Carbon Nanotubes: Optical Properties Part 2 (Nanotechnology)

OPTICAL ABSORPTION

Optical absorption of an electron occurs ”vertically” at a k point of the Brillouin zone from an occupied to an unoccupied energy band. An optical transition from an occupied 2p to an unoccupied 2p atomic orbital in the same carbon atom is forbidden. However, the transition from the 2p orbital of one atom to the 2p orbital of another atom (mainly a nearest neighbor atom) is possible, which makes p to p transitions possible in graphite and carbon nanotubes.

First-order time-dependent perturbation theory gives the following formula for the absorption probability per unit time W(k) for an electron with wave vector k,

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where Cc and Cv (Ec and Ev) are the wave functions (energies) of the valence and conduction bands, respectively, P is the polarization vector of light, and Elaser, I, e, m, and t are the incident laser energy and intensity, dielectric constant, mass of the electron, and the time used for taking the average, respectively. The value of t=Ax/ c=2p/Ao is determined by the uncertainty relation and the width of the incident laser frequency Ao. Typical values of t for Ao =10 cm— 1 correspond to 0.5 psec. If we take a sufficiently large value for t, the function sin2X/ X2{X =[Ec(k) —Ev(k) —Elaser]t/{2h} appearing in the second line of Eq. 7, becomes the delta function d[Ec(k) — Ev(k) —Elaser], and Eq. 7 is known as Fermi’s golden rule. Cc and Cv can be written as a sum of the two Bloch functions consisting of the two carbon atoms, A and B in the graphite unit cell, F,(k, r), (j=A, B)


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where Cj(k) is the coefficient of the Bloch functions that is obtained by solving the 2-D Hamiltonian and overlap matrices (H and S) for 2-D graphite.

In the case of graphite, simple zone folding of the energy bands gives the wavefunction of the nanotube in which C!(k, r) of graphite is changed to C’p(k||, r) for the nanotubes, where the subband index p and the 1-D wave vectors k? and k| are given by Eqs. 3 and 4.

The dipole selection rule is given by group theory applied to the G point (zone center) of the Brillouin zone, because each eigenfunction belongs to an irreducible representation of the space group. This rule can be extended smoothly to lower symmetry k points, even though there is no longer strict selection rule for the lower symmetry k points. The optical intensity can be directly obtained by calculating the dipole matrix element, as shown in Eq. 7. When the light polarization is parallel to the nanotube axis, the optical transition occurs between valence and conduction subbands that come from the same cutting line. When the light polarization is perpendicular to the nanotube axis, the optical transition occurs between the two subbands that belong to nearest neighbor cutting lines in the 2-D Brillouin zone.

Hereafter we mainly consider the case of light polarization parallel to the nanotube axis, because the optical absorption for perpendicular polarization is relatively much weaker. The reason why the perpendicular polarization is weak is that 1) the nanotubes have a high aspect ratio of nanotube diameter to length, 2) the van Hove singular k points are not the same for the valence and conduction energy subbands for different cutting lines, and 3) there is a strong screening effect of the electric field perpendicular to the nanotube axis [depolarization effect, see H. Ajiki and T. Ando, Physica 201 (1994) 349]. Especially for isolated single nanotube spectroscopy, no Raman signals are observed for light polarization perpendicular to the nanotube axis.

In Fig. 8a the energy separations Eu between the ith valence and the ith conduction band van Hove singular energies, which are dipole allowed for parallel light polarization, are plotted as a function of the nanotube diameter, dt. Here the subband label i is counted from the subband closest to the Fermi energy. Each point of this figure corresponds to van Hove singularities for a different (n, m) nanotube. For a given tube diameter, we can obtain optical transitions for the first and the second van Hove singularities in the joint density of states (JDOS) for semiconducting nanotubes, En, E|2 when we increase the laser energy. The next band in Fig. 8a is for metallic nanotubes EM, followed by E33 for semiconducting nanotubes. For a given laser energy, we can find possible diameter regions of nanotubes that have an optical transition between two van Hove singularities. In this sense, the excitation laser light selects the diameter of a nanotube within a mixed sample containing nanotubes with a diameter distribution.

(a) Energy separations Eii between valence and conduction band van Hove singularities are plotted as a function of nanotube diameter. (b) Raman spectra (radial breathing mode) from three different (n, m) isolated single-wall nanotubes.

Fig. 8 (a) Energy separations Eii between valence and conduction band van Hove singularities are plotted as a function of nanotube diameter. (b) Raman spectra (radial breathing mode) from three different (n, m) isolated single-wall nanotubes.

In Fig. 8b, typical Raman signals at the single nanotube level are shown for 3 different (n, m) SWNTs at three different locations on the sample obtained by illuminating an individual SWNT with a 1-^m-diameter laser light spot on a Si/SiO2 substrate. The Raman feature at around 300 and 225 cm— 1 come from the nonresonant Si substrate. Although the number of carbon atoms is much smaller than the number of Si atoms (Si/C-106), the Raman signal of one nanotube is comparable to that of Si. This is because the Raman signal in the nanotube comes from a resonance effect in which the resonant signal is enhanced by more than 103 times relative to the nonresonant signal. This enhancement factor can be observed by changing the laser excitation energy on the same light spot. Because a SWNT is too small to see with an optical microscope, the signal from a resonant SWNT is obtained by putting many light spots on the Si/SiO2 substrate that has previously been scanned by an AFM probe to determine the location of the SWNTs in the sample.

The three spectra correspond to the radial breathing mode (RBM) of a SWNT in which the carbon atoms of the nanotube are vibrating in the radial direction. The RBM frequency ®rbM for an isolated SWNT is inversely proportional to the tube diameter and is independent of chirality:

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The (n, m) assignment is done by fitting two empirical theoretical parameters, g0 in Eq. 5 and a in Eq. 9, to reproduce many RBM frequencies observed in the experiment. The values of the parameters that were obtained for isolated SWNT sample on a Si/SiO2 substrate are g0=2.89 eV and a=248 cm— 1 nm. These parameters work well to three effective digits for a laser excitation energy Elaser= 1.58 eV and in the nanotube diameter range 1<dt<2 nm for isolated SWNTs on a Si/SiO2 substrate. In other cases, a few possible (n, m) identifications are possible. Then, using the same two parameters, more accurate Raman techniques, such as the Stokes/anti-Stokes intensity ratio or a tunable laser approach, are necessary (see Ref.[4] for details. This is a review article on Raman spectroscopy in single-carbon nanotubes and ”Resonant Raman Process” and is relevant to this article). However, the accuracy of the (n, m) assignments might decrease in the following situations: 1) for SWNT diameters smaller than 1 nm, where we expect departures to occur from the calculation of Eu described above, 2) for thick SWNT bundles in which we expect nanotube-nanotube interactions, and 3) for higher excitation energies (Elaser>3 eV) where s-band contributions become more important.

 Optical absorption spectra are taken for single-wall nanotubes synthesized using four different catalysts, each yielding a different diameter distribution, namely, NiY (1.241.58 nm), NiCo (1.06-1.45 nm), Ni (1.06-1.45 nm), and RhPd (0.68-1.00 nm). Peaks at 0.55 and 0.9 eV are due to absorption by the quartz substrate. The inset shows the corresponding RBM modes of Raman spectroscopy obtained using 488-nm laser excitation with the same four catalysts, and the tube diameters corresponding to RBM peaks are given for two catalysts.

Fig. 9 Optical absorption spectra are taken for single-wall nanotubes synthesized using four different catalysts, each yielding a different diameter distribution, namely, NiY (1.241.58 nm), NiCo (1.06-1.45 nm), Ni (1.06-1.45 nm), and RhPd (0.68-1.00 nm). Peaks at 0.55 and 0.9 eV are due to absorption by the quartz substrate. The inset shows the corresponding RBM modes of Raman spectroscopy obtained using 488-nm laser excitation with the same four catalysts, and the tube diameters corresponding to RBM peaks are given for two catalysts.

In Fig. 9, we show the optical absorption spectra of SWNT bundles synthesized using the laser ablation method and four different catalysts. In the laser ablation method, the pulsed YAG laser hits a graphite rod containing the indicated catalysts in a furnace kept at about 1200°C with an Ar gas flow, and the smoke emerging from the rod and containing the SWNT bundles is transported by Ar gas flow out of the furnace. By controlling the temperature of the furnace and the selection of catalysts, a narrow diameter distribution of nanotubes is obtained. Although there are SWNTs with many different chiralities mixed in the SWNT bundle sample, we can see three peaks in the optical absorption in Fig. 9, where each peak corresponds to either En, E|2, or Em. By comparing the energies where the peaks in Fig. 9 occur with the results of Fig. 8 a, we can estimate the diameter distribution produced by each catalyst and these estimates are confirmed by the RBM spectra shown in the inset of Fig. 9.

When such a bundle of SWNTs is doped with an alkali metal, a charge transfer of electrons occurs from the alkali metal donor ions to the nanotube or from the nanotube to acceptor ions. In the case of donor ions, the unoccupied states of the pristine sample become occupied with electrons starting from the lowest available conduction band energies. This causes a disappearance of the optical absorption peaks starting from the lower-energy side as the doping concentration increases, because of a lack of unoccupied excited states to allow optical transitions to occur. An electrochemical doping or a field effect transistor (FET) type doping method would be good for observing this phenomenon since the Fermi energy can be modified by changing the gate voltage. Although direct optical absorption measurements of one nanotube will be very difficult because there is no special enhancement mechanism available such as resonance Raman spec-troscopy, a combination of optical absorption with resonance Raman spectroscopy should make it possible to observe the Fermi energy shift associated with donor or acceptor doping.

Optical emission or fluorescence signals from a nanotube are not easy to observe, because most SWNT samples contain bundles of nanotubes in which one of three nanotubes are metallic. In a metallic nanotube, the electron-hole recombination occurs nonradiatively along the metallic energy dispersion. Furthermore, optical or direct coupling between nanotubes in the bundle suppresses fluorescence in semiconducting nanotubes, but if a semiconducting nanotube is isolated from other nano-tubes, it is then possible to measure the fluorescence. Such spectra have been obtained for nanotubes separated from one another in a zeolite or when covered by a micelle (soap) structure [see Z. M. Li et al. Phys. Rev. Lett. 87 (2001 ) 127401 and M. J. O'Connell et al. Science 297 (2002) 593]. The fluorescence of nanotubes occurs at the real energy gap ES1 of semiconducting nanotubes and provides a good probe of excitonic effects in SWNTs. If high spatial resolution greater than the average distance between two nanotubes is available, we can assign (n, m) values from the fluorescence measurement.

Resonance Raman Process

One of the most powerful optical measurements for carbon nanotubes is resonance Raman spectroscopy. The resonance enhancement effect which is caused by a real optical absorption or emission process makes it possible to observe a signal from even one nanotube, as shown in Fig. 8b. Recently much progress in Raman spectroscopy has been made, and therefore it is not possible to explain the many important advances in detail (see Ref. [4]) within the available space of this review. The present summary therefore only provides an overview of Raman spectros-copy of carbon nanotubes, and only a few essential points that are important for analyzing the many features of the observed resonance Raman spectra of SWNTs are discussed (Fig. 10).

Room-temperature Raman spectra for purified single-wall carbon nanotube bundles excited at five different laser frequencies, showing RBM modes around 200 cm— \ G band (~1590 cm— J), and D band (1250-1350 cm— J). We can also see some weak features in the intermediate frequency region around 800 cm.

Fig. 10 Room-temperature Raman spectra for purified single-wall carbon nanotube bundles excited at five different laser frequencies, showing RBM modes around 200 cm— \ G band (~1590 cm— J), and D band (1250-1350 cm— J). We can also see some weak features in the intermediate frequency region around 800 cm.

In carbon nanotubes, the Raman spectra show a variety of features associated with first-order processes, as well as combination or overtone modes up to 3200 cm—1 The characteristic mode that is not observed in other sp2 carbons, but is observed only in nanotube samples, is the radial breathing mode (RBM). The RBM frequency appears from 100 to 550 cm— \ depending on the SWNT diameter, ranging from 2.5 to 0.4 nm, respectively. To check if a sample contains any nanotubes, the RBM spectra provides an easy way to do just that. It is further noted that if no RBM spectra are observed, this does not always mean that there are no SWNTs in the sample, because we can imagine the case that the resonance condition for a given diameter distribution of SWNTs might be far from the available laser excitation energy. In some cases, because of the large noise signal from the Rayleigh (elastically) scattered light, the RBM signal cannot be resolved in the noisy Rayleigh background signal. Use of a notch filter is a simple way to avoid this problem, but use of a notch filter in practice limits observations to oRBM>100 cm—1

Van Hove singular energies in the JDOS, Eii as a function of 1/dt are shown in (b) where circles and crosses correspond, respectively, to metallic and semiconducting (n, m) chirality tubes. For 1.579-eV laser excitation, expansion of the rectangular section in (b) is shown in (a). The resonance condition satisfies only a limited number of (n, m) tubes whose RBM frequencies can almost always be distinguished from one another.

Fig. 11 Van Hove singular energies in the JDOS, Eii as a function of 1/dt are shown in (b) where circles and crosses correspond, respectively, to metallic and semiconducting (n, m) chirality tubes. For 1.579-eV laser excitation, expansion of the rectangular section in (b) is shown in (a). The resonance condition satisfies only a limited number of (n, m) tubes whose RBM frequencies can almost always be distinguished from one another.

In Fig. 11b we plot Eii as a function of inverse tube diameter 1/dt. Circles and crosses correspond, respectively, to metallic and semiconducting (n, m) tubes. For a laser energy of 1.579 eV, only a few metallic and semiconducting (n, m) tubes with E^f and E|z van Hove singularities satisfy the resonance condition, as seen in Fig. 11a. Since the RBM frequency can be measured to the accuracy of 1 cm— 1 out of 100-200 cm— 1, the difference between the peak frequencies of different (n, m) tubes are easily distinguished.

The second important Raman feature is the G-band mode that appears near 1590 cm— 1 in SWNTs. This mode is the Raman-active mode of sp2 carbons (graphite), and thus many graphitic materials show this spectral feature. What is special in nanotubes is that the G band mainly splits into two peaks, denoted by G+ and G—, and this is because of the properties of the six (or three) phonon modes that are active for chiral (or achiral) nanotubes around this phonon frequency region. Among Raman-active modes, two A (or A1g) modes are considered to be the strongest and account for the two peaks, G+ and G—. Regarding the other four Raman-active modes, two have E1 and two have E2 symmetries (or two Raman-active modes E1g or E2g for achiral tubes), and the Ej and E2 modes are relatively weak. The presence of the E1 and E2 modes are mainly seen from the Lorentzian decompositions of the line shape of the observed spectra using polarized light when carrying out polarization studies. The frequency difference between the two G band spectral features G+—G—, is inversely proportional to dt2, which comes from the curvature effect that is responsible for the splitting of the degenerate in-plane optic phonon modes, E2g, of 2-D graphite at the G point.

Around 1250-1350 cm— \ defect-induced Raman modes, called the D band, appear only for defect-rich sp2 carbons. The appearance of a D band in nanotubes means that a nanotube (or a non-SWNT carbon impurity in the sample) has many defects. The nanotubes synthesized by CVD are known to be defective relative to other nanotubes. The D-band mode is known as a dispersive phonon mode in which the D-band frequency increases by 53 cm— 1 with a 1 eV increase in the laser excitation energy. The physical origin of this very large dispersive-ness has been recently explained by double resonance Raman theory[5] in which the photon-absorbed electron is scattered twice before final recombination. Although this process is a second-order light-scattering process, the intensity is comparable to that of the first-order resonance process when two of the three intermediate states are in resonance with the real electronic states.

 (a) First-order Raman processes that are resonant with (a1) the incident and (a2) the scattered laser light. Solid and open dots denote resonance and nonresonance scattering processes, respectively. Crossed lines show the linear energy dispersion of 2-D graphite around the K point. (b) Second-order Raman processes that are resonant with (b1, b2) the incident laser light, and with (b3, b4) the scattered laser light. The solid scattered vectors and dashed scattered vectors with wave vector q, respectively, denote inelastic and elastic scattering processes.

Fig. 12 (a) First-order Raman processes that are resonant with (a1) the incident and (a2) the scattered laser light. Solid and open dots denote resonance and nonresonance scattering processes, respectively. Crossed lines show the linear energy dispersion of 2-D graphite around the K point. (b) Second-order Raman processes that are resonant with (b1, b2) the incident laser light, and with (b3, b4) the scattered laser light. The solid scattered vectors and dashed scattered vectors with wave vector q, respectively, denote inelastic and elastic scattering processes.

In Fig. 12, inequivalent (a) first-order and (b) second-order Raman processes are shown. The crossed lines for each figure denote the electronic energy states of graphite around the K points in the Brillouin zone about which the optical transition occurs. The upper figures are for the incident resonance condition for laser light in which the laser excitation is at almost the same energy as the energy required for optical absorption. The lower figures are for the scattered resonance condition, in which the scattered electron energy, (Elaser—Ephonon), is almost the same as the optical emission energy. Both resonance conditions give a similar Raman intensity, but the resonance laser energy differs in the two cases by the phonon energy. More precisely, the laser energy for the scattered resonance energy is higher by Ephonon than that for the incident resonance energy in a Stokes scattering process, which creates a phonon. Note that for all Raman-active phonon modes, the incident resonance condition gives the same resonance laser energy, whereas the scattered resonance condition gives different laser energies, depending on the energy of phonons. Thus, by observing more than one Raman spectral feature for the same tube or not, respectively, we know whether the observed resonance Raman process is with an incident photon or with a scattered photon.

(a) Calculated Raman frequencies for the double resonance condition as a function of Elaser (bottom axis) and q vector along G-K (top axis). Solid and open circles correspond to phonon modes around the K and G points, respectively. (b) The six graphite phonon dispersion curves (lines) and experimental Raman observations (symbols) are placed according to double resonance theory.

Fig. 13 (a) Calculated Raman frequencies for the double resonance condition as a function of Elaser (bottom axis) and q vector along G-K (top axis). Solid and open circles correspond to phonon modes around the K and G points, respectively. (b) The six graphite phonon dispersion curves (lines) and experimental Raman observations (symbols) are placed according to double resonance theory.

In the first-order process, in order to recombine an electron and a hole at the original k point, the phonon wave vector q should be sufficiently small. This is why we see in most solid-state textbooks that the Raman spectra of solids can be observed only for zone-center (q=0) Raman-active phonon modes. In the second-order double resonance process, it is clear from Fig. 12 that the phonon wave vector q is not zero. Furthermore, when the laser energy increases, the electron k vector moves further from the K point to satisfy the requirements for optical absorption, and thus the corresponding q vectors become longer. Here we use the requirement of the double resonance process that the intermediate k+q states should be unoccupied electronic states, too. This is the reason why the dispersive phonon modes change their Raman frequencies when the laser excitation energy is changed, since the phonon wave vector q that satisfies the double resonance Raman condition changes when the excitation laser energy changes.

In 2-D graphite, there are two inequivalent corners of the Brillouin zone, K and K’, as shown in Fig. 7. In the electron scattering by phonons or impurities (elastic scattering), there are two possibilities. One is an intra-valley scattering process in which an electron scatters within the same region of the K (or K’) point. The other is intervalley scattering in which an electron is scattered from the K to the K region (or K to K). The corresponding phonon q vectors for intravalley and intervalley scattering are phonons around the G and K points, respectively. Thus, we try to find the phonon dispersion point around either the G or K points that satisfies the double resonance condition.

In Fig. 13a the phonon q vectors for the double resonance condition are shown as a function of Elaser (bottom axis) and of the q vector along G-K (top axis). Solid and open circles correspond to phonon modes around the K and G points, respectively. In Fig. 13b are collected the many experimental Raman signals that have been observed in many disordered graphitic materials for many years, and these are here plotted in the Brillouin zone of 2D graphite. By specifying either G or K point phonons, all of the experimental points could be assigned to one of the phonon energy dispersion relations (six solid lines), thus providing clear evidence that double resonance theory can work well for the dispersive phonon modes of 2-D graphite. The resulting dispersion relations show how resonance Raman scattering can be used to obtain phonon dispersion relations experimentally based on double resonance theory.

All second-order phonon processes shown here are one-phonon emission processes. Thus, one of the two scattering processes shown in Fig. 12b should not be a phonon-scattering process. We consider that the nonpho-non scattering process is an elastic electron scattering caused by an impurity or a defect in which electronic states with momentum k are mixed with each other. This is the reason why the dispersive phonon modes appear in a frequency region lower than 1600 cm~1 only for defective carbon materials. If the two scattering processes are both phonon-scattering processes, we do not need the defect scattering process, and thus strong dispersive overtone phonon modes appear. An example of a two-phonon process is the G’ band appearing around 2700 cm~1 with its large dispersion of 106 cm~ 1/eV.

CONCLUSION

In summary, the optical properties of carbon nanotubes show rich spectra because of low-dimensional physics phenomena, such as van Hove singularities in the density of states. The resonance Raman spectra, in particular, provide a powerful tool for observing individual nano-tubes in the sample, and especially for specifying (n, m) values of individual nanotubes. Double resonance theory explains the dispersive Raman phonon modes, and can be used to provide information about the dispersion relations for submicron-sized samples, too small for study by neutron scattering techniques.

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