k=0处的类狄拉克锥

狄拉克锥在电子和经典波体系中分别被发现, 由于其线性能带关系, 伴随着很多独特的现象. 除了一般存在于布里渊区边界处的狄拉克锥, k=0处也存在包含线性能带关系的类狄拉克锥. 这个类狄拉克锥可以由单极子和偶极子的偶然简并而形成. k=0处的类狄拉克锥可以通过两维电介质光子晶体来实现, 利用等效媒质理论, 此时的光子晶体在类狄拉克点频率可以等效为介电常数和磁导率都为零的材料. 电介质双零折射率材料既可以避免阻抗的不匹配, 也可以避免体系推广到高频所引起的强烈损耗. 此外, k=0处的类狄拉克锥与双零折射率的概念可以从两维体系拓展到三维体系, 而且还可以从电磁波体系推广到声波和弹性波体系. 利用具有类狄拉克点的两维光子晶体, 在材料参数都偏离类狄拉克点条件的两个半无限大光子晶体所构成的界面中, 一定存在界面态. 这些界面态的存在可以通过层状多重散射理论得到的表面阻抗以及体能带的几何相位来彻底解释.

Dirac-like cones at k=0

Dirac cones and Dirac points are found at the K (K') points in the Brillouin zones of electronic and classical waves systems with hexagonal or triangular lattices. Accompanying the conical dispersions, there are many intriguing phenomena including quantum Hall effect, Zitterbewegung and Klein tunneling. Such Dirac cones at the Brillouin zone boundary are the consequences of the lattice symmetry and time reversal symmetry. Conical dispersions are difficult to form in the zone center because of time reversal symmetry, which generally requires the band dispersions to be quadratic at k=0. However, the conical dispersions with a triply degenerate state at k=0 can be realized in two dimensional (2D) photonic crystal (PC) using accidental degeneracy. The triply degenerate state consists of two linear bands that generate Dirac cones and an additional flat band intersecting at the Dirac point. If the triply degenerate state is derived from the monopolar and dipolar excitations, effective medium theory can relate this 2D PC to a double zero-refractive-index material with effective permittivity and permeability equal to zero simultaneously. There is hence a subtle relationship between two seemingly unrelated concepts: Dirac-like cone and zero-refractive index. The all-dielectric “double zero”-refractive-index material has advantage over metallic zero-index metamaterials which are usually poorly impedance matched to the background and are lossy in high frequencies. The Dirac-like cone zero-index materials have impedances that can tune to match the background material and the loss is small as the system has an all-dielectric construction, enabling the possibility of realizing zero refractive index in optical frequencies. The realization of Dirac-like cones at k=0 can be extended from the electromagnetic wave system to acoustic and elastic wave systems and effective medium theory can also be applied to relate these systems to zero-index materials. The concept of Dirac/Dirac-like cone is intrinsically 2D. However, using accidental degeneracy and special symmetries, the concept of Dirac-like point can be extended from two to three dimensions in electromagnetic and acoustic waves. Effective medium theory is also applicable to these systems, and these systems can be related to isotropic media with effectively zero refractive indices. One interesting implication of Dirac-like cones in 2D PC is the existence of robust interface states. The existence of interface states is not a trivial problem and there is usually no assurance that localized state can be found at the boundary of photonic or phononic crystal. In order to create an interface state, one usually needs to decorate the interface with strong perturbations. Recently, it is found that interface state can always be found at the boundary separating two semi-infinite PCs which have their system parameters slightly perturbed from the Dirac-like cone formation condition. The assured existence of interface states in such a system can be explained by the sign of the surface impedance of the PCs on either side of the boundary which can be derived using a layer-by-layer multiple scattering theory. In a deeper level, the existence of the interface state can be accounted for by the geometric properties of the bulk band. It turns out that the geometric phases of the bulk band determine the surface impedance within the frequency range of the band gap. The geometric property of the momentum space can hence be used to explain the existence of interface states in real space through a bulk-interface correspondence.