A remarkable enhancement of light extraction efficiency in GaN-based blue light-emitting diodes (LEDs) with rough beveled ZnO nanocone arrays grown on the planar indium tin oxide (ITO) layer is reported. The light output power of LEDs with rough beveled ZnO nanocone arrays was increased by about 110% at 20 mA compared with conventional LEDs with planar ITO. The light extraction efficiency of GaN-based LEDs with rough-beveled ZnO nanocones is measured much greater than with smooth-surface hexagonal ZnO nanorods. The light-ray tracing analysis showed that ZnO nanocones with rough surfaces enlarge the light escape cone of GaN-based LEDs and have a greater advantage for extracting light compared with ZnO nanorods.
© 2012 OSA
Solid-state lighting devices have many advantages, such as impressive energy, environmental savings, more reliable performance, and exceptionally long lifetime. Gallium nitride (GaN)-based blue light emitting diodes (LEDs), one of the most promising solid-state lighting devices, have recently attracted a large amount of interest [1–3]. The light output efficiency (LOE) of GaN-based LEDs is dominated by the internal quantum efficiency (IQE) and light extraction efficiency (LEE) . The IQE of high-quality GaN-based LEDs is very high, typically exceeding 70%. However, the LEE of GaN-based LED is very low due to total internal reflection of the light originating from the active layer . Most of the generated photons are confined inside the LED, absorbed by the LED-chip and lost, degrading the device efficiency. Currently, much effort has been devoted to improving the LEE of GaN-based LEDs for high-brightness, including photonic crystals [5,6], surface texturing [7–10], flip-chip LEDs , and conductive omnidirectional reflectors [12,13]. However, these methods involve complex and expensive processes or produce limited enhancement of the LEE.
Recently, a novel and promising method using ZnO nanostructures has been applied to improving the LEE of GaN-based LEDs. Since the refractive index of ZnO (n = 2.0) is close to that of GaN (n = 2.5), and ZnO has a transmittance of over 90% in the visible region , the ZnO nanostructures can be used as a graded-refractive index layer to effectively suppress Fresnel reflection . An et al.  have grown ZnO nanotips on the p-GaN layer of GaN based LEDs using metal-organic chemical vapor deposition (MOCVD) to enhance the light output power (LOP) by up to 1.7 times compared with conventional Ni/Au p-metal LEDs. Zhong et al.  have grown ZnO nanorods on the p-GaN layer of GaN based LEDs using metal-organic vapor phase epitaxy to enhance the LOP by up to 1.5 times compared with conventional Ni/Au p-metal LEDs. However, high temperature growth processes can lead to thermal damage to the electrodes and a significant increase in the forward voltage. Many studies have focused on the aqueous solution growth technique to fabricate ZnO nanorods on the ITO layer or on the p-GaN layer to enhance the LEE of GaN-based LEDs [18–21]. Adjusting the morphology of the ZnO nanorod arrays to control the LEE of GaN-based LEDs has been studied [22–24]. Chiu et al.  have grown ZnO nanorod arrays on the n-GaN layer of high power GaN-based laser lift-off vertical LEDs, giving a 40% enhancement of LEE. Patterned ZnO nanorod arrays have been formed on the ITO layer/p-GaN of GaN-based LEDs to increase the LEE of the LED by chemical bath deposition [26,27]. Kim et al. have further researched the wave-guide effect of ZnO sub-microrods in InGaN blue LEDs .
Most of the previous research on ZnO nanorod modification was focused on the growth methods, diameter, length, or density of the nanorods. An important yet neglected factor, the geometric morphology of the ZnO nanostructure which has close relation to the LEE, has been barely studied. Obviously the smooth surface of the ZnO rods can also produce total internal reflection and limit the LEE . According to the conclusion of Masui  that among the polygons, the triangle has the least chance to trap light rays, we propose that the cone structure has a higher LEE than hexagonal rods. In this paper, we demonstrate that the LEE of GaN-based LED with rough-bevel ZnO nanocones is much higher than that with smooth-surface hexagonal ZnO nanorods.
The geometry and morphology of ZnO nanostructure are sensitive to reaction parameters, such as growth temperature, pH of the precursor, growth time, seed crystals, impurities, etc . We precisely controlled the growth process in this experiment to obtain ZnO nanocone arrays with a roughened surface. Considering the vulnerability of the Ni/Au electrodes to temperature, we chose a low temperature hydrothermal method to grow rough-surface ZnO nanocone arrays on the ITO transparent electrode . A photoresist lift-off process was employed to deposit ZnO nanorod arrays only on the ITO electrode of the p-GaN layer, leaving the rest of the area of the LED-chip free from damage . In order to obtain the ZnO nanocone arrays, a 100 nm thick transparent compact crystalline ZnO seed layer was uniformly sputtered on the ITO transparent electrodes by a magnetron sputtering apparatus before depositing the ZnO nanostructures. 10 mL 50 mM zinc nitrate [Zn(NO3)2] and 10 ml 50 mM hexamethylenetetramine (HMT) [C6H12N4] were mixed and transferred to a 25 ml Teflon autoclave which was placed into a sealed reaction vessel. The prepared LED wafer was tilted facing down and immersed into the above solution. The reaction vessel was sealed to avoid evaporation of the reaction solution and the reaction system was kept at a stable pressure. The system was heated from ambient temperature to 95 °C for 0.5 h and maintained at 95 °C for 2.5 h in a standard laboratory grade oven. At the end of the reaction, the system was cooled to ambient temperature naturally. The LED wafer was immediately rinsed with de-ionized water to remove any residual salts and dried in high-purity nitrogen at room temperature. Photoresist masking was dissolved by photoresist remover at 55 °C for about 0.5 h. The p and n electrodes, and the lateral faces of the LED dies were exposed to open space. The ZnO nanocone array pattern was grown only on the surface of the ITO transparent electrodes.
For comparison, ZnO nanorod arrays were also prepared by suspending the sample upside-down in a glass beaker filled with the aqueous solution of 50 mM Zn(NO3)2 and 50 mM HMT at 90 °C for 3 h . The LED wafer was also rinsed with de-ionized water and dried in high-purity nitrogen at room temperature.
Field-emission scanning electron microscopy (FE-SEM) was used to study the geometry and morphology of the ZnO samples grown on the ITO electrodes. The LOP and electrical characteristic of the GaN-based LEDs were measured with an IPT 6000 LED Chip/Wafer Probing System using an on-wafer testing configuration. The system is comprised of an optical parameter analyzer and integrating half-sphere system mounted above the LED chip. In addition, the measuring system can collect most of the light emitted from the LED. The LOP and electrical characteristic was measured on the unseparated LED chips and the EL was measured on the separated LED chips.
3. Results and discussion
Figure 1 shows a schematic of the GaN-based LED with ZnO nanocone arrays on the ITO transparent electrode. The GaN-based LED epitaxial layers were grown on the c-plane of sapphire by MOCVD. From the bottom to the top there are sapphire (400 μm), undoped GaN (2 μm), n-GaN (2 μm), five periods of InGaN/GaN multiple-quantum-Well (MQW), p-GaN (250 nm), indium tin oxide (200 nm), p and n Ni/Au electrodes, ZnO seed layer, and ZnO nanocone arrays. The size of the LED die is 10 × 23 mil2.
Field emission scanning electron microscopy (FE-SEM) images of ZnO nanocone or nanorod arrays on the ITO layer of GaN-based LEDs are shown in Fig. 2 . ZnO nanocone arrays with relatively homogeneous size and well-aligned distribution can be clearly observed (Fig. 2(a)). The average length and bottom diameter of the ZnO nanocones are 1 μm and 200 nm, respectively. Figure 2(b) shows that all surfaces of the ZnO nanocones are rough and covered with small bulges. Enlarged tilted view of the ZnO nanocones is shown in Fig. 2(c). It can be seen that the gaps among the cusps of the ZnO cones are large and the bottoms of ZnO nanocones are almost connected. This structure has a large surface area. The vertex angle α of the ZnO nanocones lies between . As illustrated in Fig. 2(d), well aligned ZnO nanorods are clearly hexagonal prisms with smooth surfaces. The average length and diameter of the relatively uniform ZnO nanorods are 1μm and 100 nm, respectively. The distance from the bottom to the top of the ZnO nanorods is nearly the same.
As shown in Fig. 3(a) , the current-voltage (I-V) characteristic curves of a conventional LED (C-LED), an LED with rough bevel ZnO nanocones (NC-LED) and an LED with ZnO nanorods (NR-LED) nearly overlap. No obvious increase of forward voltage is observed. The operating voltages of the C-LED, NC-LED and NR-LED at an injection current of 20mA are 3.284, 3.285 and 3.284V, respectively. The voltages of the C-LED, NC-LED and NR-LED at 100 mA are 4.415V, 4.432 V and 4.451V, respectively. I-V curves of the three LEDs exhibit good rectifying behavior with turn-on voltages of 2.67 V and leakage currents of 10−5 mA at −5 V. Apparently these data indicate that the deposition of ZnO nanocone arrays and ZnO nanorod arrays cause no degradation of the electrical properties of GaN-based LEDs.
Figure 3(b) shows the LOP curves of the GaN-based C-LED, NC-LED and NR-LED as a function of injection current. Every point on the curves is the average of five measured LED chips. The LOP of the NC-LEDs and NR-LEDs are obviously both higher than that of the C-LED at the same injection current within the measurement region. The LOP of the NC-LEDs are greater than for the C-LED by 109.9% at 20mA and by 110.1% at 100mA. The LOP of the NR-LEDs are greater than those of the C-LED by 60.6% at 20mA and 55.6% at 100mA. These results indicate that both the ZnO nanocone arrays and ZnO nanorod arrays can increase the LEE of GaN-based LEDs. Furthermore, the LOP of the NC-LEDs is obviously higher than that of the NR-LEDs at the same injection current. This proves that the LEE of the NC-LED is higher than that of the NR-LED.
Figure 4 shows electroluminescence (EL) spectrum of the C-LED, NC-LED and NR-LED at injection currents of 20 mA and 100 mA. There are no significant differences in the EL peak position at 445.5 nm for any of the three LEDs, with the same full width at half maximum of 19 nm at 20 mA. The three LEDs have an EL peak position at 444.0 nm with the same full width at half maximum of 22 nm at 100 mA. Obviously EL intensities of the NC-LED and NR-LED are larger than those of the C-LED at injection currents of 20 mA and 100 mA. However, EL intensities obtained from the NC-LED are larger than those achieved from the NR-LED at injection currents of 20 mA and 100 mA. This also indicates that the LEE of the NC-LED is greater than that of the NR-LED.
The increased LEE of the NC-LEDs and NR-LEDs over the C-LED is mainly due to the formation of a textured surface on the LEDs produced by the ZnO nanocones or nanorods. Because the NC-LED and NR-LED have a large number of sidewalls and rough surfaces, photons generated in the MQW region can be guided into the ZnO nanorods or nanocones, experience multiple scattering at the rod or cone surfaces, and readily escape from the device. This can be illustrated in Fig. 5 . In other words, the ZnO nanocone and nanorod arrays enlarge the escape cone for the photons in the NC-LED and NR-LED.
The LOP and EL intensity of the NC-LED are measured to be higher than that of the NR-LED. This can be attributed to the fact that the light extraction ability of the rough beveled ZnO nanocones is higher than that of the ZnO nanorods. In order to study light extraction by the ZnO nanocones and nanorods, we reduce the three-dimensional hexagonal rods and cones to a two-dimensional rectangle and an isosceles triangle, respectively, following the symmetry of the rays travelling in a cone or a rod. To further simplify the problem we only discuss the light extraction from a single rectangle and a single isosceles triangle using light-ray tracing analysis. Finally, we take into account the interaction of adjacent and nearby rectangles or isosceles triangles. Considering the practical situation with ZnO rods and ZnO cones, we impose the following conditions. The height of the rectangle is larger than its width. The vertex angle of the isosceles triangle is α (13° <α <17°). The critical angle of ZnO is θc (θc = π/6). We assume that light rays only penetrate from the base into the rectangle and isosceles triangle, because ZnO rods and cones are grown on the ZnO seed layer.
(Case 1) The process of rays in rectangle: entering, propagating and leaving
Rays with incident angle θ0 transmit from the base into the rectangle. The angle of incidence θ0 of the rays can vary from 0 to π/2. The rays with incident angle θ1 strike the right side of the rectangle. As shown in Fig. 6(a) , rays with an incident angle θ1 less than θc () to the surface will leave the rectangle after the first impact. In the rectangle, and , we obtain . This implies
Figure 6(b) shows that rays incident at an angle between will experience internal reflection and impact on the right side with an incident angle of θ2. With sides that are perfectly reflective, a set of successive reflections in the rectangle is categorized by the bounce mode n. When rays impact on the top side, this is defined as n. Rays experience (n-1) internal reflections and impact on the top with incident angle θn. If the incident angle θn satisfies , the rays will exit. From , and , we can derive ,that is to say, , the rays will exit at the n-th incidence. This implies
If the incident angle θn satisfies , rays will reflect back into the rectangle. Then, as illustrated in Fig. 6(c), they will experience a set of successive reflections in the rectangle and reflect back into the ZnO seed layer and LED epitaxial materials. Because , ,, we can derive and . This implies
(Case 2) The process of rays in isosceles triangle: entering, propagating and leaving
The vertex angle of the isosceles triangle is α (). The critical angle of ZnO is θc (π/6). Rays with incident angle θ0 transmit from the base into the isosceles triangle. The angle of incidence θ0 of the rays can vary from 0 to π/2. The rays with incident angle θ1 strike the right edge of the isosceles triangle. If rays strike vertically on the edge and exit toward the incident direction, we derive. Rays with incident angle θ1 less than θc () to the surface will exit at the first impact. As shown in Fig. 6(d), we first discuss refracted rays that travel upward relative to the incident rays. From the geometrical equation , we derive and . Figure 6(e) shows that rays with an incident angle θ0 between will exit and refract downward relative to the incident rays.
Figure 6(f) shows that rays incident at an angle between will experience internal reflection and impact on the left edge with an incident angle θ2. Because , we derive and . If the ray has an incident angle θ2 less than θc () to the surface it will exit at the second incidence. From , we derive . If the incident angle θ2 is larger than the critical angle θc () internal reflection will occur. We can conclude that a set of successive reflections in an isosceles triangle is categorized by the bounce mode n until the incident angle θn is smaller than the critical angle θc () . We obtain two conditions: and . We observe that at every internal reflection, the angle of incidence is reduced by α until , and the ray exits.
Summarizing the above analysis, we conclude that rays with an incident angle θ0 between can be extracted from the isosceles triangle experiencing multiple (n = 0, 1, 2…n is finite integer) internal reflections. Rays with incident angle θ0 given by and can be extracted from the rectangle, however, rays with incident angle θ0 given by will be trapped. We conclude that the isosceles triangle provides more of an advantage for light extraction than the rectangle. We deduce that light extraction from the cone structure is greater than that from hexagonal rods in the three-dimensional problem.
As shown in Fig. 5(a), for arrays composed of multiple ZnO nanorods, photons extracted from the top of the nanorods can travel directly into open space and photons extracted from the sidewalls of the nanorods mostly pass into adjacent nanorods, experiencing more dissipation . However, for ZnO nanocone arrays the property that the separation between the cones from the bottom to the top gradually becomes larger decreases the loss of photons into adjacent nanocones, and allows the extraction of more photons to propagate upward into open space. This is shown in Fig. 5(b). Furthermore, the rough beveled surfaces of the nanocones also increase the surface roughness of the LED and increase photon scattering, so that more photons are emitted from the LED surface. In other words, rough beveled ZnO nanocone arrays reduce the Fresnel loss and enlarge the photon escape cone of GaN-based LED compared with ZnO nanorod arrays. We conclude that rough beveled ZnO nanocones have a greater advantage for extracting light than ZnO nanorods.
In summary, we have demonstrated that the LEE of GaN-based blue LEDs with rough beveled ZnO nanocone arrays on the planar ITO layer is greater than that with ZnO nanorods with smooth surface. The LOP of NC-LEDs is greater by about 110% at 20 mA and 100 mA compared with C-LEDs with planar ITO. The LOP of NR-LEDs is greater than that of C-LEDs by 60% at 20mA and 55% at 100mA, respectively. The increased light extraction efficiency of NC-LEDs is much higher than that of NR-LEDs. We have analyzed an isosceles triangle has a greater advantage for light extraction than a rectangle by light-ray tracing analysis. In general, light extraction from the geometry of cone is greater than from that of hexagonal rod. ZnO nanocones can reduce the absorbtion of light propagating to adjacent nanocones compared with ZnO nanorods. Furthermore, the rough beveled surface of the nanocones also increases the surface roughness of the LED and increases photon scattering, so that more photons are emitted from the LED surface. This simple and low-cost damage free ZnO nanocone growth process will be valuable for fabricating high-brightness solid-state lighting and other photoelectric devices.
This work was financially supported by NSFC (Contract Nos. 50801042,50823009), National Basic Research Program of China (2009CB930503), the Fund for the Natural Science of Shandong Province (ZR2010EM020, ZR2010EM049), the Key Project of Chinese Ministry of Education (No. 109096) and IIFSDU (2009TS041).
References and links
1. N. Holonyak, “Is the light emitting diode (LED) an ultimate lamp?” Am. J. Phys. 68(9), 864–866 (2000). [CrossRef]
3. E. Fred, Schubert, Light-Emitting Diodes (Cambridge University Press, 2003).
4. M.-K. Kwon, J.-Y. Kim, I.-K. Park, K. S. Kim, G.-Y. Jung, S.-J. Park, J. W. Kim, and Y. C. Kim, “Enhanced emission efficiency of GaN/InGaN multiple quantum well light-emitting diode with an embedded photonic crystal,” Appl. Phys. Lett. 92(25), 251110 (2008). [CrossRef]
5. I. Schnitzer, E. Yablonovitch, C. Caneau, T. J. Gmitter, and A. Scherer, “30% external quantum efficiency from surface textured thin-film light-emitting diodes,” Appl. Phys. Lett. 63(16), 2174 (1993). [CrossRef]
6. J. J. Wierer, A. David, and M. M. Megens, “III-nitride photonic-crystal light-emitting diodes with high extraction efficiency,” Nat. Photonics 3(3), 163–169 (2009). [CrossRef]
7. R. H. Horng, C. C. Yang, J. Y. Wu, S. H. Huang, C. E. Lee, and D. S. Wuu, “GaN-based light-emitting diodes with indium tin oxide texturing window layers using natural lithography,” Appl. Phys. Lett. 86(22), 221101 (2005). [CrossRef]
8. S. J. Chang, C. F. Shen, W. S. Chen, C. T. Kuo, T. K. Ko, S. C. Shei, and J. K. Sheu, “Nitride-based light emitting diodes with indium tin oxide electrode patterned by imprint lithography,” Appl. Phys. Lett. 91(1), 013504 (2007). [CrossRef]
9. C. H. Kuo, C. M. Chen, C. W. Kuo, C. J. Tun, C. J. Pan, B. J. Pong, and G. C. Chi, “Improvement of near-ultraviolet nitride-based light emitting diodes with mesh indium tin oxide contact layers,” Appl. Phys. Lett. 89(20), 201104 (2006). [CrossRef]
10. Y. Gao, T. Fujii, R. Sharma, K. Fujito, S. P. Denbaars, S. Nakamura, and E. L. Hu, “Roughening hexagonal surface morphology on laser lift-off (LLO) n-face GaN with simple photo-enhanced chemical wet etching,” Jpn. J. Appl. Phys. 43(No. 5A), L637–L639 (2004). [CrossRef]
11. J. J. Wierer, D. A. Steigerwald, M. R. Krames, J. J. O’Shea, M. J. Ludowise, G. Christenson, Y. C. Shen, C. Lowery, P. S. Martin, S. Subramanya, W. Götz, N. F. Gardner, R. S. Kern, and S. A. Stockman, “High-power AlGaInN flip-chip light-emitting diodes,” Appl. Phys. Lett. 78(22), 3379 (2001). [CrossRef]
12. J. K. Sheu, I. H. Hung, W. C. Lai, S. C. Shei, and M. L. Lee, “Enhancement in output power of blue gallium nitride-based light-emitting diodes with omnidirectional metal reflector under electrode pads,” Appl. Phys. Lett. 93(10), 103507 (2008). [CrossRef]
13. J. K. Kim, T. Gessmann, E. F. Schubert, J. Q. Xi, H. Luo, J. Cho, C. Sone, and Y. Park, “GaInN light-emitting diode with conductive omnidirectional reflector having a low-refractive-index indium-tin oxide layer,” Appl. Phys. Lett. 88(1), 013501 (2006). [CrossRef]
14. S. J. An, J. H. Chae, G. C. Yi, and G. H. Park, “Enhanced light output of GaN-based light-emitting diodes with ZnO nanorod arrays,” Appl. Phys. Lett. 92(12), 121108 (2008). [CrossRef]
15. Y.-C. Chao, C.-Y. Chen, C.-A. Lin, Y.-A. Dai, and J.-H. He, “Antireflection effect of ZnO nanorod arrays,” J. Mater. Chem. 20(37), 8134–8138 (2010). [CrossRef]
16. S. J. An, J. H. Chae, G.-C. Yi, and G. H. Park, “Enhanced light output of GaN-based light-emitting diodes with ZnO nanorod arrays,” Appl. Phys. Lett. 92(12), 121108 (2008). [CrossRef]
17. J. Zhong, H. Chen, G. Saraf, Y. Lu, C. K. Choi, J. J. Song, D. M. Mackie, and H. Shen, “Integrated ZnO nanotips on GaN light emitting diodes for enhanced emission efficiency,” Appl. Phys. Lett. 90(20), 203515 (2007). [CrossRef]
18. K.-K. Kim, S.- Lee, H. Kim, J.-C. Park, S.-N. Lee, Y. Park, S.-J. Park, and S.-W. Kim, “Enhanced light extraction efficiency of GaN-based light-emitting diodes with ZnO nanorod arrays grown using aqueous solution,” Appl. Phys. Lett. 94(7), 071118 (2009). [CrossRef]
19. M.-K. Lee, C.-L. Ho, and P.-C. Chen, “Light extraction efficiency enhancement of GaN blue LED by liquid-phase-deposited ZnO rods,” IEEE Photon. Technol. Lett. 20(4), 252–254 (2008). [CrossRef]
20. S. Dalui, C.-C. Lin, H.-Y. Lee, C.-H. Chao, and C.-T. Lee, “Light output enhancement of GaN-based light-emitting diodes using ZnO nanorod arrays produced by aqueous solution growth technique,” IEEE Photon. Technol. Lett. 22(16), 1220–1222 (2010). [CrossRef]
21. J.-W. Kang, M.-S. Oh, Y.-S. Choi, C.-Y. Cho, T.-Y. Park, C. W. Tu, and S.-J. Park, “Improved light extraction of GaN-based green light-emitting diodes with an antireflection layer of ZnO nanorod arrays,” Electrochem. Solid-State Lett. 14(3), H120–H123 (2011). [CrossRef]
22. P. Uthirakumar, B. D. Ryu, J. H. Kang, H. G. Kim, J. H. Ryu, and C.-H. Hong, “Impact of layer thickness and light transmission of ZnO nanomaterials on GaN-based light emitting diodes,” Vacuum 85(2), 198–202 (2010). [CrossRef]
23. C. H. Chao, W. H. Lin, C. H. Chen, C. H. Changjean, and C. F. Lin, “Tunable light extraction efficiency of GaN light emitting diodes by ZnO nanorod arrays,” Semicond. Sci. Technol. 24(10), 105017 (2009). [CrossRef]
24. C. B. Soh, C. B. Tay, S. J. Chua, H. Q. Le, N. S. S. Ang, and J. H. Teng, “Optimization of hydrothermal growth ZnO Nanorods for enhancement of light extraction from GaN blue LEDs,” J. Cryst. Growth 312(11), 1848–1854 (2010). [CrossRef]
25. C. H. Chiu, C. E. Lee, C. L. Chao, B. S. Cheng, H. W. Huang, H. C. Kuo, T. C. Lu, S. C. Wang, W. L. Kuo, C. S. Hsiao, and S. Y. Chen, “Enhancement of light output intensity by integrating ZnO nanorod arrays on GaN-based LLO vertical LEDs,” Electrochem. Solid-State Lett. 11(4), H84–H87 (2008). [CrossRef]
27. M.-S. Lin, C.-C. Chen, W.-C. Wang, C.-F. Lin, and S.-Y. Chang, “Fabrication of the selective-growth ZnO nanorods with a hole-array pattern on a p-type GaN:Mg layer through a chemical bath deposition process,” Thin Solid Films 518(24), 7398–7402 (2010). [CrossRef]
28. K. S. Kim, S.-M. Kim, H. Jeong, M. S. Jeong, and G. Y. Jung, “Enhancement of light extraction through the wave-guiding effect of ZnO sub-microrods in InGaN blue light-emitting diodes,” Adv. Funct. Mater. 20(7), 1076–1082 (2010). [CrossRef]
29. H. Masui, S. Nakamura, and S. P. DenBaars, “Analytical light-ray tracing in two-dimensional objects for light-extraction problems in light-emitting diodes,” Appl. Opt. 47(1), 88–92 (2008). [CrossRef] [PubMed]
30. K. Govender, D. S. Boyle, P. B. Kenway, and P. O’Brien, “Understanding the factors that govern the deposition and morphology of thin films of ZnO from aqueous solution,” J. Mater. Chem. 14(16), 2575–2591 (2004). [CrossRef]
31. K.-K. Kim, S.- Lee, H. Kim, J.-C. Park, S.-N. Lee, Y. Park, S.-J. Park, and S.-W. Kim, “Enhanced light extraction efficiency of GaN-based light-emitting diodes with ZnO nanorod arrays grown using aqueous solution,” Appl. Phys. Lett. 94(7), 071118 (2009). [CrossRef]
32. B. S. Kang, S. J. Pearton, and F. Ren, “Low temperature (<100°C) patterned growth of ZnO nanorod arrays on Si,” Appl. Phys. Lett. 90(8), 083104 (2007). [CrossRef]