Superluminal plasmons with resonant gain in population inverted bilayer graphene

AB-stacked bilayer graphene with a tunable electronic band gap in excess of the optical phonon energy presents an interesting active medium, and we consider such a theoretical possibility in this Rapid Communication. We argue the possibility of a highly resonant optical gain in the vicinity of the asymmetry gap. Associated with this resonant gain are strongly amplified plasmons, plasmons with negative group velocity and superluminal effects, as well as directional leaky modes.

Figure 1

Optical conductivity. (a) Schematic illustrating the singularity in the joint optical density of states in gapped bilayer graphene under population inversion. This produces a resonant optical gain, leading to amplified plasmons with superluminal effects. (b) Complex optical conductivity of AB-stacked bilayer graphene with an asymmetry gap $\mathrm{\Delta }=0.2\mathrm{eV}$. Carriers assumed to be in equilibrium with ${\mu }_e={\mu }_h=0.3$ eV, referenced with respect to the Dirac point. (c) Similar to (b), but for the inverted state, ${\mu }_e=-{\mu }_h=0.3$ eV.

Figure 2

Plasmon. (a) Loss function for bilayer graphene with an asymmetry gap $\mathrm{\Delta }=0.2\mathrm{eV}$, and for the inverted state, ${\mu }_e=-{\mu }_h=0.3$ eV. Calculation assumed $T=300\mathrm{K}$ and $\eta \approx 10\mathrm{meV}$. For the color scale, red (blue) denotes loss (gain). Plasmon dispersions for the equilibrium (black dashed) and inverted (white solid) state are calculated using Eq. (3) using the optical conductivity displayed in Fig. 1. Shaded regions are electron-hole continua ($\text{Im}\left[\epsilon \right(q,\omega \left)\right]\ne 0$) where the plasmon mode would be Landau damped. (b) Transient responses of a short plasmon pulse with a central frequency at $0.165\mathrm{eV}$ in bilayer graphene under equilibrium, initiated at position $x=0$ and then recorded at several plasmon wavelengths. Results indicate a positive group delay. (c) Similar to (b), but for the case with population inversion, revealing a negative group delay due to the negative group velocity. (d) Complex dispersion of the plasmon, $\beta \left(\omega \right)$ and $\alpha \left(\omega \right)$, for the inverted state for varying dopings of ${\mu }_e=-{\mu }_h=0.1$ (black dotted), 0.2 (red dashed), and $0.3\text{eV}$ (black solid). The quality factor $\beta /\alpha$ is $\times 10$ for better readability across the spectral range $\omega >0.4\mathrm{eV}$.

Figure 3

Leaky modes. (a) Radiation patterns for the graphene bilayer excited by a magnetic line source at a photon energy of $0.42\text{eV}$, varying the length of the graphene sheet. (b) Similar to (a), but for the pumped graphene bilayers at a photon energy of $0.26\text{eV}$ (active leaky-mode regime).