Terahertz signatures of ultrafast Dirac fermion relaxation at the surface of topological insulators at room temperature

Topologically-protected surface states present rich physics and promising spintronic, optoelectronic and photonic applications that require a proper understanding of their ultrafast carrier dynamics. Here, we investigate these dynamics in topological insulators (TIs) of the bismuth and antimony chalcogenide family, where we isolate the response of Dirac fermions at the surface from the response of bulk carriers by combining photoexcitation with below-bandgap terahertz (THz) photons with TI samples with varying Fermi level, including one sample with the Fermi level located within the bandgap. We identify distinctly faster relaxation of charge carriers in the topologically-protected Dirac surface states (few hundred femtoseconds), compared to bulk carriers (few picoseconds). In agreement with such fast cooling dynamics, we observe THz harmonic generation without any saturation effects for increasing incident fields, unlike graphene which exhibits strong saturation. This opens up promising avenues for increased THz nonlinear conversion efficiencies, and high-bandwidth optoelectronic and spintronic information and communication applications.


Abstract
Topologically-protected surface states present rich physics and promising spintronic, optoelectronic and photonic applications that require a proper understanding of their ultrafast carrier dynamics. Here, we investigate these dynamics in topological insulators (TIs) of the bismuth and antimony chalcogenide family, where we isolate the response of Dirac fermions at the surface from the response of bulk carriers by combining photoexcitation with below-bandgap terahertz (THz) photons with TI samples with varying Fermi level, including one sample with the Fermi level located within the bandgap. We identify distinctly faster relaxation of charge carriers in the topologically-protected Dirac surface states (few hundred femtoseconds), compared to bulk carriers (few picoseconds). In agreement with such fast cooling dynamics, we observe THz harmonic generation without any saturation effects for increasing incident fields, unlike graphene which exhibits strong saturation. This opens up promising avenues for increased THz nonlinear conversion efficiencies, and high-bandwidth optoelectronic and spintronic information and communication applications.
Nowadays, due to their unique transport properties, topological insulators (TIs) attract great attention. 1,2 Due to the symmetry-protected Dirac fermions with non-trivial topology on their surface, they are highly prospective as functional materials in future electronic and spintronic devices. 3 Moreover, various optoelectronic and thermoelectric applications can benefit from the combination of protected charge transport in surface states, and large Seebeck coefficients. For many of these applications it is crucial to understand the characteristic timescales of the relaxation dynamics of excited carriers, and in particular determine if these dynamics are different for topological surface states compared to the bulk.
The carrier dynamics in TIs have been addressed using various pump-probe techniques. [4][5][6][7][8][9][10][11] However, despite these experimental efforts, the exact mechanism and timescale of carrier relaxation of the Dirac fermions in surface states of TIs are still under debate. The reason for this is that it is challenging to experimentally disentangle excitation and relaxation channels of surface and bulk states. This is caused by the relatively small bandgap of TIs of the bismuth and antimony chalcogenide family, which is a few hundred meV, 12 meaning that optical excitation with near-infrared and visible light leads to interband transitions between the bulk bands. Thus, in order to optically address the surface states, energetically located within the bandgap, photons in the mid-infrared or terahertz regimes are required. Furthermore, the Fermi energy is typically located either in the valence or the conduction band for binary TI compounds, e.g. Bi 2 Se 3 and Bi 2 Te 3 , such that bulk states are already populated. Whereas separating surface dynamics from bulk dynamics has recently been achieved at the cryogenic temperature of 5 K, 11 this is not the case at more technologically relevant temperatures. The main reason for this is the occurrence of phonon-assisted surface-to-bulk scattering above the Debye temperature (∼180 K for Bi 2 Se 3 ). 7 Here, we overcome these challenges by combining optical excitation with low photon energies and a TI sample with the Fermi energy located inside the bandgap. As a result, we are able to isolate the response of Dirac electrons in the surface states, without the contribution of bulk states. In particular, we use THz pulses with photon energies below 4 meV, and verify whether the observed carrier dynamics originate from surface states (SS), bulk We first measure the relaxation dynamics of these samples using transient reflectivity spectroscopy. We use laser pulses with 800 nm central wavelength and 100 fs pulse duration for probing, while single cycle THz pulses serve as a pump in THz-pump optical-probe (TPOP) experiments. As a reference, we perform optical-pump optical-probe (OPOP) measurements, where 800 nm pulses also serve as pump, instead of THz light. The experimental setups used for TPOP and OPOP measurements are described in the Supplementary information. Ultrafast pump-probe reflectivity measurements in the near-and mid-infrared ranges have been used earlier to study the dynamics of charge carriers 4,8-10 and phonons [17][18][19] in topological insulators. In these experiments, however, the pump pulses had photon energies above or close to the energy of the bandgap, such that the dynamics of bulk and surface carriers both contribute to the observed signals. In a few pump-probe studies, THz light was used as the pump, in order to study carriers, 11 and phonons. 19 However, in both cases bulk-metallic TIs were used. Therefore, the carrier relaxation dynamics observed in all of these studies always contained contributions from both the surface and bulk electronic systems. Measurements performed on the Bi 2 Te 3 sample are presented in Fig. 1b. Here, we observe similar dynamics as for Bi 2 Se 3 : ultrafast carrier build-up and a bi-exponential decay.
The Raman-excited optical phonon, seen as periodic oscillation, is more pronounced than in Bi 2 Se 3 and slightly red-shifted due to the heavier Te atoms (Supplementary information In comparison with Bi 2 Te 3 and Bi 2 Se 3 , the carrier dynamics in BSTS (Fig. 1c) show a much faster decay after the ultrafast build-up caused by THz excitation. The slow response observed in the other samples immediately after pump excitation is almost negligible. The relaxation time in BSTS is short -a few hundred fs -which is very different compared to Bi 2 Se 3 , Bi 2 Te 3 and also with respect to the dynamics in BSTS observed in studies employing above-bandgap excitation. 10,23 In all these cases, a pronounced relaxation component with a timescale of a few picoseconds is present. We ascribe the observed fast dynamics to the isolated response of Dirac fermions in the surface states of our TI system. We suggest that these fast surface-state relaxation dynamics we unveiled are the result of efficient coupling of carriers to phonons 5,24 and ultrafast surface-to-bulk scattering, 7 and could hence be a general property of this material class. We note that the observed decay is significantly faster than the decay of surface-state carriers reported at 5 K, which was ∼1.5 ps. 11 To verify independently that the ultrafast relaxation in BSTS originates from intraband dynamics within the topological SS, we have performed degenerate optical pump-probe measurements using pump pulses with a photon energy of 1.5 eV (Fig. 1c). This scheme leads to the simultaneous excitation of bulk and surface states in all samples. In the case of Bi 2 Te 3 and Bi 2 Se 3 ( Fig. 1a-b), we see almost identical relaxation dynamics for optical and THz excitation, confirming that the dominant contribution to the dynamics originates from bulk states in bulk-metallic TIs. For BSTS we, in contrast to the below band-gap THz excitation, observe also a much slower dynamic on picosecond time scales in agreement with earlier observations for above band-gap excitation. 25 The significantly faster decay after photoexcitation with below-bandgap THz photons, in comparison with excitation with above-bandgap photons, clearly suggests that the surface-state Dirac fermions have a much shorter lifetime. In order to obtain additional insights into the relaxation dynamics of Dirac fermions in TI surface states, we study THz high-harmonic generation (HHG). Strong THz high-harmonic generation (HHG) has been observed in a number of Dirac materials: in graphene, [26][27][28] in the 3D Dirac semimetal Cd 3 As 2 , 29,30 and in the prototypical TI Bi 2 Se 3 , where it was shown that the THz harmonics originate from the Dirac electrons at the surface. 31 The underlying mechanism for THz HHG by Dirac fermions can be described by a thermodynamic nonlinearity mechanism that relies crucially on the ultrafast modulation of the THz absorption.
This mechanism is enabled by efficient heating and subsequent cooling of the electronic system on fs to ps timescales upon interaction with strong THz fields. 26,27,32 Therefore, the THz HHG behaviour is intricately related to the characteristic timescales of carrier dynamics of Dirac fermions. In particular, we point out that when cooling takes several picoseconds, this leads to a reduction of the nonlinearity coefficient and strong saturation effects for increasing incident field strengths, as observed recently for graphene under strong THz excitation 28 (see also Supplementary information).
In Fig. 2, we show measurements of fundamental and third-harmonic fields measured by electro-optic sampling for Bi 2 Te 3 , together with the respective amplitude spectra. The details of the THz HHG experimental setup are show in the Supplementary information.
We observe a clear third-harmonic signal from Bi 2 Te 3 at 1.5 THz (Fig. 2b)  ment with earlier publications. 26,28,30 The strong saturation at the highest field strengths has been partially ascribed to a slowing down of the cooling of hot Dirac carriers, decreasing the nonlinear conversion efficiency. 28 Compared to graphene, the TI samples show fundamentally different behaviour: all topological insulators demonstrate a clear cubic dependence of the third-harmonic field across the whole accessible field regime meaning that THz-induced nonlinear processes are far away from the saturation regime. In order to verify this perturbative behavior further, we show in Fig. 3b the field strength of the fifth harmonic as a function of the fundamental field strength for the BSTS sample. Using a power law fit, the fifth-harmonic fields scale with an exponent of 5.2 ± 0.3 as a function the fundamental field, which is clear evidence that the FHG process is also in the perturbative regime. This purely perturbative behavior all the way up to incident field strengths of 140 kV/cm can be ascribed (partially) to the sub-picosecond cooling dynamics we observed in our TPOP experiments: the ultrafast surface-state relaxation of the TI samples prevents heat accumulation effects within the comparably long duration (several picoseconds) of the THz excitation pulse. An additional reason for the extended perturbative regime for the tested TIs could be the lower nonlinearity, which is likely due to the lower Fermi velocity compared with graphene. 33 This allows for driving the TIs samples with higher fields before saturation is observed.
The observation of perturbative HHG in TIs is highly interesting, because it could mean that a higher conversion efficiency can be obtained for increased incident field strengths. We observe that the highest overall conversion efficiency for graphene is about 0.5 % in field, which is consistent with other studies. 26,28 However, saturation effects at high fields may impose an efficiency limit. For the TI samples, we observe maximum third-order conversion efficiencies of 0.13, 0.08, and 0.03 % in field for Bi 2 Te 3 , BSTS, and Bi 2 Se 3 , respectively. A maximum conversion efficiency for fifth harmonic generation (FHG) of around 0.014 % at 140 kV/cm fundamental field strength is observed. These efficiencies are lower than the one obtained for graphene. However, the purely perturbative scaling indicates that higher conversion efficiencies than for graphene are achievable by further increasing the incident field, which is of great technological interest.
In conclusion, we experimentally isolated the dynamical response of Dirac fermions in surface states of TIs that are excited by THz light, and observe ultrafast relaxation dynamics on a timescale of a few hundred femtoseconds. This decay is significantly faster than the dynamics of photoexcited carriers in bulk states, and likely originates from efficient phonon-assisted scattering either via surface intraband or surface-bulk interband transitions.
In agreement with such fast relaxation, we observe no saturation effects in THz harmonic generation up to the highest field of 140 kV/cm, where the graphene harmonic response is already strongly saturated. Our findings are of high technological relevance since they indicate that the nonlinear conversion efficiencies in the investigated TI's can, in contrast to graphene, potentially be scaled to unprecedented values e.g. by employing metamaterials to locally enhance the electric fields. 28

Data availability
The data that support the findings of this study are available from the corresponding authors upon reasonable request.