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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 11 — May. 28, 2007
  • pp: 6670–6676

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Light extraction analysis of GaN-based light-emitting diodes with surface texture and/or patterned substrate

Tsung-Xian Lee, Ko-Fon Gao, Wei-Ting Chien, and Ching-Cherng Sun  »View Author Affiliations


Optics Express, Vol. 15, Issue 11, pp. 6670-6676 (2007)
http://dx.doi.org/10.1364/OE.15.006670


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Abstract

Light extraction analysis of GaN-based light-emitting diodes (LEDs) with Monte Carlo ray tracing is presented. To obtain high light extraction efficiency, periodic structures introduced on the top surface and/or on the substrate of various types of LED are simulated, including wire bonding, flip chip and Thin GaN. Micro pyramid array with an apex angle from 20° to 70° is shown to effectively improve the light extraction efficiency. In addition, for an LED encapsulated within an epoxy lens, the patterned substrate with pyramid array is found to be a more effective way to increase light extraction efficiency than the surface texture.

© 2007 Optical Society of America

1. Introduction

LEDs are regarded as the most important light source in next-generation solid-state lighting owing to advantages in energy efficiency, long life, vivid colors, high reliability, environmental protection, safety and multiple applications [1–3

A. Zukauskas, M. S. Shur, and R. Caska, Introduction to Solid-state Lighting , (John Wiley & Sons, New York, 2002).

]. Among various types of LED, GaN-based LEDs attract the most attentions for blue-light emission, which is the base of white-light LEDs with yellow phosphors. However, the external quantum efficiency of GaN-based LEDs is not high enough, so that high-power LEDs cannot meet practical needs. The external quantum efficiency equals the multiplication of the internal quantum efficiency and the light extraction efficiency (LEE). The former relates to the substrate properties and epitaxy quality, and the latter relates to chip processing, die geometry and package. Since the external quantum efficiency, rather than the internal quantum efficiency and the LEE, is the only measurable term, an effective way to simulate the LEE is quite important to figure out the light efficiency of an LED.

Due to the randomness of the spontaneous photon emission from the active layer, Monte Carlo ray tracing is regarded as one of the most suitable ways to simulate the light propagation in LED dies [4–6

D. Z. Ting and T. C. McGill, “Monte Carlo simulation of. light-emitting diode light extraction characteristics,” Opt. Eng. 34, 3545–3553 (1995). [CrossRef]

]. Consequently, Monte-Carlo ray tracing has been used in studying the LEE as well as lighting design of LEDs [7–9

C. C. Sun, C. Y. Lin, T. X. Lee, and T. H. Yang, “Enhancement of light extraction of GaN-based LED with introducing micro-structure array,“ Opt. Eng. 43, 1700–1701 (2004). [CrossRef]

]. Recent studies report analyses of LEE, and some of them propose to increase the LEE through chip shaping, surface texture or patterned substrate [10–16

I. Schnitzer, E. Yablonovitch, C. Carneau, T. J. Gmitter, and A. Scherer, “30% external quantum efficiency from surface textured, thin-film lightemitting diodes,” Appl. Phys. Lett. 63, 2174–2176 (1993). [CrossRef]

]. However, a detailed study for the optimum micro-structure introduced in a die to increase the LEE is not available. In this paper, we present the study of the LEE with introducing a micro-structure on the top surface and/or on the substrate for the three main types of LED : wire bonding, flip chip, and ThinGaN [17

W. S. Wong, T. Sands, N. W. Cheung, M. Kneissl, D. P. Bour, P. Mei, L. T. Romano, and N. M. Johnson, “Fabrication of thin-film InGaN light-emitting diode membranes by laser lift-off,” Appl. Phys. Lett. 72, 1360–1362 (1999). [CrossRef]

,18

Y. Gao, T. Fujii, R. Sharma, K. Fujito, S. P. Danbaars, and S. Nakamura, “Roughening Hexagonal surface morphology on Laser lift-off (LLO) N face GaN with simple photo-enhanced chemical wet etching,” Jap. J. Appl. Phys. 43, L637–L639, (2004). [CrossRef]

], as shown in Fig. 1. Besides, we perform a comparison of the LEE with and without an epoxy lens.

Fig. 1. Schematic diagrams of wire bonding, flip chip and ThinGaN LEDs.

2. Light extraction analysis and optimization of pyramid structure

The structure parameters for the simulated GaN-based LEDs are described in Table 1. The die size is 300×300 μm2 and the absorption coefficient of active region is assumed to be 10-4 cm-1 [19

J. F. Muth, J. D. Brown, M. A. L. Johnson, Z. Yu, R. M. Kolbas, J. W. Cook Jr., and J. F. Schetzina, ”Absorption coefficient and refractive index of GaN, AlN and AlGaN alloys,” MRS Internet J. Nitride Semicond. Res. 4S1, G5.2 (1999).

,20

A. B. Djuriié, Y. Chan, and B. H. Li, “Calculations of the refractive index of AlGaN/GaN quantum well,” Proc. SPIE 4283, 630–637 (2001). [CrossRef]

]. Since the electrode may vary from one LED to another, for simplicity, we do not consider the effects of the electric pad as well as current spreading. In considering the LEE in an encapsulated LED with an epoxy lens (simplified EEL-LED), the refractive index is set to 1.5, and the diameter of the half-sphere lens is set to be 1 cm to ensure that most of the lights can escape from the lens without multiple reflections within the lens. In the Monte Carlo method, we assume that rays are generated randomly within the active region, and that are emitted isotropically. For Simplicity, we also assume monochromatic, unpolarized light emission. For each ray, the trajectory and the energy are determined by Snell’s law, Fresnel losses and material absorption. Fig. 2 shows the simulated LEE in function of the reflection coefficient of the bottom reflector for the three types of bare and encapsulated LED. We find that the LEE of an encapsulated LED is 200% more than that in a bare LED. In the case of an encapsulated LED, there is almost no difference in the LEE between the wire-bonding type and the flip-chip type, and the LEE is always within 50% to 55%, depending on the reflectivity of the bottom surface. The LEE in ThinGaN LEDs is always smaller than that in the other two types. The reason is that in comparison with the other cases, the thickness of the four sides in a Thin-GaN LED is only several microns. In comparison with others (hundreds of microns in thickness), the thickness of the side windows is too small to escape large amount of light so that the LEE is below 40% even if the LED is encapsulated in an epoxy lens. In comparison with Ref. [16

R. Windisch, C. Rooman, B. Dutta, A. Knobloch, G. Borghs, G. H. Döhler, and P. Heremans, “Light-extraction mechanisms in high-efficiency surface-textured light-emitting diodes,” J. Select. Topics Quantum Electronics 8, 248–255 (2002). [CrossRef]

], the LEE in our simulation shows less dependence of reflectivity. This is caused by less multiple reflections required for light escaping. Owing to less reflection, optical path inside a chip is shorter, and absorption by the active layer and bottom reflector is weaker. Therefore, the design of a chip to decrease the number of reflections before the light escapes from the chip is important.

Table 1  Parameters of each layer in the simulated LED.
LayerSapphireGaNAlGaNN-typeMQWP-type
Thickness (μm)33020.0520.10.05
Refractive Index1.782.402.402.422.542.45

Then we study the LEE with introducing special micro structures on the top surface of the LED and/or on the top surface of the substrate. The former is the case of surface texture, and the latter is so-called patterned substrate. Since the micro array structure in inversed pyramid form (IPF) has been reported to perform larger LEE than that one with a well structure [7

C. C. Sun, C. Y. Lin, T. X. Lee, and T. H. Yang, “Enhancement of light extraction of GaN-based LED with introducing micro-structure array,“ Opt. Eng. 43, 1700–1701 (2004). [CrossRef]

,20

A. B. Djuriié, Y. Chan, and B. H. Li, “Calculations of the refractive index of AlGaN/GaN quantum well,” Proc. SPIE 4283, 630–637 (2001). [CrossRef]

], we study the LEE with respect to the apex angle of the 4-facet inverted pyramid of the micro-structure introduced in the top surface. Fig. 3 shows the simulated LEE with respect to the pyramid angle in a bare LED and encapsulated LED with IPF structure without considering bottom reflection. The simulation result shows that the angle in a range of 20° to 70° of the slanted surfaces in the pyramid structure may cause larger LEE in both cases. We find that the slanted angle of 30° could be the best angle in enhancing the LEE. Besides, According to our simulation, some typical surface texturing in GaN-based LEDs produces 6-facet pyramids that enhance LEE in a similar way to the 4-facet pyramids. Thus in the following simulation, we introduce the micro-array structure in IPF with slanted surfaces at 30° in all cases.

Fig. 2. The LEE vs. reflectivity of the bottom surface for three LED Types: wire bonding (WB); flip chip (FC); and ThinGaN (TGaN).
Fig. 3. The LEE vs. the angle of the slanted surface in an inversed pyramid structure.

It is noted that “surface texture” refers to the pattern array introduced on the top surface of the die so the textured surface is on the top surface of the sapphire in a flip-chip LED. In contrast, in a Thin-GaN case, the patterned substrate means that the pattern array is introduced in the bottom reflector of a Thin-GaN LED. Fig. 4 shows the simulation result for wire-bonded LEDs. Surface texture and patterned substrate may effectively increase the LEE in both bare LEDs and encapsulated LEDs. When both surface texture and patterned substrate are applied at the same time, the increase of LEE in a bare LED is more dominant than that in an encapsulated LED. The reason is that the epoxy lens reduces the effect by surface texture. In such a case, the LEE of an encapsulated LED is near twice of that in a bare LED.

Figure 5 shows the result for flip-chip LEDs. The effect of surface texture of an encapsulated LED is almost eliminated since the refractive index difference between the sapphire and the epoxy lens is only about 0.2. However, the patterned substrate can effectively increase the LEE to around 70% in an encapsulated LED, where the LEE is about twice of that in a bare LED.

Fig. 4. The LEE vs. reflectivity of the bottom surface for wire-bonding LEDs: Conventional (C); surface texture (ST); patterned substrate (PS) and both (B).
Fig. 5. The LEE vs. reflectivity of the bottom surface for flip-chip LEDs.
Fig. 6. The LEE vs. reflectivity of the bottom surface for Thin-GaN LEDs.

The simulation of the LEE of Thin-GaN LEDs is shown in Fig. 6, where approaches by both surface texture and patterned substrate can effectively increase LEE. However, patterned substrate is still better than surface texture. Even both of them are applied to an encapsulated LED, the increase of LEE by surface texture is less than that by patterned substrate. The LEE with pattern array in a Thin-GaN type in an encapsulated LED is also near twice of that in a bare LED. In the case of Thin-GaN LEDs, the LEE of an encapsulated LED is nearly twice that of a bare LED.

3. Result and discussion

The effects of increasing the LEE by both encapsulating with an epoxy lens and introducing a pattern array are summarized in Fig. 7, where the reflectivity of the bottom surface is set 90%. The simulation shows that both the encapsulation and the patterned substrate with 30° pyramid array effectively increase the LEE from 200% to 300% with respect to that of a bare LED without pattern array. When both of them are applied at the same time, the LEE may be as large as 350% of that in a bare one for all three types. The simulation shows that, except flip-chip LEDs, surface texture induces an obvious effect in improving LEE in both wire-bonding and Thin-GaN LEDs. But the effect in the Thin-GaN case is not as dominant as that in the wire-bonding case because the sapphire substrate is removed so that more absorption occurs when the light travels in the thin layer with multiple reflections. In contrast, the patterned substrate causes dominant effect in improving LEE in all cases with an epoxy lens. It is interesting to note that once patterned substrate is introduced, surface texture is almost no more necessary in an encapsulated LED.

Owing to the advantage in heat dissipation, LEDs in flip chip and Thin GaN attract more attentions. From the optical simulation presented above, we may have two conclusions in enhancing the LEE for such two LED types. The first is that patterned substrate should be regarded as the most effective way in enhancing the LEE in an encapsulated flip-chip LED. Second, both patterned substrate and surface texture schemes are useful in enhancing the LEE in a Thin-GaN type of encapsulated LED, though patterned substrate scheme seems a little bit better. Another important result of the simulation is that we do not find any case to perform an LEE larger than 80% when the absorption coefficient in the active layer is as high as 104 cm-1, and the pad of the electrode as well as current spreading are not considered. Since the simulation parameters cannot be applied to all LEDs, we cannot conclude that it is impossible to perform an LEE larger than 80% in the approaches presented in this paper. However, we may regard 80% as an upper limit to reach in LEE.

Fig. 7. A summary of the LEE for different types of LED with and without an epoxy lens.

4. Summary

In this paper, we have applied Monte Carlo ray tracing to simulate enhancement of LEE in three different types of bare LEDs and encapsulated LEDs with introducing pyramid array. The simulation showed that the pyramid with slanted surface at an angle from 20° to 70° can increase LEE effectively. Moreover, the LEE can be increased to as high as 350% when a patterned substrate is introduced to all three types of encapsulated LEDs in comparison with an unpatterned bare LED. The simulation also shows that in encapsulated LEDs, surface texture contributes less in a flip-chip type than in the other two LED types, while the patterned substrate is useful in increasing the LEE in all types.

Acknowledgments

This study was sponsored by the Ministry of Economic Affairs of the Republic of China with the contract no. 94-EC-17-A-07-S1-043. The authors thank Dr. Ivan Moreno for his helpful suggestions.

References and links

1.

A. Zukauskas, M. S. Shur, and R. Caska, Introduction to Solid-state Lighting , (John Wiley & Sons, New York, 2002).

2.

D. A. Steigerwald, J. C. Bhat, D. Collins, R. M. Fletcher, M. O. Holcomb, M. J. Ludowise, P. S. Martin, and S. L. Rudaz, “Illumination with Solid State Lighting Technology,” IEEE J. Selected Topics in Quantum Electron 8, 310–320 (2002). [CrossRef]

3.

E. F. Schubert and J. K. Kim, “Solid-state light sources becoming smart,” Science 308, 1274–1278 (2005). [CrossRef] [PubMed]

4.

D. Z. Ting and T. C. McGill, “Monte Carlo simulation of. light-emitting diode light extraction characteristics,” Opt. Eng. 34, 3545–3553 (1995). [CrossRef]

5.

S. J. Lee, “Analysis of light-emitting diode by Monte Carlo photo simulation,” Appl. Opt. 40, 1427–1437 (2001). [CrossRef]

6.

A. Badano and J. Kanicki, “Monte Carlo analysis of the spectral photon emission and extraction efficiency of organic light-emitting devices,” J. Appl. Phys. 90, 1827–1830 (2001). [CrossRef]

7.

C. C. Sun, C. Y. Lin, T. X. Lee, and T. H. Yang, “Enhancement of light extraction of GaN-based LED with introducing micro-structure array,“ Opt. Eng. 43, 1700–1701 (2004). [CrossRef]

8.

T. X. Lee, C. Y. Lin, S. H. Ma, and C. C. Sun, “Analysis of position-dependent. light extraction of GaN-based LEDs,“ Opt. Express 13, 4175–4179 (2005). [CrossRef] [PubMed]

9.

C. C. Sun, T. X. Lee, S. H. Ma, Y. L. Lee, and S. M. Huang, “Precise optical. modeling for LED lighting based on cross-correlation in mid-field region,” Optics Letters 31, 2193–2195 (2006). [CrossRef] [PubMed]

10.

I. Schnitzer, E. Yablonovitch, C. Carneau, T. J. Gmitter, and A. Scherer, “30% external quantum efficiency from surface textured, thin-film lightemitting diodes,” Appl. Phys. Lett. 63, 2174–2176 (1993). [CrossRef]

11.

M. R. Krames, M. Ochiai-Holcomb, G. E. Hofler, C. Carter-Coman, E. I. Chen, I.-H. Tan, P. Grillot, N. F. Gardner, H. C. Chui, J-W. Huang, S.A. Stockman, F. A. Kish, and M. G. Craford, “High-power truncated-pyramid (Al0.5Ga1-x )0.5 In0.5 P/GaP light-emitting diodes exhibiting >50% external quantum efficiency,” Appl. Phys. Lett. 75, 2365–2367 (1999). [CrossRef]

12.

R. Windisch, C. Rooman, S. Meinlschmidt, P. Kiesel, D. Zipperer, G. H. Döhler, B. Dutta, M. Kuijk, G. Borghs, and P. Heremans, “Impact of texture-enhanced transmission on high-efficiency surface-textured light-emitting diodes,” Appl. Phys. Lett. 79, 2315–2317 (2001). [CrossRef]

13.

C. Huh, K. S. Lee, E. J. Kang, and S. J. Park, “Improved light-output and electrical performance of InGaN-based light-emitting diode by microroughening of the p-GaN surface,” J. Appl. Phys. 93, 9383–9385 (2003). [CrossRef]

14.

T. Fujii, Y. Gao, R. Sharma, E. L. Hu, S. P. Danbaars, and S. Nakamura, “Increase in the extraction efficiency of GaN-base light emitting diodes via surface roughening,” Appl. Phys. Lett. 84, 855–857 (2004). [CrossRef]

15.

J. J. Wierer, D. A. Steigerwald, M. R. Krames, J. J. O’Shea, M. J. Ludowise, N. F. Gardner, R. S. Kern, and S. A. Stockman, “High-power AlGaInN flip-chip light-emitting diodes,” Appl. Phys. Lett. 78, 3379–3381 (2001). [CrossRef]

16.

R. Windisch, C. Rooman, B. Dutta, A. Knobloch, G. Borghs, G. H. Döhler, and P. Heremans, “Light-extraction mechanisms in high-efficiency surface-textured light-emitting diodes,” J. Select. Topics Quantum Electronics 8, 248–255 (2002). [CrossRef]

17.

W. S. Wong, T. Sands, N. W. Cheung, M. Kneissl, D. P. Bour, P. Mei, L. T. Romano, and N. M. Johnson, “Fabrication of thin-film InGaN light-emitting diode membranes by laser lift-off,” Appl. Phys. Lett. 72, 1360–1362 (1999). [CrossRef]

18.

Y. Gao, T. Fujii, R. Sharma, K. Fujito, S. P. Danbaars, and S. Nakamura, “Roughening Hexagonal surface morphology on Laser lift-off (LLO) N face GaN with simple photo-enhanced chemical wet etching,” Jap. J. Appl. Phys. 43, L637–L639, (2004). [CrossRef]

19.

J. F. Muth, J. D. Brown, M. A. L. Johnson, Z. Yu, R. M. Kolbas, J. W. Cook Jr., and J. F. Schetzina, ”Absorption coefficient and refractive index of GaN, AlN and AlGaN alloys,” MRS Internet J. Nitride Semicond. Res. 4S1, G5.2 (1999).

20.

A. B. Djuriié, Y. Chan, and B. H. Li, “Calculations of the refractive index of AlGaN/GaN quantum well,” Proc. SPIE 4283, 630–637 (2001). [CrossRef]

21.

J. Q. Xi, H. Luo, A. J. Pasquale, J. K. Kim, and E. F. Schubert, “Enhanced light extraction in GaInN light-emitting diode with pyramid reflector,” IEEE Photon. Tech. Lett. 18, 2347–2349 (2006). [CrossRef]

OCIS Codes
(230.3670) Optical devices : Light-emitting diodes
(230.3990) Optical devices : Micro-optical devices

ToC Category:
Optical Devices

History
Original Manuscript: February 12, 2007
Revised Manuscript: May 8, 2007
Manuscript Accepted: May 11, 2007
Published: May 16, 2007

Citation
Tsung-Xian Lee, Ko-Fon Gao, Wei-Ting Chien, and Ching-Cherng Sun, "Light extraction analysis of GaN-based light-emitting diodes with surface texture and/or patterned substrate," Opt. Express 15, 6670-6676 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-11-6670


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References

  1. A. Zukauskas, M. S. Shur, and R. Caska, Introduction to Solid-state Lighting, (John Wiley & Sons, New York, 2002).
  2. D. A. Steigerwald, J. C. Bhat, D. Collins, R. M. Fletcher, M. O. Holcomb, M. J. Ludowise, P. S. Martin, and S. L. Rudaz, "Illumination with Solid State Lighting Technology," IEEE J. Selected Topics in Quantum Electron 8, 310-320 (2002). [CrossRef]
  3. E. F. Schubert and J. K. Kim, "Solid-state light sources becoming smart," Science 308, 1274-1278 (2005). [CrossRef] [PubMed]
  4. D. Z. Ting and T. C. McGill, "Monte Carlo simulation of. light-emitting diode light extraction characteristics," Opt. Eng. 34, 3545-3553 (1995). [CrossRef]
  5. S. J. Lee, "Analysis of light-emitting diode by Monte Carlo photo simulation," Appl. Opt. 40, 1427-1437 (2001). [CrossRef]
  6. A. Badano and J. Kanicki, "Monte Carlo analysis of the spectral photon emission and extraction efficiency of organic light-emitting devices," J. Appl. Phys. 90, 1827-1830 (2001). [CrossRef]
  7. C. C. Sun, C. Y. Lin, T. X. Lee, and T. H. Yang, "Enhancement of light extraction of GaN-based LED with introducing micro-structure array, " Opt. Eng. 43, 1700-1701 (2004). [CrossRef]
  8. T. X. Lee, C. Y. Lin, S. H. Ma and C. C. Sun, "Analysis of position-dependent. light extraction of GaN- based LEDs," Opt. Express 13, 4175-4179 (2005). [CrossRef] [PubMed]
  9. C. C. Sun, T. X. Lee, S. H. Ma Y. L. Lee and S. M. Huang, "Precise optical. modeling for LED lighting based on cross-correlation in mid-field region," Opt. Lett. 31, 2193-2195 (2006). [CrossRef] [PubMed]
  10. I. Schnitzer, E. Yablonovitch, C. Carneau, T. J. Gmitter, and A. Scherer, "30% external quantum efficiency from surface textured, thin-film lightemitting diodes," Appl. Phys. Lett. 63, 2174-2176 (1993). [CrossRef]
  11. M. R. Krames, M. Ochiai-Holcomb, G. E. Hofler, C. Carter-Coman, E. I. Chen, I.-H. Tan, P. Grillot, N. F. Gardner, H. C. Chui, J-W. Huang, S.A. Stockman, F. A. Kish, and M. G. Craford, "High-power truncated- pyramid (AlGa)InP/GaP light-emitting diodes exhibiting >50% external quantum efficiency," Appl. Phys. Lett. 75, 2365-2367 (1999). [CrossRef]
  12. R. Windisch, C. Rooman, S. Meinlschmidt, P. Kiesel, D. Zipperer, G. H. Döhler, B. Dutta, M. Kuijk, G. Borghs, and P. Heremans, "Impact of texture-enhanced transmission on high-efficiency surface-textured light-emitting diodes," Appl. Phys. Lett. 79, 2315-2317 (2001). [CrossRef]
  13. C. Huh, K. S. Lee, E. J. Kang, and S. J. Park, "Improved light-output and electrical performance of InGaN-based light-emitting diode by microroughening of the p-GaN surface," J. Appl. Phys. 93, 9383-9385 (2003). [CrossRef]
  14. T. Fujii, Y. Gao, R. Sharma, E. L. Hu, S. P. Danbaars, and S. Nakamura, "Increase in the extraction efficiency of GaN-base light emitting diodes via surface roughening," Appl. Phys. Lett. 84, 855-857 (2004). [CrossRef]
  15. J. J. Wierer, D. A. Steigerwald, M. R. Krames, J. J. O’Shea, M. J. Ludowise, N. F. Gardner, R. S. Kern, and S. A. Stockman, "High-power AlGaInN flip-chip light-emitting diodes," Appl. Phys. Lett. 78, 3379-3381 (2001). [CrossRef]
  16. R. Windisch, C. Rooman, B. Dutta, A. Knobloch, G. Borghs, G. H. Döhler, and P. Heremans, "Light- extraction mechanisms in high-efficiency surface-textured light-emitting diodes," J. Select. Topics Quantum Electronics 8, 248-255 (2002). [CrossRef]
  17. W. S. Wong, T. Sands, N. W. Cheung, M. Kneissl, D. P. Bour, P. Mei, L. T. Romano, and N. M. Johnson, "Fabrication of thin-film InGaN light-emitting diode membranes by laser lift-off," Appl. Phys. Lett. 72, 1360-1362 (1999). [CrossRef]
  18. Y. Gao, T. Fujii, R. Sharma, K. Fujito, S. P. Danbaars and S. Nakamura, "Roughening Hexagonal surface morphology on Laser lift-off (LLO) N face GaN with simple photo-enhanced chemical wet etching," Jap. J. Appl. Phys. 43, L637-L639, (2004). [CrossRef]
  19. J. F. Muth, J. D. Brown, M. A. L. Johnson, Z. Yu, R. M. Kolbas, J. W. Cook, Jr., and J. F. Schetzina, " Absorption coefficient and refractive index of GaN, AlN and AlGaN alloys," MRS Internet J. Nitride Semicond. Res. 4S1, G5.2 (1999).
  20. A. B. Djuriié, Y. Chan, and B. H. Li, "Calculations of the refractive index of AlGaN/GaN quantum well," Proc. SPIE 4283, 630-637 (2001). [CrossRef]
  21. J. Q. Xi, H. Luo, A. J. Pasquale, J. K. Kim, and E. F. Schubert, "Enhanced light extraction in GaInN light-emitting diode with pyramid reflector," IEEE Photon. Tech. Lett. 18, 2347-2349 (2006). [CrossRef]

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