Gapped out-of-phase plasmon modes in alternating-twist multilayer graphene

TL;DR

This work addresses plasmon modes in moiré-engineered alternating-twist multilayer graphene (ATMG) using the random-phase approximation in a Coulomb-eigenvector basis derived from Kac-Murdock-Szegő Toeplitz matrices. The in-phase plasmon exhibits the conventional $\omega \propto \sqrt{q}$ dispersion, while out-of-phase modes acquire gaps governed by interband transitions between Dirac cones of different velocities; in the weak Coulomb limit these gaps persist as undamped modes above a critical twist angle (approximately $2.75^\circ$ for AT3G) and remain largely density-independent, with the gaps tunable by a perpendicular electric field. The authors develop analytic expressions within a first-shell continuum model and validate them against full numerical calculations for $N=3$ and $N=4$, showing gate-tunable plasmon gaps via Dirac-cone shifts under an interlayer bias $U$. Overall, the study reveals a robust platform for engineering undamped, gate-tunable out-of-phase plasmons in moiré graphene through twist-angle control, interlayer tunneling, and external fields, supported by explicit Coulomb-eigenvector formalism and RPA analyses.

Abstract

We theoretically investigate the plasmon modes of alternating-twist multilayer graphene. In multilayer systems, interlayer coupling gives rise to distinctive plasmon modes, but calculations in moiré systems remain challenging due to their complex tunneling structures. Using the Kac-Murdock-Szegő Toeplitz formalism, we derive that the in-phase mode exhibits the conventional $\sqrt{q}$ behavior, while the out-of-phase modes acquire plasmon gaps determined by specific interband transitions between Dirac cones with different velocities in the long-wavelength limit. We demonstrate that these out-of-phase modes remain undamped in the weak Coulomb-interaction limit when the twist angle exceeds a critical value ($θ\gtrsim 2.75^\circ$ for the alternating-twist trilayer case), regardless of the carrier density as long as the low-energy effective Dirac Hamiltonian remains valid. Furthermore, we consider the effect of a perpendicular electric field, and demonstrate how plasmon modes can be tuned by a gate voltage.

Figures (8)

• Figure 1: Schematic illustrations of (a) the ATMG with $N=3$ layers and (b) a front view of the ATMG embedded in a dielectric environment with dielectric constants $\epsilon_{\rm in}$ (inside) and $\epsilon_{\rm out}$ (outside). Here, $z$ represents the out-of-plane direction.
• Figure 2: Schematic illustrations of the band structures and associated particle-hole continua of ATMG. (a) Linear band dispersions for two Dirac cones with velocities $v_1$ and $v_2$. Dashed black lines denote the Fermi energy $\varepsilon_{\rm{F}}$ and gray arrows represent the Fermi wave vector $k_{\text{F}, r}$. Black arrows depict representative interband transitions at $q = 0$ ($\omega_{\rm{A}}$-$\omega_{{\rm{C}}}$), given by $(v_1 + v_2)k_{\rm{F}, 2}$, $v_{21} k_{\rm{F}, 1}$, and $v_{21} k_{\rm{F}, 2}$, respectively. (b) The resulting particle-hole continuum in the $(q, \omega)$ plane for (a). The orange region indicates transitions between different Dirac cones, whereas the sky-blue regions denote transitions within the same Dirac cone. The boundaries are linear in $v_r q$, with slopes corresponding to the velocities indicated by the colors in (a). (c) Linear band dispersions for three Dirac cones with velocities $v_1$, $v_2$, and $v_3$. Black arrows (labeled $\omega_{\rm{A}}$-$\omega_{\rm{D}}$) depict interband transitions given by $(v_1 + v_3)k_{\rm{F}, 3}$, $v_{31} k_{\rm{F}, 1}$, $v_{31} k_{\rm{F}, 3}$, and $v_{21}k_{\rm{F}, 1}$, respectively. (d) The resulting particle-hole continuum for (c), following the same color scheme as in (b).
• Figure 3: Loss function $L(\bm{q}, \omega)$ of unbiased ATMG systems at a twist angle $\theta = 5^{\circ}$ with $\varepsilon_{\text{F}} = 0.2$ eV for (a) $N=3$ ($n_{\rm{tot}} = 1.3 \times 10^{13}$ cm$^{-2}$) and (b) $N=4$ ($n_{\rm{tot}} = 1.8 \times 10^{13}$ cm$^{-2}$). The insets show the band structures of ATMG, where the colored lines highlight the low-energy effective Dirac cones near $\bar{K}$ and $\bar{K}'$. Coulomb eigenvectors of the in-phase and out-of-phase modes are illustrated. The red dashed lines indicate the analytically calculated in-plane plasmon dispersions, and the red stars mark the approximately obtained out-of-phase plasmon gaps using Eq. (15) for $N=3$ and Eq. (16) for $N=4$. The dotted gray lines represent the boundaries of the particle-hole continuum.
• Figure 4: Loss function $L(\bm{q}, \omega)$ of a biased AT3G system at a twist angle $\theta = 5^{\circ}$ under a perpendicular eletric field $U = 0.1$ eV with $\varepsilon_{\text{F}} = 0.2$ eV ($n_{\rm{tot}} =1.5 \times 10^{13}$ cm$^{-2}$). The inset shows the band structure of a biased AT3G, where the colored lines highlight the perturbative Hamiltonian with energy splitting $\Delta_\pm$ near $\bar{K}$ and $\bar{K}'$.
• Figure 5: Loss function $L(\bm{q}, \omega)$ of an unbiased AT5G system at a twist angle $\theta = 7^{\circ}$ with $\varepsilon_{\text{F}} = 0.3$ eV ($n_{\rm{tot}} = 4.1 \times 10^{13}$ cm$^{-2}$).