ISSN 0030400X, Optics and Spectroscopy, 2010, Vol. 108, No. 3, pp. 376–384. © Pleiades Publishing, Ltd., 2010. QUANTUM OPTICS OF META, MICRO, AND NANOSYSTEMS Surface Exciton–Plasmons and Optical Response of SmallDiameter Carbon Nanotubes1 I. V. Bondareva, K. Taturb, and L. M. Woodsb a Physics Department, North Carolina Central University, Durham, NC 27707 USA b Physics Department, University of South Florida, Tampa, FL 33620 USA email: [email protected] Received August 3, 2009 Abstract—We study theoretically the interactions of excitonic states with surface electromagnetic modes of smalldiameter (Շ1 nm) semiconducting singlewalled carbon nanotubes. We show that these interactions can result in strong excitonsurfaceplasmon coupling. The exciton absorption lineshape exhibits the line (Rabi) splitting ~0.1–0.3 eV as the exciton energy is tuned to the nearest interband surface plasmon reso nance of the nanotube so that the mixed strongly coupled surface plasmonexciton excitations are formed. We discuss possible ways to bring the exciton in resonance with the surface plasmon. The excitonplasmon Rabi splitting effect we predict here for an individual carbon nanotube is close in its magnitude to that previ ously reported for hybrid plasmonic nanostructures artificially fabricated of organic semiconductors depos ited on metallic films. We expect this effect to open up paths to new tunable optoelectronic device applica tions of semiconducting carbon nanotubes. DOI: 10.1134/S0030400X10030100 1 INTRODUCTION interactions can result in a strong excitonsurface Singlewalled carbon nanotubes (CNs) are quasi plasmon coupling due to the presence of lowenergy onedimensional (1D) cylindrical wires consisting of (~0.5–2 eV) weaklydispersive interband plasmon graphene sheets rolledup into cylinders with diame modes [22, 23] and large exciton excitation energies ~1 eV [24, 25] in smalldiameter CNs. Previous stud ters ~1–10 nm and lengths ~1–104 μm [1–3]. CNs are shown to be useful for miniaturized electronic, elec ies have been focused on artificially fabricated hybrid tromechanical, chemical and scanning probe devices plasmonic nanostructures, such as dye molecules in and as materials for macroscopic composites [4]. The organic polymers deposited on metallic films [26], area of their potential applications has been recently semiconductor quantum dots coupled to metallic expanded towards nanophotonics and optoelectronics nanoparticles [27], or nanowires [28], where one [5, 6] after the experimental demonstration of control material carries the exciton and another one carries lable singlesatom incapsulation into singlewalled the plasmon. Our results are particularly interesting CNs [7, 8]. since they reveal the fundamental EM phenome non—the strong excitonplasmon coupling—in an For pristine (undoped) singlewalled CNs, the individual quasionedimensional (1D) nanostruc numerical calculations predicting large exciton bind ture, a carbon nanotube. ing energies (~0.3–0.6 eV) in semiconducting CNs [9–11] and even in some smalldiameter (~0.5 nm) metallic CNs [12], followed by the results of various measurements of the excitonic photoluminescence THE MODEL [13–16], have become available. These works, together with other reports investigating the role of We consider the vacuumtype EM interaction of an effects such as intrinsic defects [15], excitonphonon exciton with surface EM fluctuations of a single interactions [16–19], external magnetic and electric walled semiconducting CN. No external EM field is fields [20, 21], reveal the variety and complexity of the assumed to be applied. Since the problem has the intrinsic optical properties of carbon nanotubes. cylindrical symmetry, the orthonormal cylindrical basis {er, eϕ, ez} is used with the vector ez directed along Here we develop a theory for the interactions the nanotube axis. The total Hamiltonian of the cou between excitonic states and surface electromagnetic pled excitonphoton system is of the form (we use (EM) modes in smaldiameter (Շ1 nm) semiconduct Gaussian units) ing singlewalled CNs. We demonstrate that such 1 ˆ ˆ ˆ ˆ The article is published in the original. H = HF ++Hex Hint, (1) 376 SURFACE EXCITON–PLASMONS AND OPTICAL RESPONSE 377 Ee Eex ε E e Eg b ε h 0 kz Eexc 0 kz Eh Fig. 1. (Color online) Schematic of the transversely quantized azimuthal electronhole subbands (left), and the firstinterband groundinternalstate exciton energy (right) in a smalldiameter semiconducting carbon nanotube. See text for notations. where the three terms represent the free field, the free where the operators B † and B create and annihi exciton, and their interaction, respectively. More n, f n, f explicitly [29–31], the second quantized field Hamil late, respectively, an exciton with energy Ef(n) at the tonian is lattice site n of the CN surface. The index f (≠ 0) refers ∞ to the internal degrees of freedom of the exciton. ˆ ωបωˆ†(), ω ˆ(), ω Alternatively, HF = ∑∫d f n f n , (2) n 0 † † ikn ˆ† B , = B , e / N where the scalar bosonic field operators f (n, ω) and k f ∑ n f n ˆf (n, ω) create and annihilate, respectively, the surface EM excitation of frequency ω at an arbitrary point n = creates (N is the number of the lattice sites) and Bk, f = R = {R , ϕ , z } associated with a carbon atom (rep n CN n n ()† † resenting a lattice site) on the surface of the CN of Bk, f annihilates the finternalstate exciton with radius RCN. The summation is made over all the car the quasimomentum k = {kϕ, kz}, where kϕ is quan bon atoms. In the following it is replaced by the inte tized due to the transverse confinement effect and kz is gration over the entire nanotube surface according to continuous. The exciton total energy is of the form the rule 2π ∞ ()f 2 2 E ()k = E ()kϕ + ប k /2M (4) … () … ()ϕ … f exc z ex ∑ ==1/S0 ∫dRn 1/S0 ∫ d nRCN ∫ dzn , n 0 –∞ with the first term representing the excitation energy, ()f () () ()f 2 Eexc kϕ = Eg kϕ + Eb , where S0 = ()33/4 b is the area of an elementary equilateral triangle selected around each carbon atom in a way to cover the entire surface of the nanotube of the finternalstate exciton with the (negative) with = 1.42 Å being the carboncarbon interatomic ()f b binding energy Eb , created via the interband transi distance. tion with the band gap E (kϕ) = ε (kϕ) + ε (kϕ), where The second quantized Hamiltonian of the free g e h ε are transversely quantized azimuthal electron exciton (see, e.g., [32]) is of the following form e, h hole subbands (see the schematic in Fig. 1). The sec ˆ () † Hex = ∑ Ef n Bnm+ , f Bm, f ond term represents the kinetic energy of the transla ,, tional longitudinal movement of the exciton with the nmf (3) effective mass Mex = me + mh, where me and mh are the () † = ∑Ef k Bk, f Bk, f, electron and hole effective masses, respectively. The k, f two equivalent freeexciton Hamiltonian representa OPTICS AND SPECTROSCOPY Vol. 108 No. 3 2010 378 BONDAREV et al. tions are related to one another via the obvious orthog and onality relationships ∞ ∇ ϕˆ () ω ˆ ||(), ω – n n = ∫d E n + h.c. –i()kk– ' n δ –i()nm– k δ ∑e /N = kk', ∑e /N = nm 0 n k ∞ (7) 4iω || = dω πបωReσ ()R , ω G ()nm,,ω with the summation running over the first Brillouin ∑∫ 2 zz CN zz k c zone of the nanotube. The bosonic field operators in m 0 ˆ ˆ HF are transformed to the krepresentation in the × f()m, ω + h.c. same way. with the total electric field operator given by The most general (nonrelativistic, electric dipole) ∞ excitonphoton interaction on the nanotube surface ˆ () ω[]ˆ ⊥(), ω ˆ ||(), ω can be written in the form (see [30, 31]) E n = ∫d E n + E n + h.c., 0 ˆ ˆ ()1 ˆ ()2 Hint = Hint + Hint ⊥(||) ω Gzz(n, m, ) being the zzcomponent of the trans (5) verse (longitudinal) Green tensor (with respect to the e ˆ ()⋅ ˆ e ˆ () ˆ ⋅ ∇ ϕˆ () σ ω = –∑A n pn – A n + ∑dn n n , first variable) of the EM subsystem, and zz(RCN, ) mec 2c n n representing the CN dynamic surface axial conductiv ity per unit length. where Equations (1)–(7) form the complete set of equa tions describing the excitonphoton coupled system on the CN surface in terms of the EM field Green ten ˆ 〈|ˆ |〉 pn = ∑ 0 pn f Bn, f + h.c. sor and the CN surface axial conductivity. The con f ductivity is found beforehand from the realistic band structure of a particular CN. The Green tensor is is the total electron momentum operator at the lattice derived by expanding the solution of the Green equa site n under the optical dipole transition resulting in tion in cylindrical coordinates and determining the the exciton formation at the same site, Wronskian normalization constant from the appropri ately chosen boundary conditions on the CN surface ˆ ˆ (see [29–31, 33]). dn = 〈|0 dn|〉f B , + h.c. ∑ n f It is important to realize that the transversely polar f ized surface EM mode contribution to the interaction is the corresponding transition dipole moment opera Hamiltonian from Eq. (5) (first term) is negligible compared to the longitudinally polarized surface EM ˆ 〈|ˆ |〉 tor (related to pn via the equation 0 pn f = mode contribution (second term). The point is that, ˆ because of the nanotube quasionedimensionality, imeEf(n)/〈|0 dn|〉f បe), c and e are the speed of light and the electron charge, respectively. The vector potential the exciton quasimomentum vector and all the rele ˆ vant vectorial matrix elements of the momentum and operator A()n (the Coulomb gauge is assumed) and dipole moment operators are directed predominantly the scalar potential operator ϕˆ (n) represent, respec along the CN axis (the longitudinal exciton). This pre tively, the nanotube’s transversely polarized surface vents the exciton from the electric dipole coupling to EM modes and longitudinally polarized surface EM transversely polarized surface EM modes as they prop modes which the exciton interacts with.