## 3 Result and Discussion
We used a three-electrode system for the TMTe\({}_{2}\) working electrode (WE), platinum wire for the counter electrode (CE), and Ag/AgCl for the reference electrode (RE). We applied an external voltage between the WE and RE and measured the current density between the WE and CE. _TM_Te\({}_{2}\) single crystals were used for the WE and the sample area of the WE was estimated by an analyzing program (Leopard 2.5) with optical images. The electrochemical HER performance of the three _TM_Te\({}_{2}\) electrodes was examined by linear sweep voltammetry (LSV) and electrochemical impedance spectroscopy (EIS). Figure 2b shows that the LSV curves of three _TM_Te\({}_{2}\) electrodes and the overpotentials of \(-\)441, \(-\)668, and \(-\)692 mV were required to obtain 10 mA/cm\({}^{2}\) for VTe\({}_{2}\), MoTe\({}_{2}\), and WTe\({}_{2}\), respectively. The overpotential of VTe\({}_{2}\) was found to be the smallest among three _TM_Te\({}_{2}\) electrodes. Figure 2c shows the Tafel slopes of the three _TM_Te\({}_{2}\) electrodes were 70, 137, and 169 mV/dec for VTe\({}_{2}\), MoTe\({}_{2}\), and WTe\({}_{2}\), respectively, and VTe\({}_{2}\) exhibited the lowest Tafel slope among them. The overpotential and the Tafel slope of VTe\({}_{2}\) single crystal were consistent with reported values in micro-sized VTe\({}_{2}\) flakes, which were transferred to the glassy carbon working electrode by drop-cast method, indicating that the basal plane of VTe\({}_{2}\) can be active for HER [35]. We extracted the exchange current densities of three _TM_Te\({}_{2}\) from extended linear lines of Tafel slopes down to zero value in the overpotential, 0.2, 0.7, and 1.0 \(\mu\)A/cm\({}^{2}\) for VTe\({}_{2}\), MoTe\({}_{2}\), and WTe\({}_{2}\), respectively. To analyze the catalytic activity of the \(\alpha\) site in the _TM_Te\({}_{2}\), we calculated the turn over frequency (TOF), which is proportional to the conversion rate from proton to hydrogen gas at one active site for one second. The TOF was calculated by equation TOF = \(\frac{\
u\times\lambda\times N_{A}}{S\times\lambda\times N_{A}}\), where \(j\) is the exchange current density, \(n\) is the number of electrons transferred per molecule (2 for HER), \(F\) is the Faraday constant (96,458 C mol\({}^{-1}\) electrons), \(A\) is the superficial electrode area, and \(N_{A}\) is Avogadro's number (6.022 \(\times\) 10\({}^{23}\) mol\({}^{-1}\)). The TOF values were 2.14, 4.85, and 6.92 ms\({}^{-1}\) for VTe\({}_{2}\), MoTe\({}_{2}\), and WTe\({}_{2}\), respectively, and WTe\({}_{2}\) had the largest TOF value among the three. It seems that the TOF results contradict the reported Gibbs free energies of \(\alpha\) sites in MoTe\({}_{2}\) (\(\Delta\)G\({}_{\rm{H}}\) = 0.77 eV) and WTe\({}_{2}\) (\(\Delta\)G\({}_{\rm{H}}\) = 1.14 eV) [17; 18]. This is explained by the fact that the hydrogen adsorption at the catalytic active sites is less critical in determining the hydrogen conversion rate during the HER process. To study the charge transfer process for the HER, we measured the electrochemical impedance spectroscopy (EIS) with the three _TM_Te\({}_{2}\) electrodes. Figure 2d shows the Nyquist plots of the three _TM_Te\({}_{2}\) electrodes with the complex impedance, the real part \(Z_{\rm{Re}}\), and the imaginary part \(-Z_{\rm{Im}}\) with various frequencies from 100 kHz to 100 mHz. We included the Warburg impedance (w) on the \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Catalyst**} & \multirow{2}{*}{**Overpotential** **(mV vs. RHE)**} & **Tafel Slope** & \multirow{2}{*}{**Exchange Current** **(mV/dec)**} & \multirow{2}{*}{**\(\Phi\) site** **(m/cm\({}^{2}\))**} & **TOF** **(ms-)** & **Double Layer** **(m/cm\({}^{2}\))** & **Charge Transfer Resistance** \\  & & & & & & & **Resistance** \\ \hline VTe\({}_{2}\) & \(-\)441 & 70 & 0.2 & 2.92 & 2.14 & 0.03 & 5.8 \\ MoTe\({}_{2}\) & \(-\)668 & 137 & 0.7 & 4.50 & 4.85 & 0.08 & 13.0 \\ WTe\({}_{2}\) & \(-\)692 & 169 & 1.0 & 4.51 & 6.92 & 0.06 & 21.6 \\ \hline \end{tabular}
\end{table}