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

Abstract

By using the $\stackrel{\ensuremath{\rightharpoonup}}{k}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightharpoonup}}{p}$ 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 $\ensuremath{-}\ensuremath{\pi}$. 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.