Polarizability and screening in chiral multilayer graphene

We calculate the static polarizability of multilayer graphene and study the effect of stacking arrangement, carrier density, and on-site energy difference on graphene screening properties. At low densities, the energy spectrum of multilayer graphene is described by a set of chiral two-dimensional electron systems, and the associated chiral nature determines the screening properties of multilayer graphene, showing very different behavior depending on whether the chirality index is even or odd. As density increases, the energy spectrum follows that of the monolayer graphene and thus the polarizability approaches that of monolayer graphene. The qualitative dependence of graphene polarizability on chirality and layering indicates the possibility of distinct graphene quantum phases as a function of the chirality index.

Figure 1

(a) The Thomas-Fermi wave vector, $q_{\mathrm{TF}}\propto N_0=\Pi (q=0)$, as a function of density $n$ with $\kappa =2.5$, where $a=0.246$ nm is the lattice constant of graphene. (b) Same as (a) for multilayer graphene with several stacking sequences, where MLG and BLG represent monolayer and bilayer graphene, respectively. (c) and (d) show the total static polarizability $\Pi \left(q\right)$ and its first derivative $d\Pi \left(q\right)/dq$, respectively, as a function of wave vector for C2DES with chiralities $J=1,2,3,4,5$. Note that the first derivative at $2k_{\mathrm{F}}$ is continuous (discontinuous) for odd (even) $J$. (e) and (f) show the intraband and interband polarizabilities, respectively.

Figure 2

(a) Static polarizability of bilayer graphene with the zero energy gap as a function of wave vector for several carrier densities in units of ${10}^{12}$ cm${}^{-2}$. In (b) we include the on-site energy difference $U$ between the layers to open the energy gap. The polarizability is calculated for a carrier density $n={10}^{12}$ cm${}^{-2}$ and for different values of $U$. In both figures the black solid line corresponds to the result of C2DES with $J=2$. The inset of (a) shows the energy band structure of bilayer graphene for $U=0$ (solid line) and $U=0.1$ eV (dotted line). The inset of (b) shows the static polarizability of monolayer graphene with energy differences between the two sublattices of $U=0,0.1,0.2,0.3$ eV.

Figure 3

Calculated polarizability as a function of wave vector with several densities in units of ${10}^{12}$ cm${}^{-2}$ for (a) ABC trilayer graphene and (b) ABA trilayer graphene. The black line in (a) and (b) indicates the result of C2DES with $J=3$ and $J=2$, respectively. Insets show the energy band structure of each stacking.

Figure 4

(a) Density oscillations of C2DES with chiralities $J=1,2,3,4,5$ using the RPA for $n={10}^{12}$ cm${}^{-2}$ with $\kappa =2.5$. In (b) $\delta n\left(r\right)$ is rescaled by $r^{\beta }$, where $\beta =2$ ($\beta =3$) for even (odd) $J$, which clearly shows the decaying pattern at large distances with the periodicity of $\pi /k_{\mathrm{F}}$.