Femtosecond time-evolution of mid-infrared spectral line shapes of Dirac fermions in topological insulators

Mid-infrared (MIR) light sources have much potential in the study of Dirac-fermions (DFs) in graphene and topological insulators (TIs) because they have a low photon energy. However, the topological surface state transitions (SSTs) in Dirac cones are veiled by the free carrier absorption (FCA) with same spectral line shape that is always seen in static MIR spectra. Therefore, it is difficult to distinguish the SST from the FCA, especially in TIs. Here, we disclose the abnormal MIR spectrum feature of transient reflectivity changes (ΔR/R) for the non-equilibrium states in TIs, and further distinguish FCA and spin-momentum locked SST using time-resolved and linearly polarized ultra-broadband MIR spectroscopy with no environmental perturbation. Although both effects produce similar features in the reflection spectra, they produce completely different variations in the ΔR/R to show their intrinsic ultrafast dynamics.

The ARPES image of a Sb 2 TeSe 2 single crystal measured with 24 eV photon energy. All single crystals were the same pieces as those used in ultrafast experiments for the consistency of all measurements. The single crystals were in-situ cleaved under a base pressure 5.1 × 10 −11 torr at 85 K just before measurements. ARPES experiment was conducted National Synchrotron Radiation Research Center in Taiwan using BL21B1 beamline. The photoemission spectra were recorded with a Scienta R4000 hemispherical analyzer. The polarization vector was always in the angular dispersion plane. The overall energy resolution is about 12 meV. The green dash lines represent as the TSS of crystals, and the blue dash lines show the bulk-conduction-band (BCB) and bulkvalance-band (BVB). The Dirac point in Sb 2 TeSe 2 was estimated at 189 meV above the Fermi level (see S1 in Supplementary information). A notable difference of band structure exists between Bi 2 Te 2 Se and Sb 2 TeSe 2 , the Dirac point of Bi 2 Te 2 Se is embedded in the BVB. In contrast to Bi 2 Te 2 Se, Sb 2 TeSe 2 has an isolated Dirac cone and surface carriers cannot be scattered easily by bulk carriers. This difference in their band structure makes a significant difference in optical measurement results.
www.nature.com/scientificreports www.nature.com/scientificreports/ Quantitative analysis of the ultra-broadband MIR ΔR/R spectra. To quantitatively reveal the hidden mechanism, the Drude model and the SST-Kubo model are used to fit the ultra-broadband MIR ΔR/R spectra for n-type Bi 2 Te 2 Se and p-type Sb 2 TeSe 2 TIs. It is initially assumed that before and after pumping, all reflectivity R p (gray solid-line, before pumping) and R p * (red solid-line, after pumping) have similarly shaped spectra for both the Drude model (Fig. 2c) and the SST-Kubo model (Fig. 2d). After pumping, the reflection spectrum shifts because there is an increase in the free carrier concentration. In terms of the Drude model, the dielectric function ε D is: where ε ∞ is the permittivity at an infinite frequency, ω is the frequency, ω p is the plasma frequency and Γ is the plasma scattering rate.  where μ is the chemical potential, T is the carrier temperature, G is the Fermi-Dirac distribution function, τ −1 is the collision rate for TSSs, which depends on the density of impurities, and d TSS is the optical penetration depth of the TSSs. The first and second terms respectively represent the intra-band transitions and the inter-band transitions in Dirac cone. Both models are applied under the "quasi-equilibrium" state in a view of sub-10 fs probe pulse (see S2 of Supplementary information). The penetration depth of ultra-broadband MIR in TIs is few μm (see S3 of Supplementary information).
As previously mentioned, the increase of N in the Drude model represents the change in the electronic population after pumping. In Fig. 2c, the estimated value of N for R p * is larger than that for R p , which results in a blue-shift in the plasma edge. In the SST-Kubo model, the photo-excitation has a significant impact on μ and T and induces changes in the reflection spectrum. In terms of the ground state of p-type Sb 2 TeSe 2 , both the smaller number of carriers in the vicinity of the Dirac point and the higher electron temperature result in a reduction in μ 20 . Therefore, after pumping, the reduction in the chemical potential μ causes a change in the reflection spectrum from R p to R p * , as shown in Fig. 2d. This result is in good qualitative agreement with the ΔR/R spectrum in Fig. 2b.
Ultrafast time-evolution of the ultra-broadband MIR ΔR/R spectra. Figure 3 shows the typical time-evolution of the MIR ΔR/R spectrum and the fitted curves. As mentioned previously, the photoexcited carrier dynamics in n-type Bi 2 Te 2 Se is dominated by FCA and can be fitted well with the Drude model, as shown in Fig. 3a. For Sb 2 TeSe 2 , the contribution of FCA to the photoexcited carrier dynamics cannot be neglected.

T h e r e f o r e , t h e Δ R / R s p e c t r a a r e f i t t e d w i t h t h e m o d i f i e d d i e l e c t r i c f u n c t i o n o f ε ω δω ε ω δω ε ω δω
, where δ ω is a shifted frequency in fitting (This is called the Drude-SST-Kubo model). Figure 3b shows that this model fits the MIR ΔR/R spectrum at various delay times quite well. The details of the fitting are presented in the Method section.

Discussion
The fitting results in Fig. 4a,b are of interest, in particular the time evolutions of ω p , Γ, N, μ, and T in TI's. During the pumping process, the 1.55-eV pump photons excite the electrons to a higher BCB from the occupied states 1 . For Bi 2 Te 2 Se, both ω p and Γ respectively exhibit growth and relaxation dynamics. Although it is difficult to obtain the real value of N because there is no m * , it is still possible to obtain the temporal evolution of N through ω ∝ N p 2 , as shown in Fig. 4c,d. The seriously shift of ω p (~3.7 times after photo-excitation) equivalents to the dramatic enhance of photo-excited concentration (see S4 in Supplementary information). This photoexcited carrier mainly experiences FCA in bulk states (BSs), as shown by the notation of probe(1) and probe (2) in Fig. 4e, or in TSSs, as shown by the notation of probe (3). A bi-exponential decay function is further used to obtain the reduction times for the concentration of photoexcited carriers. This has a maximum within ~2.2 ps and then undergoes two relaxation processes for 1.5 ps and 8.4 ps. The fast relaxation process is caused by the thermal diffusion in BCB and TSS 27,28 , or the acoustic-phono assistant process 3 . The slow one is consistent with the results of time-resolved www.nature.com/scientificreports www.nature.com/scientificreports/ ARPES 1,2 . Additionally, the appearance of a Moss-Burstien shift (until ~6 ps) near the bulk band gap (see the inset in Fig. 3a) also indicates the recombination in BSs 15 . However, the value of N (i.e., N (0) in Fig. 4d) does not recover to its original value (i.e., N unex (0) in Fig. 4c) within the limited delay time (~ 100 ps). This inconsistency between N (0) and N unex (0) is explained by the long-lived recombination process (see S5 in Supplementary information). There are several scenarios proposed for this long relaxation process. First, it is generally assigned to the photo-voltage effect 29 . Moreover, huge Rashba-splitting effect has been clearly observed in BCB 30,31 , which might cause the long-time relaxation processes like indirect band-gap semiconductors 25 .
For p-type Sb 2 TeSe 2 , the Drude-SST-Kubo model is used to fit the results in Fig. 4f-i. It is worth emphasizing that the relative changes in μ and T are more distinct than the changes in ω p and Γ. Even though the MIR probe-pulses can also detect FCA (even though it originates from TSSs or BSs as shown by the notation of probe(4) in Fig. 4j), the ΔR/R spectra are significantly dominated by SST's in the Dirac cone (see the notations of probe(5) and probe (6)). Figure 4h shows that after the deep valance electrons are excited to the upper Dirac cone 5 , μ reaches a minimum at ~1 ps and it takes ~1.28 ps for the recombination process according to the fitting for a single exponential decay function. Besides, the hot-carrier temperature reaches ~600 K, and recovers to room temperature after 1.68 ps, which results are consistent with the time-resolved ARPES results for Sb 2 Te 3 5-7 . Therefore, this hot-carrier temperature decay would be resulted from the thermal diffusion between BVB and TSS 27 .
By taking account of the difference of the number of states between bulk and surface, when the electrons are photo-excited, the chemical potential should shift towards the higher energy direction. In Drude-SST-Kubo model, the chemical potential (μ) and the carrier temperature (T) are associated with the surface state, and the SST-Kubo term is consisted by inter-transition and intra-transition of Dirac cone. For μ ≫ T, the intra-transition term could be derived to the form 20  ε μ π ω ωΓ which coincides with the Drude expression, and the effective plasma frequency ,S could be further expressed as  μ π e / 2 . More precisely, the contribution of excited charges to " " in Dirac cone is considered by the intra-transition term. In other words, the Drude term in Drude-SST-Kubo model represents the excited carriers which are out of the surface state. For p-type Sb 2 TeSe 2 with the Fermi level locating at the lower energy part of the Dirac cone, after photoexcitation, the chemical potential shifts to the higher energy direction, and further indicates the redshift of plasma edge and decreasing of the density of states. From the fitting result of smaller (~1.3 × 10 3 cm −1 ) on Sb 2 TeSe 2 , it shows the lower contribution from the excited bulk carriers, which is consistent with the results in Fig. 2b. Summary. The ultrafast dynamics of Dirac fermions and bulk free carriers in the TIs, n-type Bi 2 Te 2 Se and p-type Sb 2 TeSe 2 single crystals, are studied using time-resolved ultra-broadband MIR spectroscopy. The dynamics www.nature.com/scientificreports www.nature.com/scientificreports/ in the n-type Bi 2 Te 2 Se is dominated by bulk carriers because the ΔR/R spectra show a blue-shift in the plasma edge due to FCA. For p-type Sb 2 TeSe 2, the dynamics is dominated by the Dirac fermion from the red-shift of the plasma edge in the ΔR/R spectra. This study shows that the MIR absorption peaks for FCA and SST in TIs can be distinguished and demonstrates the importance of time-resolved ultra-broadband MIR spectroscopy for gapless or small band gap exotic materials.

Methods
Experimental setup. Optical pump and ultra-broadband MIR probe spectroscopy 23 consists of three stages: (i) 800-nm optical pulses with a duration of 30 fs were generated, (ii) ultra-broadband MIR probe pulses were generated in nitrogen and (iii) chirped pulses were generated for detection. The fundamental pulses (800 nm) and the second harmonic pulses (400 nm, which were generated by a type I β-BaB 2 O 4 crystal with a thickness of 0.1 mm) from a Ti:sapphire amplifier (790 nm, 30 fs, 0.85 mJ at 1 kHz, Femtopower compactPro, FEMTOLASERS) were focused into nitrogen gas to generate MIR pulses. The filamentation occurred via four-wave DFG when the pulse was focused using a concave mirror (r = 1 m). The length of the filament was ~3 cm. The bandwidth and the duration of the generated MIR pulses were 200-5000 cm −1 and 8.2 fs, respectively. When the MIR pulses were reflected from the sample with an incident angle of 45°, they were converted to ~400-nm pulses for the detection using a chirped-pulse up conversion (CPU) in nitrogen gas. A third 800-nm beam was transmitted through dispersive materials, including four BK7 glass plates (thickness: 10 mm) and one ZnSe plate (thickness: 5 mm), to produce chirped pulses. The converted visible (VIS) spectrum was measured by a spectrometer with an electron-multiplying CCD camera (SP-2358 and ProEM+1600, Princeton Instruments). The time resolution was estimated to be ~60 fs. To prevent significant absorption from vapor, the system was placed in boxes whose interior was purged with nitrogen.
Retrieving the MIR spectra from an up-converted spectra and calibrating the spectra of the VIS pulse to MIR region. The MIR spectrum form, especially the sharp absorption peaks, can be seriously distorted after CPU measurements. That is to say, the dispersion of chirped pulses causes additional oscillations in the spectrum 32,33 ) and the MIR pulse (E t ( ) MIR ). The chirped pulse is written as: represents the envelope, ω (0) is the central angular frequency, and ω (1) is a chirp parameter. The MIR pulse can be divided into a main part E t ( ) MIR (0) and a free induction decay part E t ( ) MIR (1) . yields: can be assumed to be the Dirac delta function δ t ( ) due to the short duration of MIR pulse. Using the Wiener-Khinchin theorem and these assumptions, the autocorrelation C t ( ) Therefore, the original MIR spectrum with shift ω t 2 (0) is acquired using the measured up-converted power spectrum and the known value of ω (1) for the chirped pulse. Finally, the wavenumber is calibrated using a binomial fitting of the three absorption peaks, including carbon dioxide (~2300 cm −1 ) and water vapor (~1600 cm −1 and ~3700 cm −1 ). The transient ΔR/R is obtained by: www.nature.com/scientificreports www.nature.com/scientificreports/ where the superscripts "*" and "0" of R p respectively represent the reflectivity with and without optical pumping. The fitting with the Drude model is performed using the software, RefFIT 34 . The fitting with the Drude-SST-Kubo model uses 4 parameters: ω Γ μ , , p and T. To limit the computational load without losing the accuracy, the grid search method and an interval search algorithm with few iterations are used. After obtaining all possible values for these 4 parameters, the most appropriate parameter set P j is selected by calculating the minimum root-mean-square deviation between the data and the calculated results at the j th iteration. More specifically, using the grid search method, the value of P j at the j th iteration can be obtained. The best interval is decided using the neighboring points of P j . In this analysis, 4 parameters produce the 8 neighboring points. Using this interval, the next iteration j+1 of the grid search is undertaken. Therefore, the accuracy is exponentially increased.

Analyses using the Drude, SST-Kubo and
The conditions, R p 0 , are determined using the ARPES results and the FTIR spectra. For Bi 2 Te 2 Se, R p 0 is calculated using the Drude model with ε ∞ = 23.7, ω p = 1880 cm −1 and Γ = 272 cm −1 , which values are obtained by fitting the FTIR spectra using the RefFIT program 34 . For Sb 2 TeSe 2 , R p 0 is determined using the Drude-SST-Kubo model with ε ∞ = 19.4, ω p = 1320 cm −1 , Γ = 253 cm −1 , d TSS = 1.4 nm, μ = 72 meV and T = 297 K. The former 4 parameters are obtained by fitting with fixed values of μ and T using the grid search method and an interval search algorithm, as described previously. If μ is sufficiently large, it can be estimated as: where N TSS is the surface carrier concentration (~2.2 × 10 12 cm −2 ). The parameter N TSS is expressed as: where A FS is the area of the Fermi surface, A BZ is the area per Brillouin zone, A UC is the area per unit cell and K F is the Fermi-wavenumber (~5.2 × 10 6 cm −1 from ARPES). The parameter υ TSS = 4.12 × 10 7 cm/s is estimated from the gradient of Dirac cone from ARPES. More ARPES information of TIs is shown in S1 of Supplementary information.