Bismuth and antimony-based chalcogenides have been extensively publicized in recent years owing to their intrinsic characteristics and inherent topological character. Such a system contains Bi2Se3, Bi2Te3, Sb2Te3, etc. The single crystalline facets of these samples were discovered to have a generation of ~2 THz while having a giant magneto-resistance of around ~300%. These inherent and dynamical features of the system make it resilient for several applications in optoelectronics and spintronics. The temperature-dependent assessment of conductivity, terahertz generation, and charge carrier dynamics aids in understanding the fundamental phenomena in the carrier mechanism of the chalcogenides. This chapter contains the essential fundamental knowledge of the single crystal chalcogenides via charge carrier & phonon dynamics and their response in the terahertz frequency domain.
- topological material
- temperature-dependent carrier dynamics
- transport properties
The quantum interaction of charge carriers with external and internal forces in different materials remains an open question for the condensed matter community. The enigma began with Edwin Hall’s discovery of the classical Hall Effect in 1879 . The Hall effect shows a voltage difference formed by injecting steady current and magnetic fields across a conductor or semiconductor. Due to the current and external magnetic fields interplay, voltage generation occurs. This voltage difference is due to charge confinement. In the 1980s, Von Klitzing discovered the quantum hall effect (QHE) in a two-dimensional system . Due to the intense magnetic fields, charge carriers are constrained into two dimensions and exhibit topologically ordered states. The edge states at the surface cause current to flow at the superiorities, and these states are formed as a result of the high external magnetic field. These geometrical states give birth to a new phase of matter known as topological insulators (TI) . Charles Kane and Eugene Mele anticipated the development of TI in 2005 [3, 4, 5, 6, 7, 8]. It is very similar to QHE, except that no external magnetic field is necessary since the inherent characteristics of materials generate the magnetic field. The spin-orbit interaction generates this magnetic field. TI is unusual because it is insulating in bulk yet conducts at the surface. The electronic wave function of a charge carrier is dependent on its shape, which varies from bulk to surface. As a result, it is referred to as a topological insulator. The SS renders the system impervious to non-magnetic doping due to protected SS. These SS are protected by time-reversal symmetry (TRS). This small property enables many unique applications due to the impervious topological states to non-magnetic disruption and their dissipation-free spin current transit. Spintronic, thermoelectric, magnetic memory storage, magnetoelectric devices, next-generation batteries, THz generators, transistors, photodetectors, and sensor applications are only a few of these applications, which is possible in these TI [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13].
This article is focused on topological insulators, providing an overview of the topological phases and states found in TIs. Additionally, the dynamics of the carrier and phonon scattering are also discussed. TI’s surface and bulk states are probed using various optical methods. The ultrafast laser pulse is employed in particular to characterize the functional characteristics of Fermions in TI. These pulses have also been used to examine the phonon vibration mode. Finally, it establishes the existence of the coherent optical phonon (COP) and coherent acoustic phonon (CAP) modes. The temperature-dependent evolution of these modes has also been examined as these phonon vibrations progress with the charge carrier dynamics. Transient Reflectance Ultrafast Spectroscopy (TRUS) was utilized to explore the non-conventional conducting surface charge carriers. A femtosecond pump beam was employed to stimulate the sample. The material was probed with a wide beam ranging from visible to near-infrared. TRUS measurements aid in determining the charge carrier dynamics and the capacity of terahertz production. Additionally, investigating carrier and phonon dynamics in a temperature-dependent manner aids in the understanding of crucial transitions associated with surface states.
2. Transient reflectance ultrafast spectroscopy (TRUS)
J.Qi. et al. performed the first TRUS measurements on Bi2Se3 crystals , where they demonstrated the time-resolved behavior of the crystal at room temperature. Three distinct relaxations of carriers induced by photons were identified in this experiment. The first two are phonon interactions between excited charge carriers of coherent optical and acoustic phonon interactions. In contrast, the third is a negative amplitude process generated by ultrafast carrier trapping of selenium. They also observed the frequency of 2.13 THz from their optical phonon oscillations. They also concluded that the atmosphere impacts these charge carriers and phonons since air promotes band bending, which causes an elevation in the Fermi level . Nardeep Kumar et al.  utilized the ultrafast pulse to analyze the carrier’s behavior by pump-probe spectroscopy. The Bi2Se3 TI exhibits two distinct oscillations, with the rapid oscillations occurring at 2.167 THz and the slow oscillations occurring at 0.033 THz. Coherent optical and acoustic phonons are responsible for this frequency . In these systems, the terahertz frequency is not dependent on the size of the laser spot, and the ambipolar carrier diffusion coefficient is also determined to be 500 cm2/s . Nardeep et al. have shown that the COP-induced terahertz frequency production has an uneven dependency.
The ultrafast carrier dynamics were further investigated in the limited thickness of the TI, using pump-probe spectroscopy in thin films by Yuri D. Glinka et al. . The thicker films have similar relaxation lifetimes to bulk crystals, while the thinner films exhibit fastened relaxation of excited charge carriers. The longevity of SS and bulk states might be related to their contribution. The shorter carrier lifetimes in thin films are associated with reducing bulk contributions of carriers in dominant surface states. A resonance-like property is also seen in 10 nm films, but the study does not provide conclusive evidence for how the property is generated. Yuri D. Glinka et al. also investigated the thin films of Bi2Se3 (6–40 nm), in which they confirmed the presence of radiative and non-radiative processes and described resonance phenomena at 10 nm films in terms of these processes . This article establishes unequivocally that bismuth selenide contains a second SS. After the detection of the second SS, the primary trend observed in TRUS is the presence of three distinct processes: (a) electron–electron and electron longitudinal optical phonons in the 1–8 ps range, (b) a metastable bulk conduction band that continuously feeds charge carriers to the second SS for approximately 10 ps, and (c) a quasi-equilibrium carrier population. The thin layer is activated by 1.51 eV photons that excite carriers from the bulk conduction band (BCB) to the second surface states (SS). From the second surface to the first SS, carriers undergo intra-band and inter-band relaxation. The relaxations occur between the second SS and BCB and first SS, indicating that the ultimate recombination happened in these states. The relaxation of the excitation charge carriers results in forming a new Fermi level that is displaced away from the original Fermi energy. This relocation is a result of carrier localization. Again, the resonance phenomenon in a 10 nm film is found in this pump-probe spectroscopy, which is explained by the depletion of electrons caused by the connection between two film surfaces. The confinement of these 3D electrons is due to the existence of surface defects caused by selenium vacancies; however, they grew the 10 nm film many times and obtained this resonance in each of these films. This repeatability lends credence to the selenium vacancy theory. This resonance effect in 10 nm TI is unknown at the moment. However, recent work by Glinka et al.; explains the discrepancy by suggesting several surface state amendments .
3. Terahertz generation at room temperature
Because of the expansion of the communication systems, information processing, and transmission fields are the most susceptible [19, 20, 21, 22]. In terms of the frequency range, the lower Terahertz range may benefit air transmission, while the higher frequency range enables faster signal transfer and can be used to create ultra-high bandwidth data links. A fundamental component of numerous fields, including information processing and transmission, security screening, and biomedical applications, are emerging from Terahertz research [19, 21, 23, 24]. The future applications of Terahertz radiations have prompted an influx of scientific and technical research into creating Terahertz pulses for bio-material imaging, ultrafast dynamics, and nonlinear Terahertz optics [21, 25], among other things. This frequency may be generated in various ways, one of which is by using an ultrafast and ultrashort laser pulse. The interaction of short laser pulses with various objects makes it more challenging to produce adjustable, compact, coherent, and high-magnitude Terahertz radiation sources. The two-color filamentation in gases is a simpler way for a generation since it can scale the magnitude to incredible levels of interest and complexity. However, the scaling was limited by the laser pulse energy constraints, and as a result, the search for alternate target materials was essential for these applications to succeed. It is necessary to search in a varied and uncluttered study region to locate the target, which exhibits Terahertz generations of high energies. The different crystals are the subject of this study due to their crystalline nature, as they can be used for large-scale broadband and high-frequency terahertz generation. Accordingly, phonon dynamics of topological insulators single crystals provide an attractive benefit in terahertz generator performance.
The terahertz frequency range of 0.03 to 5.2 THz is created by a single crystal of Bi2Se3 [14, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38]. In summary, phonons, which are the COP in bismuth selenide, are responsible for generating the terahertz frequency, as mentioned in the TRUS section above. The terahertz spectroscopy directly validates the 2 THz frequency generation, while the TRUS was the initial method by which these oscillations were originally seen [14, 18, 26, 30, 32, 34, 36, 37, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53] and was used to confirm the frequency of the 2 THz frequency. In addition, Prince Sharma et al. in 2020 contrasted assessing the frequency determined through TRUS, revealing the COP oscillations [34, 48] to evaluate the frequency from the kinetic spectrum. Figure 1 depicts oscillations obtained from the kinetics of a flake cleaved out of the large single crystal of Bi2Se3 that was recorded at the CSIR-NPL. The FFD (filtering the high-frequency component and fitting the data) analysis investigates the generated frequency. These oscillations are removed from the kinetic data profile of charge carriers by applying a high pass filter in the 1 to 10 ps period. The data that has been filtered out is fitted with the sinusoidal damped function to determine the frequency associated with these oscillations. The frequency associated with them is determined to be ~2.42 terahertz (THz) from FFD analysis.
A significant expansion on the assessment of terahertz production is also carried out, where the terahertz frequency may be changed just by altering the interaction of electromagnetic radiation in a single crystal. First and foremost, the probe energy is monitored while the excitation energy is maintained consistently. We detected strong oscillations at 1100 nm with a frequency of 2.42 THz in this system. A continuous probe at 1100 nm is then monitored by changing the excitation wavelength, which is the second step. In Figure 2, the tunable nature of the single-crystalline flake of bismuth selenide  is shown in detail. It is discovered that the frequency of phonon vibrations may be controlled by tuning the carriers in the crystal by appropriate doping, as well [45, 54]. Consequently, the adjustable nature of terahertz in single crystals might be beneficial in the growth of optoelectronics and Terahertz applications.
Although the primary emphasis is on the COP dynamics, Yuri D. Glinka et al. return to the distinct relaxations, namely, CAP . The paper noted an increase in CAP frequency from 35 to 70 GHz. When the thickness of TI was reduced, the interaction of two processes caused this frequency difference. As the film thickness goes below 15 nm, lamb wave excitations (elastic waves whose propagation is plane to plane) become visible. Above this critical thickness, the system operates in the bulk acoustic resonator mode (indirect inter-surface coupling). Apart from performing TRUS measurements at ambient temperature, Yi-Ping Lai et al. conducted a temperature-dependent examination between 11–294 K . To begin, they summarized the physical processes occurring at various time scales predicted by pump-probe experiments as fast oscillations (1012 Hz–COP), slow oscillations (1010 Hz–CAP), non-oscillatory signal (1011 Hz – electrons and incoherent phonons), and constant residual (1011 Hz – slow electron dynamics).
The temperature-dependent evolution of the COP revealed that when the sample approaches room temperature, the optical phonon frequency reduces from 2.25 to 2.17 THz . This investigation also determines the electron–phonon coupling constant, demonstrating that the observed signal dominates the bulk. Additionally, Sung Kim et al.  studied the compatibility of the thin films with various polycrystalline and crystalline substrates and the resonance impact of the same at various thicknesses ranging from 3 to 30 nm. At 2.1 and 5.2 THz, the intensity of the differential reflectance (DR) and two distinct phonon modes exhibited some resonant behavior within a specific range of a crucial number of quintuple layers (QL). This resonant effect is substrate-independent, and the crucial QL value is between 6 and 9 QLs. Apart from this resonant effect, Jianbo Hu et al.  regulated the strong Raman mode of bismuth selenide using a two-pump laser in TRUS studies. They demonstrated phonon chirping due to the two-pump arrangement, which confirms the earlier thesis of carrier-lattice coupling. These two pumps effectively tune the amplitude, but the frequency stays constant.
Pump-probe spectroscopy is used to investigate single excitations with a specific energy. However, Giriraj Jnawali et al. employed the mid-infrared femtosecond TRUS on a nanoflake of TI for the first time . The energy range is 0.3–1.2 eV. This range includes both the original BCB and the extended BCB. As a result, the paper modeled the ultrafast photoexcited carriers and holes across the BCB. Theoretical modeling of 10 K DR indicates a strong probability of the Fermi level being present much above the lowest conduction band. These experiments established a firm knowledge of the carriers inside TI’s BCB, establishing a direct connection between carriers and Dirac SS .
Nonetheless, the TRUS provides no evidence for spin-polarized charges in SS through these broad energy probes. M.C. Wang investigated spin-dependent transitions that occurred solely in topologically protected SS using time-resolved Kerr rotation measurements . Transitions between topologically protected SS may induce net spin polarization due to the presence of spin-protected carriers inside these SS. However, the study demonstrates that a circularly polarized pump may create net polarized spin, but the transitions responsible for this polarized spin are not just between the two SS but also between the first SS and the second BCB located near the second SS . While the TRUS predicts just the energy levels of the different bulk conduction bands at ambient temperature. The surface state-related transitions are readily seen when investigating the temperature-dependent dynamics.
4. Temperature-dependent charge and phonon dynamics in Bi2Se3, Sb2Te3, and Bi2Te3
TRUS is used to investigate the exotic topological quantum characteristics of Bi2Se3, Sb2Te3, and Bi2Te3. This particular regime probing aids in comprehending the TI’s enigmatic behavior. Considering the bismuth selenide, the PL emission indicates a significant 2 eV optical transition  caused by the state bunching effect . The inert nature of these transitions is explained using density functional theory (DFT) calculations on the band structure and Kramer’s Kroning method on reflectance data from crystal flake. Additionally, TRUS measurements are performed with a variety of pump excitation energies (3.02, 2.61, 1.91, and 1.4 eV) to obtain a spectrum in the VIS–NIR region (2.58–0.77 eV) . These wide regime experiments on carrier dynamics demonstrate that the Moss–Burstein and shielding effects exist in bismuth selenide. Additionally, these studies demonstrate a variety of relaxation mechanisms, including thermalization of hot carriers, COP and CAP relaxation, and recombination.
All of the phenomena mentioned above are observed at room temperature, and there is a strong need to explore the TIs at low temperatures to assess surface state-induced transitions. Essentially, there are two distinct ways for probing surface states or surface-related transitions. We already know that TIs are conducted at the surface while insulating bulk. Thus, the surface states of TIs are located on the uppermost layer. Therefore, in order to explore these surface states, the sample thickness should be reduced to the ultrathin regime [17, 18, 55]. The sole disadvantage of this strategy is the complexity of the systems, required to develop this kind of ultrathin film. Alternatively, the low-temperature technique is used to investigate the temperature-dependent dynamics of charge carriers and phonons between 5 and 300 K. The perceptions of observing the surface states or surface states associated with transitions at low temperatures may be explained by performing a magneto-resistance analysis using the HLN equation. The literature demonstrates that at extremely low temperatures and magnetic fields, the surface states of TIs prevail over the bulk states [57, 58, 59, 60]. Thus, it may be interesting to examine the charge and phonon dynamics of TIs at low temperatures in order to see surface states or surface state-related transitions.
The TRUS is carried out at low temperatures on micro flakes of single-crystalline bismuth selenide to observe the associated transitions in the temperature range of 5–300 K . Figure 3a illustrates the DR spectra of crystalline flake across a wide range of NIR wavelengths at various temperatures. The wide DR signal demonstrates TI’s ability to exhibit a broad range of optically allowed transitions, distinguished as 0.7, 1.1, and 1.4 eV. DFT band structure calculations anticipated that these specific transitions are permissible in bismuth selenide . As seen in Figure 3a, a DR signal is denoted by a B peak that is only present above 200 K. At low temperatures, this specific transition is suppressed, indicating that it is associated with certain phonon-assigned carriers. Essentially, these carriers have initial energy of below 200 K, caused by thermalization at high temperatures.
Additionally, when the temperature decreases below 200 K, a tiny DR signal of 1.2 eV is formed at 100 K. The level of these DR signals increases when the temperature is lowered and becomes more noticeable below 5 K. These DR signals correspond to the same as the second strategy of exploring surface states associated transitions mentioned above. There are two possible explanations for this transition: defect-induced peak and surface state-related transition. If we consider the first option of a defect-induced peak, then this kind of transition has a common property, i.e., the carrier relaxation lifetime must be very short. However, the lifetimes in this situation are in the picosecond range, which eliminates the likelihood of a defect-induced peak . Additionally, the kinetic profile of the same does not alter with temperature, corroborating the preceding explanation. Thus, one thing is evident that this specific DR signal is not the result of a defect.
After establishing that the DR signal at 1.2 eV is not attributable to defects, it is not incorrect to assert that the signal is predicted to be due to a surface state-related transition. DFT calculations are used to determine the band structure using the effective SOC inclusion to resolve this particular uncertainty. After learning about the band structure, it was relatively simple to formulate the 1.2 eV transition. This transition occurs when the carrier is excited from its ground state to its second surface state. Additionally, the kinetic profile during this transition is fitted using three lifetimes, and the fitting of this kinetic profile indicates that carriers relax in the picosecond time domain when they migrate from this energy level to a lower energy level. In the case of noble metals, when the relaxation time is in the picosecond range, it is widely known that the surface states reveal metallic nature. However, in our case, too, the surface states exhibit metallic properties. As a result, the relaxation lifetime suggests the occurrence of a surface state-related transition in TI .
Band structure calculations utilizing DFT in conjunction with actual pump-probe spectroscopy may be used to predict various optically allowed transitions in bismuth selenide. First surface state below Fermi level (SS1), second surface state above second valence band (SS2), first bulk valence band (BVB1), second bulk valence band (BVB2), first bulk conduction band (BCB1), and second bulk conduction band (BCB2) of TI are shown in Figure 3b. The temperature-dependent TRUS response is shown in Figure 3a. The most prominent optical transitions in TRUS are ~0.7, ~1.0, ~1.4, and ~ 2.0 eV. Charge carriers are stimulated to the second bulk valance band, exhibiting ~0.7 and ~ 1.0 eV DR signal peaks, whereas the ~2.0 eV transition occurs in the second bulk conduction band. Additionally, the low-temperature investigation reveals the existence of a DR peak of about 1.2 eV, which corresponds to the transition to the second surface state. Thus, the temperature-dependent study of charge carrier dynamics enables the investigation of various bands and surface-related transitions.
4.2 Bi2Te3 & Sb2Te3
Temperature-dependent TRUS measurements are also performed in the near-infrared region on micro flakes of bismuth telluride. At room temperature, many optically allowed transitions are detected in TIs. Additionally, it was discovered that the DR response below 100 K exhibits a noticeable transition that is connected to the surface state of bismuth telluride and antimony telluride. At 500 fs, Figure 4 depicts the temperature-dependent DR signal, where two distinct NIR regimes are established. One is the bulk zone, while the other is the topological regime associated with surface states. The bulk regime exhibits a variety of optically allowed transitions between valence and conduction bands. Simultaneously, the surface regime denotes the shift from a surface to a more excited state .
5. Temperature-dependent terahertz frequency generation
The kinetic profile is used in order to investigate the temperature dependence of the oscillation frequencies generated in the terahertz regime. In addition to being in the terahertz frequency range, other frequency modes are also observed in the TI. These modes lie in the gigahertz frequency range due to CAP. While the terahertz frequency is generated because of the vibration of COP. These modes are analyzed using filtering the high-frequency component and fitting the data (FFD) . This detailed analysis helps exclude the charge carrier’s relaxation from phonon-associated vibrations. The COP-associated frequency is found to have lied in the terahertz regime. Moreover, these frequency modes are dependent on excitation and probe energy, as mentioned above. In order to visualize the changes of these modes with temperature, the kinetic profile at 100 nm is considered. The micro flakes of different TI are excited with pump energy of 3.02 eV at 1.0 mW average power in order to observe the kinetic profile at this particular wavelength. The temperature-dependent investigation is carried out at these pumps and probe energy at different temperatures. With the use of this FFD analysis, it is possible to identify the terahertz frequency at which the phonons oscillate. Each of the TIs, namely Bi2Te3, Sb2Te3, and Bi2Se3, exhibits oscillations of the order of terahertz frequency in the range of 1–3 THz, as seen in Figure 5.
The low-temperature investigation convincingly demonstrates the presence of a second surface state and a wide absorbance of 1.2 eV in bismuth selenide using TRUS. While the surface state-related transition occurs at ~1.2 eV, ~1.1 eV, and ~ 1.0 eV above the Fermi surface in bismuth selenide, bismuth telluride, and antimony telluride, respectively. Thus, this work demonstrates unequivocally that temperature-dependent analysis of charge carrier and phonon dynamics aids in the extraction of surface states associated transitions within TI. Additionally, the temperature dependency of the COP mode is established in all these TIs. At 100 K, 50 K, and 5 K, the charge carrier dynamics of Bi2Te3 and Sb2Te3 show a shift from their surface state to an excited state in the conduction band, which is due to increasing carriers in the surface conduction channel. Thus, investigation of TIs at low temperatures reveals the emergence of TSS-related transitions and their dominance at low temperatures, which is repressed at room temperature due to bulk carriers’ thermalization.
The director of NPL strongly supports this work. Mr. Prince Sharma wishes to express his gratitude to CSIR-UGC for financial assistance and AcSIR for admitting him as a research scholar to its Ph.D. program.
Conflict of interest
The authors declare no conflict of interest.
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