Optimal Thickness for Charge Transfer in Multilayer Graphene Electrodes

We study the charge transfer in multilayer graphene from first principles. We find that highly oriented (Bernal) and misoriented (turbostratic) multilayers show similar charge distributions despite their different electronic structure. We quantify the charge transfer and doping distribution in turbostratic graphene layers, where the screening is affected by vanishing density of states near the Fermi level. The results are in good agreement with an analytic model accounting for the electrostatic interaction and the band filling and point out the importance of system-specific interactions between the surface and the first layer. We find that graphene is an outstanding material for ultrathin electrodes, as most of the benefits of multilayer graphene can be captured with bilayers.

Figure 1

(a) Schematic of a metal-MLG ($N=6$) system. (b) Charge density profiles for the isolated metal $⟨{\rho }_M⟩$ (blue), isolated ($N=6$) TS MLG $⟨{\rho }_{\mathrm{TS}}⟩$ (green), and metal-TS system $⟨{\rho }_{M+\mathrm{TS}}⟩$ (red dashed line) for $d_{\mathrm{mg}}=3.5\AA$. Vertical dotted lines indicate 3.4-Å-wide regions centered around each graphene layer. (c)–(e) Induced charge density $⟨\mathrm{\Delta }\rho ⟩$ [Eq. (1)] for AB (blue dashed line) and TS (red line) MLG cases with $N=6$, 3, and 2, respectively.

Figure 2

Band structure (thin black lines) of bilayer graphene separated by 3.5 Å from a Mg surface: (a) AB and (b) TS stacking order, respectively. Circles superposed on bands show the projected weight of these bands onto carbon atomic orbitals of the first (blue) and second (red) graphene layers.

Figure 3

Band structure (thin grey lines) of $N=6$ MLG located at 3.5 Å from the metal surface for different stacking orders: (a) AB and (b) TS. Superposed symbols in the subplots denote the weight of the projections of bands onto carbon orbitals of each individual graphene layer (left, layer nearest to the metal; right, farthest layer).

Figure 4

Charge transfer to each of the six layers for different distances $d_{\mathrm{mg}}=3.0$ (blue), 3.5 (red), and 4.0 Å (green). Triangles show the results from first-principles calculations, with two different types of $k$-point meshes: one that includes the $K$ point (triangle up) and one that straddles it (triangle down). In the former case, the charge in the last layer actually reverses sign. For clarity, the different distances have been plotted with a small horizontal shift to avoid superposition of symbols, and a vertical dashed line is drawn between the first-principles results for different $k$ samples. Open symbols show an analytical model [24]. Inset: Projection onto layer orbitals for states at the $K$ point ($d_{\mathrm{mg}}=3.5\AA$). Each dot denotes a state with energy indicated along the vertical scale with projection onto states in the $i$th layer (horizontal axis). The magnitude of the projection is represented by the area of the dot.

Figure 5

(a) Total charge transfer (on logarithmic scale) to MLG as a function of the number of graphene layers, for values of $d_{\mathrm{mg}}=3.0$, 3.5, and 4.0 Å. In each case, the maximum charge transfer to the graphene slab occurs for two layers. (b) Estimate of the charge transfer to monolayer (∘), AB bilayer (□), and TS bilayer (⋄) as computed from the FC (open symbols) and the DP position (solid symbols).