Abstract

We present a comprehensive model for In0.18Ga0.82N/GaN self-assembled quantum dot (QD) active material. The strain distribution in the QD structure is studied using linear elastic theory with the application of the shrink-fit boundary condition at the material interface. Subsequent calculations also predict the strain-induced quantum-confined Stark effect (QCSE). Under carrier injection, the overall effect of band bending and charge screening is studied by solving the Schrödinger and Poisson equations self-consistently. The optical gain spectrum of the InGaN/GaN QD active material is calculated based on the electronic states solved from the Schrödinger-Poisson equation, and both the calculated material gain peak and emission wavelength agree well with the measured experimental data.

© 2014 Optical Society of America

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  1. C. R. Eddy, N. Nepal, J. K. Hite, and M. A. Mastro, “Perspectives on future directions in III-N semiconductor research,” J. Vac. Sci. Technol. A 31, 058501 (2013).
    [Crossref]
  2. V. A. Dmitriev, K. G. Irvine, C. H. Carter, N. I. Kuznetsov, and E. V. Kalinina, “Electric breakdown in GaN p-n junctions,” Appl. Phys. Lett. 68, 229–231 (1996).
    [Crossref]
  3. O. Ambacher, J. Smart, J. R. Shealy, N. G. Weimann, K. Chu, M. Murphy, W. J. Schaff, L. F. Eastman, R. Dimitrov, L. Wittmer, M. Stutzmann, W. Rieger, and J. Hilsenbeck, “Two-dimensional electron gases induced by spontaneous and piezoelectric polarization charges in N- and Ga-face AlGaN/GaN heterostructures,” Appl. Phys. Lett. 85, 3222–3233 (1999).
  4. Y. Arakawa, “Progress in GaN-based quantum dots for optoelectronics applications,” IEEE J. Sel. Top. Quantum Electron. 8, 823–832 (2002).
    [Crossref]
  5. N. Grandjean and M. Ilegems, “Visible InGaN/GaN quantum-dot materials and devices,” Proc. IEEE. 95, 1853–1865 (2007).
    [Crossref]
  6. A. Banerjee, T. Frost, and P. Bhattacharya, “Nitride-based quantum dot visible lasers,” J. Phys. D: Appl. Phys. 46, 264004 (2013).
    [Crossref]
  7. S. Schulz and E. P. O’Reilly, “Theory of reduced built-in polarization field in nitride-based quantum dots,” Phys. Rev. B 82, 033411 (2010).
    [Crossref]
  8. V. Ranjan, G. Allan, C. Priester, and C. Delerue, “Self-consistent calculations of the optical properties of GaN quantum dots,” Phys. Rev. B 68, 115305 (2003).
    [Crossref]
  9. Y.-R. Wu, Y.-Y. Lin, H.-H. Huang, and J. Singh, “Electronic and optical properties of InGaN quantum dot based light emitters for solid state lighting,” J. Appl. Phys. 105, 013117 (2009).
    [Crossref]
  10. S.-H. Park and W.-P. Hong, “Polarization potentials in InGaN/GaN semiconductor quantum dots,” J. Korean Phys. Soc. 57, 1308–1311 (2010).
    [Crossref]
  11. A. Banerjee, T. Frost, E. Stark, and P. Bhattacharya, “Continuous-wave operation and differential gain of InGaN/ GaN quantum dot ridge waveguide lasers (λ=420 nm) on c-plane GaN substrate,” Appl. Phys. Lett. 101, 041108 (2012).
    [Crossref]
  12. S. L. Chuang, Physics of Photonic Devices, 2nd ed. (Wiley, 2009), Chaps. 9 and 11.
  13. D. Bimberg, M. Grundmann, and N. N. Ledentsov, Quantum Dot Heterostructures (Wiley, 1999), Chap. 5.
  14. M. Grundmann, O. Stier, and D. Bimberg, “InAs/GaAs pyramidal quantum dots: Strain distribution, optical phonons, and electronic structure,” Phys. Rev. B 52, 11969–11981 (1995).
    [Crossref]
  15. N. Nakamura, H. Ogi, and M. Hirao, “Elastic, anelastic, and piezoelectric coefficients of GaN,” J. Appl. Phys. 111, 013509 (2012).
    [Crossref]
  16. R. Calarco, “InN nanowires: Growth and optoelectronic properties,” Materials 5, 2137–2150 (2012).
    [Crossref]
  17. I. Vurgaftman and J. R. Meyer, “Band parameters for nitrogen-containing semiconductors,” J. Appl. Phys. 94, 3675–3696 (2003).
    [Crossref]
  18. H. Morkoç, Handbook of Nitride Semiconductors and Devices, GaN-based Optical and Electronic Devices (Wiley, 2008), Chap. 1.
  19. Y. P. Varshni, “Temperature dependence of the energy gap in semiconductors,” Physica 34, 149–154 (1967).
    [Crossref]
  20. C.-L. Wu, H.-M. Lee, C.-T. Kuo, C.-H. Chen, and S. Gwo, “Cross-sectional scanning photoelectron microscopy and spectroscopy of wurtzite InN/GaN heterojunction: Measurement of “intrinsic” band lineup,” Appl. Phys. Lett. 92, 162106 (2008).
    [Crossref]
  21. A. Bykhovski, B. Gelmont, and M. Shur, “The influence of the strain-induced electric field on the charge distribution in GaN-AlN-GaN structure,” J. Appl. Phys. 74, 6734–6739 (2008).
    [Crossref]
  22. J. S. Im, H. Kollmer, J. Off, A. Sohmer, F. Scholz, and A. Hangleiter, “Reduction of oscillator strength due to piezoelectric fields in GaN/AlxGa1−xN quantum wells,” Phys. Rev. B 57, R9435(R) (1998).
    [Crossref]
  23. M. Suzuki and T. Uenoyama, “First-principles calculations of effective-mass parameters of A1N and GaN,” Phys. Rev. B 52, 8132–8139 (1995).
    [Crossref]
  24. S.-H. Park and S. L. Chuang, “Piezoelectric effects on electrical and optical properties of wurtzite GaN/AlGaN quantum well lasers,” Appl. Phys. Lett. 72, 3103–3105 (1998).
    [Crossref]
  25. J. M. Li, Y. W. Lü, D. B. Li, X. X. Han, Q. S. Zhu, L. Liu, and Z. G. Wang, “Effect of spontaneous and piezo-electric polarization on intersubband transition in AlxGa1−xN–GaN quantum well,” J. Vac. Sci. Technol. B 22, 2568–2573 (2004).
    [Crossref]
  26. G. Liu and S. L. Chuang, “Modeling of Sb-based type-II quantum cascade lasers,” Phys. Rev. B 65, 165220 (2002).
    [Crossref]
  27. L. A. Coldren and S. W. Corzine, Diode Lasers and Photonic Integrated Circuits (Wiley, 1995)
  28. S. L. Chuang and C. S. Chang, “k·p method for strained wurtzite semiconductors,” Phys. Rev. B 54, 2491–2504 (1996).
    [Crossref]

2013 (2)

C. R. Eddy, N. Nepal, J. K. Hite, and M. A. Mastro, “Perspectives on future directions in III-N semiconductor research,” J. Vac. Sci. Technol. A 31, 058501 (2013).
[Crossref]

A. Banerjee, T. Frost, and P. Bhattacharya, “Nitride-based quantum dot visible lasers,” J. Phys. D: Appl. Phys. 46, 264004 (2013).
[Crossref]

2012 (3)

N. Nakamura, H. Ogi, and M. Hirao, “Elastic, anelastic, and piezoelectric coefficients of GaN,” J. Appl. Phys. 111, 013509 (2012).
[Crossref]

R. Calarco, “InN nanowires: Growth and optoelectronic properties,” Materials 5, 2137–2150 (2012).
[Crossref]

A. Banerjee, T. Frost, E. Stark, and P. Bhattacharya, “Continuous-wave operation and differential gain of InGaN/ GaN quantum dot ridge waveguide lasers (λ=420 nm) on c-plane GaN substrate,” Appl. Phys. Lett. 101, 041108 (2012).
[Crossref]

2010 (2)

S.-H. Park and W.-P. Hong, “Polarization potentials in InGaN/GaN semiconductor quantum dots,” J. Korean Phys. Soc. 57, 1308–1311 (2010).
[Crossref]

S. Schulz and E. P. O’Reilly, “Theory of reduced built-in polarization field in nitride-based quantum dots,” Phys. Rev. B 82, 033411 (2010).
[Crossref]

2009 (1)

Y.-R. Wu, Y.-Y. Lin, H.-H. Huang, and J. Singh, “Electronic and optical properties of InGaN quantum dot based light emitters for solid state lighting,” J. Appl. Phys. 105, 013117 (2009).
[Crossref]

2008 (2)

C.-L. Wu, H.-M. Lee, C.-T. Kuo, C.-H. Chen, and S. Gwo, “Cross-sectional scanning photoelectron microscopy and spectroscopy of wurtzite InN/GaN heterojunction: Measurement of “intrinsic” band lineup,” Appl. Phys. Lett. 92, 162106 (2008).
[Crossref]

A. Bykhovski, B. Gelmont, and M. Shur, “The influence of the strain-induced electric field on the charge distribution in GaN-AlN-GaN structure,” J. Appl. Phys. 74, 6734–6739 (2008).
[Crossref]

2007 (1)

N. Grandjean and M. Ilegems, “Visible InGaN/GaN quantum-dot materials and devices,” Proc. IEEE. 95, 1853–1865 (2007).
[Crossref]

2004 (1)

J. M. Li, Y. W. Lü, D. B. Li, X. X. Han, Q. S. Zhu, L. Liu, and Z. G. Wang, “Effect of spontaneous and piezo-electric polarization on intersubband transition in AlxGa1−xN–GaN quantum well,” J. Vac. Sci. Technol. B 22, 2568–2573 (2004).
[Crossref]

2003 (2)

V. Ranjan, G. Allan, C. Priester, and C. Delerue, “Self-consistent calculations of the optical properties of GaN quantum dots,” Phys. Rev. B 68, 115305 (2003).
[Crossref]

I. Vurgaftman and J. R. Meyer, “Band parameters for nitrogen-containing semiconductors,” J. Appl. Phys. 94, 3675–3696 (2003).
[Crossref]

2002 (2)

Y. Arakawa, “Progress in GaN-based quantum dots for optoelectronics applications,” IEEE J. Sel. Top. Quantum Electron. 8, 823–832 (2002).
[Crossref]

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

1999 (1)

O. Ambacher, J. Smart, J. R. Shealy, N. G. Weimann, K. Chu, M. Murphy, W. J. Schaff, L. F. Eastman, R. Dimitrov, L. Wittmer, M. Stutzmann, W. Rieger, and J. Hilsenbeck, “Two-dimensional electron gases induced by spontaneous and piezoelectric polarization charges in N- and Ga-face AlGaN/GaN heterostructures,” Appl. Phys. Lett. 85, 3222–3233 (1999).

1998 (2)

J. S. Im, H. Kollmer, J. Off, A. Sohmer, F. Scholz, and A. Hangleiter, “Reduction of oscillator strength due to piezoelectric fields in GaN/AlxGa1−xN quantum wells,” Phys. Rev. B 57, R9435(R) (1998).
[Crossref]

S.-H. Park and S. L. Chuang, “Piezoelectric effects on electrical and optical properties of wurtzite GaN/AlGaN quantum well lasers,” Appl. Phys. Lett. 72, 3103–3105 (1998).
[Crossref]

1996 (2)

S. L. Chuang and C. S. Chang, “k·p method for strained wurtzite semiconductors,” Phys. Rev. B 54, 2491–2504 (1996).
[Crossref]

V. A. Dmitriev, K. G. Irvine, C. H. Carter, N. I. Kuznetsov, and E. V. Kalinina, “Electric breakdown in GaN p-n junctions,” Appl. Phys. Lett. 68, 229–231 (1996).
[Crossref]

1995 (2)

M. Suzuki and T. Uenoyama, “First-principles calculations of effective-mass parameters of A1N and GaN,” Phys. Rev. B 52, 8132–8139 (1995).
[Crossref]

M. Grundmann, O. Stier, and D. Bimberg, “InAs/GaAs pyramidal quantum dots: Strain distribution, optical phonons, and electronic structure,” Phys. Rev. B 52, 11969–11981 (1995).
[Crossref]

1967 (1)

Y. P. Varshni, “Temperature dependence of the energy gap in semiconductors,” Physica 34, 149–154 (1967).
[Crossref]

Allan, G.

V. Ranjan, G. Allan, C. Priester, and C. Delerue, “Self-consistent calculations of the optical properties of GaN quantum dots,” Phys. Rev. B 68, 115305 (2003).
[Crossref]

Ambacher, O.

O. Ambacher, J. Smart, J. R. Shealy, N. G. Weimann, K. Chu, M. Murphy, W. J. Schaff, L. F. Eastman, R. Dimitrov, L. Wittmer, M. Stutzmann, W. Rieger, and J. Hilsenbeck, “Two-dimensional electron gases induced by spontaneous and piezoelectric polarization charges in N- and Ga-face AlGaN/GaN heterostructures,” Appl. Phys. Lett. 85, 3222–3233 (1999).

Arakawa, Y.

Y. Arakawa, “Progress in GaN-based quantum dots for optoelectronics applications,” IEEE J. Sel. Top. Quantum Electron. 8, 823–832 (2002).
[Crossref]

Banerjee, A.

A. Banerjee, T. Frost, and P. Bhattacharya, “Nitride-based quantum dot visible lasers,” J. Phys. D: Appl. Phys. 46, 264004 (2013).
[Crossref]

A. Banerjee, T. Frost, E. Stark, and P. Bhattacharya, “Continuous-wave operation and differential gain of InGaN/ GaN quantum dot ridge waveguide lasers (λ=420 nm) on c-plane GaN substrate,” Appl. Phys. Lett. 101, 041108 (2012).
[Crossref]

Bhattacharya, P.

A. Banerjee, T. Frost, and P. Bhattacharya, “Nitride-based quantum dot visible lasers,” J. Phys. D: Appl. Phys. 46, 264004 (2013).
[Crossref]

A. Banerjee, T. Frost, E. Stark, and P. Bhattacharya, “Continuous-wave operation and differential gain of InGaN/ GaN quantum dot ridge waveguide lasers (λ=420 nm) on c-plane GaN substrate,” Appl. Phys. Lett. 101, 041108 (2012).
[Crossref]

Bimberg, D.

M. Grundmann, O. Stier, and D. Bimberg, “InAs/GaAs pyramidal quantum dots: Strain distribution, optical phonons, and electronic structure,” Phys. Rev. B 52, 11969–11981 (1995).
[Crossref]

D. Bimberg, M. Grundmann, and N. N. Ledentsov, Quantum Dot Heterostructures (Wiley, 1999), Chap. 5.

Bykhovski, A.

A. Bykhovski, B. Gelmont, and M. Shur, “The influence of the strain-induced electric field on the charge distribution in GaN-AlN-GaN structure,” J. Appl. Phys. 74, 6734–6739 (2008).
[Crossref]

Calarco, R.

R. Calarco, “InN nanowires: Growth and optoelectronic properties,” Materials 5, 2137–2150 (2012).
[Crossref]

Carter, C. H.

V. A. Dmitriev, K. G. Irvine, C. H. Carter, N. I. Kuznetsov, and E. V. Kalinina, “Electric breakdown in GaN p-n junctions,” Appl. Phys. Lett. 68, 229–231 (1996).
[Crossref]

Chang, C. S.

S. L. Chuang and C. S. Chang, “k·p method for strained wurtzite semiconductors,” Phys. Rev. B 54, 2491–2504 (1996).
[Crossref]

Chen, C.-H.

C.-L. Wu, H.-M. Lee, C.-T. Kuo, C.-H. Chen, and S. Gwo, “Cross-sectional scanning photoelectron microscopy and spectroscopy of wurtzite InN/GaN heterojunction: Measurement of “intrinsic” band lineup,” Appl. Phys. Lett. 92, 162106 (2008).
[Crossref]

Chu, K.

O. Ambacher, J. Smart, J. R. Shealy, N. G. Weimann, K. Chu, M. Murphy, W. J. Schaff, L. F. Eastman, R. Dimitrov, L. Wittmer, M. Stutzmann, W. Rieger, and J. Hilsenbeck, “Two-dimensional electron gases induced by spontaneous and piezoelectric polarization charges in N- and Ga-face AlGaN/GaN heterostructures,” Appl. Phys. Lett. 85, 3222–3233 (1999).

Chuang, S. L.

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

S.-H. Park and S. L. Chuang, “Piezoelectric effects on electrical and optical properties of wurtzite GaN/AlGaN quantum well lasers,” Appl. Phys. Lett. 72, 3103–3105 (1998).
[Crossref]

S. L. Chuang and C. S. Chang, “k·p method for strained wurtzite semiconductors,” Phys. Rev. B 54, 2491–2504 (1996).
[Crossref]

S. L. Chuang, Physics of Photonic Devices, 2nd ed. (Wiley, 2009), Chaps. 9 and 11.

Coldren, L. A.

L. A. Coldren and S. W. Corzine, Diode Lasers and Photonic Integrated Circuits (Wiley, 1995)

Corzine, S. W.

L. A. Coldren and S. W. Corzine, Diode Lasers and Photonic Integrated Circuits (Wiley, 1995)

Delerue, C.

V. Ranjan, G. Allan, C. Priester, and C. Delerue, “Self-consistent calculations of the optical properties of GaN quantum dots,” Phys. Rev. B 68, 115305 (2003).
[Crossref]

Dimitrov, R.

O. Ambacher, J. Smart, J. R. Shealy, N. G. Weimann, K. Chu, M. Murphy, W. J. Schaff, L. F. Eastman, R. Dimitrov, L. Wittmer, M. Stutzmann, W. Rieger, and J. Hilsenbeck, “Two-dimensional electron gases induced by spontaneous and piezoelectric polarization charges in N- and Ga-face AlGaN/GaN heterostructures,” Appl. Phys. Lett. 85, 3222–3233 (1999).

Dmitriev, V. A.

V. A. Dmitriev, K. G. Irvine, C. H. Carter, N. I. Kuznetsov, and E. V. Kalinina, “Electric breakdown in GaN p-n junctions,” Appl. Phys. Lett. 68, 229–231 (1996).
[Crossref]

Eastman, L. F.

O. Ambacher, J. Smart, J. R. Shealy, N. G. Weimann, K. Chu, M. Murphy, W. J. Schaff, L. F. Eastman, R. Dimitrov, L. Wittmer, M. Stutzmann, W. Rieger, and J. Hilsenbeck, “Two-dimensional electron gases induced by spontaneous and piezoelectric polarization charges in N- and Ga-face AlGaN/GaN heterostructures,” Appl. Phys. Lett. 85, 3222–3233 (1999).

Eddy, C. R.

C. R. Eddy, N. Nepal, J. K. Hite, and M. A. Mastro, “Perspectives on future directions in III-N semiconductor research,” J. Vac. Sci. Technol. A 31, 058501 (2013).
[Crossref]

Frost, T.

A. Banerjee, T. Frost, and P. Bhattacharya, “Nitride-based quantum dot visible lasers,” J. Phys. D: Appl. Phys. 46, 264004 (2013).
[Crossref]

A. Banerjee, T. Frost, E. Stark, and P. Bhattacharya, “Continuous-wave operation and differential gain of InGaN/ GaN quantum dot ridge waveguide lasers (λ=420 nm) on c-plane GaN substrate,” Appl. Phys. Lett. 101, 041108 (2012).
[Crossref]

Gelmont, B.

A. Bykhovski, B. Gelmont, and M. Shur, “The influence of the strain-induced electric field on the charge distribution in GaN-AlN-GaN structure,” J. Appl. Phys. 74, 6734–6739 (2008).
[Crossref]

Grandjean, N.

N. Grandjean and M. Ilegems, “Visible InGaN/GaN quantum-dot materials and devices,” Proc. IEEE. 95, 1853–1865 (2007).
[Crossref]

Grundmann, M.

M. Grundmann, O. Stier, and D. Bimberg, “InAs/GaAs pyramidal quantum dots: Strain distribution, optical phonons, and electronic structure,” Phys. Rev. B 52, 11969–11981 (1995).
[Crossref]

D. Bimberg, M. Grundmann, and N. N. Ledentsov, Quantum Dot Heterostructures (Wiley, 1999), Chap. 5.

Gwo, S.

C.-L. Wu, H.-M. Lee, C.-T. Kuo, C.-H. Chen, and S. Gwo, “Cross-sectional scanning photoelectron microscopy and spectroscopy of wurtzite InN/GaN heterojunction: Measurement of “intrinsic” band lineup,” Appl. Phys. Lett. 92, 162106 (2008).
[Crossref]

Han, X. X.

J. M. Li, Y. W. Lü, D. B. Li, X. X. Han, Q. S. Zhu, L. Liu, and Z. G. Wang, “Effect of spontaneous and piezo-electric polarization on intersubband transition in AlxGa1−xN–GaN quantum well,” J. Vac. Sci. Technol. B 22, 2568–2573 (2004).
[Crossref]

Hangleiter, A.

J. S. Im, H. Kollmer, J. Off, A. Sohmer, F. Scholz, and A. Hangleiter, “Reduction of oscillator strength due to piezoelectric fields in GaN/AlxGa1−xN quantum wells,” Phys. Rev. B 57, R9435(R) (1998).
[Crossref]

Hilsenbeck, J.

O. Ambacher, J. Smart, J. R. Shealy, N. G. Weimann, K. Chu, M. Murphy, W. J. Schaff, L. F. Eastman, R. Dimitrov, L. Wittmer, M. Stutzmann, W. Rieger, and J. Hilsenbeck, “Two-dimensional electron gases induced by spontaneous and piezoelectric polarization charges in N- and Ga-face AlGaN/GaN heterostructures,” Appl. Phys. Lett. 85, 3222–3233 (1999).

Hirao, M.

N. Nakamura, H. Ogi, and M. Hirao, “Elastic, anelastic, and piezoelectric coefficients of GaN,” J. Appl. Phys. 111, 013509 (2012).
[Crossref]

Hite, J. K.

C. R. Eddy, N. Nepal, J. K. Hite, and M. A. Mastro, “Perspectives on future directions in III-N semiconductor research,” J. Vac. Sci. Technol. A 31, 058501 (2013).
[Crossref]

Hong, W.-P.

S.-H. Park and W.-P. Hong, “Polarization potentials in InGaN/GaN semiconductor quantum dots,” J. Korean Phys. Soc. 57, 1308–1311 (2010).
[Crossref]

Huang, H.-H.

Y.-R. Wu, Y.-Y. Lin, H.-H. Huang, and J. Singh, “Electronic and optical properties of InGaN quantum dot based light emitters for solid state lighting,” J. Appl. Phys. 105, 013117 (2009).
[Crossref]

Ilegems, M.

N. Grandjean and M. Ilegems, “Visible InGaN/GaN quantum-dot materials and devices,” Proc. IEEE. 95, 1853–1865 (2007).
[Crossref]

Im, J. S.

J. S. Im, H. Kollmer, J. Off, A. Sohmer, F. Scholz, and A. Hangleiter, “Reduction of oscillator strength due to piezoelectric fields in GaN/AlxGa1−xN quantum wells,” Phys. Rev. B 57, R9435(R) (1998).
[Crossref]

Irvine, K. G.

V. A. Dmitriev, K. G. Irvine, C. H. Carter, N. I. Kuznetsov, and E. V. Kalinina, “Electric breakdown in GaN p-n junctions,” Appl. Phys. Lett. 68, 229–231 (1996).
[Crossref]

Kalinina, E. V.

V. A. Dmitriev, K. G. Irvine, C. H. Carter, N. I. Kuznetsov, and E. V. Kalinina, “Electric breakdown in GaN p-n junctions,” Appl. Phys. Lett. 68, 229–231 (1996).
[Crossref]

Kollmer, H.

J. S. Im, H. Kollmer, J. Off, A. Sohmer, F. Scholz, and A. Hangleiter, “Reduction of oscillator strength due to piezoelectric fields in GaN/AlxGa1−xN quantum wells,” Phys. Rev. B 57, R9435(R) (1998).
[Crossref]

Kuo, C.-T.

C.-L. Wu, H.-M. Lee, C.-T. Kuo, C.-H. Chen, and S. Gwo, “Cross-sectional scanning photoelectron microscopy and spectroscopy of wurtzite InN/GaN heterojunction: Measurement of “intrinsic” band lineup,” Appl. Phys. Lett. 92, 162106 (2008).
[Crossref]

Kuznetsov, N. I.

V. A. Dmitriev, K. G. Irvine, C. H. Carter, N. I. Kuznetsov, and E. V. Kalinina, “Electric breakdown in GaN p-n junctions,” Appl. Phys. Lett. 68, 229–231 (1996).
[Crossref]

Ledentsov, N. N.

D. Bimberg, M. Grundmann, and N. N. Ledentsov, Quantum Dot Heterostructures (Wiley, 1999), Chap. 5.

Lee, H.-M.

C.-L. Wu, H.-M. Lee, C.-T. Kuo, C.-H. Chen, and S. Gwo, “Cross-sectional scanning photoelectron microscopy and spectroscopy of wurtzite InN/GaN heterojunction: Measurement of “intrinsic” band lineup,” Appl. Phys. Lett. 92, 162106 (2008).
[Crossref]

Li, D. B.

J. M. Li, Y. W. Lü, D. B. Li, X. X. Han, Q. S. Zhu, L. Liu, and Z. G. Wang, “Effect of spontaneous and piezo-electric polarization on intersubband transition in AlxGa1−xN–GaN quantum well,” J. Vac. Sci. Technol. B 22, 2568–2573 (2004).
[Crossref]

Li, J. M.

J. M. Li, Y. W. Lü, D. B. Li, X. X. Han, Q. S. Zhu, L. Liu, and Z. G. Wang, “Effect of spontaneous and piezo-electric polarization on intersubband transition in AlxGa1−xN–GaN quantum well,” J. Vac. Sci. Technol. B 22, 2568–2573 (2004).
[Crossref]

Lin, Y.-Y.

Y.-R. Wu, Y.-Y. Lin, H.-H. Huang, and J. Singh, “Electronic and optical properties of InGaN quantum dot based light emitters for solid state lighting,” J. Appl. Phys. 105, 013117 (2009).
[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]

Liu, L.

J. M. Li, Y. W. Lü, D. B. Li, X. X. Han, Q. S. Zhu, L. Liu, and Z. G. Wang, “Effect of spontaneous and piezo-electric polarization on intersubband transition in AlxGa1−xN–GaN quantum well,” J. Vac. Sci. Technol. B 22, 2568–2573 (2004).
[Crossref]

Lü, Y. W.

J. M. Li, Y. W. Lü, D. B. Li, X. X. Han, Q. S. Zhu, L. Liu, and Z. G. Wang, “Effect of spontaneous and piezo-electric polarization on intersubband transition in AlxGa1−xN–GaN quantum well,” J. Vac. Sci. Technol. B 22, 2568–2573 (2004).
[Crossref]

Mastro, M. A.

C. R. Eddy, N. Nepal, J. K. Hite, and M. A. Mastro, “Perspectives on future directions in III-N semiconductor research,” J. Vac. Sci. Technol. A 31, 058501 (2013).
[Crossref]

Meyer, J. R.

I. Vurgaftman and J. R. Meyer, “Band parameters for nitrogen-containing semiconductors,” J. Appl. Phys. 94, 3675–3696 (2003).
[Crossref]

Morkoç, H.

H. Morkoç, Handbook of Nitride Semiconductors and Devices, GaN-based Optical and Electronic Devices (Wiley, 2008), Chap. 1.

Murphy, M.

O. Ambacher, J. Smart, J. R. Shealy, N. G. Weimann, K. Chu, M. Murphy, W. J. Schaff, L. F. Eastman, R. Dimitrov, L. Wittmer, M. Stutzmann, W. Rieger, and J. Hilsenbeck, “Two-dimensional electron gases induced by spontaneous and piezoelectric polarization charges in N- and Ga-face AlGaN/GaN heterostructures,” Appl. Phys. Lett. 85, 3222–3233 (1999).

Nakamura, N.

N. Nakamura, H. Ogi, and M. Hirao, “Elastic, anelastic, and piezoelectric coefficients of GaN,” J. Appl. Phys. 111, 013509 (2012).
[Crossref]

Nepal, N.

C. R. Eddy, N. Nepal, J. K. Hite, and M. A. Mastro, “Perspectives on future directions in III-N semiconductor research,” J. Vac. Sci. Technol. A 31, 058501 (2013).
[Crossref]

O’Reilly, E. P.

S. Schulz and E. P. O’Reilly, “Theory of reduced built-in polarization field in nitride-based quantum dots,” Phys. Rev. B 82, 033411 (2010).
[Crossref]

Off, J.

J. S. Im, H. Kollmer, J. Off, A. Sohmer, F. Scholz, and A. Hangleiter, “Reduction of oscillator strength due to piezoelectric fields in GaN/AlxGa1−xN quantum wells,” Phys. Rev. B 57, R9435(R) (1998).
[Crossref]

Ogi, H.

N. Nakamura, H. Ogi, and M. Hirao, “Elastic, anelastic, and piezoelectric coefficients of GaN,” J. Appl. Phys. 111, 013509 (2012).
[Crossref]

Park, S.-H.

S.-H. Park and W.-P. Hong, “Polarization potentials in InGaN/GaN semiconductor quantum dots,” J. Korean Phys. Soc. 57, 1308–1311 (2010).
[Crossref]

S.-H. Park and S. L. Chuang, “Piezoelectric effects on electrical and optical properties of wurtzite GaN/AlGaN quantum well lasers,” Appl. Phys. Lett. 72, 3103–3105 (1998).
[Crossref]

Priester, C.

V. Ranjan, G. Allan, C. Priester, and C. Delerue, “Self-consistent calculations of the optical properties of GaN quantum dots,” Phys. Rev. B 68, 115305 (2003).
[Crossref]

Ranjan, V.

V. Ranjan, G. Allan, C. Priester, and C. Delerue, “Self-consistent calculations of the optical properties of GaN quantum dots,” Phys. Rev. B 68, 115305 (2003).
[Crossref]

Rieger, W.

O. Ambacher, J. Smart, J. R. Shealy, N. G. Weimann, K. Chu, M. Murphy, W. J. Schaff, L. F. Eastman, R. Dimitrov, L. Wittmer, M. Stutzmann, W. Rieger, and J. Hilsenbeck, “Two-dimensional electron gases induced by spontaneous and piezoelectric polarization charges in N- and Ga-face AlGaN/GaN heterostructures,” Appl. Phys. Lett. 85, 3222–3233 (1999).

Schaff, W. J.

O. Ambacher, J. Smart, J. R. Shealy, N. G. Weimann, K. Chu, M. Murphy, W. J. Schaff, L. F. Eastman, R. Dimitrov, L. Wittmer, M. Stutzmann, W. Rieger, and J. Hilsenbeck, “Two-dimensional electron gases induced by spontaneous and piezoelectric polarization charges in N- and Ga-face AlGaN/GaN heterostructures,” Appl. Phys. Lett. 85, 3222–3233 (1999).

Scholz, F.

J. S. Im, H. Kollmer, J. Off, A. Sohmer, F. Scholz, and A. Hangleiter, “Reduction of oscillator strength due to piezoelectric fields in GaN/AlxGa1−xN quantum wells,” Phys. Rev. B 57, R9435(R) (1998).
[Crossref]

Schulz, S.

S. Schulz and E. P. O’Reilly, “Theory of reduced built-in polarization field in nitride-based quantum dots,” Phys. Rev. B 82, 033411 (2010).
[Crossref]

Shealy, J. R.

O. Ambacher, J. Smart, J. R. Shealy, N. G. Weimann, K. Chu, M. Murphy, W. J. Schaff, L. F. Eastman, R. Dimitrov, L. Wittmer, M. Stutzmann, W. Rieger, and J. Hilsenbeck, “Two-dimensional electron gases induced by spontaneous and piezoelectric polarization charges in N- and Ga-face AlGaN/GaN heterostructures,” Appl. Phys. Lett. 85, 3222–3233 (1999).

Shur, M.

A. Bykhovski, B. Gelmont, and M. Shur, “The influence of the strain-induced electric field on the charge distribution in GaN-AlN-GaN structure,” J. Appl. Phys. 74, 6734–6739 (2008).
[Crossref]

Singh, J.

Y.-R. Wu, Y.-Y. Lin, H.-H. Huang, and J. Singh, “Electronic and optical properties of InGaN quantum dot based light emitters for solid state lighting,” J. Appl. Phys. 105, 013117 (2009).
[Crossref]

Smart, J.

O. Ambacher, J. Smart, J. R. Shealy, N. G. Weimann, K. Chu, M. Murphy, W. J. Schaff, L. F. Eastman, R. Dimitrov, L. Wittmer, M. Stutzmann, W. Rieger, and J. Hilsenbeck, “Two-dimensional electron gases induced by spontaneous and piezoelectric polarization charges in N- and Ga-face AlGaN/GaN heterostructures,” Appl. Phys. Lett. 85, 3222–3233 (1999).

Sohmer, A.

J. S. Im, H. Kollmer, J. Off, A. Sohmer, F. Scholz, and A. Hangleiter, “Reduction of oscillator strength due to piezoelectric fields in GaN/AlxGa1−xN quantum wells,” Phys. Rev. B 57, R9435(R) (1998).
[Crossref]

Stark, E.

A. Banerjee, T. Frost, E. Stark, and P. Bhattacharya, “Continuous-wave operation and differential gain of InGaN/ GaN quantum dot ridge waveguide lasers (λ=420 nm) on c-plane GaN substrate,” Appl. Phys. Lett. 101, 041108 (2012).
[Crossref]

Stier, O.

M. Grundmann, O. Stier, and D. Bimberg, “InAs/GaAs pyramidal quantum dots: Strain distribution, optical phonons, and electronic structure,” Phys. Rev. B 52, 11969–11981 (1995).
[Crossref]

Stutzmann, M.

O. Ambacher, J. Smart, J. R. Shealy, N. G. Weimann, K. Chu, M. Murphy, W. J. Schaff, L. F. Eastman, R. Dimitrov, L. Wittmer, M. Stutzmann, W. Rieger, and J. Hilsenbeck, “Two-dimensional electron gases induced by spontaneous and piezoelectric polarization charges in N- and Ga-face AlGaN/GaN heterostructures,” Appl. Phys. Lett. 85, 3222–3233 (1999).

Suzuki, M.

M. Suzuki and T. Uenoyama, “First-principles calculations of effective-mass parameters of A1N and GaN,” Phys. Rev. B 52, 8132–8139 (1995).
[Crossref]

Uenoyama, T.

M. Suzuki and T. Uenoyama, “First-principles calculations of effective-mass parameters of A1N and GaN,” Phys. Rev. B 52, 8132–8139 (1995).
[Crossref]

Varshni, Y. P.

Y. P. Varshni, “Temperature dependence of the energy gap in semiconductors,” Physica 34, 149–154 (1967).
[Crossref]

Vurgaftman, I.

I. Vurgaftman and J. R. Meyer, “Band parameters for nitrogen-containing semiconductors,” J. Appl. Phys. 94, 3675–3696 (2003).
[Crossref]

Wang, Z. G.

J. M. Li, Y. W. Lü, D. B. Li, X. X. Han, Q. S. Zhu, L. Liu, and Z. G. Wang, “Effect of spontaneous and piezo-electric polarization on intersubband transition in AlxGa1−xN–GaN quantum well,” J. Vac. Sci. Technol. B 22, 2568–2573 (2004).
[Crossref]

Weimann, N. G.

O. Ambacher, J. Smart, J. R. Shealy, N. G. Weimann, K. Chu, M. Murphy, W. J. Schaff, L. F. Eastman, R. Dimitrov, L. Wittmer, M. Stutzmann, W. Rieger, and J. Hilsenbeck, “Two-dimensional electron gases induced by spontaneous and piezoelectric polarization charges in N- and Ga-face AlGaN/GaN heterostructures,” Appl. Phys. Lett. 85, 3222–3233 (1999).

Wittmer, L.

O. Ambacher, J. Smart, J. R. Shealy, N. G. Weimann, K. Chu, M. Murphy, W. J. Schaff, L. F. Eastman, R. Dimitrov, L. Wittmer, M. Stutzmann, W. Rieger, and J. Hilsenbeck, “Two-dimensional electron gases induced by spontaneous and piezoelectric polarization charges in N- and Ga-face AlGaN/GaN heterostructures,” Appl. Phys. Lett. 85, 3222–3233 (1999).

Wu, C.-L.

C.-L. Wu, H.-M. Lee, C.-T. Kuo, C.-H. Chen, and S. Gwo, “Cross-sectional scanning photoelectron microscopy and spectroscopy of wurtzite InN/GaN heterojunction: Measurement of “intrinsic” band lineup,” Appl. Phys. Lett. 92, 162106 (2008).
[Crossref]

Wu, Y.-R.

Y.-R. Wu, Y.-Y. Lin, H.-H. Huang, and J. Singh, “Electronic and optical properties of InGaN quantum dot based light emitters for solid state lighting,” J. Appl. Phys. 105, 013117 (2009).
[Crossref]

Zhu, Q. S.

J. M. Li, Y. W. Lü, D. B. Li, X. X. Han, Q. S. Zhu, L. Liu, and Z. G. Wang, “Effect of spontaneous and piezo-electric polarization on intersubband transition in AlxGa1−xN–GaN quantum well,” J. Vac. Sci. Technol. B 22, 2568–2573 (2004).
[Crossref]

Appl. Phys. Lett. (5)

V. A. Dmitriev, K. G. Irvine, C. H. Carter, N. I. Kuznetsov, and E. V. Kalinina, “Electric breakdown in GaN p-n junctions,” Appl. Phys. Lett. 68, 229–231 (1996).
[Crossref]

O. Ambacher, J. Smart, J. R. Shealy, N. G. Weimann, K. Chu, M. Murphy, W. J. Schaff, L. F. Eastman, R. Dimitrov, L. Wittmer, M. Stutzmann, W. Rieger, and J. Hilsenbeck, “Two-dimensional electron gases induced by spontaneous and piezoelectric polarization charges in N- and Ga-face AlGaN/GaN heterostructures,” Appl. Phys. Lett. 85, 3222–3233 (1999).

A. Banerjee, T. Frost, E. Stark, and P. Bhattacharya, “Continuous-wave operation and differential gain of InGaN/ GaN quantum dot ridge waveguide lasers (λ=420 nm) on c-plane GaN substrate,” Appl. Phys. Lett. 101, 041108 (2012).
[Crossref]

C.-L. Wu, H.-M. Lee, C.-T. Kuo, C.-H. Chen, and S. Gwo, “Cross-sectional scanning photoelectron microscopy and spectroscopy of wurtzite InN/GaN heterojunction: Measurement of “intrinsic” band lineup,” Appl. Phys. Lett. 92, 162106 (2008).
[Crossref]

S.-H. Park and S. L. Chuang, “Piezoelectric effects on electrical and optical properties of wurtzite GaN/AlGaN quantum well lasers,” Appl. Phys. Lett. 72, 3103–3105 (1998).
[Crossref]

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

Y. Arakawa, “Progress in GaN-based quantum dots for optoelectronics applications,” IEEE J. Sel. Top. Quantum Electron. 8, 823–832 (2002).
[Crossref]

J. Appl. Phys. (4)

Y.-R. Wu, Y.-Y. Lin, H.-H. Huang, and J. Singh, “Electronic and optical properties of InGaN quantum dot based light emitters for solid state lighting,” J. Appl. Phys. 105, 013117 (2009).
[Crossref]

N. Nakamura, H. Ogi, and M. Hirao, “Elastic, anelastic, and piezoelectric coefficients of GaN,” J. Appl. Phys. 111, 013509 (2012).
[Crossref]

I. Vurgaftman and J. R. Meyer, “Band parameters for nitrogen-containing semiconductors,” J. Appl. Phys. 94, 3675–3696 (2003).
[Crossref]

A. Bykhovski, B. Gelmont, and M. Shur, “The influence of the strain-induced electric field on the charge distribution in GaN-AlN-GaN structure,” J. Appl. Phys. 74, 6734–6739 (2008).
[Crossref]

J. Korean Phys. Soc. (1)

S.-H. Park and W.-P. Hong, “Polarization potentials in InGaN/GaN semiconductor quantum dots,” J. Korean Phys. Soc. 57, 1308–1311 (2010).
[Crossref]

J. Phys. D: Appl. Phys. (1)

A. Banerjee, T. Frost, and P. Bhattacharya, “Nitride-based quantum dot visible lasers,” J. Phys. D: Appl. Phys. 46, 264004 (2013).
[Crossref]

J. Vac. Sci. Technol. A (1)

C. R. Eddy, N. Nepal, J. K. Hite, and M. A. Mastro, “Perspectives on future directions in III-N semiconductor research,” J. Vac. Sci. Technol. A 31, 058501 (2013).
[Crossref]

J. Vac. Sci. Technol. B (1)

J. M. Li, Y. W. Lü, D. B. Li, X. X. Han, Q. S. Zhu, L. Liu, and Z. G. Wang, “Effect of spontaneous and piezo-electric polarization on intersubband transition in AlxGa1−xN–GaN quantum well,” J. Vac. Sci. Technol. B 22, 2568–2573 (2004).
[Crossref]

Materials (1)

R. Calarco, “InN nanowires: Growth and optoelectronic properties,” Materials 5, 2137–2150 (2012).
[Crossref]

Phys. Rev. B (7)

M. Grundmann, O. Stier, and D. Bimberg, “InAs/GaAs pyramidal quantum dots: Strain distribution, optical phonons, and electronic structure,” Phys. Rev. B 52, 11969–11981 (1995).
[Crossref]

S. Schulz and E. P. O’Reilly, “Theory of reduced built-in polarization field in nitride-based quantum dots,” Phys. Rev. B 82, 033411 (2010).
[Crossref]

V. Ranjan, G. Allan, C. Priester, and C. Delerue, “Self-consistent calculations of the optical properties of GaN quantum dots,” Phys. Rev. B 68, 115305 (2003).
[Crossref]

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

J. S. Im, H. Kollmer, J. Off, A. Sohmer, F. Scholz, and A. Hangleiter, “Reduction of oscillator strength due to piezoelectric fields in GaN/AlxGa1−xN quantum wells,” Phys. Rev. B 57, R9435(R) (1998).
[Crossref]

M. Suzuki and T. Uenoyama, “First-principles calculations of effective-mass parameters of A1N and GaN,” Phys. Rev. B 52, 8132–8139 (1995).
[Crossref]

S. L. Chuang and C. S. Chang, “k·p method for strained wurtzite semiconductors,” Phys. Rev. B 54, 2491–2504 (1996).
[Crossref]

Physica (1)

Y. P. Varshni, “Temperature dependence of the energy gap in semiconductors,” Physica 34, 149–154 (1967).
[Crossref]

Proc. IEEE. (1)

N. Grandjean and M. Ilegems, “Visible InGaN/GaN quantum-dot materials and devices,” Proc. IEEE. 95, 1853–1865 (2007).
[Crossref]

Other (4)

S. L. Chuang, Physics of Photonic Devices, 2nd ed. (Wiley, 2009), Chaps. 9 and 11.

D. Bimberg, M. Grundmann, and N. N. Ledentsov, Quantum Dot Heterostructures (Wiley, 1999), Chap. 5.

H. Morkoç, Handbook of Nitride Semiconductors and Devices, GaN-based Optical and Electronic Devices (Wiley, 2008), Chap. 1.

L. A. Coldren and S. W. Corzine, Diode Lasers and Photonic Integrated Circuits (Wiley, 1995)

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Figures (10)

Fig. 1
Fig. 1 (a) A breakdown of our InGaN/GaN QD model. (b) Modeling flow chart for the published experimental work on a InGaN/GaN QD-based ridge waveguide laser [11].)

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Fig. 2
Fig. 2 (a) The radial displacement field in an embedded spherical structure (inset) as a function of radial distance. (b) A simple illustration of the superposition method for calculating the global stress field in our truncated hexagonal pyramid QD structure. All the points inside the QD should be considered in the real calculation.

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Fig. 3
Fig. 3 (a)–(c) The biaxial strain (εii) distribution in the hexagonal pyramidal QD at different viewing angles. εxx and εyy in (a) and (b) are plotted in the dot-wetting layer interface.

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Fig. 4
Fig. 4 (a), (b) The polarization-induced charge at different viewing angles. (c) The overall polarization-induced electrical potential Vpol. along the z direction, as specified in the inset.

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Fig. 5
Fig. 5 (a) The energy diagram of the InGaN/GaN QD heterostructure along the z and x direction, as specified in the inset. (b) The energy diagram of the same QD heterostructure with a quantum-well-like strain distribution and without the polarization-induced potential.

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Fig. 6
Fig. 6 The squared moduli of the wavefunctions of the bottommost three electron states in the conduction band in the InGaN/GaN QD at different viewing angles.

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Fig. 7
Fig. 7 The squared moduli of the wavefunctions the topmost three HH states in the valence band in the InGaN/GaN QD at different viewing angles.

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Fig. 8
Fig. 8 (a) The screening potential solved from the self-consistent Schorödinger-Poisson equation. The injection level is N2D = 8n2D. The inset shows the convergence of the solution, which meets the criterion after 22 iterations. (b) A comparison between the unscreened (black) and screened (blue) band diagram in both z and x directions.

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Fig. 9
Fig. 9 (a) A fitting of the EL peak emission wavelength using our QD model (s0 ≈ 0.65). The active region temperature is adjustable in the model while the substrate temperature is held at 10 °C. (b) The C1–HH1 wavefunction overlap as a function of injection level.

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Fig. 10
Fig. 10 (a) The profile of the power flow |P z (x, y)| of the fundamental mode in the ridge waveguide InGaN/GaN QD laser. The zoomed-in part shows the overlap between the fundamental mode and the 8 QD layers. The portion of the overlap determines the optical confinement factor. (b) The calculated material gain spectrum of the InGaN/GaN QD active material. The threshold material gain is specified as well.

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Tables (1)

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Table 1 Material parameters used in this work.

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Equations (24)

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ε = 1 2 [ ∇ u + ( ∇ u ) T ] σ = C ̿ : ε ∇ ⋅ σ + f = 0
C 44 ∇ 2 u ( r ) + ( C 11 − C 44 ) ∇ ∇ ⋅ u ( r ) = 0
u ρ ( r 0 − ) − u ρ ( r 0 + ) = ε ( 0 ) r 0 σ r r ( r 0 − ) = σ r r ( r 0 + )
u r ( r ) = { 2 C 11 − C 12 − C 13 3 C 11 ε ( 0 ) r , r < r 0 − C 11 + C 12 + C 13 3 C 11 ε ( 0 ) r 0 3 r 2 , r > r 0
σ i j = P . V . ∫ QD σ i j ( e ) ( r − r ′ ) d 3 r ′ + δ i j ∫ QD A 3 ( 2 C 11 − C 12 − C 13 ) ε ( 0 ) δ ( r − r ′ ) d 3 r ′
σ i j ( e ) ( r ) = { ε ( 0 ) A 4 π C 11 ( 3 x 2 − r 2 ) + C 12 ( 3 y 2 − r 2 ) + C 13 ( 3 z 2 − r 2 ) r 5 , i j = x x ε ( 0 ) A 4 π 3 C 13 ( x 2 + y 2 ) + 3 C 33 z 2 − ( 2 C 13 + C 33 ) r 2 r 5 , i j = z z 3 ε ( 0 ) A 4 π C 44 y z r 5 , i j = z x 3 ε ( 0 ) A 4 π C 11 − C 12 2 x y r 5 , i j = x y
ε ( 0 ) = s 0 ( ε x x ( 0 ) + ε y y ( 0 ) + ε z z ( 0 ) )
ε x x ( 0 ) = ε y y ( 0 ) = a ( GaN ) − a ( InGaN ) a ( InGaN ) ε z z ( 0 ) = c ( GaN ) − c ( InGaN ) c ( InGaN )
[ P p z , x P p z , y P p z , z ] = [ 0 0 0 0 e 15 0 0 0 0 e 15 0 0 e 31 e 31 e 33 0 0 0 ]   [ ε x x ε y y ε z z ε y z ε z x ε x y ]
V strain ( r ) = v 1 ε z z + v 2 ( ε x x + ε y y )
− { ∇ t ⋅ ε t ( r ) ∇ t + ∂ ∂ z ε z ( r ) ∂ ∂ z } V pol . ( r ) = ρ pol . ( r ) = − ∇ ⋅ ( P pz + P sp )
− h ¯ 2 2 { ∇ t ⋅ [ m t * ( r ) ] − 1 ∇ t + ∂ ∂ z [ m z * ( r ) ] − 1 ∂ ∂ z } ψ ( r ) + V ( r ) ψ ( r ) = E ψ ( r )
E g ( T ) = E g ( T = 0 ) − α T 2 β + T
Δ E v ( T ) = 850   meV Δ E g ( T = 298 ) Δ E g ( T ) Δ E c ( T ) = Δ E g ( T ) − Δ E v ( T )
ε x x ( r ) = ε y y ( r ) = a ( GaN ) − a ( InGaN ) a ( InGaN ) = − C 33 2 C 13 ε z z ( r ) ,     r ∈ QD
n 2 D = 2 N 2 D ∑ m = 1 M ∫ − ∞ ∞ G c ( E c − E em ) f c ( E c , T ) d E c + ∫ E c , b ∞ L D ρ 3 D e ( E c − E c , b ) f c ( E c , T ) d E c
ρ − ( r ) = − 2 q ∑ m = 1 M | ψ em ( r ) | 2 f c ( E em , T )
− { ∇ t ⋅ ε t ( r ) ∇ t + ∂ ∂ z ε z ( r ) ∂ ∂ z } V sc . ( r ) = ρ sc . ( r ) = ρ − ( r ) + ρ + ( r )
− h ¯ 2 2 { ∇ t ⋅ [ m t * ( r ) ] − 1 ∇ t + ∂ ∂ z [ m z * ( r ) ] − 1 ∂ ∂ z } ψ ( r ) + V tot . ( r ) ψ ( r ) = E ψ ( r )
V tot . ( r ) = V 0 ( r ) + V strain ( r ) + V pol . ( r ) + V sc . ( r )
| λ − 〈 λ 〉 last   5 | λ < 0.002 %
g TE ( h ¯ ω ) = 2 N 2 D f L D π q 2 n r , t ε 0 c 0 ω m 0 2 | 〈 i S | p X | X 〉 | 2 ∑ n , m N , M | I hm en | 2 | F hm | 2 ×                 ∫ − ∞ ∞ G c v ( E − E hm en ) [ f c ( E en , T ) − f c ( E hm , T ) ] L ( E − h ¯ ω ) d E
Γ = f ∑ QD ∬ | P z ( x , y ) | d x d y ∬ | P z ( x , y ) | d x d y
g th = α i + 1 2 L ln 1 R 1 R 2 Γ

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