optical excitations of quasi-one-dimensional systems: carbon nanotubes versus polymers and...

phys. stat. sol. (a) 203, No. 14, 3602–3610 (2006) / DOI 10.1002/pssa.200622407

© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

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Optical excitations of quasi-one-dimensional systems:

carbon nanotubes versus polymers and semiconductor wires

D. Prezzi*, 1, 2

and E. Molinari1, 2

1 National Research Center on nanoStructures and bioSystems at Surfaces (S3), INFM-CNR,

via G. Campi 213/a, 41100 Modena, Italy 2 Dipartimento di Fisica, Università di Modena e Reggio Emilia, via G. Campi 213/a, 41100 Modena,

Italy

Received 19 May 2006, revised 9 August 2006, accepted 17 August 2006

Published online 10 November 2006

PACS 71.15.Qe, 78.40.Ri, 78.67.Ch

We review the main characteristics of optical excitations of semiconductor nanotubes, as obtained from

accurate ab-initio theories and model calculations as well as experimental evidence, and discuss them in

light of the previous understanding of other quasi-one-dimensional semiconducting systems. We point out

striking similarities of nanotubes with III–V quantum wires and conjugated polymers, especially (i) the

clear excitonic nature of absorption, very far from the single-particle behaviour; (ii) its manifestations in

optical spectra, where excitonic peaks are accompanied by a strong intensity reduction at the onset of the

free-particle continuum; (iii) the strategies that allow experimental access to exciton binding energies. The

recent theoretical and experimental evidence obtained on semiconducting single-walled nanotubes con-

verges quantitatively to a picture of strongly bound excitons (about 0.3–1.0 eV for nanotubes with

0.4–1.0 nm diameter). We discuss its implications and list a few open issues of relevance to fundamental

understanding and optoelectronic applications.

© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

Single-walled carbon nanotubes (SWCNTs) are quasi-one-dimensional (quasi-1D) graphitic nano-

cylinders, which can be either semiconducting or metallic depending on diameter and chirality (see

Fig. 1). Since their discovery in early nineties [1, 2], carbon nanotubes have attracted an unprecedented

interest for their remarkable physical features: from structural and elastic properties to electronic and

thermal transport characteristics [3, 4]. Despite the intense research effort devoted to these nanostruc-

tures, their optical properties received a comparatively small attention, since no measurable light emis-

sion could be observed until few years ago. This scenario radically changed with the successful fabrica-

tion of isolated SWCNTs, which were found to fluoresce brightly in the 0.8–1.6 µm wavelength range

even at room temperature [5, 6].

These advances have established optical spectroscopies as powerful noninvasive tools for nanotube

characterization [7–9], and hold promise for novel nanoscale opto-electronic and photonic applica-

tions [10–13]. As a consequence, a renewed interest has grown about the nature of elementary optical

excitations in carbon nanotubes, and their reduced dimensionality is thought to play a central role in

determining both the electronic bandstructure and the optical spectra.

* Corresponding author: e-mail: [email protected]

phys. stat. sol. (a) 203, No. 14 (2006) 3603

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Fig. 1 Geometry and notations for single-walled carbon nanotubes. a) Following the standard notation [3], a nano-

tube can be identified by two indices, (n, m). They univocally define the direction Ch, along which the graphene

sheet is rolled-up to build the nanotube, in terms of the to unit vectors 1a and

2a of the hexagonal lattice:

h 1 2n m= +C a a . The chiral angle, α (angle between

1a and C

h), and the translational vector, T, are also determined by

n and m (see [3] for further details). The rectangle defined by Ch and T represents the unit cell of the tube. A nano-

tube is metallic if (n–m) is divisible by 3; semiconductor otherwise. b) Examples of chiral and achiral nanotubes.

Carbon nanotubes are classified according to three families: zig-zag (n, 0), armchair (n, n), chiral (n, m). Zig-zag and

armchair tubes are also called achiral.

Several theoretical studies, based on different approaches, have predicted the excitonic nature of opti-

cal excitations of semiconducting nanotubes [14], as well as large excitonic binding energies [15–21].

At the same time, most experimental data have been interpreted for a long time by assuming one-electron

interband transitions and graphene-like single-particle bands (e.g. [5–7]). The main reason for such a

controversial interpretation of a fundamental issue was probably the initial absence of direct experiments

which could settle the problem. In fact the standard techniques used for ordered systems of higher di-

mensionality are not accessible for usual nanotube samples. An experimental determination of exciton

binding energies has been obtained only very recently via different experimental approaches, ruling out

definitively the single-particle picture [25–29].

In the present work we will review the main results for optical excitations in single-walled carbon

nanotubes, and compare them with the optical properties of other real quasi-one-dimensional systems,

like semiconductor wires and polymer chains. We will underline the fundamental similarities and differ-

ences among them and with the ideal one-dimensional hydrogenic model [30, 31].

2 Key features in linear absorption of quasi-1D systems:

dimensionality and excitonic effects

Coulomb interactions are well-known to be strongly dependent on the dimensionality of the system, and

the low-dimensional features should be reflected in optical responses. In fact, the optically excited carri-

ers will experience the reduced spatial confinement, and form strongly bound electron–hole pairs, exci-

tons. Hence, optical spectroscopies are in principle good means to probe the low-dimensional character

of such systems.

The excitonic nature of excitations is known to give rise to peculiar features in both density of states

and optical spectra for model systems [30, 32]. For both three- and two-dimensional systems, this leads

to series of discrete excitonic states below the onset of the single-particle continuum in the density of

states. In one-dimensional systems the effects are even more dramatic: the presence of individual exciton

peaks below the band edge is accompanied with a marked suppression of one-particle singularities.

Whether or not the properties found for ideal low-dimensional objects can be applied to the real ones

has been a fundamental issue in the study of several systems, such as semiconductor wires, conjugated

polymer chains and, in last, semiconducting single-walled carbon nanotubes. In all these cases, the inclu-

sion of the electron–hole interaction in the theoretical treatment has resulted in major corrections. More-

a) b)

3604 D. Prezzi and E. Molinari: Optical excitations of quasi-one-dimensional systems

© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.pss-a.com

Fig. 2 Calculated optical absorption spectra of different quasi-1D systems. a) A semiconductor quantum wire.

After Rossi and Molinari [33]. b) A single-chain conjugated polymer, poly-para-phenylene-vinylene (PPV) chain.

After Ruini et al. [34–36]. c) A semiconducting single-walled carbon nanotube (6, 4). After Chang et al. [16, 37],

Maultzsch et al. [26]. Note the different scale of the three panels.

over, similar features in the optical spectra can be recognized, confirming the quasi-one-dimensional

character of all these structures, as will be discussed below.

Panels (a) and (b) of Fig. 2 show the calculated absorption spectra of a semiconductor quantum wire, a

GaAs V-shaped wire [33, 38], and of a single-chain conjugated polymer (poly-para-phenylene-vinylene,

PPV) [34–36], respectively. The single-particle spectra (dashed lines) exhibit the typical behaviour of a

one-dimensional joint density of states (JDOS). In fact the absorption spectrum is proportional to the

JDOS, which diverges as 1 2( )g n

Eω�- /

,- at the n-th subband edge g n c n v n

E E E, , ,

= - .

When electron–hole excitonic interactions are included (solid lines), the spectra drastically change.

First of all, we can see the formation of individual bound exciton states below the single-particle contin-

uum, with a symmetric shape that differs from the 1 E/ behaviour in the non-interacting spectra.

Note that in all spectra of Fig. 2 an artificial broadening was introduced for computational conven-

ience. The broadening accounts for the deviations of the single-particle spectra from the singular 1 E/

dependence.

The exciton binding energies, defined as the differences between the excitonic peaks and the one-

particle continuum onset, can be considered as a measure of the electron–hole interaction. While in

polymers it has values of the order of one eV, in semiconductor wires it is about two orders of magnitude

lower. This is related to the fact that in the studied wires (about 10 nm diameter) the lateral dimension is

much larger than in polymers (1 nm or less), and the overlap between electron and hole wavefunctions

smaller. Moreover, the confining medium in quantum wires is a different semiconductor material rather

than vacuum, with screening effects that lower the binding energy further. Recent calculations on germa-

nium nanowires show binding energies comparable with those of polymers, when the lateral confinement

in vacuum is of the same order of magnitude [39]. This trend is consistent with the diverging binding

energy expected for an ideal 1D hydrogenic atom, i.e. an interacting electron–hole system in a wire with

infinitesimal diameter and perfect confinement [30].

The second striking observation is the strong reduction of the absorption spectra at the band edge,

where the noninteracting spectra instead reveal a divergent behaviour. It is worth noting that this sup-

pression of the band-edge singularity is not the result of a lowered density of states in this energy region,

which is indeed the same as in the noninteracting case above the gap, but is related to vanishingly small

oscillator strengths. Since the oscillator strengths are proportional to the probability of finding the elec-

tron and the hole at the same position, this seems to indicate some sort of effective repulsion between

electron and hole which may appear quite surprising. As shown in Ref. [38], this is an effect of the or-

thogonality constraint on the set of excitonic eigenstates: when combined with the reduced degrees of

freedom typical of strong 1D confinement, such constraint for the lowest unbound states dominates over

the electron–hole Coulomb attraction. This picture also explains why there is no suppression of the sin-

gle-particle continuum in the case of two- and three-dimensional systems [32, 33].

a) b) c)

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3 The case of carbon nanotubes: theoretical predictions

In the case of C nanotubes a 1D-like behaviour is far from obvious. Indeed, SWCNT can be described

starting from a rolled sheet of graphene, and a two-dimensional character could be expected at least for

large diameters. Intuitively, one would expect excitonic spectra of 1D nature as long as the nanotube is

thin enough to ensure electron–hole binding across its diameter. If this is the case, the excitonic enve-

lope function would have cylindrical symmetry resulting in an overall one-dimensional behaviour.

In practice, however, a quantitative guess of the range of interaction for a given nanotube cannot be

obtained without detailed calculations. Indeed, both nanotube diameter and chirality strongly affect the

single-particle band structure, and hence screening. This is an independent effect that combines with the

average electron–hole distance across the tube in determining the nature of excitonic interactions. Given

this complexity, most of the early discussions of the experimental optical spectra of nanotubes adopted

the simplifying assumption that Coulomb interactions could be neglected for the typical diameters under

investigation.

The pioneering work of T. Ando [14], based on a two-dimensional Dirac equation with a free param-

eter characterizing the strength of the electron–hole interaction, was the first to emphasize theoretically

that excitonic effects may be important. Several further theoretical studies based on semiempirical ap-

proaches have been performed to estimate the role that excitons play in the optical spectra of carbon

nanotubes. These include an orthogonal and nonorthogonal tight-binding model [18], a scheme based on

an accurate fitting of Pariser–Parr–Pople electron Hamiltonian [19], a model valid for large tubes where

the expression for the Coulomb interaction is taken from 2D graphene [20], a semiempirical excitonic

calculation based on a tight-binding modeling of quasiparticle energies and an Ohno potential for the elec-

tron–hole interaction [21], and a variational approach [15] where a Gaussian trial wave function was

used to estimate the excitonic binding energies. While all these approaches indicated significant excitonic

effects, their magnitude and influence on the spectra turned out to be very sensitive to the particular model.

The first ab-initio study [40] employed time-dependent density functional theory (TDDFT). In this

work, the drastic suppression of the optical absorption by local field effects generated by the exchange

interaction between electron–hole was described. However, the direct electron–hole Coulomb interac-

tion could not be treated.

Many-body perturbation theory then proved to be a key tool in the study of the optical properties and

excitons, as had previously occurred for other one-dimensional systems such as polymers [35, 41]. In

Ref. [16, 37], we used this approach for carbon nanotubes in combination with a localized basis set that

allows to take full advantage of the screw symmetry of these structures. Quasiparticle effects were

treated within the GW scheme, followed by the solution of the Bethe–Salpeter equation (BSE), which

yields the excitonic energies of a system by isolating the poles of a two-particle Green’s function. The

application of many-body perturbation theory to carbon nanotubes [16, 17] provided conclusive proof

that 1D-like excitonic effects are crucial in the optical spectra of these systems, and offered a quantitative

prediction of the size of exciton binding energies.

Figure 2(c) shows the linear absorption of the semiconducting SWCNT (6, 4). As observed for wires

and polymers, the 1 2E

- / singularities are strongly suppressed. In addition, bound excitons arise below the

onset of the single-particle continuum, labelled in figure as 1u and 2u. The binding energies, of respec-

tively 0.52 and 0.16 eV for a 0.7-nm-diameter tube, show a strong electron–hole interaction.

Figure 3(a) shows the real-space electron–hole wavefunctions of the four lowest excitonic states for

the (6, 4) SWCNT, 1g, 1u, 2u and 2g. They represent the probability of finding an electron when the

hole position is fixed at the centre of each panel. We notice a complete delocalization along the circum-

ference: as mentioned above, a signature of the quasi-1D nature of the interaction in this tube. The exci-

ton wavefunctions are found to be localized along the direction of the nanotube axis, with an average

extension of several nanometers. A maximum in the exciton wavefunction at zero electron–hole distance

along the nanotube will lead to a strong optical peak in linear absorption (1u peak in Fig. 2(c)), while a

node corresponds to a vanishing oscillator strength, and hence to almost complete optical inactivity (2u

peak in Fig. 2(c)). The same behaviour is seen in both wires and polymers [41].



Fig. 3 (online colour at: www.pss-a.com) a) Lowest-energy excitonic wavefunctions of (6, 4) SWCNT. The top two panels correspond to the 1g, 1u and 2g, 2u states, respectively, and show the probability of finding the electron on the tube surface when the hole is fixed at the center of the panel, that is 2

e h| ( )|r rψ , . The vertical direction corre-

sponds to the circumference (2.1 nm) and the horizontal direction to the tube axis (15.9 nm). The bottom four panels are blowups of the same states, when the hole position is in a highly symmetric point (center of the bond). They show the odd (u)/even (g) symmetry with respect to π-rotation about the U-axis. b) Rotational U-symmetry. Carbon nanotubes are symmetric under rotation of 180 degrees about the U-axis, which pierces the center of a bond and is perpendicular to the tube axis. This symmetry takes one of the two inequivalent atoms (A and B) of the nanotube into the other, and viceversa. This leads to a doubling of the excitonic states (nu, ng), which reflects the symmetries of the tube. Note that, in spite of the different implementations, the existing ab-initio descriptions have led to a

consistent picture of 1D excitonic binding for all the semiconducting nanotubes that have been studied

sofar, with binding energies in the range 0.5–1.0 eV for nanotubes with diameters between 0.4–0.9 nm.

On the theoretical side, a consensus is therefore emerging on the excitonic nature of optical excitations in

SWCNTs.

4 Approaches to the measurement of the exciton binding energy

On the experimental side, the excitonic nature of optical excitations has been debated since the discovery

of fluorescence from individual semiconducting nanotubes. The main features in optical experiments

have been initially explained within the indipendent-particle theory. On the other hand, some discrepan-

cies between the experimental findings and one-particle tight-binding predictions [7, 42, 43] have raised

questions about this interpretation, and indirect experimental evidences of the correlation effects began to

appear [44–46].

A quantitative determination of the importance of excitonic effects is of course the measurement of

the exciton binding energies. However, this measurement is non trivial for experimentally available

nanotubes: this is due to the suppression of the single-particle band edge singularity, that is expected for

one-dimensional systems and predicted theoretically for nanotubes. Indeed, for 2D or 3D systems exci-

ton binding energies can be extracted from a single one-photon optical experiment, because the contin-

uum onset and the excitonic peaks are both visible in absorption. On the contrary, the same is not possi-

ble in one-dimensional structures, where the continuum is suppressed. Other approaches making use of

combined transport and optical experiments on the same sample (see e.g. Ref. [47]) are still quite com-

plicated for typical carbon nanotube samples.

A conclusive experimental measure of excitonic effects was obtained very recently by comparing

linear and nonlinear optical properties [25, 26]. A combination of one- and two-photon spectroscopies

had been used before for the determination of the binding energies of other low-dimensional systems,

such as V-shaped semiconductor wires [48] and conjugated polymers (see e.g. [49, 50]). The comple-

mentarity of the selection rules for allowed optical transitions in one- and two-photon process allows the

direct measurement of the energy splitting between the main active exciton states in the two different

spectra (see Fig. 4(c)). In the case of semiconductor wires, this splitting was used to evaluate the exciton

binding energy, on the basis of the Rydberg series which governs the perfectly one-dimensional exci-

tonic states [48].

a) b)

phys. stat. sol. (a) 203, No. 14 (2006) 3607


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Fig. 4 (online colour at: www.pss-a.com) a) One-photon and b) two-photon absorption spectra for the (7, 5) tube [26]. Note that energies are relative to the first one-photon peak

1uE . c) Schematic picture of one- and two-

photon processes. 11

E indicates the single-particle transition between the lowest subbands. One-photon excitations couple to excitonic states with odd (u) symmetry with respect to π-rotations about the U-axis. Emission occurs from the lowest one-photon active u state. Two-photon absorption results in the excitation of exciton states with even (g) symmetry under the U-axis operation. d) Two-photon luminescence spectra of carbon nanotubes. The measured luminescence intensity is plotted as a function of excitation and detection wavelength [26].

In carbon nanotubes, ab-initio theoretical predictions of one- and two-photon experiments [26] al-

lowed to conclude that the energy difference between the lowest peak of each spectrum provides a lower

bound to the exciton binding energy Eb of a given tube, and turns out to be of the same order of mag-

nitude of Eb, i.e. several tenths of eV for small semiconducting nanotubes (see Fig. 5). The experiments of

a) c)

b) d)

Fig. 5 Ab-initio calculated one-photon (top) and two-photon (bottom) absorption spectra for the (6, 4) nanotube. Black (light blue) lines are with (without) electron–hole (e–h) inter-action. Bottom: probability amplitude of two-photon scattering for the lowest excitonic states 1g, 1u, 2g, 2u with e–h (black circles); probability amplitude of scattering to the final state without e–h (diamond). Higher-energy states with negligible amplitude are omitted for clarity.



Wang et al. (Ref. [25]) and Maultzsch et al. (Ref. [26]) consistently yielded values of Eb in the range

0.3–0.4 eV.

Other approaches for the experimental determination of exciton binding energies have been proposed

very recently for nanotubes. They are based on femtosecond transient absorption spectroscopy [27],

combined resonant Raman scattering and electrochemical doping [29], electroluminescence from

SWCNT field-effect transistors [28]. While less direct, these experiment yield results that are generally

consistent with those of two-photon vs one-photon absorption [25, 26], and confirm the picture of strong

excitonic binding in small semiconducting nanotubes.

5 Perspectives

Starting from the above understanding, a number of relevant open issues concerning optical excitations

of semiconducting nanotubes are now in front of the scientific community. Here is a subjective and par-

tial list of basic questions:

(i) Dark states and radiative lifetimes of excitations. What are the main intrinsic mechanisms that are

predicted to control radiative emission? Of interest here is the ongoing discussion on the general pres-

ence and origin of dark states [19, 51, 52] and on the possibility of artificially modifying these states and

their influence on radiative emission [52–54].

(ii) Recombination and relaxation processes. What are the pathways leading to excitonic relaxation of

nanotubes, as observed in their ultrafast optical response? (see e.g. [55, 56]) A deeper understanding of

higher-energy excitonic states, and their interaction with continuum states and vibrations [57] will also

be relevant to address this issue.

(iii) Electronic interactions beyond excitonic effects. Biexcitons may be stable and observable in

nanotubes [58]. We expect that interactions beyond excitons could be especially relevant in the case of

further confinement (nanotube quantum dots) or for very high photoexcitation intensities. Quantitatively,

how will this affect spectrocopies, as well as emission or gain in high density regimes?

Some of these questions have been explored in the past for other quasi-one-dimensional systems, and

are now becoming accessible for nanotubes thanks to the recent advancements in fundamental under-

standing. We believe that they will be not only stimulating for scientific curiosity, but also relevant in

view of nano-optoelectronics (and-possibly-lasing) in these systems.

Acknowledgements This paper is dedicated to Gerhard Abstreiter, a pioneer in the field of excitations and interac-tions in low dimensional semiconductors, on the occasion of his sixtieth birthday. It has been an honour and a pleas-ure to collaborate or just discuss with him along the years. We are grateful to all the collegues and friends who contributed to different aspects of the theoretical work in Modena on optical excitations of quasi one-dimensional systems, from III–V wires to polymers and nanotubes: among them Fausto Rossi, Guido Goldoni, Alice Ruini, Marilia J. Caldas, Giovanni Bussi, and especially Eric Chang, for his fundamental contribution to our studies on nanotubes. We are also grateful to C. Lienau, J. Kono, S. Mazumdar and G. Lanzani for interesting discussions on nanotube spectroscopies. This paper was supported in part by the EU (“Exciting” Network), and by the Italian Ministry of Education, University and Research (MIUR) under Grant FIRB “Nomade”.

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