- Email: [email protected]
Analysis on the effect of amorphous photonic crystals on light extraction efficiency enhancement for GaN-based thin-film-flip-chip light-emitting diodes
Optics Communications 367 (2016) 72–79
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Analysis on the effect of amorphous photonic crystals on light extraction efficiency enhancement for GaN-based thin-film-flip-chip light-emitting diodes Qingyang Yue a, Kang Li b,n, Fanmin Kong b, Jia Zhao b, Meng Liu b a b
School of Physics and Electronics, Shandong Normal University, Jinan 250014, China School of Information Science and Engineering, Shandong University, Jinan 250100, China
art ic l e i nf o
a b s t r a c t
Article history: Received 12 October 2015 Received in revised form 10 December 2015 Accepted 31 December 2015 Available online 4 February 2016
This work showed the liquid-like amorphous photonic crystals (PhCs) can effectively enhance the light extraction efficiency of GaN-based thin-film-flip-chip light-emitting diodes (TFFC LEDs) by the light scattering effect. The finite-difference time-domain (FDTD) method was employed to analyze the light scattering characteristics, numerical studies revealed that the amorphous PhCs can provide omnidirectional scattering, and the transmittances of amorphous PhCs is superior to that of triangular lattice PhCs when the incident angle is in the region 20° ≤ θ ≤ 35°. The influence of p-GaN layer thickness and the amorphous PhCs feature size on the light extraction efficiency was also studied by 3D-FDTD method systematically. For the proposed amorphous PhCs structure in n-GaN layer, the light extraction efficiency is enhanced by a factor of 1.49 as compared to conventional TFFC LEDs, which shows 1.07 times enhancement in comparison to that of triangular lattice PhCs. & 2016 Elsevier B.V. All rights reserved.
Keywords: GaN-based LEDs Amorphous photonic crystals Light extraction efficiency Forward scattering Finite-difference time-domain (FDTD) method
1. Introduction Recently, high performance GaN-based light-emitting diodes (LEDs) have attracted much attention for use in back lighting in large liquid-crystal displays [1] and general lighting [2]. GaN-based LEDs are promising candidates for the application of next-generation general lighting source because their wall-plug efficiency is comparable to conventional fluorescent lamps [3]. In general, the conventional GaN-based LEDs are grown on a sapphire substrate and emit photons from the p-side indium-tin-oxide (ITO) contact. The top-emitting configuration results that a considerable fraction of light is absorbed by the metal contacts, and it still has severe current-crowding and heat-conducting problems at high current injection due to poor thermal conductivity of sapphire. A flip-chip LEDs (FC LEDs) [4,5] configuration that can simultaneously satisfy thermal management and light extraction has been developed. A typical FC LEDs chip is commonly bonded to a submount with high thermal conductivity, and downward-propagating light can be reflected upward by placing a reflector (e.g. silver) on the p-GaN layer [6,7]. Recently, improvements in the nitride light emitters efficiency can also be achieved by thin-film-flip-chip (TFFC) structure, it can be fabricated with the help of FC and laser n
Corresponding author. E-mail address: [email protected] (K. Li).
http://dx.doi.org/10.1016/j.optcom.2015.12.072 0030-4018/& 2016 Elsevier B.V. All rights reserved.
lift-off (LLO) technique [8,9]. The thickness of TFFC LEDs is ∼1 μm [10,11], which can also reduce the number of waveguide modes. This method has been proven to be effective in boosting the optical power. For illumination applications, GaN-based TFFC LEDs is still necessary to further improve output efficiency. It has been found that efficiency of GaN-based LEDs is mainly limited by external quantum efficiency also known as light extraction efficiency (LEE) [12,13]. The limitation occurred in LEE of GaN-based LEDs is attributed to the large refractive index difference between GaN and air/epoxy resin [14]. Large portion of generated photons in the active region are trapped inside the LEDs device by total internal reflection at the interface. As a result, currently the major challenge associated with light extraction is enhancing the LEE of LEDs. Several approaches have been proposed to extract the light confined in waveguide modes, including surface roughness [15–17], photonic crystals (PhCs) and so on. The rough-surface thin-film LEDs can offer high LEE ( 80%) for an encapsulated device [18], but it offers no control over the direction of the emitted light, resulting in a Lambertian far field emission. Using 2D PhCs is an effective approach for improving the LEE of LEDs [19–23]. It has been proven that the bandgap of PhCs can inhibit the light emission into guided modes and hence increasing the LEE of LEDs in which the PhCs penetrates through the entire device [24]. However, this approach is difficult to achieve a high performance working device when the PhCs penetrates through the active layer
Q. Yue et al. / Optics Communications 367 (2016) 72–79
[25]. Most scientific efforts have therefore focused on the diffractive properties of PhCs [25–27]. In this scheme, PhCs can be used to diffract guided modes confined in LEDs into air. In general, the PhCs are located on the surfaces of the LEDs and do not penetrates through the active layer, and current injection is less problematic. Compared with triangular lattice PhCs, the amorphous PhCs can provide omnidirectional scattering. In this work, the influence of the amorphous PhCs on the LEE of TTFC LEDs was first studied by using the three-dimensional finite difference time domain (3D-FDTD) method [28], the FDTD method has also been used to analyze the efficiency of superluminescent diodes [29,30]. The light scattering characteristics of liquid-like amorphous PhCs were analyzed by FDTD method, numerical studies revealed that the liquid-like amorphous PhCs can provide larger LEE than triangular lattice PhCs. The influence of p-GaN layer thickness and the amorphous PhCs feature size on LEE was also systematically studied by 3D-FDTD method.
2. Numerical model and numerical method A schematic diagram of the TFFC LEDs structure is shown in the Fig. 1(a). The typical TFFC LEDs is multi-layer planer structure with a silver reflector, a P-GaN layer, a multiple quantum wells (MQWs) layer and an N-GaN layer. The liquid-like amorphous PhCs is located on the top surface of the N-GaN layer. Our liquid-like amorphous PhCs are different from that quasi PhCs [31–33] and Archimedean PhCs [34], which is lack of symmetry. We distributed the cylindrical air holes as a snapshot of atoms in a liquid by the use of the Metropolis Monte Carlo method [35] with a repulsive interatomic force. Fig. 1(b) shows the refractive index distribution of the amorphous PhCs, which has only short-range order but no long-range order. And the average distance of adjacent air holes is equal to a. As illustrated in Fig. 1(c), the modulus of the Fourier transform of amorphous PhCs shows an isotropic ring pattern, which has no Bragg peaks [36]. Next, the 3-D FDTD method was employed to calculate the LEE of the TFFC LEDs. The thickness of total GaN layer included the P-GaN layer, MQWs layer and N-GaN layer was set to 700 nm, the thickness of silver layer was set as 200 nm, and the lateral dimension of the computational domain was set as 8 μm in the simulation. The complex refractive index of GaN material was assumed to 2.5 + 0.0013i at the wavelength of 460 nm [37]. In addition, the homogeneous mesh was used during the simulation, and the grid was set to Δx = Δy = Δz = 10 nm . Furthermore, a perfectly matched layer (PML) enclosing the entire simulation domain was used to absorb outgoing waves. It has been proven that the electron-hole-recombination can classically be represented by a dipole. The conventional InGaN/
73
GaN quantum well grows along the c-axis (z direction), and it is polar and can be modeled by a uniform distribution x and y polarization orientation dipoles placed in the MQWs plane that is perpendicular to the growing axis (z direction). In the z direction, the dipole source is located in the middle of the MQWs layer. In the x–y plane, the typical locations of the dipole sources in the PhCs or amorphous PhCs TFFC LEDs are shown in Fig. 2. And thus, there are 6 kinds of dipole source states in the 3D-FDTD simulation, the result of Pan [38] proved that that the accurate averaged efficiency can be obtained by these typical dipole sources. Nevertheless, the FDTD intrinsically is a coherent simulation method. In order to avoid generating an unphysical coherent effect between different dipole sources, only one dipole source was located in the MQWs plane once in the simulation. And the center wavelength of the dipole source in the FDTD simulation is 460 nm. In the FDTD simulation, the LEE of one dipole can be calculated from the power flux extracted from the LEDs with respect to the overall emitted power from the dipole source
ηextr = Pout /Psource
(1)
where Pout is the power flow integrated over a plane just above the LEDs structure and Psource the integrated power flux through the closed surface enclosing the source. And thus, the LEE of LEDs can be obtained by N
η=
n ∑n = 1 ηextr
N
(2)
where N is the number of dipole source states. Finally, the LEE enhancement factor F is defined as follows:
F = η/η0
(3)
where η is the vertical LEE of the LEDs while η0 is the LEE of planar TFFC LEDs.
3. Numerical analysis and discussion 3.1. Influence of gap distance d between the active layer and reflector on LEE for planar TFFC LEDs Note that in the TFFC LEDs, the MQWs layer is placed close enough to the silver reflector ( 100 nm). The light emitted from QWs will interfere with the reflected waves, and such interference can modulate the radiation's profile. It is important for high performance TFFC LEDs to choose an appropriate gap distance between the active layer and reflector. To systematically study the interference effect, we first calculated the LEE of the TFFC LEDs structure while varying the positions of the dipole sources from
Fig. 1. (a) The schematic diagram of TFFC LEDs structure, The thickness of total GaN layer included the P-GaN layer, MQWs layer and N-GaN layer is 700 nm, the thickness of silver layer is set as 200 nm, and the center wavelength of the dipole source in the FDTD simulation is 460 nm. The detection plane is set as 460 nm away from the emission surface of n-GaN. (b) Schematic view of the transverse cross section in the amorphous PhCs region. (c) Spatial Fourier spectrum of the structure (b).
74
Q. Yue et al. / Optics Communications 367 (2016) 72–79
Fig. 2. Schematic diagram of three kinds of typical dipole sources located in the x–y plane, and the plane is located in the middle of the MQWs layer in the z direction. (a) triangular lattice PhCs, (b) amorphous PhCs. Comment: A, B and C represents the position that is under one hole, between the two holes, and in the center of three holes, respectively.
Fig. 3. (a) LEE for the conventional TFFC LEDs at wavelength 460 nm with flat surface as a function of the gap distance d between the active layer and reflector (silver mirror or perfect electric conductor (PEC)). (b) and (c) Electric filed intensity profiles of GaN-based TFFC LEDs with PEC when d is 140 nm (b), 90 nm (c), respectively. The red dotted lines indicate the critical angle ( 23.7°) of the total internal reflection between GaN and air. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
the silver layer by carrying out the FDTD simulation. The total thickness of the GaN layer was set to 700 nm. The LEE exhibited an oscillatory behavior as a function of the gap distance d, as shown in Fig. 3a. The peaks of LEE were found at gap distances d = 30 nm and 120 nm respectively, and the periodicity of d is 90 nm, it agrees with the result of kim et al. [39]. According to the coherent theory, constructive interference coupling happens when d = mλ /2n (m is integer, n is the refractive index of GaN material), which indicates the periodicity of d is λ /2n = 90 nm . Furthermore, the lowest order requirement of constructive interference condition at d = 0 nm . In the case of a silver reflector, the phase change for reflected light is given by π + 2α , where α = 2π /l and l is the skin depth of silver. And thus, the first maxima was obtained at d ≈ 40 nm . Next, we investigated the interference effect between the dipole and perfect electric conductor (PEC). The LEE of conventional TFFC LEDs with PEC shows similar trend to that of silver, as illustrated in Fig. 3a. From Fig. 3a, the second maxima of 28.51% were obtained at d = 140 nm , which is much more than that of silver (23.67%). This can be explained by that the PEC is lack of the absorption and the skin depth. From Fig. 3b, it is clearly observed that the vertical radiation becomes predominant and a large proportion of light is emitted inside a small angle ( −40° < θ < 40°), if the gap distance meets the constructive conditions. On the other hand, the vertical radiation vanishes and most of light is emitted outside the critical angle, as shown in From Fig. 3c. Compared with the absolute value of LEE, we take more interest in the LEE enhancement factor F. Consequently, the PEC was
chose as the bottom reflector for saving calculation time and the optimized gap distance d = 140 nm were used in the next calculation. 3.2. Comparison of the scattering effect of PhCs and amorphous PhCs on LEE for TFFC LEDs It has been proven the triangular lattice PhCs with the period p¼ 380 nm, 450 nm and radius r = 0.3p (filling factor f = 2πr 2/( 3 p2) ≈ 0.33) can effectively improve the LEE of TFFC LEDs [40]. For simplicity, the depth of PhCs/amorphous PhCs was fixed at 250 nm in the simulation of this section, it is deep enough to ensure that the interaction of these guided modes with PhCs is expected to be good [26]. Therefore, this two standard PhCs with the period of 380 nm and 450 nm were chose to calculate the LEEs. From Table 1, it can be seen that the LEE of TFFC LEDs is related to the dipole source state (location and polarization direction). And thus, the LEE of LEDs can be obtained by the Eq. (2). The LEE of 37.75% is obtained when the period d is 450 nm, which is 1.32 times that of planar TFFC LEDs. And an enhancement factor of 1.24 is also obtained when d ¼380 nm, as shown in Table 1. Next, let us to calculate the LEE of amorphous PhCs TFFC LEDs. The average distance of adjacent air holes was set to a = 380 nm and a = 450 nm , respectively. The ratio of radius r over a was fixed at r /a = 0.3 , so that filling factor of air holes did not change ( f ≈ 0.30). The LEEs of amorphous PhCs TFFC LEDs with period p = 450 nm and p = 380 nm can reach up to 40.57% and 39.30%,
Q. Yue et al. / Optics Communications 367 (2016) 72–79
75
Table 1 LEE (%) of different dipole source states for PhCs/amorphous PhCs TFFC LEDs with the depth D ¼ 250 nm. Dipole source state PhCs
p¼ 380 nm p¼ 450 nm a¼ 380 nm a¼ 450 nm
Amorphous PhCs
A(x)
A(y)
B(x)
B(y)
C(x)
C(y)
Average
38.49 45.22 42.79 43.49
38.18 44.99 40.55 42.19
33.72 29.57 38.65 38.58
34.25 37.66 38.45 37.30
35.05 34.57 37.72 42.05
32.12 34.30 37.64 40.01
35.30 37.75 39.30 40.57
Table 2 LEE (%) of different positions for amorphous PhCs TFFC LEDs with the depth D ¼250 nm. Dipole source position a ¼380 nm a ¼450 nm
x y x y
1
2
3
4
5
6
Average
38.79 37.76 41.14 39.42
41.18 39.86 42.92 41.47
40.34 38.87 40.36 37.30
41.42 41.54 40.37 38.50 40.35 37.30 39.57 44.29 39.01 39.53 44.16 37.44
39.70 40.36
Table 3 LEE (%) of different amorphous PhCs patterns for TFFC LEDs with the depth D ¼250 nm. Dipole source state
A(x)
A(y)
B(x)
B(y)
C(x)
C(y)
Average
Pattern A
44.37 45.72 39.11 42.05 38.18 38.39
42.80 43.11 39.69 42.80 38.35 37.20
39.94 39.25 36.85 44.57 42.66 45.27
38.64 38.01 36.16 44.03 43.01 46.73
35.85 37.53 41.89 36.72 34.97 35.31
35.08 36.91 42.24 36.55 34.63 35.56
39.45 40.09 39.32 41.12 38.63 39.71
Pattern B Pattern C
p¼ 380 nm p¼ 450 nm a¼ 380 nm a¼ 450 nm a¼ 380 nm a¼ 450 nm
respectively. Considering that the amorphous PhCs is non-periodic, more LEEs of different dipole sources for amorphous PhCs LEDs were calculated, as illustrated in Table 2. The result showed that the LEE calculated by all the dipole sources is approximately equal to that by the typical dipole sources that was shown in Fig. 2 (b). It means that these typical locations were enough to give an accurate LEE for amorphous PhCs. The LEEs of different amorphous PhCs pattern that was the snapshot at a different time were also calculated, as illustrated in Table 3. From Table 3, it can be seen that the LEE of different amorphous PhCs pattern is nearly equal. For saving time, the LEE of amorphous PhCs LEDs was calculated by using 6 dipole sources (they located in 3 typical locations with two different polarization directions, as illustrated in Fig. 2) in a amorphous PhCs pattern in the next simulation. The LEE of amorphous PhCs with period p = 450 nm is 1.07 times that triangular lattice PhCs, as illustrated in Table 1. And about 1.11 times enhancement was also observed in the amorphous PhCs with period p = 380 nm It means that amorphous PhCs can prove a much larger LEE than that of PhCs with the same pitch (p or a) and radius. This phenomenon can be explained by that the amorphous PhCs can provide omnidirectional diffraction. Triangular lattice PhCs offer more direction of diffraction into air than grating and square lattice PhCs, but the extraction of light is still non-omnidirectional, some directions are still not be extracted [26]. On the contrary, the diffraction points of amorphous PhCs become a circle of radius 2π /a (as shown in Fig. 1(c)), diffraction toward to air can occur for any azimuthal angle. The far-field pattern of LEDs in air can be obtained by the Nearto-Far-Field transformation (NTFF) method [41], its’ main procedure was describe as follows. First, the near-field electromagnetic field in frequency domain can be obtained by performing the Fourier Transform on the near-field electromagnetic field in time domain. And then, we can get the equivalent electromagnetic current in frequency domain using the equivalence principle.
Finally, the far-field electromagnetic field can be obtained by using free space Green function. As illustrated in Fig. 4a and c, the farfield pattern of PhCs TFFC LEDs has a six-fold symmetry characteristic of the triangular lattice, which is caused by the diffraction (scattering) of PhCs. The scattering on a periodic structure is usually called diffraction rather than scattering. On other hand, the far-field pattern of amorphous PhCs TFFC LEDs is lack of symmetry due to the amorphous PhCs is not periodic, and it is a typical scattering pattern, as shown in Fig. 4b and d. It revealed that the amorphous PhCs play a role of forward scattering. And thus, these guided modes confined in TFFC LEDs can escape from the LEDs with the help of amorphous PhCs. In the following, we study scattering transmission efficiency of PhCs/amorphous PhCs. Rigorous coupled waves analysis (RCWA) method was usually employed to analyze the scattering efficiency of plane waves in a PhCs structure [42,43]. However, RCWA simulation for amorphous PhCs is insoluble because the amorphous PhCs is non-periodic. And thus, 3D-FDTD method was used for analyzing the light scattering characteristics of amorphous PhCs [44]. Fig. 5 shows the schematic diagram of simulating the transmission intensity (transmittances) of various structures. The dipole is decomposed into polarized light (s- and p-polarized light) [45]. The PhCs/amorphous PhCs structure is illuminated by an incident wave (s- or p-polarized), under incident angle θ , that incident from the x-direction or y-direction. First, the FDTD result was successfully validated with the result of RCWA for PhCs structure with the period of p ¼450 nm, as show in Fig. 6a. Next, the average distance of amorphous PhCs was fixed at a = 450 nm , and the transmittances of PhCs/amorphous PhCs with light incident from the x-direction were obtained by using FDTD method. From Fig. 6b, it can be observed that the scattering efficiency of PhCs is larger than that of amorphous PhCs in x-direction, especially when the incident angle is in the region 25° ≤ θ ≤ 50°. On the contrary, the scattering efficiency of PhCs is smaller than that of amorphous PhCs in y-direction, when the incident angle is between 20° and 35°, as illustrated in Fig. 6c. It also can be found that the scattering efficiency of triangular lattice PhCs in x-direction is superior to that in y-direction. According to Bloch's theorem [26], the horizontal wave vector of incident light k// can be expressed as k// + G (G = mG0, G0 = 2π /p, m is the integer, and p is the period of PhCs ) in the PhCs region. When it meets the condition k// + G < k 0 ( k 0 is wave vector of incident light in air), the light propagates into air. And thus, the scattering efficiency of PhCs is closely related to G0 . The period of triangular lattice PhCs is equal to 3 p ( G = mG0, G0 = 2π / 3 p) in y-direction rather than p (G0 = 2π /p) in x-direction. This caused that the scattering efficiency of triangular lattice PhCs in x-direction is larger to that in y-direction. For amorphous PhCs structure, the average distance of adjacent air holes a is approximately equal p, but it has only shortrange order and is not rigorous PhCs. So, its’ scattering efficiency is poorer than that of PhCs in x-direction, while it is larger than that in y-direction. The light emitted from LEDs is randomly polarized, therefore the transmittance of the randomly polarized light can be computed by averaging the transmittances of s- and p-polarized light. From Fig. 6d, it can be seen that the transmittances of amorphous PhCs is larger than that of PhCs when the incident
76
Q. Yue et al. / Optics Communications 367 (2016) 72–79
Fig. 4. Polar projection of far-field intensities of TFFC LEDs, (a) PhCs with period p ¼ 380 nm, (b) amorphous PhCs with period p ¼380 nm, (c) PhCs with period p ¼ 450 nm and (d) amorphous PhCs with period p ¼450 nm. All the far-field patterns were normalized.
angle is in the region 20° ≤ θ ≤ 35°. And the most of light emitted from LEDs is concentrated in the region 0° ≤ θ ≤ 40° (Fig. 3b), this
results that the LEE of amorphous PhCs slightly larger than the PhCs.
Fig. 5. The schematic diagram of calculating the transmittances of TFFC LEDs with the structure of PhCs/amorphous PhCs, using 3D-FDTD method.
Q. Yue et al. / Optics Communications 367 (2016) 72–79
77
Fig. 6. The transmittances of TFFC LEDs with the structure of PhCs/amorphous PhCs as a function of the incident angle θ , and θ is the angle from normal direction. (a) and (b) the structure is illuminated by an incident light that incident from the x-direction, (c) the structure is illuminated by an incident light from the − y-direction, (d) the incident light is from − x-direction or − y-direction.
3.3. Influence of the average distance of adjacent air holes on LEE for TFFC LEDs To simplify, the filling factor f and depth D of amorphous PhCs are fixed, except for the average distance a of adjacent air holes. The LEE enhancement factors F of the different structures were plotted in Fig. 7(a). The average distance a varies from 300 nm to 750 nm. The LEE enhancement factors F increases dramatically
when a varies from 300 nm to 450 nm, and the first maxima of 1.42 was obtained at a¼ 450 nm. The LEE enhancement factors F keeps a higher value about 1.4, when a varies from 450 nm to 600 nm. The second maxima of 1.43 was obtained at a = 600 nm . However, the value of F keeps descending with a increasing, when a > 450 nm . The distribution of far-field patterns is also observed for various different average distance a of adjacent air holes: 450 nm and 600 nm. From the Fig. 7(b), we find that the far-field pattern has collimated distribution and the power was mainly concentrated in the region 0° ≤ θ ≤ 30°, when the average distance a is 450 nm. This structure was usually applied in a head-mounted display system [30]. When the average distance a ¼600 nm, the power was mainly distributed in the region 0° ≤ θ ≤ 60°, and it is a very promising candidate for general lighting. 3.4. Influence of the depth of amorphous PhCs on LEE for TFFC LEDs
Fig. 7. (a) The enhancement factors of LEE as a function of the average distance a of adjacent air holes with filling factor f = 0.3 (r /a = 0.3) and depth D = 250 nm . Inset: (a) and (b) the far-field pattern of amorphous PhCs TFFC LEDs with a¼ 450 nm and a¼ 600 nm, respectively. And all the far-field patterns were normalized.
The LEE enhancement factors F dependence on the depth of amorphous PhCs for the TFFC LEDs with the average distance of adjacent air holes a ¼450 nm and radius r = 135 nm was calculated and plotted, as shown in Fig. 8. From , it can be seen that LEEs of the LEDs increases with the amorphous PhCs etched, especially when the depth D is larger than 50 nm, the enhancement factors F increases dramatically from approximately 1.01 ( D = 50 nm ) to more than 1.32 ( D = 125 nm ). This can be explained by that the interaction between the amorphous PhCs and guided modes becomes stronger and stronger with the depth of amorphous PhCs increasing. The maxima of 1.45 was obtained at D = 275 nm , and the significantly higher intensity was observed in far-field
78
Q. Yue et al. / Optics Communications 367 (2016) 72–79
3.5. Optimized parameters of amorphous PhCs for TFFC LEDs As a multimode waveguide structure, the LEE of amorphous PhCs LEDs can be obtained by averaging all the guide modes extraction efficiency. The extraction efficiency of mono-mode is related closely to the average distance a and the depth D of amorphous PhCs, as discussed in above. The enhancement factor of LEE which is depicted as a function of the average distance a (from 300 nm to 800 nm) and the depth D (from 0 nm to 400 nm) was calculated as shown in Fig. 9. Fig. 9 shows that the LEE of TFFC LEDs was influenced by the average distance a and the depth D of amorphous PhCs simultaneously. And the maximum of enhancement factor F was obtained at a = 500 nm and D = 350 nm , it was enhanced by a factor of 1.49 as compared to planar TFFC LEDs. The LEE of PhCs reached the maximum when the period p¼500 nm and the depth D¼250 nm, which is 1.39 times that of planar TFFC LEDs. The result agrees with the result of Long [47]. And the LEE of amorphous PhCs with optimized parameters can reach 1.07 times of that of PhCs. Fig. 8. LEE enhancement factors F for TFFC LEDs with the amorphous PhCs structures as a function of the depth of amorphous PhCs D with the average distance of adjacent air holes a = 450 nm . Inset: Polar projection of far-field intensities of amorphous PhCs TFFC LEDs with D ¼50 nm (a) and D ¼275 nm (b).
4. Conclusion The LEE for GaN-based TFFC LEDs with the structure of amorphous PhCs was studied by using 3-D FDTD method. First, the influence of gap distance d between the active layer and reflector on LEE for TFFC LEDs was studied, and the optimized gap distance d = 140 nm was given in this paper. Furthermore, the light scattering characteristics of amorphous PhCs was analyzed by FDTD method, the numerical results revealed that the amorphous PhCs can provide omnidirectional scattering, and the transmittances of amorphous PhCs is superior to that of PhCs especially when the incident angle is in the region 20° ≤ θ ≤ 35°. Finally, the influence of amorphous PhCs feature size on the LEE was studied systematically. The optimized parameters amorphous PhCs was given as follows, average distance a~500 nm , and depth D~350 nm . We conclude that the amorphous PhCs structure can be a very promising candidate for high performance GaN-based TFFC LEDs.
Fig. 9. Relationship between the enhancement factor F and the average distance a and depth D of amorphous PhCs.
radiation pattern as illustrated in the inset of Fig. 8. As pointed on by David [27], when the amorphous PhCs depths reaches 275 nm ( ∼λ /nPhCs , nPhCs is the average refractive index of PhCs), the interaction between the amorphous PhCs and most guided modes becomes strongest (these modes are strongly evanescent in the etched region), so that the scattering becomes strongest and the LEE reaches maximum. However, this ascending trend stops at this depth and changes to fluctuation as the PhCs are etched deeper. This implies that one has no additional benefit from drilling the holes deeper and deeper. The fluctuation phenomenon can be explained by the Fabry-Perot resonator [46]. The resonant condition can be written as
k⊥ × 2dGaN + φPhCs ( θm ) + φPEC ( θm ) = 2mπ
(4)
where m is the mode number ( m = 0, 1, 2, … is an integer), φPhCs (θm ) and φPEC (θm ) are the phase changes reflected from the GaN-amorphous PhCs and GaN-PEC interfaces respectively, dGaN is the thickness of unetched GaN layer and k⊥ = nGaN (2π /λ ) cos θ is the perpendicular wave vector. As the PhCs are etched deeper, the thickness of unetched GaN layer decreases and it cause that the LEE of amorphous PhCs TFFC LEDs appears periodical fluctuation.
Acknowledgment This work was supported by the National Natural Science Foundation of China (61475084) and the Fundamental Research Funds of Shandong University (2014JC032).
References [1] S.-I. Chang, J.-B. Yoon, H. Kim, J.-J. Kim, B.-K. Lee, D.H. Shin, Microlens array diffuser for a light-emitting diode backlight system, Opt. Lett. 31 (2006) 3016–3018. [2] M.H. Crawford, LEDs for solid-state lighting: performance challenges and recent advances, IEEE J. Sel. Top. Quant. 15 (2009) 1028–1040. [3] G. Chen, M. Craven, A. Kim, A. Munkholm, S. Watanabe, M. Camras, W. Götz, F. Steranka, Performance of high‐power III‐nitride light emitting diodes, Phys. Status Solidi A 205 (2008) 1086–1092. [4] J. Wierer, D. Steigerwald, M. Krames, J. O’shea, M. Ludowise, G. Christenson, Y.C. Shen, C. Lowery, P. Martin, S. Subramanya, High-power AlGaInN flip-chip light-emitting diodes, Appl. Phys. Lett. 78 (2001) 3379–3381. [5] O. Shchekin, J. Epler, T. Trottier, T. Margalith, D. Steigerwald, M. Holcomb, P. Martin, M. Krames, High performance thin-film flip-chip InGaN–GaN lightemitting diodes, Appl. Phys. Lett. 89 (2006), pp.071109071103. [6] J.-Y. Kim, M.-K. Kwon, I.-K. Park, C.-Y. Cho, S.-J. Park, D.-M. Jeon, J.W. Kim, Y. C. Kim, Enhanced light extraction efficiency in flip-chip GaN light-emitting diodes with diffuse Ag reflector on nanotextured indium-tin oxide, Appl. Phys. Lett. 93 (2008) 021121. [7] H. Guo, X. Zhang, H. Chen, P. Zhang, H. Liu, H. Chang, W. Zhao, Q. Liao, Y. Cui, High performance GaN-based LEDs on patterned sapphire substrate with patterned composite SiO2/Al2O3 passivation layers and TiO2/Al2O3 DBR backside reflector, Opt. Express 21 (2013) 21456–21465.
Q. Yue et al. / Optics Communications 367 (2016) 72–79
[8] W. Wong, T. Sands, N. Cheung, M. Kneissl, D. Bour, P. Mei, L. Romano, N. Johnson, Fabrication of thin-film InGaN light-emitting diode membranes by laser lift-off, Appl. Phys. Lett. 75 (1999) 1360–1362. [9] C.-F. Chu, F.-I. Lai, J.-T. Chu, C.-C. Yu, C.-F. Lin, H.-C. Kuo, S. Wang, Study of GaN light-emitting diodes fabricated by laser lift-off technique, J. Appl. Phys. 95 (2004) 3916–3922. [10] J.J. Wierer, A. David, M.M. Megens, III-nitride photonic-crystal light-emitting diodes with high extraction efficiency, Nat. Photonics 3 (2009) 163–169. [11] M. Lachab, F. Asif, B. Zhang, I. Ahmad, A. Heidari, Q. Fareed, V. Adivarahan, A. Khan, Enhancement of light extraction efficiency in sub-300 nm nitride thin-film flip-chip light-emitting diodes, Solid-State Electron. 89 (2013) 156–160. [12] A.I. Zhmakin, Enhancement of light extraction from light emitting diodes, Phys. Rep. 498 (2011) 189–241. [13] P. Zhu, G. Liu, J. Zhang, N. Tansu, FDTD analysis on extraction efficiency of GaN light-emitting diodes with microsphere arrays, J. Disp. Tech. 9 (2013) 317–323. [14] P. Zhu, N. Tansu, “Resonant cavity effect optimization of III-nitride thin-film flip-chip light-emitting diodes with microsphere arrays, Appl. Opt. 54 (2015) 6305–6312. [15] S.-J. Chang, L. Chang, D. Kuo, T.K. Ko, S. Hon, S. Li, GaN-based LEDs with rough surface and selective KOH etching, J. Disp. Technol. 10 (2014) 27–32. [16] A. Mao, R.F. Karlicek, Surface patterning of nonscattering phosphors for light extraction, Opt. Lett. 38 (2013) 2796–2799. [17] X.-H. Li, R. Song, Y.-K. Ee, P. Kumnorkaew, J.F. Gilchrist, N. Tansu, Light extraction efficiency and radiation patterns of III-nitride light-emitting diodes with colloidal microlens arrays with various aspect ratios, IEEE Photonics J. 3 (2011) 489–499. [18] M.R. Krames, O.B. Shchekin, R. Mueller-Mach, G.O. Mueller, L. Zhou, G. Harbers, M.G. Craford, Status and future of high-power light-emitting diodes for solid-state lighting, J. Disp. Technol. 3 (2007) 160–175. [19] D.-H. Kim, C.-O. Cho, Y.-G. Roh, H. Jeon, Y.S. Park, J. Cho, J.S. Im, C. Sone, Y. Park, W.J. Choi, Enhanced light extraction from GaN-based light-emitting diodes with holographically generated two-dimensional photonic crystal patterns, Appl. Phys. Lett. 87 (2005) 203508. [20] X. Chen, F. Kong, K. Li, Q. Ding, M. Zhang, W. Li, Study of light extraction efficiency of flip-chip GaN-based LEDs with different periodic arrays, Opt. Commun. 314 (2014) 90–96. [21] P.-H. Fu, G.-J. Lin, H.-P. Wang, K.-Y. Lai, J.-H. He, Enhanced light extraction of light-emitting diodes via nano-honeycomb photonic crystals, Nano Energy 8 (2014) 78–83. [22] S.K. Patra, S. Adhikari, S. Pal, Design and analysis of “Chess Board” like photonic crystal structure for improved light extraction in GaN/InGaN LEDs, J. Disp. Technol. 9 (2013) 339–345. [23] J.-G. Kim, C.-H. Hsieh, H.J. Choi, J. Gardener, B. Singh, A. Knapitsch, P. Lecoq, G. Barbastathis, Conical photonic crystals for enhancing light extraction efficiency from high refractive index materials, Opt. Express 23 (2015) 22730–22739. [24] M. Fujita, S. Takahashi, Y. Tanaka, T. Asano, S. Noda, Simultaneous inhibition and redistribution of spontaneous light emission in photonic crystals, Science 308 (2005) 1296–1298. [25] A. David, H. Benisty, C. Weisbuch, Photonic crystal light-emitting sources, Rep. Prog. Phys. 75 (2012) 126501. [26] A. David, H. Benisty, C. Weisbuch, Optimization of light-diffracting photoniccrystals for high extraction efficiency LEDs, J. Disp. Technol. 3 (2007) 133–148. [27] A. David, T. Fujii, R. Sharma, K. McGroddy, S. Nakamura, S.P. DenBaars, E.L. Hu, C. Weisbuch, H. Benisty, Photonic-crystal GaN light-emitting diodes with tailored guided modes distribution, Appl. Phys. Lett. 88 (2006) 061124.
79
[28] K.S. Yee, Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media, IEEE Trans. Antennas Propag. 14 (1966) 302–307. [29] Z. Zang, K. Mukai, P. Navaretti, M. Duelk, et al., Thermal resistance reduction in high power superluminescent diodes by using active multi-mode interferometer, Appl. Phys. Lett. 100 (2012) 031108. [30] Z. Zang, T. Minato, P. Navaretti, Y. Hinokuma, et al., High-power (4 110 mW) superluminescent diodes by using active multimode interferometer, IEEE Photonics Technol. Lett. 10 (2010) 721–723. [31] K.-D. Chang, C.-Y. Li, J.-W. Pan, K.-Y. Cheng, A hybrid simulated method for analyzing the optical efficiency of a head-mounted display with a quasi-crystal OLED panel, Opt. Express 22 (2014) A567–A576. [32] L. Jia, I. Bita, E.L. Thomas, Photonic density of states of two-dimensional quasicrystalline photonic structures, Phys. Rev. A 84 (2011) 023831. [33] C.-E. Lee, C.-F. Lai, Y.-C. Lee, H.-C. Kuo, T.-C. Lu, S.-C. Wang, Nitride-based thinfilm light-emitting diodes with photonic quasi-crystal surface, IEEE Photonics Technol. Lett. 21 (2009) 331–333. [34] M. Rattier, H. Benisty, E. Schwoob, C. Weisbuch, T. Krauss, C. Smith, R. Houdre, U. Oesterle, Omnidirectional and compact guided light extraction from Archimedean photonic lattices, Appl. Phys. Lett. 83 (2003) 1283–1285. [35] D. Frenkel, B. Smit, Understanding Molecular Simulation: From Algorithms to Applications, Academic Press, United States, 2001. [36] M. Rechtsman, A. Szameit, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, M. Segev, Amorphous photonic lattices: band gaps, effective mass, and suppressed transport, Phys. Rev. Lett. 106 (2011) 193904. [37] M. Bass, E.W. Van Stryland, D.R. Williams, W.L. Wolfe, Handbook of Optics, McGraw-Hill, United States, 2001. [38] J.-W. Pan, P.-J. Tsai, K.-D. Chang, Y.-Y. Chang, Light extraction efficiency analysis of GaN-based light-emitting diodes with nanopatterned sapphire substrates, Appl. Opt. 52 (2013) 1358–1367. [39] S.-K. Kim, H.-G. Park, Coherent vertical beaming using Bragg mirrors for highefficiency GaN light-emitting diodes, Opt. Express 21 (2013) 14566–14572. [40] C.-F. Lai, C.-H. Chao, H.-C. Kuo, P. Yu, H.-H. Yen, W.-Y. Yeh, GaN thickness effect on directional light enhancement from GaN-based film-transferred photonic crystal light-emitting diodes, Jpn. J. Appl. Phys. 49 (2010) 04DG09. [41] A. Taflove, S.C. Hagness, Computational Electrodynamics, Artech House, United States, 2005. [42] Y.-C. Lee, C.-C. Chang, Y.-Y. Chou, Experimental and simulation studies of antireflection sub-micron conical structures on a GaAs substrate, Opt. Express 21 (2013) A36–A41. [43] C. Xiong, B. Zhang, X. Kang, T. Dai, G. Zhang, Diffracted transmission effects of GaN and polymer two-dimensional square-lattice photonic crystals, Opt. Express 17 (2009) 23684–23689. [44] M.K. Callens, H. Marsman, L. Penninck, P. Peeters, et al., RCWA and FDTD modeling of light emission from internally structured OLEDs, Opt. Express 22 (2014) A589–A600. [45] D. Delbeke, P. Bienstman, R. Bockstaele, R. Baets, Rigorous electromagnetic analysis of dipole emission in periodically corrugated layers: the grating-assisted resonant-cavity light-emitting diode, J. Opt. Soc. Am. A 19 (2002) 871–880. [46] W.-L. Yeh, C.-M. Fang, Y.-P. Chiou, Enhancing LED light extraction by optimizing cavity and waveguide modes in grating structures, J. Disp. Technol. 9 (2013) 359–364. [47] D.H. Long, I.-K. Hwang, S.-W. Ryu, Design optimization of photonic crystal structure for improved light extraction of GaN LED, IEEE J. Sel. Top. Quant. Electron. 15 (2009) 1257–1263.
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Analysis on the effect of amorphous photonic crystals on light extraction efficiency enhancement for GaN-based thin-film-flip-chip light-emitting diodes Qingyang Yue a, Kang Li b,n, Fanmin Kong b, Jia Zhao b, Meng Liu b a b
School of Physics and Electronics, Shandong Normal University, Jinan 250014, China School of Information Science and Engineering, Shandong University, Jinan 250100, China
art ic l e i nf o
a b s t r a c t
Article history: Received 12 October 2015 Received in revised form 10 December 2015 Accepted 31 December 2015 Available online 4 February 2016
This work showed the liquid-like amorphous photonic crystals (PhCs) can effectively enhance the light extraction efficiency of GaN-based thin-film-flip-chip light-emitting diodes (TFFC LEDs) by the light scattering effect. The finite-difference time-domain (FDTD) method was employed to analyze the light scattering characteristics, numerical studies revealed that the amorphous PhCs can provide omnidirectional scattering, and the transmittances of amorphous PhCs is superior to that of triangular lattice PhCs when the incident angle is in the region 20° ≤ θ ≤ 35°. The influence of p-GaN layer thickness and the amorphous PhCs feature size on the light extraction efficiency was also studied by 3D-FDTD method systematically. For the proposed amorphous PhCs structure in n-GaN layer, the light extraction efficiency is enhanced by a factor of 1.49 as compared to conventional TFFC LEDs, which shows 1.07 times enhancement in comparison to that of triangular lattice PhCs. & 2016 Elsevier B.V. All rights reserved.
Keywords: GaN-based LEDs Amorphous photonic crystals Light extraction efficiency Forward scattering Finite-difference time-domain (FDTD) method
1. Introduction Recently, high performance GaN-based light-emitting diodes (LEDs) have attracted much attention for use in back lighting in large liquid-crystal displays [1] and general lighting [2]. GaN-based LEDs are promising candidates for the application of next-generation general lighting source because their wall-plug efficiency is comparable to conventional fluorescent lamps [3]. In general, the conventional GaN-based LEDs are grown on a sapphire substrate and emit photons from the p-side indium-tin-oxide (ITO) contact. The top-emitting configuration results that a considerable fraction of light is absorbed by the metal contacts, and it still has severe current-crowding and heat-conducting problems at high current injection due to poor thermal conductivity of sapphire. A flip-chip LEDs (FC LEDs) [4,5] configuration that can simultaneously satisfy thermal management and light extraction has been developed. A typical FC LEDs chip is commonly bonded to a submount with high thermal conductivity, and downward-propagating light can be reflected upward by placing a reflector (e.g. silver) on the p-GaN layer [6,7]. Recently, improvements in the nitride light emitters efficiency can also be achieved by thin-film-flip-chip (TFFC) structure, it can be fabricated with the help of FC and laser n
Corresponding author. E-mail address: [email protected] (K. Li).
http://dx.doi.org/10.1016/j.optcom.2015.12.072 0030-4018/& 2016 Elsevier B.V. All rights reserved.
lift-off (LLO) technique [8,9]. The thickness of TFFC LEDs is ∼1 μm [10,11], which can also reduce the number of waveguide modes. This method has been proven to be effective in boosting the optical power. For illumination applications, GaN-based TFFC LEDs is still necessary to further improve output efficiency. It has been found that efficiency of GaN-based LEDs is mainly limited by external quantum efficiency also known as light extraction efficiency (LEE) [12,13]. The limitation occurred in LEE of GaN-based LEDs is attributed to the large refractive index difference between GaN and air/epoxy resin [14]. Large portion of generated photons in the active region are trapped inside the LEDs device by total internal reflection at the interface. As a result, currently the major challenge associated with light extraction is enhancing the LEE of LEDs. Several approaches have been proposed to extract the light confined in waveguide modes, including surface roughness [15–17], photonic crystals (PhCs) and so on. The rough-surface thin-film LEDs can offer high LEE ( 80%) for an encapsulated device [18], but it offers no control over the direction of the emitted light, resulting in a Lambertian far field emission. Using 2D PhCs is an effective approach for improving the LEE of LEDs [19–23]. It has been proven that the bandgap of PhCs can inhibit the light emission into guided modes and hence increasing the LEE of LEDs in which the PhCs penetrates through the entire device [24]. However, this approach is difficult to achieve a high performance working device when the PhCs penetrates through the active layer
Q. Yue et al. / Optics Communications 367 (2016) 72–79
[25]. Most scientific efforts have therefore focused on the diffractive properties of PhCs [25–27]. In this scheme, PhCs can be used to diffract guided modes confined in LEDs into air. In general, the PhCs are located on the surfaces of the LEDs and do not penetrates through the active layer, and current injection is less problematic. Compared with triangular lattice PhCs, the amorphous PhCs can provide omnidirectional scattering. In this work, the influence of the amorphous PhCs on the LEE of TTFC LEDs was first studied by using the three-dimensional finite difference time domain (3D-FDTD) method [28], the FDTD method has also been used to analyze the efficiency of superluminescent diodes [29,30]. The light scattering characteristics of liquid-like amorphous PhCs were analyzed by FDTD method, numerical studies revealed that the liquid-like amorphous PhCs can provide larger LEE than triangular lattice PhCs. The influence of p-GaN layer thickness and the amorphous PhCs feature size on LEE was also systematically studied by 3D-FDTD method.
2. Numerical model and numerical method A schematic diagram of the TFFC LEDs structure is shown in the Fig. 1(a). The typical TFFC LEDs is multi-layer planer structure with a silver reflector, a P-GaN layer, a multiple quantum wells (MQWs) layer and an N-GaN layer. The liquid-like amorphous PhCs is located on the top surface of the N-GaN layer. Our liquid-like amorphous PhCs are different from that quasi PhCs [31–33] and Archimedean PhCs [34], which is lack of symmetry. We distributed the cylindrical air holes as a snapshot of atoms in a liquid by the use of the Metropolis Monte Carlo method [35] with a repulsive interatomic force. Fig. 1(b) shows the refractive index distribution of the amorphous PhCs, which has only short-range order but no long-range order. And the average distance of adjacent air holes is equal to a. As illustrated in Fig. 1(c), the modulus of the Fourier transform of amorphous PhCs shows an isotropic ring pattern, which has no Bragg peaks [36]. Next, the 3-D FDTD method was employed to calculate the LEE of the TFFC LEDs. The thickness of total GaN layer included the P-GaN layer, MQWs layer and N-GaN layer was set to 700 nm, the thickness of silver layer was set as 200 nm, and the lateral dimension of the computational domain was set as 8 μm in the simulation. The complex refractive index of GaN material was assumed to 2.5 + 0.0013i at the wavelength of 460 nm [37]. In addition, the homogeneous mesh was used during the simulation, and the grid was set to Δx = Δy = Δz = 10 nm . Furthermore, a perfectly matched layer (PML) enclosing the entire simulation domain was used to absorb outgoing waves. It has been proven that the electron-hole-recombination can classically be represented by a dipole. The conventional InGaN/
73
GaN quantum well grows along the c-axis (z direction), and it is polar and can be modeled by a uniform distribution x and y polarization orientation dipoles placed in the MQWs plane that is perpendicular to the growing axis (z direction). In the z direction, the dipole source is located in the middle of the MQWs layer. In the x–y plane, the typical locations of the dipole sources in the PhCs or amorphous PhCs TFFC LEDs are shown in Fig. 2. And thus, there are 6 kinds of dipole source states in the 3D-FDTD simulation, the result of Pan [38] proved that that the accurate averaged efficiency can be obtained by these typical dipole sources. Nevertheless, the FDTD intrinsically is a coherent simulation method. In order to avoid generating an unphysical coherent effect between different dipole sources, only one dipole source was located in the MQWs plane once in the simulation. And the center wavelength of the dipole source in the FDTD simulation is 460 nm. In the FDTD simulation, the LEE of one dipole can be calculated from the power flux extracted from the LEDs with respect to the overall emitted power from the dipole source
ηextr = Pout /Psource
(1)
where Pout is the power flow integrated over a plane just above the LEDs structure and Psource the integrated power flux through the closed surface enclosing the source. And thus, the LEE of LEDs can be obtained by N
η=
n ∑n = 1 ηextr
N
(2)
where N is the number of dipole source states. Finally, the LEE enhancement factor F is defined as follows:
F = η/η0
(3)
where η is the vertical LEE of the LEDs while η0 is the LEE of planar TFFC LEDs.
3. Numerical analysis and discussion 3.1. Influence of gap distance d between the active layer and reflector on LEE for planar TFFC LEDs Note that in the TFFC LEDs, the MQWs layer is placed close enough to the silver reflector ( 100 nm). The light emitted from QWs will interfere with the reflected waves, and such interference can modulate the radiation's profile. It is important for high performance TFFC LEDs to choose an appropriate gap distance between the active layer and reflector. To systematically study the interference effect, we first calculated the LEE of the TFFC LEDs structure while varying the positions of the dipole sources from
Fig. 1. (a) The schematic diagram of TFFC LEDs structure, The thickness of total GaN layer included the P-GaN layer, MQWs layer and N-GaN layer is 700 nm, the thickness of silver layer is set as 200 nm, and the center wavelength of the dipole source in the FDTD simulation is 460 nm. The detection plane is set as 460 nm away from the emission surface of n-GaN. (b) Schematic view of the transverse cross section in the amorphous PhCs region. (c) Spatial Fourier spectrum of the structure (b).
74
Q. Yue et al. / Optics Communications 367 (2016) 72–79
Fig. 2. Schematic diagram of three kinds of typical dipole sources located in the x–y plane, and the plane is located in the middle of the MQWs layer in the z direction. (a) triangular lattice PhCs, (b) amorphous PhCs. Comment: A, B and C represents the position that is under one hole, between the two holes, and in the center of three holes, respectively.
Fig. 3. (a) LEE for the conventional TFFC LEDs at wavelength 460 nm with flat surface as a function of the gap distance d between the active layer and reflector (silver mirror or perfect electric conductor (PEC)). (b) and (c) Electric filed intensity profiles of GaN-based TFFC LEDs with PEC when d is 140 nm (b), 90 nm (c), respectively. The red dotted lines indicate the critical angle ( 23.7°) of the total internal reflection between GaN and air. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
the silver layer by carrying out the FDTD simulation. The total thickness of the GaN layer was set to 700 nm. The LEE exhibited an oscillatory behavior as a function of the gap distance d, as shown in Fig. 3a. The peaks of LEE were found at gap distances d = 30 nm and 120 nm respectively, and the periodicity of d is 90 nm, it agrees with the result of kim et al. [39]. According to the coherent theory, constructive interference coupling happens when d = mλ /2n (m is integer, n is the refractive index of GaN material), which indicates the periodicity of d is λ /2n = 90 nm . Furthermore, the lowest order requirement of constructive interference condition at d = 0 nm . In the case of a silver reflector, the phase change for reflected light is given by π + 2α , where α = 2π /l and l is the skin depth of silver. And thus, the first maxima was obtained at d ≈ 40 nm . Next, we investigated the interference effect between the dipole and perfect electric conductor (PEC). The LEE of conventional TFFC LEDs with PEC shows similar trend to that of silver, as illustrated in Fig. 3a. From Fig. 3a, the second maxima of 28.51% were obtained at d = 140 nm , which is much more than that of silver (23.67%). This can be explained by that the PEC is lack of the absorption and the skin depth. From Fig. 3b, it is clearly observed that the vertical radiation becomes predominant and a large proportion of light is emitted inside a small angle ( −40° < θ < 40°), if the gap distance meets the constructive conditions. On the other hand, the vertical radiation vanishes and most of light is emitted outside the critical angle, as shown in From Fig. 3c. Compared with the absolute value of LEE, we take more interest in the LEE enhancement factor F. Consequently, the PEC was
chose as the bottom reflector for saving calculation time and the optimized gap distance d = 140 nm were used in the next calculation. 3.2. Comparison of the scattering effect of PhCs and amorphous PhCs on LEE for TFFC LEDs It has been proven the triangular lattice PhCs with the period p¼ 380 nm, 450 nm and radius r = 0.3p (filling factor f = 2πr 2/( 3 p2) ≈ 0.33) can effectively improve the LEE of TFFC LEDs [40]. For simplicity, the depth of PhCs/amorphous PhCs was fixed at 250 nm in the simulation of this section, it is deep enough to ensure that the interaction of these guided modes with PhCs is expected to be good [26]. Therefore, this two standard PhCs with the period of 380 nm and 450 nm were chose to calculate the LEEs. From Table 1, it can be seen that the LEE of TFFC LEDs is related to the dipole source state (location and polarization direction). And thus, the LEE of LEDs can be obtained by the Eq. (2). The LEE of 37.75% is obtained when the period d is 450 nm, which is 1.32 times that of planar TFFC LEDs. And an enhancement factor of 1.24 is also obtained when d ¼380 nm, as shown in Table 1. Next, let us to calculate the LEE of amorphous PhCs TFFC LEDs. The average distance of adjacent air holes was set to a = 380 nm and a = 450 nm , respectively. The ratio of radius r over a was fixed at r /a = 0.3 , so that filling factor of air holes did not change ( f ≈ 0.30). The LEEs of amorphous PhCs TFFC LEDs with period p = 450 nm and p = 380 nm can reach up to 40.57% and 39.30%,
Q. Yue et al. / Optics Communications 367 (2016) 72–79
75
Table 1 LEE (%) of different dipole source states for PhCs/amorphous PhCs TFFC LEDs with the depth D ¼ 250 nm. Dipole source state PhCs
p¼ 380 nm p¼ 450 nm a¼ 380 nm a¼ 450 nm
Amorphous PhCs
A(x)
A(y)
B(x)
B(y)
C(x)
C(y)
Average
38.49 45.22 42.79 43.49
38.18 44.99 40.55 42.19
33.72 29.57 38.65 38.58
34.25 37.66 38.45 37.30
35.05 34.57 37.72 42.05
32.12 34.30 37.64 40.01
35.30 37.75 39.30 40.57
Table 2 LEE (%) of different positions for amorphous PhCs TFFC LEDs with the depth D ¼250 nm. Dipole source position a ¼380 nm a ¼450 nm
x y x y
1
2
3
4
5
6
Average
38.79 37.76 41.14 39.42
41.18 39.86 42.92 41.47
40.34 38.87 40.36 37.30
41.42 41.54 40.37 38.50 40.35 37.30 39.57 44.29 39.01 39.53 44.16 37.44
39.70 40.36
Table 3 LEE (%) of different amorphous PhCs patterns for TFFC LEDs with the depth D ¼250 nm. Dipole source state
A(x)
A(y)
B(x)
B(y)
C(x)
C(y)
Average
Pattern A
44.37 45.72 39.11 42.05 38.18 38.39
42.80 43.11 39.69 42.80 38.35 37.20
39.94 39.25 36.85 44.57 42.66 45.27
38.64 38.01 36.16 44.03 43.01 46.73
35.85 37.53 41.89 36.72 34.97 35.31
35.08 36.91 42.24 36.55 34.63 35.56
39.45 40.09 39.32 41.12 38.63 39.71
Pattern B Pattern C
p¼ 380 nm p¼ 450 nm a¼ 380 nm a¼ 450 nm a¼ 380 nm a¼ 450 nm
respectively. Considering that the amorphous PhCs is non-periodic, more LEEs of different dipole sources for amorphous PhCs LEDs were calculated, as illustrated in Table 2. The result showed that the LEE calculated by all the dipole sources is approximately equal to that by the typical dipole sources that was shown in Fig. 2 (b). It means that these typical locations were enough to give an accurate LEE for amorphous PhCs. The LEEs of different amorphous PhCs pattern that was the snapshot at a different time were also calculated, as illustrated in Table 3. From Table 3, it can be seen that the LEE of different amorphous PhCs pattern is nearly equal. For saving time, the LEE of amorphous PhCs LEDs was calculated by using 6 dipole sources (they located in 3 typical locations with two different polarization directions, as illustrated in Fig. 2) in a amorphous PhCs pattern in the next simulation. The LEE of amorphous PhCs with period p = 450 nm is 1.07 times that triangular lattice PhCs, as illustrated in Table 1. And about 1.11 times enhancement was also observed in the amorphous PhCs with period p = 380 nm It means that amorphous PhCs can prove a much larger LEE than that of PhCs with the same pitch (p or a) and radius. This phenomenon can be explained by that the amorphous PhCs can provide omnidirectional diffraction. Triangular lattice PhCs offer more direction of diffraction into air than grating and square lattice PhCs, but the extraction of light is still non-omnidirectional, some directions are still not be extracted [26]. On the contrary, the diffraction points of amorphous PhCs become a circle of radius 2π /a (as shown in Fig. 1(c)), diffraction toward to air can occur for any azimuthal angle. The far-field pattern of LEDs in air can be obtained by the Nearto-Far-Field transformation (NTFF) method [41], its’ main procedure was describe as follows. First, the near-field electromagnetic field in frequency domain can be obtained by performing the Fourier Transform on the near-field electromagnetic field in time domain. And then, we can get the equivalent electromagnetic current in frequency domain using the equivalence principle.
Finally, the far-field electromagnetic field can be obtained by using free space Green function. As illustrated in Fig. 4a and c, the farfield pattern of PhCs TFFC LEDs has a six-fold symmetry characteristic of the triangular lattice, which is caused by the diffraction (scattering) of PhCs. The scattering on a periodic structure is usually called diffraction rather than scattering. On other hand, the far-field pattern of amorphous PhCs TFFC LEDs is lack of symmetry due to the amorphous PhCs is not periodic, and it is a typical scattering pattern, as shown in Fig. 4b and d. It revealed that the amorphous PhCs play a role of forward scattering. And thus, these guided modes confined in TFFC LEDs can escape from the LEDs with the help of amorphous PhCs. In the following, we study scattering transmission efficiency of PhCs/amorphous PhCs. Rigorous coupled waves analysis (RCWA) method was usually employed to analyze the scattering efficiency of plane waves in a PhCs structure [42,43]. However, RCWA simulation for amorphous PhCs is insoluble because the amorphous PhCs is non-periodic. And thus, 3D-FDTD method was used for analyzing the light scattering characteristics of amorphous PhCs [44]. Fig. 5 shows the schematic diagram of simulating the transmission intensity (transmittances) of various structures. The dipole is decomposed into polarized light (s- and p-polarized light) [45]. The PhCs/amorphous PhCs structure is illuminated by an incident wave (s- or p-polarized), under incident angle θ , that incident from the x-direction or y-direction. First, the FDTD result was successfully validated with the result of RCWA for PhCs structure with the period of p ¼450 nm, as show in Fig. 6a. Next, the average distance of amorphous PhCs was fixed at a = 450 nm , and the transmittances of PhCs/amorphous PhCs with light incident from the x-direction were obtained by using FDTD method. From Fig. 6b, it can be observed that the scattering efficiency of PhCs is larger than that of amorphous PhCs in x-direction, especially when the incident angle is in the region 25° ≤ θ ≤ 50°. On the contrary, the scattering efficiency of PhCs is smaller than that of amorphous PhCs in y-direction, when the incident angle is between 20° and 35°, as illustrated in Fig. 6c. It also can be found that the scattering efficiency of triangular lattice PhCs in x-direction is superior to that in y-direction. According to Bloch's theorem [26], the horizontal wave vector of incident light k// can be expressed as k// + G (G = mG0, G0 = 2π /p, m is the integer, and p is the period of PhCs ) in the PhCs region. When it meets the condition k// + G < k 0 ( k 0 is wave vector of incident light in air), the light propagates into air. And thus, the scattering efficiency of PhCs is closely related to G0 . The period of triangular lattice PhCs is equal to 3 p ( G = mG0, G0 = 2π / 3 p) in y-direction rather than p (G0 = 2π /p) in x-direction. This caused that the scattering efficiency of triangular lattice PhCs in x-direction is larger to that in y-direction. For amorphous PhCs structure, the average distance of adjacent air holes a is approximately equal p, but it has only shortrange order and is not rigorous PhCs. So, its’ scattering efficiency is poorer than that of PhCs in x-direction, while it is larger than that in y-direction. The light emitted from LEDs is randomly polarized, therefore the transmittance of the randomly polarized light can be computed by averaging the transmittances of s- and p-polarized light. From Fig. 6d, it can be seen that the transmittances of amorphous PhCs is larger than that of PhCs when the incident
76
Q. Yue et al. / Optics Communications 367 (2016) 72–79
Fig. 4. Polar projection of far-field intensities of TFFC LEDs, (a) PhCs with period p ¼ 380 nm, (b) amorphous PhCs with period p ¼380 nm, (c) PhCs with period p ¼ 450 nm and (d) amorphous PhCs with period p ¼450 nm. All the far-field patterns were normalized.
angle is in the region 20° ≤ θ ≤ 35°. And the most of light emitted from LEDs is concentrated in the region 0° ≤ θ ≤ 40° (Fig. 3b), this
results that the LEE of amorphous PhCs slightly larger than the PhCs.
Fig. 5. The schematic diagram of calculating the transmittances of TFFC LEDs with the structure of PhCs/amorphous PhCs, using 3D-FDTD method.
Q. Yue et al. / Optics Communications 367 (2016) 72–79
77
Fig. 6. The transmittances of TFFC LEDs with the structure of PhCs/amorphous PhCs as a function of the incident angle θ , and θ is the angle from normal direction. (a) and (b) the structure is illuminated by an incident light that incident from the x-direction, (c) the structure is illuminated by an incident light from the − y-direction, (d) the incident light is from − x-direction or − y-direction.
3.3. Influence of the average distance of adjacent air holes on LEE for TFFC LEDs To simplify, the filling factor f and depth D of amorphous PhCs are fixed, except for the average distance a of adjacent air holes. The LEE enhancement factors F of the different structures were plotted in Fig. 7(a). The average distance a varies from 300 nm to 750 nm. The LEE enhancement factors F increases dramatically
when a varies from 300 nm to 450 nm, and the first maxima of 1.42 was obtained at a¼ 450 nm. The LEE enhancement factors F keeps a higher value about 1.4, when a varies from 450 nm to 600 nm. The second maxima of 1.43 was obtained at a = 600 nm . However, the value of F keeps descending with a increasing, when a > 450 nm . The distribution of far-field patterns is also observed for various different average distance a of adjacent air holes: 450 nm and 600 nm. From the Fig. 7(b), we find that the far-field pattern has collimated distribution and the power was mainly concentrated in the region 0° ≤ θ ≤ 30°, when the average distance a is 450 nm. This structure was usually applied in a head-mounted display system [30]. When the average distance a ¼600 nm, the power was mainly distributed in the region 0° ≤ θ ≤ 60°, and it is a very promising candidate for general lighting. 3.4. Influence of the depth of amorphous PhCs on LEE for TFFC LEDs
Fig. 7. (a) The enhancement factors of LEE as a function of the average distance a of adjacent air holes with filling factor f = 0.3 (r /a = 0.3) and depth D = 250 nm . Inset: (a) and (b) the far-field pattern of amorphous PhCs TFFC LEDs with a¼ 450 nm and a¼ 600 nm, respectively. And all the far-field patterns were normalized.
The LEE enhancement factors F dependence on the depth of amorphous PhCs for the TFFC LEDs with the average distance of adjacent air holes a ¼450 nm and radius r = 135 nm was calculated and plotted, as shown in Fig. 8. From , it can be seen that LEEs of the LEDs increases with the amorphous PhCs etched, especially when the depth D is larger than 50 nm, the enhancement factors F increases dramatically from approximately 1.01 ( D = 50 nm ) to more than 1.32 ( D = 125 nm ). This can be explained by that the interaction between the amorphous PhCs and guided modes becomes stronger and stronger with the depth of amorphous PhCs increasing. The maxima of 1.45 was obtained at D = 275 nm , and the significantly higher intensity was observed in far-field
78
Q. Yue et al. / Optics Communications 367 (2016) 72–79
3.5. Optimized parameters of amorphous PhCs for TFFC LEDs As a multimode waveguide structure, the LEE of amorphous PhCs LEDs can be obtained by averaging all the guide modes extraction efficiency. The extraction efficiency of mono-mode is related closely to the average distance a and the depth D of amorphous PhCs, as discussed in above. The enhancement factor of LEE which is depicted as a function of the average distance a (from 300 nm to 800 nm) and the depth D (from 0 nm to 400 nm) was calculated as shown in Fig. 9. Fig. 9 shows that the LEE of TFFC LEDs was influenced by the average distance a and the depth D of amorphous PhCs simultaneously. And the maximum of enhancement factor F was obtained at a = 500 nm and D = 350 nm , it was enhanced by a factor of 1.49 as compared to planar TFFC LEDs. The LEE of PhCs reached the maximum when the period p¼500 nm and the depth D¼250 nm, which is 1.39 times that of planar TFFC LEDs. The result agrees with the result of Long [47]. And the LEE of amorphous PhCs with optimized parameters can reach 1.07 times of that of PhCs. Fig. 8. LEE enhancement factors F for TFFC LEDs with the amorphous PhCs structures as a function of the depth of amorphous PhCs D with the average distance of adjacent air holes a = 450 nm . Inset: Polar projection of far-field intensities of amorphous PhCs TFFC LEDs with D ¼50 nm (a) and D ¼275 nm (b).
4. Conclusion The LEE for GaN-based TFFC LEDs with the structure of amorphous PhCs was studied by using 3-D FDTD method. First, the influence of gap distance d between the active layer and reflector on LEE for TFFC LEDs was studied, and the optimized gap distance d = 140 nm was given in this paper. Furthermore, the light scattering characteristics of amorphous PhCs was analyzed by FDTD method, the numerical results revealed that the amorphous PhCs can provide omnidirectional scattering, and the transmittances of amorphous PhCs is superior to that of PhCs especially when the incident angle is in the region 20° ≤ θ ≤ 35°. Finally, the influence of amorphous PhCs feature size on the LEE was studied systematically. The optimized parameters amorphous PhCs was given as follows, average distance a~500 nm , and depth D~350 nm . We conclude that the amorphous PhCs structure can be a very promising candidate for high performance GaN-based TFFC LEDs.
Fig. 9. Relationship between the enhancement factor F and the average distance a and depth D of amorphous PhCs.
radiation pattern as illustrated in the inset of Fig. 8. As pointed on by David [27], when the amorphous PhCs depths reaches 275 nm ( ∼λ /nPhCs , nPhCs is the average refractive index of PhCs), the interaction between the amorphous PhCs and most guided modes becomes strongest (these modes are strongly evanescent in the etched region), so that the scattering becomes strongest and the LEE reaches maximum. However, this ascending trend stops at this depth and changes to fluctuation as the PhCs are etched deeper. This implies that one has no additional benefit from drilling the holes deeper and deeper. The fluctuation phenomenon can be explained by the Fabry-Perot resonator [46]. The resonant condition can be written as
k⊥ × 2dGaN + φPhCs ( θm ) + φPEC ( θm ) = 2mπ
(4)
where m is the mode number ( m = 0, 1, 2, … is an integer), φPhCs (θm ) and φPEC (θm ) are the phase changes reflected from the GaN-amorphous PhCs and GaN-PEC interfaces respectively, dGaN is the thickness of unetched GaN layer and k⊥ = nGaN (2π /λ ) cos θ is the perpendicular wave vector. As the PhCs are etched deeper, the thickness of unetched GaN layer decreases and it cause that the LEE of amorphous PhCs TFFC LEDs appears periodical fluctuation.
Acknowledgment This work was supported by the National Natural Science Foundation of China (61475084) and the Fundamental Research Funds of Shandong University (2014JC032).
References [1] S.-I. Chang, J.-B. Yoon, H. Kim, J.-J. Kim, B.-K. Lee, D.H. Shin, Microlens array diffuser for a light-emitting diode backlight system, Opt. Lett. 31 (2006) 3016–3018. [2] M.H. Crawford, LEDs for solid-state lighting: performance challenges and recent advances, IEEE J. Sel. Top. Quant. 15 (2009) 1028–1040. [3] G. Chen, M. Craven, A. Kim, A. Munkholm, S. Watanabe, M. Camras, W. Götz, F. Steranka, Performance of high‐power III‐nitride light emitting diodes, Phys. Status Solidi A 205 (2008) 1086–1092. [4] J. Wierer, D. Steigerwald, M. Krames, J. O’shea, M. Ludowise, G. Christenson, Y.C. Shen, C. Lowery, P. Martin, S. Subramanya, High-power AlGaInN flip-chip light-emitting diodes, Appl. Phys. Lett. 78 (2001) 3379–3381. [5] O. Shchekin, J. Epler, T. Trottier, T. Margalith, D. Steigerwald, M. Holcomb, P. Martin, M. Krames, High performance thin-film flip-chip InGaN–GaN lightemitting diodes, Appl. Phys. Lett. 89 (2006), pp.071109071103. [6] J.-Y. Kim, M.-K. Kwon, I.-K. Park, C.-Y. Cho, S.-J. Park, D.-M. Jeon, J.W. Kim, Y. C. Kim, Enhanced light extraction efficiency in flip-chip GaN light-emitting diodes with diffuse Ag reflector on nanotextured indium-tin oxide, Appl. Phys. Lett. 93 (2008) 021121. [7] H. Guo, X. Zhang, H. Chen, P. Zhang, H. Liu, H. Chang, W. Zhao, Q. Liao, Y. Cui, High performance GaN-based LEDs on patterned sapphire substrate with patterned composite SiO2/Al2O3 passivation layers and TiO2/Al2O3 DBR backside reflector, Opt. Express 21 (2013) 21456–21465.
Q. Yue et al. / Optics Communications 367 (2016) 72–79
[8] W. Wong, T. Sands, N. Cheung, M. Kneissl, D. Bour, P. Mei, L. Romano, N. Johnson, Fabrication of thin-film InGaN light-emitting diode membranes by laser lift-off, Appl. Phys. Lett. 75 (1999) 1360–1362. [9] C.-F. Chu, F.-I. Lai, J.-T. Chu, C.-C. Yu, C.-F. Lin, H.-C. Kuo, S. Wang, Study of GaN light-emitting diodes fabricated by laser lift-off technique, J. Appl. Phys. 95 (2004) 3916–3922. [10] J.J. Wierer, A. David, M.M. Megens, III-nitride photonic-crystal light-emitting diodes with high extraction efficiency, Nat. Photonics 3 (2009) 163–169. [11] M. Lachab, F. Asif, B. Zhang, I. Ahmad, A. Heidari, Q. Fareed, V. Adivarahan, A. Khan, Enhancement of light extraction efficiency in sub-300 nm nitride thin-film flip-chip light-emitting diodes, Solid-State Electron. 89 (2013) 156–160. [12] A.I. Zhmakin, Enhancement of light extraction from light emitting diodes, Phys. Rep. 498 (2011) 189–241. [13] P. Zhu, G. Liu, J. Zhang, N. Tansu, FDTD analysis on extraction efficiency of GaN light-emitting diodes with microsphere arrays, J. Disp. Tech. 9 (2013) 317–323. [14] P. Zhu, N. Tansu, “Resonant cavity effect optimization of III-nitride thin-film flip-chip light-emitting diodes with microsphere arrays, Appl. Opt. 54 (2015) 6305–6312. [15] S.-J. Chang, L. Chang, D. Kuo, T.K. Ko, S. Hon, S. Li, GaN-based LEDs with rough surface and selective KOH etching, J. Disp. Technol. 10 (2014) 27–32. [16] A. Mao, R.F. Karlicek, Surface patterning of nonscattering phosphors for light extraction, Opt. Lett. 38 (2013) 2796–2799. [17] X.-H. Li, R. Song, Y.-K. Ee, P. Kumnorkaew, J.F. Gilchrist, N. Tansu, Light extraction efficiency and radiation patterns of III-nitride light-emitting diodes with colloidal microlens arrays with various aspect ratios, IEEE Photonics J. 3 (2011) 489–499. [18] M.R. Krames, O.B. Shchekin, R. Mueller-Mach, G.O. Mueller, L. Zhou, G. Harbers, M.G. Craford, Status and future of high-power light-emitting diodes for solid-state lighting, J. Disp. Technol. 3 (2007) 160–175. [19] D.-H. Kim, C.-O. Cho, Y.-G. Roh, H. Jeon, Y.S. Park, J. Cho, J.S. Im, C. Sone, Y. Park, W.J. Choi, Enhanced light extraction from GaN-based light-emitting diodes with holographically generated two-dimensional photonic crystal patterns, Appl. Phys. Lett. 87 (2005) 203508. [20] X. Chen, F. Kong, K. Li, Q. Ding, M. Zhang, W. Li, Study of light extraction efficiency of flip-chip GaN-based LEDs with different periodic arrays, Opt. Commun. 314 (2014) 90–96. [21] P.-H. Fu, G.-J. Lin, H.-P. Wang, K.-Y. Lai, J.-H. He, Enhanced light extraction of light-emitting diodes via nano-honeycomb photonic crystals, Nano Energy 8 (2014) 78–83. [22] S.K. Patra, S. Adhikari, S. Pal, Design and analysis of “Chess Board” like photonic crystal structure for improved light extraction in GaN/InGaN LEDs, J. Disp. Technol. 9 (2013) 339–345. [23] J.-G. Kim, C.-H. Hsieh, H.J. Choi, J. Gardener, B. Singh, A. Knapitsch, P. Lecoq, G. Barbastathis, Conical photonic crystals for enhancing light extraction efficiency from high refractive index materials, Opt. Express 23 (2015) 22730–22739. [24] M. Fujita, S. Takahashi, Y. Tanaka, T. Asano, S. Noda, Simultaneous inhibition and redistribution of spontaneous light emission in photonic crystals, Science 308 (2005) 1296–1298. [25] A. David, H. Benisty, C. Weisbuch, Photonic crystal light-emitting sources, Rep. Prog. Phys. 75 (2012) 126501. [26] A. David, H. Benisty, C. Weisbuch, Optimization of light-diffracting photoniccrystals for high extraction efficiency LEDs, J. Disp. Technol. 3 (2007) 133–148. [27] A. David, T. Fujii, R. Sharma, K. McGroddy, S. Nakamura, S.P. DenBaars, E.L. Hu, C. Weisbuch, H. Benisty, Photonic-crystal GaN light-emitting diodes with tailored guided modes distribution, Appl. Phys. Lett. 88 (2006) 061124.
79
[28] K.S. Yee, Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media, IEEE Trans. Antennas Propag. 14 (1966) 302–307. [29] Z. Zang, K. Mukai, P. Navaretti, M. Duelk, et al., Thermal resistance reduction in high power superluminescent diodes by using active multi-mode interferometer, Appl. Phys. Lett. 100 (2012) 031108. [30] Z. Zang, T. Minato, P. Navaretti, Y. Hinokuma, et al., High-power (4 110 mW) superluminescent diodes by using active multimode interferometer, IEEE Photonics Technol. Lett. 10 (2010) 721–723. [31] K.-D. Chang, C.-Y. Li, J.-W. Pan, K.-Y. Cheng, A hybrid simulated method for analyzing the optical efficiency of a head-mounted display with a quasi-crystal OLED panel, Opt. Express 22 (2014) A567–A576. [32] L. Jia, I. Bita, E.L. Thomas, Photonic density of states of two-dimensional quasicrystalline photonic structures, Phys. Rev. A 84 (2011) 023831. [33] C.-E. Lee, C.-F. Lai, Y.-C. Lee, H.-C. Kuo, T.-C. Lu, S.-C. Wang, Nitride-based thinfilm light-emitting diodes with photonic quasi-crystal surface, IEEE Photonics Technol. Lett. 21 (2009) 331–333. [34] M. Rattier, H. Benisty, E. Schwoob, C. Weisbuch, T. Krauss, C. Smith, R. Houdre, U. Oesterle, Omnidirectional and compact guided light extraction from Archimedean photonic lattices, Appl. Phys. Lett. 83 (2003) 1283–1285. [35] D. Frenkel, B. Smit, Understanding Molecular Simulation: From Algorithms to Applications, Academic Press, United States, 2001. [36] M. Rechtsman, A. Szameit, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, M. Segev, Amorphous photonic lattices: band gaps, effective mass, and suppressed transport, Phys. Rev. Lett. 106 (2011) 193904. [37] M. Bass, E.W. Van Stryland, D.R. Williams, W.L. Wolfe, Handbook of Optics, McGraw-Hill, United States, 2001. [38] J.-W. Pan, P.-J. Tsai, K.-D. Chang, Y.-Y. Chang, Light extraction efficiency analysis of GaN-based light-emitting diodes with nanopatterned sapphire substrates, Appl. Opt. 52 (2013) 1358–1367. [39] S.-K. Kim, H.-G. Park, Coherent vertical beaming using Bragg mirrors for highefficiency GaN light-emitting diodes, Opt. Express 21 (2013) 14566–14572. [40] C.-F. Lai, C.-H. Chao, H.-C. Kuo, P. Yu, H.-H. Yen, W.-Y. Yeh, GaN thickness effect on directional light enhancement from GaN-based film-transferred photonic crystal light-emitting diodes, Jpn. J. Appl. Phys. 49 (2010) 04DG09. [41] A. Taflove, S.C. Hagness, Computational Electrodynamics, Artech House, United States, 2005. [42] Y.-C. Lee, C.-C. Chang, Y.-Y. Chou, Experimental and simulation studies of antireflection sub-micron conical structures on a GaAs substrate, Opt. Express 21 (2013) A36–A41. [43] C. Xiong, B. Zhang, X. Kang, T. Dai, G. Zhang, Diffracted transmission effects of GaN and polymer two-dimensional square-lattice photonic crystals, Opt. Express 17 (2009) 23684–23689. [44] M.K. Callens, H. Marsman, L. Penninck, P. Peeters, et al., RCWA and FDTD modeling of light emission from internally structured OLEDs, Opt. Express 22 (2014) A589–A600. [45] D. Delbeke, P. Bienstman, R. Bockstaele, R. Baets, Rigorous electromagnetic analysis of dipole emission in periodically corrugated layers: the grating-assisted resonant-cavity light-emitting diode, J. Opt. Soc. Am. A 19 (2002) 871–880. [46] W.-L. Yeh, C.-M. Fang, Y.-P. Chiou, Enhancing LED light extraction by optimizing cavity and waveguide modes in grating structures, J. Disp. Technol. 9 (2013) 359–364. [47] D.H. Long, I.-K. Hwang, S.-W. Ryu, Design optimization of photonic crystal structure for improved light extraction of GaN LED, IEEE J. Sel. Top. Quant. Electron. 15 (2009) 1257–1263.