Optical Imaging and Characterization of Graphene and Other 2D Materials Using Quantitative Phase Microscopy

Abstract

This article introduces an optical microscopy technique for the characterization of two-dimensional (2D) materials. The technique is based on the use of quadriwave lateral shearing interferometry (QLSI), a quantitative phase imaging technique that allows the imaging of both the intensity and the phase of an incoming light beam. The article shows how QLSI can be used to (i) image 2D materials with high contrast on transparent substrates, (ii) detect the presence of residues coming from the fabrication process and (iii) map the 2D complex optical conductivity and complex refractive index by processing the intensity and phase images of a light beam crossing the 2D material of interest. To illustrate the versatility of this approach for 2D material imaging and characterization, measurements have been performed on graphene and MoS2.

Full text

Optical Imaging and Characterization of Graphene and Other 2D
Materials Using Quantitative Phase Microscopy
Samira Khadir,*
,†
Pierre Bon,
‡
Dominique Vignaud,
§
Elizabeth Galopin,
§
Niall McEvoy,
⊥,∥,¶
David McCloskey,
∥,¶,#
Serge Monneret,
†
and Guillaume Baffou*
,†
†
Institut Fresnel, CNRS, Aix Marseille Univ, Centrale Marseille, Marseille, France
‡
Laboratoire Photonique Numé
rique et Nanoscience (LP2N), CNRS UMR5298, Bordeaux University, Institut d’Optique Graduate
School, Talence 33405, France
§
IEMN, University of Lille, Avenue PoincaréCS 60069, Villeneuve d’Ascq Cedex 59652, France
⊥
School of Chemistry, Trinity College Dublin, Dublin 2, Ireland
∥
Centre for Research on Adaptive Nanostructures and Nanodevices (CRANN), Trinity College Dublin, Dublin 2, Ireland
¶
Advanced Materials and BioEngineering Research (AMBER), Trinity College Dublin, Dublin 2, Ireland
#
School of Physics, Trinity College Dublin, Dublin 2, Ireland
*
SSupporting Information
ABSTRACT: This article introduces an optical microscopy technique for the character-
ization of two-dimensional (2D) materials. The technique is based on the use of quadriwave
lateral shearing interferometry (QLSI), a quantitative phase imaging technique that allows
the imaging of both the intensity and the phase of an incoming light beam. The article shows
how QLSI can be used to (i) image 2D materials with high contrast on transparent
substrates, (ii) detect the presence of residues coming from the fabrication process, and (iii)
map the 2D complex optical conductivity and complex refractive index by processing the
intensity and phase images of a light beam crossing the 2D material of interest. To illustrate
the versatility of this approach for 2D material imaging and characterization, measurements
have been performed on graphene and MoS2.
KEYWORDS: wavefront sensing, phase imaging, 2D materials, graphene, MoS2, optical conductivity, refractive index
Graphene is a single-atom-thick, two-dimensional (2D)
material composed of carbon atoms arranged in a
honeycomb lattice structure. Due to its fascinating properties
including massless Dirac fermions,
1
anomalous quantum Hall
effect,
1,2
and high intrinsic strength,
3
graphene is already at the
basis of various promising applications, such as spintronics,
4
transparent electrodes for solar cells and light display
technologies,
5
supercapacitors,
6
among others.
7−9
Since the
first isolation of graphene in 2004,
10
a large variety of 2D
materials such as semiconducting transition metal dichalcoge-
nides (MoS2, MoSe2, WSe2, ...) have been synthesized. All these
2D materials are considered as promising candidates for
generating novel technological applications in various do-
mains.
11,12
The geometrical and optical characterizations of 2D materials
are not straightforward, as they are quite transparent and their
thickness does not exceed the nanometer scale. Several
methods have been used for determining the number of
graphene layers. For instance, atomic force microscopy (AFM)
is widely used to probe and image the thickness of 2D
materials.
10
However, this approach is delicate and time-
consuming. Raman spectroscopy is another widely used
technique to characterize graphene.
13
Although this technique
allows the identification of single-layer (SL) graphene from the
far field, it is not robust to identify the number of layers if
greater than 2. In addition, this method can be invasive, as it
requires the use of a focused laser beam that can sometimes
reach 1 mW.
14
Such a large power results in a large irradiance
(around 107mW·cm−2) that can induce the formation of
defects modifying the optical properties of the material.
14,15
Recently, other optical methods have been proposed to identify
the number of layers of 2D materials. Among them, one can
simply use an optical microscope and measure the image
contrast in reflection or in transmission.
16,17
But this method is
not reliable, as the contrast differs from one experiment to
another and the contrast is very weak in the case of SL
graphene on a transparent substrate.
18
SiO2-coated silicon
wafers are usually mandatory to make SL graphene visible.
19−21
The surface plasmon polariton (SPP) property of noble metals
Special Issue: 2D Materials for Nanophotonics
Received: July 28, 2017
Published: September 27, 2017
Article
pubs.acs.org/journal/apchd5
© 2017 American Chemical Society 3130 DOI: 10.1021/acsphotonics.7b00845
ACS Photonics 2017, 4, 3130−3139

has also been proposed as a means to determine the number of
graphene layers.
22
This method requires the deposition of a
thin layer of noble metal under the graphene sheets to generate
the SPP in the visible range. Additional technological steps are
needed, and the presence of the metal layer makes the sample
opaque and may modify the properties of the graphene
deposited on top.
In addition to their limitations cited above, these methods
only allow the identification of the number of layers of the 2D
material without giving information about the physical
properties of the studied material (complex refractive index,
dielectric function, etc.). Other techniques are generally
required to measure the optical response of 2D materials,
e.g., frequency-dependent transmittance and reflectance meas-
urements,
23,24
picometrology,
25
and spectroscopic ellipsome-
try.
26
Spectroscopic ellipsometry can also operate in an imaging
mode.
27,28
This approach is powerful to image, classify, and
determine the optical properties of 2D materials on the
microscale. However, this technique uses an unconventional,
specific tilted illumination that cannot be easily implemented in
standard microscopes. In addition, such an illumination limits
the lateral resolution to about 1 μm. In 2010, Wang et al.
studied SL graphene using spatial light interference microscopy
(a technique similar to that reported herein) but did not report
any quantification of the optical properties mainly due to
reconstruction artifacts that are inherent to this technique.
29
In this article, we introduce a simple, fast, and accurate
optical method based on high-resolution quantitative phase
microscopy to image and determine the number of layers and
characterize the optical properties of 2D materials. This
technique uses a wavefront-sensing camera based on quad-
riwave lateral shearing interferometry (QLSI). This camera is
mounted onto a lateral video port of an inverted transmission
microscope to recover simultaneously the phase and intensity
of an incoming light beam crossing the sample, using only one
measurement. In the first part, we introduce the basic principle
of QLSI. In the second part, we explain how QLSI
measurements can be processed to retrieve the complex
refractive index and the complex optical conductivity of 2D
materials. The third and last part of the article focuses on
experimental results. Results on single and multilayer graphene
are shown to explain how the measured QLSI phase image can
be used to (i) easily visualize transparent 2D materials, (ii)
determine the number of layers of 2D materials, (iii) evidence
the presence of a residual molecular layer, e.g., stemming from
the fabrication process, and (iv) retrieve the complex refractive
index and the complex optical conductivity of 2D materials.
Results on MoS2are subsequently presented to show the
versatility of the technique and to explain that caution has to be
used when working with 2D materials with a high refractive
index.
■QUADRIWAVE LATERAL SHEARING
INTERFEROMETRY
We introduce the basic principle of QLSI. We have chosen to
use this phase and intensity measurement technique, but other
techniques that quantitatively measure the scalar electro-
magnetic field in a nonbiased and halo-free manner could
also be applied for investigation of the optical properties of 2D
materials including, for example, digital holography and related
techniques
30−33
or external reference-free approaches.
34−36
Quadriwave lateral shearing interferometry is an optical
imaging technique capable of mapping both the intensity and
wavefront profiles of a light beam. A QLSI wavefront-sensing
camera consists of a modified Hartmann grating (MHG)
located just in front of a regular CCD camera.
37
The
interferogram produced by the MHG and recorded by the
CCD camera is subsequently processed to retrieve both the
intensity and the wavefront profiles of the incoming light.
38
As
this interferometric technique only involves a grating that is
fixed onto a CCD sensor, it does not require complicated and
sensitive alignments, like other interferometric techniques. The
interferometric nature of the measurements makes this
technique highly sensitive; QLSI can easily detect wavefront
distortions smaller than 1 nm. Our QLSI camera (Sid4Bio from
Figure 1. (a) Scheme of a QLSI microscope setup. A Köhler illumination is used to illuminate the sample with a light beam controlled in size and
numerical aperture. A band-pass filter is used to select the spectral range of interest. The sample is composed of microscale objects (2D materials in
the context of this article). The wavefront analyzer is constituted by a modified Hartmann grating (MHG) and a CCD camera. (b) Scheme of the
principle of the QLSI technique. A refractive object characterized by its refractive index ndeposited on a substrate having a refractive index n2and
surrounded by a medium of a refractive index n1causes a distortion of an incoming planar wavefront.
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Phasics SA) features a sensitivity of 0.3 nm·Hz−1/2. Note that
this is not a limitation. The signal-to-noise ratio could be
improved using more recent and sensitive cameras.
39
Originally introduced to characterize laser beams, our group
recently extended the scope of applications of QLSI by using
QLSI wavefront-sensing cameras on inverted optical micro-
scopes, where specific wavefront distortions are created by
micrometric objects located at the sample plane. For instance,
we have recently shown how QLSI cameras can be
implemented on optical microscopes to image single living
cells with high contrast
40
or microscale temperature profiles
with a sensitivity of around 1 K.
41
The experimental
configuration combining a QLSI camera and an optical
microscope is depicted in Figure 1a. In this configuration, a
Köhler illumination is used to illuminate the sample with a
controlled optical plane wave (controlled illuminated area and
controlled numerical aperture). In any measurement, a
reference image is first taken over a clear area (without any
object), prior to taking an image with the object of interest
within the field of view. Let us name t0the transmission of the
sample without any object and tthe transmission in the
presence of the object of interest. We define the normalized
complex transmission coefficient t/t0as
̲
̲=
φ
t
tTe
0
i
(1)
φπ
λδ=
lw
here 2
(2)
Tis the transmittance of the object, φis the phase retardation it
induces, and δlis the wavefront distortion caused by the object,
a physical quantity that is usually called the optical path
difference (OPD). In this article, underlined letters, such as t,
mean they are complex numbers. In this configuration, when
considering the simplest case of an imaged object featuring a
uniform refractive index n, embedded in a surrounding medium
of refractive index n1as shown in Figure 1b, the OPD reads in
first approximation (without taking into account multiple
reflections within the slab)
δ
=−lx y n n dx y(, ) ( ) (, )
1(3)
This simple expression is widely used in quantitative phase
imaging, for instance when studying biological samples.
39,42
However, we will explain in the next section that for some 2D
materials this simple expression may be inaccurate.
■QLSI APPLIED TO THE STUDY OF 2D MATERIALS
The scheme presented in the previous section suggests a
straightforward use of QLSI for the study of 2D materials and
the determination of their refractive index by using eq 3.
However, this equation may not be appropriate in the case of
2D materials. Its validity depends mostly on the refractive index
of the material and to a lesser extent on its thickness. For
materials with large refractive indices, the multiple reflections
inside the layer cannot be neglected anymore and the OPD is
no longer as simple as in eq 3. For instance, eq 3 leads to an
error in the determination of refractive index of 1% for
graphene and 40% for MoS2, as will be seen hereinafter. We
explain in the following how this equation has to be refined to
accurately apply to 2D materials.
Note that refractive indices, just like optical permittivities, are
defined only for bulk materials. For this reason, the
determination of the refractive index of a 2D material requires
the consideration of a 2D material as a slab with a defined
physical thickness. This can be relevant for thick 2D
materialssuch as multilayer graphene or MoS2but
questionable for graphene. The thickness of a graphene layer
is usually considered to be the interlayer distance of graphite.
However, this choice is arbitrary since a SL has no well-defined
boundaries (the electronic density is decaying exponentially
from the atomic plane). Thus, the refractive index can only be
defined as an effective quantity once an ef fective thickness of the
material is chosen. For this reason, the refractive index of
graphene will be noted neff in this article, corresponding to an
assumed graphene thickness of deff = 0.335 nm.
The fact that the refractive index is not a relevant quantity for
2D materials has already been noted in the literature.
43
This is
the reason that another quantity is often considered: the two-
dimensional optical conductivity σ2D. It is defined such that the
2D electronic current density J2D (charge per unit length and
time) within the 2D materials reads
σ
̲
=̲ ̲
J
E
2D 2D (4)
where Eis the electric field in the 2D material, in complex
notation. This definition considers a normal incidence of the
incoming light beam.
In the following, we explain how both the complex refractive
index and the complex 2D optical conductivity can by
measured by QLSI. Three models are described: the first
model focuses on the complex 2D optical conductivity, and
models 2 and 3 explain how the complex refractive index can be
Figure 2. Three descriptions to model the optical response of a 2D material. Model 1 considers the 2D material as an infinitely thin object
characterized by a 2D optical conductivity σ2D. The transmission coefficient tis the ratio between the transmitted electric field Etand the incident
electric field Ei. Models 2 and 3are equivalent. They are meant to model a 2D material as a slab of refractive index n, and they take into account the
occurrence of multiple reflections within the slab. In model 2, the transmission coefficient tis obtained as an infinite sum of terms stemming from
multiple reflections (ti). The transmission coefficient tin model 3 is the ratio between the transmitted electric field Etin the substrate and the
incident electric field Ei.
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retrieved. Albeit equivalent, models 2 and 3 are both important
to mention because they correspond to different physical
pictures of the underlying physics.
Model 1. In the first model, we characterize the 2D material
via its 2D optical conductivity σ2D. Here, the 2D material is
considered as an infinitely thin layer with surface charges that
interact with the incoming light. This picture is well adapted to
the study of an atom-thick 2D material such as graphene. We
consider that the 2D material is placed at the interface between
two semi-infinite media represented by their refractive indices
n1and n2. As presented in Figure 2, the propagation direction of
the incoming light is normal to the interface and oriented from
medium 1 (incident medium) to medium 2 (substrate). By
using Maxwell’s equations with the appropriate electromagnetic
boundary conditions, one can obtain the complex transmission
coefficient tas a function of the complex 2D optical
conductivity σ2D as follows:
̲
=
++
σ
ε
̲
tn
nn
2
c
1
12
2D
0
(5)
This expression is derived in the Supporting Information. By
normalizing this quantity by the transmission coefficient in the
absence of the 2D material =+
t
n
nn
0
21
12
, one obtains
̲=
+σ
ε
̲
+
t
t
1
1nn c
0()
2D
120 (6)
Using eq 1, one can derive the expressions of the real σrand
imaginary σiparts of σ2D as a function of the measured
quantities by QLSI, T, and δl:
σε π
λδ=+ −
⎜⎟
⎡
⎣
⎢⎛
⎝
⎞
⎠
⎤
⎦
⎥
cn n Tl()
1cos 21
r012 (7)
σε π
λδ=−+
⎜⎟
⎛
⎝
⎞
⎠
cn n Tl()
1sin 2
i012 (8)
Models 2 and 3. We consider now the 2D material as a slab
with a physical thickness dand characterized by a complex
refractive index n=n+iκ, where nis the refractive index and κ
the extinction coefficient. To refine eq 3, we now take into
account the multiple reflections of the incident light within the
slab. For this purpose, two different models corresponding to
two physical pictures can be considered (models 2 and 3 in
Figure 2). Although these two models lead to exactly the same
expression of the normalized transmission coefficient, we
believe it is worth mentioning them both for the sake of
comprehensiveness. Albeit equivalent, these two models
correspond to totally different physical pictures that are both
valuable.
Model 2 consists in writing the transmitted electric field as an
infinite sum of terms stemming from the multiple reflections. A
fraction of the transmitted light from medium 1 to the 2D
material layer undergoes multiple reflections inside the layer.
For each round-trip, a fraction of the light is transmitted to the
third medium. The total transmission coefficient is obtained by
the interference of all the transmitted beams to the third
medium. The result of the calculation is given by eq 9, and the
derivation of this expression is given in the Supporting
Information.
Model 3 considers that each of the three media contains a
planar electromagnetic wave (see Figure 2). In the first
medium, the electromagnetic wave is decomposed into an
incident and a reflected wave. The second medium (the 2D
material layer) contains a transmitted and a reflected wave. The
third medium contains only a transmitted wave. Using the
Maxwell−Faraday equation for each medium and considering
the appropriate electromagnetic boundary conditions, one can
obtain the total transmission coefficient of the system (see eq
9). The derivation is given in the Supporting Information.
Interestingly, this approach amounts to taking into account
multiple reflections.
The expression of the complex transmission coefficient as a
function of n1,n2, and nobtained from models 2 and 3 reads
̲
=̲
̲+̲+−
̲−̲−
̲
̲
tnn
nnnn nnnn
4e
()()()( )e
kdn
kdn
1
i
12 12
i2
0
0
(9)
where =π
λ
k0
2is the wavevector of the light in vacuum and dis
the thickness of the 2D material. By normalizing this quantity
by the transmission coefficient in the absence of the 2D
material (i. e., n=n1), which reads
̲
=
+
te
n
nn
kdn
0
2i
1
12
0
1
, we get
̲
̲=̲+
̲+̲+−
̲−̲−
̲−
̲
t
t
nn n
nnnn nnnn
2( )e
()()()( )e
kd n n
kdn
0
12
i( )
12 12
i2
01
0
(10)
The complex quantity t/t0can be determined by QLSI from the
measurements of Tand δl, following eq 1. Thus, the only
unknown in eq 10 is n, the physical quantity of interest.
However, unlike eq 6, which can be easily inverted to express
σ2D,eq 10 cannot be simply inverted to express nas a function
of t/t0. Hence, eq 10 has to be solved numerically using an
inversion algorithm. In our study, we used the function
FindRoot of Wolfram Mathematica. The corresponding
Mathematica notebook is provided in the Supporting
Information.
Noteworthily, models 2 and 3 are relatively simple when
dealing with two interfaces, which is the case in this work.
However, these models become tedious when dealing with a
larger number of interfaces, i.e., constituted by a stack of layers
having different refractive indices. In this case, another model,
named the transfer matrix formalism, is more suited to calculate
the transmission coefficient of the system.
44,45
■EXPERIMENTAL RESULTS AND DISCUSSION
This section is devoted to the study of graphene and MoS2by
the QLSI technique. For clarity, we show in Figure 3 the atomic
structures of graphene (right) and MoS2(left). Graphene is
Figure 3. 3D scheme of the atomic structures of graphene (right) and
MoS2(left).
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composed of a single layer of carbon atoms. MoS2is composed
of a layer of molybdenum atoms sandwiched between two
layers of sulfur atoms.
Graphene. Experimentally, we studied graphene structures
fabricated by two different methods: mechanical exfoliation and
chemical vapor deposition (CVD).
With exfoliated graphene, flakes are randomly scattered on a
glass substrate, with thicknesses ranging from that of a single
layer to a few tens of layers. As an illustrative example, we chose
to study a particular graphene structure ideally composed of
two different regions corresponding to SL and bilayer graphene,
as supported by Raman measurement (see Figure 4d). Figure
4a and b show the OPD and intensity images of the graphene
structure taken by QLSI, illuminated over a wavelength range
of 625 ±10 nm. As shown in Figure 4c, the OPD for a SL
graphene measured by QLSI is δl= 0.5 ±0.1 nm. The error of
0.1 nm corresponds to the standard deviation of the
background signal. In addition to the OPD, the transmittance
of our sample can be determined from the normalized intensity
image (see Figure 4b,c) defined as the intensity image divided
by the reference intensity image. For SL graphene, we
measured a transmittance of T=98.5±0.1%. The
corresponding opacity is 1 −T= 1.5 ±0.1%. Note that this
value is lower than the reported value for suspended graphene
(substrate-free graphene), which is 2.3%,
46
because graphene
opacity is also dependent on the refractive index of the
surrounding medium,
47
as evidenced by eq 5.
Let us first consider a SL graphene as a slab with an effective
thickness deff = 0.335 nm. One can work out an effective
complex refractive index of graphene by inverting eq 10. Taking
n1= 1 (air) and n2= 1.50 (glass substrate), it yields neff = (2.46
±0.16) + i(1.14 ±0.14). If we use the simplistic expression of
the OPD introduced in eq 3 to retrieve the real part of the
refractive index, we get n= 2.49 ±0.3. One can see that in the
case of graphene the error between the two approaches is only
1%. The reason that the simple eq 3 works well for graphene is
that its refractive index is not too large. Figure 5 shows a
comparison between the measured refractive index (n) and
extinction coefficient (κ) in this work with those reported in
Figure 4. (a) OPD, (b) intensity images, and (c) their corresponding profiles for a particular graphene structure obtained by mechanical exfoliation
and deposited on a glass substrate. The illumination source is centered at a wavelength of 625 nm. This observation was made with 40×
magnification and NA = 0.75. (d) Raman image for the same structure (integrated intensity of the G band).
Figure 5. Complex refractive indices of SL graphene given in the
literature and reported in this work at a wavelength close to 625 nm
(Ni et al.,
48
Wang et al.,
25
Bruna and Borini,
23
Kravets et al.,
49
Wurstbauer et al.,
27
Matkovićet al.,
52
Ye et al.,
50
Cheon et al.
51
).
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the literature at a wavelength close to 625 nm.
23,25,27,48−51
One
can see that the effective complex refractive index determined
by the QLSI technique is in general agreement with the
literature. The variations from one value to another observed in
this figure do not necessarily stem from systematic errors of the
different techniques. Different graphene samples may have
different doping levels, defects, or residues on top that may
modify their optical properties.
As discussed in the previous section, characterizing graphene
via its 2D optical conductivity instead of a refractive index
makes more sense. By using eqs 7 and 8, the complex optical
conductivity of a SL graphene at a wavelength of 625 nm is
found to be σ2D = (0. 82 ±0.2)σ0−i(0.55 ±0.1)σ0, where σ0=
e2/4ℏis the universal conductivity in the limit of a massless
Dirac−Fermion band structure. Table 1 lists measurements of
the optical conductivity of graphene reported in the literature.
Most of the measurements in the visible range focused on the
real part. Few studies measured the imaginary part. Note that
Wang et al.,
25
Ni et al.,
48
and Bruna et al.
23
did not directly
measure optical conductivities but permittivities of graphene.
The conductivities of these studies mentioned in Table 1 were
subsequently computed by Skulason et al.
53
from the values of
their permittivities. The complex optical conductivity measured
by QLSI is in general agreement with the literature. It may be
surprising to observe that the measured values are so dispersed.
As aforementioned, we believe this problem does not
necessarily come from systematic errors related to the different
experimental techniques. It can stem from the nature of the
graphene samples, which may vary from one fabrication
method to another. As will be shown in the next paragraph,
some residue can remain on the sample and be difficult to
detect. The Fermi level can also vary from one sample to
another depending on the quality of the sample. For this
reason, the values we report here have to be seen as specific
values for the sample we observed, not as an attempt to
measure a universal optical conductivity of graphene on glass.
The study of the dispersion of graphene optical constants from
one sample to another could form the basis of future studies.
We now present results on SL graphene samples fabricated
by CVD.
57−59
These results are meant to show how QLSI can
efficiently determine the presence and the nature of residue on
graphene flakes. Experimentally, an extended SL graphene was
grown on a copper substrate. The SL graphene has been
subsequently transferred on a glass substrate by a wet chemical
transfer process involving the use of a PMMA polymer layer,
300 nm thick, as a temporary graphene supporting layer. The
polymer was then removed by acetone under heating.
59
The
fabricated graphene layer has been eventually structured by O2
plasma etching in order to get a 2D array of SL graphene disks,
around 8 μm in diameter. The phase and intensity images of
this sample are shown in Figure 6a and b, respectively. With
such a sample, the measured OPD of the SL graphene discs was
about 2 nm (see Figure 6c). This OPD value is 4 times as large
as that of exfoliated graphene, while the transmittance remains
approximately the same (98.2%) as shown in the histogram of
Figure 6d. The thicker measured OPD for SL CVD graphene
suggests that this sample includes some other material in
addition to graphene. The fact that some residual polymer layer
can remain on graphene using this fabrication process is a well-
known problem,
59
but it is often difficult to evidence and to
characterize. QLSI enables one to retrieve both the thickness dL
and the refractive index nLof a possible residual layer. To
retrieve these two quantities, it is sufficient to perform two
QLSI measurements with two different surrounding media,
such as air and water (i.e., n1= 1 and n1= 1.33). Using eq 3 (a
good approximation for graphene), one can derive a system of
two equations where the two unknowns are nLand dL(see
Supporting Information for the demonstration). We found nL=
1.4 and dL= 3.3 nm. The thickness is consistent with AFM
measurements,
60
and the refractive index is consistent with
PMMA’s. This example illustrates how QLSI can tell whether
2D materials obtained using wet transfer are clean and free
from remaining chemical residues and highlights the difficulty
to achieve complete removal of polymer used for the transfer
process.
Molybdenum Disulfide. In order to demonstrate the
versatility of QLSI and explain how QLSI can apply to thicker
2D materials, we conducted experiments on molybdenum
disulfide (MoS2). It belongs to the family of transition metal
dichalcogenides and does not consist of a single layer of atoms.
It is thus thicker than graphene (see Figure 3). Figure 7a and b
show the OPD and intensity images, respectively, for a SL
Table 1. Comparison of the Complex Optical Conductivity
of a SL Graphene Obtained in This Work to the Reported
Values
reference
σr(in unit of
σ0)
−σi(in unit of
σ0)wavelength
(nm)
this work 0.82 ±0.2 0.55 ±0.1 625
Chang et al.
54
1.16 0.39 625
Wang et al.
25
0.80 0.63 550
Ni et al.
48
0.74 0.3 550
Bruna and Borini
23
1 1.17 550
Mak et al.
55
1.26 625
Gogoi et al.
56
1.06 625
Figure 6. (a) Phase and (b) intensity images of SL graphene fabricated
by CVD technique and transferred on a glass substrate and then
structured by plasma etching. The illumination source is centered at a
wavelength of 625 nm. This observation was made with 60×
magnification and NA = 0.7. (c, d) Pixel-by-pixel histogram of the
OPD and intensity images, respectively.
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MoS2flake fabricated by CVD and transferred on a glass
substrate.
61
The transfer process of MoS2on the glass substrate
is described in the Supporting Information. These measure-
ments are done with an illumination source centered at 625
nm. The obtained OPD and transmittance of SL MoS2are 5.70
±0.22 nm and 96.5 ±0.2% respectively. The corresponding
effective complex refractive index obtained by inverting eq 10 is
neff = (4.98 ±0.10) + i(0.66 ±0.06) using a thickness of SL
MoS2of dMoS2= 0.62 nm. Figure 7c shows a comparison of the
determined effective complex refractive index in this work with
those reported in the literature at a wavelength close to 625 nm.
One can see considerable variations in the refractive index and
extinction coefficient obtained from the literature. This can be
attributed to the fabrication process of the samples, which may
modify the properties of MoS2.
It is worth noting that, unlike graphene, eq 3 fails in the
estimation of the refractive index of MoS2by an error larger
than 40% (neff = 2.91 ±0.40). This is due to the fact that the
refractive index of MoS2is higher than that of graphene, which
makes the multiple reflections more efficient.
In addition to the complex refractive index, one can also
estimate the complex optical conductivity of a SL MoS2by
using eqs 7 and 8. This leads to a value of σ2D = [(1.40 ±0.35)
−i(5.0 ±0.2)]2e2/hat a wavelength close to 625 nm.
Unlike graphene, MoS2is very dispersive in the visible range
(i.e., its optical properties strongly depend on the wavelength).
We performed QLSI measurements at different wavelengths to
investigate the optical properties of MoS2in the whole visible
range. For this purpose, we have used a white light illumination
source combined with bandpass filters, every 50 nm, with a
bandwidth of 40 nm. As an example, we show in Figure 7d the
real (σr) and imaginary (σi) parts of the optical conductivity of
MoS2measured at different wavelengths. We also plotted in the
same figure the complex optical conductivity estimated by
Morozov and Kuno using transmission and reflection measure-
ments.
43
Comparisons of the complex refractive indices are
shown in Supporting Information.Figure 7d shows a good
agreement for the values of σr.σihas the same appearance but
features lower values in comparison with those of Morozov and
Kuno.
43
However, the MoS2samples used by Morozov and
Kuno were fabricated by mechanical exfoliation, leading to a
different quality of MoS2in comparison with CVD.
The studies reported above show that the QLSI is a powerful
technique to characterize different 2D materials on transparent
substrates. The only limitation of QLSI regarding the nature of
the investigated 2D material is its transparency. Too thick
materials may become opaque, and in this case one can no
longer measure any wavefront distortion. However, we can
overcome this limitation by working in the reflection geometry.
This should be suited to image and characterize 2D materials in
nontransparent samples.
65
This modality will be addressed in
future work.
Figure 7. (a) Phase and (b) intensity images of SL MoS2fabricated by CVD on a glass substrate. The illumination source is centered at a wavelength
of 625 nm. This observation was made with 100×magnification and NA = 1.3. (c) Comparison of the estimated effective complex refractive index in
this work (refractive index and extinction coefficient of a SL MoS2) with those reported in the literature at a wavelength close to 625 nm (Liu et al.,
62
Morozov and Kuno,
43
Yu et al.,
63
Zhang et al.,
64
Mukherjee et al.,
24
Funke et al.
28
). (d) Complex optical conductivity of MoS2determined by our
technique and compared to the values given by Morozov and Kuno.
43
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ACS Photonics 2017, 4, 3130−3139
3136

■CONCLUSION
This article introduces a novel optical microscopy technique to
characterize 2D materials. This method involves the use of a
QLSI camera, which maps both the phase and the intensity of
an incoming light beam, when implemented on a microscope.
We show how such a simple scheme allows both the
geometrical and the optical characterization for 2D materials
like no other technique before. In particular, the main purpose
of QLSI we evidenced is to determine the number of stacked
layers and measure the complex optical conductivity and
complex refractive index of 2D material structures.
The results presented in this article highlight different
benefits of this technique compared to other characterization
techniques of 2D materials. First, QLSI is noninvasive. For
example, an irradiance of typically 100 mW·cm−2is used for
QLSI measurements. This is 5 orders of magnitude lower than
that used in Raman spectroscopy, where a laser of typically 1
mW is focused on the sample. Modifications of the optical
properties of 2D materials have been evidenced after Raman
characterization induced by the large intensity of the probe
laser.
14,15
Then, compared with Raman spectroscopy, QLSI is
fast. Single-layer graphene can be observed with an acquisition
time of around 1 s, but it can be even faster using more
advanced QLSI cameras.
39
These advantages make it possible
to easily look for and visualize single-layer graphene, like with a
regular camera. QLSI also enables a mapping of the optical
properties, while other techniques such as transmittance or
reflectance measurements rather perform ensemble measure-
ment over spatially extended and uniform samples. For this
reason, QLSI should be particularly powerful when applied to
2D material heterostacks. We have also shown how QLSI can
detect and characterize the presence of an unwanted residual
polymer thin film of a few nanometers on top of graphene,
stemming from the fabrication process. Finally, QLSI allows for
both morphological and optical characterizations, while other
techniques usually achieve one or the other, which make QLSI
particularly convenient and versatile.
For all these reasons, we expect QLSI to impact both the
field of research working on 2D materials, by providing an
effective and simple characterization technique, and the
community working in quantitative phase imaging, by high-
lighting a new topic of interest for them: the study of 2D
materials.
■ASSOCIATED CONTENT
*
SSupporting Information
The Supporting Information is available free of charge on the
ACS Publications website at DOI: 10.1021/acsphoto-
nics.7b00845.
Derivation of the expressions of models 1, 2, and 3;
additional results and details concerning the estimation
of the residual layer on CVD-graphene; Raman spectrum
of CVD-graphene sample; description of the transfer
process of MoS2sample; wavelength-dependent complex
refractive index of MoS2measured in this work and
compared with the literature (PDF)
The mathematica notebook allowed the determination of
the complex refractive index from models 2 and 3 (ZIP)
■AUTHOR INFORMATION
Corresponding Authors
*E-mail: samira.khadir@Fresnel.fr.
*E-mail: guillaume.baffou@Fresnel.fr.
ORCID
Samira Khadir: 0000-0002-2280-4099
Guillaume Baffou: 0000-0003-0488-1362
Notes
The authors declare no competing financial interest.
■ACKNOWLEDGMENTS
D.V. and E.G. acknowledge the financial support of the
Renatech network. The authors thank J. R. Huntzinger for
helpful discussions, in particular for pointing out that eq 3 was a
priori not applicable for 2D materials. R. Parret and M. Paillet
are also acknowledged for preparation and characterization of
mechanically exfoliated graphene. N.M. acknowledges support
from Science Foundation Ireland (15/SIRG/3329).
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