Surface State Dynamics of Topological Insulators Investigated by Femtosecond Time- and Angle-Resolved Photoemission Spectroscopy

Abstract

Topological insulators (TI) are known for striking quantum phenomena associated with their spin-polarized topological surface state (TSS). The latter in particular forms a Dirac cone that bridges the energy gap between valence and conduction bands, providing a unique opportunity for prospective device applications. In TI of the Bi_xSb_2−xTe_ySe_3−y (BSTS) family, stoichiometry determines the morphology and position of the Dirac cone with respect to the Fermi level. In order to engineer specific transport properties, a careful tuning of the TSS is highly desired. Therefore, we have systematically explored BSTS samples with different stoichiometries by time- and angle-resolved photoemission spectroscopy (TARPES). This technique provides snapshots of the electronic structure and discloses the carrier dynamics in surface and bulk states, providing crucial information for the design of electro-spin current devices. Our results reveal the central role of doping level on the Dirac cone structure and its femtosecond dynamics. In particular, an extraordinarily long TSS lifetime is observed when the the vertex of the Dirac cone lies at the Fermi level.

1. Introduction

Due to their unconventional electronic structure, topological insulators (TI) are potential candidates for next-generation electronic and spintronic devices [1,2,3]. They behave like standard insulators or semiconductors in the bulk, but stimulating physics takes place at their surface. Here, the fermion gas forms a conductive state with a Dirac cone structure in the Brillouin zone and a helical spin texture. Recent studies have shown several unique features of TI, including observations of the quantum spin Hall effect [4], Majorana fermions [5], the topological magnetoelectric effect [6], magnetic monopoles [7], nonlinear modulation of the electromagnetic field [8], and the anomalous quantum Hall effect [9]. In practice, some difficulties have prevented the widespread use of TI in mainstream electronic components. The major obstacle is the large contribution of bulk electrons that hamper surface transport properties [10]. It has been shown that a reduction of bulk effects leads to better control of the interesting surface phenomena. In particular, focusing on the Bi${}_x$Sb${}_{2-x}$Te${}_y$Se${}_{3-y}$ (BSTS) family, some of the most promising ternary and quaternary compounds were identified as Bi${}_2$Te${}_2$Se [11], Bi${}_x$Sb${}_{x-2}$Te${}_3$ [12], Bi${}_{0.5}$Sb${}_{1.5}$Te${}_{1.8}$Se${}_{1.2}$ [13] and BiSbTeSe${}_2$ [14]. These investigations have shown that fine-tuning the BSTS stoichiometry allows one to control the Dirac cone structure and, consequently, the electronic and spin properties of the TI [15,16,17,18].

From the experimental point of view, angle-resolved photoemission spectroscopy (ARPES) has been a key tool to disclose the main features of TI, as it provides a direct picture of the occupied electronic states in the Brillouin zone [19,20,21]. Being conductive, the topological surface states (TSS) are only partially filled. However, complete knowledge of their structure would be highly desirable, especially if one wishes to tailor specific charge and spin transport properties. To overcome this limitation an advanced approach combines ARPES and pump-probe techniques, resulting in the so-called time-resolved ARPES (TARPES). Here, the sample is perturbed by a short laser pulse—the pump—which promotes electrons to unoccupied states above the Fermi level. Recording ARPES images during the excitation and relaxation processes, one can access the electronic structure of both occupied and empty states, and also reveal the main electronic scattering processes as thermodynamic equilibrium is restored after photoexcitation [22,23,24,25,26].

In this paper, we investigate the ultrafast electronic dynamics of selected BSTS samples by means of TARPES with sub-100 femtosecond time resolution. The binding energy of the Dirac node (DN), i.e., the energy where upper and lower branches of the Dirac cone meet, can be tuned from $-0.3$ eV to $+0.2$ eV, depending on the BSTS stoichiometry. The electron density of states (DOS) is drastically reduced in the vicinity of the DN, leading to a strong alteration of electronic interactions [27]. Therefore, BSTS samples offer an unprecedented opportunity to investigate the electronic behavior of the TSS in relation to its filling, i.e., to the position of the DN with respect to the Fermi level. Real applications require long-lasting electronic dynamics at a TI’s surface (which usually only last a few picoseconds). Our results show that Dirac carriers relax on an extraordinarily long timescale when the binding energy of the DN is vanishingly small. We believe this novel observation might provide critical information to guide the fabrication of innovative TI-based spintronic devices.

2. Materials and Methods

Bi${}_x$Sb${}_{2-x}$Te${}_y$Se${}_{3-y}$ (BSTS) single crystals were grown by a self-flux method. The high-purity raw materials were weighed and mixed according to the desired stoichiometry. The specimens were heated up to 1000 ${}^{\circ }$C for 12 h, then slowly cooled down to 500 ${}^{\circ }$C for over 100 h, before finally being cooled to room temperature. Single crystals of several millimeters in size were obtained (see Reference [28] for more details). Here, we focus on the following compounds: Sb${}_2$Te${}_3$, Bi${}_{0.5}$Sb${}_{1.5}$TeSe${}_2$, Bi${}_{0.8}$Sb${}_{1.2}$Te${}_3$, Bi${}_2$Te${}_3$, Bi${}_2$Te${}_{2.65}$Se${}_{0.35}$, and Bi${}_2$Te${}_{0.5}$Se${}_{2.5}$.

Figure 1 schematically illustrates our TARPES experimental apparatus. The laser source is a commercial Yb-based system (Pharos, Light Conversion) generating 300 fs pulses with photon energy $h\nu =1.2$ eV (wavelength $\lambda =1030$ nm) at 100 kHz repetition rate (for details about the optical system see Reference [29]). The laser feeds a home-built non-collinear optical parametric amplifier (NOPA) that provides ultrashort pump photons of energy $h\nu =1.85$ eV ($\lambda =670$ nm) with a pulse duration of 30 fs. A sum frequency generation (SFG) stage generates UV probe photons with energy $h\nu =6$ eV ($\lambda =206$ nm) and a pulse duration of 65 fs. The instrumental response function (IRF) given by the cross-correlation of pump and probe pulses, thus corresponds to an overall time resolution of about 72 fs. Pump and probe beams, focused respectively on spots of about 100 $\mathsf{\mu }$m and 50 $\mathsf{\mu }$m in diameter, converge on a sample placed inside an ultrahigh vacuum chamber (pressure $\approx {10}^{-10}$ mbar) and cleaved in situ before each measurement.

Photo-emitted electrons are detected by a home-built time of flight (ToF) spectrometer equipped with a micro channel plate (MCP). Rotation of the sample’s normal with respect to the analyzer axis provides the angular resolution (for more details of the ToF analyzer and acquisition system see Reference [30]). All spectra were recorded at room temperature.

3. Results and Discussion

Figure 2 reports TARPES measurements of selected BSTS samples. Each panel shows the spectra at three illustrative pump-probe delays: before pump arrival (left, “–” label), at zero delay (center, “0” label) and one picosecond after pumping (right, “+” label). The linear dispersion of the TSS is clearly observed for all compounds and marked with dashed lines to guide the eye. Moving from panel (a) to panel (f) in Figure 2, the amounts of Bi and Se increase at the expenses of Sb and Te and the samples become progressively n-doped. The Dirac cone of the most p-doped sample, Sb${}_2$Te${}_3$ in panel (a), is unoccupied at negative delays. Upon pumping, electrons are optically transferred above the Fermi energy (FE), filling the TSS. At the opposite extreme, the most n-doped sample, Bi${}_2$Te${}_{0.5}$Se${}_{2.5}$ in panel (f), has an almost completely occupied TSS even under equilibrium conditions (negative delay). It should be noticed that the DN of the Bi${}_{0.8}$Sb${}_{1.2}$Te${}_3$ sample, panel (c), has vanishing binding energy. As we will see in the next section, this feature has important consequences on the dynamics of Dirac fermions. We also point out that, regardless of the stoichiometry, at zero pump-probe delay two bulk bands (named BB1 and BB2 in Figure 2) are transiently populated, while after one picosecond most of the excess spectral weight induced by photoexcitation has been transferred from BB1 (higher energy) to BB2 (lower energy) and the TSS.

In order to better visualize the photo-excited band structure and dynamics reported in Figure 2, we have subtracted the equilibrium TARPES maps (panels marked with negative sign) from the pumped spectra (marked with zero and positive sign). The results are plotted in Figure 3 (same sample sequence as Figure 2). The upper and lower rows refer to the band structure at zero and at +1 ps delays, respectively. Photo-induced electronic population is represented in red, while band depletion is displayed in green. The large gray arrows connect all DN, revealing a regular 100 meV decrease of the TSS binding energy for adjacent samples. It is also interesting to observe that in the samples where the DN lies above the Fermi level (panels a and b), even one picosecond past the optical excitation (bottom row) the lower branch of the Dirac cone is not significantly populated. This dynamical behavior has been previously observed in Sb${}_2$Te${}_3$ [27], but seems to be a rather general feature due to the so-called bottleneck effect related to the vanishing phase space at the DN.

The electronic dynamics of each band (BB1, BB2, and TSS) were extracted by integrating the spectral intensity within the small energy-momentum boxes shown in the upper row in Figure 3: boxes 1 and 2 refer to BB1 and BB2, respectively, while boxes 3 and 4 refer to the TSS. The results are reported in Figure 4. We have followed the same color scheme used for the boxes in Figure 3: as the DN decreases in energy (i.e., p- to n-doping), the line color turns lighter. All curves are normalized to their respective maxima.

Figure 4a shows the dynamics of BB1 (the high-energy bulk band). Within experimental noise all curves are identical and can be fit by a simple phenomenological function: an exponential decay convoluted with the IRF of the instrument (red dashed line). This fit is reproduced in all other panels (b–f) for comparison. Previous investigations reported similar fast dynamics of BB1 [31]. Here we notice that, upon pumping, electrons are optically transferred to BB1 states and subsequently relax to lower energy states with a characteristic time of 220 fs, regardless of the doping level. Therefore, bulk relaxation seems to be unaffected by sample doping and in particular by the filling degree of the TSS. Moreover, in all analyzed BSTS samples, the BB2 state (the low-energy bulk band) is largely unoccupied before pumping and it is only mildly photo-populated. Thus, it can easily accommodate most of the electrons decaying from higher energy levels. According to this observation, it seems that BB1-to-BB2 is the dominant interband scattering process leading to the electronic relaxation of BB1 states. In other words, the TSS does not significantly affect the BB1 dynamics.

On the other hand, there seems to be an interplay between BB2 and TSS. Their electronic dynamics are shown in panels (b–c) of Figure 4. Several relaxation mechanisms have been proposed in the past, including electron-phonon interactions [22,31], spin-mediated channels [32], bulk diffusion [18], the photovoltage effect [23,33] and surface resonant states [34]. By comparing Figure 4b, the BB2, and Figure 4c, the TSS, on the same time scale we can clearly see that: (i) both relaxation dynamics are sensibly slower than BB1; (ii) there is a clear intensity build-up for p-doped samples (i.e., darker lines) (Figure 4b); and (iii) the slowest relaxation, especially for the TSS, takes place when the DN is at the Fermi level (i.e., the Bi${}_{0.8}$Sb${}_{1.2}$Te${}_3$ sample). These three considerations point out the similarities, and thus suggest a possible entanglement, between the dynamics of BB2 and TSS. In fact, some authors have already pointed out that BB2 might have a strong surface character, sharing the spin texture of the TSS [34,35]. This aspect is beyond the scope of the present study, but each of the three previous points deserve some comments.

Point (i) is probably the most trivial. The fact that relaxation times in BB2 and TSS are longer than in BB1 can be simply understood in terms of Fermi’s golden rule: the higher the energy of the excited particle, the larger the number of available (empty) states it can decay into, and consequently a larger decay rate (i.e., faster decay time).

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 $\mathsf{\Gamma }$ (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 $\mathsf{\Gamma }$. 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 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 ${\tau }_r$ 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 ${\tau }_r$ 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]). In fact, when the sample is highly p-doped (n-doped), the bulk valence (bulk conduction) band might intercept the Fermi level, providing additional states that could enhance the relaxation process, in the spirit of Fermi’s golden rule. Therefore, a more isolated (i.e., bulk-free) TSS shows a longer decay. However, when the DN is at the the Fermi level, an exceptionally long electron persistence is observed in the TSS. This is the case of the Bi${}_{0.8}$Sb${}_{1.2}$Te${}_3$ sample (see the corresponding dark cyan curve in Figure 4c), where the relaxation time is much longer than the investigated time windows (i.e., $>>$7 ps). In this particular case, not only the absence of bulk contributions, but also the vanishing DOS in the TSS as the FE is approached is important. Under equilibrium conditions, the bulk conduction bands and the upper TTS branch are empty, while the bulk valence bands and the lower TSS branch are fully occupied. Upon laser pumping, electrons are excited to BB1 and subsequently decay to lower energy states. Since bulk diffusion is strongly hindered, electrons eventually accumulate on the lowest available energy levels that is the upper branch of the TSS. The extremely limited number of available states as the Fermi level is approached (keeping in mind that the DOS at the Fermi level is vanishingly small), leads to the extraordinarily long fermions lifetime, which can be interpreted as a strongly enhanced bottleneck effect.

It should be mentioned that some authors have claimed an additional effect that could increase the electronic lifetime in TSS; a photo-induced surface band bending (also known as surface photovoltage effect) [23,33]. This occurs when the TSS is well isolated from bulk states at the FE and leads to a different transient chemical potential of electrons and holes which creates a barrier for relaxation. However, its relevance to the observed enhancement of the electronic lifetime in TSS is still debated [36].

4. Conclusions

We have studied the electronic dynamics of the TSS in the BSTS family of compounds with different doping levels. We have focused on the relative position of the Dirac node (DN) with respect to the Fermi level. In particular, the close-to-intrinsic TI, where the DN is at the Fermi energy, shows a reduced effect of bulk states on the electronic properties of the TSS with an extraordinarily long lifetime of Dirac fermions. Our investigation reveals that TSS can be efficiently manipulated, providing an excellent platform to novel devices based on topological insulators.