# Coherent Dynamics of Charge Carriers in $\rm\gamma$-InSe Revealed by
Ultrafast Spectroscopy

Jianwei Shen1 Jiayu Liang1 Qixu Zhao1 Menghui Jia1 Jinquan Chen1 Haitao Sun1
Qinghong Yuan1 Hong-Guang Duan2 Ajay Jha3,4,5 Yan Yang1 & Zhenrong Sun1

###### Abstract

For highly efficient ultrathin solar cells, layered indium selenide (InSe), a
van der Waals solid, has shown a great promise. In this paper, we study the
coherent dynamics of charge carriers generation in $\rm\gamma$-InSe single
crystals. We employ ultrafast transient absorption spectroscopy to examine the
dynamics of hot electrons after resonant photoexcitation. To study the effect
of excess kinetic energy of electrons after creating A exciton (VB1 to CB
transition), we excite the sample with broadband pulses centered at 600, 650,
700 and 750 nm, respectively. We analyze the relaxation and recombination
dynamics in $\rm\gamma$-InSe by global fitting approach. Five decay associated
spectra with their associated lifetimes are obtained, which have been assigned
to intraband vibrational relaxation and interband recombination processes. We
extract characteristic carrier thermalization times from 1 to 10 ps. To
examine the coherent vibrations accompanying intraband relaxation dynamics, we
analyze the kinetics by fitting to exponential functions and the obtained
residuals are further processed for vibrational analysis. A few key phonon
coherences are resolved and ab-initio quantum calculations reveal the nature
of the associated phonons. The wavelet analysis is employed to study the time
evolution of the observed coherences, which show that the low-frequency
coherences last for more than 5 ps. Associated calculations reveal that the
contribution of the intralayer phonon modes is the key determining factor for
the scattering between free electrons and lattice. Our results provide
fundamental insights into the photophysics in InSe and help to unravel their
potential for high-performance optoelectronic devices.

State Key Laboratory of Precision Spectroscopy, School of Physics and
Electronic Science, East China Normal University, Shanghai 200241, P.R. China

Department of Physics and Institute of Modern Physics, Ningbo University,
Ningbo 315211, China

Rosalind Franklin Institute, Harwell, Oxfordshire OX11 0QX, United Kingdom

Department of Pharmacology, University of Oxford, Oxford OX1 3QT, United
Kingdom

Research Complex at Harwell, Rutherford Appleton Laboratory, Didcot OX11 0QX,
United Kingdom

The van der Waals chalcogenides show a variety of different specific functions
that depend on their composition and number of layers. Weak interaction of
materials along the stacking direction facilitates the realization of
heterostructures with different functionalities. Recent reports demonstrate
possible ways for fine tuning the band gap [1, 2, 3], controlling valley
polarization [4], and the realization of high charge mobilities [5]. Two-
dimensional InSe has attracted extensive attention due to its large tunable
band gap from 1.4 to 2.6 eV and high charge carrier mobility. A few-layer InSe
has been synthesized via physical and chemical methods, which exhibits
promising characteristics for optoelectronic application [6]. Tamalampudi et
al. showed that devices based on few-layer InSe obtained by mechanical
exfoliation can be used as photosensors with high photo-responsivity [6]. The
experimental measurement has shown the carrier mobilities reaching (or
exceeding) $10^{3}$ cm2 V-1 s-1 at room temperature [7, 8], which makes it a
champion material in transition-metal dichalcogenides, TMDCs. To further
optimize the optoelectronics devices based on InSe, a detailed insight to the
photophysics of the material is eminent.

A schematic of lowest energy electronic transition leading to A (VB1 to CB
transition) exction in single-crystal InSe is depicted here in Fig. 1. To have
a comprehensive understanding towards photoinduced processes in InSe after
resonant excitation to generate A excitons, one needs to follow the electronic
and nuclear evolution with ultrafast time-resolution. Weiss and colleagues
have studied the hot carrier and surface recombination dynamics in layered
InSe crystals after photoexciation to generation A and B excitions using
transient reflectivity measurements. The hot carrier lifetime was found to be
dependent on excited carrier density, indicating that the cooling process is
caused by phonon scattering [9]. Lu et al. have studied the photogenerated
carrier relaxation dynamics using optical pump-terahertz probe measurements in
InSe multilayer. The intraband relaxation showed biexponential decay rates
followed by recombination in $\sim$100 ps timescale [10]. Chen et al. employed
two-photon photoemission spectroscopy to study the ultrafast electron dynamics
in the material of InSe [11]. They observed that the dissipation of hot
electrons occurs only via Fröhlich coupling to optical phonons, which
indicates the InSe has an excellent potential for hot-carrier optoelectronics.
The subsequent studies of hot carriers in InSe show many-body screening of
quasi-two-dimensional electron gas dramatically reduces the Fröhlich
scattering strength [12]. Desipate these extensive studies, the intrinsic
mechanism for the high charge mobility and the initial phonon-coupled quantum
dynamics of charge carriers in InSe is still elusive.

In this work, we focus our attention on quantum coherent dynamics after
subjecting the $\rm\gamma$-InSe single crystals to the lowest energy
electronic transition, which involves generation of A exciton. We employ
ultrafast transient absorption spectroscopy using broadband probe in the
visible region of the spectrum to capture charge carrier relaxation dynamics
after resonant photoexcitation. We selectively excite the sample within
different regions of the lowest energy transition envelop using various pump
photon wavelengths (600, 650, 700 and 750 nm) to generate carriers with
different kinetic energies. The spectral and temporal dynamics of hot carriers
have been analyzed by global fitting approach. The systematic variation in
carrier relaxation pathways and lifetimes are observed. We also extract
vibrational coherences from the kinetic data of ground-state bleach (GSB)
band. The time-evolution of these vibrational coherences has been examined by
wavelet analysis. To reveal the nature of the observed long-lived coherences,
ab-initio calculations have been performed, which reveals the important
contribution of intralayer phonon modes in carrier relaxation processes. Our
study provides a detailed understanding towards charge carrier dynamics near
the band edge in $\rm\gamma$-InSe single crystals.

## Results

The $\rm\gamma$-InSe sample was directly purchased from 2D semiconductor (home
page: www.2dsemiconductors.com) and use it without further modification. To
perform transient spectroscopic studies in transmission mode, ultrathin slices
of the sample have been prepared using microtoming machine (Ultramicrotome
Leica EM UC7). To check for the thickness dependence on transient dyanmics, we
microtome samples with three different thicknesses, 30, 50 and 80 nm.

###  0.1 Femtosecond Transient absorption measurements

Prepared thin slices of the $\rm\gamma$-InSe sample are placed over quartz
substrate and then mounted on a XYZ translation sample-stage. The pump/probe
laser pulses are focused on the sample spot. The optical measurements have
been performed after optimizing the focus sizes of pump and probe beams with
minimized optical scattering. All the measurements reported in this work have
been performed at room temperature.

With the experimental condition described above, we preformed the optical
measurements of InSe and show the transient absorption (TA) spectra for 50 nm
thin microtomed crystals in Fig. 2. We also measure the TA data with the
dependence of sample thickness and we do not observe any significant
difference of results with different sample thickness. The TA data with sample
thickness are shown in the Supporting Information (SI). In Fig. 2(a),
transient absorption data is shown for pump excitation at 600 nm. The data
shows ground state bleach feature (GSB) centered at 510 nm, which lives long.
Additionally, the TA spectrum has been measured by pumping at 650 nm (shown in
Fig. 2(b)), the associated broadband pulses of excitation have been shown in
Fig. 1(b)). Similar to the TA data for 600 nm, it presents a positive and a
negative band centered at 510 nm and 470 nm, respectively. The positive
magnitude indicates the GSB and stimulated emission (SE) signals in TA
spectrum, the negative one originates from excited state absorption (ESA).
Compare to the GSB, the ESA signal is much weaker in the measurement.
Moreover, we show the TA spectrum pumped at 700 nm in Fig. 2(c). It presents
the GSB and ESA bands at 510 and 480 nm, respectively. The basic features is
very similar to the TA spectrum in (b). The TA spectra pumped at 750 nm is
shown in Fig. 2(d). It shows the GSB and ESA bands at 510 and 480 nm as well.
However, it shows, at initial time, the fast decay of magnitude in (b) compare
to the case in (d). To observe the effect of carrier densities on relaxation
dynamics, we have also performed transient absorption measurements using
different pump powers (shown in the SI). It can be seen that the relaxation
dynamics is strongly dependent on the pump power and hence, the carrier
densities. This signifies that the hot carrier cooling process occur through
phonon scattering. All the data shown in Fig. 2 has been measured at 80
$\mu$W.

###  0.2 Global fitting results

To study the time-evolved magnitude in TA, we further employ the global
fitting approach to examine the kinetics of TA data. The detailed description
of global fitting method has been described in the SI. In global fitting
algorithm, the TA data has been fitted by multi-exponential functions to
retrieve the decay components and the associated lifetimes at each probing
wavelengths. By this, we obtain the decay-associated spectrum (DAS) with decay
lifetime of each spectrum and we show them in Fig. 3. The DAS of pumping at
600 nm is shown in Fig. 3(a), 5 components with different lifetimes are
presented. The fastest DAS shows a broadband profile and centered at 510 nm.
It shows the decay lifetime of 0.93 ps and plotted as blue circles. The
negative magnitude of this DAS indicates the fast growing of amplitude in TA
in Fig. 2(a). Moreover, the fastest components from Fig. 3(b) to (d) show the
similar spectral profiles and lifetimes of 0.68, 1.0 and 0.65 ps,
respectively, which imply the generation of charge carriers in InSe. The
second component shows the decay lifetime of 7.52 ps (red circles). It
presents a broadband profile with positive amplitude and the spectrum is
centered at 500 nm. The positive magnitude of DAS indicates the decay of
magnitude in TA in Fig. 2(a). This components in Fig. 3(b) to (d) shows the
lifetimes of 10.6, 6.9 and 5.8 ps, respectively. Moreover, the spectral
profiles from Fig. 3(a) to (c) are roughly identical, but the amplitudes are
gradually decay from (a) to (c), which imply that the contribution of the
first decay component decrease with varying of the pumping wavelength. The
broadband profiles in (a) to (c) indicate the decay of magnitude from 480 nm
to 560 nm. This can also be confirmed by directly observations of the TA
spectra in Fig. 2. The spectral profile of 5.84 ps in Fig. 3(d), however,
shows quite different shape. It presents a positive and negative peaks at 500
and 520 nm, which imply a signature of population transfer from higher
conduction band to lower band in InSe. In addition, the magnitude of this
component is dramatically reduced. The decay associated component of 20.17 ps
(green circles) shows a broad peak at 510 nm in Fig. 3(a), however, it
gradually change the profile from (b) to (d). The positive and negative
features of this component indicate the decay and grow of amplitude in TA
spectra. Moreover, the component of 414 ps (light blue circles) shows a
broadband from 500 nm to 560 nm. This component has been repeated from Fig.
3(b) to (d). The last component shows the spectral profile of residuals and
plotted as pink circles from (a) to (d). To compare the spectral profiles of
the time components that might represent hot carrier relaxation dynamics, we
have overlayed the spectral components that we believe to represent the same
physical processes after photexcitation with different pump excitations as
Fig. 4. A spectrally narrowing of the ground state bleach features on reducing
the pump excitation frequency can clearly be observed in Fig. 4(a). It is a
very interesting observation, which will need further investigation to reveal
the physical nature of this observed narrowing. Additionally, we observe
reduction in the amplitude of the second time components on decrease of the
pump excitation frequency, shown in Fig. 4(b). This can be attributed to the
reduction in the excess kinetic energy given to the carriers on pump
excitation near the edge of the bandgap. The overlay of the other time
components representing recombination dynamics are provided in the supporting
information.

###  0.3 Vibrational coherences

To examine the coherent dynamics in InSe, we extract the kinetic profiles of
GSB bands in TA spectra obtained at different pump excitation wavelengths,
which are shown here in the Fig. 5. To improve the signal-to-noise, the
plotted data are averaged over the kinetic profiles from 510 nm to 514 nm of
the respective transient data, shown as red circles in Fig. 5. To retrieve the
vibrational coherences convoluted with population dynamics in the kinetic
data, we firstly perform the global fitting approach and show the fitting data
of the GSB kinetic profiles as black dashed lines in Fig. 5. The obtained
residuals from the kinetic fitting are shown as red solid lines in Fig. 5. The
amplitude of residuals are magnified by 3 times to show the clear oscillatory
dynamics. Then, we perform the Fourier transform of these residuals and the
resulted vibrational modes are plotted as red solid lines in Fig. 6 (a), (b),
(c) and (d) for TA data at different pump excitations 600, 650, 700 and 750
nm, respectively. Many low-frequency modes at frequencies 18, 38, 60, 78 and
110 cm-1 are prominently visible in the obtained vibrational power spectra.
Intensity profiles of the observed vibrations varies for different pump
excitation data. Assignment of the observed vibrational coherences have been
described later in the discussion section. To examine the vibrational
lifetimes of the observed normal modes, we employ the wavelet analysis on the
residuals and plot the respective results from (e) to (h) in Fig. 6. The
details of the wavelet analysis have been described in the SI. The results
show that, in the low-frequency region, the vibrational mode at ($<20$cm-1)
has a clear oscillatory dynamics even after waiting time of 5 ps. Lifetime of
these modes is dramatically reduced with the increase of vibrational
frequencies. The mode at 38 cm-1 decays within 2 ps and the frequency at $>60$
cm-1 shows the lifetime shorter than 1 ps.

## Discussions

Understanding the electron-phonon-coupling dynamics upon ultrafast
photoexcitation of InSe is necessary for understanding its optoelectronic
properties. The wavelength dependent data discussed earlier in the result
sections presents information on how different phonons play role in carrier
relaxation after photoexcitation from VB1 to CB. To retrieve the origin of
these observed phonons in the spectroscopic measurements, we perform the ab-
initio calculations of $\rm\gamma$-InSe. The details of theoretical
calculations are presented in the Method section. We have presented the
relevant vibrational modes at 15, 38, 117 and 227 cm-1 in Fig. 7. These
calculated vibrational frequencies agree to the observed vibrational
coherences in our experimental data. Importantly, based on our vibrational
analysis and associated wavelet analysis, we observe two timescales of phonon
dynamics after photoexcitation: 1 ps and 5-10 ps. Our wavelet analysis reveal
that the lifetime of vibrational coherences of a mode at ($<$20 cm-1) is
around 5 ps, while the mode of 38 cm-1 decays within a timescale of 2 ps.
Thus, we can infer that the carrier relaxation dynamics is mainly contributed
by the interaction with the low-frequency phonons, which results into two
different timescales of the experimentally observed relaxation timescales. So,
the phonon mode at $<$20 cm-1 with a lifetime of 5 ps may contribute towards
the carrier relaxation dynamics observed at 5-10 ps, while the phonon modes
with shorter lifetime may have contribution towards the earlier time
relaxation processes. Theoretical calculations reveal that the motion of mode
15 and 38 cm-1 strongly relate to the E’(1) and E”(1) motion between In and Se
atoms. This particular motion of low-frequency modes is assigned to the
intrelayer phonons [13]. Moreover, the vibrational coherences obtained for
different pump excitation wavelengths show that the magnitude of the
contribution of these phonons towards carrier relaxation hugely varies with
initial kinetic energy of the carriers. Hence, the thermal energy
redistribution of the charge carriers is dependent on the initial energy level
that they are excited to. Additionally, it must be noted that the amplitudes
of these wave-packet motions report on the magnitude of the excited-state
structural changes. Hence, the structure-based dynamics of motion of each mode
has been plotted in the Fig. 7, which can be used to interpret the initial
photo-dynamics in an atomic level.

## Conclusions

We have studied the excitation wavelength dependence charge carrier dynamics
in InSe single crystals using broadband femtosecond transient absorption
spectroscopy. The excitation pump wavelength dependence near the band edge (A
exciton; VB1 to CB transition) shows the faster cooling of the hot carriers
with increasing pump wavelength. Additionally, our measurements show that the
carrier thermalization process is biexpoential and varies from 1 to 10 ps with
different excitations. This slow cooling has been explained to be caused by
the large phononic band gap created by the large mass ratio between the anion
(Se2-) and cation (In2+). The recombination shows two timescales: 400 ps and
$>$1 ns (beyond the available time-delays available in the current
experimental setup). The recombination is reported to be primarily caused by
the surface recombination [9]. Our vibrational analysis of the observed
coherences in the transient absorption data reveals the important low
frequency lattice motions, which is coupled to electron relaxation dynamics.
The nature of the long-lived phonons have been deciphered using ab-initio
quantum calculations. Importantly, our mesurements distinctly captures short
lived ( 1 ps) and long-lived ( 5 ps) vibrational coherences that might be
contributing to the observed biexpoential thermal-relaxation dynamics occuring
in the similar time-scales. It is a very unique observation since interactions
of hot carriers to different phonons lead to differential thermalization
dynamics in the system. Thus, our work provides a detailed mechanism of
electrons interacting to the lattice phonons during relaxation dynamics after
resonant photoexcitation of InSe. The detail photophysical understanding
provided by this work will help to realize high-performance optoelectronic
devices using InSe crystals.

## Materials and Methods

###  0.4 Sample preparation.

The bulk InSe crystal sample was purchased from 2D semiconductor (home page:
www.2dsemiconductors.com) and used it without further modification. The thin
slices of the InSe sample 50 nm have been prepared using microtome machine
(Ultramicrotome Leica EM UC7). Prepared thin slides of samples have been
directly attached to the quartz substrate and mounted on a XYZ translation
stage, the pump/probe laser pulses are focused in the same spot on the surface
of sample.

###  0.5 Femtosecond transient absorption measurements.

A detailed descrption of the optical setup is provided in an earlier reported
work from our group [14]. Briefly, transient absorption measurements are
performed in a Helios-EOS fire spectrometer from Ultrafast System. The primary
femtosecond laser source is Astrella (800 nm, 100 fs, 7 mJ/pulse, and 1 kHz
repetition rate, Coherent Inc.). To generate probe beam in visible region of
the spectrum, a white light supercontinuum is generated by focusing on 3 mm
thick sapphire crystal. The tunable pump beam is generated using a commercial
OPA (OPerA Solo, Coherent Inc.). Following a 200 nm pump focus at the sample,
the pump and probe pulses are spatially overlapped on the sample. For this
system, a 120 fs instrument response function has been determined. The optical
measurements have been performed after optimizing the measuring spot on sample
surface. The XYZ translation stage has been modulated to find the best spot
and reduce the scattering from the pump and probe pulses. The real
measurements have been performed at room temperature after these refining
procedures. The equally distributed time step of 200 fs has been used at
initial waiting time up to 40 ps. The larger time step has been used for the
rest of measurements up to 2 ns. MATLAB has been used to process the TA data
using custom-written codes.

###  0.6 Theoretical calculations.

We perform the theoretical calculations to examine the structural origin of
observed vibrational modes. For this, we firstly use the crystal structure of
$\gamma$-InSe from the X-ray diffraction measurement and the final structure
used in the calculation has been refined according to literature
(111-InSe.cif) [15, 16]. The density functional theoretical calculations for
this study are performed by using the projector-augmented-wave (PAW) method
and the Perdew-Burke-Ernzerhof (PBE) generalized gradient exchange
approximation correlational functional implemented in Vienna ab initio
simulation package (VASP) [17, 18, 19]. For the initial geometrical
optimization, the energy convergence criteria for electronic and ionic
iterations are set to be 10-4eV and 10-5eV, respectively. The energy cutoff
for the plane-wave basis is set to 300 eV for all calculations. The Brillouin
zone is represented by a Monkhorst–Pack special k-point mesh of
8$\times$8$\times$2 for the geometry optimization. To give an accurate
explanation of the weak interactions such as London dispersion interaction and
long-range van der Walls (vdW) intermolecular interactions, DFT-D3 method was
applied [20]. The phonon dispersion spectra and Raman spectral lines according
to group theory are calculated based on the density functional perturbation
theory (DFPT) method [21]. In theory, the larger the supercell, the more
accurate the dispersion curves that are obtained. We adopt the
2$\times$2$\times$2 rectangle supercell for all phonon calculations.

## References

  * [1] K. F. Mak, C. Lee, J. hone, J. Shan, and T. F. Heinz, Atomically Thin MoS2: A New Direct-Gap Semiconductor. Phys. Rev. Lett. 105, 136805 (2010).
  * [2] V. Tran, R. Soklaski, Y. Liang, and L. Yang, Layer-controlled band gap and anisotropic excitons in few-layer black phosphorus. Phys. Rev. B 89, 235319 (2014).
  * [3] G. W. Mudd, S. A. Svatek, T. Ren, A. Patané, O. Makarovsky, L. Eaves, P. H. Beton, Z. D. Kovalyuk, G. V. Lashkarev, Z. R. Kudrynskyi, and A. I. Dmitriev, Tuning the bandgap of exfoliated InSe nanosheets by quantum confinement. Adv. Mater. 25, 5714 (2013).
  * [4] K. F. Mak, K. He, J. Shan, and T. F. Heinz, Control of valley polarization in monolayer MoS2 by optical helicity. Nat. Nanotechnol. 7, 494 (2012).
  * [5] B. Radisavljevic, A. Radenovic, J. Brivio, V. Giacometti, and A. Kis, Single-layer MoS2 transistors. Nat. Nanotechnol. 6, 147 (2011).
  * [6] S. R. Tamalampudi, et al. High Performance and Bendable Few-Layered InSe Photodetectors with Broad Spectral Response. Nano Lett. 14, 2800 (2014).
  * [7] W. Feng, W. Zhang, W. Cao, and P. Hu, Back gated multilayer InSe transistors with enhanced carrier mobilities via the suppression of carrier scattering from a dielectric interface. Adv. Mater. 26, 6587 (2014).
  * [8] D. A. Bandurin, et al. High electron mobility, quantum Hall effect and anomalous optical response in atomically thin InSe. Nature Nanotechnol. 12, 223 (2017).
  * [9] C. Zhong, et al. Hot Carrier and Surface Recombination Dynamics in Layered InSe Crystals. J. Phys. Chem. Lett. 10, 493 (2019).
  * [10] W. Lu, et al. Terahertz relaxation dynamics of a two-dimensional InSe multilayer. Phys. Rev. B 102, 014314 (2020).
  * [11] Z. -S. Chen, C. Giorgetti, J. Sjakste, R. Cabouat, V. Véniard, and Z. Zhang, Ultrafast electron dynamics reveal the high potential of InSe for hot-carrier optoelectronics. Phys. Rev. B 97, 241201 (2018).
  * [12] Z. -S. Chen, et al. Ultrafast dynamics of hot carriers in a qusi-two-dimensional electron gas on InSe. Proc. Natl. Acad. Sci. (USA) 117, 21962 (2020).
  * [13] Y-J. Sun, S.-M. Pang, and J. Zhang, Layer Number-Dependent Raman Spectra of $\rm\gamma$‑InSe. J. Phys. Chem. Lett. 13, 3691 (2022).
  * [14] J. Shen, A. Jha, M. Lv, G. Jiang, Q. Zhao, Z. Liu, J. Chen, Y. Yang, H. Sun, H.-G. Duan, and Z. Sun, Ultrafast photoinduced dynamics of a donor-(pi)bridge-acceptor based merocyanine dye. Sci. Rep. 12, 18216 (2022).
  * [15] P. Gomes da Costa, R. G. Dandrea, R. F. Wallis, and M. Balkanski. First-principles study of the electronic structure of y-InSe and P-InSe. Phys. Rev. B 48, 14135 (1993).
  * [16] D. Olguín, A. Cantarero, C. Ulrich, K. Syassen, Effect of pressure on structural properties and energy band gaps of $\gamma$-InSe. I Phys. Status Solid 235, 456 (2003).
  * [17] G. Kresse, D. Joubert, From ultrasoft pseudopotentials to the projector augmented-wave method. Phys. Rev. B 59, 1758 (1999).
  * [18] J. Paier, R. Hirschl, M. Marsman, G. Kresse, The Perdew-Burke-Ernzerhof exchange-correlation functional applied to the G2-1 test set using a plane-wave basis set. J. Chem. Phys. 122, 234102 (2005).
  * [19] J. Hafner, Ab‐initio simulations of materials using VASP: Density‐functional theory and beyond. Journal of computational chemistry 29, 2044 (2008).
  * [20] S. Ehrlich, J. Moellmann, W. Reckien, T. Bredow, S. Grimme, System-Dependent Dispersion Coefficients for the DFT-D3 Treatment of Adsorption Processes on Ionic Surfaces. Chem. Phys. Chem. 12, 3414 (2011).
  * [21] X. Gonze, C. Lee, Dynamical matrices, born effective charges, dielectric permittivity tensors, and interatomic force constants from density-functional perturbation theory. Phys. Rev. B 55, 10355 (1997).
  * [22] M. R. Molas, A. V. Tyurnina, V. Zólyomi, A. K. Ott, D. J. Terry, M. J. Hamer, C. Yelgel, A. Babiński, A. G. Nasibulin, A. C. Ferrari, V. I. Falḱo, and Roman Gorbachev, Raman spectroscopy of GaSe and InSe post-transition metal chalcogenides layers. Faraday Discussion 227, 163 (2021).

This work was supported by the research grant of State Key Laboratory
Spectroscopy, School of Physics and Electronic Science, East China Normal
University and the starting grant of Ningbo University. The work was also
supported by NSFC grant with NO. 12274247 and the foundation of national
excellent young scientist.

The supplementary Information includes the global fitting approach and
obtained results, the TA measurement with sample thickness and the details of
wavelet analysis. The TA data measured with excitation at 400 nm. The TA data
measured with different excitation energies.

The authors declare that they have no competing financial interests.

Correspondence of paper should be addressed to A. J. (email:
ajay.Jha@rfi.ac.uk), Y. Y. (email<EMAIL_ADDRESS>and Z. R. S. (email:
<EMAIL_ADDRESS>
H. -G. D. and A. J. conceived the research and discussed with J. W. S. and Z.
R. S.. J. W. S. and M. H. J. performed the spectroscopic measurements. J. Y.
L. H. T. S. and Q. H. Y. constructed the model and performed theoretical
calculations. H. G. D. and A. J. wrote the initial draft and refined by all
authors. Z. R. S. and A. J. supervised this project.

Figure 1:  (a) schematic showing the lowest energy electronic transitions of
InSe probed in this study. Exciton A is created after transition from valence
band 1 (VB1) to conduction band (CB) (bandgap is 1.25 eV i.e. 991 nm). (b) the
absorption spectrum (blue dots and solid line) of $\rm\gamma$ phase InSe
crystal and the center wavelengths of excitation pulses used in this study.

Figure 2:  Femtosecond transient absorption (TA) measurements: TA spectra of
microtomed InSe crystals obtained after excitation at 600 (a), 650 (b), 700
(c) and 750 nm (d), respectively. The positive (red) and negative (blue)
amplitude indicate the GSB, SE and ESA, respectively.

Figure 3:  Decay associated spectra (DAS) obtained by employing global fitting
approach. The DAS components with their respective lifetimes are presented for
different pump excitation wavelengths: (a) 600 nm, (b) 650 nm, (c) 700 nm and
(d) 750 nm, respectively.

Figure 4:  Overlay of the first and the second DAS components obtained using
global fitting of the transient absorption data for different pump excitation
wavelengths. The first and second components represents the carrier relaxation
dynamics. The fastest components show the lifetime of 0.93, 0.68, 1.0 and 0.65
ps for the excitation of 600, 650, 700 and 750 nm in (a). The second
components show the lifetime of 7.52, 10.66, 6.89 and 5.84 ps in (b).

Figure 5:  Time-resolved traces of GSB band (red circles) and the exponential
fitting results (black dashed lines) for the transient absorption data
obtained at pump excitation of 600, 650, 700 and 750 nm in (a), (b), (c) and
(d), respectively. The residuals are shown as red solid lines and the
magnitude of oscillations is magnified 3 times.

Figure 6:  Fourier transform of residuals and the results are shown in (a),
(b), (c) and (d) for the excitation at 600, 650, 700 and 750 nm, respectively.
The identified modes are marked as black numbers. The obtained results of
associated wavelet analysis are shown from (e) to (h) with clarified
frequencies of vibrational coherences.

Figure 7:  The calculated vibrational modes of 15 cm-1, 38 cm-1, 117 cm-1 and
227 cm-1 in (a), (b), (c) and (d), respectively. The motions of each atom in
crystal are marked by the red and blue arrows.