Figure 3: Density plots showing the bulk modes in the miniband continuum and the line traces of plasmon dispersion for N = 4 graphene layers on a conducting substrate.

The line plots show damped (dashed blue lines) and undamped (red curves) plasmon excitations. In (a–d), the layers are gapless, and in (e–h) each layer has a gap Δ = 0.6 μ. The layers are equally spaced with inter-layer spacing . The separation between the first layer and the surface was chosen as a₁k_F = 2.0 . If the gap or number of layers is increased, the lowest branch does not re-appear for large q_||.