Disentangling contributions of point and line defects in the Raman spectra of graphene-related materials

The transition from graphene to a fully disordered sp2 carbon material can be idealized by either cutting graphene into smaller and smaller pieces, or adding more and more point defects. In other words, from the dimensionality standpoint, defects in two-dimensional (2D) systems can be either one- (1D) or zero-dimensional (0D). From an application point of view, both in terms of bottom-up as well as top-down approaches, the discrimination between these two structural disorder in two-dimensional systems is urgently desired. In graphene, both types of defects produce changes in the Raman spectrum, but identifying separately the contribution from each defect-type has not yet been achieved. Here we show that a diagram can be built for disentangling contributions of point-like and line-like defects to the Raman spectra of graphene-related materials embracing, from the topology point of view, all possible structures from perfect to fully disordered sp2 bonded carbons. Two sets of graphene-related samples, produced by well-established protocols that generate either 0D or 1D defects in a controlled way, are analysed with our model and used to parameterize the limiting values of the phase space. We then discuss the limitations and apply our new methodology to analyse the structure of two-dimensional nanocarbons generated from renewable gas, used to produce inks and conducting coatings.


Introduction
In two-dimensional (2D) lattices, defects can be either zero-dimensional (0D), such as vacancies, dopants or functional chemical groups, or one-dimensional (1D), such as dislocations or crystallite borders, the latter appearing during growth and enclosing a crystallite area [1,2]. This simple geometrical distinction determines defect functionality and their influence on materials properties [3][4][5][6][7]. Defects in the 2D sp 2 honeycomb carbon lattice dictate structural amorphization from pristine graphene or graphite down to more complex structures, such as amorphous carbon, black carbon, charcoal, biochar or, to a larger extent, organic molecules. One can idealize the transition from graphene to amorphous sp 2 carbon materials either by cutting graphene into smaller and smaller pieces, or by adding more and more point defects.
From the synthesis procedures to the integration of graphene into devices, from amorphous carbon to highly oriented pyrolytic graphite (HOPG), Raman spectroscopy is the preferred tool to identify and quantify defects . Structural characterization of defects in graphene by Raman spectroscopy has already produced well-established protocols for the quantification of both point-defect concentrations [19,20,24] and crystallite sizes [14,30] separately. However, most graphene samples, e.g. generated by chemical vapor deposition (CVD) or liquid phase exfoliation, are likely to exhibit both types of defects, and a method to disentangle and differentiate between 0D and 1D defects in such systems has not been developed until now. Here we show it is possible to disentangle the contribution of defects with distinct dimensionalities in the sp 2 carbon Raman spectra. We develop a procedure that enables researchers to use Raman spectroscopy to Disentangling contributions of point and line defects in the Raman spectra of graphene-related materials 2 L G Cançado et al determine quantitatively and simultaneously pointdefect concentration and crystallite size in graphenerelated materials, a necessary tool to study the effect of point and line defects on the properties of carbon sp 2 materials. The methodology has limitations, mostly for samples with low defect concentrations, where doping, strain and number of layers add significant additional parameters that have to be taken into account to properly parameterize the protocol proposed here. Otherwise, the procedure may help to identify suitable starting carbon materials for dispersion, or for mat erials applications, and also to adjust process parameters to boost properties such as electrical and thermal conductivity in deposited graphene films.

Statement of the problem
To identify the spectral features that allow the disentanglement of 0D and 1D defects in the Raman spectrum of graphitic materials, we studied 25 different graphene samples, grouped in two sets. These two sets correspond to standard reference materials varying from pristine 2D hexagonal sp 2 carbon lattices to highly disordered structures, following two different routes: • Sample set 1 starts as pristine graphene prepared by the mechanical exfoliation method of HOPG, and it is ion-bombarded with different ion doses, generating an increasing number of 0D defects, down to a fully disordered structure [19,20,24]. From now on these samples are referred to as 'samples with point-defects', and they are characterized by the average distance between nearest defects (L D ) or by the defect density . This type of system is illustrated in figure 1(a). Pristine graphene has → ∞ L D , and fully disordered graphene has → L 0 D ; • Sample set 2 starts as a fully amorphous carbon material prepared by laser ablation of HOPG, and it is heat treated at different temperatures, thus generating sp 2 crystalline structures of increasingly larger crystallite sizes, up to a highly crystalline turbostratic graphitic structure for the highest heat treatment temperature [14,30]. Geometrically, this sample can be thought as a graphene layer 'cut' by several 1D defects, or an ensemble of nanographite crystallites delimited by their borders, as illustrated in figure 1(b). From now on these samples are referred to as 'samples with line-defects', and they are characterized by their average crystallite size (L a ), or by the crystallite area (L a 2 ). Pristine graphene has → ∞ L a , and fully disordered graphene has → L 0 a .
To date, these two sets of samples have been treated separately in Raman spectroscopy studies. The most ordinary protocols are based on the intensity ratio between the disorder-induced D band (∼ 1350 cm −1 ) and the firstorder allowed bond-stretching G band (∼ 1580 cm −1 ), namely / I I D G . As in [14,30] we use the integrated intensity (peak area) ratio rather than the intensity ratio, because the area under each peak represents the probability of the process, and the differences between the spectral information from samples with point versus line defects are better represented. Different from our previous publications [14,30], here we use / A A D G rather than / I I D G to make it clear we are accounting for the area ratio.
These protocols are summarized in figures 1(c) and (d), which present theoretical plots of / A A D G as a function of L D [19,24] and L a [14,30], respectively, for three distinct values of excitation laser energies, E L (see figure legends). Although the spectral differences between the two sets are clearly shown in the plots, one can expect ambiguities when the two parameters of disorder, L a and L D , come into play simultaneously in samples containing both 1D and 0D defects, as illustrated in figure 1(e). The reason is simple: two parameters of disorder (L a and L D ) cannot be univocally extracted from one spectral information ( / A A D G ). Indeed, the plots shown in figures 1(c) and (d) are based on the assumptions → ∞ L a , and → ∞ L D , respectively. However, none of these two assumptions apply for samples containing both 1D and 0D defects. Therefore, to disentangle the information about point and line defects in such a system, a second different spectral information has to be taken into account. In the next section, we show that the G-band line width, Γ G , serves the purpose. , where the values converge. Therefore, the two spectral features, / A A D G and Γ G , which are precisely the features that have been broadly studied to quantify defects in graphitic materials [8-15, 17-21, 23, 24, 26-30], form a pair of variables that can be used to disentangle the contribution from point and line defects in the Raman spectra of samples where these two types of structural disorder are intermixed, as illustrated in figure 1(e). In this sense, the plot presented in figure 2(a) can be referred as a Raman diagram for graphene.

Definition of the relevant parameters
Lines and down-triangles in figure 2 are results from the theoretical modeling that explains how to build the Raman diagram. Modeling the amorphization routes shown in figure 2 requires the definition of the parameters ruling the inelastic light scattering process via phonons in graphene: • Structural parameters: point-defects-r S is the radius of the structurally-damaged area (S-region, illustrated as red circles in figures 1(a) and (e)) / as a function of L D [19,24] and L a [14,30], respectively, for three distinct values of excitation laser energies, E L , as indicated in the legends. (e) Illustrations of a graphene sample containing both point and line defects. The red circles in ((a) and (e)) define the structurally-damaged area (S-region) surrounding a point defect [19]. These circles have radii r S . The red lines in ((b) and (e)) are structurally-damaged ribbons (S-regions) of width l S . The ribbons define the borders of crystallites in polycrystalline graphene [30]. The green circles and lines in ((a), (b) and (e)) are activated area (A-regions) surrounding structural defects where the D band is active [19,30]. The extents of these A-regions (radii of circles or widths of lines) are defined by the electron coherence length e [19,22,30]. around a point-like defect, important for L D determination [19,20]; line-defects-l S is the width of the structurally-damaged ribbon (illustrated as red lines in figures 1(b) and (e)) near a crystallite border in a polycrystalline sample, important for L a determination [30].
• Dynamical parameters: electron [19,22] and phonon [30,[34][35][36] coherence lengths in graphene, labeled here as e and ph , respectively. These two quantities are important for the Raman linewidth broadening, due the combined effects of quantum confinement and the uncertainty principle, and also for the D to G intensity ratio, by defining the region surrounding a structural defect where the D band is active (A-region, illustrated as green areas The G band linewidth (Γ G ) increases exponentially as the phonon localization length ξ decreases with respect to the phonon coherence length ph , as proposed by Ribeiro-Soares et al [30]:  [19] and [30] , respectively. The filled up-triangles were taken from [24]. The empty and filled down-triangles are data obtained from numerical simulation with ∞ L a → and ∞ L D → , respectively. Exemplary Raman spectra can be found in appendix A.
For samples with only line defects ξ = L a (since phonons are confined within a crystallite of size L a ), while for samples with pure point defects ξ α = L D . It will be shown that α > 1, which is expected because point defects are less effective in localizing phonons when compared to line defects.
4 ratio for each sample is computed by summing contributions for the D and G bands from S− and A−regions. S−regions are either circles of radius r S at each point defect, or ribbons of widths l S at crystallite edges. These two regions define the area that contribute ( are calculated by assuming that the probability of D band scattering decreases exponentially with the distance from the S−region (see appendix B), with decay length e (electron coherence length) [22,30].

Simulations
Next, we determine all these parameters from simulations for the reference cases of samples with only point defects or samples with only line defects, and use them to predict the behavior of the more general case of samples containing both kinds of defects.

Simulations for samples with line defects
In this case, for a given average crystallite size L a , we take an ensemble of 20 square crystallites with sizes L a randomly chosen from a Gaussian distribution centered at L a , and with a standard deviation of / L 4 a . This is roughly the width of crystallite size distributions of polycrystalline graphite [30]. The G band linewidth for each crystallite is calculated using equation (1), with ξ = L a . In figure 2 table 1). An analytical approximation to the simulation data can be obtained (see appendix B), and it is given by This analytical function fits both experimental and simulation data, as shown by the solid line in figure 2(c).

Simulations for samples with point defects
In this case, we consider borderless graphene with periodic boundary conditions and a target point defect concentration σ, which defines the average distance between defects L D as We consider 20 realizations of random defect distributions for each value of L D . The G band linewidth for a given L D is given by equation (1), with ξ α = L D . We find that α = 10 reproduces the experimental data, showing that pointdefect disorder leads to G band phonon localization lengths that are approximately 10 times larger than the average distance between defects. This is consistent with calculated localization lengths for K-point phonons in graphene with a disordered distribution of vacancies [37]. The remaining parameters ( ( ) Γ ∞ G , Γ C and ph ) are the same as in the case of line defects. Simulation data are the filled down-triangles in figure 2(b), and once again an analytical approximation given by the substitution of L D by the average value L D in equation (1)  nm (these parameters are summarized in table 1). Notice that the values of e obtained by fitting the experimental data for samples with line and point defects are quite similar, which gives us confidence to associate them with the same physical quantity (electron coherence length) [19,22,30].
Again, an analytical approximation to the simulation data can be obtained by solving the rate equations for the evolution of S− and A−regions, in a similar manner to [19]. The resulting equation is Table 1. Numerical values (central column) for the fitting parameters found in this work (left column), after fitting experimental data with the proposed equations (indicated in the last column).

Parameter
Value (unit) Equation  Once again, an analytical approximation to the ( / ) A A E D G L 4 simulation data can be obtained by solving the approximate rate equations for evolution of S− and A−regions, as described in detail in appendix C, resulting in the following equation:  (4) and (1), respectively) for different values of L a , with L D fixed at 500 nm ( L D e , reproducing pure line defects). These two curves delimit a phase space that embraces samples with 0D and 1D defects, and the plots in figure 3 provide a user-friendly diagram for the defects quantification.
Notice that, in the largest portion of the Raman diagram shown in figure 3, a given point [ determines unambiguously the pair ( ) L L , a D and therefore allows disentanglement of the contributions from point and line defects, fully based on the D and G Raman spectral information. However, in small parts of the Raman diagram, where different line trajectories cross, there is ambiguity in defining (L a , L D ) and the disentanglement is not possible.
In addition, researchers have to be careful when using figure 3 and related equations for estimating the concentrations of point and line defects in the extreme-left side of the diagram. Within this region, Γ G changes between 15 and 20 − cm 1 for L a between infinity and 45 nm, or L D between infinity and 4.5 nm. Definitive (L a ,L D ) assignments are not accurate here because Γ G changes within this range also because of strain, doping, and number of layers [38,39]. At this extreme, the Tuinstra-Koenig\Cançado relation [8,14],

Application of the method
An example of how the methodology developed here can enhance significantly the importance of using Raman spectroscopy to characterize graphene-related technologies is given now. Hof et al [40] reported the production of graphitic nanoparticles from a sustainable carbon feedstock for ink and conductive coating applications. This work is specifically interesting to be analyzed here, not only because of its technological importance, but also because it brings an extensive characterization of their samples, performed with different techniques. More specifically, five different samples were produced by the cracking of methane/ CO 2 mixtures into graphitic carbon and hydrogen, via the cold microwave plasma method, by setting different initial CO 2 contents, namely 0% (NC 0 ), 0.4% (NC 1 ), 1.7% (NC 2 ), 4.9% (NC 3 ) and 7.4% (NC 4 ). The synthesis was followed by a controlled heat-treatment at 500° C for 6 h, for sample purification [40]. Besides Raman spectroscopy, the authors performed resistivity measurements, and demonstrated that the lower average resistivity of the films produced from these nanocarbon materials was obtained for NC 2 , followed by NC 1 . They also performed thermogravimetric analysis, showing that the mass loss was minimized for NC 1 , followed by NC 2 . X-ray diffraction experiments indicated the highest crystallinity for NC 1 , followed by NC 0 and then NC 2 . Sample NC 4 exhibited the highest average volume resistance, highest mass loss, and lowest crystallinity. ), shown in figure 4(b), where the respective (L a ,L D ) values are extracted from figure 3. The samples occupy different positions on the 0D versus 1D defect diagram, and this information can be used to get insights into the quality of the resulting material. Sample NC 4 is the most defective, exhibiting the worse compromise between density of point defects (highest) and crystallite size (lowest). This is in agreement with the results reported by Hof et al [40]. With respect to crystallinity, sample NC 0 appears as being superior to NC 2 and NC 3 , and this is not because the density of point defects is small, but because the crystallite sizes for the samples obtained without the CO 2 additive are considerably higher than the ones obtained with the CO 2 additive. The mass loss observed by Hof et al [40] scales very nicely with the density of point defects. Finally, the average volume resistivity is observed to be best for sample NC 2 , which shows the best compromise between lower amount of defects and larger crystallite sizes, i.e. the best compromise on minimizing the amount of one and two-dimensional defects together. On this sense, the Raman diagram shown in figure 4(b) suggests that a fine tunning should be performed with samples where the CO 2 additive varies between 0% (NC 0 ) and 1.7% (NC 2 ), trying to populate the low-right quadrant of the Raman diagram in figure 4(b).

Conclusions
In summary, figure 3 and the related equations contain clear specifications for the quantification of defects, establishing a protocol for disentangling the contributions of point-like and line-like defects to the Raman spectra of 2D sp 2 carbon materials. It can be useful for understanding and optimizing processes of synthesis, purification, and functionalization, where both L a and L D can change, as demonstrated here for graphitic nanocarbon made for inks and conductive coatings. L a and L D appear as the significant structural parameters that rule transition between perfect graphene to amorphous carbon, and figure 3 can be used to identify their values. calculated by assuming that the probability of D band scattering decreases exponentially with the distance from the S−region, with decay length e (electron coherence length) [30]. Polarization effects are taken into account by considering electric fields polarized along y, which means that only distances from the left and right vertical S−region ribbons are relevant [30]. The contribution to ( / ) A A E D G L 4 from a square crystallite of size L a is then given by: where x is the horizontal coordinate of a point inside the crystallite and the integral is taken over the A-region (which is the whole crystallite except the S-region).

Appendix A. Exemplary Raman spectra along the amorphization trajectory for samples with line and point defects
The analytical approximation given in equation (2) of the main text is obtained by substituting L a for the average value L a and performing the integration in equation (B.2).

Appendix C. Calculation of (A D /A G ) for single crystallites with point defects
We start our analysis with the assumption that the ( / ) A A E D G L 4 intensity ratio is composed by the sum of three contributions S is the contribution from S-regions S is the contribution from S-regions near borders, and ( / ) ( ) A A D G 0D,1D A is the contribution from A-regions (including activation due to both, point defects and borders).

C.1. Calculation of
This is exactly the same as in the case of an infinite graphene sheet [19] ( ) where ( ) σ f S,1D is the ratio between the S-region near the borders (A B ), and the total area A T , which is The σ-dependency of ( ) σ f S,1D comes from the fact that each point defect near a border removes an area πr S 2 from the A B area generating D band scattering, that is with N being the total number of point defects. This equation can be readly integrated to give where we have used / σ = N A T . Dividing both sides of equation (C.7) by A T yields Here, the cutoff radius r c is determined by the condition that the area of integration is the same for a circular region and a square cluster of side − L l 2 a S , that is  We propose a solution to (C.14) of the type