Background Issues in X-Ray Diffraction and Raman Spectroscopy of Carbon Materials

Abstract

Removing background signals is a common preprocessing step, but it is not without drawbacks. In X-ray diffraction data, background correction can artificially symmetrize diffraction peaks, which becomes a critical issue for lamellar materials such as graphenic carbon when the Laue indices lie in the plane (e.g., the 10 and 11 peaks). We discuss several approaches to background correction and their implications for the resulting data. In Raman spectroscopy, defects activate the phonon density of states, leading to higher intensity below the D band than above the G band, with respect to the Raman shift. After discussing the linear and circular polarization on the Raman selection rules, we show how flattening the background—a widely used measure of disorder—alters the I_D/I_G ratio. Finally, principal component analysis (PCA) provides a useful preliminary exploration of data structure; however, because its components may include negative contributions, it cannot be directly applied to spectral decomposition. In contrast, non-negative component decomposition offers an optimal way to preserve the Raman background, even in the presence of luminescence. We confirm our analysis with ANOVA p-values.

1. Introduction

In carbon materials, the signal originating from the ordered phase is superimposed on that arising from the disordered phase. In both X-ray diffraction and Raman spectroscopy, this additional contribution of the latter is present and commonly referred to as the background. However, in case this background corresponds to a genuine physical signal, it may interfere with the determination of structural parameters, particularly L_a, the in-plane crystallite size. Background subtraction is therefore routinely applied, either as a post-processing operation or directly during data acquisition, as proposed by instrument manufacturers. Various methodologies have been developed, including linear, quadratic, exponential, and adaptive background corrections. Among them, the rolling-ball algorithm can be implemented, and in many cases, a control parameter allows the user to choose between aggressive or conservative subtraction. Following this correction, peak height and full-width at half-maximum (FWHM) can be easily extracted, simplifying phase analysis. Nevertheless, while most authors do not report the uncertainties associated with background removal, this step is rarely considered a potential source of error, even though it can significantly affect the reliability of structural parameter values obtained from the analyses.

In X-ray diffraction, individual peaks can be analyzed separately or through dedicated software specifically designed for poorly ordered graphenic carbons exhibiting crystallites with turbostratic stacking. The quantitative analysis of such materials traces back to the pioneering work of Warren (1941) [1]. It subsequently became widely accepted that the FWHM of the 10 or 11 bands typical of the turbostratic structure can be used to estimate the in-plane crystallite size, L_a, according to the Scherrer equation with a larger shape-factor K, typically 1.84 instead of 0.94 [1,2]. The so-called “bottom-up approach”, a rigorous method reconstructing the diffractograms by mixing the right proportions of calculated diffractograms generated by basic stacking sequences, further refined the value of K for small crystallite sizes as well as the apparent lattice parameters [3]. Nevertheless, the background arising from (i) coherent scattering (such as Rayleigh-type diffusion), (ii) the presence of voids and of nanometric domains at low scattering angles, and (iii) Compton scattering at high angles, complicates this analysis. As a result, a highly localized fitting strategy, restricted to a narrow 2θ range, is often adopted. Despite its minimalist nature, this approach is generally sufficient since only a limited number of parameters are intended to be obtained. In all XRD studies carried out since the forties and until recently, only two graphene-stacking configurations have been considered, turbostratic (random stacking with no correlation between the graphene layers) and graphitic (ordered stacking ABA etc. as in graphite, with full correlation between the graphene layers). Recently, while using the bottom-up approach, AB-stacked graphene pairs (two graphene layers) were used as a third possible elementary structural component of the crystallites [4]. With these three components, the asymmetry of the 10 and 11 bands and their progressive transformation (in graphitizable carbons) into 100/101 and 110/112 peaks, respectively, are fully explained and modelled, demonstrating full consistency between the parameters derived from either the 10 or 11 bands [4]. Importantly, the procedure used enabled the elimination of the requirements to determine the K value. Conversely, stacking faults were shown not to affect the 110 reflection, as opposed to the 100 one [5]. Instead of dealing with isolated peaks, the entire diffractogram can also be modeled. This approach, developed by several research groups, has led to the creation of dedicated fitting software. Among them, CarbonXS [6], one of the earliest tools, allows fitting with up to two phases, while more recent programs such as OctCarb [7] have been developed for poorly ordered graphenic carbons.

Although these fitting software programs provide accurate fits, the large number of adjustable parameters can make the interpretation of the results particularly complex.

The situation is even more complex in Raman spectroscopy. As an optical technique, Raman spectra are often superimposed with luminescence background arising from localized electronic states, which may be related to structural defects, disorder, or charge-transfer processes. When present, luminescence often modifies the background significantly. As the Raman process is faster than luminescence, time-resolved spectroscopy is attractive [8], but due to the repetition with large pulse power, it is limited to transparent materials or materials with a large Raman cross-section, which is not usually the case with carbon materials. Thus, in regular cartography, there is no easy solution to suppress this luminescence. In graphenic carbons with low crystallinity (small L_a and L_c), phonon activation through double-resonance processes [9] results in the appearance of only two main bands [10], whose positions and intensities may vary with the excitation wavelength [11]. Consequently, the fitting of these bands remains challenging. Spectral decomposition is often problematic, whether using a two-band or a five-band model. In the two-band approach, Ferrari and Robertson employed Fano-type line shapes to fit both bands [10], while in our group, we use a modified Fano profile for the G band combined with two Voigt functions centered at the same position for the D band [12]. Because the Fano function does not return to zero intensity below or above the position of the maximum of the band intensity, setting the background level becomes essential to achieve a physically meaningful fit [13] and to avoid non-physical coupling leading to artificial minima in the error function. In the five-band decomposition model, as described by Sadezky et al. [14], the number of adjustable parameters is large, necessitating certain assumptions, such as fixing at least the position of one band, typically near 1500 cm⁻¹. In this case, the background must be carefully treated, usually until complete removal, to obtain reliable results. However, such multi-band fittings are often empirical in nature and should be regarded as mathematical curve-fitting rather than true physical deconvolutions; the term “decomposition” is therefore more appropriate. After such an empirical multi-band fitting, the FWHM values of the D and G bands can be determined. Some authors, however, prefer to measure the parameters directly from the raw spectra, such as the heights of the D and G bands, or the upper linewidth of the G band [15], to bypass the complexity of both the fitting and the background correction. The FWHM of the D and G bands are key indicator of the degree of crystallinity, as they are directly related to both the crystallite size and the occurrence of in-plane defects (both being related but not strictly equal), and exhibit distinct behaviors depending on whether the broadening originates from point defects or from size confinement effects [15].

Multivariate data analysis methods are also an option to explore. Principal component analysis (PCA) is commonly applied to identify dominant sources of variance and reduce dimensionality, facilitating the comparison of large series of spectra. However, its unconstrained nature may hinder the direct physical interpretation of the resulting components. In contrast, non-negative decompositions enforce positivity constraints that are well suited to spectral intensities, often yielding components that can be directly related to physically meaningful contributions. Beyond linear decompositions, recent advances in statistical learning for high-dimensional data provide a broader framework for addressing variability, noise, and model uncertainty in spectroscopic analyses [16,17,18].

In this work, we demonstrate how removing the background influences the choice of fitting methodology and why a change in strategy requires a complete re-evaluation of the analytical approach. For X-ray diffraction, the implementation of bottom-up modeling is now straightforward, allowing a more appropriate treatment of bidimensional materials. In contrast, the situation is more complex for Raman spectroscopy, as no predictive model currently allows the calculation of spectra for low-crystallinity graphenic materials due to the strong coupling between electronic and phononic states. As a result, spectral analysis is often operator-dependent. Nevertheless, recent statistical methods offer promising tools to improve data interpretation. All MATLAB interfaces developed for data processing are provided as Supplementary Materials.

2. Materials and Methods

Commercial carbon black N300 (CB-N330) was selected for data processing as a representative material of low-crystallinity graphenic carbon. For the study of circular polarization selection rules, a piece of HOPG was used.

X-ray diffraction patterns were recorded on a Bruker D8 Advance diffractometer equipped with a non-monochromated Cu Kα radiation source (average λ = 1.5406 Å).

Raman spectra were acquired using 2 instruments: a T64000 for polarization selection rules at 515 nm (acquisition time of 200 s with one accumulation), and an XploRA Plus spectrometer (green excitation wavelength of 532 nm) for Raman mapping without autofocus, followed by data treatment and statistical analysis (acquisition time of 10 s with two accumulations). Measurements were performed with a low laser power (1.4 mW) and a Long Working distance ×50 objective. For Raman mapping, the sample surface is not perfectly flat at the micrometer-scale, and residual local height-variations may contribute to the observed point-to-point intensity variations.

All X-ray and Raman data were processed using three customized MATLAB (version 2025b) interfaces, both provided in the Supplementary Information.

3. Experiments and Discussion

3.1. X-Ray Diffraction

In Figure 1a, the raw diffractogram of carbon black (CB) is shown together with the different background-removal approaches. The Bruker DIFFRA.EVA software outputs data with modified intensities, so only the peak shapes, not the absolute values, can be directly compared with other treatments. This procedure symmetrizes the peaks, particularly the 10 and 11 reflections. As a result, the FWHM can be easily extracted, allowing the use of K = 1.84 for crystallite size determination. However, by doing so, a further lack of accuracy is introduced, because this K factor value is no longer adapted to the hk peaks for which the asymmetry is minimized by calculation. With our example, using the 10 band, we found an overestimated L_a = 3.0 nm. However, the genuine turbostratic line-shape is lost, as evidenced in Figure 1b, which shows a simulation of the diffractogram obtained with the XaNSoNS freeware [19].

The rolling-ball method performs even worse: in this case, the peak becomes completely symmetric.

We therefore propose a monotonically decreasing background based on the atomic form factor [20], which catches the decrease for all existing atoms, involving 9 adjustable parameters as follows:

$$f\left(q\right)=\sum _{i=1}^4{a}_i exp\left(-b_{i }{\left({\displaystyle \frac{q}{4\pi }}\right)}^2\right)+c$$

For purely amorphous carbon materials, we expect a signal that approximately corresponds to this profile. Since the background has multiple origins, as discussed in the Introduction section, this formulation has the advantage of not artificially symmetrizing the peaks when they are intrinsically asymmetric. As reported in Figure 1, the advantage is that the signal in the region of the 10 peak closely matches the exact simulated profile. Indeed, for intermediate 2θ values, additional low-angle (Rayleigh, …) and high-angle (Compton) contributions do not significantly distort the signal. For the other peaks, selecting appropriate fitting ranges makes removing the background easier. With the MATLAB-2025b program given in the Supplementary Information, we obtain L_c = 1.5 nm and L_a = 2.0 nm.

This background-removal method is better suited to lamellar materials, such as graphenic carbon, than a more “aggressive” approach, which would symmetrize the peaks. It should be noted that this is not an exact solution; it is merely the least undesirable option for retaining the large asymmetry of the 10 and 11 bands.

Obtaining the correct signal also implies using the appropriate peak functions. Since these functions are not implemented in commercial software, our approach remains restricted to home-made routines and to diffractogram simulations, such as those shown in Figure 1b using XaNSoNS software [19] or others such as Debyer [21]. With the latter, due to the large number of adjustable options, some misused and non-physical results (notably the intensity deficits at low angles) are sometimes published. From Figure 1b, one can see that the decrease in intensity after the 10 peak is slight. At low angles, when the number of layers is small, interference effects lead to a complex broad band [22], whereas the background rises significantly when only a single layer is considered.

For X-ray diffractograms, it is important to approach the signal using good initial values. A low signal to noise (S/N) ratio can be compensated for by starting close to the solution value to ensure a proper fit. As L_a and L_c are determined by the peak or band width, and large widths are associated with small crystallite sizes, the uncertainty in L_a and L_c is small even if the S/N ratio is large. We can calculate the standard deviation between the fit and the curve from the deviation for 10 (std = 70, amplitude of typically 800), but unfortunately, it depends strongly on the selected 2θ range (for example, the broad 002_t/004_g peak impacts the signal coming from 10 and should be removed from the fit). Depending on the user choices, the L_a value is, with our approach, between 19 and 21.5 nm. As 2000 points are used for each band, the uncertainty due to the S/N ratio (70/800/$\sqrt{2000}$ = 0.002) is less significant than the issue of both the background and the 002_t/004_g peak removal.

3.2. Raman Spectroscopy

3.2.1. Utility of Raman Selection Rules

In crystalline materials, the D and D′ bands, arising from double-resonance processes near 1350 and 1620 cm⁻¹ and exhibiting A_1g symmetry, are only weakly activated. By contrast, the G mode, with E_2g symmetry, is intense. Linear selection rules are often reported, whereas circular polarization is more commonly used with 2D materials other than graphene.

The G band corresponds to the doubly degenerate E_2g mode, described by two Raman tensors associated with orthogonal in-plane phonon polarizations, denoted E_2g(x) and E_2g(y). The tensors are:

$$R_{E2g}(\mathrm{x})=\left(\begin{array}{ccc}a& 0& 0\\ 0& -a& 0\\ 0& 0& 0\end{array}\right) R_{E2g}(\mathrm{y})=\left(\begin{array}{ccc}0& a& 0\\ a& 0& 0\\ 0& 0& 0\end{array}\right)$$

For the D mode:

$$R_{A1g} = \left(\begin{array}{ccc}b& 0& 0\\ 0& b& 0\\ 0& 0& b\end{array}\right)$$

Using circular polarization within the Jones formalism, the right-hand circular state is: ${\sigma }_{+}={\displaystyle \frac{1}{\sqrt{2}}}(1,i,0)$.

In a ${\sigma }_{+}{\sigma }_{+}$ configuration, the intensity reads $I={\sigma }_{+}^{†}R{\sigma }_{+}$, which yields a signal for A_1g modes but not for E_2g as already reported [23]. This is the only possible configuration that provides a way to enhance the visibility of the D′ band. Unfortunately, for low-crystallinity carbons, selection rules are partially lost [24] as reported in Figure 2. In addition, circular polarization is rarely compatible with modern spectrometers, since notch or edge filters distort the polarization state; reflection at 45° also degrades the circular polarization (explaining why it is not perfect in our case, as a microscope was used). Figure 2b shows an example for CB-N330, compared to graphite (Figure 2a). In this case, the D′ band merges with the G band [13], and while the G band is not strongly attenuated, the D band is altered in a complex manner. Consequently, for low-crystallinity carbon materials, selection rules, which reduce the Raman signal by roughly a factor of two, do not provide a clear advantage.

3.2.2. Taking into Account the Background

With standard acquisition software, linear background removal is available. It flattens the spectrum and is very practical in cases of luminescence. The same post-treatment can be applied before fitting the data. While Ferrari and Robertson [10] used a Fano lineshape to handle the asymmetry, other authors like Schwan et al. [25] adopt other approaches, giving different dependence at small crystallite sizes (typically $L_a<2$ nm), whereas for $L_a>2$ nm, the Tuinstra–Koenig $1/L_a$ dependence [26] still applies. These discrepancies are often attributable to the background subtraction procedure and to the specific fitting strategy employed [27]. As we will see, many of these discrepancies originate solely from the way the data are treated. Figure 3 shows two different approaches for data processing. In Figure 3a, the raw data are fitted with a double asymmetric Voigt profile for the D band, whereas the G band is described by a left-modified Fano line-shape complemented by a Voigt function on the high-wavenumber (right) side. Although this procedure is quite complex, it allows a nearly perfect fit of the spectrum to be obtained using only two physically positioned bands. Since the background level is not identical below and above the D and G features with respect to the Raman shift, intensities can also be measured directly on the spectrum. The reported values (Figure 3a) yield an I_D/I_G ratio of 1.03 (from band heights), while the ratio obtained from the fitted intensities is only 0.91.

In Figure 3b, the spectrum after linear background removal is shown. This approach is often combined with a five-band fitting. To ensure consistency, we fixed the position of two bands, one at 1500 cm⁻¹ with a Gaussian profile and one at 1620 cm⁻¹ with a Voigt profile, while all remaining bands were free to adjust their position and also fitted with Voigt functions. Two of these bands can be assigned to the D and G bands, respectively, giving an I_D/I_G ratio of 1.28. If the heights are directly measured instead, clearly independently from the fitting model, the ratio is 1.10.

One can note that the fitting with five bands is more a decomposition of a complex shape than a deconvolution of true symmetric bands. Nevertheless, there is a consensual attribution of the five Raman bands commonly used to model graphenic carbon materials [9,14,22]. The G band, centered around 1580 cm⁻¹, is associated with the in-plane stretching mode of sp²-bonded carbon and is modeled using a Voigt function to account for both homogeneous and inhomogeneous broadening. The D1 band, which corresponds to the well-known D band, located near 1350 cm⁻¹, is also fitted with a Voigt profile and reflects the presence of defects in the sp² network, including point defects, edges, and any modification of the polarizability of the bonds. A secondary defect-related contribution, the D2 band at approximately 1620 cm⁻¹ (also called D′), is described by a Voigt function and is introduced to account for disorder-activated scattering when the G-band wavenumber is constrained during fitting. In practice, this band merges with the G band for very defective carbon materials. The D3 band, centered around 1500 cm⁻¹ and modeled with a Gaussian profile, is attributed to amorphous carbon contributions. This decomposition provides a physically meaningful framework for discriminating Raman signals coming from ordered sp²-carbon and those coming from defect-induced and amorphous components. Finally, the D4 band is often observed around 1200 cm⁻¹ in very low crystallinity graphenic carbons. This band has many origins, due to the double resonance effect, including one attribution to vibrations involving sp²–sp³ mixed bonding, polyene-like structures, or carbon atoms associated with heteroatoms and ionic impurities. In our case, as CB is made from polycyclic compounds (tar), the most probable origin is also double resonance.

Consequently, a large dispersion of I_D/I_G values emerges, depending entirely on the chosen background removal and fitting procedures. Applying the Ferrari–Robertson laws (L_a lower or larger than 2 nm) requires a two-band fit with the background left intact [10]. Conversely, in the Tuinstra and Koenig pioneering work [26], peak heights were used. Removing a supposed linear background is therefore the simplest and most commonly applied option, but when L_a is very small, meaning that the phonon density of states is activated, this procedure clearly overestimates the I_D/I_G ratio. In such cases, correlations with other properties remain possible, but the extracted values no longer represent a universal, accurate I_D/I_G ratio.

After fitting the Raman spectrum the standard deviation between the two- and five-band fits to the data can be calculated, giving values of 14.05 and 14.03, respectively, which are nearly identical. For the uncertainty in the I_D/I_G ratio, as the height is around 600 and 30 points are used for fitting, the uncertainty for each intensity is approximately 14/600/$\sqrt{30}$ = 0.004. This leads to an uncertainty of less than ±0.01 for the I_D/I_G ratio. Consequently, the reported differences in the I_D/I_G ratio are significant.

Thus, we show that even from a single spectrum, multiple values of the I_D/I_G ratio can be obtained depending on the data treatment. It is therefore crucial to recognize that the main Raman signal originates from phonons coupled with electrons (obviously concerning the D band but also the G band, in some cases), and that the background is non-monotonic due to the activation of several disorder-related modes (with respect to the signal from perfect graphenic systems). Consequently, any comparison between published spectra requires a careful analysis of the fitting and background treatments used, rather than a simple comparison of the I_D/I_G ratio values.

This means that the fitting and background removal procedures must be clearly described in any publication; otherwise, it becomes impossible for readers to reproduce the analysis, reliably interpret the reported findings, and compare samples accurately.

The fitting with 2 or 5 bands, as well as the fitting of the diffractograms in XRD, can be found in the SI of previous works [22,28,29] showing the robustness of these approaches, even if the 5 bands fitting is often arbitrary because of the dependence on the operator choices [21].

3.2.3. Use of Statistics

When a sufficiently large number of spectra is available, statistical analysis becomes interesting to objectively identify significant spectral components (often due to several crystalline phases). In this context, separating luminescence contributions from the Raman signal is therefore a legitimate issue. In cases where the background increases due to luminescence, removing it while keeping the physical contributions of low crystallinity and defects can remain challenging. During mapping, hundreds of spectra are typically collected, which often vary from one location to another. Figure 4a shows Raman spectra of CB-N330, in which the background clearly depends on the location of the specimen preparation. Considering a 25 µm × 25 µm map, Figure 4b displays the principal-component analysis (PCA), which identifies two principal components (PCs). The method is explained and implemented in MATLAB by Tharwat [30]. The PCA-1 is typical of spectra with strong luminescence, while PCA-2 has a negative part in the high wavenumber range, preventing meaningful analysis. Figure 4c presents the non-negative matrix factorization (NNMF), also referred to as non-negative matrix decomposition (NNMD), with two components, implemented following Kasai [31]. NNMF-1 is close to the spectrum with the maximum luminescence, while NNMF-2 is close to the spectrum with no luminescence. To assess the relationship between the extracted components and the experimental groups, a one-way ANOVA was performed on the component scores obtained from both PCA and NNMF. We consider two populations of similar sizes for the analysis. For PCA, the analysis revealed that the first principal component (PC1) was strongly dependent on the group factor (p-score = 1.9 × 10⁻¹⁰⁶), whereas the second component (PC2) showed no statistically significant effect (p-score = 0.67), indicating that PC2 mainly captures intra-group variability. In contrast, ANOVA applied to the NNMF weight matrix showed that both components were highly significant, with p-score values of 9.7 × 10⁻¹⁰⁵ for component 1 and 4.7 × 10⁻⁶⁰ for component 2. This result indicates that, unlike PCA, NNMF distributes the group-related information across multiple non-orthogonal components. Beyond its statistical relevance, NNMF is particularly attractive from a physical standpoint, as its non-negativity and additive nature yield components that can be directly interpreted as meaningful spectral contributions, the NNMF-2 being really close to the spectra with the lowest background, in contrast to PCA components, which often mix positive and negative values.

4. Conclusions

Correcting Raman spectra from background signals remains a challenging task, in particular when dealing with low crystallinity graphenic carbons. When the background exhibits non-monotonic variations, there is no reliable a priori method to extract it. In X-ray diffraction of carbon materials, only a limited number of diffractograms are typically available, and certain assumptions must therefore be made. Assuming a monotonically decreasing background is the best approach when asymmetric functions are employed to fit the 10 and 11 bands. One can note that using a commercial background correction procedure partially symmetrizes the bands and thus introduces errors in the crystallinity parameter L_a value calculated from the bands. A similar situation arises in Raman spectroscopy. When different background levels are present before and after the main bands, fitting the signal, even with low crystallinity, becomes possible, allowing for determining the two-phonon contribution of the D band with a double Lorentzian component and the G band with a single Fano-shaped component. Conversely, the five-symmetric-band approach requires flattening the baseline to zero on both sides of the spectral range. To access the true Raman spectral information, non-negative component decomposition offers a promising and powerful alternative and should be adopted more widely in spectral analysis when possible.