Control of multiple excited image states around segmented carbon nanotubes
CControl of multiple excited image states around segmented carbon nanotubes
J. Kn¨orzer, ∗ C. Fey, † H. R. Sadeghpour, and P. Schmelcher
1, 3 Zentrum f¨ur Optische Quantentechnologien, Universit¨at Hamburg,Luruper Chaussee 149, 22761 Hamburg, Germany ITAMP, Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA The Hamburg Centre for Ultrafast Imaging, Luruper Chaussee 149, 22761 Hamburg, Germany (Dated: November 8, 2018)Electronic image states around segmented carbon nanotubes can be confined and shaped alongthe nanotube axis by engineering the image potential. We show how several such image states can beprepared simultaneously along the same nanotube. The inter-electronic distance can be controlled apriori by engineering tubes of specific geometries. High sensitivity to external electric and magneticfields can be exploited to manipulate these states and their mutual long-range interactions. Thesebuilding blocks provide access to a new kind of tailored interacting quantum systems.
PACS numbers: 32.10.-f, 32.80.-t, 32.60.+i
I. INTRODUCTION
Image-potential states above metallic surfaces thatarise from the interplay between an induced attractiveimage potential and a repulsive surface barrier have beenof considerable interest in surface studies of conductingmaterials [1–4]. Experimentally, these states can be ob-served with the aid of high-resolution time-resolved spec-troscopy techniques [5] and their lifetimes were found tobe on the order of tens of ps [6, 7]. The energy levels ofsuch states form a Rydberg series in the Coulomb-like im-age potential V ( z ) ≈ − [0 . e ( ε − / [( ε + 1)4 z ] [8] in adistance z above the flat surface with dielectric constant ε . In contrast, electronic states equipped with a non-zeroangular momentum l around infinitely long nanowiressuch as carbon nanotubes (CNTs) [9], or fullerenes [10],stem from the balance between image potential and cen-trifugal motion around nanostructure and are thus muchmore detached from the metallic surface. Tubular im-age states (TIS) surrounding CNTs were predicted someyears ago, [9] and experimentally observed shortly after-ward [11]. High-angular momentum states with (cid:96) > ∼ τ ≈ − µ s [12], but even low- (cid:96) TIS [13] have beenshown to decay considerably slower than their counter-parts above flat surfaces [11]. Extensive work has beendone on Bloch states in lattices of CNTs aligned parallel[14] and periodically arranged finite segments combinedby junctions constituting a single, long CNT [15]. Fur-thermore, image-potential states in other systems [16, 17]have been considered in the context of quantum infor-mation processing. Since they are long-lived states andhighly controllable by external fields [18, 19], Rydberg-like TISs turn out to be interesting candidates for suchpurposes. ∗ e-mail: [email protected] † e-mail: [email protected] Our aim is to propose a scheme for the simultaneouspreparation of multiple excited states around CNTs, al-lowing for subsequent studies of the long-range interac-tions in those systems. We consider single CNTs whoseelectronic properties give rise to confinement potentialsalong the longitudinal axis of the nanotube. Thereby,two external charges outside the CNT can be held at acontrollable distance. We have found that specifically en-gineered CNTs in the presence of an external charge giverise to highly adjustable image potentials.The present work is organized as follows: in Sec. II wedescribe the general properties of tubular image statesaround infinitely long nanowires. We address the ques-tion of how they are controllable in Sec. III A. Subse-quently, in Sec. III B, the properties of finite and infinitegeometries are interlinked. Sec. III C is dedicated to ageneral description of trapping two or more electrons intubular image states around one segmented CNT, wherethe inter-electronic distance is, in principle, a priori con-trollable by geometry. Furthermore, in Sec. III D we in-vestigate segmented arrays [15] of nanotubes and showthat by considering also not perfectly conducting CNTs[20], in principle arbitrary potentials can be generatedalong the longitudinal axis of the nanotube. As an out-look, we sketch potential applications and the descrip-tion of long-range interactions between two image statesin Sec. IV.
II. CYLINDRICAL IMAGE STATES ABOVENANOWIRES
As for charged particles in the vicinity of flat metallicsurfaces, the method of images [21] can be applied in or-der to describe the attractive force between an infinitely-long hollow metallic cylinder of radius a and a charge q at r = ( ρ > a, z , ϕ ). The polarization of the cylindricalsurface can be described by the induced scalar potentialΦ ind at r = ( ρ, z, ϕ ) via an expansion in terms of regularand irregular Bessel functions [22], K α and I α , α ∈ Z ,respectively: a r X i v : . [ c ond - m a t . m e s - h a ll ] A ug Φ ind ( r , r ) = − qπ ∞ (cid:88) α = −∞ ∞ (cid:90) dk cos[ k ( z − z )] × exp[ iα ( ϕ − ϕ )] I α ( ka ) K α ( ka ) K α ( kρ ) K α ( kρ ) . (1)Evaluated at the position of the external charge r = r ,this yields the image potential which the charge feels: V im ( ρ ) = q ind (cid:12)(cid:12) ( ρ , , = − q π ∞ (cid:88) α = −∞ ∞ (cid:90) dk I α ( ka ) K α ( ka ) K α ( kρ ) . (2)Asymptotic forms of this potential were derived andan analytical expression was informed by the asymptotics[9]. In most studies thereafter, the analytical form wasused in calculations in order to reduce numerical costs.Recently it was pointed out [20] that quantitative differ-ences in eigenenergies and eigenstates existed, using theexact, cf. Eq. (2), and approximate [9] potentials, respec-tively.Introducing the repulsive angular momentum barrier,in the absence of external fields, the one-dimensional ef-fective potential reads V (cid:96) ( ρ ) = V im ( ρ ) + (cid:96) − / mρ , (3)the parameter (cid:96) being the angular momentum and m denoting the mass of the charged particle, in the followingassumed to be an electron. Hereafter, we use atomicunits, i.e. (cid:126) = m e = e = (4 πε ) − = 1. Whether ornot detached bound states exist primarily depends onthe angular momentum quantum number. For too low (cid:96) the effective potential is overall attractive everywhereand for too high (cid:96) there is no detached local potentialminimum accompanying the angular momentum barrier.In earlier works (cid:96) min = 5 was already found to be theminimum angular momentum which supports bound TIS[20] around infinite nanowires.The total electronic wavefunction separates in cylin-drical coordinates ( ρ, z, ϕ ),Ψ (cid:96)nk ( ρ, z, ϕ ) = ψ (cid:96)n ( ρ ) √ πρ e i(cid:96)ϕ φ k ( z ) , (4)where ρ = 0 refers to the center of the tube. This allowsus to express the total energy as a sum of transverse andlongitudinal energies, E (cid:96)nk = E (cid:96)n + E k , where the trans-verse part scales for fixed n as E (cid:96)n ∝ (cid:96) − [9]. Typicalbinding energies range between 15 meV and a few meV.The radial wavefunction ψ (cid:96)n ( ρ ) satisfies the radialSchr¨odinger equation (cid:18) ∂ ∂ρ + 2[ E (cid:96)n − V (cid:96) ( ρ )] (cid:19) ψ (cid:96)n ( ρ ) = 0 . (5)States with the same radial quantum number n , i.e. num-ber of nodes of ψ (cid:96)n , and different (cid:96) are non-degenerateand are, in general, energetically farther apart than ad-jacent states of same (cid:96) but different n . The maximumelectronic density is found at a distance ρ max from theCNT, which scales as ∝ (cid:96) [9]. This makes a certainrange of angular momenta (cid:96) > ∼ (cid:96) min especially interestingfor our purposes since farther away from the surface thestates are experimentally hard to protect against extra-neous influence by the surrounding medium [11]. III. CONTROL OF IMAGE STATES
TISs may be controlled by means of external fields.However, these do not serve as the only control parame-ters for the states. Furthermore, the radius of the nan-otube can be adjusted by rolling up a graphene sheetaccordingly [23]. So far, studies have focused on (10,10)armchair nanotubes ( a = 0 .
68 nm). Tubes with smaller(larger) diameters tend to support states with smaller(greater) spatial extent. Moreover, the length and con-ducting properties may alter the TIS properties as well.
A. Impact of magnetic and electric fields
Tubular image states can be strongly affected by ex-ternal magnetic fields applied parallel to the longitudinalaxis z [12]. Neglecting any contributions stemming fromthe electron’s spin due to its global character, two addi-tional terms, H Z and H d , a Zeeman and a diamagneticterm, respectively, enter the Hamiltonian and thus theSchr¨odinger Eq. (5), H Z = − B L z ,H d = B ρ , (6)where L z denotes the z -component of the angular mo-mentum operator. For TISs, being highly extended, thediamagnetic term can dominate. Higher- (cid:96) states are far-ther away from the tube’s surface, e.g. (cid:104) ρ (cid:105) n =1 (cid:96) =5 ≈ (cid:104) ρ (cid:105) n =1 (cid:96) =7 ≈
18 nm for the n = 1 ground states of a (10,10)CNT of radius a = 0 .
68 nm. Hence, states possessinghigher angular momenta are more strongly affected bythe presence of a magnetic field. The respective groundstates of Eq. (5) are shown for (cid:96) = 5 , ..., (cid:96) = 5, the states with or without an external mag-netic field of B = 20 T essentially coincide whereas the (a)(b) FIG. 1. (a) Radial wavefunctions ψ (cid:96) ( ρ ) for angular momenta (cid:96) = 5 − (from left to right) in the case of no external fields (solid) and with a magnetic field of B = 20 T applied (dashed) parallel to the longitudinal axis of the nanotube. (b) Meandisplacement (cid:104) y (cid:105) perpendicular to the tube for an electric fieldapplied along the y -axis for a ( l = 5 , n = 1)-state. Theinsets show the probability densities of the perturbed statesfor the field strengths E = 50 V/cm, 250 V/cm and 500 V/cm,respectively. White dots indicate the nanotube. higher- (cid:96) states differ significantly. Also, the spectrum isaltered, again more significantly for higher-lying states.Under the application of external electric fields perpen-dicular to the longitudinal axis of the nanotube, TISslose their rotational symmetry, as states of different (cid:96) mix. The consequent electric-field induced decenteringand distortion of the image states makes the position ofmaximum probability density vary with the field strength E , cf. Fig. 1(b). This decentering is reminiscent of a field-induced electric dipole. Longitudinally confined statescan be selectively addressed by an external electric field.Hence, electric fields can be exploited to tune the inter-action between two image states. B. Impact of finite-length segmentation
The wavefunctions in Eq. (4) reflect that the exter-nal electron is spatially not confined along the longitu-dinal axis z . In order to induce a confining potentialalong z , we make use of the versatile electronic prop-erties of CNTs which, above all, depend on the geom-etry of the CNT. Finite CNT segments with differentelectronic properties can be merged into a single CNTof finite length which inherits its conducting propertieslocally from its constituents. Moreover, the individualsegments can differ in length and diameter. Periodic ar-rays of such finite-sized metallic nanotubes have beeninvestigated with a focus on Bloch states in systems ofidentical segments connected by junctions [15]. Thesejunctions are constructed by a rotationally asymmetricseries of pentagon-heptagon defects in the lattice struc-ture [24, 25]. In contrast, we explore CNTs consistingof non-periodically arranged segments which eventuallygive rise to confinement potentials for the external elec-tron along the axis of the nanotube. In a next step, wedevelop a scheme that allows the preparation of severalsuch excited states around one suspended CNT, which inturn eventually may consist of a series of metallic, semi-conducting and insulating CNT segments.As the standard method of images in electrostatics is,for flat surfaces as for cylindrical geometries, grounded onthe assumption of an infinitely extended metal [21], theattractive portion of the potential , binding the electronto a CNT of finite length, has to be computed differently.Since the external electron induces a charge distributionon the nanotube surface with which it interacts, the cal-culation of the image potential in the presence of a finiteCNT reduces to the determination of the induced surfacecharge density. A discretized charge density, consisting of N = 10 to N = 10 tile charges { q i } ≤ i ≤ N at positions { r j } ≤ j ≤ N on the surface, distributed along and aroundthe CNT, yields the image potential [15] as a sum overCoulombic interactions between the external electron andthe tile charges: V im ( r ) = − N (cid:88) i =1 q i | r − r i | . (7)The effective potential in Eq. (3) is thus the sum of theabove potential and the centrifugal barrier potential. Weare now interested in the numerical solutions of the wave-functions χ (cid:96)n ( ρ, z ) from Eq. (8). Due to the longitudinalconfinement which couples ρ - and z -motion, they now de-pend on both ρ and z and are labeled by (cid:96) and n , a quan-tum number counting the energy levels. The Schr¨odingerequation, For the sake of simplicity, we will call the attractive part of thepotential also the image potential for finite CNTs. L CNT (cid:96) . µ m -2.80 . µ m -7.1 -1.9 -0.40 . µ m -10.3 -3.9 -1.7 -0.71 . µ m -13.5 -6.9 -4.2 -2.3 ∞ -14.0 -7.7 -4.8 -3.3TABLE I. Binding energies of degenerate bound states for(10,10) nanotubes of various lengths in meV. Blank cells indi-cate that no bound states exist. Our L CNT = ∞ results agreeup to less than a percent with [20]. (cid:20) − (cid:18) ∂ ∂ρ + ∂ ∂z (cid:19) + V (cid:96) ( ρ, z ) − E (cid:96)n (cid:21) χ (cid:96)n ( ρ, z ) = 0 , (8)can be solved, e.g., by employing a two-dimensional fi-nite differences method [26] or by setting up a discretevariable representation [27].Since the gapped excitation spectrum of longitudinalexcitations features much smaller transition energies thanradial excitations, an adiabatic approximation in z maybe employed, reducing the numerical efforts of solvingEq. (8) to the evaluation of two one-dimensional eigen-value equations [15]. Since the primary computationalcost stems from the calculation of the surface tile chargesand thus the image potential, i.e. Eq. (7), whether or notthe adiabatic approximation is employed is of minor im-portance. Therefore, we have in most calculations solvedthe full problem of Eq. (8), because especially for higher-lying states the resulting binding energies of the exactand approximative calculations do differ up to a few per-cent.As to be expected, the image potential becomes moreattractive as the length of the CNT, L CNT , increases, cf.Fig. 2(a). In the depicted range, i.e. 5 nm ≤ ρ <
40 nm,the finite-tube results converge to the image potential ofan infinite nanowire as L CNT is increased. Of course, inthe long-range limit ρ → ∞ the electron feels the dif-ference in attraction no matter the length of the nan-otube. The lowest-lying bound states of Eq. (5) extendto less than 50 nm away from the tube’s surface. Hence,finite-size effects affect the image potential considerablyfor moderate lengths L CNT < ∼ µ m. The binding en-ergies of the resulting TIS are also profoundly affectedespecially in this length regime. This is demonstratedin Table I for the respective n = 1 ground states. Theusual (cid:96) − scaling of the binding energies with the angu-lar momentum is, for high- (cid:96) states, distorted by finite-size effects, since these states are to be expected far out-side and therefore witness the finite spatial extent of theCNT. Note that finite geometries lead to edge effectsthat are also present here. Therefore, for the potentialcurves in Fig. 2(a), V im ( ρ, z ) is evaluated at z = L CNT / (a)(b)(c) FIG. 2. (a) Image potentials for (10,10) carbon nanotubesof various lengths. The potential is evaluated at z = L CNT / L CNT is the length of the respectivenanotube. (b) Image potential at ρ = 10 nm and evaluatedalong the longitudinal axis z for a (10,10) carbon nanotube oflength L CNT = 1600 nm. (c) Sketch of CNT of length L CNT .The electronic orbit at z = L CNT / cf. Fig. 2(c). Around this value, V im is symmetric, i.e. V im ( ρ, L CNT / − z ) = V im ( ρ, L CNT / z ). Fairly long nan-otubes almost generate box potentials, with shallow wellsat the ends, along their axes at radial distances wherethe image-potential states are most likely to be found,cf. Fig. 2(b). In the presence of a single segment, theenergetically lowest-lying states tend to be localized atthe ends of the segment. An intuitive understanding ofthis can be obtained by looking at the induced chargedistribution on the CNT. This charge distribution willbe symmetric about the center of the tube if the ex-ternal electron is located above the center of the CNT.Therefore, on both sides of the electron in longitudinaldirections, charges drag the electron towards the edges.The longitudinal components of these forces cancel. In (a)(b) A BFIG. 3. (a) Scheme of three-segment nanotube with two con-fined electrons in tubular image states. d is the length ofthe insulating segment that connects two metallic segmentsat whose ends the states are localized. The distance R de-notes the mean-value distance of the electronic orbits. (b)The corresponding effective potential is evaluated along z atthe radial distance of maximum probability density. Energylevels of the first three single-electron eigenstates of even par-ity ( n = 1 , ,
5) in the radial ground states (red) as well asthe first three single-electron eigenstates of even parity in thefirst-excited (blue) radial-state manifolds are shown for bothsubsystems. contrast, if the electron is located closer to one of theedges, the induced charge density is also located closerto this edge and as the external electron approaches theedge, the induced charge distribution will be localized atthis edge. Therefore, at fixed radial distance from theCNT, there is a total attraction towards the edges. Aperfectly symmetric image potential around z = L CNT / C. Localization of multiple excited states
An electron above a non-metallic surface does eithernot feel the feedback of the material in form of an in-duced image potential at all, if the material is insulating,or sees a weaker image potential than in the metalliccase, depending on the precise electronic properties ofthe semi-conducting CNT. Therefore, by connecting twoconducting segments by an insulator, two TISs can bespatially separated, cf. Fig. 3(a). In this way, the dis-tance between two states can be controlled a priori viathe length of the insulating segment. In Fig. 3(b), theeffective potential V ABeff ( r A , r B ) ≈ V Aeff ( r A ) + V Beff ( r B ) as afunction of z at the radial distance of maximum proba-bility ρ = ρ max is shown for a system consisting of twospatially separated electrons both in TISs. Along the in-sulating segment, the effective potential is set to zero.The two external electrons are confined around metal-lic segments of L ACNT = 0 . µ m and L BCNT = 0 . µ m, re-spectively, with angular momenta (cid:96) A = 6, (cid:96) B = 5. Thedistance R between the electronic orbits is, most of all,dictated by the length d = 0 . µ m of the insulating andthe lengths of the metallic segments, since these stateshave significant amplitudes only near the edges. Also,Fig. 3(b) shows the energy levels of the three lowest-lying even parity one-electron states in both the radialground and first excited state, respectively. The odd andeven parity states lie energetically so close that they can-not be resolved in Fig. 3(b). Initializing a single-electronstate at one of the edges of its segment in a superposition | L (cid:105) σ = 1 / √ | (cid:105) σ + | (cid:105) σ ) or | R (cid:105) σ = 1 / √ | (cid:105) σ − | (cid:105) σ ), σ = A, B labeling the two TISs around their segmentsand | (cid:105) σ ( | (cid:105) σ ) denoting the corresponding even (odd)parity state, the interaction with a second TIS can be al-tered significantly since thereby, the mean interelectronicdistance R can be controlled. Since the odd and even par-ity states lie energetically very close, the lifetimes of the | L (cid:105) σ and | R (cid:105) σ states are comparable with the estimatedlifetimes due to other decay channels.The states can also be manipulated a priori by insert-ing semi-conducting segments. For finite dielectric con-stant ε , i.e. the low-frequency limit of the permittivity ε ( ω ), the leading-order correction to the image poten-tial in Eqs. (2) and (7) scales as 1 /ε [28]. Because ofthe sparse literature available on dielectric functions ofCNTs in the static limit [29], a two-fluid model [30] wasapplied to infinitely long CNTs with the aim of findinga way to properly describe the image potentials due tosemi-conducting CNTs [20]. However, for the sake ofsimplicity, we have calculated image potentials of semi-conducting CNTs in the present work by assuming amacroscopically defined dielectric constant. For CNTs,it might be obtained from projecting the dielectric tensorof graphite onto a cylinder [31]. The smaller the dielec-tric constant is, the fainter is the attractive inward forceon the electron due to the image potential. It has beenremarked [20] that the band gap of a semi-conductingCNT should not exceed a value of 0 . D. Engineering of image potentials
The electronic properties of carbon nanotubes are de-termined by their chiral indices ( n, m ) [33, 34]. There-fore, the dielectric constants vary from nanotube to nan-otube. Hence, by merging different CNTs, step-wise po-tentials can be generated along the longitudinal axis of asegmented CNT. The dielectric constants of the individ-ual segments act as scaling factors for the attractive partof the potential which is thus varied along the CNT fromsegment to segment. The only limiting factor for generat-ing arbitrary step-wise potentials along z is the availabil-ity of CNTs with the desired electronic properties. How-ever, nanotubes do possess versatile electronic propertiesand these may even be altered by doping techniques [35].Two nanotubes can be fused by imprinting pentagon-heptagon defects into the lattice structure [24, 25] whichserve as junctions between both tubes. To a certain de-gree, the chiral indices of the intial nanotube segmentsdetermine the conductivity of the joint CNT which in-herits the electronic properties from its constituents. Inaddition, the relative rotational symmetry of the junc-tion plays a crucial role. The rotational symmetry ofa CNT is related to the lines of allowed k vectors inthe Brillouin zone, and the rotational symmetry of thejunction refers to how the pentagon-heptagon defects andhexagons are ordered along the circumference of the nan-otube. Now considering a scattering process of a surfaceelectron from one segment to another, the outcome of thisstrongly depends on the relative rotational symmetriesof the segments as well as on the rotational symmetryof the junction. For example, a rotationally symmetricjunction cannot mediate between two electronic surfacestates of different symmetries due to energy and angularmomentum conservation laws [36]: it cannot impart anyadditional angular momentum to the initial state which isthen totally reflected at the junction. Therefore, depend-ing on how two CNT segments are connected, the wholenanotube may feature different electronic properties.To illustrate the influence of the junctions’ symme-tries on the image potential of a series of joint CNT seg-ments, Fig. 4 shows two image potentials of the same nan-otubes but connected with different junctions. Twentysegments, all 20 nm in length, are stitched together bytwo distinct sorts of junctions: In the first case, segmentswith identical rotational symmetry properties, connectedby a junction with a rotational symmetry mismatch rel-ative to the segments it merges, constitute a nanotubewhich is overall not conducting. The resulting effective FIG. 4. Image potential V im ( ρ, z ) due to a carbon nanotube oflength L CNT = 400 nm consisting of twenty identical 20-nm-long segments fused by (a) rotationally asymmetric or (b)rotationally symmetric junctions, respectively. potential V eff will not be smooth but feature ripples, cf.Fig. 4(a). In the second case, an almost smooth poten-tial is the outcome as the junctions possess the samesymmetry as the semi-conducting segments. At fixedradial distance ρ , the resulting potential is of approxi-mately harmonic nature, cf. Fig. 4(b), and yields almostequidistantly spaced energy levels. The potentials werecalculated by adjusting the dielectric constants of the in-dividual segments accordingly in Eq. (7). IV. SUMMARY AND OUTLOOK
In summary, we have shown how multiple excited im-age states can be confined and arranged along a compos-ite nanotube, comprised of non-periodically aligned finitenanotube segments. A framework for creating highly ad-justable image potentials has been proposed, wideningthe range of potential control mechanisms for these exoticstates. This work paves the way for subsequent studies ofmany-electron systems around CNTs and their interac-tions. The Hamiltonian of a bipartite system consistingof two image states can be represented as a sum of twoisolated image-state Hamiltonians and their mutual in-teraction which contains the electrostatic interaction V ,expressed via the two-body density ˆ ρ AB as V = (cid:90) ˆ ρ AB ( r − r (cid:48) ) | r − r (cid:48) | d r d r (cid:48) . (9)Spin-dependent electron exchange and magnetic dipole-dipole interactions may be considered. We expect strongTIS-TIS interactions at distances < µ m. The R -dependent interaction energy to second order can be com-puted efficiently with the wavefunctions in hand. Weplan to extend these studies to interacting TISs, spin-orbit interaction, and design of practical tube settings.Image states on liquid helium have already been demon-strated as manipulable and strongly interacting sets ofqubits [16, 17]. Their versatility, especially the rather long lifetimes and moderate binding energies, but alsotheir tunability makes TIS advantageous candidates forquantum information applications. ACKNOWLEDGMENTS
The authors thank D. Segal for valuable comments aswell as S. Kr¨onke and S. Markson for helpful remarks onthis manuscript. J. K. and P. S. gratefully acknowledgesupport by the National Science Foundation through agrant for the Institute for Theoretical Atomic, Molecularand Optical Physics at Harvard University and Smith-sonian Astrophysical Observatory. J. K. acknowledgesfinancial support by the German Academic ExchangeService and C. F. is grateful for support by the Studi-enstiftung des Deutschen Volkes in the framework of ascholarship. [1] P. M. Echenique and J. B. Pendry, J. Phys. C: Solid StatePhys. , 2065 (1978).[2] P. M. Echenique and J. B. Pendry, Surf. Sci. , 125(1990).[3] P. M. Echenique, J. M. Pitarke, E. V. Chulkov, and V. M.Silkin, J. Electr. Spectr. Rel. Phen. , 163 (2002).[4] S. Bose, V. M. Silkin, R. Ohmann, I. Brihuega1, L. Vi-tali1, C. H. Michaelis, P. Mallet, J. Y. Veuillen, M. A.Schneider, E. V. Chulkov, et al., New J. Phys. (2010).[5] U. H¨ofer, I. L. Shumay, C. Reu, U. Thomann, W. Wal-lauer, and T. Fauster, Science , 1480 (1997).[6] S. Link, H. A. D¨urr, G. Bihlmayer, S. Bl¨ugel, W. Eber-hardt, E. V. Chulkov, V. M. Silkin, and P. M. Echenique,Phys. Rev. B , 115420 (2001).[7] W. Berthold, U. H¨ofer, P. Feulner, E. V. Chulkov, V. M.Silkin, and P. M. Echenique, Phys. Rev. Lett. , 056805(2002).[8] M. W. Cole and M. H. Cohen, Phys. Rev. Lett. , 1238(1969).[9] B. E. Granger, P. Kr´al, H. R. Sadeghpour, andM. Shapiro, Phys. Rev. Lett. , 135506 (2002).[10] G. Gumbs, A. Balassis, A. Iurov, and P. Fekete, TheScientific World Journal , 726303 (2014).[11] M. Zamkov, N. Woody, S. Bing, H. S. Chakraborty,Z. Chang, U. Thumm, and P. Richard, Phys. Rev. Lett. , 156803 (2004).[12] D. Segal, P. Kr´al, and M. Shapiro, Surf. Sci. , 86(2005).[13] M. Zamkov, H. S. Chakraborty, A. Habib, N. Woody,U. Thumm, and P. Richard, Phys. Rev. B , 115419(2004).[14] D. Segal, B. E. Granger, H. R. Sadeghpour, P. Kr´al, andM. Shapiro, Phys. Rev. Lett. , 016402 (2005).[15] D. Segal, P. Kr´al, and M. Shapiro, Phys. Rev. B ,153405 (2004).[16] P. M. Platzman and M. I. Dykman, Science , 1967(1999).[17] M. Dykman and P. Platzman, Fortschritte der Physik , 1095 (2000).[18] D. Segal, P. Kr´al, and M. Shapiro, Chem. Phys. Lett. , 314 (2004).[19] D. Segal, P. Kr´al, and M. Shapiro, J. Chem. Phys. ,134705 (2005).[20] S. Segui, C. Celed´on L´opez, G. A. Bocan, J. L. Gervasoni,and N. R. Arista, Phys. Rev. B , 235441 (2012).[21] J. D. Jackson, Classical electrodynamics (Wiley, NewYork, NY, 1999), 3rd ed.[22] N. R. Arista and M. A. Fuentes, Phys. Rev. B , 165401(2001).[23] M. Terrones, Ann. Rev. Mat. Res. , 419 (2003).[24] J.-C. Charlier, T. W. Ebbesen, and P. Lambin, Phys.Rev. B , 11108 (1996).[25] L. Chico, V. H. Crespi, L. X. Benedict, S. G. Louie, andM. L. Cohen, Phys. Rev. Lett. , 971 (1996).[26] R. LeVeque, Finite Difference Methods for Ordinary andPartial Differential Equations: Steady-State and Time-Dependent Problems (Society for Industrial and AppliedMathematics, 2007).[27] J. C. Light and T. Carrington, Adv. Chem. Phys ,263 (2000).[28] J. L. Gervasoni and N. R. Arista, Phys. Rev. B ,235302 (2003).[29] W. Lu, D. Wang, and L. Chen, Nano Lett. , 2729 (2007).[30] D. J. Mowbray, S. Segui, J. Gervasoni, Z. L. Miˇskovi´c,and N. R. Arista, Phys. Rev. B , 035405 (2010).[31] T. St¨ockli, Z. L. Wang, J.-M. Bonard, P. Stadelmann,and A. Chˆatelain, Phil. Mag. B , 1531 (1999).[32] M. Saffman, T. G. Walker, and K. Mølmer, Rev. Mod.Phys. , 2313 (2010).[33] R. Saito, G. Dresselhaus, and M. S. Dresselhaus, PhysicalProperties of Carbon Nanotubes (World Scientific Pub-lishing, 1998).[34] J.-C. Charlier, X. Blase, and S. Roche, Rev. Mod. Phys. , 677 (2007).[35] X. Li, L. M. Guard, J. Jiang, K. Sakimoto, J.-S. Huang,J. Wu, J. Li, L. Yu, R. Pokhrel, G. W. Brudvig, et al., Nano Lett. , 3388 (2014). [36] L. Chico, L. X. Benedict, S. G. Louie, and M. L. Cohen,Phys. Rev. B54