Open Access
Add to BibSonomy
Abstract
This study comprehensively investigated the coherent lattice dynamics in Bi2Se3 by ultrafast optical pump–probe spectroscopy with tunable near-infrared probe pulses. Sample-thickness- and probe-wavelength-dependent experiments revealed the key role of Bi2Se3 optical property in the generation and detection of photoinduced strain waves, whose confinement initiated coherent interlayer vibrations. Furthermore, the frequency and lifetime of the interlayer vibrations could be quantitatively explained with a modified linear chain and an acoustic mismatch model considering elastic coupling at sample–substrate interfaces. The results of this work provide insights for analyzing and interpreting, through ultrafast optical spectroscopy, nanomechanical interactions in layered materials.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Bismuth selenide (Bi2Se3) is a layered electronic material that draws attention as a topological insulator [1,2]. The fascinating coexistence of insulating bulk and conducting surface states with spin-momentum locking in it enables ultrafast carrier transport and broadband light absorption, allowing the manufacturing of high-performance electronic [2], spintronic [3], and optoelectronic devices [4,5]. Furthermore, the manipulation of its electronic properties has been spotlighting the study of fundamental material properties for advanced applications. For this manipulation, diverse methods have been introduced, including chemical substitution and thickness control [6,7], mechanical strain [8,9], infrared excitation [10], and nanostructured fabrication [11,12]. Among them, out-of-plane mechanical strain can efficiently modify the bandgap and subsequently change the topological phase of Bi2Se3 [8,9,13,14]; similarly, optically generated out-of-plane interlayer vibrations in Bi2Se3 can be a predominant strategy to control its material property in an ultrafast timescale because of their high-speed tunable frequency and coherent controllability [15–17].
The interlayer vibrational modes in Bi2Se3 have been investigated through various techniques, including Raman spectroscopy [15], ultrafast time-resolved X-ray diffraction (UTXRD) [16], and optical pump–probe (OPP) spectroscopy [17,18]. Due to the lack of translation symmetry along the c-axis and the layered structure with quintuple layers (QLs, ∼1-nm thickness), quantized interlayer vibrational modes in thin films differ from their bulk counterpart, as observed through traveling acoustic waves [18,19]. According to previous studies [15–17], the frequency of interlayer vibrations increases with decreasing the number of QLs (N); this is explained by a linear chain model considering each QL as a unit mass with van der Waals (vdW) force and a restoring force. In an N-QL system, the solution of this model with a substrate indicates that the out-of-plane interlayer vibrational modes consist of a substrate-induced interface mode and N−1 breathing modes, which are the nondegenerate A-symmetry modes of each QL along the c-axis. Their estimated lifetimes are of about tens of ps, sufficiently long for investigations in the time domain [16–18]. In particular, a recent UTXRD study [16] has directly demonstrated such out-of-plane interlayer vibrations by monitoring the photoinduced changes of the (006) Bragg peak position, which corresponds to the interlayer spacing modulations. On the other hand, in some OPP experiments, involving also transient reflectivity (TR) [17–19], the main detection mechanism was the photoelastic effect, which is attributed to local changes in the refractive index by lattice vibrations; thus, these studies typically identified the interlayer vibrations according to the frequency of the measured signals. However, for a specific sample thickness and probe wavelength, the observed frequency can be explained by both the traveling acoustic waves geometry (bulk) and the quantized interlayer vibrations (thin films) [18]. This suggests that further studies are needed to properly identify the characteristics of vibration signals with similar frequencies. Moreover, when the penetration depth of the probe beam is too short compared to the sample thickness, acoustic echo signals moving back and forth are detected [20–22]. This condition also limits the precise optical investigation of interlayer vibrations in relatively thick samples. Therefore, the dynamics of interlayer vibrations based on OPP spectroscopy in samples within a broad thickness range covering dimensional crossovers are still unclear.
In the present work, we comprehensively studied the sample-thickness and probe-wavelength dependence of interlayer vibration dynamics in Bi2Se3 via the OPP spectroscopy technique. With the probe-wavelength-dependent measurements, we could distinguish whether the measured coherent oscillations were attributed to traveling acoustic waves in the bulk or interlayer vibrations in thin films within a broad frequency range (∼7–180 GHz). By choosing probe beams with a large penetration depth compared to the sample thickness, we could identify low-frequency interlayer vibrations even in a relatively thick sample (<100 nm), consistently with the abovementioned UTXRD study [16]. The detected frequency and decay time of oscillatory signals in the bulk strongly depended on the probe pulse wavelength; conversely, in the thin-film regime, the frequency and lifetime of the interlayer vibrations depended on N and not on the probe wavelength. Our results demonstrate that the initiation of interlayer vibrations in Bi2Se3 by pulsed laser excitation results from the confinement of photoinduced strain waves by acoustic mismatch at the sample-substrate interfaces.
2. Experimental method
The OPP spectroscopy measurements were performed on Bi2Se3 samples with different thicknesses (d) of 7, 16, 26, 43, and 78 QLs and bulk crystals. The Bi2Se3 films were grown on an Al2O3 (0001) substrate by molecular beam epitaxy. The detailed growth process and characterization results can be found in a previous report [16]. Figure 1(a) schematizes the OPP spectroscopy measurements. The experiment was conducted by using a mode-locked Ti:sapphire oscillator, which generated 100-fs pump pulses at a center wavelength of 830 nm and a repetition rate of 80 MHz; the probe pulses, which were tunable 150-fs pulses, were generated by a synchronously pumped optical parametric oscillator in the wavelength range of 1100–1550 nm at the same repetition rate. After collinearly combining with a dichroic mirror, the pump and probe pulses were focused to a beam spot diameter of ∼2.5 and ∼4 µm, respectively, on the sample by a single microscope objective lens. The time delay between the pump and probe pulses was controlled with a motorized linear stage. The excitation fluence of the pump pulses was ∼380 µJ/cm2, and the probe fluence was sufficiently weak compared to it. The changes in probe reflectivity (ΔR/R0 = (R0-R(t))/R0, where R0 and R(t) are the reflectivity without and with pump pulse according to the time delay) were recorded with a photodetector and a lock-in amplifier. All the measurements were conducted at room temperature and normal pressure.
Fig. 1. (a) Schematic of the optical pump–probe measurements. (b) Crystal structure of layered Bi2Se3 consisting of quintuple layers (QLs) formed by Se2–Bi–Se1–Bi–Se2 atomic sequences; each QL is ∼1-nm thick and repeated periodically with van der Waals gaps. (c) Experimental generation and detection of traveling strain waves propagating through the sample with a longitudinal sound speed (${{\boldsymbol v}_{\boldsymbol s}}$). (d) Transient reflectivity signal of the bulk sample at different probe wavelengths (${\lambda _{pr}}$); the curved lines crossing the negative and positive peaks indicate the wavelength dependence of the oscillation period. (e) Frequency and (f) decay time of the oscillatory signal, obtained by fitting the data in (d) with Eq. (1); the blue curves represents the results derived from Eq. (2) and Eq. (3).
3. Results and discussion
Figure 1(b) illustrates the crystal structure of Bi2Se3 with an atomic QL sequence between vdW gaps. The coherent oscillation dynamics in bulk Bi2Se3 were recorded by reflection geometry (Fig. 1(c)). The carriers excited by optical pump pulses (1.5 eV) can generate lattice stress in samples mainly through thermoelasticity and deformation potential, which induce thermal and electronic stress, respectively [23,24]. According to previous ellipsometry measurements [25], the penetration depth (ξ) in bulk Bi2Se3 can be obtained as ξ830 nm ∼ 30 nm at the pump wavelength (830 nm), which is comparable with the value of ξ800 nm ∼ 26 nm in Ref. [26]. The values for the probe beams (ξprobe), according to Ref. [25], are 72 nm (ξ1100 nm), 85 nm (ξ1200 nm), 116 nm (ξ1300 nm), 153 nm (ξ1400 nm), 169 nm (ξ1500 nm), and 176 nm (ξ1550 nm). In the opaque case, when ξpump << d (i.e., bulk crystal), photoinduced stress can be formed at the sample surface and propagate to its opposite side with a longitudinal sound speed (${{\boldsymbol v}_{\boldsymbol s}}$) (Fig. 1(c)).
These traveling strain waves were measured through the TR signals as a coherent oscillation pattern (Fig. 1(d)). The nonoscillatory fast-rising and decay signals at the initial time delay represent the photoexcited carrier dynamics. The oscillation period and decay time of the oscillatory signals seemed to increase along with the probe wavelength (${\lambda _{pr}}$). The frequency (${f_{osc}})$ and decay time (${\tau _{osc}})$ of the oscillatory signals were obtained by fitting the following ΔR/R0 equation:
For the Bi2Se3 thin-film (∼few or tens of nm) samples, the acoustic waves would bounce at each interface considering the reflection at a boundary and, thus, be confined in the sample; out-of-plane standing waves can subsequently form in such samples (Fig. 2(a)). After their formation, the observed frequency depends on N and no more on ${\lambda _{pr}}$ [18,33]. Figure 2(b) shows the TR signals of the 78 QL sample at different ${\lambda _{pr}}$ with the condition of ξpump < d. The initial time of acoustic dip increased along with ${\lambda _{pr}}$, as indicated by the tilted black arrow in Fig. 2(b); this tendency is like that of the signals in the bulk sample, as represented by the red arrow in Fig. 1(d). The subsequent acoustic shoulder was located near 36 ps regardless of ${\lambda _{pr}}$, as shown by the red dashed line in Fig. 2(b). The value of 36 ps well agrees with the transit time (d/${v_s}$) of the traveling strain waves for the 78 QL sample. After the transit time, the oscillation period became independent of ${\lambda _{pr}}$ as indicated by blue, black, and green vertical dashed lines, implying the confinement of strain waves. Furthermore, the obtained waveform was nearly identical to the UTXRD signal reported in a previous study [16], which directly probed the out-of-plane interlayer spacing modulations. This suggests that the generation of interlayer vibrations by ultrafast laser pulses was mainly caused by the confinement of photoinduced strain waves. A long penetration depth for the probe light (ξprobe ∼ d or > d) allows one to observe the lattice modulation throughout the sample induced by interlayer vibrations, even for thick samples. If ξprobe << d, acoustic echoes would be observed when the acoustic wave reaches the interface of the air-sample surface [20,21]. Given the relation ξ = $\lambda /4\pi \kappa $, the slightly small ${\kappa _{pr}}$ in Bi2Se3 thin films compared to the bulk [25] assured the experimental condition of ξprobe > d in this work.
Fig. 2. (a) Excitation profiles in the 78 quintuple layer (QL) Bi2Se3 thin film, showing the propagation of photoinduced strain waves and the formation of standing waves, and (b) corresponding transient reflectivity (TR) signals at different probe wavelengths (${\lambda _{pr}}$), where the black arrow indicates the shift in acoustic dip position, which is attributed to the traveling strain waves. (c) Excitation profiles in the 16 QL sample and (d) corresponding TR signals. The dash lines in (b) and (d) indicate the independence of oscillation frequency on ${\lambda _{pr}}$.
In the thinner sample case (16 QL) with ξpump > d, the pump pulses can excite the whole film, and the photoinduced lattice stress would be almost homogeneous along the normal direction of the sample; accordingly, out-of-plane standing waves can be instantaneously formed (Fig. 2(c)). As a result, the observed oscillation signal was independent of ${\lambda _{pr}}$ during the entire time delay (Fig. 2(d)). The oscillation period of the 16 QL sample, as indicated by the vertical dashed lines, was significantly short compared to that of the 78 QL one. Note that the waveform was different above and below the ${\lambda _{pr}}$ of 1300 nm (0.95 eV). This implies that the amplitude of the oscillation signal depends on ${\lambda _{pr}}$, which may not be observed in the 78 QL sample. The dielectric function near the bandgap changes rapidly from the absorption onset [20,27]. Therefore, the ${\lambda _{pr}}$-dependent amplitudes suggest the existence of a bandgap near 0.95 eV, similar to the reported value of ∼1 eV in Bi2Se3 thin films [34], which corresponds to the interband transition from the second and third valence bands to the second conduction band at the gamma point. Because of the ${\lambda _{pr}}$ dependence of the vibration amplitude, subsequent analysis was conducted by selecting the wavelength at which the vibration is most visible in the TR signal. Furthermore, the observed waveform in the thin films does not seem to fit a single damped oscillation, implying the existence of more than one oscillation mode.
Figure 3 illustrates the procedure of frequency analysis from the measured TR signals; each signal was fitted with Eq. (1) and the best fit was obtained with two oscillation modes, as displayed in the left panels by blue and beige curves. For the thin films, the uncertainties of the ${f_{osc}}$, ${\tau _{osc}}$, and $\phi $ obtained from the fitting were about 4%, 10%, and 1%, respectively. The fast Fourier transform (FFT) results in the right panel of Fig. 3 clearly show that the two vibrational modes are well-matched with the fitting results from Eq. (1). The frequency of these two vibrational modes pronouncedly indicates the N dependence, consistently with Ref. [16]. Furthermore, the initial phases in TR are similar to those reported by the previous UTXRD study [16]. The increase in inter-QL spacing accordingly resulted in a decrease in optical reflectivity. Moreover, the initial traveling strain waves were only observed in the TR of the 78 QL sample, but the waveform after the transit time is almost identical to the UTXRD signal [16]. Note that the frequency of interlayer vibration with ∼20 GHz (blue curves and circles in Fig. 3) for the 78 QL sample is the same as the one observed in the bulk sample with a ∼1400-nm probe. This suggests the importance of ${\lambda _{pr}}$-dependent experiments for properly analyzing interlayer vibration dynamics in vdW thin films. The oscillation amplitude of the slow mode was larger than the fast oscillations in the 7, 16, 26, and 43 QL samples. For the 78 QL one, however, the amplitudes of the fast and slow modes were comparable, as shown in Fig. 3. This tendency of vibration frequency and amplitude may be attributed to the frequency spectrum of the photoinduced strain waves $\tilde{\eta }(\omega )$, which is proportional to $\omega \delta /({1 + {\omega^2}{\delta^2}} )$, where $\delta = {v_\textrm{s}}/{\xi _{\textrm{pump}}}$ [35]. The estimated central frequency of the strain waves was 13 GHz, close to the observed vibration frequencies of 7 and 20 GHz in the 78 QL sample. As represented by the dashed red curves in the right panels of Fig. 3, the spectrum of the photoinduced strain waves, which were rescaled for comparison with the FFT results, well explain our results for the vibrational mode amplitudes. In terms of acoustic wave packet, the pump pulses generate an acoustic wave packet with a specific spectrum, which is determined by $\delta = {v_\textrm{s}}/{\xi _{\textrm{pump}}}$ in bulk or thick samples (ξpump < d). For thin samples (ξpump > d), instead, the generated strain waves can be confined and only a few vibration modes satisfying the boundary conditions will survive as standing waves. This implies that the N-dependent vibrational modes act as a spectral filter of photoinduced strain waves, which are determined by the sound velocity and penetration depth of the sample at the used pump wavelength.
Fig. 3. Transient reflectivity (TR) signals of Bi2Se3 samples with different thicknesses (left panels) and corresponding fast Fourier transform results at each TR signal (right panels), showing two oscillation modes. The wavelength of the probe beam was 1500 nm for the 7, 16, 26, and 43 quintuple layer (QL) samples and 1300 nm for the 78 QL one. The fit curves (black) and the initial phase of oscillations (${\phi _{1,2}}$ for fast and slow modes, respectively) in the left panel were obtained from Eq. (1). The blue and beige curves represent the decomposed oscillation from the fitting results. The solid and dashed red curve in the bottom-left panel is the fit curve with the parameters obtained from the bulk at the same probe wavelength, corresponding to the traveling strain waves before forming the standing wave. The blue and beige circles in the right panels indicate the obtained frequencies by fitting the measured signals in the left ones.
The observed interlayer vibration frequencies could be described with a linear chain model [15–17,36], where a finite one-dimensional chain of N-QLs with a specific mass (M) is connected by vdW gaps having a spring constant (${K_0}$). The elastic coupling between sample and substrate was also considered with an additional force constant (${K_i}$) (Fig. 4(a)). The eigenmode solution of this linear chain model with a substrate effect is expressed as follows:
and where ${f_n}$ is the frequency of n-th eigenmodes (n = 1, 2, … N) and ${q_{n,z}} = {k_{n,z}}a$ (where ${k_{n,z}}$ is the wave vector along with the z-direction (out-of-plane) and $a \cong 1\; \textrm{nm}$ is the inter-chain distance), $\mu $ and ${K_0}$ are the mass density (7.5 × 10−6 kg/m2) and force constant (5.26 × 1019 N/m3) of Bi2Se3, respectively [15], and ${K_r} = {K_i}/{K_0}$ is the ratio of force constants between QL–QL and QL–substrate. For non-zero ${K_r}$, n = 1 corresponds to the interface mode (IM) and n = 2, 3, … N corresponds to N−1 breathing modes (BM) [15]. Satisfying Eq. (5), which is in the form of a transcendental equation, provides the quantized solution for the interlayer vibrational modes. As shown on the right side of Eq. (5), the boundary condition is changed by ${K_i}$. The special cases of ${K_i}$ = 0 and 2 represent, respectively, the free boundary condition (open-pipe mode) and the fixed boundary condition (organ-pipe mode) at the sample/substrate interface.Fig. 4. (a) Schematic of a linear chain model with substrate coupling. (b) Dispersion relation of the linear chain model calculated using Eq. (4). (c,d) Ω(N) derived from the left-hand side of Eq. (5) for (c) 10 and (d) 20 quintuple layer (QL) Bi2Se3 samples (N = number of QLs); the black and red dashed lines in (c) indicate the values of the right-hand side of Eq. (5) with Kr as 0 and 0.5, respectively, and the circles indicate the quantized solution of interlayer vibrations. (e) Thickness-dependent frequency of interlayer vibrations with different Kr values (IM: out-of-plane interface mode; BM1: first breathing mode). (f) Thickness-dependent lifetime of interlayer vibrations; the gray line with a slope of ∼0.8 ps/nm illustrates the linear dependence.
Figure 4(b) shows the dispersion relation calculated using Eq. (4). Figure 4(c) illustrates the procedure for finding the eigenmode of out-of-plane vibrational modes in the 10 QL case. The left-hand side of Eq. (5) was replaced by Ω(N) ≡ $\tan ({{q_{n,z}}/2} )\tan ({{q_{n,z}}N} )$. The horizontal dashed lines in Fig. 4(c) indicate the value of ${K_r}$/(2−${K_r}$). The lowest and second lowest vibrational mode (q1 and q2) correspond to IM and first BM (BM1), respectively (inset of Fig. 4(c)). The IM corresponds to the out-of-plane vibration mode where all the QLs move in the same direction based on the restoring force between sample and substrate, while the BM1 corresponds to the lowest A-symmetry vibrational mode where one node is formed in the center of the sample. We did not observe a higher mode from q3. Figure 4(d) shows Ω(20 QL) for comparison with Ω(10 QL) (Fig. 4(c)). The qn value decreased as the thickness increased, showing the tendency of the linear chain model of frequency lowering at higher N. Furthermore, the periodicity of Ω(20 QL) is much shorter than that of Ω(10 QL), which suggests that the solution for bulk samples is continuous rather than discrete. Based on Eqs. (4) and (5) with different ${K_r}$ values, the calculated eigenmode frequency according to N is plotted in Fig. 4(e). The observed fast and slow vibrational modes correspond the out-of-plane IM and BM1, respectively. The slow IM can be obtained only at Ki = 0 since the first eigenmode for Ki = 0 always gives a zero-frequency solution, implying just the transverse mode without oscillation. The ${K_r}$ range of 0.1–0.2 well matches the experimental data, meaning that the sample and substrate were loosely coupled. The obtained Ki results are also in good agreement with the previous UTXRD study [16] and TR measurements [17].
Figure 4(f) shows the lifetime (τosc) of the interlayer vibrations as a function of N. In previous studies on the lifetime of confined acoustic phonons in silicon membranes [37] and BM in free-standing MoSe2 membranes [38], the intrinsic lifetime has been determined by several factors based on the scattering, including three-phonon scattering, Akhiezer-type scattering, and boundary scattering. In our case, the lifetimes of IM and BM1 exhibit a linear dependence on the sample thickness, which is conceptually the same as the boundary scattering model. In this model, τosc is determined by the mean free path (Λ) of acoustic waves and ${v_s}$ as ${\tau _{osc}}\; = \; \Lambda /{v_s}$. For a free-standing sample [34,35], Λ is limited by the surface roughness as Λ = bΛ0, where b is the factor considering the mean surface specularity and Λ0 is the intrinsic phonon Λ for a perfectly flat surface, which is proportional to d. In our case with sample/substrate geometry (not free-standing), the acoustic Λ can be limited by the reflection coefficient (Rac) of acoustic waves at the sample/substrate interfaces. Rac is obtained from the acoustic impedance mismatch model [33,39]:
4. Conclusion
Through ultrafast OPP spectroscopy measurements with widely tunable probe pulses, we comprehensively studied the interlayer vibration dynamics in Bi2Se3 for a wide thickness range, from a few QLs to bulk samples. By tuning the optical probe wavelength, we could clearly distinguish the coherent oscillation in the TR signal between bulk (traveling strain waves) and thin films (quantized interlayer vibrations) generated by ultrafast optical pulses. For the bulk samples, the observed frequency and decay time of the oscillatory signals strongly depended on the probe wavelength due to interferometric effects and different penetration depths of the probe pulses. In contrast, for the thin films, the frequency and lifetime of the interlayer vibrations (whose modes were IM and BM1) depended only on the QL number due to the strain wave confinement through acoustic mismatch. The proposed method can be further applied to optically measure the electronic/optical property changes in Bi2Se3 by interlayer vibrations and observe low-frequency vibrations in other layered systems and their heterostructures, in particular, bulk-like thick regimes.
Funding
National Research Council of Science and Technology (CAP18054-000); National Research Foundation of Korea (2019R1A2C3003504, 2020R1A4A2002828, 2021R1A3B1077076).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. M. Z. Hasan and C. L. Kane, “Colloquium: topological insulators,” Rev. Mod. Phys. 82(4), 3045–3067 (2010). [CrossRef]
2. H. Zhang, C.-X. Liu, X.-L. Qi, X. Dai, Z. Fang, and S.-C. Zhang, “Topological insulators in Bi2Se3, Bi2Te3 and Sb2Te3 with a single Dirac cone on the surface,” Nat. Phys. 5(6), 438–442 (2009). [CrossRef]
3. J. E. Moore, “The birth of topological insulators,” Nature 464(7286), 194–198 (2010). [CrossRef]
4. F. Wang, L. Li, W. Huang, L. Li, B. Jin, H. Li, and T. Zhai, “Submillimeter 2D Bi2Se3 Flakes toward High-Performance Infrared Photodetection at Optical Communication Wavelength,” Adv. Funct. Mater. 28(33), 1802707 (2018). [CrossRef]
5. M. Yang, Q. Han, X. Liu, J. Han, Y. Zhao, L. He, J. Gou, Z. Wu, X. Wang, and J. Wang, “Ultrahigh stability 3D TI Bi2Se3/MoO3 thin film heterojunction infrared photodetector at optical communication waveband,” Adv. Funct. Mater. 30(12), 1909659 (2020). [CrossRef]
6. L. Wu, M. Brahlek, R. Valdés Aguilar, A. Stier, C. Morris, Y. Lubashevsky, L. Bilbro, N. Bansal, S. Oh, and N. Armitage, “A sudden collapse in the transport lifetime across the topological phase transition in (Bi1− xInx)2Se3,” Nat. Phys. 9(7), 410–414 (2013). [CrossRef]
7. M. Brahlek, N. Bansal, N. Koirala, S.-Y. Xu, M. Neupane, C. Liu, M. Z. Hasan, and S. Oh, “Topological-metal to band-insulator transition in (Bi1− xInx)2Se3 thin films,” Phys. Rev. Lett. 109(18), 186403 (2012). [CrossRef]
8. Y. Liu, Y. Li, S. Rajput, D. Gilks, L. Lari, P. Galindo, M. Weinert, V. Lazarov, and L. Li, “Tuning Dirac states by strain in the topological insulator Bi2Se3,” Nat. Phys. 10(4), 294–299 (2014). [CrossRef]
9. T.-H. Kim, K. Jeong, B. C. Park, H. Choi, S. H. Park, S. Jung, J. Park, K.-H. Jeong, J. W. Kim, J. H. Kim, and M.-H. Cho, “Tuning the Fermi level with topological phase transition by internal strain in a topological insulator Bi2Se3 thin film,” Nanoscale 8(2), 741–751 (2016). [CrossRef]
10. L. Luo, X. Yang, X. Liu, Z. Liu, C. Vaswani, D. Cheng, M. Mootz, X. Zhao, Y. Yao, C.-Z. Wang, K.-M. Ho, I. E. Perakis, M. Dobrowolska, J. K. Furdyna, and J. Wang, “Ultrafast manipulation of topologically enhanced surface transport driven by mid-infrared and terahertz pulses in Bi2Se3,” Nat. Commun. 10(1), 1–9 (2019). [CrossRef]
11. L. Alegria, M. Schroer, A. Chatterjee, G. Poirier, M. Pretko, S. Patel, and J. R. Petta, “Structural and electrical characterization of Bi2Se3 nanostructures grown by metal–organic chemical vapor deposition,” Nano Lett. 12(9), 4711–4714 (2012). [CrossRef]
12. J. J. Cha, J. R. Williams, D. Kong, S. Meister, H. Peng, A. J. Bestwick, P. Gallagher, D. Goldhaber-Gordon, and Y. Cui, “Magnetic doping and Kondo effect in Bi2Se3 nanoribbons,” Nano Lett. 10(3), 1076–1081 (2010). [CrossRef]
13. S. M. Young, S. Chowdhury, E. J. Walter, E. J. Mele, C. L. Kane, and A. M. Rappe, “Theoretical investigation of the evolution of the topological phase of Bi2Se3 under mechanical strain,” Phys. Rev. B 84(8), 085106 (2011). [CrossRef]
14. D. Flötotto, Y. Bai, Y.-H. Chan, P. Chen, X. Wang, P. Rossi, C.-Z. Xu, C. Zhang, J. A. Hlevyack, J. D. Denlinger, H. Hong, M.-Y. Chou, E. J. Mittemeijer, J. N. Eckstein, and T.-C. Chiang, “In situ strain tuning of the Dirac surface states in Bi2Se3 films,” Nano Lett. 18(9), 5628–5632 (2018). [CrossRef]
15. Y. Zhao, X. Luo, J. Zhang, J. Wu, X. Bai, M. Wang, J. Jia, H. Peng, Z. Liu, S. Y. Quek, and Q. Xiong, “Interlayer vibrational modes in few-quintuple-layer Bi2Te3 and Bi2Se3 two-dimensional crystals: Raman spectroscopy and first-principles studies,” Phys. Rev. B 90(24), 245428 (2014). [CrossRef]
16. S. Kim, Y. Kim, J. Kim, S. Choi, K. Yun, D. Kim, S. Y. Lim, S. Kim, S. H. Chun, J. Park, I. Eom, K. S. Kim, T.-Y. Koo, Y. Ou, F. Katmis, H. Wen, A. DiChiara, D. A. Walko, E. C. Landahl, H. Cheong, E. Sim, J. Moodera, and H. Kim, “Ultrafast Carrier–Lattice Interactions and Interlayer Modulations of Bi2Se3 by X-ray Free-Electron Laser Diffraction,” Nano Lett. 21(20), 8554–8562 (2021). [CrossRef]
17. T. G. Park, H. R. Na, S.-H. Chun, W. B. Cho, S. Lee, and F. Rotermund, “Coherent control of interlayer vibrations in Bi 2 Se 3 van der Waals thin-films,” Nanoscale 13(45), 19264–19273 (2021). [CrossRef]
18. Y. D. Glinka, S. Babakiray, T. A. Johnson, M. B. Holcomb, and D. Lederman, “Acoustic phonon dynamics in thin-films of the topological insulator Bi2Se3,” J. Appl. Phys. 117(16), 165703 (2015). [CrossRef]
19. N. Kumar, B. A. Ruzicka, N. Butch, P. Syers, K. Kirshenbaum, J. Paglione, and H. Zhao, “Spatially resolved femtosecond pump-probe study of topological insulator Bi2Se3,” Phys. Rev. B 83(23), 235306 (2011). [CrossRef]
20. C. Thomsen, H. T. Grahn, H. J. Maris, and J. Tauc, “Surface generation and detection of phonons by picosecond light pulses,” Phys. Rev. B 34(6), 4129–4138 (1986). [CrossRef]
21. M. Lejman, V. Shalagatskyi, O. Kovalenko, T. Pezeril, V. V. Temnov, and P. Ruello, “Ultrafast optical detection of coherent acoustic phonons emission driven by superdiffusive hot electrons,” J. Opt. Soc. Am. B 31(2), 282–290 (2014). [CrossRef]
22. A. Devos, “Colored ultrafast acoustics: From fundamentals to applications,” Ultrasonics 56, 90–97 (2015). [CrossRef]
23. P. Ruello and V. E. Gusev, “Physical mechanisms of coherent acoustic phonons generation by ultrafast laser action,” Ultrasonics 56, 21–35 (2015). [CrossRef]
24. H. Park, X. Wang, S. Nie, R. Clinite, and J. Cao, “Mechanism of coherent acoustic phonon generation under nonequilibrium conditions,” Phys. Rev. B 72(10), 100301 (2005). [CrossRef]
25. M. Eddrief, F. Vidal, and B. Gallas, “Optical properties of Bi2Se3: from bulk to ultrathin films,” J. Phys. D: Appl. Phys. 49(50), 505304 (2016). [CrossRef]
26. J. McIver, D. Hsieh, S. G. Drapcho, D. H. Torchinsky, D. R. Gardner, Y. S. Lee, and N. Gedik, “Theoretical and experimental study of second harmonic generation from the surface of the topological insulator Bi2Se3,” Phys. Rev. B 86(3), 035327 (2012). [CrossRef]
27. J. Miller, J. Qi, Y. Xu, Y.-J. Cho, X. Liu, J. Furdyna, I. Perakis, T. Shahbazyan, and N. Tolk, “Near-bandgap wavelength dependence of long-lived traveling coherent longitudinal acoustic phonons in GaSb-GaAs heterostructures,” Phys. Rev. B 74(11), 113313 (2006). [CrossRef]
28. W. Wu and Y. Wang, “Ultrafast carrier dynamics and coherent acoustic phonons in bulk CdSe,” Opt. Lett. 40(1), 64–67 (2015). [CrossRef]
29. O. Matsuda, T. Pezeril, I. Chaban, K. Fujita, and V. Gusevet, Phy. Rev. B 97, 064301 (2018).
30. S. Giraud, A. Kundu, and R. Egger, “Electron-phonon scattering in topological insulator thin films,” Phys. Rev. B 85(3), 035441 (2012). [CrossRef]
31. H.-J. Chen, K.-H. Wu, C.-W. Luo, T. Uen, J.-Y. Juang, J.-Y. Lin, T. Kobayashi, H.-D. Yang, R. Sankar, F. Chou, H. Berger, and J. M. Liu, “Phonon dynamics in CuxBi2Se3 (x= 0, 0.1, 0.125) and Bi2Se3 crystals studied using femtosecond spectroscopy,” Appl. Phys. Lett. 101(12), 121912 (2012). [CrossRef]
32. X. Lin, B. Wu, W. Ning, M. Feng, Q. Xu, F. Gao, T. C. Sum, and G. Zhou, “Effect of alloying on the dynamics of coherent acoustic phonons in bismuth double perovskite single crystals,” Opt. Express 29(5), 7948–7955 (2021). [CrossRef]
33. F. Vialla and N. D. Fatti, “Time-domain investigations of coherent phonons in van der waals thin films,” Nanomaterials 10(12), 2543 (2020). [CrossRef]
34. M. Fang, Z. Wang, H. Gu, M. Tong, B. Song, X. Xie, T. Zhou, X. Chen, H. Jiang, T. Jiang, and S. Liu, “Layer-dependent dielectric permittivity of topological insulator Bi2Se3 thin films,” Appl. Surf. Sci. 509, 144822 (2020). [CrossRef]
35. P.-A. Mante, C. C. Stoumpos, M. G. Kanatzidis, and A. Yartsev, “Electron–acoustic phonon coupling in single crystal CH3NH3PbI3 perovskites revealed by coherent acoustic phonons,” Nat. Commun. 8(1), 1–7 (2017). [CrossRef]
36. N. Luo, P. Ruggerone, and J. P. Toennies, “Theory of surface vibrations in epitaxial thin films,” Phys. Rev. B 54(7), 5051–5063 (1996). [CrossRef]
37. J. Cuffe, O. Ristow, E. Chávez, A. Shchepetov, P.-O. Chapuis, F. Alzina, M. Hettich, M. Prunnila, J. Ahopelto, T. Dekorsy, and C. M. Sotomayor Torres, “Lifetimes of confined acoustic phonons in ultrathin silicon membranes,” Phys. Rev. Lett. 110(9), 095503 (2013). [CrossRef]
38. P. Soubelet, A. A. Reynoso, A. Fainstein, K. Nogajewski, M. Potemski, C. Faugeras, and A. E. Bruchhausen, “The lifetime of interlayer breathing modes of few-layer 2H-MoSe2 membranes,” Nanoscale 11(21), 10446–10453 (2019). [CrossRef]
39. G. Benetti, M. Gandolfi, M. J. Van Bael, L. Gavioli, C. Giannetti, C. Caddeo, and F. Banfi, “Photoacoustic sensing of trapped fluids in nanoporous thin films: device engineering and sensing scheme,” ACS Appl. Mater. Interfaces 10(33), 27947–27954 (2018). [CrossRef]
40. H. Wang, Y. Xu, M. Shimono, Y. Tanaka, and M. Yamazaki, “Computation of interfacial thermal resistance by phonon diffuse mismatch model,” Mater. Trans. 48(9), 2349–2352 (2007). [CrossRef]
41. M. Guo, Z. Wang, Y. Xu, H. Huang, Y. Zang, C. Liu, W. Duan, Z. Gan, S.-C. Zhang, K. He, X. Ma, Q. Xue, and Y. Wang, “Tuning thermoelectricity in a Bi2Se3 topological insulator via varied film thickness,” New J. Phys. 18(1), 015008 (2016). [CrossRef]
42. L. Paulatto, D. Fournier, M. Marangolo, M. Eddrief, P. Atkinson, and M. Calandra, “Thermal conductivity of Bi2Se3 from bulk to thin films: Theory and experiment,” Phys. Rev. B 101(20), 205419 (2020). [CrossRef]
