Interference effects in one-dimensional moir\'e crystals

Interference effects in finite sections of one-dimensional moir\'e crystals are investigated using a Landauer-B\"uttiker formalism within the tight-binding approximation. We explain interlayer transport in double-wall carbon nanotubes and design a predictive model. Wave function interference is visible at the mesoscale: in the strong coupling regime, as a periodic modulation of quantum conductance and 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.

Quantum materials 1,2 are a class of materials that exhibit quantum effects at the macroscopic scale. They offer the opportunity to realize paradigm-shifting quantum electronics.
The controlled generation and manipulation of quantum states by electrical, magnetic, or optical means is a key challenge in bringing quantum materials to applications. Twisted bilayer graphene (tBLG) is a prime example of a tunable quantum material. The twist angle, and the resulting bi-dimensional moiré pattern, controls the emerging material properties: for specific magic angles, strong interlayer coupling and flat bands arise, resulting in superconductivity and strongly correlated phases. 3,4 One-dimensional moiré systems are realized in double-wall carbon nanotubes (DWNT) and are determined by two parameters: the angle between the inner and outer tubes' chiral vectors (like tBLG) and the difference between their radii. These degrees of freedom control the effective interlayer interaction and the resulting electronic states. While the physics emerging in 2D moiré crystals has been studied extensively, the understanding of its one-dimensional counterpart is limited to the ideal infinite nanotube case [5][6][7] and the commensurate, telescopic nanotubes. [8][9][10][11][12][13] Koshino et al. 5 devised a continuum model for the idealized infinite DWNT. They identify three regimes with unique electronic properties (localized insulating, strong and weak coupling) determined by the relative alignment of the chiral vectors of the tubes. Dispersionless flat bands appear in the localized insulating regime. The electronic structure of DWNT with weak coupling is given by the superposition of the constituent nanotubes, whereas it is heavily perturbed in the strong coupling regime: semiconducting tube combinations can produce a finite density of states in their gap, and metallic tubes can become semiconducting. Experimental evidence of DWNTs with strong interlayer coupling has been recently reported, 7 but the relation between the measurements and the idealized infinite DWNT model by Koshino is not trivial. Experimental conditions impose a finite tube length and electrical contacts on the outer tube only. As the overlap region between inner and outer tubes is finite, strong confinement effects can arise, as in SWNTs with finite length. 14,15 It is not clear a priori whether the regimes will be visible, in what limits of nanotube overlap they could be recov-ered, and whether the regime separation is even valid for finite tubes. 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 transmission between two concentric nanotubes that overlap in a finite region and extend infinitely in opposite directions (telescopic double wall nanotubes, tDWNT, Fig 1a). We predict the effects of the length of the overlap region in the different regimes, going beyond previous works that only consider commensurate tDWNT. The inter-layer transmission in tDWNTs composed of two armchair nanotubes (armchair@armchair) oscillates as a function of the overlap length, giving rise to regions with zero and maximal transmission. The overlap length modulates the tDWNT electronic properties. We demonstrate that this tunability extends to strongly coupled, chiral and incommensurate tDWNts. We explain that dips in the transmission spectra of armchair@armchair tDWNTs emerge due to back-scattering by localized states in the overlap region. These transmission dips are absent in strongly coupled chiral tDWNT due the lack of rotational symmetries. We devise a predictive model for the transmission, based on wave interference in one dimension, that reproduces the tunability and position of transmission dips. In the localized insulating regime we show that the interlayer conductance is very small irrespective of the overlap length, in agreement with Ref. 17. We recover the Koshino limit for sufficiently long (but finite) overlap lengths, and provide benchmarks for experimental realizations of 1D moirés and correlated states in DWNT. In the weak coupling regime, we show that interlayer transmission for chiral or incommensurate tDWNTs is suppressed. tDWNTs composed of metallic zigzag nanotubes (zigzag@zigzag) also belong to the weak coupling regime, but are an exception to this rule: the inter-layer transmission can be significant if states on the inner and outer tube with different angular momentum couple. This coupling is subject to selection rules based on the chiral indices of the tubes involved. 8

Results and Discussion
Conductance simulations are performed for two concentric nanotubes that overlap over a finite length L, connected to semi-infinite SWNT leads (Fig. 1a). Electron transmission is only allowed at energies where conduction channels are available in both electrodes (only metallic nanotube contacts allow transmission close to the Fermi level). When electrodes consist of semi-conducting tubes, transmission can be achieved for chemical potentials within the valence or conduction bands of both nanotubes. In either case, the total transmission T through this asymmetrically contacted system can not exceed the smallest of the two electrode transmissions. The magnitude of T depends on how electrode states couple through the overlap region, probing sensitively the interlayer interaction.

Strong coupling regime
DWNTs are in the strong coupling regime when the constituting tubes' chiral vectors are nearly parallel and their difference points along the armchair direction. DWNTs consisting of armchair tubes (n,n)@(m,m) fulfill both conditions. Armchair single-wall nanotubes (SWNT) present two nearly linear bands crossing at the Fermi energy, resulting in a metallic system with 2 G 0 conductance. 18,19 Without interlayer interaction, the band structure and conductance of an ideal DWNT would simply be the sum of the individual SWNT ones.
Deviations from the ideal case can only occur for perturbations of the sidewalls, for instance due to defects 20,21 or functionalization. 22   highest, and they decay towards the electrode regions (SI Fig. ??).
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 k F and velocity v F . We further assume that the energy separation of the bands with negative v F in the DWNT is symmetric, giving two new vectors at each energy k ± (E) = k(E) ± 1 2 δk F . An incoming electron with energy E = −v F (k − k F ) 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 propagates with two wave vector components: the average wave vector k, and the wave vector difference δk F /2 modulating the incoming wave. As this superposition propagates through the scattering device its weight oscillates between the two tubes. If the wavelength of the modulation is commensurate with the overlap region (δk F L = 2nπ) the incoming electron will be reflected at the termination and scatters back into the electrode. However, if the overlap length fits (2n + 1)/4 wavelengths of the modulation (L = (2n + 1)π/2δk F ) 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: Furthermore, the shape of localized eigenstates matches the expected character (Fig. 3).
The sine functions make the dips smoother compared to the LB-calculation, but the general trends are accurately reproduced. In fact this simple expression is an approximations of the more accurate one derived in Ref. 8 when the linear-band approximation is applied. Numerical interpolation of the band structure allows for very accurate reproduction of the LB+TB transmission (SI Fig. ??). However, the simple approach is qualitatively correct and can intuitively be connected to the image of 1D-waves.
We now consider an example of an incommensurate and chiral, strongly coupled, metallic DWNT: (18,15)@ (23,20). The band structure can be computed by artificially imposing periodicity using a commensurate supercell consisting of 4 and 3 repetitions of (18,15) and (23,20), respectively, and straining both tubes by ±1% (SI Fig ??). For the telescopic setup, the maximum conductance (2 G 0 ) can be attained due to the strong hybridization of both linear bands between the tubes, resulting in two simultaneously available conduction channels. The maximum of the conductance near E F (Fig 4a) oscillates with the overlap length between 2 G 0 and 0, with a period of 90 ± 1 A (Fig 4b).   nanotubes. The suppression of transmission at discrete energies observed in (10,10)@ (15,15) results from achirality, and does not occur in chiral nanotubes. The wave functions of chiral tubes possess no rotational symmetry around the tube axis. Therefore they can only have nodes at few, specific points along the tube circumference, and cannot form localized states in the overlap region (SI Fig ??). Avoided crossings in the band structure 5 result in sharp dips at 0 eV and 0.31 eV when the conductance is substantial (L ≈ 44 A, L ≈ 20 A), and peaks at ±0.31 eV when conductance is suppressed (L ≈ 88 A) (Fig. 4).

Localized insulating regime
The and the onset of the parabolic bands is clearly visible at ±0.35 eV (Fig. 6). A series of peaks are observed in the DOS of both cases ( Fig. 5 and 6). This is consistent with the emergence of flat bands (SI Fig. ??) predicted by Koshino et al., who attribute them to the localization of electrons in an effective potential with long spatial period (≈ 1200 A for these two cases).
However, our simulations indicate that localized states already emerge in much shorter tube segments (L ≈ 244 A), and could be experimentally observable even without requiring long, pristine DWNT samples. While some of the localized states emerge at these short overlap lengths, the peak density in the DOS increases with overlap length and is significantly lower than predicted by Koshino. This shows that the minima of effective potential that cause  Fig. ??), which are more pronounced with increasing overlap length. The transport gap of the outer tube is highlighted in striped grey.
which matches the observed periods in the DOS (δE DOS ) quite well (Table 1). Given the small size of these oscillations, they will likely be challenging to observe experimentally.

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 transmission near the Fermi-level. 8  These tubes preserve the oscillating behavior found in all strongly coupled tDWNTs, while the absence of rotational symmetries prevents back-scattering at localized states.

Tight-binding model
We model inter-and intra-layer interactions using a non-orthogonal tight-binding (TB) model with one orbital per carbon site, based on Reich et al., 24 Laissardiere et al., 25 and Bonnet et al. 6 We extend the model to include curvature effects, using ppσ interactions in addition to the ppπ interactions, and interlayer hopping to third nearest neighbors. The interlayer interaction ranges up to a cut-off distance of 5 A and is described by exponential decays: H ppσ/π (d) = H inter ppσ/π e (d−a 0 ) g ppσ/π (3) S ppσ/π (d) = S inter ppσ/π e (d−a 0 ) where a 0 = 3.35 A is the interlayer spacing of bulk graphite. The TB parameters were fit to electronic band structures of single and double-wall carbon nanotubes obtained from first-principles density functional theory. We first fix the ppπ parameters of the intra-layer interaction by fitting to the band structure of graphene, to reproduce correct behavior in the limit of very large nanotubes. Next, we fix the ppσ parameters of the intra-layer interaction to reproduce the ab initio bands of nanotubes with radii between 5.2 A to 6.2 A: (9,9), (16,0) and (9,6) (SI- Fig. ??). Lastly, we optimize the the interlayer parameters to reproduce the ab initio bands of two DWNTs: (16,0)@(24,0) and (9,6)@ (15,10). The optimized parameters are summarized in Table 2. The transmission and density of states (DOS) are extracted using the Landauer-Büttiker formalism (LB) formalism 26 as implemented in TBTrans. 27 All calculations are performed at zero voltage between the two electrodes.

First principles calculations
First principles calculations were done with the siesta 28,29 implementation of the density functional theory (DFT) method. We employed the local density approximation (LDA) exchange-correlation functional as parametrized J. P. Perdew and Y. Wang 30 in conjunction with optimized norm-conserving Vanderbilt (ONCV) scalar-relativistic pseudopotentials 31,32 with 2s and 2p valence electrons. The Brillouin zone was sampled using a Γ-centered, onedimensional grid with 78 k-points for the pristine armchair nanotubes.
basis of implementation for our tight-binding model.