## 3 Results and Discussion  
### Analysis of Pd-InN Monolayer  
For Pd-doping on the InN monolayer, four possible sites have been recognized, namely T\({}_{\text{B}}\) (the bridge site on the top of the In-N bond), T\({}_{\text{H}}\) (in the hollow center of the hexatomic In-N ring), T\({}_{\text{N}}\) (on the top of the N atom), and T\({}_{\text{In}}\) (on the top of the In atom). As reported in Reference [17], Pd prefers to be doped on the T\({}_{\text{N}}\) site, with the binding energy of \(-2.47\) eV. Thus, in this work, we directly employed the T\({}_{\text{N}}\) site for Pd-doping on the intrinsic InN monolayer, with a calculated binding energy (\(E_{\text{b}}\)) of \(-2.55\) eV.  
The geometric structure associated with the deformation charge density (EDD) of our optimized Pd-InN monolayer is shown in Figure 1. According to our calculations, the bandgap obtained by the PBE function (0.606) is much closer to the previous report (0.62 eV [16]) than the HSE06 function (0.670 eV). Thus, in the following calculations, we employed the PBE functional to discuss the properties. One can see from Figure 1a that the Pd dopant captured on the right top of the N atom is bonded with it and three neighboring In atoms of the InN monolayer. The new formed Pd-N and Pd-In bonds were measured as 2.08 and 2.84 A, respectively. After doping, the Pd atom was positively charged by 0.139 e, which indicated its electron-donating property transferring charge to the InN surface. As shown in Figure 1b, it was found that, in the Pd dopant, three bonded In atoms were encompassed with electron depletion, while the N atom under the Pd dopant were somewhat surrounded by electron accumulation. Thus, there is no wonder that strong electron accumulation is localized on the Pd-In and Pd-N bonds that account for their formation. These findings imply the electron-transfer from the Pd dopant to the InN layer and the remarkable electron hybridization on the Pd-In and Pd-N bonds.  
Figure 2 depicts the density of state (DOS) distribution of the Pd-InN system. As can be seen in Figure 2a, several novel peaks appeared that around \(-3\), \(-1\), and \(2.5\) eV in the Pd-doped surface, compared with the total DOS of the intrinsic InN counterpart. This could be attributed to the introduction of the Pd atom on the InN surface., which induces several impurity states around the Fermi level. As a consequence of this, the DOS peaks in the Pd-InN monolayer below the Fermi level, where the Pd dopant contributes remarkably, are dramatically increased. From Figure 2b, one can find that the state contributions of the Pd dopant are mainly from its \(4d\) orbital, which behaves actively and hybridizes strongly with the N \(2p\) and In \(5p\) orbitals of the bonding atoms. One can see that the hybridization between the Pd \(4d\) orbital and the In \(5p\) orbital is much stronger than that with N \(2p\), indicating the stronger binding force of Pd-In bonds. This is in agreement with the larger electron accumulation on the Pd-In bonds, which verifies the stable adsorption of Pd dopant on the InN monolayer with the formation of chemical bonds. In addition, from the atomic DOS, we could also conclude that the top of the valence band is mainly localized on the Pd atom and the bottom of the conduction band is mainly on the In atoms [31], which supports the electron-donating behavior of the Pd dopant.