Dielectric Screening inside Carbon Nanotubes

Dielectric screening plays a vital role in determining physical properties at the nanoscale and affects our ability to detect and characterize nanomaterials using optical techniques. We study how dielectric screening changes electromagnetic fields and many-body effects in nanostructures encapsulated inside carbon nanotubes. First, we show that metallic outer walls reduce the scattering intensity of the inner tube by 2 orders of magnitude compared to that of air-suspended inner tubes, in line with our local field calculations. Second, we find that the dielectric shift of the optical transition energies in the inner walls is greater when the outer tube is metallic than when it is semiconducting. The magnitude of the shift suggests that the excitons in small-diameter inner metallic tubes are thermally dissociated at room temperature if the outer tube is also metallic, and in essence, we observe band-to-band transitions in thin metallic double-walled nanotubes.

eh + e - h + E k E 1.9 2.0 2.1 2.2 2.3 2. 4  RBM intensity (arb.units) Excitation energy (eV) (11,2)@S (11,2)@M exciton single particle Key words: Dielectric screening, excitons, one-dimensional heterostructures, double-walled nanotubes, resonant Raman, carbon nanotubes 2][3][4][5][6][7][8] A CNT container may be exploited as a drug carrier and local sensor. 9,10It also provides a unique environment to tailor and study encapsulated materials.For example, water changes its dielectric behavior and viscosity inside tubes and can adopt new phases. 11,12Such water@CNT channels were used for ion transport to potentially model biological systems like transmembrane proteins. 13In another direction, CNTs act as templates to order and align molecules, which led to micron-sized single file J aggregates or molecular chains with huge optical nonlinearities. 3,14Despite extensive studies on filled nanotubes and hybrid one-dimensional (1D) systems, the environment produced by an encapsulating CNT remains mysterious.For instance, 1D chains of dye molecules or carbon atoms inside a nanotube yield record-high Raman cross sections. 3,15This enhancement may be due to intrinsic effects, molecule-molecule interaction, molecule-wall coupling such as state hybridization, or dielectric effects by the nanotube wall.In case of 1D molecular and carbon chains it is impossible to discriminate between the different effects, because they only exist inside the CNTs and cannot be extracted and studied under ambient conditions.
There are some indications for systematic changes of materials inside CNTs.For example, the nanotube walls affect the electromagnetic (EM) field inside the CNT.C 60 @CNT demonstrated different depolarization ratios, 16 but it remained unclear whether this was related to the depolarization of the EM field or strain.On the other hand, single-walled nanotubes are stable in ambient conditions or can be an inner part of a double-walled CNT (DWCNT). 17e DWCNTs can potentially serve as ideal probes for the environment produced by a nanotube, since an inner wall nanotube may be easily referenced to an SWCNT 18 and will probe the environment produced by the outer wall.
The nanotube walls may also alter many-body effects of encapsulated materials, because electron-hole interactions are subjected to the exterior screening. 19This may fundamentally change how collective states form, for example, for 1D J aggregates inside tubes.It has often been suggested that nanotube excitons are tuned by interior filling, 4,20 but the inverse effect, where a CNT affects the excitons of encapsulated species has not received much attention.For example, the excitonic series of carbyne are screened by the CNT wall, 21 but the intrinsic exciton energies remain unknown.This effect may become particularly noteworthy for metallic outer CNTs, because the metallic species are expected to provide a much denser dielectric environment. 22For DWCNTs the screening by an outer tube was predicted to change the excitons of the inner tube to single-particle excitation in small diameter CNTs, 23 but experimental evidence has not been reported so far.
In this work we study dielectric screening by metallic CNTs using resonant Raman scattering on DWCNTs.The EM screening by the outer wall reduces the inner tube Raman intensity in metallic@metallic DWCNTs by a factor of ∼100, in agreement with a dielectric model of a hollow cylinder in the quasi-static approximation and a dielectric constant of ϵ oT = 10.8 for the outer metallic wall.We compare the inner tube excitonic transition energies for semiconducting and metallic hosts.The transition energies shift to lower energies compared to SWCNTs, which we analyze in the framework of dielectric screening.The magnitude of the shift in the transitions energies of small inner metallic tubes is compatible with a complete dissociation of the exciton due to dielectric screening.

II. METHODS
To study the dielectric screening effects in the inner walls, we need to sort as-grown DWC-NTs into fractions according to the electronic character of the inner and outer wall. 24We sorted the (inner@outer) DWCNTs using three-step technique into the electronic fractions M@M,M@S, S@M, and M@M, where M indicates metallic and S semiconducting character of the wall.The DWCNTs were filtered first in a gel permeation chromatography column by monitoring Raman intensity.Second, the DWCNTs were re-suspended in toluene and chlorobenzene with PFO-BPy polymer for improved outer-wall separation.At the final step pellets were extracted from solutions by 1 hour centrifugation at up to 10 6 g and deposited onto silicon substrates. 24After drying, the samples were used for resonant Raman experiments.
Transition energies and electromagnetic screening were analyzed with resonant Raman spectroscopy of the radial breathing mode (RBM). 25,26Two excitation-tunable lasers were used as excitation sources, for the visible excitation range 570-670nm a Radiant dye laser (DCM, R6G) and a Coherent Ti-Sa laser for near infra-red (700-850nm).The laser was focused onto the sample using a 100x microscope objective (N.A. 0.9) with position and focus optically controlled by a camera.The back-scattered light was filtered by a Horiba triple grating t64000 system to remove the Rayleigh light and dispersed by 600 and 900 grooves/mm gratings.A Peltier cooled charge-coupled device was detecting the Raman signals.The spectra of 532 nm laser line were acquired with a Horiba Xplora, singlegrating spectrometer equipped with a dichroic mirror.The (n, m) chiralities were identified and RBM shifts were investigated systematically using multi-peak fitting.The concept of laola family facilitated chiral identification, the (n, m) from the same laola group share the parameter l = 2n + m.In M@M and M@S samples we found the l = 2m + n = 24 laola groups -containing the (n, m) chiralities (9,6), (10,4), (11,2), and (12,0) -plus the l =27 and 30 laola groups.A CaF 2 single crystal reference spectrum was measured for each laser wavelength.The integrated area of the RBM peaks was divided by the area of CaF 2 peak at 320 cm −1 in order to account for changes in the sensitivity of optical components in the measurement. 25

III. RESULTS
A. Theory of dielectric screening in one dimension.
We consider a situation where a single-walled carbon nanotubes is embedded in another tube, Fig. 1a.Neglecting direct tube-tube coupling, the outer tube creates a dielectric environment that changes the optical response of the inner tube in two ways: The outer tube reduces the amplitude of an externally applied electric field for the inner wall and the field orientation so that it is predominantly polarized along the axis. 19The change in polarization direction arises from what has been coined the antenna effect, 19 i.e., the fact that where f loc is the local field factor, E 0 external electric field, and ϵ env is a dielectric constant of the environment around the outer tube.The effective dielectric constant ϵ eff arises from the combined dielectric effect of the outer wall itself with ϵ oT and the inner core ϵ i (ϵ i = 1 for an empty tube).We calculate it according to Maxwell-Garnet mixing [28][29][30] where The effective dielectric constant ϵ eff changes the exciton binding energies E b , the electronelectron interaction, and, therefore, the optical transition energy E ii of the inner tube.The exciton binding energy scales as 31 For example E ϵ=1 b ≈ 100 meV for metallic tubes with d = 1−2.2nm 22 is the intrinsic binding energy for the unscreened CNT and α = 1.2 − 1.4 is a semi-empirical scaling factor [31][32][33] .
The change of the optical transition energy is given by 31,34 The single particle band gap E sp is independent of ϵ eff .E BGR is the electron-electron interaction energy for the unscreened system. 31The electron-electron correlation E ee scales as ϵ −1 eff for small electron wave vectors according to the Coulomb potential. 35The electron-hole E eh interaction is more complex and leads back to a hydrogen arom problem in one dimension, where a cutoff potential is typically introduced to converge the ground state ∼ 1/ |z 0 + z|.
This yields E eh = R * h /λ 2 , where the effective Rydberg radius R * h depends on ϵ eff and λ also varies with the potential cutoff z 0 as a function of ϵ eff .Combining these two factors one gets 31 The different scaling of the electron-electron and the electron-hole interaction yields an overall shift of the optical excitation energy as given by Eq. ( 4).
The optical response of the inner tube is sensitive to the change of the local field and the exciton transition energies caused by the outer tube.In principle, the optical effects can be studied by any optical techniques such as photoluminescence-excitation 36 , direct absorption 37,38 , or resonance Raman spectroscopy 25,39,40 .However, photoluminescence is only present in semiconducting inner tubes and is strongly quenched by the outer wall.
Optical absorption is challenging to measure experimentally due to the low cross sections and signal overlap in chiral mixtures.On the other hand, resonant Raman scattering provides sufficient signal for metallic and semiconducting walls up to the single tube level. 41sonance Raman spectroscopy of the radial breathing modes (RBM) is a key method to follow optical and vibrational changes in carbon tubes. 19,25,39,40The phonon energy of the RBM ℏω RBM (d) = c 1 /d + c 2 depends on tube diameter allowing one to distinguish between nanotubes of different size (e.g., inner-and outer-tube).Empirical parameters c 1 and c 2 depend on to the filling, exterior functionalization, and wall-to-wall interactions. 4,17,26,39,42e optical transitions of CNTs are probed via resonance Raman profiles, i.e., the dependence of the scattering intensity on laser excitation. 19,25,26,39The screening by the outer tube manifests in a shift of the resonant Raman profile that is determined by the optical transition energy E ii , Fig. 1b.The change in field intensity reaching the inner tubes reduces the amplitude of the resonance Raman profile varying with laser excitation energy E l as where M R is the combined Raman matrix element.It is given by M R = M 2 ex−pt M ex−RBM the product of the exciton-photon M ex−pt and the exciton-phonon matrix elements M ex−RBM . 43ex−pt(in) depends linearly on the local electric field intensity and thus scales with f 2 loc which leads to I R ∝ f 4 loc .The broadening factor γ is inversely proportional to the exciton lifetime.The pre-factor E 4 l in Eq. ( 5) is eliminated experimentally by calibrating on a Raman reference with a known and constant cross section, see Methods. Figure 1b compares the expected resonant Raman profiles for same (n 0 , m 0 ) SWCNT and (n 0 , m 0 )@M inner wall of a DWCNT.The energetic positions of the resonance are red shifted due to screening, Eqs.,(4) and (5).The effective matrix element M R for the inner tube is smaller than for the free-standing SWCNT due to the electromagnetic field factor f loc , Eqs. (1) and (5).We expect a smaller RBM intensity from an inner tube of a DWCNT compared to a SWCNT.
We now examine the resonance Raman profiles of sorted DWCNTs for signs of the predicted screening effects.

B. Inner tube RBM frequency and intensity
Figure 2a shows the RBM of M@M and M@S DWCNT samples, i.e., all inner tubes are metallic and the outer tubes are metallic in the M@M but semiconducting in the M@S sample.We confirm the semiconducting or metallic character of the DWCNT wall from the RBM spectra.We first divide the frequencies into a range of outer ℏω RBM < 200 cm −1 and inner tubes > 200 cm −1 , see labels in Fig. 2a.We deduce the metallic and semiconducting character of the inner tube from the excitation energy dependence and frequency, since only tubes resonant show measurable intensity. 19,25,39Thereby we identify the RBM frequency ranges for metallic (labelled blue) and semiconducting (orange) species, Fig. 2a.The M@M and M@S samples contain inner metallic nanotubes with ℏω RBM = 200 − 250 cm −1 .With the laser energy E l = 2.1 eV in Fig. 2a we efficiently excite outer semiconducting walls via E 33 ; they are strong in the M@S sample and much weaker in the M@M sample where they appear due to imperfections in the chirality sorting. 24Resonantly exciting outer metallic walls requires lower laser energies E l <1.85 eV.
The RBM frequencies of the DWCNTs with metallic and mixed walls were only slightly shifted (few cm −1 and less) compared to the corresponding SWCNT and no splitting, as it is characteristic for S@S, was observed.5 The RBM frequency shifts of the inner metallic tubes results from a changed intercept of the RBM diameter dependence, which is associated with environmental effects, as a change in solvent or nanotube bundling. 25,44he EM screening effect by the outer tubes manifests in the measured inner intensities.
We first selected spectra with the highest RBM intensities for the inner and outer species in S@S, M@S, and M@M samples, Fig. 3a-c (the S@M sample had lower sorting purity and will not be considered here).When the outer tubes are semiconducting, top and middle trace, the maximum RBM intensity of the inner tube is comparable or even stronger than that of the outer species.This is expected, because the RBM intensities scale with the inverse of nanotube diameter. 18,19,45In sharp contrast, the integrated intensity of the inner tube is one order of magnitude smaller with a metallic outer tube in the M@M sample, bottom trace in Fig. 3c.To study the intensities in detail, we measured the full resonance profiles of the inner and outer walls, see Methods. Figure 3d  chemical reactions, e.g., in the fusion of molecules to graphene ribbons and inner tubes, 48,49 will strongly favor reactions inside semiconducting containers, although metallic tubes might actually be superior for that task.On the other hand, an exciting concept that follows from Eq. ( 1) is active screening when the laser frequency matches the optical transition of the outer tube.The dielectric function then follows the Lorentz resonance and creates an additional active dielectric screening.Matching an inner and outer tube resonance in DWCNTs is best achieved in S@S or mixed electronic type samples.Realizing active outer screening on individual DWCNTs with overlapping optical resonances would be of great interest for future work.(a) Resonant Raman profile of (11,2)@M compared to (b) (11,2)@S.Symbols represent experimental data and lines fits by Eq. ( 5).The positions of the transition energies are marked by vertical lines.(c) Energy shift of the transitions measured in identical inner tubes in the M@M and M@S samples, listed in Table I.The inset shows the diameter dependence ϵ eff , versus outer wall diameter for ϵ oT,i = 10, 1 C. Exciton screening and optical transition energies shift.
We determine the transition energies of inner CNTs from the resonant Raman profiles.We start with the peak at 244 cm −1 in Fig. 2a, belonging to the RBM (11,2)@S.The integrated area of the (11,2)@S RBM is plotted as a function of excitation energy in Fig. 4b.The intensity increases when the laser approaches the E 11(L) transition energy of the nanotube.
We quantify this energy by fitting the (11,2)@S Raman profile by Eq. ( 5) and find the transition energy E 11(L) =2.16 eV, marked by a vertical red line.The transition energy in the DWCNTs is comparable to the (11,2) SWCNTs (2.12 eV).In single-walled CNTs the E 11(L) may depend on many factors, surfactant type and filling, compared to DWCNTs where the exclusive environment is the outer wall.Next, we investigate the transition energy for a metallic outer wall.
The metallic outer wall induces a larger transition energy of the inner wall exciton, compared to a semiconducting outer wall.The resonance Raman profiles for (11,2)@S and (11,2)@M are plotted in Figures 4b and c, respectively.In the (11,2)@M sample, we find .Since the Raman frequencies of the inner walls are identical, both at ∼244 cm −1 , moiré coupling can be neglected 17 and the dominant origin of the energy shift must be dielectric screening, as expected from Eq. ( 4).
The red shift in transition energy is stronger for small-diameter inner nanotubes.We plot the shift of transition difference between the M@S and M@M samples in Fig. 4c.The magnitude of the shift decreases from ∼40 meV for d = 1 nm to nearly zero d = 1.2 nm.
The shift scales with the effective dielectric constant ϵ eff , plotted as a function of inner wall diameter for ϵ oT,i = 10, 1 in the inset of Figure 4c.The transition energies shifts in all other investigated chiralities are listed in Table I.The scattering of the symbols in the order of 10 meV is mainly due to the weak moiré effects, which as well induce a slight negative shift in some cases.extracted from fitting resonance Raman profiles, where ix = SW (single-walled), M@M, and M@S.Excitons in thinner M@M inner walls can be reduced to single particles by the extreme outer wall screening.The thermal dissociation of the excitons occurs when the exciton binding energy becomes comparable to the thermal energy at room temperature k B T 293 = 25 meV.With ϵ oT = 10 we obtain ϵ eff = 4.5 and a reduction in binding energy from 114 to 15 meV.This yields a E 11(L) red-shift of 19 meV, given by Eqs. ( 3) and ( 4).Such or greater red shifts are indeed observed for inner CNTs with d < 1.03 nm in our sample, Fig. 3 and Table I, indicating that we observe single particle band gaps in the inner M@M walls.
Many fundamental properties of CNTs are governed by excitons, including absorption 22 , emission 50 , and Raman scattering. 40,51It would be extremely interesting to redo such experiments on screened excitons as available in inner walls of M@M DWCNTs.For example, we expect asymmetric absorption peaks, identical energies in one-and two-photon luminescence excitation spectroscopy, and a change in the relative intensity of the incoming and outgoing G mode Raman resonances. 23,40,50,51The fragility of the excitonic states in partly or fully metallic DWCNTs makes them unlikely candidates for preparing exciton condensates and exciton insulators, where a better choice would be fully semiconducting species with strong moiré effects. 17On the other hand, the metallic walls would be better reactors due to the low-energy electronic bands, however much lower optical signals will require enriched metallic nanotube samples for detection of reaction products.
The dielectric effects may be used to control materials encapsulated inside carbon nanotubes via screening.The optical transition energies of the molecules and carbon chains are ruled by excitonic effects. 52Such effects would manifest in an optical energy shift when confined inside metallic nanotubes compared to semiconducting ones.For example, in linear carbon chains, we would expect a deviation from the linear behavior between the Raman mode frequency and the transition energy. 53Up to now, individual single carbon chains have only been reported in semiconducting CNT containers. 15,54This is likely related to their localization method, where first the lateral Raman maps are analyzed for the highest Raman signal.As we showed, the Raman signals inside the metallic shells are much smaller, therefore, improved localization methods are required to target excitonic effects in one-dimensional chains.
We also need to account for chiral variation of matrix elements 4 .For SWCNTs can use empirical formulas proposed by Pesce et al. where • eV ) deduced by correlating transmission electron microscopy (TEM) and resonance Raman scattering, 5 d is the CNT diameter and Θ its chiral angle. 6e full profile of Raman matrix elements can be described as Transition energies of the CNTs can be calculated using following empirical formula 7 In Figure S1a an b we show the waterfall plots for SWCNT and M@M sample, the RBMs belonging to different chiralities move inside and outside the resonance conditions, depending on the laser energy.The SWCNT spectra were obtained using known intensity and transition energies behaviors 5,7 and the M@M are the experimental.The spectra are normalized to one for better visibility and the normalization factors are shown in the right Experimental spectra of M@M sample, with c) histograms of the maximum Raman intensity.
panels.In the M@M sample we find only metallic walls, compared to the SWCNT sample, shown in Figure S1 b,a.The M@M and S@S samples are the cleanest in the separation process. 1 Interestingly, the intensities distributions are inverse.In the single walled CNTs a highest intensity is found for smaller diameter tubes 5 , whereas in the M@M sample we find largest intensity for the large diameter walls, red region in Figure S1b.This inversion already indicates dielectric effects.When analyzing the M@S sample we find some smaller semiconducting inner walls, see Figure S1c and overall intensity behavior is more erratic.This is caused by S@S impurities and also different E ii transitions at play.Hence, we concentrate only on the M@M sample for studying the EM screening.The M@M an ideal sample, since both inner and outer walls are excited at the same transition.However, first, we need to identify individual (n,m) chiralities from the groups the RBM peaks in Figure M@S-fit M@M-fit M@M M@S SFig.S2.Fit of the experimental RBM frequencies with Eq. ( 7) S1.

III. RBM FREQUENCY DIAMETER DEPENDENCE
The dependence of the RBM frequency depends on the effective mass of the CNT cylinder, traditionally the expression was used: where c 1 and c 2 are constants determinned from an experiment.Later, a more physical expression was proposed: .Figure S2 shows the fit by Eq. (7).We obtain c ms a = 2.85, c mm a = 2.5, both slightly higher than c swcnts a 2.2. 8TABLE I. Summary of RBM frequencies ℏω M @S and transition energies E M @X 11(L) , where X = M for metallic inner wall and X = S for semiconducting inner wall.∆E M @X 11(L) = E SW 11 − E DW 11 .

IV. MAXWELL-GARNETT MIXING
The formula Maxwell-Garnet mixing used 9 where V (d) = 1 − 4/(d + t) 2 is the volume fraction of the empty part of the cylinder.Eq (8)is plotted in Figure S3 for various V parameters.
The volume fraction V refers to the ratio of the outer nanotube empty V emp part to the outer nanotube 'full' volume V f ull .The scheme is shown in figure S3, with empty part the volume fraction of the empty part of the cylinder.For a CNT with d = 2 nm we find V = 0.54.Assuming ϵ i = 1 and ϵ o T = 10 we obtain the effective dielectric constant ϵ eff ≈ 5.

FIG. 2 .
FIG. 2. Radial breathing modes in M@M (blue) and M@S (purple) nanotubes excited with a 2.1 eV laser.(a) RBM spectra in the region between 100 − 400 cm −1 .The top line roughly divides expected RBMs from metallic (blue) and semiconducting (orange) walls.(b) Inner wall RBM fitting in M@M sample and (c) M@S sample.The vertical lines divide different 2n + m laola families.The arrow indicates the increase of the chiral angle within a laola group.
We fitted the diameter dependence of the RBM frequencies and obtained a constant c 1 = 215 nm •cm −1 for SW and DWCNTs inner metallic tubes, but a c 2 that varied between the three samples [c M@M 2 = 19.3nm and c M@S 2 = 20.2nm compared to c SWCNTs 2 = 18 nm].

FIG. 3 .
Fig.3d.Similar differences were observed for the other M@M walls as shown in Fig.3efor inner and outer tube diameters 0.8 − 1.8 nm.The experimentally observed variation of Raman intensity is reproduced by Eq. (1) with dielectric constants ϵ oT = 9 − 10 and ϵ in = 1.For fitting our experimental data we represent inner and outer walls populations by two Gaussian like distributions with equal amplitudes, see Supplementary information.The intrinsic diameter-dependent Raman intensity M 2 R (d) for these populations was corrected by the local field factor in Eq. (1) and plotted in Fig.3eby the red line, see Supplementary information for more details.Overall, we see good agreement in terms of intensities, with an outer-to-inner intensity ratio of ∼ 36.The dependence of the M 2 R without local field correction (grey line in Fig.3e) unambiguously disagrees with experiment.The experimental dielectric constant ϵ oT = 10, agrees reasonably well with ϵ th oT = 16 predicted by Malic et al.22 The screening by a typical metallic CNT with d = 1 − 2 nm and ϵ oT = 10 reduces the amplitude of the electric field within the tube by f loc ≈ 0.36 compared to the far field.This has important consequences when CNTs are used as containers or reactors and the encapsulated species are monitored by optical methods.The linear optical response of encapsulated molecules will drop by a factor of f 2 loc ≈ 0.13; in the case of non-linear techniques such as Raman scattering, the reduction amounts to two orders of magnitude (1.6 • 10 −2 ), Fig.3e.The sharp drop in intensity, may be the reason why carbyne chains so far were only observed in individual semiconducting DWCNTs.46,47An in-situ monitoring of

7 FIG. 4 .
FIG.4.Exciton screening in the S@M and M@M samples studied by resonant Raman scattering.
Dielectric screening plays an important role in DWCNT; its effects are two-fold as it modulates many-body effects and alternates the electric field inside the outer tube.Manybody effects manifest in the energetic position of the excitons.We measured inner walls exciton energies by means of resonant Raman spectroscopy for semiconducting and metallic outer walls.In metallic outer walls we found an additional red-shift by up to 40 meV, compared to semiconducting outer walls.The optical resonances of inner metallic walls most likely originate from band-to-band excitations, because the excitons dissociate thermally.The electric field is also strongly altered by the electronic type of the outer wall.The metallic walls act as a dense dielectric shield blocking a substantial fraction of the electromagnetic field.That manifests in up to 30 times weaker Raman signals of the inner metallic walls compared to the outer metallic walls.These results open interesting prospects for dielectric cloaking and active dielectric screening.We believe that in all types of one-dimensional heterostructures one will find strong dielectric effects altering many-body interactions and electromagnetic fields.at 320 cm −1 in order to account for changes in the sensitivity of optical components in the measurement.2II.VARIATION OF RAMAN INTENSITY IN SWCNTS AND IN THE M@M SAMPLEIn this section we explain intensities variation in the single walled carbon nanotubes and how local factor can be implemeted.The population is modelled by a standard Gaussian function:
a l i z e d i n t e n s i t y ( a r b .u n i t s ) m e t a l l i c s i n g l e w a l l e d N T s SFig.S1.Resonant Raman spectra of nanotubes excited between of 2.3 eV (blue) and 1.55 eV (red), the intermediate colors are mixed in rainbow-like pattern.The panel (a) represents the single-walled CNTs, generated for the same populations as M@M walls using eqs.(5),(4), (2).5,7  .The spectra are normalized and offset for clarity, (b) the normalization factors for each laser wavelength.(c) CNTs environment.We combine these two constants into c 2 to simplify the expression:ω RM B (d) = 227 ( 1 d 2 + c a ),(7)where c a = 6(1−v 2 )

TABLE I .
Summary of RBM frequencies ℏω ix and transition energies E ix 11(L)