Recently, radially collapsed single-walled carbon nanotubes (SWCNTs) have been recognized as an analog of closed-edged bilayer graphene nanoribbons (GNRs). To explore this analogy, we first make detailed analyses of the radial deformation and collapse of armchair SWCNTs using the density functional theory with van der Waals corrections. The traditional threshold diameters dividing SWCNTs into the three regimes are obtained as $D_{\mathrm{meta}}=2.2\mathrm{nm}$ and $D_{\mathrm{abs}}=5.1\mathrm{nm}$, where deformed configurations can be stabilized (metastable) for tubes with diameter $D>D_{\mathrm{meta}}$ and are energetically more favorable than the cylindrical tube for $D>D_{\mathrm{abs}}$ (absolute stability). We find that the present result for $D_{\mathrm{abs}}$ is marginally in excellent agreement with the most plausible experimental result. We also identify, for the first time, other threshold diameters given by $D_{\mathrm{flat}1}=3.3\mathrm{nm}$ and $D_{\mathrm{flat}2}=4.4\mathrm{nm}$ in between $D_{\mathrm{meta}}$ and $D_{\mathrm{abs}}$, where the cross-sectional shape of a collapsed SWCNT is peanutlike for $D<D_{\mathrm{flat}1}$, either peanutlike or dumbbell-like for $D_{\mathrm{flat}1}<D<D_{\mathrm{flat}2}$, and dumbbell-like for $D>D_{\mathrm{flat}2}$. This bistability for tubes with $D$ between $D_{\mathrm{flat}1}$ and $D_{\mathrm{flat}2}$ implies that the most stable configuration of these collapsed tubes cannot necessarily be achieved by molecular dynamics simulation. Electronic structures of collapsed armchair SWCNTs are also investigated to find that substantial band gaps develop in the flattened armchair tubes with Bernal stacked opposing faces. These band gap openings of flattened SWCNTs are explored by exploiting their analogies to bilayer graphene and bilayer GNRs. We find in particular that band gaps of flattened armchair SWCNTs with dumbbell-like cross sections, whose width is denoted $W$, show a scale behavior, $\sim 1/W_{\mathrm{flat}}$, with $W_{\mathrm{flat}}=W-3.38\left(\mathrm{nm}\right)$, where $W_{\mathrm{flat}}$ is found to be the width of the flat region, consistent with recent theoretical analyses for a model of collapsed SWCNTs [T. Nakanishi and T. Ando, Phys. Rev. B 91, 155420 (2015)]. This behavior of band gaps is similar to that for monolayer zigzag GNRs (ZGNRs), $1/W$, where $W$ is the ribbon width, and is indicative of quantum confinement. The largest band gap of $130$ meV, which develops in the flattened armchair (30,30) SWCNT, is comparable to that for bilayer ZGNRs ($160$ meV), indicating that flattened armchair tubes could be used in place of ZGNRs in electronic device applications.

• Received 26 June 2015
• Revised 2 December 2015

DOI:https://doi.org/10.1103/PhysRevB.92.245429