Exciton-Plasmon Interactions in Individual Carbon Nanotubes
aa r X i v : . [ c ond - m a t . m e s - h a ll ] N ov Exciton-Plasmon Interactions inIndividual Carbon Nanotubes Igor V. Bondarev
Department of Physics, North Carolina Central UniversityDurham, NC 27707, USA
E-mail: [email protected]
Lilia M. Woods and
Adrian Popescu
Department of Physics, University of South FloridaTampa, FL 33620, USA
We use the macroscopic quantum electrodynamics approach suitable for ab-sorbing and dispersing media to study the properties and role of collectivesurface excitations — excitons and plasmons — in single-wall and double-wall carbon nanotubes. We show that the interactions of excitonic stateswith surface electromagnetic modes in individual small-diameter ( . ∼ . Keywords:
Carbon nanotubes, Near-field effects, Excitons, Plasmons To appear in ”Plasmons: Theory and Applications”, ed. K.Helsey (Nova Publishers, NY, USA) xciton-Plasmon Interactions in Individual Carbon Nanotubes Single-walled carbon nanotubes (CNs) are quasi-one-dimensional (1D) cylindricalwires consisting of graphene sheets rolled-up into cylinders with diameters ∼ −
10 nm and lengths ∼ − µ m [1, 2, 3, 4]. CNs are shown to be useful asminiaturized electronic, electromechanical, and chemical devices [5], scanning probedevices [6], and nanomaterials for macroscopic composites [7]. The area of theirpotential applications was recently expanded to nanophotonics [8, 9, 10, 11, 12, 13]after the demonstration of controllable single-atom incapsulation into CNs [14, 15,16, 17], and even to quantum cryptography since the experimental evidence wasreported for quantum correlations in the photoluminescence spectra of individualnanotubes [18].For pristine (undoped) single-walled CNs, the numerical calculations predictinglarge exciton binding energies ( ∼ . − . ∼ . ∼ . − I.V.Bondarev, L.M.Woods, and A.Popescu vidual small-diameter ( . ∼ . hy-brid plasmonic nanostructures, such as dye molecules in organic polymers depositedon metallic films [53], semiconductor quantum dots coupled to metallic nanoparti-cles [54], or nanowires [55], where semiconductor material carries the exciton andmetal carries the plasmon. Our results are particularly interesting since they revealthe fundamental electromagnetic (EM) phenomenon — the strong exciton-plasmoncoupling — in an individual quasi-1D nanostructure, a carbon nanotube, as well asits tunability feature by means of the quantum confined Stark effect. We expectthese results to open up new paths for the development of tunable optoelectronicdevice applications with optically excited carbon nanotubes, including the strongexcitation regime with optical non-linearities.Next, we turn to the double-wall carbon nanotubes to investigate the effect ofcollective surface excitations on the inter-tube Casimir interaction in these systems.The Casimir interaction is a paradigm for a force induced by quantum EM fluctua-tions. The fundamental nature of this force has been studied extensively ever sincethe prediction of the existence of an attraction between neutral metallic mirrors invacuum [49, 56]. In recent years, the Casimir effect has acquired a much broaderimpact due to its importance for nanostructured materials and devices. The de-velopment and operation of micro- and nano-electromechanical systems are limiteddue to unwanted effects, such as stiction, friction, and adhesion, originating from theCasimir force [57]. This interaction is also an important component for the stabilityof nanomaterials. Here, we show that at tube separations similar to their equilib-rium distances interband surface plasmons have a profound effect on the inter-tubeCasimir force. Strong overlapping plasmon resonances from both tubes warranttheir stronger attraction. Nanotube chiralities possessing such collective excitationfeatures will result in forming the most favorable inner-outer tube combination indouble-wall carbon nanotubes. This theoretical understanding is important for thedevelopment of nano-electromechanical devices with CNs.This Chapter is organized as follows. Section 2 introduces the general Hamilto-nian of the exciton interaction with vacuum-type quantized surface EM modes ofa single-walled CN. No external EM field is assumed to be applied. The vacuum–type–field we consider is created by CN surface EM fluctuations. Section 3 explainshow the interaction introduced results in the coupling of the excitonic states tothe nanotube’s surface plasmon modes. Here we derive and discuss the character-istics of the coupled exciton–plasmon excitations, such as the dispersion relation, xciton-Plasmon Interactions in Individual Carbon Nanotubes We consider the vacuum-type EM interaction of an exciton with the quantizedsurface electromagnetic fluctuations of a single-walled semiconducting CN by usingour recently developed Green function formalism to quantize the EM field in thepresence of quasi-1D absorbing bodies [58, 59, 60, 61, 62, 9]. No external EM fieldis assumed to be applied. The nanotube is modelled by an infinitely thin, infinitelylong, anisotropically conducting cylinder with its surface conductivity obtained fromthe realistic band structure of a particular CN. Since the problem has the cylindricalsymmetry, the orthonormal cylindrical basis { e r , e ϕ , e z } is used with the vector e z directed along the nanotube axis as shown in Fig. 1. Only the axial conductivity, σ zz ,is taken into account, whereas the azimuthal one, σ ϕϕ , being strongly suppressedby the transverse depolarization effect [63, 64, 65, 66, 67, 68], is neglected. I.V.Bondarev, L.M.Woods, and A.Popescu
The total Hamiltonian of the coupled exciton-photon system on the nanotubesurface is of the form ˆ H = ˆ H F + ˆ H ex + ˆ H int , (1)where the three terms represent the free (medium-assisted) EM field, the free (non-interacting) exciton, and their interaction, respectively. More explicitly, the secondquantized field Hamiltonian isˆ H F = X n Z ∞ dω ~ ω ˆ f † ( n , ω ) ˆ f ( n , ω ) , (2)where the scalar bosonic field operators ˆ f † ( n , ω ) and ˆ f ( n , ω ) create and annihilate,respectively, the surface EM excitation of frequency ω at an arbitrary point n = R n = { R CN , ϕ n , z n } associated with a carbon atom (representing a lattice site –Fig. 1) on the surface of the CN of radius R CN . The summation is made over all thecarbon atoms, and in the following it is replaced by the integration over the entirenanotube surface according to the rule X n . . . = 1 S Z d R n . . . = 1 S Z π dϕ n R CN Z ∞−∞ dz n . . . , (3)where S = (3 √ / b is the area of an elementary equilateral triangle selectedaround each carbon atom in a way to cover the entire surface of the nanotube, b = 1 .
42 ˚A is the carbon-carbon interatomic distance.The second quantized Hamiltonian of the free exciton (see, e.g., Ref. [69]) on theCN surface is of the formˆ H ex = X n , m ,f E f ( n ) B † n + m ,f B m ,f = X k ,f E f ( k ) B † k ,f B k ,f , (4)where the operators B † n ,f and B n ,f create and annihilate, respectively, an excitonwith the energy E f ( n ) in the lattice site n of the CN surface. The index f ( = 0)refers to the internal degrees of freedom of the exciton. Alternatively, B † k ,f = 1 √ N X n B † n ,f e i k · n and B k ,f = ( B † k ,f ) † (5)create and annihilate the f -internal-state exciton with the quasi-momentum k = { k ϕ , k z } , where the azimuthal component is quantized due to the transverse con-finement effect and the longitudinal one is continuous, N is the total number of thelattice sites (carbon atoms) on the CN surface. The exciton total energy is thenwritten in the form E f ( k ) = E ( f ) exc ( k ϕ ) + ~ k z M ex ( k ϕ ) (6) xciton-Plasmon Interactions in Individual Carbon Nanotubes E ( f ) exc ( k ϕ ) = E g ( k ϕ ) + E ( f ) b ( k ϕ ) (7)of the f -internal-state exciton with the (negative) binding energy E ( f ) b , created viathe interband transition with the band gap E g ( k ϕ ) = ε e ( k ϕ ) + ε h ( k ϕ ) , (8)where ε e,h are transversely quantized azimuthal electron-hole subbands (see theschematic in Fig. 2). The second term in Eq. (6) represents the kinetic energy of thetranslational longitudinal movement of the exciton with the effective mass M ex = m e + m h , where m e and m h are the (subband-dependent) electron and hole effectivemasses, respectively. The two equivalent free-exciton Hamiltonian representationsare related to one another via the obvious orthogonality relationships1 N X n e − i ( k − k ′ ) · n = δ kk ′ , N X k e − i ( n − m ) · k = δ nm (9)with the k -summation running over the first Brillouin zone of the nanotube. Thebosonic field operators in ˆ H F are transformed to the k -representation in the sameway.The most general (non-relativistic, electric dipole) exciton-photon interaction onthe nanotube surface can be written in the form (we use the Gaussian system ofunits and the Coulomb gauge; see details in Appendix A)ˆ H int = X n , m ,f Z ∞ dω [ g (+) f ( n , m , ω ) B † n ,f − g ( − ) f ( n , m , ω ) B n ,f ] ˆ f ( m , ω ) + h.c., (10)where g ( ± ) f ( n , m , ω ) = g ⊥ f ( n , m , ω ) ± ωω f g k f ( n , m , ω ) (11)with g ⊥ ( k ) f ( n , m , ω ) = − i ω f c p π ~ ω Re σ zz ( R CN , ω ) ( d f n ) z ⊥ ( k ) G zz ( n , m , ω ) (12)being the interaction matrix element where the exciton with the energy E ( f ) exc = ~ ω f is excited through the electric dipole transition ( d f n ) z = h | (ˆ d n ) z | f i in the lattice site n by the nanotube’s transversely (longitudinally) polarized surface EM modes. Themodes are represented in the matrix element by the transverse (longitudinal) part ofthe Green tensor zz -component G zz ( n , m , ω ) of the EM subsystem (Appendix B).This is the only Green tensor component we have to take into account. All the othercomponents can be safely neglected as they are greatly suppressed by the strongtransverse depolarization effect in CNs [63, 64, 65, 66, 67, 68]. As a consequence, I.V.Bondarev, L.M.Woods, and A.Popescu
Figure 2: Schematic of the two transversely quantized azimuthal electron-hole sub-bands ( left ), and the first-interband ground-internal-state exciton energy ( right ) ina small-diameter semiconducting carbon nanotube. Subbands with indices j = 1and 2 are shown, along with the optically allowed (exciton-related) interband tran-sitions [67]. See text for notations.only σ zz ( R CN , ω ), the axial dynamic surface conductivity per unit length, is presentin Eq.(12).Equations (1)–(12) form the complete set of equations describing the exciton-photon coupled system on the CN surface in terms of the EM field Green tensorand the CN surface axial conductivity. For the following it is important to realize that the transversely polarized sur-face EM mode contribution to the interaction (10)–(12) is negligible compared tothe longitudinally polarized surface EM mode contribution. As a matter of fact, ⊥ G zz ( n , m , ω ) ≡ ⊥ f ( n , m , ω ) ≡ , g ( ± ) f ( n , m , ω ) = ± ωω f g k f ( n , m , ω ) (13)in Eqs. (10)–(12). The point is that, because of the nanotube quasi-one-dimen-sionality, the exciton quasi-momentum vector and all the relevant vectorial matrixelements of the momentum and dipole moment operators are directed predominantlyalong the CN axis (the longitudinal exciton; see, however, Ref. [70]). This prevents xciton-Plasmon Interactions in Individual Carbon Nanotubes π -plasmon at ∼ ∼ . − ∼ . To obtain the dispersion relation of the coupled exciton-plasmon excitations, wetransfer the total Hamiltonian (1)–(10) and (13) to the k -representation usingEqs. (5) and (9), and then diagonalize it exactly by means of Bogoliubov’s canonicaltransformation technique (see, e.g., Ref. [74]). The details of the procedure are givenin Appendix C. The Hamiltonian takes the formˆ H = X k , µ =1 , ~ ω µ ( k ) ˆ ξ † µ ( k ) ˆ ξ µ ( k ) + E . (14)Here, the new operatorˆ ξ µ ( k ) = X f h u ∗ µ ( k , ω f ) B k ,f − v µ ( k , ω f ) B †− k ,f i (15)+ Z ∞ dω h u µ ( k , ω ) ˆ f ( k , ω ) − v ∗ µ ( k , ω ) ˆ f † ( − k , ω ) i annihilates and ˆ ξ † µ ( k ) = [ ˆ ξ µ ( k )] † creates the exciton-plasmon excitation of branch µ , the quantities u µ and v µ are appropriately chosen canonical transformation co-efficients. The ”vacuum” energy E represents the state with no exciton-plasmonsexcited in the system, and ~ ω µ ( k ) is the exciton-plasmon energy given by the solu-tion of the following (dimensionless) dispersion relation x µ − ε f − ε f π Z ∞ dx x ¯Γ f ( x ) ρ ( x ) x µ − x = 0 . (16)Here, x = ~ ω γ , x µ = ~ ω µ ( k )2 γ , ε f = E f ( k )2 γ (17) I.V.Bondarev, L.M.Woods, and A.Popescu with γ = 2 . σ zz ( R CN , ω ). The function¯Γ f ( x ) = 4 | d fz | x ~ c (cid:18) γ ~ (cid:19) (18)with d fz = P n h | (ˆ d n ) z | f i represents the (dimensionless) spontaneous decay rate, and ρ ( x ) = 3 S παR CN Re 1¯ σ zz ( x ) (19)stands for the surface plasmon density of states (DOS) which is responsible for theexciton decay rate variation due to its coupling to the plasmon modes. Here, α = e / ~ c = 1 /
137 is the fine-structure constant and ¯ σ zz = 2 π ~ σ zz /e is the dimensionlessCN surface axial conductivity per unit length.Note that the conductivity factor in Eq. (19) equalsRe 1¯ σ zz ( x ) = − αcR CN (cid:18) ~ γ x (cid:19) Im 1 ǫ zz ( x ) − σ zz ( x ) = − iω πSρ T [ ǫ zz ( x ) −
1] (21)representing the Drude relation for CNs, where ǫ zz is the longitudinal (along theCN axis) dielectric function, S and ρ T are the surface area of the tubule and thenumber of tubules per unit volume, respectively [59, 62, 64]. This relates very closelythe surface plasmon DOS function (19) to the loss function − Im(1 /ǫ ) measured inElectron Energy Loss Spectroscopy (EELS) experiments to determine the propertiesof collective electronic excitations in solids [50].Figure 3 shows the low-energy behaviors of ¯ σ zz ( x ) and Re[1 / ¯ σ zz ( x )] for the(11,0) and (10,0) CNs ( R CN = 0 .
43 nm and 0 .
39 nm, respectively) we study here.We obtained them numerically as follows. First, we adapt the nearest-neighbor non-orthogonal tight-binding approach [75] to determine the realistic band structure ofeach CN. Then, the room-temperature longitudinal dielectric functions ǫ zz are cal-culated within the random-phase approximation [76, 77], which are then convertedinto the conductivities ¯ σ zz by means of the Drude relation. Electronic dissipationprocesses are included in our calculations within the relaxation-time approximation(electron scattering length of 130 R CN was used [30]). We did not include excitonicmany-electron correlations, however, as they mostly affect the real conductivityRe(¯ σ zz ) which is responsible for the CN optical absorption [20, 22, 67], whereas weare interested here in Re(1 / ¯ σ zz ) representing the surface plasmon DOS according toEq. (19). This function is only non-zero when the two conditions, Im[¯ σ zz ( x )] = 0and Re[¯ σ zz ( x )] →
0, are fulfilled simultaneously [72, 73, 76]. These result in the xciton-Plasmon Interactions in Individual Carbon Nanotubes
Energy ]/2 γ ,according to Eq. (17).peak structure of the function Re(1 / ¯ σ zz ) as is seen in Fig. 3. It is also seen fromthe comparison of Fig. 3 (b) with Fig. 3 (a) that the peaks broaden as the CNdiameter decreases. This is consistent with the stronger hybridization effects insmaller-diameter CNs [78].Left panels in Figs. 4(a) and 4(b) show the lowest-energy plasmon DOS res-onances calculated for the (11,0) and (10,0) CNs as given by the function ρ ( x )in Eq. (19). Also shown there are the corresponding fragments of the functionsRe[¯ σ zz ( x )] and Im[¯ σ zz ( x )]. In all graphs the lower dimensionless energy limits areset up to be equal to the lowest bright exciton excitation energy [ E (11) exc = 1 .
21 eV( x = 0 . .
00 eV ( x = 0 . ρ ( x ) are seen to coincide in energy with zeros of Im[¯ σ zz ( x )] { or zeros of Re[ ǫ zz ( x )] } ,clearly indicating the plasmonic nature of the CN surface excitations under con-sideration [72, 79]. They describe the surface plasmon modes associated with thetransversely quantized interband electronic transitions in CNs [72]. As is seen inFig. 4 (and in Fig. 3), the interband plasmon excitations occur in CNs slightly abovethe first bright exciton excitation energy [67], in the frequency domain where theimaginary conductivity (or the real dielectric function) changes its sign. This is aunique feature of the complex dielectric response function, the consequence of thegeneral Kramers-Kr¨onig relation [47].We further take advantage of the sharp peak structure of ρ ( x ) and solve thedispersion equation (16) for x µ analytically using the Lorentzian approximation ρ ( x ) ≈ ρ ( x p )∆ x p ( x − x p ) + ∆ x p . (22)0 I.V.Bondarev, L.M.Woods, and A.Popescu
Figure 4: (a),(b) Surface plasmon DOS and conductivities (left panels), and lowestbright exciton dispersion when coupled to plasmons (right panels) in (11,0) and(10,0) CNs, respectively. The dimensionless energy is defined as [
Energy ]/2 γ , ac-cording to Eq. (17). See text for the dimensionless quasi-momentum.Here, x p and ∆ x p are, respectively, the position and the half-width-at-half-maximumof the plasmon resonance closest to the lowest bright exciton excitation energy inthe same nanotube (as shown in the left panels of Fig. 4). The integral in Eq. (16)then simplifies to the form2 π Z ∞ dx x ¯Γ f ( x ) ρ ( x ) x µ − x ≈ F ( x p )∆ x p x µ − x p Z ∞ dx ( x − x p ) + ∆ x p = F ( x p )∆ x p x µ − x p (cid:20) arctan (cid:18) x p ∆ x p (cid:19) + π (cid:21) with F ( x p ) = 2 x p ¯Γ f ( x p ) ρ ( x p ) /π . This expression is valid for all x µ apart from thoselocated in the narrow interval ( x p − ∆ x p , x p +∆ x p ) in the vicinity of the plasmon res-onance, provided that the resonance is sharp enough. Then, the dispersion equationbecomes the biquadratic equation for x µ with the following two positive solutions(the dispersion curves) of interest to us xciton-Plasmon Interactions in Individual Carbon Nanotubes x , = s ε f + x p ± q ( ε f − x p ) + F p ε f . (23)Here, F p = 4 F ( x p )∆ x p ( π − ∆ x p /x p ) with the arctan-function expanded to linearterms in ∆ x p /x p ≪ f ( x p ) [Eq.(18)] from theequation | d f | = 3 ~ λ / τ radex according to Hanamura’s general theory of the excitonradiative decay in spatially confined systems [80], where τ radex is the exciton intrinsicradiative lifetime, and λ = 2 πc ~ /E with E being the exciton total energy given inour case by Eq. (6). For zigzag-type CNs considered here, the first Brillouin zone ofthe longitudinal quasi-momentum is given by − π ~ / b ≤ ~ k z ≤ π ~ / b [1, 2]. Thetotal energy of the ground-internal-state exciton can then be written as E = E exc +(2 π ~ / b ) t / M ex with − ≤ t ≤ E (11) exc = 1 .
21 eV and 1 .
00 eV, τ radex = 14 . . M ex = 0 . m and 0 . m ( m is the free-electron mass) for the (11,0) CN and (10,0) CN, respectively, asreported in Ref.[51] by directly solving the Bethe-Salpeter equation.Both graphs in the right panels in Fig. 4 are seen to demonstrate a clear an-ticrossing behavior with the (Rabi) energy splitting ∼ . F p factor in Eq. (23). In other words, as long asthe plasmon resonance is sharp enough (which is always the case for interband plas-mons), so that the Lorentzian approximation (22) applies, the effect is determinedby the area under the plasmon peak in the DOS function (19) rather than by thepeak height as one would expect.However, the formation of the strongly coupled exciton-plasmon states is onlypossible if the exciton total energy is in resonance with the energy of a surface plas-mon mode. The exciton energy can be tuned to the nearest plasmon resonance inways used for excitons in semiconductor quantum microcavities — thermally [81,82, 83] (by elevating sample temperature), or/and electrostatically [84, 85, 86, 87](via the quantum confined Stark effect with an external electrostatic field appliedperpendicular to the CN axis). As is seen from Eqs. (6) and (7), the two possibilitiesinfluence the different degrees of freedom of the quasi-1D exciton — the (longitudi-2 I.V.Bondarev, L.M.Woods, and A.Popescu nal) kinetic energy and the excitation energy, respectively. Below we study the (lesstrivial) electrostatic field effect on the exciton excitation energy in CNs.
The optical properties of semiconducting CNs in an external electrostatic field di-rected along the nanotube axis were studied theoretically in Ref. [40]. Strong oscil-lations in the band-to-band absorption and the quadratic Stark shift of the excitonabsorption peaks with the field increase, as well as the strong field dependence of theexciton ionization rate, were predicted for CNs of different diameters and chiralities.Here, we focus on the perpendicular electrostatic field orientation. We study howthe electrostatic field applied perpendicular to the CN axis affects the CN band gap,the exciton binding/excitation energy, and the interband surface plasmon energy,to explore the tunability of the strong exciton-plasmon coupling effect predictedabove. The problem is similar to the well-known quantum confined Stark effect firstobserved for the excitons in semiconductor quantum wells [84, 85]. However, thecylindrical surface symmetry of the excitonic states brings new peculiarities to thequantum confined Stark effect in CNs. In what follows we will generally be inter-ested only in the lowest internal energy (ground) excitonic state, and so the internalstate index f in Eqs. (6) and (7) will be omitted for brevity.Because the nanotube is modelled by a continuous, infinitely thin, anisotropicallyconducting cylinder in our macroscopic QED approach, the actual local symmetryof the excitonic wave function resulted from the graphene Brillouin zone structure isdisregarded in our model (see, e.g., reviews [41, 67]). The local symmetry is implic-itly present in the surface axial conductivity though, which we calculate beforehandas described above. We start with the Schr¨odinger equation for the electron and hole on the CNsurface, located at r e = { R CN , ϕ e , z e } and r h = { R CN , ϕ h , z h } , respectively. Theyinteract with each other through the Coulomb potential V ( r e , r h ) = − e /ǫ | r e − r h | , where ǫ = ǫ zz (0). The external electrostatic field F = { F, , } is directedperpendicular to the CN axis (along the x -axis in Fig. 1). The Schr¨odinger equationis of the form h ˆ H e ( F ) + ˆ H h ( F ) + V ( r e , r h ) i Ψ( r e , r h ) = E Ψ( r e , r h ) (24) In real CNs, the existence of two equivalent energy valleys in the 1st Brillouin zone, the K -and K ′ -valleys with opposite electron helicities about the CN axis, results into dark and brightexcitonic states in the lowest energy spin-singlet manifold [88]. Since the electric interaction doesnot involve spin variables, both K - and K ′ -valleys are affected equally by the electrostatic field inour case, and the detailed structure of the exciton wave function multiplet is not important. Thisis opposite to the non-zero magnetostatic field case where the field affects the K - and K ′ -valleysdifferently either to brighten the dark excitonic states [39], or to create Landau sublevels [67] forlongitudinal and perpendicular orientation, respectively. xciton-Plasmon Interactions in Individual Carbon Nanotubes H e,h ( F ) = − ~ m e,h R CN ∂ ∂ϕ e,h + ∂ ∂z e,h ! ∓ e r e,h · F (25)We further separate out the translational and relative degrees of freedom of theelectron-hole pair by transforming the longitudinal (along the CN axis) motion ofthe pair into its center-of-mass coordinates given by Z = ( m e z e + m h z h ) /M ex and z = z e − z h . The exciton wave function is approximated as followsΨ( r e , r h ) = e ik z Z φ ex ( z ) ψ e ( ϕ e ) ψ h ( ϕ h ) . (26)The complex exponential describes the exciton center-of-mass motion with the lon-gitudinal quasi-momentum k z along the CN axis. The function φ ex ( z ) representsthe longitudinal relative motion of the electron and the hole inside the exciton. Thefunctions ψ e ( ϕ e ) and ψ h ( ϕ h ) are the electron and hole subband wave functions,respectively, which represent their confined motion along the circumference of thecylindrical nanotube surface.Each of the functions is assumed to be normalized to unity. Equations (24) and(25) are then rewritten in view of Eqs. (6)–(8) to yield (cid:20) − ~ m e R CN ∂ ∂ϕ e − eR CN F cos( ϕ e ) (cid:21) ψ e ( ϕ e ) = ε e ψ e ( ϕ e ) , (27) (cid:20) − ~ m h R CN ∂ ∂ϕ h + eR CN F cos( ϕ h ) (cid:21) ψ h ( ϕ h ) = ε h ψ h ( ϕ h ) , (28) (cid:20) − ~ µ ∂ ∂z + V eff ( z ) (cid:21) φ ex ( z ) = E b φ ex ( z ) , (29)where µ = m e m h /M ex is the exciton reduced mass, and V eff is the effective longitu-dinal electron-hole Coulomb interaction potential given by V eff ( z ) = − e ǫ Z π dϕ e Z π dϕ h | ψ e ( ϕ e ) | | ψ h ( ϕ h ) | V ( ϕ e , ϕ h , z ) (30)with V being the original electron-hole Coulomb potential written in the cylindricalcoordinates as V ( ϕ e , ϕ h , z ) = 1 { z + 4 R CN sin [( ϕ e − ϕ h ) / } / . (31)The exciton problem is now reduced to the 1D equation (29), where the excitonbinding energy does depend on the perpendicular electrostatic field through theelectron and hole subband functions ψ e,h given by the solutions of Eqs. (27) and(28) and entering the effective electron-hole Coulomb interaction potential (30).4 I.V.Bondarev, L.M.Woods, and A.Popescu
The set of Eqs. (27)-(31) is analyzed in Appendix D. One of the main resultsobtained in there is that the effective Coulomb potential (30) can be approximatedby an attractive cusp-type cutoff potential of the form V eff ( z ) ≈ − e ǫ [ | z | + z ( j, F )] , (32)where the cutoff parameter z depends on the perpendicular electrostatic fieldstrength and on the electron-hole azimuthal transverse quantization index j = 1 , , ... (excitons are created in interband transitions involving valence and conduction sub-bands with the same quantization index [67] as shown in Fig. 2). Specifically, z ( j, F ) ≈ R CN π − − ∆ j ( F )] π + 2 ln 2 [1 − ∆ j ( F )] (33)with ∆ j ( F ) given to the second order approximation in the electric field by∆ j ( F ) ≈ µM ex e R CN w j ~ F , (34) w j = θ ( j − − j + 11 + 2 j , where θ ( x ) is the unit step function. Approximation (32) is formally valid when z ( j, F ) is much less than the exciton Bohr radius a ∗ B (= ǫ ~ /µe ) which is estimatedto be ∼ R CN for the first ( j = 1 in our notations here) exciton in CNs [19]. As isseen from Eqs. (33) and (34), this is always the case for the first exciton for thosefields where the perturbation theory applies, i. e. when ∆ ( F ) < E (11) b for the first exciton we are interested inhere from the transcendental equationln (cid:20) z (1 , F ) ~ q µ | E (11) b | (cid:21) + 12 s | E (11) b | Ry ∗ = 0 . (35)In doing so, we first find the exciton Rydberg energy, Ry ∗ (= µe / ~ ǫ ), from thisequation at F = 0. We use the diameter- and chirality-dependent electron and holeeffective masses from Ref. [90], and the first bright exciton binding energy of 0.76 eVfor both (11,0) and (10,0) CN as reported in Ref. [21] from ab initio calculations. Weobtain Ry ∗ = 4 .
02 eV and 0 .
57 eV for the (11,0) tube and (10,0) tube, respectively.The difference of about one order of magnitude reflects the fact that these are thesemiconducting CNs of different types — type-I and type-II, respectively, based on xciton-Plasmon Interactions in Individual Carbon Nanotubes
Energy ]/2 γ , accordingto Eq. (17).(2 n + m ) families [90]. The parameters Ry ∗ thus obtained are then used to find | E (11) b | as functions of F by numerically solving Eq. (35) with z (1 , F ) given byEqs. (33) and (34).The calculated (negative) binding energies are shown in Fig. 5(a) by the solidlines. Also shown there by dashed lines are the functions E (11) b ( F ) ≈ E (11) b [1 − ∆ ( F )] (36)with ∆ ( F ) given by Eq. (34). They are seen to be fairly good analytical (quadraticin field) approximations to the numerical solutions of Eq. (35) in the range of not toolarge fields. The exciton binding energy decreases very rapidly in its absolute valueas the field increases. Fields of only ∼ . − . µ m are required to decrease | E (11) b | by a factor of ∼ I.V.Bondarev, L.M.Woods, and A.Popescu part comes from the band gap energy (8), where ε e and ε h are given by the solutionsof Eqs. (27) and (28), respectively. Solving them to the leading (second) orderperturbation theory approximation in the field (Appendix D), one obtains E ( jj ) g ( F ) ≈ E ( jj ) g (cid:20) − m e ∆ j ( F )2 M ex j w j − m h ∆ j ( F )2 M ex j w j (cid:21) , (37)where the electron and hole subband shifts are written separately. This, in view ofEq. (34), yields the first band gap field dependence in the form E (11) g ( F ) ≈ E (11) g (cid:20) −
32 ∆ ( F ) (cid:21) , (38)The bang gap decrease with the field in Eq. (38) is stronger than the opposite effectin the negative exciton binding energy given (to the same order approximation infield) by Eq. (36). Thus, the first exciton excitation energy (7) will be graduallydecreasing as the perpendicular field increases, shifting the exciton absorption peakto the red. This is the basic feature of the quantum confined Stark effect observedpreviously in semiconductor nanomaterials [84, 85, 86, 87]. The field dependencesof the higher interband transitions exciton excitation energies are suppressed by therapidly (quadratically) increasing azimuthal quantization numbers in the denomi-nators of Eqs. (34) and (37).Lastly, the perpendicular field dependence of the interband plasmon resonancescan be obtained from the frequency dependence of the axial surface conductivitydue to excitons (see Ref. [67] and refs. therein). One has σ exzz ( ω ) ∼ X j =1 , ,... − i ~ ωf j [ E ( jj ) exc ] − ( ~ ω ) − i ~ ω/τ , (39)where f j and τ are the exciton oscillator strength and relaxation time, respectively.The plasmon frequencies are those at which the function Re[1 /σ exzz ( ω )] has max-ima. Testing it for maximum in the domain E (11) exc < ~ ω < E (22) exc , one finds the firstinterband plasmon resonance energy to be (in the limit τ → ∞ ) E (11) p = s [ E (11) exc ] + [ E (22) exc ] . (40)Using the field dependent E (11) exc given by Eqs. (7), (36) and (38), and neglecting thefield dependence of E (22) exc , one obtains to the second order approximation in the field E (11) p ( F ) ≈ E (11) p " − E (11) g / E (11) exc E (22) exc /E (11) exc ∆ ( F ) . (41)Figure 6 shows the results of our calculations of the field dependences for thefirst bright exciton parameters in the (11,0) and (10,0) CNs. The energy is measured xciton-Plasmon Interactions in Individual Carbon Nanotubes Energy ]/2 γ , accordingto Eq. (17). The energy is measured from the top of the first unperturbed holesubband.from the top of the first unperturbed hole subband (as shown in Fig. 2, right panel).The binding energy field dependence was calculated numerically from Eq. (35) asdescribed above [shown in Fig. 5 (a)]. The band gap field dependence and the plas-mon energy field dependence were calculated from Eqs. (37) and (41), respectively.The zero-field excitation energies and zero-field binding energies were taken to bethose reported in Ref. [51] and in Ref. [21], respectively, and we used the diameter-and chirality-dependent electron and hole effective masses from Ref. [90]. As isseen in Fig. 6 (a) and (b), the exciton excitation energy and the interband plas-mon energy experience red shift in both nanotubes as the field increases. However,the excitation energy red shift is very small (barely seen in the figures) due to thenegative field dependent contribution from the exciton binding energy. So, E (11) exc ( F )and E (11) p ( F ) approach each other as the field increases, thereby bringing the totalexciton energy (6) in resonance with the surface plasmon mode due to the non-zerolongitudinal kinetic energy term at finite temperature. Thus, the electrostatic fieldapplied perpendicular to the CN axis (the quantum confined Stark effect) may beused to tune the exciton energy to the nearest interband plasmon resonance, to putthe exciton-surface plasmon interaction in small-diameter semiconducting CNs tothe strong-coupling regime. We are based on the zero-exciton-temperature approximation in here [91], which is well justifiedbecause of the exciton excitation energies much larger than k B T in CNs. The exciton Hamiltonian(4) does not require the thermal averaging over the exciton degrees of freedom then, yielding thetemperature independent total exciton energy (6). One has to keep in mind, however, that theexciton excitation energy can be affected by the enviromental effect not under consideration in here(see Ref. [92]). I.V.Bondarev, L.M.Woods, and A.Popescu
Here, we analyze the longitudinal exciton absorption line shape as its energy is tunedto the nearest interband surface plasmon resonance. Only longitudinal excitons (ex-cited by light polarized along the CN axis) couple to the surface plasmon modesas discussed at the very beginning of this section (see Ref. [70] for the perpendic-ular light exciton absorption in CNs). We start with the linear (weak) excitationregime where only single-exciton states are excited, and follow the optical absorp-tion/emission lineshape theory developed recently for atomically doped CNs [10].(Obviously, the absorption line shape coincides with the emission line shape if themonochromatic incident light beam is used in the absorption experiment.) Then, thenon-linear (strong) excitation regime is considered with the photonduced excitationof biexciton states.When the f -internal state exciton is excited and the nanotube’s surface EMfield subsystem is in vacuum state, the time-dependent wave function of the wholesystem ”exciton+field” is of the form | ψ ( t ) i = X k ,f C f ( k , t ) e − i ˜ E f ( k ) t/ ~ |{ f ( k ) }i ex |{ }i (42)+ X k Z ∞ dω C ( k , ω, t ) e − iωt |{ }i ex |{ k , ω ) }i . Here, |{ f ( k ) }i ex is the excited single-quantum Fock state with one exciton and |{ k , ω ) }i is that with one surface photon. The vacuum states are |{ }i ex and |{ }i for the exciton subsystem and field subsystem, respectively. The coefficients C f ( k , t )and C ( k , ω, t ) stand for the population probability amplitudes of the respectivestates of the whole system. The exciton energy is of the form ˜ E f ( k ) = E f ( k ) − i ~ /τ with E f ( k ) given by Eq. (6) and τ being the phenomenological exciton relaxationtime constant [assumed to be such that ~ /τ ≪ E f ( k )] to account for other possibleexciton relaxation processes. From the literature we have τ ph ∼ −
100 fs for theexciton-phonon scattering [40], τ d ∼
50 ps for the exciton scattering by defects [25,28], and τ rad ∼
10 ps −
10 ns for the radiative decay of excitons [51]. Thus, thescattering by phonons is the most likely exciton relaxation mechanism.Using Eqs.(5) and (9), we transform the total Hamiltonian (1)–(10) to the k -representation (see Appendix A), and apply it to the wave function in Eq. (42).We obtain the following set of the two simultaneous differential equations for thecoefficients C f ( k , t ) and C ( k , ω, t ) from the time dependent Schr¨odinger equation ˙C f ( k , t ) e − i ˜ E f ( k ) t/ ~ = − i ~ X k ′ Z ∞ dω g (+) f ( k , k ′ , ω ) C ( k ′ , ω, t ) e − iωt , (43) ˙C ( k ′ , ω, t ) e − iωt δ kk ′ = − i ~ X f [g (+) f ( k , k ′ , ω )] ∗ C f ( k , t ) e − i ˜ E f ( k ) t/ ~ . See the footnote on page 12 above. xciton-Plasmon Interactions in Individual Carbon Nanotubes δ -symbol on the left in Eq. (44) ensures that the momentum conservation isfulfilled in the exciton-photon transitions, so that the annihilating exciton creates thesurface photon with the same momentum and vice versa. In terms of the probabilityamplitudes above, the exciton emission intensity distribution is given by the finalstate probability at very long times corresponding to the complete decay of allinitially excited excitons, I ( ω ) = | C ( k , ω, t → ∞ ) | = 1 ~ X f | g (+) f ( k , k , ω ) | × (cid:12)(cid:12)(cid:12)(cid:12)Z ∞ dt ′ C f ( k , t ′ ) e − i [ ˜ E f ( k ) − ~ ω ] t ′ / ~ (cid:12)(cid:12)(cid:12)(cid:12) . (44)Here, the second equation is obtained by the formal integration of Eq. (44) overtime under the initial condition C ( k , ω,
0) = 0. The emission intensity distributionis thus related to the exciton population probability amplitude C f ( k , t ) to be foundfrom Eq. (43).The set of simultaneous equations (43) and (44) [and Eq. (44), respectively]contains no approximations except the (commonly used) neglect of many-particleexcitations in the wave function (42). We now apply these equations to the exciton-surface-plasmon system in small-diameter semiconducting CNs. The interactionmatrix element in Eqs. (43) and (44) is then given by the k -transform of Eq. (13),and has the following property (Appendix C)12 γ ~ | g (+) f ( k , k , ω ) | = 12 π ¯Γ f ( x ) ρ ( x ) (45)with ¯Γ f ( x ) and ρ ( x ) given by Eqs. (18) and (19), respectively. We further substitutethe result of the formal integration of Eq. (44) [with C ( k , ω,
0) = 0] into Eq. (43), useEq. (45) with ρ ( x ) approximated by the Lorentzian (22), calculate the integral overfrequency analytically, and differentiate the result over time to obtain the followingsecond order ordinary differential equation for the exciton probability amplitude[dimensionless variables, Eq. (17)] ¨C f ( β ) + [∆ x p − ∆ ε f + i ( x p − ε f )] ˙C f ( β ) + ( X f / C f ( β ) = 0 , where X f = [2∆ x p ¯Γ f ( x p )] / with ¯Γ f ( x p ) = ¯Γ f ( x p ) ρ ( x p ), ∆ ε f = ~ / γ τ , β = 2 γ t/ ~ is the dimensionless time, and the k -dependence is omitted for brevity. When thetotal exciton energy is close to a plasmon resonance, ε f ≈ x p , the solution of thisequation is easily found to be C f ( β ) ≈ δx q δx − X f e − (cid:16) δx − q δx − X f (cid:17) β/ (46)+ 12 − δx q δx − X f e − (cid:16) δx + q δx − X f (cid:17) β/ , I.V.Bondarev, L.M.Woods, and A.Popescu where δx = ∆ x p − ∆ ε f > X f = [2∆ x p ¯Γ f ( ε f )] / . This solution is valid when ε f ≈ x p regardless of the strength of the exciton-surface-plasmon coupling. It yieldsthe exponential decay of the excitons into plasmons, | C f ( β ) | ≈ exp[ − ¯Γ f ( ε f ) β ],in the weak coupling regime where the coupling parameter ( X f /δx ) ≪
1. If,on the other hand, ( X f /δx ) ≫
1, then the strong coupling regime occurs, andthe decay of the excitons into plasmons proceeds via damped Rabi oscillations, | C f ( β ) | ≈ exp( − δxβ ) cos ( X f β/ δx in the coupling parameter. Inother words, the phonon scattering broadens the (longitudinal) exciton momentumdistribution [93], thus effectively increasing the fraction of the excitons with ε f ≈ x p .In view of Eqs. (45) and (46), the exciton emission intensity (44) in the vicinityof the plasmon resonance takes the following (dimensionless) form¯ I ( x ) ≈ ¯ I ( ε f ) X f (cid:12)(cid:12)(cid:12)(cid:12)Z ∞ dβ C f ( β ) e i ( x − ε f + i ∆ ε f ) β (cid:12)(cid:12)(cid:12)(cid:12) , (47)where ¯ I ( x ) = 2 γ I ( ω ) / ~ and ¯ I = ¯Γ f ( ε f ) / π . After some algebra, this results in¯ I ( x ) ≈ ¯ I ( ε f ) [( x − ε f ) + ∆ x p ][( x − ε f ) − X f / + ( x − ε f ) (∆ x p + ∆ ε f ) , (48)where ∆ x p > ∆ ε f . The summation sign over the exciton internal states is omittedsince only one internal state contributes to the emission intensity in the vicinity ofthe sharp plasmon resonance.The line shape in Eq. (48) is mainly determined by the coupling parameter( X f / ∆ x p ) . It is clearly seen to be of a symmetric two-peak structure in the strongcoupling regime where ( X f / ∆ x p ) ≫
1. Testing it for extremum, we obtain thepeak frequencies to be x , = ε f ± X f vuuts (cid:18) ∆ x p X f (cid:19) − (cid:18) ∆ x p X f (cid:19) [terms ∼ (∆ x p ) (∆ ε f ) /X f are neglected], with the Rabi splitting x − x ≈ X f . Inthe weak coupling regime where ( X f / ∆ x p ) ≪
1, the frequencies x and x becomecomplex, indicating that there are no longer peaks at these frequencies. As thistakes place, Eq. (48) is approximated with the weak coupling condition, the factthat x ∼ ε f , and X f = 2∆ x p ¯Γ f ( ε f ), to yield the Lorentzian˜ I ( x ) ≈ ¯ I ( ε f ) / [1 + (∆ ε f / ∆ x p ) ]( x − ε f ) + h ¯Γ f ( ε f ) / q ε f / ∆ x p ) i xciton-Plasmon Interactions in Individual Carbon Nanotubes x = ε f , whose half-width-at-half-maximum is slightly narrower, however,than ¯Γ f ( ε f ) / χ (3) ( x ) ≈ ˜ I ( x ) " x − ε f + i (Γ f / ε f ) − x − ( ε f − | ε XXf | ) + i (Γ f / ε f ) , (49)where ε XXf is the (negative) dimensionless binding energy of the biexciton composedof two f -internal state excitons, and χ is the frequency-independent constant. Thefirst and second terms in the brackets represent bleaching due to the depopula-tion of the ground state and photoinduced absorption due to exciton-to-biexcitontransitions, respectively.The binding energy of the biexciton in a small-diameter ( ∼ H ( z , z , ∆ Z ) = − (cid:18) ∂ ∂z + ∂ ∂z (cid:19) (50) − (cid:20) | z | + z + 1 | z + ∆ Z | + z + 1 | z | + z + 1 | z − ∆ Z | + z (cid:21) − | ( z + z ) / Z | + z − | ( z + z ) / − ∆ Z | + z + 1 | ( z − z ) / Z | + z + 1 | ( z − z ) / − ∆ Z | + z . Here, z , = z e , − z h , is the electron-hole relative motion coordinates of the two1D excitons, z is the cut-off parameter of the effective longitudinal electron-hole2 I.V.Bondarev, L.M.Woods, and A.Popescu
Figure 7: (a) Schematic (arbitrary units) of the exchange coupling of two ground-state 1D excitons to form a biexcitonic state. (b) The coupling occurs in the con-figuration space of the two independent longitudinal relative electron-hole motioncoordinates, z and z , of each of the excitons, due to the tunneling of the systemthrough the potential barriers formed by the two single-exciton cusp-type poten-tials [bottom, also in (a)], between equivalent states represented by the isolatedtwo-exciton wave functions shown on the top.Coulomb potential (32), and ∆ Z = Z − Z is the center-of-mass-to-center-of-massinter-exciton separation distance. Equal electron and hole effective masses m e,h areassumed [90] and ”atomic units” are used [95, 96, 97], whereby distance and energyare measured in units of the exciton Bohr radius a ∗ B and in units of the doubledground-state-exciton binding energy 2 E b = − Ry ∗ /ν , respectively. The first twolines in Eq. (50) represent two isolated non-interacting 1D excitons [see Fig. 7 (a)].The last two lines are their exchange Coulomb interactions — electron-hole andelectron-electron + hole-hole, respectively.The Hamiltonian (50) is effectively two dimensional in the configuration space ofthe two independent relative motion coordinates, z and z . Figure 7 (b), bottom,shows schematically the potential energy surface of the two closely spaced non- xciton-Plasmon Interactions in Individual Carbon Nanotubes z , z ) space. The surface hasfour symmetrical minima [representing equivalent isolated two-exciton states shownin Fig. 7 (b), top], separated by the potential barriers responsible for the tunnelexchange coupling between the two-exciton states in the configuration space. Thecoordinate transformation x = ( z − z − ∆ Z ) / √ y = ( z + z ) / √ U g,u (∆ Z ) − E b = ∓ J (∆ Z ) , (51)where U g,u are the ground and excited state energies, respectively, of the two cou-pled excitons (the biexciton) as functions of their center-of-mass-to-center-of-massseparation, and J (∆ Z ) = 23! Z ∆ Z/ √ − ∆ Z/ √ dy (cid:20) ψ ( x, y ) ∂ψ ( x, y ) ∂x (cid:21) x =0 (52)is the tunnel exchange coupling integral, where ψ ( x, y ) is the solution to the Schr¨o-dinger equation with the Hamiltonian (50) transformed to the ( x, y ) coordinates.The factor 2 /
3! comes from the fact that there are two equivalent tunnel channelsin the problem, mixing three equivalent indistinguishable two-exciton states in theconfiguration space [one state is given by the two minima on the y -axis, and twomore are represented by each of the minima on the x -axis — compare Figs. 7 (a)and (b)].The function ψ ( x, y ) in Eq. (52) is sought in the form ψ ( x, y ) = ψ ( x, y ) exp[ − S ( x, y )] , (53)where ψ = ν − exp[ − ( | z ( x, y, ∆ Z ) | + | z ( x, y, ∆ Z ) | ) /ν ] is the product of twosingle-exciton wave functions representing the isolated two-exciton state centeredat the minimum z = z = 0 (or x = − ∆ Z/ √ y = 0) of the configuration spacepotential [Fig. 7 (b)], and S ( x, y ) is a slowly varying function to take into accountthe deviation of ψ from ψ due to the tunnel exchange coupling to another equiv-alent isolated two-exciton state centered at z = ∆ Z , z = − ∆ Z (or x = ∆ Z/ √ y = 0). Substituting Eq. (53) into the Schr¨odinger equation with the Hamiltonian(50) pre-transformed to the ( x, y ) coordinates, one obtains in the region of interest ∂S∂x = ν (cid:18) x + 3∆ Z/ √ − x − ∆ Z/ √ y − √ Z − y + √ Z (cid:19) , This is an approximate solution to the Shr¨odinger equation with the Hamiltonin given by thefirst two lines in Eq. (50), where the cut-off parameter z is neglected [89]. This approximationgreatly simplifies problem solving here, while still remaining adequate as only the long-distance tailof ψ is important for the tunnel exchange coupling. I.V.Bondarev, L.M.Woods, and A.Popescu up to (negligible) terms of the order of the inter-exciton van der Waals energy andup to second derivatives of S . This equation is to be solved with the boundary condi-tion S ( − ∆ Z/ √ , y ) = 0 originating from the natural requirement ψ ( − ∆ Z/ √ , y ) = ψ ( − ∆ Z/ √ , y ), to result in S ( x, y ) = ν ln (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x +3∆ Z/ √ x − ∆ Z/ √ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + 2 √ Z ( x +∆ Z/ √ y − Z ! . (54)After plugging Eqs. (54) and (53) into Eq. (52), and retaining only the leadingterm of the integral series expansion in powers of ν subject to ∆ Z >
1, Eq. (51)becomes U g,u (∆ Z ) ≈ E b (cid:20) ± ν (cid:16) e (cid:17) ν ∆ Z e − Z/ν (cid:21) . (55)The ground state energy U g of two coupled 1D excitons is now seen to go through thenegative minimum (biexcitonic state) as the inter-exciton center-of-mass-to-center-of-mass separation ∆ Z increases (Fig. 8). The minimum occurs at ∆ Z = ν / E XX ≈ (2 E b / ν )( e/ ν − , or, expressing ν in terms of E b and measuring the energy in units of Ry ∗ , E XX [in Ry ∗ ] ≈ − | E b | / (cid:16) e (cid:17) / √ | E b | − . (56)The energy E XX can be affected by the quantum confined Stark effect since | E b | decreases quadratically with the perpendicular electrostatic field applied as shownin Fig. 5 (a). Since e/ ∼
1, the field dependence in Eq. (56) mainly comes fromthe pre-exponential factor. So, | E XX | will be decreasing quadratically with the fieldas well, for not too strong perpendicular fields. At the same time, the equilibriuminter-exciton separation in the biexciton, ∆ Z = ν / ∼ | E b | − / , will be slowlyincreasing with the field consistently with the lowering of | E XX | . In the zero field,one has roughly E XX ∼ | E b | / ∼ R − . CN for the biexciton binding energy versus theCN radius R CN ( | E b | ∼ R − . CN as reported in Ref. [19] from variational calculations),pretty consistent with the R − CN dependence obtained numerically [32]. Interestingly,as R CN goes down, | E XX | goes up faster than | E b | does. This is partly due to thefact that ∆ Z slowly decreases as R CN goes down, — a theoretical argument insupport of experimental evidence for increased exciton-exciton annihilation in smalldiameter CNs [98, 99, 100].Figure 8 shows the ground state energy U g (∆ Z ) of the coupled pair of the firstbright excitons, calculated from Eq. (55) for the semiconducting (11,0) CN exposedto different perpendicular electrostatic fields. The inset shows the field dependencesof E XX [as given by Eq. (56)] and of ∆ Z . All the curves are calculated using thefield dependence of E b obtained as described in the previous subsection (Figs. 5and 6). They exhibit typical behaviors discussed above.Figure 9 compares the linear response lineshape (48) with the imaginary partof Eq. (49) representing the non-linear optical response function under resonant xciton-Plasmon Interactions in Individual Carbon Nanotubes U g of the coupled pair of the first brightexcitons in the (11,0) CN as a function of the center-of-mass-to-center-of-mass inter-exciton distance ∆ Z and perpendicular electrostatic field applied. Inset showsthe biexciton binding energy E XX and inter-exciton separation ∆ Z ( y - and x -coordinates, respectively, of the minima in the main figure) as functions of the field.pumping, both calculated for the 1st bright exciton in the (11,0) CN as its energyis tuned (by means of the quantum confined Stark effect) to the nearest plasmonresonance (vertical dashed line in the figure). The biexciton binding energy inEq. (49) was taken to be E XX ≈
52 meV as given by Eq. (56) in the zero field.[Weak field dependence of E XX (inset in Fig. 8) plays no essential role here as | E XX | ≪ | E b | ≈ .
76 eV regardless of the field strength.] The phonon relaxationtime τ ph = 30 fs was used as reported in Ref. [29], since this is the shortest oneout of possible exciton relaxation processes, including exciton-exciton annihilation( τ ee ∼ ∼ . I.V.Bondarev, L.M.Woods, and A.Popescu
Figure 9: [(a), (b), and (c)] Linear (top) and non-linear (bottom) response functionsas given by Eq. (48) and by the imaginary part of Eq. (49), respectively, for thefirst bright exciton in the (11,0) CN as the exciton energy is tuned to the nearestinterband plasmon resonance (vertical dashed line). Vertical lines marked as Xand XX show the exciton energy and biexciton binding energy, respectively. Thedimensionless energy is defined as [
Energy ]/2 γ , according to Eq. (17).the experimental observation of the non-linear absorption line splitting predictedhere would help identify the presence and study the properties of biexcitonic states(including biexcitons formed by excitons of different subbands [33]) in individualsingle-walled CNs, due to the fact that when tuned close to a plasmon resonancethe exciton relaxes into plasmons at a rate much greater than τ − ph ( ≫ τ − ee ), totallyruling out the role of the competing exciton-exciton annihilation process. Here, we consider the Casimir interaction between two concentric cylindrical gra-phene sheets comprising a double-wall CN, using the macroscopic QED approachemployed above to study the exciton-surface-plasmon interactions in single wall xciton-Plasmon Interactions in Individual Carbon Nanotubes The method is fully adequate in this case as the Casimir force isknown to originate from quantum EM field fluctuations. The fundamental natureof this force has been studied for many years since the prediction of the attractionforce between two neutral metallic plates in vacuum (see, Refs. [49, 56]). After thefirst report of observation of this spectacular effect [101], new measurements withimproved accuracy have been done involving different geometries [102, 103, 104].The Casimir force has also been considered theoretically with methods primarilybased on the zero-point summation approach and Lifshitz theory [105, 106].The Casimir effect has acquired a much broader impact recently due to its im-portance for nanostructured materials, including graphite and graphitic nanostruc-tures [56] which can exist in different geometries and with various unique electronicproperties. Moreover, the efficient development and operation of modern micro- andnano-electromechanical devices are limited due to effects such as stiction, friction,and adhesion, originating from or closely related to the Casimir effect [57].The mechanisms governing the CN interactions still remain elusive. It is knownthat the system geometry [107, 108] and dielectric response [45, 62] have a profoundeffect on the interaction, in general, but their specific functionalities have not beenqualitatively and quantitatively understood. Since CNs of virtually the same radialsize can possess different electronic properties, investigating their Casimir interac-tions presents a unique opportunity to obtain insight into specific dielectric responsefeatures affecting the Casimir force between metallic and semiconducting cylindri-cal surfaces. This can also unveil the role of collective surface excitations in theenergetic stability of multi-wall CNs of various chiral combinations.Since Lifshitz theory cannot be easily applied to geometries other than parallelplates, researchers have used the Proximity Force Approximation (PFA) to calculatethe Casimir interaction between CNs [107, 109] (see also Ref. [56] for the latestreview). The method is based on approximating the curved surfaces at very closedistances by a series of parallel plates and summing their energies using the Lifshitzresult. Thus, the PFA is inherently an additive approach, applicable to objects atvery close separations (still to be greater than objects inter-atomic distances) underthe assumption that the CN dielectric response is the same as the one for the plates.This last assumption is very questionable as the quasi-1D character of the electronicmotion in CNTs is known to be of principal importance for the correct descriptionof their electronic and optical properties [1, 59, 64].We model the double-wall CN by two infinitely long, infinitely thin, continu-ous concentric cylinders with radii R , , immersed in vacuum. Each cylinder ischaracterized by the complex dynamic axial dielectric function ǫ zz ( R , , ω ) with the z -direction along the CN axis as shown in Fig. 10. The azimuthal and radial com-ponents of the complete CN dielectric tensor are neglected as they are known to bemuch less than ǫ zz for most CNs [64]. The QED quantization scheme in the presence In this Section only, the International System of units is used to make the comparison easier ofour theory with other authors’ results. I.V.Bondarev, L.M.Woods, and A.Popescu
Figure 10: Schematic of the two concentric CNs in vacuum. The CN radii are R and R . The regions between the CN surfaces are denoted as (1), (2), and (3).of CNs [49, 62] generates the second-quantized Hamiltonianˆ H = X i =1 , Z ∞ dω ~ ω Z d R i ˆ f † ( R i , ω ) ˆ f ( R i , ω )of the vacuum-type medium assisted EM field, with the bosonic operators ˆ f † andˆ f creating and annihilating, respectively, surface EM excitations of frequency ω atpoints R , = { R , , ϕ , , z , } of the double-wall CN system. The Fourier-domainelectric field operator at an arbitrary point r = ( r, ϕ, z ) is given byˆ E ( r , ω ) = iωµ X i =1 , Z d R i G ( r , R i , ω ) · ˆ J ( R i , ω ) , where G ( r , R i , ω ) is the dyadic EM field Green’s function (GF), andˆ J ( R i , ω ) = ωµ c s ~ Im ǫ zz ( R i , ω ) πε ˆ f ( R i , ω ) e z is the surface current density operator selected in such a way as to ensure thecorrect QED equal-time commutation relations for the electric and magnetic fieldoperators [49, 62]. Here, e z is the unit vector along the CN axis, ε , µ , and c are thedielectric constant, magnetic permeability, and vacuum speed of light, respectively.The dyadic GF satisfies the wave equation ∇ × ∇ × G ( r , r ′ , ω ) − ω c G ( r , r ′ , ω ) = δ ( r − r ′ ) I (57)with I being the unit tensor. The GF can further be decomposed as follows G ( s,f ) = G (0) δ sf + G ( s,f ) scatt xciton-Plasmon Interactions in Individual Carbon Nanotubes G (0) and G ( s,f ) scatt represent the contributions of the direct and scattered waves,respectively [110, 111], with a point-like field source located in region s and thefield registered in region f (see Fig. 10). The boundary conditions for Eq. (57)are obtained from those for the electric and magnetic field components on the CNsurfaces [45, 59], which result in e r × h G ( r , r ′ , ω ) (cid:12)(cid:12) R +1 , − G ( r , r ′ , ω ) (cid:12)(cid:12) R − , i = 0 , (58) e r ×∇× h G ( r , r ′ , ω ) (cid:12)(cid:12) R +1 , − G ( r , r ′ , ω ) (cid:12)(cid:12) R − , i = iωµ σ (1 , ( r , ω ) · G ( r , r ′ , ω ) (cid:12)(cid:12) R , (59)where e r is the unit vector along the radial direction. The discontinuity in Eq. (59)results from the full account of the finite absorption and dispersion for both CNs bymeans of their conductivity tensors σ (1 , approximated by their largest components σ (1 , zz ( R , , ω ) = − iωε Sρ T [ ǫ (1 , zz ( R , , ω ) −
1] (60)[compare with Eq. (21)].Following the procedure described in Refs. [110, 111], we expand G (0) and G ( s,f ) scatt into series of even and odd vector cylindrical functions with unknown coefficients tobe found from Eqs. (58) and (59). This splits the EM modes in the system into TEand TM polarizations, with Eqs. (58) and (59) yielding a set of 32 equations (16for each polarization) with 32 unknown coefficients. The unknown coefficients arefound determining the dyadic GF in each region. Using the expressions for the electric and magnetic fields, the electromagneticstress tensor is constructed [49, 112] T ( r , r ′ ) = T ( r , r ′ ) + T ( r , r ′ ) − I T r (cid:2) T ( r , r ′ ) + T ( r , r ′ ) (cid:3) (61) T ( r , r ′ ) = ~ π Z ∞ dω ω c Im (cid:2) G ( r , r ′ , ω ) (cid:3) (62) T ( r , r ′ ) = − ~ π Z ∞ dω Im (cid:20) ∇ × G ( r , r ′ , ω ) × ← ∇ ′ (cid:21) (63)We are interested in the radial component T rr which describes the radiation pressureof the virtual EM field on each CN surface in the system. The Casimir force perunit area exerted on the surfaces is then given by [49] F i = lim r → R i (cid:26) lim r ′ → r h T ( i ) rr ( r , r ′ ) − T ( i +1) rr ( r , r ′ ) i(cid:27) , i = 1 , Due to the lengthy and tedious algebra, this derivation will be presented in a separate longercommunication. I.V.Bondarev, L.M.Woods, and A.Popescu
The forces F , calculated from Eq. (64) are of equal magnitude and opposite di-rection, indicating the attraction between the cylindrical surfaces. The Casimir forcethus obtained accounts simultaneously for the geometrical curvature effects (throughthe GF tensor) and the finite absorption and dissipation of each CN [through theirdielectric response functions (60)]. The dielectric response functions of particu-lar CNs were calculated from the CN realistic band structure as described above,in Section 3. We decomposed them into the Drude contribution and the contri-bution originating from (transversely quantized) interband electronic transitions, ǫ zz = ǫ Dzz + ǫ interzz , in order to be able to see how much each individual contributionaffects the inter-tube Casimir attraction.It is interesting to consider the case of infinitely conducting parallel plates firstusing Eq. (64). This is obtaned by taking the limits σ (1 , zz → ∞ and R , → ∞ while keeping constant the inter-tube distance, R − R = d . We find F = − ~ c π R Z ∞ dx x ∞ X n =0 (2 − δ n ) I n ( x ) K n ( x ) − I n ( x ) K n ( x ) × (cid:8)(cid:2) x K ′ n ( x ) + (cid:0) n + x (cid:1) K n ( x ) (cid:3) (cid:2) I n ( x ) K n ( x ) /K n ( x ) − I n ( x ) I n ( x ) (cid:3) − (cid:2) x I ′ n ( x ) + (cid:0) n + x (cid:1) I n ( x ) (cid:3) K n ( x ) K n ( x ) − (cid:2) x I ′ n ( x ) K ′ n ( x ) + (cid:0) n + x (cid:1) I n ( x ) K n ( x ) (cid:3) I n ( x ) K n ( x ) (cid:9) where x , = xR , , I n ( x ) and K n ( x ) are the modified Bessel functions of the firstand second kind, respectively. The above expression is obtained by making the tran-sition to imaginary frequencies ω → iω , and using the Euclidean rotation techniqueas described in Refs. [112, 113]. This can further be evaluated by summing up theseries over n using the large-order Bessel function expansions [114]. This resultsin F ∼ ( − / ~ cπ / d ) which is 1 / ǫ zz = 0 only and the remainingdielectric tensor components being zero in our model.Figure 11 presents results from the numerical calculations of F as a function ofthe inter-tube surface-to-surface distance for various pairs of CNs with their realisticchirality dependent dielectric responses taken into account. We have chosen theinner CN to be the achiral (12 ,
12) metallic nanotube, and to change the outertubes. As R is varied, one can envision double wall CNs consisting of metal/metalor metal/semiconductor combinations of different chiralities but of similar radialdimensions.Figure 11 shows that F decreases in strength as the surface-to-surface distanceincreases. This dependence is monotonic for the zigzag ( m,
0) and armchair ( n, n )outer tubes, but it happens at different rates. The attraction is stronger if the outerCN is an armchair ( n, n ) one as compared to the attraction for the outer ( m, d in a rather irregular fashion. It is seen that for relatively small d , xciton-Plasmon Interactions in Individual Carbon Nanotubes d , for different pairs of CNs. The inset shows force found with the full dielectricfunction and the Drude contribution only for the same CN pairs indicated in thefigure.the interaction force can be quite different. For example, the attraction between(27 , ,
12) and (21 , ,
12) differ by ∼
20 % in favor of the second pair,even though the radial difference is only 0 . ǫ Dzz ( ω ) contribution alone in eachdielectric function. The inset in Fig. 11 indicates that the attraction is strongerwhen the interband transitions are neglected. The decay of F as a function of d is monotonic. Including the ǫ interzz ( ω ) term not only reduces the force, but alsointroduces non-linearities due to the chirality dependent optical excitations. At largesurface-to-surface separations, the discrepancies between the force calculated withthe full dielectric response, and those obtained with the Drude term only becomeless significant. We find that for d ∼
15 ˚A, this difference is less than 10 %.To investigate further the important functionalities originating from the cylindri-cal geometry and the CN dielectric response properties, F is calculated for differentachiral inner/outer nanotube pairs. Studying zigzag and armchair CNs allows track-ing generalities from ǫ ( ω ) in a more controlled manner. The results are presented in2 I.V.Bondarev, L.M.Woods, and A.Popescu
Figure 12: The Casimir force per unit area as a function of the inter-tube separation d for selected CN pairs. The insets show the EELS spectra for several CNs.Fig. 12. We have chosen representatives of three inner CN types – metallic (12 , , ,
0) tubules. They are of similar radii,8 .
14 ˚A, 8 .
22 ˚A, and 7 .
83 ˚A, respectively. We see that depending on the outer nan-otube types, the F versus d curves are positioned in three groups. The weakestinteraction is found when there are two zigzag concentric CNs (top two curves).The fact that some of these are semi-metallic and others are semiconducting doesnot seem to influence the magnitude and monotonic decrease of the Casimir force.The attraction is stronger when there is a combination of an armchair and azigzag CNT as compared to the previous case. The curves for ( m, , n, n )@(21 , n, n )@(20 ,
0) are practically overlapping, meaning that the spe-cific location of the zigzag and armchair tubes (inner or outer) is of no significanceto the force. The small deviations can be attributed to the small differences in theinner CN radii. Finally, we see that the strongest interaction occurs between twoarmchair CNs (red curve). These functionalities are not unique just for the con-sidered CNs. We have performed the same calculations for many different achiraltubes, and we always find that the strongest interaction occurs between two arm-chair CNs and the weakest — between two zigzag CNs (provided that their radialdimensions are similar). xciton-Plasmon Interactions in Individual Carbon Nanotubes − Im[1 /ǫ ( ω )], and compare them forvarious inner and outer CNs combinations — Fig. 12 (inset).Considering F as a function of d and the specific form of the EELS spectra, itbecomes clear from the inset in Fig. 12 that the low frequency plasmon excitations,given by peaks in − Im[1 /ǫ ( ω )], are key to the strength of the Casimir force. Wealways find that the strongest force is between the tubules with well pronounced over-lapping low frequency plasmon excitations. This is consistent with the conclusionof Ref. [117] for generic 1D-plasmonic structures. However, in our case we deal withthe interband plasmons originating from the space quantization of the transverseelectronic motion, and, therefore, having quite a different frequency-momentumdispersion law (constant) as compared to that normally assumed (linear) for plas-mons [50]. A weaker force is obtained if only one of the CNs supports strong lowfrequency interband plasmon modes. The weakest interaction happens when neitherCN has strong low frequency plasmons. For the cases shown in Fig 12, one findswell pronounced overlapping plasmon transitions in the (12 ,
12) CN at ω = 2 .
18 eVand ω = 3 .
27 eV, and at ω = 1 .
63 eV and ω = 2 .
45 eV in the (17 ,
17) CN. Atthe same time, no such well defined strong low frequency excitations in the (21 , ,
0) CNs are found. Figure 12 shows that the attraction in (17 , , , , , ,
0) there is only one such low frequency excitation coming from thearmchair tube and, consequently, the Casimir force has an intermediate value ascompared to the above discussed two cases.We performed calculations of the Casimir force between many CN pairs andmade comparisons between the relevant regions of the EELS spectra. It is foundthat, in general, armchair tubes always have strong, well pronounced interbandplasmon excitations in the low frequency range. Zigzag and most chiral CNs havelow frequency interband plasmons [37], too, but they are not as near as well pro-nounced as those in armchair tubes; their stronger plasmon modes are found athigher frequencies.These studies are indicative of the significance of the collective response prop-erties of the involved CNs. Specifically, the collective low energy plasmon excita-tions and their relative location can result in nanotube attraction with differentstrengths. We further investigate this point by considering a double wall CN withradii R = 11 .
63 ˚Aand R = 8 .
22 ˚A. The dielectric function of each tube is taken4
I.V.Bondarev, L.M.Woods, and A.Popescu
Figure 13: The Casimir force per unit area as a function of the outer CN plasmonfrequency, while the inner CN plasmon peak ω is constant. Results are shown forfour values of ω . The dielectric functions are modeled by a generic Lorentzian asgiven by Eq. (65).to be of the generic Lorentzian form ǫ zz ( R , , ω ) = 1 − Ω ω − ω , + iω Γ (65)with the typical for nanotubes values Ω = 2 . .
03 eV [45]. Then, theEELS spectrum has only one plasmon resonance at ω , for each tube. This genericform allows us to change the relative position and strength of the plasmon peaksand uncover more characteristic features originating from the EELS spectra.In Fig. 13, the force as a function of plasmon frequency resonances of the outerCN is shown when the plasmon transition for the inner CNT is kept constant (fourvalues are chosen for ω ). One sees that the local minima in F versus ω occurwhen ω and ω coincide. In fact, the strongest attraction happens when both CNshave the lowest plasmon excitations at the same frequency ω = ω = 0 .
81 eV. Itis evident that the existence of relatively strong low frequency EELS spectrum and an overlap between the relevant plasmon peaks of the two structures is necessary toachieve a strong interaction. xciton-Plasmon Interactions in Individual Carbon Nanotubes
We have shown that the strong exciton-surface-plasmon coupling effect with char-acteristic exciton absorption line (Rabi) splitting ∼ . . ∼ . − . ∼
180 meV [53]). It is much larger than the exciton-polariton Rabi splitting insemiconductor microcavities ( ∼ − µ eV [81, 82, 83]), or the exciton-plasmonRabi splitting in hybrid semiconductor-metal nanoparticle molecules [54].Since the formation of the strongly coupled mixed exciton-plasmon excitationsis only possible if the exciton total energy is in resonance with the energy of aninterband surface plasmon mode, we have analyzed possible ways to tune the ex-citon energy to the nearest surface plasmon resonance. Specifically, the excitonenergy may be tuned to the nearest plasmon resonance in ways used for the ex-citons in semiconductor quantum microcavities — thermally (by elevating sampletemperature) [81, 82, 83], and/or electrostatically [84, 85, 86, 87] (via the quan-tum confined Stark effect with an external electrostatic field applied perpendicularto the CN axis). The two possibilities influence the different degrees of freedom ofthe quasi-1D exciton — the (longitudinal) kinetic energy and the excitation energy,respectively.We have studied how the perpendicular electrostatic field affects the excitonexcitation energy and interband plasmon resonance energy (the quantum confinedStark effect). Both of them are shown to shift to the red due to the decrease inthe CN band gap as the field increases. However, the exciton red shift is muchless than the plasmon one because of the decrease in the absolute value of thenegative binding energy, which contributes largely to the exciton excitation energy.The exciton excitation energy and interband plasmon energy approach as the fieldincreases, thereby bringing the total exciton energy in resonance with the plasmonmode due to the non-zero longitudinal kinetic energy term at finite temperature.The noteworthy point is that the strong exciton-surface-plasmon coupling wepredict here occurs in an individual CN as opposed to various artificially fabricated6 I.V.Bondarev, L.M.Woods, and A.Popescu hybrid plasmonic nanostructures mentioned above. We strongly believe this phe-nomenon, along with its tunability feature via the quantum confined Stark effect wehave demonstrated, opens up new paths for the development of CN based tunableoptoelectronic device applications in areas such as nanophotonics, nanoplasmonics,and cavity QED. One straightforward application like this is the CN photolumi-nescence control by means of the exciton-plasmon coupling tuned electrostaticallyvia the quantum confined Stark effect. This complements the microcavity controlledCN infrared emitter application reported recently[27], offering the advantage of lessstringent fabrication requirements at the same time since the planar photonic micro-cavity is no longer required. Electrostatically controlled coupling of two spatiallyseparated (weakly localized) excitons to the same nanotube’s plasmon resonancewould result in their entanglement [11, 12, 13], the phenomenon that paves the wayfor CN based solid-state quantum information applications. Moreover, CNs com-bine advantages such as electrical conductivity, chemical stability, and high surfacearea that make them excellent potential candidates for a variety of more practicalapplications, including efficient solar energy conversion [7], energy storage [14], andoptical nanobiosensorics [42]. However, the photoluminescence quantum yield ofindividual CNs is relatively low, and this hinders their uses in the aforementionedapplications. CN bundles and films are proposed to be used to surpass the poor per-formance of individual tubes. The theory of the exciton-plasmon coupling we havedeveloped here, being extended to include the inter-tube interaction, complementscurrently available ’weak-coupling’ theories of the exciton-plasmon interactions inlow-dimensional nanostructures [54, 121] with the very important case of the strongcoupling regime. Such an extended theory (subject of our future publication) willlay the foundation for understanding inter-tube energy transfer mechanisms thataffect the efficiency of optoelectronic devices made of CN bundles and films, as wellas it will shed more light on the recent photoluminescence experiments with CNbundles [43, 44] and multi-walled CNs [122], revealing their potentialities for thedevelopment of high-yield, high-performance optoelectronics applications with CNs.In addition, we have first applied the macroscopic QED approach suitable fordispersing and absorbing media to study the Casimir interaction in a double-wallcarbon nanotube systems with the realistic dielectric response taken into account.We found that at distances similar to the equilibrium separations between graphiticsurfaces ( ∼ xciton-Plasmon Interactions in Individual Carbon Nanotubes Acknowledgements
I.V.B. is supported by the US National Science Foundation, Army Research Of-fice and NASA (grants ECCS-1045661 & HRD-0833184, W911NF-10-1-0105, andNNX09AV07A). L.M.W. and A.P. are supported by the US Department of En-ergy contract DE-FG02-06ER46297. Helpful discussions with Mikhail Braun (St.-Peterburg U., Russia), Jonathan Finley (WSI, TU Munich, Germany), and Alexan-der Govorov (Ohio U., USA) are gratefully acknowledged.
Appendix A
Exciton interaction with the surface EM field
We follow our recently developed QED formalism to describe vacuum-type EMeffects in the presence of quasi-1D absorbing and dispersive bodies [58, 59, 60, 61,62, 9]. The treatment begins with the most general EM interaction of the surfacecharge fluctuations with the quantized surface EM field of a single-walled CN. Noexternal field is assumed to be applied. The CN is modelled by a neutral, infinitelylong, infinitely thin, anisotropically conducting cylinder. Only the axial conductivityof the CN, σ zz , is taken into account, whereas the azimuthal one, σ ϕϕ , is neglectedbeing strongly suppressed by the transverse depolarization effect [63, 64, 65, 66, 67,68]. Since the problem has the cylindrical symmetry, the orthonormal cylindricalbasis { e r , e ϕ , e z } is used with the vector e z directed along the nanotube axis asshown in Fig. 1. The interaction has the following form (Gaussian system of units)ˆ H int = ˆ H (1) int + ˆ H (2) int (66)= − X n ,i q i m i c ˆ A ( n + ˆ r ( i ) n ) · h ˆ p ( i ) n − q i c ˆ A ( n + ˆ r ( i ) n ) i + X n ,i q i ˆ ϕ ( n + ˆ r ( i ) n ) , where c is the speed of light, m i , q i , ˆ r ( i ) n , and ˆ p ( i ) n are, respectively, the masses,charges, coordinate operators and momenta operators of the particles (electronsand nucleus) residing at the lattice site n = R n = { R CN , ϕ n , z n } associated with acarbon atom (see Fig. 1) on the surface of the CN of radius R CN . The summationis taken over the lattice sites, and may be substituted with the integration over theCN surface using Eq. (3). The vector potential operator ˆ A and the scalar potentialoperator ˆ ϕ represent the nanotube’s transversely polarized and longitudinally po-larized surface EM modes, respectively. They are written in the Schr¨odinger picture8 I.V.Bondarev, L.M.Woods, and A.Popescu as follows ˆ A ( n ) = Z ∞ dω ciω ˆ E ⊥ ( n , ω ) + h.c., (67) − ∇ n ˆ ϕ ( n ) = Z ∞ dω ˆ E k ( n , ω ) + h.c.. (68)We use the Coulomb gauge whereby ∇ n · ˆ A ( n ) = 0, or [ˆ p ( i ) n , ˆ A ( n + ˆ r ( i ) n )] = 0.The total electric field operator of the CN-modified EM field is given for anarbitrary r in the Schr¨odinger picture byˆ E ( r ) = Z ∞ dω ˆ E ( r , ω ) + h.c. = Z ∞ dω [ ˆ E ⊥ ( r , ω ) + ˆ E k ( r , ω )] + h.c. (69)with the transversely (longitudinally) polarized Fourier-domain field componentsdefined as ˆ E ⊥ ( k ) ( r , ω ) = Z d r ′ δ ⊥ ( k ) ( r − r ′ ) · ˆ E ( r ′ , ω ) , (70)where δ k αβ ( r ) = −∇ α ∇ β πr , (71) δ ⊥ αβ ( r ) = δ αβ δ ( r ) − δ k αβ ( r )are the longitudinal and transverse dyadic δ -functions, respectively. The total fieldoperator (69) satisfies the set of the Fourier-domain Maxwell equations ∇ × ˆ E ( r , ω ) = ik ˆ H ( r , ω ) , (72) ∇ × ˆ H ( r , ω ) = − ik ˆ E ( r , ω ) + 4 πc ˆ I ( r , ω ) , (73)where ˆ H = ( ik ) − ∇ × ˆ E is the magnetic field operator, k = ω/c , andˆ I ( r , ω ) = X n δ ( r − n ) ˆ J ( n , ω ) , (74)is the exterior current operator with the current density defined as followsˆ J ( n , ω ) = r ~ ω Re σ zz ( R CN , ω ) π ˆ f ( n , ω ) e z (75)to ensure preservation of the fundamental QED equal-time commutation relations(see, e.g., [47]) for the EM field components in the presence of a CN. Here, σ zz is theCN surface axial conductivity per unit length, and ˆ f ( n , ω ) along with its counter-part ˆ f † ( n , ω ) are the scalar bosonic field operators which annihilate and create, xciton-Plasmon Interactions in Individual Carbon Nanotubes ω at the lattice site n of the CN surface. They satisfy the standard bosonic commutation relations[ ˆ f ( n , ω ) , ˆ f † ( m , ω ′ )] = δ nm δ ( ω − ω ′ ) , (76)[ ˆ f ( n , ω ) , ˆ f ( m , ω ′ )] = [ ˆ f † ( n , ω ) , ˆ f † ( m , ω ′ )] = 0 . One further obtains from Eqs. (72)–(75) thatˆ E ( r , ω ) = ik πc X n G ( r , n , ω ) · ˆ J ( n , ω ) , (77)and, according to Eqs. (69) and (70),ˆ E ⊥ ( k ) ( r , ω ) = ik πc X n ⊥ ( k ) G ( r , n , ω ) · ˆ J ( n , ω ) , (78)where ⊥ G and k G are the transverse part and the longitudinal part, respectively,of the total Green tensor G = ⊥ G + k G of the classical EM field in the presence ofthe CN. This tensor satisfies the equation X α = r,ϕ,z (cid:0) ∇ × ∇ × − k (cid:1) zα G αz ( r , n , ω ) = δ ( r − n ) (79)together with the radiation conditions at infinity and the boundary conditions onthe CN surface.All the ’discrete’ quantities in Eqs. (74)–(79) may be equivalently rewritten incontinuous variables in view of Eq. (3). Being applied to the identity 1 = P m δ nm ,Eq. (3) yields δ nm = S δ ( R n − R m ) . (80)This requires to redefineˆ f ( n , ω ) = p S ˆ f ( R n , ω ) , ˆ f † ( n , ω ) = p S ˆ f † ( R n , ω ) (81)in the commutation relations (76). Similarly, from Eq. (77), in view of Eqs. (3), (75)and (81), one obtains G ( r , n , ω ) = p S G ( r , R n , ω ) , (82)which is also valid for the transverse and longitudinal Green tensors in Eq. (78).Next, we make the series expansions of the interactions ˆ H (1) int and ˆ H (2) int in Eq. (66)about the lattice site n to the first non-vanishing terms,ˆ H (1) int ≈ − X n ,i q i m i c ˆ A ( n ) · ˆ p ( i ) n + X n ,i q i m i c ˆ A ( n ) , (83)ˆ H (2) int ≈ X n ,i q i ∇ n ˆ ϕ ( n ) · ˆ r ( i ) n , (84)0 I.V.Bondarev, L.M.Woods, and A.Popescu and introduce the single-lattice-site Hamiltonianˆ H n = ε | ih | + X f ( ε + ~ ω f ) | f ih f | (85)with the completeness relation | ih | + X f | f ih f | = ˆ I. (86)Here, ε is the energy of the ground state | i (no exciton excited) of the carbon atomassociated with the lattice site n , ε + ~ ω f is the energy of the excited carbon atomin the quantum state | f i with one f -internal-state exciton formed of the energy E ( f ) exc = ~ ω f . In view of Eqs. (85) and (86), one hasˆ p ( i ) n = m i d ˆ r ( i ) n dt = m i i ~ [ˆ r ( i ) n , ˆ H n ] = m i i ~ ˆ I [ˆ r ( i ) n , ˆ H n ] ˆ I ≈ m i i ~ X f ~ ω f (cid:16) h | ˆ r ( i ) n | f i B n ,f − h f | ˆ r ( i ) n | i B † n ,f (cid:17) (87)and ˆ r ( i ) n = ˆ I ˆ r ( i ) n ˆ I ≈ X f (cid:16) h | ˆ r ( i ) n | f i B n ,f + h f | ˆ r ( i ) n | i B † n ,f (cid:17) , (88)where h | ˆ r ( i ) n | f i = h f | ˆ r ( i ) n | i in view of the hermitian and real character of the coor-dinate operator. The operators B n ,f = | ih f | and B † n ,f = | f ih | create and annihilate,respectively, the f -internal-state exciton at the lattice site n , and exciton-to-excitontransitions are neglected. In addition, we also have δ ij δ αβ = i ~ [(ˆ p ( i ) n ) α , (ˆ r ( j ) n ) β ] , (89)where α, β = r, ϕ, z . Substituting these into Eqs. (83) and (84) [commutator (89)goes into the second term of Eq. (83) which is to be pre-transformed as follows P i,j,α,β q i q j ˆ A ( n ) α ˆ A ( n ) β δ ij δ αβ / m i c ], one arrives at the following (electric dipole)approximation of Eq. (66) ˆ H int = ˆ H (1) int + ˆ H (2) int (90)= − X n ,f iω f c d f n · ˆ A ( n ) (cid:20) B † n ,f − B n ,f + i ~ c d f n · ˆ A ( n ) (cid:21) + X n ,f d f n · ∇ n ˆ ϕ ( n ) (cid:16) B † n ,f + B n ,f (cid:17) xciton-Plasmon Interactions in Individual Carbon Nanotubes d f n = h | ˆ d n | f i = h f | ˆ d n | i , where ˆ d n = P i q i ˆ r ( i ) n is the total electric dipolemoment operator of the particles residing at the lattice site n .The Hamiltonian (90) is seen to describe the vacuum-type exciton interactionwith the surface EM field (created by the charge fluctuations on the nanotube sur-face). The last term in the square brackets does not depend on the exciton operators,and therefore results in the constant energy shift which can be safely neglected. Wethen arrive, after using Eqs. (67), (68), (75), and (78), at the following secondquantized interaction Hamiltonianˆ H int = X n , m ,f Z ∞ dω [ g (+) f ( n , m , ω ) B † n ,f − g ( − ) f ( n , m , ω ) B n ,f ] ˆ f ( m , ω ) + h.c., (91)where g ( ± ) f ( n , m , ω ) = g ⊥ f ( n , m , ω ) ± ωω f g k f ( n , m , ω ) (92)withg ⊥ ( k ) f ( n , m , ω ) = − i ω f c p π ~ ω Re σ zz ( R CN , ω ) X α = r,ϕ,z ( d f n ) α ⊥ ( k ) G αz ( n , m , ω ) , (93)and ⊥ ( k ) G αz ( n , m , ω ) = Z d r δ ⊥ ( k ) αβ ( n − r ) G βz ( r , m , ω ) . (94)This yields Eqs. (10)–(12) after the strong transverse depolarization effect in CNsis taken into account whereby d f n ≈ ( d f n ) z e z . Appendix B
Green tensor of the surface EM field
Within the model of an infinitely thin, infinitely long, anisotropically conductingcylinder we utilize here, the classical EM field Green tensor is found by expandingthe solution to the Green equation (79) in series in cylindrical coordinates, andthen imposing the appropriately chosen boundary conditions on the CN surface todetermine the Wronskian normalization constant (see, e.g., Ref. [118]).After the EM field is divided into the transversely and longitudinally polarizedcomponents according to Eqs. (69)–(71), the Green equation (79) takes the form X α = r,ϕ,z (cid:0) ∇ × ∇ × − k (cid:1) zα h ⊥ G αz ( r , n , ω ) + k G αz ( r , n , ω ) i = δ ( r − n ) (95)with the two additional constraints, X α = r,ϕ,z ∇ α ⊥ G αz ( r , n , ω ) = 0 (96)2 I.V.Bondarev, L.M.Woods, and A.Popescu and X β,γ = r,ϕ,z ǫ αβγ ∇ β k G γz ( r , n , ω ) = 0 , (97)where ǫ αβγ is the totally antisymmetric unit tensor of rank 3. Equations (96) and(97) originate from the divergence-less character (Coulomb gauge) of the transverseEM component and the curl-less character of the longitudinal EM component, re-spectively. The transverse ⊥ G αz and longitudinal k G αz Green tensor componentsare defined by Eq. (94) which is the corollary of Eq. (70) using the Eqs. (77) and(78). Equation (95) is further rewritten in view of Eqs. (96) and (97), to give thefollowing two independent equations for ⊥ G zz and k G zz we need (cid:0) ∆ + k (cid:1) ⊥ G zz ( r , n , ω ) = − δ ⊥ zz ( r − n ) , (98) k k G zz ( r , n , ω ) = − δ k zz ( r − n ) (99)with the transverse and longitudinal delta-functions defined by Eq. (71).We use the differential representations for the transverse ⊥ G zz and longitudinal k G zz Green functions of the following form [consistent with Eq. (94)] ⊥ G zz ( r , n , ω ) = (cid:18) k ∇ z ∇ z + 1 (cid:19) g ( r , n , ω ) , (100) k G zz ( r , n , ω ) = − k ∇ z ∇ z g ( r , n , ω ) , (101)where g ( r , n , ω ) is the scalar Green function of the Helmholtz equation (98), satis-fying the radiation condition at infinity and the finiteness condition on the axis ofthe cylinder. Such a function is known to be given by the following series expansion g ( r , n , ω ) = √ S π e ik | r − R n | | r − R n | = √ S (2 π ) ∞ X p = −∞ e ip ( ϕ − ϕ n ) (102) × Z C dh I p ( vr ) K p ( vR CN ) e ih ( z − z n ) , r ≤ R CN , where I p and K p are the modified cylindric Bessel functions, v = v ( h, ω ) = √ h − k ,and we used the property (82) to go from the discrete variable n to the correspondingcontinuous variable. The integration contour C goes along the real axis of thecomplex plane and envelopes the branch points ± k of the integrand from below andfrom above, respectively. For r ≥ R CN , the function g ( r , n , ω ) is obtained fromEq. (102) by means of a simple symbol replacement I p ↔ K p in the integrand.The scalar function (102) is to be imposed the boundary conditions on the CNsurface. To derive them, we represent the classical electric and magnetic field com- xciton-Plasmon Interactions in Individual Carbon Nanotubes E α ( r , ω ) = ik ⊥ G αz ( r , n , ω ) , (103) H α ( r , ω ) = − ik X β,γ = r,ϕ,z ǫ αβγ ∇ β E γ ( r , ω ) . (104)These are valid for r = n under the Coulomb-gauge condition. The boundary condi-tions are then obtained from the standard requirements that the tangential electricfield components be continuous across the surface, and the tangential magneticfield components be discontinuous by an amount proportional to the free surfacecurrent density, which we approximate here by the (strongest) axial component, σ zz ( R CN , ω ), of the nanotube’s surface conductivity. Under this approximation,one has E z | + − E z | − = E ϕ | + − E ϕ | − = 0 , (105) H z | + − H z | − = 0 , (106) H ϕ | + − H ϕ | − = 4 πc σ zz ( ω ) E z | R CN , (107)where ± stand for r = R CN ± ε with the positive infinitesimal ε . In view of Eqs. (103),(104) and (100), the boundary conditions above result in the following two boundaryconditions for the function (102) g | + − g | − = 0 , (108) ∂g∂r (cid:12)(cid:12)(cid:12)(cid:12) + − ∂g∂r (cid:12)(cid:12)(cid:12)(cid:12) − = − πi σ zz ( ω ) ω (cid:18) ∂ ∂z + k (cid:19) g | R CN . (109)We see that Eq. (108) is satisfied identically. Eq. (109) yields the Wronskian ofmodified Bessel functions on the left, W [ I p ( x ) , K p ( x )] = I p ( x ) K ′ p ( x ) − K p ( x ) I ′ p ( x ) = − /x , which brings us to the equation − R CN = 4 πi σ zz ( ω ) ω v I p ( vR CN ) K p ( vR CN ) . (110)This is nothing but the dispersion relation which determines the radial wave num-bers, h , of the CN surface EM modes with given p and ω . Since we are interestedhere in the EM field Green tensor on the CN surface [see Eq. (93)], not in particularsurface EM modes, we substitute I p ( vR CN ) K p ( vR CN ) from Eq. (110) into Eq. (102)with r = R CN . This allows us to obtain the scalar Green function of interest withthe boundary conditions (108) and (109) taken into account. We have g ( R , n , ω ) = − iω √ S δ ( ϕ − ϕ n )8 π σ zz ( ω ) R CN Z C dh e ih ( z − z n ) k − h , (111)4 I.V.Bondarev, L.M.Woods, and A.Popescu where R = { R CN , ϕ, z } is an arbitrary point of the cylindrical surface. Usingfurther the residue theorem to calculate the contour integral, we arrive at the finalexpression of the form g ( R , n , ω ) = − c √ S δ ( ϕ − ϕ n )8 πσ zz ( ω ) R CN e iω | z − z n | /c , (112)which yields ⊥ G zz ( R , n , ω ) ≡ , (113) k G zz ( R , n , ω ) = g ( R , n , ω ) , (114)in view of Eqs. (100) and (101).The fact that the transverse Green function (113) identically equals zero on theCN surface is related to the absence of the skin layer in the model of the infinitelythin cylinder (see, e.g., Ref. [118]). In this model, the transverse Green functionis only non-zero in the near-surface area where the exciton wave function goes tozero. Thus, only longitudinally polarized EM modes with the Green function (114)contribute to the exciton surface EM field interaction on the nanotube surface. Appendix C
Diagonalization of the Hamiltonian (1)–(13)
We start with the transformation of the total Hamiltonian (1)–(13) to the k -repre-sentation using Eqs. (5) and (9). The unperturbed part presents no difficulties. Spe-cial care should be given to the interaction matrix element g ( ± ) f ( n , m , ω ) in Eq. (13).In view of Eqs. (114), (112) and (3), one has explicitlyg ( ± ) f ( k , k ′ , ω ) = 1 N X n , m g ( ± ) f ( n , m , ω ) e − i k · n + i k ′ · m (115)= ± iω p π ~ ω Re σ zz ( ω )2 πc σ zz ( ω ) R CN d fz N p S R CN N S × Z π dϕ n dϕ m δ ( ϕ n − ϕ m ) e − ik ϕ ϕ n + ik ′ ϕ ϕ m Z ∞−∞ dz n dz m e iω | z n − z m | /c − ik z z n + ik ′ z z m , where we have also taken into account the fact that the dipole matrix element( d f n ) z = h | (ˆ d n ) z | f i is the same for all the lattice sites on the CN surface in view oftheir equivalence. As a consequence, ( d f n ) z = d fz /N with d fz = P n h | (ˆ d n ) z | f i .The integral over ϕ in Eq. (115) is taken in a standard way to yield Z π dϕ n dϕ m δ ( ϕ n − ϕ m ) e − ik ϕ ϕ n + ik ′ ϕ ϕ m = 2 πδ k ϕ k ′ ϕ . (116) xciton-Plasmon Interactions in Individual Carbon Nanotubes z is performed by first writing the integral in the form Z ∞−∞ dz n dz m ... = lim L →∞ Z L/ − L/ dz n Z L/ − L/ dz m ... ( L being the CN length), then dividing it into two parts by means of the equation e iω | z n − z m | /c = θ ( z n − z m ) e iω ( z n − z m ) /c + θ ( z m − z n ) e − iω ( z n − z m ) /c , and finally by taking simple exponential integrals with allowance made for the for-mula δ k z k ′ z = lim L →∞ L ( k z − k ′ z ) / L ( k z − k ′ z ) . After some simple algebra we obtain the result Z ∞−∞ dz n dz m e iω | z n − z m | /c − ik z z n + ik ′ z z m = lim L →∞ L (cid:26) − iω/cL [ k z − ( ω/c ) ] (cid:27) δ k z k ′ z . (117)In view of Eqs. (116) and (117), the function (115) takes the formg ( ± ) f ( k , k ′ , ω ) = ± iω d fz p πS ~ ω Re σ zz ( ω )(2 π ) c σ zz ( ω ) R CN lim L →∞ (cid:26) − iω/cL [ k z − ( ω/c ) ] (cid:27) δ kk ′ . (118)We have taken into account here that δ k ϕ k ′ ϕ δ k z k ′ z = δ kk ′ , as well as the fact that( R CN L/N S ) = 1 / (2 π ) . This can be further simplified by noticing that onlyabsolute value squared of the interaction matrix element matters in calculations ofobservables. We then have (cid:12)(cid:12)(cid:12)(cid:12) − iω/cL [ k z − ( ω/c ) ] (cid:12)(cid:12)(cid:12)(cid:12) = 1 + αu ≈ αu + α with u = ( ck z /ω ) −
1, and α = (2 c/Lω ) being the small parameter which tends tozero as L → ∞ . Using further the formula (see, e.g., Ref. [74]) δ ( u ) = 1 π lim α → αu + α , and the basic properties of the δ -function, we arrive atlim L →∞ (cid:12)(cid:12)(cid:12)(cid:12) − iω/cL [ k z − ( ω/c ) ] (cid:12)(cid:12)(cid:12)(cid:12) = 1 + πc | k z | δ ( ω + ck z ) + δ ( ω − ck z )] (119)We also have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p Re σ zz ( ω ) σ zz ( ω ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = Re 1 σ zz ( ω ) . (120)6 I.V.Bondarev, L.M.Woods, and A.Popescu
Equation (118), in view of Eqs. (119) and (120), is rewritten effectively as followsg ( ± ) f ( k , k ′ , ω ) = ± iD f ( ω ) δ kk ′ (121)with D f ( ω ) = ω d fz p πS ~ ω Re[1 /σ zz ( ω )](2 π ) c R CN r πc | k z | δ ( ω + ck z ) + δ ( ω − ck z )] . (122)In terms of the simplified interaction matrix element (121), the k -representationof the Hamiltonian (1)–(13) takes the following (symmetrized) formˆ H = 12 X k ˆ H k , (123)where ˆ H k = X f E f ( k ) (cid:16) B † k ,f B k ,f + B †− k ,f B − k ,f (cid:17) (124)+ Z ∞ dω ~ ω h ˆ f † ( k , ω ) ˆ f ( k , ω ) + ˆ f † ( − k , ω ) ˆ f ( − k , ω ) i + X f Z ∞ dω iD f ( ω ) (cid:16) B † k ,f + B − k ,f (cid:17) h ˆ f ( k , ω ) − ˆ f † ( − k , ω ) i + h.c. with D f ( ω ) given by Eq. (122). To diagonalize this Hamiltonian, we follow Bo-goliubov’s canonical transformation technique (see, e.g., Ref. [74]). The canonicaltransformation of the exciton and photon operators is of the form B k ,f = X µ =1 , h u µ ( k , ω f ) ˆ ξ µ ( k ) + v µ ( k , ω f ) ˆ ξ † µ ( − k ) i , (125)ˆ f ( k , ω ) = X µ =1 , h u ∗ µ ( k , ω ) ˆ ξ µ ( k ) + v ∗ µ ( k , ω ) ˆ ξ † µ ( − k ) i , (126)where the new operators, ˆ ξ µ ( k ) and ˆ ξ † µ ( k ) = [ ˆ ξ µ ( k )] † , annihilate and create, respec-tively, the coupled exciton-photon excitations of branch µ on the nanotube surface.They satisfy the bosonic commutation relations of the form h ˆ ξ µ ( k ) , ˆ ξ † µ ′ ( k ′ ) i = δ µµ ′ δ kk ′ , (127)which, along with the reversibility requirement of Eqs. (125) and (126), impose the xciton-Plasmon Interactions in Individual Carbon Nanotubes u µ and v µ X f (cid:2) u ∗ µ ( k , ω f ) u µ ′ ( k , ω f ) − v µ ( k , ω f ) v ∗ µ ′ ( k , ω f ) (cid:3) + Z ∞ dω (cid:2) u µ ( k , ω ) u ∗ µ ′ ( k , ω ) − v ∗ µ ( k , ω ) v µ ′ ( k , ω ) (cid:3) = δ µµ ′ , X µ (cid:2) u ∗ µ ( k , ω f ) u µ ( k , ω f ′ ) − v ∗ µ ( k , ω f ) v µ ( k , ω f ′ ) (cid:3) = δ ff ′ , X µ (cid:2) u ∗ µ ( k , ω ) u µ ( k , ω ′ ) − v ∗ µ ( k , ω ) v µ ( k , ω ′ ) (cid:3) = δ ( ω − ω ′ ) . Here, the first equation guarantees the fulfilment of the commutation relations (127),whereas the second and the third ensure that Eqs. (125) and (126) are inverted toyield ˆ ξ µ ( k ) as given by Eq. (15). Other possible combinations of the transformationfunctions are identically equal to zero.The proper transformation functions that diagonalize the Hamiltonian (124) tobring it to the form (14), are determined by the identity ~ ω µ ( k ) ˆ ξ µ ( k ) = h ˆ ξ µ ( k ) , ˆ H k i . (128)Putting Eqs. (15) and (124) into Eq. (128) and using the bosonic commutationrelations for the exciton and photon operators on the right, one obtains ( k -argumentis omitted for brevity)( ~ ω µ − E f ) u ∗ µ ( ω f ) = − i Z ∞ dω D f ( ω ) (cid:2) u µ ( ω ) − v ∗ µ ( ω ) (cid:3) , ( ~ ω µ + E f ) v µ ( ω f ) = i Z ∞ dω D f ( ω ) (cid:2) u µ ( ω ) − v ∗ µ ( ω ) (cid:3) , ~ ( ω µ − ω ) u µ ( ω ) = i X f D f ( ω ) (cid:2) u ∗ µ ( ω f ) + v µ ( ω f ) (cid:3) , ~ ( ω µ + ω ) v ∗ µ ( ω ) = i X f D f ( ω ) (cid:2) u ∗ µ ( ω f ) + v µ ( ω f ) (cid:3) . These simultaneous equations define the complex transformation functions u µ and v µ uniquely. They also define the dispersion relation (the energies ~ ω µ , µ = 1 , u µ and v ∗ µ from the third and forth equations intothe first one, one has (cid:20) ~ ω µ − E f − E f ~ ω µ + E f Z ∞ dω ω | D f ( ω ) | ~ ( ω µ − ω ) (cid:21) u ∗ µ ( ω f ) = 0 , I.V.Bondarev, L.M.Woods, and A.Popescu whereby, since the functions u ∗ µ are non-zero, the dispersion relation we are interestedin becomes ( ~ ω µ ) − E f − E f Z ∞ dω ω | D f ( ω ) | ~ ( ω µ − ω ) = 0 . (129)The energy E of the ground state of the coupled exciton-plasmon excitations isfound by plugging Eq. (15) into Eq. (14) and comparing the result with Eqs. (123)and (124). This yields E = − X k , µ =1 , ~ ω µ ( k ) X f | v µ ( k , ω f ) | + Z ∞ dω | v µ ( k , ω ) | . Using further D f ( ω ) as explicitly given by Eq. (122), the dispersion relation(129) is rewritten as follows( ~ ω µ ) − E f = E f S | d fz | π c R CN (cid:26)Z ∞ dω ω Re[1 /σ zz ( ω )] ω µ − ω + π ( c | k z | ) Re[1 /σ zz ( c | k z | )] ω µ − ( c | k z | ) (cid:27) . Here we have taken into account the general property σ zz ( ω ) = σ ∗ zz ( − ω ), whichoriginates from the time-reversal symmetry requirement, in the second term on theright hand side. This term comes from the two delta functions in | D f ( ω ) | , anddescribes the contribution of the spatial dispersion (wave-vector dependence) to theformation of the exciton-plasmons. We neglect this term in what follows becausethe spatial dispersion is neglected in the nanotube’s axial surface conductivity in ourmodel, and, secondly, because it is seen to be very small for not too large excitonicwave vectors. Thus, converting to the dimensionless variables (17), we arrive at thedispersion relation (16) with the exciton spontaneous decay (recombination) rateand the plasmon DOS given by Eqs. (18) and (19), respectively.Lastly, bearing in mind that the delta functions in | D f ( ω ) | are responsiblefor the spatial dispersion which we neglect in our model, and therefore droppingthem out from the squared interaction matrix element (121), we arrive at the prop-erty (45). Appendix D
Effective longitudinal potential in the presenceof the perpendicular electrostatic field
Here we analyze the set of equations (27)–(29), and show that the attractive cusp-type cutoff potential (32) with the field dependent cutoff parameter (33) is a uni-formly valid approximation for the effective electron-hole Coulomb interaction po-tential (30) in the exciton binding energy equation (29). xciton-Plasmon Interactions in Individual Carbon Nanotubes (cid:18) d dϕ + q + p cos ϕ (cid:19) ψ ( ϕ ) = 0 . (130)Here, ϕ = ϕ e,h , ψ = ψ e,h , q = R CN p m e,h ε e,h / ~ , and p = ± em e,h R CN F/ ~ withthe (+)-sign to be taken for the electron and the (–)-sign to be taken for the hole. Weare interested in the solutions to Eq. (130) which satisfy the 2 π -periodicity condition ψ ( ϕ ) = ψ ( ϕ + 2 π ). The change of variable ϕ = 2 t transfers this equation to thewell known Mathieu’s equation (see, e.g., Refs. [119, 114]), reducing the solution’speriod by the factor of two. The exact solutions of interest are, therefore, givenby the odd Mathieu functions se m +2 ( t = ϕ/
2) with the eigen values b m +2 , where m is a nonnegative integer (notations of Ref. [119]). These are the solutions tothe Sturm-Liouville problem with boundary conditions on functions, not on theirderivatives.It is easier to estimate the z -dependence of the potential (30) if the functions ψ e,h ( ϕ e,h ) are known explicitly. So, we do solve Eq. (130) using the second orderperturbation theory in the external field (the term p cos ϕ ). The second order fieldcorrections are also of practical importance in the most of experimental applications.The unperturbed problem yields the two linearly independent normalized eigenfunctions and the eigen values as follows ψ (0) j ( ϕ ) = exp( ± ijϕ ) √ π , q = j = R CN ~ q m e,h ε (0) e,h (131)with j being a nonnegative integer. The energies ε (0) e,h ( j ) are doubly degeneratewith the exception of ε (0) e,h (0) = 0, which we will discard since it results in thezero unperturbed band gap according to Eq. (8). The perturbation p cos ϕ doesnot lift the degeneracy of the unperturbed states. Therefore, we use the standardnondegenerate perturbation theory with the basis wave functions set above (plussign selected for definiteness) to calculate the energies and the wave functions to thesecond order in perturbation. The standard procedure (see, e.g., Ref. [95]) yields ψ j e,h ( ϕ e,h ) = − (cid:26) ϑ ( j − j − − j ] + 1[( j + 1) − j ] (cid:27) m e,h e R CN ~ F ! ψ (0) j e,h ( ϕ e,h ) ± ϑ ( j − ψ (0) j − e,h ( ϕ e,h )( j − − j + ψ (0) j +1 e,h ( ϕ e,h )( j + 1) − j m e,h eR CN ~ F (132)+ ϑ ( j − ϑ ( j − ψ (0) j − e,h ( ϕ e,h )[( j − − j ][( j − − j ] + ψ (0) j +2 e,h ( ϕ e,h )[( j + 1) − j ][( j + 2) − j ] m e,h e R CN ~ F . I.V.Bondarev, L.M.Woods, and A.Popescu
Here, j is a positive integer, and the theta-functions ensure that j = 1 is the groundstate of the system. The corresponding energies are as follows ε e,h = ~ j m e,h R CN − m e,h e R CN w j ~ F (133)with w j given by Eq. (34), thus, according to Eq. (8), resulting in the nanotube’sband gap as given by Eq. (37).From Eq. (132), in view of Eq. (131), we have the following to the second orderin the field | ψ e ( ϕ e ) | | ψ h ( ϕ h ) | ≈ π (cid:20) − m h cos ϕ h − m e cos ϕ e ) eR CN w j ~ F (134)+2 (cid:0) m h cos 2 ϕ h + m e cos 2 ϕ e (cid:1) e R CN v j ~ F − µM ex cos ϕ e cos ϕ h e R CN w j ~ F , where v j = ϑ ( j − j − − j (cid:26) ϑ ( j − j − − j + 1( j + 1) − j (cid:27) + 1[( j + 1) − j ][( j + 2) − j ] . Plugging Eqs. (134) and (31) into Eq. (30) and noticing that the integrals involv-ing linear combinations of the cosine-functions are strongly suppressed due to theintegration over the cosine period, and are therefore negligible compared to the oneinvolving the quadratic cosine-combination, we obtain V eff ( z ) = − e π ǫ Z π dϕ e Z π dϕ h − ϕ e cos ϕ h ∆ j ( F ) { z + 4 R CN sin [( ϕ e − ϕ h ) / } / (135)with ∆ j ( F ) given by Eq. (34).The next step is to perform the double integration in Eq. (135). We have toevaluate the two double integrals. They are I = Z π dϕ e Z π dϕ h { z + 4 R CN sin [( ϕ e − ϕ h ) / } / (136)and I = Z π dϕ e Z π dϕ h cos ϕ e cos ϕ h { z + 4 R CN sin [( ϕ e − ϕ h ) / } / . (137)We first notice that both I and I can be equivalently rewritten as follows Z π dϕ e Z π dϕ h ... = 2 Z π dϕ e Z ϕ e dϕ h ... (138) xciton-Plasmon Interactions in Individual Carbon Nanotubes ϕ e = ϕ h )-line. Using thisproperty, we substitute ϕ h with the new variable t = sin[( ϕ e − ϕ h ) /
2] in Eqs. (136)and (137). This, after simplifications, yields I = 4 Z π dϕ e Z sin( ϕ e / dt [(1 − t )( z + 4 R CN t )] / (139)and I = 4 Z π dϕ e cos ϕ e Z sin( ϕ e / dt (1 − t )[(1 − t )( z + 4 R CN t )] / . (140)Here, the inner integrals are reduced to the incomplete elliptical integrals of the firstand second kinds (see, e.g., Ref. [114]).We continue the evaluation of Eqs. (139) and (140) by expanding the denom-inators of the integrands in series at large and small | z | as compared to the CNdiameter 2 R CN . One has1( z + 4 R CN t ) / ≈ | z | " − (cid:18) R CN t | z | (cid:19) + 38 (cid:18) R CN t | z | (cid:19) − (cid:18) R CN t | z | (cid:19) + ... for | z | / R CN ≫
1, and Z sin( ϕ e / dt f ( t )[(1 − t )( z + 4 R CN t )] / = 12 R CN lim ( | z | / R CN ) → Z sin( ϕ e / | z | / R CN dt f ( t ) t √ − t for | z | / R CN ≪ f ( t ) is a polynomial function]. Using these in Eqs. (139) and(140), we arrive at I ≈ πR CN " ln (cid:18) R CN | z | (cid:19) − (cid:18) | z | R CN (cid:19) , | z | R CN ≪ π | z | " − (cid:18) R CN | z | (cid:19) + 964 (cid:18) R CN | z | (cid:19) , | z | R CN ≫ I ≈ πR CN "
12 ln (cid:18) R CN | z | (cid:19) − (cid:18) | z | R CN (cid:19) , | z | R CN ≪ π | z | (cid:18) R CN | z | (cid:19) " − (cid:18) R CN | z | (cid:19) , | z | R CN ≫ I and I into Eq. (135) and retaining only leading expansion termsyields V eff ( z ) ≈ − e [1 − ∆ j ( F )] πǫR CN ln (cid:18) R CN | z | (cid:19) , | z | R CN ≪ − e ǫ | z | , | z | R CN ≫ I.V.Bondarev, L.M.Woods, and A.Popescu
Figure 14: The dimensionless function (142) with the zero-field cutoff parameter(143). See text for details.We see from Eq. (141) that, to the leading order in the series expansion parame-ter, the perpendicular electrostatic field does not affect the longitudinal electron-holeCoulomb potential at large distances | z | ≫ R CN , as one would expect. At shortdistances | z | ≪ R CN the situation is different, however. The potential decreaseslogarithmically with the field dependent amplitude as | z | goes down. The amplitudeof the potential decreases quadratically as the field increases [see Eq. (34)], therebyslowing down the potential fall-off with decreasing | z | , or, in other words, making thepotential shallower as the field increases. Such a behavior can be uniformly approx-imated for all | z | by an appropriately chosen attractive cusp-type cutoff potentialwith the field dependent cutoff parameter. Indeed, consider the dimensionless func-tion f ( y ) = − R CN ǫ V eff /e of the dimensionless variable y = | z | / R CN . Then,according to Eq. (141), one has f ( y ) = Φ ( y ) = 2 π [1 − ∆ j ( F )] ln (cid:16) y (cid:17) , < y ≪ ( y ) = 1 y , y ≫ y ) = 1 y + y (142)with the cutoff parameter y selected in such a way as to satisfy the condition xciton-Plasmon Interactions in Individual Carbon Nanotubes (1) + Φ (1)] /
2. This yields y = π − − ∆ j ( F )] π + 2 ln 2 [1 − ∆ j ( F )] . (143)Figure 14 shows the zero-field behavior of the Φ( y ) function as compared to thecorresponding Φ ( y ) and Φ ( y ) functions. We see that Φ( y ) gradually approachesΦ ( y ) = 1 /y for increasing y >
1. For decreasing y <
1, on the other hand, Φ( y )is very close to the logarithmic behavior as given by Φ ( y ), with the exceptionthat there is no divergence at y ∼ − e / R CN ǫ and putting y = | z | / R CN , we obtain the attractivelongitudinal cusp-type cutoff potential (32) we build our analysis on in this paper.4 I.V.Bondarev, L.M.Woods, and A.Popescu
References [1] R.Saito, G.Dresselhaus, and M.S.Dresselhaus,
Science of Fullerens and CarbonNanotubes (Imperial College Press, London, 1998).[2] H.Dai, Surf. Sci. , 218 (2002).[3] L.X.Zheng, M.J.O’Connell, S.K.Doorn, X.Z.Liao, Y.H. Zhao, E.A.Akhadov,M.A.Hoffbauer, B.J.Roop, Q.X.Jia, R.C.Dye, D.E.Peterson, S.M.Huang, J.Liuand Y.T.Zhu, Nature Materials , 673 (2004).[4] S.M.Huang, B.Maynor, X.Y.Cai, and J.Liu, Advanced Materials , 1651(2003).[5] R.H.Baughman, A.A.Zakhidov, and W.A.de Heer, Science , 787 (2002).[6] A.Popescu, L.M.Woods, and I.V.Bondarev, Nanotechnology , 435702 (2008).[7] J.E.Trancik, S.C.Barton, and J.Hone, Nano Lett. , 982 (2008).[8] I.V.Bondarev, J. Comput. Theor. Nanosci. , 1673 (2010) (invited review articlefor the special issue on ”Technology Trends and Theory of Nanoscale Devicesfor Quantum Applications”, American Scientific Publishers, USA).[9] I.V.Bondarev and Ph.Lambin, in: Trends in Nanotubes Reasearch (Nova Sci-ence, NY, 2006). Ch.6, pp.139-183.[10] I.V.Bondarev and B.Vlahovic, Phys. Rev. B , 073401 (2006).[11] I.V.Bondarev and B.Vlahovic, Phys. Rev. B , 033402 (2007).[12] I.V.Bondarev, J. Electron. Mater. , 1579(2007).[13] I.V.Bondarev, Optics & Spectroscopy (Springer) , 366 (2007).[14] H.Shimoda, B.Gao, X.P.Tang, A.Kleinhammes, L.Fleming, Y.Wu, and O.Zhou,Phys. Rev. Lett. , 015502 (2001).[15] G.-H. Jeong, A.A.Farajian, R.Hatakeyama, T.Hirata, T.Yaguchi, K.Tohji,H.Mizuseki, and Y.Kawazoe, Phys. Rev. B , 075410 (2003)[16] G.-H.Jeong, A.A.Farajian, T.Hirata, R.Hatakeyama, K. Tohji, T.M.Briere,H.Mizuseki, and Y.Kawazoe, Thin Solid Films , 307 (2003).[17] M.Khazaei, A.A.Farajian, G.-H.Jeong, H.Mizuseki, T. Hirata, R.Hatakeyama,and Y.Kawazoe, J. Phys. Chem. B , 15529 (2004).[18] A.H¨ogele, Ch.Galland, M.Winger, and A.Imamo˘glu, Phys. Rev. Lett. ,217401 (2008). xciton-Plasmon Interactions in Individual Carbon Nanotubes , 073401 (2003).[20] T.G.Pedersen, Carbon , 1007 (2004).[21] R.B.Capaz, C.D.Spataru, S.Ismail-Beigi, and S.G.Louie, Phys. Rev. B ,121401(R) (2006).[22] C.D.Spataru, S.Ismail-Beigi, L.X.Benedict, and S.G. Louie, Phys. Rev. Lett. , 077402 (2004).[23] F.Wang, G.Dukovic, L.E.Brus, and T.F.Heinz, Phys. Rev. Lett. , 177401(2004).[24] F.Wang, G.Dukovic, L.E.Brus, and T.F.Heinz, Science , 838 (2005).[25] A.Hagen, M.Steiner, M.B.Raschke, C.Lienau, T.Hertel, H.Qian, A.J.Meixner,and A.Hartschuh, Phys. Rev. Lett. , 197401 (2005).[26] F.Plentz, H.B.Ribeiro, A.Jorio, M.S.Strano, M.A.Pimenta, Phys. Rev. Lett. , 247401 (2005).[27] F.Xia, M.Steiner, Y.-M.Lin, and Ph.Avouris, Nature Nanotechnology , 609(2008).[28] B.F.Habenicht and O.V.Prezhdo, Phys. Rev. Lett. , 197402 (2008).[29] V.Perebeinos, J.Tersoff, and Ph.Avouris, Phys. Rev. Lett. , 027402 (2005).[30] M.Lazzeri, S.Piscanec, F.Mauri, A.C.Ferrari, and J.Robertson, Phys. Rev. Lett. , 236802 (2005).[31] S.Piscanec, M.Lazzeri, J.Robertson, A.C.Ferrari, and F. Mauri, Phys. Rev. B75, 035427 (2007).[32] T.G.Pedersen, K.Pedersen, H.D.Cornean, and P.Duclos, NanoLett. , 291(2005).[33] D.J.Styers-Barnett, S.P.Ellison, B.P.Mehl, B.C.Westlake,R.L.House, C.Park,K.E.Wise, and J.M.Papanikolas, J. Phys. Chem. C , 4507 (2008).[34] I.V.Bondarev, H.Qasmi, Physica E , 2365 (2008).[35] I.V.Bondarev, K.Tatur, and L.M.Woods, Optics Commun. , 661 (2009).[36] I.V.Bondarev, K.Tatur, and L.M.Woods, Optics and Spectroscopy , 376(2010).[37] I.V.Bondarev, L.M.Woods, and K.Tatur, Phys. Rev. B , 085407 (2009).6 I.V.Bondarev, L.M.Woods, and A.Popescu [38] S.Zaric, G.N.Ostojic, J.Shaver, J.Kono, O.Portugall, P.H.Frings,G.L.J.A.Rikken, M.Furis, S.A.Crooker, X. Wei, V.C.Moore, R.H.Hauge,and R.E.Smalley, Phys. Rev. Lett. , 016406 (2006).[39] A.Srivastava, H.Htoon, V.I.Klimov, and J.Kono, Phys. Rev. Lett. , 087402(2008).[40] V.Perebeinos and Ph.Avouris, Nano Lett. , 609 (2007).[41] M.S.Dresselhaus, G.Dresselhaus, R.Saito, and A.Jorio, Annu. Rev. Phys.Chem. , 719 (2007).[42] A.Goodsell, T.Ristroph, J.A.Golovchenko, and L.V.Hau, Phys. Rev. Lett. ,133002 (2010).[43] H.Qian, C.Georgi, N.Anderson, A.A.Green, M.C.Hersam, L.Novotny, andA.Hartschuh, Nano Lett. , 1363 (2008).[44] P.H.Tan, A.G.Rozhin, T.Hasan, P.Hu, V.Scardaci, W.I.Milne, and A.C.Ferrari,Phys. Rev. Lett. , 137402 (2007).[45] R.Fermani, S.Scheel, and P.L.Knight, Phys. Rev. A , 062905 (2007).[46] A.Popescu and L.M.Woods, Appl. Phys. Lett. , 203507 (2009).[47] W.Vogel and D.-G.Welsch, Quantum Optics (Wiley-VCH, NY, 2006). Ch.10,p.337.[48] L.Kn¨oll, S.Scheel, and D.-G.Welsch, in:
Coherence and Statistics of Photonsand Atoms , edited by J.Peˇrina (Wiley, NY, 2001).[49] S.Y.Buhmann and D.-G.Welsch, Prog. Quantum Electron. , 51 (2007).[50] T.Pichler, M.Knupfer, M.S.Golden, J.Fink, A.Rinzler, and R.E.Smalley, Phys.Rev. Lett. , 4729 (1998).[51] C.D.Spataru, S.Ismail-Beigi, R.B.Capaz, and S.G.Louie, Phys. Rev. Lett. ,247402 (2005).[52] Y.-Z.Ma, C.D.Spataru, L.Valkunas, S.G.Louie, and G.R. Fleming, Phys. Rev.B , 085402 (2006).[53] J.Bellessa, C.Bonnand, J.C.Plenet, and J.Mugnier, Phys. Rev. Lett. , 036404(2004).[54] W.Zhang, A.O.Govorov, and G.W.Bryant, Phys. Rev. Lett. , 146804 (2006).[55] Y.Fedutik, V.V.Temnov, O.Sch¨ops, U.Woggon, and M.V. Artemyev, Phys. Rev.Lett. , 136802 (2007). xciton-Plasmon Interactions in Individual Carbon Nanotubes ,1827 (2009).[57] H.B.Chan, V.A.Aksyuk, R.N.Kleiman, D.J.Bishop, and F.Capasso, Phys. Rev.Lett. , 211801 (2001); Science , 1941 (2001).[58] I.V.Bondarev, G.Ya.Slepyan and S.A.Maksimenko, Phys. Rev. Lett. , 115504(2002).[59] I.V.Bondarev and Ph.Lambin, Phys. Rev. B , 035407 (2004).[60] I.V.Bondarev and Ph.Lambin, Phys. Lett. A , 235 (2004).[61] I.V.Bondarev and Ph.Lambin, Solid State Commun. , 203 (2004).[62] I.V.Bondarev and Ph.Lambin, Phys. Rev. B , 035451 (2005).[63] L.X.Benedict, S.G.Louie, and M.L.Cohen, Phys. Rev. B , 8541 (1995).[64] S.Tasaki, K.Maekawa, and T.Yamabe, Phys. Rev. B , 9301 (1998).[65] Z.M.Li, Z.K.Tang, H.J.Liu, N.Wang, C.T.Chan, R.Saito, S.Okada, G.D.Li,J.S.Chen, N.Nagasawa, and S.Tsuda, Phys. Rev. Lett. , 127401 (2001).[66] A.G.Marinopoulos, L.Reining, A.Rubio, and N.Vast, Phys. Rev. Lett. ,046402 (2003).[67] T.Ando, J. Phys. Soc. Jpn. , 777 (2005).[68] B.Kozinsky and N.Marzari, Phys. Rev. Lett. , 166801 (2006).[69] H.Haken, Quantum Field Theory of Solids , (North-Holland, Amsterdam, 1976).[70] S.Uryu and T.Ando, Phys. Rev. B , 155411 (2006).[71] L.D.Landau and E.M.Lifshits, The Classical Theory of Fields (Pergamon, NewYork, 1975).[72] K.Kempa and P.R.Chura, special edition of the Kluwer Academic Press Jour-nal, edited by L.Liz-Marzan and M.Giersig (2002).[73] K.Kempa, D.A.Broido, C.Beckwith, and J.Cen, Phys. Rev. B , 8385 (1989).[74] A.S.Davydov, Quantum Mechanics (Pergamon, New York, 1976).[75] V.N.Popov, L.Henrard, Phys. Rev. B , 115407 (2004).[76] M.F.Lin, D.S.Chuu, and K.W.-K.Shung, Phys. Rev. B , 1430 (1997).[77] H.Ehrenreich and M.H.Cohen, Phys. Rev. , 786 (1959).8 I.V.Bondarev, L.M.Woods, and A.Popescu [78] X.Blase, L.X.Benedict, E.L.Shirley, and S.G.Louie, Phys. Rev. Lett. , 1878(1994).[79] K.Kempa, Phys. Rev. B , 195406 (2002).[80] E.Hanamura, Phys. Rev. B , 1228 (1988).[81] J.P.Reithmaier, G.S¸ek, A.L¨offler, C.Hofmann, S.Kuhn, S.Reitzenstein,L.V.Keldysh, V.D.Kulakovskii, T.L.Reinecke, and A.Forchel, Nature , 197(2004).[82] T.Yoshie, A.Scherer, J.Hendrickson, G.Khitrova, H.M. Gibbs, G.Rupper, C.Ell,O.B.Shchekin, and D.G.Deppe, Nature , 200 (2004).[83] E.Peter, P.Senellart, D.Martrou, A.Lemaˆıtre, J.Hours, J.M.G´erard, andJ.Bloch, Phys. Rev. Lett. , 067401 (2005).[84] D.A.B.Miller, D.S.Chemla, T.C.Damen, A.C.Gossard, W.Wiegmann,T.H.Wood, and C.A.Burrus, Phys. Rev. Lett. , 2173 (1984).[85] D.A.B.Miller, D.S.Chemla, T.C.Damen, A.C.Gossard, W.Wiegmann,T.H.Wood, and C.A.Burrus, Phys. Rev. B , 1043 (1985).[86] A.Zrenner, E.Beham, S.Stufler, F.Findeis, M.Bichler, and G.Abstreiter, Nature , 612 (2002).[87] H.J.Krenner, E.C.Clark, T.Nakaoka, M.Bichler, C.Scheurer, G.Abstreiter, andJ.J.Finley, Phys. Rev. Lett. , 076403 (2006).[88] T.Ando, J. Phys. Soc. Jpn. , 024707 (2006).[89] T.Ogawa and T.Takagahara, Phys. Rev. B , 8138 (1991).[90] A.Jorio, C.Fantini, M.A.Pimenta, R.B.Capaz, Ge.G.Samsonidze, G.Dressel-haus, M.S.Dresselhaus, J.Jiang, N. Kobayashi, A.Gr¨uneis, and R.Saito, Phys.Rev. B , 075401 (2005).[91] A.Suna, Phys. Rev. , A111 (1964).[92] Y.Miyauchi, R.Saito, K.Sato, Y.Ohno, R.Iwasaki, T.Mizutani, J.Jiang, andS.Maruyama, Chem. Phys. Lett. , 394 (2007).[93] I.V.Bondarev, Y.Nagai, M.Kakimoto, T.Hyodo, Phys. Rev. B , 012303(2005).[94] R.Mukamel, Principles of Nonlinear Optical Spectroscopy (Oxford UniversityPress, New York, 1995). xciton-Plasmon Interactions in Individual Carbon Nanotubes
Quantum Mechanics: Non-Relativistic Theory (Pergamon, New York, 1977).[96] L.P.Gor’kov and L.P.Pitaevski, Dokl. Akad. Nauk SSSR , 822 (1963) [En-glish transl.: Soviet Phys.—Dokl. , 788 (1964)].[97] C.Herring and M.Flicker, Phys. Rev. , A362 (1964); C.Herring, Rev. Mod.Phys. , 631 (1962).[98] F.Wang, et. al., Phys. Rev. B , 241403(R) (2004).[99] D.Abramavicius, et. al., Phys. Rev. B , 195445 (2009).[100] A.Srivastava and J.Kono, Phys. Rev. B , 205407 (2009).[101] M.J.Sparnaay, Physica , 751 (1958).[102] S.K.Lamoreaux, Phys. Rev. Lett. , 5 (1997).[103] U.Mohideen and A.Roy, Phys. Rev. Lett. , 4549 (1998); F.Chen andU.Mohideen, Rev. Sci. Instrum. , 3100 (2001).[104] J.N.Munday, F.Capasso, and V.A.Parsegian, Nature , 170 (2009).[105] M.Bordag, U.Mohideen, and V.M.Mostepanenko, Phys. Rep. , 1 (2001).[106] V.A.Parsegian, van der Waals forces (Cambridge University Press, Cityplace-Cambridge, 2005).[107] R.F.Rajter, R.Podgornik, V.A.Parsegian, R.H.French, and W.Y.Ching, Phys.Rev. B , 045417 (2007).[108] A.Popescu, L.M.Woods, and I.V.Bondarev, Phys. Rev. B , 115443 (2008).[109] M.Bordag, B.Geyer, G.L.Klimchitskaya, and V.M.Mostepanenko, Phys. Rev.B 74, 205401 (2006).[110] C.T.Tai, Dyadic Green Functions in Electromagnetic Theory , 2nd Ed. (IEEEPress, Piscatawnay, NY, 1994).[111] L.W.Li, M.S.Leong, T.S.Yeo, and P.S.Kooi, J. Electr. Waves Appl. , 961(2000).[112] I.Cavero-Pelaez and K.A.Milton, Annals of Phys. , 108 (2005).[113] K.A.Milton, L.L.DeRaad, Jr., and J.Schwinger, Ann. Phys. , 388 (1978).[114] Handbook of Mathematical Functions , edited by M.Abramovitz and I.A.Stegun(Dover, New York, 1972).0
I.V.Bondarev, L.M.Woods, and A.Popescu [115] N.G. van Kampen, B.R.A.Nijboer, and K.Schram, Phys. Lett. A , 307(1968).[116] F.Intravaia and A.Lambrecht, Phys. Rev. Lett. , 110404 (2005).[117] J.F.Dobson, A.White, and A.Rubio, Phys. Rev. Lett. , 073201 (2006).[118] J.D.Jackson, Classical Electrodynamics (Wiley, New York, 1975).[119]
Higher Transcendental Functions , edited by H.Bateman and A.Erd´elyi (McGraw-Hill, New York, 1955). Vol. 3.[120] S.Liu, J.Li, Q.Shen, Y.Cao, X.Guo, G.Zhang, C.Feng, J.Zhang, Z.Liu,M.L.Steigerwald, D.Xu, and C.Nuckolls, Angew. Chem. , 4759 (2009).[121] P.L.Hern´andes-Martinez and A.O.Govorov, Phys. Rev. B , 035314 (2008).[122] H.Hirori, K.Matsuda, and Y.Kanemitsu, Phys. Rev. B78