Abstract

A comprehensive numerical model for distributed Bragg reflectors (DBRs) based on thin-film optics is developed. Detailed refractive-index calculations for GaN, AlN, AlGaN, and InGaN can also be included in this numerical model. This model can predict DBR performances for refractive-index variations, layer-thickness fluctuations, and the number of quarter-wave stack pairs in DBR as well different light polarizations.

© 2007 Optical Society of America

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References

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  1. H. Ng, T. Moustakas, and S. Chu, "High reflectivity and bandwidth AlN/GaN distributed Bragg reflectors grown by molecular-beam epitaxy," Appl. Phys. Lett. 76, 2818-2820 (2000).
    [CrossRef]
  2. N. Nakada, M. Nakaji, H. Ishikawa, T. Egawa, M. Umeno, and T. Jimbo, "Improved characteristics of InGaN multiple-quantum-well light-emitting diode by GaN/AlGaN distributed Bragg reflector grown on sapphire," Appl. Phys. Lett. 76, 1804-1806 (2000).
    [CrossRef]
  3. T. Someya and Y. Arakawa, "Highly reflective GaN/A10.34Ga0.66N quarter-wave reflectors grown by metal organic chemical vapor deposition," Appl. Phys. Lett. 73, 3653-3655 (1998).
    [CrossRef]
  4. R. Langer, A. Barski, J. Simon, N. Pelekanos, O. Konovalov, R. Andre, and L. Dang, "High-reflectivity GaN/GaAlN Bragg mirrors at blue/green wavelengths grown by molecular beam epitaxy," Appl. Phys. Lett. 74, 3610-3612 (1999).
    [CrossRef]
  5. M. Diage, Y. He, H. Zhou, E. Makarona, A. Nurmikko, J. Han, K. Waldrip, J. Figiel, T. Takeuchi, and M. Krames, "Vertical violet light emitting diode incorporating an aluminum gallium nitride distributed Bragg mirror and a tunnel junction," Appl. Phys. Lett. 79, 3720-3722 (2001).
    [CrossRef]
  6. P. Yeh, Optical Waves in Layered Media (Wiley, 1988).
  7. D. Brunner, H. Angerer, E. Bustarret, F. Freudenberg, R. Hopler, R. Dimitrov, O. Ambacher, and M. Stutzmann, "Optical constants of epitaxial AlGaN films and their temperature dependence," J. Appl. Phys. 82, 5090-5096 (1997).
    [CrossRef]
  8. A. Adachi, Physical Properties of III-V Semiconductor Compounds (Wiley, 1992).
    [CrossRef]
  9. J. Piprek, Semiconductor Optoelectronic Devices, Introduction to Physics and Simulation (Academic, 2003).
  10. T. Peng and J. Piprek, "Refractive index of AlGaInN alloys," Electron. Lett. 32, 2285-2286 (1996).
    [CrossRef]
  11. G. Laws, E. Larkins, and I. Harrison, "Improved refractive index formulas for the AlxGa1-xN and InyGa1-yN alloys," J. Appl. Phys. 89, 1108-1115 (2001).
    [CrossRef]
  12. M. Leung, A. Djurisic, and E. Li, "Refractive index of InGaN/GaN quantum well," J. Appl. Phys. 84, 6312-6317 (1998).
    [CrossRef]

2001 (2)

M. Diage, Y. He, H. Zhou, E. Makarona, A. Nurmikko, J. Han, K. Waldrip, J. Figiel, T. Takeuchi, and M. Krames, "Vertical violet light emitting diode incorporating an aluminum gallium nitride distributed Bragg mirror and a tunnel junction," Appl. Phys. Lett. 79, 3720-3722 (2001).
[CrossRef]

G. Laws, E. Larkins, and I. Harrison, "Improved refractive index formulas for the AlxGa1-xN and InyGa1-yN alloys," J. Appl. Phys. 89, 1108-1115 (2001).
[CrossRef]

2000 (2)

H. Ng, T. Moustakas, and S. Chu, "High reflectivity and bandwidth AlN/GaN distributed Bragg reflectors grown by molecular-beam epitaxy," Appl. Phys. Lett. 76, 2818-2820 (2000).
[CrossRef]

N. Nakada, M. Nakaji, H. Ishikawa, T. Egawa, M. Umeno, and T. Jimbo, "Improved characteristics of InGaN multiple-quantum-well light-emitting diode by GaN/AlGaN distributed Bragg reflector grown on sapphire," Appl. Phys. Lett. 76, 1804-1806 (2000).
[CrossRef]

1999 (1)

R. Langer, A. Barski, J. Simon, N. Pelekanos, O. Konovalov, R. Andre, and L. Dang, "High-reflectivity GaN/GaAlN Bragg mirrors at blue/green wavelengths grown by molecular beam epitaxy," Appl. Phys. Lett. 74, 3610-3612 (1999).
[CrossRef]

1998 (2)

M. Leung, A. Djurisic, and E. Li, "Refractive index of InGaN/GaN quantum well," J. Appl. Phys. 84, 6312-6317 (1998).
[CrossRef]

T. Someya and Y. Arakawa, "Highly reflective GaN/A10.34Ga0.66N quarter-wave reflectors grown by metal organic chemical vapor deposition," Appl. Phys. Lett. 73, 3653-3655 (1998).
[CrossRef]

1997 (1)

D. Brunner, H. Angerer, E. Bustarret, F. Freudenberg, R. Hopler, R. Dimitrov, O. Ambacher, and M. Stutzmann, "Optical constants of epitaxial AlGaN films and their temperature dependence," J. Appl. Phys. 82, 5090-5096 (1997).
[CrossRef]

1996 (1)

T. Peng and J. Piprek, "Refractive index of AlGaInN alloys," Electron. Lett. 32, 2285-2286 (1996).
[CrossRef]

Appl. Phys. Lett. (5)

H. Ng, T. Moustakas, and S. Chu, "High reflectivity and bandwidth AlN/GaN distributed Bragg reflectors grown by molecular-beam epitaxy," Appl. Phys. Lett. 76, 2818-2820 (2000).
[CrossRef]

N. Nakada, M. Nakaji, H. Ishikawa, T. Egawa, M. Umeno, and T. Jimbo, "Improved characteristics of InGaN multiple-quantum-well light-emitting diode by GaN/AlGaN distributed Bragg reflector grown on sapphire," Appl. Phys. Lett. 76, 1804-1806 (2000).
[CrossRef]

T. Someya and Y. Arakawa, "Highly reflective GaN/A10.34Ga0.66N quarter-wave reflectors grown by metal organic chemical vapor deposition," Appl. Phys. Lett. 73, 3653-3655 (1998).
[CrossRef]

R. Langer, A. Barski, J. Simon, N. Pelekanos, O. Konovalov, R. Andre, and L. Dang, "High-reflectivity GaN/GaAlN Bragg mirrors at blue/green wavelengths grown by molecular beam epitaxy," Appl. Phys. Lett. 74, 3610-3612 (1999).
[CrossRef]

M. Diage, Y. He, H. Zhou, E. Makarona, A. Nurmikko, J. Han, K. Waldrip, J. Figiel, T. Takeuchi, and M. Krames, "Vertical violet light emitting diode incorporating an aluminum gallium nitride distributed Bragg mirror and a tunnel junction," Appl. Phys. Lett. 79, 3720-3722 (2001).
[CrossRef]

Electron. Lett. (1)

T. Peng and J. Piprek, "Refractive index of AlGaInN alloys," Electron. Lett. 32, 2285-2286 (1996).
[CrossRef]

J. Appl. Phys. (3)

G. Laws, E. Larkins, and I. Harrison, "Improved refractive index formulas for the AlxGa1-xN and InyGa1-yN alloys," J. Appl. Phys. 89, 1108-1115 (2001).
[CrossRef]

M. Leung, A. Djurisic, and E. Li, "Refractive index of InGaN/GaN quantum well," J. Appl. Phys. 84, 6312-6317 (1998).
[CrossRef]

D. Brunner, H. Angerer, E. Bustarret, F. Freudenberg, R. Hopler, R. Dimitrov, O. Ambacher, and M. Stutzmann, "Optical constants of epitaxial AlGaN films and their temperature dependence," J. Appl. Phys. 82, 5090-5096 (1997).
[CrossRef]

Other (3)

A. Adachi, Physical Properties of III-V Semiconductor Compounds (Wiley, 1992).
[CrossRef]

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

P. Yeh, Optical Waves in Layered Media (Wiley, 1988).

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Figures (9)

Fig. 1
Fig. 1

DBR diagram.

Fig. 2
Fig. 2

Refractive index of GaN (solid curve) and Al 0 .34 Ga 0 .66 N (dashed curve).

Fig. 3
Fig. 3

Reflections of AlGaN∕GaN DBR with normal incident TE light (a) from air and (b) from GaN.

Fig. 4
Fig. 4

Reflections of AlGaN∕GaN DBR with incident light from GaN. The solid curve represents normal incidence, the connected circles represent 5 degrees slanted incidence, and the dashed curve represents 10 degrees slanted incidence.

Fig. 5
Fig. 5

Reflection versus the number of pairs with incident light from air:the dotted curve represents 20 pairs, the connected circles represent 35 pairs, and the solid curve represents 50 pairs.

Fig. 6
Fig. 6

Performance of a 35 pair DBR with light incident from air for different layer-thickness variation:the solid curve is for zero variance, the dotted curve is for 2 nm variance, and the connected circles are for 5 nm variance.

Fig. 7
Fig. 7

Performance of a 50 pair DBR for different layer-thickness variation:the solid curve is for zero variance, the dotted curve is for 2 nm variance, and the connected dots are for 5 nm variance.

Fig. 8
Fig. 8

35 pair DBR performance under different layer refractive-index variations with light incident from air:zero variance (solid curve), 0.01 variance (connected circles), and 0.02 variance (dashed curve).

Fig. 9
Fig. 9

50 pair DBR performance under different layer refractive-index variations with light incident from air:zero variance (solid curve), 0.01 variance (connected circles), and 0.02 variance (dashed curve).

Equations (13)

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k n 1 d 1 = k n 2 d 2 = 1 2 π ,
k = 2 π λ = ω c ,
Δ ω = ω 4 π sin 1 ( | n 1 n 2 | n 1 + n 2 ) ω 2 π Δ n n ,
( E 0 + E 0 ) = M ( E s + E s ) = ( m 11 m 12 m 21 m 22 ) ( E s + E s ) .
r ( θ ) = [ E 0 E 0 + ] E s = 0 = m 21 m 11 .
D i = [ 1 1 n i   cos   θ i n i   cos   θ i ] .
P i = [ exp ( j φ i ) 0 0 exp ( j φ i ) ] ,
φ i = 2 π n i d i λ   cos   θ i .
n ( λ ) 2 = a ( x ) ( h c λ E g ) 2 [ 2 ( 1 + h c λ E g ) 1 / 2 ( 1 h c λ E g ) 1 / 2 ] + b ( x ) .
E g ( x ) = 6.13 x + 3.42 ( 1 x ) + 1.3 x ( 1 x ) .
a ( x ) = 3.17 x + 9.98 ,
b ( x ) = 2.20 x + 2.66.
d = 1 2 π σ   exp [ ( d d 0 ) 2 2 σ 2 ] ,

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