Plasmonic excitations in Coulomb-coupled $N$-layer graphene structures

We study Dirac plasmons and their damping in spatially separated $N$-layer graphene structures at finite doping and temperatures. The plasmon spectrum consists of one optical excitation with square-root dispersion and $N-1$ acoustical excitations with linear dispersion, which are undamped at zero temperature and finite doping within a triangular energy region outside the electron-hole continuum. In the long-wavelength limit the energy and weight of the optical plasmon modes increase, respectively, as the square root and linearly with $N$ in agreement with recent experimental findings. The energy and weight of the upper-lying acoustical branches also increase with $N$. This increase is strongest for the uppermost acoustical mode, and we find that its energy can exceed at some value of momentum the plasmon energy in an individual graphene sheet. Meanwhile, the energy of the low-lying acoustical branches decreases weakly with $N$ as compared with the single acoustical mode in double-layer graphene structures. Our numerical calculations provide a detailed understanding of the overall behavior of the wave-vector dependence of the optical and acoustical multilayer plasmon modes and show how their dispersion and damping are modified as a function of temperature, interlayer spacing, and inlayer carrier density in (un)balanced graphene multilayer structures.

Figure 1

The plasmon dispersions at finite temperature $T=0.1T_F$ in GMLS consisting of $N=2,3,4$, and 5 graphene monolayers (shown in bold parallel lines in each panel) for the interlayer separation $d=5$ nm. The upper solid curve represents the optical plasmon mode with the square-root dispersion; the lower dotted, dashed, dotted-dashed, and long-dashed curves represent acoustical branches of the plasmon spectrum with linear dispersions. The thin black line represents the plasmon mode in an individual graphene layer. The doping level in each layer corresponds to the in-layer carrier density $n_i=n_0={10}^{12}$ cm${}^{-2}$. The thick dot-dashed lines show the boundaries of inter- and intra-sub-band electron-hole continua. The thick dot-dashed lines show the boundaries of the electron-hole continuum. All quantities in this figure are dimensionless.

Figure 2

The damping function of multilayer plasmon modes at temperature $T=0.1T_F$ in GMLS consisting of $N=2,3,4$, and 5 graphene monolayers. The curves plotted here correspond to the plasmon branches presented in Fig. 1 with the same formatting. All system parameters are taken the same as in Fig. 1.

Figure 3

The plasmon dispersions in GMLS consisting of $N=4$ graphene monolayers for four different temperatures, $T=T_F$. All other parameters and notations are the same as in Fig. 1.

Figure 4

The broadening of the plasmon dispersions. The different curves here correspond to the multilayer plasmon branches, with the same formatting and the same parameters as in Fig. 3.

Figure 5

The plasmon dispersions at finite temperatures $T=0.1T_F$ in GMLS consisting of $N=2,3,4$, and 5 graphene monolayers (shown in bold parallel lines in each panel) for the interlayer separation $d=20$ nm. All other notations and parameters and notations are the same as in Fig. 1.

Figure 6

The plasmon dispersions in an unbalanced $N=3$-layer graphene structure. The different panels correspond to the in-layer carrier density $n_i=0,0.4,0.8,1\times {10}^{12}$ cm${}^{-2}$ of the middle layer graphene sheet. All other parameters and notations are the same as in Fig. 1.

Figure 7

The plasmon dispersions in an unbalanced $N=5$-layer graphene structure with two unroped layers. The different panels correspond to the different situations of $n_1=n_3=0$, $n_1=n_4=0$, $n_2=n_3=0$, and $n_2=n_4=0$. All other parameters and notations are the same as in Fig. 1.