Dynamical screening in bilayer graphene

We calculate one-loop polarization in bilayer graphene in the four-band approximation for arbitrary values of frequency, momentum, and doping. At low and high energy our results reduce to the polarization functions calculated in the two-band approximation and in the case of single-layer graphene, respectively. The special cases of static screening and plasmon modes are analyzed.

Figure 1

Normalized polarization function for $\mu /t=0.6$. (a) and (b) show density plots of the real and imaginary parts of the normalized polarization bubble defined in Eq. (2.18), respectively. (c) and (d) present constant-frequency cuts for $\omega /\mu =0.5,1.0,1.5,2.0,2.5$.

Figure 2

The static polarization. (a), (b), and (c) show plots of the normalized static polarization given by (3.1) at $\mu /t=0.01$, $0.1,$ and $0.6$, respectively. Dotted lines correspond to the asymptotic values $\Pi \left(k\right)=\pi k/2(t+2\mu )$. In (e) and (f) we show, respectively, the static limit of the polarization function for the bilayer graphene in the two-band approximation obtained in Ref. 17 and monolayer graphene calculated in the Dirac approximation in Refs. 15,25, and3. The dotted line in (e) corresponds to the asymptotic value $\Pi \left(k\right)=\ln 4$ while the asymptote in (f) is $\Pi \left(k\right)=\pi k/8\mu$.

Figure 3

The RPA improved Coulomb potential at finite and zero doping. In (a) the dashed line corresponds to the potential at $\mu =0.4t$ and the solid line to zero $\mu$. (b) and (c) show asymptotes of the RPA improved potential at zero and finite doping, respectively.

Figure 4

(a), (b), and (c) present the dispersion relations (black solid lines) for plasmons in free-standing graphene at densities $\mu /t=0.6$, 0.3, and 0.05 respectively. Black dashed lines describe the classical plasmon (3.8) and the high-energy plasmon (3.11). Black dotted lines describe the additional low-energy plasmon given by (3.10). Filled areas show domains with nonzero imaginary part of the polarization whose boundaries (red dotted lines) are given by Eqs. (3.12).