Out-of-Plane Dielectric Susceptibility of Graphene in Twistronic and Bernal Bilayers

Abstract

We describe how the out-of-plane dielectric polarizability of monolayer graphene influences the electrostatics of bilayer graphene-both Bernal (BLG) and twisted (tBLG). We compare the polarizability value computed using density functional theory with the output from previously published experimental data on the electrostatically controlled interlayer asymmetry potential in BLG and data on the on-layer density distribution in tBLG. We show that monolayers in tBLG are described well by polarizability α_exp = 10.8 ų and effective out-of-plane dielectric susceptibility ϵ_z = 2.5, including their on-layer electron density distribution at zero magnetic field and the interlayer Landau level pinning at quantizing magnetic fields.

Keywords: bilayer graphene excitons; dielectric susceptibility; gating; graphene; screening; twisted bilayer graphene.

Figure 1

(a) Sketches illustrating how the dielectric polarizability of each monolayer enters in the electrostatics analysis of bilayers in eq 1. (b) Characteristic electron dispersion in tBLG (here, θ = 3°; u = 100 meV). Electron state amplitude on the top/bottom layer is shown by red/blue. (c) Minivalley carrier densities n_κ/κ′ in a single-gated tBLG calculated for various misalignment angles outside the magic angle range, in comparison with the densities corresponding to SdHO measured in a tBLG flake with an unknown twist angle (black dots).

Figure 2

(a) Resistance map for a double-gated tBLG with a 30° twist angle, computed with ϵ_z = 2.5 and d = 3.44 Å (left) and measured (right) as a function of the total carrier density, n, and vertical displacement field, D, at B = 0 and T = 2 K. (b) Computed density of states of pinning LLs (left) and the measured resistance, ρ_xx, (right) in a 30° tBLG at B = 2 T, plotted as a function of displacement field and filling factor. Bright regions correspond to the marked N_t/N_b LL pinning conditions.

Figure 3

Interlayer asymmetry potential (dashed lines) and band gap (solid lines) in an undoped BLG, self-consistently computed with various values of ϵ_z = 1 (green), 2.6 (blue), and 2.35 (red) and compared to the optical gap measured in ref (16) (circles) and the transport gap (crosses). Here, we use^,v = 10.2 · 10⁶ m/s, γ₁ = 0.38 eV, v₃ = 1.23 · 10⁵ m/s, v₄ = 4.54 · 10⁴ m/s, δ = 22 meV, and d = 3.35 Å. Dotted lines show the values of the gap computed with γ₁ = 0.35 eV and the same other parameters. The sketch illustrates four BLG bands (1,2 below and 3,4 above the gap) highlighting a small difference between u and Δ.