Surface antireflection properties of GaN nanostructures with various effective refractive index profiles

GaN nanostructures with various effective refractive index profiles (Linear, Cubic, and Quintic functions) were numerically studied as broadband omnidirectional antireflection structures for concentrator photovoltaics by using three-dimensional finite difference time domain (3D-FDTD) method. Effective medium theory was used to design the surface structures corresponding to different refractive index profiles. Surface antireflection properties were calculated and analyzed for incident light with wavelength, polarization and angle dependences. The surface antireflection properties of GaN nanostructures based on six-sided pyramid with both uniform and non-uniform patterns were also investigated. Results indicate a significant dependence of the surface antireflection on the refractive index profiles of surface nanostructures as well as their pattern uniformity. The GaN nanostructures with linear refractive index profile show the best performance to be used as broadband omnidirectional antireflection structures. ©2014 Optical Society of America OCIS codes: (040.5350) Photovoltaic; (310.1210) Anti-reflection coatings; (310.6628) Subwavelength structures, nanostructures; (050.2065) Effective medium theory. References and links 1. J. Wu, W. Walukiewicz, K. M. Yu, W. Shan, J. W. Ager, E. E. Haller, H. Lu, W. J. Schaff, W. K. Metzger, and S. Kurtz, “Superior radiation resistance of In1−xGaxN alloys: full-solar-spectrum photovoltaic material system,” J. Appl. Phys. 94(10), 6477–6482 (2003). 2. R. M. Farrell, C. J. Neufeld, S. C. Cruz, J. R. Lang, M. Iza, S. Keller, S. Nakamura, S. P. DenBaars, U. K. Mishra, and J. S. Speck, “High quantum efficiency InGaN/GaN multiple quantum well solar cells with spectral response extending out to 520 nm,” Appl. Phys. Lett. 98(20), 201107 (2011). 3. E. Trybus, G. Namkoong, W. Henderson, S. Burnham, W. A. Doolittle, M. Cheung, and A. Cartwright, “InN: a material with photovoltaic promise and challenges,” J. Cryst. Growth 288(2), 218–224 (2006). 4. L. Sang, M. Liao, N. Ikeda, Y. Koide, and M. Sumiya, “Enhanced performance of InGaN solar cell by using a super-thin AlN interlayer,” Appl. Phys. Lett. 99(16), 161109 (2011). 5. I. G. Kavakli and K. Kantarli, “Single and double-layer antireflection coatings on silicon,” Turk. J. Phys. 26, 349–354 (2002). 6. S. E. Lee, S. W. Choi, and J. Yi, “Double-layer anti-reflection coating using MgF2 and CeO2 films on a crystalline silicon substrate,” Thin Solid Films 376(1-2), 208–213 (2000). 7. J.-Q. Xi, M. F. Schubert, J. K. Kim, E. F. Schubert, M. Chen, S.-Y. Lin, W. Liu, and J. A. Smart, “Optical thinfilm materials with low refractive index for broadband elimination of Fresnel reflection,” Nat. Photonics 1, 176– 179 (2007). 8. Y. Zhao, F. Chen, Q. Shen, and L. Zhang, “Optimal design of graded refractive index profile for broadband omnidirectional antireflection coatings using genetic programming,” Prog. Electromagnetics Res. 145, 39–48 (2014). 9. Y. F. Huang, S. Chattopadhyay, Y. J. Jen, C. Y. Peng, T. A. Liu, Y. K. Hsu, C. L. Pan, H. C. Lo, C. H. Hsu, Y. H. Chang, C. S. Lee, K. H. Chen, and L. C. Chen, “Improved broadband and quasi-omnidirectional antireflection properties with biomimetic silicon nanostructures,” Nat. Nanotechnol. 2(12), 770–774 (2007). 10. W. Zhou, M. Tao, L. Chen, and H. Yang, “Microstructured surface design for omnidirectional antireflection coatings on solar cells,” J. Appl. Phys. 102(10), 103105 (2007). #224132 $15.00 USD Received 30 Sep 2014; revised 26 Nov 2014; accepted 1 Dec 2014; published 17 Dec 2014 (C) 2014 OSA 29 Dec 2014 | Vol. 22, No. 26 | DOI:10.1364/OE.22.031907 | OPTICS EXPRESS 31907 11. H. K. Raut, V. A. Ganesh, A. S. Nair, and S. Ramakrishna, “Anti-reflective coatings: a critical, in-depth review,” Energy Environ. Sci. 4(10), 3779–3804 (2011). 12. S. L. Diedenhofen, G. Vecchi, R. E. Algra, A. Hartsuiker, O. L. Muskens, G. Immink, E. P. A. M. Bakkers, W. L. Vos, and J. G. Rivas, “Broad-band and omnidirectional antireflection coatings based on semiconductor nanorods,” Adv. Mater. 21(9), 973–978 (2009). 13. C. H. Chiu, P. Yu, H. C. Kuo, C. C. Chen, T. C. Lu, S. C. Wang, S. H. Hsu, Y. J. Cheng, and Y. C. Chang, “Broadband and omnidirectional antireflection employing disordered GaN nanopillars,” Opt. Express 16(12), 8748–8754 (2008). 14. J. Zhu, Z. Yu, G. F. Burkhard, C.-M. Hsu, S. T. Connor, Y. Xu, Q. Wang, M. McGehee, S. Fan, and Y. Cui, “Optical absorption enhancement in amorphous silicon nanowire and nanocone arrays,” Nano Lett. 9(1), 279– 282 (2009). 15. W. Zhou, M. Tao, L. Chen, and H. Yang, “Microstructured surface design for omnidirectional antireflection coatings on solar cells,” J. Appl. Phys. 102(10), 103105 (2007). 16. C.-H. Sun, W.-L. Min, N. C. Linn, P. Jiang, and B. Jiang, “Templated fabrication of large area subwavelength antireflection gratings on silicon,” Appl. Phys. Lett. 91(23), 231105 (2007). 17. L. Han and H. Zhao, “Simulation analysis of GaN microdomes with broadband omnidirectional antireflection for concentrator photovoltaics,” J. Appl. Phys. 115(13), 133102 (2014). 18. L. Han, T. A. Piedimonte, and H. Zhao, “Experimental exploration of the fabrication of GaN microdome arrays based on a self-assembled approach,” Opt. Mater. Express 3(8), 1093 (2013). 19. W. Guo, J. Xie, C. Akouala, S. Mita, A. Rice, J. Tweedie, I. Bryan, R. Collazo, and Z. Sitar, “Comparative study of etching high crystalline quality AlN and GaN,” J. Cryst. Growth 366, 20–25 (2013). 20. W. H. Southwell, “Gradient-index antireflection coatings,” Opt. Lett. 8(11), 584–586 (1983). 21. S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–474 (1956). 22. D. L. Brundrett, E. N. Glytsis, and T. K. Gaylord, “Homogeneous layer models for high-spatial-frequency dielectric surface-relief gratings: conical diffraction and antireflection designs,” Appl. Opt. 33(13), 2695–2706 (1994). 23. W. H. Southwell, “Pyramid-array surface-relief structures producing antireflection index matching on optical surfaces,” J. Opt. Soc. Am. A 8(3), 549–553 (1991). 24. T. Fujii, Y. Gao, R. Sharma, E. L. Hu, S. P. DenBaars, and S. Nakamura, “Increase in the extraction efficiency of GaN-based light-emitting diodes via surface roughening,” Appl. Phys. Lett. 84(6), 855–857 (2004). 25. T. W. Kuo, S. X. Lin, Y. Y. Hung, J. H. Horng, and M. P. Houng, “Improved extraction efficiency of lightemitting diodes by wet-etching modifying AZO surface roughness,” IEEE Photon. Technol. Lett. 23(6), 362–364 (2011). 26. J. W. Seo, H. S. Oh, and J. S. Kwak, “Improved light-extraction efficiency of the AlGaInP-based light-emitting diodes fabricated using a chemical wet etch of n-AlGaInP layer,” J. Korean Phys. Soc. 55(1), 314–317 (2009). 27. Y. J. Lee, H. C. Kuo, S. C. Wang, T. C. Hsu, M. H. Hsieh, M. J. Jou, and B. J. Lee, “Increasing the extraction efficiency of AlGaInP LEDs via n-side surface roughening,” IEEE Photon. Technol. Lett. 17(11), 2289–2291 (2005). 28. A. Lundskog, J. Palisaitis, C. W. Hsu, M. Eriksson, K. F. Karlsson, L. Hultman, P. O. A. Persson, U. Forsberg, P. O. Holtz, and E. Janzén, “InGaN quantum dot formation mechanism on hexagonal GaN/InGaN/GaN pyramids,” Nanotechnology 23(30), 305708 (2012). 29. D. Zhuang and J. H. Edgar, “Wet etching of GaN, AlN, and SiC: a review,” Mater. Sci. Eng. Rep. 48(1), 1–46 (2005). 30. S. Han, J. Kim, J. Y. Kim, K. Kim, H. Tampo, S. Niki, and J. Lee, “Formation of hexagonal pyramids and pits on V-/VI-polar and III-/II-polar GaN/ZnO surfaces by wet etching,” J. Electrochem. Soc. 157(1), D60–D64 (2010).


Introduction
III-nitride (In, Al, Ga, -N) based semiconductor alloys with direct band gap and wide band gap tuning from deep ultraviolet (UV) to near infrared have a great potential to be applied for high efficiency concentrator photovoltaics (CPV) [1][2][3][4].One key challenge associated with the III-nitride based PV is from the high surface reflection due to the large refractive index contrast between III-nitride semiconductor (n~2.5) and air (n~1).Various antireflection technologies have been applied to suppress surface reflection in different types of solar cells, including 1) single-or multi-layer antireflection coatings (ARCs) [5,6], 2) gradient refractive index ARC [7,8], and 3) sub-wavelength textured surface [9,10].The perfect ARC for a single incident wavelength with normal incident angle is a quarter wavelength thick coating with appropriate refractive index [11].However, antireflection for solar cell device application requires broadband and omnidirectional antireflection, as well as polarization insensitivity.There is no perfect surface coating or surface structure that simultaneously satisfies all of the requirements with zero surface reflection.Among the proposed antireflection technologies, surface texturing is an effective approach to reduce surface reflection which has been widely implemented in various solar cell devices.Surface texturing such as nanorods [12], nanopillars [13], nanocones [14], nanohemisphere [15] and nanopyramids [16] have been used.Very recently, we have studied the surface antireflection properties of GaN based hemi-ellipsoid microdome structures, which could significantly reduce surface reflection over a broad wavelength range and a wide incidence angle [17].Experimentally, we have also demonstrated the feasibility to tune the surface shape and size of the GaN microdome structures by controlling the top-down plasma dry etching conditions [18].Note that essentially the different surface texturing structures provide different effective refractive index profiles that affect the propagation, reflection and transmission of electromagnetic fields.
In this paper, we studied and compared the surface antireflection properties of GaN nanostructures with various effective refractive index profiles (Linear, Cubic, and Quintic functions) by using three-dimensional finite difference time domain (3D-FDTD) method.Effective medium theory was used to design the surface structures corresponding to different refractive index profiles.Comprehensive studies of the surface reflections on GaN nanostructures with various effective refractive index take into account the incident wavelength, incident angle and incident light polarizations.The surface antireflection properties were compared with that of GaN six-sided pyramids which are readily formed by wet etching process [19].The effect of the uniformity of GaN pyramids on surface reflection was also investigated.

Numerical simulation method
In this work, the numerical studies of surface antireflection properties of GaN nanostructures with various effective refractive index profiles were performed by using a 3D-FDTD method.Based on our previous studies [17], the surface reflection significantly depends on the surface structure size and shape.GaN hemi-ellipsoid microdome with dimension of diameter D = 200 nm and height H = 250 nm, and arranged in a periodic hexagonal close-packed pattern provides a relatively low surface reflection over a broad wavelength range (300nm~1200nm) and a wide incidence angle (0°~80°).Here, based on the GaN nanostructure with diameter D = 200 nm and height H = 250 nm with hexagonal pattern, we comprehensively studied the effect of the surface structure effective refractive index profile on the surface reflection properties.
Utilizing the symmetric property of the hexagonal pattern, the simulation region was reduced to a unit cell with area of a × 3 a, where a represents the center-to-center distance between two nanostructures within the close-packed pattern.Perfectly matched layers (PML) boundary condition was applied in the normal direction to the surface interface.Bloch boundary condition was used at the interface along the in-plane direction.

Structure design and formula
In order to design surface structures corresponding to different refractive index profiles, effective medium theory was applied here.Three different surface structures corresponding to effective index profiles of Linear, Cubic and Quintic functions were designed.The continuous gradient refractive index profiles with the three functions are expressed as following [20]: ( ) where n i and n s are the refractive indices of air and substrate, respectively, t is the normalized thickness of the nanostructures ranging from 0 to 1.

Effective medium theory (EMT)
The effective medium theory is most accurate when the feature size is smaller than that of the incident wavelength.Here, the maximum feature size is in the range of 200nm, which is considerably small as compared to the incident wavelength in the near-UV to the near infrared range.The effective refractive index for both transverse electric (TE, or S) and transverse magnetic (TM, or P) polarizations are expressed as following [21,22]: (1 ) (1 ) where is the effective index for TE (or S) polarization, , eff TE n is the effective index for TM (or P) polarization, and f is the filling factor of the surface nanostructures.

Filling factor
In this study, filling factor (f) is defined as the filling percentage of the structure cross section area over one hexagon area on the substrate [23].Due to the rotational symmetric property of the structures with cone shapes, the cross section of the structure has a circular shape with radius (r) which varies as a function of the structure height (h).The cone shape structures (D = 200nm) with hexagonal close-packed lattice pattern occupied the hexagon area ( 22 3L ) with side length L on the substrate surface, which is shown in Fig. 1(a).For this pattern, the filling factor of nanostructure varies from 0 to 0.907, which can be described by the following equation:

Six-sided pyramid structure
Wet etching process has been widely implemented in GaN based light-emitting diodes (LEDs) to enhance its light extraction efficiency [24][25][26][27].Six-sided pyramid structures are typically obtained from the wet etching of N-polar GaN.Observed from experiments, the formed GaN pyramid structures contain a 62° angle between the basal plane {000-1} and the edge of bounding pyramids plane {10-1-1} [19,[28][29][30].The wet etching process of N-polar GaN is based on the defect-selective etching behavior, which leads to non-uniform dimension and distribution of the etched pyramid structures.
In this study, we investigated the surface antireflection properties of the GaN six-sided pyramid nanostructures with both uniform and non-uniform patterns, as shown in Figs.1(b) and 1(c).For the uniform pattern, the same dimension of D = 200 nm was used, the height of the structure was calculated to be H = 188 nm based on the angle of 62° between the basal plane and the edge of bounding pyramids planes.The filling factor for the six-sided pyramid structure with hexagonal pattern can be obtained from the following equation:

Antireflections of GaN nanostructures with linear, cubic, and quintic index profiles
Based on the effective medium theory, the surface nanostructures with linear, cubic and quintic refractive index profiles were obtained as shows in Fig. 2. Note that here we utilized the S mode function to obtain these surface structures.The nanostructures were arranged with close-packed hexagonal patterns.Figures 2(d), 2(e) and 2(f) plot the linear, cubic and quintic refractive index profiles as a function of the surface structure height.From Fig. 2, the surface feature shapes vary according to the refractive index profile functions.3(a)] and P polarizations [Fig.3(b)].Note that, the fixed incident wavelength of 500nm represents the wavelength region with the highest irradiance power in solar spectrum.For S polarization, all of the three GaN nanostructures with different refractive index profiles show significant reduction of the surface reflection at different incident angles from 0° (normal incidence) to 80°.For P polarization, all three structures show reduced surface reflection with incident angle <60°.The nanostructures lead to a shift of the Brewster angle from 67° for the conventional GaN flat surface to 38°-42° for the GaN nanostructures.By comparing between the three GaN nanostructures, the surface structure with cubic index profile function shows lower surface reflection than that of the one with quintic index profile function, and the structure with the linear index profile function shows the lowest surface reflection at all incident angles for both S and P polarizations.Specifically, the surface reflection could be reduced to <2% (for S) and <1% (for P) with incident angle <50° by using the surface structures with linear index profile function.Figures 4(a) and 4(b) plot the wavelength dependence of the surface reflection for GaN nanostructures at normal incidence (0°) for both S and P polarizations as compared to that of the conventional GaN flat surface.The surface reflection as a function of the incident wavelength for the GaN nanostructures shows oscillations, which is due to the interference between the reflected waves from the surface nanostructures.For S polarization, the three GaN based nanostructures show significant reduction of surface reflection over a broad wavelength range (300 nm<λ<1200 nm).The GaN nanostructure with linear refractive index profile shows the lowest surface reflection over the entire wavelength range.For P polarization, the general trend is similar as that of the S polarization, except in some particular wavelength region the surface reflection of the GaN nanostructure with the cubic index profile shows lower reflection than that of the one with linear index profile.This is due to the oscillation of the surface reflection.Specifically, the results indicate that the GaN nanostructure with linear refractive index profile leads to surface reflection < 1% over the wavelength range from 340nm to 880 nm for both S and P polarizations.

Antireflections of GaN nanostructures with six-sided pyramids (uniform Vs. non-uniform)
GaN six-sided pyramid structures are typically obtained during the wet-etching process of Npolar GaN.Here, we studied the surface reflection properties of the GaN six-sided pyramid structures with both uniform and non-uniform hexagonal patterns as shown in Fig. 5. Figures 5(c) and 5(d) show the top view schematics of the corresponding structures, in which the simulation unit cells are identified as red dash lines (2a × 3 a).Figures 5(e) and 5(f) plot the effective refractive index profiles for the GaN pyramids with uniform and non-uniform patterns, respectively.For the uniform GaN pyramid structures, the filling factor varies from 0 to 1.For the non-uniform GaN pyramid structures, here we used the mixture of two types (D = 200 nm, H = 188nm and D = 100 nm, H = 94 nm) of pyramids.Thus, the filling factor varies from 0 to 0.4375.The non-close-packed pattern at the interface of the substrate leads to the discontinuous step of the effective refractive index as shown in Fig. 5(f).
Figures 6(a polarizations.For S polarization, the GaN pyramid structures with uniform pattern show the lowest surface reflection over the broad incident angle range.At normal incidence, the surface reflection of the S polarization reaches as low as 0.03%.For P polarization, the surface reflection of the GaN pyramid structures with uniform pattern shows better performance as compared to that of the structure with non-uniform pattern.The surface reflection of the GaN pyramid structures have higher value with relative large incident angle (>45°) as compared to that of the GaN flat surface.This is due to the Brewster angle of the GaN flat surface at 68°, which gives the lowest surface reflection.At normal incidence, the surface reflection of the P polarization reaches as low as 0.05%.Note that no Brewster angle is observed for the GaN pyramid structures, which is due to the mixture of multiple side walls of the pyramid structures with oblique incidence.The wavelength dependence of the surface reflection of the GaN pyramid structures at normal incidence is shown in Fig. 7.The surface reflection for the GaN pyramid structures with uniform pattern shows oscillation which is due to the interference between the reflected waves from the surface structure.The general trend indicates that the surface reflection of the GaN pyramids with uniform pattern has lower surface reflection as compared to that of the one with non-uniform pattern over a broad wavelength range for both S and P polarizations.
In order to compare the surface reflection properties of the above studied five structures with GaN flat surface, Fig. 8 plots the comparison of the surface reflection for these structures as a function of incident angle with fixed wavelength of 500 nm.The surface reflection is obtained by averaging both S and P polarizations.From the comparison, the surface reflection of the GaN pyramid structures with uniform pattern is the lowest with incident angle < 30°.For larger incident angle, the surface reflection of the GaN nanostructures with linear index profile shows the lowest reflection.Note that the uniformity of the pyramid structures affects the surface reflection significantly.Non-uniformity of the GaN pyramid structures brings up the reflection significantly as shown in Fig. 8. Experimentally, it is very challenging to achieve uniform GaN pyramid pattern.Figure 9 plots the wavelength dependence surface reflection for the five structures as compared to that of the conventional GaN flat surface with normal incidence.Similarly, the surface reflection values are averaged for both S and P polarizations.Over the broad wavelength range for 300nm −1200 nm, the GaN nanostructure with linear index profile shows the best performance as compared to the other structures.Experimentally, it is very feasible to form uniform GaN nanostructures with desired surface feature and size by using self-assembled nanolithography and reactive ion etching method [18].By optimizing the reactive ion etching condition, 3D GaN nano/micro structures with desired shape can be fabricated.Thus, the GaN nanostructure with linear index profile is a very promising structure to be implemented in concentrator photovoltaics for enhancing solar cell device efficiency.Similar concept can be transformed to be implemented in other material systems such as silicon and III-V based solar cell devices.

Summary
In summary, surface reflection properties of GaN nanostructures with linear, cubic and quintic effective refractive index profiles were studied and compared with that of the conventional GaN flat surface.The results indicate significant reduction of surface reflection for these GaN nanostructures over a wide incident angle and broad wavelength range for both S and P polarizations.Studies of the surface reflection properties of GaN pyramid structures indicate its significant dependence on the nanostructure uniformity.By systematically comparing the surface reflections between the five structures as well as the GaN flat surface, the GaN nanostructure with linear index profile shows great promise to be implemented in solar cell devices to enhance the light collection efficiency.Similar concept could be used in other material system such as silicon and III-V based solar cells.

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224132 -$15.00USD Received 30 Sep 2014; revised 26 Nov 2014; accepted 1 Dec 2014; published 17 Dec 2014 (C) 2014 OSA shown in Fig.1(b), L is the side length of one hexagon as unit-cell configuration to be repeated, r is the radius of pyramids nanostructures.Hence, the filling factor of pyramid structures with uniform pattern is ranging from 0 to 1 (the interface between nanostructures and substrate).For the non-uniform pattern six-sided pyramid nanostructures, we use the mixture of two types of pyramids, D = 200nm with H = 188nm and D = 100nm with H = 94nm to represent the non-uniformity, as shown in Fig.1(c).

Fig. 1 .
Fig. 1.Schematics of nanostructures (a) rotational symmetric dome shape (b) six-sided pyramids shape with uniform hexagonal close-packed pattern and (c) six-sided pyramids shape with non-uniform hexagonal close-packed pattern.Red dash lines indicate nanostructures occupied in hexagonal area, with length of D. L is side length of hexagonal area, and r is the radius of nanostructures.

Fig. 3 .
Fig. 3. Surface reflection of the flat GaN surface, GaN nanostructures (D = 200nm, H = 250nm) with linear, cubic, and quintic index profiles, as a function of light incidence angle for both (a) S and (b) P polarizations with fixed incidence wavelength λ = 500nm.

Fig. 4 .
Fig. 4. Surface reflection of the flat GaN surface, GaN nanostructures (D = 200nm, H = 250nm) with linear, cubic, quintic index profiles, as a function of light incidence wavelength for both (a) S polarization and (b) P polarization at normal incidence.
) and 6(b) plot the surface reflection as a function of the incident angle with fixed wavelength of 500 nm for the following three structures: (i) conventional GaN with flat surface; (ii) GaN six-sided pyramid nanostructures with uniform pattern (D = 200nm, H = 188nm); and (iii) GaN six-sided pyramids nanostructures with non-uniform pattern (D = 200nm, H = 188nm and D = 100nm, H = 94nm) for both S [Fig.6(a)] and P [Fig.6(b)]

Fig. 5 .
Fig. 5. Schematics of 3D GaN six-sided pyramid structures with (a) (c) close-packed and (b) (d) non-close-packed refractive index profiles, respectively.(c) and (d) show the top view of the pyramid structures with the corresponding effective refractive index profiles of (e) and (f).

Fig. 7 .
Fig. 7. Surface reflection of the flat GaN surface, GaN based six-sided uniform close-packed pyramid structures (L = 200nm, H = 188nm) and non-uniform pyramid structures (L = 200nm, H = 188nm; L = 100nm, H = 94nm) as a function of light incidence wavelength for both (a) S polarization and (b) P polarization at normal incidence.