Fig. 2. Influence of metallic quantum surface-response on the dispersion of acoustic graphene plasmons (AGPs).

Three increasingly sophisticated tiers of response models are considered: (i) classical, local response for both the graphene and the metal [gray dot-dashed line]; (ii) nonlocal RPA and local Drude response for the graphene and the metal, respectively [black dashed line]; and (iii) nonlocal RPA and d-parameter-augmented response for the graphene and the metal, respectively [red solid line]. a AGP dispersion diagram, ω/(2π) versus $\mathrm{Re}q$. The hatched region indicates the graphene’s electron–hole continuum. b Associated imaginary part of the AGP wavevector, $\mathrm{Im}q$. c Corresponding quality factor $Q\equiv \mathrm{Re}q/\mathrm{Im}q$. The inset shows a zoom of the indicated region. System parameters: we take a graphene–metal separation of t = 1 nm; for concreteness and simplicity, we consider an unscreened jellium metal with plasma frequency $ħ$ω_p ≈ 9.07 eV (corresponding to r_s = 3) where ζ ≈ 0.8 Å and ξ ≈ 0.3 Ų⁴, with Drude-type damping $ħ$γ_m = 0.1 eV; for graphene, we take E_F = 0.3 eV and $ħ$γ = 8 meV; finally, we have assumed ϵ_d ≡ ϵ₁ = ϵ₂ = 1 (for consistency with the d-parameter data which assumes a metal–vacuum interface²⁴).