## 3 Results
### Effects of TiO\({}_{2}\) NS Monolayer on the Device Performance of LED
To design an adequate TiO\({}_{2}\) NS monolayer array on LEDs, various parameters should be considered, such as the refractive index, crystal structure, and the packing density of spheres. Among the considered parameters described here, we paid special attention to the effects of the crystal structure of the spheres on the LEE of LED chips. In order to theoretically evaluate LEEs of LEDs with respect to changes in the refractive indices of used spheres monolayers, which can be modified by changes in the crystal phases of TiO\({}_{2}\), we used the 3-D FDTD simulation method. Figure 6a-c shows the far-field intensity mapping of the reference LED and LED chip with An- and Am-TiO\({}_{2}\) sphere monolayer arrays. It is apparent that emitted light was trapped in the inner LED chip due to TIR resulting from the abrupt change in the refractive index across p-GaN and air in the reference LED chip. However, the LED with An-TiO\({}_{2}\) monolayer arrays shows that light was effectively diffused from the LED chip through the An-TiO\({}_{2}\) NSs, as clearly shown in Figure 6b. Furthermore, the Am-TiO\({}_{2}\) monolayer array can more effectively diffuse a significant amount of light from the LED through the NSs, as depicted in Figure 6c. Similar to far-field intensity, Figure S2a-c shows the cross-section profiles of the power field for reference LED and LED chips with An- and Am-TiO\({}_{2}\) sphere monolayer arrays. It is clear from Figure S2a that emitted light was trapped in LEDs due to the abrupt change in the refractive index across p-GaN and air in the reference LED chips. However, the LEDs with An-TiO\({}_{2}\) and Am-TiO\({}_{2}\) NSs monolayer arrays demonstrate that light was effectively diffused from the LED chips through the NSs, as shown in Figure S2b and S2c, respectively. The enhancement of light extraction from LEDs with Am-TiO\({}_{2}\) NSs arrays can be attributed to the increase in the escape probability of photons from the LED as observed by simulation results. In addition, in terms of the escape angle of light, light emission of LED mainly occurred between 10\({}^{\circ}\) and 20\({}^{\circ}\) as denoted by the red dotted line in Figure 6a-c. The Am-TiO\({}_{2}\) monolayer array/LED shows the highest light-emission intensity and brightest area among all samples. In addition, by comparison of the integrated power field intensity for LEDs with different spheres of TiO\({}_{2}\) monolayers, LED chips with An-TiO\({}_{2}\) monolayers and Am-TiO\({}_{2}\) monolayer arrays exhibited, respectively, 1.4 times and 1.76 times higher intensity than LED chips without any structures. To further investigate the increased light extraction via Am-TiO\({}_{2}\) NS monolayers as compared to An-TiO\({}_{2}\) NSs, we captured the evolution of traveling waves showing light propagation with respect to changes in time, as shown in Figure 7. When we compared the light extraction via different Am- and An-TiO\({}_{2}\) NSs, light propagation modes in Am-TiO\({}_{2}\) NSs were more strongly concentrated on the apex than were those of An-TiO\({}_{2}\) as denoted by yellow arrows in Figure 7j,d, respectively. The backscattering portions of light in An-TiO\({}_{2}\) spheres remaining from escaping via the top of TiO\({}_{2}\) spheres toward GaN are clearly larger than those of Am-TiO\({}_{2}\) as denoted by the red arrow in Figure 7j,l, respectively. Therefore, more photons can effectively escape from the apex of Am-TiO\({}_{2}\) NS concentrated by the convex lens-focusing effects and propagated via Am- TiO\({}_{2}\) NSs to air from the LEDs. This result further proved that the Am-TiO\({}_{2}\) NSs are more effective than An-TiO\({}_{2}\) NSs at improving the LEE of LEDs.