## 3 Results  
The atomic structure of the predicted Cu\({}_{2}\)TeO\({}_{6}\) monolayer is shown in Figure 1a. It has a monoclinic unit cell, similarly to its bulk counterpart. Structural parameters of the predicted Cu\({}_{2}\)TeO\({}_{6}\) monolayer can be found in Table 1. Firstly, to show a kinetic stability of the predicted Cu\({}_{2}\)TeO\({}_{6}\) monolayer, the phonon dispersion spectra along the high symmetry directions \(\Gamma\)\(\rightarrow\)Y\(\rightarrow\)M\(\rightarrow\)A\(\rightarrow\)\(\Gamma\)\(\rightarrow\)L2\(\rightarrow\)\(\Gamma\)\(\rightarrow\)V2 of the Brillouin zone for the Cu\({}_{2}\)TeO\({}_{6}\) monolayer were calculated. According to Figure 1b, the phonon dispersion curves of the Cu\({}_{2}\)TeO\({}_{6}\) monolayer are positive in whole of the Brillouin zone, and the transverse, longitudinal, and out-of-plane acoustic modes show a normal linear dispersion around the \(\Gamma\) point, which confirms the kinetic stability of the Cu\({}_{2}\)TeO\({}_{6}\) monolayer. Figure 1: The atomic structure of the unit cell **(a)** and phonon dispersion curve **(b)** for the Cu\({}_{2}\)TeO\({}_{6}\) monolayer. Thermal stability of the Cu\({}_{2}\)TeO\({}_{6}\) monolayer is verified though the estimation of its formation energy \(E_{\text{form}}\):  
\[E_{\text{form}}=E_{\text{tot}}-N_{\text{Cu}}\cdot E_{\text{Cu}}-N_{\text{Te}} \cdot E_{\text{Te}}-N_{\text{O}}\cdot E_{\text{O}} \tag{1}\]  
where \(E_{\text{tot}}\) is the total energy of the Cu\({}_{2}\)TeO\({}_{6}\) monolayer, \(E_{\text{Cu}}\), \(E_{\text{Te}}\), and \(E_{\text{O}}\) are the energies of single Cu, Te, and O atoms, and \(N_{\text{Cu}}\), \(N_{\text{Te}}\), and \(N_{\text{O}}\) are the numbers of the Cu, Te, and O atoms. The negative \(E_{\text{form}}=-1.72\) eV/atom that was found suggests the thermodynamic stability of the Cu\({}_{2}\)TeO\({}_{6}\) monolayer. To confirm the thermal stability of the materials, AIMD simulations conducted at a room temperature of 300 K are usually used. Figure 2a shows the total energy fluctuation of the Cu\({}_{2}\)TeO\({}_{6}\) monolayer system at 300 K for the time of 4 ps. It can be seen that there were no artificial energy fluctuations. In addition, according to Figure 2b, there were no visible changes in the structure of the Cu\({}_{2}\)TeO\({}_{6}\) monolayer happening after 4 ps, which suggests its thermal stability. To clarify the application temperature range of Cu\({}_{2}\)TeO\({}_{6}\) monolayer, its stability at higher temperature of 320 K for the time of 4 ps was also evaluated via an AIMD simulation. Figure 2c presents the total energy fluctuation of the Cu\({}_{2}\)TeO\({}_{6}\) monolayer system at 320 K for the time of 4 ps. It is seen that there are also no artificial energy fluctuations. In addition, according to Figure 2d, there were no structural changes in the Cu\({}_{2}\)TeO\({}_{6}\) monolayer after 4 ps, which suggests its thermal stability at 320 K.  
To identify the bonding type in the Cu\({}_{2}\)TeO\({}_{6}\) monolayer, the electron localization function (ELF) is simulated and analyzed. It is known that the ELF value, which is in the range from 0 to 1, reflects the degree of charge localization in the real space, where the values close to 0 represent the existence a free electronic state, while the values close to 1 represent a perfect electron localization [37; 38; 39]. The ELF plot for the Cu\({}_{2}\)TeO\({}_{6}\) monolayer with the isosurface value of 0.70 is presented in Figure 3a. According to Figure 3a, the electron localization is high at the regions around the sites of the O anions, while at the Cu and Te cations sites the value of ELF is low. A large difference found in the ELF plot proposes the existence of an ionic bonding in the Cu\({}_{2}\)TeO\({}_{6}\) monolayer. This conclusion is \begin{table}
\begin{tabular}{l c} \hline a & 5.272 Å \\ b & 5.272 Å \\ \(\alpha\) & 100.25 \\ \(\beta\) & 100.25 \\ \(\gamma\) & 112.43 \\ \hline \end{tabular}
\end{table} 
Table 1: Structural parameters of the Cu\({}_{2}\)TeO\({}_{6}\) monolayer.