Line of Dirac Nodes in Hyperhoneycomb Lattices

We propose a family of structures that have “Dirac loops,” closed lines of Dirac nodes in momentum space, on which the density of states vanishes linearly with energy. Those lattices all possess the planar trigonal connectivity present in graphene, but are three dimensional. We show that their highly anisotropic and multiply connected Fermi surface leads to quantized Hall conductivities in three dimensions for magnetic fields with toroidal geometry. In the presence of spin-orbit coupling, we show that those structures have topological surface states. We discuss the feasibility of realizing the structures as new allotropes of carbon.

Figure 1

Simple lattice structures where all atoms are connected by three coplanar bonds spaced by 120°. (a) The hyperhoneycomb lattice ($\mathcal{H}\text{−}0$), with a four atom unit cell. Atoms 1, 2, and 3 ($xy$ plane); atoms 2, 3, and 4 ($yz$ plane). Atoms 1 and 2 form a vertical chain (black links); atoms 3 and 4 form a horizontal chain (blue links). The chains are connected by links (red) in the $z$ direction. (b) An eight atom unit cell ($\mathcal{H}\text{−}1$). Atoms 1–4 create a vertical chain of hexagons along the $x$ direction. Atoms 5–7 create a horizontal chain when repeated in the $y$ direction.

Figure 2

(a) BZ in the $k_z=0$ plane showing the Dirac loop lines (solid white). Black line, boundary of the BZ, centered at the $\mathrm{\Gamma }$ point. (b) 3D Brillouin zone. Black arrows, directions of the reciprocal lattice vectors ${\mathbf{b}}_i$, $i=1,2,3$. (c) Energy spectra of the four bands of Eq. (1) in units of $t$ plotted along the path shown in the red line of panels (a) and (b). The low energy bands cross along the Dirac loop. (d) Velocity of the quasiparticles at the Dirac line in units of $ta$, as a function of the cylindrical polar angle $\varphi$ with respect to $\mathrm{\Gamma }$. In cylindrical coordinates are red, $v_z(\varphi )$; black, $v_{\rho }(\varphi )$; blue, $v_{\varphi }(\varphi )$. The orange and violet curves describe $v_x(\varphi )$ and $v_y(\varphi )$, respectively. (e) Red arrows, cylindrical moving basis around the line of Dirac nodes. Toroidal Fermi surfaces for energies $E/t=0.1,0.2,0.3$, and 0.4 around the Dirac loop.

Figure 3

Energy bands in units of $t$ in the presence of a large spin-orbit coupling $t_2=0.1t$ (see text). The crossed lines at zero energy are topological surface states along the [100] direction of the crystal. Left, $\mathcal{H}\text{−}0$ crystal; right, $\mathcal{H}\text{−}1$ crystal.