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. We express agate predominantly along the CN axis with their them in terms of our earlier developed EM field quan electric vectors orthogonal to the propagation direc tization formalism in the presence of quasi1D tion. The longitudinally polarized surface EM modes absorbing bodies to obtain [30, 31] are generated by the electronic Coulomb potential (see, e.g., [34]), and therefore represent the CN sur ∞ face plasmon excitations. These have their electric c ⊥ Aˆ ()n = d ωEˆ ()n, ω + h.c. vectors directed along the propagation direction. They ∫ iω do couple to the longitudinal excitons on the CN sur 0 face. Such modes were observed in [22]. They occur in ∞ (6) CNs both at high energies (wellknown πplasmon at ω4 πបω σ (), ω ⊥ (),,ω = ∑∫d Re zz RCN Gzz nm ~6 eV) and at comparatively low energies of ~0.5– c 2 eV. The latter ones are related to transversely quan m 0 ˆ tized interband (intervan Hove) electronic transi × f()m, ω + h.c. tions. These weaklydispersive modes [22, 23] are sim

OPTICS AND SPECTROSCOPY Vol. 108 No. 3 2010 SURFACE EXCITON–PLASMONS AND OPTICAL RESPONSE 379

(а) (11,0) 10 (b) (10,0)

10

0 0

− −

Dimensionless conductivity 10 10 0 0.2 0.4 Dimensionless conductivity 0 0.2 0.4 Dimensionless energy Dimensionless energy

Fig. 2. (Color online) (a, b) Calculated (see text) dimensionless axial surface conductivities for the (11,0) and (10,0) nanotubes: Re[σzz] (dotted line), Re[1/σzz] (solid line), Im[σzz] (dashed line). Dimensionless energy is defined as [Energy]/2γ0, according to Eq. (10). ilar to the intersubband plasmons in quantum wells Here, [35]. They occur in the same energy range of ~1 eV x ==បω/2γ , xμ បωμ()k /2γ , where the exciton excitation energies are located in 0 0 (10) smalldiameter (Շ1 nm) semiconducting CNs [24, ε () γ f = Ef k /2 0 25]. In what follows we focus our consideration on the γ exciton interactions with these particular surface plas with 0 = 2.7 eV being the carbon nearest neighbor mon modes. overlap integral entering the CN surface axial conduc tivity σ . The function To obtain the dispersion relation of the coupled zz excitonplasmon excitations, we utilize Bogoliubov’s f 2 3 γ 2 f 4 dz x ⎛⎞2 0 canonical transformation technique (see, e.g., [36]) Γ0()x = (11) 3 ⎝⎠ប and diagonalize the Hamiltonian (1)–(7) exactly to 3បc have represents the (dimensionless) exciton spontaneous decay rate, where ˆ បω ()ξˆ† ()ξˆ () H = ∑ μ k μ k μ k + E0. (8) f 〈|()ˆ |〉 k, μ = 12, dz = ∑ 0 dn z f n Here, the new operator is the longitudinal exciton transition dipole moment matrix element. The function ξˆ () []*(), ω v (), ω † μ k = ∑ uμ k f Bk, f – μ k f B–k, f 3S 1 ρ()x = 0 Re (12) f 2 σ () πα zz x ∞ 4 RCN ˆ ˆ † + dω[]uμ()k, ω f ()k, ω – vμ*()k, ω f ()–k, ω stands for the surface plasmon density of states (DOS) ∫ which is responsible for the exciton decay rate varia 0 tion due to the coupling to surface plasmon modes. Here, α = e2/បc = 1/137 is the finestructure constant ξˆ† ξˆ † annihilates and μ (k) = [μ (k)] creates the exciton and σ = 2πបσ /e2 is the dimensionless CN surface μ v zz zz plasmon excitation of branch (= 1, 2), uμ and μ are axial conductivity per unit length. the (appropriately chosen) canonical transformation Note that the conductivity factor in Eq. (12) equals coefficients, ωf = Ef/ប. The “vacuum” energy E0 rep resents the state with no excitonplasmons excited in 1 4αc⎛⎞ប 1 បω Re = – Im (13) the system, and μ(k) is the excitonplasmon energy σ ()x R ⎝⎠2γ x ⑀ ()x – 1 given by the solution of the following (dimensionless) zz CN 0 zz dispersion relation [37] in view of Eq. (10) and the equation

∞ σ = –iω()⑀ – 1 /4πSρ (14) ε ρ() zz zz T 2 ε 2 f Γf () x xμ – f – dxx 0 x = 0. (9) representing the Drude relation for CNs, where ⑀ is 2π∫ 2 2 zz xμ – x ρ 0 the longitudinal dielectric function, S and T are the

OPTICS AND SPECTROSCOPY Vol. 108 No. 3 2010 380 BONDAREV et al.

4 (11,0) 100 0.32 (11,0) σ Re[ zz] (а) σ 0.30 Im[ zz] 80 2 0.28 60 0.26 40 0 0.24 −1 20 0.22 0 0.20 0.224 0.24 0.26 0.28 0.30 0 0.10 0.20 0.30 σ (10,0) (10,0) Re[ zz] 30 (b) 6 0.28 Im[σzz] Plasmon DOS 4 20 0.26 Dimensionless energy

Dimensionless conductivity 0.24 2 10 0.22 0 0.20 0 0.18 0.185 0.21 0.24 0.27 0.30 0 0.10 0.20 Dimensionless energy Dimensionless quasimomentum

Fig. 3. (Color online) (a, b) Surface plasmon DOS and conductivities (left panels), and lowest bright exciton dispersion when coupled to plasmons (right panels) in the (11,0) and (10,0) CN, respectively. Dimensionless energy is defined as [Energy]/2γ0, according to Eq. (10). See text for dimensionless quasimomentum. surface area of the tubule and the number of tubules according to Eq. (12). This function is only nonzero per unit volume, respectively [29–31, 38]. This relates σ when the two conditions, Im[zz (x)] = 0 and very closely the surface plasmon DOS function (12) to σ the loss function –Im(1/⑀) measured in electron Re[zz (x)] 0, are fulfilled simultaneously [23, 35, energy loss spectroscopy (EELS) experiments to 40]. These result in the peak structure of the function σ determine the properties of collective electronic exci Re(1/zz ) as is seen in Fig. 2. It is also seen from the tations in solids [22]. comparison of Fig. 2b with Fig. 2a that the peaks broaden as the CN diameter decreases. This is consis tent with the stronger hybridization effects in smaller RESULTS AND DISCUSSION diameter CNs [43]. Figure 2 shows the lowenergy behaviors of the σ σ Left panels in Figs. 3a and 3b show the lowest functions zz (x) and Re[1/zz (x)] for the (11,0) and energy plasmon DOS resonances calculated for the (10,0) CNs (RCN = 0.43 and 0.39 nm, respectively) we (11,0) and (10,0) CNs as given by the function ρ(x) in study here. We obtained them numerically as follows. Eq. (12). The corresponding fragments of the func First, we adapt the nearestneighbor nonorthogonal tions Re[σ (x)] and Im[σ (x)] are also shown there. tightbinding approach [39] to determine the realistic zz zz band structure of each CN. Then, the roomtempera In all graphs the lower dimensionless energy limits are ture longitudinal dielectric functions ⑀ are calculated set up to be equal to the lowest bright exciton excita zz tion energy ( = 1.21 eV ( = 0.224) and 1.00 eV within the randomphase approximation [40, 41], Eexc x σ (x = 0.185) for the (11,0) and (10,0) CN, respectively, which are then converted into the conductivities zz as reported in [24] by directly solving the Bethe–Sal by means of the Drude relation (14). Electronic dissi peter equation). Peaks in ρ(x) are seen to coincide in pation processes are included in our calculations energy with zeros of Im[σ (x)] {or zeros of within the relaxationtime approximation (electron zz Re[⑀ (x)]}, clearly indicating the plasmonic nature of scattering length of 130RCN was used [18]). We did not zz include excitonic manyelectron correlations, how the CN surface excitations under consideration [23, ever, as they mostly affect the real conductivity 44]. They describe the surface plasmon modes associ Re(σ ) which is responsible for the CN optical ated with the transversely quantized interband elec zz tronic transitions in CNs [23]. As is seen in Fig. 3 (and absorption [10, 12, 42], whereas we are interested here in Fig. 2), the interband plasmon excitations occur in σ in Re(1/zz ) representing the surface plasmon DOS CNs slightly above the first bright exciton excitation

OPTICS AND SPECTROSCOPY Vol. 108 No. 3 2010 SURFACE EXCITON–PLASMONS AND OPTICAL RESPONSE 381 ប ≤ πប energy [42], which is a unique feature of the complex kz 2 /3b [1, 2]. The total energy of the ground dielectric response function—the consequence of the internalstate exciton can then be written as general Kramers–Krönig relation [45]. E = E + (2πប/3b)2t2/2M We further take advantage of the sharp peak struc exc ex ≤ ≤ ture of ρ(x) and solve the dispersion equation (9) for xµ with –1 t 1 representing the dimensionless longi analytically using the Lorentzian approximation tudinal quasimomentum. In our calculations we used the lowest bright exciton parameters Eexc = 1.21 and 2 ρ()Δ rad xp xp τ ρ()x ≈ . (15) 1.00 eV, ex = 14.3 and 19.1 ps, Mex = 0.44m0 and ()2 Δ 2 xx– p + xp 0.19m0 (m0 is the freeelectron mass) for the (11,0) CN Δ and (10,0) CN, respectively, as reported in [24] by Here, xp and xp are, respectively, the position and the directly solving the Bethe–Salpeter equation. halfwidthathalfmaximum of the plasmon reso Both graphs in the right panels in Fig. 3 are seen to nance closest to the lowest bright exciton excitation demonstrate a clear anticrossing behavior with the energy in the same nanotube (as shown in the left pan (Rabi) energy splitting ~0.1 eV. This indicates the for els of Fig. 3). The integral in Eq. (9) then simplifies to mation of the strongly coupled surface plasmonexci the form ton excitations in the nanotubes under consideration. ∞ ∞ f 2 It is important to realize that here we deal with the 1 xΓ ()ρx ()x Fx()Δx dx0 ≈ p p dx strong excitonplasmon interaction supported by an 2π∫ 2 2 2 2 ∫()2 Δ 2 individual quasi1D nanostructure—a singlewalled xμ – x xμ – xp xx– p + xp 0 0 (16) (smalldiameter) semiconducting carbon nanotube, Fx()Δx ⎛⎞x π as opposed to the artificially fabricated metalsemi = p p arctan p + 2 2 ⎝⎠Δx 2 conductor nanostructures studied previously [26–28] xμ – xp p where the metallic component normally carries the with plasmon and the semiconducting one carries the exci ton. It is also important that the effect comes not only () Γf ()ρ()π from the height but also from the width of the plasmon Fxp = xp 0 xp xp /2 . resonance as is seen from the definition of the Fp factor This expression is valid for all xµ apart from those in Eq. (17). In other words, as long as the plasmon res located in the narrow interval (xp – Δxp, xp + Δxp) in onance is sharp enough (which is always the case for the vicinity of the plasmon resonance, provided that interband plasmons), so that the Lorentzian approxi the resonance is sharp enough. Then, the dispersion mation (15) applies, the effect is determined by the equation becomes the biquadratic equation for xµ with area under the plasmon peak in the DOS function (12) the following two positive solutions (the dispersion rather then by the peak height as one would expect. curves) of interest to us: We are now in a position to derive the exciton absorption lineshape function. To do that, we follow ε 2 2 f + xp 1 2 2 2 the optical absorption lineshape theory that one of us x , = ± ()ε – x + F ε . (17) 12 2 2 f p p f developed recently for atomically doped CNs [5]. In doing so, we take into account the excitonphonon Here scattering in the relaxation time approximation. The (dimensionless) exciton absorption lineshape function ()Δ()πΔ Fp = 4Fxp xp – xp/xp I(x) in the vicinity of the plasmon resonance is then of and the arctanfunction of Eq. (16) is expanded into the form Δ Ӷ 2 2 series to linear terms in xp/xp 1. ()ε Δ (18) () ()ε x – f + xp The dispersion curves (17) are shown in the right Ix = I0 f , []()ε 2 2 2 ()ε 2()Δ 2 Δε2 panels in Figs. 3a, 3b as functions of the dimensionless x – f – Xf /4 + x – f xp + f longitudinal quasimomentum. In these calculations, where we estimated the interband transition matrix element 2 ()ε Γf ()ρεε () π πΔ ()ε Γf () f I0 f ==0 f f /2 , Xf 4 xpI0 f , in 0 xp (Eq. (11)) from the equation dz = rad with Δε = ប/2γ τ being the exciton energy broaden 3បλ3/4τ according to Hanamura’s general theory of f 0 ph ex ing due to the phonon scattering with the relaxation the exciton radiative decay in spatially confined sys time τph. rad tems [46], where τex is the exciton intrinsic radiative The calculated exciton absorption lineshapes for lifetime, and λ = 2πcប/E with E being the exciton total the CNs under consideration are shown in Fig. 4 as the energy given in our case by Eq. (4). For zigzagtype exciton energies are tuned to the nearest plasmon res CNs we here consider, the first Brillouin zone of the onances. We used τph = 30 fs as reported in [17]. The longitudinal quasimomentum is given by –2πប/3b ≤ line (Rabi) splitting effect is seen to be ~0.1–0.3 eV,

OPTICS AND SPECTROSCOPY Vol. 108 No. 3 2010 382 BONDAREV et al.

(а) (11,0) (b) (10,0) Lineshape, arb. units Lineshape, arb. units

0.26 0.28 0.30 0.20 0.24 0.28 Dimensionless energy Dimensionless energy

Fig. 4. (Color online) (a, b) Exciton absorption lineshapes as the exciton energies are tuned to the nearest plasmon resonance energies (vertical dashed lines here; see Fig. 3, left panels) in the (11,0) and (10,0) CN, respectively. Dimensionless energy is defined as [Energy]/2γ0, according to Eq. (10). indicating the strong excitonplasmon coupling with an external electrostatic field applied perpendicular to the formation of the mixed surface plasmonexciton the CN axis). The two possibilities influence the dif excitations. The splitting is larger in the smaller diam ferent degrees of freedom of the quasi1D exciton— eter nanotubes, and is not masked by the exciton the (longitudinal) kinetic energy and the excitation phonon scattering. energy, respectively (see Eq. (4)). In the latter case, in spite of the fact that the cylindrical surface symmetry Obviously, the formation of the strongly coupled of the excitonic states brings new peculiarities, the mixed excitonplasmon states is only possible if the general qualitative behavior of the quantum confined exciton total energy is in resonance with the energy of Stark effect in CNs should be similar to what was pre an interband surface plasmon mode. The exciton viously observed and theoretically analyzed for semi energy might be tuned to the nearest plasmon reso conductor quantum wells [50, 51]. One should expect nance in ways used for the excitons in semiconductor that the exciton excitation energy Eexc and the inter quantum microcavities—thermally (by elevating band plasmon energy E both shift to the red due to sample temperature) [47–49], and/or electrostatically pl the decrease in the CN band gap Eg as perpendicular [50–53] (via the quantum confined Stark effect with electrostatic field increases. However, the exciton red shift is expected to be much less due to the decrease in (10,0) the absolute value Eb of the exciton (negative) bind ing energy, which is estimated to be ~0.5Eexc in small 0.3 diameter CNs [9–11, 14] and thus contributes largely Eb to the exciton excitation energy. So, Eexc and Epl will be approaching as the field increases, bringing the total exciton energy (see Eq. (4)) in resonance with the 0.2 plasmon mode due to the nonzero longitudinal kinetic energy term at finite temperature. Figure 5

Eg shows the results of our calculations of the quantum Epl confined Stark effect for the (10,0) CN which confirm Eexc the qualitative expectations just discussed. More 0.1 details on these calculations and the complete theory Dimensionless energy of the quantum confined Stark effect in carbon nano tubes will be published elsewhere.

0 0.04 0.08 0.12 Electric field, V/μm CONCLUSION We have shown the strong excitonsurfaceplas mon coupling effect with the characteristic exciton Fig. 5. (Color online) Calculated dependences of the first bright exciton parameters in the (10,0) CN on the electro absorption line (Rabi) splitting ~0.1–0.3 eV in small static field applied perpendicular to the CN axis. Dimen diameter (Շ1 nm) semiconducting CNs. This is sionless energy is defined as [Energy]/2γ0, according to almost as large as the typical exciton binding energies Eq. (10). See text for notations. in such CNs (~0.3–0.8 eV [9–11, 14]), and of the

OPTICS AND SPECTROSCOPY Vol. 108 No. 3 2010 SURFACE EXCITON–PLASMONS AND OPTICAL RESPONSE 383 same order of magnitude as the excitonplasmon Rabi 15. A. Hagen, M. Steiner, M. B. Raschke, C. Lienau, splitting in organic semiconductors (~180 meV [26]). T.Hertel, H. Qian, A. J. Meixner, and A. Hartschuh, Also, this is much larger than the excitonpolariton Phys. Rev. Lett. 95, 197401 (2005). Rabi splitting in semiconductor microcavities (~140– 16. F. Plentz, H. B. Ribeiro, A. Jorio, M. S. Strano, and 400 μeV [47–49]), or the excitonplasmon Rabi split M. A. Pimenta, Phys. Rev. Lett. 95, 247401 (2005). ting in hybrid semiconductormetal nanoparticle 17. V. Perebeinos, J. Tersoff, and Ph. Avouris, Phys. Rev. molecules [27]. We have discussed possible ways to Lett. 94, 027402 (2005). tune the exciton energy to the nearest surface plasmon 18. M. Lazzeri, S. Piscanec, F. Mauri, A. C. Ferrari, and resonance and emphasized an important role of the J. Robertson, Phys. Rev. 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