Van derWaals-coupled electronic states in incommensurate double-walled carbon nanotubes

Non-commensurate two-dimensional materials such as a twisted graphene bilayer or graphene on boron nitride, consisting of components that have no finite common unit cell, exhibit emerging moiré physics such as novel Van Hove singularities1–3, Fermi velocity renormalization4,5, mini Dirac points6 and Hofstadter butterflies7–11. Here we use double-walled carbonnanotubes as amodel system for probing moiréphysics in incommensurateone-dimensional systems, by combining structural and optical characterizations. We show that electron wavefunctions between incommensurate innerand outer-wall nanotubes can hybridize strongly, contrary to the conventional wisdom of negligible electron hybridization due to destructive interference12,13. The chirality-dependent inter-tube electronic coupling is described by one-dimensional zone folding of the electronic structure of twisted-andstretched graphene bilayers. Our results demonstrate that incommensurate van der Waals interactions can be important for engineering the electronic structure and optical properties of one-dimensional materials. Engineering and tailoring the properties of materials is of central importance in modern science and technology. In bulk materials this is realized mainly through modifying the strong covalent bond by changing the crystalline structure or doping chemical elements. However, in low-dimensional materials, electronic properties can also be engineered through modifying the weak non-covalent coupling, because the electron waves are not buried in the bulk and the inter-molecular electronic coupling becomes important. This is exemplified in artificially twisted bilayers of graphene/graphene and graphene/boron nitride, where new phenomena ranging from Van Hove singularities1–6 to Hofstadter butterflies7–11 emerge from van der Waals coupling between atomically thin two-dimensional (2D) layers. In this letter we investigate coupled electronic states in double-walled carbon nanotubes (DWNTs), a prototypical van der Waals-coupled one-dimensional (1D) material14–19, and reveal structure-dependent electron wave hybridization between incommensurate innerand outer-wall carbon nanotubes. DWNTs provide an ideal family of 1D structures to explore van der Waals electronic coupling because each DWNT is precisely defined at the atomic level and there are hundreds of different DWNT varieties. Specifically, a DWNT is uniquely characterized by the chiral indices (no,mo)/(ni,mi) of the constituent outer and inner single-walled carbon nanotubes (SWNTs), and a constituent SWNT can have a different electronic structure varying from metallic to semiconducting depending on its chiral index20. Electronic coupling between van der Waals-coupled DWNTs has long fascinated researchers21–26. Theoretical studies of commensurate DWNTs, where the inner and outer walls have commensurate carbon lattices and therefore are amenable to theoretical calculations, predict strong inter-tube coupling26. However, commensurate DWNTs have never been observed experimentally because it is almost impossible to have two commensurate SWNTs with the radius difference matching the tube–tube separation in a DWNT (refs 16–25). Electronic structure calculations of incommensurate DWNTs, on the other hand, are challenging because a finite unit cell does not exist. Several theoretical attempts studying incommensurate DWNTs suggest that inter-tube electronic coupling is negligible between the incommensurate innerand outer-wall carbon lattices because couplings at different carbon atom sites oscillate with random phases and cancel each other12,13. Here we show experimentally that, contrary to previous theoretical predictions, electronic coupling from van derWaals interactions can be surprisingly strong in incommensurate DWNTs. The coupled electronic states can lead to either a blueshift or redshift of the optical transition energy by up to 150meV, with the exact energy shift depending sensitively on the DWNT chirality. In addition, we develop a theory based on 1D zone folding of twisted-and-stretched graphene bilayers that successfully describes the coupled electronic states in incommensurate DWNT systems. Inter-tube electronic coupling can be probed through the shift in optical transition energies of a DWNT compared with those from isolated constituent SWNTs. It requires accurate determination of the chiral structure and optical transition of individual DWNTs, which is achieved in our study by combining electron diffraction measurements18,19 and single-tube absorption spectroscopy27,28 on the same suspended DWNTs (Fig. 1a and Methods). Figure 1b,c shows, respectively, the electron diffraction pattern and absorption spectrum of a representative DWNT. The electron diffraction pattern (Fig. 1b) unambiguously determines the DWNT chiral indices (nomo)/(nimi) to be (22,9)/(11,11), which corresponds to a semiconducting outer-wall nanotube with a diameter of 2.16 nm and an armchair metallic inner-wall nanotube with a diameter of 1.49 nm. Its absorption spectrum (Fig. 1c) shows four prominent optical resonances at 1.58, 1.66, 2.08 and 2.41 eV. In comparison, the isolated (22,9) outer SWNT has three optical transitions S33, S44 and S55 at 1.66, 2.17 and 2.58 eV, respectively, and the isolated (11,11) inner SWNT has one M i 11 transition at 1.77 eV in the experimental spectral range29. (Transition energies of the isolated SWNTs are indicated by vertical lines in Fig. 1c). There is a oneto-one correspondence between the DWNT optical resonances and those from the constituent SWNTs, but the resonance energies are shifted by −80, −90, −170 and −110meV for the S33, S44, S55 and M i 11 transitions, respectively. (The larger oscillator strength of the metallic M i 11 transition is because this transition