Strong exciton-plasmon coupling in semiconducting carbon nanotubes

Transcription

1 PHYSICAL REVIEW B 80, Strong exciton-plasmon coupling in semiconducting carbon nanotubes I. V. Bondarev* Department o Physics, North Carolina Central University, 1801 Fayetteville Street, Durham, North Carolina 7707, USA L. M. Woods and K. Tatur Department o Physics, University o South Florida, 40 E. Fowler Avenue, Tampa, Florida 3360, USA Received 7 April 009; revised manuscript received 3 July 009; published 6 August 009 We study theoretically the interactions o excitonic states with surace electromagnetic modes o smalldiameter 1 nm semiconducting single-walled carbon nanotubes. We show that these interactions can result in strong exciton-surace-plasmon coupling. The exciton absorption line shape exhibits Rabi splitting 0.1 ev as the exciton energy is tuned to the nearest interband surace-plasmon resonance o the nanotube. We also show that the quantum conined Stark eect may be used as a tool to control the exciton binding energy and the nanotube band gap in carbon nanotubes in order, e.g., to bring the exciton total energy in resonance with the nearest interband plasmon mode. The exciton-plasmon Rabi splitting we predict here or an individual carbon nanotube is close in its magnitude to that previously reported or hybrid plasmonic nanostructures artiicially abricated o organic semiconductors on metallic ilms. We expect this eect to open up paths to new tunable optoelectronic device applications o semiconducting carbon nanotubes. DOI: /PhysRevB PACS numbers: Ri, 73.., Fg, Ch I. INTRODUCTION Single-walled carbon nanotubes CNs are quasi-onedimensional 1D cylindrical wires consisting o graphene sheets rolled up into cylinders with diameters 1 10 nm and lengths m. 1 4 CNs are shown to be useul as miniaturized electronic, electromechanical, and chemical devices, 5 scanning probe devices, 6 and nanomaterials or macroscopic composites. 7 The area o their potential applications was recently expanded to nanophotonics 8 11 ater the demonstration o controllable single-atom encapsulation into CNs Res and even to quantum cryptography since the experimental evidence was reported or quantum correlations in the photoluminescence spectra o individual nanotubes. 16 For pristine undoped single-walled CNs, the numerical calculations predicting large exciton binding energies ev in semiconducting CNs Res and even in some small-diameter 0.5 nm metallic CNs, 0 ollowed by the results o various exciton photoluminescence measurements, 16,1 5 have become available. These works, together with other reports investigating the role o eects such as intrinsic deects, 3,6 exciton-phonon interactions, 4,6 9 external magnetic and electric ields, 30 3 reveal the variety and complexity o the intrinsic optical properties o CNs. 33 Here, we develop a theory or the interactions between excitonic states and surace electromagnetic EM modes in small-diameter 1 nm semiconducting single-walled CNs. We demonstrate that such interactions can result in a strong exciton-surace-plasmon coupling due to the presence o low-energy 0.5 ev weakly dispersive interband plasmon modes 34 and large exciton excitation energies 1 ev Res. 35 and 36 in small-diameter CNs. Previous studies have been ocused on artiicially abricated hybrid plasmonic nanostructures, such as dye molecules in organic polymers deposited on metallic ilms, 37 semiconductor quantum dots coupled to metallic nanoparticles, 38 or nanowires, 39 where one material carries the exciton and another one carries the plasmon. Our results are particularly interesting since they reveal the undamental EM phenomenon the strong exciton-plasmon coupling in an individual quasi-1d nanostructure, a carbon nanotube. The paper is organized as ollows. Section II presents the general Hamiltonian o the exciton interaction with vacuumtype quantized surace EM modes o a single-walled CN. No external EM ield is assumed to be applied. The vacuumtype-ield we consider is created by CN surace EM luctuations. In describing the exciton-ield interaction on the CN surace, we use our recently developed Green s unction ormalism to quantize the EM ield in the presence o quasi-1d absorbing bodies The ormalism ollows the original line o the macroscopic quantum electrodynamics QED approach developed by Welsch and co-workers to correctly describe medium-assisted electromagnetic vacuum eects in dispersing and absorbing media also reerences therein. Section III explains how the interaction introduced in Sec. II results in the coupling o the excitonic states to the nanotube s surace-plasmon modes. Here, we derive, calculate, and discuss the characteristics o the coupled excitonplasmon excitations, such as the dispersion relation, the plasmon density o states DOS, and the optical-absorption line shape, or particular semiconducting CNs o dierent diameters. We also analyze how the electrostatic ield applied perpendicular to the CN axis aects the CN band gap, the exciton binding energy, and the surace-plasmon energy to explore the tunability o the exciton-surace-plasmon coupling in CNs. The summary and conclusions o the work are given in Sec. IV. All the technical details about the construction and diagonalization o the exciton-ield Hamiltonian, the EM ield Green s tensor derivation, and the perpendicular electrostatic ield eect are presented in the Appendixes in order not to interrupt the low o the arguments and results /009/808/ The American Physical Society

2 BONDAREV, WOODS, AND TATUR PHYSICAL REVIEW B 80, FIG. 1. Color online The geometry o the problem. II. EXCITON-ELECTROMAGNETIC-FIELD INTERACTION ON THE NANOTUBE SURFACE We consider the vacuum-type EM interaction o an exciton with the quantized surace electromagnetic luctuations o a single-walled semiconducting CN by using our recently developed Green s unction ormalism to quantize the EM ield in the presence o quasi-1d absorbing bodies No external EM ield is assumed to be applied. The nanotube is modeled by an ininitely thin, ininitely long, anisotropically conducting cylinder with its surace conductivity obtained rom the realistic band structure o a particular CN. Since the problem has the cylindrical symmetry, the orthonormal cylindrical basis e r,e,e z is used with the vector e z directed along the nanotube axis as shown in Fig. 1. Only the axial conductivity, zz, is taken into account, whereas the azimuthal one,, being strongly suppressed by the transverse depolarization eect, is neglected. The total Hamiltonian o the coupled exciton-photon system on the nanotube surace is o the orm Ĥ = Ĥ F + Ĥ ex + Ĥ int, 1 where the three terms represent the ree medium-assisted EM ield, the ree noninteracting exciton, and their interaction, respectively. More explicitly, the second-quantized ield Hamiltonian is Ĥ F = n 0 dˆ n,ˆn,, where the scalar bosonic ield operators ˆ n, and ˆn, create and annihilate, respectively, the surace EM excitation o requency at an arbitrary point n=r n =R CN, n,z n associated with a carbon atom representing a lattice site, Fig. 1 on the surace o the CN o radius R CN. The summation is made over all the carbon atoms, and in the ollowing it is replaced by the integration over the entire nanotube surace according to the rule n = 1 S 0 dr n = 1 S 0 0 d n R dz n, CN where S 0 =3 3/4b is the area o an elementary equilateral triangle selected around each carbon atom in a way to cover the entire surace o the nanotube and b = 1.4 Å is the carbon-carbon interatomic distance. The second-quantized Hamiltonian o the ree exciton see, e.g., Re. 55 on the CN surace is o the orm 3 FIG.. Color online Schematic o the two transversely quantized azimuthal electron-hole subbands let and the irst-interband ground-internal-state exciton energy right in a small-diameter semiconducting carbon nanotube. Subbands with indices j =1 and are shown, along with the optically allowed exciton-related interband transitions Re. 53. See text or notations. Ĥ ex = n,m, E nb n+m, B m, = E kb k, B k,, 4 k, where the operators B n, and B n, create and annihilate, respectively, an exciton with the energy E n in the lattice site n o the CN surace. The index 0 reers to the internal degrees o reedom o the exciton. Alternatively, B k, = 1 N n B n, e ik n and B k, = B k, 5 create and annihilate the -internal-state exciton with the quasimomentum k=k,k z, where the azimuthal component is quantized due to the transverse coninement eect and the longitudinal one is continuous, N is the total number o the lattice sites carbon atoms on the CN surace. The exciton total energy is then written in the orm E k = E k exc k + z M ex k. 6 Here, the irst term represents the excitation energy E exc k = E g k + E b k 7 o the -internal-state exciton with the negative binding energy E b, created via the interband transition with the band gap E g k = e k + h k, 8 where e,h are transversely quantized azimuthal electron-hole subbands see the schematic in Fig.. The second term in Eq. 6 represents the kinetic energy o the translational longitudinal movement o the exciton with the eective mass M ex =m e +m h, where m e and m h are the subband-dependent electron and hole eective masses, respectively. The two equivalent ree-exciton Hamiltonian representations are related to one another via the obvious orthogonality relationships

3 STRONG EXCITON-PLASMON COUPLING IN 1 e ik k n 1 = kk, e in m k = nm, 9 N n N k with the k summation running over the irst Brillouin zone o the nanotube. The bosonic ield operators in Ĥ F are transormed to the k representation in the same way. The most general nonrelativistic, electric dipole excitonphoton interaction on the nanotube surace can be written in the orm we use the Gaussian system o units and the Coulomb gauge; see details in Appendix A Ĥ int = dg + n,m,b n, n,m,0 where with g n,m,b n, ˆm, + H.c., 10 g n,m, = g n,m, g n,m, 11 g n,m, = i 4 c Re zz R CN, d n z G zz n,m, 1 being the interaction matrix element where the exciton with the energy E exc = is excited through the electric-dipole transition d n z =0dˆ n z in the lattice site n by the nanotube s transversely longitudinally polarized surace EM modes. The modes are represented in the matrix element by the transverse longitudinal part o the Green s tensor zz component G zz n,m, o the EM subsystem Appendix B. This is the only Green s tensor component we have to take into account. All the other components can be saely neglected as they are greatly suppressed by the strong transverse depolarization eect in CNs As a consequence, only zz R CN,, the axial dynamic surace conductivity per unit length, is present in Eq. 1. Equations 1 1 orm the complete set o equations describing the exciton-photon coupled system on the CN surace in terms o the EM ield Green s tensor and the CN surace axial conductivity. III. EXCITON-SURFACE-PLASMON COUPLING For the ollowing, it is important to realize that the transversely polarized surace EM mode contribution to the interaction 10 1 is negligible compared to the longitudinally polarized surace EM mode contribution. As a matter o act, G zz n,m,0 in the model o an ininitely thin cylinder we use here Appendix B, thus yielding g n,m,0, g n,m, = g n,m, 13 in Eqs The point is that, because o the nanotube quasi-one dimensionality, the exciton quasimomentum vector and all the relevant vectorial matrix elements o the momentum and dipole moment operators are directed predominantly along the CN axis the longitudinal exciton; see, however, Re. 56. This prevents the exciton rom the electric-dipole coupling to transversely polarized surace EM modes as they propagate predominantly along the CN axis with their electric vectors orthogonal to the propagation direction. The longitudinally polarized surace EM modes are generated by the electronic Coulomb potential see, e.g., Re. 57 and thereore represent the CN surace-plasmon excitations. These have their electric vectors directed along the propagation direction. They do couple to the longitudinal excitons on the CN surace. Such modes were observed in Re. 34. They occur in CNs both at high energies well-known plasmon at 6 ev and at comparatively low energies o 0.5 ev. The latter ones are related to the transversely quantized interband inter van Hove electronic transitions. These weakly dispersive modes 34,58 are similar to the intersubband plasmons in quantum wells. 59 They occur in the same energy range o 1 ev where the exciton excitation energies are located in small-diameter 1 nm semiconducting CNs. 35,36 In what ollows, we ocus our consideration on the exciton interactions with these particular surace-plasmon modes. A. Dispersion relation To obtain the dispersion relation o the coupled excitonplasmon excitations, we transer the total Hamiltonian 1 10 and 13 to the k representation using Eqs. 5 and 9 and then diagonalize it exactly by means o Bogoliubov s canonical transormation technique see, e.g., Re. 60. The details o the procedure are given in Appendix C. The Hamiltonian takes the orm Ĥ = k,=1, Here, the new operator ˆk = +0 kˆ kˆk + E 0. u k, B k, v k, B k, PHYSICAL REVIEW B 80, du k,ˆk, v k,ˆ k, annihilates and ˆ k=ˆk creates the exciton-plasmon excitation o branch. The quantities u and v are appropriately chosen canonical transormation coeicients. The vacuum energy E 0 represents the state with no excitonplasmons excited in the system, and k is the excitonplasmon energy given by the solution o the ollowing dimensionless dispersion relation Here, x 0 dx x 0 xx x x =

4 BONDAREV, WOODS, AND TATUR PHYSICAL REVIEW B 80, x = 0, x = k 0, = E k 0, 17 with 0 =.7 ev being the carbon nearest-neighbor overlap integral entering the CN surace axial conductivity zz R CN,. The unction 0 x = 4d z x 3 3c 3 0, 18 with d z = n 0dˆ n z, represents the dimensionless spontaneous decay rate, and 3S 0 1 x = Re 19 16R CN zz x stands or the surace-plasmon DOS which is responsible or the exciton decay rate variation due to its coupling to the plasmon modes. Here, =e /c=1/137 is the ine-structure constant and zz = zz /e is the dimensionless CN surace axial conductivity per unit length. Note that the conductivity actor in Eq. 19 equals 1 Re zz x = 4c R CN 1 0 xim zz x 1, 0 in view o Eq. 17 and equation zz = i zz 1/4S T representing the Drude relation or CNs, where zz is the longitudinal along the CN axis dielectric unction, S and T are the surace area o the tubule and the number o tubules per unit volume, respectively. 41,44,50 This relates very closely the surace-plasmon DOS unction 19 to the loss unction Im1/ measured in electron energy-loss spectroscopy EELS experiments to determine the properties o collective electronic excitations in solids. 34 Figure 3 shows the low-energy behaviors o the unctions zz x and Re1/ zz x or the 11,0 and 10,0 CNs R CN =0.43 and 0.39 nm, respectively we study here. We obtained them numerically as ollows. First, we adapt the nearestneighbor nonorthogonal tight-binding approach 61 to determine the realistic band structure o each CN. Then, the roomtemperature longitudinal dielectric unctions zz are calculated within the random-phase approximation, 6,63 which are then converted into the conductivities zz by means o the Drude relation. Electronic dissipation processes are included in our calculations within the relaxation-time approximation electron-scattering length o 130R CN was used 8. We did not include excitonic many-electron correlations, however, as they mostly aect the real conductivity Re zz which is responsible or the CN optical absorption, 18,0,53 whereas we are interested here in Re1/ zz representing the surace-plasmon DOS according to Eq. 19. This unction is only nonzero when the two conditions, Im zz x=0 and Re zz x 0, are ulilled simultaneously. 58,59,6 These result in the peak structure o the unction Re1/ zz as is seen in Fig. 3. It is also seen rom the comparison o Fig. 3b to Fig. 3a that the peaks broaden as the CN diameter decreases. This is consistent with the stronger hybridization eects in smaller-diameter CNs. 64 FIG. 3. Color online a and b Calculated dimensionless see text axial surace conductivities or the 11,0 and 10,0 CNs. The dimensionless energy is deined as Energy/ 0, according to Eq. 17. Let panels in Figs. 4a and 4b show the lowest-energy plasmon DOS resonances calculated or the 11,0 and 10,0 CNs as given by the unction x in Eq. 19. Also shown there are the corresponding ragments o the unctions Re zz x and Im zz x. In all graphs, the lower dimensionless energy limits are set up to be equal to the lowest FIG. 4. Color online a and b Surace-plasmon DOS and conductivities let panels and lowest bright exciton dispersion when coupled to plasmons right panels in 11,0 and 10,0 CNs, respectively. The dimensionless energy is deined as Energy/ 0, according to Eq. 17. See text or the dimensionless quasimomentum

5 STRONG EXCITON-PLASMON COUPLING IN bright exciton excitation energy E 11 exc =1.1 ev x=0.4 and 1.00 ev x=0.185 or the 11,0 and 10,0 CN, respectively, as reported in Re. 35 by directly solving the Bethe- Salpeter equation. Peaks in x are seen to coincide in energy with zeros o Im zz x or zeros o Re zz x, clearly indicating the plasmonic nature o the CN surace excitations under consideration. 58,65 They describe the surace-plasmon modes associated with the transversely quantized interband electronic transitions in CNs. 58 As is seen in Fig. 4 and in Fig. 3, the interband plasmon excitations occur in CNs slightly above the irst bright exciton excitation energy, 53 in the requency domain where the imaginary conductivity or the real dielectric unction changes its sign. This is a unique eature o the complex dielectric-response unction, the consequence o the general Kramers-Krönig relation. 46 We urther take advantage o the sharp peak structure o x and solve the dispersion Eq. 16 or x analytically using the Lorentzian approximation x x px p x x p + x. 1 p Here, x p and x p are, respectively, the position and the halwidth-at-hal-maximum o the plasmon resonance closest to the lowest bright exciton excitation energy in the same nanotube as shown in the let panels o Fig. 4. The integral in Eq. 16 then simpliies to the orm 0 dx x 0 xx x x Fx px p x x p 0 = Fx px p x x p dx x x p + x p arctan x p x p +, with Fx p =x p 0 x p x p /. This expression is valid or all x apart rom those located in the narrow interval x p x p,x p +x p in the vicinity o the plasmon resonance, provided that the resonance is sharp enough. Then, the dispersion equation becomes the biquadratic equation or x with the ollowing two positive solutions the dispersion curves o interest to us: x 1, = + x p 1 x p + F p. Here, F p =4Fx p x p x p /x p with the arctan unction expanded to linear terms in x p /x p 1. The dispersion curves are shown in the right panels in Figs. 4a and 4b as unctions o the dimensionless longitudinal quasimomentum. In these calculations, we estimated the interband transition matrix element in 0 x p Eq. 18 rom the equation d z =3 3 /4 rad ex according to Hanamura s general theory o the exciton radiative decay in spatially conined systems, 66 rad where ex is the exciton intrinsic radiative lietime, and =c/e with E being the exciton total energy given in our case by Eq. 6. For zigzagtype CNs considered here, the irst Brillouin zone o the longitudinal quasimomentum is given by /3bk z /3b. 1, The total energy o the ground-internal-state exciton can then be written as PHYSICAL REVIEW B 80, E=E exc +/3b t /M ex with 1t1 representing the dimensionless longitudinal quasimomentum. In our calculations, we used the lowest bright exciton parameters E exc 11 =1.1 and 1.00 ev, rad ex =14.3 and 19.1 ps, and M ex =0.44m 0 and 0.19m 0 m 0 is the ree-electron mass or the 11,0 CN and 10,0 CN, respectively, as reported in Re. 35 by directly solving the Bethe-Salpeter equation. Both graphs in the right panels in Fig. 4 are seen to demonstrate a clear anticrossing behavior with the Rabi energy splitting 0.1 ev. This indicates the ormation o the strongly coupled surace-plasmon-exciton excitations in the nanotubes under consideration. It is important to realize that here we deal with the strong exciton-plasmon interaction supported by an individual quasi-1d nanostructure a single-walled small-diameter semiconducting carbon nanotube, as opposed to the artiicially abricated metalsemiconductor nanostructures studied previously, where the metallic component normally carries the plasmon and the semiconducting one carries the exciton. It is also important that the eect comes not only rom the height but also rom the width o the plasmon resonance as it is seen rom the deinition o the F p actor in Eq.. In other words, as long as the plasmon resonance is sharp enough which is always the case or interband plasmons, so that the Lorentzian approximation 1 applies, the eect is determined by the area under the plasmon peak in the DOS unction 19 rather than by the peak height as one would expect. However, the ormation o the strongly coupled excitonplasmon states is only possible i the exciton total energy is in resonance with the energy o a surace-plasmon mode. The exciton energy can be tuned to the nearest plasmon resonance in ways used or excitons in semiconductor quantum microcavities thermally by elevating sample temperature and/or electrostatically via the quantum conined Stark eect with an external electrostatic ield applied perpendicular to the CN axis. As is seen rom Eqs. 6 and 7, the two possibilities inluence the dierent degrees o reedom o the quasi-1d exciton the longitudinal kinetic energy and the excitation energy, respectively. Below, we study the less trivial electrostatic ield eect on the exciton excitation energy in carbon nanotubes. B. Perpendicular electrostatic ield eect The optical properties o semiconducting CNs in an external electrostatic ield directed along the nanotube axis were studied theoretically in Re. 31. Strong oscillations in the band-to-band absorption and the quadratic Stark shit o the exciton absorption peaks with the ield increase, as well as the strong-ield dependence o the exciton ionization rate, were predicted or CNs o dierent diameters and chiralities. Here, we ocus on the perpendicular electrostatic ield orientation. We study how the electrostatic ield applied perpendicular to the CN axis aects the CN band gap, the exciton binding/excitation energy, and the interband surace-plasmon energy to explore the tunability o the strong excitonplasmon coupling eect predicted above. The problem is similar to the well-known quantum conined Stark eect irst observed or the excitons in semiconductor quantum

6 BONDAREV, WOODS, AND TATUR wells. 70,71 However, the cylindrical surace symmetry o the excitonic states brings new peculiarities to the quantum conined Stark eect in CNs. In what ollows, we will generally be interested only in the lowest internal energy ground excitonic state and so the internal-state index in Eqs. 6 and 7 will be omitted or brevity. Because the nanotube is modeled by a continuous, ininitely thin, anisotropically conducting cylinder in our macroscopic QED approach, the actual local symmetry o the excitonic wave unction resulted rom the graphene Brillouin-zone structure is disregarded in our model see, e.g., reviews 33,53. The local symmetry is implicitly present in the surace axial conductivity though, which we calculate beorehand as described above. 74 We start with the Schrödinger equation or the electron located at r e =R CN, e,z e and the hole located at r h =R CN, h,z h on the nanotube surace. They interact with each other through the Coulomb potential Vr e,r h = e /r e r h, where = zz 0. The external electrostatic ield F=F,0,0 is directed perpendicular to the CN axis along the x axis in Fig. 1. The Schrödinger equation is o the orm Ĥ e F + Ĥ h F + Vr e,r h r e,r h = Er e,r h, 3 with Ĥ e,h F = m e,h 1 + R CN e,h z e,h er e,h F. 4 We urther separate out the translational and relative degrees o reedom o the electron-hole pair by transorming the longitudinal along the CN axis motion o the pair into its center-o-mass coordinates given by Z=m e z e +m h z h /M ex and z=z e z h. The exciton wave unction is approximated as ollows: r e,r h = e ik z Z ex z e e h h. 5 The complex exponential describes the exciton center-omass motion with the longitudinal quasimomentum k z along the CN axis. The unction ex z represents the longitudinal relative motion o the electron and the hole inside the exciton. The unctions e e and h h are the electron and hole subband wave unctions, respectively, which represent their conined motion along the circumerence o the cylindrical nanotube surace. Each o the unctions is assumed to be normalized to unity. Equations 3 and 4 are then rewritten in view o Eqs. 6 8 to yield m e R CN m h R CN e er CNF cos e e e = e e e, h + er CNF cos h h h = h h h, 6 7 z + V ez ex z = E b ex z, 8 where =m e m h /M ex is the exciton reduced mass and V e is the eective longitudinal electron-hole Coulomb interaction potential given by V e z = e d 0 e0 d h e e h h V e, h,z 9 with V being the original electron-hole Coulomb potential written in the cylindrical coordinates as V e, h,z = PHYSICAL REVIEW B 80, z +4R CN 1 sin e h / 1/. 30 The exciton problem is now reduced to the 1D equation 8, where the exciton binding energy does depend on the perpendicular electrostatic ield through the electron and hole subband unctions e,h given by the solutions o Eqs. 6 and 7 and entering the eective electron-hole Coulomb interaction potential 9. The set o Eqs is analyzed in Appendix D. One o the main results obtained in there is that the eective Coulomb potential 9 can be approximated by an attractive cusp-type cuto potential o the orm e V e z z + z 0 j,f, 31 where the cuto parameter z 0 depends on the perpendicular electrostatic ield strength and on the electron-hole azimuthal transverse quantization index j=1,,... excitons are created in interband transitions involving valence and conduction subbands with the same quantization index 53 as shown in Fig.. Speciically, ln 1 j F z 0 j,fr CN 3 + ln 1 j F with j F given to the second-order approximation in the electric ield by e R 6 CN w j j FM ex 4 F, 33 j 1 w j = + 1 j 1+j, where x is the unit step unction. Approximation 31 is ormally valid when z 0 j,f is much less than the exciton Bohr radius a B = /e which is estimated to be 10R CN or the irst j=1 in our notations here exciton in CNs. 17 As is seen rom Eqs. 3 and 33, this is always the case or the irst exciton or those ields where the perturbation theory applies, i.e., when 1 F1 in Eq. 33. Equation 8 with the potential 31 ormally coincides with the one studied by Ogawa and Takagahara in their treatments o excitonic eects in 1D semiconductors with no external electrostatic ield applied. 76 The only dierence in our case is that our cuto parameter 3 is ield dependent. We thereore ollow Re. 76 and ind the ground-state binding

7 STRONG EXCITON-PLASMON COUPLING IN PHYSICAL REVIEW B 80, The exciton binding energy decreases very rapidly in its absolute value as the ield increases. Fields o only V/m are required to decrease E b 11 by a actor o or the CNs considered here. The reason is the perpendicular ield shits up the bottom o the eective potential 31 as shown in Fig. 5b or the 11,0 CN. This makes the potential shallower and pushes bound excitonic levels up, thereby decreasing the exciton binding energy in its absolute value. As this takes place, the shape o the potential does not change and the longitudinal relative electron-hole motion remains inite at all times. As a consequence, no tunnel exciton ionization occurs in the perpendicular ield, as opposed to the longitudinal electrostatic ield Franz-Keldysh eect studied in Re. 31 where the nonzero ield creates the potential barrier separating out the regions o inite and ininite relative motions and the exciton becomes ionized as the electron tunnels to ininity. The binding energy is only the part o the exciton excitation energy 7. Another part comes rom the band-gap energy 8, where e and h are given by the solutions o Eqs. 6 and 7, respectively. Solving them to the leading second order perturbation-theory approximation in the ield Appendix D, one obtains FIG. 5. Color online a Calculated binding energies o the irst bright exciton in the 11,0 and 10,0 CNs as unctions o the perpendicular electrostatic ield applied. Solid lines are the numerical solutions to Eq. 34, dashed lines are the quadratic approximations as given by Eq. 35. b Field dependence o the eective cuto Coulomb potential 31 in the 11,0 CN. The dimensionless energy is deined as Energy/ 0, according to Eq. 17. energy E 11 b or the irst exciton we are interested in here rom the transcendental equation ln z 01,F Eb E b Ry =0. 34 In doing so, we irst ind the exciton Rydberg energy, Ry =e 4 /, rom this equation at F=0. We use the diameter- and chirality-dependent electron and hole eective masses rom Re. 77, and the irst bright exciton binding energy o 0.76 ev or both 11,0 and 10,0 CN as reported in Re. 19 rom ab initio calculations. We obtain Ry =4.0 and 0.57 ev or the 11,0 tube and 10,0 tube, respectively. The dierence o about 1 order o magnitude relects the act that these are the semiconducting CNs o dierent types type-i and type-ii, respectively based on n+m amilies. 77 The parameters Ry thus obtained are then used to ind E 11 b as unctions o F by numerically solving Eq. 34 with z 0 1,F given by Eqs. 3 and 33. The calculated negative binding energies are shown by the solid lines in Fig. 5a. Also shown there by dashed lines are the unctions E b 11 FE b F, 35 with 1 F given by Eq. 33. They are seen to be airly good analytical quadratic in ield approximations to the numerical solutions o Eq. 34 in the range o not too large ields. E jj g FE g jj1 m e j F M ex j m h j F w j M ex j 36 w j, where the electron and hole subband shits are written separately. This, in view o Eq. 33, yields the irst band-gap ield dependence in the orm E g 11 FE g F. 37 The band gap decrease with the ield in Eq. 37 is stronger than the opposite eect in the negative exciton binding energy given to the same order approximation in ield by Eq. 35. Thus, the irst exciton excitation energy 7 will be gradually decreasing as the perpendicular ield increases, shiting the exciton absorption peak to the red. This is the basic eature o the quantum conined Stark eect observed previously in semiconductor nanomaterials The ield dependences o the higher interband transitions exciton excitation energies are suppressed by the rapidly quadratically increasing azimuthal quantization numbers in the denominators o Eqs. 33 and 36. Lastly, the perpendicular ield dependence o the interband plasmon resonances can be obtained rom the requency dependence o the axial surace conductivity due to excitons see Re. 53 and reerences therein. One has ex zz j=1,,... i j jj i /, E exc 38 where j and are the exciton oscillator strength and relaxation time, respectively. The plasmon requencies are those at which the unction Re1/ ex zz has maxima. Testing it or maximum in the domain E 11 exc E exc, one inds the irst-interband plasmon resonance energy to be in the limit

8 BONDAREV, WOODS, AND TATUR PHYSICAL REVIEW B 80, Figure 6 shows the results o our calculations o the ield dependences or the irst bright exciton parameters in the 11,0 and 10,0 CNs. The energy is measured rom the top o the irst unperturbed hole subband as shown in Fig., right panel. The binding-energy ield dependence was calculated numerically rom Eq. 34 as described above shown in Fig. 5a. The band-gap ield dependence and the plasmon energy ield dependence were calculated rom Eqs. 36 and 40, respectively. The zero-ield excitation energies and zero-ield binding energies were taken to be those reported in Res. 35 and 19, respectively, and we used the diameter- and chirality-dependent electron and hole eective masses rom Re. 77. As is seen in Figs. 6a and 6b, the exciton excitation energy and the interband plasmon energy experience redshit in both nanotubes as the ield increases. However, the excitation energy red shit is very small barely seen in the igures due to the negative ield-dependent contribution rom the exciton binding energy. So, E 11 exc F and E 11 p F approach each other as the ield increases, thereby bringing the total exciton energy 6 in resonance with the surace-plasmon mode due to the nonzero longitudinal kinetic-energy term at inite temperature. 78 Thus, the electrostatic ield applied perpendicular to the CN axis the quantum conined Stark eect may be used to tune the exciton energy to the nearest interband plasmon resonance to put the exciton-surace-plasmon interaction in small-diameter semiconducting CNs to the strong-coupling regime. C. Optical absorption FIG. 6. Color online a and b Calculated dependences o the irst bright exciton parameters in the 11,0 and 10,0 CNs, respectively, on the electrostatic ield applied perpendicular to the nanotube axis. The dimensionless energy is deined as Energy/ 0, according to Eq. 17. The energy is measured rom the top o the irst unperturbed hole subband. E 11 p = E 11 exc + E exc. 39 Using the ield dependent E 11 exc given by Eqs. 7, 35, and 37 and neglecting the ield dependence o E exc, one obtains to the second-order approximation in the ield E 11 p FE E g /E exc p 11 1F. 1+E exc /E exc 40 Here, we analyze the longitudinal exciton absorption line shape as its energy is tuned to the nearest interband suraceplasmon resonance. Only longitudinal excitons excited by light polarized along the CN axis couple to the suraceplasmon modes as discussed at the very beginning o this section see Re. 56 or the perpendicular light exciton absorption in CNs. We ollow the optical absorption-emission line shape theory developed recently or atomically doped CNs. 8 Obviously, the absorption line shape coincides with the emission line shape i the monochromatic incident light beam is used in the absorption experiment. When the -internal state exciton is excited and the nanotube s surace EM ield subsystem is in vacuum state, the time-dependent wave unction o the whole system exciton+ield is o the orm 74 t = C k,te ie kt/ 1 k ex 0 k, + k 0 dck,,te it 0 ex 1k,. 41 Here, 1 k ex is the excited single-quantum Fock state with one exciton and 1k, is that with one surace photon. The vacuum states are 0 ex and 0 or the exciton subsystem and ield subsystem, respectively. The coeicients C k,t and Ck,,t stand or the population probability amplitudes o the respective states o the whole system. The exciton energy is o the orm Ẽ k=e k i/, with E k given by Eq. 6 and being the phenomenological exciton relaxation-time constant assumed to be such that /E k to account or other possible exciton relaxation processes. From the literature, we have ph s or the exciton-phonon scattering, 31 d 50 ps or the exciton scattering by deects, 3,6 and rad 10 ps 10 ns or the radiative decay o excitons. 35 Thus, the scattering by phonons is the most likely exciton relaxation mechanism. We transorm the total Hamiltonian 1 10 to the k representation using Eqs. 5 and 9 see Appendix A and apply it to the wave unction in Eq. 41. We obtain the ollowing set o the two simultaneous dierential equations or the coeicients C k,t and Ck,,t rom the time-dependent Schrödinger equation:

9 STRONG EXCITON-PLASMON COUPLING IN Ċ k,te ie kt/ = i k 0 Ċk,,te it kk = i dg + k,k,ck,,te it, 4 g + k,k, C k,te ie kt/. 43 The symbol on the let in Eq. 43 ensures that the momentum conservation is ulilled in the exciton-photon transitions, so that the annihilating exciton creates the surace photon with the same momentum and vice versa. In terms o the probability amplitudes above, the exciton emission intensity distribution is given by the inal-state probability at very long times corresponding to the complete decay o all initially excited excitons I = Ck,,t = 1 k t/ g + k,k, 0 dtc k,te ie. 44 Here, the second equation is obtained by the ormal integration o Eq. 43 over time under the initial condition Ck,,0=0. The emission intensity distribution is thus related to the exciton population probability amplitude C k,t to be ound rom Eq. 4. The set o simultaneous equations 4 and 43 and Eq. 44, respectively contains no approximations except the commonly used neglect o many-particle excitations in the wave unction 41. We now apply these equations to the exciton-surace-plasmon system in small-diameter semiconducting CNs. The interaction matrix element in Eqs. 4 and 43 is then given by the k transorm o Eq. 13 and has the ollowing property Appendix C: 1 0 g + k,k, = 1 0 xx, 45 with 0 x and x given by Eqs. 18 and 19, respectively. We urther substitute the result o the ormal integration o Eq. 43 with Ck,,0=0 into Eq. 4, use Eq. 45 with x approximated by the Lorentzian 1, calculate the integral over requency analytically, and dierentiate the result over time to obtain the ollowing second-order ordinary dierential equation or the exciton probability amplitude dimensionless variables, Eq. 17 C + x p + ix p Ċ + X / C =0, where X =x p x p 1/ with x p = 0 x p x p, =/ 0, = 0 t/ is the dimensionless time, and the k dependence is omitted or brevity. When the total exciton energy is close to a plasmon resonance, x p, the solution o this equation is easily ound to be C x x X e x x X / x x X e x+ x X /, 46 where x=x p 0 and X =x p 1/. This solution is valid when x p regardless o the strength o the exciton-surace-plasmon coupling. It yields the exponential decay o the excitons into plasmons, C exp, in the weak-coupling regime where the coupling parameter X /x 1. I, on the other hand, X /x 1, then the strong-coupling regime occurs and the decay o the excitons into plasmons proceeds via damped Rabi oscillations, C exp xcos X /. This is very similar to what was earlier reported or an excited twolevel atom near the nanotube surace. 40 4,45 Note, however, that here we have the exciton-phonon scattering as well, which acilitates the strong exciton-plasmon coupling by decreasing x in the coupling parameter. In other words, the phonon scattering broadens the longitudinal exciton momentum distribution, 81 thus eectively increasing the raction o the excitons with x p. In view o Eqs. 45 and 46, the exciton emission intensity 44 in the vicinity o the plasmon resonance takes the ollowing dimensionless orm: ĪxĪ 0 dc e 0 ix +i, 47 where Īx= 0 I/ and Ī 0 = /. Ater some algebra, this results in Īx PHYSICAL REVIEW B 80, Ī 0 x + x p x X /4 + x x p +, 48 where x p. The summation sign over the exciton internal states is omitted since only one internal state contributes to the emission intensity in the vicinity o the sharp plasmon resonance. The line shape in Eq. 48 is mainly determined by the coupling parameter X /x p. It is clearly seen to be o a symmetric two-peak structure in the strong-coupling regime where X /x p 1. Testing it or extremum, we obtain the peak requencies to be 1+8 x 1, = X x p X 4 x p X, terms x p /X 4 are neglected with the Rabi splitting x 1 x X. In the weak-coupling regime where X /x p 1, the requencies x 1 and x become complex, indicating that there are no longer peaks at these requencies. As this takes place, Eq. 48 is approximated with the weakcoupling condition, the act that x and X =x p, to yield the Lorentzian

10 BONDAREV, WOODS, AND TATUR PHYSICAL REVIEW B 80, the smaller-diameter nanotubes and is not masked by the exciton-phonon scattering. IV. CONCLUSIONS FIG. 7. Color online a and b Exciton absorption-emission line shapes as the exciton energies are tuned to the nearest plasmon resonance energies vertical dashed lines in here; see Fig. 3 and let panels in Fig. 4 in the 11,0 and 10,0 nanotubes, respectively. The dimensionless energy is deined as Energy/ 0 according to Eq. 17. Ī 0 /1+ /x p Īx x + / 1+ /x p peaked at x=, whose hal width at hal maximum is slightly narrower, however, than / it should be i the exciton-plasmon relaxation were the only relaxation mechanism in the system. The reason is that the competing phonon scattering takes excitons out o resonance with plasmons, thus decreasing the exciton-plasmon relaxation rate. We thereore conclude that the phonon scattering does not aect the exciton emission-absorption line shape when the excitonplasmon coupling is strong it acilitates the strong-coupling regime to occur, however, as was noticed above and it narrows the Lorentzian emission-absorption line when the exciton-plasmon coupling is weak. Calculated exciton emission-absorption line shapes, as given by Eq. 48 or the CNs under consideration, are shown in Figs. 7a and 7b. The exciton energies are assumed to be tuned, e.g., by means o the quantum conined Stark eect discussed in Sec. III B, to the nearest plasmon resonances shown by the vertical dashed lines in the igure. We used ph =30 s as reported in Re. 7. The line Rabi splitting eect is seen to be 0.1 ev, indicating the strong exciton-plasmon coupling with the ormation o the mixed surace plasmon-exciton excitations. The splitting is larger in We have shown that the strong exciton-surace-plasmon coupling eect with characteristic exciton absorption line Rabi splitting 0.1 ev exists in small-diameter 1 nm semiconducting CNs. The splitting is almost as large as the typical exciton binding energies in such CNs ev Res and and o the same order o magnitude as the exciton-plasmon Rabi splitting in organic semiconductors 180 mev Re. 37. It is much larger than the excitonpolariton Rabi splitting in semiconductor microcavities ev Res or the exciton-plasmon Rabi splitting in hybrid semiconductor-metal nanoparticle molecules. 38 Since the ormation o the strongly coupled mixed exciton-plasmon excitations is only possible i the exciton total energy is in resonance with the energy o an interband surace plasmon mode, we have analyzed possible ways to tune the exciton energy to the nearest surace plasmon resonance. Speciically, the exciton energy may be tuned to the nearest plasmon resonance in ways used or the excitons in semiconductor quantum microcavities thermally by elevating sample temperature and/or electrostatically via the quantum conined Stark eect with an external electrostatic ield applied perpendicular to the CN axis. The two possibilities inluence the dierent degrees o reedom o the quasi-1d exciton the longitudinal kinetic energy and the excitation energy, respectively. We have studied how the perpendicular electrostatic ield aects the exciton excitation energy and interband plasmon resonance energy the quantum conined Stark eect. Both o them are shown to shit to the red due to the decrease in the CN band gap as the ield increases. However, the exciton redshit is much less than the plasmon one because o the decrease in the absolute value o the negative binding energy, which contributes largely to the exciton excitation energy. The exciton excitation energy and interband plasmon energy approach as the ield increases, thereby bringing the total exciton energy in resonance with the plasmon mode due to the nonzero longitudinal kinetic-energy term at inite temperature. Lastly, the noteworthy message we would like to deliver in this paper is that the strong exciton-surace-plasmon coupling we predict here occurs in an individual CN as opposed to various artiicially abricated hybrid plasmonic nanostructures mentioned above. We strongly believe this phenomenon, along with its tunability eature via the quantum conined Stark eect we have demonstrated, opens up new paths or the development o CN-based tunable optoelectronic device applications in areas such as nanophotonics, nanoplasmonics, and cavity QED. One straightorward application like this is the CN photoluminescence control by means o the exciton-plasmon coupling tuned electrostatically via the quantum conined Stark eect. This complements the microcavity-controlled CN inrared emitter application reported recently, 5 oering the advantage o less stringent ab

11 STRONG EXCITON-PLASMON COUPLING IN rication requirements at the same time since the planar photonic microcavity is no longer required. Electrostatically controlled coupling o two spatially separated weakly localized excitons to the same nanotube s plasmon resonance would result in their entanglement, 9 11 the phenomenon that paves the way or CN-based solid-state quantum inormation applications. Moreover, CNs combine advantages such as electrical conductivity, chemical stability, and high surace area that make them excellent potential candidates or a variety o more practical applications, including eicient solar energy conversion, 7 energy storage, 1 and optical nanobiosensorics. 86 However, the photoluminescence quantum yield o individual CNs is relatively low and this hinders their uses in the aorementioned applications. CN bundles and ilms are proposed to be used to surpass the poor perormance o individual tubes. The theory o the excitonplasmon coupling we have developed here, being extended to include the intertube interaction, complements currently available weak-coupling theories o the exciton-plasmon interactions in low-dimensional nanostructures 38,87 with the very important case o the strong-coupling regime. Such an extended theory subject o our uture publication will lay the oundation or understanding intertube energy-transer mechanisms that aect the eiciency o optoelectronic devices made o CN bundles and ilms, as well as it will shed more light on the recent photoluminescence experiments with CN bundles 88,89 and multiwalled CNs, 90 revealing their potentialities or the development o high-yield, highperormance optoelectronics applications with CNs. ACKNOWLEDGMENTS The work is supported by NSF Grants No. ECS and No. HRD L.M.W. and K.T. acknowledge support rom DOE Grant No. DE-FG0-06ER4697. Helpul discussions with Mikhail Braun St. Peterburg U., Russia, Jonathan Finley WSI, TU Munich, Germany, and Alexander Govorov Ohio U., USA are grateully acknowledged. APPENDIX A: EXCITON INTERACTION WITH THE SURFACE ELECTROMAGNETIC FIELD We ollow our recently developed QED ormalism to describe vacuum-type EM eects in the presence o quasi-1d absorbing and dispersive bodies The treatment begins with the most general EM interaction o the surace charge luctuations with the quantized surace EM ield o a singlewalled CN. No external ield is assumed to be applied. The CN is modeled by a neutral, ininitely long, ininitely thin, anisotropically conducting cylinder. Only the axial conductivity o the CN, zz, is taken into account, whereas the azimuthal one,, is neglected being strongly suppressed by the transverse depolarization eect Since the problem has the cylindrical symmetry, the orthonormal cylindrical basis e r,e,e z is used with the vector e z directed along the nanotube axis as shown in Fig. 1. The interaction has the ollowing orm Gaussian system o units: Ĥ int = Ĥ 1 int + Ĥ q int = i n,i m i c Ân + rˆn i pˆ ni q i i Ân + rˆn + q i ˆ n + c rˆni, A1 n,i where c is the speed o light, m i, q i, rˆn i, and pˆ i n are, respectively, the masses, charges, coordinate operators, and momenta operators o the particles electrons and nucleus residing at the lattice site n=r n =R CN, n,z n associated with a carbon atom see Fig. 1 on the surace o the CN o radius R CN. The summation is taken over the lattice sites and may be substituted with the integration over the CN surace using Eq. 3. The vector potential operator  and the scalar potential operator ˆ represent the nanotube s transversely polarized and longitudinally polarized surace EM modes, respectively. They are written in the Schrödinger picture as ollows: Ân =0 n ˆ n =0 d c i Ê n, + H.c., dê n, + H.c. A A3 We use the Coulomb gauge whereby n Ân=0 or, equivalently, pˆ i n,ân+rˆn i =0. The total electric ield operator o the CN-modiied EM ield is given or an arbitrary r in the Schrödinger picture by Êr =0 =0 dêr, + H.c. dê r, + Ê r, + H.c., A4 with the transversely longitudinally polarized Fourierdomain ield components deined as where PHYSICAL REVIEW B 80, Ê r, = dr r r Êr,, 1 r = 4r, A5 r = r r A6 are the longitudinal and transverse dyadic -unctions, respectively. The total ield operator A4 satisies the set o the Fourier-domain Maxwell equations Êr, = ikĥr,, Ĥr, = ikêr, + 4 c Îr,, A7 A8 where Ĥ=ik 1 Ê is the magnetic ield operator, k=/c, and

12 BONDAREV, WOODS, AND TATUR Îr, = r nĵn, A9 n is the exterior current operator with the current density deined as ollows: Ĵn, = Re zzr CN, ˆn,ez, A10 to ensure preservation o the undamental QED equal-time commutation relations see, e.g., Re. 46 or the EM ield components in the presence o a CN. Here, zz is the CN surace axial conductivity per unit length, and ˆn, along with its counterpart ˆ n, are the scalar bosonic ield operators which annihilate and create, respectively, singlequantum EM ield excitations o requency at the lattice site n o the CN surace. They satisy the standard bosonic commutation relations ˆn,, ˆ m, = nm, ˆn,, ˆm, = ˆ n,, ˆ m, =0. One urther obtains rom Eqs. A7 A10 that Êr, = ik 4 Gr,n, Ĵn, c n and, according to Eqs. A4 and A5, A11 A1 Ê r, = ik 4 Gr,n, Ĵn,, A13 c n where G and G are the transverse part and the longitudinal part, respectively, o the total Green s tensor G= G+ G o the classical EM ield in the presence o the CN. This tensor satisies the equation k z G z r,n, = r n, =r,,z A14 together with the radiation conditions at ininity and the boundary conditions on the CN surace. All the discrete quantities in Eqs. A9 A14 may be equivalently rewritten in continuous variables in view o Eq. 3. Being applied to the identity 1= m nm, Eq. 3 yields nm = S 0 R n R m. This requires redeining A15 ˆn, = S0 ˆR n,, ˆ n, = S0 ˆ R n, A16 in the commutation relations A11. Similarly, rom Eq. A1, in view o Eqs. 3, A10, and A16, one obtains Gr,n, = S0 Gr,R n,, A17 which is also valid or the transverse and longitudinal Green s tensors in Eq. A13. Next, we make the series expansions o the interactions Ĥ 1 int and Ĥ int in Eq. A1 about the lattice site n to the irst nonvanishing terms Ĥ 1 int n,i Ĥ int q i m i c Ân pˆ n i + n,i q i n ˆ n rˆni, n,i q i m i c  n, A18 A19 and introduce the single-lattice-site Hamiltonian Ĥ n = , A0 with the completeness relation 00 + = Î. A1 Here, 0 is the energy o the ground state 0 no exciton excited o the carbon atom associated with the lattice site n and 0 + is the energy o the excited carbon atom in the quantum state with one -internal-state exciton ormed o the energy E exc =. In view o Eqs. A0 and A1, one has and pˆ ni = m i drˆni dt m i i = m i i rˆn i,ĥ n = m i i Îrˆn i,ĥ n Î 0rˆni B n, rˆni 0B n, A rˆni = Îrˆni Î 0rˆni B n, + rˆni 0B n,, A3 where 0rˆn i =rˆn i 0 in view o the Hermitian and real character o the coordinate operator. The operators B n, =0 and B n, =0 create and annihilate, respectively, the -internal-state exciton at the lattice site n, and excitonto-exciton transitions are neglected. In addition, we also have ij = i pˆ n i,rˆni, A4 where,=r,,z. Substituting these into Eqs. A18 and A19 commutator A4 goes into the second term o Eq. A18 which is to be pretransormed as ollows: i,j,, q i q j  nâ n ij /m i c, one arrives at the ollowing electric-dipole approximation o Eq. A1: Ĥ int = Ĥ 1 int + Ĥ i int = n, c d n ÂnB n, B n, + i c d n Ân + PHYSICAL REVIEW B 80, n, d n n ˆ nb n, + B n, A5 with d i n =0dˆ n=dˆ n0, where dˆ n= i q i rˆn is the total electric-dipole moment operator o the particles residing at the lattice site n. The Hamiltonian A5 is seen to describe the vacuumtype exciton interaction with the surace EM ield created by the charge luctuations on the nanotube surace. The last term in the square brackets does not depend on the exciton