This review discusses advances that have been made in the study of defect-induced double-resonance processes in nanographite, graphene and carbon nanotubes, mostly coming from combining Raman spectroscopic experiments with microscopy studies and from the development of new theoretical models. The disorder-induced peak frequencies and intensities are discussed, with particular emphasis given to how the disorder-induced features evolve with increasing amounts of disorder. We address here two systems, ion-bombarded graphene and nanographite, where disorder is represented by point defects and boundaries, respectively. Raman spectroscopy is used to study the ‘atomic structure’ of the defect, making it possible, for example, to distinguish between zigzag and armchair edges, based on selection rules of phonon scattering. Finally, a different concept is discussed, involving the effect that defects have on the lineshape of Raman-allowed peaks, owing to local electron and phonon energy renormalization. Such effects can be observed by near-field optical measurements on the G′ feature for doped single-walled carbon nanotubes.
1. Introduction: an historical overview
In general, disorder-induced symmetry-breaking effects play a very important role in the determination of several material properties, such as transport properties and the relaxation of photo-excited carriers. In particular, sp2 carbons, which have high symmetry, are highly sensitive to symmetry-breaking defects. Disorder and symmetry breaking are observed sensitively by spectroscopy, which depends strongly on crystal symmetry (Ashcroft & Merman 1976; Kittel 1986). The presence of disorder in sp2 hybridized carbon systems leads to rich and intriguing phenomena in their resonance Raman spectra, thus making Raman spectroscopy one of the most sensitive and informative techniques to characterize disorder in sp2 carbon materials. Raman spectroscopy has thus become a key tool and is widely used to identify disorder in the sp2 network of different carbon structures, such as diamond-like carbon (DLC), amorphous carbon, nanostructured carbon, as well as carbon nanofibres, nanotubes and nanohorns and most recently graphene (Ferrari & Robertson 2004; Novoselov et al. 2005; Zhang et al. 2005; Pimenta et al. 2007; Jorio et al. 2008).
The Raman spectrum of crystalline graphite is marked by the presence of two strong peaks centred at 1580 and 2700 cm−1, named the G and G′ bands, respectively, where the G label refers to graphite (figure 1). The G band is a first-order Raman-allowed feature originating from the zone centre (phonon-wavevector q=0), corresponding to the in-plane optical phonon modes (Tuinstra & Koenig 1970a,b). The G′ band is a second-order (two phonon) feature that allows exploration of q≠0 phonons (Nemanich & Solin 1977, 1979).
In the Raman spectra obtained from samples with small crystallite size La (less than 0.5 μm, i.e. smaller than the wavelength of light), the presence of an additional dispersive peak centred at approximately 1350 cm−1 is observed when using laser excitation at 2.41 eV. This feature is assigned to the breathing of the carbon hexagons that become Raman active at the borders of the crystallite areas owing to the loss of translational symmetry (Tuinstra & Koenig 1970a,b). The frequency of this feature is about half of the second-order G′ frequency, and is related to the phonon of the first-order process of the G′ peak. Since the approximately 1350 cm−1 peak is observed in the presence of defects in an otherwise perfect infinite graphite structure, it has been named the D band (D for defect or disorder). Therefore, its intensity is proportional to the amount of disorder (crystallite boundary) in the sample, and the ratio between the intensities of the disorder-induced D band and the first-order graphite G band (ID/IG) provides a parameter that can be used for quantifying disorder.
In 1970, it was shown that the D to G band intensity ratio (Tuinstra & Koenig 1970a,b) is correlated with crystallite size La by the relation ID/IG=A/La, where A is a constant for a fixed laser excitation energy. This relation can be applied to large enough carbon sp2 crystallites (Knight & White 1989), while the complete amorphization trajectory for ID/IG going down to small La values was proposed. The ID/IG ratio starts to decrease for very small values of La (a few nanometers) because the sp2 carbon hexagonal structure then starts to disappear. Furthermore, this ID/IG intensity ratio was shown to be excitation laser energy dependent (Mernagh et al. 1984). Another disorder-induced band centred at 1620 cm−1 is usually observed in the Raman spectra of disordered graphitic materials, although with smaller intensity than the D band. This feature, reported by Tsu et al. (1978), has been named the D′ band and also depends on La and Elaser.
In 1981, it was shown (Vidano et al. 1981) that the D and G′ bands are dispersive, i.e. their frequencies change with the incident laser energy Elaser, with ΔωD/ΔElaser∼53 cm−1 eV−1 and ΔωG′/ΔElaser∼106 cm−1 eV−1. The out-of-plane stacking order has also been shown to affect the G′ Raman lineshape (Lespade et al. 1982, 1984; Wilhelm et al. 1998). Baranov et al. (1987) proposed that the dispersive behaviour of the D band comes from the coupled resonance between the excited electrons and the scattered phonons, as previously discussed in semiconductor physics (Martin & Falicov 1975; Cardona 1982). The full appreciation of the double-resonance model came in 2000, as discussed by Thomsen & Reich (2000), and their model was extended by Saito et al. (2002) to explain the mechanism behind many other dispersive Raman peaks usually observed in the literature, yielding an explanation for all the features observed in the Raman spectra of ordered and disordered graphite (Ferrari & Robertson 2000). Many of the weak Raman features are dispersive and can be used to determine the graphite phonon dispersion (Saito et al. 2002). In fact, the atomic vibrations with different wavevectors are usually observed by inelastic neutron scattering, making use of momentum conservation requirements. Another interesting result obtained by the double-resonance model (discussed below) was the identification of the atomic structure at graphite and graphene ribbon edges (Cançado et al. 2004).
2. Double-resonance Raman scattering
Defects break the momentum conservation requirement for first order q=0 Raman-allowed phonons, so that, in principle, any scattering event involving phonons in the interior of the Brillouin zone (q≠0) would then be allowed. However, in sp2 carbon materials the resonant electron–phonon scattering processes connecting real electronic states (figure 2a) minimize the denominators in the resonance Raman intensity (or cross section) which are given by (Dresselhaus et al. 2005) 2.1 where ΔEai=(Elaser−(Ea−Ei)−iγr) and γr denotes a broadening factor. Here subscripts i,a,b and c, respectively, denote the initial state, the excited state, the first scattered state of an electron by a phonon and the second scattered state of an electron by a defect. Mop, Mep and Md denote the electron–photon, electron–phonon and electron-defect scattering matrix elements, respectively, and we use the fact that Eb=Ec since the scattering from b to c is an elastic scattering process. These resonant processes enhance the scattering amplitude much more than for non-resonant processes, so that the spectra are dominated by the double-resonance D and D′ band scattering processes, fulfilling energy and momentum conservation. While momentum conservation in a perfect lattice can only be fulfilled by two-phonon scattering processes (G′ band), momentum conservation in the presence of disorder can be satisfied through an elastic scattering process by a defect, represented by dashed arrows in figure 2a.
The dispersion of the frequency of the D, D′ and G′ bands is shown in figure 2b by plotting their frequency dependence on Elaser, and these features are well explained by the double-resonance model. The slope associated with the G′ band is about 106 cm−1 eV−1 and, as stated above, is two times larger than the slope of the D band (53 cm−1 eV−1). The D′ band also exhibits a weak dispersive behaviour, the slope being approximately 10 cm−1 eV−1 (Pimenta et al. 2007).
The D and D′ bands are not the only disorder-induced one-phonon peaks in the Raman spectra for disordered sp2 materials. Any combination or overtone of the six dispersive phonon energy branches in graphene can occur and disorder-induced Raman frequencies can be related to any of the six phonon branches of two-dimensional graphite with the appropriate wavevector which fulfils the double-resonance condition. The intra-valley and inter-valley double-resonance processes are mediated by phonons near the Γ and K (or K′) points, respectively, and we can vary both the resonant k and q values by changing Elaser, as determined by conservation of energy and momentum requirements. Thus by using electronic band structure information, we can determine the phonon dispersion relations around the K and the Γ points, by considering inter-valley and intra-valley processes, respectively. This approach has been used for obtaining the graphene phonon dispersion relations using Raman spectroscopy, as reported by Saito et al. (2002).
3. Quantifying disorder in graphene and nanographite from Raman intensity analysis
The frequencies of the disorder-induced features in the Raman spectra for sp2 carbons are well explained by the double-resonance model. What remains to be established for making Raman spectroscopy an even more powerful tool to characterize disorder in sp2 materials is how to relate specific defects to their corresponding disordering processes, and how to obtain quantitative information about the amount of each type of lattice defect.
To achieve a quantitative description of these phenomena, the effects of disorder on the electron and phonon properties have to be probed in both momentum space (k-space) and real space (r-space), which means Raman spectroscopy has to be combined with microscopy experiments. Transmission electron microscopy (TEM) or scanning tunnelling microscopy (STM) can characterize disorder of the crystal in r-space by probing the local surface density of electronic states, with atomic-level resolution. Simultaneous in situ TEM and Raman measurements are possible (Meyer et al. 2005). However, a special experimental set-up and special sample preparation methods would be needed. Usually, STM and Raman spectroscopy cannot be easily correlated with each other, since optical spectroscopy probes a volume that is limited by the light penetration depth, while STM is mostly sensitive to surfaces. In this context, the possibility of exfoliating graphite to pull out a single graphene sheet provides an ideal situation in which microscopy and spectroscopy can be correlated to probe disorder effects in both r-space and k-space for the same sample. The initial efforts in this research direction are now discussed.
(a) Point defects induced by ion bombardment
The controlled use of ion implantation to study defects in sp2 carbons is a well-established technique (Dresselhaus & Kalish 1992). These experiments are normally carried out as a function of ion dose and for different ion species and different ion energies. Low-mass ions at low ion fluence introduce point defects. Increasing the ion dose causes an increasing density of point defects and eventually causes the damaged regions to overlap, as discussed for ion-bombarded highly ordered pyrolytic graphite (HOPG) (Jorio et al. 2009) and graphene (Jorio et al. 2010a,b, in press; Lucchese et al. 2010; Ferreira et al. in press).
Consecutive Ar+ ion bombardment and Raman spectroscopy experiments were performed on monolayer graphene samples (Lucchese et al. 2010) using low-energy ions to avoid cascade effects. Cascade effects are effects whereby a scattered C atom with a large energy hits another C atom iteratively, similar to the chain reaction of dominoes. The bombarding ion doses vary from 1011 to 1015Ar+ cm−2, which correspond, respectively, to one defect per 4×104 C atoms and to the onset of full disorder in graphene. The defect density in real space was monitored by STM images, which allowed the extraction of the defect density or alternatively values of the average distances between defects (Lucchese et al. 2010). For the highest ion doses of 1015 Ar+ cm−2, where the defect-induced regions start to coalesce, we consider that the defect density increases linearly with bombardment time.
Figure 3 shows the Raman spectra of a graphene monolayer subjected to different ion bombardment intensities. From the pristine sample (bottom spectrum) to the lowest bombardment dose in figure 3a (1011 Ar+ cm−2), the D-band process is activated, showing a very small intensity relative to the G peak. Within the bombardment dose range 1011–1013 Ar+ cm−2, the intensities of the disorder-induced peaks increase. The second disorder-induced peak around approximately 1620 cm−1 (the D′ band) also becomes evident. Above 1013 Ar+ cm−2, the Raman spectra start to broaden significantly and end up exhibiting a profile similar to the graphene phonon density of states (PDOS) for the highest ion dose (Lucchese et al. 2010).
By plotting the ID/IG data as a function of the average distance between defects LD, as shown in figure 3b, it is possible to quantify the disorder evolution in the graphene layer. The ID/IG ratio has a non-monotonic dependence on LD, increasing initially with increasing LD up to LD∼3.5 nm, where ID/IG in figure 3b has a peak value, and then decreasing for LD>3.5 nm. This result is similar to the proposed amorphization trajectory for graphitic nanocrystallites, and such a behaviour suggests the existence of two disorder-induced competing mechanisms contributing to the Raman D-band intensity, which we describe next.
(b) Model for the D-band activated region
The results of figure 3 are modelled in figure 4 by assuming that a single impact of an ion on the graphene sheet causes modifications on two length scales, here denoted by rA and rS (with rA>rS), which are the radii of two circular areas measured from the ion impact point (see figure 4). Within the shorter radius rS, a structurally disordered S-region occurs relative to the point of impact. For distances larger than rS but shorter than rA, the lattice structure is preserved, but the proximity between defects causes a mixing of Bloch states near the K and K′ valleys of the graphene Brillouin zone, thus causing a breakdown of the selection rules, and leading to an enhancement of the D-band intensity. We call this the activated or A-region. In qualitative terms, an electron–hole excitation will only be able to ‘see’ the structural defect if the electron–hole pair is created sufficiently close to the defect site and if the excited electron (or hole) lives long enough for the defective region to be probed by Raman spectroscopy. If the Raman scattering process occurs at distances larger than ℓ=rA−rS from the ion impact point, the wavevector k is a good quantum number for analysing the scattering selection rules and those regions that remain well ordered will only contribute significantly to the G-band feature in the Raman spectra, and not to symmetry-breaking features. Our phenomenological model for the ID/IG ratio is given as a function of the average distance between two defects, LD, 3.1 where IG is considered to be a constant (independent of LD), and fA and fS are simply the fractions of the A and S areas, respectively, with respect to the total area in the graphene sheet. Although both the A and S regions can break momentum conservation, giving rise to a D band, the A regions will contribute most strongly to the D band, while the S regions will make less contribution to the D band owing to the breakdown of the lattice structure itself.
Stochastic simulations were performed for implementing the phenomenological model and they are discussed in detail in Lucchese et al. (2010). Snapshots of the disorder evolution are shown in figure 4b–e for the same argon ion concentrations as in figure 3a. The stochastic simulations of the bombardment process, with the impact points for the ions chosen at random, combined with equation (3.1) and using the parameters CA=4.56, CS=0.86, rA=3 nm and rS=1 nm, give the full line curve in figure 3b, which is in excellent agreement with the experimental results (points) in this figure (Lucchese et al. 2010).
The non-monotonic behaviour in figure 3b can be understood by considering that, for low defect concentrations (large LD), the total area contributing to D-band scattering is proportional to the number of defects, thus giving rise to a dependence that works well for LD>2rA. Upon increasing the defect concentration, the activated regions start to overlap and these regions eventually saturate. The D-band intensity then reaches a maximum and a further increase in the defect concentration decreases the D-band intensity because the graphene sheet starts to be dominated by the structurally disordered areas.
The length scale rS=1 nm, which defines the structurally disordered area, is in excellent agreement with the average size of the disordered structures seen in the STM images. This parameter should, however, not be universal, but it should be specific to the bombardment process and ion bombardment conditions. The Raman relaxation length ℓ for the defect-induced resonant Raman scattering in graphene for Elaser= 2.41 eV (514 nm) is found to be ℓ=rA−rS=2 nm. Here, we discuss the relaxation length for the excited electron, and this should not be confused with the relaxation length for the phonons. The CA parameter in equation (3.1) is a measure of the maximum possible value of the ID/IG ratio in graphene, which would occur in a hypothetical situation in which K–K′ wavevector mixing would be allowed everywhere, but no damage would be made to the hexagonal network of carbon atoms. CA should then be defined by the electron–phonon matrix elements, and the value CA=4.56 is then in rough agreement with the ratio between the electron–phonon coupling for the iTO phonons evaluated between the Γ and K points in the Brillouin zone (Lazzeri et al. 2008). The CS parameter is then the value of the ID/IG ratio in the highly disordered limit, which has not yet been addressed theoretically.
Finally, for practical use, it is important to have an equation relating ID/IG to LD, and such an equation can be obtained by solving the rate equations for the bombardment process. The entire regime () can be fitted using 3.2 Fitting the data in figure 3b with equation (3.2) gives CA=(4.2±0.1), CS=(0.87±0.05), rA=(3.00±0.03) nm and rS=(1.00±0.04) nm, also in excellent agreement with experiment and consistent with the parameters obtained by computational modelling (Lucchese et al. 2010).
The model of equation (3.2) provides a method to accurately quantify the density of defects σ or, equivalently, the average distance between defects (LD=σ−1/2) in graphene. As the defects start to coalesce, the expected behaviour occurs, and here , where A=(102±2) nm2 and LD>6 nm. When the defects start to coalesce, there is a competition between the two disorder mechanisms, and equation (3.2) can be used to determine the relative importance of each mechanism quantitatively. The present results discussed for graphene are similar to what has been observed in ion-bombarded graphite (HOPG) (Jorio et al. 2009), although some details are different. First, for HOPG a more intense G band is always observed owing to the contribution from the undisturbed under-layers. Second, for graphene, above 1015 Ar+ cm−2, the Raman spectra show a decreased intensity, indicating full amorphization or partial sputtering of the graphene layer. For HOPG, above 1015 Ar+ cm−2, ID/IG saturates and no further change is observed in the Raman spectra because of the large number of layers that have been amorphitized and/or sputtered. This behaviour is seen in the inset to figure 3b, which shows the ID/IG evolution for two HOPG samples of different thicknesses. Despite differences in absolute values, which depend on the number of undisturbed under-layers (the data given by the diamond-shaped points in the inset to figure 3b coming from a thicker HOPG sample were scaled by ×3.5), the ID/IG values increase and saturate at about LD=9 nm when increasing the ion dose.
(c) One-dimensional defects represented by the boundaries of nanocrystallites
Next let us discuss the D-band image at the edges of nanographite. The defect associated with an edge has a one-dimensional character and it is only able to transfer momentum in the direction perpendicular to the edge. Figure 5a,b shows two confocal Raman images of a 6 nm high HOPG crystallite deposited on a glass substrate. Figure 5a shows a Raman image of the crystallite, obtained by plotting a spatial dependence map of the G-band intensity, while in figure 5b a similar spatial dependence of the intensity of the disorder-induced D band is shown and here the boundary of the crystallite is highlighted. Figure 5c shows two Raman spectra, one taken at an interior point of the crystallite and the other at the edge. It is clear from figure 5a–c that the G-band intensity is uniform over the whole graphite surface, while the D-band intensity is localized where the crystalline structure is not perfect, mostly at the edges of the crystallite. The edge also behaves like a symmetry-breaking effect and causes D-band intensity. Note also that the D-band intensity varies from edge to edge, and this D-band intensity is dependent on the light polarization direction and the atomic structure at the edge, as discussed in this section.
For evaluation of the ID/IG dependence on the crystallite dimensions, one can consider a square of side La, for which the intensity of the G band will vary as , where La is the mean crystallite size. The intensity of the D band will, however, depend on the width δ of the ‘border’ where the D band is activated, and is given by . The intensity ratio will then be given by 3.3 where α is dependent on the appropriate matrix elements (Pimenta et al. 2007).
In the limit La≫δ, equation (3.3) can be simplified to yield the Tuinstra–Koenig relation 3.4 where the value of the empirical constant C(Elaser) depends on Elaser. One could then expect that, once the relaxation length and matrix element ratio were measured for the D-band scattering in ion-bombarded graphene, these values could just be used to obtain α and δ. However, the relaxation length and matrix element ratio depend on the structurally disordered area (SS), which is not well defined for nanographite. Figure 6 shows two STM images with atomic resolution obtained from the surface of a crystallite in a nanographite sample with La=65 nm. The atomic arrangement of the carbon atoms observed in these images indicates that the samples are formed by nanographitic crystallites, with a disordered grain boundary between crystallites (Cançado et al. 2007). Variability associated with grain boundaries such as in figure 6 may also be responsible for the different ID/IG versus La results observed in the literature.
Furthermore, the empirical constant C(Elaser) has been known to depend on Elaser since 1984 (Mernagh et al. 1984), but C(Elaser) has been quantitatively determined only more recently (Cançado et al. 2008), using experimental results from nanographites with different La values prepared from DLC films heat treated at different temperatures Thtt (Cançado et al. 2008). The La sizes were determined by using both STM and X-ray measurements and these La values were correlated with the ID/IG intensity ratios measured at different laser energies leading to a general equation for determining ID/IG independent of laser excitation energy (Cançado et al. 2008). This empirical equation was developed as follows. Shown in figure 7a are the results from Raman scattering experiments performed at room temperature with different Elaser values, showing spectra of the D, G and D′ bands for the Thtt=2000°C sample (La=35 nm) for five different Elaser values. The spectra are normalized to the G band intensity, and clearly the ratio (ID/IG) is strongly dependent on Elaser. Figure 7b shows the Raman spectra using Elaser=1.92 eV for samples with different Thtt values, thereby giving rise to samples with different crystallite sizes La (Cançado et al. 2008).
Figure 8a shows a plot of (ID/IG) versus 1/La for all samples shown in figure 7. Noting that ID/IG for a given sample is strongly dependent on Elaser, we see that all these curves collapse onto the same curve in the ( versus La plot shown in figure 8b, demonstrating that a general equation is obtained for the determination of the nanographite crystallite size La using any laser line in the visible range (Cançado et al. 2008) 3.5 where the laser excitation is given in terms of both Elaser (eV) and wavelength λlaser (nm). The samples used for establishing the empirical equation (3.5) can be considered as model samples and the use of equation (3.5) for any general sp2 carbon-based material would probably need to be generalized. One example of such a study has been made for carbon foams (Barros et al. 2005).
(d) Absolute Raman cross section
Measuring the absolute cross section for the Raman scattering processes is challenging, since the Raman signal depends strongly on the specific set-up (specific optics), on alignment and on Elaser. This is the reason why the intensity ratio ID/IG has been used systematically for quantifying disorder. Some argue that the intensity ratio ID/IG′ should instead be chosen because both the D and G′ bands involve a very similar phonon (inter-valley iTO, q≈2k). However, the D and G′ bands differ strongly in energy, and different Raman set-ups would give different responses for a given sample.
However, Cançado et al. (2007) have done all the calibration procedures for measuring the absolute Raman cross section of the D, G, D′ and G′ bands. In this work, the dependence of ID/IG on La was shown to come from ID, while IG was found to be independent of La within the measured La range (from 20 to 500 nm). For the Elaser dependence, the double-resonance features were shown to be Elaser independent, while IG shows the E4laser dependence expected from scattering theory, as discussed above (Cançado et al. 2007). It is not yet known whether the E4laser dependence will also be observed for zero-dimensional (e.g. ion-induced) defective graphene. Actually, one of the open fields in the Raman spectroscopy of sp2 carbons is what rules determine the intensity of the double-resonance features. Some theoretical work has been done (Sato et al. 2006; Basko 2008) on this topic, but the results are still not at the level of explaining actual experimental observations.
4. Defect-induced selection rules: dependence on the edge atomic structure
Besides defect quantification, it is important to discuss how disorder depends on the specific defect. An example of a result that was successful in distinguishing different defects from one another is the study of the edge of a graphite sample, analysing the orientation of the carbon hexagons with respect to the edge axis, thereby distinguishing the so-called zigzag edge arrangements from the armchair or random atomic edge structures (Cançado et al. 2004). As discussed here, the armchair (zigzag) structure can be identified spectroscopically by the presence (absence) of the D band. This effect can be understood by applying double-resonance theory to a semi-infinite graphite crystal and by considering the one-dimensional character of the defect, as discussed below.
Figure 9a shows the edges with zigzag (top) and armchair (bottom) atomic structure, separated from each other by 150° in real space. The wavevectors of the defects associated with these edges are represented by da for the armchair edge and dz for the zigzag edge. Figure 9b shows the first Brillouin zone of two-dimensional graphite oriented according to the lattice in real space shown in figure 9a. Note that, for inter-valley scattering, only the armchair da vector is able to connect points belonging to circles centred at two inequivalent K and K′ points. Considering usual laser energies (less than 3 eV), the radii of the circles around the K′ and K points are not large enough to allow the connection of any k′ and k states by a zigzag dz vector. Therefore, the inter-valley double-resonance process associated with this defect cannot occur for a perfect zigzag edge. The mechanism depicted in figure 9b thus shows that the D-band scattering process is forbidden in a zigzag edge (Cançado et al. 2004).
On the other hand, the D′ band, around 1620 cm−1, is given by an intra-valley process, which connects points belonging to the same circle around the K (or K′) point (see K point in figure 9b). Therefore, the intra-valley process for the D′ band can be satisfied by both da and dz vectors and, for this reason, the observation of the D′ band should be largely independent of the edge structure.
Finally, when measuring an armchair edge by Raman spectroscopy, the D-band intensity will depend strongly on the light polarization direction with respect to the edge direction. The D-band intensity will be a maximum when the light is polarized along the edge, and zero when the light polarization is crossed with respect to the edge direction. This result is related to the theoretical calculations that predict an anisotropy in the optical absorption (emission) coefficient of two-dimensional graphite, given by (Grüneis et al. 2003) 4.1 where P is the polarization of the incident (scattered) light for the absorption (emission) process and k is the wavevector of the electron measured from the K point. The D-band intensity is then dependent on the light polarization direction relative to the edge direction.
Recently, Sasaki et al. (2010) showed that the phonon softening effect of the G can characterize the graphene edge structure. In the presence of free electrons, the G band and the radial breathing mode (RBM) phonon in single-walled carbon nanotubes (SWNTs) becomes soft as a function of the Fermi energy position, through an effect which is known as the Kohn anomaly (Sasaki et al. 2008a,b; Farhat et al. 2009; Kalbac et al. 2009; Park et al. 2009). These phenomena occur preferentially at graphene edges, since the electron wave function forms a standing wave by mixing elastically scattered waves at the edge where the momentum along the edge (perpendicular to the edge) is conserved (changes its sign). Here, we select the x and y axes for zigzag and armchair edges, respectively, the directions in a rectangular-shaped graphene sample of which can be selected to be perpendicular to each other. Then the scattering event results in a sign change in ky and kx for zigzag and armchair edges, respectively. At the edges, the electron wave function forms a standing wave which consists of the initial and scattered k states. Then the electron–phonon matrix elements at a graphene edge for the G-band phonon can be calculated by using the standing wave eigenfunction for each of the LO and TO phonon modes for which the vibration direction of the LO and TO phonon modes are parallel and perpendicular to the edge direction. The details of the calculation (Sasaki et al. 2010) show that only the LO (TO) phonon mode is Raman active for armchair (zigzag) edges, thereby showing a polarization dependence of the light. On the other hand, only LO phonon modes show a Kohn anomaly effect for both types of edges. Thus, we can distinguish between LO and TO phonon modes by the phonon softening phenomena, and in this way we can use Raman spectroscopy to characterize the edge-structure type. These phenomena should appear for a restricted spatial region near the edge where the concept of the standing wave is well established. The theoretical prediction has already been observed partially by experiment (Sasaki et al. 2009). However, more precise experimental observations are still needed as a function of gate voltage or by the polarization direction of the light at a well-defined edge structure, and such work remains for future investigation.
5. Specificities of the disorder in the Raman spectra of carbon nanotubes
When moving to carbon nanotubes, the quantum confinement of the electronic structure will constrain the double-resonance processes in SWNTs, similarly to what has been discussed in §4 for graphene.
(a) The D band in SWNT bundles
A multi-peak structure (Zolyomi et al. 2003) and an oscillatory dispersive behaviour (Brown et al. 2001) has been observed for the D band in SWNT bundles. For isolated tubes, unusually sharp features can be seen in the D band with a full width half maximum (FWHM) intensity linewidth down to 7 cm−1 (Jorio et al. 2002) owing to quantum confinement effects. Furthermore, the D-band frequency also depends on the nanotube diameter, with the following result for 514 nm (2.41 eV) laser excitation (Souza Filho et al. 2003): 5.1 Furthermore, the D band is a second-order process with a phonon frequency ωD related to that of the second order G′ band by 2ωD≃ωG′.
Some efforts have been made to quantify disorder in SWNTs, by studying irradiated SWNT samples (Hulman et al. 2005), and in SWNTs cut to different lengths (Chou et al. 2007). Clear enhancement of the D band is observed with increasing defect density or with decreasing nanotube lengths. However, because of the lack of direct real-space characterization, the results are not as quantitative as the results obtained in graphene (Lucchese et al. 2010) and in nanographite (Cançado et al. 2008) and in irradiated double-walled carbon nanotubes (DWNTs), as discussed in §5b. Some aspects are still unclear such as why the D band in metallic SWNTs is usually more intense than in semiconducting tubes. Although some theoretical efforts based on the double-resonance process were directed to address this problem (Maultzsch et al. 2001), the predictions (Maultzsch et al. 2002, 2003) regarding the D band are still very incomplete.
(b) Si+ and C+ bombarded DWNTs
Resonance Raman scattering spectroscopy has been used to monitor the implantation of Si and C ions into highly purified DWNTs (Saraiva et al. 2009). The data indicate that the outer tubes of the DWNTs are more affected by the ion bombardment than the inner tubes. As the ion beam fluence increases, the tubes get disordered, as indicated by the increase in the D to G band intensity ratio (ID/IG), and this effect is much stronger for implantation with Si than with C. In figure 10a, we show the plot of ID/IG as a function of the C and Si ion bombardment dose for two different Elaser values. We can see that the ID/IG ratio increases as the dose increases, thus confirming the increasing density of defects in the DWNT samples with increasing ion fluence. It is also clear that the ID/IG ratio depends strongly on the ion species used for the bombardment, as well as on the Elaser used for characterization (Saraiva et al. 2009).
In order to separate the contributions from the dose and the ion species at each laser excitation energy, we normalized the ID/IG curves from figure 10a by E4laser in accordance with the model of Cançado et al. (figure 10b). Although the model was developed for graphite, this model should apply to some extent to other sp2-based materials such as carbon nanotubes. After carrying out this normalization, we can see that the curves for a given ion-implanted sample show a unique curve only for ion doses up to 2×1014 ions cm−2. Therefore, we can use Cançado’s empirical equation for estimating the characteristic length La in this range of ion dose for the bombarded DWNTs. Here, the La should be interpreted as a characteristic distance between the defects on the carbon nanotube lattice introduced by the ion beam. The La values are shown in figure 10c. Here, we can see that this characteristic length dependence on ion fluence is stronger for Si bombardment than for C ion bombardment at the same fluence, but both datasets show a tendency towards saturation. It has been demonstrated in the literature that carbon nanotubes when irradiated with electrons exhibit pronounced healing effects. It is possible that the same healing mechanism is operative in the case of ion bombardment, with the healing effect being more efficient in the case of the less damaging carbon irradiation (when compared with silicon irradiation).
The minimum crystallite size used by Cançado et al. for obtaining the empirical equation (equation 5.1) was 20 nm. For these ion-bombarded samples, the lowest characteristic length determined from the range of La where the ID/IG ratio follows the E4laser dependence also was about 20 nm (see figure 10b). For doses larger than 40×1013 ions cm−2, the experimental data do not exhibit the E4laser dependence any more, and this is attributed to the very large density of defects which translates into a very small crystallite size La. In this limit, the carbon lattice is likely to lose its sp2-like character, thus forming amorphous carbon and carbon clusters with a substantial amount of sp3 bonding. It seems that, for Si-bombarded samples, the fraction of amorphous carbon and sp3-like carbon is higher than that of C-bombarded samples. This argument is based on the observation that the Raman cross section for sp3 carbon is much lower than that for sp2 carbon. Amorphous carbon nanowires were obtained in multiwalled carbon nanotubes (MWNTs) by high doses (1015–1017 ions cm−2) of a 40 keV Si ion beam. An amorphous carbon nanowire is completely formed for fluences of 1017 ions cm−2. The lower fluence 1015 ions cm−2 only make the MWNT layers very disordered, which indicates a higher stability against damage by the ion beam than in DWNTs, owing to the considerably larger number of layers in MWNTs.
Finally, the behaviour of the ID/IG ratio as a function of ion fluence rules out the possibility of bombardment-induced nanographite because in the 1015 ions cm−2 regime it is expected that the ID/IG ratio would exhibit an dependence following the Tuinstra model (Tuinstra & Koenig 1970a). This dependence would imply a monotonic decrease in intensity of the curves shown in figure 7, instead of the observed saturation behaviour, thus indicating that the bombardment of C ions is not leading to the formation of nanocrystalline graphite.
6. Local effects revealed by near-field measurements
Besides disturbing the momentum conservation of the ordered system, the presence of disorder is expected to change the local electron and phonon structure. The G′-band feature is not a disorder-induced feature but it can nevertheless be used to probe changes in the electronic and vibrational structure related to disorder. The two-dimensional versus three-dimensional stacking order of graphene layers is one example where the G′ band provides important information. Highly crystalline three-dimensional graphite shows two G′ peaks (see figure 11g). When the interlayer stacking order is lost, a one-peak feature, identified with two-dimensional graphite, starts to develop, and this peak is centred near the middle of the two peaks in the G′ lineshape from ordered graphite (see figure 11g). More interesting, localized emission of a red-shifted G′ band was observed and this emission is related to the local distortion of the nanotube lattice by a negatively charged defect. The defect position was initially located by studying local variations in the D-band intensity as described below (Maciel et al. 2008b).
Figure 11 shows near-field Raman and near-field photoluminescence spectra and spatial maps for an individual SWNT. The near-field technique can generate spectral information with spatial resolution Δx below the diffraction limit (Δx∼λlaser/2) (Novotny & Hecht 2006). In particular, figure 11a,b shows the photoluminescence and Raman spectra of defects in a SWNT, respectively, with Δx∼30 nm. The near-field microscopy measurements from the same SWNT are shown in figure 11c–f. In particular, figure 11c represents the near-field photoluminescence image of this SWNT, where the image contrast is provided by spectral integration over the photoluminescence peak centred at λem=900 nm (see figure 11a). The most striking feature in this image is the high degree of spatial localization of the photoluminescence emission along the SWNT. This is evident by inspection of the extended topography image of the nanotube shown in figure 11f, and also of the near-field Raman image of the G band, with a peak intensity near 1590 cm−1 shown in figure 11d. While we observe from figure 11d that the G-band Raman scattering is present along the entire length of the nanotube, from figure 11e we observe an increased defect-induced D band at 1300 cm−1 with Raman scattering intensity localized in the same spatial region where exciton emission was detected. Defects are known to act as trapping states for electron–hole recombination (i.e. exciton emission), thereby providing insights into the correlations observed between figure 11a,c.
Interestingly, when measuring the Raman spectra across the defective spot, sudden changes in many Raman features are observed. It has further been shown by Maciel et al. (2008b) that substitutional doping in SWNTs causes changes in the G′-band spectra owing to charge-induced renormalization of the electronic and vibrational energies. Figure 11g shows the G′ band measured on the same SWNT, moving along the position where the local D band and photoluminescence emission is observed (circle in figure 11e). The two spectra marked by ‘*’ in figure 11g were obtained at this defect location, and a new peak is observed in the G′ band region. The frequency and intensity of this new peak depend on the type and level of doping, respectively (Maciel et al. 2008a, 2009a,b). This makes the Raman G′ band a sensitive probe for studying and quantifying doping, which is a more accurate probe than the D band, since the D band can also be related to amorphous carbon and any other defective sp2 structure. Also the D-band frequency is not so sensitive to the type of defect or dopant. The one-dimensional edge defect selects the direction of the electron and phonon wavevectors associated with the disorder-induced Raman process, and causes a dependence of the Raman D-band intensity on the atomic structure of the edge (strong for an armchair edge and weak for a zigzag edge).
The first experimental evidence for the selection rules discussed here was developed for a graphite boundary (Cançado et al. 2004). However, similar results have already been shown for monolayer graphene (Casiraghi et al. 2009; Gupta et al. 2009). Interestingly, up to now variations in the D-band intensity have been observed, but never has a complete absence of the D band together with the presence of the D′ band been seen experimentally. This result indicates that, up to now, no perfect zigzag edge structure has been measured by Raman spectroscopy. It should be emphasized that Raman spectroscopy provides one easy method for distinguishing between armchair and zigzag edges. Of course, high-resolution TEM provides another experimental method for distinguishing armchair from zigzag edges (Jia et al. 2009).
This review focuses on two types of defects in sp2 carbon nanostructures, namely point defects and edges. Point defects are explicitly studied by controlled argon ion implantation studies of graphene as a function of fluence. Near-field studies at the individual nanotube level using both Raman scattering and photoluminescence probes are used to give high spatial resolution to defect studies. A simple model is given to account for the major findings of these ion implantation studies of point defects. Boundaries and sample edges are shown to behave similarly to defects insofar as they disturb the crystal symmetry and lead to the observation of symmetry-breaking spectral features, such as the D band and the D′ band in the Raman spectra. Differences in the selection rules for zigzag and armchair edges lead to the important result that armchair edges show strong D-band intensity while zigzag edges do not, thereby providing a sensitive tool for distinguishing between armchair and zigzag edges spectroscopically.
A.J. acknowledges support from CNPq, Brazil, and AFOSR/SOARD (award no. FA9550-08-1-0236). M.S.D. acknowledges support from NSF-DMR 07-04197 and the ONR GATE MURI: project monitored by Dr Chagaan Baatar under N00014-09-1-1063. R.S. acknowledges support from MEXT grant (no. 20241023). A.G.S.F. acknowledges support from CNPq grants no. 306335/2007-7, no. 503956/2007-4, and no. 577489/2008-9, Rede Nacional de Pesquisa em Nanotubos de Carbono, European Union Seventh Framework Programme (FP7) Small Collaborative project, Neuronano (NMP4-SL-2008-214547), and INCT NanoBioSimes.
One contribution of 12 to a Theme Issue ‘Electronic and photonic properties of graphene layers and carbon nanoribbons’.
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