## 3 Results and Discussions
Thus, the one of the main reasons for the decrease in \(S\) by Hf-doping (Figure 2b) is increased \(n_{\rm H}\). On the other hand, \(\mu_{\rm H}\) is slightly decreased by Hf-doping due to the increase both in electron-electron scattering and carrier scattering by point defects (Hf\({}_{\rm{Bi}}\)). The significant increase in the power factor (inset of Figure 2b) observed in Cu\({}_{0.01}\)Bi\({}_{1.95}\)Hf\({}_{0.05}\)Te\({}_{2.7}\)Se\({}_{0.3}\) suggests that the band structure would be modified by Hf-doping. Because the DOS effective mass (\(m_{d}\)\({}^{*}\)) is the critical parameter that determines \(S\), we estimated \(m_{d}\)\({}^{*}\) based on the following Equation (1), assuming a single parabolic band model [15], where \(e\), \(h\), and \(k_{\rm B}\) are the elementary charge, [PERSON]'s constant, and Boltzmann constant, respectively. \[S=\frac{8\pi^{2}\,k_{B}{}^{2}}{3e\hbar^{2}}\bigg{(}\frac{\pi}{3n_{H}}\bigg{)}^ {2/3}m_{d}^{*}T \tag{1}\]  
Figure 3: (a) Hall carrier concentration (\(n_{\rm H}\)) and Hall mobility (\(\mu_{\rm H}\)) as a function of Hf-doping content. (b) Pisarenko plot (ISI-\(n_{\rm H}\)) for the SPSed Cu\({}_{0.01}\)Bi\({}_{2.7}\)Hf\({}_{x}\)Te\({}_{2.7}\)Se\({}_{0.3}\) (\(x\) = 0, 0.05, 0.075, 0.1, 0.125) bulks. Figure 2: Temperature dependences of the (**a**) electrical conductivity (\(\sigma\)) and (**b**) Seebeck coefficient (\(S\)) for the SPSed Cu\({}_{0.01}\)Bi\({}_{2.7}\)Hf\({}_{x}\)Te\({}_{2.7}\)Se\({}_{0.3}\) (\(x\) = 0, 0.05, 0.075, 0.1, 0.125) bulks. Insets in (**a**) and (**b**) show the thermoelectric measurement orientations and the temperature-dependent power factor (\(S^{2}\sigma\)), respectively. Figure 3b shows the Pisarenko plot (ISI-\(n_{\rm H}\)) for the SPSed Cu\({}_{0.01}\)Bi\({}_{2\cdot x}\)Hf\({}_{x}\)Te\({}_{2.7}\)Se\({}_{0.3}\) bulks. Interestingly, the \(m_{d}\) value at 300 K of the pristine Cu\({}_{0.01}\)Bi\({}_{2}\)Te\({}_{2.7}\)Se\({}_{0.3}\) (\(\sim\)0.92 \(m_{0}\), where \(m_{0}\) is the free electron mass) is increased to \(\sim\)1.24 \(m_{0}\) by Hf-doping, indicating that the electronic structure of the conduction band is modified. The possible mechanisms for band modification are band convergence, band flattening, or resonant state formation by Hf-doping [16]. These results observed in the electronic transport parameters suggest that the improved power factor of Cu\({}_{0.01}\)Bi\({}_{1.925}\)Hf\({}_{0.075}\)Te\({}_{2.7}\)Se\({}_{0.3}\) is mainly due to the increase in \(n_{\rm H}\) without severe deterioration in \(\mu_{\rm H}\). Doping of Hf at the Bi-site generates point defects (Hf\({}_{\rm Bi}\)), and it can act as a scattering center, especially for a high-frequency phonon. To confirm the effect of Hf-doping on the thermal transport properties of Cu\({}_{0.01}\)Bi\({}_{2}\)Te\({}_{2.7}\)Se\({}_{0.3}\), \(\kappa_{\rm lat}\) was calculated by subtracting \(\kappa_{\rm ele}\) from the measured \(\kappa\) (Figure 4a). Figure 4b shows the temperature dependence of \(\kappa_{\rm ele}\), which is estimated based on the Wiedemann-Franz law (\(\kappa_{\rm ele}=L\sigma T\)). The temperature-dependent Lorenz number (\(L\) in W \(\Omega\) K\({}^{-2}\)) was calculated by using the following Equation (2) [17] and represented in the inset of Figure 4b. \[L=1.5+\exp\biggl{(}-\frac{|S|}{116}\biggr{)} \tag{2}\]  
As shown in Figure 4a, the acquired temperature-dependent behavior is not ruled by a typical \(T^{-1}\) relationship, suggesting that the bipolar thermal conduction (\(\kappa_{\rm bp}\)) is included, and its contribution increases with temperature. To clarify the point defect phonon scattering effect, we compared the room temperature \(\kappa_{\rm lat}\) values since the \(\kappa_{\rm bp}\) can be neglected at low temperatures.