Dynamic screening and low-energy collective modes in bilayer graphene

We theoretically study the dynamic screening properties of bilayer graphene within the random-phase approximation assuming quadratic band dispersion and zero gap for the single-particle spectrum. We calculate the frequency-dependent dielectric function of the system and obtain the low-energy plasmon dispersion and broadening of the plasmon modes from the dielectric function. We also calculate the optical spectral weight (i.e., the dynamical structure factor) for the system. We find that the leading-order long-wavelength limit of the plasmon dispersion matches with the classical result for two-dimensional electron gas. However, contrary to electron-gas systems, the nonlocal plasmon dispersion corrections decrease the plasmon frequency. The nonlocal corrections are also different from the single-layer graphene. Finally, we also compare our results with the double-layer graphene system (i.e., a system of two independent graphene monolayers).

Figure 1

(a)–(c) Imaginary part of intraband susceptibility as a function of frequency for (a) $q=0.8k_F$, (b) $q=1.5k_F$, and (c) $q=2.5k_F$. (d)–(f) Real part of intraband susceptibility as a function of frequency for (d) $q=0.8k_F$, (e) $q=1.5k_F$, and (f) $q=2.5k_F$.

Figure 2

(a)–(c) Imaginary part of interband susceptibility as a function of frequency for (a) $q=0.8k_F$, (b) $q=1.5k_F$, and (c) $q=2.5k_F$. (d)–(f) Real part of interband susceptibility as a function of frequency for (d) $q=0.8k_F$, (e) $q=1.5k_F$, and (f) $q=2.5k_F$.

Figure 3

(Color online) Optical plasmon dispersion in bilayer graphene systems for (a) $r_s=14$, (b) $r_s=5.6$, and (c) $r_s=4.3$. The thick line (black) is the plasmon dispersion while the dashed lines (blue and red) show the upper edge of the intraband continuum and the lower edge of the interband continuum.

Figure 4

Imaginary part of inverse dielectric function for (a) $q=0.15k_F$ (undamped plasmon), (b) $q=0.8k_F$ (damped plasmon), and (c) $q=2.2k_F$ (no plasmon pole). All figures are for $r_s=4.3$.

Figure 5

(Color online) Plasmon frequency for BLG (solid black) and SLG (dashed red) systems for two different values of substrate dielectric constant $\kappa =1$ and $\kappa =10$. The two figures are for different densities (a) $n={10}^{12}{\text{cm}}^{-1}$ and (b) $n={10}^{13}{\text{cm}}^{-1}$.

Figure 6

(a) The quasiparticle residue $Z$ and (b) The width $\Gamma$ of the plasmon pole in the inverse dielectric function for $r_s=4.3$.

Figure 7

(Color online) The scaling of the leading-order plasmon dispersion in the long-wavelength limit with density on a log-log scale. The leading-order result is obtained by evaluating the plasmon dispersion at a wave vector $q=5\times {10}^4{\text{cm}}^{-1}$. The red line is the BLG result with parabolic dispersion. The blue line is the SLG result. The black line is the BLG plasmon dispersion with hyperbolic dispersion.

Figure 8

(Color online) Low-energy plasmon dispersion for three different bilayer systems: (a) an $n\text{-GaAs}$ double quantum well with $r_s=1$ (corresponding to a density of $n=3.2\times {10}^{11}{\text{cm}}^{-2}$ and a layer separation of $d=9.5\text{nm}$, which corresponds to the effective Bohr radius of $n\text{-GaAs}$ quantum well), (b) a double-layer graphene with a density of $n={10}^{12}{\text{cm}}^{-2}$ and a layer separation of $d=3\text{Å}$ (which corresponds to the layer separation of bilayer graphene), and (c) a bilayer graphene system with a density of $n={10}^{12}{\text{cm}}^{-2}$ and $r_s=4.3$. The $\text{sqrt}q$ mode is in blue and the linear mode is in red solid lines. The black lines indicate interband and intraband continuum. The undamped linear acoustic mode is absent in bilayer graphene.