Interlayer screening effect in graphene multilayers with $ABA$ and $ABC$ stacking

We study the effect of perpendicular electric fields on the band structures of $ABA$ and $ABC$ graphene multilayers, and find that the electronic screening effect is significantly different between them. In $ABA$ multilayers, the field produces a band overlap and gives a linear screening, while in $ABC$ multilayers, in contrast, it opens an energy gap in the surface-state band at low energy, leading to a strong screening effect essentially nonlinear to the field amplitude. The energy gap of a large $ABC$ stack sharply rises when the external field exceeds a certain critical value.

Figure 1

Atomic structures of multilayer graphenes with (a) $ABA$ (Bernal) stacking and (b) $ABC$ (rhombohedral) stacking

Figure 2

(Color online) (a) Self-consistent band structures of $ABA$ (Bernal) multilayer graphenes with several layer number $N$’s at external field $e{F}_{0}d/{\gamma }_1=0.2$ (solid, red online) and 0 (dotted). (b) Potential distribution and (c) electron density of $ABA$ multilayers with several $N$’s at $e{F}_{0}d/{\gamma }_1=0.2$.

Figure 3

Potential difference between outermost layers as a function of external field, in $ABA$ multilayer graphenes with several $N$’s.

Figure 4

(Color online) (a) Self-consistent band structures of $ABC$ (rhombohedral) multilayer graphenes with several layer number $N$’s at external field $e{F}_{0}d/{\gamma }_1=0.2$ (red, solid) and 0 (black, dotted). (b) Potential distribution and (c) electron density of $ABC$ multilayers with several $N$’s at $e{F}_{0}d/{\gamma }_1=0.2$.

Figure 5

Potential difference between outermost layers of $ABC$ multilayers as a function of external field. Vertical broken line indicates the critical field $F_c$. Lower four panels compare the same results (solid curves) to the approximate expressions of Eq. (21) (dashed).

Figure 6

Energy gap width in the self-consistent band structures as function of external field for $ABC$ multilayers with several $N$’s. Vertical broken line indicates the critical field $F_c$.