Manifestation of a Second Dirac Surface State and Bulk Bands in THz Radiation from Topological Insulators

Topological insulators (TIs) are interesting quantum matters that have a narrow bandgap for bulk and a Dirac-cone-like conducting surface state (SS). The recent discovered second Dirac surface state (SS) and bulk bands (BBs) located ~1.5 eV above the first SS are important for optical coupling in TIs. Here, we report on the time-domain measurements of THz radiation generated from TIs n-type Cu0.02Bi2Se3 and p-type Bi2Te3 single crystals by ultrafast optical pulse excitation. The observed polarity-reversal of the THz pulse originated from transient current is unusual, and cannot be reconciled with the photo-Dember effect. The second SS and BBs are found to be indispensable for the explanation of the unusual phenomenon. Thanks to the existence of the second SS and BBs, TIs manifest an effective wide band gap in THz generation. The present study demonstrates that time-domain THz spectroscopy provide rich information of the optical coupling and the electronic structure of TIs.

strong spin-orbital coupling and is expected to keep the same topological and spin characteristics as the 1st SS 23,24 . However, the role of the 2nd SS and BBs on the interactions between TIs and photons has been much less investigated. Further explorations on the relevant subjects are indispensable to the development of TI optical devices.
This study reports the time domain measurements of THz radiation generated from the surface and bulk of n-type Cu 0.02 Bi 2 Se 3 and p-type Bi 2 Te 3 single crystals. Two different mechanisms of THz generation: optical rectification (OR) and transient current, have been identified by different polarization configurations of both optical excitation and THz radiation. With this time-domain technique, the 2nd SS and BBs manifest themselves in the spectrum, and prove vital in generating the THz radiation through a transient drift current.

Results
P-polarized THz radiation from TIs. The experimental setup was a reflective type, where the optical pulse illuminated the (111) surfaces of the TI crystals at an incident angle of 45°, to generate THz radiation, as shown in Fig. 1(a). Two types of TI single crystals were used in this study: Cu-doped n-type Bi 2 Se 3 (Cu 0.02 Bi 2 Se 3 ) with a carrier concentration of 3.66 × 10 18 cm −3 and p-type Bi 2 Te 3 with a carrier concentration of 1.32 × 10 19 cm −3 . Both were determined using the Hall measurements and the ARPES images of the employed single crystals are shown in Fig. 1(b,c). Figure 2(a,b) show the typical time domain THz waveforms with several cycles for the n-type TIs. In these two cases, the P-and S-polarized optical pulse illuminated the surface of an n-type Cu 0.02 Bi 2 Se 3 single crystal and a wire-grid polarizer was set to detect P-polarized THz waves (denoted by P Opt -P THz and S Opt -P THz ), respectively. Figure 2(e) shows the dependence of the azimuthal angle (φ-scan) of the peak-to-peak value of the THz pulse (A THz ) from the (111) surface of Cu 0.02 Bi 2 Se 3 in P Opt -P THz configuration and a corresponding x-ray diffraction (XRD) φ-scan of the (1 1 15) peak of TI crystals. The result in S Opt -P THz configuration is shown in Fig. 2(f). In these two configurations, the φ-dependent THz peak-to-peak value (THz φ-scan) does not show a significant three-fold symmetry associated with that of the crystalline structure of Cu 0.02 Bi 2 Se 3 . Because the P-polarized THz radiation contains both in-plane and out-of-plane electric-field components, the THz  (a-d) Time-domain P-polarized THz waveforms radiated from the (111) surface of n-type Cu 0.02 Bi 2 Se 3 and p-type Bi 2 Te 3 by P-and S-polarized optical pulse excitation. In (a), the thin-red line represents the 0.015-times smaller P-polarized THz wave that is generated from an n-type InAs under the same conditions. In both P Opt -P THz and S Opt -P THz configurations, the polarities of the P-polarized THz waveforms from p-type Bi 2 Te 3 (c,d) are in the reverse of that from n-type Cu 0.02 Bi 2 Se 3 (a,b). A THz is the peak-to-peak value of the THz waveform. (e-h): the plots of A THz for the P-polarized THz waveforms as a function of azimuthal angles (φ-scan) of Cu 0.02 Bi 2 Se 3 (a,b) and Bi 2 Te 3 (c,d) single crystals (black). The corresponding XRD φ-scans for the (1 1 15) peak for both single crystals are also shown in (e-h) (green). φ-scan results show that the P-polarized THz radiation is dominated by the out-of-plane component from the bulk of the TIs, as shown in Fig. 1(a). Obviously, the THz waveforms from Cu 0.02 Bi 2 Se 3 are similar to that from InAs (thin-red line in Fig. 2(a)) but the amplitude of the P-polarized THz pulse from Cu 0.02 Bi 2 Se 3 is much smaller than that from InAs under the same conditions. Figure 2(c,d) show the typical time-domain THz waveforms from a p-type Bi 2 Te 3 single crystal in the configurations of P Opt -P THz and S Opt -P THz , respectively. The shape and amplitude are almost the same as those for the n-type Cu 0.02 Bi 2 Se 3 in Fig. 2(a,b). The THz φ-scans in Fig. 2(g,h) do not show a significant three-fold symmetry associated with the crystalline structure of Bi 2 Te 3 , either. Nevertheless, the polarities of the P-polarized THz pulse from p-type Bi 2 Te 3 are opposite to those from the n-type Cu 0.02 Bi 2 Se 3 . This intriguing contrast provides a revealing insight into TIs' optical coupling and novel electronic structure as we shall show later.

S-polarized THz radiation from TIs.
In the experiments of the P Opt -P THz and S Opt -P THz configurations, the in-plane contribution to THz radiation cannot be separated. To obtain the in-plane contribution to THz radiation, the wire-grid polarizer and the EO sampling setup were rotated to detect the S-polarized THz radiation under P-and S-polarized optical excitation (P Opt -S THz and S Opt -S THz ). Figure 3(a) shows the time-domain THz waveforms from the (111) surface of n-type Cu 0.02 Bi 2 Se 3 in the P Opt -S THz configuration. The results in S Opt -S THz configuration is also shown in Fig. 3(b). Although the signal-to-noise ratio is low (about 2 to 3), it still can be observed that the polarity of the THz waveform is reversed if the azimuth angle of samples is rotated by 180°. The similar results of p-type Bi 2 Te 3 are shown in Fig. 3(c,d). The origins of S-polarized THz radiation shall be discussed later.

Discussion
For THz generation from common semiconductors, there are three possible mechanisms: (1) optical rectification (OR), (2) the photo-Dember effect and (3) the surface field effect 26 . In the experimental results, the P-polarized THz radiation from both Cu 0.02 Bi 2 Se 3 and Bi 2 Te 3 do not show a significant three-fold symmetry that is related to the crystalline structure, and this excludes OR as the mechanism for P-polarized THz radiation from TIs. However, both the photo-Dember effect and the surface field effect are associated with transient current from photocarriers that are excited by an optical pump pulse inside the material. The far-field electrical field  E THz for THz radiation is described by where  J is the transient current created by an optical pulse, and it is composed of diffusion current  J diffusion and drift current  J drift . The amplitude of  E THz is proportional to the time-derivative of  J . In general, the direction of  J is normal to the surface of the material, and it contributes to the P-polarized THz radiation if the pump beam is obliquely incident. The polarity of the THz radiation also depends on the direction of  J , whether it is inward to or outward from the surface of the material. The photo-Dember effect is a result of the concentration gradient of the photoexcited carriers near the semiconductor surface. Because there is a large difference between photoexcited electrons and holes in the diffusion coefficient and the mobility, photoexcited electrons in the bulk of a material diffuse much faster than holes, which causes the diffusion current  J diffusion to be the main component of  J in Eq. (1). Therefore, the photo-Dember effect is the dominant mechanism for THz generation from narrow bandgap semiconductors 26 , such as InAs (bandgap ∼ 0.36 eV) and InSb (bandgap ∼ 0.17 eV). In this case, the direction of  J diffusion is always outward from the surface and the polarity of the THz pulses must be the same for both n-and p-type semiconductors. Although both n-type Cu 0.02 Bi 2 Se 3 (bandgap ∼ 0.3 eV) and p-type Bi 2 Te 3 (bandgap ∼ 0.15 eV) could be classified as narrow bandgap semiconductors, the photo-Dember effect cannot explain the reversed polarity of P-polarized THz radiation between n-and p-type TIs.
For wide bandgap semiconductors, such as GaAs (bandgap ∼ 1.43 eV), THz radiation can be explained by the surface field effect which is due to the acceleration of photoexcited carriers by the surface field near the semiconductor surface. The surface field is caused by band-bending due to Fermi-level pinning by the surface state 26 . The photoexcited electrons and holes are accelerated in opposite directions by the surface field to form a transient drift current  J drift . In general, the direction of the surface field in n-type semiconductors is opposite to that in p-type semiconductors. Therefore, the directions of the induced transient currents are opposite and the polarity of THz pulses from n-type semiconductors is also opposite to that from p-type semiconductors.
For Bi 2 Se 3 and Bi 2 Te 3 , surface band-bending has been observed by previous studies, using angle-resolved photoemission spectroscopy, scanning tunneling spectroscopy, and hard x-ray photoelectron spectroscopy [27][28][29][30] . Theoretical calculations also predict that the sign of band-bending could be affected by the type of doping: electron doping or hole doping 31 . Therefore, the polarity of P-polarized THz radiation from TIs is affected by the TIs' doping. The experimental results indicate that  J drift predominates over the photoexcited current inside p-type Bi 2 Te 3 , which further implies that the excess energy of the photoexcited carrier is too low to form a sufficiently strong photo-Dember field. However, from an energy point of view, the excess energy that the photoexcited carriers gain from the pumping photon energy is larger than that given by the surface field for narrow band semiconductors as in TIs. In this regard,  J diffusion from the photo-Dember field should have been the dominant component of the photoexcited current 32 . This paradox can be beautifully resolved in the context of the 2nd SS and BBs in TIs. Very recently, using two-photon photoemission in ARPES measurements 24,25 , an unoccupied 2nd SS and BBs located 1.1 to 1.5 eV above the 1st SS were observed in the material family of Bi 2 Te x Se 3-x . For Bi 2 Se 3 , the energy difference between the 2nd band (2nd SS and BBs) and 1st band (1st SS, bulk conduction band: BCB and bulk valance band: BVB) matches the photon energy of 1.55 eV. It has also been confirmed that direct transitions to the 2nd SS and BBs can be driven by 1.55 eV optical pulses in n-type samples 25 . This indicates the excess energy of the photoexcited carriers would be reduced dramatically. For Bi 2 Te 3 , the 2nd SS is closer to the 1st SS 24 than that of Bi 2 Se 3 . For the p-type Bi 2 Te 3 samples, however, the energy difference between the 2nd band (2nd SS and BBs) and the 1st band (1st SS, BCB and BVB) would still lead to sufficient energy loss of the photoexcited carriers. As shown in Fig. 4, to have a brief summary, thanks to the existence of 2nd SS and BBs, TIs manifest an effective wide band gap in THz generation. The 2nd SS and BBs could result in energy loss for these photoexcited carriers that generate THz radiation in TIs. Therefore,  J drift , rather than  J diffusion , may predominate over the photoexcited current in TIs and produce THz pulses with opposite polarity, due to the doping dependent band-bending.
The band-to-band transitions may contribute to the absorption of pumping photon to cause the reduction of energy. In general, this phenomenon can be described by the Fermi Golden rule, which is a way to calculate the transition probability between the initial and final density of states. This indicates that the band-to-band transitions could be further predicted by the statuses of initial and final density of states. For both cases of n-type Cu 0.02 Bi 2 Se 3 and p-type Bi 2 Te 3 , the Fermi surface locates in the 1st band and no carriers fill the 2nd band and higher energy bands. While the samples were pumped by a femtosecond laser pulse with photon energy of 1.55 eV, the transition would predominantly occur between the 1st and 2nd band at the initial stage according to the Fermi Golden rule. Even there are some photoexcited carriers on 2nd band after pumping, the band-to-band transitions from the 2nd band and higher energy bands can be neglected due to the femtosecond ultrashort pulse (< 75 fs), which doesn't provide enough time for the further absorption of photoexcited carriers on the 2nd band or higher energy bands.
On the other hand, if we use a long pulse for pumping, the band-to-band transitions should be considered for the THz generation on TIs.
In the cases of S-polarized THz radiation from TIs, as shown in Fig. 3(a,b), although the signal-to-noise ratio of the measured THz waveforms is low (about 2 to 3), it still can be observed that the polarity of the THz waveform is reversed if the azimuth angle of samples is rotated by 180°. Further, the THz φ -scans in Fig. 3(e,f) also show a three-fold symmetry, which is consistent with the surface crystalline structure of Cu 0.02 Bi 2 Se 3 . A similar symmetry was also recently observed in SHG signal using ultrafast optical pulse illumination in Bi 2 Se 3 16,17 . For p-type Bi 2 Te 3 , the same phenomena are also observed, as shown in Fig. 3(c,d,g,h). Consequently, it is concluded that the OR is the main mechanism for S-polarized THz radiation from a TI surface. In general, the contribution of OR is not negligible in P-polarized THz radiation from the semiconductors, such as InAs and GaAs 26 . Because there is weak S-polarized THz radiation for Cu 0.02 Bi 2 Se 3 and Bi 2 Te 3 , there is no significant modulation in the φ-scan results for P-polarized THz radiation, as shown in Fig. 2(e-h).
Second-order nonlinear optical process is a simultaneous process in fs time scale and it means that THz pulse originated from OR is generated simultaneously with the incidence of optical pulse. On the other hand, it takes time for the photoexcited electrons, in the 2nd SS and BBs, to relax to the 1st SS or BCB 33 . This indicates that THz pulse originated from this complex process would lag behind that from OR. Comparing with the P-polarized THz pulses in Fig. 2(a-d), the main peak of the S-polarized THz pulses has a lead of ∼ 0.2 ps over TIs, as shown in Fig. 3(a-d). This value is closed to the relaxation time 0.5 ps, which is obtained by recent time-resolved ARPES measurement, for the photoexcited electrons from the 2nd SS and BBs relaxing to the 1st SS and BCB 33 . These results also demonstrate the influence of the 2nd SS and BBs on the P-polarized THz radiation from TIs. In this study, we cannot separate the contribution of the 2nd SS from that of 2nd BBs and the different-wavelength excitation may provide more insights into this unusual phenomenon. We also performed large photon-energy-excitation (central wavelength: 400 nm, E photon = 3.1 eV) and the details are presented in Supplementary information.
In summary, we have characterized THz radiation from n-type Cu 0.02 Bi 2 Se 3 and p-type Bi 2 Te 3 single crystals using ultrafast optical pulse illumination. If an n-type Cu 0.02 Bi 2 Se 3 single crystal is replaced by a p-type Bi 2 Te 3 single crystal, there is a reversal in the polarity of THz radiation. These results suggest that the drift current  J drift predominates over the transient current in these narrow bandgap materials. This notable phenomenon can be reconciled in the context of the second Dirac surface state and bulk bands. The S-polarized THz radiation results show a three-fold symmetry, which results from the second-order nonlinear optical effect of optical rectification. The present study demonstrates that the second Dirac surface state and bulk bands play the important roles in the optical coupling with the electronic structure of topological insulators.

Methods
In the experiments, a mode-locked Ti: sapphire oscillator (XL 300, FEMTOLASERS) was used as the light source to generate a 300-nJ optical pulse train with a repetition rate 5.2 MHz and a central wavelength of 800 nm. The pulse duration is ∼ 75 fs. The output beam was split into two parts by an 80/20 beam splitter. The stronger one was used for THz generation and the weaker one for standard electro-optic sampling. A retro-reflector was mounted on a linear translation stage which was incorporated into the probe-beam path as an optical delay line. A half-wave plate and a cubic polarizer were used to vary the polarization of the optical pump beam. A wire-grid polarizer was used to detect the polarization of the generated THz radiation. Optical pump pulse illuminated the (111) surfaces of the TI crystals at an incident angle of 45° to generate THz radiation. The pulse energy of the pump beam was around 10 nJ and the spot size on the samples was about 280 μ m in diameter. For 400-nm excitation, a nonlinear crystal slab, beta-BaB 2 O 4 (BBO) of 0.5 mm in thickness, was used to generate 400-nm optical pulses by second harmonic generation. The THz radiation generated from the TI crystals was collimated by two off-axis parabolic mirrors and focused on a 1-mm-thick < 110> ZnTe slab to detect THz radiation. All of the experiments were performed at room temperature and in a chamber filled with nitrogen gas to avoid THz-waveform distortion due to water vapor absorption.