## 3 Results and Discussion  
The most favorable configurations for NO\({}_{2}\) on ZnGa\({}_{2}\)O\({}_{4}\) (111) surfaces are shown in Figure 3. In Model N1, the nitrogen atom of the NO\({}_{2}\) molecule was adsorbed on the Ga\({}_{3c}\) atom of the ZnGa\({}_{2}\)O\({}_{4}\)(111) surface to yield a N-Ga bond with a bond length of 2.01 A. In Model N2, the NO\({}_{2}\) molecule moved away from the originally adsorbed Zn\({}_{3c}\) atom to approach the neighboring Ga atom of the ZnGa\({}_{2}\)O\({}_{4}\)(111) surface to form an O-Ga bond with a bond length of 1.90 A. In Model N3, the NO\({}_{2}\) molecule moved away from the originally adsorbed O\({}_{3c}\) atom toward the neighboring Ga and Zn atoms of the ZnGa\({}_{2}\)O\({}_{4}\)(111) surface to form N-Ga and O-Zn bonds with bond lengths of 2.05 A and 2.30 A, respectively. In Model N4, the nitrogen (oxygen) atom of the NO\({}_{2}\) molecule adsorbed on the Ga\({}_{3c}\) (Zn\({}_{3c}\)) atom of the ZnGa\({}_{2}\)O\({}_{4}\)(111) surface to form an N-Ga (O-Zn) bond with a bond length of 2.06 A (2.21 A). In Model O1, the two O atoms of the NO\({}_{2}\) molecule adsorbed on the Ga\({}_{3c}\) atom of the ZnGa\({}_{2}\)O\({}_{4}\)(111) surface to yield two O-Ga bonds with bond lengths of 2.07 A and 2.13 A. Similarly, in Model O\({}_{2}\) (O3), the NO\({}_{2}\) molecule moved away from the originally adsorbed Zn\({}_{3c}\) (O\({}_{3c}\)) atom toward the neighboring Ga\({}_{3c}\) atom of the ZnGa\({}_{2}\)O\({}_{4}\)(111) surface to form two O-Ga bonds with bond lengths of 2.10 (2.13) A and 2.11 (2.08) A. In Model O4, an O atom of the NO\({}_{2}\) molecule was adsorbed on the Ga\({}_{3c}\) atom of the ZnGa\({}_{2}\)O\({}_{4}\)(111) surface to form an O-Ga bond with a bond length of 1.92 A. The vacuum energy, Fermi energy, work functions of ZnGa\({}_{2}\)O\({}_{4}\)(111) with and without NO\({}_{2}\), and work function difference among NO\({}_{2}\) adsorption models are presented in Table 1. The work function increased when the NO\({}_{2}\) molecule was adsorbed on the ZnGa\({}_{2}\)O\({}_{4}\)(111) surface. The increase in work function ranged from +0.29 eV to +0.97 eV, which concurs with previously reported experimental results of similar resistance responses of ZnO(Ga) samples to NO\({}_{2}\)[8; 11]. The NO\({}_{2}\) molecule preferred bonding with the Ga\({}_{3c}\) or Zn\({}_{3c}\) atom. Model N1 had the maximum work function difference of +0.97 eV, exhibiting increased sensitivity for detecting NO\({}_{2}\) molecules. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Models** & **Adsorption Sites** & \(E_{VAC}\) & \(E_{F}\) & \(\Phi_{NO_{2}}\) & \(\Phi_{ZGO}\) & \(\Delta\Phi\) \\ \hline ZnGa\({}_{2}\)O\({}_{4}\)(111) & - & 0.66 & \(-\)3.38 & - & 4.04 & - \\ N1 & Ga\({}_{3c}\) & 1.45 & \(-\)3.56 & 5.01 & - & +0.97 \\ N2 & Ga\({}_{3c}\) & 1.04 & \(-\)3.29 & 4.33 & - & +0.29 \\ N3 & Ga\({}_{3c}\), Zn\({}_{3c}\) & 1.12 & \(-\)3.31 & 4.43 & - & +0.39 \\ N4 & Ga\({}_{3c}\), Zn\({}_{3c}\) & 1.09 & \(-\)3.31 & 4.40 & - & +0.36 \\ O1 & Ga\({}_{3c}\) & 0.98 & \(-\)3.41 & 4.39 & - & +0.35 \\ O2 & Ga\({}_{3c}\) & 1.01 & \(-\)3.36 & 4.37 & - & +0.33 \\ O3 & Ga\({}_{3c}\) & 1.07 & \(-\)3.31 & 4.38 & - & +0.34 \\ O4 & Ga\({}_{3c}\) & 1.04 & \(-\)3.29 & 4.33 & - & +0.29 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Calculated vacuum energy \(E_{VAC}\), Fermi energy \(E_{F}\), work function of NO\({}_{2}\)\(\Phi_{NO_{2}}\), work function of ZnGa\({}_{2}\)O\({}_{4}\)(111) \(\Phi_{ZGO}\), and work function difference \(\Delta\Phi\) (\(\Delta\Phi=\Phi_{NO_{2}}-\Phi_{ZGO}\) ) among the eight NO\({}_{2}\) adsorption models. All energies are presented in eV. Figure 3: Top and side views of the favorable configurations for NO\({}_{2}\) on the ZnGa\({}_{2}\)O\({}_{4}\)(111) surface. Atoms are represented by spheres: Zn (purple, large), Ga (brown, large), S (yellow, medium-sized), O (red, medium-sized), and H (white, small). Equilibrium bond lengths between molecular and surface atoms are given in Angstrom.