First-principles study of Dirac and Dirac-like cones in phononic and photonic crystals

By using the $\stackrel{\rightharpoonup }{k}·\stackrel{\rightharpoonup }{p}$ method, we propose a first-principles theory to study the linear dispersions in phononic and photonic crystals. The theory reveals that only those linear dispersions created by doubly degenerate states can be described by a reduced Hamiltonian that can be mapped into the Dirac Hamiltonian and possess a Berry phase of $-\pi$. Linear dispersions created by triply degenerate states cannot be mapped into the Dirac Hamiltonian and carry no Berry phase, and, therefore should be called Dirac-like cones. Our theory is capable of predicting accurately the linear slopes of Dirac and Dirac-like cones at various symmetry points in a Brillouin zone, independent of frequency and lattice structure.

Figure 1

(a) Band structures of a 2D phononic crystal consisting of a triangular lattice of iron rods embedded in a water host, with a rod radius $r/a=0.3203$, where $a$ is the lattice constant. Point $A$ is a Dirac-like point while point $B$ is a Dirac point. The inset shows the Brillouin zone. (b), (c) Closeup of bandstructures around the $\Gamma$ point for different rod radii, with $r/a=0.26$ in (b), and $r$/$a=0.38$ in (c), respectively. (d), (e) Closeup of band structures around the $K$ point for different rod radii, with $r/a=0.26$ in (d), and $r/a=0.38$ in (e), respectively. The red boxes in Fig. 1a mark the areas from where Figs. 1b–1e, 2d, and 3c are extended.

Figure 2

(a)–(c) Pressure field distributions of three degenerate Bloch states, ${\psi }_{1\Gamma }$, ${\psi }_{2\Gamma }$, and ${\psi }_{3\Gamma }$, respectively, at point $A$ as indicated in Fig. 1. Dark red and dark blue denote the positive and negative maxima, respectively, which imply that the wave energy is mainly concentrated in the host medium. (d) Black circles show the band structure near point $A$ obtained by finite-element calculations. Red curves show the results predicted by Eqs. (8) and (11).

Figure 3

(a), (b) Pressure field distributions of two degenerate Bloch states, ${\psi }_{1K}$ and ${\psi }_{2K}$, respectively, at point $B$ as indicated in Fig. 1. Dark red and dark blue denote the positive and negative maxima, respectively. (c) Black circles show the band structures near point B obtained by finite-element calculations. Red curves show the results predicted by Eqs. (8) and (18).