Ultrafast photoinduced band splitting and carrier dynamics in chiral tellurium nanosheets

Trigonal tellurium (Te) is a chiral semiconductor that lacks both mirror and inversion symmetries, resulting in complex band structures with Weyl crossings and unique spin textures. Detailed time-resolved polarized reflectance spectroscopy is used to investigate its band structure and carrier dynamics. The polarized transient spectra reveal optical transitions between the uppermost spin-split H4 and H5 and the degenerate H6 valence bands (VB) and the lowest degenerate H6 conduction band (CB) as well as a higher energy transition at the L-point. Surprisingly, the degeneracy of the H6 CB (a proposed Weyl node) is lifted and the spin-split VB gap is reduced upon photoexcitation before relaxing to equilibrium as the carriers decay. Using ab initio density functional theory (DFT) calculations, we conclude that the dynamic band structure is caused by a photoinduced shear strain in the Te film that breaks the screw symmetry of the crystal. The band-edge anisotropy is also reflected in the hot carrier decay rate, which is a factor of two slower along the c-axis than perpendicular to it. The majority of photoexcited carriers near the band-edge are seen to recombine within 30 ps while higher lying transitions observed near 1.2 eV appear to have substantially longer lifetimes, potentially due to contributions of intervalley processes in the recombination rate. These new findings shed light on the strong correlation between photoinduced carriers and electronic structure in anisotropic crystals, which opens a potential pathway for designing novel Te-based devices that take advantage of the topological structures as well as strong spin-related properties.


B. Band structure under uniaxial strains or a hydrostatic pressure applied
It is well known that the band structure of a solid crystal is modified by inducing strains. The band modification, however, depends on how the crystal symmetry is affected by different types of strains. In the case of Te, an anisotropic chiral semiconductor, its complex band structure near the band-edge displays unique strain effects that have been demonstrated by recent studies of ab initio electronic structure calculations. 2,3,6 Applying uniaxial strains or a hydrostatic pressure on a Te crystal (see   With ∥ , a nearly overlapped additional feature is observed at ~0.44 eV, which is not present with ⊥ , suggesting the polarization sensitive optical absorption in the Te samples.
It is known that the TR response is related to the perturbation of the dielectric function of the material induced by photoexcitation of charged carriers. For the Lorentzian form of the dielectric function under low-field modulation with the parabolic band approximation, TR spectra around the band-edge regions can be analyzed by using a derivative Lorentzian lineshape functional form appropriate for excitonic transitions, 7 where represents the number of spectral functions for the possible interband transitions involved, is the probe photon energy, and ! , ! , ! , and Γ ! are the amplitude, phase, transition energy, and the energy broadening parameter of the !! feature, respectively. In order to quantify each of these features and corresponding transition energies, the TRS spectra are fitted using Eq. lowest energy transition ! → ! !" with ⊥ is ! ! = 0.34 ± 0.002 eV, which is nearly the same value as the low temperature value shown in the main text. This behavior nicely corresponds with the peculiar temperature dependent absorption coefficient in Te previously observed. 9 One can also see the lowest energy transition with ∥ , which is blue-shifted by ∼ 20 meV to ! ∥ = 0.36 ± 0.002 eV. Such a polarization anisotropy of optical transition at the band-gap remains unchanged over entire delays, in contrast to the strong modulation up to 30 ps observed at 10 K (see main text). The ! ! transition energy as well as the polarization anisotropy, i.e., the dichroism of the optical absorption edge, agree very well with previous results obtained by linear absorption measurements on degenerately p-doped Te samples at room temperature. [9][10][11] The optical transition between ! → ! !" is observable only with ∥ and totally absent with ⊥ , which is in agreement with low temperature measurements and consistent with the expected dipole allowed transition between these two states. The ! → ! !" transition energy is estimated to be ! ∥ = 0.44 ± 0.005 eV, which is again nearly identical with the value measured at 10 K. The spinsplit VB gap or the separation between the ! and the ! VBs turns out to be ! ∥ − ! ! = 100 meV. This value, taking into account the doping-induced Fermi level shift, is nearly the same as the so-called 11micron hole absorption band previously measured in a bulk single crystal of Te. 12,13 The energy difference between the ! ∥ transition and ! ∥ transition decreases to ! ∥ − ! ∥ = 80 meV, which suggests not only the fundamental gap but also the gap between spin-split VBs is anisotropic at room temperature.
In addition to the direct transitions between uppermost VBs and the lowest CB, there is an indication of a higher energy transitions at around ! ∥,! = 0.75 ± 0.005 eV with the lineshape sensitive to polarizations.
Since the TR signal is nearly one order of magnitude weaker than the near band-edge transition, this transition might be indirect in k-space, as expected from the band structure.
We also observed a higher lying transition at ! ! ~ 1.16 ± 0.005 eV, which is blue shifted by 20 meV to ! ∥ ~ 1.18 ± 0.005 eV. Due to limitations of the spectral range of the probe pulse, a full derivative-like lineshape is not observed, which causes uncertainty in the precise energy transition.
Nevertheless, the transition is clearly sensitive to polarizations and the anisotropy is the same as in the fundamental gap. This high-energy transition is noticeably red-shifted (by ~ 80 meV for ⊥ and ~ 60 meV for ∥ ) as compared to the value observed at low temperature ! ∥,! ~ 1.24 ± 0.005 eV, suggesting fundamentally different in nature. Since the CBs above ! !" and VBs below ! !" at the Hpoint are far apart, the next higher energy transition at the H-point beyond ! ∥,! is not accessible with our probe beam. Therefore, we attribute the transitions ! ! and ! ∥ are caused by a direct transition at the Lpoint. Detail understanding of higher energy transitions in Te requires further theoretical calculations.

Supplementary Figure 4 | Polarized transient reflectance response of Te nanosheets around the band-edge region. a Two-dimensional false color map of pump-probe delay dependent transient reflectance spectra
[∆ ! ( , )] over an extended probe energies for parallel ∥ and perpendicular ⊥ relative to the c-axis of Te crystal. Narrow region of probe energies between 0.72 − 0.8 eV is missing due to technical limitations. Negative signal (red) indicates a pump-induced increase in absorption and positive signal (blue) corresponds to decrease in absorption. Overall, strong features with distinctive dynamics at different energies are clearly visible, suggesting a series of optical transition with some polarization anisotropy. b Transient spectra (blue spheres) acquired at 50 ps time delay with two orthogonal probe polarizations. Each spectrum is fitted with a simple model described in the main text (red dashed lines). Calculated moduli Δ ! of each fit are also plotted with vertical offset (black lines). Both the model fits and the moduli show transient features corresponding to the series of direct and indirect interband transitions in Te, as indicated by vertical dashed lines. The transition energy for ∥ is blueshifted by ~20 meV, suggesting band-edge optical anisotropy in Te at room temperature.

B. Carrier dynamics at 300 K
Polarization dependent carrier decay processes are investigated by measuring TR time traces at different probe energies following excitation of the sample with 1.51 eV pump pulses. Representative polarized time traces taken from the nanosheet at 300 K from two low-and high-energy regimes are shown in Supplementary Figure 5a,b. At energies intermediate between these two regimes, the TR response is very weak and fast due to lack of any direct optical transitions and, therefore we will not discuss it here. Near the band gap region (Supplementary Figure 5a), the majority of the TR decays exponentially within first 30 ps followed by a very weak (nearly 2-orders of magnitude lower than the Probe energy (eV) peak) residual signal, which persists over 300 ps. Around the high-energy regime (Supplementary Figure   5b), the time traces display initial ultrafast decay of the signal followed by a long exponential recovery of the remaining signal at later times. In order to quantify the overall decay behavior around the low and high energy regimes, each time trace is fitted using multi-exponential functions convoluted with a Gaussian response function: 14 where ! is the amplitude with decay time constant of ! of the !! exponential term and is full width half maximum of the pump laser pulse ( = 200 ). A least square fitting to Eq. (2), as shown by dashed red lines in Supplementary Figure 5a,b estimates the decay time constants ! of each !! decay channel at different bands. Near the low-energy region, the decay constant of the majority of the signal is ! ! ∼ 17 ± 1 ps for ⊥ and ! ∥ ∼ 22 ± 1 ps for ∥ , respectively, followed by long-lived ( ! ! , ! ∥ ≳ 500 ps) residual signal. Around the high-energy region, only a fraction of the peak intensity decays abruptly within a few ps ( ! ! ∼ 10 ± 1 ps for ⊥ and ! ! ∼ 12 ± 1 ps for ∥ ) and the remaining signal decays rather slowly ( ! ! , ! ∥ ≳ 300 ps) for both polarizations. The fractional signal of rapid decay is ∼ 50 % of the peak signal for ⊥ while it is only ∼ 10 % of the peak signal for ∥ . Such differences of decay dynamics with respect to the polarization and energy of the probe laser beam are qualitatively similar as observed at low temperature (see main text).
Around the low-energy region, ultrafast carrier thermalization by carrier-carrier and carrierphonon scattering is not distinguishable at room temperature. Therefore, the decay time constants of ! ! and ! ∥ for both polarizations are attributed to the interband carrier recombination time including carrier thermalization. Due to ultrathin samples, thinner than the penetration depth !" in Te around the IR-region ( !" ∼ 50 nm), 9 carrier diffusion does not play a role in the decay dynamics. The subsequent weak residual signal (less than 2 % of the peak) observed after the recombination is attributed to a constant feeding of carriers from higher lying bands through phonon-assisted intervalley processes. Around the high-energy region, the polarized TR responses display sharp transient within ∼ 10 ps, which is attributed to intervalley scattering followed by the intraband cooling of carriers via carrier-carrier and carrierphonon scattering. Since the signal after initial rapid decay is reduced substantially with ⊥ as compared to the signal with ∥ , this observation suggests intervalley scattering is effectively suppressed with ∥ . Subsequent slower decay of the signal is caused by persistent feeding of intervalley scattered carriers at neighboring valleys as well as carriers at higher lying bands to the H-valley. Note that the initial rapid recombination time at all spectral regions is noticeably slower with ∥ as compared to the times with ⊥ . This behavior is more apparent at low temperature (see main text), which is consistent with anisotropic carrier scattering times and hole mobility observed in Te crystal. 10 Overall, carrier decay dynamics at room temperature is qualitatively similar to the low-temperature dynamics, which has been described with sufficient detail in the main text.

Supplementary Figure 5 | Polarized transient reflectance response of Te nanosheets at room temperature.
Representative polarization resolved transient reflectance (TR) Δ ! traces of Te samples around the fundamental band-edge a and high-energy transition region b. Dashed red lines are the multi-exponential fits of each corresponding data over a long delay range. Around the band-edge region, majority of the TR signal relaxes within first 30 ps due to ultrafast intraband thermalization followed by interband recombination. Around the higher energy transition, about half of the peak TR signal decays rapidly ( ! ! ∼ 10 ps) with ⊥ due to intervalley scattering accompanied by intraband thermalization. In contrast, TR signal with ∥ decays only marginally (10 % of the peak) with a decay constant of ! ∥ ∼ 12 ps, suggesting suppressed intervalley scattering and weak coupling to phonon. Rest of the signal for both polarizations decay rather slowly ( ! ! , ! ∥ ≳ 300 ps) due to constant feeding of carriers from higher lying bands to the H-valley.

Supplementary Note 4: Modeling carrier recombination dynamics at 10 K
One of the major results described in the paper is that a certain fraction of carriers at high energy scatter into remote valleys away from the H-valley minimum. These long-lived carriers then can provide a long-lived source for carriers which eventually recombine at the band edge. In order to make a more quantitative analysis of the decay dynamics in this system we perform modeling of time decays using coupled rate equations: where ! and ! denote the number density of thermalized carriers at the high (presumably the indirect L-valley) and low-energy (the lowest energy H-valley) regimes. Considering a two-level system, the first equation determines the scattering time ! of which describes the slow feeding of carriers from neighboring higher lying valleys to the H-valley as well as lifetime !" of residual carriers in those higher lying indirect valleys,, which is beyond the experimental limit and assumed to be !" = 5 ns. In other words, we assume the lifetime of the indirect higher lying valleys is determined by feeding carriers to the H-valley band edge. The second equation describes the direct recombination time ! with accounting for constant feeding of carriers from higher energy bands to the band edge minima. The terms G 0 and G 1 determine the fraction of carriers excited into indirect higher lying valleys vs. the fraction excited into the H-valley minima. The solution of Eq. (3) fits the experimental time traces around both regions reasonably well, as shown by dashed red lines in Supplementary Figure 6a,b. We determine that the ratio ! !~1 .6 % as expected from the time decays. The fits, however, does not cover the early transients (below 10 ps), where the carrier relaxation mostly dominated by carrier thermalization processes. The extracted values of carrier recombination times ! are , and ~15 ps and ~25 ps for ⊥ and ∥ , respectively, and the lifetime of slow bleeding of carriers from higher lying valleys to H-valley for both polarizations is ! ≈ 750 ps. These values are nearly identical with the simple exponential fittings (shown in Supplementary Note 3) and agree with the previously described qualitative understanding of carrier decay dynamics in Te.
Where c and v denote density of states for the conduction and valence bands, ! and ! are the respective carrier densities for holes and electrons, ! and ! are the appropriate Fermi-Dirac distributions, is the carrier temperature, and ! is a constant factor fit to the data. Given the p-doped nature of our samples, we consider the total hole population ! as a sum of doped holes ! and photoexcited holes Δ ! , i.e., ! = ! + Δ ! . The electron population ! is taken to be entirely photoexcited, i.e., ! = Δ ! , and the photoexcited electron and hole densities are taken to be equal, i.e., Δ ! = Δ ! , on the basis of charge neutrality. The carrier temperature is assumed to be equal for electron and holes. Now we calculate the photoinduced absorption coefficient Δ using relevant parameters, which are known for our samples (see Supplementary Table 1).
Once the photoinduced modulation in the absorption is calculated, we can derive the modulation of the real part of index of refraction, Δ , using the Kramers-Kronig relation: where ∆ is calculated from Eq. (4). Note that although the upper bound of the integral is infinite, ∆ rapidly approaches zero above the band-edge and thus our theoretical description of the absorption need not include the high-energy regime.
Once we derive both the real part of refractive index and the absorption coefficient, we can connect to these parameters to evaluate the reflectance of the sample. Assuming normal incidence of the probe beam on the sample, which is the case in our experiment, the reflectance of the sample without excitation is given by: The fractional change of reflectance induced by pump excitation (normalized by initial reflectance ! before exciting the sample) can be expressed (in first-order expansion) as following: Where ! and ! are the background values of the real and imaginary components of the complex index of refraction, respectively. We take ! , for light polarized perpendicular to the c-axis, to be constant at the average value of 4.95 for our probe tuning range, based on measurements of single-crystal Tellurium. 17 We calculate ! = 4 • from Eq. (4) using background estimates for carrier density. All basic material parameters are tabulated in Supplementary Table 1.
We fit theoretical lineshapes from Eq. 7 to our experimental transient reflectance spectra by parameterizing the density of doped holes, density of photoexcited carriers, carrier temperature, band-gap energy, and an overall scaling factor. We numerically optimize these parameters to minimize the sum squared error for each spectrum taken at distinct delay times, keeping the time-independent parameters consistent. The resulting fit parameters are tabulated in Supplementary Table 2. Supplementary

Supplementary Note 6: Polarization anisotropy of optical reflectance in Te
The  (wavelengths). Following initial ultrafast transients, the TRS signal oscillates in time and gradually decays because of interference between light waves reflected off the surface and light waves reflected off the propagating strain wave. The TR signal, therefore, contains both the electronic response of the material as well as the oscillatory interference signal due to the CLAP generated in the sample. In order to extract information about the oscillatory signal, each trace is fitted with following empirical function:

Supplementary Note 7: Coherent longitudinal acoustic phonon in Te nanosheets
where first three terms fit the non-oscillatory electronic response including bi-exponential (initial fast ! followed by slower ! ) response and last one fits the slowly decaying oscillatory signal. Coefficients , , are amplitude of each decaying components and is the overall background signal. Least square fitting method is applied to each wavelength dependent trace, as shown by black line superimposed on each trace, and useful parameters such as the velocity of the CLAP are extracted as fit parameters. An average amplitude of the oscillation is of the order of 10 !! and damping constant !"# varies from 50-100 ps. Interestingly, the oscillation period remains unchanged for each probe wavelength. This behavior suggests that standing wave condition of CLAP propagation within the film is fulfilled, which is expected in ultrathin samples. The period of the oscillation is determined by film thickness : where ! is the velocity of CLAP (sound velocity), which depends on the polarization of probing laser pulse. An average oscillation period of 26.3 ± 0.1 ps estimates the average velocity ! of the LA phonon (sound velocity) of ∼ 1800 m/s in our ultrathin ( ~ 24 nm) Te samples. The estimated velocity ! is lower than both the sound velocity along parallel (3400 m/s) and perpendicular to c-axis (2290 m/s), but higher than the velocity along xy-plane or shear plane (1390 m/s), 23 suggesting quasi-shear wave propagation in our sample. 24 It is important to discuss the physical origin of the acoustic phonon generation in our Te samples.
There are mainly three possible mechanism of stress generation upon photoexcitation: (1) thermoelastic stress ( !" ), (2) electron-acoustic deformation potential stress, !" and (3) inverse piezoelectric stress, !" . The photoinduced !" due to rapid lattice heating by electron-phonon coupling can be estimated by following the standard model, 19 !" = −3 • • • ∆ , where is the linear thermal expansion coefficient, is the bulk modulus and ∆ is the maximum temperature increased by pump pulse. Due to anisotropic crystal, Te has negative expansion coefficient parallel to c-axis, i.e., = −5×10 !! !! , and positive coefficient perpendicular to c-axis, i.e., = 5×10 !! !! . The bulk modulus in Te is = 19 GPa. 25 It is known that electron temperature immediately after pump excitation of Te sample is extremely high due to excitation of hot carriers, which thermalizes to lattice temperature within a ps through electron-optic-phonon coupling. 26 The lattice temperature increases only few K given the low excitation density used in our low-temperature (10 K) experiment. Therefore, we can safely assume an upper bound increased lattice temperature of ∆ = 20 K within 10 ps. Now, using all these parameters the photoinduced thermoelastic stress !! can be estimated to be !" = 0.006 GPa, which will be even smaller at later delay times. Next, the electron-acoustic deformation potential stress !" can be estimated by: 19 !" = − !!!" • Δ ≈ −0.0003 GPa, with !!!" = −8.5 eV along c-axis 27 and with photoinduced carrier density of Δ ~ 2×10 !" m -3 . The stress perpendicular to c-axis increases to !! ≈ 0.001 GPa, with !!!" = 35 eV. 27 If we estimate the band gap change from these stress values, 28 these stress values are too small to induce tens of meV band gap shifts that we observed in our samples. Therefore, we argue that the piezoelectric strain induced by inverse piezoelectric effect (IPE) is the most likely mechanism of strain generation in our samples. Based on experimental observations, piezoelectric shear strain is dominant in our samples.
With the simple theoretical expression, 29, 30 ∆ !"#$% ≈ 4 • Ψ ! • ℇ !" , where ∆ !"#$% is the band splitting or lifting, Ψ ! is the shear deformation potential and ℇ !" is the shear strain, it is possible to estimate the shear strain value of the materials if we know the shear deformation potential and the resulting strain induced band splitting. The band splitting of ∆ !"#$% = ∆ !"#$ ∼ 20 meV is known from the TRS measurements at low temperature but the shear deformation potential is not known yet to the best of our knowledge. The typical value of deformation potential found in the literature is ~5 eV, which is estimated by the shift of energy gap in Te with dilation, 31 If we assume Ψ !~5 eV in our case, the required shear strain turns out to be on the order of ~ 0.1 %, which is substantially lower than the value estimated by our ab initio DFT band structure calculations, i.e., ~2 − 3 %. Since the shear deformation potential is fundamentally different and complex, it is hard to predict the exact value of shear strain using this simple expression. However, considering required maximum shear strain of ~ 2 % to induce 20 meV band splitting, we can estimate the shear deformation potential in Te to be 0.25 eV. This simple estimation of photoinduced shear strain and shear deformation potential in Te will be very helpful for future studies since photoexcitation can be an alternative and non-destructive means for inducing exotic topological phases in Te.

Supplementary Figure 8 | Coherent longitudinal acoustic phonon in Te nanosheets.
Transient reflectance signal (TRS) of Te samples measured at different probe energies following 820 nm pump excitation at room temperature. Probe beam is polarized off-axis to ab-plane as well as c-axis but fixed for each wavelength. The TRS traces display weak oscillations embedded on a slowly decaying electronic signal. The oscillatory signal is caused by interference between the probe laser beam and photoinduced strain pulse propagation in the sample. Due to ultrathin samples, standing wave condition is fulfilled, which satisfy the linear dependence of oscillation period with film thickness, but does not depend on wavelength of the probe beam (see inset). This condition allows to estimate thickness knowing the acoustic wave velocity or vice versa through: = 2 • , where is sample thickness, is the oscillation period, is the velocity of longitudinal acoustic phonon, i.e., sound velocity, in Te. An average period of the oscillations is estimated to be 26.33 ps, which estimates the sound velocity of ∼ 1800 m/s in our 24 nm thick Te sample. Period, T