Figure 1

(a) Bilayer graphene flake of width $2d$ is placed in electric field that separates $p$ and $n$ regions. Guided plasmons propagate along the $y$ axis and decay in the $x$ direction. (b) Schematic picture of the electron band structure: in equilibrium the sum of the electrostatic potential and kinetic energy of electrons (at the Fermi level) is constant. Vertical arrows indicate the possibility of optical absorption with a given frequency: while possible near the center of the flake (green non-crossed arrow), such transitions are forbidden (red crossed arrows) when both initial and final states are empty or occupied.Reuse & Permissions

Figure 2

The eigenfunctions ${\chi }_{{\alpha }_n}^{(+)}\left(\xi \right)$, Eq. (19), for three lowest values $n=1,2$, and 3 and $qa=0.2$. The first mode (red) is node-less while the second (blue) one has just one node and the third (green) has two nodes and the sharpest peak. For the sake of convenience the functions $2{\chi }_{\alpha }^{(+)}\left(\xi \right)/\cosh (\pi \alpha /2)$ are plotted. The corresponding eigenvalues for ${\alpha }_1=1.44$, ${\alpha }_2=2.49$, ${\alpha }_3=3.42$ are found from Eq. (15).Reuse & Permissions

Figure 3

The sketch of even plasmon (red, monotonic curves) and odd plasmon (blue curves with minima at around $1/d$) frequencies for $n=1$ (solid lines) and $n=2$ (dashed lines). The three regions, $q\ll 1/d$, $q\sim 1/d$, and $1/d\ll q<1/a$ are denoted by $\left(a\right)$, $\left(b\right)$, and $\left(c\right)$, respectively. With decreasing the wavelength through $\lambda \sim d$ the $n$th odd mode at $q\ll 1/d$ develops into the $n-1$th odd mode at $q\sim 1/d$, e.g., $(2-)$ becomes $(1-)$.Reuse & Permissions

Figure 4

Schematic charge distribution for the lowest plasmonic modes corresponding to different regions in Fig. 3. (a) For long wavelengths, $\lambda \gg d$, the lowest mode $(1+)$ is the quasi-one-dimensional plasmon with symmetric nodeless (across the flake) charge distribution and electric field pointed mostly along the $y$ direction. The magnitude of electric field vanishes with increasing the wavelength, in agreement with the gapless spectrum; Eq. (28). The second mode $(2-)$ is odd and gapped and has three nodes. A mode with a single node $(1-)$ is forbidden by the condition ${\int }_0^{d}dx\chi \left(x\right)=0$, valid for any gapped mode at $q\to 0$. The origin of that condition is the suppression of charge transport across the line separating $p$ and $n$ regions. The third mode $(2+)$ is even and has two nodes. The electric field in the modes $(2-)$ and $(2+)$ is mostly pointed along the $x$ direction. The field is stronger for the mode $(2+)$ (this fact could be easily inferred qualitatively from the picture of charge distribution), thus leading to ${\omega }_{2-}<{\omega }_{2+}$. (b) For intermediate wavelengths $\lambda \sim d$, the mode $(1-)$ is no longer forbidden and the $(1-)$ mode at $\lambda \sim d$ evolves from the $(2-)$ mode at $q=0$ and at $\lambda \sim d$ has the frequency below that of the $(1+)$ plasmon. This reversal happens because the checkerboard pattern of the $(1-)$ mode has lower electric field than the $(1+)$ mode.Reuse & Permissions

Figure 5

Electron band structure across the bilayer with the energy gap of $2U$. The $p$ (blue line starting from the point where the second lower peak on the left side touches the Fermi level and ending at the left end of the flake) and $n$ regions (red line on the right side of the axis) are separated by a neutral ($N$) strip (black double-headed arrow) of width $2h=2U/eE$ determined by the energy gap and the slope of the scalar potential.Reuse & Permissions

Figure 6

(a) Contour in the complex plane for the calculation of the integral (B1); the singularities in the integrand are indicated by the dots; (b) deformation of the contour from the real axis for the calculation of the integral (B4).Reuse & Permissions