32
2 Zero-Index Metamaterials
Fig. 2.4 Computational cell used for the computation of s-parameters using COMSOL Multiphysics
2.4.2 Dirac Cones—A Deep Insight into the Zero-Index
Behavior
A few exotic materials have become popular topics of research nowadays, graphene
is one of them. Graphene is a two-dimensional single-layer arrangement of carbon
atoms in a hexagonal lattice (Fig. 2.5a). The exotic material has an exotic band
structure having a conical-shaped feature in the valence and conduction bands, called
the Dirac cones, intersecting at a common point (Fig. 2.5b) [106, 107]. Due to the
conical shape of the dispersion surfaces, the dispersion in the vicinity of Dirac point
is linear E = F , where E is energy, k is wave vector, h/2π) is the reduced
Planck’s constant, and v F is the Fermi velocity. On account of the linear dispersion,
the behavior of the conduction electrons in graphene differs from that in metals and
insulators, which are parabolic dispersion materials.
In 2009, Wang et al. [85] analyzed the possibility of Dirac cones in the photonic
band structure of typical optical systems and laid down the fundamental principle of
(a)
(b)
Fig. 2.5 a Structure of graphene—hexagonal lattice; b Dirac cones in the band structure obtained
from the dispersion relation E = F
2 Zero-Index Metamaterials
Fig. 2.4 Computational cell used for the computation of s-parameters using COMSOL Multiphysics
2.4.2 Dirac Cones—A Deep Insight into the Zero-Index
Behavior
A few exotic materials have become popular topics of research nowadays, graphene
is one of them. Graphene is a two-dimensional single-layer arrangement of carbon
atoms in a hexagonal lattice (Fig. 2.5a). The exotic material has an exotic band
structure having a conical-shaped feature in the valence and conduction bands, called
the Dirac cones, intersecting at a common point (Fig. 2.5b) [106, 107]. Due to the
conical shape of the dispersion surfaces, the dispersion in the vicinity of Dirac point
is linear E = F , where E is energy, k is wave vector, h/2π) is the reduced
Planck’s constant, and v F is the Fermi velocity. On account of the linear dispersion,
the behavior of the conduction electrons in graphene differs from that in metals and
insulators, which are parabolic dispersion materials.
In 2009, Wang et al. [85] analyzed the possibility of Dirac cones in the photonic
band structure of typical optical systems and laid down the fundamental principle of
(a)
(b)
Fig. 2.5 a Structure of graphene—hexagonal lattice; b Dirac cones in the band structure obtained
from the dispersion relation E = F