Bandgap modulation in photoexcited topological insulator Bi₂Te₃ via atomic displacements

Three-dimensional topological insulators (TIs) which are characterized as bulk insulators with an electronic-conduction surface band^1–4 have yielded a variety of exotic electronic-transport responses under magnetic field or circular polarized light.^5–11 A capacitor-type device with thin-film TIs has been proposed for exciton condensates wherein quasi-particle excitation carries fractional charge.^12,13 TIs have attracted considerable attention in both scientific and technical application fields. One of the interesting challenges regarding the technical application of TIs is the artificial and transient tuning of their electronic properties. The modification of electronic bandgaps at the surface and in the bulk of TIs has been demonstrated with static methods that use dopants or the application of pressure.^13–16 However, the artificial and transient modification of bandgaps^12,13 in TIs via photoexcitation has not thus far been examined. The bulk bandgap of a TI, e.g., Bi₂Te₃, has been characterized as a direct gap at cryogenic temperatures, which is “buried” by the indirect gap formed at room temperature.¹⁷ Therefore, the temperature dependence of the Bi₂Te₃ bandgap from cryogenic temperatures to room temperature is large. However, the temperature dependence of the indirect bandgap in Bi₂Te₃ above room temperature is negligibly small (see Fig. S1 of the supplementary material¹⁸). In this context, the responses of TIs under photoexcitation should be useful not only in understanding the nature of the electronic energy excitation and relaxation pathways but also in utilizing the ensuing photoinduced atomic displacements as a means to modify the electronic properties. The electronic dynamics in TIs under interband photoexcitation have been investigated with time- and angle-resolved photoelectron spectroscopy (tr-ARPES)^19,20 and time-resolved mid-infrared (MIR) spectroscopy.²¹ The electrons photoexcited beyond the bandgap are immediately scattered into the bulk conduction band and partially into the surface state. The electrons in the bulk conduction band persistently relax into the bulk valence band through the surface state. A recent report has suggested that the balance between electrons and holes on the surface of TIs can strongly affect the electronic band structure of the system.²² In contrast, coherent phonons have been observed in both the bulk and surface state of TIs under relatively weak photoexcitation.^23,24 The correlation of the electronic state dynamics to atomic motions under strong photoexcitation, particularly in the bulk, has not been considered in the dynamics of photoexcited TIs. Herein, we show that photoinduced atomic motions can be correlated to change in the electronic bandgap of TIs. We also examine the possibility of the artificial modification of the electronic structure in Bi₂Te₃ by interband photoexcitation via atomic motions occurring along the excited state surface.

Near-ultraviolet (NUV) (3.1 eV) pump and MIR (0.12–0.22 eV) probe experiments were performed in the reflection mode on a Bi₂Te₃ bulk single-crystal. The experimental details of this optical pump-probe setup are presented elsewhere.²⁵ The incident angle of the pump and probe light was set nearly parallel to the surface normal of the sample. The pulse durations of the NUV and MIR pulses were 100 fs and <1 ps, respectively. The repetition rate and the incident fluence of the NUV pump pulse were 500 Hz and 1.7 mJ/cm², respectively. The absorption pump fluence (F_A) was calculated to be 1.0 mJ/cm², and the reflectivity of Bi₂Te₃ was measured to be 41%.

Time-resolved electron diffraction (Tr-ED) measurements were next performed in the transmission mode. The experimental setup of the compact DC-accelerated electron diffraction is provided elsewhere.²⁶ NUV light (3.1 eV) was focused to a 280-μm spot size on a single-crystalline Bi₂Te₃ ultrathin film (30 nm thick). The incident laser fluence was 0–9.0 mJ/cm². From the transmission and reflectivity of the Bi₂Te₃ thin film measured to be 15% and 41%, respectively, F_A was determined to be 0–4.0 mJ/cm². The acceleration voltage of the probe electron pulses was 75 keV under a DC electric field. Photoinduced structural changes inside the material were investigated with electron pulses containing 1.25 × 10⁴ electrons confined to a 100-μm-diameter spot incident on the Bi₂Te₃ sample. The pulse duration of the electron beam was <1.5 ps as measured by the plasma method.²⁷ Diffracted and directly transmitted electrons were focused with a magnetic lens onto a 1:2 fiber-coupled charge-coupled device (CCD) camera (iKon-L HF, Andor) coated with a P43 (Gd₂O₂S:Tb) phosphor scintillator. To acquire one electron diffraction image, 1–4 × 10⁴ shots of electron pulses were collected at repetition rates of 1 kHz below the F_A value of 2 mJ/cm² and 500 Hz above the F_A value of 2 mJ/cm² to avoid sample damage and accumulation of heat. The literature^28,29 indicates that the carrier dynamics of this class of topological insulator in 30-nm-thick-film correspond to that of a bulk sample rather than the surface of a sample. In the following discussion, we consider the Bi₂Te₃ ultrathin film to be a bulk sample.

N-type Bi₂Te₃ single crystals were grown by means of the modified Bridgman technique, as described in Ref. 6. The carrier density of the sample was found to be 5 × 10¹⁸ cm⁻³. The surface of the bulk Bi₂Te₃ single crystal was cleaved with Scotch^™ tape, and a fresh surface c-axis was used for the optical experiments. For the Tr-ED studies, the Bi₂Te₃ single crystal was cut to a 30-nm thickness perpendicular to the c-axis with a diamond knife on a microtome (UC7, Leica). The ultrathin Bi₂Te₃ film was “picked-up” with a commercial Cu TEM (#400) grid.

First-principles calculations within the framework of density functional theory (DFT) were performed with the use of the full-potential linearized augmented plane-wave (FP-LAPW) method with the generalized gradient approximation/Perdew–Burke–Ernzerhof functional (GGA/PBE) implemented in the WIEN2k code.³⁰ Spin-orbit coupling was included as a second variation step. In order to describe the photoexcited states of Bi₂Te₃, the original rhombohedral crystal structure³¹ was converted to orthorhombic symmetry (a = 14.3362 Å, b = 57.5988 Å, c = 8.277 Å) with the space group B2 (origin choice 5). The atomic coordinates at 0, 10, and 100 ps used for the calculations are listed in Tables S1–S7.¹⁸ These coordinates were measured by time-resolved electron diffraction. The plane-wave cutoff energy was set to R_MTK_max = 9, where R_MT = 2.5 for both Bi and Te. The Brillouin zone was sampled with the Monkhorst–Pack scheme³² with a Γ-centered 35 × 6 × 35 k-point mesh, which achieved reasonable convergence (<1 meV) of the total energy and the bandgap.

NUV pump and MIR probe reflective spectroscopy were used to characterize the carrier dynamics on the relevant time scales. Figure 1(a) presents the differential reflectivity (ΔR/R) as a function of the probe photon energy at various time delays. As shown in Figs. 1(a) and 1(b), the signature of ΔR/R changes at a probe photon energy of approximately 0.13 eV. The decrease and recovery time constants in the transient reflectivity from Bi₂Te₃ vary from 10 to 3 ps and 60 to 15 ps, respectively, in the probe photon energy range of 0.12–0.22 eV (Fig. S2 provides more detail on the decrease and recovery time constants¹⁸). When the ground-state electrons undergo direct excitation to the far-from-equilibrium state, holes are created at the edge of the bulk valence band, and the electrons in the surface state recombine into the bulk valence band to “fill” the holes. The excited electrons are scattered into the edge of the bulk conduction bands and then relaxed into the bulk valence band through the surface state. The variation in the time constants in the probe energy as shown in Figs. 1(a) and 1(b) reflects carrier dynamics in the surface state in a manner similar to that explained in Ref. 21.

The indirect bandgap of Bi₂Te₃ was confirmed to be 0.13 eV by transmission spectroscopy (Fig. S1¹⁸) and an optical conductivity spectrum calculated using the Kramers–Kronig (K–K) transform of the reflectivity (Figs. 2(a) and 2(b)). For the K–K transformation, we measured reflectivity in the energy range of 0.07–2.4 eV (see Fig. S3¹⁸) and assumed simulated reflectivity obtained using the conventional Drude model below 0.07 eV and ω² extrapolation above 2.4 eV. The figure shows a “bump” structure in the reflectivity spectrum at 0.14–0.15 eV (blue arrow). We obtained more details on this structure when we plot the simulated data based on the simple Drude model (dashed line). A comparison between the optical conductivity spectra calculated with the K–K transform from the experimental and simulated data reveals a kink structure (blue arrow in Fig. 2(b)) that is related to the bump structure in reflectivity. For the low-energy extrapolation method, we examined several models and parameters, but we could not find significant differences with respect to the position of the kink structure in the optical conductivity spectra shown in Fig. 2(b). The reflectivity spectrum modulation on the time scale of the first 10 ps is particularly noteworthy. Figure 3 presents the transient reflectivity in the probe-photon energy range of 0.12–0.22 eV. From the K–K analysis results, we note that the bandgap of Bi₂Te₃ is strongly related to the kink structure (0.14–0.15 eV), indicated by blue arrows in Figs. 2(a) and 3. This kink structure, marked in Fig. 3, is clearly shifted toward a lower energy (10 meV) in the time-resolved spectra. The penetration depth of the MIR light in Bi₂Te₃ is 400 nm (above the bandgap) and 700 nm (below the bandgap), whereas that of NUV light is approximately 30 nm, which is considerably different.^33–35 This mismatch of the penetration depth may affect the spectral shape of the reflectivity of the photoinduced state, however, we could not consider the effect of the mismatch on the pump-probe results because of the ambiguity of the spectral shape of the photoinduced state as well as the relatively small dispersion of the penetration depth in the photon energy range of our measurement. At present, we believe the observed shift of the kink structure in the reflectivity spectrum suggests the band-gap narrowing on the time scale of 10 ps. We present more detailed observations of the photoinduced dynamics on the results of the time-resolved electron diffraction.

We performed Tr-ED experiments that provided direct structural information on the evolution of the atomic configuration.^26,36–38 Figures 4(a) and 4(b) show an electron diffraction pattern from the 30-nm-thick 〈0001〉-oriented Bi₂Te₃ single-crystalline film and a simulated diffraction pattern obtained with the WinHREM^™ software (based on the dynamical diffraction theory with multislice methods³⁹), respectively. We analyzed the most significant diffraction spots. The resulting Tr-ED intensities are presented in Figs. 5(a)–5(c) for the ${ 1 2 ̄ 10 }$ and ${ 3 3 ̄ 00 }$ Bragg reflections from the Bi₂Te₃ film photoexcited at an F_A value of 1.5 mJ/cm². All the Tr-ED intensities are shown in Fig. S4.¹⁸ The Tr-ED intensities shown in Figs. 5(a) and 5(b) are the products of three contributions: Debye–Waller effects, acoustic phonons, and atomic displacements within the unit cell. The intensity of the electron diffraction (I) can be expressed via the following equation:^40–42 

where K is the scattering vector, G(K) is the Laue function, f_j(K) is the atomic scattering factor, D_j(K) is the Debye–Waller factor, and r_j is the equilibrium position of the jth atom in the unit cell. Because the intensity modulation by the acoustic phonon is derived from the mismatch of the Bragg condition, the contribution of the acoustic phonon is only related to (G(K)). On the other hand, the thermal effects and the effects of atomic displacements within the unit cell yield inversion-symmetric modulation in the electron diffraction intensity. The resulting temperature rise can be independently calculated from the thermal expansion coefficients (1.3 × 10⁵ K⁻¹)^43,44 and the shifts of the spot positions in the electron diffraction pattern (Fig. 5(d)). The value of the temperature rise from thermal expansion at 100 ps was calculated to be 100 K and to reach approximately 200 K on the time scale of nanoseconds. This temperature increase was also indicated by the diffuse scattering background^45,46 (see Fig. S5¹⁸). The Debye–Waller factor can be expressed as a function of the mean square displacement (U_j) as follows:

The isotropic thermal displacement (U) can be calculated using the coefficient of linear contraction (ρ), which is derived from the bulk modulus,⁴⁴ and the atomic distance (d_A = 4.38 Å) with the following equation:^47,48

where k_B and T are the Boltzmann constant and lattice temperature, respectively. Taking into account the room temperature of 300 K, the temperature rise of approximately 100 K corresponded to decreases in the electron diffraction intensity of 1% and 3% of their original values for the ${ 1 2 ̄ 10 }$ and ${ 3 3 ̄ 00 }$ reflections.

Acoustic phonon vibrations at a frequency of 35 GHz (Fig. 5(e)) were also observed in the intensity of the diffraction spot from the ${ 1 2 ̄ 10 }$ planes. In general, the sound velocity in a crystal is calculated as a function of the frequency of the acoustic phonon vibrations and film thickness.^49,50 In this study, the frequency of the acoustic mode (35 GHz) and film thickness (30 nm) yielded a sound velocity of 2100 m/s, which corresponds to the longitudinal-mode sound velocity (2000–2200 m/s) in Bi₂Te₃.^51,52 Since these acoustic phonon modulations have opposite phases in opposite planes, this effect of the acoustic phonons in the transient electron intensity can be canceled out by adding these opposite-phased oscillations. The remaining inversion six-symmetric contribution is attributed to the atomic displacements in the unit cell. As shown in Figs. 5(a) and 5(b), there is a two-step change: a decrease in the Tr-ED intensity by 5%–6% during the first 10 ps, and a subsequent gradual recovery over 10–100 ps. The symmetric-atomic-displacement-related component in the intensity of the higher-order diffraction spot ${ 3 3 ̄ 00 }$ exhibits a 25% initial drop and a slight subsequent recovery (Fig. 5(c)).

To determine the transient atomic coordinates in the photoexcited Bi₂Te₃ crystal, we performed model calculations with the simulation package WinHREM. The electron pulses pass through the sample, parallel to the c-axis of the Bi₂Te₃ crystal; therefore, the contributions along the c-axis (see Fig. S6¹⁸) are responsible for significant uncertainty in the determination of the disordered atomic coordinates. For the purpose of this discussion, we used the diffraction spots from the ${ 1 2 ̄ 10 }$ and ${ 3 3 ̄ 00 }$ planes. In principle, two atomic coordinates in the ab plane can be determined with two independent diffraction spots. A quintuple layer (QL), known as a functional periodic unit of this class of TIs, consists of five atomic layers (Te¹, Bi, Te², Bi, and Te¹). The Te¹ atom occupies a different site than the Te² atom from a crystallographic viewpoint. We assumed center-symmetric motions of Bi and Te¹ atoms toward or away from the center axis of the unit cell with fixing of the Te² atom. Simulations based on dynamic theory were performed by changing the atomic coordinates of the Bi and Te¹ atoms. The calculations yielded differential electron diffraction intensities from the ${ 1 2 ̄ 10 }$ and ${ 3 3 ̄ 00 }$ planes as Bi atoms moved in relation to the Te² atoms upon fixing of the Te¹ atoms (Fig. S7(a)) and movement of the Te¹ atoms in relation to the Te² atoms with fixing of the Bi atoms (Fig. S7(b)).¹⁸ The intensities of the diffraction spots decrease as the Bi atoms move; however, the motion of the Te¹ atoms makes no significant contribution to the intensities of the diffraction spots. Thus, the first-step changes observed in Figs. 5(a) and 5(b) (10 ps) were concluded to have been induced primarily by the motions of the Bi atoms. For the purpose of further elaboration of the atomic displaced coordinates, the two-step changes in the diffraction intensity were characterized (Figs. S7(c) and S7(d)).¹⁸ Transient atomic coordinates were determined to minimize R²(d_Bi, d_Te) as expressed by the following equation:

where d_Bi and d_Te are the displacements of the Bi and Te¹ atoms, respectively, and I_i,exp and I_i,sim(d_Bi, d_Te) are the experimental and simulated electron diffraction intensities of the ith index (i.e., ${ 1 2 ̄ 10 }$ and ${ 3 3 ̄ 00 }$⁠), respectively. Figures 6(a)–6(e) depict the R²(d_Bi, d_Te) maps as functions of the displacements of the Bi and Te¹ atoms at the representative time delays. The local minimum of the map is the transient atomic position, and the displacements of the Bi and Te¹ atoms are summarized in Fig. 6(f). As the figure shows, the Bi and Te¹ atoms move independently, and the motion observed in the first 10 ps corresponds to large Bi (∼8 pm) and small Te¹ (∼2 pm) displacements toward or away from the center axis. The atomic motion over 10–100 ps is the Te¹ (∼2 pm) motion along the same direction and reversal of the Bi (∼2 pm) displacement. The directions of these motions correspond to the fundamental-mode E¹_g and E²_g phonons (see Fig. S8).¹⁸ These atomic displacements increase linearly with the absorption laser fluence at F_A < 1.0 mJ/cm² and are saturated at an F_A value of 1.0 mJ/cm². As shown in Fig. 7, the damage threshold of the ultrathin Bi₂Te₃ sample was determined to be 2.5 mJ/cm². In the F_A range from 1.0 to 2.0 mJ/cm², the atomic dynamics of Bi and Te are identical. The atomic displacements observed with Tr-ED are summarized in Fig. 8, along with the QL structure of Bi₂Te₃. The disordered atomic coordinates are listed in Tables S1–S7.¹⁸ 

We discuss the correlation between the atomic motions and the change in the electronic structure of the Bi₂Te₃ system by comparing the results from the Tr-ED and NUV pump and MIR probe reflection spectroscopy. The transient electronic structure modulated by the disordered atoms can be calculated by DFT with the atomic coordinates obtained by Tr-ED. The DFT total energy increased by 1.0 meV per formula unit (f.u.) in 10 ps and relaxed to 0.2 meV/f.u. in 100 ps. These values correspond to excited electron densities of 4.5 × 10¹⁹ and <1 × 10¹⁹ cm⁻³, respectively, in the bulk conduction band where the photoexcited electron and hole density for NUV light is approximately 1 × 10²¹ e cm⁻³. The density of the states obtained by the DFT calculation is shown in Figs. 9(a) and 9(b). The energy bandgap is 0.12 eV, which agrees with the spectroscopic measurements. The bandgap becomes approximately 5–10 meV narrower at 10 ps after photoexcitation. These results suggest that a fraction of the photoexcited and repopulated holes in the bulk valence band (around 5% of excited electrons and holes) create new atomic coordinates, which are related to the Bi motions occurring over 10 ps. These atomic motions cause bandgap narrowing, which is consistent with the NUV pump and MIR probe reflection spectroscopy results. Atomic displacements often reduce the crystal symmetry of a sample, which leads to the split of degenerated orbital level into several levels. This effect is called “crystal field splitting.”^53,54 This crystal field splitting invokes the broadening of the conduction and valence bands, and results in bandgap narrowing unless the center value of the valence and conduction bands shifts. The bandgap renormalization effects⁵⁵ via hot electrons might also induce bandgap narrowing; however, these hot-electron effects should start to occur within 3 ps when the hot electron is in the excited state. The time duration corresponding to the phenomena observed in our case was too large (10 ps) for the phenomena to be considered to be due only to hot-electron effects. Hot-hole effects⁵⁶ triggering these atomic motions can indirectly form one of the possible mechanisms for bandgap narrowing.

In summary, we observed that photoinduced atomic displacements in Bi₂Te₃ affect its bandgap, which is reflected in the DFT calculation. The observed dynamics illustrate the very nature of the photoexcited electronic relaxation pathways and the strong coupling between the electronic structure and atomic configuration of TIs. Bulk bandgap modification of TI films with intra- and interband photoexcitation in the range from THz to UV could play an important role in newer TI applications, particularly for modulating their properties on ultrafast time scales. To better understand electron–lattice coupling in TIs, it would be useful to observe the atomic motions along the c-axis with the use of ultrabright X-ray diffraction and spectroscopic methods.^57,58 As computing methods to calculate the excited states of these complex materials are further developed, ab initio molecular dynamics will also provide a more detailed microscopic understanding of these electronically coupled atomic motions.⁵⁹ In the present context, we have characterized both the transient electronic structure and atomic motions involved in photoinduced bandgap changes and demonstrated a causal connection to the atomic motions’ amenable control. The tabletop combination of time-resolved MIR spectroscopy and electron diffraction methodologies can provide significant insight into material science and condensed matter physics in strongly correlated materials, or even in chemistry, where information on electronic and atomic dynamics is important for designing new functional materials.^60,61

This work was mainly supported by the Japan Science Technology Agency (JST), PRESTO, for a project entitled “Molecular technology and creation of new functions.” K.O. acknowledges JST-PRESTO for the project “Chemical conversion of light energy.” K.G.N. and Y.K. acknowledge JST-CREST for the project of “Enhancing Applications of Innovative Optical Science and Technologies by making Ultimate Use of Advanced Light Sources.” S.-y.K. acknowledges Grants-in-Aid for Scientific Research (Grant No. 15H02103) from the Japan Society for the Promotion of Science and the Ministry of Education, Culture, Sports, Science and Technology. S.K. and R.J.D.M. were supported by the Max Planck Society. The authors acknowledge Dr. Dongfang Zhang at the Max Planck Institute for the Structure and Dynamics of Matter and Professor Masahiro Kitajima at the Tokyo Institute of Technology for discussions.