Bulk diffusive relaxation mechanisms in optically excited topological insulators

The possibility to inject spin currents in topological insulators (TIs) by ultrashort optical pulses has stimulated intense studies of their out-of-equilibrium electronic properties. However, a comprehensive description of the electronic relaxation dynamics has been elusive, so far. In order to reveal the role of the bulk and surface states in the microscopic scattering mechanisms, we have investigated, by means of time- and angle-resolved photoemission spectroscopy, a wide set of TIs. These have been selected in order to display different positions of the Fermi energy ($E_{\text{F}}$) within the bulk band structure. When three-dimensional bulk states lie at $E_{\text{F}}$, we observe a fast relaxation dynamics with a characteristic time scale of a few picoseconds (ps). On the contrary, a long lasting excited state is recorded when only the two-dimensional surface state crosses $E_{\text{F}}$. These findings suggest the important role played by spatial diffusion in the direction orthogonal to the surface in governing the relaxation mechanisms. We propose that this electron diffusive mechanism is driven by the optically induced temperature gradient that is at play only for electrons residing in bulk states.

Figure 1

(a)–(h) ARPES images of the band structure measured along the $\overline{\mathrm{\Gamma }K}$ high symmetry direction. (a) b-${\mathrm{GeBi}}_2{\mathrm{Te}}_4$ and (b) ${\mathrm{GeBi}}_4{\mathrm{Te}}_7$ are $n$-type TIs, with the 3D bulk CB lying at $E_{\text{F}}$. The data of (a) have been measured for negative wave vectors and successively symmetrized. In the case of (c) ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ and (d) v-${\mathrm{GeBi}}_2{\mathrm{Te}}_4$, $E_{\text{F}}$ lies within the bulk band gap. $E_{\text{F}}$ is only crossed by the 2D SS and the materials are intrinsic TIs. (e) and (f) Differential images obtained from the difference between the measured band dispersion after ($+500$ fs) and before ($-500$ fs) optical excitation for (e) ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ and (f) v-${\mathrm{GeBi}}_2{\mathrm{Te}}_4$. For (g) ${\mathrm{Sb}}_2{\mathrm{Te}}_3$ and (h) ${\mathrm{Sb}}_6{\mathrm{Te}}_3$, $E_{\text{F}}$ is crossed by the 3D bulk VB and the samples are $p$-type TIs. The SS lies completely in the unoccupied density of states, and the data refer to the transient electronic population excited by the pump pulse. (i)–(n) Temporal evolution of the corresponding electronic population after optical excitation, as integrated in the three regions indicated by colored rectangles in (a)–(d), (g), and (h). The dynamics for the $n$- and $p$-type TIs are comparable and characterized by relaxation times of a few ps, regardless of the specific position of $E_{\text{F}}$ either within CB [(i) and (j)] or VB [(m) and (n)]. On the contrary, for intrinsic TIs [(k) and (l)] the dynamics is much slower, and the intensity does not recover the equilibrium value in the investigated temporal window, exceeding 60 ps for ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ (k).

Figure 2

(a)–(f) Intensity distribution curves as a function of energy (EDCs) extracted along the SS dispersion for (a) b-${\mathrm{GeBi}}_2{\mathrm{Te}}_4$, (b) ${\mathrm{GeBi}}_4{\mathrm{Te}}_7$, (c) ${\mathrm{Bi}}_2{\mathrm{Te}}_3$, and (d) v-${\mathrm{GeBi}}_2{\mathrm{Te}}_4$. For the $p$-type TIs (e) ${\mathrm{Sb}}_2{\mathrm{Te}}_3$ and (f) ${\mathrm{Sb}}_6{\mathrm{Te}}_3$, the EDCs are taken along the dispersion of the bulk VB crossing $E_{\text{F}}$. For each compound, the three EDCs are taken immediately before (blue), after (red), and 5 ps after (orange) the arrival of the optical excitation. Black dashed lines show the best fits to the data.

Figure 3

(a)–(f) Temporal evolutions of $T_e$ as extracted from the fit to the EDCs of Fig. 2. For the (a) and (b) $n$- and (e) and (f) $p$-type TIs, $T_e$ recovers the equilibrium value with characteristic time $\tau$ of a few ps. On the contrary, in intrinsic TIs [(c) and (d)] $T_e$ does not relax back to the equilibrium value, but it forms a plateau, indicated by the brawn area. (g)–(i) Schematization of the proposed diffusion mechanism in the bulk [(g) and (i)] and at surface (h).