# Impact of Pauli-blocking effect on optical limiting properties of $WSe_{2}$
thin films

Km. Surbhi  Sourav Bhakta  Anupa Kumari  Utkalika P. Sahoo  Pratap K. Sahoo
<EMAIL_ADDRESS>Ritwick Das<EMAIL_ADDRESS>School of
Physical Sciences, National Institute of Science Education and Research, An
OCC of Homi Bhabha National Institute, Jatni, Odisha - 752050, India

###### Abstract

We present a detailed investigation on thickness-dependent third-order
($\chi^{(3)}$) nonlinear optical properties of RF-sputtered $WSe_{2}$ thin-
films using ultrashort pulses centered at different excitation wavelengths.
The single-beam Z-scan based investigation in the visible spectrum reveals the
prominent role of Pauli-blocking towards inducing saturable absorption or
optical limiting in $WSe_{2}$ thin-films of various thicknesses. The study
also explores the dispersion in third-order nonlinear susceptibility
($\chi^{(3)}$) in $WSe_{2}$ thin films. Interestingly, $WSe_{2}$ thin-films of
any thickness exhibit a self-focusing effect depicting a positive nonlinear
refractive index ($n_{2}~{}>~{}0$). The frequency-dependent nonlinear
absorption ($\beta$) bear distinct correlations with the bandgap of the films
which is also investigated through density-functional-theory (DFT) based
simulations. The alteration in bandstructure is primarily due to the
$Se$-deficiency induced defect bands in $WSe_{2}$ thin-films which has
discernible impact on the optical limiting characteristics.

††preprint: APS/123-QED

##  I INTRODUCTION

Two-dimensional (2D) materials have been the center of research focus in many
fields due to their unique and versatile physical, chemical as well as opto-
mechanical properties. Graphene, a single layer semi-metallic 2D-carbon
architecture having honeycomb lattice geometry, has raised phenomenal interest
to investigate its remarkable properties and novel applications [1, 2, 3]. A
weak spin-orbit coupling and absence of other interactions in a pristine
graphene layer results in a very small, practically non-existent, electronic
bandgap. Although, gap opening have been realized through multilayer graphene
sheets, doped-graphene layers or by introducing stress, it is appreciably
small for a broad range of applications [4, 5, 6]. This motivated the search
for layered 2D semiconductors (with large molecular weight) which could be
grown to a monolayer-scale and they exhibit discernible band opening. Layered
transition metal dichalcogenides (TMDs) have received significant attention in
this context over the last few years due to their distinctive properties such
as high in-plane charge carrier mobility, high mechanical strength and weak
coupling between the layers [7, 8, 9, 10, 11]. TMDs have been investigated for
a broad range of applications which includes energy storage, gas sensing,
biochemical/biological sensing, transistor development, piezoelectric and
electronic devices and devising photo-switches [12, 13, 14, 15, 16, 17, 11].
TMDs include layered crystal structures with strong (covalent) in plane
bonding and weak (vander Waals) out of plane interactions. Recent reports have
revealed that the electronic bandstructure of TMDs is strongly dependent on
the number of monolayers and undergoes transition from indirect to direct as a
function of thickness [18, 19]. Such a possibility endows tremendous
functional flexibility from an application point of view. From the perspective
of nonlinear optical (NLO) properties, the possibility of tailoring the
bandstructure (or bandgap) allows to control the hyperpolarizability and
consequently, one could limit the thickness of TMDs in order to suit a certain
application. By utilizing the idea, TMD-based architectures have been explored
for a broad category of photonic applications such as optical switching [20,
21, 22], Q-switching and mode-locking [23, 24, 25, 26], data storage [27],
optical limiting [28, 29], as well as, for optoelectronic applications [30]. A
few comprehensive review of NLO characteristics of two-dimensional layered
materials could be found in Ref. [31, 32]. Conventionally, a single monolayer
or a few monolayer TMD has thickness which is about $1000^{th}$-times smaller
than the wavelength of light probe. Therefore, the NLO phase-shifts and
absorption by a _few monolayered_ TMD-based configuration are relatively
weaker. From a practical perspective, TMD thin-films with thicknesses ranging
between a hundred nanometer to a few micrometers would be more suitable for
photonic applications. However, it is crucial to ascertain whether the
properties of a monolayer TMD remain identical (or even similar) for TMD thin-
films which is comprised of vertically-stacked multiple monolayers. More
importantly, the growth of TMD thin-films could invariably, induce defects
which would be essentially dictated by the growth mechanism. Therefore, there
remains a strong possibility of modification in the absorption as well as
luminescence characteristics of TMD thin-films. From an application viewpoint,
it would be very important to investigate the alteration in NLO properties in
presence of such defects. In general, TMDs exhibit discernible third-order NLO
response owing to small bandgap and consequently, strong hyperpolarizability
in the visible/near-infrared spectral domain. This manifests in processes such
as third harmonic generation, nonlinear absorption (as saturable absorption
(SA), multiphoton absorption) and intensity dependent nonlinear refraction.
The dependence of third-order ($\chi^{(3)}$) NLO properties of a few TMD
species have been explored with respect to variation in structural parameters
and excitation source [33, 34, 35, 36, 37, 38, 39]. However, an investigation
aiming at exploring the impact of defects in TMD-based thin-films architecture
on $\chi^{(3)}$-induced manifestations is yet to be explored.

Our work essentially focus on investigating spectral dependence of
$\chi^{(3)}$ NLO properties of tungsten-selenide ($WSe_{2}$) thin-films which
have been grown using RF sputtering technique. The NLO investigations, which
were carried out using single-beam Z-scan technique, reveal that nonlinear
absorption exhibits a saturation behaviour in the visible spectral band owing
to _Pauli-blocking_ effect. In presence of defects, the impact of _Pauli-
blocking_ tends to weaken and in the near-infrared (NIR) spectral band, a
discernible two-photon absorption (TPA) signature is observed. Consequently,
the investigation gives rise to the possibility of deploying RF sputtered
$WSe_{2}$ thin-films for optical limiting and optical switching based opto-
electronic applications.

##  II EXPERIMENTAL DETAILS

###  II.1 Sample preparation

A set of $WSe_{2}$ thin films were prepared using a RF magnetron sputtering
system with a $WSe_{2}$ (99.9% pure) target in an argon environment at room
temperature. $WSe_{2}$ thin films with a desired thickness were sputtered onto
a $1~{}cm\times 1~{}cm$ $SiO_{2}/Si$ and glass substrate. Prior to the
deposition, the chamber was evacuated by a turbomolecular pump to a vacuum of
3.1 $\times$ $10^{-6}$ mbar. Throughout the deposition process, the $Ar$ gas
concentration was maintained at $15~{}sccm$, and the power was fixed at
$60~{}W$. The depositions were performed at a pressure of $1\times 10^{-2}$
mbar. The number of samples are prepared by varying the time for which
deposition takes place which essentially results in a thicker thin film. The
morphological features of the RF-sputtered $WSe_{2}$ thin-films is
investigated through field-emission scanning electron microscope (FESEM). The
crystal structure and phase analysis is carried out using X-ray diffraction
spectrum (from an X-ray diffractometer, Rigaku Smartlab) with Cu K$\alpha$
($\lambda$ = 1.5418 Å). The Raman spectrum were measured using a Raman
spectrometer (Jobin Yvon LabRam HREvolution, Horiba) with an excitation at
$532~{}nm$ wavelength. The linear absorption spectrum of the $WSe_{2}$ thin
films were carried out using a UV-VIS-NIR spectrometer (Agilent carry 5000 UV-
Vis-NIR).

###  II.2 Z-scan experimental set up

The spectral dependence of NLO characteristics of $WSe_{2}$ thin-films were
investigated through single-beam $Z$-scan technique which used Fourier-
transform limited (FTL) ultrashort pulses of temporal width of
$\approx~{}350~{}fs$ at $515~{}nm$, $1030~{}nm$ and $1520~{}nm$ excitation
wavelengths. In order to have discernible optical power at the excitation
wavelengths, the pulse repetition rate at $515~{}nm$ wavelength was chosen to
be $1~{}kHz$ whereas that at $1030~{}nm$ and $1520~{}nm$ wavelength was fixed
at $100~{}kHz$ and $504~{}kHz$ respectively. The optical power of the incident
beam is controlled by a combination of half-wave-plate (HWP) and a polarising
beam-splitter (PBS). The laser beam is focused to a suitable spot-size using a
thin plano-convex lens of focal length $150~{}mm$. The $WSe_{2}$ thin films
were translated by $\approx~{}10~{}cm$ symmetrically about the focal point. An
aperture of $1/5^{th}$ the size of laser beam is placed far from the sample
for recording the closed-aperture (CA) transmission from the sample. For open-
aperture (OA) transmission measurement, the aperture is removed from the beam
path. In general, the contribution of OA Z-scan transmission (or nonlinear
absorption) is eliminated from the CA transmission measurement by considering
a ratio of CA transmission to the OA transmission at a given $z$ coordinate.

##  III RESULTS AND DISCUSSION

Figure 1: Field Emission Scanning Electron Microscopy (FESEM). (a)-(b) shows
top surface images of $WSe_{2}$ thin films having thickness of $130~{}nm$ and
$242~{}nm$ respectively. Insets show cross-sectional FESEM images for
determining thickness. Figure 2: (a) X-ray diffraction spectrum of RF-
sputtered $WSe_{2}$ thin films. (b) Variation of strain and dislocation
density in $WSe_{2}$ thin films as a function of thickness.

The homogeneity of structure and thickness of the samples is estimated from
FESEM images. The cross-sectional FESEM is performed to obtain the thickness
of the deposited thin films. The thicknesses were measured to be $34~{}nm$,
$81~{}nm$, $130~{}nm$, $150~{}nm$, $242~{}nm$ and $326~{}nm$. In Fig. 1(a) and
(b), representative top-view images of $WSe_{2}$ thin films of thickness
$130~{}nm$ and $242~{}nm$ respectively are shown. Fig 1(a) appears to exhibit
a smooth surface as compared to that shown in Fig. 1(b). The insets represent
the respective cross-sectional images.

Figure 3: (a) Raman spectrum of $WSe_{2}$ thin films. (b) Variation of Raman
modes of film as a function of film thickness. (c) The multiple-peak fit for
$34~{}nm$ film. Insets show the schematic of possible vibrational modes.
Figure 4: (a) Absorption spectra of $WSe_{2}$ thin films. Inset shows the Tauc
plot for estimating the bandgap. (b) Variation of bandgap as a function of
$WSe_{2}$ fim thickness.

The study of crystal structure, crystallite sizes, strain, dislocation density
and quantitative phase analysis is carried out using X-ray diffraction (XRD)
spectrum of the $WSe_{2}$ thin films. The XRD peaks for all the thin-films
could be seen in Fig. 2(a). The measured XRD peaks of $WSe_{2}$ films match
reasonably well with JCPDS data card no 71-0600. One peak was consistently
obtained at an angle $27.5^{\circ}$ which is a consequence of (004) plane. The
crystallite sizes were estimated through the Debye- Scherrer formula while
strain and dislocation density is calculated using the recipe prescribed in
the Ref. [40]. Fig. 2(b) shows the variation of strain and dislocation density
as a function of $WSe_{2}$ film thickness. The variation of dislocation
density and strain exhibit a similar trend and maximize for the film of
thickness $326~{}nm$. This XRD spectrum distinctly depicts the presence of
defects for thicker thin-films which would be investigated later in detail.
Raman spectroscopy was employed to ascertain the vibrational modes of the
deposited thin $WSe_{2}$ ﬁlms. Fig. 3(a) shows the Raman spectra for the thin-
film samples in the thickness range of 34-326 nm. All the spectra show three
characteristic phonon modes. The peak corresponding to $E^{1}_{2g}$ mode is
associated with in-plane vibration of $W$ and $Se$ atoms, and the sharp peak
corresponds to $A_{1g}$ is related to out-of-plane vibration of $Se$ atoms.
The peak corresponding to $B^{1}_{2g}$ mode is essentially due to vibration of
$W$ and $Se$ atoms owing to the interlayer interactions [41, 42, 43, 44]. All
the peak intensities and shifts were calculated by fitting the Raman spectra
with multiple peaks as shown in fig.3(b) for the 34 nm sample. The inset shows
the schematic of displacement of W and Se atoms representing the plane of
vibration in Raman active modes. For bulk $WSe_{2}$ crystal, $E^{1}_{2g}$ and
$A_{1g}$ modes are degenerate, yielding single peak as compared to other TMDs
materials. Sahin et.al [45] demonstrated that when the crystal symmetry broken
even by a small amount of uniaxial strain, the degeneracy lifted off and two
peaks are clearly observed. We observed that nearly 9-10% of strain exist for
34 nm samples and increases with higher thickness due to large number of
dislocations calculated from XRD, as shown in Fig 2(b). Such starins are
obvious in room temperature sputter deposited thin films. This is the reason
that we have observed two distinct $E^{1}_{2g}$ and $A_{1g}$ phonon modes, in
all our samples. The variation of Raman-active modes as a function of
$WSe_{2}$ film thickness is shown in Fig. 3(c). It has been observed that the
Raman shift for all the vibrational modes reduces for the film of thickness
$242~{}nm$, which can be attributed to the surface defects and that
corroborated with the increased surface roughness observed in the FESEM image
of Fig. 1(b). The slight change in the Raman shift for a film thickness of
$326~{}nm$ arises because the surface uniformity improves with higher
thickness.

Figure 5: Simulated bandstructure of (a) pure $WSe_{2}$ crystal (b) $WSe_{2}$
crystal in presence of a defect induced by $Se$ deficiency.

The linear absorption spectrum for all the $WSe_{2}$ thin films are measured
using _UV-visible_ spectroscopy which is shown in Fig. 4(a). The absorbance
tends to reduce at longer wavelengths for all the samples. However, a small
increase in absorbance for $WSe_{2}$ films with thicknesses $\geq~{}150~{}nm$,
depicts the presence of dual bandgap. From the Tauc fitting expression given
by [46, 47],

$\alpha h\nu=A(h\nu-E_{g})^{n}$ (1)

where $\alpha$, h$\nu$, A and $E_{g}$ are the linear absorption coefficient,
photon energy, a constant and the bandgap respectively, we obtain the bandgap
of $WSe_{2}$ films by extrapolating the linear part of the plot between
$(\alpha h\nu)^{1/2}$ and $h\nu$ (see inset of Fig.4(a)). The variation of
bandgap(s) as a function of $WSe_{2}$ film thickness is shown in Fig. 4(b)
which distinctly depicts the existence of dual bandgap arising due to the
presence of defects in the films. From one perspective, the appearance of mid-
gap states (or bands) is attributed to the $Se$ (lighter atom) deficiency in
the thin-films grown using RF-sputtering technique. In other words, the
deposition method introduces stoichiometric defects (absence of $Se$ atoms)
which is distributed randomly throughout the thin-film architecture.
In order to gain a deeper insight, we simulated the electronic bandstructure
for a $WSe_{2}$ crystal using density functional theory (DFT) which utilizes
the Vienna ab-initio simulation package (VASP) code [48]. The interaction
between electrons and ions is introduced by using Projector-Augmented Wave
(PAW) method [49]. In these calculations, the structural relaxation is
accomplished by deploying generalized gradient approximation (GGA) scheme for
adopting the exchange–correlation potential with the Perdew, Burke, and
Ernzerhof (PBE) functional [50]. A cut-off energy of $350~{}eV$ is fixed for
plane-wave basis set for all the calculations. The Monkhorst–Pack method [51]
are redacted to generate the $5\times 5\times 5$ $k$-point meshes for
achieving the required convergence within $10^{-6}~{}eV$ per atom. When the
forces reached smaller than the value of $0.01$ $eV/\AA$ for each atom, the
relaxation is assumed to be completed. The band structure and density of
states (DOS) estimations are obtained by considering the geometry which
results in most stable configuration and we obtain desirable convergence using
the linear tetrahedron method with Bloch corrections [52]. Figures 5(a) shows
the simulated bandstructure for $WSe_{2}$ which distinctly depicts a bandgap
of $1.4~{}eV$ at the high-symmetry $K$-point in the Brilliouin zone. Also, it
is worth mentioning that the energy difference between the conduction band
(yellow solid curve) and the valence band (blue solid curve) at other high
symmetry points ($\Gamma$ and $M$) are greater than that at the $K$-point
which is expected for a hexagonal lattice. At this point, we introduce a $Se$
defect (absence) from a supercell comprising of 16 $W$ atoms. This results in
a bandstructure shown in Fig. 5(b) where the defect band is separated from the
valence band by about $0.7~{}eV$ at the $K$-point. Also, it is apparent that
the mid-gap defect band is degenerate at the $\Gamma$-point
($\approx~{}1.4~{}eV$) and varies between $0.7~{}eV$ to $0.8~{}eV$ between
$M$-point and $K$-point in the Brilliouin zone. It is worth recalling that the
bandgap estimation of $WSe_{2}$ films using the UV-visible absorption spectrum
in Fig. 4(b) which shows the existence of a defect band and it varies between
$0.8-1.0~{}eV$ for film thicknesses $\geq~{}150~{}nm$. Within the
computational limitation of exactly positioning and identifying the defect
sites, the numerically simulated bandstructure provides a plausible
explanation for the absorption characteristics and existence of dual bandgap
in $WSe_{2}$ thin films. The presence of the defect band (due to $Se$
deficiency) has a direct impact on the NLO properties of $WSe_{2}$ films which
is discussed below.

We measured the Z-scan OA transmission of $WSe_{2}$ thin films at an
excitation wavelength of $515~{}nm$ and a peak on-axis intensity of
$I_{0}\approx 3.20\times 10^{11}$ W/cm2 at the focal point. The OA normalized
transmittance is shown in Figs. 6(a)-6(f) respectively. It could be observed
from the figures that the OA normalized transmittance exhibited a maxima at
the focal point of the lens ($z=0$) where the laser intensity is maximum. This
is distinct signature of saturable absorption (SA) behaviour. The normalized
transmittance were theoretically fitted using the analytical expression given
by [53, 37],

$\Delta T=1-\frac{\beta I_{0}L_{eff}}{2^{3/2}(x^{2}+1)}$ (2)

where $\beta$ is the nonlinear absorption coefficient, $L_{eff}$ is the
effective length of the sample which is defined as $L_{eff}$ = (1 -
$e^{-\alpha L})/\alpha$, $L$ is the thickness of the films, $\alpha$ is the
linear absorption of the film, $x=z/z_{0}$ where $z_{0}$ is Rayleigh length
for the laser beam. The best theoretical fit to the experimental measurements
are shown by solid (black) lines in Fig. 6 which yields a negative value of
$\beta$. This is expected from the samples which exhibit a saturable
absorption behaviour. The imaginary component of $\chi^{(3)}$, namely
$\chi_{Im}^{(3)}$ could be obtained using [33]

$\chi_{Im}^{(3)}=\left[\frac{10^{-7}cn^{2}\lambda}{96\pi^{2}}\right]\beta$ (3)

where $c$ is the light speed in vacuum, $n$ is the refractive index of the
thin films and $\chi_{Im}^{(3)}$ is expressed in _esu_ units. The estimated
values of $\beta$ and $\chi_{Im}^{(3)}$ are given in Table-1.

Figure 6: Open aperture (OA) Z-scan normalized transmittance for $WSe_{2}$
thin films of thicknesses $34~{}nm$ (a), $81~{}nm$ (b), $130~{}nm$ (c),
$150~{}nm$ (d), $242~{}nm$ (e) and $326~{}nm$ (f). Red dots represent the
experimentally measured values and black solid curve represent the theoretical
fitting (solid lines). Figure 7: A schematic to show saturable absorption
mechanism of $WSe_{2}$ thin films for demonstrating the Pauli-blocking
principle. (a) shows the inter-band transition process. (b) shows the process
of thermalization of hot electrons and obtaining a thermal equilibrium. (c)
shows the filled states near the band edge which restricts the absorption of
light.

In order to eliminate the impact of linear absorption ($\alpha$) on the
nonlinear absorption ($\beta$), we define a figure of merit (FOM) as FOM =
$|\chi_{Im}^{(3)}/\alpha|$ which is also mentioned in Table-1. As expected, it
could be noted that the linear absorption monotonically increases from
$4.25~{}cm^{-1}$ to $19.07~{}cm^{-1}$ as a function of increasing $WSe_{2}$
film thickness. Also, $\beta$ exhibits an approximately $25$-fold increase
when film thickness increases from $34~{}nm$ to $326~{}nm$. Except a small
irregular behaviour for $130~{}nm$ thin-film, the FOM also exhibits a
increasing trend with the film thickness. Consequent upon this observation, it
could be safely inferred that the $WSe_{2}$ thin-films exhibit a SA behaviour
at an excitation wavelength of $515~{}nm$ which tends to increase with film
thickness. Although not shown here, the OA Z-scan measurements show a similar
SA behaviour for all excitation wavelengths in the visible band of the
electromagnetic spectrum.
In order to get a deeper insight into the SA characteristics exhibited by
$WSe_{2}$ thin-films, we note that the primary bandgap ($E_{g}$) for most of
the $WSe_{2}$ films are in $1.2-1.4~{}eV$ range (see Fig. 4(b)). Therefore,
one-photon absorption is expected to be the most dominant absorption process
for $515~{}nm$ wavelength excitation [54]. A schematic viewgraph in Fig. 7(a)
shows that the incident wave at $515~{}nm$ wavelength leads to the excitation
of electrons from valence-band (VB) to the conduction-band (CB) in $WSe_{2}$
thin films. When the laser intensity is small (at $|z|>>0$), the intra-band
transitions in CB quickly attains a distribution dictated by Fermi–Dirac
statistics for the newly-generated electron-hole pairs. In other words, the
electron-hole recombination yields an equilibrium state via intra-band phonon
scattering and the electrons (in CB) as well as the holes (in VB) exhibits a
distribution as shown in Fig. 7(b). On the other hand, the incident laser
intensity is high around the focus ($z=0$), which leads to significant
increase in the number of charge carriers in respective bands. In the thermal
equilibrium, the states near the bottom of CB (for electrons) and those near
the top of VB (for holes) are completely occupied. This also manifests into a
densely populated states (within the band) which are involved in the _two-
photon_ (nonlinear) absorption process (see Fig. 7(c)). Consequently, the
_two-photon_ transition probability reduce as a result of non-availability of
states in the relevant CB which is alternately termed as _Pauli-blocking
effect_ [55, 56]. Due to the weak _two-photon_ absorption process, the
transmission of the films (near the focus) is more than that for regions far-
from-focus which could be observed in Fig. 6. In case of $WSe_{2}$ thin-films,
the OA transmission peak at $z=0$ is higher for thicker films and therefore,
it could be inferred that the Pauli-blocking effect is more dominant in
thicker films. Further, the bandstructure in Fig. 5(a) depicts that the Pauli-
blocking effect is expected to weaken when the _one-photon_ (or linear)
absorption is small. In order to appreciate this point, we measured OA Z-scan
transmittance using ultrashort pulses (of $\approx 370~{}fs$) at excitation
wavelengths of $1030~{}nm$ and $1520~{}nm$. A representative measurement for
two $WSe_{2}$ thin-films (thickness of $130~{}nm$ and $242~{}nm$) are shown in
Fig. 8(a) and (b) which depicts a distinct transmission minima at $z=0$. The
solid curves are theoretical fit to experimental measurements (triangular and
rectangular dots). Such an OA transmittance is a signature of reverse
saturable absorption (RSA) which could be a consequence of _two-photon_
absorption (TPA) in $WSe_{2}$ thin-films. All the other $WSe_{2}$ thin-films
also exhibit similar RSA signature. It is interesting to observe that the RSA
behaviour is stronger at $1520~{}nm$ as compared to that at $1030~{}nm$
wavelength for both the $WSe_{2}$ films shown in Fig. 8. For $WSe_{2}$ film of
thickness $130~{}nm$, the absence of dual bandgap (or absence of band due $Se$
deficiency) results in a stronger RSA behaviour in comparison with that
exhibited by a $242~{}nm$ $WSe_{2}$ film which has a distinct dual bandgap. In
order to get a deeper insight, we note that the linear absorption in $WSe_{2}$
thin-films at $1030~{}nm$ and $1520~{}nm$ are relatively small which could be
observed in Fig. 4(a). This reduces charge-carrier build-up near the band
edges which is primarily responsible for Pauli-blocking effect (see Fig.
7(c)). Consequently, the energy eigenstates, which are responsible for TPA
process), are likely to have a sparse population density. The TPA probability,
therefore, increases which yields OA transmittance signature as shown in Fig.
8(a)-(b). It is apparent from the aforementioned argument that the Pauli-
blocking effect would be relatively weak when the linear absorption is small.
In consonance with this assertion, we note that the linear absorption of
$WSe_{2}$ is $\alpha=2.72~{}cm^{-1}$ at $1520~{}nm$ excitation wavelength and
$\alpha=4.48~{}cm^{-1}$ at $1030~{}nm$ excitation wavelength. Consequently,
the OA transmittance minima (at $z=0$) (in Fig. 8(a)-(b)) for $1520~{}nm$
excitation wavelength (red-solid curve) has a significantly greater magnitude
than that observed in case of $1030~{}nm$ excitation wavelength.

Figure 8: OA Z-scan normalized transmittance curves for (a) $130~{}nm$ (b) $242~{}nm$ thick $WSe_{2}$ films at $1030~{}nm$ (blue) and $1520~{}nm$ (red). Table 1: Nonlinear absorption and nonlinear refraction coefficients of the $WSe_{2}$ samples at 515 nm excitation wavelength. Sample | $T_{0}$ | $\alpha$ | $\beta\times 10^{-11}$ | $n_{2}\times 10^{-16}$ | $\chi_{Im}^{(3)}\times 10^{-14}$ | $\chi_{R}^{(3)}\times 10^{-16}$ | FOM$\times 10^{-15}$
---|---|---|---|---|---|---|---
(nm) | $(\%)$ | $(cm^{-1})$ | $(cm/W)$ | $(cm^{2}/W)$ | (esu) | (esu) | (esu cm)
34 | $65.40$ | $4.25$ | $-3.01$ | $2.81$ | $-1.39$ | $0.25$ | $3.27$
81 | $57.76$ | $5.49$ | $-4.37$ | $3.50$ | $-2.13$ | $0.56$ | $3.88$
130 | $43.65$ | $8.29$ | $-11.04$ | $17.96$ | $-2.47$ | $1.31$ | $2.98$
150 | $36.70$ | $10.02$ | $-14.64$ | $17.98$ | $-5.44$ | $2.18$ | $5.43$
242 | $28.27$ | $12.63$ | $-28.03$ | $34.79$ | $-11.13$ | $4.49$ | $8.81$
326 | $14.84$ | $19.07$ | $-76.15$ | $59.92$ | $-33.28$ | $8.53$ | $17.45$
Table 2: Nonlinear absorption coefficients of the $WSe_{2}$ thin films at 1030 nm and 1520 nm excitation wavelength. Laser | Sample | $\alpha_{0}$ | $\beta\times 10^{-11}$ | FOM$\times 10^{-14}$
---|---|---|---|---
| (nm) | $(cm^{-1})$ | $(cm/W)$ | (esu cm)
1030 nm | 130 | $4.48$ | $7.50$ | $3.64$
| 242 | $2.72$ | $2.38$ | $0.76$
1520 nm | 130 | $3.24$ | $58.94$ | $33.29$
| 242 | $4.17$ | $11.02$ | $3.45$
Figure 9: Variation of nonlinear absorption (saturable absorption) coefficient
$\beta$ as a function of thickness.

At this point, it would be interesting to observe the dependence of nonlinear
absorption coefficient ($\beta$) on $WSe_{2}$ film thickness at $515~{}nm$
excitation wavelength. This is shown in Fig. 9 which depicts an improvement in
SA characteristics as a function of thin-film thickness. In order to gain a
physical insight, we note that the nonlinear absorption exhibits a direct
correlation with bandgap ($E_{g}$) of the films under investigation. Also, a
thicker film is expected to have a higher carrier density owing to denser
stacking of $WSe_{2}$ monolayers. This essentially leads to high occupation of
states around the band-edges at high laser intensity (high photon flux),
thereby creating a situation for dominant impact of Pauli-blocking. This also
manifests into an weaker absorption at high photon-flux (high intensity) which
leads to a higher transmission peak at $z~{}=~{}0$ in Fig. 6 for thicker
$WSe_{2}$ films. Alternately, this results in stronger saturable absorption
features in thicker $WSe_{2}$ films as depicted in Fig. 9. It could also be
observed that the enhancement in SA behaviour increases significantly for
$WSe_{2}$ film thickness $\geq~{}150~{}nm$. A natural consequence of denser
stacking of $WSe_{2}$ monolayers manifests in the form of defects whose impact
was discussed in the context of dual-bandgap in Fig. 4(b). At $515~{}nm$
excitation wavelength, the defect band provide a more favorable situation for
charge carrier accumulation near the band edges, thereby fostering the
possibility of a stronger impact of Pauli-blocking effect and improved SA
characteristics. Further the presence defect band also alters the RSA features
at $1030~{}nm$ (($0.82~{}eV$)) excitation wavelength which could be observed
in Fig. 8(a) and (b). The RSA behaviour (blue curve/dots) weakens in the
$WSe_{2}$ film of $242~{}nm$ thickness as compared to that exhibited by
$130~{}nm$ $WSe_{2}$ film. This is essentially brought about by a stronger
Pauli-blocking effect induced by the presence of defect band. At $1520~{}nm$
($0.82~{}eV$) excitation wavelength, the impact of defect band (in $242~{}nm$
$WSe_{2}$ film) on the RSA characteristics weaken essentially due to closer
spacing of the defect band and the CB (see Fig. 4(b).

It is apparent that the nonlinear absorption properties of $WSe_{2}$ thin
films exhibit strong spectral sensitivity. In the near-infrared (NIR) spectral
band, $WSe_{2}$ films could play a crucial role in optical power limiting
applications as a consequence of prominent intensity dependent absorption
characteristics. For example, an optimally thick $WSe_{2}$ film could be
employed as intensity (or amplitude) modulator in the NIR spectral band and
therefore, they are highly promising candidates for developing monolithically-
integrated miniaturized high-speed ($\sim~{}GHz$ level) modulators for optical
communication. On the other hand, in the visible domain Pauli-blocking induced
strong saturable absorption characteristics in $WSe_{2}$ films allow a deep
signal modulation which facilitate desirable shaping of optical pulses. In
fact, the depth of modulation for pulse generation as well as shaping is
directly proportional to $|\beta|$ [57] and consequently, $WSe_{2}$ thin-films
exhibiting strong Pauli-blocking induced SA behaviour would serve the purpose
very efficiently. It has been demonstrated that stable high-energy nanosecond
pulses could be generated by employing a few layer $WSe_{2}$ sheets (acting as
saturable absorber) through passive Q-switching in $Pr^{3+}$-doped ZBLAN fiber
laser emitting at $635~{}nm$ wavelength [58]. In a similar investigation, B.
Chen $et~{}al.$ have employed an arbitrarily thick $WSe_{2}$ film as a
saturable absorber for generating Q-switched nanosecond pulses in erbium doped
fiber laser [59]. This idea has been extended for generating ultrashort pulses
through passive mode-locking technique as well [60]. In a recent
investigation, a few layer alternating $MoSe_{2}$ and $WSe_{2}$ nano-sheets
have shown generation of stable ultrashort pulses of a few picoseconds in the
NIR band [61]. In fact, the broadband SA characteristics exhibited by
$WSe_{2}$ films give rise to the possibility of developing highly-efficient
wavelength-tunable pulse-shaping components in the visible as well as NIR
spectrum [62, 25]. In order to compress pulses to very low time-scales (say
$\leq~{}100~{}fs$), it is essential to proportionally scale-up the nonlinear
absorption ($\beta$) of $WSe_{2}$ sheets. This flexibility, in general, is
limited in all the aforementioned illustrations which is primarily due to the
presence of a few monolayers of $WSe_{2}$. The RF-sputtered thin-films, which
are experimentally simpler to fabricate, provide a plausible route to tailor
$\beta$. As shown here, thicker films exhibit larger $|\beta|$ and
consequently, they could be employed situations requiring deeper modulation.

Figure 10: Closed aperture (CA) Z-scan normalized transmittance at $515~{}nm$
wavelength for $WSe_{2}$ thin films of thicknesses $34~{}nm$ (a) – $326~{}nm$
(f) respectively.

The present investigation also includes the measurements on nonlinear
refractive index ($n_{2}$) using CA Z-scan transmittance. The measured CA
normalized transmittance for all the $WSe_{2}$ thin films at $515~{}nm$
excitation wavelength is shown in Figs. 10(a)-(f) (red dots). The CA
transmittance exhibits a pre-focal minima followed by a post-focal maxima
which is a distinct signature of positive $n_{2}$ values for all the $WSe_{2}$
thin-films [21]. However, the asymmetry in the peak-to-valley transmittance is
essentially brought about by a strong SA characteristics exhibited by all the
thin-films at $515~{}nm$ excitation wavelength. In order to estimate $n_{2}$,
the normalized CA transmittance curves were fitted using the relation [63, 53,
64]

$\Delta
T=1-\frac{4x\Delta\phi_{0}}{(x^{2}+9)(x^{2}+1)}-\frac{2(x^{2}+3)\Delta\Psi_{0}}{(x^{2}+1)(x^{2}+9)}$
(4)

where $\Delta\phi_{0}=kn_{2}I_{0}L_{eff}$ is the phase change due to nonlinear
refraction at the focus, k=$2\pi/\lambda$ is propagation wavevector and
$\Delta\Psi_{0}=\beta I_{0}L_{eff}/2$ is phase change due to nonlinear
absorption. The real part of third order nonlinear susceptibility
($\chi^{(3)}$) and $n_{2}$ exhibit a relation given by [33]

$\chi_{R}^{(3)}=\left[\frac{n^{2}c}{12\pi^{2}}\right]n_{2}$ (5)

The values of $n_{2}$ and $\chi_{R}^{(3)}$ are estimated by fitting (solid
black curve in Fig. 10) the experimental measurements using Eq. (4) and
represented in Table-1. In case of thinnest film ($34~{}nm$) thickness,
$n_{2}$ was estimated to be $2.81\times 10^{-16}$ cm2/W whereas the thickest
film ($326~{}nm$ thickness) had $n_{2}=59.92\times 10^{-16}$ $cm^{2}/W$ which
is approximately a 21-fold increase.

##  IV CONCLUSION

In summary, we investigated the nonlinear optical properties of $WSe_{2}$ thin
films using single-beam Z-scan technique at $515~{}nm$, $1030~{}nm$ and
$1520~{}nm$ excitation wavelengths. The $WSe_{2}$ thin films exhibits a SA
behaviour and a self-focusing characteristics at $515~{}nm$ excitation
wavelength. The magnitude of nonlinear absorption tends to increase
for.thicker $WSe_{2}$ films. The same $WSe_{2}$ films exhibit RSA
characteristics at both $1030~{}nm$ and $1520~{}nm$ wavelengths. The DFT based
theoretical modeling provides a plausible route to obtain deeper physical
insight into the underlying mechanism for change in the sign of the nonlinear
absorption coefficient ($\beta$). In addition, the spectral sensitivity of
$\beta$ could be correlated with the electronic bandgap and therefore, it
could be tailored by choosing an appropriate film thickness. The $WSe_{2}$
thin-films of optimum thickness could, therefore, find applications in
realizing devices which require a varying degree of optical modulation during
operation.

##  V ACKNOWLEDGMENTS

We thank the Department of Atomic Energy (DAE), Government of India for
funding the research work through the project number _RIN-4001_. The authors
would like to thank Mrinal K. Sikdar for his help during thin film deposition.

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