Carbon Nanotubes: Electronic Structure and Spectroscopy

1.02 Carbon Nanotubes: Electronic Structure and Spectroscopy G Lanzani, Italian Institute of Technology, Milano, Italy L Lu¨er, Madrid Institute for Advanced Studies, IMDEA Nanociencia, Madrid, Spain ª 2011 Elsevier B.V. All rights reserved.

1.02.1 1.02.2 1.02.3 1.02.4 1.02.5 1.02.6 1.02.7 1.02.8 References

Introduction Geometry of the CNT Lattice Electronic and Optical Properties Characterization of the Exciton State Photoexcitation Dynamics: The Critical Role of Defects Vibrational Modes in CNTs Coherent Phonons in CNTs Conclusions

1.02.1 Introduction Carbon nanotubes (CNTs) are cylinders of carbon atoms with nanometer radius and length ranging from hundreds of nanometers to micron and even millimeters [1]. This makes them quasi-ideal onedimensional (1D) structures with highly anisotropic properties. They usually come in a variety of forms, including single-wall and multi-wall, semiconducting and metallic, bundles, networks or isolated, and in a broad distribution of chiralities. The latter, which depends on the production and purification method, regards the geometry of the atomic lattice that defines the tube family and many of the electronic properties [2]. This chapter deals with optical properties of single-wall CNT, and their electronic structure. The carbon atoms on the tube surface are in sp2 hybridization, as in most conjugated systems, and thus support -electron delocalization. This happens because even in small diameter tubes, the pyramidalization angle is enough for allowing partial overlap of near atomic orbitals. The small circle across the tube leads to quantization of the electronic states perpendicular to the tube axis, while along the tube axis the elementary excitations are fully delocalized Bloch states. The semiconducting behavior of the nanotubes derives from this quantization and can be considered a new property with respect to graphene having zero gap.

23 24 25 28 29 32 33 36 37

The first description of electronic states in CNT is provided by the tight-binding method. This approach is indeed insufficient for describing correctly the electronic interaction in the tube, yet it provides a base for the more advanced correlation models. According to the latter, the elementary excitations in CNTs are excitons of Wannier–Mott type, because the electron–hole correlation distance is about 2 nm, that is well exceeding the lattice constant. This has the important consequence that absorbed photons do not lead to charge carriers. In other words, the interacting state connecting the photon field to the tube electronic structure is the neutral exciton. Only a small fraction of the absorbed energy goes into charge carriers, with a branching ratio not yet precisely determined and through mechanisms not yet understood. Charge carriers are important in many electrical processes and can propagate rather efficiently along the tube, and possibly between tubes. The interplay of charge and neutral states is still an open question in the field. An important piece of work has been done on single tubes, leading to information on fundamental physical properties and demonstrations of nanoscale devices, for example, the single-tube field effect transistor (FET) devices, which challenge the ultimate limit in miniaturizing. In these FET devices, the photoconductivity of single nanotubes has been studied, showing excitonic characteristics [3], and polarized infrared emission has been observed [4].

23

24 Carbon Nanotubes: Electronic Structure and Spectroscopy

However, this approach has little potential for shortterm applications, due to the almost complete lack of control in the deposition process, which hampers any upscaling to industrial production. The near future of CNT is in composite materials made by blending in active or passive matrices. CNT can provide optical (e.g., saturable absorption), electrical (e.g., transport), or mechanical (strength) properties to the hosting matrix, as a result of cheap and simple processing. These systems do not require fancy nanoprobes for testing or exploiting and they are suitable for large-scale industrial production. After standard purification and separation techniques, CNT can be mixed in the matrix preserving a degree of multi-chirality and bundling. This opens up a new froutier for characterizations that regard small bundles and ensemble properties. It has recently been shown that the selection of the nanotubes on the basis of their electrical properties is not always necessary for electronic applications where semiconducting nanotubes are needed. In this case, using the fact that two-thirds of the types of nanotubes are semiconducting, a submonolayer of nanotubes was found to well behave as a semiconducting material since metallic nanotubes are found to be sufficiently diluted and determine conducting channels in the sample [5]. In order to improve handling and separation, chemical functionalization has been pursued rather extensively [6]. This has advantages, such as the control and the huge potential of the organic chemistry, but also drawbacks. Chemical functionalization introduces defects; it is a complicated process and may lead to undesired selectivity in diameters. Alternatively, a much simpler approach is that of the noncovalent functionalization. Here, aromatic interactions lead to complex nanostructures where a conjugated system wraps around the tube. It has been shown that noncovalent functionalization leads to chiral selection, sometimes to the individual tube. The composite polymer þ CNT nanostructure is a new system with characteristics by and large to be investigated. In particular, ground- and excited-state electronic or excitonic interactions could take place, leading to new excited states and new deactivation paths, interesting coupling or interface phenomena. A work apart is the attempt to produce samples that contain a large excess of a single tube type. Efforts initially targeted to semiconductor–metallic separation have now turned to single chirality. In most cases good results are obtained on tiny quantity in highly diluted solution, but there is at least one notable

difference: the case of the (6,5) tube. Starting with CoMoCAT material, rich in (6,5) and using ultracentrifuge separation, sorting can be very effective. These samples have the important advantage to be macroscopic, allowing a lot of standard characterization, and yet highly homogeneous. This allows, for instance, us to apply transient spectroscopy essentially to one kind of tube, leading to much clearer results and interpretation.

1.02.2 Geometry of the CNT Lattice Take a CNT and ideally use a nanoscissor to cut it along a line on the surface, parallel to the axis. Open it flat down. What you get is a nanoribbon, a long strip of carbon atoms with width equal to the tube circumference and the typical honeycomb structure of graphene. In Figure 1 we show this for the tube (6,5). The figure shows a portion of the long strip, corresponding to the translation unit cell. Along the axis you should imagine this as infinite. The chiral vector across the ribbon defines the tube circumference and it can be expressed in terms of the graphene Bravais vectors a1 and a2 as C ¼ n a1 þ m a2

ð1Þ

Each tube is characterized by the two chiral indexes (n,m). Zigzag tubes have m ¼ 0, armchairs have n ¼ m. Those two classes possess a symmetry mirror plane and thus are achiral. The third class contains all others, which are optical enantiomers without the mirror symmetry plane. The diameter of the tube is d¼

jC j a0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ? n2 þ nm þ m2  

ð2Þ

where a0 ¼ 2:461A˚ is the length of the basis vectors. The diameter of the (6,5) tube is then 0.75 nm. The tube diameter can be estimated by Raman spectroscopy, measuring the frequency of the so-called radial breathing mode (RBM), which typically occurs in the 200–300 cm1 range (307 cm1 for the (6,5) tube). The RBM frequency, which corresponds to the oscillation of the diameter length, is inversely proportional to the tube diameter. This result can be easily obtained assuming that the tube circumference is a linear chain of springs. Figure 1 shows the unit cell of a (6,5) tube.

Carbon Nanotubes: Electronic Structure and Spectroscopy

25

a2 a1

Figure 1 Roll-up vector of a (6,5) tube and primitive cell in real space.

1.02.3 Electronic and Optical Properties Since their discovery, CNTs were thought as covalent semiconductors, supporting electron and hole pairs as elementary excitations, similar to 3D standard inorganic semiconductors. They were thought to be nonemissive, and most experiments focused on electrical properties. The absorption spectrum of early samples shows a broad and rather unstructured band, extending from the near-infrared (IR) to the ultravoilet (UV). At that condition, the tight-binding model within single particle approximation provided a consistent description of the observed properties. CNT can be regarded as graphene sheets that are rolled up as discussed earlier. Their electronic structure should thus bear some correspondence with graphene, that has been studied and understood years ago. Graphene is a semimetal, with zero gap and small electron density near the Fermi level. Approaching the gap, electrons have linear dispersion in k-space, that is, zero mass (Dirac fermions). These particles have exotic properties that have been experimentally accessed only recently. For graphene the Brillouin zone in the reciprocal lattice has hexagonal structure, with three points of high symmetry, designated K, M and . In K (Dirac point) the  and  (valence and conduction) bands touch each other, with null gap. The unit cell of CNT is a cylindrical

surface with height T and diameter d. To first approximation we can think of the flat geometry in Figure 1. In the reciprocal space the vector kZ, parallel to the long axis, is continuous because the tube is regarded as infinite. This corresponds to the ID delocalization of the electronic states along the tube axis. In the other direction, however, the finite size leads to confinement of the electron, that is, the motion along the circumference is quantized. The kj vector, in our approximation of flat geometry, is kj ¼

2 2 ?j ¼ ?j d jC j

ð3Þ

with j ¼ – p=2 þ 1;:::; – 1; 0; 1; :::; p=2 þ 1; where p is the number of points in the primitive cell. The k-space of CNTs is then made by a series of segments, parallel to the tube axis and separated by 2/d, Within the flat approximation, CNT bands can be obtained by cutting the 2D band structure of graphene. Two situations may occur:

• •

The lines never cross the K-point, the tube behaves like a 1D semiconductor. This happens if n – m 6¼ 3p; pPN : Some of the lines cross the K-point, the tube behaves like a metal with Dirac fermions in 1D. This happens if n – m ¼ 3p; pPN :

It turns out from simple statistics based on equal chirality probability that, on average, a sample should

26 Carbon Nanotubes: Electronic Structure and Spectroscopy

have about two-thirds of semiconducting tube and one-third of metallic ones. Assuming a normal, quadratic dispersion of the electrons in k-space and the 1D space. the density of states (DOS) at the critical points near band gaps is proportional to the inverse square root of the energy. This causes sharp resonances in the joint DOS, as shown in Figure 2(a). Where the gap collapses to zero, electrons have linear dispersion in k-space and the DOS is constant. The latter situation only occurs in metals. It can be concluded that according to tightbinding theory, both S and M tubes should have sharp resonances in the DOS (called Van Hove singularities) which give rise to sharp peaks in the absorption spectrum. Due to the very small momentum of the absorbed photon compared to the line separation in k-space, 2= 2=d , the allowed transitions are only those within the same Brillouin zone line. This means that allowed transitions only occur between bands symmetric with respect to the Fermi energy, typically named E11, E22, etc. The energies for the first two transitions are given in terms of tightbinding parameters by   2acc 0 þ f n; m; d – 2 d   4acc 0 þ f n; m; d – 2 E22  d

ð4Þ

E11 

ð5Þ

(b)

(a) (...)

2nd CB Edge

Energy

(p)

1st vHs 2

nd

EX-2

(s)

vHs

st 1 CB Edge

(p)

EX-1 (s)

(...)

2nd manifold bound states

st

1 manifold bound states

G.S.

DOS(E) dE Figure 2 Optical transitions in semiconducting carbon nanotubes (CCNTs). (a) Density of states of the two bands closest to the Fermi level. The Van Hove singularities are labeled vHs. (b) Possible optical transitions from the electronic ground state. Each vHs from panel (a) contributes an excitonic manifold. The symmetry of the excitons is given in brackets. Photon energies represented by upward arrows. Transitions into s-excitons are one-photon allowed and those into p-excitons are two-photon allowed.

where  0 is the transfer integral, acc ¼ 0.142 nm is C–C bond length, and d is the diameter. The term fðn;m;d – 2 Þ adds a small chirality-dependent correction, which is usually neglected. Therefore, the theory predicts E11 =E22  0:5. Upon improving material quality (less impurity, some selectivity in chirality, and less bunding), optical properties started to attract interest. Photoluminescence was observed, the estimate on fluorescence quantum yield kept rising during the years, and fast nonlinear optics came into play. Theory first pointed out the role of electron–hole correlation [7], before experimental evidences were available, based on the well-known fact that in 1D screening is reduced and Coulomb interaction is important. Then experiments and theory developed in parallel to bring a solid evidence for the existence of excitons. From the experimental point of view, accepted evidences for excitons are: 1 1. The ratio E11/E22 is not as predicted by tight 2 binding. 2. The one-photon optical gap differs from the twophoton optical gap [8,9]. This contrasts with a valence-to-conduction band transition that involves quasi-degenerate s-like and p-like states, and thus has the two energies that coincide. 3. Upon photoexcitation transient transmission spectra show in-gap optical transitions. The lack of a Drude tail in the photo-induced absorption would also support excitons, but this is still under debate, due to contrasting reports. From the theoretical point of view, the 1D exciton can be understood starting from the early theory of the 1D hydrogen atom. In going from 3D to 1D, the exact treatment of the problem of the motion of two charges in their mutual Coulomb potential becomes impossible: the binding energy goes to 1 and the lowest wave function collapses into a delta function at zero electron and hole distance. To avoid this and resume to a real solution one should introduce a cutoff length, co. This can be understood as the length at which the 1D approximation breaks down. In CNTs, this will happen for a length of the order of the tube diameter. Introducing the cutoff length co allows finding an analytical solution of the problem. Discrete energy levels appear below the conduction band, with lower spacing approaching the band edge; the lowest wave function preserves a strong localization, with spreading comparable to the cutoff size, while the other states resemble the

Carbon Nanotubes: Electronic Structure and Spectroscopy

hydrogenoid wave functions with alternating symmetry. A sketch of the model is shown in Figure 3. This naturally explains the one-photon versus twophoton gap. It is peculiar to 1D that most of the oscillator strength is concentrated in the lowest transition, while the transitions to higher excitonic states and to the conduction band are largely suppressed. Once the exciton is formed, optical coupling to an even state near the conduction band is expected, providing a direct estimate for the binding energy [10]. Full quantum mechanical studies, with various techniques, brings about a more complex structure of the electronic levels of CNTs [11,12]. There are four lower lying excitonic states, three of them optically forbidden (dark) and only one allowed (bright), see Figure 2. The latter can be regarded as the one we qualitatively discussed above. These dark states are weakly interacting and remain unseen in most experiments. They might play a role in temperaturedependent PL, being probably within kBT from the bright state [13]. Their role is still not well known.

(a) U (x) x Eb n=1

~ξ n=0

(b) F(E) n=0

{1,2,...,∞} Continuum 0

1

E/Eg

Figure 3 One-dimensional (1D) excitons. (a) Electron– hole Coulomb potential U(x) as function of separation x. The introduction of a cutoff length co leads to hydrogen-like bound states. Wave functions are given for the lowest two states. (b) Resulting oscillator strength as function of photon energy. Most of the oscillator strength is transferred to the discrete excitonic levels while the direct band-to-band transition is nearly suppressed.

27

Due to the small energy separation, it is however feasible that upon photoexcitation, equilibrium between the bright state and the dark state is established (see later). One should note that the interpretation of optical spectra is not revolutionized by the exciton model because the sharp transitions at the Van Hove singularity gets replaced by the sharp resonances of the excitons (one hydrogen series for each band-to-band transition). This happens for a coincidence of two effects. The exciton levels are below the conduction band edge, but band-gap renormalization (BGR), also caused by Coulomb repulsion, leads to a larger energy gap compared to the free carrier picture. The net impact of these two effects is a singlet excitonic state with an energy close to calculated values based on zone-folding free-carrier approximation. What changes substantially is the dynamics of photoexcitations, now leading to neutral, currentless states. According to the exciton model, the probability of photo-generating charges is negligible at low energy, and gets sizable for the III or IV exciton. Note however that one should expect a fast recombination of the free carriers into the bound state, so that this model predicts over all negligible photocurrent. This is not however supported by experimental results due to a variety of phenomena still to be fully discovered, such as exciton ionization at impurities or defect, inter-tube charge transfer reactions, and other environmental effects. Experimental determination of absorption spectra in CNTs is complicated by the polydisperse nature of most nanotube samples, consisting of CNTs of various chiralities, with varying lengths and defect densities. Optical spectroscopy of individual CNTs can give full insight into linewidths and side bands of single chiralities; however, classical transmission spectroscopy is inhibited by the low signal to be expected from a single CNT. Therefore, two different approaches have been successfully adopted to obtain information about the optical absorption of isolated nanotubes. Elastic Rayleigh scattering from individual CNTs has been analyzed [14], providing spectroscopic information through its enhancement in the vicinity of an optical transition. In another approach, photothermal heterodyne detection (resulting purely from absorption events) has been used to present absorption spectra of individual CNTs of various metallic and semiconducting chiralities [15].

28 Carbon Nanotubes: Electronic Structure and Spectroscopy

1.02.4 Characterization of the Exciton State

The degree to which an exciton is localized can most readily be accessed through the affiliated distribution of electron (fe(k)) and hole (fh(k)) oscillator strengths, which, in the case of resonantly generated excitons, is related to exc(k), the Fourier transform of the exciton relative motion orbital wave function ˜ exc(x) by [20] 

Wannier–Mott excitons are described as bound electron–hole pairs with two characteristic properties: the center-of-mass position (RCM) and the relative electron–hole distance or correlation length (e). (see Figure 4). The latter is directly linked to the electronic structure of the material and its physical properties, such as screening, Coulomb attraction, binding energy, exchange interaction, and confinement of wave functions. For this reason, e is the key figure of merit for a better understanding of optical, optoelectronic, and photonic properties and also for validation of existing theories on excitonic effects in CNTs. Specifically, if e is comparable to the lattice constant, then the exciton is of the tightly bound Frenkel type, typical of molecular solids. In contrast, a value of e much larger than the lattice constant supports the Wannier–Mott (W–M) picture, typical of covalent semiconductors. The size of excitons, as expressed by the average electron–hole separation in the W–M envelope function, has been predicted theoretically by various models [16–18] reporting values for this critical parameter from about 1 to 2 nm. The experimental determination of the exciton size in semiconducting CNTs has been done by probing the light-induced reduction of the lowest energetic (E11) excitonic transition [19]. According to the phase space filling (PSF) model [20], the measured reduction in oscillator strength can be directly related to the exciton size e. This can be rationalized by regarding 1/e as a 1D ‘real space filling’. The reciprocity of phase and real space is developed as follows.

fe ðkÞ ¼ fh ðkÞ ¼

N  ~ exc ðkÞj2 2

where N is the density of excitons per unit length. The exclusion principle for fermions blocks transitions into final states that are already occupied. The effect of this PSF is an overall reduction of oscillator strength upon exciton creation: f – ¼ f

P

~ exc ðkÞ þ fh ðkÞ exc ðx ¼ 0Þ

k ½fe ðk Þ

exc ðx Þ ¼

qffiffiffiffiffiffiffiffiffi – 1   pffiffiffi x2 a  exp  2 2e

ð8Þ

where e is the electron–hole correlation length which can be identified with the exciton size. With this wave function, equation 7 becomes f ¼ – N e ; f

  2:05

ð9Þ

where  is a wave-function-dependent proportionality constant. The relative reduction in oscillator strength is equal to the relative transient groundstate bleaching. f/f ¼ A/A, available directly from experiment. Equation 9 is of the simple and instructive saturation form, where the saturation density, Ns1 ¼ c , is inversely proportional to the

ξe

Vg

O

ð7Þ

The relative motion exciton wave function can be approximated by a Gaussian function [21]:

E

~ RCM

ð6Þ

E

Z k = ke+kh

Figure 4 Properties and geometrical reresentation of ID Wannier–Mott weakly bound excitons.

Carbon Nanotubes: Electronic Structure and Spectroscopy

exciton size. The exciton density per unit length in our experiment is related to the macroscopic exciton density per unit area, n, by N ¼ n/L(6,5), where L(6,5) is the total length of all (6,5) tubes in the light path normalized to its cross section, as obtained from 1 L(6,5) ¼ cnc. Here,  is the C atom denC ¼ 0.01 nm sity per unit length of a (6,5) tube, and nC ¼ A/C is the macroscopic density of carbon atoms per unit area. This can be evaluated by measuring the ground-state absorbance A using the absorption cross section C ¼ 7  1018 cm 2 obtained by Zheng et al. [22]. Finally, at low excitation levels where two-photon processes are negligible, the macroscopic density of excitons per unit area is equal to the number of absorbed photons per unit area, n ¼ Ia. Inserting these relations into equation 9, and considering the anisotropy of the exciton distribution induced by the linearly polarized pump field finally yields Aani ¼

– ra  ce c Ia

ð10Þ

with ra  1.6 being the correction factor to obtain the differential absorption of the hypothetical isotropic distribution. The experimental differential absorption as a function of the absorbed light intensity Ia is shown in Figure 5(a). We find linear behavior at low Ia, while saturation is observed at higher intensities. Three different samples (evidenced by different colors) yield similar results. A zoom-out of the linear region is shown in Figure 5(b). A linear regression yields (a)

a slope of – r C e =C ¼ 4:5  10 – 15 cm 2 . From this, we calculate an exciton size of e ¼ 2:0  0:7nm, using a Gaussian envelope function. If an exponential envelope function is assumed, one obtains e ¼ 2:7  1:0 nm. These experimental values slightly exceed those predicted by semi-empirical calculations or extracted from the exciton-binding energy.

1.02.5 Photoexcitation Dynamics: The Critical Role of Defects Similar to the case of optical spectroscopy, the investigation of photoexcitation dynamics in CNT is complicated by the dispersion of tube chiralities, metallic CNT, and by the simultaneous occurrence of excitons and weakly or unbound charges, each with vastly different relaxation channels and relaxation rates. It therefore depends on the design of the experiment to extract information on specific relaxation pathways. In 2000, Hertel and Moos studied the nonequilibrium distribution of electrons (1D Dirac fermions) near the Fermi edge, induced by a femtosecond laser pulse. Since only metallic CNTs have nonzero electron density near the Fermi edge, this experiment was useful to extract electron thermalization kinetics in metallic CNTs [23]. It was found that after 200 fs from photoexcitation, the electron distribution could be described well by a Fermi– Dirac function. This suggests that electronic relaxation of the nascent nonequilibrium distribution (b)

0.00

Sample 1 Sample 2 #1 Sample 2 #2

0.00 −0.02 −0.05

Slope = 4.5x10−15 cm2

A

ani

−0.04

∇ −0.06

−0.10

−0.08

−0.15 0

2

1

|abs (10

14

−2

cm

)

29

−0.10 0

1 |abs (10

2 13

−2

cm

3 )

Figure 5 Measurement of the transient bleanch in CoMoCat CNT based on equation 10. (a, b) Redrawn from Luer L, Hoseinkhani S, Polli D, et al. (2009) Size and mobility of excitons in (6, 5) carbon nanotubes. Nature Physics 5(1), 54–58.

30 Carbon Nanotubes: Electronic Structure and Spectroscopy

occurs in less than 200 fs. The subsequent cooling of the hot electron distribution occurs in a few picoseconds, similarly to 3D metals. Photoexcitation dynamics in semiconducting nanotubes were first published by Ishida et al. in 2002, who performed femtosecond pump–probe spectroscopy on a polydisperse sample of CNT. Pumping at 3.1 eV, they obtained the first transient absorption spectrum, showing a bleach of the E11 transition in semiconducting CNT [24]. A bleach recovery in the range of 0.5–1 ps was observed. In 2004, a concomitant bleach of the E22 band was discovered, showing transient absorption features at both low and high energy sides of the E22 band [25]. It was found that bleach recovery is faster in the E22 band than in the E11 band. Using sub-20 fs pump and probe pulses, Manzoni et al. proposed that when semiconducting CNTs are excited into the Ex-2 manifold, then the population is transferred within 50 fs to the Ex-1 manifold [26]. This was based on the

observation of rapid E22 bleach recovery and concomitant buildup of E11 bleach (see Figure 6). The process of inter-exciton relaxation was later modeled by Habenicht et al. [27] and Perebeinos [28] considering two paths: the direct exciton–exciton coupling and multiphonon emission and the ionization of Ex-2 into the Ex-1 continuum, followed by electron– hole thermalization in k-space and relaxation to the lowest (bright) exciton state. Theory suggests that the latter contributes about 10% of the initial exciton population. Distinguishing by experiments between these two phenomena remains difficult. One can note that emission of G-phonons, of about 0.2 eV, would indeed account for the observed time scale. The process of Ex-2–Ex-1 conversion leads to a consistent buildup of excess energy in the tube, possibly rising the local temperature. The evidence of the effects of such energy is not clear. Perhaps one is the persistent bleaching of Ex-2 and some modulation E1 reasonance after Ex-2–Ex-1 conversion.

(a)

T/T (a.u)

E11(0.92 eV)

∇

(b) E22(2.15 eV)

−100

0

100

200

300

400

Time delay (fs) Figure 6 The intersubband recombination in HiPCO CNT, pumped by a broadband 7 fs pulse centered at 2.1 eV: (a) probe energy 0.92 eV and (b) probe energy 2.15 eV. The pump–probe data are given as colored lines. Black lines: pump–probe cross-corrections, dashed lines: fits to extract kinetic parameters. Redrawn from Manzoni C, Gambetta A, Menna E, et al. (2005). Intersubband exciton relaxation dynamics in single-walled carbon nanotubes. Physical Review Letters 94(20), 207401.

Carbon Nanotubes: Electronic Structure and Spectroscopy

Femtosecond spectroscopy of a single chirality was first demonstrated by Zhu et al. in 2007 [29], enabled by isopycnic fractionation of DNAsuspended CNT produced by the CoMoCat method; in this way, samples were obtained in which the (6.5) chirality was the dominant CNT. A narrowband pump pulse further enhanced the selectivity. Exciton decay was found to proceed with a power law, still in the 10 ps time domain, which was explained by pre-diffusive trapping, highlighting the role of impurities in exciton relaxation. More refined studies of the lifetime of the Ex-1 state in semiconducting CNT showed that it crucially depends on both excitation and defect density. In fact, single-molecule fluorescence spectroscopy in isolated CNT revealed a strong dispersion of exponential lifetimes spanning from 180 ps to below 20 ps [30], clearly showing the influence of nanotube imperfections on the exciton dynamics. A statistical analysis of the lifetime distribution is presented by Gokus et al. [31] for few selected chiralities, while the possibility that tube ends could be the nonradiative centers is investigated in [32]. Haratyunyan et al. brought evidence for photoactivation of radiative defects, tentatively assigned to activated triplet states [33]. Evidence for exciton–exciton annihilation was obtained at elevated excitation densities [34,35]. It is proposed that by this mechanism, the interaction between two Ex-1 states leads to the creation of a single higher excitonic state, which subsequently decays back to the Ex-1 state under the net loss of one Ex-1 state: k

k

21 Ex1 þ Ex1 !a Ex2 ! Ex1

ð11Þ

. Early measurements were interpreted based on the ‘ideal’ nanotube picture, where the excitonic wave function describing the center-of-mass probability distribution, occupies the whole physical nanotube length. In consequence, the presence of more than one exciton would inevitably lead to a superposition of the respective wave functions and thus to annihilation according to equation 11. This picture was contested by the observation of stepwise quenching of fluorescence when isolated nanotubes were exposed to chemical agents [36]. It could be shown that the introduction of one fluorescence quenching site did not quench the whole nanotube fluorescence, but led to a characteristic relative

31

quenching I/I of the local fluorescence intensity I. Considering the spatial resolution of the experiment, the authors obtained an excitonic diffusion length of about 90 nm, fairly independent of tube diameter. Note that the concept of diffusivity implies the assumption of incoherent excitonic motion, in contrast to coherent (wavepacket-like) excitonic motion expected for the ideal state [35]. Kinetics may help in distinguishing between these two regimes. If exciton annihilation were mediated by diffusive exciton motion, this would give rise to a time-dependent annihilation rate constant ka ðt Þ ¼ ka 9ðt =t0 Þ – 1=2 [37]. The time dependence reflects the rapid decrease over time of the probability that new sites be visited in a stochastic process, for returning to already-visited sites becomes more likely. The demonstration of a time-dependent annihilation constant requires an intensity-dependent study of the Ex-1 population relaxation kinetics. It is complicated by a number of factors. First, the time dependence of the rate coefficient can only be observed-during the initial phase of the decay, so sub-30 fs pulses are needed. Second, there is currently no spectroscopic signature specifically for Ex-1 because E11 bleach recovery is a nonselective probe for any excited states including charged ones, due to ground-state depletion. To perform the experiment, care must be taken that only Ex-1 excitons are produced, and that the E11 bleach is not superposed with PA features from different chiralities. These requirements have been fulfilled by resonantly pumping the Ex-1 state in a highly chirality-enriched sample consisting predominantly of isolated (6,5) tubes with 20 fs pulses, resulting in the first demonstration of a time-dependent annihilation coefficient [19]. Pump intensity- dependent E11 bleach recovery is shown in Figure 7, together with numerical solutions according to 1D diffusion, showing good agreement. This analysis yields an exciton diffusion constant of Dexc ¼ 0.1 cm2s1 and an exciton diffusion length, according to the exciton lifetime, below 10 nm. This particularly short diffusion length reflects the higher defect load introduced by ultrasonication in the CoMoCat samples; in contrast, the studies of Cognet et al. were performed on particurarly defectpoor, highly luminescent single nanotubes. Nonetheless, both studies, based on different methods, demonstrate that the exciton diffusion length is much shorter than a typical nanotube size, so that more than one exciton can coexist on a single nanotube without annihilation. Both studies, in fact, show that CNT photophysics is affected by the imperfect

32 Carbon Nanotubes: Electronic Structure and Spectroscopy

0.0

0.5

0.00

Pump fluences:

5.8

–0.05

46.4 μJ cm–2 29 18.3

ΔAani, normalized

18.3 29

ΔAani

–0.5

Probe energy (nm)

46.4 μJ cm–2 –0.10

1040

5.8 0.5

1020 1000 980

–1.0

960 0 1 2 3 Pump-probe delay (ps)

–0.15 0.0

0.5

1.0

1.5

2.0

0.0

0.5

1.0

1.5

2.0

Pump probe delay (ps) Figure 7 Pump intensity-dependent kinetics of E11 bleach recovery in isolated (6,5) tubes after resonant excitation of Ex-1 state. The dashed lines are numerical solutions of the ID diffusion kinetics. The insert shows time-dependent pump–probe spectra, normalized to their maxima. It is shown that there are no spectral shifts that could interfere with the kinetic model. Adapted from Luer L, Hoseinkhani S, Polli D, Crochet J, Hertel T, and Lanzani G (2009) Size and mobility of excitions in (6,5) carbon nanotubes. Nature Physics 5(1), 54–58.

nature of real nanotubes. Defects in CNT cannot break the conjugation as in conjugated polymers because of the 2D nature of the cylinder surface. Their effect is thus less dramatic on the electronic structure and gap size: yet, breaking of the translation symmetry leads to a localization of the excitonic wave function. A center-of-mass movement of the exciton across the defect can only occur by virtue of incoherent hopping. In other words, defects cause the crossover from coherent to hopping transport. The ability to control the type and the density of defects on a CNT is an important tool for the inclusion of CNT into optoelectronic devices. For photovoltaic devices, an exciton diffusion length of 90 nm is very useful because it exceeds typical light penetration depths in conjugated polymers. This makes the inclusion of polymer-sensitized CNT, acting as exciton/charge transporters, into photovoltaic cells feasible, with improved light-harvesting properties [38]. Sensor applications can benefit from an enhanced probing length of the exciton as well. On the other hand, an exciton diffusion length of 6 nm allows the stabilization of a large number of excitons on the nanotubes; this is useful for high-intensity applications such as ultrafast optical switches or lasing.

1.02.6 Vibrational Modes in CNTs As discussed earlier, CNTs can formally be derived by rolling up a graphene sheet along a rolling vector (n.m). Consequently, also the normal modes of CNTs can be derived from graphene, where a strong G band of E2g symmetry is observed at around 1590 cm1. Due to the introduction of cylindrical symmetry in CNTs, this mode is split into six nearly degenerate modes, that can be grouped into a Gþ and a G mode, which in semiconducting nanotubes correspond to the TO and LO phonon modes. In neararmchair tubes such as the (6,5) one, the dominant mode is Gþ, corresponding to the zone center LO phonon mode [39]. For a more detailed discussion on phonons in CNTs, see the overview by Graupner [40]. In Raman spectroscopy, a strong G9 mode is observed at 2700 cm1, caused by double Raman scattering in a two-phonon process. This G9 mode is strongly renormalized by the presence of defects or dopants [41]. Elastic scattering at defects also gives rise to a D band at half the G9 frequency, at 1350 cm1. The D band is frequently used to detect the efficiency of chemical functionalizations, introducing sp3 sites [6]. At low frequency, the so-called RBM plays a crucial role in CNT characterization. It describes

Carbon Nanotubes: Electronic Structure and Spectroscopy

the fully symmetric breathing of the tube circumference and its frequency depends on the diameter of the CNT according to the empirical relation vRBM ðcm1 Þ ¼ 223:5=d þ 12:5 [42], where d in nanometers is the diameter of the tube. Note however that the precise value of the constants is influenced by environmental conditions such as bundling or wrapping by surfactants [40]. The RBM frequency can be used in the chirality assignment of CNTs. This is particularly useful for metallic nanotubes, which cannot be observed by fluorescence measurements [43]. Raman spectra of CNTs show also a weak overtone band of the RBM region, which is due to exciton–phonon coupling and therefore can be used to assess its strength. The analysis also showed that non-Condon effects, that is, a nondisplacement-related coupling must be taken into account [44]. When comparing the RBM overtone region of bundled and isolated CNT, it is found that the exciton–phonon coupling is reduced upon bundling, while a series of intermediate frequency modes (IFMs) becomes apparent [45]. The G mode is only weakly dependent on chirality; however, a charge-induced renormalization has been described; at a charge density of 0.14 nm1, the G mode is shifted upward by about 5 cm1 [46].

1.02.7 Coherent Phonons in CNTs Electron–phonon coupling results from a dependence of the nuclear coordinates on the electronic state: following the Born–Oppenheimer (BO) approximation, the nuclei will adapt to the changed energetics upon creation of an excited state. This will induce the excitation of those vibrational modes, whose respective potentials undergo a shift (firstorder coupling) or a distortion (second-order coupling), upon electronic excitation, because along the respective configurational coordinate, there will be a force acting on the nuclei. Again following the BO approximation, the excitation of vibrational modes will modulate the transition energy of the respective electronic transition. When exciting an ensemble of CNT with an optical pulse whose duration is shorter than the considered vibrational period, then the whole ensemble will oscillate in phase. Using femtosecond spectroscopy, the resulting modulation of the transition energy can then be traced in real time, which is a direct measure of electron–phonon coupling.

33

Two different mechanisms of excitation of coherent oscillations can be distinguished [47]; however, both embodied in the Raman tensor [48]. Impulsive absorption leads to the creation of a displaced (with respect to equilibrium) wavepacket in the excited state, with zero initial momentum. The displacive excitation leads to periodic modulations of excitedstate transitions such as, stimulated emission and photo-induced absorption. In contrast, the resonant impulsive stimulated Raman (RISR) mechanism places a displaced wavepacket with nonzero momentum in the ground state potential. Sometimes this is called a transient hole [49]. The quasi-classical dynamics of this wavepacket modulates the groundstate transition, essentially adding a transient hotstate absorption. The Raman-like interaction responsible for ground-state coherence is well understood also in the frequency domain, considering that a broad band pulse contains both excitation and Stokes or anti-Stokes spectral component, supporting the stimulated process. When in resonance, this is however mediated by the propagation of an intermediate wavepacket in the excited state. This is a crucial difference with respect to standard Raman, that is instantaneous (practically limited by the electronic dephasing time), and leads to the displacement and momentum buildup (kick) for the wavepacket in the ground state. Due to the shorttime excited-state evolution, the wavepacket is no longer an eigenfunction of the ground state potential: therefore, it must oscillate. More details can be found in the literature [50]. The peculiar mechanism we briefly described leads to a characteristic dependence of the modulation depth on the pump–pulse duration and also a nontrivial distribution in probe wavelength. For instance, upon shortening of the pump pulse with respect to the vibrational period, the excited-state coherence remains strong, while the ground-state coherence fades off (delta-like limit). The transient spectrum of the nonstationary state comprises transitions from all the wavepacket eigenmode components to all coupled arrival status. This is cumbersome to be worked out in the frequency domain, but it can fairly simply be worked out in the Wigner space, as shown by Kumar et al. [51,52]. We adopted the latter approach for our quantum mechanical calculation of the coherent phonon profile. More details on the connection between Raman excitation profile and coherent phonon spectra can be found in literature [53]. In molecules the distinction between ground- and excited-state coherence

34 Carbon Nanotubes: Electronic Structure and Spectroscopy

has a clear meaning. In crystals, such as CNT, this is lost. Lattice phonons coexist with many electronic excitations in the tube, and they are weakly perturbed up to very high excitation density. The lifetime of the coherent oscillations is limited by two effects: first, coherence must die according to the loss of population, contained in the constant T1. A typical loss mechanism is anharmonic coupling to lower energy phonon modes, leading eventually to the equal partition rule of all degrees of freedom, defined as thermodynamical temperature. Pure population decay can be measured by detecting the ratio of Stokes versus anti-Stokes line strengths as a function of time using femtosecond pulses. In the (6,5) tube, where the dominant G-type oscillation is the Gþ mode, a value of Tl ¼ (1.1  0.2)ps has been found [39]. In addition to population decay, coherent phonons can also be destroyed by a dephasing process, contained in the lifetime T2. Such dephasing is induced by inhomogeneity, leading to a distribution of oscillation frequencies around a mean value. In addition, collision events can occur, changing phase but not population number. The width of the distribution function determines the dephasing time; it contains contributions from both dynamic and static disorder. Based on coherent phonon lifetime measurements [54], we conclude that G-mode dephasing is not limited by T1 but an efficient decoherence (transversal, nondiagonal) process is also present (pure dephasing). Note that for ground-state coherence the damping time observed in the time domain is linked to the linewidth observed in the Raman peak by v ¼ 1=cT2 . Coherent phonon (CP) dynamics of the RBM in HiPCO samples were first found by Manzoni et al. [26]. Lim et al. were able to separate the contributions from various chiralities by using a tunable narrowband pump pulse [55]. These two experiments adopted different detection schemes: narrow band the former (signal is extracted at single spectral component of the pulse band), open band the latter (signal is spectrally integrated onto the pulse band). They found that the CP signal corresponds to the chirality-dependent resonance Raman scattering signal, but with improved selectivity due to the absence of backgrounds arising from fluorescence or Rayleigh scattering. Using pulse trains, it is possible to isolate the coherent response of CNT chiralities according to their RBM frequency, destroying the coherence of any other chirality [56].

CP dynamics of the G mode were first described by Gambetta et al. [54]. In functionalized isolated HiPCO CNT, they showed that the G mode is anharmonically coupled to the RBM mode. This was demonstrated by a real-time observation of a G-mode frequency modulation with the RBM frequency. The frequency modulation was demonstrated by a sliding window Fourier transform (FT) spectrum (see Figure 8). In principle, frequency modulation can also be observed in the frequency domain by the presence of satellite peaks around the main G-mode peaks offset by  ERBM; however, such satellite peaks can also be explained by amplitude modulation or apodization artifacts; the confirmation of the presence of a frequency modulation can only be done in the time domain. The anharmonic coupling between a radial and a longitudinal mode, being geometrically orthogonal, is not a priori to be expected. However, a simple continuum model suggests it may happen, and more advanced molecular dynamics calculations showed that the coupling is induced by a corrugation of the nanotube in the excitonic state, breaking up the sharp distinction between ‘radial’ and ‘longitudinal’. In contrast to the RBM, the G-mode frequency is only weakly chirality dependent. Therefore, amplitude and phase profiles for single chiralities in a polydisperse sample cannot be selected by appropriate excitation energies as in the case of RBM. Moreover, the observation of G-mode coherent oscillations of 25 fs period requires broadband, intrinsically nonselective excitation pulses. Using chirality-enriched samples and a single shot detection technique [57], such studies became possible [58]. Nanotubes from the CoMoCat production method were further enriched by density gradient ultracentrifugation and were isolated in a Xerogel matrix. The samples consisted of about 70% of a single semiconducting chirality, the (6,5) CNT. Upon pumping with 7 fs broadband visible pulses, strong coherent oscillations were observed for both the RBM and G modes, (see Figure 9). FT maps in the chirality-sensitive RBM region showed 307 cm1 for the whole visible wavelength region, showing that RBM activity is induced only by (6,5) tubes, (see Figure 9(e)). It was also concluded that the G-mode activity is only due to the (6,5) tube. Thus, meaningful phase and amplitude profiles for the (6,5) tube could be obtained (Figure 10). For both the RBM and G modes, they show amplitude minima and  phase flips at the maximum of the E22 excitonic resonance of the (6,5) tube. The RBM amplitude

Carbon Nanotubes: Electronic Structure and Spectroscopy

(a)

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Figure 8 Anharmonic coupling of G mode and RBM, leading to a frequency modulation of the G mode. This is shown here by a sliding window Fourier transform (FT) map (a). In panel (b), an FT of the center frequency (red line in panel (a)) is given. Redrawn from Gambetta A, Manzoni C, Menna E, et al. (2006). Real-time observation of nonlinear coherent phonon dynamics in single-walled carbon nonotubes. Nature Physics 2(8), 515–520.

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Wave number (cm−1) Figure 9 Coherent oscillations in (6.5) tubes after resonant excitation of the Ex-2 state with 7 fs broadband pulses. (a) Time-resolved pump–probe spectrum. (b) Isolated contribution of coherent oscillationed after substraction of population contribution. (c) Fourier transform map. Modulation amplitude is given as a function of probe wavelength. (d) Calculated Fourier transform (FT) map, according of the model described in the text. (e) Zoom-out of the RBM region in panel (c). Redrawn from Luer L, Gadermaier C, Crochet J, et al. (2009). Coherent phonon dynamcs in semiconducting carbon nanotubes: A quantitative study of electron-phonon coupling. Physical Review Letters 102, 127401.

36 Carbon Nanotubes: Electronic Structure and Spectroscopy

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Figure 10 Fourier transform (FT) amplitude and phase maps as a function of probe wavelength: experimental RBM and G-mode profiles in panel (a) and (c), respectively, and calculated RBM and G mode profiles in panel (b) and (d), respectively. From Luer L, Gadermaier C, Crochet J, et al. (2009). Coherent phonon dynamics in semiconducting carbon nanotubes: A quantitative study of electron-phonon coupling. Physical Review Letters 102, 127401.

profile roughly follows the first derivative of the absorption spectrum of the E22 resonance. It is interesting to compare the results of Lim et al. [55] to those of Luer et al. [58]. The former group finds that the amplitude profile is narrower than the first derivative of the absorption spectrum, while in the latter case, they are slightly larger. This is explained by the different experimental sctups that have been used. Luer et al. used broadband pump and probe pulses; in this case, the amplitude profile is given as a function of probe wavelength, while the excitation density (population) is fairly constant. Lim et al., on the other hand, applied degenerate narrowband pump and probe pulses; in this case, the amplitude profile is given as a function of probe and pump wavelength, so it is expected to be proportional to the first derivative of the absorption spectrum times the population, which is proportional to the absorption. This explains the narrower amplitude profile. The G-mode profile deviates significantly from the first derivative of the absorption spectrum, showing secondary maxima and unsymmetric low- and high-energy lobes. To understand this behavior, a time-domain quantum mechanical modeling was performed, adopting the approach of Kumar et al.

[51, 52]. A two-level system was assumed, where the energy of the ground- and excited-state potential energy surfaces (PESs) was given as function of the RBM and G-mode normal coordinates. With respect to the ground-state PES, the excited state one was displaced by two unitless displacements RBM and G along the RBM and G-mode normal coordinates, respectively. No distortion was assumed (linear electron–phonon coupling regime, (see Figure 11(b)). Good agreement between the measured and calculated amplitude and phase profiles was obtained by assuming a displacement of  ¼ 0.9  0.2 (Figure 11). This is slightly higher than predicted by theory.

1.02.8 Conclusions In this chapter we provide an introduction to the study of the electronic structure in CNT, mainly based on spectroscopical experiments. The picture that we describe is rather incomplete because many questions are still open. One crucial issue is material quality. The fast improvement in CNT preparation, handling and separation, is giving a better chance for

Carbon Nanotubes: Electronic Structure and Spectroscopy

37

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Figure 11 (a) Simulation of G-mode amplitude profile for different electron–phonon coupling strengths. (b) Two-level model used for the quantum mechanical simulations. From Luer, L., Gadermaier, C., Crochet, J., et al. (2009). Coherent phonon dynamics in semiconducting carbon nanotubes: A quantitative study of electron-phonon coupling. Physical Review Letters 102, 127401.

understanding fundamental properties and will certainly result in important findings in the near future. The most debatable issues are: inter-tube energy and charge transfer, the branching of the initial photoexcitation into charged and neutral states, and the role of defects. Better chiral selectivity and metal– semiconductor separation will provide materials with high technological impact, suitable for integration into device structures that, in the near future, will be based on CNT networks and blends rather than single tubes. The decade ahead will be the one decisive for the future of CNT, distinguishing between real opportunities in nanotechnology or curiosities for graduate student classes in solid-state physics.

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