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The Optical Resonances in Carbon Nanotubes Arise from Excitons

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Science  06 May 2005:
Vol. 308, Issue 5723, pp. 838-841
DOI: 10.1126/science.1110265

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Abstract

Optical transitions in carbon nanotubes are of central importance for nanotube characterization. They also provide insight into the nature of excited states in these one-dimensional systems. Recent work suggests that light absorption produces strongly correlated electron-hole states in the form of excitons. However, it has been difficult to rule out a simpler model in which resonances arise from the van Hove singularities associated with the one-dimensional bond structure of the nanotubes. Here, two-photon excitation spectroscopy bolsters the exciton picture. We found binding energies of ∼400 millielectron volts for semiconducting single-walled nanotubes with 0.8-nanometer diameters. The results demonstrate the dominant role of many-body interactions in the excited-state properties of one-dimensional systems.

Coulomb interactions are markedly enhanced in one-dimensional (1D) systems. Single-walled carbon nanotubes (SWNTs) provide an ideal model system for studying these effects. Strong electron-electron interactions are associated with many phenomena in the charge transport of SWNTs, including Coulomb blockade (1, 2), Kondo effects (3, 4), and Luttinger liquid behavior (5, 6). The effect of Coulomb interactions on nanotube optical properties has remained unclear, in spite of its central importance both for a fundamental understanding of these model 1D systems (7-9) and for applications (7, 10, 11). Theoretical studies suggest that optically produced electron-hole pairs should, under their mutual Coulomb interaction, form strongly correlated entities known as excitons (12-18). Although some evidence of excitons has emerged from studies of nanotube optical spectra (7, 19) and excited-state dynamics (20), it is difficult to rule out an alternative and widely used picture that attributes the optical resonances to van Hove singularities in the 1D density of states (21-23). Here, we demonstrate experimentally that the optically excited states of SWNTs are excitonic in nature. We measured exciton binding energies that represent a large fraction of the semiconducting SWNT band gap. As such, excitonic interactions are not a minor perturbation as in comparable bulk semiconductors, but actually define the optical properties of SWNTs. The importance of many-body effects in nanotubes derives from their 1D character; similar excitonic behavior is also seen in organic polymers with 1D conjugated backbones (24).

We identified excitons in carbon nanotubes using two-photon excitation spectroscopy. Two-photon transitions obey selection rules distinct from those governing linear excitation processes and thereby provide complementary insights into the electronic structure of excited states, as has been demonstrated in studies of molecular systems (25) and bulk solids (26). In 1D materials like SWNTs, the exciton states show defined symmetry with respect to reflection through a plane perpendicular to the nanotube axis. A Rydberg series of exciton states describing the relative motion of the electron and hole, analogous to the hydrogenic states, is then formed with definite parity with respect to this reflection plane. The even states are denoted as 1s, 2s, 3s, and so on, and the odd wave functions are labeled as 2p, 3p, and so on (27). Because of the weak spin-orbit coupling in SWNTs, all optically active excitons are singlet states, with the allowed transitions being governed by electric-dipole selection rules. For the dominant transitions polarized along the nanotube axis, one-photon (linear) excitation requires the final and initial states to exhibit opposite symmetry. In contrast, a two-photon transition is allowed only when the final state has the same parity as the initial state. Given the symmetry of the underlying atomic-scale wave functions, one-photon excitation produces only excitons of s-symmetry, whereas two-photon excitation leads only to excitons of p-symmetry (28). Thus, one-photon transitions access the lowest lying 1s exciton; two-photon transitions access only the excited states of the exciton.

An experimental method to determine the energies of the ground and excited exciton states follows immediately from these symmetry arguments: We measured the energies needed for one-photon and two-photon transitions in semiconducting nanotubes (Fig. 1A). A comparison of these energies yields the energy difference between the ground and excited exciton states and thereby directly indicates the exciton binding strength. When the excitonic interactions were negligible, we reverted to a simple band picture in which the onset of two-photon absorption coincides with the energy of one-photon absorption (Fig. 1B). The two-photon excitation spectra reflect the qualitative difference between these two pictures in an unambiguous fashion. In contrast, conventional linear optical measurements, such as absorption and fluorescence spectroscopy, access only one-photon transitions, for which a van Hove singularity and a broadened excitonic resonance exhibit qualitatively similar features. Because the one-photon absorption and emission arise from the same electronic transition in SWNTs, there is no Stokes shift between the two, as apparent in comparison of absorption and fluorescence spectra (8).

Fig. 1.

Schematic representation of the density of states for a SWNT, showing the two-photon excitation (blue arrows) with photon energy hν and subsequent fluorescence emission (red arrows) in the exciton and band pictures. (A) In the exciton picture, the 1s exciton state is forbidden under two-photon excitation. The 2p exciton and continuum states are excited. They relax to the 1s exciton state and fluoresce through a one-photon process. (B) In the band picture, the threshold for two-photon excitation lies at the band edge, where the relaxed fluorescence emission also takes place.

In our experiment, we used isolated SWNTs in a poly(maleic acid/octyl vinyl ether) (PMAOVE) matrix. SWNTs grown by high-pressure CO synthesis were dispersed in an aqueous solution of PMAOVE by a sonication method (29). In order to minimize infrared absorption of water, we formed a film of SWNTs imbedded in polymer matrix by slowly drying a drop of the solution. The SWNT samples obtained by this procedure showed fluorescence emission comparable to that of the SWNTs in aqueous solution.

Two-photon excitation is a nonlinear optical effect that requires the simultaneous absorption of a pair of photons. Femtosecond laser pulses provided the high intensities of light necessary to drive this process. The light source, a commercial optical parametrical amplifier (Spectra Physics OPA-800C), pumped by an amplified mode-locked Ti:sapphire laser, produced infrared pulses of 130-fs duration at a 1-kHz repetition rate. Peak powers exceeding 108 W were obtained over a photon energy range from 0.6 to 1.0 eV. Because these photon energies were well below the 1-photon absorption threshold (>1.2 eV) of the relevant SWNTs, no linear excitation occurred. A laser fluence of 5 J/m2 was typically chosen for the measurements. At this fluence, we explicitly verified the expected quadratic dependence of the excitation process on laser intensity.

To detect the two-photon excitation process in the SWNTs, we did not directly measure the depletion of the pump beam. Rather, we used the more sensitive approach of monitoring the induced light emission. The scheme can thus be described as two-photon-induced fluorescence excitation spectroscopy. Prior studies have shown that rapid excited-state relaxation processes in SWNTs (20) lead to fluorescence emission exclusively from the 1s-exciton state. Measurement of the two-photon-induced fluorescence thus yielded (Fig. 1A) both two-photon absorption spectra (from the fluorescence strength as a function of the laser excitation wavelength) and the one-photon 1s-exciton spectra (from the fluorescence emission wavelength). Further, because the fluorescence peaks reflect the physical structure of the emitting nanotubes, we obtained structure-specific excitation spectroscopy even when probing an ensemble sample. We detected the fluorescence emission in a backscattering geometry, using a spectrometer with 8-nm spectral resolution and a 2D array charge-coupled device (CCD) detector. Our data sampled the infrared excitation range in 10-meV steps.

The measured two-photon excitation spectra (Fig. 2) show the strength of fluorescence emission as a function of both the (two-photon) excitation energy and the (one-photon) emission energy. From the 2D contour plot, distinct fluorescence emission features emerge at emission energies of 1.21, 1.26, 1.30, and 1.36 eV (Fig. 2, circles). These emission peaks have been assigned, respectively, to SWNTs with chiral indices of (7,5), (6,5), (8,3), and (9,1) (7). It is apparent that none of the nanotubes were excited when the two-photon excitation energy was the same as the emission energy (Fig. 2, solid line). Only when the excitation energy was substantially greater than the emission energy did two-photon absorption occur. This behavior is a signature of the presence of excitons with significant binding energy and is incompatible with a simple band picture of the optical transitions.

Fig. 2.

Contour plot of two-photon excitation spectra of SWNTs. The measured fluorescence intensity is shown in a false-color representation as a function of the (two-photon) excitation energy and the (one-photon) fluorescence emission energy. Fluorescence peaks of different SWNT species [(7,5), (6,5), (8,3), and (9,1) with increasing emission energy] can be identified (black circles). The two-photon excitation peaks are shifted substantially above the energy of the corresponding emission feature, as is apparent by comparison with the solid line describing equal excitation and emission energies. The large shift arises from the excitonic nature of SWNT optical transitions.

The two-photon excitation spectra for nanotubes of given chiral index can be obtained as a horizontal cut in the contour plot of Fig. 2, taken at an energy corresponding to 1s-exciton emission of the relevant SWNT. To enhance the quality of the data, we applied a fitting procedure (30) to eliminate background contributions from the emission of other nanotube species. The resulting two-photon excitation spectra are shown for the (7,5), (6,5), and (8,3) SWNTs in Fig. 3. For each of the SWNT structures, the energy of the 1s fluorescence emission is indicated by an arrow.

Fig. 3.

Two-photon excitation spectra of (7,5), (6,5), and (8,3) SWNTs. The traces, offset for clarity, show onset energies for two-photon transitions that are appreciably higher than the corresponding fluorescence peaks (indicated by the arrows). The solid lines are the fits to the excitation spectrum obtained from our exciton model. For comparison, we show the single-particle band model prediction for an (8,3) nanotube as the dashed line in the lower trace.

The peaks in the two-photon excitation spectra can be assigned to the energy for creation of the 2p exciton, the lowest lying symmetry-allowed state for the nonlinear excitation process. From a comparison of this energy with that of the 1s-exciton emission feature, we obtained directly the relevant energy differences for the ground and excited exciton states: E2p - E1s = 280, 310, and 300 meV, respectively, for the (7,5), (6,5), and (8,3) SWNTs.

To determine the exciton binding energy and understand the nature of the two-photon spectra more fully, we considered the two-photon excitation process in greater detail. In addition to two-photon transitions to the 2p state, higher lying bound excitons are also accessible (such as 3p and 4p). The strength of these transitions was relatively small, and they do not account for the main features of the spectrum. We also, however, have transitions to the continuum or unbound exciton states. Including the influence of electron-hole interactions on the continuum transitions, we found that the expected shape of this contribution to the two-photon excitation spectrum could be approximated by a step function near the band edge (31). The experimental two-photon excitation spectra can be fit quite satisfactorily to the sum of a Lorentzian 2p exciton resonance and the continuum transitions with a broadened onset.

A more quantitative description of the two-photon excitation spectra can be achieved with a specific model of the effective electron-hole interaction within a SWNT. In the model, we consider a truncated 1D Coulomb interaction given by the potential V(z) = -e2/[ϵ(|z| + z0)] for electron-hole separation z. The value of z0 = 0.30d is fixed to approximate the Coulomb interaction between two charges distributed as rings at a separation z on a cylindrical surface of diameter d (27); the effective dielectric screening ϵ is the only adjustable parameter in the analysis. This simple model provides a good fit to the experimental data for the different nanotube species examined when we use an effective dielectric constant of 2.5 (Fig. 3, solid line). The features predicted in the model have been broadened by 80 meV (full width at half maximum). This broadening is in part experimental, reflecting the spectral width of the short laser excitation pulses (30 meV). The main contribution, however, is the width of the excitonic transition itself. This width is ascribed to lifetime broadening associated with the rapid relaxation of the excited states to the 1s exciton state (20). From this analysis, we determined the energy of 2p for the three SWNT species in Fig. 3 to be E2p≈ -120 meV with respect to the onset of the continuum states at the band gap energy Eg.

Combining the previously determined E2p - E1s energy difference with the position of the 2p exciton relative to the continuum, we obtained an overall binding energy for the ground-state (1s) exciton of Eex = (Eg - E1s)≈ 420 meV for the investigated SWNTs. This value is comparable to recent theoretical predictions of large exciton binding energies (13, 14). The exciton binding energy thus constitutes a substantial fraction of the gap energy Eg≈ 1.3 eV for our 0.8-nm SWNTs. To put this result in context, the exciton binding energies in bulk semiconductors typically lie in the range of several meV and represent a slight correction to the band gap. Furthermore, because thermal energies at room temperature exceed typical bulk exciton binding energies, excitonic effects in bulk materials can be largely neglected under ambient conditions. This situation clearly does not prevail for SWNTs.

We can understand the strong increase in excitonic effects in the SWNTs as the consequence of two factors. The first arises from a general property of reduced dimensionality: In three dimensions, the probability of having an electron and hole separated by a displacement of r includes a phase space factor of r2, favoring larger separations over smaller ones. In one dimension, no such factor exists. Short separations are thus of greater relative importance, and the role of the Coulomb interactions is enhanced. The second factor relates to the decreased dielectric screening for a quasi-1D SWNT system. This effect arises because the electric field lines generated by the separated electron-hole pair travel largely outside of the nanotube, where dielectric screening is decreased. Because these effects are general features arsing from the 1D character, they should be widely present in 1D systems. Indeed, similar excitonic effects have been extensively studied in a large family of 1D structures of conjugated polymers (24).

To help visualize the strongly bound excitons in SWNTs, we estimated the exciton's spatial extent, i.e., the typical separation between the electron and the hole in the correlated exciton state. Assuming an exciton kinetic energy comparable to its binding energy Eex, which applies precisely for 3D excitons, we obtain the relation Embedded Image, where Embedded Image is Planck's constant h divided by 2π, m is the reduced electron-hole mass, and R is the exciton radius. For m = 0.05 m0 (21), we deduced from our experimental binding energy a ground-state exciton radius of R = 1.2 nm. This value is similar to that obtained by calculation within the truncated Coulomb model specified above. Figure 4 provides a representation of the calculated density distribution of the exciton envelope wave function. The result is a highly localized entity, with a spatial extent along the nanotube axis only slightly exceeding the nanotube radius of 0.8 nm.

Fig. 4.

Density of the 1s-exciton envelope wave function for a (6,5) SWNT. The wave function has been calculated using the experimentally determined exciton binding energy and the truncated Coulomb electron-hole interaction. The density represents the probability of finding the electron and hole composing the exciton at the indicated relative separation. The half width of the exciton along the nanotube is R = 1.2 nm, compared to the 0.8-nm diameter of the nanotube.

The importance of excitonic effects is clear for the interpretation and assignment of the observed optical spectra, as discussed in the literature on the relation of the E11 and E22 transition energies in SWNTs (7, 15, 17). The excitonic character of the optically excited state also has immediate implications for optoelectronic devices and phenomena. For example, photo-conductivity in SWNTs should have a strong dependence on the applied electric field, because charge transport requires spatial separation of the electron-hole pair. The excitonic character of optically excited SWNTs also raises the possibility of modifying the SWNT transitions through external perturbations, thus facilitating new electro-optical modulators and sensors. More broadly, the strong electron-hole interaction demonstrated in our study highlights the central role of many-body effects in 1D materials.

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