Transverse-electric surface plasmon for graphene in the Dirac equation model

We consider single-layer plane graphene with electronic excitations described by the Dirac equation. Using a known representation of the polarization tensor in terms of the spinor loop, we show the existence of surface modes, i.e., undamped in time excitations of the electromagnetic field, propagating along the graphene. These show up in the TE polarization and exist at zero temperature. We start with a finite-mass gap, which can be shrunk to zero at the end preserving the existence of the plasmon found. We also discuss the scattering modes.

Figure 1

The solutions of Eq. (23) for $v=1/2$ and several values of parameter $\alpha$. From bottom to top, the curves correspond to $\alpha =1,0.5,0.2,0.1,1/137$.

Figure 2

The solutions of Eq. (23) for $v=1/300$ and several values of parameter $\alpha$. From bottom to top, the curves correspond to $\alpha =1,0.5,0.2,0.1,1/137$.

Figure 3

The frequencies of scattering waves obey the condition $\omega >k_{\left|\right|}$. The pair creation begins when $\omega$ exceeds the threshold $\sqrt{4m^2+v^2{k}_{\left|\right|}^2}$. For completeness the dispersion of the surface plasmon for $\alpha =v=0.5$ is also shown (solid line).

Figure 4

Real and imaginary parts of the reflection coefficients $r_{TE}^{Di}$ and $r_{TM}^{Di}$ in the Dirac model as functions of frequency with $k_{\left|\right|}=m<2m/{(1-v^2)}^{1/2}$. The threshold of pair creation, $\omega ={(v^2{k}_{\left|\right|}^2+4m^2)}^{1/2}$, lies within the scattering sector.

Figure 5

Real and imaginary parts of the reflection coefficients $r_{TE}^{Di}$ and $r_{TM}^{Di}$ in the Dirac model as functions of frequency with $k_{\left|\right|}=3m>2m/{(1-v^2)}^{1/2}$. The threshold of pair creation, $\omega ={(v^2{k}_{\left|\right|}^2+4m^2)}^{1/2}$, lies beyond the scattering sector. Here $|r^{TM}|<1$ always holds.