## 3 Results and Discussion
Point (ii) can be interpreted in terms of intra-band scattering processes. Referring to Figure 3, the p-doped samples (panels a and b) have their DN well above the Fermi level. Thus, both the BB2 and the upper branch of the TSS have unoccupied band minima at \(\Gamma\) (the center of the Brillouin zone) under equilibrium conditions. As photo-excited electrons decay on BB2 and TSS states from higher levels, they accumulate towards the respective band minima through intra-band scattering. This process happens on a timescale that is clearly longer than the BB1 relaxation and it is particularly critical for the TSS since the DN corresponds to a vanishing DOS, i.e., fewer and fewer electrons can be accommodated as the DN is approached. On the other hand, for n-doped samples (panels d, e, and f of Figure 3) the band minimum lies below the Fermi level and the unoccupied portions of BB2 and TSS bands are away from \(\Gamma\). In this case, the intensity build-up time is compatible with the BB1 relaxation time (compare lighter curves of panels a and c in Figure 4). As electrons decay from BB1 to BB2 and TSS, they rapidly disperse into the Fermi sea since they cannot accumulate on any band minimum. We also point out that, according to our analysis, we have no evidence of direct, photo-induced population of BB2 and the upper branch of TSS. However, in Figure 4d we compare the electronic dynamics of the upper and lower branches of the TSS in Sb\({}_{2}\)Te\({}_{3}\) (boxes 3 and 4 of Figure 3a, respectively). Although relaxation times are quite similar, the lower branch of the TSS shows a faster intensity build-up and a reduced spectral weight (about 4 times lower) compared to the higher branch. The faster build-up is in agreement with the absence of a band minimum (actually in the lower branch there is a maximum), while the reduced spectral weight is in part attributed to the lack of electron accumulation  
Figure 4: Summary of the electronic dynamics for different bands and doping levels (same color code as Figure 3) of the BSTS samples. The curves are normalized to their respective maxima. (**a**) BB1 band dynamics (box 1 in Figure 3). The red dashed line is a phenomenological fit and it has been reproduced in all panels for comparison. (**b**) BB2 band dynamics (box 2 in Figure 3). (**c**) TSS band dynamics (box 3 in Figure 3). The intensity build-up is faster for p-doped samples (as evidenced by the red arrow). Notice the extremely long electron lifetime for the undoped samples (DN at the Fermi level). (**d**) Comparative dynamics of the upper and lower branches of the TSS in Sb\({}_{2}\)Te\({}_{3}\) (boxes 3 and 4 of Figure 3a). and in part the consequence of the bottleneck effect: intra-band scattering from the higher branch of the cone is hindered by the vanishing DOS at the Dirac node. Point (iii)--the extremely long lifetime of TSS electrons when the DN is at the Fermi level--is the most intriguing feature. Figure 5 summarizes the analysis of the TSS dynamics. Panel (a) reports a typical phenomenological fit (the data refer to the Sb\({}_{2}\)Te\({}_{3}\) sample): an exponential growth with rise time \(\tau_{t}\) followed by an exponential decay with characteristic time \(\tau_{d}\). The fitting function has been convoluted with the Gaussian IRF of our experimental setup. The obtained values of the time constants \(\tau_{t}\) and \(\tau_{d}\) are reported in panels (b) and (c), respectively, as a function of the DN binding energy (i.e., doping). As already discussed, the data of panel (b) show that p-doped samples are characterized by a longer rise time (i.e., longer build-up) compared to the n-doped ones. And the latter nicely matches the decay time of BB1 states (220 fs, see Figure 4a). Figure 5c shows the decay time constants vs. doping. First, we point out that the higher the doping level (regardless of \"n\" or \"p\"), the faster the electronic decay. This behavior is most likely related to bulk contribution in the relaxation process (bulk diffusion according to Reference [18]).