Carbon 186 (2022) 416e422 Contents lists available at ScienceDirect Carbon journal homepage: www.elsevier.com/locate/carbon  crystals Interference effects in one-dimensional moire  n a, Zeila Zanolli a, c, * Nils Wittemeier a, Matthieu J. Verstraete b, Pablo Ordejo a Catalan Institute of Nanoscience and Nanotechnology - ICN2 (CSIC, BIST), Campus UAB, 08193, Bellaterra, Spain b NanoMat/Q-Mat/CESAM and ETSF, Universit e de Li ege (B5) and ETSF, B-4000, Li ege, Belgium c Chemistry Department and, Debye Institute for Nanomaterials Science, Condensed Matter and Interfaces, Utrecht University and ETSF, PO Box 80.000, 3508 TA, Utrecht, the Netherlands a r t i c l e i n f o a b s t r a c t Article history: Interference effects in finite sections of one-dimensional moire  crystals are investigated using a Lan- Received 12 August 2021 dauer-Büttiker formalism within the tight-binding approximation. We explain interlayer transport in Received in revised form double-wall carbon nanotubes and design a predictive model. Wave function interference is visible at the 2 October 2021 mesoscale: in the strong coupling regime, as a periodic modulation of quantum conductance and Accepted 12 October 2021 Available online 15 October 2021 emergent localized states; in the localized-insulating regime, as a suppression of interlayer transport, and oscillations of the density of states. These results could be exploited to design quantum electronic devices. Keywords: Quantum transport © 2021 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license First-principles (http://creativecommons.org/licenses/by/4.0/). Tight-binding Double wall carbon nanotubes Moire Quantum interference Quantum materials [1,2] are a class of materials that exhibit model for the idealized infinite DWNT. They identify three regimes quantum effects at the macroscopic scale. They offer the opportu- with unique electronic properties (localized insulating, strong, and nity to realize paradigm-shifting quantum electronics. The weak coupling) determined by the relative alignment of the chiral controlled generation and manipulation of quantum states by vectors of the tubes. Dispersionless flat bands appear in the local- electrical, magnetic, or optical means is a key challenge in bringing ized insulating regime. The electronic structure of DWNT with quantum materials to applications. Twisted bilayer graphene (tBLG) weak coupling is given by the superposition of the constituent is a prime example of a tunable quantum material. The twist angle, nanotubes, whereas it is heavily perturbed in the strong coupling and the resulting bi-dimensional moire  pattern, control the regime: semiconducting tube combinations can produce a finite emerging material properties: for specific magic angles, strong density of states in their gap, and metallic tubes can become interlayer coupling and flat bands arise, resulting in superconduc- semiconducting. Experimental evidence of DWNTs with strong tivity and strongly correlated phases [3,4]. One-dimensional moire  interlayer coupling has been recently reported [7], but the rela- systems are realized in double-wall carbon nanotubes (DWNT) and tionship between the measurements and the idealized infinite are determined by two parameters: the angle between the inner DWNT model by Koshino is not trivial. Experimental conditions and outer tubes’ chiral vectors (like tBLG) and the difference be- impose a finite tube length and electrical contacts on the outer tube tween their radii. These degrees of freedom control the effective only. As the overlap region between inner and outer tubes is finite, interlayer interaction and the resulting electronic states. While the strong confinement effects can arise, as in SWNTs with finite length physics emerging in 2D moire  crystals has been studied extensively, [14,15]. It is not clear a priori whether the regimes will be visible, in the understanding of its one-dimensional counterpart is limited to what limits of nanotube overlap they could be recovered, and the ideal infinite nanotube case [5e7] and the commensurate, whether the regime separation is even valid for finite tubes. telescopic nanotubes [8e13]. Koshino et al. [5] devised a continuum Answering these basic questions is essential for the use of coaxial nanotubes in applications involving electronic transport [16]. In this work, we address these questions by studying interlayer * Corresponding author. Catalan Institute of Nanoscience and Nanotechnology - transmission between two concentric nanotubes that overlap in a ICN2 (CSIC, BIST), Campus UAB, 08193, Bellaterra, Spain. finite region and extend infinitely in opposite directions (telescopic E-mail address: z.zanolli@uu.nl (Z. Zanolli). https://doi.org/10.1016/j.carbon.2021.10.028 0008-6223/© 2021 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). n et al. N. Wittemeier, M.J. Verstraete, P. Ordejo Carbon 186 (2022) 416e422 double-wall nanotubes, tDWNT, Fig. 1a). We predict the effects of nanotubes that overlap over a finite length L, connected to semi- the length of the overlap region in the different regimes, going infinite SWNT leads (Fig. 1a). Nanotube edges do not the influ- beyond previous works that only consider commensurate tDWNT. ence transport properties of nanotubes longer than 5e10 nm and The interlayer transmission in tDWNTs composed of two armchair can be safely neglected for semi-infinite leads [14,15]. The main nanotubes (armchair@armchair) oscillates as a function of the mechanism affecting nanotube transport is the coupling regime. overlap length, giving rise to regions with zero and maximal Relative rotation of the nanotubes can change the moire  pattern, transmission. The overlap length modulates the tDWNT electronic but the coupling regime remains invariant because the rotation properties. We demonstrate that this tunability extends to strongly does not affect the 1D moire  lattice vector. This implies that the coupled, chiral, and incommensurate tDWNts. We explain that dips rotational alignment will have a minor effect on the results (see in the transmission spectra of armchair@armchair tDWNTs emerge Supplementary Information (SI) for details). Electron transmission due to back-scattering by localized states in the overlap region. is only allowed at energies where conduction channels are available These transmission dips are absent in strongly coupled chiral in both electrodes (only metallic nanotube contacts allow trans- tDWNT due to the lack of rotational symmetries. We devise a mission close to the Fermi level). When electrodes consist of predictive model for the transmission, based on wave interference semiconducting tubes, transmission can be achieved for chemical in one dimension, that reproduces the tunability and position of potentials within the valence or conduction bands of both nano- transmission dips. In the localized insulating regime we show that tubes. In either case, the total transmission T through this asym- the interlayer conductance is very small irrespective of the overlap metrically contacted system can not exceed the smallest of the two length, in agreement with Ref. 17. We recover the Koshino limit for electrode transmissions. The magnitude of T depends on how sufficiently long (but finite) overlap lengths and provide bench- electrode states couple through the overlap region, probing sensi- marks for experimental realizations of 1D moire s and correlated tively the interlayer interaction. states in DWNT. In the weak coupling regime, we show that interlayer transmission for chiral or incommensurate tDWNTs is 1.1. Strong coupling regime suppressed. tDWNTs composed of metallic zigzag nanotubes (zig- zag@zigzag) also belong to the weak coupling regime but are an DWNTs are in the strong coupling regime when the constituting exception to this rule: the interlayer transmission can be significant tubes’ chiral vectors are nearly parallel and their difference points if states on the inner and outer tube with different angular mo- along the armchair direction. DWNTs consisting of armchair tubes mentum couple. This coupling is subject to selection rules based on (n,n)@(m,m) fulfill both conditions. Armchair single-wall nano- the chiral indices of the tubes involved [8]. tubes (SWNT) present two nearly linear bands crossing at the Fermi energy, resulting in a metallic system with 2 G0 conductance [18,19]. Without interlayer interaction, the band structure and 1. Results and discussion conductance of an ideal DWNT would simply be the sum of the individual SWNT ones, namely 4G0. We observe, instead, a reduc- Conductance simulations are performed for two concentric tion of the conductance to 1G0 and dips at specific energy values (Fig. 1c) in the set-up of Fig. 1a (L ¼ 34.08 Å). Dips in the conduc- tance can occur for perturbations of the sidewalls, for instance, due to defects [20,21] or functionalization [22,23]. As the nanotube walls are perfect, the origin of the dips must be related to intertube interaction: The infinite (10,10)@(15,15) DWNT features a subtle band structure (Fig. 1b) of states localized on either tube (positive Fermi velocity, red/blue) and hybridized states (negative Fermi velocity, white). Each of these dips corresponds to a localized eigenstate in the overlap region (Fig. 1c horizontal lines). The number, position, and width of the dips depend on L (Fig. 2a). At L z 50 Å and L z 150 Å, the dips in transmission are barely visible. At intermediate overlap lengths, the interlayer conductance is reduced, the dips are broader, and transmission is blocked completely for L z 100 Å and L z 200 Å, following a pe- riodic behavior. This reduction of the maximum conduction as well as the oscillation with overlap length was predicted in Ref. [10], and we can now show that localized states cause the dips observed in the conductance at fixed overlap lengths. We further show that the modulation of the transmission can be observed in a wide range of energy and overlap lengths. The reduction of the maximum conductance near the Fermi energy with respect to the pristine DWNT can be explained by Fig. 1. a) Transport set-up: The scattering region consists of overlapping nanotubes, taking into account the nature of the transport setup and the screening region, and part of the electrodes (light red/blue), which extend in opposite character of the bands in the DWNT. The asymmetric setup limits semi-infinite directions, and function as leads (red/blue). b) Projected TB band struc- the conductance from 4G0 to the minimum of the electrode con- ture of an ideal (10,10)@(15,15) DWNT. The bands with negative Fermi-velocity (white) are hybridized between the nanotubes, bands with positive Fermi-velocity (red/blue) ductances (2G0). The reduction of the maximum conductance to are localized on one tube each. c) TB electron transmission from a semi-infinite (10,10) 1G0 is traced back to the three-fold rotational symmetry of the into a semi-infinite (15,15) SWNT through an overlap region with a length of 34.08 Å. outer tube, which results in the cancellation of one of the two Horizontal lines visualize energy levels for an isolated, finite section of the system with available transport channels [10]. We note here that this reduction an additional 44 Å of leads on either side. The yellow-to-blue scale indicates the weight of wave functions in the overlap region. Each dip in conductance coincides (approxi- can already be inferred from the eigenstates of the pristine DWNT. mately) with one or several localized eigenstates. (A colour version of this figure can be The electronic eigenstates with positive Fermi-velocity are unaf- viewed online.) fected by the interlayer interaction and have to scatter back into the 417 n et al. N. Wittemeier, M.J. Verstraete, P. Ordejo Carbon 186 (2022) 416e422 k, and the wave vector difference dkF/2 modulating the incoming wave. As this superposition propagates through the scattering de- vice its weight oscillates between the two tubes. If the wavelength of the modulation is commensurate with the overlap region (dkFL ¼ 2np) the incoming electron will be reflected at the termi- nation and scatters back into the electrode. However, if the overlap length fits (2n þ 1)/4 wavelengths of the modulation (L ¼ (2n þ 1)p/ 2dkF) the electron passes through the overlap region without back- scattering (Fig. 3). At intermediate lengths, the incoming wave is partially transmitted reducing the conductance without fully blocking it. Similarly, at certain energies, which depend on the overlap length, the primary component k becomes commensurate with the overlap. This allows a standing wave to form in the quantum box of the overlap region, which blocks the transmission and explains the emergence of dips at specific energies. We combine the two trends to model the transmission using sine functions:      dk E TðE; LÞ ¼ sin L F 2 ,sin L kF  2 2 vF   2pn (1) T E; ¼0 dkF  pn   T  vF þ kF ; L ¼ 0 Fig. 2. TB electron transmission from a (10,10) SWNT into a (15,15) SWNT through a L finite overlap region at different overlap lengths calculated with a) LB þ TB formalism, and b) our model (Eq.(1)). Overlaps are sampled with a high density below 50 Å (1/40 , Fig. 2 shows the impressive agreement between the trans- 2.459 Å) and lower density above (1/2.459 Å). The model reproduces both trends in LB- mission calculated with this simple model and the tight-binding transmission: the energy-independent modulation of the transmission with overlap results. The model reproduces the energy-independent global length on long spatial period, and the secondary modulation of the transmission oscillation of the transmission with the overlap length and allows dependent on energy and overlap. (A colour version of this figure can be viewed us to predict the periodicity of the transmission oscillation from the online.) shift of the Fermi vector (dkF z 0.071 Å1) for any nanotube com- bination. For this tube shift (dkF z 0.065 Å1) the corresponding electrode, effectively limiting the possible available channels to the periodicity is 89 Å. The periodicity observed in our simulations is electrode bands with negative Fermi velocity. These electrons only slightly larger (100 Å). The curvature of regions without couple to the two hybridized DWNT states, and have the possibility transmission (dark blue) can be recovered when the non-linearity to be transmitted through the device. Depending on the overlap of the bands is included in the model (Supplementary length this transmission occurs with almost no back-scattering (L z 50 Å, 150 Å) or is completely blocked (L z 100 Å). In order to understand the origin of the transmission dips, we calculate the eigenstates of finite segments of the open system including addi- tional nanotube rings on each side of the scattering region. In the limit of infinitely long electrode regions, we would recover the states of the open system. We find that each dip corresponds to a localized state (Fig. 1c) and that the agreement between eigene- nergies and dip energies varies slightly, depending on the exact number of additional nanotube rings. The eigenstates localized in the overlap region (L ¼ 34.08 Å) are all similar in shape: the wave function weight on the atoms at the tube terminations are the highest, and they decay towards the electrode regions (SI Fig. 6). We remark that nanotube edge states do not affect the transport properties in the relevant energy window [0.8 eV, 0.8 eV], as demonstrated by the computed wavefunctions in this energy range (SI Fig. 6). To explain the origin of the localized states and the modulation of the transmission, we construct a simple model, assuming linear dispersion of the SWNT and DWNT bands with one common Fermi wave vector kF and velocity vF. We further assume that the energy separation of the bands with negative vF in the DWNT is symmetric, giving two new vectors at each energy k± ðEÞ ¼ kðEÞ± 12 dkF . An incoming electron with energy E ¼  vF(k  kF) needs to couple to the hybridized states with the same energy to pass from one layer into the other. The resulting superposition of overlap states prop- agates with two wave vector components: the average wave vector Fig. 3. Schematic of standing waves in the overlap region between two semi-infinite nanotubes. (A colour version of this figure can be viewed online.) 418 n et al. N. Wittemeier, M.J. Verstraete, P. Ordejo Carbon 186 (2022) 416e422 Information). The model also reproduces the E and L dependent at few, specific points along the tube circumference, and cannot position of the smaller dips observed in the transmission. form localized states in the overlap region (SI Fig. 8). Avoided Furthermore, the shape of localized eigenstates matches the ex- crossings in the band structure [5] results in sharp dips at 0 eV and pected character (Fig. 3). The sine functions make the dips 0.31 eV when the conductance is substantial (L z 44 Å, L z 20 Å), smoother compared to the LB-calculation, but the general trends and peaks at ±0.31 eV when conductance is suppressed (L z 88 Å) are accurately reproduced. This simple expression is an approxi- (Fig. 4). mation of the more accurate one derived in Ref. 8 when the linear- band approximation is applied. Numerical interpolation of the band 1.2. Localized insulating regime structure allows for very accurate reproduction of the LB þ TB transmission (SI Fig. 5). However, the simple approach is qualita- The localized insulating regime is achieved when the chiral tively correct and can intuitively be connected to the image of 1D vectors of the inner and outer tube are nearly parallel and their waves. difference points along the zig-zag direction. We consider two We now consider an example of an incommensurate and chiral, representatives of this family, (27,3)@(36,3) and (27,3)@(35,3), strongly coupled, metallic DWNT: (18,15)@(23,20). The band which are metallic-metallic (M@M) and metallic-semiconducting structure can be computed by artificially imposing periodicity using (M@SC), respectively. Interference effects are observable as peaks a commensurate supercell consisting of 4 and 3 repetitions of in the (27,3)@(35,3) DOS in Fig. 5. The small (3.12 Å) interlayer (18,15) and (23,20), respectively, and straining both tubes by ± 1% spacing enhances the coupling and suppresses the increase in DOS (SI Fig. 7). For the telescopic setup, the maximum conductance near the onset of the outer tube parabolic bands (gray area in Fig. 5 (2G0) can be attained due to the strong hybridization of both linear and SI Fig. 9). This effect can be observed already in short DWNT bands between the tubes, resulting in two simultaneously available segments (L < 81 Å) and is not exclusive to the infinite tubes of conduction channels. The maximum of the conductance near EF Ref. 5. In (27,3)@(36,3), instead, the interference is weaker due to (Fig. 4a) oscillates with the overlap length between 2G0 and 0, with the larger interlayer spacing (3.51 Å) and the onset of the parabolic a period of 90(1) Å (Fig. 4b). Our simple model of wave function bands is clearly visible at ±0.35 eV (Fig. 6). A series of peaks are interference in a quantum box predicts a spatial period of 96 Å, observed in the DOS of both cases (Figs. 5 and 6). This is consistent using a shift of the Fermi wave vectors dkF z 0.065 Å1 extracted with the emergence of flat bands (SI Fig. 9) predicted by Koshino from the band structure in Ref. [5]. Despite its simplicity, our model et al., who attribute them to the localization of electrons in an is semi-quantitative and catches the main physical mechanisms at effective potential with a long spatial period (z1200 Å for these play: the oscillations of the transmission are caused by the differ- two cases). However, our simulations indicate that localized states ence in the Fermi wave vectors of the two tubes and will appear in already emerge in much shorter tube segments (L z 244 Å), and all combinations of strong coupling, metallic@metallic nanotubes. could be experimentally observable even without requiring long, The suppression of transmission at discrete energies observed in pristine DWNT samples. While some of the localized states emerge (10,10)@(15,15) results from achirality and does not occur in chiral at these short overlap lengths, the peak density in the DOS nanotubes. The wave functions of chiral tubes possess no rotational symmetry around the tube axis. Therefore they can only have nodes Fig. 5. Localized insulating DWNT (M@SC): density of states in the scattering region (a) Fig. 4. Strong coupling (M@M): TB electron transmission from (18,15) into (23,20) a) and electron transmission (b) from (27,3) into (35,3) for different overlap lengths. for three selected overlap length L showcasing perfect (L ¼ 44 Å), partially blocked Interference between the tubes results in sharp spikes and flat bands (SI Fig. 9), which (L ¼ 20 Å), and blocked transmission (L ¼ 88 Å). b) For a wide range of overlap lengths. are more pronounced with increasing overlap length. The transport gap of the outer The transmission oscillates as a function of L. At 0 eV and 0.31 eV sharp features occur tube is highlighted in striped gray. (A colour version of this figure can be viewed irrespective of L. (A colour version of this figure can be viewed online.) online.) 419 n et al. N. Wittemeier, M.J. Verstraete, P. Ordejo Carbon 186 (2022) 416e422 the overlap. This quantization effect is absent in the (27,3)@(35,3) DWNT, because its outer tube is semiconducting and, therefore, only one set of linear bands is present. In both localized insulating DWNTs considered here, the interlayer conductance is very small (0.01 G0) around the Fermi energy (0.35 eVe0.35 eV) and only becomes relevant in those regions where non-linear bands are present in the inner and outer nanotubes. This demonstrates that not only the metallic/semiconducting nature of the individual tubes is important for the conductance between layers, but that the coupling regime also plays a significant role. Consequently, metallic shells in multi-wall carbon nanotubes are likely to contribute less to the overall conductance than previously expected due to the low interlayer conductance, unless the strong coupling condition is fulfilled. This explains the low interlayer conductivity reported in Ref. [11]. Different examples of localized-insulating tDWNT (met- allic@metallic) with shorter interlayer spacing (dR) can be found in SI Fig. 10 (dR ¼ 3.4 Å) and SI Fig. 11 (dR ¼ 3.1 Å). 1.3. Weak coupling regime In the weak coupling regime, there is a distinction between DWNTs composed of two zig-zag nanotubes (zig-zag@zig-zag) and all others. In zig-zag@zig-zag tDWNTs the rotational symmetry plays an important role, specifically, the three-fold rotational symmetry of (9,0)@(18,0) leads to significant interlayer trans- mission near the Fermi-level [8]. Similar to (10,10)@(15,15), the Fig. 6. Localized quasi-insulating (M@M): density of states in the scattering region (a) commensurability allows the formation of localized states in the and electron transmission (b) from (27,3) into (36,3) for different overlap lengths. Interference between the tubes results in sharp spikes and flat bands at EF and at the overlap region and causes the transmission to be blocked at the edge of the quasi-insulating gap, which are more pronounced with increasing overlap corresponding energies. For other tDWNTs in this regime, length. (A colour version of this figure can be viewed online.) composed of either incommensurable or chiral nanotubes, the interlayer transmission is negligible as a result of weak coupling (SI Fig. 13-16). increases with overlap length and is significantly lower than pre- In the Supplementary Information, we present the results ob- dicted by Koshino. This shows that the minima of effective potential tained for ten different combinations of SWNT, covering all possible that cause localization, emerge successively with increasing over- combinations of different chirality, commensurabilities, and lap. The full set localized states will be observable only if the coupling regimes. The strong and localized insulating coupling overlap length is larger than the period of the effective potential. conditions are very restrictive and do not permit the mixing of We observe additional oscillations of lower magnitude in the chiral and achiral nanotubes. DOS of (27,3)@(36,3) DWNT, which can not be attributed to flat bands. These oscillations occur in the energy ranges between  1.4. Conclusion 0.35 eV to  0.05 eV and 0.1 eVe0.35 eV and are a result of the finite overlap length which causes a quantization of the wave vectors of In conclusion, we show that the two main coupling regimes states in the finite overlap region. Analogous to the discussion in predicted for double-wall carbon nanotubes (strong coupling and the strong coupling regime, we assume linear dispersion of the low localized insulating) are present already in finite DWNT segments. energy bands and determine the energy spacing of states These regimes should be identifiable in experimental set-ups even commensurate with the overlap length: when DWNT segments are short. For strongly coupled arm- chair@armchair metallic nanotubes, the interlayer transmission is ̊ p significant and modulated by two phenomena. It is blocked by p dE ¼ vF z5:6 eV A ; (2) localized states at discrete energies for any given overlap region L L length, and it oscillates with the overlap length due to wave which matches the observed periods in the DOS (dEDOS) quite well interference in the overlap region. The rotational symmetry of the (Table 1). tDWNT can reduce the maximal transmission, blocking trans- Given the small size of these oscillations, they will likely be mission through one of the electrode channels completely, as seen challenging to observe experimentally. These oscillations are not for (10,10)@(15,15), or partially [10]. In other strongly coupled predicted by Koshino, since they are caused by the finite length of tDWNTs, where at least one nanotube is chiral, wave interference also leads to an oscillation of the maximal transmission. However, the absence of rotational symmetry prevents the formation of Table 1 localized states and the corresponding back-scattering. Based on Oscillation periods in DOS of a (27,3)@(36,3) tDWNT in the energy the picture of 1D waves in a quantum box, we derive an expression ranges  0.35 eV to  0.05 eV and 0.1 eVe0.35 eV for three different overlap length extracted from TB þ LB calculations (dEDOS) and calculated using Eq. (2) (dE). for the interlayer transmission of strongly coupled nanotubes. The simplified model describes the modulation of transmission with L [Å] dEDOS[eV] dE[eV] excellent accuracy. In weakly coupled zigzag@zigzag metallic 1016 0.016 0.017 tDWNTs, the interlayer transmission is strongly affected by the 528 0.031 0.033 rotational symmetry. This leads to significant transmission be- 284 0.058 0.062 tween the layers of (9,0)@(18,0), while the interlayer transmission 420 n et al. N. Wittemeier, M.J. Verstraete, P. Ordejo Carbon 186 (2022) 416e422 Table 2 Optimized tight-binding parameters. onsite intra-layer inter-layer pps ppp pps ppp 1nn inter Ep[eV] 2.04 H [eV] 3.93 2.81 H [eV] 0.505 0.709 S1nn 0.573 0.301 Sinter 0.003 0.062 H2nn[eV] 1.17 0.679 g[Å] 0.408 0.387 S2nn 0.018 0.047 h[Å] 1.12 0.620 H3nn[eV] 1.11 0.298 S3nn 0.074 0.040 in (18,0)@(27,0) is negligible. In other weak coupling tDWNTs, provides a good description of carbon nanostructures [28]. We used where at least one tube is chiral or the two tubes are incommen- an extended basis set that is large enough to describe interlayer surate, the interlayer transmission is negligible. Lastly, in the interaction in our ab initio simulations. Our tight-binding model is localized insulating regime the interlayer transmission is also based on these ab initio simulations and therefore retains this ac- heavily suppressed near the Fermi-level. In this regime, the emer- curacy to model double-wall carbon nanotubes, even without gence of flat bands causes strong oscillations in the DOS. Notably, explicitly including vdW corrections. the flat bands already appear in segments significantly shorter than the predicted spatial periodicity of the underlying effective po- 2.2. First-principles calculations tential (z1200 Å for tubes discussed above). The number of localized states is directly linked to the overlap length. Our results First-principles calculations were done with the SIESTA [29,30] highlight that applications like nano-electronic switches based on implementation of the density functional theory (DFT) method. We tDWNTs are very sensitive to their structure. Chiral tDWNTs are the employed the local density approximation (LDA) exchange- most promising candidates for such applications. These tubes correlation functional as parametrized J. P. Perdew and Y. Wang preserve the oscillating behavior found in all strongly coupled [31] in conjunction with optimized norm-conserving Vanderbilt tDWNTs, while the absence of rotational symmetries prevents (ONCV) scalar-relativistic pseudopotentials [32,33] with 2s and 2p back-scattering at localized states. valence electrons, and a custom double-zeta polarized basis set (details in the Supplementary Information). The Brillouin zone was sampled using a G-centered, one-dimensional grid with 78 k-points 2. Methods for the pristine armchair nanotubes. 2.1. Tight-binding model CRediT authorship contribution statement We model interlayer and intralayer interactions using a non- Nils Wittemeier: Methodology, Software, Validation, Formal orthogonal tight-binding (TB) model with one orbital per carbon analysis, Investigation, Writing e original draft, Visualization. site, based on Reich et al. [24], Laissardiere et al. [25], and Bonnet Matthieu J. Verstraete: Methodology, Writing e review & editing, et al. [6]. We extend the model to include curvature effects, using Resources, Supervision. Pablo Ordejo n: Software, Writing e review pps interactions in addition to the ppp interactions, and interlayer & editing, Funding acquisition, Supervision. Zeila Zanolli: hopping to third nearest neighbors. The interlayer interaction Conceptualization, Resources, Writing e original draft, Supervision, ranges up to a cut-off distance of 5 Å and is described by expo- Project administration. nential decays: ðda0 Þ Declaration of competing interest Hpps=p ðdÞ ¼ H inter pps=p e gpps=p (3) The authors declare that they have no known competing ðda0 Þ financial interests or personal relationships that could have Spps=p ðdÞ ¼ Sinter pps=p e hpps=p ; (4) appeared to influence the work reported in this paper. where a0 ¼ 3.35 Å is the interlayer spacing of bulk graphite. The TB Acknowledgments parameters were fit to electronic band structures of single and double-wall carbon nanotubes obtained from first-principles den- The authors acknowledge enriching discussions on the tight- sity functional theory. We first fix the ppp parameters of the intra- binding parametrization with L. Henrard and P. Lambin. This layer interaction by fitting to the band structure of graphene, to work was supported by Spanish MINECO (the Severo Ochoa Centers reproduce correct behavior in the limit of very large nanotubes. of Excellence Program under Grant No. SEV- 2017-0706), Spanish Next, we fix the pps parameters of the intra-layer interaction to MICIU, AEI and EU FEDER (Grants No. PGC2018-096955-B-C43), reproduce the ab initio bands of nanotubes with radii between 5.2 Generalitat de Catalunya (Grant No. 2017SGR1506 and the CERCA Åto 6.2 Å: (9,9), (16,0), and (9,6) (SI Fig. 1). Lastly, we optimize the Programme), and the European Union MaX Center of Excellence interlayer parameters to reproduce the ab initio bands of two (EU-H2020 Grant No. 824143). ZZ acknowledges financial support DWNTs: (16,0)@(24,0) and (9,6)@(15,10). The optimized parame- by the Ramon y Cajal program RYC-2016-19344 (MINECO/AEI/FSE, ters are summarized in Table 2. The transmission and density of UE) and the Netherlands Sector Plan program 2019e2023. MJV states (DOS) are extracted using the Landauer-Büttiker formalism acknowledges funding by the Belgian FNRS (PDR G.A. T.1077.15e1/7 (LB) formalism [26] as implemented in TBTrans [27]. All calcula- and a sabbatical “OUT” grant at ICN2), and computational resources tions are performed at zero voltage between the two electrodes. from the Consortium des Equipements de Calcul Intensif (CECI, Our ab initio and tight-binding simulations are sufficiently ac- FRS-FNRS G.A. 2.5020.11) and Zenobe/CENAERO funded by the curate to describe the interlayer interaction. The LDA functional Walloon Region under G.A. 1117545. NW has received funding from 421 n et al. N. Wittemeier, M.J. Verstraete, P. Ordejo Carbon 186 (2022) 416e422 the European Union's Horizon 2020 research and innovation pro- in a finite carbon nanotube: a one-dimensional quantum box, Phys. Rev. Lett. 82 (1999) 3520, https://doi.org/10.1103/PhysRevLett.82.3520. gramme under the Marie Skłodowska-Curie grant agreement No. [15] A. Rochefort, D.R. Salahub, P. Avouris, Effects of finite length on the electronic 754558. We acknowledge computer resources at MareNostrum4 at structure of carbon nanotubes, J. Phys. Chem. B 103 (4) (1999) 641e646, Barcelona Supercomputing Center (BSC), provided through the https://doi.org/10.1021/jp983725m. PRACE Project Access (OptoSpin project 2020225411) and RES [16] D. Nakar, G. Gordeev, L.D. Machado, R. Popovitz-Biro, K. Rechav, E.F. Oliveira, P. Kusch, A. Jorio, D.S. Galva ~o, S. Reich, E. Joselevich, Few-wall carbon nano- (activity FI-2020-1-0014), and technical support provided by the tube coils, Nano Lett. 20 (2020) 953, https://doi.org/10.1021/ Barcelona Supercomputing Center. The authors acknowledge the acs.nanolett.9b03977. use of the open-source project sisl [34] used to generate atomic [17] S. 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