These two bands have a linear dispersion E(k)around the. K point in the Brillouin zone and, as shown in Fig. 1, and the valence and conduction bands meet at ...
Semiconducting Carbon Nanotubes M. S. Dresselhaus , R. Saito† and A. Jorio £
MIT, Cambridge, MA 02139 USA Dept. of Physics, Tohoku University, Sendai Japan ££ Dept. of Physics, UFMG, Belo Horizonte, Brazil
†
Abstract. The use of Raman spectroscopy to reveal the remarkable structure and properties of carbon nanotubes is briefly reviewed. Particular emphasis is given to the fact that a nanotube can be semiconducting or metallic depending on its diameter dt and chirality θ , and how Raman spectroscopy at the single nanotube level reveals such information. Some of the implications of the unusual properties of carbon nanotubes are summarized. It is shown how the vibrational spectra of one tiny tube, only about 1 nm in diameter, can be observed experimentally. Raman spectroscopy normally measures vibrational frequencies. What is unique about carbon nanotubes is that for this one-dimensional system, resonance Raman spectroscopy (RRS) also determines, in addition, the geometrical structure of the resonant nanotube, that is its diameter and chirality. Some of the recent advances in single nanotube spectroscopy are briefly discussed. The use of RRS for characterizing carbon nanotube samples is also discussed. The connection between Raman spectroscopy and photoluminescence for studying carbon nanotubes is mentioned. Some of the current research challenges facing the field are briefly summarized.
INTRODUCTION Single-wall carbon nanotubes (SWNTs) provide a model system for a nanostructured one dimensional (1D) semiconductor and at the same time are promising for ultrasmall semiconductor device applications. Since SWNTs can be either semiconducting (S) or metallic (M), it is important to have a probe such as a photon, which can distinguish between S and M SWNTs. The optical properties of SWNTs provide an especially sensitive tool for probing the remarkable 1D semiconducting properties of SWNTs and for this reason are the focus of this review. The unique optical properties observed in SWNTs are due to the one-dimensional (1D) confinement of their electronic states, resulting in so-called van Hove singularities (vHSs) in the nanotube density of states (DOS) [1, 2]. These singularities in the DOS, and correspondingly in the joint density of states (JDOS) for optical transitions, are of great relevance for a variety of optical phenomena. Whenever the energy of incident photons matches a vHS in the JDOS of the valence and conduction bands (subject to selection rules for optical transitions), one expects to find resonant enhancement of the corresponding photophysical process. Owing to the sharpness of these singularities, optical spectra can be strongly confined in energy, so that the optical spectra from SWNTs show sharp spectral features, similar to what is seen from single molecules, yet showing behavior related to the quasi-continuum of states along the SWNT axis. Furthermore, we can distinguish one
SWNT from another by its diameter and chirality. Thus semiconducting single wall carbon nanotubes are of particular interest as a model system which in some ways behaves like a molecule and in other ways like a semiconducting solid. We here review some of the findings obtained by resonance Raman spectroscopy.
GEOMETRIC AND ELECTRONIC STRUCTURE A single-walled carbon nanotube can be considered to be a rolled up 2D graphene sheet, which is a single layer of a 3D graphite crystal. The structure of each SWNT can be specified by a pair of integers (n m) denoting the chi ral vector C na1 ma2 that forms the circumference h of the nanotube where a 1 and a2 are basis vectors of the hexagonal honeycomb graphene lattice. This chiral vec also defines a chiral angle θ , which is the angle betor C h . For the tubes with n 0 called zigzag tween a1 and C h tubes (θ 0) and the tubes with n n called armchair tubes (θ 30Æ ) [1], the translation and rotation symmetry operations are independent of each other, while for all other n m tubes, called chiral SWNTs, the rotations and translations are coupled [1]. Although a graphene sheet (2D graphite) is a zero-gap semiconductor [1], carbon nanotubes can be metals or semiconductors with different size energy gaps, depending very sensitively on the indices (n m) or equivalently on the diameter and helicity of the tubes. The unique
band structure of a graphene sheet has states crossing the Fermi level at only 2 inequivalent points in k-space, denoted by the point K and K , which are located at adjacent corners of the hexagonal Brillouin zone. The quantization of the electron wave-vector along the circumferential direction, resulting from the periodic boundary conditions, leads to the formation of energy subbands associated with cutting lines k µ K1 separated from one another by a distance K1 2dt . Here dt is the nanotube diameter, µ is an integer denoting each quantum state in the circumferential direction for each cutting line. Associated with each quantum state is a quasi-continuum of states along the length of the cutting lines [3] described by a reciprocal lattice vector K2 . The K1 and K2 reciprocal lattice vectors form the basis vectors of the nanotube Brillouin zone. An isolated sheet of graphite is a zero-gap semiconductor whose electronic structure near the Fermi energy is given by an occupied π band and an empty π band. These two bands have a linear dispersion E k around the K point in the Brillouin zone and, as shown in Fig. 1, and the valence and conduction bands meet at the Fermi level at the K point. The Fermi surface of an ideal graphene sheet consists of the three corner K points interlaced with three K points. The allowed set of k’s in SWNTs, indicated by the lines in Fig. 1, depends on the diameter and helicity of the tube. Whenever the allowed k’s include the point K or K , the system is a (1D) metal with a nonzero density of states at the Fermi level, resulting in a one-dimensional metal and with 2 linear dispersing bands. When the point K is not included in the set of allowed states, the system is a semiconductor with different size energy gaps depending on both the diameter dt and the chiral angle θ . The d t dependence comes from the K1 2dt relation, and the θ dependence comes from the fact that the equi-energies around the K point are not circles, but rather exhibit a trigonally warped shape [1]. It is important to note that the states near the Fermi energy for both metallic and semiconducting tubes are all from states near the K point, and hence their transport, electronic and optical properties are related to the properties of the states on the cutting lines, with the conduction band and valence band states of a semiconducting tube coming from states along the cutting line closest to the K point. The general rules coming from symmetry considerations tell us that SWNTs come in three varieties: armchair (n n) tubes which are always metals; (n m) tubes with n m 3 j, where j is a nonzero integer, which are very tiny-gap semiconductors, but they behave like metals at 300 K; all others are large-gap semiconductors. As the tube diameter dt increases, the band gaps of the large-gap and tiny-gap varieties decrease with a 1d t and 1dt2 dependence, respectively. The 1d t2 dependence for the tiny gap SWNT is due to a curvature effect and de-
pends on θ , the tiny gap being a maximum for zigzag and zero for armchair SWNTs. Thus, the (4,2) tube shown in Fig. 1, would on the basis of simple arguments based on zone folding considerations be expected to be a semiconductor [see Fig. 1(b)] since there is no cutting line going through the K point of the 2D Brillouin zone. Although the 1D electronic band structure of this small diameter tube, shown in Fig. 1(b), appears to be complex, it becomes clear when considering the density of electronic states, as shown in Fig. 1(c), that the optical absorption or emission rate in nanotubes is related primarily to the electronic states at the vHSs, thereby greatly simplifying the analysis of the optical experiments. The distance between two neighboring cutting lines in Fig. 1(a) is related to the nanotube diameter (K 1 2dt ), and their direction relative to the hexagonal 2D Brillouin zone depends on the rolling up direction relative to the 2D-graphite sheet, i.e., depends on the nanotube chiral angle θ . It is, therefore, easy to see that each n m SWNT exhibits a different set of vHSs in its valence and conduction bands, and a different set of electronic transition energies between its valence and conduction band vHSs. For this reason, optics experiments can be used for structural determination of a given n m carbon nanotube. By calling E ii the electronic transition energies between electronic valence and conduction bands with the same symmetry, with the subscript i 1 2 3 labeling the Eii values for a given SWNT as their energy magnitude increases [1], we see that a set of measured E ii values will be specific to each n m nanotube. SWNTs can be classified in three different families according to whether mod2n m 3 0 1 or 2, where the integers 0 1 2 denote the remainders r when 2n m is divided by 3. Here r 1 2 are for semiconducting SWNTs, while r 0 SWNTs (n m) are metallic at room temperature. Because of the large curvature of the (4,2) SWNT, the simple tight binding model needs to be extended, and this has become a topic of current research interest.[4, 5] The main point to be emphasized is that because of the trigonal warping effect, the energy eigenvalues for every n m SWNT obtained from the cutting lines (shown in Fig. 1) are unique, and therefore each n m nanotube is considered as a distinct molecule with potentially different properties. Van Hove singularities (vHSs), characteristic of the density of states for 1D systems, are found for both semiconducting and metallic SWNTs [see Fig. 1(c)], and are characteristic of molecular systems, while the asymmetric tails in the DOS are characteristic of the quasi-continuum of states along the nanotube axis. Three synthesis methods [arc discharge, laser vaporization and chemical vapor deposition (CVD)] are used to produce carbon atoms in a hot gaseous form by evaporation from a condensed phase and these highly energetic carbon atoms then reassemble themselves to form carbon
(a)
(b)
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10
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0
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-5
-π/T
0 Wave vector
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FIGURE 1. (a) The calculated constant energy contours for the conduction and valence bands of a graphene layer in the first Brillouin zone using the π -band nearest-neighbor tight-binding model [1]. Solid curves show the cutting lines for the 4 2 nanotube [3]. (b) Electronic energy band diagram for the 4 2 nanotube obtained by zone-folding from (a). (c) Density of electronic states for the band diagram shown in (b).
nanotubes. All methods use nanosize catalytic particles to induce the synthesis process and to control the diameter of the nanotubes that are synthesized. In the past year, rapid progress has been made to increase the control of the synthesis process, steadily narrowing the diameter and chirality range of the nanotubes that are produced, decreasing their defect and impurity content, increasing their production efficiency and yield, while expanding the functionality of nanotubes. The main directions in the pursuit of controlled nanotube synthesis include the synthesis of molecular catalytic clusters with atomically well defined size and shape, the development of mild catalytic synthesis conditions at reduced temperatures, the development of patterned growth with a high degree of control in nanotube location and orientation, and the synthesis of complex and organized networks or arrays of nanotubes on substrates [6]. At present, all known synthesis processes yield a mixture of semiconducting and metallic SWNTs within a single SWNT bundle. If the growth process produces semiconducting (S) and metallic (M) SWNTs with equal a priori probability, an approximate 2:1 ratio is expected for S:M tubes. For some high precision semiconductor device applications, a single n m species might be required, while for other applications it might be sufficient to produce a sample containing only S SWNTs (e.g., for transistor applications) or only M SWNTs (e.g., for interconnect applications) to the standards of the semiconductor industry. At present, some success
has been achieved in synthesizing SWNT samples with a narrow distribution of small diameter SWNTs with a preponderance of two species (6,5) and (7,5).[7] Further enhancement of specific tubes within this narrow distribution can be accomplished by a variety of chemical and physical processes [8, 9]. The controlled synthesis of semiconducting nanotubes to match the standards of the semiconductor industry remains a grand challenge for nanotube applications to electronics.
RAMAN SPECTROSCOPY The Raman spectra of SWNTs have been particularly valuable for providing both insights into the detailed 1D physics of SWNTs and detailed information for characterizing SWNT samples from more of a chemistry standpoint [10]. To discuss these fundamental and practical issues, a brief account of the main features of the Raman spectra are presented. Referring to the Raman spectra in Fig. 2, we see that there are two dominant features, namely the radial breathing mode (RBM) which is observed at a low frequency ( 300 cm 1 ), and the tangential band for vibrations along the surface of the SWNT, which appears in the frequency range from 1520–1620 cm 1 . Because of the strong connection of this tangential band to the corresponding mode in 2D graphite, this higher frequency band for SWNTs is commonly called the G-band. In the first paper on the Raman effect in SWNTs,[10] the strong and non-monotonic dependence of the SWNT
Raman spectra on the laser excitation energy E laser established Raman scattering to be a resonance process occurring when E laser matches the optical transition energy Eii , between van Hove singularities [see Fig. 1(c)] in the valence band and conduction bands. Raman scattering in SWNTs is very important for the characterization of SWNTs because it relates the photon excitation directly to both phonon and electron processes in SWNTs. Since the Eii value for a particular SWNT is dependent on its diameter dt , so are the Raman spectra that are observed. Because of the very small diameters of SWNTs ( 1 nm), the joint density of states for this optical process exhibits very strong and sharp singularities, with associated large enhancements in Raman intensity, allowing the observation of Raman spectra from an individual SWNT that is in strong resonance with E laser .[11] From the point of view of nanotube characterization, the resonance Raman effect allows the selection of only the few SWNTs for detailed examination in a sample that are within the resonance window of a given E laser . For example, the radial breathing mode (RBM) for which all carbon atoms in a given nanotube are vibrating in phase in the radial direction is only observed for carbon nanotubes and not for other carbon materials. Furthermore, this mode has a frequency ω RBM that is highly sensitive to the nanotube diameter as discussed below, thereby providing information on both the mean diameter and the diameter distribution of a particular SWNT sample [12]. The G-band feature (see Fig. 2) consists of a superposition of two dominant components, shown at 1593 cm 1 (G ) and at 1567 cm 1 (G ) in the initial paper on SWNTs taken with Elaser 241 eV,[10] in which the G component exhibits a weak diameter dependence. The G feature is associated with carbon atom vibrations along the nanotube axis and its frequency ω G is sensitive to charge transfer from dopant additions to SWNTs (up-shifts in ωG for acceptors, and downshifts for donors). The G feature, in contrast, is associated with vibrations of carbon atoms along the circumferential direction of the nanotube, and its line-shape is highly sensitive to whether the SWNT is metallic (Breit– Wigner–Fano line-shape) or semiconducting (Lorentzian line-shape) [13, 14]. All-in-all there are 6 modes contributing to the G-band, with two each having A, E 1 and E2 symmetries, and each symmetry mode can be distinguished from one another by their behavior in polarization-sensitive Raman experiments [15]. The features of highest intensity are the G and G features of A symmetry, and it is these components that are normally used to characterize carbon nanotubes with regard to their metallicity and involvement in charge transfer. Polarization studies on SWNTs are best done at the single nanotube level [16]. Also commonly found in the Raman spectra in
SWNT bundles are the D-band (with ω D at 1347 cm 1 for Elaser 241 eV[10], stemming from the disorderinduced mode in graphite, and its second harmonic, the G band (see Fig. 2) occurring at 2ω D . Both the D-band and the G -band are associated with a double resonance process [17], and are sensitive to the nanotube diameter and chirality. The large dispersion of ω D and ωG with Elaser has been very important in revealing detailed information about the electronic and phonon properties of SWNTs as well as of the parent material graphite [18, 19, 20]. Because of the sharp van Hove singularities occurring in carbon nanotubes with diameters less than 2 nm, the Raman intensities for the resonance Raman process can be so large that it is possible to observe the Raman spectra from one individual SWNT [11], as shown in Fig. 2, where the differences in the G-band spectra between semiconducting and metallic SWNTs can be seen at the single nanotube level. Because of the trigonal warping effect, whereby the constant energy contours in reciprocal k space for 2D graphite are not circles but show three-fold distortion, every n m carbon nanotube has a different electronic structure and a unique density of states (see Fig. 1). Therefore the energies E ii of the van Hove singularities in the joint density of states for each SWNT are different, as shown in Fig. 3, where the E ii (i 1 2 3 ) values for all n m nanotubes in the diameter range up to 3.0 nm are shown for E ii up to 3.0 eV [21]. In this so-called Kataura plot (Fig. 3), which is constructed within the framework of the simple tight binding approximation, we see that for small diameter SWNTs (dt 17 nm) and for the first few electronic transitions for semiconducting and metallic SWNTs, the E ii values are arranged in bands, whose spread in energy at constant nanotube diameter d t is determined by the trigonal warping effect,[19] so that if there were no trigonal warping, the spread in energy at constant d t would go to zero. A general chiral semiconducting SWNT will have S S S , E22 , E33 bands, while a chiral metalvHSs in the E11 M, EM lic SWNT will have two vHSs in each of the E 11 22 bands. The tight binding approximation predicts values for the Eii energies for the vHSs for SWNTs with diameters in the 1.1–2.0 nm range (to an accuracy of better than 20 meV) [22]. This Kataura plot can be used to estimate the appropriate laser energy that could be used to resonate with a particular n m SWNT. The Raman effect furthermore provides a determination of Eii values, either by measurement of the relative intensities of the radial breathing mode for the Stokes (phonon emission) and anti-Stokes (phonon absorption) processes, or by measurements of the RBM Raman intensity for the Stokes process for a particular n m SWNT relative to the intensity for many SWNTs measured under similar conditions [23]. The frequency of the RBM has been used to determine the diameter of ¼
FIGURE 2. Raman spectra from a metallic (top) and a semiconducting (bottom) SWNT at the single nanotube level using 785 nm (1.58 eV) laser excitation, showing the radial breathing mode (RBM), D-band, G-band, and G¼ band features, in addition to weak double resonance features associated with the M-band and the iTOLA second-order modes. Insets on the left and the right show, respectively, atomic displacements associated with the RBM and G-band normal mode vibrations. The isolated carbon nanotubes are sitting on an oxidized silicon substrate which provides contributions to the Raman spectra denoted by ‘*’ which are used for calibration purposes.
an isolated SWNT sitting on an oxidized Si surface, using the relation ω RBM (cm1 ) = 248/dt (nm), which is established by measurements on many SWNTs. Moreover, from a knowledge of the (E ii , dt ) values for an individual SWNT (Fig. 3), the n m indices for that SWNT can be determined from the Kataura plot. Resonance Raman spectroscopy at the single nanotube level can determine both the n m indices from study of the RBM frequency and intensity, and the E ii values for individual SWNTs to about 10 meV accuracy, using a single laser line and with the Stokes-anti Stokes intensity analysis method [23]. Resonance Raman measurements with a tunable laser system can give E ii values with an accuracy of 5 meV.[24, 25] The higher precision for the n m assignment using Stokes and anti-Stokes Raman measurements depends on the determination of the shape of the resonance window (Raman intensity as a function of excitation laser energy E laser ). The resonance window could change for nanotubes experiencing different environments, sample preparation methods, and systematic work is still needed to study these environmental effects in detail. For example, we would expect perturbations to the Eii values and changes in the width of the resonance window, depending upon whether the SWNTs are freely suspended, sitting on an Si/SiO 2 substrate, are isolated or in bundles, are encased in a surfactant like SDS (sodium dodecyl sulfate) or are wrapped with DNA, are in solution or are dry, etc.[26] From inspection of the Kataura plot in Fig. 3 it is easy to see how the Raman effect at a given Elaser can be used to separate the contribution of semiconducting and metallic tubes in a given sample, making use of the diameter dependence of the RBM. Thus Raman spectroscopy can provide a powerful probe
for studying the properties of semiconducting SWNTs at the single SWNT level, or when in bundles containing both M and S SWNTs. Raman spectra of carbon nanotubes, particularly at the single nanotube level, have been especially rich. Because of the simplicity of the geometrical structure of nanotubes, detailed analysis of the Raman spectra have yielded much information about the phonon dispersion relations, such as information about the trigonal warping effect for phonons. Such information was not yet available for 2D graphite, but can now be studied in nanotubes because of their one-dimensionality [20]. Because of the close coupling between electrons and phonons under resonance conditions, Raman spectra have also provided valuable and detailed information about the electronic structure (such as an evaluation of the magnitude of the trigonal warping effect for electrons [18]), thus yielding valuable information about the electronic structure that can be used in the characterization of nanotube samples. At present, the use of many laser lines (over 50) provides the capability of carrying out Raman spectroscopy measurements with a quasi-continuous source [25, 27]. This capability promises to provide a means to access essentially all the SWNTs in a given sample, thereby providing a powerful new characterization technique for SWNTs. With this capability, the intermediate frequency modes (between 600–1200 cm 1 ) have been investigated for semiconducting SWNTs showing the possibility of observing combination modes and infrared modes in this frequency range. Such combination modes and infrared modes cannot be observed in the Raman spectra for crystalline graphite. The ability to use a gate [28], an externally applied
FIGURE 3. Calculated [21] energy separations Eii between van Hove singularities i in the 1D electronic density of states of the conduction and valence bands for all possible n m values vs. nanotube diameter in the range 04 dt 30 nm, using a value for the carbon-carbon energy overlap integral of γ0 29 eV and a nearest neighbor carbon-carbon distance aC C 0142 nm.[19, 30]. Semiconducting (S) and metallic (M) tubes are indicated by crosses and open circles, respectively.
potential [29] or a tunable laser [24, 25, 27] to move the van Hove singularity for an individual SWNT into and out of resonance with the laser offers great promise for future detailed studies of the 1D physics of single wall carbon nanotubes using resonance Raman spectroscopy. The ability to characterize individual SWNTs for their n m identification allows the determination of the dependence on nanotube diameter and chirality of many physical phenomena in SWNTs that can be measured at the single nanotube level. Near field microscopy has also proven to provide a powerful tool for examining the Raman spectra on a spatially resolved basis (to 20 nm resolution),[31, 32] thus providing detailed information on the effect of specific defects on the vibrational spectra. The disorder-induced D-band provides a sensitive probe of symmetry-breaking agents, such as defects, in carbon nanotubes.[31, 32] The observation of photo-luminescence (PL) from isolated SWNTs made possible the observation of the S energy gap for semiconducting SWNTs. Intense PL E11 S , peaks, indicating strong optical absorption at a given E 22 S and emission at E11 , have been related to specific n m SWNTs. Such PL experiments were used to construct a plot of the E11 and E22 electronic transition energies for many n m, thus providing an empirical expression for Eii as a function of nanotube diameter and chiral angle for isolated SWNTs dispersed in aqueous solution with SDS surfactant,[33] used to separate one SWNT from another. Analysis of these PL data showed the impor-
tance of 2n m family behavior as the nanotube diameter is reduced, thereby indicating that modification to the simple tight binding model that had been successfully used for analysis of the RRS spectra for larger d t SWNTs at the single nanotube level had to be modified.[4] These PL experiments led to new directions for RRS and to advances in theoretical modeling. The experimental programs in many laboratories were recently directed toward investigating the effect of wrapping nanotubes with DNA and with surfactants such as SDS (sodium dodecyl sulfate) on the Raman spectra, comparing transition energies observed in PL and RRS, looking for 2n m family effects and in general focusing more on studies of smaller diameter nanotubes. Consistent results were obtained when PL and RRS techniques were applied to the same sample. Wrapping and bundling of SWNTs has a minor effect on the RBM frequencies and a larger effect on the E ii in the small dt limit. Work is still on-going to see how single nanotube spectroscopy of individual SWNTs on a Si/SiO 2 substrate fit into the picture. Extensions of the tight binding calculations have been made to include curvature effects (by allowing bond angle and bond length variations), long range interactions between carbon atoms, followed by structural optimization and the inclusion of electron-electron interactions and exciton binding effects. With all these extensions of the simple tight binding model, good agreement is now obtained with experimental PL and RRS measurements for SDS wrapped SWNTs. Further efforts to account for observations on isolated SWNTs on Si/SiO 2 substrates are in progress, and studies on individual suspended SWNTs are also being carried out.[26]
Acknowledgments The authors acknowledge M. A. Pimenta for helpful discussions. A.J acknowledges financial support by PRPq-UFMG, Profix-CNPq and the Instituto de Nanociências (Millennium Institute Program), Brazil. R.S. acknowledges a Grant-in-Aid (No. 13440091) from the Ministry of Education, Japan. M.S.D. acknowledges support under NSF Grants DMR 04-05538, and INT 00-00408.
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