A new phase from compression of carbon nanotubes with anisotropic Dirac fermions

Searching for novel functional carbon materials is an enduring topic of scientific investigations, due to its diversity of bonds, including sp-, sp2-, and sp3-hybridized bonds. Here we predict a new carbon allotrope, bct-C12 with the body-centered tetragonal I4/mcm symmetry, from the compression of carbon nanotubes. In particular, this structure behaviors as the Dirac fermions in the kz direction and the classic fermions in the kx and ky directions. This anisotropy originates from the interaction among zigzag chains, which is inherited from (n, n)-naotubes.

presented (Fig. 1a,c). Firstly the compression makes the original cylinder tubes (Fig. 1a) to become into the rounded squares (Fig. 1b). Then the newly formed sp 3 carbon atoms (cyan) force the neighboring atoms (red) away from each other and the four edges of the rounded square become concaved (Fig. 1c). Meanwhile, the sp 2 atoms favor to stay in a plane, which further exacerbates the concavity of the tubes (Fig. 1d). And then the tubes transform to pinwheel shapes (Fig. 1e). Finally, some bonds are broken and some new bonds are rebuilt, resulting in the formation of a new phase (Fig. 1f). Such a new structure can be considered as the second step product of the CNT compression with sp 2 and sp 3 hybridization reconfiguring the geometry, while the SWCNT polymers can be the first order product. Surprisingly, we also decompress to the atmospheric pressure, and find that it has low energy, good mechanic characters, and fantastic band structure, as the following discussion.
This new phase is identified to exhibit the body-centered tetragonal I4/mcm symmetry and is designated as the bct-C12, in which its conventional cell contains 24 atoms (Fig. 2,c) and its primitive cell has 12 atoms (Fig. 2b). At zero pressure, it has lattice parameters of a = b = 8.5 Å and c = 2.45 Å. Carbon atoms occupy the Wyckoff positions 8h (0.21, 0.29, 0) and 16k (0.73, 0.96, 0), respectively. In particular, the atoms located at the 8h Wyckoff positions have sp 3 hybridization (painted in grey in Fig. 2e) and those occupying the 16k have sp 2 hybridization (painted in yellow in Fig. 2e). Due to the same D 4h point group, the bct-C12 has much structural similarity to bct-C4 (refs. 5,6). It can be found that if we use zigzag carbon chains to replace the 4-ring of the bct-C4, the bct-C4 (Fig. 2d) becomes into the bct-C12 (Fig. 2d) in geometric configuration. However, the atoms in the additional zigzag chain are sp 2 hybridization, while the atoms in the original bct-C4 are sp 3 hybridization.
Stability. The bct-C12, as a metastable phase of carbon at atmospheric pressure, has a formation energy of 0.18 eV/atom relative to graphite, while the bct-C4 is 0.26 eV/atom, the M-carbon 0.17 eV/ atom, the (6,6)-CNT 0.13 eV/atom, and the fullerene C 60 0.39 eV/atom, respectively. One should be emphasized that the above results we calculated are similar to the previous calculations 3,5,6,23 . This means that the bct-C12 has the similar stability to the precursor (6, 6)-CNT and the superhard post-graphite phases, M-carbon, and the lower formation energy than bct-C4 and C 60 . In addition, the calculated phonon spectrum clearly indicates its dynamical stability (Fig. 3). Clearly, this phase is a novel material, even though the pressure is decompressed to the atmospheric pressure. Mechanical property. As shown in the calculation results, the bct-C12 has its bulk modulus of 315.9 GPa and shear modulus of 225.4 GPa. Based on the modified microscope model [24][25][26] , the calculated theoretical Vickers hardness is 31.6 GPa for the bct-C12. Although this hardness is much lower than 93.6 GPa of diamond, the bct-C12 is still a hard material with its Vickers hardness being slightly higher than 30.2 GPa of α -SiO 2 . In particular, the metallic property will dramatically reduce the hardness of material 26 , hence this hardness of the metallic bct-C12 should be quite high among the metallic materials. In addition, the bct-C12 has much interspace and its sp 3 parts are not in a perfect tetrahedron. As a result, the structure is easy to slip. This is the reason why the bulk modulus of this structure is about 1.5 times of the shear modulus, which implies that this structure has better malleability than the superhard materials, such as diamond.
Electronic property. The band structure (Fig. 4a) indicates the bct-C12 to be metallic. Since the σ bands of the sp 3 atoms have the lower energy, the bands cross the Fermi level are the π bands around the sp 2 zigzag chain (red lines in Fig. 4a). In particular, the band structure has a novel property that, along the k z axis, the linear valence and conduction bands meet at a single point, which is similar to the Dirac point in graphene. These points of intersection depend on both k x and k y , and show the energy fluctuation in Brillouin zone. The maximum and minimum of the points of intersection are 0.57 eV at F A (0, 0, 0.172) × 2πÅ −1 along the G-A line and − 0.71 eV at F B (0.117, 0, 0.174) × 2π Å −1 along the Z-B line, respectively. It should be noted that since F A and F B have nearly the same z coordinates, the surface composed of these Dirac points is close to a plane, which can be referred to as the Dirac surface.
We find a "weird" quasiparticle that exhibits a linear Dirac fermion behavior in the k z direction, while a classic two-order dispersion in the k x -k y plane, meaning that the quasiparticle can be changed from the Dirac fermions to the normal fermions by gradually changing the wavevector. As shown in Fig. 4c, in the k x -k z plane, the band exhibits an anisotropic wedge instead of the cone in graphene.
In particular, at the F A point, which is the top of this band, the slopes in the k z direction is ±39.1 eV.Å, equivalent to a velocity Meanwhile, the band bottom, F B point, is also close to the boundary of Brillouin zone, which distorts its Dirac behavior in the z direction. The slopes are −51.4 and 37.1 eV•Å, correspondingly the velocity v z = 0.90 × 10 6 m/s in the +k z direction and −1.24 × 10 6 m/s (more than 1.5 times of that in graphene) in the −k z direction, while v x = v y = 0. In the k x -k y plane, the curvatures of the valance and conductive bands are 308.0 and 707.0 eV.Å 2 , respectively, and then the corresponding effective mass is m xx = m yy = 0.98m e for the valence band and m xx = m yy = 0.43m e for the conductive band, while m zz = 0.
Since the Dirac surface crosses the Fermi surface (E = 0), which is of great importance to electronic transport, we also analyze the band property of their intersections, for instance in F F (0.054, 0, 0.170) × 2π Å − 1 . In the k z direction, the slopes are −59. 4   By analyzing F A , F B and F F , we find the anisotropy of the massless Dirac behavior in the k z direction and the classic behavior in the k x and k y directions, is the intrinsic property of the Dirac surface in the Brillouin zone of bct-C12. It distributes in the energy space from −0.71 to 0.51 eV and is quite close to the Fermi level.

Discussions
The unexpected anisotropic Dirac fermion originates from the anisotropy of effective mass of the quasiparticle. True particles have spatial rotation invariance, so their masses are scalar constants. But three dimensional periodic system breaks the rotation invariance, so the effective mass of the quasiparticle in crystals is described by a 3 × 3 matrix tensor. In the general case, the three eigenvalues approach to each other and we can average them to give the averaged effective mass of the quasiparticle. By contrast, the bct-C12 is an extreme case. Especially, the bct-C12 inherits the zigzag carbon chain from its precursor (6, 6)-CNT, so it is massless Dirac fermion in the z direction. Unlike the normal CNT, however, the distance between the zigzag chains is only 3.02 Å which is close enough for the interaction between the 2p electrons around the sp 2 C atoms beyond the tubes, and the chain makes a common tetragonal lattice to have the classical band structure. As a result, it shows a strong anisotropy between the Dirac and classical behaviors. Interestingly, although the bct-C12 is metallic, it should have the transport behavior like the semiconductor. Particularly, the F A point (0.57 eV) is slightly above the Fermi level, while the F B point (−0.71 eV) lies slightly below the Fermi level. So the Fermi level cross the Dirac surface (Fig. 4c), and the bct-C12 possesses the spontaneous electrons at F A and holes at F B in its ground state. Due to different signs of effective mass in the x-y plane, F A and F B are different doping and will obviously contribute differently to the electronic properties.
In conclusion, by compressing the (6, 6)-CNT, we discover a new tetragonal carbon phase, which can be quenchable when decompressing to zero pressure. It is metallic and as hard as α -quartz. Most strikingly, this structure has the anisotropy of the Dirac behavior in the k z direction and the classic behavior in the k x and k y directions, which originates from the interaction between the Dirac zigzag chains. To our knowledge, this should be the first reported a system that has anisotropy in both Dirac and classic fermions. This research provides a new member of the big family of carbon allotropes and novel insight to their transport behaviors. Of course, this new phase maybe have some unexpected electronic behaviors.

Methods
Carbon nanotubes packed in periodic crystal lattices with a standard intertube spacing of 3.4 Å were constructed using the Materials Studio package 27 . Structural relaxations and property calculations were performed based on the density functional theory (DFT) as implemented in the CASTEP code 27 . The Vanderbilt ultrasoft pseudopotential was used and the electron-electron exchange interaction was described by the local density approximation (LDA) exchange-correlation functional of Ceperley and Alder, as parameterized by Perdew and Zunger (CA-PZ) 28,29 . The plane-wave cutoff energy with 800 eV, and a k-point spacing (2π × 0.03 Å −1 ) was used to generate Monkhorst-Pack k-points grids for Brillouin zone sampling 30 . Primitive cells were used to calculate the band structures and the bulk modulus, and shear modulus. Vickers hardness is calculated using the modified microscopic model [24][25][26] . All the results were also confirmed by the all-electron projector augmented wave (PAW) method 31 as implemented in the VASP code 32 . Phonon calculations were performed using the PHONOPY code 33 .