Wake potential in graphene-insulator-graphene composite systems

Abstract

We study the wake potential produced by an external charged particle that moves parallel to various ${\mathrm{sy}}_1-{\mathrm{Al}}_2{\mathrm{O}}_3-{\mathrm{sy}}_2$ sandwich-like composites, where the system ${\mathrm{sy}}_i$ (with $i=1,2$) may be vacuum, pristine graphene, or doped graphene. The effective dielectric function of the composites is obtained using three complementary methods for graphene's electronic response, based on the massless Dirac fermions (MDF) method, the extended hydrodynamic (eHD) model, and the ab initio approach. Three velocity regimes are explored with respect to the threshold for excitations of the Dirac plasmon in graphene, given by its Fermi velocity $v_F$. In the low-velocity regime (below $v_F$), only the transverse optical (TO) phonons in the ${\mathrm{Al}}_2{\mathrm{O}}_3$ layer contribute to the wake potential in the surface with ${\mathrm{sy}}_2$ (which is nearest to the charged particle), in a manner that is only sensitive to the composition of that system: if ${\mathrm{sy}}_2$ is vacuum, the TO phonons give rise to intense oscillations in the wake potential, which are strongly suppressed if ${\mathrm{sy}}_2$ is pristine or doped graphene. For intermediate velocities (above $v_F$), the hybridized plasmon–TO phonon modes on both surfaces contribute to the wake potential in the surface with ${\mathrm{sy}}_2$, with the most dominant contribution coming from the hybridized Dirac-like plasmonic modes. In the high-velocity regime (well above $v_F$), the highest-lying hybridized Dirac plasmon gives the dominant contribution to the wake potential, which exhibits a typical $\mathsf{V}$-shaped wave-front pattern that lags behind the charged particle. It is found that the MDF method agrees very well with the results of the ab initio method for small and intermediate velocities. However, in the high-velocity regime, the high-energy $\pi$ plasmon in graphene introduces new features in the wake potential in the form of fast oscillations, just behind the charged particle. Those oscillations in the wake potential are well described by both the eHD and the ab initio method, proving that the $\pi$ plasmon indeed behaves as a collective mode.