## 3 Results and Discussions
The lattice parameters and the formation energy obtained from the first-principles calculations are shown in Table 3. The only increase in the lattice parameter of the c-axis was for Ag-interstitial (Agi), while this value decreased for the other models. It is fairly difficult to compare the absolute value of the differences in the lattice parameters with the experimental calculations, because the amount of Ag in the modeled alloys is much larger than that in the experiment. The results of the formation energy also support Agi in Bi\({}_{2}\)Te\({}_{3}\). The lowest energy of formation was obtained for Agi. From the results of the lattice parameters and the first-principles calculations, it may be thought that the reduction in electron concentrations is due to the Ag-interstitial. However, it can be expected that metal atoms at the interstitial site will donate electrons, following the reaction equation  
\[\text{Ag}_{\text{s}}\xrightarrow[\text{Bi}_{2}(\text{Te},\text{Se})_{3}]\text {Ag}_{\text{i}}^{\circ}+e^{\prime} \tag{1}\]  
where Ag\({}_{\text{s}}\) means solid and nonactivated Ag atoms. Accordingly, if Ag is interstitial, electron concentrations should increase with Ag, which is contrary to the results of the transport properties. To understand the mechanism for the change in carrier concentration with Ag addition, we then considered the intrinsic defects, because the carrier concentrations in Bi\({}_{2}\)Te\({}_{3}\) related compounds are mainly determined by such defects. For n-type transport properties, it is known that the n-type TeBi antisite defects (Te at Bi sites) are the first major defects, and the second major defects are the n-type vacancies of Te (VTe) [7; 30; 31; 32]. Because the concentration of antisite defects is highly dependent on the average difference in electronegativity between cations and anions [7; 9], in Bi\({}_{2}\)Te\({}_{3}\) dopants such as In and Pb interact with the native defects, and this changes the concentration of the defects [9; 13; 33]. It was thought that the addition of Ag might also interact with the antisite defects or vacancies, resulting in a reduction in electron concentration. We subsequently confirmed the interaction of Ag with the defects. The Te metal phase in the samples with Ag was clearly observed in SEM images, as shown in Figure 4. The interaction of Ag with TeBi decreased the concentration of TeBi so that the remaining Te, which could not be included in the lattice, was then expelled as a metal phase. \begin{table}
\begin{tabular}{c c c c} \hline
**Sample** & \(R_{\text{wp}}\) **(\%)** & **a (Å)** & **c (Å)** \\ \hline Ag = 0.0 & 8.44 & 4.3377(1) & 30.2611(1) \\ Ag = 0.01 & 8.61 & 4.3375(1) & 30.2641(1) \\ Ag = 0.03 & 7.82 & 4.3380(1) & 30.2768(1) \\ Ag = 0.05 & 7.97 & 4.3387(1) & 30.2774(1) \\ \hline \end{tabular}
\end{table} 
Table 2: Refined lattice parameters and R-weighted factors (\(R_{\text{wp}}\)) for the heat-treated Bi\({}_{2}\)Te\({}_{3}\) and heat-treated Ag doped Bi\({}_{2}\)Te\({}_{3}\) samples obtained from the Rietveld refinement using X-ray powder diffraction data at 297 K. \begin{table}
\begin{tabular}{c c c c c} \hline
**Sample** & \multicolumn{2}{c}{**Lattice Parameters (Å)**} & \multicolumn{2}{c}{**Formation Energy (eV)**} \\ \cline{2-5}  & **a** & **c** & **Te-Rich** & **Bi-Rich** \\ \hline Bi\({}_{2}\)Te\({}_{3}\) & 4.421 & 30.478 & - & - \\ Agi & 4.443 & 30.660 & 0.68 & 0.68 \\ AgBi & 4.408 & 30.211 & 1.07 & 1.47 \\ AgTe(6c) & 4.422 & 30.046 & 1.83 & 1.57 \\ AgTe(3a) & 4.415 & 30.323 & 1.89 & 1.63 \\ \hline \end{tabular}
\end{table} 
Table 3: The optimized lattice parameters and formation energy from the first-principles calculation. The values of pristine Bi\({}_{2}\)Te\({}_{3}\), Ag interstitial (Agi)-doped Bi\({}_{2}\)Te\({}_{3}\), Bi\({}_{2}\)Te\({}_{3}\) with Ag substituted for Bi (AgBi), and Bi\({}_{2}\)Te\({}_{3}\) with Ag substituted for Te sites (6c and 3a) are listed. However, it is also important to consider the possibility of interaction between the Ag interstitials and V\({}_{\text{Te}}\). [PERSON] et al.