Fermi energy and carrier concentrations of bulk and surface states of various Bi2Se3 single crystals. All samples are
Abstract
Ultrafast dynamics of carriers and phonons in topological insulator Bi2Se3, CuxBi2Se3 (x = 0, 0.1, 0.125) single crystals were studied by time-resolved pump-probe spectroscopy. The coherent optical phonon (A1g1) is found via the damped oscillation in the transient reflectivity changes (∆R/R) for CuxBi2Se3. The observed red shift of A1g1 phonon frequency suggests the intercalation of Cu atoms between a pair of the quintuple layers of Bi2Se3 crystals. Moreover, the relaxation processes of Dirac fermion near the Dirac point of Bi2Se3 are studied by optical pump and mid-infrared probe spectroscopy through analyzing the negative peak of the ∆R/R. The Dirac fermion-phonon coupling strength was found in the range of 0.08–0.19 and the strength is reduced as it gets closer to the Dirac point. The ultrafast dynamics and fundamental parameters revealed by time-resolved pump-probe spectroscopy are important for designing the optoelectronics in the mid-IR and THz ranges.
Keywords
- topological insulators
- ultrafast dynamics
- pump-probe spectroscopy
- Dirac fermion
1. Introduction
Recently, topological insulators (TIs) [1, 2, 3, 4, 5, 6, 7, 8] and two-dimensional (2D) materials such as graphene [9], MoS2, WS2, and MoSe2 [10] are of great interests because of their unique physical properties and applications. These materials have a band structure that is linearly dispersed with respect to momentum, in which the transportation of electrons in these materials is essentially governed by Dirac’s (relativistic) equation with zero rest mass and an effective “speed of light”—c* ≈ 106 m/s [9]. In TIs, a novel electronic state called the topological surface state (TSS) has been predicted and observed [1, 2, 3, 4, 5, 6, 7, 8]. Unlike the trivial insulator, TIs have a spin degenerate and fully gapped bulk state but exhibit a spin polarized and gapless electronic state on the surface [8]. This metallic surface state has a linear energy-momentum dispersion relation in the low-energy region, which is known as a Dirac cone. Unlike the Dirac cone of graphene, the Dirac cone of a TI is protected by the time-reversal symmetry. This robust TSS can survive under time-reversal invariant perturbations, such as surface pollution, crystalline defects, and distortions of the surface [6]. Additionally, because of the fully spin-polarized characteristics of the surface state, TIs have a high potential for the development of spintronic devices and quantum computation [6, 11].
The optoelectronic properties of TIs are important subjects for the development of optoelectronic devices. Therefore, the issues associated with electron–phonon interaction, carrier lifetime, carrier dynamics, energy loss rate, and low-energy electronic responses are very important for optimizing device performance. These ultrafast dynamic properties of the materials can be resolved by pump-probe spectroscopy. This chapter provides a brief introduction to the materials, time-resolved pump-probe spectroscopy, and some ultrafast dynamic properties of Bi-based topological insulators.
2. Bismuth-based topological insulators
Bismuth chalcogenide compounds (Bi2Ch3, Ch = Se, Te) have been extensively investigated in material science and condensed-matter physics because of their intriguing properties regarding thermoelectricity [12, 13, 14] and three-dimensional TIs [15, 16, 17, 18]. Bi2Ch3 is a narrow bandgap semiconductor with a rhombohedral crystal structure belonging to the
In 2009, Zhang et al. predicted that the Bi2Ch3 crystal is a strong TI [15]. A calculation of the electronic structure with spin-orbit coupling in the Bi2Se3 crystal has also been performed [15]. By tuning the spin-orbit coupling in the system, band inversion occurred around the Γ point. As these two levels, which are closest to the Fermi energy, have opposite parity, the inversion between them drives the system into a TI phase [15]. Figure 2 shows the calculated energy and momentum dependence of the local density of states (LDOS) for Sb2Se3, Sb2Te3, Bi2Se3, and Bi2Te3. All of these materials have the same rhombohedral crystal structure with the space group
Figure 3(a) and (b) shows the ARPES results of the surface electronic structure on a Bi2Se3 (111) surface [20]. Around the
3. Principle of femtosecond spectroscopy
3.1. Degenerate pump-probe spectroscopy
Highly temporal resolution is one of the unique characteristics in femtosecond optics. By the pump-probe technique, the photoexcited carrier dynamics and phonon dynamics in solid state materials can be clearly resolved. Additionally, the interband and intraband relaxation processes can be also obtained.
The basic understanding of time-resolved pump-probe spectroscopy is introduced as follows. The pump pulses are served as a perturbation which leads to the changes of the electronic population in materials. The probe pulses are used for the detection of the optical property changes of the materials. By controlling the time interval between the pump and probe pulses, the transient changes of the optical properties can be recorded. In pump-probe spectroscopies, the transient reflectivity changes (∆
Here, we explain more experimental details about the detection of ∆
where the
3.2. Optical pump and mid-infrared probe spectroscopy
The plasma edge of the doped n-type semiconductor usually lies in the mid-infrared (MIR) regime. By measuring the reflectivity around the plasma edge, many characteristics of carriers such as scattering rate and carrier concentration can be obtained [21]. The development of a pulsed mid-infrared light source provides the opportunities for understanding the dynamics of carriers. The mid-infrared pump-probe spectroscopy has been already applied on various materials (i.e., oxides, semiconductors, superconductors, graphene, and topological insulators) [22, 23, 24, 25, 26, 27, 28]. In the reflection-type mid-infrared pump-probe spectroscopy, the effect of multiple reflections should be considered in the analysis, and the dynamical characteristics of carriers can be further obtained through modeling the measured data with the Drude-Lorentz model.
Figure 5 shows a schematic diagram of our optical pump and mid-infrared probe (OPMP) spectroscopy. The light source of the pump-probe system is a regenerative amplifier with 800 nm central wavelength, 5 kHz repetition rate, and 30 fs pulse duration. The beam is split into a pump beam (40% of the incident light) and a probe beam (60% of the incident light). The probe beam passes through a 0.7-mm-thick GaSe crystal to generate mid-infrared (MIR) pulses, in which the MIR wavelength can be tuned from 9.0 μm (138 meV) to 14.1 μm (88 meV) through differential frequency generation (DFG). The optical pump beam with the fluence of 68 μJ/cm2 and a spot size of 485 μm (in diameter) is focused on the sample using a 150 mm lens. An Au-coated off-axis parabolic mirror with
4. Ultrafast dynamics in topological insulators
4.1. Time-resolved spectroscopy in a topological insulators
The dynamic properties of photoexcited TIs have attracted a great deal of attention. For example, the relaxation behavior of a carrier near the Fermi surface has been observed by the time-resolved angle-resolved photoemission spectroscopy (Tr-ARPES) [29, 30, 31, 32]. Figure 6(c) shows that the 1.55 eV photons excite the electrons from the bulk valence band to a higher-lying state in the bulk materials. Then, the photoexcited carriers fall into the bulk conduction band (BCB) and the surface state within 1 ps [31]. In Figure 6(a), we can see the rise time of curve 10 is ~1 ps. This means that after photoexcitation, the carriers in the higher lying band are rapidly relaxed into the BCB, then cooled to the bottom of the BCB via intraband scattering. These interband transitions and intraband scattering are shown in Figure 6(d) and (e) [31].
Furthermore, the relaxation time of curve 10 in Figure 6(a) is longer than 10 ps. This slow relaxation indicates the metastable behavior of the population of carriers in the BCB [30, 31]. Meanwhile, as curves 6–9 shown in Figure 6(b), the population of surface states also exhibits an unusually long-lived existence [31]. Here, the relaxation bottleneck is attributed to the scattering processes between the BCB and the surface state [31]. As Figure 6(f) shows, the photoexcited carriers first relax via surface-bulk scattering and then cooling via surface-state intraband scattering. This scattering channel is mainly in response to the acoustic phonon-mediated surface-bulk coupling and the acoustic phonon scattering of the surface-state Dirac fermions [32]. The Tr-ARPES can directly deliver information about the population changes of the electronic state near the Fermi level. However, reports on the transition processes occurring in the early stages after photoexcitation are rare. To fully understand the photoexcited carrier dynamics, studies for the interband transition and the intraband cooling are needed, which can be revealed using optical pump/optical probe spectroscopy (OPOP) and optical-pump/mid-infrared probe (OPMP) spectroscopy.
4.2. Interband relaxations in topological insulators
The interband relaxation of photoexcited carriers in topological insulator (TI) single crystals is examined by the optical pump and optical probe spectroscopy [33]. In this section, we present the phonon and carrier dynamics in doped TI CuxBi2Se3 (x = 0, 0.1, 0.125) single crystals. Figure 7(a) shows the typical Δ
The slow oscillation components, as shown in Figure 7(a), are attributed to the coherent acoustic phonons (CAPs) generated by ultrafast laser pulses. This damped slow oscillation in Δ
The fast oscillation components of CuxBi2Se3 crystals are presented in Figure 7(b), which can be extracted by removing the relaxation background from the Δ
Figure 8(a) shows the fast oscillation component for CuxBi2Se3 (x = 0, 0.1, 0.125) crystals. In order to quantitatively analyze these oscillations, a damped oscillation function,
4.3. Intraband relaxations in topological insulators
The femtosecond snapshots of the relaxation processes and Dirac fermion-phonon coupling strength of 3D TI Bi2Se3 were revealed by OPMP spectroscopy [26]. In this study, several selected Bi2Se3 single crystals with a wide range of carrier concentrations (
Code | (meV) | Carrier concentration | ||
---|---|---|---|---|
(1018 cm−3) | (1013 cm−2) | |||
#1 | 422 | −51.5 ± 0.84 | −1.45 | 0.11 |
#2 | 325 | −13.9 ± 0.26 | −0.83 | 0.20 |
#3 | 284 | −5.58 ± 0.25 | −0.72 | 0.35 |
#4 | 260 | −0.25 ± 0.01 | −0.47 | 0.89 |
Based on the ARPES image and the energy band structure of TI Bi2Se3, a model is proposed [in Figure 10(a)] for the optical pumping (1.55 eV) and mid-infrared probing processes to elucidate the origins of both positive and negative signals. The band gap of Bi2Se3 is ~300 meV, as shown in the ARPES images of Figure 9(b), which is much larger than the probe photon energy (87~153 meV) of the mid-infrared (mid-IR). Thus, it does not allow the occurring of the interband transitions between the valence band (VB) and the conduction band (CB) of the bulk. Meanwhile, the free-carrier absorption in the CB [the probe (1) in Figure 10(a)] and Dirac cone surface state [the probe (2) in Figure 10(a)] will dominate the probe processes, which can be assigned to the positive and negative peaks in Δ
As found in Figure 9 and Table 1, the amplitude of positive peak in Δ
As shown in Figure 11(d), both rising time (
The relaxation of Dirac fermions has been demonstrated via phonon medium [38, 39]. The coupling strength (λ) between Dirac fermions and phonons varies at different positions of the Dirac cone, which can be revealed from the photon energy-dependent rising time. Based on the second moment of the Eliashberg function [40], the coupling strength (λ) is inversely proportional to the relaxation time (
where
5. Conclusion
We report the ultrafast dynamics of carriers and phonons in topological insulator Bi2Se3, CuxBi2Se3 (x = 0, 0.1, 0.125) single crystals. By time-resolved pump-probe spectroscopy, one damped fast oscillation was clearly observed in the transient reflectivity changes (∆
Acknowledgments
Financial support from Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 103.99–2015.17, the Ministry of Science and Technology of the Republic of China, Taiwan (Grant No. 103-2628-M-009-002-MY3, 103-2119-M-009-004-MY3, 106-2119-M-009-013-FS, 106-2628-M-009-003-MY3) and the Grant MOE ATU Program at NCTU are gratefully acknowledged.
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