8822 | J. Mater. Chem. C, 2016, 4, 8822--8828 This journal is
©
The Royal Society of Chemistry 2016
Cite this: J. Mater. Chem. C, 2016,
4, 8822
Excitonic quantum confinement modified optical
conductivity of monolayer and few-layered MoS
2
†
Guang Yi Jia,*
ab
Yue Liu,
a
Jing Yu Gong,
a
Dang Yuan Lei,*
b
Dan Li Wang
a
and
Zhen Xian Huang
a
Optical conductivity plays an important role in characterizing the optoelectronic properties of two-
dimensional materials. Here we derive the complex optical conductivities for monolayer and few-layered
MoS
2
films from their reflectance and transmittance responses. We show that the excitonic quantum
confinement effect significantly modifies both the peak energy and magnitude of their optical conductivity,
manifested by a gradual blueshift in energy (consistent with two well-known models for quantum well
systems) and exponential attenuation in magnitude with decreasing layer number. More importantly, the
C excition induced optical conductivity peak exhibits the strongest dependence on the MoS
2
layer number
because of its largest Bohr radius among the A, B and C excitons. This unambiguously confirms the strong
influence of quant um conf inement e ffect in the optical conductivity of MoS
2
, shedding important insights
into understanding its rich exciton-related optical properties and therefore facilitating potential applications
in optoelectronic devices.
Introduction
In recent years, two-dimensional (2D) transition metal dichalco-
genide (TMDC) materials with the generalized formula MX
2
(M = Mo, W; X = S, Se, and Te) have received burgeoning research
interest due to their peculiar physicochemical properties.
1–5
Their electronic states generally are subject to strong interlayer
coupling and undergo transitions from the indirect bandgap in
the bulk form to the direct bandgap in monolayers. In particular,
the direct bandgap energy in many monolayer TMDCs lies in the
visible and near infrared range, making them ideal candidates for
2D optoelectronic applications.
1,2
Among various TMDCs, MoS
2
is
a widely-studied example, which consists of a sandwich structure
of S–Mo–S in each layer.
2
Few-layered MoS
2
is an excellent light
absorber, with the absorption spectrum generally composed of
three characteristic peaks due to excitonic resonance and inter-
band transitions.
6
In addition, with the abundance of molybdenite
in nature, MoS
2
is chemically more stable and relatively cheaper
than other TMDCs. Previous studies have proven that monolayer
MoS
2
-based photodetectors and phototransistors exhibit high
photo-responsivity and a large photo-thermoelectric effect.
7,8
Few-layered MoS
2
with thickness-dependent bandgap energy is
also promising for fiber lasers, solar cells, optical lenses, gratings,
etc.
2,9–11
In all the aforementioned optoelectronic applications,
the optical conductivity of MoS
2
plays a key role in characterizing
the electronic states of the system and links the current density to
an externally applied electric field.
12
In a typical 2D TMDC material, optical conductivity stems
from interband transitions due to electron–photon interaction.
When the material absorbs one photon, the generated electron
and hole will propagate with the same velocity amplitude but
opposite direction, resulting in the so called band nesting.
13
In
the nesting region of the band structure, the conduction and
valence bands are parallel to each other in energy. The band
nesting gives rise to a singularity in the joint density of states,
producing a greatly enhanced optical conductivity.
6,13
Recently,
Wang et al. have successfully measured the photoexcited carrier
lifetimes in monolayer and few-layered MoS
2
flakes on the basis of
frequency-dependent optical conductivity, without carrying out a
systematic analysis of the thickness-dependent optical conductivity
of MoS
2
.
14,15
Apart from the reports by Wang et al.,otherresearch
studies on investigating the complex optical conductivity of MoS
2
are mainly limited to monolayers.
16–18
Thus far, the relationship
between complex optical conductivity and MoS
2
film thickness
has not yet been clearly understood. In particular, the detailed
influence of the A, B and C excitons on the optical conductivities
of monolayer and few-layered MoS
2
remains elusive.
In the present work, we calculate the reflectance and trans-
mittance spectra for MoS
2
with the number of layers varying from
one (1L) to ten (10L) and subsequently derive the corresponding
a
School of Science, Tianjin University of Commerce, Tianjin 300134, PR China.
E-mail: gyjia8 7@163.com
b
Department of Applied Physics, The Hong Kong Polytechnic University, Hong Kong,
PR China. E-mail: dylei@polyu.edu.hk
† Electronic supplementary information (ESI) available: Experimentally measured
complex optical conductivity of a monolayer MoS
2
film, and calculated thickness-
dependent reflectance intensity and magnitude of optical conductivity without
considering the excitonic effect. See DOI: 10.1039/c6tc02502a
Received 17th June 2016,
Accepted 26th August 2016
DOI: 10.1039/c6tc02502a
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complex optical conductivity and compare the results with the
extracted excitonic resonance energies. Our results reveal that
the quantum confinement effect in the excitonic resonance has a
significant impact on the corresponding induced optical con-
ductivity of MoS
2
, manifested by different thickness dependences
of peak energy and magnitude of optical conductivity peaks
associated with different excitonic resonances.
Model system and the
theoretical method
As shown in Fig. 1, our model system consists of a MoS
2
film
of thickness L deposited on a semi-infinite sapphire substrate.
Yu et al. systematically measured the refractive index n and the
extinction coefficient k of atomically thin MoS
2
films. They
found that both values of n and k are dependent on the film
thickness, with the results tabulated in their work and used in
our calculation.
19
In the structure of a multilayer MoS
2
film, the
Mo atoms of an upper layer locate right above the S atoms of a
lower layer, following an AB stacking sequence and having the
crystal lattice of bulk 2H-MoS
2
.
19,20
The sapphire substrate is
considered as a non-absorbing medium of refractive index
n
s
= B1.77 in the wavelength range of l = 400–900 nm.
21
The
upper medium of the model system is air and the incident light
is perpendicular to the sample surface. The reflectance R and
transmittance T of the MoS
2
film can then be calculated as
18,22
where
a =2pkL/l, g =2pnL/l
g
1
¼
n
2
n
s
2
þ k
2
n
s
þ nðÞ
2
þk
2
; h
1
¼
2n
s
k
n
s
þ nðÞ
2
þk
2
g
2
¼
1 n
2
k
2
1 þ nðÞ
2
þk
2
; h
2
¼
2k
1 þ nðÞ
2
þk
2
Note that an alternative approach to calculate the reflectance
and transmittance of the MoS
2
film involves the use of optical
conductivity s = s
1
+is
2
of MoS
2
. By Maxwell’s equations of the
system with appropriate electromagnetic boundary conditions,
R and T of the MoS
2
film can be expressed as functions of optical
conductivity
18,23
R ¼
n
0
n
s
s= e
0
cðÞjj
2
n
0
þ n
s
þ s
=
e
0
cðÞ
jj
2
(3)
T ¼
4n
0
n
s
n
0
þ n
s
þ s= e
0
cðÞjj
2
(4)
where n
0
stands for the refractive index of the incident medium
(here n
0
= 1.0 for air), e
0
and c are the free-space permittivity and
light velocity, respectively. As a result, one can calculate the
thickness-dependent reflectance and transmittance spectra of
the MoS
2
film by eqn (1) and (2). The real and imaginary parts
of MoS
2
optical conductivity can then be deduced via solving
eqn (3) and (4) with the calculated R and T. According to the
previous thickness measurement for few-layered MoS
2
using an
atomic force microscope,
19
the thicknesses of 1L, 2L, 3L, 4L, 5L,
6L, 7L, 8L, 9L and 10L MoS
2
films in our calculations are set as
0.65, 1.30, 1.90, 2.50, 3.10, 3.50, 4.30, 5.20, 5.60 and 6.20 nm,
respectively.
Results and discussion
Fig. 2a and b show the reflectance and transmittance spectra of
MoS
2
films with various layer numbers. According t o energy
conservation, the frequency-dependent absorption A of the MoS
2
films can also be calculated by R + T + A = 1. One can clearly
see that monolayer MoS
2
shows three peaks at 1.879, 2.016 and
2.867 eV in the reflectance spectra (corresponding to dips in the
transmittance spectra), which originate respectively from the
absorption of A, B and C excitons. The A and B excitonic peaks
come from the direct transitions from two valence bands (which
are split due to spin–orbit coupling) to the lowest conduction band
at the K(K
0
) points, while the C excitonic peak arises from the
indirect transition between the valence band maximum located at
the G point and the conduction band minimum located at the
L pointoftheBrillouinzone.
6,19
As the layer number increases,
the excitonic peaks gradually shift to lower energies. To clearly
illustrate the layer dependence, Fig. 3a plots the peak energies of
the A, B and C excitons as a function of the MoS
2
film thickness.
It is estimated that the energy difference between the A and B
excitonic energies is approximately 140 meV for all thicknesses,
being an indication of the strength of spin–orbit interaction.
Fig. 1 Schematic depiction of a MoS
2
film with the refractive index n and
the extinction coefficient k. The MoS
2
film is deposited on a semi-infinite
sapphire substrate.
R ¼
g
2
2
þ h
2
2
expð2aÞþ g
1
2
þ h
1
2
expð2aÞþ2 g
1
g
2
þ h
1
h
2
ðÞcos 2g þ 2 g
2
h
1
g
1
h
2
ðÞsin 2g
exp 2aðÞþg
1
2
þ h
1
2
ðÞg
2
2
þ h
2
2
ðÞexpð2aÞþ2 g
1
g
2
h
1
h
2
ðÞcos 2g þ 2 g
2
h
1
þ g
1
h
2
ðÞsin 2g
(1)
T ¼
n
s
1 þ g
1
ðÞ
2
þh
1
2
hi
1 þ g
2
ðÞ
2
þh
2
2
hi
exp 2aðÞþg
1
2
þ h
1
2
ðÞg
2
2
þ h
2
2
ðÞexpð2aÞþ2 g
1
g
2
h
1
h
2
ðÞcos 2g þ 2 g
2
h
1
þ g
1
h
2
ðÞsin 2g
(2)
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Fig. 2c and d present the real and imaginary parts of the
derived optical conductivities for the MoS
2
films with various
layer numbers. Possibly due to the band nesting effect as men-
tioned earlier,
6,13
the s
1
(s
2
) spectra exhibit peaks at energies
which are slightly larger (smaller) than the respective A, B and C
excitonic energies. From low to high energies, the three peaks in
s
1
(s
2
) spectra are labelled as s
1,A
, s
1,B
and s
1,C
(s
2,A
, s
2,B
and s
2,C
),
indicating that they are associated with the corresponding exci-
tonic transitions in MoS
2
, respectively. In our results, both s
1
and
s
2
are positive, consistent with that reported by Morozov et al.
18
To further confirm this observation, we measured the reflectance
and transmittance of a monolayer MoS
2
sample and derived its
complex optical conductivity which has positive numbers for
both s
1
and s
2
(see Fig. S1 in the ESI†). However, the sign of s
2
is contrary to the result reported by Wang et al.
15
The negative
s
2
observed by Wang et al. could partly be attributed to the
significantly decreased mobility of charge carriers, because
recombination of photoexcited carriers under ultrafast optical
pump–probe beams can increase the carrier effective mass and
the scattering probability.
Fig. 4a shows the peak energies induced by the A, B and C
excitons in the optical conductivity for the MoS
2
films with
various layer numbers. One can see that the peaks of s
1,C
and
s
2,C
gradually shift towards lower energies as the layer number
increases, which appears to mimic the C excitonic resonance
energies as observed in Fig. 3a. In particular, when the film
thickness is larger than 4L, the energy difference between s
1,C
(s
2,C
) and C excitonic resonance energy exhibits a nearly linear
dependence on L as shown in Fig. 4b. However, the peak
energies of s
1,A
and s
2,A
(s
1,B
and s
2,B
) do not exhibit a similar
blueshift with decreasing layer number down to 3L (2L) as shown
in Fig. 4a and Table 1. As a result, no such linear dependence of
energy difference between the optical conductivity peak and
exciton energy on L can be observed for either the A or the B
exciton for layer numbers 43L or 2L. These different effects of
the A, B and C excitons in the optical conductivity spectra could be
explained by the quantum confinement effect as discussed later.
In the following text, we first evaluate the evolution of
excitonic resonance peaks of MoS
2
with film thickness, and fit
the peak positions extracted from Fig. 2a by using the models of
infinite quantum wells and the quantum wells in fractional
dimensional space.
19,24,25
For the MoS
2
films thicker than 7L,
the excitonic peak position E
e
linearly depends on 1/L
2
as
presented in Fig. 3b. This matches well with the model of infinite
quantum wells, and E
e
can be fitted by E
e
= E
g
+ p
2
h
2
/2mL
2
,
where E
g
is the excitonic resonance energy in bulk MoS
2
flake
(E
g
= 1.812, 1.966 and 2.616 eV for A, B and C excitons,
respectively, according to Fig. 2a) and m is the reduced electron–
hole effective mass. The fitting results (dashed lines in Fig. 3b)
Fig. 2 (a) Reflectance and (b) transmittance spectra, (c) real and (d) imaginary
parts of the optical conductivities for MoS
2
films with different layer numbers.
Legends are given in (a) and (b). Optical conductivity spectra in the dashed box
are magnified by a factor of two.
Fig. 3 Peak energies of the A, B and C excitons extracted from the reflectance
spectra vs. (a) the MoS
2
film thick ness L and (b) 1/L
2
. Dashed lines in (a) are the
fitting results obtained by using eqn (5). Dashed lines in (b) are simulated by
using the model of infinite quantum wells.
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give E
e
=1.812+0.781/L
2
, 1.966 + 0.577/L
2
and 2.616 + 1.834/L
2
for
the A, B and C excitons, respectively. With these results we can derive
the reduced masses for the A, B and C excitons in MoS
2
films thicker
than 7L, which are m = 0.482m
0
,0.652m
0
and 0.205m
0
, respectively.
We can also derive the Bohr radii of excitons by using
a
B
= 0.0529m
0
e/m,wherem
0
is the free electron mass and e =10.71
is the static dielectric constant of bulk MoS
2
.
26
The Bohr radii of A,
B and C excitons are calculated to be 1.177, 0.869 and 2.763 nm,
respectively. Since in the literature it lacks a comprehensive
comparison about the physical parameters of the A, B and C
excitons in MoS
2
films, it may be difficult to thoroughly cross-
check all our calculated results. However, the derived m values
for the A and C excitons are reasonably in line with previously
reported results which are 0.42m
0
and 0.25m
0
for bulk MoS
2
,
respectively.
19,27
HerethederivedBohrradiiinthick/bulkMoS
2
are also comparable with the previous experimental results which
are B2.0 and B0.8 nm for the A and B excitons, respectively.
28,29
For MoS
2
films with layer numbers r7L, the excitonic peak
positions deviate from the model of infinite quantum wells (see
Fig. 3b). Instead, they could be fitted by the model of confined
quantum wells with
19,24,25
E
e
¼ E
g
þ R
b
þ
p
2
h
2
2mL
2
a 1ðÞ=2½
2
R
b
a 1ðÞ=2½
2
(5)
where R
b
= 13.6m/m
0
e
2
is the exciton binding energy in bulk
MoS
2
and a is a parameter to describe the dimensionality of the
confined exciton. By using eqn (5) to fit the excitonic peak
positions in Fig. 3a, we can get the effective dimensionality a
for the A, B and C excitons in 1L B 7L MoS
2
films, as shown in
Table 2. It is seen that the a value gradually increases from B2
for 1L to 3 for 8L, reasonably depicting the evolution of MoS
2
films from the 2D to 3D form. When a equals to 3 for layer
numbers Z8L, eqn (5) is naturally reduced to the equation for
infinite quantum wells.
Using only the imaginary part of the dielectric function of
MoS
2
,Yuet al. quantitatively fitted the peak position of the
C exciton and extracted the related physical parameters as a
function of the layer number.
19
Nevertheless, they encountered
bottleneck problems when dealing with the A and B excitons
due to the fact that the peak positions of the A and B excitons
exhibit no substantial layer dependence in the dielectric function.
Interestingly, by substituting their measured dielectric functions
into eqn (1), the calculated reflectance spectra (Fig. 2a) show that
all the peaks of the A, B and C excitons gradually shift to lower
energies with increasing layer number. Although the underlying
physics for this observation needs further investigation, our
calculation method can be used to extract the relevant physical
Fig. 4 (a) Excitonic resonance induced peak positions in optical conduc-
tivity spectra. Refer to the right (left) vertical axis for s
1,C
and s
2,C
(the others).
(b) Energy differences between optical conductivity peaks and respective
excitonic resonance energies. Dashed lines are just drawn as a guide to
the eye.
Table 1 Peak positions of optical conductivities in MoS
2
films with
different layer numbers
Layer
number
Peak position (eV)
s
1,A
s
2,A
s
1,B
s
2,B
s
1,C
s
2,C
1 1.884 1.852 2.028 1.981 2.894 2.693
2 1.878 1.833 2.018 1.968 2.841 2.622
3 1.877 1.829 2.024 1.965 2.818 2.605
4 1.877 1.829 2.026 1.965 2.812 2.583
5 1.863 1.825 2.013 1.967 2.790 2.582
6 1.879 1.833 2.028 1.968 2.787 2.581
7 1.870 1.828 2.024 1.967 2.785 2.565
8 1.873 1.825 2.021 1.960 2.783 2.559
9 1.869 1.825 2.021 1.962 2.777 2.549
10 1.863 1.825 2.020 1.962 2.771 2.551
Table 2 Effective dimensionalities a and Bohr radii a
B
of excitons in
different layer numbers of MoS
2
films
Layer
number
a a
B
(nm)
A exciton B exciton C exciton A exciton B exciton C exciton
1 1.845 1.955 1.649 0.497 0.415 0.897
2 2.182 2.303 2.005 0.695 0.566 1.389
3 2.420 2.524 2.237 0.836 0.662 1.709
4 2.615 2.695 2.510 0.950 0.737 2.086
5 2.720 2.822 2.692 1.012 0.792 2.338
6 2.806 2.895 2.780 1.063 0.824 2.459
7 2.939 2.980 2.935 1.141 0.861 2.673
Z8 3.000 3.000 3.000 1.177 0.869 2.763
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parameters of the A, B and C excitons, and open up the oppor-
tunity to have a comprehensive comparison between them. For
example, by using the a values given in Table 2, one can derive the
effective exciton binding energy from R
b
0
=4R
b
/(a 1)
2
.
25
Additionally, the effective Bohr radii of the excitons can also be
obtained by utilizing a
B
0
= a
B
(a 1)/2,
25
asshowninTable2.
Here we should point out that there are strong interface
couplings, e.g., van der Waals interaction, in the few-layered
MoS
2
and between the monolayer MoS
2
and the substrate.
Moreover, the functional form of electron–hole interaction could
be modified by the nonlocal nature of dielectric screening.
30,31
Thus the borrowing of eqn (5) which assumes a homogeneous
quantum well to 2D materials is essentially an approximation.
This may also be the reason why the values of a in Table 2 for the
three excitons of the monolayer MoS
2
are smaller than 2. By
using the dielectric screened hydrogen model,
30,31
the effective
binding energy of the C exciton in the monolayer MoS
2
is
calculated to be 0.72 eV. Substituting this value instead of R
b
0
into eqn (5), we deduce the value of a to be 1.934. It is clear that
the incorporation of a dielectric screening effect makes a more
reasonable value for a. In order to obtain an accurate dimen-
sionality, taking into account the interface couplings as well
as the effect of dielectric screening on the confinement energy
[the third term in eqn (5)] may be necessary. This may need more
complicated theoretical studies. Even so, we believe that the
simplicity is the merit of eqn (5). Besides, the model of confined
quantum wells leaves out many other influence factors, making
it feasible to solely inspect the quantum confinement effect.
Fig. 5 shows the extracted Bohr radii a
B
0
as a function of the
film thickness. One can see that the C exciton has the largest
Bohr radius while the B exciton gives the smallest size in the
same thickness of MoS
2
film. Furthermore, the Bohr radius of
the C exciton rapidly decreases with reducing thickness after
the film thins to 7L, whereas, the decreasing rates of the A and
B excitonic radii are relatively slower than C. In a semiconductor
crystal, an exciton will experience strong or weak quantum con-
finement when the semiconductor thickness is less or greater
than double the Bohr radius of the exciton.
32
The sizes of the A,
B and C excitons in the thick/bulk MoS
2
are 2a
B
= B2.35, B1.74
and B5.53 nm, respectively. They are close to the thicknesses
of 3L, 2L and 8L MoS
2
films, i.e., 1.90, 1.30 and 5.20 nm,
respectively. As a consequence, the A, B and C excitons in thin
MoS
2
films are expected to experience a strong quantum con-
finement effect. However, the C exciton is more tightly confined
in most of the films studied in our work while the A and B
excitons can be confined only in films thinner than 4L and 3L,
respectively. The variation in the C excitonic radius presents the
largest range as the layer number decreases to 1L (see Fig. 5),
implying that the excitonic quantum confinement plays the
strongest effect on the C exciton in comparison with the A and
B excitons. As a result, the peaks of s
1,C
and s
2,C
gradually shift
towards lower energies with increasing layer number (Fig. 4a)
and the energy difference between the s
1,C
(s
2,C
) peak and C
excitonic resonance energy linearly depends on the thickness
for films thicker than 4L (see Fig. 4b). In sharp contrast, the
peak positions of s
1,A
and s
2,A
(s
1,B
and s
2,B
) fluctuate with
the layer number when the MoS
2
film is thicker than 3L (2L),
as shown in Fig. 4. Note that the fluctuation of the energy
difference s
1,C,peak
C
peak
(or C
peak
s
2,C,peak
) in films thinner
than 5L could be a result of substrate-induced van der Waals
interaction and/or the local doping effect.
33,34
Previous studies
have shown that the energy variation in the excitonic peak
position induced by the influence of the substrate could be up
to B80 meV for a bilayer MoS
2
film, and this kind of influence
decreases significantly with increasing the layer number.
33,34
In addition to the effect in the peak position of MoS
2
optical
conductivity, our results further show that the excitonic quan-
tum confinement phenomenon also has a strong influence in
the excitonic peak intensity and the magnitudes of s
1,A
, s
1,B
and s
1,C
(s
2,A
, s
2,B
and s
2,C
). Intuitively, it is reasonable that the
reflectance intensity and the optical conductivity magnitude
decrease upon reducing the MoS
2
film thickness. By using the
measured dielectric function, e.g., the n and k values for bulk
MoS
2
, the calculation results (see Fig. S2 in the ESI†) show that
both reflectance intensity and optical conductivity decrease
Fig. 5 Bohr radii of A, B and C excitons vs. the MoS
2
film thickness. Dashed
lines are just drawn as a guide to the eye.
Fig. 6 (a) Reflectance intensities and (b) magnitudes of optical conductivity
induced by A, B and C excitons vs. the MoS
2
film thickness. Symbols are the
results extracted from Fig. 2. Dashed lines are the fitting results obtained by
using eqn (6).
Journal of Materials Chemistry C Paper
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