## 3 Results and Discussion  
The striped grooves can be seen luminously in Figure 2 by scanning electron microscopy (SEM) measurement. Figure 3 shows the AFM (atomic force microscope) images of surface three-dimensional topography of the striped grooves. There are many rugged bumps at the bottom of the groove. The average depth and width of the grooves are approximately 6 \(\upmu\)m and 10 \(\upmu\)m.  
Figure 4a illustrates the XRD pattern of c-plane GaN films grown on striped PSS, compared with those grown on planar substrate. From Figure 4a, red line represents the XRD pattern on striped PSS while black line corresponds to that on planar substrate and their peaks locate at the same position. The sharp peaks emerging at 34.28\({}^{\circ}\) and 72.63\({}^{\circ}\) have been attached to the GaN (0002) plane and GaN (0004) plane, respectively, which illustrates that the GaN grown on sapphire substrates is predominately c-plane orientation [19]. The peak located at 37.50\({}^{\circ}\) is corresponded to Al\({}_{2}\)O\({}_{3}\) (123) substrate. No other phases have been observed in this pattern, so it illustrates the GaN epilayers grown on both substrates have a relatively high purity. Crystalline quality of the GaN (0002) can be accessed by the full width at half maximum (FWHM) values and crystal size. From Figure 4b rocking curves, the FWHM values of the two samples can be calculated to be 0.302\({}^{\circ}\) and 0.370\({}^{\circ}\), corresponding to the striped sample  
Figure 3: AFM (atomic force microscope) images of surface three-dimensional topography of the striped PSS. Figure 2: SEM (scanning electron microscopy) images of surface microstructure of the striped U-shaped groove. (**a**) magnification is 200\(\times\); (**b**) magnification is 5000\(\times\). and planar sample symmetrically. And the average crystallite size of the GaN can be deployed to calculated by Scherrer equation [20]:  
\[D=\frac{K\lambda}{\beta\cos\theta} \tag{1}\]  
where \(D\) is the crystallite size relative to the peak width in radians, \(K\) is a constant, \(\lambda\) represents the X-ray wavelength, \(\beta\) corresponds to the FWHM value of the diffraction peak while \(\theta\) is relative to the Bragg diffraction angle of c-plane GaN. By calculation, the crystallite size of GaN films grown on striped sample and planar substrate are 31.3 and 25.6 nm, respectively. Thus, the crystallite size of striped sample is much larger than that of planar sample. Additionally, in the nucleation process of GaN, as the crystal size increases, the crystal boundaries decrease, and there are more opportunities to generate binary and ternary phases within the crystal boundaries. Thus, compared with the planar sample, the GaN epilayers grown on striped PSS have less crystal boundary, which results in that smoothness and compactness of the c-plane GaN epilayers will be moderately improved. Therefore, crystalline quality of c-plane GaN epilayers have been improved by patterning striped sapphire substrate. Figure 4: (a) X-ray diffraction patterns of GaN films grown on striped PSS and planar substrate, and (b) rocking curves for GaN films grown on striped PSS and planar substrate. Moreover, XRD rocking curves are employed to calculate the threading dislocation density by the following equation [21]:  
\[\rho=\frac{\beta^{2}}{4.35\times b^{2}} \tag{2}\]  
where \(\rho\) represents the threading dislocation density, \(\beta\) is the FWHM of the GaN (0002) XRD rocking curve, and \(b\) is the value of the Burgers vector. From Figure 4, the FWHM of planar sample is much larger than that of the striped PSS which gives a result that striped PSS will have less threading dislocation density according to equation above. Less threading dislocation density of striped PSS can also indicate the higher crystalline quality of c-plane GaN epilayers grown on striped PSS. The Raman spectroscopy is also used to demonstrate the residual stress in GaN epitaxial layers [21]. Figure 5 illustrates the Raman spectra of different samples grown on planar substrate and striped PSS.