Abstract

An improved rate equation model for GaN-based LEDs considering the effective volume of the active region is proposed. Through numerical simulations, it is confirmed that the IQE, especially efficiency droop is related with small effective volume. Also, we confirmed that the effective volume is controlled by polarization charge, the barriers between the quantum wells, and current density. We also developed a fast and reliable method for extracting the recombination coefficients and the IQE of the GaN-based LEDs by measuring transient characteristics and considering the effective volume.

© 2014 Optical Society of America

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  1. H. Jia, L. Guo, W. Wang, H. Chen, “Recent progress in GaN‐based light‐emitting diodes,” Adv. Mater. 21(45), 4641–4646 (2009).
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  2. M. F. Schubert, J. Xu, J. K. Kim, E. F. Schubert, M. H. Kim, S. Yoon, S. M. Lee, C. Sone, T. Sakong, Y. Park, “Polarization-matched GaInN/ AlGaInN multi-quantum-well light-emitting diodes with reduced efficiency droop,” Appl. Phys. Lett. 93(4), 041102 (2008).
    [CrossRef]
  3. J. Xie, X. Ni, Q. Fan, R. Shimada, U. Ozgur, H. Morkoc, “On the efficiency droop in InGaN multiple quantum well blue light emitting diodes and its reduction with p-doped quantum well barriers,” Appl. Phys. Lett. 93(12), 121107 (2008).
    [CrossRef]
  4. S.-H. Han, D.-Y. Lee, S.-J. Lee, C.-Y. Cho, M.-K. Kwon, S. Lee, D. Noh, D.-J. Kim, Y. C. Kim, S.-J. Park, “Effect of electron blocking layer on efficiency droop in InGaN/GaN multiple quantum well light-emitting diodes,” Appl. Phys. Lett. 94(23), 231123 (2009).
    [CrossRef]
  5. J. Hader, J. Moloney, S. Koch, “Density-activated defect recombination as a possible explanation for the efficiency droop in GaN-based diodes,” Appl. Phys. Lett. 96(22), 221106 (2010).
    [CrossRef]
  6. V. K. Malyutenko, S. S. Bolgov, A. D. Podoltsev, “Current crowding effect on the ideality factor and efficiency droop in blue lateral InGaN/GaN light emitting diodes,” Appl. Phys. Lett. 97(25), 251110 (2010).
    [CrossRef]
  7. Q. Dai, Q. F. Shan, J. Wang, S. Chhajed, J. Cho, E. F. Schubert, M. H. Crawford, D. D. Koleske, M. H. Kim, Y. Park, “Carrier recombination mechanisms and efficiency droop in GaInN/GaN light-emitting diodes,” Appl. Phys. Lett. 97(13), 133507 (2010).
    [CrossRef]
  8. N. Gardner, G. Muller, Y. Shen, G. Chen, S. Watanabe, W. Gotz, M. Krames, “Blue-emitting InGaN–GaN double-heterostructure light-emitting diodes reaching maximum quantum efficiency above 200,” Appl. Phys. Lett. 91(24), 243506 (2007).
    [CrossRef]
  9. W. Guo, M. Zhang, P. Bhattacharya, J. Heo, “Auger recombination in III-Nitride nanowires and its effect on nanowire light-emitting diode characteristics,” Nano Lett. 11(4), 1434–1438 (2011).
    [CrossRef] [PubMed]
  10. G.-B. Lin, D. Meyaard, J. Cho, E. F. Schubert, H. Shim, C. Sone, “Analytic model for the efficiency droop in semiconductors with asymmetric carrier-transport properties based on drift-induced reduction of injection efficiency,” Appl. Phys. Lett. 100(16), 161106 (2012).
    [CrossRef]
  11. J. Piprek, “Efficiency droop in nitride-based light-emitting diodes,” Phys. Status Solidi A 207(10), 2217–2225 (2010).
    [CrossRef]
  12. J. Cho, E. F. Schubert, J. K. Kim, “Efficiency droop in light‐emitting diodes: Challenges and countermeasures,” Laser Photonics Rev. 7(3), 408–421 (2013).
    [CrossRef]
  13. S. Nakamura, N. Senoh, N. Iwasa, S. I. Nagahama, “High-brightness InGan blue, green and yellow light-emitting-diodes with quantum-well structures,” Jpn. J. Appl. Phys. 34(7A), L797–L799 (1995).
    [CrossRef]
  14. A. David, M. J. Grundmann, “Influence of polarization fields on carrier lifetime and recombination rates in InGaN-based light-emitting diodes,” Appl. Phys. Lett. 97(3), 033501 (2010).
    [CrossRef]
  15. B. Santic, “On the hole effective mass and the free hole statistics in wurtzite GaN,” Semicond. Sci. Technol. 18(4), 219–224 (2003).
    [CrossRef]
  16. K. Kumakura, T. Makimoto, N. Kobayashi, “Mg-acceptor activation mechanism and transport characteristics in p-type InGaN grown by metalorganic vapor phase epitaxy,” J. Appl. Phys. 93(6), 3370–3375 (2003).
    [CrossRef]
  17. H. Y. Ryu, D. S. Shin, J. I. Shim, “Analysis of efficiency droop in nitride light-emitting diodes by the reduced effective volume of InGaN active material,” Appl. Phys. Lett. 100(13), 131109 (2012).
    [CrossRef]
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  21. E. Kioupakis, P. Rinke, K. T. Delaney, C. G. Van de Walle, “Indirect Auger recombination as a cause of efficiency droop in nitride light-emitting diodes,” Appl. Phys. Lett. 98(16), 161107 (2011).
    [CrossRef]
  22. Q. Dai, Q. Shan, J. Cho, E. F. Schubert, M. H. Crawford, D. D. Koleske, M.-H. Kim, Y. Park, “On the symmetry of efficiency-versus-carrier-concentration curves in GaInN/GaN light-emitting diodes and relation to droop-causing mechanisms,” Appl. Phys. Lett. 98(3), 033506 (2011).
    [CrossRef]

2013 (1)

J. Cho, E. F. Schubert, J. K. Kim, “Efficiency droop in light‐emitting diodes: Challenges and countermeasures,” Laser Photonics Rev. 7(3), 408–421 (2013).
[CrossRef]

2012 (2)

G.-B. Lin, D. Meyaard, J. Cho, E. F. Schubert, H. Shim, C. Sone, “Analytic model for the efficiency droop in semiconductors with asymmetric carrier-transport properties based on drift-induced reduction of injection efficiency,” Appl. Phys. Lett. 100(16), 161106 (2012).
[CrossRef]

H. Y. Ryu, D. S. Shin, J. I. Shim, “Analysis of efficiency droop in nitride light-emitting diodes by the reduced effective volume of InGaN active material,” Appl. Phys. Lett. 100(13), 131109 (2012).
[CrossRef]

2011 (3)

W. Guo, M. Zhang, P. Bhattacharya, J. Heo, “Auger recombination in III-Nitride nanowires and its effect on nanowire light-emitting diode characteristics,” Nano Lett. 11(4), 1434–1438 (2011).
[CrossRef] [PubMed]

E. Kioupakis, P. Rinke, K. T. Delaney, C. G. Van de Walle, “Indirect Auger recombination as a cause of efficiency droop in nitride light-emitting diodes,” Appl. Phys. Lett. 98(16), 161107 (2011).
[CrossRef]

Q. Dai, Q. Shan, J. Cho, E. F. Schubert, M. H. Crawford, D. D. Koleske, M.-H. Kim, Y. Park, “On the symmetry of efficiency-versus-carrier-concentration curves in GaInN/GaN light-emitting diodes and relation to droop-causing mechanisms,” Appl. Phys. Lett. 98(3), 033506 (2011).
[CrossRef]

2010 (5)

J. Hader, J. Moloney, S. Koch, “Density-activated defect recombination as a possible explanation for the efficiency droop in GaN-based diodes,” Appl. Phys. Lett. 96(22), 221106 (2010).
[CrossRef]

V. K. Malyutenko, S. S. Bolgov, A. D. Podoltsev, “Current crowding effect on the ideality factor and efficiency droop in blue lateral InGaN/GaN light emitting diodes,” Appl. Phys. Lett. 97(25), 251110 (2010).
[CrossRef]

Q. Dai, Q. F. Shan, J. Wang, S. Chhajed, J. Cho, E. F. Schubert, M. H. Crawford, D. D. Koleske, M. H. Kim, Y. Park, “Carrier recombination mechanisms and efficiency droop in GaInN/GaN light-emitting diodes,” Appl. Phys. Lett. 97(13), 133507 (2010).
[CrossRef]

A. David, M. J. Grundmann, “Influence of polarization fields on carrier lifetime and recombination rates in InGaN-based light-emitting diodes,” Appl. Phys. Lett. 97(3), 033501 (2010).
[CrossRef]

J. Piprek, “Efficiency droop in nitride-based light-emitting diodes,” Phys. Status Solidi A 207(10), 2217–2225 (2010).
[CrossRef]

2009 (2)

S.-H. Han, D.-Y. Lee, S.-J. Lee, C.-Y. Cho, M.-K. Kwon, S. Lee, D. Noh, D.-J. Kim, Y. C. Kim, S.-J. Park, “Effect of electron blocking layer on efficiency droop in InGaN/GaN multiple quantum well light-emitting diodes,” Appl. Phys. Lett. 94(23), 231123 (2009).
[CrossRef]

H. Jia, L. Guo, W. Wang, H. Chen, “Recent progress in GaN‐based light‐emitting diodes,” Adv. Mater. 21(45), 4641–4646 (2009).
[CrossRef]

2008 (2)

M. F. Schubert, J. Xu, J. K. Kim, E. F. Schubert, M. H. Kim, S. Yoon, S. M. Lee, C. Sone, T. Sakong, Y. Park, “Polarization-matched GaInN/ AlGaInN multi-quantum-well light-emitting diodes with reduced efficiency droop,” Appl. Phys. Lett. 93(4), 041102 (2008).
[CrossRef]

J. Xie, X. Ni, Q. Fan, R. Shimada, U. Ozgur, H. Morkoc, “On the efficiency droop in InGaN multiple quantum well blue light emitting diodes and its reduction with p-doped quantum well barriers,” Appl. Phys. Lett. 93(12), 121107 (2008).
[CrossRef]

2007 (1)

N. Gardner, G. Muller, Y. Shen, G. Chen, S. Watanabe, W. Gotz, M. Krames, “Blue-emitting InGaN–GaN double-heterostructure light-emitting diodes reaching maximum quantum efficiency above 200,” Appl. Phys. Lett. 91(24), 243506 (2007).
[CrossRef]

2003 (3)

B. Santic, “On the hole effective mass and the free hole statistics in wurtzite GaN,” Semicond. Sci. Technol. 18(4), 219–224 (2003).
[CrossRef]

K. Kumakura, T. Makimoto, N. Kobayashi, “Mg-acceptor activation mechanism and transport characteristics in p-type InGaN grown by metalorganic vapor phase epitaxy,” J. Appl. Phys. 93(6), 3370–3375 (2003).
[CrossRef]

I. Vurgaftman, J. Meyer, “Band parameters for nitrogen-containing semiconductors,” J. Appl. Phys. 94(6), 3675–3696 (2003).
[CrossRef]

1995 (1)

S. Nakamura, N. Senoh, N. Iwasa, S. I. Nagahama, “High-brightness InGan blue, green and yellow light-emitting-diodes with quantum-well structures,” Jpn. J. Appl. Phys. 34(7A), L797–L799 (1995).
[CrossRef]

Bhattacharya, P.

W. Guo, M. Zhang, P. Bhattacharya, J. Heo, “Auger recombination in III-Nitride nanowires and its effect on nanowire light-emitting diode characteristics,” Nano Lett. 11(4), 1434–1438 (2011).
[CrossRef] [PubMed]

Bolgov, S. S.

V. K. Malyutenko, S. S. Bolgov, A. D. Podoltsev, “Current crowding effect on the ideality factor and efficiency droop in blue lateral InGaN/GaN light emitting diodes,” Appl. Phys. Lett. 97(25), 251110 (2010).
[CrossRef]

Chen, G.

N. Gardner, G. Muller, Y. Shen, G. Chen, S. Watanabe, W. Gotz, M. Krames, “Blue-emitting InGaN–GaN double-heterostructure light-emitting diodes reaching maximum quantum efficiency above 200,” Appl. Phys. Lett. 91(24), 243506 (2007).
[CrossRef]

Chen, H.

H. Jia, L. Guo, W. Wang, H. Chen, “Recent progress in GaN‐based light‐emitting diodes,” Adv. Mater. 21(45), 4641–4646 (2009).
[CrossRef]

Chhajed, S.

Q. Dai, Q. F. Shan, J. Wang, S. Chhajed, J. Cho, E. F. Schubert, M. H. Crawford, D. D. Koleske, M. H. Kim, Y. Park, “Carrier recombination mechanisms and efficiency droop in GaInN/GaN light-emitting diodes,” Appl. Phys. Lett. 97(13), 133507 (2010).
[CrossRef]

Cho, C.-Y.

S.-H. Han, D.-Y. Lee, S.-J. Lee, C.-Y. Cho, M.-K. Kwon, S. Lee, D. Noh, D.-J. Kim, Y. C. Kim, S.-J. Park, “Effect of electron blocking layer on efficiency droop in InGaN/GaN multiple quantum well light-emitting diodes,” Appl. Phys. Lett. 94(23), 231123 (2009).
[CrossRef]

Cho, J.

J. Cho, E. F. Schubert, J. K. Kim, “Efficiency droop in light‐emitting diodes: Challenges and countermeasures,” Laser Photonics Rev. 7(3), 408–421 (2013).
[CrossRef]

G.-B. Lin, D. Meyaard, J. Cho, E. F. Schubert, H. Shim, C. Sone, “Analytic model for the efficiency droop in semiconductors with asymmetric carrier-transport properties based on drift-induced reduction of injection efficiency,” Appl. Phys. Lett. 100(16), 161106 (2012).
[CrossRef]

Q. Dai, Q. Shan, J. Cho, E. F. Schubert, M. H. Crawford, D. D. Koleske, M.-H. Kim, Y. Park, “On the symmetry of efficiency-versus-carrier-concentration curves in GaInN/GaN light-emitting diodes and relation to droop-causing mechanisms,” Appl. Phys. Lett. 98(3), 033506 (2011).
[CrossRef]

Q. Dai, Q. F. Shan, J. Wang, S. Chhajed, J. Cho, E. F. Schubert, M. H. Crawford, D. D. Koleske, M. H. Kim, Y. Park, “Carrier recombination mechanisms and efficiency droop in GaInN/GaN light-emitting diodes,” Appl. Phys. Lett. 97(13), 133507 (2010).
[CrossRef]

Crawford, M. H.

Q. Dai, Q. Shan, J. Cho, E. F. Schubert, M. H. Crawford, D. D. Koleske, M.-H. Kim, Y. Park, “On the symmetry of efficiency-versus-carrier-concentration curves in GaInN/GaN light-emitting diodes and relation to droop-causing mechanisms,” Appl. Phys. Lett. 98(3), 033506 (2011).
[CrossRef]

Q. Dai, Q. F. Shan, J. Wang, S. Chhajed, J. Cho, E. F. Schubert, M. H. Crawford, D. D. Koleske, M. H. Kim, Y. Park, “Carrier recombination mechanisms and efficiency droop in GaInN/GaN light-emitting diodes,” Appl. Phys. Lett. 97(13), 133507 (2010).
[CrossRef]

Dai, Q.

Q. Dai, Q. Shan, J. Cho, E. F. Schubert, M. H. Crawford, D. D. Koleske, M.-H. Kim, Y. Park, “On the symmetry of efficiency-versus-carrier-concentration curves in GaInN/GaN light-emitting diodes and relation to droop-causing mechanisms,” Appl. Phys. Lett. 98(3), 033506 (2011).
[CrossRef]

Q. Dai, Q. F. Shan, J. Wang, S. Chhajed, J. Cho, E. F. Schubert, M. H. Crawford, D. D. Koleske, M. H. Kim, Y. Park, “Carrier recombination mechanisms and efficiency droop in GaInN/GaN light-emitting diodes,” Appl. Phys. Lett. 97(13), 133507 (2010).
[CrossRef]

David, A.

A. David, M. J. Grundmann, “Influence of polarization fields on carrier lifetime and recombination rates in InGaN-based light-emitting diodes,” Appl. Phys. Lett. 97(3), 033501 (2010).
[CrossRef]

Delaney, K. T.

E. Kioupakis, P. Rinke, K. T. Delaney, C. G. Van de Walle, “Indirect Auger recombination as a cause of efficiency droop in nitride light-emitting diodes,” Appl. Phys. Lett. 98(16), 161107 (2011).
[CrossRef]

Fan, Q.

J. Xie, X. Ni, Q. Fan, R. Shimada, U. Ozgur, H. Morkoc, “On the efficiency droop in InGaN multiple quantum well blue light emitting diodes and its reduction with p-doped quantum well barriers,” Appl. Phys. Lett. 93(12), 121107 (2008).
[CrossRef]

Gardner, N.

N. Gardner, G. Muller, Y. Shen, G. Chen, S. Watanabe, W. Gotz, M. Krames, “Blue-emitting InGaN–GaN double-heterostructure light-emitting diodes reaching maximum quantum efficiency above 200,” Appl. Phys. Lett. 91(24), 243506 (2007).
[CrossRef]

Gotz, W.

N. Gardner, G. Muller, Y. Shen, G. Chen, S. Watanabe, W. Gotz, M. Krames, “Blue-emitting InGaN–GaN double-heterostructure light-emitting diodes reaching maximum quantum efficiency above 200,” Appl. Phys. Lett. 91(24), 243506 (2007).
[CrossRef]

Grundmann, M. J.

A. David, M. J. Grundmann, “Influence of polarization fields on carrier lifetime and recombination rates in InGaN-based light-emitting diodes,” Appl. Phys. Lett. 97(3), 033501 (2010).
[CrossRef]

Guo, L.

H. Jia, L. Guo, W. Wang, H. Chen, “Recent progress in GaN‐based light‐emitting diodes,” Adv. Mater. 21(45), 4641–4646 (2009).
[CrossRef]

Guo, W.

W. Guo, M. Zhang, P. Bhattacharya, J. Heo, “Auger recombination in III-Nitride nanowires and its effect on nanowire light-emitting diode characteristics,” Nano Lett. 11(4), 1434–1438 (2011).
[CrossRef] [PubMed]

Hader, J.

J. Hader, J. Moloney, S. Koch, “Density-activated defect recombination as a possible explanation for the efficiency droop in GaN-based diodes,” Appl. Phys. Lett. 96(22), 221106 (2010).
[CrossRef]

Han, S.-H.

S.-H. Han, D.-Y. Lee, S.-J. Lee, C.-Y. Cho, M.-K. Kwon, S. Lee, D. Noh, D.-J. Kim, Y. C. Kim, S.-J. Park, “Effect of electron blocking layer on efficiency droop in InGaN/GaN multiple quantum well light-emitting diodes,” Appl. Phys. Lett. 94(23), 231123 (2009).
[CrossRef]

Heo, J.

W. Guo, M. Zhang, P. Bhattacharya, J. Heo, “Auger recombination in III-Nitride nanowires and its effect on nanowire light-emitting diode characteristics,” Nano Lett. 11(4), 1434–1438 (2011).
[CrossRef] [PubMed]

Iwasa, N.

S. Nakamura, N. Senoh, N. Iwasa, S. I. Nagahama, “High-brightness InGan blue, green and yellow light-emitting-diodes with quantum-well structures,” Jpn. J. Appl. Phys. 34(7A), L797–L799 (1995).
[CrossRef]

Jia, H.

H. Jia, L. Guo, W. Wang, H. Chen, “Recent progress in GaN‐based light‐emitting diodes,” Adv. Mater. 21(45), 4641–4646 (2009).
[CrossRef]

Kim, D.-J.

S.-H. Han, D.-Y. Lee, S.-J. Lee, C.-Y. Cho, M.-K. Kwon, S. Lee, D. Noh, D.-J. Kim, Y. C. Kim, S.-J. Park, “Effect of electron blocking layer on efficiency droop in InGaN/GaN multiple quantum well light-emitting diodes,” Appl. Phys. Lett. 94(23), 231123 (2009).
[CrossRef]

Kim, J. K.

J. Cho, E. F. Schubert, J. K. Kim, “Efficiency droop in light‐emitting diodes: Challenges and countermeasures,” Laser Photonics Rev. 7(3), 408–421 (2013).
[CrossRef]

M. F. Schubert, J. Xu, J. K. Kim, E. F. Schubert, M. H. Kim, S. Yoon, S. M. Lee, C. Sone, T. Sakong, Y. Park, “Polarization-matched GaInN/ AlGaInN multi-quantum-well light-emitting diodes with reduced efficiency droop,” Appl. Phys. Lett. 93(4), 041102 (2008).
[CrossRef]

Kim, M. H.

Q. Dai, Q. F. Shan, J. Wang, S. Chhajed, J. Cho, E. F. Schubert, M. H. Crawford, D. D. Koleske, M. H. Kim, Y. Park, “Carrier recombination mechanisms and efficiency droop in GaInN/GaN light-emitting diodes,” Appl. Phys. Lett. 97(13), 133507 (2010).
[CrossRef]

M. F. Schubert, J. Xu, J. K. Kim, E. F. Schubert, M. H. Kim, S. Yoon, S. M. Lee, C. Sone, T. Sakong, Y. Park, “Polarization-matched GaInN/ AlGaInN multi-quantum-well light-emitting diodes with reduced efficiency droop,” Appl. Phys. Lett. 93(4), 041102 (2008).
[CrossRef]

Kim, M.-H.

Q. Dai, Q. Shan, J. Cho, E. F. Schubert, M. H. Crawford, D. D. Koleske, M.-H. Kim, Y. Park, “On the symmetry of efficiency-versus-carrier-concentration curves in GaInN/GaN light-emitting diodes and relation to droop-causing mechanisms,” Appl. Phys. Lett. 98(3), 033506 (2011).
[CrossRef]

Kim, Y. C.

S.-H. Han, D.-Y. Lee, S.-J. Lee, C.-Y. Cho, M.-K. Kwon, S. Lee, D. Noh, D.-J. Kim, Y. C. Kim, S.-J. Park, “Effect of electron blocking layer on efficiency droop in InGaN/GaN multiple quantum well light-emitting diodes,” Appl. Phys. Lett. 94(23), 231123 (2009).
[CrossRef]

Kioupakis, E.

E. Kioupakis, P. Rinke, K. T. Delaney, C. G. Van de Walle, “Indirect Auger recombination as a cause of efficiency droop in nitride light-emitting diodes,” Appl. Phys. Lett. 98(16), 161107 (2011).
[CrossRef]

Kobayashi, N.

K. Kumakura, T. Makimoto, N. Kobayashi, “Mg-acceptor activation mechanism and transport characteristics in p-type InGaN grown by metalorganic vapor phase epitaxy,” J. Appl. Phys. 93(6), 3370–3375 (2003).
[CrossRef]

Koch, S.

J. Hader, J. Moloney, S. Koch, “Density-activated defect recombination as a possible explanation for the efficiency droop in GaN-based diodes,” Appl. Phys. Lett. 96(22), 221106 (2010).
[CrossRef]

Koleske, D. D.

Q. Dai, Q. Shan, J. Cho, E. F. Schubert, M. H. Crawford, D. D. Koleske, M.-H. Kim, Y. Park, “On the symmetry of efficiency-versus-carrier-concentration curves in GaInN/GaN light-emitting diodes and relation to droop-causing mechanisms,” Appl. Phys. Lett. 98(3), 033506 (2011).
[CrossRef]

Q. Dai, Q. F. Shan, J. Wang, S. Chhajed, J. Cho, E. F. Schubert, M. H. Crawford, D. D. Koleske, M. H. Kim, Y. Park, “Carrier recombination mechanisms and efficiency droop in GaInN/GaN light-emitting diodes,” Appl. Phys. Lett. 97(13), 133507 (2010).
[CrossRef]

Krames, M.

N. Gardner, G. Muller, Y. Shen, G. Chen, S. Watanabe, W. Gotz, M. Krames, “Blue-emitting InGaN–GaN double-heterostructure light-emitting diodes reaching maximum quantum efficiency above 200,” Appl. Phys. Lett. 91(24), 243506 (2007).
[CrossRef]

Kumakura, K.

K. Kumakura, T. Makimoto, N. Kobayashi, “Mg-acceptor activation mechanism and transport characteristics in p-type InGaN grown by metalorganic vapor phase epitaxy,” J. Appl. Phys. 93(6), 3370–3375 (2003).
[CrossRef]

Kwon, M.-K.

S.-H. Han, D.-Y. Lee, S.-J. Lee, C.-Y. Cho, M.-K. Kwon, S. Lee, D. Noh, D.-J. Kim, Y. C. Kim, S.-J. Park, “Effect of electron blocking layer on efficiency droop in InGaN/GaN multiple quantum well light-emitting diodes,” Appl. Phys. Lett. 94(23), 231123 (2009).
[CrossRef]

Lee, D.-Y.

S.-H. Han, D.-Y. Lee, S.-J. Lee, C.-Y. Cho, M.-K. Kwon, S. Lee, D. Noh, D.-J. Kim, Y. C. Kim, S.-J. Park, “Effect of electron blocking layer on efficiency droop in InGaN/GaN multiple quantum well light-emitting diodes,” Appl. Phys. Lett. 94(23), 231123 (2009).
[CrossRef]

Lee, S.

S.-H. Han, D.-Y. Lee, S.-J. Lee, C.-Y. Cho, M.-K. Kwon, S. Lee, D. Noh, D.-J. Kim, Y. C. Kim, S.-J. Park, “Effect of electron blocking layer on efficiency droop in InGaN/GaN multiple quantum well light-emitting diodes,” Appl. Phys. Lett. 94(23), 231123 (2009).
[CrossRef]

Lee, S. M.

M. F. Schubert, J. Xu, J. K. Kim, E. F. Schubert, M. H. Kim, S. Yoon, S. M. Lee, C. Sone, T. Sakong, Y. Park, “Polarization-matched GaInN/ AlGaInN multi-quantum-well light-emitting diodes with reduced efficiency droop,” Appl. Phys. Lett. 93(4), 041102 (2008).
[CrossRef]

Lee, S.-J.

S.-H. Han, D.-Y. Lee, S.-J. Lee, C.-Y. Cho, M.-K. Kwon, S. Lee, D. Noh, D.-J. Kim, Y. C. Kim, S.-J. Park, “Effect of electron blocking layer on efficiency droop in InGaN/GaN multiple quantum well light-emitting diodes,” Appl. Phys. Lett. 94(23), 231123 (2009).
[CrossRef]

Lin, G.-B.

G.-B. Lin, D. Meyaard, J. Cho, E. F. Schubert, H. Shim, C. Sone, “Analytic model for the efficiency droop in semiconductors with asymmetric carrier-transport properties based on drift-induced reduction of injection efficiency,” Appl. Phys. Lett. 100(16), 161106 (2012).
[CrossRef]

Makimoto, T.

K. Kumakura, T. Makimoto, N. Kobayashi, “Mg-acceptor activation mechanism and transport characteristics in p-type InGaN grown by metalorganic vapor phase epitaxy,” J. Appl. Phys. 93(6), 3370–3375 (2003).
[CrossRef]

Malyutenko, V. K.

V. K. Malyutenko, S. S. Bolgov, A. D. Podoltsev, “Current crowding effect on the ideality factor and efficiency droop in blue lateral InGaN/GaN light emitting diodes,” Appl. Phys. Lett. 97(25), 251110 (2010).
[CrossRef]

Meyaard, D.

G.-B. Lin, D. Meyaard, J. Cho, E. F. Schubert, H. Shim, C. Sone, “Analytic model for the efficiency droop in semiconductors with asymmetric carrier-transport properties based on drift-induced reduction of injection efficiency,” Appl. Phys. Lett. 100(16), 161106 (2012).
[CrossRef]

Meyer, J.

I. Vurgaftman, J. Meyer, “Band parameters for nitrogen-containing semiconductors,” J. Appl. Phys. 94(6), 3675–3696 (2003).
[CrossRef]

Moloney, J.

J. Hader, J. Moloney, S. Koch, “Density-activated defect recombination as a possible explanation for the efficiency droop in GaN-based diodes,” Appl. Phys. Lett. 96(22), 221106 (2010).
[CrossRef]

Morkoc, H.

J. Xie, X. Ni, Q. Fan, R. Shimada, U. Ozgur, H. Morkoc, “On the efficiency droop in InGaN multiple quantum well blue light emitting diodes and its reduction with p-doped quantum well barriers,” Appl. Phys. Lett. 93(12), 121107 (2008).
[CrossRef]

Muller, G.

N. Gardner, G. Muller, Y. Shen, G. Chen, S. Watanabe, W. Gotz, M. Krames, “Blue-emitting InGaN–GaN double-heterostructure light-emitting diodes reaching maximum quantum efficiency above 200,” Appl. Phys. Lett. 91(24), 243506 (2007).
[CrossRef]

Nagahama, S. I.

S. Nakamura, N. Senoh, N. Iwasa, S. I. Nagahama, “High-brightness InGan blue, green and yellow light-emitting-diodes with quantum-well structures,” Jpn. J. Appl. Phys. 34(7A), L797–L799 (1995).
[CrossRef]

Nakamura, S.

S. Nakamura, N. Senoh, N. Iwasa, S. I. Nagahama, “High-brightness InGan blue, green and yellow light-emitting-diodes with quantum-well structures,” Jpn. J. Appl. Phys. 34(7A), L797–L799 (1995).
[CrossRef]

Ni, X.

J. Xie, X. Ni, Q. Fan, R. Shimada, U. Ozgur, H. Morkoc, “On the efficiency droop in InGaN multiple quantum well blue light emitting diodes and its reduction with p-doped quantum well barriers,” Appl. Phys. Lett. 93(12), 121107 (2008).
[CrossRef]

Noh, D.

S.-H. Han, D.-Y. Lee, S.-J. Lee, C.-Y. Cho, M.-K. Kwon, S. Lee, D. Noh, D.-J. Kim, Y. C. Kim, S.-J. Park, “Effect of electron blocking layer on efficiency droop in InGaN/GaN multiple quantum well light-emitting diodes,” Appl. Phys. Lett. 94(23), 231123 (2009).
[CrossRef]

Ozgur, U.

J. Xie, X. Ni, Q. Fan, R. Shimada, U. Ozgur, H. Morkoc, “On the efficiency droop in InGaN multiple quantum well blue light emitting diodes and its reduction with p-doped quantum well barriers,” Appl. Phys. Lett. 93(12), 121107 (2008).
[CrossRef]

Park, S.-J.

S.-H. Han, D.-Y. Lee, S.-J. Lee, C.-Y. Cho, M.-K. Kwon, S. Lee, D. Noh, D.-J. Kim, Y. C. Kim, S.-J. Park, “Effect of electron blocking layer on efficiency droop in InGaN/GaN multiple quantum well light-emitting diodes,” Appl. Phys. Lett. 94(23), 231123 (2009).
[CrossRef]

Park, Y.

Q. Dai, Q. Shan, J. Cho, E. F. Schubert, M. H. Crawford, D. D. Koleske, M.-H. Kim, Y. Park, “On the symmetry of efficiency-versus-carrier-concentration curves in GaInN/GaN light-emitting diodes and relation to droop-causing mechanisms,” Appl. Phys. Lett. 98(3), 033506 (2011).
[CrossRef]

Q. Dai, Q. F. Shan, J. Wang, S. Chhajed, J. Cho, E. F. Schubert, M. H. Crawford, D. D. Koleske, M. H. Kim, Y. Park, “Carrier recombination mechanisms and efficiency droop in GaInN/GaN light-emitting diodes,” Appl. Phys. Lett. 97(13), 133507 (2010).
[CrossRef]

M. F. Schubert, J. Xu, J. K. Kim, E. F. Schubert, M. H. Kim, S. Yoon, S. M. Lee, C. Sone, T. Sakong, Y. Park, “Polarization-matched GaInN/ AlGaInN multi-quantum-well light-emitting diodes with reduced efficiency droop,” Appl. Phys. Lett. 93(4), 041102 (2008).
[CrossRef]

Piprek, J.

J. Piprek, “Efficiency droop in nitride-based light-emitting diodes,” Phys. Status Solidi A 207(10), 2217–2225 (2010).
[CrossRef]

Podoltsev, A. D.

V. K. Malyutenko, S. S. Bolgov, A. D. Podoltsev, “Current crowding effect on the ideality factor and efficiency droop in blue lateral InGaN/GaN light emitting diodes,” Appl. Phys. Lett. 97(25), 251110 (2010).
[CrossRef]

Rinke, P.

E. Kioupakis, P. Rinke, K. T. Delaney, C. G. Van de Walle, “Indirect Auger recombination as a cause of efficiency droop in nitride light-emitting diodes,” Appl. Phys. Lett. 98(16), 161107 (2011).
[CrossRef]

Ryu, H. Y.

H. Y. Ryu, D. S. Shin, J. I. Shim, “Analysis of efficiency droop in nitride light-emitting diodes by the reduced effective volume of InGaN active material,” Appl. Phys. Lett. 100(13), 131109 (2012).
[CrossRef]

Sakong, T.

M. F. Schubert, J. Xu, J. K. Kim, E. F. Schubert, M. H. Kim, S. Yoon, S. M. Lee, C. Sone, T. Sakong, Y. Park, “Polarization-matched GaInN/ AlGaInN multi-quantum-well light-emitting diodes with reduced efficiency droop,” Appl. Phys. Lett. 93(4), 041102 (2008).
[CrossRef]

Santic, B.

B. Santic, “On the hole effective mass and the free hole statistics in wurtzite GaN,” Semicond. Sci. Technol. 18(4), 219–224 (2003).
[CrossRef]

Schubert, E. F.

J. Cho, E. F. Schubert, J. K. Kim, “Efficiency droop in light‐emitting diodes: Challenges and countermeasures,” Laser Photonics Rev. 7(3), 408–421 (2013).
[CrossRef]

G.-B. Lin, D. Meyaard, J. Cho, E. F. Schubert, H. Shim, C. Sone, “Analytic model for the efficiency droop in semiconductors with asymmetric carrier-transport properties based on drift-induced reduction of injection efficiency,” Appl. Phys. Lett. 100(16), 161106 (2012).
[CrossRef]

Q. Dai, Q. Shan, J. Cho, E. F. Schubert, M. H. Crawford, D. D. Koleske, M.-H. Kim, Y. Park, “On the symmetry of efficiency-versus-carrier-concentration curves in GaInN/GaN light-emitting diodes and relation to droop-causing mechanisms,” Appl. Phys. Lett. 98(3), 033506 (2011).
[CrossRef]

Q. Dai, Q. F. Shan, J. Wang, S. Chhajed, J. Cho, E. F. Schubert, M. H. Crawford, D. D. Koleske, M. H. Kim, Y. Park, “Carrier recombination mechanisms and efficiency droop in GaInN/GaN light-emitting diodes,” Appl. Phys. Lett. 97(13), 133507 (2010).
[CrossRef]

M. F. Schubert, J. Xu, J. K. Kim, E. F. Schubert, M. H. Kim, S. Yoon, S. M. Lee, C. Sone, T. Sakong, Y. Park, “Polarization-matched GaInN/ AlGaInN multi-quantum-well light-emitting diodes with reduced efficiency droop,” Appl. Phys. Lett. 93(4), 041102 (2008).
[CrossRef]

Schubert, M. F.

M. F. Schubert, J. Xu, J. K. Kim, E. F. Schubert, M. H. Kim, S. Yoon, S. M. Lee, C. Sone, T. Sakong, Y. Park, “Polarization-matched GaInN/ AlGaInN multi-quantum-well light-emitting diodes with reduced efficiency droop,” Appl. Phys. Lett. 93(4), 041102 (2008).
[CrossRef]

Senoh, N.

S. Nakamura, N. Senoh, N. Iwasa, S. I. Nagahama, “High-brightness InGan blue, green and yellow light-emitting-diodes with quantum-well structures,” Jpn. J. Appl. Phys. 34(7A), L797–L799 (1995).
[CrossRef]

Shan, Q.

Q. Dai, Q. Shan, J. Cho, E. F. Schubert, M. H. Crawford, D. D. Koleske, M.-H. Kim, Y. Park, “On the symmetry of efficiency-versus-carrier-concentration curves in GaInN/GaN light-emitting diodes and relation to droop-causing mechanisms,” Appl. Phys. Lett. 98(3), 033506 (2011).
[CrossRef]

Shan, Q. F.

Q. Dai, Q. F. Shan, J. Wang, S. Chhajed, J. Cho, E. F. Schubert, M. H. Crawford, D. D. Koleske, M. H. Kim, Y. Park, “Carrier recombination mechanisms and efficiency droop in GaInN/GaN light-emitting diodes,” Appl. Phys. Lett. 97(13), 133507 (2010).
[CrossRef]

Shen, Y.

N. Gardner, G. Muller, Y. Shen, G. Chen, S. Watanabe, W. Gotz, M. Krames, “Blue-emitting InGaN–GaN double-heterostructure light-emitting diodes reaching maximum quantum efficiency above 200,” Appl. Phys. Lett. 91(24), 243506 (2007).
[CrossRef]

Shim, H.

G.-B. Lin, D. Meyaard, J. Cho, E. F. Schubert, H. Shim, C. Sone, “Analytic model for the efficiency droop in semiconductors with asymmetric carrier-transport properties based on drift-induced reduction of injection efficiency,” Appl. Phys. Lett. 100(16), 161106 (2012).
[CrossRef]

Shim, J. I.

H. Y. Ryu, D. S. Shin, J. I. Shim, “Analysis of efficiency droop in nitride light-emitting diodes by the reduced effective volume of InGaN active material,” Appl. Phys. Lett. 100(13), 131109 (2012).
[CrossRef]

Shimada, R.

J. Xie, X. Ni, Q. Fan, R. Shimada, U. Ozgur, H. Morkoc, “On the efficiency droop in InGaN multiple quantum well blue light emitting diodes and its reduction with p-doped quantum well barriers,” Appl. Phys. Lett. 93(12), 121107 (2008).
[CrossRef]

Shin, D. S.

H. Y. Ryu, D. S. Shin, J. I. Shim, “Analysis of efficiency droop in nitride light-emitting diodes by the reduced effective volume of InGaN active material,” Appl. Phys. Lett. 100(13), 131109 (2012).
[CrossRef]

Sone, C.

G.-B. Lin, D. Meyaard, J. Cho, E. F. Schubert, H. Shim, C. Sone, “Analytic model for the efficiency droop in semiconductors with asymmetric carrier-transport properties based on drift-induced reduction of injection efficiency,” Appl. Phys. Lett. 100(16), 161106 (2012).
[CrossRef]

M. F. Schubert, J. Xu, J. K. Kim, E. F. Schubert, M. H. Kim, S. Yoon, S. M. Lee, C. Sone, T. Sakong, Y. Park, “Polarization-matched GaInN/ AlGaInN multi-quantum-well light-emitting diodes with reduced efficiency droop,” Appl. Phys. Lett. 93(4), 041102 (2008).
[CrossRef]

Van de Walle, C. G.

E. Kioupakis, P. Rinke, K. T. Delaney, C. G. Van de Walle, “Indirect Auger recombination as a cause of efficiency droop in nitride light-emitting diodes,” Appl. Phys. Lett. 98(16), 161107 (2011).
[CrossRef]

Vurgaftman, I.

I. Vurgaftman, J. Meyer, “Band parameters for nitrogen-containing semiconductors,” J. Appl. Phys. 94(6), 3675–3696 (2003).
[CrossRef]

Wang, J.

Q. Dai, Q. F. Shan, J. Wang, S. Chhajed, J. Cho, E. F. Schubert, M. H. Crawford, D. D. Koleske, M. H. Kim, Y. Park, “Carrier recombination mechanisms and efficiency droop in GaInN/GaN light-emitting diodes,” Appl. Phys. Lett. 97(13), 133507 (2010).
[CrossRef]

Wang, W.

H. Jia, L. Guo, W. Wang, H. Chen, “Recent progress in GaN‐based light‐emitting diodes,” Adv. Mater. 21(45), 4641–4646 (2009).
[CrossRef]

Watanabe, S.

N. Gardner, G. Muller, Y. Shen, G. Chen, S. Watanabe, W. Gotz, M. Krames, “Blue-emitting InGaN–GaN double-heterostructure light-emitting diodes reaching maximum quantum efficiency above 200,” Appl. Phys. Lett. 91(24), 243506 (2007).
[CrossRef]

Xie, J.

J. Xie, X. Ni, Q. Fan, R. Shimada, U. Ozgur, H. Morkoc, “On the efficiency droop in InGaN multiple quantum well blue light emitting diodes and its reduction with p-doped quantum well barriers,” Appl. Phys. Lett. 93(12), 121107 (2008).
[CrossRef]

Xu, J.

M. F. Schubert, J. Xu, J. K. Kim, E. F. Schubert, M. H. Kim, S. Yoon, S. M. Lee, C. Sone, T. Sakong, Y. Park, “Polarization-matched GaInN/ AlGaInN multi-quantum-well light-emitting diodes with reduced efficiency droop,” Appl. Phys. Lett. 93(4), 041102 (2008).
[CrossRef]

Yoon, S.

M. F. Schubert, J. Xu, J. K. Kim, E. F. Schubert, M. H. Kim, S. Yoon, S. M. Lee, C. Sone, T. Sakong, Y. Park, “Polarization-matched GaInN/ AlGaInN multi-quantum-well light-emitting diodes with reduced efficiency droop,” Appl. Phys. Lett. 93(4), 041102 (2008).
[CrossRef]

Zhang, M.

W. Guo, M. Zhang, P. Bhattacharya, J. Heo, “Auger recombination in III-Nitride nanowires and its effect on nanowire light-emitting diode characteristics,” Nano Lett. 11(4), 1434–1438 (2011).
[CrossRef] [PubMed]

Adv. Mater. (1)

H. Jia, L. Guo, W. Wang, H. Chen, “Recent progress in GaN‐based light‐emitting diodes,” Adv. Mater. 21(45), 4641–4646 (2009).
[CrossRef]

Appl. Phys. Lett. (12)

M. F. Schubert, J. Xu, J. K. Kim, E. F. Schubert, M. H. Kim, S. Yoon, S. M. Lee, C. Sone, T. Sakong, Y. Park, “Polarization-matched GaInN/ AlGaInN multi-quantum-well light-emitting diodes with reduced efficiency droop,” Appl. Phys. Lett. 93(4), 041102 (2008).
[CrossRef]

J. Xie, X. Ni, Q. Fan, R. Shimada, U. Ozgur, H. Morkoc, “On the efficiency droop in InGaN multiple quantum well blue light emitting diodes and its reduction with p-doped quantum well barriers,” Appl. Phys. Lett. 93(12), 121107 (2008).
[CrossRef]

S.-H. Han, D.-Y. Lee, S.-J. Lee, C.-Y. Cho, M.-K. Kwon, S. Lee, D. Noh, D.-J. Kim, Y. C. Kim, S.-J. Park, “Effect of electron blocking layer on efficiency droop in InGaN/GaN multiple quantum well light-emitting diodes,” Appl. Phys. Lett. 94(23), 231123 (2009).
[CrossRef]

J. Hader, J. Moloney, S. Koch, “Density-activated defect recombination as a possible explanation for the efficiency droop in GaN-based diodes,” Appl. Phys. Lett. 96(22), 221106 (2010).
[CrossRef]

V. K. Malyutenko, S. S. Bolgov, A. D. Podoltsev, “Current crowding effect on the ideality factor and efficiency droop in blue lateral InGaN/GaN light emitting diodes,” Appl. Phys. Lett. 97(25), 251110 (2010).
[CrossRef]

Q. Dai, Q. F. Shan, J. Wang, S. Chhajed, J. Cho, E. F. Schubert, M. H. Crawford, D. D. Koleske, M. H. Kim, Y. Park, “Carrier recombination mechanisms and efficiency droop in GaInN/GaN light-emitting diodes,” Appl. Phys. Lett. 97(13), 133507 (2010).
[CrossRef]

N. Gardner, G. Muller, Y. Shen, G. Chen, S. Watanabe, W. Gotz, M. Krames, “Blue-emitting InGaN–GaN double-heterostructure light-emitting diodes reaching maximum quantum efficiency above 200,” Appl. Phys. Lett. 91(24), 243506 (2007).
[CrossRef]

G.-B. Lin, D. Meyaard, J. Cho, E. F. Schubert, H. Shim, C. Sone, “Analytic model for the efficiency droop in semiconductors with asymmetric carrier-transport properties based on drift-induced reduction of injection efficiency,” Appl. Phys. Lett. 100(16), 161106 (2012).
[CrossRef]

A. David, M. J. Grundmann, “Influence of polarization fields on carrier lifetime and recombination rates in InGaN-based light-emitting diodes,” Appl. Phys. Lett. 97(3), 033501 (2010).
[CrossRef]

H. Y. Ryu, D. S. Shin, J. I. Shim, “Analysis of efficiency droop in nitride light-emitting diodes by the reduced effective volume of InGaN active material,” Appl. Phys. Lett. 100(13), 131109 (2012).
[CrossRef]

E. Kioupakis, P. Rinke, K. T. Delaney, C. G. Van de Walle, “Indirect Auger recombination as a cause of efficiency droop in nitride light-emitting diodes,” Appl. Phys. Lett. 98(16), 161107 (2011).
[CrossRef]

Q. Dai, Q. Shan, J. Cho, E. F. Schubert, M. H. Crawford, D. D. Koleske, M.-H. Kim, Y. Park, “On the symmetry of efficiency-versus-carrier-concentration curves in GaInN/GaN light-emitting diodes and relation to droop-causing mechanisms,” Appl. Phys. Lett. 98(3), 033506 (2011).
[CrossRef]

J. Appl. Phys. (2)

I. Vurgaftman, J. Meyer, “Band parameters for nitrogen-containing semiconductors,” J. Appl. Phys. 94(6), 3675–3696 (2003).
[CrossRef]

K. Kumakura, T. Makimoto, N. Kobayashi, “Mg-acceptor activation mechanism and transport characteristics in p-type InGaN grown by metalorganic vapor phase epitaxy,” J. Appl. Phys. 93(6), 3370–3375 (2003).
[CrossRef]

Jpn. J. Appl. Phys. (1)

S. Nakamura, N. Senoh, N. Iwasa, S. I. Nagahama, “High-brightness InGan blue, green and yellow light-emitting-diodes with quantum-well structures,” Jpn. J. Appl. Phys. 34(7A), L797–L799 (1995).
[CrossRef]

Laser Photonics Rev. (1)

J. Cho, E. F. Schubert, J. K. Kim, “Efficiency droop in light‐emitting diodes: Challenges and countermeasures,” Laser Photonics Rev. 7(3), 408–421 (2013).
[CrossRef]

Nano Lett. (1)

W. Guo, M. Zhang, P. Bhattacharya, J. Heo, “Auger recombination in III-Nitride nanowires and its effect on nanowire light-emitting diode characteristics,” Nano Lett. 11(4), 1434–1438 (2011).
[CrossRef] [PubMed]

Phys. Status Solidi A (1)

J. Piprek, “Efficiency droop in nitride-based light-emitting diodes,” Phys. Status Solidi A 207(10), 2217–2225 (2010).
[CrossRef]

Semicond. Sci. Technol. (1)

B. Santic, “On the hole effective mass and the free hole statistics in wurtzite GaN,” Semicond. Sci. Technol. 18(4), 219–224 (2003).
[CrossRef]

Other (2)

SILVACO International, Atlas User’s Manual (SILVACO International, 2011).

J. Piprek, Semiconductor Optoelectronic Devices: Introduction to Physics and Simulation (Academic, 2003).

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Figures (8)

Fig. 1
Fig. 1

(a) Calculated recombination rates (normalized) in five MQW of GaN-based LEDs at current density of 2 A/cm2 and 200 A/cm2. (b) Calculated electron and hole concentration in the quantum well near the p-GaN layer.

Fig. 2
Fig. 2

Calculated volume correction factor and the IQE of polar c-plane GaN-based LED using 5 MQW structure.

Fig. 3
Fig. 3

Effects of the polarization field and barriers of the MQW on the volume correction factor and the IQE. In order to confirm the effect of the polarization field, the calculated effective volume and the IQE of non-polar m-plane GaN-based LED using five MQW structure are analyzed (a), and the distribution of the recombination rate in the MQW is shown (b). To analyze the effect of the barriers, the same simulations are conducted for non-polar m-plane GaN-based LED using single quantum well structure (c), (d).

Fig. 4
Fig. 4

Calculated volume correction factor by using a numerical simulation when carrier concentration in the active region of the GaN-based LED increases.

Fig. 5
Fig. 5

Experimental setup for measuring transient characteristics of GaN-based LEDs.

Fig. 6
Fig. 6

Measured transient characteristics of the light output power (a) and the calculated carrier concentration (b) after the current pulse is applied to the LED sample. When the effective volume is used, the calculated carrier concentration becomes much larger.

Fig. 7
Fig. 7

Calculated IQE applying the physical volume or the effective volume of the active region and the IQE obtained by light output power measurement.

Fig. 8
Fig. 8

Calculated IQE applying the effective volume of the active region as a function of the square root of light output power.

Tables (1)

Tables Icon

Table 1 Calculated recombination coefficients by measuring transient characteristics

Equations (6)

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

Volume correction factor = Total recombination in MQW Highest recombination rate × Volume of the active region
Effective volume ( V effective ) = Volume correction factor × Physical volume ( V physical )
dn dt =An+B n 2 +C n 3
t= 1 2A [ 2B tan 1 ( B+2Cn 4AC B 2 ) 4AC B 2 +ln( A+n( B+Cn) ) )2lnn ] t 0
t 0 = 1 2A [ 2B tan 1 ( B+2C n 0 4AC B 2 ) 4AC B 2 +ln( A+ n 0 ( B+C n 0 ) ) )2ln n 0 ]
I=q V active ( An+B n 2 +C n 3 )

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