and R. B. James, “Saturation of intersubband transitions in p-type semiconductor quantum wells,” Phys. Rev. B39, 12672–12681 (1989).
    [CrossRef]
  27. P. Lawaetz, “Valence-band parameters in cubic semiconductors,” Phys. Rev. B4, 3460–3467 (1971).
    [CrossRef]
  28. X. Cartoixa, D. Z.-Y. Ting, and T. C. McGill, “Numerical spurious solutions in the effective mass approximation,” J. Appl. Phys.93, 3974–3981 (2003).
    [CrossRef]
  29. G. B. Liu, S.-L. Chuang, and S.-H. Park, “Optical gain of strained GaAsSb/GaAs quantum-well lasers: A self-consistent approach,” J. Appl. Phys.88, 5554–5561 (2000).
    [CrossRef]
  30. Y. Tsou, A. Ichii, and E. M. Garmire, “Improving InAs double heterostructure lasers with better confinement,” IEEE J. Quantum Electron.28, 1261–1268 (1992).
    [CrossRef]
  31. P. K. W. Vinsome and D. Richardson, “The dielectric function in zincblende semiconductors,” J. Phys. C: Solid St. Phys.4, 2650–2657 (1971).
    [CrossRef]
  32. C. G. Van de Walle, “Band lineups and deformation potentials in the model-solid theory,” Phys. Rev. B39, 1871–1883 (1989).
    [CrossRef]
  33. R. E. Nahory, M. A. Pollack, J. C. DeWinter, and K. M. Williams, “Growth and properties of liguid-phase epitaxial GaAs1−x Sbx,” J. Appl. Phys.48, 1607–1614 (1977).
    [CrossRef]
  34. D. Ahn, S. L. Chuang, and Y.-C. Chang, “Valence-band mixing effects on the gain and the refractive index change of quantum-well lasers,” J. Appl. Phys.64, 4056–4064 (1988).
    [CrossRef]

2012 (1)

2008 (1)

T.-Y. Zhang and W. Zhao, “Magnetoexcitonic optical absorption in semiconductors under strong magnetic fields and intense terahertz radiation in the Voigt configuration,” EPL82, 67001 (2008).
[CrossRef]

2006 (1)

M. Karner, A. Gehring, and H. Kosina, “Efficient calculation of lifetime based direct tunneling through stacked dielectrics,” J. Comput. Electron.5, 161–165 (2006).
[CrossRef]

2005 (1)

S. Odermatt, M. Luisier, and B. Witzigmann, “Bandstructure calculation using the k· p method for arbitrary potentials with open boundary conditions,” J. Appl. Phys.97, 046104 (2005).
[CrossRef]

2003 (2)

S. Kim, B. Fisher, H.-J. Eisler, and M. Bawendi, “Type-II quantum dots: CdTe/CdSe(core/shell) and CdSe/ZnTe(core/shell) heterostructures,” J. Am. Chem. Soc.125, 11466–11467 (2003).
[CrossRef] [PubMed]

X. Cartoixa, D. Z.-Y. Ting, and T. C. McGill, “Numerical spurious solutions in the effective mass approximation,” J. Appl. Phys.93, 3974–3981 (2003).
[CrossRef]

2002 (1)

G. Liu and S.-L. Chuang, “Modeling of Sb-based type-II quantum cascade lasers,” Phys. Rev. B65, 165220 (2002).
[CrossRef]

2000 (2)

A. Ahland, M. Wiedenhaus, D. Schulz, and E. Voges, “Calculation of exciton absorption in arbitrary layered semiconductor nanostructures with exact treatment of the coulomb singularity,” IEEE J. Quantum Electron.36, 842–848 (2000).
[CrossRef]

G. B. Liu, S.-L. Chuang, and S.-H. Park, “Optical gain of strained GaAsSb/GaAs quantum-well lasers: A self-consistent approach,” J. Appl. Phys.88, 5554–5561 (2000).
[CrossRef]

1999 (1)

A. Ahland, D. Schulz, and E. Voges, “Accurate mesh truncation for Schrodinger equations by a perfectly matched layer absorber: Application to the calculation of optical spectra,” Phys. Rev. B60, R5109–R5112 (1999).
[CrossRef]

1998 (2)

J. R. Meyer, C. L. Felix, W. W. Bewley, I. Vurgaftman, E. H. Aifer, L. J. Olafsen, J. R. Lindle, C. A. Hoffman, M.-J. Yang, B. R. Bennett, B. V. Shanabrook, H. Lee, C.-H. Lin, S. S. Pei, and R. H. Miles, “Auger coefficients in type-II InAs/Ga1−x Inx Sb quantum wells,” Appl. Phys. Lett.73, 2857–2859 (1998).
[CrossRef]

F. L. Teixeira and W. C. Chew, “General closed-form PML constitutive tensors to match arbitrary bianisotropic and dispersive linear media,” IEEE Microw. Guided Wave Lett.8, 223–225 (1998).
[CrossRef]

1997 (1)

S. L. Chuang and C. S. Chang, “A band-structure model of strained quantum-well wurtzite semiconductors,” Semicond. Sci. Technol.12, 252–263 (1997).
[CrossRef]

1996 (1)

S. Glutsch, D. S. Chemla, and F. Bechstedt, “Numerical calculation of the optical absorption in semiconductor quantum structures,” Phys. Rev. B54, 11592–11601 (1996).
[CrossRef]

1995 (2)

C.-S. Chang and S. L. Chuang, “Modeling of strained quantum-well lasers with spin-orbit coupling,” IEEE J. Sel. Topics 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. B51, 1613–1623 (1995).
[CrossRef]

1994 (2)

J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys.114, 185–200 (1994).
[CrossRef]

W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates,” Microw. Opt. Technol. Lett.7, 599–604 (1994).
[CrossRef]

1992 (2)

Y. Tsou, A. Ichii, and E. M. Garmire, “Improving InAs double heterostructure lasers with better confinement,” IEEE J. Quantum Electron.28, 1261–1268 (1992).
[CrossRef]

C. Y.-P. Chao and S. L. Chuang, “Spin-orbit-coupling effects on the valence-band structure of strained semiconductor quantum wells,” Phys. Rev. B46, 4110–4122 (1992).
[CrossRef]

1991 (1)

S.-L. Chuang, S. Schmitt-Rink, D. A. B. Miller, and D. S. Chemla, “Exciton Green’s-function approach to optical absorption in a quantum well with an applied electric field,” Phys. Rev. B43, 1500–1509 (1991).
[CrossRef]

1989 (2)

C. G. Van de Walle, “Band lineups and deformation potentials in the model-solid theory,” Phys. Rev. B39, 1871–1883 (1989).
[CrossRef]

Y.-C. Chang and R. B. James, “Saturation of intersubband transitions in p-type semiconductor quantum wells,” Phys. Rev. B39, 12672–12681 (1989).
[CrossRef]

1988 (2)

D. Ahn and S.-L. Chuang, “Optical gain in a strained-layer quantum-well laser,” IEEE J. Quantum Electron.24, 2400–2406 (1988).
[CrossRef]

D. Ahn, S. L. Chuang, and Y.-C. Chang, “Valence-band mixing effects on the gain and the refractive index change of quantum-well lasers,” J. Appl. Phys.64, 4056–4064 (1988).
[CrossRef]

1977 (1)

R. E. Nahory, M. A. Pollack, J. C. DeWinter, and K. M. Williams, “Growth and properties of liguid-phase epitaxial GaAs1−x Sbx,” J. Appl. Phys.48, 1607–1614 (1977).
[CrossRef]

1971 (2)

P. Lawaetz, “Valence-band parameters in cubic semiconductors,” Phys. Rev. B4, 3460–3467 (1971).
[CrossRef]

P. K. W. Vinsome and D. Richardson, “The dielectric function in zincblende semiconductors,” J. Phys. C: Solid St. Phys.4, 2650–2657 (1971).
[CrossRef]

1961 (1)

U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev.124, 1866–1878 (1961).
[CrossRef]

1955 (1)

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

Ahland, A.

A. Ahland, M. Wiedenhaus, D. Schulz, and E. Voges, “Calculation of exciton absorption in arbitrary layered semiconductor nanostructures with exact treatment of the coulomb singularity,” IEEE J. Quantum Electron.36, 842–848 (2000).
[CrossRef]

A. Ahland, D. Schulz, and E. Voges, “Accurate mesh truncation for Schrodinger equations by a perfectly matched layer absorber: Application to the calculation of optical spectra,” Phys. Rev. B60, R5109–R5112 (1999).
[CrossRef]

Ahn, D.

D. Ahn and S.-L. Chuang, “Optical gain in a strained-layer quantum-well laser,” IEEE J. Quantum Electron.24, 2400–2406 (1988).
[CrossRef]

D. Ahn, S. L. Chuang, and Y.-C. Chang, “Valence-band mixing effects on the gain and the refractive index change of quantum-well lasers,” J. Appl. Phys.64, 4056–4064 (1988).
[CrossRef]

Aifer, E. H.

J. R. Meyer, C. L. Felix, W. W. Bewley, I. Vurgaftman, E. H. Aifer, L. J. Olafsen, J. R. Lindle, C. A. Hoffman, M.-J. Yang, B. R. Bennett, B. V. Shanabrook, H. Lee, C.-H. Lin, S. S. Pei, and R. H. Miles, “Auger coefficients in type-II InAs/Ga1−x Inx Sb quantum wells,” Appl. Phys. Lett.73, 2857–2859 (1998).
[CrossRef]

Bawendi, M.

S. Kim, B. Fisher, H.-J. Eisler, and M. Bawendi, “Type-II quantum dots: CdTe/CdSe(core/shell) and CdSe/ZnTe(core/shell) heterostructures,” J. Am. Chem. Soc.125, 11466–11467 (2003).
[CrossRef] [PubMed]

Bechstedt, F.

S. Glutsch, D. S. Chemla, and F. Bechstedt, “Numerical calculation of the optical absorption in semiconductor quantum structures,” Phys. Rev. B54, 11592–11601 (1996).
[CrossRef]

Bennett, B. R.

J. R. Meyer, C. L. Felix, W. W. Bewley, I. Vurgaftman, E. H. Aifer, L. J. Olafsen, J. R. Lindle, C. A. Hoffman, M.-J. Yang, B. R. Bennett, B. V. Shanabrook, H. Lee, C.-H. Lin, S. S. Pei, and R. H. Miles, “Auger coefficients in type-II InAs/Ga1−x Inx Sb quantum wells,” Appl. Phys. Lett.73, 2857–2859 (1998).
[CrossRef]

Berenger, J.-P.

J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys.114, 185–200 (1994).
[CrossRef]

Bewley, W. W.

J. R. Meyer, C. L. Felix, W. W. Bewley, I. Vurgaftman, E. H. Aifer, L. J. Olafsen, J. R. Lindle, C. A. Hoffman, M.-J. Yang, B. R. Bennett, B. V. Shanabrook, H. Lee, C.-H. Lin, S. S. Pei, and R. H. Miles, “Auger coefficients in type-II InAs/Ga1−x Inx Sb quantum wells,” Appl. Phys. Lett.73, 2857–2859 (1998).
[CrossRef]

Bir, G. L.

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

Cartoixa, X.

X. Cartoixa, D. Z.-Y. Ting, and T. C. McGill, “Numerical spurious solutions in the effective mass approximation,” J. Appl. Phys.93, 3974–3981 (2003).
[CrossRef]

Chang, C. S.

S. L. Chuang and C. S. Chang, “A band-structure model of strained quantum-well wurtzite semiconductors,” Semicond. Sci. Technol.12, 252–263 (1997).
[CrossRef]

Chang, C.-S.

C.-S. Chang and S. L. Chuang, “Modeling of strained quantum-well lasers with spin-orbit coupling,” IEEE J. Sel. Topics 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. B39, 12672–12681 (1989).
[CrossRef]

D. Ahn, S. L. Chuang, and Y.-C. Chang, “Valence-band mixing effects on the gain and the refractive index change of quantum-well lasers,” J. Appl. Phys.64, 4056–4064 (1988).
[CrossRef]

Chao, C. Y.-P.

C. Y.-P. Chao and S. L. Chuang, “Spin-orbit-coupling effects on the valence-band structure of strained semiconductor quantum wells,” Phys. Rev. B46, 4110–4122 (1992).
[CrossRef]

Chemla, D. S.

S. Glutsch, D. S. Chemla, and F. Bechstedt, “Numerical calculation of the optical absorption in semiconductor quantum structures,” Phys. Rev. B54, 11592–11601 (1996).
[CrossRef]

S.-L. Chuang, S. Schmitt-Rink, D. A. B. Miller, and D. S. Chemla, “Exciton Green’s-function approach to optical absorption in a quantum well with an applied electric field,” Phys. Rev. B43, 1500–1509 (1991).
[CrossRef]

Chew, W. C.

F. L. Teixeira and W. C. Chew, “General closed-form PML constitutive tensors to match arbitrary bianisotropic and dispersive linear media,” IEEE Microw. Guided Wave Lett.8, 223–225 (1998).
[CrossRef]

W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates,” Microw. Opt. Technol. Lett.7, 599–604 (1994).
[CrossRef]

Chuang, S. L.

P.-F. Qiao, S. Mou, and S. L. Chuang, “Electronic band structures and optical properties of type-II superlattice photodetectors with interfacial effect,” Opt. Express20, 2319–2334 (2012).
[CrossRef] [PubMed]

S. L. Chuang and C. S. Chang, “A band-structure model of strained quantum-well wurtzite semiconductors,” Semicond. Sci. Technol.12, 252–263 (1997).
[CrossRef]

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

C. Y.-P. Chao and S. L. Chuang, “Spin-orbit-coupling effects on the valence-band structure of strained semiconductor quantum wells,” Phys. Rev. B46, 4110–4122 (1992).
[CrossRef]

D. Ahn, S. L. Chuang, and Y.-C. Chang, “Valence-band mixing effects on the gain and the refractive index change of quantum-well lasers,” J. Appl. Phys.64, 4056–4064 (1988).
[CrossRef]

S. L. Chuang, Physics of Optoelectronic Devices (John Wiley & Sons, New York, 1995).

S. L. Chuang, Physics of Photonic Devices, 2 (John Wiley & Sons, New Jersey, 2009).

Chuang, S.-L.

G. Liu and S.-L. Chuang, “Modeling of Sb-based type-II quantum cascade lasers,” Phys. Rev. B65, 165220 (2002).
[CrossRef]

G. B. Liu, S.-L. Chuang, and S.-H. Park, “Optical gain of strained GaAsSb/GaAs quantum-well lasers: A self-consistent approach,” J. Appl. Phys.88, 5554–5561 (2000).
[CrossRef]

S.-L. Chuang, S. Schmitt-Rink, D. A. B. Miller, and D. S. Chemla, “Exciton Green’s-function approach to optical absorption in a quantum well with an applied electric field,” Phys. Rev. B43, 1500–1509 (1991).
[CrossRef]

D. Ahn and S.-L. Chuang, “Optical gain in a strained-layer quantum-well laser,” IEEE J. Quantum Electron.24, 2400–2406 (1988).
[CrossRef]

DeWinter, J. C.

R. E. Nahory, M. A. Pollack, J. C. DeWinter, and K. M. Williams, “Growth and properties of liguid-phase epitaxial GaAs1−x Sbx,” J. Appl. Phys.48, 1607–1614 (1977).
[CrossRef]

Eisler, H.-J.

S. Kim, B. Fisher, H.-J. Eisler, and M. Bawendi, “Type-II quantum dots: CdTe/CdSe(core/shell) and CdSe/ZnTe(core/shell) heterostructures,” J. Am. Chem. Soc.125, 11466–11467 (2003).
[CrossRef] [PubMed]

Fano, U.

U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev.124, 1866–1878 (1961).
[CrossRef]

Felix, C. L.

J. R. Meyer, C. L. Felix, W. W. Bewley, I. Vurgaftman, E. H. Aifer, L. J. Olafsen, J. R. Lindle, C. A. Hoffman, M.-J. Yang, B. R. Bennett, B. V. Shanabrook, H. Lee, C.-H. Lin, S. S. Pei, and R. H. Miles, “Auger coefficients in type-II InAs/Ga1−x Inx Sb quantum wells,” Appl. Phys. Lett.73, 2857–2859 (1998).
[CrossRef]

Fisher, B.

S. Kim, B. Fisher, H.-J. Eisler, and M. Bawendi, “Type-II quantum dots: CdTe/CdSe(core/shell) and CdSe/ZnTe(core/shell) heterostructures,” J. Am. Chem. Soc.125, 11466–11467 (2003).
[CrossRef] [PubMed]

Garmire, E. M.

Y. Tsou, A. Ichii, and E. M. Garmire, “Improving InAs double heterostructure lasers with better confinement,” IEEE J. Quantum Electron.28, 1261–1268 (1992).
[CrossRef]

Gehring, A.

M. Karner, A. Gehring, and H. Kosina, “Efficient calculation of lifetime based direct tunneling through stacked dielectrics,” J. Comput. Electron.5, 161–165 (2006).
[CrossRef]

Glutsch, S.

S. Glutsch, D. S. Chemla, and F. Bechstedt, “Numerical calculation of the optical absorption in semiconductor quantum structures,” Phys. Rev. B54, 11592–11601 (1996).
[CrossRef]

Hoffman, C. A.

J. R. Meyer, C. L. Felix, W. W. Bewley, I. Vurgaftman, E. H. Aifer, L. J. Olafsen, J. R. Lindle, C. A. Hoffman, M.-J. Yang, B. R. Bennett, B. V. Shanabrook, H. Lee, C.-H. Lin, S. S. Pei, and R. H. Miles, “Auger coefficients in type-II InAs/Ga1−x Inx Sb quantum wells,” Appl. Phys. Lett.73, 2857–2859 (1998).
[CrossRef]

Ichii, A.

Y. Tsou, A. Ichii, and E. M. Garmire, “Improving InAs double heterostructure lasers with better confinement,” IEEE J. Quantum Electron.28, 1261–1268 (1992).
[CrossRef]

James, R. B.

Y.-C. Chang and R. B. James, “Saturation of intersubband transitions in p-type semiconductor quantum wells,” Phys. Rev. B39, 12672–12681 (1989).
[CrossRef]

Karner, M.

M. Karner, A. Gehring, and H. Kosina, “Efficient calculation of lifetime based direct tunneling through stacked dielectrics,” J. Comput. Electron.5, 161–165 (2006).
[CrossRef]

Kim, S.

S. Kim, B. Fisher, H.-J. Eisler, and M. Bawendi, “Type-II quantum dots: CdTe/CdSe(core/shell) and CdSe/ZnTe(core/shell) heterostructures,” J. Am. Chem. Soc.125, 11466–11467 (2003).
[CrossRef] [PubMed]

Kohn, W.

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

Kosina, H.

M. Karner, A. Gehring, and H. Kosina, “Efficient calculation of lifetime based direct tunneling through stacked dielectrics,” J. Comput. Electron.5, 161–165 (2006).
[CrossRef]

Lawaetz, P.

P. Lawaetz, “Valence-band parameters in cubic semiconductors,” Phys. Rev. B4, 3460–3467 (1971).
[CrossRef]

Lee, H.

J. R. Meyer, C. L. Felix, W. W. Bewley, I. Vurgaftman, E. H. Aifer, L. J. Olafsen, J. R. Lindle, C. A. Hoffman, M.-J. Yang, B. R. Bennett, B. V. Shanabrook, H. Lee, C.-H. Lin, S. S. Pei, and R. H. Miles, “Auger coefficients in type-II InAs/Ga1−x Inx Sb quantum wells,” Appl. Phys. Lett.73, 2857–2859 (1998).
[CrossRef]

Lin, C.-H.

J. R. Meyer, C. L. Felix, W. W. Bewley, I. Vurgaftman, E. H. Aifer, L. J. Olafsen, J. R. Lindle, C. A. Hoffman, M.-J. Yang, B. R. Bennett, B. V. Shanabrook, H. Lee, C.-H. Lin, S. S. Pei, and R. H. Miles, “Auger coefficients in type-II InAs/Ga1−x Inx Sb quantum wells,” Appl. Phys. Lett.73, 2857–2859 (1998).
[CrossRef]

Lindle, J. R.

J. R. Meyer, C. L. Felix, W. W. Bewley, I. Vurgaftman, E. H. Aifer, L. J. Olafsen, J. R. Lindle, C. A. Hoffman, M.-J. Yang, B. R. Bennett, B. V. Shanabrook, H. Lee, C.-H. Lin, S. S. Pei, and R. H. Miles, “Auger coefficients in type-II InAs/Ga1−x Inx Sb quantum wells,” Appl. Phys. Lett.73, 2857–2859 (1998).
[CrossRef]

Liu, G.

G. Liu and S.-L. Chuang, “Modeling of Sb-based type-II quantum cascade lasers,” Phys. Rev. B65, 165220 (2002).
[CrossRef]

Liu, G. B.

G. B. Liu, S.-L. Chuang, and S.-H. Park, “Optical gain of strained GaAsSb/GaAs quantum-well lasers: A self-consistent approach,” J. Appl. Phys.88, 5554–5561 (2000).
[CrossRef]

Luisier, M.

S. Odermatt, M. Luisier, and B. Witzigmann, “Bandstructure calculation using the k· p method for arbitrary potentials with open boundary conditions,” J. Appl. Phys.97, 046104 (2005).
[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]

Mahan, G. D.

G. D. Mahan, Many-particle Physics, 3 (Kluwer Academic/Plenum Publishers, New York, 2000).

McGill, T. C.

X. Cartoixa, D. Z.-Y. Ting, and T. C. McGill, “Numerical spurious solutions in the effective mass approximation,” J. Appl. Phys.93, 3974–3981 (2003).
[CrossRef]

Meyer, J. R.

J. R. Meyer, C. L. Felix, W. W. Bewley, I. Vurgaftman, E. H. Aifer, L. J. Olafsen, J. R. Lindle, C. A. Hoffman, M.-J. Yang, B. R. Bennett, B. V. Shanabrook, H. Lee, C.-H. Lin, S. S. Pei, and R. H. Miles, “Auger coefficients in type-II InAs/Ga1−x Inx Sb quantum wells,” Appl. Phys. Lett.73, 2857–2859 (1998).
[CrossRef]

Miles, R. H.

J. R. Meyer, C. L. Felix, W. W. Bewley, I. Vurgaftman, E. H. Aifer, L. J. Olafsen, J. R. Lindle, C. A. Hoffman, M.-J. Yang, B. R. Bennett, B. V. Shanabrook, H. Lee, C.-H. Lin, S. S. Pei, and R. H. Miles, “Auger coefficients in type-II InAs/Ga1−x Inx Sb quantum wells,” Appl. Phys. Lett.73, 2857–2859 (1998).
[CrossRef]

Miller, D. A. B.

S.-L. Chuang, S. Schmitt-Rink, D. A. B. Miller, and D. S. Chemla, “Exciton Green’s-function approach to optical absorption in a quantum well with an applied electric field,” Phys. Rev. B43, 1500–1509 (1991).
[CrossRef]

Mou, S.

Nahory, R. E.

R. E. Nahory, M. A. Pollack, J. C. DeWinter, and K. M. Williams, “Growth and properties of liguid-phase epitaxial GaAs1−x Sbx,” J. Appl. Phys.48, 1607–1614 (1977).
[CrossRef]

Odermatt, S.

S. Odermatt, M. Luisier, and B. Witzigmann, “Bandstructure calculation using the k· p method for arbitrary potentials with open boundary conditions,” J. Appl. Phys.97, 046104 (2005).
[CrossRef]

Olafsen, L. J.

J. R. Meyer, C. L. Felix, W. W. Bewley, I. Vurgaftman, E. H. Aifer, L. J. Olafsen, J. R. Lindle, C. A. Hoffman, M.-J. Yang, B. R. Bennett, B. V. Shanabrook, H. Lee, C.-H. Lin, S. S. Pei, and R. H. Miles, “Auger coefficients in type-II InAs/Ga1−x Inx Sb quantum wells,” Appl. Phys. Lett.73, 2857–2859 (1998).
[CrossRef]

Park, S.-H.

G. B. Liu, S.-L. Chuang, and S.-H. Park, “Optical gain of strained GaAsSb/GaAs quantum-well lasers: A self-consistent approach,” J. Appl. Phys.88, 5554–5561 (2000).
[CrossRef]

Pei, S. S.

J. R. Meyer, C. L. Felix, W. W. Bewley, I. Vurgaftman, E. H. Aifer, L. J. Olafsen, J. R. Lindle, C. A. Hoffman, M.-J. Yang, B. R. Bennett, B. V. Shanabrook, H. Lee, C.-H. Lin, S. S. Pei, and R. H. Miles, “Auger coefficients in type-II InAs/Ga1−x Inx Sb quantum wells,” Appl. Phys. Lett.73, 2857–2859 (1998).
[CrossRef]

Pikus, G. E.

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

Pollack, M. A.

R. E. Nahory, M. A. Pollack, J. C. DeWinter, and K. M. Williams, “Growth and properties of liguid-phase epitaxial GaAs1−x Sbx,” J. Appl. Phys.48, 1607–1614 (1977).
[CrossRef]

Qiao, P.-F.

Richardson, D.

P. K. W. Vinsome and D. Richardson, “The dielectric function in zincblende semiconductors,” J. Phys. C: Solid St. Phys.4, 2650–2657 (1971).
[CrossRef]

Schmitt-Rink, S.

S.-L. Chuang, S. Schmitt-Rink, D. A. B. Miller, and D. S. Chemla, “Exciton Green’s-function approach to optical absorption in a quantum well with an applied electric field,” Phys. Rev. B43, 1500–1509 (1991).
[CrossRef]

Schulz, D.

A. Ahland, M. Wiedenhaus, D. Schulz, and E. Voges, “Calculation of exciton absorption in arbitrary layered semiconductor nanostructures with exact treatment of the coulomb singularity,” IEEE J. Quantum Electron.36, 842–848 (2000).
[CrossRef]

A. Ahland, D. Schulz, and E. Voges, “Accurate mesh truncation for Schrodinger equations by a perfectly matched layer absorber: Application to the calculation of optical spectra,” Phys. Rev. B60, R5109–R5112 (1999).
[CrossRef]

Shanabrook, B. V.

J. R. Meyer, C. L. Felix, W. W. Bewley, I. Vurgaftman, E. H. Aifer, L. J. Olafsen, J. R. Lindle, C. A. Hoffman, M.-J. Yang, B. R. Bennett, B. V. Shanabrook, H. Lee, C.-H. Lin, S. S. Pei, and R. H. Miles, “Auger coefficients in type-II InAs/Ga1−x Inx Sb quantum wells,” Appl. Phys. Lett.73, 2857–2859 (1998).
[CrossRef]

Szmulowicz, F.

F. Szmulowicz, “Derivation of a general expression for the momentum matrix elements within the envelope-function approximation,” Phys. Rev. B51, 1613–1623 (1995).
[CrossRef]

Teixeira, F. L.

F. L. Teixeira and W. C. Chew, “General closed-form PML constitutive tensors to match arbitrary bianisotropic and dispersive linear media,” IEEE Microw. Guided Wave Lett.8, 223–225 (1998).
[CrossRef]

Ting, D. Z.-Y.

X. Cartoixa, D. Z.-Y. Ting, and T. C. McGill, “Numerical spurious solutions in the effective mass approximation,” J. Appl. Phys.93, 3974–3981 (2003).
[CrossRef]

Tsou, Y.

Y. Tsou, A. Ichii, and E. M. Garmire, “Improving InAs double heterostructure lasers with better confinement,” IEEE J. Quantum Electron.28, 1261–1268 (1992).
[CrossRef]

Van de Walle, C. G.

C. G. Van de Walle, “Band lineups and deformation potentials in the model-solid theory,” Phys. Rev. B39, 1871–1883 (1989).
[CrossRef]

Vinsome, P. K. W.

P. K. W. Vinsome and D. Richardson, “The dielectric function in zincblende semiconductors,” J. Phys. C: Solid St. Phys.4, 2650–2657 (1971).
[CrossRef]

Voges, E.

A. Ahland, M. Wiedenhaus, D. Schulz, and E. Voges, “Calculation of exciton absorption in arbitrary layered semiconductor nanostructures with exact treatment of the coulomb singularity,” IEEE J. Quantum Electron.36, 842–848 (2000).
[CrossRef]

A. Ahland, D. Schulz, and E. Voges, “Accurate mesh truncation for Schrodinger equations by a perfectly matched layer absorber: Application to the calculation of optical spectra,” Phys. Rev. B60, R5109–R5112 (1999).
[CrossRef]

Vurgaftman, I.

J. R. Meyer, C. L. Felix, W. W. Bewley, I. Vurgaftman, E. H. Aifer, L. J. Olafsen, J. R. Lindle, C. A. Hoffman, M.-J. Yang, B. R. Bennett, B. V. Shanabrook, H. Lee, C.-H. Lin, S. S. Pei, and R. H. Miles, “Auger coefficients in type-II InAs/Ga1−x Inx Sb quantum wells,” Appl. Phys. Lett.73, 2857–2859 (1998).
[CrossRef]

Weedon, W. H.

W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates,” Microw. Opt. Technol. Lett.7, 599–604 (1994).
[CrossRef]

Wiedenhaus, M.

A. Ahland, M. Wiedenhaus, D. Schulz, and E. Voges, “Calculation of exciton absorption in arbitrary layered semiconductor nanostructures with exact treatment of the coulomb singularity,” IEEE J. Quantum Electron.36, 842–848 (2000).
[CrossRef]

Williams, K. M.

R. E. Nahory, M. A. Pollack, J. C. DeWinter, and K. M. Williams, “Growth and properties of liguid-phase epitaxial GaAs1−x Sbx,” J. Appl. Phys.48, 1607–1614 (1977).
[CrossRef]

Witzigmann, B.

S. Odermatt, M. Luisier, and B. Witzigmann, “Bandstructure calculation using the k· p method for arbitrary potentials with open boundary conditions,” J. Appl. Phys.97, 046104 (2005).
[CrossRef]

Yang, M.-J.

J. R. Meyer, C. L. Felix, W. W. Bewley, I. Vurgaftman, E. H. Aifer, L. J. Olafsen, J. R. Lindle, C. A. Hoffman, M.-J. Yang, B. R. Bennett, B. V. Shanabrook, H. Lee, C.-H. Lin, S. S. Pei, and R. H. Miles, “Auger coefficients in type-II InAs/Ga1−x Inx Sb quantum wells,” Appl. Phys. Lett.73, 2857–2859 (1998).
[CrossRef]

Zhang, T.-Y.

T.-Y. Zhang and W. Zhao, “Magnetoexcitonic optical absorption in semiconductors under strong magnetic fields and intense terahertz radiation in the Voigt configuration,” EPL82, 67001 (2008).
[CrossRef]

Zhao, W.

T.-Y. Zhang and W. Zhao, “Magnetoexcitonic optical absorption in semiconductors under strong magnetic fields and intense terahertz radiation in the Voigt configuration,” EPL82, 67001 (2008).
[CrossRef]

Appl. Phys. Lett. (1)

J. R. Meyer, C. L. Felix, W. W. Bewley, I. Vurgaftman, E. H. Aifer, L. J. Olafsen, J. R. Lindle, C. A. Hoffman, M.-J. Yang, B. R. Bennett, B. V. Shanabrook, H. Lee, C.-H. Lin, S. S. Pei, and R. H. Miles, “Auger coefficients in type-II InAs/Ga1−x Inx Sb quantum wells,” Appl. Phys. Lett.73, 2857–2859 (1998).
[CrossRef]

EPL (1)

T.-Y. Zhang and W. Zhao, “Magnetoexcitonic optical absorption in semiconductors under strong magnetic fields and intense terahertz radiation in the Voigt configuration,” EPL82, 67001 (2008).
[CrossRef]

IEEE J. Quantum Electron. (3)

D. Ahn and S.-L. Chuang, “Optical gain in a strained-layer quantum-well laser,” IEEE J. Quantum Electron.24, 2400–2406 (1988).
[CrossRef]

Y. Tsou, A. Ichii, and E. M. Garmire, “Improving InAs double heterostructure lasers with better confinement,” IEEE J. Quantum Electron.28, 1261–1268 (1992).
[CrossRef]

A. Ahland, M. Wiedenhaus, D. Schulz, and E. Voges, “Calculation of exciton absorption in arbitrary layered semiconductor nanostructures with exact treatment of the coulomb singularity,” IEEE J. Quantum Electron.36, 842–848 (2000).
[CrossRef]

IEEE J. Sel. Topics Quantum Electron. (1)

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

IEEE Microw. Guided Wave Lett. (1)

F. L. Teixeira and W. C. Chew, “General closed-form PML constitutive tensors to match arbitrary bianisotropic and dispersive linear media,” IEEE Microw. Guided Wave Lett.8, 223–225 (1998).
[CrossRef]

J. Am. Chem. Soc. (1)

S. Kim, B. Fisher, H.-J. Eisler, and M. Bawendi, “Type-II quantum dots: CdTe/CdSe(core/shell) and CdSe/ZnTe(core/shell) heterostructures,” J. Am. Chem. Soc.125, 11466–11467 (2003).
[CrossRef] [PubMed]

J. Appl. Phys. (5)

S. Odermatt, M. Luisier, and B. Witzigmann, “Bandstructure calculation using the k· p method for arbitrary potentials with open boundary conditions,” J. Appl. Phys.97, 046104 (2005).
[CrossRef]

R. E. Nahory, M. A. Pollack, J. C. DeWinter, and K. M. Williams, “Growth and properties of liguid-phase epitaxial GaAs1−x Sbx,” J. Appl. Phys.48, 1607–1614 (1977).
[CrossRef]

D. Ahn, S. L. Chuang, and Y.-C. Chang, “Valence-band mixing effects on the gain and the refractive index change of quantum-well lasers,” J. Appl. Phys.64, 4056–4064 (1988).
[CrossRef]

X. Cartoixa, D. Z.-Y. Ting, and T. C. McGill, “Numerical spurious solutions in the effective mass approximation,” J. Appl. Phys.93, 3974–3981 (2003).
[CrossRef]

G. B. Liu, S.-L. Chuang, and S.-H. Park, “Optical gain of strained GaAsSb/GaAs quantum-well lasers: A self-consistent approach,” J. Appl. Phys.88, 5554–5561 (2000).
[CrossRef]

J. Comput. Electron. (1)

M. Karner, A. Gehring, and H. Kosina, “Efficient calculation of lifetime based direct tunneling through stacked dielectrics,” J. Comput. Electron.5, 161–165 (2006).
[CrossRef]

J. Comput. Phys. (1)

J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys.114, 185–200 (1994).
[CrossRef]

J. Phys. C: Solid St. Phys. (1)

P. K. W. Vinsome and D. Richardson, “The dielectric function in zincblende semiconductors,” J. Phys. C: Solid St. Phys.4, 2650–2657 (1971).
[CrossRef]

Microw. Opt. Technol. Lett. (1)

W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates,” Microw. Opt. Technol. Lett.7, 599–604 (1994).
[CrossRef]

Opt. Express (1)

Phys. Rev. (2)

U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev.124, 1866–1878 (1961).
[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 (9)

C. Y.-P. Chao and S. L. Chuang, “Spin-orbit-coupling effects on the valence-band structure of strained semiconductor quantum wells,” Phys. Rev. B46, 4110–4122 (1992).
[CrossRef]

G. Liu and S.-L. Chuang, “Modeling of Sb-based type-II quantum cascade lasers,” Phys. Rev. B65, 165220 (2002).
[CrossRef]

S.-L. Chuang, S. Schmitt-Rink, D. A. B. Miller, and D. S. Chemla, “Exciton Green’s-function approach to optical absorption in a quantum well with an applied electric field,” Phys. Rev. B43, 1500–1509 (1991).
[CrossRef]

S. Glutsch, D. S. Chemla, and F. Bechstedt, “Numerical calculation of the optical absorption in semiconductor quantum structures,” Phys. Rev. B54, 11592–11601 (1996).
[CrossRef]

A. Ahland, D. Schulz, and E. Voges, “Accurate mesh truncation for Schrodinger equations by a perfectly matched layer absorber: Application to the calculation of optical spectra,” Phys. Rev. B60, R5109–R5112 (1999).
[CrossRef]

C. G. Van de Walle, “Band lineups and deformation potentials in the model-solid theory,” Phys. Rev. B39, 1871–1883 (1989).
[CrossRef]

F. Szmulowicz, “Derivation of a general expression for the momentum matrix elements within the envelope-function approximation,” Phys. Rev. B51, 1613–1623 (1995).
[CrossRef]

Y.-C. Chang and R. B. James, “Saturation of intersubband transitions in p-type semiconductor quantum wells,” Phys. Rev. B39, 12672–12681 (1989).
[CrossRef]

P. Lawaetz, “Valence-band parameters in cubic semiconductors,” Phys. Rev. B4, 3460–3467 (1971).
[CrossRef]

Semicond. Sci. Technol. (1)

S. L. Chuang and C. S. Chang, “A band-structure model of strained quantum-well wurtzite semiconductors,” Semicond. Sci. Technol.12, 252–263 (1997).
[CrossRef]

Other (4)

S. L. Chuang, Physics of Photonic Devices, 2 (John Wiley & Sons, New Jersey, 2009).

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

G. D. Mahan, Many-particle Physics, 3 (Kluwer Academic/Plenum Publishers, New York, 2000).

S. L. Chuang, Physics of Optoelectronic Devices (John Wiley & Sons, New York, 1995).

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Figures (5)

Fig. 1
Fig. 1

The band diagram of type-II GaAs0.7Sb0.3 coupled QWs. The strained conduction bandedge Ec,s and valence counterpart Ev,s are marked in red lines. The blue lines indicate energies of a few bound valence subbands at the BZ center. The PMLs (green regions) at two ends of the computation domain effectively model the open regions.

Fig. 2
Fig. 2

(a) The valence subband structures E n , k t U versus kt. There are six bound subbands which are doubly degenerate. Four of them are HH-like (blue lines), and others are LH-like (red lines). (b) The envelope functions ϕ v ( U , n , k t = 0 ) ( z ) [v = 1 – 4; conduction (red), HH (blue), LH (green), and SO (cyan)] at the BZ center for the first (n′ = 1) valence subband (HH1). Only the HH component is present. (c) The counterpart of the third (n′ = 3) valence subband (LH1). The noticeable band mixing exists despite a significant LH part.

Fig. 3
Fig. 3

(a) Lineshapes of Im[2Vn,kt,xx(ω′)] corresponding to the HH1 subband at kt = kt, where kt ranges from 0 to 0.4 nm−1 at an interval of 0.04 nm−1. All the lineshapes are skew symmetric and exhibit coherent dips at blue sides of their peaks. (b) The fitting function f (ω′) (red) and Im[2Vn,kt,xx(ω′)] (blue) of the HH1 subband at the BZ center. The background one fb(ω′) (green) is also depicted.

Fig. 4
Fig. 4

Parameters of the Fano lineshapes Im[2Vn,kt,xx] (kt = kt) for the HH1 (blue) and LH1 (red) subbands as a function of kt: (a) the resonant energy h̄ωr; (b) linewidth γF; and (c) the Fano parameter q.

Fig. 5
Fig. 5

(a) The tunneling rate (linewidth) γF versus the width of GaAs0.7Sb0.3 layers at the BZ center. The rate decreases exponentially as the barrier width increases. (b) The polarized absorption spectra for type-II coupled QWs with a 2 nm GaAs0.7Sb0.3 layer. The TE absorption (blue) is more significant than TM one (red) at low photon energies.

Tables (2)

Tables Icon

Table 1 Material parameters in eight-band calculations

Tables Icon

Table 2 The eight Bloch parts adopted in the eight-band effective Hamiltonian. Symbols S, X, Y, and Z indicate that the corresponding spatial distributions are similar to the s, x, y, and z orbitals in hydrogen atoms, and ↑ and ↓ are the two spin states.

Equations (37)

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

Ψ ( n , k t ) ( r ) = r | n , k t e i k t ρ A QW l = 1 8 ϕ l ( n , k t ) ( z ) u l ( r ) ,
l = 1 8 H l l [ k t , k z = i z ] ϕ l ( n , k t ) ( z ) = E n , k t ϕ l ( n , k t ) ( z ) ,
A ( z ) [ i z ] 2 z A ( z ) z , B ( z ) [ i z ] i 2 [ B ( z ) z + z B ( z ) ] ,
ε r , i j ( ω ) = ε r , b , i j ( ω ) + 1 ε 0 V QW k t n , n ( f n , k t f n , k t ) × ( e m 0 ω ) 2 ( p n n , k t i p n , n , k t j E n , n , k t h ¯ ω i ϒ k t + p n n , k t j p n n , k t i E n n , k t + h ¯ ω + i ϒ k t ) ,
ε r , i j ( ω ) = ε r , b , i j ( ω ) + 1 ε 0 V QW k t , n X n , k t , i j ( ω ) ,
X n , k t , i j ( ω ) = ( e m 0 ω ) 2 n 1 2 ( f n , k t f n , k t ) ( p n n , k t i p n n , k t j + p n n , k t j p n n , k t i ) × ( 1 E n n , k t h ¯ ω i ϒ k t + 1 E n n , k t + h ¯ ω + i ϒ k t ) .
α i ( ω ) 1 Re [ ε r , b , i i ( ω ) ] ( ω c ) { Im [ ε r , b , i i ( ω ) ] + 1 ε 0 V QW k t , n Im [ X n , k t , i i ( ω ) ] } .
Im [ X n , k t , i j ( ω ) ] = ( e m 0 ω ) 2 [ I n , k t , i j ( ω ) I n , k t , i j ( ω ) ] ,
I n , k t , i j ( ω ) = 0 d ω 1 π ϒ k t / h ¯ [ ( ω ω ) 2 + ( ϒ k t / h ¯ ) 2 ] Im [ V n , k t , i j ( ω ) + V n , k t , j i ( ω ) ] ,
V n , k t , i j ( ω ) = 1 2 [ f FD ( E n , k t ) f FD ( E n , k t + h ¯ ω ) ] n p n n , k t i p n n , k t j E n , k t ( E n , k t + h ¯ ω ) i η ,
p l l [ k ] = m 0 h ¯ k H l l [ k ] = Γ l l + h ¯ k z Q l l + h ¯ Λ l l [ k t ] ,
p n n , k t j = n , k t | p j | n , k t = l , l d z ϕ l ( n , k t ) * ( z ) p j , l l [ k t , k z = i z ] ϕ l ( n , k t ) ( z ) l d z ϕ l ( n , k t ) * ( z ) J j , l ( n , k t ) ( z ) ,
J j , l ( n , k t ) ( z ) l p j , l l [ k t , k z = i z ] ϕ l ( n , k t ) ( z ) = l { Γ j , l l ( z ) ϕ l ( n , k t ) ( z ) i h ¯ 2 ( Q j , l l ( z ) z [ ϕ l ( n , k t ) ( z ) ] + z [ Q j , l l ( z ) ϕ l ( n , k t ) ] ) + h ¯ Λ j , l l [ k t , z ] ϕ l ( n , k t ) ( z ) } .
V n , k t , i j ( ω ) = 1 2 [ f FD ( E n , k t ) f FD ( E n , k t + h ¯ ω ) ] × l , l d z d z J i , l ( n , k t ) * ( z ) G k t , l l ( R ) ( z , z , E n , k t + h ¯ ω ) J j , l ( n , k t ) ( z ) ,
G k t , l l ( R ) ( z , z , E ) = n ϕ l ( n , k t ) ( z ) ϕ l ( n , k t ) * ( z ) E n , k t E i η .
F j , l ( n , k t ) ( z , ω ) = l d z G k t , l l ( R ) ( z , z , E n , k t + h ¯ ω ) J j , l ( n , k t ) ( z ) ,
V n , k t , i j ( ω ) = 1 2 [ f FD ( E n , k t ) f FD ( E n , k t + h ¯ ω ) ] l d z J i , l ( n , k t ) * ( z ) F j , l ( n , k t ) ( z , ω ) .
l = 1 8 { H l ˜ l [ k t , k z = i z ] ( E + i η ) δ l ˜ l } G k t , l l ( R ) ( z , z , E ) = δ l ˜ l δ ( z z ) ,
l = 1 8 { H l ˜ l [ k t , k z = i z ] ( E n , k t + h ¯ ω + i η ) δ l ˜ l } F j , l ( n , k t ) ( z , ω ) = J j , l ˜ ( n , k t ) ( z ) .
z s ( s ) = z + i z I ( z ) = { z i κ ( z L z h z ) m , z out , L < z < z L , z , z L < z < z R , z + i κ ( z z R h z ) m , z R < z < z out , R ,
F j , l ( n , k t , s ) ( z , ω ) = F ^ j , l ( n , k t ) ( z s ( z ) , ω ) , J j , l ˜ ( n , k t , s ) ( z ) = J ^ j , l ˜ ( n , k t ) ( z s ( z ) ) ,
l = 1 8 { H l ˜ l [ k t , k z s = i [ s z ( z ) ] 1 z ] ( E n , k t + h ¯ ω + i η ) δ l ˜ l } F j , l ( n , k t , s ) ( z , ω ) = J j , l ˜ ( n , k t , s ) ( z ) ,
f ( ω ) = f F ( ω ) f b ( ω ) , f F ( ω ) = ( ω ω r + γ F q / 2 ) 2 ( ω ω r ) 2 + ( γ F / 2 ) 2 , f b ( ω ) = { f 0 ( ω / ω b 1 ) n b , ω > ω b , 0 , ω ω b ,
H [ k ] = ( E c + P c 0 3 V + 2 U V 0 U 2 V 0 E c + P c 0 V + 2 U 3 V 2 V + U 3 V 0 E v P Q S R 0 1 2 S 2 R 2 U V S * E v P + Q 0 R 2 Q 3 2 S V + 2 U R * 0 E v P + Q S 3 2 S * 2 Q 0 3 V + 0 R * S * E v P Q 2 R * 1 2 S * U 2 V 1 2 S * 2 Q 3 2 S 2 R E v P Δ 0 2 V + U 2 R * 3 2 S * 2 Q 1 2 S 0 E v P Δ ) .
P c = P c , k + P c , ε , P = P k + P ε , Q = Q k + Q ε , R = R k + R ε , S = S k + S ε ,
P c , k = ( h ¯ 2 2 m 0 ) γ c ( k t 2 + k z 2 ) , P k = ( h ¯ 2 2 m 0 ) γ 1 ( k t 2 + k z 2 ) , Q k = ( h ¯ 2 2 m 0 ) γ 2 ( k t 2 + 2 k z 2 ) , R k = ( h ¯ 2 2 m 0 ) 3 [ γ 2 ( k x 2 k y 2 ) + i 2 γ 3 k x k y ] , S k = ( h ¯ 2 2 m 0 ) 2 3 γ 3 k k z , k ± = k x ± i k y = k t e ± i θ , θ = arg [ k + ] ,
P c , ε = a c ( ε x x + ε y y + ε z z ) , P ε = a v ( ε x x + ε y y + ε z z ) , Q ε = b 2 ( ε x x ε y y + 2 ε z z ) , R ε = 3 b 2 ( ε x x ε y y ) i d ε x y , S ε = d ( ε x z i ε y z ) .
V ± = 1 6 ( h ¯ m 0 ) P c v k ± , U = 1 3 ( h ¯ m 0 ) P c v k z , P c v = i S | p x | X = i S | p y | Y = i S | p z | Z ,
γ 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 ) ,
U ( θ ) = 1 2 ( e i θ / 2 0 0 0 e i θ / 2 0 0 0 i e i θ / 2 0 0 0 i e i θ / 2 0 0 0 0 e i 3 θ / 2 0 0 0 e i 3 θ / 2 0 0 0 0 i e i θ / 2 0 0 0 i e i θ / 2 0 0 0 e i θ / 2 0 0 0 e i θ / 2 0 0 i e i 3 θ / 2 0 0 0 i e i 3 θ / 2 0 0 0 0 0 i e i θ / 2 0 0 0 i e i θ / 2 0 0 0 e i θ / 2 0 0 0 e i θ / 2 ) .
H [ k t , k z ] = U ( θ ) H [ k ] U ( θ ) = ( H U [ k t , k z ] 0 4 × 4 0 4 × 4 H L [ k t , k z ] ) ,
H U [ k t , k z ] = ( E c + P c 3 V ρ V ρ + i 2 U 2 V ρ i U 3 V ρ E v P Q R ρ + i S ρ 2 R ρ i 1 2 S ρ V ρ i 2 U R ρ i S ρ E v P + Q 2 Q i 3 2 S ρ 2 V ρ + i U 2 R ρ + i 1 2 S ρ 2 Q + i 3 2 S ρ E v P Δ ) ,
V ρ = 1 6 ( h ¯ m 0 ) P c v k t , R ρ = ( h ¯ 2 2 m 0 ) 3 ( γ 2 + γ 3 2 ) k t 2 , S ρ = ( h ¯ 2 2 m 0 ) 2 3 γ 3 k t k z .
Φ ( n , k t , k z ) = ( Φ ( U , n , k t , k z ) 0 4 × 1 ) or ( 0 4 × 1 Φ ( L , n , k t , k z ) ) ,
v = 1 4 H v v σ [ k t , k z = i z ] ϕ v ( σ , n , k t ) ( z ) = E n , k t σ ϕ v ( σ , n , k t ) ( z ) ,
Γ = Γ + e ^ + + Γ e ^ + Γ z z ^ , Q = Q + e ^ + + Q e ^ + Q z z ^ , Λ ( k t ) = Λ t k t + [ Λ 1 + Λ 2 ] k + e ^ + + [ Λ 1 + Λ 2 ] k e ^ + [ Λ z , + k + + Λ z , k ] z ^ .
Γ + = P c v 3 ( 0 0 3 0 0 0 0 0 0 0 0 1 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 ) , Γ z = P c v 3 ( 0 0 0 2 0 0 1 0 0 0 0 0 2 0 0 1 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 ) , Q + = γ 3 ( 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 6 0 0 3 0 0 3 0 0 0 0 0 0 0 0 3 0 0 0 0 ) , Λ 1 = γ 2 + γ 3 2 ( 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 3 0 0 6 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 ) , Λ t = ( γ c 0 0 0 0 0 0 0 0 γ c 0 0 0 0 0 0 0 0 γ 1 γ 2 0 0 0 0 0 0 0 0 γ 1 + γ 2 0 0 2 γ 2 0 0 0 0 0 γ 1 + γ 2 0 0 2 γ 2 0 0 0 0 0 γ 1 γ 2 0 0 0 0 0 2 γ 2 0 0 γ 1 0 0 0 0 0 2 γ 2 0 0 γ 1 ) ,

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