Nonlocal quantum gain facilitates loss compensation and plasmon amplification in graphene hyperbolic metamaterials

Graphene-based hyperbolic metamaterials have been predicted to transport evanescent fields with extraordinarily large vacuum wave vectors. It is particularly at much higher wave vector values that the commonly employed descriptional models involving structure homogenization and assumptions of an approximatively local graphene conductivity start breaking down. Here, we combine a nonlocal quantum conductivity model of graphene with an exact mathematical treatment of the periodic structure to develop a toolset for determining the hyperbolic behavior of these graphene-based hyperbolic metamaterials. The quantum conductivity model of graphene facilitates us to predict the plasmonic amplification in graphene sheets of the considered structures. This allows us to reverse the problem of Ohmic and temperature losses, making this simple yet powerful arrangement practically applicable. We analyze the electric field distribution inside the finite structures, concluding that Bloch boundary solutions can be used to predict their behavior. With the transfer-matrix method, we show that at finite temperature and collision loss, we can compensate for losses, restoring imaging qualities of the finite structure via an introduction of chemical imbalance.

Figure 1

Replacing the metal layers of traditional hyperbolic metamaterials (a) by single sheets of graphene leads to graphene-based hyperbolic metamaterials (b). In case of the conductive sheets of the finite thickness (a), an incident TM wave couples to carrier oscillations in both $x$ and $z$ directions. In case of infinitesimally thin conductive layers (b), the carriers oscillations are confined to the plain of the layer.

Figure 2

Plasmon frequency dispersion of a doped, air-suspended single graphene sheet without photo inversion ($\overline{\mu }={\mu }_e$) (a) and with photoinversion ($\overline{\mu }={\mu }_e+{\mu }_h$) (b). The nonlocal quantum conductivity model is used for obtaining the solid green curves, local Drude model for the dotted curves, and Drude model with added interband transitions for the dashed curves. The excitation frequency ${\omega }_0$ is scaled as ${\omega }_0\hslash /\overline{\mu }$ and plotted on the vertical axis. The in-plane component of the excitation wave vector on the horizontal axis is scaled with the vacuum wave vector of the excitation frequency to transform the light cone and the Dirac cone into vertical lines at $k_x/k_0=1$ and $k_x/k_{}=c_0/v_{\text{Fermi}}=300$, respectively. The plasmons experience gain within region I due to interband electron-hole recombination; loss within regions III and IV due to the inter- and intraband absorption, respectively; and propagate without loss or gain within region II due to the absence of phase space for any inter- or intraband transitions.

Figure 3

(a) Real part of the Bloch wave vector $Re\left[\text{K}D\right]$ for a passive infinite structure is represented as a density map, where solid green line marks the single sheet plasmon dispersion. The black solid and dashed lines bound the region of plasmonic operation of the structure. (b) The isofrequency contours [horizontal cross sections of (a) along the dashed lines] at three different $\tilde{\omega }$ values. (c) $Re\left[{\varepsilon }_{x-\text{eff}}\right]$, calculated from Eq. (5), is shown as a density plot. Here the condition $Re\left[\text{K}D\right]=Im\left[\text{K}D\right]=0$ is marked by the solid black line, the condition $Re\left[\text{K}D\right]=Im\left[\text{K}D\right]=\pi$ by the dashed black line, and the condition $Re\left[\text{K}D\right]=\pi$ and $Im\left[\text{K}D\right]=0$ by the dotted black line. Above calculations were performed for the structures with the following parameters: ${\mu }_e=-{\mu }_h=0.031\mathrm{eV}$, $T=0\mathrm{eV}$, $\gamma =0$, ${\varepsilon }_d=1$, $d=300\mathrm{nm}$.

Figure 4

(a) Real (teal) and imaginary (brown) parts of the isofrequency curves are plotted for various values of $\tilde{\omega }$. (b) The imaginary part of the Bloch wave vector is shown for the HMM, based on doped graphene with photo excitation. Thin gray lines mark the boundaries of regions of behavior, corresponding to those in Fig. 2. The black curve bounds the region of ${\varepsilon }_{x-\text{eff}}\le 0$. Dashed gray lines correspond to the cross sections in $\tilde{\omega }$, for which the isofrequency curves are shown in (a). For reference, the green line is the frequency dispersion of the single graphene sheet, suspended in a dielectric with ${\varepsilon }_d$. Parameters of the structure: ${\mu }_e=0.023\mathrm{eV}$, ${\mu }_h=0.008\mathrm{eV}$, ${\varepsilon }_d=1$, $d=50\mathrm{nm}$.

Figure 5

${\left|E_x\right|}^2$ field distribution, corresponding to the propagation of a Gaussian envelope through the finite graphene-dielectric stack, composed of 14 layers with $d_1=50\mathrm{nm}$, is shown as density maps. (a) and (b) are calculated for the passive case, with the graphene sheets doped to ${\mu }_e=-{\mu }_h=0.031\mathrm{eV}$. For (a) and (b) $T=0$, $\gamma \to 0$, and $T=0.017\mathrm{eV}$, $\gamma =0.015\mathrm{eV}$, respectively. (c) shows the active case with the same losses as in (b): $T=0.017\mathrm{eV}$, $\gamma =0.015\mathrm{eV}$, and doping ${\mu }_e=0.023\mathrm{eV}$, ${\mu }_h=0.008\mathrm{eV}$.

Figure 6

Plasmon frequency dispersion of an air-suspended single sheet of doped graphene without photo inversion ($\overline{\mu }={\mu }_e$) (a); and with photo inversion ($\overline{\mu }={\mu }_e+{\mu }_h$) (b). The nonlocal quantum conductivity model is used for obtaining the solid green curves, local Drude model for the dotted curves, and Drude model with added interband transitions for the dashed curves. The excitation frequency ${\omega }_0$ is scaled as ${\omega }_0\hslash /\overline{\mu }$ and plotted on the vertical axis. Plasmons experience gain in region I, loss in regions III and IV due to the inter- and intraband excitations, respectively, and no loss or gain within region II.