Density functional theory calculations of the electric-field-induced Dirac cones and quantum valley Hall state in ABA-stacked trilayer graphene

We have investigated ABA-stacked trilayer graphene under a perpendicular electric field by using the density functional theory (DFT) calculations, which may contribute to the resolution of the discrepancies between experimental and theoretical results on the electric-field-induced band gap and topological phase transition. We found that the electric field opens a band gap at a low field and closes the gap at a high field, supporting one of the experimental results. While the seven electric-field-induced Dirac cones with mass gaps predicted in recent tight-binding (TB) models are confirmed, our DFT calculations demonstrate a phase transition from a quantum valley Hall insulator to a semimetal, contrasting to the TB model prediction of a topological phase transition between topologically nontrivial insulators at a high electric field.

Figure 1

Electronic screening effect. (a) Schematic structure of the ABA-TG. (b) Band structure near the $K$ point in the absence of electric field. Black and red lines correspond to the valence and conduction bands, respectively. (c) Average potential energy difference between the adjacent layers ${\mathrm{\Delta }}_1$ as a function of the external potential energy difference between the adjacent layers $U_{\mathrm{ext}}$, where $U_{\mathrm{ext}}=0.335D$ in units of eV and V/nm with a perpendicular electric field $D$. $\delta \vec{k}=(\delta k_x,\delta k_y)=1000(\vec{k}-K)$ in units of $2\pi /a$.

Figure 2

Band structure near the $K$ point in the presence of the perpendicular electric field $D$. In each panel, band structure and energy map for the valence band are shown at the top and bottom, respectively. (a) $D=0.02$ V/nm, (b) $D=0.51$ V/nm, (c) $D=1.65$ V/nm, and (d) $D=2.06$ V/nm, respectively. $\delta \vec{k}=(\delta k_x,\delta k_y)=1000(\vec{k}-K)$ in units of $2\pi /a$. Black and red lines correspond to the valence and conduction bands, respectively. In the energy map, high energy appears dark and the darkest spots (energy maxima) correspond to Dirac cones.

Figure 3

Contour map of the Berry curvature near the $K$ point in the presence of the perpendicular electric field $D$. (a) $D=0.02$ V/nm, (b) $D=0.51$ V/nm, (c) $D=1.65$ V/nm, and (d) $D=2.06$ V/nm, respectively. $\delta \vec{k}=(\delta k_x,\delta k_y)=1000(\vec{k}-K)$ in units of $2\pi /a$. Red and blue colors correspond to the positive and negative values of the Berry curvature, respectively. $p_o$ and $p\pm$ indicate the Dirac points centered and off-centered on the $K$ point, respectively.

Figure 4

Band gap $E_g$ as a function of the perpendicular electric field $D$. Inset shows $E_g$ at low fields.