Graphyne: Hexagonal network of carbon with versatile Dirac cones

We study α, β, and γ graphyne, a class of graphene allotropes with carbon triple bonds, using a first-principles density-functional method and tight-binding calculation. We find that graphyne has versatile Dirac cones and it is due to remarkable roles of the carbon triple bonds in electronic and atomic structures. The carbon triple bonds modulate effective hopping matrix elements and reverse their signs, resulting in Dirac cones with reversed chirality in α graphyne, momentum shift of the Dirac point in β graphyne, and switch of the energy gap in γ graphyne. Furthermore, the triple bonds provide chemisorption sites of adatoms which can break sublattice symmetry while preserving planar $s{p}^2$-bonding networks. These features of graphyne open new possibilities for electronic applications of carbon-based two-dimensional materials and derived nanostructures.

Figure 1

Schematic representations of (a) graphene, and (b) α, (c) β, and (d) γ graphyne. Red quadrangles indicate unit cells. (e) The hopping matrix elements along a carbon triple bond in graphyne. The carbon atoms 1 and 4 are at vertices of a hexagon in graphyne. (f) Effective direct hopping matrix element between the two carbon atoms 1 and 4 in (e).

Figure 2

(a) Electronic band structure of α graphyne along $M$-Γ-$K$ lines. (b) Comparison of Dirac cones and pseudospins in graphene and α graphyne. Red arrows indicate pseudospins. (c) Electronic band structure of β graphyne. (d) Electronic band structure of γ graphyne. In (a) and (c) Fermi energy is set zero. In (d) the middle of the energy gap is set zero.

Figure 3

(a) Definition of the triple bond length ($d_t$) of γ graphyne. (b) Energy gap in γ graphyne as a function of $d_t$. (c) Electronic band structure of γ graphyne for $d_t=1.2780$ Å. (d) Electronic wave function at the valence band maximum of γ graphyne for the equilibrium $d_t$ of 1.2214 Å. (e) Electronic wave function at the valence band maximum of γ graphyne for the elongated length $d_t=1.2780$ Å.

Figure 4

(a) Honeycomb lattice geometry with three nearest-neighbor hopping matrix elements (${\tilde{t}}_1$, ${\tilde{t}}_2$, ${\tilde{t}}_3$) and three nearest-neighbor relative positions (${\vec{d}}_1$, ${\vec{d}}_2$, ${\vec{d}}_3$). A and B denote two sublattices. (b) 2D plot of the energy dispersion, $E(k_x,k_y)/\left|{\tilde{t}}_1\right|$ with three equivalent hoppings (${\tilde{t}}_1={\tilde{t}}_2={\tilde{t}}_3$). (c) 2D plot of the energy dispersion $E(k_x,k_y)/\left|{\tilde{t}}_1\right|$ with ${\tilde{t}}_1={\tilde{t}}_2$ and ${\tilde{t}}_3=2.1\times {\tilde{t}}_1$. (d) The dispersion $E(0,k_y)/\left|{\tilde{t}}_1\right|$ for various values of ${\tilde{t}}_3$/${\tilde{t}}_1$. In (d) the energy-gap opening can be clearly seen when ${\tilde{t}}_3$/${\tilde{t}}_1>2$. In (b)–(d) the lattice constant $a$ is $\sqrt{3}$ times the C-C bond length.

Figure 5

Optimized atomic structures of (a) H${}_2$ added α graphyne and (b) HF added α graphyne. Blue, green, and purple dots represent C, H, and F, respectively. (c) Electronic band structure of (a). (d) Electronic band structures of (b). Note that the energy gap opens in (d).