PHYSICAL REVIEW B 88, 035426 (2013)
Raman study on defective graphene: Effect of the excitation energy, type, and amount of defects
Axel Eckmann,
1
Alexandre Felten,
2
Ivan Verzhbitskiy,
2
Rebecca Davey,
2
and Cinzia Casiraghi
1,2,*
1
School of Chemistry, University of Manchester, United Kingdom
2
Physics Department, Free University Berlin, D-14195 Berlin, Germany
(Received 11 February 2013; revised manuscript received 29 March 2013; published 15 July 2013)
We present a detailed Raman study of defective graphene samples containing specific types of defects. In
particular, we compared sp
3
sites, vacancies, and substitutional Boron atoms. We find that the ratio between
the D and G peak intensities, I(D)/I(G), does not depend on the geometry of the defect (within the Raman
spectrometer resolution). In contrast, in the limit of low defect concentration, the ratio between the D
and G
peak intensities is higher for vacancies than sp
3
sites. By using the local activation model, we attribute this
difference to the term C
S,x
,representingtheRamancrosssectionofI(x)/I(G) associated with the distortion of the
crystal lattice after defect introduction per unit of damaged area, where x = D or D
. We observed that C
S,D
= 0
for all the defects analyzed, while C
S,D
of vacancies is 2.5 times larger than C
S,D
of sp
3
sites. This makes
I(D)/I(D
) strongly sensitive to the nature of the defect. We also show that the exact dependence of I(D)/I(D
)
on the excitation energy may be affected by the nature of the defect. These results can be used to obtain further
insights into the Raman scattering process (in particular for the D
peak) in order to improve our understanding
and modeling of defects in graphene.
DOI: 10.1103/PhysRevB.88.035426 PACS number(s): 81.05.ue, 78.30.Ly, 78.67.Wj, 63.22.Rc
I. INTRODUCTION
Graphene has attracted enormous interest because of its
unique properties.
1–4
Near-ballistic transport at room temper-
ature and high mobility
1,5–8
make it a potential material for
nanoelectronics.
9–13
Furthermore, its optical and mechanical
properties, combined with its high charge mobility, allow
the use of this material for other applications, such as thin-
film transistors, transparent and conductive composites and
electrodes, and opto-electronics.
14–21
Graphene is usually considered as a perfect honeycomb
crystal. However, real samples may contain defects. The
amount and nature of defects strongly depend on the produc-
tion method and may change from sample to sample.
16,22–30
Defects can also be introduced in pristine graphene through
ion bombardment,
31–36
e-beam irradiation,
37,38
soft x-ray
irradiation,
39
covalent modification,
40–53
and implantation of
substitutional atoms.
54–57
The possibility of introducing only
well-defined defects and carefully control their amount allows
fine tuning of the properties of graphene: defect lines can be
used as metallic wire interconnectors for nanoelectronics,
58,59
while sp
3
-site defects can be used to turn the electronic
properties of graphene from metallic to insulating, leading to
the creation of stoichiometric graphene-based derivatives.
40,41
These can be easily obtained by exposing the crystal to a
plasma. This method is very attractive for industrial appli-
cations because it is a simple, fast and scalable process.
Depending on the plasma gas used, various species such as
oxygen, fluorine, nitrogen, and chlorine can be grafted at the
graphene scaffold. The bonding with the out-of-plane atom
changes the carbon hybridization from sp
2
to sp
3
, leading to
changes in the electronic and optical properties.
40,41
Raman spectroscopy is a fast and nondestructive technique
for investigating the properties of graphene.
60–63
This is able
to identify graphene from graphite and few-layer graphene,
64
and to probe doping level,
65–67
strain,
68,69
disorder,
31–33,70,71
chemical derivatives,
40–42
and the atomic arrangement at the
edges.
72,73
Due to its sensitivity to defects, Raman spectroscopy has
been used since more than 40 years to study disorder in
carbon-based materials, from nanocrystalline graphite,
74–82
to
disordered carbons,
83–85
and to carbon nanotubes.
86–88
These
works provided important advances in understanding disorder
in sp
2
-bonded carbon materials and they strongly contributed
to the widespread use of Raman spectroscopy for general
characterization of these samples. However, the correlation
between defect-activated Raman features and geometry of
defects is still missing. We do not know if Raman spectroscopy
is sensitive to every type of defect, i.e., if and how the
disorder-activated Raman intensities depend on the nature of
the defects. For instance, Raman spectroscopy could be more
sensitive to certain defects rather than others. If so, then a
Raman spectrum without a D peak would not be necessarily
associated with a defect-free material.
To answer these questions, one needs to be able to carefully
introduce defects in the material and control their nature, in
order to compare the corresponding Raman spectra. In the
case of a three-dimensional (3-D) material such as graphite,
this is extremely challenging. The only defect that can be
carefully controlled is the grain boundary, which depends on
the size of the crystalline grains. Other 3-D materials such
as disordered carbons do not appear suitable for this type of
study because they contain both topological and structural
disorder, that is disorder is not defined only by the sp
3
content, but also by the different amount and type of sp
2
clustering.
83–85
In general, structural and topological disorders
are not correlated with each other (e.g., two disordered carbons
films may have the same sp
3
content, but a different degree
of sp
2
clustering) and they strongly depend on the deposition
conditions.
83–85
This makes the Raman spectrum of disordered
carbon rather difficult to analyze and correlate with the specific
nature of disorder. Moving to low-dimensional carbon forms,
nanotubes are difficult to manipulate and characterize, due to
their one-dimensional nature. In contrast, graphene is an ideal
material to study the Raman sensitivity to defects because its
035426-1
1098-0121/2013/88(3)/035426(11) ©2013 American Physical Society
AXEL ECKMANN et al. PHYSICAL REVIEW B 88, 035426 (2013)
two-dimensional nature makes it easy to add, remove, or move
carbon atoms, i.e., to carefully introduce only a specific type
of disorder.
Here we present a detailed analysis of the evolution of
the Raman spectra of graphene samples containing specific
defects, such as sp
3
sites, substitutional atoms, and vacancies.
We show that the general trend of the Raman spectrum of
defective graphene does not depend on the nature of defects: a
two-stage disordering evolution is always observed, no matter
the geometry of the defect. In each stage the Raman fit param-
eters have a different dependence on the excitation energy. By
comparing the Raman spectra of graphene containing sp
3
sites,
substitutional atoms and vacancies, we found that in the limit
of small defect concentration the D peak is not sensitive to the
defect geometry, but only to the amount of disorder (at least
within the Raman resolution and for the type of defects studied
here). In contrast, the D
peak shows a strong dependence
on the type of defect introduced in the lattice, e.g., the D
peak intensity is higher for vacancies than sp
3
sites. Within
the local activation model,
33
we attribute this to the term C
S
,
representing the Raman cross section of I(x)/I(G) associated
with the distortion of the crystal lattice after defect introduction
per unit of damaged area, where x = D or D
. Thus, this
parameter is expected to strongly depend on the nature of the
defect.
33
We found that for the D peak C
S
∼ 0, no matter the
nature of the defect; while for the D
peak, C
S
= 0.33 for sp
3
sites, and C
S
= 0.82 for vacancies. This makes the D
peak
more sensitive to vacancies than sp
3
sites.
This paper is organized as follows: Sec. II presents the
Raman scattering background, while Sec. III describes the
experiential setup and the sample preparation. Section IV
shows the evolution of the Raman spectrum with the amount
of defects, its dependence on the excitation energy and on the
nature of defects.
II. BACKGROUND
The Raman spectrum of graphene is composed of two main
features, the G and the 2D peaks, which lay at around 1580 and
2680 cm
−1
, respectively, when taken at an excitation energy
of 2.4 eV (514 nm).
64
The G peak corresponds to the E
2g
phonon at the Brillouin zone center ( point).
74
The 2D peak
is an overtone peak, associated with the breathing modes of
six-atom rings.
64
It comes from TO phonons in the vicinity
of the K point
74
and it is activated by a resonant intervalley
scattering process.
89,90
Raman spectroscopy is able to probe defects in graphitic
materials because, in addition to the G and 2D peaks that
always satisfy the Raman selection rule,
91
the otherwise
forbidden D and D
bands appear.
74,78
They correspond to
single phonon intervalley and intravalley scattering events,
respectively, where the defect provides the missing momentum
in order to satisfy momentum conservation during the Raman
scattering process.
89,90,92
Another (weak) defect-activated
peak is observed at about 3000 cm
−1
, corresponding to the
combination mode of the D and D
modes. It is therefore
called the D + D
peak.
The activation mechanism of the defect-activated fea-
tures, their overtones, and combination modes involves res-
onant electronic transitions.
89,90,92
Hence the frequency of
these peaks are intimately correlated to the electronic band
structure making the peaks dispersive with the excitation
energy.
80,89,93,94
Previous works on defective graphene introduced the local
activation model in order to explain the evolution of the Raman
spectrum for increasing amount of defects.
31–33
Within this
model the intensity of any defect activated peak I(x), where
x = D or D
, as compared to the G peak intensity I(G), is
given by
33
I (x)
I (G)
= C
A
r
2
A
− r
2
S
r
2
A
− 2r
2
S
e
−πr
2
S
/L
2
D
− e
−π(r
2
A
−r
2
S
)/L
2
D
+ C
S
[1 − e
−πr
2
S
/L
2
D
]. (1)
This equation shows that the intensity of the defect-
activated peak depends on two length scales, r
S
and r
A
, which
are the radii of two circular areas measured from the defect site.
The first length, r
S
, is the radius of the structurally disordered
area around the defect, so it is expected to change from defect
to defect.
33
For distances larger than r
S
but shorter than r
A
,
the lattice structure is preserved, but the proximity to a defect
causes a mixing of Bloch states near the K and K
valleys of the
graphene Brillouin zone, thus causing a breaking of selection
rules, and leading to an enhancement of the D band.
31,33
r
A
defines the disk where the D peak scattering takes place and
it defines the activated area.
33
From a microscopic point of
view, an electron/hole excitation will only be able to “see” the
structural defect if it is created sufficiently close to it and if the
excited electron (or hole) lives long enough for the defective
region to be probed by Raman spectroscopy.
31,33
Therefore,
the distance r
A
− r
S
= l
x
represents the length traveled over
the lifetime of the electron-hole pair, roughly given by v
F
/ω
x
,
where v
F
is the graphene Fermi velocity and ω
x
is the peak
frequency of either the D or D
peak.
31,33
C
A
depends only on the Raman mode, being roughly given
by the ratio of the electron-phonon coupling between the two
phonons considered.
31,33
C
S
is a factor assumed to depend only
on the geometry of the defect for a fixed phonon mode.
Note that Eq. (1) can also be used for the intensity
measured as integrated area. In any case, in the limit of low
defect concentration, the use of intensity or integrated area is
equivalent.
31,33,71
Only in the high disorder regime it is more
informative to decouple the peak intensity from the full width
at half maximum.
32
There are also physical models based on first principles
and quantum mechanics that calculate the intensities of the
Raman resonant features. In particular, a recent work
92
has
been able to successfully reproduce numerous features of the
Raman spectrum of graphene. We compare our experimental
data with the results presented in Ref. 92, in particular the
dependence of the Raman features on the excitation energy and
amount of defects. However, the simple on-site and hopping
perturbations used in Ref. 92 to simulate defects in graphene
are not suitable to describe real defects.
71
Because of that,
we will use the activation model to explain our results of the
dependence of the Raman intensities on the geometry of the
defect.
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RAMAN STUDY ON DEFECTIVE GRAPHENE: EFFECT OF ... PHYSICAL REVIEW B 88, 035426 (2013)
A. Dependence on the amount of defects
The only parameter that changes with defect concentration
(n
D
)inEq.(1) is L
D
, being n
D
= 10
14
/πL
2
D
.
32
This equation
gives a nonmonotonic evolution of I(x)/I(G). This is the result
of two competing mechanisms for the increase and decrease
in the defect-activated bands. The increase in the activated
area gives rise to an increase of the defect activated peak
intensities; on the other side, an increase in the defect-activated
area produces a decrease of the intensities. Therefore, one
can identify two stages, where one mechanism dominates the
other.
31,33
The transition from Stage 1 to Stage 2 is typically
observed when the mean distance between two defects (L
D
)is
comparable to l
x
.
31
Note that the stage terminology was first
introduced for disordered carbons.
83–85
Within the two-stage model, C
S
is the value of I(x)/I(G)
measured in the highly disordered limit.
33
This is difficult to
measure since both the D and D
peak intensities decreases in
Stage 2 and the D
peak, being close to the G peak, merges
with this peak.
If we now focus on Stage 1 (i.e., low defect concentration),
the model shows that both the D and D
peak intensities
increase with increasing amount of defects. In particular, in
the limit of low defect concentration, a Taylor expansion of
Eq. (1) to the first order gives:
I (x)
I (G)
= C
A
π
r
2
A
− r
2
S
L
2
D
+ C
S
πr
2
S
L
2
D
. (2)
The defect concentration for this stage is then given by
n
D
(cm
−2
) =
10
14
π
2
C
A
r
2
A
− r
2
S
+ C
S
r
2
S
I (x)
I (G)
. (3)
B. Dependence on the geometry of the defect
By looking at Eq. (2), the dependence of the peak intensities
on the nature of the defects is given by r
S
and C
S
, being
r
A
= r
S
+ l
x
, where l
x
is fixed by the phonon mode and the
excitation energy.
The only experimental works reporting data for those
parameters are based on vacancies.
31–33
In these works, the
D peak was extensively studied. The following parameters
were reported for intensity measured as height: r
A
∼ 3nm,
r
s
∼ 1nm,C
A
= 4.2. C
S
has been reported being 0 or 0.87
in Refs. 32 and 33, respectively. Reference 32 claims that
C
S
= 0 for the ideal case where the defect is the breakdown
of the C-C bonds. However, C
S
may be different for other
types of defects such as sp
3
sites, which do not break the
network, but just produce a different arrangement of the carbon
atoms. In any case, one should observe that the term C
S
has
a minor influence in Stage 1, in particular in the limit of very
low defect concentration. A change in C
S
from 0 to 0.87
produces variations well smaller than 10% on I(D)/I(G)at
the beginning of Stage 1. Therefore, if we focus on the low
defect concentration regime, it is correct to assume C
S
∼ 0, so
Eq. (2) becomes:
I (D)
I (G)
πC
A
r
2
A
− r
2
S
L
2
D
. (4)
Therefore, the D peak depends on the defect geometry only
through r
S
.
In this work, we analyze sp
3
sites, vacancies, and implanted
atoms, so we do not expect r
S
to strongly change with the type
of defect. Consequently, we expect the D intensity not to be
able to probe differences in the geometry of the defects because
the Raman spectrometer is not enough sensitive (the typical
error bar on a Raman intensity ratio is 10%–15%). Thus, in the
following we will use Eq. (1) to find the defect concentration
from I(D)/I(G).
In the case of the D
peak, its intensity follows qualitatively
the same behavior as the D peak, i.e., it increases in Stage
1 and decreases in Stage 2. However, the exact dependence
on defect concentration is different.
71
Indeed Ref. 31 shows
that C
S
cannot be neglected for the D
peak, and that small
variations on C
S
can produce strong changes in I(D
)/I(G),
even at low defect concentrations.
In this work we aim at comparing the results obtained
for vacancies with other type of defects, such as Boron
substitutional atoms and sp
3
sites. In particular, we will
investigate the intensity dependence of the D and D
peak
on the parameter C
S
, which we will refer to as C
S,D
and C
S,D
,
respectively.
III. EXPERIMENTAL METHOD
We studied three types of defects in graphene:
(i) sp
3
defects. Pristine graphene samples were prepared
by micromechanical exfoliation of single-crystal graphite
flakes (Nacional de Graphite LTDA) on Si/SiO
x
substrates.
The flakes were then placed in a purpose-built chamber,
where they underwent an inductively coupled plasma at
RF of 13.56 MHz.
42
Defects were introduced by exposing
pristine graphene to a mild O
2
and CF
4
plasma.
42
The plasma
treatments were performed at a power of 10 W and a pressure
of 0.1 Torr. The amount of defects was tuned by changing the
treatment time (between 5 s and 300 s). More details of the
process are described in Ref. 42. In addition, we fluorinated
some exfoliated flakes by using the technique described in
Ref. 41. In all cases, chemical modification was performed on
pristine s amples with no detectable D peak.
(ii) Vacancylike defects. The samples were produced by
anodic bonding, as reported in Refs. 24,71.Wealsousedthe
results reported in Refs. 31–33, where graphene was exposed
to Ar
+
ion bombardment.
(iii) Substitutional atoms. We used B-doped graphene
samples, as reported in Ref. 54.
In the following discussion, we will group our data based
on the nature of defects, so we will refer to “vacancies” as the
data obtained by anodic bonding and ion bombardment, while
we will refer to “sp
3
” as the data obtained for oxidized and
fluorinated samples. The nature of the defects in these samples
has been verified in Refs. 33,42,71.
Raman measurements were performed in a backscattering
configuration with a confocal WITec spectrometer equipped
with 2.54 eV, 2.41 eV, and 1.96 eV laser lines. The WITec
spectrometer is also equipped with a piezoelectric stage that
allows Raman mapping of areas up to 200 × 200 μm
2
.
Because of the inhomogeneity of the fluorinated and anodic
bonded flakes, we used Raman mapping to collect a large
035426-3
AXEL ECKMANN et al. PHYSICAL REVIEW B 88, 035426 (2013)
amount of spectra with varying I(D)/I(G) ratios, typically
between 0.5 and 4. Multiwavelength analysis was performed
with a Dilor triple-monochromator Raman spectrometer,
equipped with an Ar-Kr laser with excitation lines between
647 and 457 nm. In all cases we used a 100× objective giving
a laser spot size of about 400 nm. The laser power was kept
well below 1 mW to avoid damage or heating, which could
induce desorption of the adatoms from graphene. The spectral
resolution is 2cm
−1
.TheD, G, and 2D peaks were fitted
with Lorentzian functions and the D
peak by a Fano line shape.
A Fano line shape was preferred to a Lorentzian because for
a defect concentration close to the transition between Stage 1
and Stage 2 and beyond this point, the G and D
peak start to
merge. This does not allow using a fully symmetric line for
fitting the G peak. In any case, the use of Fano or Lorentzian
line does not change the results relying on the peaks’ amplitude
as the peak height is the same for the two lines. We analyze
the following Raman fit parameters: position (POS) and full
width at half maximum (FWHM), and intensities. Here, we
refer to peak intensity as the height of the peaks and it will be
denoted as I(D), I(G), I(D
), I(2D)fortheD, G, D
, and 2D
peaks, respectively. The i ntegrated areas will be labeled A(D),
A(G), A(D
), and A(2D).
Note that the D
peak has a small intensity compared to the
D peak; often the peak appears just as a small shoulder of the
G peak. However, at low and moderate defect concentration,
the D
peak can be clearly distinguished from the G peak and
it can have relatively large intensity (up to 1/3 of the intensity
of the G peak).
IV. RESULTS AND DISCUSSION
A. Evolution of the Raman spectrum with the amount of defects
Let us start by looking at the evolution of the Raman
spectrum of graphene for several types of defective graphene
with increasing defect concentration.
Figures 1(a) and 1(b) show a collection of first and second
order Raman spectra, measured at 2.41 eV of graphene
containing sp
3
sites obtained by partial fluorination (a) and
oxidization (b). Figure 1(c) shows a collection of first and sec-
ond order Raman spectra for increasing defect concentration
(from bottom to top), measured at 2.41 eV of graphene with
vacancylike defects obtained by anodic bonding as described
in Ref. 24.
The defect-activated features (D, D
, and D + D
peaks)
appear in all the spectra. Qualitatively, one can see that
as the defect concentration increases, the D peak increases
at first and then decreases while broadening. The D
peak
increases and eventually merges with the G peak and the 2D
peak monotonously decreases until it almost disappears. The
D + D
peak increases in intensity and broadens. The same
general evolution is observed in ion-bombarded graphene.
31–33
Figure 2 shows the evolution of the Raman fit parameters
against exposure time for the oxidized samples. Although we
do not know the exact relation between plasma exposure time
and defect concentration, we expect the amount of defects to
increase for increasing time.
If we focus on the D peak intensity [Fig. 2(a), top], we can
clearly see a two-stage evolution: at low defect concentration
(between 0 and 40 s), I(D) and I(D
) increase for increasing
FIG. 1. Representative spectra of (a) fluorinated graphene,
(b) oxidized graphene, and (c) anodic bonded graphene, with increas-
ing defect concentrations (from bottom to top). All measurements are
taken at 2.41 eV.
time. This corresponds to Stage 1. In contrast, I(G) and I(2D)
show very little variation. At higher defect concentration
(between 40 and 300 s), I(D) decreases with time. This
corresponds to Stage 2. Furthermore, I(G) decreases, but more
slowly than I(D) and I(2D) strongly decreases. The transition
between the two stages corresponds to the maximum I(D)/I(G)
035426-4
RAMAN STUDY ON DEFECTIVE GRAPHENE: EFFECT OF ... PHYSICAL REVIEW B 88, 035426 (2013)
FIG. 2. (Color online) Evolution of (a) intensity and FWHM of
the D, D
, G,and2D peaks for increasing defect concentration.
(b) Absolute position of the D, D
, G,2D peak and relative shift
of these peaks, as compared with the position measured on pristine
graphene. Measurements taken at 2.41 eV.
(5). This two-stage evolution agrees with the experimental
results obtained in vacancy-defective graphene.
31–33
A two-stage evolution is also visible when looking at the
peaks’ FWHM [Fig. 2(a), bottom]. In Stage 1, the FWHM
of all peaks remains constant, while it strongly increases in
Stage 2. The FWHM of any peak can consequently be used
to distinguish between the two stages. Conclusions drawn
about the defect concentration when only considering the ratio
I(D)/I(G) is indeed ambiguous since a given D peak intensity
[or I(D)/I(G) ratio] may correspond to two different defects
concentrations.
The introduction of disorder also changes the peak posi-
tions, as shown in Fig. 2(b) (top). In order to better visualize
these changes, we plotted the shift of the position with
respect to the positions measured on the pristine graphene
[Fig. 2(b), bottom]. This figure shows that the D, D
, and 2D
peak positions down-shift for increasing defect concentration,
where Pos(2D) experiences the largest shift (well above
15 cm
−1
at 300 s). In contrast, the G peak shifts to higher
wave numbers, up to 10 cm
−1
at 300 s. We expect the G
peak to shift with disorder: This effect is well reported for
disordered carbons.
83–85
Changes in the G peak position occur
because the introduction of defects relaxes the Raman selection
rule (q 0). Due to the Kohn anomaly,
95
the phonon energy
strongly increases with the phonon wave vector, resulting in
a blue-shift of the G peak position for increasing disorder
in the hexagonal rings. Note that at 300 s, the D
and the
G peak have merged, so the uncertainty related to the fit is
large. The down-shift of both the D and 2D peaks is assigned
to the TO phonon dispersion branch in the vicinity of the K
point while Pos(D
) decreases due to the LO phonon branch
dispersion near .
31
These observations agree with the results
from Refs. 31,47.
FIG. 3. (Color online) A(D)/A(G) against FWHM(G) for a wide
collection of two-dimensional (2-D) defective graphene,
27,28,31,96,97
as compared with the three-stage evolution of disordered 3-D
carbons.
78,85
The dotted lines are guides to the eyes.
It is now interesting to compare the disordering trajec-
tory of the Raman spectrum of disordered graphene and
disordered carbons, i.e., to compare disorder in two- and
three-dimensional carbon-based materials.
Figure 3 plots the ratio A(D)/A(G)againstFWHM(G).
This allows one to decouple the amount of defective hexagonal
rings from the overall disorder.
85
Indeed, the FWHM always
increases for increasing disorder because this parameter is
sensitive to all types of defects, either in the sp
2
rings or
chains; in contrast, A(D)/A(G) is sensitive only to defects in
the rings. Note that we decided to plot the ratio A(D)/A(G)
because it allows comparison of a large set of data available in
the literature, which is reported in area only.
The two-dimensional materials group includes fluorinated
graphene (defect = sp
3
site), ion-bombarded graphene (de-
fect = vacancies),
31
and graphene oxide (GO) and reduced
graphene oxide (rGO).
27,28,96,97
These last two materials
have been selected because in contrast to hydrogenated
and fluorinated graphene that contain only sp
3
sites, GO
and rGO contain different types of defects, whose nature and
corresponding amount is not completely known. From this
point of view, GO and rGO can be seen as the two-dimensional
equivalent of disordered carbons. For the three-dimensional
materials group, we used the data reported for disordered
carbons in Ref. 85.
Figure 3 shows that both two- and three-dimensional
disordered materials have a similar “bell-like” disordering
trajectory, in agreement with the two-stage evolution described
in Sec. II. However, disordered carbon material extends into
a third stage [for FWHM(G) >200 cm
−1
], which corresponds
to the conversion of the rings into sp
2
chains.
83–85
It does
not seem to happen for graphene, even in highly fluorinated
samples, so the defects may stretch the rings but do not open
them into chains. In the following discussion, we will use
Eq. (1) to calculate the defect concentration from I(D)/I(G)
for all defective two-dimensional samples based on the fact
that they observe the same disordering trajectory, as seen in
Fig. 3.
035426-5
