## 2 Experiment  
In experiments, our suggested SiO\({}_{2}\) NPs embedded PSS LEDs were compared with reference of a conventional PSS LED structure. First, a template of unintentionally doped u-GaN grown on PSS was prepared. A schematic illustration of the InGaN/GaN LED embedded with SiO\({}_{2}\) NPs and process thereof are shown in Figure 1. With metal organic chemical vapor deposition (MOCVD), 3-\(\upmu\)m-thick undoped GaN (u-GaN) films were grown on c-plane PSS in conventional way. To make nano-pillars in GaN film, a 100-nm-thick SiO\({}_{2}\) layer was deposited as an interlayer on top of the u-GaN surface by plasma-enhanced chemical vapor deposition. With e-beam evaporation, a 10-nm-thick Ni layer was deposited on the SiO\({}_{2}\) interlayer. To form the self-assembled Ni metal clusters, the Ni/SiO\({}_{2}\)/GaN samples were treated with rapid thermal annealing (RTA) under N\({}_{2}\) flow at 850 \({}^{\circ}\)C for 1 min. Inductively coupled plasma (ICP) etching with the gases of O\({}_{2}\) and CF\({}_{4}\) was applied to pattern the SiO\({}_{2}\) layer using the Ni nano-dots as an etch mask and then, the u-GaN was further etched down to 1.5 \(\upmu\)m with Cl\({}_{2}\) gas [22]. After that, the Ni nano-dots on SiO\({}_{2}\), remaining on the tops of nano-pillars, were removed by buffered oxide etchant. As a result, a GaN template with vertically arranged nano-pillars was prepared. On the nano-pillar GaN template, the SiO\({}_{2}\) NPs were deposited by spin coating (5000 rpm, 30 s). We used commercial colloidal SiO\({}_{2}\) NPs with 100 nm diameter. This SiO\({}_{2}\) embedded nano-pillar template was used for ELO overgrowth. The u-GaN templates grown on PSS without the nano-pillars (reference) were loaded at the same time. After well-coalesced GaN seed layer was formed on top of the nano-pillars, a Si-doped 3-\(\upmu\)m-thick n-GaN layer, a multiquantum-well (MQW) region, and a 200-nm-thick Mg-doped p-GaN top contact layer were grown in sequence. The MQW region consisted of five undoped InGaN (2 nm)/GaN (10 nm) quantum wells.  
The cross-section images of the SiO\({}_{2}\) NP LED were obtained using a field emission scanning electron microscopy (FE-SEM). Room temperature photoluminescence (PL) mapping measurements were carried out with a 325 nm line of a 25 mW He-Cd laser as an excitation source. The current-voltage (I-V) and optical output (L-I) characteristics were measured by means of an on-wafer probing with indium contacts. The results were compared to those obtained from the reference sample--the growth conditions for both types of samples were the same, but for the fact of the presence of the nano-pillar region embedded with SiO\({}_{2}\) NPs in the modified structure.  
The experimentally observed performance increase is compared with the simulation result. Because the structure of this experiment and PSS are intended to increase LEE with different geometries, the simulation is planned to understand how to further increase the efficiency, which means geometrical differences are compatible and complementary to each other even when the functions are overlapped. Detailed conditions for the simulation are described in Appendix A.  
Figure 1: (**a**) GaN light emitting diode (LED) structure consisting of nano-pillars and air-void filled with SiO\({}_{2}\) nano-particles (NPs) formed on patterned sapphire substrate (PSS). (**b**) Schematic illustration of the process flow to use Ni dots as a mask to form nano-pillars and fill air-void with SiO\({}_{2}\) NPs, and regrow GaN. ICP: inductively coupled plasma; MQW: multi quantum well.