## 2 Materials and Methods  
The synthesis of GaN/Ga\({}_{2}\)O\({}_{3}\) core-shell NW heterostructures is composed of two successive steps: 1-GaN-core NW growth: Self-assembled free-catalyst GaN-core NWs were grown on conductive oxide-free Si (111) substrate (resistivity of the order of 0.007 \(\Omega\) cm) in a molecular beam epitaxy (MBE) chamber (RIBER Compact 12 PA-MBE system, Bezons, France), equipped with a radio-frequency N plasma source. After a chemical cleaning and a thermal deoxidation of the substrate surface to remove the organic pollutants and the native oxide, a 2.5-nm-thick AlN buffer layer was deposited at 620 \({}^{\circ}\)C following a previously reported procedure [42]. This seed layer allows a better control of the NW nucleation and density, a reduction of the NW twist and an improvement of the NW vertical orientation [42; 43; 44; 45]. Then, the temperature was ramped to 790 \({}^{\circ}\)C to grow the GaN NWs under nominally N-rich conditions with an N/Ga flux ratio of 1.36. Following this growth procedure, the GaN NWs are vertically oriented with a hexagonal shape delimited by [10-10] plane, are characterized by a quasi-crystalline perfection (absence of dislocations) [46], a prevalent N-polarity [42; 47] and present reproducible dimensions and densities (Figure 1). GaN NWs considered in this study are characterized by a density of 5-10\({}^{9}\) NW/cm\({}^{2}\), an average diameter of 40 \(\pm\) 5 nm and an average height of 650 \(\pm\) 100 nm, these structural characteristics being extracted from high-resolved SEM images (plan-view and cross-section view). 2-Ga\({}_{2}\)O\({}_{3}\) shell synthesis: The Ga\({}_{2}\)O\({}_{3}\)-shell was formed around GaN NW by thermal oxidation post-treatment. The as-grown GaN NWs were oxidized at high temperature following the procedure described in Figure 1. The adjustment of the oxidation temperature was performed under Nitrogen to avoid any alteration of the NW surfaces. After the stabilization of the targeted oxidation temperature, the controlled environment was injected, and the oxidation time was considered. Finally, the temperature was decreased to the ambient one under Nitrogen. During the oxidation step, due to the high temperature, the GaN is decomposed at the surface. The Nitrogen is removed in the environment, while the Ga, with its free dangling bonds, is linked with oxygen, then forming Ga\({}_{2}\)O\({}_{3}\) oxide (Figure 1). After the formation of the first oxide monolayer, the oxide shell plays the role of a barrier separating the oxygen-based environment and the GaN NWs. To continue further the oxidation, the oxygen species should diffuse through the Ga\({}_{2}\)O\({}_{3}\) oxide to form an additional oxide monolayer. As a function of the oxidation time, the shell surrounding the GaN-core becomes thicker. The oxidation mechanisms, the rate of oxidation, the quality of Ga\({}_{2}\)O\({}_{3}\) layer and the GaN/Ga\({}_{2}\)O\({}_{3}\) interface can be affected by many factors such as temperature, oxidized agents and their flow rate, pressure, GaN structure and the concentration of impurities. In order to investigate the formation of the Ga\({}_{2}\)O\({}_{3}\)-shell, the oxidation of the GaN-core NWs has been tested under different conditions summarized in Table 1, such as oxidation time, temperature, N\({}_{2}\)/O\({}_{2}\) ratio and humid or dry atmosphere. We note here that the N\({}_{2}\)/O\({}_{2}\) ratio was chosen to be approximately equivalent to the normal air composition. The \"humid\" term refers to oxidation environment equivalent to normal air (i.e., with water vapor), while the \"dry\" term refers to the same oxidation conditions (N\({}_{2}\)/O\({}_{2}\) ratio) but without water vapor. The structural properties of the core-shell NWs have been characterized by combining different techniques of characterization. To evaluate the shell formation, the diameters of NWs were