## 4 Simulation and Discussion  
In the experiment, the air-void surrounding nano-pillars, the air-gap among SiO\({}_{2}\) NPs, and SiO\({}_{2}\) NPs themselves in the nano-pillar layer were designed to act as scattering centers together. To maximize light extraction performance, it is required to optimize conditions including the periodicity of the GaN pillars, the radius of the pillars, and the radius of SiO\({}_{2}\) spheres of the structure demonstrated in this paper. As a basic consideration, to increase LEE toward the top surface, air-voids between the nano-pillars can be advantageous in terms of improved scattering than air-voids filled with SiO\({}_{2}\) NPs because the refractive index gap between GaN and air is bigger than the one between GaN and SiO\({}_{2}\), but only if the air-voids are as effective in ELO suppression of dislocation density as the voids filled with SiO\({}_{2}\) (Appendix C). However, this is not the only reason to adopt air to act as a scattering center. There was a case of jointly using air and SiO\({}_{2}\) at the same time that demonstrated effective light output enhancement with SiO\({}_{2}\) nano-pattern layer which was placed over the air-void layer [27]. To understand a mechanism of LEE increase in the experiment, we numerically investigated the four types of LED structures by full three-dimensional (3-D) finite-difference time-domain (FDTD) method. The schematic structures for FDTD simulation are shown in the left of each four groups in Figure 4b; Type0 (conventional LED on CSS); Type1 (conventional LED on PSS, reference LED in the experiment); Type2 (SiO\({}_{2}\) NP LED on CSS); Type3 (SiO\({}_{2}\) NP LED on PSS in the experiment). Figure 4a shows the top and bottom direction LEE, respectively, as a function of time after source excitations. In the case of the top direction LEE, the SiO\({}_{2}\) NP LED on PSS shows the highest LEE during most of the time. In contrast, in the case of the bottom direction LEE, at initial time, the LEE of the conventional LED on PSS is higher than that of the SiO\({}_{2}\) NP LED on PSS. However, the LEE of SiO\({}_{2}\) NP LED on PSS shows the highest LEE after 0.7 ps. Figure 4b-e show the vertical cut-views of intensity distribution of the electric field at different times after source excitations in four different LEDs to illustrate the LEE increase mechanism of the experiment. According to Figure 4a, the top emission of Type3 is 10% higher than that of Type1 at the beginning and then converges to a level similar to Type1 later. It is considered that the increase at the beginning of Type3 is due to the structure playing two roles of confining and scattering photons within the upper volume above the nano-pillars. Photons of Type3 exist densely within about half size volume of Type1, which raises probability of scattering at the top of the nano-pillar layer and increases top emission. Figure 4: (Simulation data) (**a**) Top and bottom direction light extraction efficiency (LEE) of four types as a function of time after source excitation. (**b–e**) Cut-views of intensity distributions of the electric field at different times after source excitations shown at the bottom. (**b**) Reference (**c**) PSS (**d**) SiO\({}_{2}\), and (**e**) PSS + SiO\({}_{2}\). The dotted lines indicate the boundary of the materials. This analysis is matched with the simulation results shown in Figure 4b,e. It is worth noting that absorption of the materials including MQW was not considered in the simulation, therefore, the LEE might be saturated earlier than the simulation results. According to the field intensity distribution of Type3 at 3 ps in Figure 4e, the amount of photons in the volume below the nano-pillar layer becomes smaller than that in the volume above the nano-pillar layer, which leads us to conclude that light extraction toward the backside would decrease because the layer works as a light barrier.