Observation of a Dirac point in microwave experiments with a photonic crystal modeling graphene

We present measurements of transmission and reflection spectra of a microwave photonic crystal composed of 874 metallic cylinders arranged in a triangular lattice. The spectra show clear evidence of a Dirac point, a characteristic of a spectrum of relativistic massless fermions. In fact, Dirac points are a peculiar property of the electronic band structure of graphene, whose properties consequently can be described by the relativistic Dirac equation. In the vicinity of the Dirac point, the measured reflection spectra resemble those obtained by conductance measurements in scanning tunneling microscopy of graphene flakes.

Figure 1

(Color online) Left: triangular lattice of metallic cylinders. The arrows indicate the two different directions $\Gamma \text{M}$ and $\Gamma \text{K}$ in the triangular lattice; the radius $R$ of the metallic cylinders equals $R=0.25a$, where $a$ is the lattice constant. Right: the plot of the numerically determined band structure $f\left(k_x,k_y\right)$, as function of the quasi momentum components $\left(k_x,k_y\right)$. The solid lines indicate the hexagonal Brillouin zone.

Figure 2

Scheme of the experimental setup. The photonic crystal consists of altogether 874 metallic cylinders arranged in a triangular lattice. Electromagnetic waves are transmitted from the horn antenna through the lattice to the wave guide antenna. The antennae and the lattice are oriented with respect to each other such that the waves impinge the lattice in $\Gamma \text{M}$ direction (see Fig. 1).

Figure 3

Left: the measured transmission spectrum from the horn to the wave guide antenna. Right: numerically calculated band structure $f\left(\left|\vec{k}\right|\right)$ with $\vec{k}$ chosen along the $\Gamma \text{MK}\Gamma$ path. The inset shows the Brillouin zone of the lattice and the irreducible Brillouin zone $\Gamma \text{MK}$.

Figure 4

Left: a reflection spectrum measured with the wire antenna located in the middle of the crystal in the center of a triangle of cylinders as schematically shown in the inset. The sharp resonances at the edges of the bands are related to the so-called van Hove singularities. Right: for comparison the computed band diagram from Fig. 3 is shown.

Figure 5

Two measured reflection spectra in the vicinity of the Dirac frequency. The upper spectrum was measured with an antenna placed in the middle of the crystal, the lower one with an antenna placed six rows apart from the boundary of the crystal. The smooth solid lines are fits of Eq. (8) to the experimental spectra.

Figure 6

Reflection spectra measured with a wire antenna for photonic crystals of different sizes. The upper panel shows the reflection spectrum for an empty space between two conducting plates. The other spectra correspond to systems containing 3, 7, 37, 91, and 874 cylinders, respectively. The icons to the right of the spectra show the shape of the corresponding sample. The Dirac frequency is marked by a dashed line.