## 3 Results and Discussion
### Structural Characterization of the Ga\({}_{2}\)O\({}_{3}\)-Shell
To further investigate the formation of this shell, we have analyzed samples by XRD using in-plane Phi scan configuration. In this configuration, the samples are rotated by 360\({}^{\circ}\), while the angle between the X-ray source and the detector (2 ThetaChi) is fixed at 57.8\({}^{\circ}\), 61.1\({}^{\circ}\) and 64.3\({}^{\circ}\) corresponding respectively to the GaN (11-20), \(\beta\)-Ga\({}_{2}\)O\({}_{3}\) (020) and \(\beta\)-Ga\({}_{2}\)O\({}_{3}\) (403) peak positions. Figure 5 presents the results for pure GaN NWs and GaN/Ga\({}_{2}\)O\({}_{3}\) NWs. On Figure 5a, six peaks correspond to the six (11-20) planes of GaN, demonstrating the hexagonal symmetry of GaN. Figure 5b,c show also six peaks, indicating that the (020) and (403) planes of \(\beta\)-Ga\({}_{2}\)O\({}_{3}\) present symmetries along the hexagonal structure of GaN. These results, in agreement with the result from Phi/2 ThetaChi scan (Figure 4), demonstrate that the \(\beta\)-Ga\({}_{2}\)O\({}_{3}\) shell synthesized by post-thermal treatment is grown in an epitaxial relationship with the GaN core. The XRD analyses have evidenced that the Ga\({}_{2}\)O\({}_{3}\) shell crystallize in the \(\beta\)-Ga\({}_{2}\)O\({}_{3}\), the most-stable phase, but is characterized by a lack of uniformity. To investigate this point, we have performed highly resolved TEM and STEM analyses combined with EDX mappings. Figure 6a,c presents bright field and dark field images of GaN/\(\beta\)-Ga\({}_{2}\)O\({}_{3}\) NWs. The crystal structure of the heterostructures is characterized by different patterns, corresponding to the structure of GaN for the core, and the structure of Ga\({}_{2}\)O\({}_{3}\) oxide for the shell. The TEM images evidence a ripple at the GaN/oxide interface and show variations in the shell lattice. This could be caused by the mismatch between GaN and Ga\({}_{2}\)O\({}_{3}\) structures, creating internal stress located in the shell as shown by the XRD analyses (Figure 4), and the appearance of defects in Ga\({}_{2}\)O\({}_{3}\). The zoom in the NW core (Figure 6c) indicates that the pattern responds to the (11-20) plane of hexagonal GaN. The perfect lattice of GaN in the figure indicates that its monocrystalline structure is preserved after the oxidation treatment. Concerning the shell, the TEM images reveal that it presents a different crystalline structure from the NW core and is non-uniform with different patterns. According to the result from thr XRD analyses, the (11-20) plane of GaN is parallel to the (020) one of \(\beta\)-Ga\({}_{2}\)O\({}_{3}\). This means that the crystal structure of this last plane, or of any other parallel ones, can be observed in the same images as the one containing the (11-20) plane of GaN. Figure 6d presents STEM analysis performed at the GaN/Ga\({}_{2}\)O\({}_{3}\) interface. The shell presents the same structure as the structure of \(\beta\)-Ga\({}_{2}\)O\({}_{3}\) (020) plane [63]. Combined with the fact that their parameters correspond to the theoretical ones, we can confirm that the oxide shell formed by thermal oxidation of MBE-grown GaN NWs presents the monoclinic \(\beta\)-Ga\({}_{2}\)O\({}_{3}\) crystalline phase [63]. The non-uniformity of the shell results from the mismatch caused by hexagonal structure of GaN, which disarranges the formation of monoclinic structure of Ga\({}_{2}\)O\({}_{3}\) at the angles of hexagon. Since Ga\({}_{2}\)O\({}_{3}\) structure presents an epitaxial relation with GaN structure, the angles of hexagonal structure change the direction of monoclinic structure in an inappropriate way, leading to the observed mixing direction of Ga\({}_{2}\)O\({}_{3}\) lattice. Finally, the Figure 7 presents EDX mappings performed on GaN/\(\beta\)-Ga\({}_{2}\)O\({}_{3}\) NWs. The different elements composing the heterostructure can be well distinguished between the GaN-core and the Ga\({}_{2}\)O\({}_{3}\)-shell. At this stage, it is important to note that blue dots of oxygen atoms can be seen in the GaN-core area. However, this is not the signature of oxygen incorporation inside the GaN-core.