Gate-induced interlayer asymmetry in ABA-stacked trilayer graphene

We calculate the electronic band structure of ABA-stacked trilayer graphene in the presence of external gates, using a self-consistent Hartree approximation to take account of screening. In the absence of a gate potential, there are separate pairs of linear and parabolic bands at low energy. A gate field perpendicular to the layers breaks mirror reflection symmetry with respect to the central layer and hybridizes the linear and parabolic low-energy bands, leaving a chiral Hamiltonian essentially different from that of monolayer or bilayer graphene. Using the self-consistent Born approximation, we find that the density of states and the minimal conductivity in the presence of disorder generally increase as the gate field increases, in sharp contrast with bilayer graphene.

Figure 1

(a) Schematic of the ABA-stacked trilayer lattice containing six sites in the unit cell, A (white circles) and B (black circles) on each layer, showing the Slonczewski-Weiss-McClure parametrization (Ref. 24) of relevant couplings ${\gamma }_0–{\gamma }_5$. (b) The conductivity versus external asymmetry ${\Delta }_1^{\left(\text{ext}\right)}$, calculated for trilayer and bilayer graphene using the self-consistent Born approximation and the band model including ${\gamma }_0$ and ${\gamma }_1$. (c) Schematic of trilayer graphene (three thin black lines at $x=-d,0,d$) with top and bottom gates (thick black lines at $x=L_t,-L_b$) separated from the trilayer by dielectric media (gray shaded areas).

Figure 2

Self-consistently calculated band structures in trilayer graphene near the $K$ point, with $n_{\text{tot}}=0$. Left plots are for the model including only ${\gamma }_0$ and ${\gamma }_1$, with (a) no asymmetry ${\Delta }_1^{\left(\text{ext}\right)}=0$ and (b) finite external asymmetry ${\Delta }_1^{\left(\text{ext}\right)}=0.5{\gamma }_1$. Right plots are for the full-parameter model including ${\gamma }_i$ $\left(i=1,2,\dots ,5\right)$ and $\delta$, with (c) ${\Delta }_1^{\left(\text{ext}\right)}=0$ and (d) $0.5{\gamma }_1$. Dashed and solid curves represent $\theta =0$ and $\pi /6$. The self-consistently calculated value of ${\Delta }_1$ is shown in the lower side of each plot. The thin horizontal line shows the Fermi energy.

Figure 3

[(a) and (c)] Density of states and [(b) and (d)] conductivity at $n_{\text{tot}}=0$ as functions of ${\Delta }_1^{\left(\text{ext}\right)}$ for the model including only ${\gamma }_0$ and ${\gamma }_1$ (left) and the full-parameter model (right). Solid, dotted, and dashed lines are for progressively larger disorder strengths.