Transfer Matrix Approach to Identify the Effect of Interlayer Distances of Graphene Multilayers in The Far and Near Field

We probe theoretically how sensitive the optical properties of multilayered graphene (MG) structures are to the sub-nanometer interlayer separation (D). This separation is often neglected, which is less of an issue for only a few layers of graphene. We use absorption and dipole lifetime as the far- and near-field observables, respectively. Assuming that the layers are electronically non-interacting, we use the transfer-matrix formalism in combination with Chebyshev relations for the lossy medium. This provides an analytical expression for the effective reflection and transmission coefficients for the pristine and doped MG structures (that follow a Drude response). We find that the known theoretical maximum limit of 50% absorption for infinitely thin absorbers only holds for pristine MG until a threshold value of the interlayer distance D = D_lim, after which the absorption increases monotonically with D for a fixed number of graphene layers N, as shown in fig. 1(a). This Dlim is far below the experimental interlayer distances in pristine MG (0.50-0.77 nm). For D → 0 we indeed find that the effective conductivity of MG becomes N times the conductivity of the single graphene sheet and that the absorption does not exceed 50%. The minimum number of pristine graphene layers required to attain this was found to be fixed by the fine structure constant α and given by N = 2/(πα). The doped MG also has this maximum limit of 50% absorption, but unlike in pristine graphene, absorption depends on the damping constant γ and the frequency ω. Our method also allows us to determine the dispersion relation for arbitrary interlayer distances and number of graphene layers and gives an important insight into the guided modes in such structures, as shown in fig. 1(b). A simplified expression for the reflection was also found to be very useful to evaluate the reflected Green’s function in a computationally more efficient way. We compare dipole lifetimes near the MG structure for pristine and doped graphene. In the near field the sub-nanometer interlayer distances do change the dipole emission lifetimes, with more pronounced changes in the doped MG (plotted in fig. 1(c)) than in the pristine MG structures.