Two-dimensional carbon allotrope with strong electronic anisotropy

Two two-dimensional carbon allotropes comprised of octagons and pentagons are proposed based on the first-principles calculations. The two carbon allotropes, named OPG-L and OPG-Z, are found to have distinct properties. OPG-L is metallic, while OPG-Z is a gapless semimetal. Remarkably, OPG-Z exhibits pronounced electronic anisotropy with highly anisotropic Dirac points at the Fermi level. A tight-binding model is suggested to describe the low-energy quasiparticles, which clarifies the origin of the anisotropic Dirac points. The electronic anisotropy of OPG-Z is expected to have interesting potential applications in electronic devices.

Figure 1

The structures of (a) OPG-L and (b) OPG-Z. The black dashed frames are the orthogonal unit cells of OPG-L and OPG-Z, where OA and OB are lattice vectors. These two structures can be tiled by the red-colored 558 structure by copying it along the green arrows. The primitive cell of OPG-L is shown in blue dashed frame while the primitive cell of OPG-Z is the same as the crystal cell. Here we give the structural information of unit cells from the DFT calculations. OPG-L: space group $Cmmm$, OA $=$ 3.68 Å, OB = 9.12 Å, two atoms in the asymmetric unit cell, (0, 0.42) and (0.69, 0.33); OPG-Z: space group $Pmam$, OA $=$ 6.90 Å, OB $=$ 4.87 Å, four atoms in asymmetric unit cell, (0.45,0.87), (0.56,0.62), (0.25,0.48), (0.25,0.78).

Figure 2

(a) The formation energies of OPG-L, OPG-Z, graphdiyne, T graphene, and pentaheptite as a function of area ratio in comparison to graphene, where $A_0$ is the optimized lattice area. The formation energy of graphene is set to 0. (b) The Helmholtz free energy as a function of temperature for the above-mentioned 2D carbon allotropes. The phonon spectra of (c) OPG-L and (d) OPG-Z. Inset: The first Brillouin zones and high symmetry points of OPG-L and OPG-Z. The high symmetry points are: $\Gamma$(0,0), M(0,0.5), K(0.419,0.709), H(0.581,0.291) in OPG-L, and $\Gamma$(0,0), X(0.5,0), M(0.5,0.5), Y(0,0.5) in OPG-Z (fractional coordinates in reciprocal space).

Figure 3

Electronic band structures and density of states for (a) OPG-L and (b) OPG-Z, respectively. The inset figures are locations of high symmetry points. The red dashed lines represent Fermi levels, which are set to 0 eV. Blue lines are the results of DFT calculations, while orange dotted lines are the results of tight-binding model.

Figure 4

(a) The blow-up band structures of Kohn-Sham quasiparticles of OPG-Z near the Fermi level around $\Gamma$ point. (b) The partial charge density (PCD) at $\kappa$ point near Fermi level, calculated by DFT. (c) The wave function in a unit cell at the same point, calculated by the tight-binding model [Eq. (1)]. (d) The component of group velocity parallel to the k vector ($v_k$) measured from the $\kappa$ point in units of the Fermi velocity along $k_y$ ($v_0$) versus the angle (${\theta }_k$) of the k vector. Inset: The isosurface of the band structure of OPG-Z within the range of $k_x\in [0.139,0.141]$ and $k_y\in [-0.0001,0.0001]$. The black dots in (b) and (c) are carbon atoms.

Figure 5

(a) OPG-Z as a distorted honeycomb lattice. Gray atoms are the atoms with zero wave-function amplitude around Fermi level in OPG-Z. The supercell of graphene consists of eight atoms with its TB hopping matrix elements $t_1$ and $t_2$ shown. (b) The tight-binding band structure of graphene in the supercell shown in (a), with $t_2=3$ eV. (c) The TB band structure of graphene of the same primitive cell, with $t_2=0$ eV. Inset: high symmetry points of the graphene supercell. (d) The different bond lengths of the zigzag ribbon in OPG-Z.