## 3 Results and Discussion
### Application as Anode Materials for Li, Na, K, Mg and Ca-Ions Batteries
The ion conductivity of electrode materials is among the most crucial parameters for the optimal charging and discharging rate. Fast charging processes are needed for electric vehicles and mobile devices. Therefore, we studied the ion conductivity over the P-TpG monolayer to explore Li, Na, K, and Ca ion diffusion rates. The climbing image NEB method is conducted for two hoping paths using most stable adsorption sites (Figure 3). Figure 9 shows the corresponding hoping diffusion paths and barrier energies for considered ions. Path 1 and path 2 start from the S1 adsorption site while it has the highest adsorption energy value for all considered ions. Path1 follows the hoping of the ions to the second most stable adsorption site (S2 adsorption site in Figure 3) over the center of the C4P2 hexagonal ring jumping over the full carbon hexagonal ring and finally to the next C4P2 hexagonal ring. Path2 tracks the hoping of the ions to the second most stable adsorption site then jumps to the S4 adsorption site in the hallowed part and follows the path to the top left site over the C4P2 hexagonal ring. Figure 9 depicts the predicted images between two hopping targets and the correspo  
\[E_{Ad}=E_{S}-E_{P}-E_{At} \tag{1}\]0 Planar and lateral view of the most stable or lowest-energy structure of P-TpG nanosheet with (**a**) 38 Li; (**b**) 30 Na; (**c**) 18 K; (**d**) 18 Ca adatoms content. \[E_{Ad}=E_{S}-E_{P}-E_{At} \tag{1}\]1 Electronic density of states (DOS) of P-TpG nanosheets with and without diffusive ions. The vertical dashed line represents the Fermi level and is modified to zero. \[E_{Ad}=E_{S}-E_{P}-E_{At} \tag{1}\]2 Energy barriers along Path1 and Path2 are predicted by the nudged-elastic band method for a single Li, Na, K, Ca ions diffusion over the monolayer P-TpG. According to Table 2, the paths start from the first most stable site. The maximum barrier energies for Li, Na, K, and Ca ions take values of 0.75, 1, 0.78, 1.42 eV, respectively. The K ion has the lowest barrier energy for the rest of the hopping paths, which is also lower than the Li-ion barrier energy of graphene (0.37 eV) [69]. The Li-ion diffusion over the hexagonal ring in path 1 N-TpG (0.37 eV) [37] has a lower energy barrier than P-TpG (0.44 eV). Diffusion energy barriers play an important role in the diffusion rate of intercalation electrodes. The optimized charging/discharging rate strongly depends on the diffusive ions kinetic on an electrode. Intuitively lower energy barriers deliver a fast diffusion rate. However, diffusion rates can be affected by multiple diffusive ion interactions and stacking electrode materials [70; 71]. As shown in Table 2, Li and K ions are energetically preferential ions to deliver a high diffusion rate in the first and second pathways. The stacking structure is an important influence on ions diffusion within changing/discharging and is one of the possible ways to enhance anode materials performance [70; 71]. Bilayer structures of the graphene [72] considered as anode material for Li-ion batteries. Their results confirmed the higher diffusion rate in AB-stacking structure than AA-stacking structures. The dominant potential analysis indicated stacking structure highly affected the height of potential well. In other study, stacking structure effects of bilayer black phosphorene were investigated on Li-ions adsorption and diffusion [73]. The AC-stacking exhibited most favorable results for Li adsorption and diffusion. Their results revealed stacking structure has very important effects on ion adsorption, open-circuit voltage, and diffusion and indicated appropriate stacking structure improves the anodic behaviour of the material. Figure 9 shows various position-dependent energy barriers for different ions. The ions choose the paths with the lowest barrier energies for passing during charging/discharging. The porous structure of the TpG materials helps the ions' in-plane paths, resulting in a faster charging process.