Dmitry I Levshov
email: dmitry.levshov@uantwerpen.be

Marina V Avramenko

Maksiem Erkens

Huy-Nam Tran

Thi Thanh Cao

Van Chuc Nguyen

Emmanuel Flahaut

Valentin N Popov

Ahmed-Azmi Zahab

Jean- Louis Sauvajol

Raul Arenal

Wim Wenseleers

Sofie Cambré

Matthieu Paillet

Partial quenching of electronic Raman scattering in double-wall carbon nanotubes by interlayer coupling

Keywords: double-wall carbon nanotubes, electronic Raman scattering, van der Waals interaction, interlayer coupling, structural characterization

Measuring electronic Raman scattering (ERS) has become an efficient method for structural characterization of metallic single-wall carbon nanotubes (SWCNT). However, applying this method to other types of SWCNT-based structures, e.g., those with strong van der Waals (VDW) coupling, is currently not well studied. In this work, we combine electron diffraction, Rayleigh and Raman spectroscopies to investigate the ERS process near 36 metallic transitions in 21 individual double-wall carbon nanotubes (DWCNTs) with all types of electronic configurations.

We observe the partial suppression of ERS intensity in DWCNTs compared to SWCNTs and mainly attribute it to the effect of dielectric screening of Coulomb interactions. We probe ultrapure macroscopic multi-chirality DWCNT solutions and identify the role of inhomogeneous broadening in observing ERS peaks in Raman spectra. Based on the experimental findings, we propose an adapted method for the structural identification of DWCNT samples from the ERS data. The obtained results can be generalized to the characterization of the emerging 1D VDW heterostructures based on metallic SWCNTs.

Introduction

Knowing the atomic structure of nanomaterials is essential for their applications in modern science and technology, as even slight structural variations will result in distinct physical properties. One typical example is a single-wall carbon nanotube (SWCNT), a quasi-onedimensional cylindrical crystal, conceptually obtained by rolling up a sheet of graphene [1]. Two geometrically similar SWCNTs can be either metallic (M) or semiconducting (SC), differing only in one carbon atom along their circumferences. This difference, however, essentially defines the area of application: whether in digital electronics requiring at least 99.99% purity of SC-SWCNTs [2], as high-performance interconnects based on M-SWCNTs [3], or as nanoscale electromagnets, inductors, and transformers demanding particular coaxial combinations of both M-and SC-SWCNTs [4].

Identifying the SWCNT structure is possible either through microscopy-based techniques [5][6][7] or through spectroscopic methods [8][9][10]. One of the most powerful and non-invasive approaches relies on optical spectroscopy [8]. It is routinely performed using infrared photoluminescence excitation (PLE) spectroscopy [8,11], absorption spectroscopy [12], resonant Raman spectroscopy (RRS) [9,13], or their combinations [14]. RRS is particularly important since it provides both excitonic and vibrational information about SWCNTs [15,16] but requires measuring Raman spectra at multiple excitation energies for full characterization accuracy.

Recently, an alternative SWCNT identification method was proposed providing information about excitonic resonances from a single wide-range Raman spectrum, which is based on registering the so-called electronic Raman scattering (ERS) in individual metallic [17][18][19] or semiconducting [20] SWCNTs. The ERS is associated with electronic transitions [21] as opposed to vibrational transitions in conventional RRS and has been employed extensively in other fields, e.g., as an ultra-sensitive probe for strain effects in semiconductors [22] or for determining band gap magnitudes and nodes in superconductors [START_REF] Vazquez-Zuniga | New insights into the phase diagram of the copper oxide superconductors from electronic Raman scattering[END_REF][START_REF] Boyd | Probing the pairing symmetry of the iron pnictides with electronic Raman scattering[END_REF]. In SWCNTs, such electronic transitions originate from the inelastic scattering of photoexcited excitons by the low-energy excitons from the linear band of M-SWCNTs or by high-energy E11 excitons of SC-SWCNTs. The ERS manifests itself in Raman spectra as a broad dispersive line located between 0 and 4000 cm -1 for M-SWCNTs [19] or around 6000 -9000 cm -1 for SC-SWCNTs [20]. Although the Raman shift of the ERS varies with the laser excitation wavelength, its absolute scattered energy is always equal to the energy of the excited Eii transition. Saito et al. proposed a detailed theoretical description of the ERS process, highlighting the spectral changes originating from the interference between ERS and Raman-active phonons [START_REF] Hasdeo | Electronic Raman scattering and the Fano resonance in metallic carbon nanotubes[END_REF][START_REF] Saito | Electronic and Optical Properties of Single Wall Carbon Nanotubes[END_REF]. Note that ERS peaks have also been observed in the lowfrequency region (between 0-10 cm -1 ) of the anti-Stokes Raman spectrum of SC-SWCNTs, but they were assigned to a different mechanism, namely, scattering by single-particle excitation [START_REF] Chen | Electronic Raman scattering on individual semiconducting single walled carbon nanotubes[END_REF].

The ERS-based structural characterization of SWCNTs was shown to be robust due to its weak sensitivity to local variations in the nanotube environment [13,19]. However, it is unclear whether it can be extended for characterizing the emerging one-dimensional van-der-Waals (VDW) heterostructures [START_REF] Xiang | Heteronanotubes: Challenges and Opportunities[END_REF][START_REF] Xiang | One-dimensional van der Waals heterostructures[END_REF][START_REF] Cambré | Nanotube-Based 1D Heterostructures Coupled by van der Waals Forces[END_REF] based on SWCNTs. For instance, the VDW coupling was shown to significantly affect the electronic, optical and vibrational properties of double-wall carbon nanotubes (DWCNT) [START_REF] Zhao | Interlayer Interactions in 1D Van der Waals Moiré Superlattices[END_REF], which can be viewed as the simplest 1D VDW structure comprised of two concentric SWCNTs [START_REF] Cambré | Nanotube-Based 1D Heterostructures Coupled by van der Waals Forces[END_REF]. The electronic VDW coupling modulates the excitonic energies [START_REF] Liu | Van der Waals-coupled electronic states in incommensurate double-walled carbon nanotubes[END_REF] or leads to the appearance of entirely new optical transitions [START_REF] Zhao | Observation of Drastic Electronic-Structure Change in a One-Dimensional Moiré Superlattice[END_REF][START_REF] Popov | Theoretical evidence of a significant modification of the electronic structure of double-walled carbon nanotubes due to the interlayer interaction[END_REF][START_REF] Chalin | Band structure and inter-tube optical transitions in double-walled carbon nanotubes[END_REF][START_REF] Paillet | Optically active cross-band transition in double-walled carbon nanotube and its impact on Raman resonances[END_REF], while efficient interlayer energy transfer is proposed to quench the inner layer photoluminescence (PL) [START_REF] Levshov | Photoluminescence from an individual double-walled carbon nanotube[END_REF][START_REF] Erkens | Efficient Inner-to-Outer Wall Energy Transfer in Highly Pure Double-Wall Carbon Nanotubes Revealed by Detailed Spectroscopy[END_REF]. On the other hand, the mechanical VDW coupling is responsible for new collective vibrations [10,[START_REF] Liu | Quantum-coupled radial-breathing oscillations in double-walled carbon nanotubes[END_REF] and frequency shifts due to internal pressure [START_REF] Levshov | Interlayer Interaction Effects on the G Modes in Double-Walled Carbon Nanotubes With Different Electronic Configurations[END_REF][START_REF] Popov | Computational study of the shift of the G band of double-walled carbon nanotubes due to interlayer interactions[END_REF].

The current knowledge of the effect of VDW coupling on the ERS is incomplete. ERS has so far not been detected in DWCNTs due to the general difficulty of its observation and the lack of individual samples with precisely defined chirality. Although much progress has been made recently in the sorting of DWCNTs [START_REF] Li | Inner-and outer-wall sorting of double-walled carbon nanotubes[END_REF], e.g., sorting the DWCNTs by metallicity of the constituting layers, it is not yet possible to achieve single-chirality purity of DWCNTs on a macroscopic scale. In this work, we therefore measure the ERS in 21 well-identified individual suspended DWCNTs with different electronic configurations, structurally characterized by electron diffraction (ED) or all-optical methods. We observe the partial suppression of ERS intensity in DWCNTs compared to SWCNTs and discuss it in terms of the effect of dielectric screening, interlayer energy and charge transfers. Afterwards, we investigate the ERS in ultra-pure macroscopic and mixed-chirality DWCNT solutions where ERS is hardly observed due to the effects of inhomogeneous broadening and polydispersity in chiral structures. Finally, based on the experimental findings, we propose an adapted method for the structural characterization of DWCNT samples from the ERS data.

Materials and methods

Synthesis of individual free-standing DWCNTs:

Ultralong individual DWCNTs were grown by the CCVD method directly across open slit structures (~50×1000 µm²) fabricated on silicon substrates ((100) orientation, 200 µm thick) with a 90 nm thick SiO2 layer, according to the method described in Ref. [START_REF] Tinh | Synthesis of individual ultra-long carbon nanotubes and transfer to other substrates[END_REF].

Preparation of macroscopic UP-DWCNT sample:

The pristine DWCNT powder (batch R281010) was synthesized following the method of Ref. [START_REF] Flahaut | Gram-scale CCVD synthesis of double-walled carbon nanotubes[END_REF]. It was purified from MgO-supported catalytic particles using the acid treatment and then solubilized in D2O (99.8%, Cortecnet) with 2 wt./V% of sodium deoxycholate (DOC, 99%, Acros Organics) at a concentration of 4 mg CNT powder per ml surfactant solution, using only gentle magnetic stirring, avoiding ultrasonication.

Finally, the DWCNT sample was purified from SWCNT impurities remaining from the synthesis by density gradient ultracentrifugation. More details are given in Ref. [START_REF] Erkens | Ultrasonicationinduced extraction of inner shells from double-wall carbon nanotubes characterized via in situ spectroscopy after density gradient ultracentrifugation[END_REF].

High-resolution electron microscopy and Electron Diffraction: HRTEM images and ED

patterns were recorded in a FEI Titan Cube microscope (equipped with a spherical aberration corrector) operating at 80 kV and within short acquisition times (less than 5 s for ED) to reduce damage induced by electron diffraction.

Rayleigh spectroscopy:

Rayleigh spectra of individual free-standing DWCNTs were measured in a backscattering geometry using a cross-polarization scheme with a Fianium supercontinuum laser as a light source and a fiber-fitted QE-Pro Ocean Optics spectrometer for detection.

Raman spectroscopy of free-standing DWCNTs:

Resonant micro-Raman scattering measurements were performed in a backscattering configuration on two separate homemade setups including an iHR550 Jobin-Yvon spectrometer (gratings with 1200 or 1800 grooves/mm) and an Acton SP2500 spectrometer (grating with 1800 grooves/mm), both equipped with a liquidnitrogen-cooled silicon CCD detectors. The first system was used for the NIR tunable measurements, while the second for the measurements in the visible range. The home-made microscopes in both systems were equipped with 100x objectives (numerical aperture = 0.95). The samples were mounted on a three-axis piezo-electric stage (PIMars P-563, Physik Instrumente) to ensure the precise positioning and focusing of the laser spot. Incident excitations from various continuous-wave lasers were used: HeNe at 632.8 nm (1.96 eV), diode-pumped solid-state lasers at 457 nm (2.71 eV), 532 nm (2.33 eV) and 561 nm (2.21 eV), and a Ti-Sapphire with a tunable laser line filter in the NIR (700 -885 nm; 1.4 -1.77 eV). To avoid heating effects, we kept the laser intensity impinging the sample below 40 kW/cm 2 .

2.6

Raman spectroscopy of macroscopic UP-DWCNTs: Resonant Raman spectroscopy was performed with a Dilor XY800 triple spectrometer (gratings with 1800 grooves/mm) in backscattering geometry equipped with a liquid nitrogen-cooled CCD detector. For the nearinfrared excitation wavelengths (695 nm -855 nm), a tunable Ti:sapphire laser (Spectraphysics, model 3900S) pumped by a 5 W Ar + -laser (Spectra Physics, model 2020) was used. The range 670 -690 nm was covered by a dye laser (Spectra Physics, model 375) equipped with the dye DCM

(4-(Dicyanomethylene)-2-methyl-6-(4-dimethylaminostyryl)-4H-pyran).
The excitation wavelength was increased in approximately 5 nm steps. The laser power variations between the spectra were corrected using a tetrachloromethane (CCl4) reference sample, which is transparent throughout the entire wavelength range and thus exhibits non-resonant Raman scattering. In addition, all of the Raman spectra were corrected for the setup's sensitivity and the spectrometer's wavelength calibration.

Results

Experimental observation of ERS in individual free-standing DWCNTs

The optical signals present in the Raman spectra of DWCNTs may vary depending on the purity and environmental interaction in the probed samples. The latter can be in the form of as-grown

DWCNT powders [START_REF] Kuzmany | Raman scattering from double-walled carbon nanotubes[END_REF], mixed-chirality or partially sorted DWCNT dispersions [START_REF] Li | Inner-and outer-wall sorting of double-walled carbon nanotubes[END_REF][START_REF] Green | Properties and application of double-walled carbon nanotubes sorted by outer-wall electronic type[END_REF],

individual DWCNTs on substrates [START_REF] Villalpando-Paez | Raman spectroscopy study of isolated double-walled carbon nanotubes with different metallic and semiconducting configurations[END_REF] or DWCNTs suspended over open slits [16,[START_REF] Rochal | Chirality manifestation in elastic coupling between the layers of double-walled carbon nanotubes[END_REF]. The first two sample types often contain impurities like SWCNTs, whose optical response may be confused with the signal from DWCNTs [START_REF] Tsyboulski | Do inner shells of double-walled carbon nanotubes fluoresce?[END_REF][START_REF] Green | Processing and properties of highly enriched double-wall carbon nanotubes[END_REF]. In addition, the intrinsic DWCNT optical properties can be drastically changed through the interaction with the environment, e.g., due to unintentional doping of nanotubes lying on substrates [START_REF] Nguyen | Fano Lineshape and Phonon Softening in Single Isolated Metallic Carbon Nanotubes[END_REF] or through strain [START_REF] Hertel | Deformation of carbon nanotubes by surface van der Waals forces[END_REF]. In this work, to exclude the majority of the extrinsic effects and thoroughly control the probed samples' purity, we investigate individual DWCNTs grown by the catalytic chemical vapor deposition technique (CCVD) over open slits. As demonstrated previously [START_REF] Rochal | Chirality manifestation in elastic coupling between the layers of double-walled carbon nanotubes[END_REF], this approach allowed us to combine Rayleigh spectroscopy, Raman spectroscopy, and ED measurements on the same free-standing parts of a DWCNT (Fig. 1a).

First, we identify the atomic structure of individual suspended DWCNTs by analyzing the ED patterns or by combining Rayleigh and Raman spectroscopic information [10,14] (further details in the Supporting Information (SI) Section S1). Having determined the DWCNT chiral indices, we then estimate the layers' optical resonance energies by referring to the atlas of SWCNT electronic transitions [START_REF] Liu | An atlas of carbon nanotube optical transitions[END_REF]. The transition energy shifts induced by the interlayer VDW electronic coupling in the inner and outer DWCNT layers were calculated using the approach of Chalin et al. [START_REF] Chalin | Band structure and inter-tube optical transitions in double-walled carbon nanotubes[END_REF]. This coupling modulates optical resonances by shifting them from -200 to +50 meV [START_REF] Liu | Van der Waals-coupled electronic states in incommensurate double-walled carbon nanotubes[END_REF][START_REF] Zhao | Rayleigh scattering studies on inter-layer interactions in structure-defined individual double-wall carbon nanotubes[END_REF] or even creating new intertube resonances absent in isolated layers [START_REF] Zhao | Observation of Drastic Electronic-Structure Change in a One-Dimensional Moiré Superlattice[END_REF][START_REF] Popov | Theoretical evidence of a significant modification of the electronic structure of double-walled carbon nanotubes due to the interlayer interaction[END_REF][START_REF] Chalin | Band structure and inter-tube optical transitions in double-walled carbon nanotubes[END_REF]. Using the calculated data as input parameters, we then fitted the DWCNT Rayleigh spectra with a complex optical susceptibility function 𝜔 3 |𝜒| 2 (see Section S5 1c). The observation of a broad LO mode [START_REF] Levshov | Interlayer Interaction Effects on the G Modes in Double-Walled Carbon Nanotubes With Different Electronic Configurations[END_REF][START_REF] Michel | About the indexing of the structure of single-walled carbon nanotubes from resonant Raman scattering[END_REF] proved the presence of the metallic layer, as is the case for the G-band spectra of (15,12) metallic layer at 1.46 and 2.71 eV (incoming resonance) and 1.65 eV (outgoing resonance) in Figure 1c. The complete set of fitted Rayleigh spectra and RBLM and G-band spectra for all studied DWCNTs are presented in the SI Section S2.

We then probed the ERS signals of the identified metallic transitions by measuring widerange Raman spectra from ~80 to ~3000 cm -1 at multiple excitation energies from near-infrared (NIR) to ultra-violet for each characterized DWCNT. Overall, we investigated 36 metallic resonances in 15 DWCNTs with different electronic configurations, summarized in SI Section S1.

Some typical examples of ERS peaks are shown in Figure 1d for SC@M, M@SC, and M@M DWCNTs. The Raman spectra are multiplied to highlight the ERS features relative to the intense phonon peaks (in blue in comparison with the original black spectra; the multiplication factor is provided). SC@M (14,1)@(15,12), M@SC (14,8)@ (19,14), M@M (14,2)@(22,4) (inner layer in resonance), M@M (11,8)@(19,10) (outer layer in resonance) excited at 1.60, 1.66, 1.96 and 1.60 eV, respectively. The Raman spectra are shown, either normalized on the intensity of the strongest peak (black) or scaled (blue), in order to highlight the ERS signal (with the given multiplication).

We summarize the observed behavior of ERS in all studied DWCNTs in Figure 2. It is best illustrated by two distinctive cases of (14,1)@(15,12) and (22,13)@(39,2) DWCNTs. For the former (Figure 2a), we detect almost all metallic transitions as ERS peaks in the Raman spectra:

the 𝑀 11𝐿 𝑜𝑢𝑡 is visible when exciting at 1. 2b). To quantitatively describe this suppression, we fitted ERS peaks in 79 available wide-range Raman spectra and presented the ERS intensity (peak height) in relation to the intensity of the broad LO mode of the metallic layer, i.e., IERS/ILO (arguments for representing the data in this form and details of the fitting procedure are given in the Sections S6-S7 in the SI). We find that the IERS/ILO ratio in DWCNTs rapidly decays when increasing the difference between the laser energy Elaser and the energy Eii of the excited transition (Figure 2c, bottom panel). Therefore, it becomes challenging to resolve ERS features within the noise level or distinguish them from the intense Raman modes when Raman spectra are excited more than 200-300 meV above the optical transition, i.e., when the ERS peak falls in the range between G-and 2D-bands. Although analyzing the behavior of ERS lines in terms of IERS/ILO intensity ratios provides a quick practical way to compare SWCNT and DWCNT data, it has a certain drawback, namely the ratio depends not only on the intrinsic properties of the ERS process but also on the intensity variations of the LO phonon peaks. For instance, quantum interference between exciton states in the G-band excitation profiles was previously reported for SWCNTs [START_REF] Duque | Quantum Interference between the Third and Fourth Exciton States in Semiconducting Carbon Nanotubes Using Resonance Raman Spectroscopy[END_REF] and DWCNTs [START_REF] Tran | Quantum interference effects on the intensity of the G modes in double-walled carbon nanotubes[END_REF],

manifesting in the peculiar intensity variation of the G-band lines.

To avoid the influence of G-band properties, we analyze the evolution of the ERS peak's integrated intensity (peak area) instead of intensity ratios with the excitation energy. We measured the Raman excitation maps (REM) in the range between RBLM and G-band, as shown in Figure 3a,c for DWCNTs (14,1)@(15,12) and (18,6)@ (21,14), respectively. The maximum intensities in Figures 3a,c are chosen in such a way to highlight the ERS peaks; hence RBLM and G-band regions appear saturated (dark red color). We fitted the spectra of the REM simultaneously using a set of Lorentzians (peak parameters were shared between the spectra) and neglecting the interference effects [START_REF] Hasdeo | Electronic Raman scattering and the Fano resonance in metallic carbon nanotubes[END_REF] between the ERS and phonon lines for simplicity (see details in Section S6). The ERS, RBLM and LO integrated intensities extracted from the REM fits are presented as filled symbols in Figures 3b,d.

In addition, we calculated the absorption spectra of (14,1)@(15,12) and (18,6)@ (21,14) DWCNTs within an ab initio-based non-orthogonal tight-binding (NTB) approach [START_REF] Popov | Theoretical evidence of a significant modification of the electronic structure of double-walled carbon nanotubes due to the interlayer interaction[END_REF], which allowed us to clarify the positions of the Mii transitions within the measured excitation range (see Section S4, Tables S4 3b). Based on these transition energy assignments, we fitted the RBLM and LO Raman excitation profiles (REP) using the approach of Tran et al. [START_REF] Tran | Quantum interference effects on the intensity of the G modes in double-walled carbon nanotubes[END_REF] accounting for the interference between M11L and M11H transitions. The results of the fit are shown as solid lines in the bottom panels of Figures 3b,d. In contrast to RBLM and LO modes, we fitted the ERS REP using a single Lorentzian component. We find that the evolution of the ERS integrated intensity has one common feature for two investigated DWCNTs: it rapidly decays with the increase of the laser excitation energy. Hence, the ratio of the ERS and LO integrated intensities should tend to very small values with the increase of Elaser -Mii, which reflects well the behavior observed in Figure 2c for

Discussion

Origin of ERS intensity suppression

We now discuss the possible origins of the ERS signal attenuation (or occasional complete suppression) in DWCNTs compared to SWCNTs. In the previous studies [17][18][19][START_REF] Hasdeo | Electronic Raman scattering and the Fano resonance in metallic carbon nanotubes[END_REF], the ERS intensity in SWCNTs was assumed to depend on both intrinsic and extrinsic factors. The latter comprises heating by the laser beam [18], electrostatic or electrochemical doping [17], and environmental effects (increased dielectric screening and charge transfer) when SWCNTs are located on substrates or in bundles [18]. In this work, we tried to reduce such extrinsic factors. For example, we minimized the heating of DWCNTs by employing low laser powers (P < 0.4 nW/nm 2 ), which should lead to negligible effects according to Ref. [18]. Doping from the substrate was excluded due to the suspended character of individual DWCNT samples. An effect of the water molecules adsorbed on the outer surface of DWCNTs [START_REF] Chiashi | Adsorption effects on radial breathing mode of single-walled carbon nanotubes[END_REF][START_REF] Homma | Photoluminescence measurements and molecular dynamics simulations of water adsorption on the hydrophobic surface of a carbon nanotube in water vapor[END_REF] is hard to estimate but should be minor since we do not observe any notable difference between ERS behavior in the outer and inner layers (Figure 2e). Indeed, the inner layers are known to be chemically well shielded from the external environment and should behave differently in case of strong interactions with water vapor [START_REF] Moore | Double-Walled Carbon Nanotube Processing[END_REF]. Moreover, the latter should also be present in reference SWCNTs since they were also studied in air [17][18][19]. As for the influence of defects in the DWCNT crystal structure on the ERS intensity quenching, it should be negligible since we observe almost no D-band (see Raman spectra in SI Section S3) and hence we have a minimal amount of defects in all our studied DWCNTs.

Therefore, we argue that intrinsic effects contribute more significantly to the observed change of the ERS signal in DWCNTs.

The intrinsic aspects affecting the ERS intensity in SWCNTs include the chirality-and diameter-dependent exciton-exciton coupling, the curvature-induced bandgap opening and the resonance factor governed by the difference between laser energy EL and the Mii transition energy [START_REF] Hasdeo | Electronic Raman scattering and the Fano resonance in metallic carbon nanotubes[END_REF]. In multi-layer systems, like DWCNTs, other types of intrinsic effects can be present, namely, a reduced Coulomb (exciton-exciton) interaction resulting from an increase in the environmental dielectric screening [18] (Figure 4a), the interlayer energy or exciton transfer (IET) according to the Forster-Dexter mechanism [START_REF] Koyama | Ultrafast Exciton Energy Transfer between Nanoscale Coaxial Cylinders: Intertube Transfer and Luminescence Quenching in Double-Walled Carbon Nanotubes[END_REF] (Figure 4b) or interlayer charge transfer (ICT) predicted theoretically for commensurate DWCNTs due to the orbital mixing [START_REF] Zólyomi | Intershell interaction in double walled carbon nanotubes: Charge transfer and orbital mixing[END_REF].

We first consider the SWCNT-specific effects. In principle, observing inconsistent IERS/ILO ratios in Refs. [17][18][19] and this work may be explained by purely statistical reasons, e.g., due to different probed structural types of CNTs. Hence, we analyzed the dependence of IERS/ILO ratios on diameter d and chiral angle θ for all studied DWCNTs and SWCNTs in Figure S9. It turns out that the diameters of the SWCNTs from the literature overlap mostly with the diameters of the inner layers of our DWCNTs, but apart from this, the structural distributions of the probed SWCNTs and DWCNT layers are rather similar. The latter implies, for instance, that the curvatureinduced bandgap opening (Section S11) should have the same effect for both SWCNTs and DWCNTs, and is therefore disregarded. Furthermore, the different IERS/ILO ratios obtained from comparing the same chiralities in SWCNT and DWCNT forms (Table 1) suggest that the partial quenching of ERS is intrinsic for DWCNTs and unrelated to statistical effects.

As for the influence of interlayer coupling, we first considered the ICT by calculating its dependence on the interlayer distance Δr using the formula from Zólyomi et al. [START_REF] Zólyomi | Intershell interaction in double walled carbon nanotubes: Charge transfer and orbital mixing[END_REF] (Section S12).

It turns out that the calculated ICT for inner and outer layers is five times lower than the one required for the complete suppression of the ERS signal due to doping, according to Ref. [17].

Although it is still possible to interpret the partial suppression of the ERS features as a result of doping due to non-zero ICT, this effect should be most efficient in commensurate DWCNTs with strong coupling (so far studied only theoretically [START_REF] Zólyomi | Intershell interaction in double walled carbon nanotubes: Charge transfer and orbital mixing[END_REF]) and less in the incommensurate DWCNTs from this work. Alternatively, the partial suppression of ERS intensity can be related to the Forster-Dexter IET predicted for both commensurate and incommensurate DWCNTs [START_REF] Koyama | Ultrafast Exciton Energy Transfer between Nanoscale Coaxial Cylinders: Intertube Transfer and Luminescence Quenching in Double-Walled Carbon Nanotubes[END_REF][START_REF] Postupna | Time-Domain Ab Initio Simulation of Energy Transfer in Double-Walled Carbon Nanotubes[END_REF]. The fast IET process (lifetime τ ~ 70-220 fs [START_REF] Koyama | Ultrafast Exciton Energy Transfer between Nanoscale Coaxial Cylinders: Intertube Transfer and Luminescence Quenching in Double-Walled Carbon Nanotubes[END_REF][START_REF] Postupna | Time-Domain Ab Initio Simulation of Energy Transfer in Double-Walled Carbon Nanotubes[END_REF]) can act as a non-radiative decay channel for photoexcited excitons competing with the fast exciton-exciton or electron-electron scattering processes (τ ~ 10 fs [START_REF] Liu | Highthroughput optical imaging and spectroscopy of individual carbon nanotubes in devices[END_REF] for individual SWCNTs or ~200 fs for SWCNT ropes [START_REF] Hagen | Electronic structure and dynamics of optically excited single-wall carbon nanotubes[END_REF]) responsible for the ERS peak appearance (Figure 4b). The IET efficiency is known to depend on the distance between the interacting layers and the spectral overlap between the involved optical transitions [START_REF] Koyama | Ultrafast Exciton Energy Transfer between Nanoscale Coaxial Cylinders: Intertube Transfer and Luminescence Quenching in Double-Walled Carbon Nanotubes[END_REF], however, revealing these dependencies in the experimental distributions of IERS/ILO values (Section S10) turned out impossible because the data points are significantly scattered due to the resonance effect (i.e., dependence of IERS/ILO value on the |Elaser -Mii|). Therefore, further experimental and theoretical studies are necessary to estimate the exact contribution of IET to ERS intensity suppression. Another factor affecting the ERS intensity can be the reduction of Coulomb interaction between photoexcited excitons and low-energy excitons due to increased dielectric screening in DWCNTs compared to SWCNTs. Indeed, this screening was used to explain the ERS peak intensity suppression in SWCNTs deposited on substrates or for SWCNTs in bundles [18,19].

Since previous optical studies on suspended DWCNTs reported the existence of dielectric screening manifesting in a constant redshift of layer transition energies by about 50-60 meV [START_REF] Liu | Van der Waals-coupled electronic states in incommensurate double-walled carbon nanotubes[END_REF][START_REF] Zhao | Rayleigh scattering studies on inter-layer interactions in structure-defined individual double-wall carbon nanotubes[END_REF], we suppose it should also have a significant influence on the ERS process in DWCNTs with metallic layers leading to the suppression of ERS intensity. We also analyzed the Raman spectra of DWCNTs with semiconducting layers to investigate previously reported semiconducting-to-metal transitions [START_REF] Zolyomi | Semiconductor-to-metal transition of double walled carbon nanotubes induced by inter-shell interaction[END_REF][START_REF] Okada | Curvature-Induced Metallization of Double-Walled Semiconducting Zigzag Carbon Nanotubes[END_REF] and the appearance of mini-Dirac points in DWCNTs [START_REF] Bonnet | Charge transport through one-dimensional Moiré crystals[END_REF]. Such changes in the electron band structure can, in principle, lead to the observation of ERS signals in the Raman spectra of semiconducting DWCNTs. In total, we probed 20 semiconducting layers (including six S@S DWCNTs), represented by blue hexagons in Figure 5. We did not find any ERS signals in such DWCNTs, which may be partly explained by the difficulty in finding DWCNT combinations with strong electronic coupling. Further systematic studies are necessary to observe such semiconducting-to-metal transitions in strongly coupled DWCNTs.

Interestingly, analyzing the dependence of the IERS/ILO ratio on the structural parameters in DWCNTs can give additional insights into the SWCNT intrinsic properties. In particular, Hasdeo et al. [START_REF] Hasdeo | Electronic Raman scattering and the Fano resonance in metallic carbon nanotubes[END_REF] previously argued that the ERS signal should be absent in large diameter nanotubes (> 2 nm) due to the reduced Coulomb interaction and, hence, smaller exciton-exciton coupling strength. The CVD-grown DWCNTs act as perfect systems for exploring these effects since outer layers can reach 3 nm or more. In our study, there are nine outer SWCNTs (layers) with d > 2 nm (see Figure 2b,d), and we indeed observed almost complete quenching of ERS in the range above 2.3 nm.

Indexation of metallic DWCNTs based on the ERS features.

We now address the practical use of the ERS effect for indexing metallic layers of DWCNTs. Previously, Yang et al. [13,19] proposed a procedure for the structural characterization of individual SWCNTs that relied on measuring distinct MiiL and MiiH energies from ERS peaks and an RBM frequency in the Raman spectra. Due to the small dependence of the |MiiH -MiiL| energy difference on the environmental interaction, the index-assignment of the SWCNT can be made with high accuracy by combining optical and vibrational information in the so-called bilayer plots.

In principle, the interlayer coupling between DWCNT layers can make some of the characterization methods established for SWCNTs highly challenging, as is the case of PL-based indexation. For instance, we previously estimated the PL from inner semiconducting layers of DWCNTs, including the (14,1)@(15,12) studied here, to be four orders of magnitude smaller than in isolated SWCNTs [START_REF] Levshov | Photoluminescence from an individual double-walled carbon nanotube[END_REF]. In comparison, the ERS signal from metallic layers of DWCNTs is much less quenched and therefore can still be used for structural identification of DWCNTs.

Based on our experimental findings and former theoretical predictions [START_REF] Hasdeo | Electronic Raman scattering and the Fano resonance in metallic carbon nanotubes[END_REF][START_REF] Saito | Electronic and Optical Properties of Single Wall Carbon Nanotubes[END_REF], we divide the available SWCNT chirality map into three zones (Figure 5). DWCNT layers with the highest expected ERS signal are located in Zone II with diameters up to ~2.3 nm. Zone I represents those CNTs where the curvature-induced bandgap can complicate the detection of ERS features above 100 cm -1 as described in Section S11. Zone III corresponds to the large-diameter layers, where the ERS intensity should be affected by the reduced exciton-exciton interaction. Of course, the partition of the chirality space into three zones is somewhat arbitrary, but it can nevertheless guide future research until more statistical data about the ERS effect are available. In practice, it is first crucial to estimate the diameter of the involved DWCNTs from the RBLM frequencies. This can be done using the coupled-oscillator model from Refs. [10,16]. The knowledge of the layers' diameters points to the zones in Figure 5 and the expected strength of the ERS signal. For instance, for large DWCNT layers, we can expect that even if the ERS signal is not quenched, it can be hard to measure due to reduced exciton-exciton interaction and IET. When the information of layer chirality is available, e.g., from G-band lineshape analysis [10,[START_REF] Wu | Variable Electron-Phonon Coupling in Isolated Metallic Carbon Nanotubes Observed by Raman Scattering[END_REF] or from additional methods (like ED or HRTEM), one can position the studied CNT more precisely within Zones I and II. Then, for layers from Zone I, the curvature-induced bandgap opening can influence the ERS, and the measurement should be performed with caution.

Based on the data from Figure 2c, we find that it is best to measure ERS peaks when they are located between the RBLM and G-band of the metallic layer. It corresponds to the DWCNT excitation with laser energy from a few meV to 200 meV above the Mii transition. However, at large energy separation (e.g., ERS peak near the G-band), it may be difficult to spot the small and broad ERS peaks among much stronger and narrow phonon lines due to the fast ERS intensity decay with energy separation (Figure 3b,d). In this case, it is necessary to zoom in on the spectrum

Probing ERS in macroscopic samples of ultra-pure DWCNTs.

Recent progress in sorting DWCNTs by electronic type [START_REF] Li | Inner-and outer-wall sorting of double-walled carbon nanotubes[END_REF] and the development of macroscopic chirality-sorting procedures [START_REF] Yang | Chirality Pure Carbon Nanotubes: Growth, Sorting, and Characterization[END_REF] call for fast structural characterization methods of the macroscopic ensemble samples sensitive to specific SWCNT and DWCNT chiralities. ERS may become in this regard a useful tool for characterizing metallic layers, however it has been so far studied and applied only on microscopic individual SWCNTs and DWCNTs. Here, we investigate its behavior in a mixed-chirality ultra-pure DWCNT (UP-DWCNT) sample for the first time. The as-grown DWCNT sample was carefully purified from SWCNT and MWCNT species by a twostep density gradient ultracentrifugation approach, as discussed in Ref. [START_REF] Erkens | Ultrasonicationinduced extraction of inner shells from double-wall carbon nanotubes characterized via in situ spectroscopy after density gradient ultracentrifugation[END_REF][START_REF] Erkens | Efficient Inner-to-Outer Wall Energy Transfer in Highly Pure Double-Wall Carbon Nanotubes Revealed by Detailed Spectroscopy[END_REF]. We then measured its RBLM REM in the region of the outer layers' M11 transitions by continuously varying the excitation energy between 1.45 and 1.85 eV. The map is shown in Figure 6a, where we observe a series of maxima, each corresponding to several non-resolved DWCNT configurations. Each such maximum can potentially lead to a separate ERS peak in the Raman spectra. However, in contrast to ERS from individual free-standing DWCNT layers, we expect these peaks to be broadened in We then measured Raman spectra in the range from RBLM to G-band by exciting the UP-DWCNT sample at 737 nm (1.68 eV), as shown in Figure 6c (black curve). The spectrum is also compared with the reference solvent spectrum (blue curve) of a 2% solution of sodium deoxycholate (DOC) in D2O measured in identical conditions. The UP-DWCNT spectrum contains the contributions from the RBLMs (100 -400 cm -1 ), intermediate frequency modes [START_REF] Inaba | Effects of Chirality and Defect Density on the Intermediate Frequency Raman Modes of Individually Suspended Single-Walled Carbon Nanotubes[END_REF] (400 -1100 cm -1 ), D-band (~1300 cm -1 ) and the intense signal from largely overlapping G-band, M-band [START_REF] Vierck | ZA-derived phonons in the Raman spectra of single-walled carbon nanotubes[END_REF] and D′-band above 1500 cm -1 . Using the colored dashed lines, we also show in Figure 6c the predicted positions of ERS peaks corresponding to three M11 transitions from Figures 6a,b.

Interestingly, we do not observe any distinctive ERS features in the spectrum. This fact may be explained if we simulate the ERS peaks by using the linewidth and intensity values obtained from REP fits (see blue, red and yellow solid curves at the bottom of Figure 6c). Superimposing these ERS curves results in a featureless background (red solid line), which illustrates the effect of multichirality in macroscopic samples. We want to point out that the experimental DWCNT spectrum in Figure 6c is not proof of the ERS signal detection, but on the contrary, emphasizes the difficulty of the measurements on the mixed-chirality macroscopic DWCNT samples.

To get more insights into the ERS behavior in macro samples containing just one DWCNT chirality, we again modeled ERS lines using the example of individual free-standing DWCNT (14,8)@(19,14) (top solid blue line in Figure 6d). The effect of inhomogeneous broadening on the ERS was estimated using the Voigt functions with a varying Gaussian linewidth from 0 to 150 meV (see bottom four curves in Figure 6d). It becomes evident that above 50 meV, the ERS peak becomes too broad for practical detection. We, therefore, conclude that the successful observation of ERS features in macroscopic DWCNT samples should be possible only when they are sufficiently sorted, ideally to the single chirality level, or have well-separated electronic transitions and minimum possible inhomogeneous broadening (below ~ 50 meV). These conclusions should apply both for macroscopic mixed-chirality DWCNT solutions and DWCNT powders and thin films since the major difficulties in detecting ERS signals are common for both solid and liquid samples.

Conclusions

In this work, we report the observation of the ERS signal in individual index-identified DWCNTs having all types of electronic configurations with metallic layers. By comparing SWCNT and DWCNT data, we show that the increased dielectric screening in DWCNTs is most likely responsible for the suppression of the intensity of ERS peaks, in some cases leading to their complete disappearance from the Raman spectra.

Besides, we probe the ERS process in a macroscopic mixed-chirality ultra-pure DWCNT sample. It is found that the ERS lines are not distinguishable due to the large overlap between ERS peaks from different DWCNT chiralities present in the multi-chirality sample leading to a broad featureless background. We propose that observing ERS lines at the macroscale should be possible when the DWCNT samples can be sufficiently sorted, ideally to the single chirality level, or have well-separated electronic transitions and minimal inhomogeneous broadening.

Finally, we adapt a structural characterization procedure based on the ERS for DWCNTs with the dielectric screening and IET effects. The obtained results can be generalized to the

Section S1. Details of the DWCNT structure identification procedure

We characterized the structure of individual free-standing DWCNTs following either the electron diffraction (ED) based method from Ref. [1] or by combining Rayleigh and Raman spectroscopic information [2]. Tables S1.1 and S1.2 summarize the (n,m) indices of the studied DWCNTs and the indexation method used. The first column introduces the notation, e.g., DWCNT-1, that will be used throughout this SI for referring to specific DWCNTs. The ED patterns of the studied DWCNTs were previously provided in the Supplementary Materials of Ref.

[3]. Table S2.1. Optical transition energies for isolated SWCNTs and van-der-Waals coupled layers of DWCNT (14,1)@(15,12) with equal and opposite handedness, shown as DW(1) and DW(-1), respectively. SWCNT energies were estimated using the formula from Ref. [6], while those of DWCNT layers were calculated using the adapted code from Ref. [7]. The transition energy values are given in eV.

( Table S2.2. Optical transition energies for isolated SWCNTs and van-der-Waals coupled layers of DWCNT (14,2)@(22,4) with equal and opposite handedness, shown as DW(1) and DW(-1), respectively. SWCNT energies were estimated using the formula from Ref. [6], while those of DWCNT layers were calculated using the adapted code from Ref. [7]. The transition energy values are given in eV.

( Table S2.3. Optical transition energies for isolated SWCNTs and van-der-Waals coupled layers of DWCNT (11,8)@(19,10) with equal and opposite handedness, shown as DW(1) and DW(-1), respectively. SWCNT energies were estimated using the formula from Ref. [6], while those of DWCNT layers were calculated using the adapted code from Ref. [7]. The transition energy values are given in eV.

( Table S2.4. Optical transition energies for isolated SWCNTs and van-der-Waals coupled layers of DWCNT (14,8)@(19,14) with equal and opposite handedness, shown as DW(1) and DW(-1), respectively. SWCNT energies were estimated using the formula from Ref. [6], while those of DWCNT layers were calculated using the adapted code from Ref. [7]. The transition energy values are given in eV. Table S2.5. Optical transition energies for isolated SWCNTs and van-der-Waals coupled layers of DWCNT (14,5)@(26,1) with equal and opposite handedness, shown as DW( 1) and DW(-1), respectively. SWCNT energies were estimated using the formula from Ref. [6], while those of DWCNT layers were calculated using the adapted code from Ref. [7]. The transition energy values are given in eV. Table S2.6. Optical transition energies for isolated SWCNTs and van-der-Waals coupled layers of DWCNT (18,2)@(19,13) with equal and opposite handedness, shown as DW( 1) and DW(-1), respectively. SWCNT energies were estimated using the formula from Ref. [6], while those of DWCNT layers were calculated using the adapted code from Ref. [7]. The transition energy values are given in eV. Table S2.7. Optical transition energies for isolated SWCNTs and van-der-Waals coupled layers of DWCNT (18,6)@(21,14) with equal and opposite handedness, shown as DW( 1) and DW(-1), respectively. SWCNT energies were estimated using the formula from Ref. [6], while those of DWCNT layers were calculated using the adapted code from Ref. [7]. The transition energy values are given in eV.

( Table S2.8. Optical transition energies for isolated SWCNTs and van-der-Waals coupled layers of DWCNT (22,13)@(39,2) with equal and opposite handedness, shown as DW( 1) and DW(-1), respectively. SWCNT energies were estimated using the formula from Ref. [6], while those of DWCNT layers were calculated using the adapted code from Ref. [7]. The transition energy values are given in eV.

( Table S2.9. Optical transition energies for isolated SWCNTs and van-der-Waals coupled layers of DWCNT (17,14)@(26,14) with equal and opposite handedness, shown as DW( 1) and DW(-1), respectively. SWCNT energies were estimated using the formula from Ref. [6], while those of DWCNT layers were calculated using the adapted code from Ref. [7]. The transition energy values are given in eV. Table S2.10. Optical transition energies for isolated SWCNTs and van-der-Waals coupled layers of DWCNT (17,14)@(28,12) with equal and opposite handedness, shown as DW( 1) and DW(-1), respectively. SWCNT energies were estimated using the formula from Ref. [6], while those of DWCNT layers were calculated using the adapted code from Ref. [7]. The transition energy values are given in eV. Table S2.11. Optical transition energies for isolated SWCNTs and van-der-Waals coupled layers of DWCNT-11 with equal and opposite handedness, shown as DW(1) and DW(-1), respectively. SWCNT energies were estimated using the formula from Ref. [6], while those of DWCNT layers were calculated using the adapted code from Ref. [7]. The "std" and "mad" stand for standard deviation and mean absolute deviation. All energy values are given in eV. Table S2.12. Optical transition energies for isolated SWCNTs and van-der-Waals coupled layers of DWCNT-12 with equal and opposite handedness, shown as DW(1) and DW(-1), respectively. SWCNT energies were estimated using the formula from Ref. [6], while those of DWCNT layers were calculated using the adapted code from Ref. [7]. The "std" and "mad" stand for standard deviation and mean absolute deviation. All energy values are given in eV.

SC1@M

In

M@SC1

In Table S2. 13. Optical transition energies for isolated SWCNTs and van-der-Waals coupled layers of DWCNT-13 with equal and opposite handedness, shown as DW(1) and DW(-1), respectively. SWCNT energies were estimated using the formula from Ref. [6], while those of DWCNT layers were calculated using the adapted code from Ref. [7]. The "std" and "mad" stand for standard deviation and mean absolute deviation. All energy values are given in eV. Table S2.14. Optical transition energies for isolated SWCNTs and van-der-Waals coupled layers of DWCNT-14 with equal and opposite handedness, shown as DW(1) and DW(-1), respectively. SWCNT energies were estimated using the formula from Ref. [6], while those of DWCNT layers were calculated using the adapted code from Ref. [7]. The "std" and "mad" stand for standard deviation and mean absolute deviation. All energy values are given in eV. The absorption spectrum of (11,6)@(18,9) DWCNT was previously reported in Ref. [8] (see Fig. 4.23 and Table 4.10). Its fit gives the following DWCNT transition energies: S22 in = 1.36 eV, M11L out = 1.46 eV, M11H out = 1.52 eV, M22L out = 2.60 eV, S33 in + S44 in = 2.77 ± 0.04 eV, M22H out = 2.88 eV. Table S2. 15. Optical transition energies for isolated SWCNTs and van-der-Waals coupled layers of DWCNT (11,6)@(18,9) with equal and opposite handedness, shown as DW (1) and DW(-1), respectively. SWCNT energies were estimated using the formula from Ref. [6], while those of DWCNT layers were calculated using the adapted code from Ref. [7]. The transition energy values are given in eV. Table S4.1. Optical transition energies for DWCNT-1 (14,1)@(15,12) with equal and opposite handedness calculated within an ab initio-based NTB approach [10]. The energy values are given in eV and represent the raw NTB data from the recursion method, i.e., no upshift is applied and no dielectric downshift is added. S4.1.

SC1@M

In

SC1@M

In

On the transition energy assignment of (14,1)@(15,12) DWCNT

In our previous work on the detection of PL from individual DWCNTs [5], we assigned two peaks at ~1.44 and ~1.54 eV in the Rayleigh spectrum of (14,1)@(15,12) DWCNT (e.g., Figure 1b in the main text) to M11L and M11H + S22 transitions, respectively. The main arguments for this assignment were as follows: i) the RBM REP of the (15,12) layer has two maxima at ~1.44 and ~1.54 eV (Figure S4.3, upper panels) which may originate from M11L and M11H transitions, given their energies for noninteracting layers are expected at 1.47 and 1.5 eV, respectively (see Table S2.1); ii) the best fit of the TO REP of the (15,12) layer can be obtained only when M11L and M11H transitions are well separated (compare bottom panels in Figure S4.3a,b). However, new absorption spectrum calculations (Table S4.1) suggest an alternative assignment: M11L and M11H transitions of the outer (15,12) layer are overlapping at 1.44 eV, while the peak at 1.54 eV is solely due to the S22 transition. This interpretation can be supported by previous studies on isolated metallic SWCNTs [11] which established that the value of |MiiH -MiiL| is least sensitive to environmental interaction. As a first approximation, it may be true also for the interlayer interactions in DWCNTs at least in what concerns the dielectric screening effect. Therefore, considering the transition energies of air-suspended (15,12) SWCNT (Table S2.1), we obtain M11H -M11L = 0.032 eV. Given the experimental error, we should expect M11H in the DWCNT-1 at 1.47~1.50 eV. Combined with the fact that the intensity of M11H should be much lower than M11L, the former can correspond to the side peak of the 1.44 eV peak. In this interpretation, the appearance of the second maxima in the RBM REP at ~1.54 eV (Figure S4.3, upper panels) can be due to the mechanical coupling of inner and outer layers as previously reported for DWCNTs [12,13]. The behavior of the TO REP of the (15,12) layer is more difficult to understand but can also be related to the coupling between the layers.

In summary, we carefully considered all arguments in favor of both interpretations, but at this moment we cannot discriminate unambiguously between them. Therefore, we marked the possibility of two assignments with asterisks in Figure 1b, Figure 2a, and Figure 3b. We note that considering one or the other doesn't change the conclusions drawn here or in Ref. [5]. S4.2. Optical transition energies for DWCNT-7 (18,6)@(21,14) with equal and opposite handedness calculated within ab initio-based NTB approach [10]. The energy values are given in eV and represent the raw NTB data from the recursion method, i.e., no upshift is applied and no dielectric downshift is added. S4.2.

Section S5. Details on the Rayleigh data processing and fitting procedures

The experimental Rayleigh spectra were first converted from wavelength to energy scale using the Jacobian transformation, as described in Ref. [14]. Then, each spectrum was fitted by a least-squares optimization procedure with the following complex optical susceptibility function [15]:

𝐼 𝑅𝑎𝑦𝑙𝑒𝑖𝑔ℎ = 𝜔 3 (|𝜒 𝑖𝑛 | 2 + |𝜒 𝑜𝑢𝑡 | 2 ) 𝜒 𝑖𝑛 = ∑ 𝐴 𝑖 𝑖𝑛 𝜔 𝑖 𝑖𝑛 -𝜔 -𝑖𝛾 𝑖 𝑖𝑛 2 ⁄ 𝑁1 𝑖=1 + 𝜒 𝑛𝑜𝑛-𝑟𝑒𝑠 𝑖𝑛 𝜒 𝑜𝑢𝑡 = ∑ 𝐴 𝑖 𝑜𝑢𝑡 𝜔 𝑖 𝑜𝑢𝑡 -𝜔 -𝑖𝛾 𝑖 𝑜𝑢𝑡 2 ⁄ 𝑁2 𝑖=1 + 𝜒 𝑛𝑜𝑛-𝑟𝑒𝑠 𝑜𝑢𝑡
where Ak, ωk, γk (k = in, out) correspond to the amplitude, resonance frequency and linewidth of the complex Lorentzian function, respectively. The total number of transitions N1 and N2 and their approximated energies were estimated based on the SWCNT energies from the Atlas [6] and the shifts due to van der Waals electronic coupling [7] (see Tables in Section S2). The 𝜒 𝑖𝑛 𝑛𝑜𝑛-𝑟𝑒𝑠 and 𝜒 𝑜𝑢𝑡 𝑛𝑜𝑛-𝑟𝑒𝑠 modeled by real linear functions represent the effective non-resonant contributions of the transitions outside of the measured experimental range. These contributions were shown to be essential to account for the observed asymmetry of Rayleigh spectrum features [15].

Section S6. Details on the ERS data processing and fitting procedures

We present here the details of the processing and fitting of the wide-range Raman spectra presented in Sections S3-S4.

Data processing. First, we subtracted the background spectrum from all Raman spectra measured on the Acton SP2500 spectrometer to remove the artifact at ~ 400-500 cm -1 (Figure S6.1). Besides, we correct all Raman spectra measured in the NIR (shown as REMs in Figure 3 and in Section S4) for the instrument response and laser power variation during the measurements. Data fitting. We fitted the wide-range Raman spectra for the given DWCNT simultaneously, i.e., by sharing peak parameters between all spectra, with the following function:

𝐼 𝑗 𝑅𝑎𝑚𝑎𝑛 = ∑ 𝐹 𝑖 𝑝ℎ𝑜𝑛𝑜𝑛 + ∑ 𝐹 𝑘 𝐸𝑅𝑆 + 𝐹 𝑏𝑘𝑔 𝑁2 𝑘=1 𝑁1 𝑖=1
where 𝐼 𝑗 𝑅𝑎𝑚𝑎𝑛 is the intensity of the Role of interference effects between ERS and phonon lines. Saito et al. [16] previously predicted that the quantum interference between ERS and RBLM (or G-band) leads to the frequency shifts and changes in the lineshape (from symmetric to asymmetric) of the phonon lines. These changes become more pronounced if the intensity of ERS peaks is large relative to phonon peaks. However, we find that in DWCNTs, the ERS line due to its suppressed intensity leads to minor changes in the phonon properties. For instance, it is evidenced in Figure S6

Section S7. Table with results of the ERS fitting for DWCNTs

We note that we chose to present the results of the ERS data fit in the form of intensity (height) ratio IERS/ILO because it allows the direct comparison with the available literature data on SWCNTs. The broad LO mode was chosen for reference since it is considered representative of metallic SWCNT [17], and from our experience it is much easier to identify it in the rich G-band spectra of DWCNTs. The only problem may occur with the armchair SWCNTs, where the LO mode is absent, but we did not observe such chiralities in our experimental data set. On the other hand, choosing the narrow TO mode for the reference may complicate the analysis of (close) zigzag metallic DWCNT layers, where the narrow TO component is absent or hard to identify due to the overlap with the narrow LO modes of semiconducting layers.

Section S12. Estimation of the interlayer charge transfer between the layers of DWCNTs

The interlayer charge transfer density (in units of e/Å) for the probed DWCNTs was calculated using the formula from Zólyomi et al. [22]:

𝑄 𝐶𝑇 = -0.06 • ∆𝑟 + 0.23

The calculated QCT values for inner and outer layers were then converted into units of e -/C atom and are shown in Table S12.1 and Figure S12. The charge transfer necessary for the complete quenching of ERS peaks in individual metallic SWCNTs (QCT ≤ 0.005 e/C atom) was previously estimated by Farhat et al. [20] and is represented by the dashed line in Figure S12. Fig. S12. Estimation of the charge transfer between the layers of DWCNTs according to Zólyomi et al. [22]

Figure 1 .

 1 Figure 1. (a) Schematics of the experimental approach. (b) Experimental (black) and fitted (red)

Figure 2 .

 2 Figure 2. Details of the Electronic Raman scattering process in individual DWCNTs. The

  DWCNTs. The only deviation from this trend may occur when the LO integrated intensity vanishes between the incoming and outgoing resonances (e.g., at 1.59 eV in Figure3bor 1.65 eV in Figure3d) leading to the high values of the IERS/ILO ratio despite the low ERS intensity.

Figure 3 .

 3 Figure 3. Tunable wavelength study of ERS in individual index-identified DWCNTs (14,1)@(15,12) and (18,6)@(21,14). (a) REM for (14,1)@(15,12) DWCNT. The dashed line

Figure 4 .

 4 Figure 4. The ERS schematics and principal effects leading to the suppression of ERS intensity in DWCNTs. (a) The screening of Coulomb interactions between two excitons upon an increase of

Figure 5 .

 5 Figure 5. Experimental chirality distribution of studied DWCNT layers. Yellow and blue hexagons correspond to the metallic and semiconducting DWCNT layers, respectively, indexassigned by ED in this work. Green hexagons represent DWCNT layers characterized by the alloptical approach [14], hence several candidates are possible. Grey hexagons denote all geometrically possible metallic CNTs. Dashed lines tentatively partition the chirality map into three zones with the different expected intensity of ERS signal: Zone Ilayers affected by curvature-induced bandgap; Zone IIlayers with the expected maximum ERS intensity; Zone III layers with suppressed ERS intensity due to reduced exciton-exciton interaction at large diameters.

10- 20

 20 times to observe ERS unambiguously, as shown in Figure 2a (orange spectra). Repeating the ERS peak measurement at several laser excitation energies separated by 200 meV should allow the detection of most Mii transitions of the probed metallic layer.It is worth noting that the ERS can be useful for determining Mii energies in case of strong overlap between the optical transitions of the layers, which is quite often the situation for DWCNTs. A good example is the DWCNT (14,8)@(19,14) (see DWCNT-4 in SI Section S3), where M11L of the inner layer and S33 of the outer layer constitute a single broad peak centered at around 1.614 eV in the Rayleigh spectrum. Nevertheless, measuring the M11L energy from the ERS line in Raman spectra allows identifying the exact position of the metallic transition at 1.590 eV.

  the UP-DWCNT sample due to different local environments and multiple contributing DWCNT configurations. In consequence, we first estimate the inhomogeneous broadening of the Mii transitions in the macroscopic UP-DWCNT sample. For this purpose, we obtained the RBLM REPs by summing up the Raman spectrum intensities between 141-153.5, 153.5-163 and 163-177 cm -1 (see yellow, red and blue data points in Figure 6b labeled P3, P2 and P1, respectively). Fitting these RBLM REPs (solid curves in Figure 6b) allowed us to estimate the M11 linewidths, which are approximately equal to ERS linewidths in the Raman spectra according to the studies on individual free-standing DWCNTs.

Figure 6 .

 6 Figure 6. ERS process in macroscopic samples of mixed-chirality DWCNTs. (a) REM for inphase RBLM of UP-DWCNTs in the range of M11 optical transitions. Different maxima correspond to different DWCNT outer layers present in the macroscopic solution. Dashed horizontal lines show the position of the M11 maxima determined from REP fits. The white solid vertical line between 1.45 and 1.68 eV denotes the range covered by the Raman spectrum in (c). (b) RBLM REPs for the three intense maxima present in (a). The yellow, red and blue data points are obtained by summing up Raman spectrum intensity between 141-153.5, 153.5-163 and 163-177 cm -1 , respectively. The solid lines correspond to the results of the REP fits with Voigt functions, while dashed lines and P1, P2 and P3 labels show fitted M11 transition energies. (с) Experimental Raman spectra of UP-DWCNTs (black) and DOC/D2O (blue) excited at 1.68 eV (737 nm) with vertical dashed lines showing the expected positions of ERS peaks corresponding to M11 transitions identified in (b). The three colored Voigt curves below the experimental spectrum are the calculated ERS peaks with the inhomogeneous broadening obtained from REP

Figure S2. 1 .

 1 Figure S2.1. Rayleigh spectrum (left) and RBLM and G-band Raman spectra (right) for DWCNT (14,1)@(15,12). The peak at ~1.72 eV in the Rayleigh spectrum corresponds to the electronphonon sideband of the 𝑆 22 𝑖𝑛 transition.

Figure S2. 2 .

 2 Figure S2.2. Rayleigh spectrum (left) and RBLM and G-band Raman spectra (right) for DWCNT (14,2)@(22,4).

DWCNT- 3 :

 3 Figure S2.3. Rayleigh spectrum (left) and RBLM and G-band Raman spectra (right) for DWCNT (11,8)@(19,10).

Figure S2. 4 .

 4 Figure S2.4. Rayleigh spectrum (left) and RBLM and G-band Raman spectra (right) for DWCNT (14,8)@(19,14).

DWCNT- 5 :

 5 Figure S2.5. Rayleigh spectrum (left) and RBLM and G-band Raman spectra (right) for DWCNT (14,5)@(26,1).

DWCNT- 6 :

 6 Figure S2.6. Rayleigh spectrum (left) and RBLM and G-band Raman spectra (right) for DWCNT (18,2)@(19,13).

Figure S2. 7 .

 7 Figure S2.7. Rayleigh spectrum (left) and RBLM and G-band Raman spectra (right) for DWCNT (18,6)@(21,14).

Figure S2. 8 .

 8 Figure S2.8. Rayleigh spectrum (left) and RBLM and G-band Raman spectra (right) for DWCNT (22,13)@(39,2).

DWCNT- 9 :

 9 Figure S2.9. Rayleigh spectrum (left) and RBLM and G-band Raman spectra (right) for DWCNT (17,14)@(26,14).

Figure S2. 10 .

 10 Figure S2.10. Rayleigh spectrum (left) and RBLM and G-band Raman spectra (right) for DWCNT (17,14)@(28,12).

Figure S2. 11 .

 11 Figure S2.11. Rayleigh spectrum (left) and RBLM and G-band Raman spectra (right) for DWCNT-11.

Figure S2. 12 .

 12 Figure S2.12. Rayleigh spectrum (left) and RBLM and G-band Raman spectra (right) for DWCNT-12.

Figure S2. 13 .

 13 Figure S2.13. Rayleigh spectrum (left) and RBLM and G-band Raman spectra (right) for DWCNT-13.

Figure S2. 14 .

 14 Figure S2.14. Rayleigh spectrum (left) and RBLM and G-band Raman spectra (right) for DWCNT-14.

Section S3 .Figure S3. 1 .

 S31 Figure S3.1. Comparison of Rayleigh spectrum (top) and wide-range Raman spectra (bottom) plotted as a function of scattered photon energy for (14,1)@(15,12) DWCNT.

Figure S3. 2 .Figure S3. 3 .

 23 Figure S3.2. All wide-range Raman spectra for (14,1)@(15,12) DWCNT. Labels show the excitation wavelengths of the spectra.

Figure S3. 4 .

 4 Figure S3.4. All wide-range Raman spectra for (14,2)@(22,4) DWCNT. Labels show the excitation wavelengths of the spectra.

Figure S3. 6 .

 6 Figure S3.6. All wide-range Raman spectra for (11,8)@(19,10) DWCNT. Labels show the excitation wavelengths of the spectra.

Figure S3. 8 .

 8 Figure S3.8. All wide-range Raman spectra for (14,8)@(19,14) DWCNT. Labels show the excitation wavelengths of the spectra.

Figure S3. 10 .

 10 Figure S3.10. All wide-range Raman spectra for (14,5)@(26,1) DWCNT. Labels show the excitation wavelengths of the spectra.

Figure S3. 12 .

 12 Figure S3.12. All wide-range Raman spectra for (18,2)@(19,13) DWCNT. Labels show the excitation wavelengths of the spectra.

Figure S3. 14 .

 14 Figure S3.14. All wide-range Raman spectra for (18,6)@(21,14) DWCNT. Labels show the excitation wavelengths of the spectra.

Figure S3. 16 .

 16 Figure S3.16. All wide-range Raman spectra for (22,13)@(39,2) DWCNT. Labels show the excitation wavelengths of the spectra.

Figure S3. 18 .

 18 Figure S3.18. All wide-range Raman spectra for (17,14)@(26,14) DWCNT. Labels show the excitation wavelengths of the spectra.

Figure S3. 20 .

 20 Figure S3.20. All wide-range Raman spectra for (17,14)@(28,12) DWCNT. Labels show the excitation wavelengths of the spectra.

Figure S3. 22 .

 22 Figure S3.22. All wide-range Raman spectra for DWCNT-11. Labels show the excitation wavelengths of the spectra.

Figure S3. 24 .

 24 Figure S3.24. All wide-range Raman spectra for DWCNT-12. Labels show the excitation wavelengths of the spectra.

Figure S3. 26 .

 26 Figure S3.26. All wide-range Raman spectra for DWCNT-13. Labels show the excitation wavelengths of the spectra. The sharp peaks between 2500 and 4500 cm -1 in the Raman spectra measured at 492 nm correspond to the multi-phonon Raman lines[9].

Figure S3. 28 .

 28 Figure S3.28. All wide-range Raman spectra for DWCNT-14. Labels show the excitation wavelengths of the spectra.

Figure S3. 30 .Figure S4. 1 .

 301 Figure S3.30. The wide-range Raman spectrum for DWCNT-15. The label shows the excitation wavelength.

Figure S4. 2

 2 Figure S4.2 (a) Electronic DOS and (b) absorption coefficient of DWCNT-1 (14,1)@(15,12) with equal (DW(1), black line) and opposite (DW(-1), red line) handedness calculated within an ab initio-based NTB approach [10]. The dotted line in (b) represents the weighted sum of absorption coefficients of DWCNT layers in the absence of van der Waals interactions. The transition energies are summarized in TableS4.1.

Figure S4. 3 .

 3 Figure S4.3. REPs of RBM, LO-and TO-modes for (15,12) layer of the DWCNT-1 (14,1)@(15,12). The fit of the LO and TO REPs with (a) two close M11L and M11H transitions and (b) two separated M11L and M11H transitions.

Figure S4. 5 .

 5 Figure S4.5. (a) Electronic DOS and (b) absorption coefficient of DWCNT-7 (18,6)@(21,14) with equal (DW(1), black line) and opposite (DW(-1), red line) handedness calculated within ab initiobased NTB approach [10]. The dotted line in (b) represents the weighted sum of absorption coefficients of DWCNT layers in the absence of van der Waals interactions. The transition energies are summarized in TableS4.2.

Figure S6. 1 .

 1 Figure S6.1. Result of the background subtraction for a DWCNT.

  The examples of the simultaneous fit for DWCNT-4 and DWCNT-10 are shown in FigureS6.2.

Figure S6. 2 .

 2 Figure S6.2. Simultaneous fit results for DWCNT-4 (upper) and DWCNT-10 (bottom). Solid black and red curves represent experimental and fitted spectra, respectively. Solid and dashed colored lines correspond to the phonon and ERS peaks, respectively.

  .3, where the frequencies of RBLM and G-band barely change as a function of excitation wavelengths.

Figure S6. 3 .

 3 Figure S6.3. RBLM (left) and G-band (right) regions of Raman spectra of DWCNT-1 measured with 300 grooves/mm grating at 699.9, 750, 768.2, 771, 780, 785.2, 792, 800, 807, 815, 822, 837, 845, 852, 861.4, 865.9, 872.9 and 885 nm (from top to bottom).

). The spectra were measured at 1.60, 1.65, 2.52 and 2.71 eV for (a) and at 1.96,

	suspended SWCNTs (top panel) and DWCNTs (bottom panel). White crosses indicate data points diameter layers (> 2.3 nm). The dependences of the IERS/ILO ratio on the other DWCNT structural
	with ELaser -Eii < -0.01 eV obtained through a simultaneous fit of ERS peak tails in Stokes Raman parameters, namely interlayer distance Δr = (dout -din)/2, chiral angle θ and interlayer twist angle
	spectra (SI Section S6; peak linewidths and positions in these cases were fixed by the parameters Δθ = |θin|-|θout|, are given in the Section S10 with similar conclusions as made from Figure 2e.
	of the same ERS lines in other spectra with ELaser -Eii > 0) and do not correspond to Anti-Stokes
	Raman spectra. The SWCNT data points were taken from Refs. [17-19]. The error bars for
	DWCNTs correspond to two standard deviations; (d) Intensity distribution of IERS/ILO ratio for
	SWCNTs (gray bars) and DWCNTs (orange bars) (data from (c) except for data points with ELaser
	-Eii < -0.01 eV); (e) Dependence of the IERS/ILO ratio on the diameter for inner and outer layers
	(orange and blue dots, respectively). The white crosses have the same meaning as in (c).
	In general, we find that the ERS signal in DWCNTs has lower intensities than in isolated
	metallic SWCNTs, as is clear from comparing the intensity distributions in Figure 2d. For instance,
	the IERS/ILO ratios in DWCNTs (orange bars) are mainly distributed below 0.5, while those in
	SWCNTs (gray bars) have a more slowly-varying distribution up to a value of 1.5. Moreover, we
	can directly compare IERS/ILO ratios for three DWCNT layers, which were probed as individual
	SWCNTs in Refs. [17-19], namely (11,8), (14,2) and (14,5) (see Table 1). When compared under
	similar excitation (resonance) conditions, the IERS/ILO ratios for six optical transitions are at least a
	2.21, 2.33 and 2.52 eV for (b), counting from bottom to top. These laser excitation energies are factor of two lower in DWCNTs than in SWCNTs (see Section S9 for further discussion of this
	marked by black vertical bars on the right side of each spectrum. The Raman spectra are plotted comparison). In one case, namely the M11L transition of the (14,5) layer, the ERS peak was
	in absolute energy scale so that the ERS peak positions can be directly compared to the metallic completely quenched in the DWCNT.
	transitions in the corresponding Rayleigh spectra. Raman spectra are multiplied (orange lines) by Furthermore, we find no noticeable difference between the ERS behavior in the inner and
	a certain factor to emphasize the ERS features among stronger phonon peaks. The asterisk in (a) outer DWCNT layers, represented by orange and blue dots in Figure 2e, respectively. Both cases
	shows two possible positions of 𝑀 11𝐻 𝑜𝑢𝑡 transition (see SI Section S4); (c) Dependence of the IERS/ILO are characterized by the strong influence of the resonance effect (represented by the vertical spread
	ratio on the difference between laser energy Elaser and optical transition energy Eii in individual of the points for the given diameter) and the significant suppression of the ERS signal in large

Table 1 .

 1 Comparison

	of IERS/ILO ratios for selected (n,m) chiralities measured as individual
	SWCNTs [18,19] or as layers of DWCNTs. For each (n,m) chirality, the diameter d, chiral angle
	θ, type of optical transition, laser excitation energy Elaser, experimental transition energy Eii and
	energy separation dE = Elaser -Eii are given. RDW/SW corresponds to the ratio of (IERS/ILO) in
	DWCNT to the (IERS/ILO) in SWCNT.

# (

n,m) d (nm) θ (°) Trans. CNT Elaser (eV) Eii (eV) dE (eV) IERS/ILO RDW/SW

	1 (11,8)	1.29	24.8	M11L	SW DW	2.33 2.33	1.94 1.90	0.39 0.43	2.67 ± 0.67 0.16 ±0.07	0.06 (±0.03)
	2 (11,8)	1.29	24.8	M11H	SW DW	2.33 2.33	2.03 1.96	0.30 0.37	1.50 ±0.43 0.34 ±0.13	0.23 (±0.11)
	3 (14,2)	1.18	6.6	M11L	SW DW	2.33 2.21	1.95 1.86	0.30 0.35	0.28 ±0.01 0.12 ±0.01	0.43 (±0.05)
	4 (14,5)	1.34	14.7	M11L	SW DW	2.33 2.21	1.83 1.81	0.50 0.40	1.38 ±0.06 0.18 ±0.10	0.13 (±0.07)
	5 (14,5)	1.34	14.7	M11L	SW DW	1.96 1.96	1.84 1.81	0.12 0.15	0.83 ±0.02 0.00	0.00
					SW	2.33	2.07	0.26	0.44 ±0.04
	6 (14,5)	1.34	14.7	M11H	DW	2.21	1.96	0.25	0.20 ±0.03	0.46 (±0.09)
	3

.2 Tunable wavelength study of the ERS process in individual free-standing DWCNTs.

  The characterization of the macroscopic UP-DWCNT samples was financially supported by a PhD fellowship for M. Erkens of the Research Foundation of Flanders (FWO, grant number: 11C9220N). D.L. acknowledges the financial support from the University of Antwerp Research

	Fund (grant BOF KP 43770). D.L., M.A., W.W. and S.C. also acknowledge funding from FWO
	projects (G040011N, G02112N, G035918N and G036618N).
	Supporting information
	Supporting Information is available online at ….. and includes (1) Details of the DWCNT

structure identification procedure, (2) Complete set of Rayleigh, RBLM and G-band spectra for investigated DWCNTs, (3) Complete set of wide-range Raman and ERS data for investigated DWCNTs, (4) Complete set of tunable Raman spectra and results of absorption spectrum calculations for DWCNTs (14,1)@(15,12) and (18,6)@(21,14),

(5)

, Details on the Rayleigh data processing and fitting procedures, (6) Details on the ERS data processing and fitting procedures,

(7)

Table with results of the ERS fitting for DWCNTs, (8) Table with results of the ERS data

digitization for SWCNTs, (9) Comparison of IERS/ILO ratios in individual free-standing SWCNTs and DWCNTs, (10) Dependence of IERS/ILO ratio on structural parameters of DWCNT layers, (11) Effect of curvature-induced bandgap on ERS lines, (12) Estimation of the interlayer charge transfer between the layers of DWCNTs.

Table S1 . 1 .

 S11 List of the studied DWCNTs with metallic layers.

	#	(nin,min)@(nout,mout)	Indexation Method	Electronic Configuration	Source
	DWCNT-1	(14,1)@(15,12)	ED	SC@M	Ref. [3]
	DWCNT-2	(14,2)@(22,4)	ED	M@M	Ref. [3]
	DWCNT-3	(11,8)@(19,10)	Rayleigh+Raman	M@M	Ref. [4]
	DWCNT-4	(14,8)@(19,14)	ED	M@SC	Ref. [3]
	DWCNT-5	(14,5)@(26,1)	ED	M@SC	Ref. [3]
	DWCNT-6	(18,2)@(19,13)	ED	SC@M	Ref. [3]
	DWCNT-7	(18,6)@(21,14)	ED	M@SC	Ref. [3]
	DWCNT-8	(22,13)@(39,2)	ED	M@SC	Ref. [3]
	DWCNT-9	(17,14)@(26,14)	ED	M@M	Ref. [3]
	DWCNT-10	(17,14)@(28,12)	ED	M@SC	Ref. [3]
	DWCNT-11	multiple candidates	Rayleigh+Raman	SC@M	this work
	DWCNT-12	multiple candidates	Rayleigh+Raman	M@SC	this work
	DWCNT-13	multiple candidates	Rayleigh+Raman	SC@M	this work
	DWCNT-14	multiple candidates	Rayleigh+Raman	SC@M	this work
	DWCNT-15	(11,6)@(18,9)	ED	SC@M	Ref. [3]

Table S1 . 2 .

 S12 List of the studied DWCNTs with purely semiconducting layers.

	#	(nin,min)@(nout,mout)	Indexation Method	Electronic Configuration	Source
	DWCNT-16	(10,6)@(14,13)	Rayleigh+Raman	SC@SC	Ref. [5]
	DWCNT-17	(16,2)@(16,14)	ED	SC@SC	Ref. [3]
	DWCNT-18	(14,1)@(21,4)	Rayleigh+Raman	SC@SC	Ref. [3]
	DWCNT-19	(30,1)@(27,19)	ED	SC@SC	Ref. [3]
	DWCNT-20	(13,5)@(18,11)	Rayleigh+Raman	SC@SC	this work
	DWCNT-21	(22,17)@(30,20)	ED	SC@SC	this work

,2)@(22,4) M@M

	In	𝐌 𝟏𝟏𝐋 𝒊𝒏	𝐌 𝟏𝟏𝐇 𝒊𝒏	𝐌 𝟐𝟐𝐋 𝒊𝒏	𝐌 𝟐𝟐𝐇 𝒊𝒏	𝐌 𝟑𝟑𝐋 𝒊𝒏	𝐌 𝟑𝟑𝐇 𝒊𝒏
	SWCNT	1.941	2.307	2.900	4.150	-	-
	DW(1)	1.872	2.238	2.786	4.064	-	-
	DW(-1)	1.853	2.207	3.62	4.182	-	-
	Out	𝐌 𝟏𝟏𝐋 𝐨𝐮𝐭	𝐌 𝟏𝟏𝐇 𝐨𝐮𝐭	𝐌 𝟐𝟐𝐋 𝐨𝐮𝐭	𝐌 𝟐𝟐𝐇 𝐨𝐮𝐭	𝐌 𝟑𝟑𝐋 𝐨𝐮𝐭	𝐌 𝟑𝟑𝐇 𝐨𝐮𝐭
	SWCNT	1.370	1.508	2.426	2.894	3.038	3.891
	DW(1)	1.312	1.455	2.385	2.842	3.017	3.839
	DW(-1)	1.316	1.451	2.369	2.839	2.974	3.837

11,8)@(19,10) M@M

	In	𝐌 𝟏𝟏𝐋 𝒊𝒏	𝐌 𝟏𝟏𝐇 𝒊𝒏	𝐌 𝟐𝟐𝐋 𝒊𝒏	𝐌 𝟐𝟐𝐇 𝒊𝒏	𝐌 𝟑𝟑𝐋 𝒊𝒏	𝐌 𝟑𝟑𝐇 𝒊𝒏
	SWCNT	1.936	2.024	3.228	3.527	3.626	4.171
	DW(1)	1.856	1.984	3.098	3.454	3.24	4.081
	DW(-1)	1.863	1.941	3.167	3.413	3.516	3.981
	Out	𝐌 𝟏𝟏𝐋 𝐨𝐮𝐭	𝐌 𝟏𝟏𝐇 𝐨𝐮𝐭	𝐌 𝟐𝟐𝐋 𝐨𝐮𝐭	𝐌 𝟐𝟐𝐇 𝐨𝐮𝐭	𝐌 𝟑𝟑𝐋 𝐨𝐮𝐭	𝐌 𝟑𝟑𝐇 𝐨𝐮𝐭
	SWCNT	1.342	1.411	2.447	2.682	3.159	3.589
	DW(1)	1.269	1.357	2.417	2.629	3.15	3.534
	DW(-1)	1.289	1.357	2.378	2.629	3.133	3.534

DWCNT-4: (14,8)@(19,14), M@SC

(14,8)@(19,14) M@SC2

	In	𝐌 𝟏𝟏𝐋 𝒊𝒏	𝐌 𝟏𝟏𝐇 𝒊𝒏	𝐌 𝟐𝟐𝐋 𝒊𝒏	𝐌 𝟐𝟐𝐇 𝒊𝒏	𝐌 𝟑𝟑𝐋 𝒊𝒏	𝐌 𝟑𝟑𝐇 𝒊𝒏	-
	SWCNT 1.693	1.801	2.912	3.280	3.462	4.132	-
	DW(1)	1.651	1.794	2.862	3.224	3.406	4.203	-
	DW(-1)	1.634	1.737	2.862	3.207	3.4	4.043	-
	Out	𝐒 𝟏𝟏 𝐨𝐮𝐭	𝐒 𝟐𝟐 𝐨𝐮𝐭	𝐒 𝟑𝟑 𝐨𝐮𝐭	𝐒 𝟒𝟒 𝐨𝐮𝐭	𝐒 𝟓𝟓 𝐨𝐮𝐭	𝐒 𝟔𝟔 𝐨𝐮𝐭	𝐒 𝟕𝟕 𝐨𝐮𝐭
	SWCNT 0.468	0.830	1.686	1.981	2.700	2.837	3.507
	DW(1)	0.447	0.766	1.599	1.917	2.633	2.772	3.437
	DW(-1)	0.407	0.77	1.625	1.917	2.639	2.786	3.445

(14,5)@(26,1) M@SC1

	In	𝐌 𝟏𝟏𝐋 𝒊𝒏	𝐌 𝟏𝟏𝐇 𝒊𝒏	𝐌 𝟐𝟐𝐋 𝒊𝒏	𝐌 𝟐𝟐𝐇 𝒊𝒏	𝐌 𝟑𝟑𝐋 𝒊𝒏	𝐌 𝟑𝟑𝐇 𝒊𝒏
	SWCNT 1.821	2.040	2.948	3.695	-	-
	DW(1)	1.761	1.976	2.873	3.669	-	-
	DW(-1)	1.76	1.976	2.869	3.62	-	-
	Out	𝐒 𝟏𝟏 𝐨𝐮𝐭	𝐒 𝟐𝟐 𝐨𝐮𝐭	𝐒 𝟑𝟑 𝐨𝐮𝐭	𝐒 𝟒𝟒 𝐨𝐮𝐭	𝐒 𝟓𝟓 𝐨𝐮𝐭	𝐒 𝟔𝟔 𝐨𝐮𝐭
	SWCNT 0.487	0.943	1.684	2.294	2.562	3.407
	DW(1)	0.427	0.881	1.623	2.233	2.497	3.346
	DW(-1)	0.427	0.881	1.623	2.233	2.499	3.346

(18,2)@(19,13) SC1@M

	In	𝐒 𝟏𝟏 𝐢𝐧	𝐒 𝟐𝟐 𝐢𝐧	𝐒 𝟑𝟑 𝐢𝐧	𝐒 𝟒𝟒 𝐢𝐧	𝐒 𝟓𝟓 𝐢𝐧	𝐒 𝟔𝟔 𝐢𝐧
	SWCNT 0.660	1.290	2.140	3.018	-	-
	DW(1)	0.609	1.188	1.982	2.881	-	-
	DW(-1)	0.611	1.192	2.054	2.887	-	-
	Out	𝐌 𝟏𝟏𝐋 𝐨𝐮𝐭	𝐌 𝟏𝟏𝐇 𝐨𝐮𝐭	𝐌 𝟐𝟐𝐋 𝐨𝐮𝐭	𝐌 𝟐𝟐𝐇 𝐨𝐮𝐭	𝐌 𝟑𝟑𝐋 𝐨𝐮𝐭	𝐌 𝟑𝟑𝐇 𝐨𝐮𝐭
	SWCNT 1.255	1.292	2.339	2.463	3.096	3.322
	DW(1)	1.201	1.236	2.283	2.349	3.035	3.287
	DW(-1)	1.202	1.238	2.283	2.407	3.031	3.251

DWCNT-7: (18,6)@(21,14), M@SC

,6)@(21,14) M@SC1

	In	𝐌 𝟏𝟏𝐋 𝒊𝒏	𝐌 𝟏𝟏𝐇 𝒊𝒏	𝐌 𝟐𝟐𝐋 𝒊𝒏	𝐌 𝟐𝟐𝐇 𝒊𝒏	𝐌 𝟑𝟑𝐋 𝒊𝒏	𝐌 𝟑𝟑𝐇 𝒊𝒏	-
	SWCNT 1.516	1.658	2.636	3.119	3.209	4.089	-
	DW(1)	1.574	1.56	2.605	2.983	3.162	3.83	-
	DW(-1)	1.457	1.572	2.54	3.011	3.223	3.949	-
	Out	𝐒 𝟏𝟏 𝐨𝐮𝐭	𝐒 𝟐𝟐 𝐨𝐮𝐭	𝐒 𝟑𝟑 𝐨𝐮𝐭	𝐒 𝟒𝟒 𝐨𝐮𝐭	𝐒 𝟓𝟓 𝐨𝐮𝐭	𝐒 𝟔𝟔 𝐨𝐮𝐭	𝐒 𝟕𝟕 𝐨𝐮𝐭
	SWCNT 0.434	0.803	1.554	1.953	2.471	2.882	3.125
	DW(1)	0.371	0.73	1.489	1.919	2.402	2.843	3.049
	DW(-1)	0.374	0.741	1.492	1.891	2.403	2.818	3.017

DWCNT-8: (22,13)@(39,2), M@SC

,13)@(39,2) M@SC1

	In	𝐌 𝟏𝟏𝐋 𝒊𝒏	𝐌 𝟏𝟏𝐇 𝒊𝒏	𝐌 𝟐𝟐𝐋 𝒊𝒏	𝐌 𝟐𝟐𝐇 𝒊𝒏	𝐌 𝟑𝟑𝐋 𝒊𝒏	𝐌 𝟑𝟑𝐇 𝒊𝒏	-
	SWCNT 1.150	1.191	2.163	2.301	2.901	3.153	-
	DW(1)	1.092	1.132	2.098	2.237	2.825	3.081	-
	DW(-1)	1.094	1.132	2.101	2.238	2.83	3.083	-
	Out	𝐒 𝟏𝟏 𝐨𝐮𝐭	𝐒 𝟐𝟐 𝐨𝐮𝐭	𝐒 𝟑𝟑 𝐨𝐮𝐭	𝐒 𝟒𝟒 𝐨𝐮𝐭	𝐒 𝟓𝟓 𝐨𝐮𝐭	𝐒 𝟔𝟔 𝐨𝐮𝐭	𝐒 𝟕𝟕 𝐨𝐮𝐭
	SWCNT 0.330	0.631	1.201	1.586	1.945	2.435	2.492
	DW(1)	0.269	0.57	1.14	1.524	1.884	2.373	2.428
	DW(-1)	0.269	0.57	1.14	1.525	1.884	2.374	2.429

(17,14)@(26,14) M@M

	In	𝐌 𝟏𝟏𝐋 𝒊𝒏	𝐌 𝟏𝟏𝐇 𝒊𝒏	𝐌 𝟐𝟐𝐋 𝒊𝒏	𝐌 𝟐𝟐𝐇 𝒊𝒏	𝐌 𝟑𝟑𝐋 𝒊𝒏	𝐌 𝟑𝟑𝐇 𝒊𝒏
	SWCNT	1.304	1.325	2.432	2.502	3.214	3.341
	DW(1)	1.161	1.14	2.203	2.489	2.801	3.243
	DW(-1)	1.215	1.182	1.98	2.292	3.24	3.046
	Out	𝐌 𝟏𝟏𝐋 𝐨𝐮𝐭	𝐌 𝟏𝟏𝐇 𝐨𝐮𝐭	𝐌 𝟐𝟐𝐋 𝐨𝐮𝐭	𝐌 𝟐𝟐𝐇 𝐨𝐮𝐭	𝐌 𝟑𝟑𝐋 𝐨𝐮𝐭	𝐌 𝟑𝟑𝐇 𝐨𝐮𝐭
	SWCNT	1.016	1.051	1.939	2.060	2.649	2.869
	DW(1)	0.946	0.984	2.091	1.993	NA	2.790
	DW(-1)	0.960	0.988	1.843	1.995	3.051	2.800

DWCNT-10: (17,14)@(28,12), M@SC

(17,14)@(28,12) M@SC1

	In	𝐌 𝟏𝟏𝐋 𝒊𝒏	𝐌 𝟏𝟏𝐇 𝒊𝒏	𝐌 𝟐𝟐𝐋 𝒊𝒏	𝐌 𝟐𝟐𝐇 𝒊𝒏	𝐌 𝟑𝟑𝐋 𝒊𝒏	𝐌 𝟑𝟑𝐇 𝒊𝒏	-
	SWCNT 1.304	1.325	2.432	2.502	3.214	3.341	-
	DW(1)	1.210	1.246	2.302	2.315	3.036	3.273	-
	DW(-1) 1.237	1.232	2.318	2.378	3.201	3.179	-
	Out	𝐒 𝟏𝟏 𝐨𝐮𝐭	𝐒 𝟐𝟐 𝐨𝐮𝐭	𝐒 𝟑𝟑 𝐨𝐮𝐭	𝐒 𝟒𝟒 𝐨𝐮𝐭	𝐒 𝟓𝟓 𝐨𝐮𝐭	𝐒 𝟔𝟔 𝐨𝐮𝐭	𝐒 𝟕𝟕 𝐨𝐮𝐭
	SWCNT 0.372	0.701	1.348	1.737	2.170	2.623	2.773
	DW(1)	0.317	0.634	1.281	1.671	1.984	2.556	2.738
	DW(-1) 0.313	0.637	1.284	1.673	2.094	2.558	2.617

DWCNT-11: SC@M

  j-th Raman spectrum, Fbkg is a linear function representing the

	remaining background, 𝐹 𝑖 𝑝ℎ𝑜𝑛𝑜𝑛 and 𝐹 𝑘 𝐸𝑅𝑆 are Lorentzian functions for phonon and ERS lines,
	respectively defined as:
	𝐹 𝑖 𝑝ℎ𝑜𝑛𝑜𝑛 =	𝑝ℎ𝑜𝑛𝑜𝑛 𝐴 𝑖 𝑝ℎ𝑜𝑛𝑜𝑛 ) (𝑥 -𝜔 𝑖 2 + (0.5𝛾 𝑖 𝑝ℎ𝑜𝑛𝑜𝑛 )	2
	𝐹 𝑘 𝐸𝑅𝑆 =	𝐸𝑅𝑆 𝐴 𝑘 𝐸𝑅𝑆 ) 2 + (0.5𝛾 𝑘 (𝑥 -𝜔 𝑘 𝐸𝑅𝑆 ) 2
	where 𝜔 𝑖 𝑝ℎ𝑜𝑛𝑜𝑛 , 𝛾 𝑖 𝑝ℎ𝑜𝑛𝑜𝑛 , 𝐴 𝑖 𝑝ℎ𝑜𝑛𝑜𝑛 , 𝜔 𝑘 𝐸𝑅𝑆 = 𝑓(𝜆 𝑙𝑎𝑠𝑒𝑟 ), 𝛾 𝑘 𝐸𝑅𝑆 , 𝐴 𝑘

𝐸𝑅𝑆

are phonon frequency, phonon linewidth, phonon line amplitude, ERS frequency at the given excitation wavelength λlaser, ERS linewidth and ERS amplitude, respectively. The ERS frequency (in cm -1 ) for a given spectrum (i.e. at the excitation wavelength λlaser in nm) is determined from the Mii energy in eV by:

Table S7 . 1 .

 S71 Results of the ERS data fitting for investigated individual DWCNTs. For each (n,m) chirality, the diameter d, uncertainty in diameter δd (relevant for DWCNTs characterized by optical spectroscopy), interlayer distance Δr, chiral angle θ, the cutting line index p (L or H stand for the lower and higher transitions), laser excitation energy EL, experimental transition energy EERS and energy separation dE = EL -EERS are given. RH and δRH correspond to the ratio (IERS/ILO) and its uncertainty (two standard deviations), respectively; ωLO is the frequency of the LO phonon. RH δRH EL(eV) ωLO (cm -1 ) (14,1)@(15,12) 12 1.834 0.000 26.33 out 0.3486 3 L 1.44 0.212 0.218 0.091 1.650 1568.6 (14,1)@(15,12) 12 1.834 0.000 26.33 out 0.3486 3 L 1.44 0.022 16.066 10.186* 1.460 1568.6 (14,1)@(15,12) 12 1.834 0.000 26.33 out 0.3486 6 L 2.63 0.079 0.228 0.029 2.710 1568.6 (14,1)@(15,12) 12 1.834 0.000 26.33 out 0.3486 6 H 2.68 0.029 0.180 0.069 2.710 1568.6 (14,2)@(22,4) 2 1.182 0.000 6.59 in 0.3581 3 L 1.86 0.351 0.124 0.014 2.210 1547.4 (14,2)@(22,4) 2 1.182 0.000 6.59 in 0.3581 3 L 1.86 0.101 0.215 0.025 1.960 1547.4 (14,2)@(22,4) 2 1.182 0.000 6.59 in 0.3581 3 L 1.86 0.021 )@(19,13) 13 2.182 0.000 23.82 out 0.3443 6 H 2.37 0.152 0.063 0.004 2.520 1572.4 (14,8)@(19,14) 8 1.510 0.000 21.05 in 0.3680 3 L 1.59 0.372 0.000 -1.960 1542.2 (14,8)@(19,14) 8 1.510 0.000 21.05 in 0.3680 3 L 1.59 0.292 0.130 0.012 1.880 1542.2 (14,8)@(19,14) 8 1.510 0.000 21.05 in 0.3680 3 L 1.59 0.192 0.373 0.044 1.780 1542.2 (14,8)@(19,14) 8 1.510 0.000 21.05 in 0.3680 3 L 1.59 0.072 0.920 0.047 1.660 1542.2

		0.734	0.219 1.880 1547.4
	(14,2)@(22,4) 2 1.182 0.000 6.59 in 0.3581 3 H 2.15 0.369	0.039	0.017 2.520 1547.4
	(14,2)@(22,4) 2 1.182 0.000 6.59 in 0.3581 3 H 2.15 0.179	0.018	0.013 2.330 1547.4
	(14,2)@(22,4) 2 1.182 0.000 6.59 in 0.3581 3 H 2.15 0.059	0.038	0.017 2.210 1547.4
	(14,2)@(22,4) 4 1.898 0.000 8.21 out 0.3581 3 L 1.35 0.254	0.259	0.022 1.600 1577.0
	(14,2)@(22,4) 4 1.898 0.000 8.21 out 0.3581 3 L 1.35 0.114	1.313	0.253 1.460 1577.0
	(14,2)@(22,4) 4 1.898 0.000 8.21 out 0.3581 3 H 1.47 0.133	0.217	0.023 1.600 1577.0
	(14,2)@(22,4) 4 1.898 0.000 8.21 out 0.3581 3 H 1.47 -0.007 2.778	0.564 1.460 1577.0
	(14,2)@(22,4) 4 1.898 0.000 8.21 out 0.3581 3 H 2.34 0.182	0.019	0.006 2.520 1577.0
	(14,2)@(22,4) 4 1.898 0.000 8.21 out 0.3581 3 H 2.34 -0.008 0.083	0.021 2.330 1577.0
	(11,8)@(19,10) 8 1.294 0.000 24.79 in 0.3520 3 L 1.90 0.433	0.156	0.069 2.330 1521.9
	(11,8)@(19,10) 8 1.294 0.000 24.79 in 0.3520 3 L 1.90 0.063	0.365	0.162 1.960 1521.9
	(11,8)@(19,10) 8 1.294 0.000 24.79 in 0.3520 3 H 1.96 0.373	0.340	0.129 2.330 1521.9
	(11,8)@(19,10) 8 1.294 0.000 24.79 in 0.3520 3 H 1.96 0.003	1.286	0.448 1.960 1521.9
	(11,8)@(19,10) 10 1.998 0.000 19.84 out 0.3520 3 H 1.36 0.244	0.546	0.064 1.600 1572.3
	(11,8)@(19,10) 10 1.998 0.000 19.84 out 0.3520 6 L 2.38 0.334	0.047	0.015 2.710 1572.3
	(11,8)@(19,10) 10 1.998 0.000 19.84 out 0.3520 6 L 2.38 0.154	0.092	0.008 2.530 1572.3
	(11,8)@(19,10) 10 1.998 0.000 19.84 out 0.3520 6 L 2.38 -0.046 2.404	0.243 2.330 1572.3
	(14,5)@(26,1) 5 1.336 0.000 14.70 in 0.3701 3 L 1.81 0.401	0.175	0.102 2.210 1545.5
	(14,5)@(26,1) 5 1.336 0.000 14.70 in 0.3701 3 L 1.81 0.151	0.000	-	1.960 1545.5
	(14,5)@(26,1) 5 1.336 0.000 14.70 in 0.3701 3 H 1.96 0.249	0.202	0.034 2.210 1545.5
	(14,5)@(26,1) 5 1.336 0.000 14.70 in 0.3701 3 H 1.96 -0.001 0.087	0.007 1.960 1545.5
	(18,2)@(19,13) 13 2.182 0.000 23.82 out 0.3443 6 L 2.25 0.275	0.013	0.003 2.520 1572.4
	(18,2)@(19,13) 13 2.182 0.000 23.82 out 0.3443 6 L 2.25 0.085	0.113	0.006 2.330 1572.4
	(18,2)@(19,13) 13 2.182 0.000 23.82 out 0.3443 6 L 2.25 -0.035 0.600	0.055 2.210 1572.4
	(18,2

DWCNT

n m d (nm) δd (nm) θ (°) Layer Δr (nm) p EERS dE (eV)

Section S8. Table with results of the ERS data digitization for SWCNTsTable S8 . 1 .

 S81 Results of the ERS data digitization for SWCNTs from the literature[18][19][20]. For each (n,m) chirality, the diameter d, chiral angle θ, the cutting line index p (L or H stand for the lower and higher transitions), laser excitation energy EL, experimental transition energy EERS and energy separation dE = EL -EERS are given. RH and δRH correspond to the ratio (IERS/ILO) and its uncertainty (determined from the pixels in the image), respectively.

							RH	δRH EERS (eV)	p	dE (eV)
	23 14	2.53	22.01	2.14	134	505	0.265 0.006	2.095	6 L	0.045
	12	1.43	25.28	1.96	52	62	0.839 0.060	1.888	3 H	0.071
	12	1.43	25.28	1.96	69	62	1.113 0.068	1.817	3 L	0.142
	13	1.38	20.17	1.96	92	148	0.622 0.023	1.836	3 L	0.123
	14	1.34	14.70	1.96	124	150	0.827 0.024	1.837	3 L	0.122
	15	1.31	8.95	1.96	144	176	0.818 0.021	1.835	3 L	0.124
	16	1.29	3.00	1.96	141	161	0.876 0.023	1.834	3 L	0.125
	12	1.43	25.28	2.33	18	17	1.059 0.242	1.884	3 H	0.446
	14	1.34	14.70	2.33	41	40	1.025 0.101	2.044	3 H	0.286
	11	1.29	24.79	2.33	18	12	1.500 0.425	2.028	3 H	0.302
	11	1.29	24.79	2.33	32	12	2.667 0.671	1.936	3 L	0.394
	12	1.24	19.11	2.33	62	241	0.257 0.012	2.155	3 H	0.175
	12	1.24	19.11	2.33	32	241	0.133 0.012	1.956	3 L	0.374
	13	1.21	13.00	2.33	250	492	0.508 0.006	2.244	3 H	0.086
	13	1.21	13.00	2.33	43	492	0.087 0.006	1.942	3 L	0.388
	14	1.18	6.59	2.33	117	426	0.275 0.007	1.949	3 L	0.381
	17	1.56	12.52	1.96	34	130	0.262 0.022	1.793	3 H	0.166
	17	1.56	12.52	1.96	22	130	0.169 0.022	1.624	3 L	0.335
	13 10	1.56	25.69	1.96	10	135	0.074 0.021	1.750	3 H	0.209
	13 10	1.56	25.69	1.96	21	135	0.156 0.021	1.688	3 L	0.271
	16	1.29	3.00	2.33	52	229	0.227 0.013	2.182	3 H	0.148
	16	1.29	3.00	2.33	25	229	0.109 0.012	1.823	3 L	0.507
	13	1.38	20.17	2.33	39	29	1.345 0.163	1.979	3 H	0.351
	13	1.38	20.17	2.33	44	29	1.517 0.177	1.826	3 L	0.504
	13	1.38	20.17	1.96	111	216	0.514 0.015	1.820	3 L	0.139
	14	1.34	14.70	2.33	37	84	0.440 0.037	2.066	3 H	0.264
	14	1.34	14.70	2.33	116	84	1.381 0.057	1.825	3 L	0.505

n m d (nm) θ (°) EL (eV) IERS (pix) ILO (pix)

Table S12 . 1 .

 S121 Charge transfer density for investigated DWCNTs calculated using the formula from Zólyomi et al.[22].

												Qin	Qout
	#	nin	min	Din (nm)	Type	nout	mout	Dout (nm)	Type	<D> (nm)	Δr (nm)	(10 -5 e/C	(10 -5 e/C
												atom)	atom)
	1	14	1	1.137	1	15	12	1.834	0	1.486	0.349	88	55
	2	11	6	1.169	2	18	9	1.864	0	1.517	0.348	89	56
	3	14	2	1.182	0	22	4	1.898	0	1.540	0.358	62	38
	4	11	8	1.294	0	19	10	1.998	0	1.646	0.352	70	45
	5	14	5	1.336	0	26	1	2.076	1	1.706	0.370	29	18
	6	18	2	1.494	1	19	13	2.182	0	1.838	0.344	75	52
	7	14	8	1.510	0	19	14	2.246	2	1.878	0.368	29	20
	8	18	6	1.694	0	21	14	2.389	1	2.041	0.348	61	43
	9	17	14	2.105	0	26	14	2.752	0	2.429	0.324	82	63
	10	17	14	2.105	0	28	12	2.783	1	2.444	0.339	61	46
	11	22	13	2.399	0	39	2	3.134	1	2.767	0.368	19	14

ARAID Foundation, 50018 Zaragoza, Spain

Acknowledgment DL thanks Prof. S. Rochal and D. Chalin for providing MAPLE code for calculating DWCNT transition energy shifts and W. Van Werveke and M. De Clercq for providing MATLAB code for creating chirality distribution maps. A part of the experimental data (i.e., microscopic Rayleigh and Raman spectra, ED patterns) was obtained and preliminary processed within the framework of Russian Foundation for Basic Research grant 18-32-00397 mol_a. Further data analysis (i.e., developing models for ERS, Raman and Rayleigh spectra fitting) was performed by DL with the support of the postdoctoral grant (12ZP720N) of the Fund for Scientific Research Flanders (FWO). The TEM measurements were performed in the Laboratorio de Microscopias Avanzadas (LMA), Universidad de Zaragoza (Spain). R.A. gratefully acknowledges the support from the Spanish MICINN (PID2019-104739GB-100/AEI/10.13039/501100011033), Government of Aragon (projects DGA E13-20R (FEDER, EU)) and from the European Union H2020 program "ESTEEM3" (Grant number 823717). V.N.P. acknowledges financial support for NTB calculations from the National Science Fund of Bulgaria under grant KP-06-N38/10 from 06.12.2019. T.T.C acknowledges financial support from Vietnam Academy of Science and Technology (grant number: VAST03.06/22-23). H.N.T, E.F., A.A.Z, J.L.S. and M.P. acknowledge financial support by the ANR GAMBIT project, grant ANR-13-BS10-0014 of the French Agence Nationale de la Recherche.

structural description of other 1D VDW heterostructures based on metallic SWCNTs [START_REF] Cambré | Nanotube-Based 1D Heterostructures Coupled by van der Waals Forces[END_REF] if they have similar dielectric properties as DWCNT systems.

Declaration of competing interest

The authors declare that they have no conflicts of interest.

SUPPORTING INFORMATION FOR:

Partial quenching of electronic Raman scattering in double-wall carbon nanotubes by interlayer coupling Dmitry I. Levshov 1,2,3,* , Marina V. Avramenko 1,3 , Maksiem Erkens 1 , Huy-Nam Tran 2 , Thi Thanh Cao 4 , Van Chuc Nguyen 4 , Emmanuel Flahaut 5 , Valentin N. Popov 6 , Ahmed-Azmi Zahab 2 , Jean-Louis Sauvajol 2 , Raul Arenal 7,8,9 , Wim Wenseleers 1  SWCNT and DWCNT data should be analyzed by keeping in mind that they were obtained under (possibly) different lab conditions (e.g., humidity, room temperature), which may affect the presented values. In the future, the best approach would be to probe the same chiralities in the SWCNT and DWCNT forms under the same lab conditions.  Some of the (n,m) chiralities in SWCNT and DWCNT from Table 1 (main text) were measured at slightly different resonance conditions (e.g., values of dE = EL -Eii) despite the same laser excitation wavelengths were used. It is explained by the fact that electronic transition energies in DWCNTs are shifted by interlayer van der Waals electronic coupling. Nevertheless, for large dE values, the resonance effect should be less important relative to intrinsic factors like dielectric screening and IET (see slowly changing ERS intensity values away from resonance in Figure 3b,d) and should not change the conclusions of the work.  Finally, we discuss the possible origin of high IERS/ILO values in SWCNTs [18][19][20] for EL -Eii > 300 meV (Figure 2c, top panel) compared to DWCNT data. Since the ERS lines in this spectral region overlap with the intense higher-order modes (e.g., 2D or G+2D), they could be misinterpreted with some of the phonon modes during the fit. In the future, the behavior of ERS lines in this spectral region should be studied further.

Dependence of IERS/ILO ratios on d and θ. We present in Figure S9 the dependence of the IERS/ILO ratios on the diameter and chiral angle for SWCNTs from the literature [18][19][20] and DWCNTs from this work.

Section S11. Effect of curvature-induced bandgap on ERS lines

We calculated the curvature-induced bandgap values for all geometrically possible metallic SWCNTs with diameters below 3 nm using the equation [21]:

The dependences of this bandgap value on the metallic SWCNT diameter and chiral angle are