Ultrafast electron dynamics at the Dirac node of the topological insulator Sb2Te3

Topological insulators (TIs) are a new quantum state of matter. Their surfaces and interfaces act as a topological boundary to generate massless Dirac fermions with spin-helical textures. Investigation of fermion dynamics near the Dirac point (DP) is crucial for the future development of spintronic devices incorporating topological insulators. However, research so far has been unsatisfactory because of a substantial overlap with the bulk valence band and a lack of a completely unoccupied DP. Here, we explore the surface Dirac fermion dynamics in the TI Sb2Te3 by time- and angle-resolved photoemission spectroscopy (TrARPES). Sb2Te3 has an in-gap DP located completely above the Fermi energy (EF). The excited electrons in the upper Dirac cone stay longer than those below the DP to form an inverted population. This was attributed to a reduced density of states (DOS) near the DP.


Fig
. S1(a) shows the difference image along the Γ−K line measured at t = 0.26 ps. Here the bulk conduction band and two branches of UDC are represented as Bulk, Surface A and Surface B, respectively. We set energy and momentum frames and plotted the normalized intensity variation in each frame as a function of t [see Fig. S1(b)-(i)]. The intensities in the same energy range are normalized. The intensity variations of bulk and surface states do not show any significant differences. That tells us that the bulk and surface electron dynamics only depend on the energy.

II Simulation of decay behavior from thermal distribution
Here we show how the excited electrons decay via electronic temperature cooling.
First of all, we show the Fermi Dirac distribution function with a temperature T = T 0 at the 'zero' delay time (t=0), as shown in the inset panel of Fig. S2. Then we assume that T exponentially decays as described with T = T 1 + (T 0 − T 1 ) * exp(−t/τ ), where T 0 , T 1 denote the initial and the equilibrium temperatures, and τ expresses the decay rate. We plot the electron occupation probabilities at five binding energies E1, E2, E3, E4 and E5 (marked in the inset panel) as shown in Fig. S2. We can find that the decay rate increases as the energy approaches the Fermi energy. The result of this simulation shows that the thermal decay cannot explain the observed population inversion.

III Simulation of population inversion
To qualitatively demonstrate the population inversion, we draw the decay lines for the uniform DOS with the non-uniform one. Here we consider a simple model with 10 energy windows, marked as S0-S9. We assume that the electron transfer takes place only between the adjacent windows. The binding energy S9 is the highest and that of S0 is the lowest. The intensity of S0 is assumed to be caused only by a direct excitation.
The intensity of S9 is considered to follow an exponential decay.
Considering a uniform DOS, as shown in the upper panel of Fig. S3 (c), we can simulate the decay lines of different windows. As shown in Fig. S3 (a), the high binding energy window shows the earlier rising edge Therefore the uniform DOS cannot be the origin of the population inversion.
The non-uniform DOS, where the DOS in the region S7 is reduced down to to 10% of the others is assumed for the simulation as shown in the lower panel of Fig. S3 (c).
Here, all the other parameters are set to the same values as those used for the uniform DOS. As shown in Fig. S3 (b), a big change occurs for the regions S6-S8, while the higher energy regions do not change significantly. The intensity of S8 shifts to earlier delay time than that of S6, just like what we experimentally observed for UDC and LDC. This result can reasonably explain that the observed population inversion takes place due to the bottleneck effect near Dirac node.