## 4 Conclusions  
We have developed a synthesis procedure to form GaN/\(\beta\)-Ga\({}_{2}\)O\({}_{3}\) NWs composed of two successive steps: (1) the growth of GaN NWs by plasma-assisted MBE, followed by (2) their thermal oxidation post-treatment, leading to the formation of a Ga\({}_{2}\)O\({}_{3}\)-shell surrounding the GaN-core. Due to the larger specific volume of Ga\({}_{2}\)O\({}_{3}\) by regarding the one of GaN, the oxidation of the GaN-core is associated  
Figure 6: STEM images of GaN/Ga\({}_{2}\)O\({}_{3}\) NWs in (a) bright field mode; (b) dark field mode. (c) Magnification of the GaN-core. (d) Magnification at the GaN/Ga\({}_{2}\)O\({}_{3}\) interface. The \(\beta\)-Ga\({}_{2}\)O\({}_{3}\) structural representation is extracted from [63].  
Figure 7: STEM (a) and EDX mappings (b) of GaN/Ga\({}_{2}\)O\({}_{3}\) NW.  
with an expansion of its diameter. By analyzing the diameter expansion of the shell as a function of the oxidation conditions (temperature, O\({}_{2}\) flux, atmosphere and time of oxidation), we have established that the thickness of the Ga\({}_{2}\)O\({}_{3}\)-shell can be modulated with the oxidation conditions, from 1 to 14 nm. We have also shown that the oxidation process follows the classical model with an initial rapid oxide-shell growth, followed by a second stage of reduced but continuous oxidation rate. In the first regime, the GaN oxidation process is controlled by the surface reaction leading to a linear oxidation, while in the second regime, the oxidation process is controlled by the reagent transport through the growing gallium oxide layer. The influence of the oxidation environment on the oxidation rate has also been discussed. Then, the enhancement of the oxidation rate under a wet environment is associated with a faster diffusion through the oxide-shell of the OH\({}^{-}\) species, resulting from the oxygen decomposition under a wet atmosphere, than the O\({}^{2-}\) species, resulting from the O\({}_{2}\) decomposition under dry atmosphere.  
The structural properties of the shell have been investigated by combining various techniques: XRD, TEM-STEM and EDX. We have demonstrated that the oxide shell formed by thermal oxidation is synthesized in epitaxial relationship with the GaN core, and presents the \(\beta\)-Ga\({}_{2}\)O\({}_{3}\) crystalline phase, the most stable one.  
By controlling the synthesis of GaN/\(\beta\)-Ga\({}_{2}\)O\({}_{3}\) NWs, this paper constitutes a building block for developing high-efficient CO-gas sensors. GaN/\(\beta\)-Ga\({}_{2}\)O\({}_{3}\) core-shell NW based sensor devices with different shell thicknesses (synthesis under dry and humid atmosphere) have been fabricated and are under investigation to establish the gas sensor performances as a function of the shell characteristics.  
Finally, we would like to comment that GaN/\(\beta\)-Ga\({}_{2}\)O\({}_{3}\) core-shell NW based sensors are promising for CO detection in harsh environments and under high temperatures. However, these nanostructures can also exhibit sensitivity to other gas at low temperature, such as oxygen [5] or hydrogen [65].  
NW growth, M.M., L.T., N.G., and Q.C.B.; Thermal oxidation, Q.C.B., X.L., and N.G.; Methodology, Q.C.B., M.M. and N.G.; NW structural characterizations, L.L., Q.C.B, C.D., O.M., and N.G.; Investigation, Q.C.B., L.L., and N.G.; Data Analysis, Q.C.B., M.M., N.J., L.L., M.T., and N.G.; Validation, M.T., and N.G.; Funding Acquisition, M.T., J.-C.H., and N.G.; Writing-Original Draft, Q.C.B., M.T., L.L., and N.G.; Supervision, M.T., J.-C.H., and N.G.  
This work was financially supported by the French National Research Agency through the GANEX program (ANR-11-LABX-0014), and the EU Horizon 2020 ERC project 'NanoHarvesf' (Grant 639052).  
The authors declare no conflict of interest.