## 3 Results and Discussion
### The Relationship Between Surface-to-Volume Ratio and Sensitivity  
During the gas detection step upon exposure to a reducing target gas, a reduction reaction occurs on the surface of ZnO gas sensors. Hereby, the following sensing reaction takes place between oxygen negative ions pre-existing on the sensor surface from the set operation and the reducing target gas molecules, as shown in Equation (7) [15]. \[X+O^{\delta-}\leftrightarrow X^{\prime}+\delta e^{-} \tag{7}\]  
where \(X\) is the target reducing gas, \(X^{\prime}\) is the byproduct after the reaction and \(\delta\) is the number of electrons. Based on Equation (7), the electron density adsorbed to the surface can be calculated by Equation (8) [16]. \[n=\Gamma_{l}k_{gas}[O]^{b}[X]^{b}+n_{0} \tag{8}\]  
where \(n\) is the electron density under the target gas atmosphere, \(\Gamma_{l}\) is a time constant, \(n_{0}\) is the electron density under the air atmosphere, \(b\) is a variable value (equal to 1 for \(O^{-}\) and 0.5 for \(O^{2-}\)), [O] is the adsorbed oxygen ion density, \([X]\) is the target gas molecule density  
Figure 13: Gained sensing response enhancement of ZnO nanorod with nested coaxial nanotubes as a function of operating temperature. and \(k_{gas}\) is the reaction rate constant [13]. Therefore, based on Equation (4), the sensitivity relation is shown in Equation (9). \[S_{g}=\left(\frac{R_{air}-R_{gas}}{R_{gas}}\right)=\frac{n_{0}-n}{n_{0}}=\frac{n_ {0}-\left(T_{l}k_{gas}[O]^{b}[X]^{b}+n_{0}\right)}{n_{0}}=\frac{\Gamma_{l}k_{ gas}[O]^{b}[X]^{b}}{n_{0}} \tag{9}\]  
where \(S_{g}\) is the sensing response of the gas sensor. Conventionally, the surface-to-volume ratio can be defined by the adsorbed oxygen ion density [13]. Therefore, the adsorbed oxygen ion density can be defined by Equation (10). \[[O]=\frac{\sigma_{0}\varphi V_{m}}{V_{S}} \tag{10}\]  
where \(\sigma_{0}\) is the number of oxygen ions per unit area, \(V_{m}\) is the volume of the gas-sensitive material, \(\varphi\) is the ratio of surface area per volume of material (\(V_{m}\)) and \(V_{S}\) is the volume of the reaction chamber of the gas testing system [13]. After substituting Equation (10) into Equation (9), we can obtain the sensitivity of the gas sensor related to the surface-to-volume ratio from Equation (11). \[S_{g}=\frac{\Gamma_{l}k_{gas}\left[\frac{\sigma_{0}\varphi V_{m}}{V_{S}} \right]^{b}[X]^{b}}{n_{0}} \tag{11}\]  
Equation (11) clearly explains the relationship between the sensitivity and the surface-to-volume ratio of MOS solid-state gas sensors. These equations provide much of the motivation background for this work and the concept of the novel nested ZnO thin-film sensor design architecture. With increased surface-to-volume ratio (\(\sigma_{0}\) and \(\varphi\)) introduced by additional layers coating the coaxial ZnO nanotube, among them the sacrificial Al\({}_{2}\)O\({}_{3}\) layer, the sensitivity of ZnO gas sensors to the ethanol vapor concentration detection correspondingly increases, based on Equation (11). Consequently, the sensitivity of ZnO nanorod gas sensors can be improved with an increased surface-to-volume ratio, which in our case is being realized by coaxial ZnO nanotubes, which expose two additional reaction surfaces after removal of the sacrificial alumina layer. Compared to the benchmarked experimental ZnO nanorods sample, the additional nanotube increased the reaction areas by about 150%. Approximately 70% of all nested nanotube structures were successfully opened up with the removal of the sacrificial layer. Therefore, after substituting 1.5\(\sigma_{0}\) and 1.5\(\varphi\) into the formula, there would be around 100% enhancement in the sensitivity, which is in agreement with the experiment result. Figure 6e reveals some wrap-around nested nanotubes that did not expose the sacrificial layer, because they did not reach the average height of 750 nm and consequently their top ALD capping layers were not cleared by the PIPS ion milling.