Tight-binding theory of spin-orbit coupling in graphynes

We investigate the effects of Rashba and intrinsic spin-orbit couplings (SOC) in graphynes. First, we develop a general method to address spin-orbit couplings within the tight-binding theory. Then, we apply this method to $\alpha$-, $\beta$-, and $\gamma$-graphyne, and determine the SOC parameters in terms of the microscopic hopping and onsite energies. We find that for $\alpha$-graphyne, as in graphene, the intrinsic SOC opens a nontrivial gap, whereas the Rashba SOC splits each Dirac cone into four. In $\beta$- and $\gamma$-graphyne, the Rashba SOC can lead to a Lifshitz phase transition, thus transforming the zero-gap semiconductor into a gapped system or vice versa, when pairs of Dirac cones annihilate or emerge. The existence of internal (within the benzene ring) and external SOC in these compounds allows us to explore a myriad of phases not available in graphene.

Figure 1

Lattice structure of $\alpha$-, $\beta$-, and $\gamma$-graphyne, shown in panels (a), (b), and (c), respectively. Atoms at the vertices are denoted by capital letters, whereas atoms located at the edges are denoted by lowercase letters. The hopping parameters $t_i$ are also shown, with the subscripts 1, 2, and 3 corresponding to vertex-vertex, vertex-edge, and edge-edge hoppings, respectively.

Figure 2

Dispersion relation for $\alpha$-, $\beta$-, and $\gamma$-graphyne along high-symmetry lines, shown in (a), (b), and (c), respectively. The (blue) solid lines correspond to the dispersion relation obtained from the full TB Hamiltonian, whereas the (red) dashed lines correspond to the dispersion relation obtained from the low-energy approximation.

Figure 3

Sketch of the BZ for $\alpha$-, $\beta$-, and $\gamma$-graphyne, shown in panels (a), (b), and (c), respectively. The crosses correspond to Dirac cones.

Figure 4

Parameters used for the $\sigma$ TB models for $\alpha$-, $\beta$-, and $\gamma$-graphyne, shown in panels (a), (b), and (c). The hopping parameters $V_1,...,V_4$ correspond to NN hoppings, whereas $V_5,...,V_9$ correspond to onsite hoppings, and ${\varepsilon }_1,...,{\varepsilon }_5$ denote the onsite energies. Other hopping parameters have been set to zero.

Figure 5

Phase diagram for the effective Hamiltonian (51) with internal Rashba SOC. In region I, the system exhibits 12 Dirac cones, region II corresponds to the gapped phase, region III corresponds to the system where only pairs of Dirac cones along the line $\Gamma \text{−}M$ are present, and region IV corresponds to a system where there are six pairs of cones along the line $\Gamma \text{−}M$ and six pairs on lines perpendicular to $\Gamma$-M.

Figure 6

Sketches of the phases corresponding to the regime $t_{\mathrm{ext}}/t_{\mathrm{int}}<-2^{1/3}$. Each cross denotes a Dirac cone and we have taken $t_{\mathrm{ext}}/t_{\mathrm{int}}=-1.3$. (a) Brillouin zone for ${\lambda }_{\mathrm{int},\mathrm{R}}={\lambda }_{\mathrm{ext},\mathrm{R}}=0$, (b) Brillouin zone for ${\lambda }_{\mathrm{int},\mathrm{R}}/t_{\mathrm{int}}=0.1$ or ${\lambda }_{\mathrm{ext},\mathrm{R}}/t_{\mathrm{int}}=0.1$, and (c) Brillouin zone for ${\lambda }_{\mathrm{int},\mathrm{R}}/t_{\mathrm{int}}=0.4$ or ${\lambda }_{\mathrm{ext},\mathrm{R}}/t_{\mathrm{int}}=0.8$.

Figure 7

Phase diagram for the effective Hamiltonian (51) with external Rashba SOC. Region I corresponds to a gapped system, region II exhibits six pairs of Dirac cones along a line perpendicular to the line connecting $\Gamma$ and $M$, region III exhibits six Dirac cones along a line perpendicular to the line connecting $\Gamma$ and $M$, and six pairs of Dirac cones along the lines connecting $\Gamma$ and $M$, and region IV shows the system with only six pairs of Dirac cones along the line connecting $\Gamma$ and $M$.

Figure 8

Phase diagram for the effective Hamiltonian (51) with internal and external Rashba SOC, where ${\lambda }_R={\lambda }_{\mathrm{ext},\mathrm{R}}=-{\lambda }_{\mathrm{int},\mathrm{R}}$. In region I, the system exhibits 12 Dirac cones, region II corresponds to the gapped phase, region III corresponds to the system where only pairs of Dirac cones along the line $\Gamma \text{−}M$ are present, and region IV corresponds to a system where there are six pairs of cones along the line $\Gamma \text{−}M$ and six pairs on lines perpendicular to $\Gamma \text{−}M$.

Figure 9

Band structure for $\beta$-graphyne for different values of ${\lambda }_{\mathrm{int},\mathrm{I}}$ and ${\lambda }_{\mathrm{ext},\mathrm{I}}$, along the line $k_x=0$. (a) ${\lambda }_{\mathrm{ext},\mathrm{I}}=0.15$ eV and ${\lambda }_{\mathrm{int},\mathrm{I}}=0$ eV. (b) ${\lambda }_{\mathrm{ext},\mathrm{I}}=0$ eV and ${\lambda }_{\mathrm{int},\mathrm{I}}=0.3$ eV. (c) ${\lambda }_{\mathrm{ext},\mathrm{I}}=0.15$ eV and ${\lambda }_{\mathrm{int},\mathrm{I}}=0.3$ eV. (d) ${\lambda }_{\mathrm{ext},\mathrm{I}}=0.15$ eV and ${\lambda }_{\mathrm{int},\mathrm{I}}=-0.3$ eV.

Figure 10

Topological band-gap opening at the $K$ point in $\alpha$-graphyne due to the intrinsic SOC, for ${\lambda }_I\approx 0.05$ eV.

Figure 11

Band structure for $\alpha$-graphyne including the Rashba SOC for ${\lambda }_R\approx 0.5$ eV, zoomed in on the $K$ point.

Figure 12

Band structure of $\gamma$-graphyne in the presence of external Rashba SOC, zoomed in on the $M$ point in the BZ. (a) Gapped system, for ${\lambda }_{\mathrm{ext},\mathrm{R}}\approx 0.9$ eV, (b) transition from gapped system into a zero-gap semiconductor, for ${\lambda }_{\mathrm{ext},\mathrm{R}}\approx 1.04$ eV, and (c) two pairs of Dirac cones, for ${\lambda }_{\mathrm{ext},\mathrm{R}}\approx 1.2$ eV.

Figure 13

Labeling used for the $\alpha$-graphyne TB model.

Figure 14

Labeling of orbitals used in the $\beta$-graphyne TB model.

Figure 15

Labeling used for the $\gamma$-graphyne TB model.