Nonlinear density waves on graphene electron fluids

Abstract

In graphene, where the electron-electron scattering is dominant, electrons collectively act as a fluid. This hydrodynamic behaviour of charge carriers leads to exciting nonlinear phenomena such as solitary waves and shocks, among others. In the future, such waves might be exploited on plasmonic devices, either for modulation or signal propagation along graphene waveguides. We study the nature of nonlinear perturbations by performing the reductive perturbation method on the hydrodynamic description of graphene electrons, taking into consideration the effect of Bohm quantum potential and odd viscosity. Thus, deriving a dissipative Kadomtsev-Petviashvili equation for the bidimensional flow as well as its unidimensional limit in the form of Korteweg-de Vries-Burgers. The stability analysis of these equations unveils the existence of unstable modes that can be excited and launched through graphene plasmonic devices.

Figures (5)

• Figure 1: Transcritical bifurcation diagram of Eq. \ref{['eq:Equation_of_motion']}, showing the position and nature of the fixed points and the turning point of the separatrix (blue dotted-dashed line). The bifurcation swaps the equilibrium (black solid line) and saddle (red dashed line) points.
• Figure 2: (a-b) Phase space of the Hamiltonian \ref{['eq:hamiltonian']} for $\mathcal{B}=1$, $S/v_F=2$ and $c/v_F=4$ (a) or $S/v_F=1.2$ (b). The fixed points $(n_0,0)$ and $(n_c,0)$ are marked by highlighted dots, and the initial conditions of the oscillatory numerical solutions are indicated by the arrow tip. Bounded orbits exist inside the separatrix (red dashed line). (c-f) Numerical solutions of orbits on the phase space. The solitary (c-d) and oscillatory (e-f) numerical solutions (red solid line) are compared against cnoidal analytical expressions of the same amplitude and wavelength (black dashed line).
• Figure 3: Parameter space regions with distinct qualitative behavior, bounded by $\varepsilon^4=16\beta^2$. Regions II, III, VI and VII only sustain bounded solutions along the heteroclinic orbit connecting the two fixed points. Whilst the remaining areas (labelled with o.) feature oscillatory solutions, either decaying or growing in time. Shaded region $\varepsilon\geq0$ indicating the achievable region of positive shear viscosity.
• Figure 4: Phase space (left, streamline plots) and numerical solutions (right, red solid line)of equation \ref{['eq:adimKdvB']} for the positive viscosity regions, showing: (a) the growing oscillations $\varepsilon=0.1$, $\beta=1$, (b) shock propagation $\varepsilon=0.1$, $\beta=1$(c) idem $\varepsilon=5/\sqrt{6}$, $\beta=-1$, (d) decaying oscillations $\varepsilon=0.1$, $\beta=-1$. On panel (c) the analytical solution is superimposed (black dashed line) and on (b) a solution of the same form is also plotted for comparison. At the phase space plots the fixed points $(\varphi_\pm,\varphi_\pm')$ are highlighted (red dots).
• Figure 5: Region of instability for the nonlinear Schrödinger equation \ref{['eq:NLSE']}. In the shaded regions, the positive Kerr term leads to a self-focusing (unstable) mode.