## 3 Results and Discussion
### Ga\({}_{2}\)O\({}_{3}\)-Shell Thickness Evolution and Formation  
By comparing the high-resolved SEM images of NWs before and after the oxidation, we can observe that the as-grown MBE GaN-core NWs are characterized by smooth surfaces, while the GaN/Ga\({}_{2}\)O\({}_{3}\) core-shell NW heterostructures present rough surfaces (Figure 1). The formation of Ga\({}_{2}\)O\({}_{3}\) shell is also characterized by an increase of the NW diameter. Due to the self-assembled growth mode of the NWs (cf. Materials and Methods), we have a modulation of the as-grown NW diameter from wire to wire. To realize a representative study, each sample has been measured before and after oxidation, in a statistical view with the measurement of a large number of nanostructures (around 400 NWs). The diameter follows a single Gaussian distribution, within the limits of statistical error where the error bars correspond to the full width at half maximum of the Gaussian fit in each case (Figure 2a). The shell thickness is estimated from the difference between the mean values of the two distributions (before and after oxidation) with standard error calculated as follows: \(\sigma_{2-1}=\sqrt{\frac{\sigma_{1}^{2}}{N_{1}}+\frac{\sigma_{2}^{2}}{N_{2}}}\). The diameter expansion is caused by the difference between the specific volume of the GaN (13.5 cm\({}^{3}\)/mol) and the \(\beta\)-Ga\({}_{2}\)O\({}_{3}\) (31.8 cm\({}^{3}\)/mol), calculated based on their molar masses and densities (for GaN: 83.7 g/mol and 6.2 g/cm\({}^{3}\)[48]; for \(\beta\)-Ga\({}_{2}\)O\({}_{3}\) 187.4 g/mol and 5.9 g/cm\({}^{3}\)[49]). The specific volume of \(\beta\)-Ga\({}_{2}\)O\({}_{3}\) being higher than the one of GaN, the final diameter of Ga\({}_{2}\)O\({}_{3}\)/GaN NWs is higher than the initial one of GaN NWs. We want to note that even if the formation of the oxide mainly explains the expansion of the NW diameter, other reasons can accompany this one, such as the defects of Ga\({}_{2}\)O\({}_{3}\) structure or the vacancies in the shell. It should also be noted that the difference between the mean values of the two distributions before and after oxidation is the average expansion of the NW diameter. However, it is not exactly equivalent to two times the thickness of the formed oxide. In fact, when GaN is oxidized, GaN layers are converted into the oxide layer. This means that the oxide thickness is equal to the thickness expansion plus the thickness of GaN layers that have been oxidized, as schematized on Figure 2b. Figure 2c,d establish the direct correlation between the diameter expansion and the oxidation conditions, with an evolution of the Ga\({}_{2}\)O\({}_{3}\) shell thickness between 1 and 14 nm. Figure 2c shows the diameter expansion at 850 \({}^{\circ}\)C (blue curve) and 900 \({}^{\circ}\)C (green curve) as a function of the oxidation time, for O\({}_{2}\) and N\({}_{2}\) fluxes of 1.3 sccm and 5 sccm, respectively, and under dry atmosphere (Table 1, Series 1 and 3). The shell thickness increases with the time and the temperature. This behavior is similar to the one observed for oxidized GaN or AlN films [50; 51; 52; 53; 54]. For the lower temperature (850 \({}^{\circ}\)C), two different oxidation processes can be distinguished: an initial rapid oxide-shell growth, followed by a second stage of reduced but continuous oxide growth (after about 15-20 min of oxidation). For the higher temperature (900 \({}^{\circ}\)C), the diameter expansion is more pronounced and can only be described by the first oxidation process, then demonstrating that the initial oxidation stage is dependent on the temperature. These descriptions of the oxidation process are consistent with the ones theoretically predicted [55] and experimentally observed [51; 54; 56]. Then, the oxidation process is characterized by two different regimes: The interfacial reaction-controlled and the oxidation diffusion-controlled regimes. In interfacial reaction-controlled regime, the GaN oxidation process is controlled by the surface reaction rate, leading to a linear oxidation [57].