Twisted bilayer graphene as a terahertz plasmonic crystal

TL;DR

This work shows that minimally twisted bilayer graphene forms a plasmonic crystal built from 1D domain-wall plasmons along AB-BA walls, coupled through a moiré network. By developing both phase-coherent RPA and phase-incoherent plasmon network models, the authors predict a rich plasmon band structure featuring multiple gapless branches, flat bands, and dissipationless modes at high-symmetry points in the moiré Brillouin zone, with frequencies in the THz range. The study demonstrates quantitative agreement in gross features between RPA and PNM while highlighting stronger damping in the network picture and provides near-field imaging simulations that link theory to potential experiments. The results establish mTBG domain-wall networks as natural, fabricable THz plasmonic crystals with potential applications in nanophotonics and plasmonic devices.

Abstract

We study surface plasmons in minimally-twisted gapped bilayer graphene that contains a triangular network of partial dislocations (or AB-BA domain walls) hosting topologically protected one-dimensional electronic states. We show that this system behaves as a plasmonic crystal and we calculate its band structure by solving classical equations of motion for charge dynamics on the network links with impedance boundary conditions at the network nodes. The plasmon dispersion exhibits several notable features such as multiple gapless branches, flat bands, and dissipationless modes at high-symmetry points. We compare our network-based formalism with the conventional random phase approximation and discuss when each approach is valid. Calculations of plasmon waves launched by local scatterers are presented to simulate terahertz nano-imaging experiments.

Figures (12)

• Figure 1: A schematic of the mTBG structure. The AB and BA stacking domains are separated by domain walls (links), which intersect at AA regions (nodes). When an interlayer bias is applied, the AB and BA regions are gapped while the domain walls host 1D electron states. The wiggly lines represent 1D plasmons that propagate along the links and scatter at the nodes. The red arrows indicate the primitive vectors $\mathbf l_1$, $\mathbf l_2$, $\mathbf l_3$ of the moiré superlattice.
• Figure 2: Effective circuit for a junction of six leads (the thick solid lines). The numbers indicate two different schemes of labeling the leads. For clarity, only the connections originating from lead $1$ ($1-$) are shown, e.g., admittance connecting it to the ground is $y_0$, the ones wiring it to the nearest leads are $y_1$, etc. The remaining leads are wired the same way.
• Figure 3: (a) Band structure along high-symmetry lines in the the moiré Brillouin zone (left) and corresponding Drude weight [Eq. \ref{['eqn:drude']}] vs. Fermi energy (right) for $P_f = 1$ (top), $P_f = 0.8$ (middle), and $P_f = 0.4$ (bottom). Only the three bands of the $\mathrm{K}$ valley are shown. The vertical dashed lines indicate the average Drude weight $\overline D$. The horizontal dashed line in the bottom left panel denotes the Fermi energy $\varepsilon_F$ used in the calculation of Figs. \ref{['fig:bands']} and \ref{['fig:bands2']}. Inset at top right: the moiré Brillouin zone. Arrows indicate the reciprocal lattice vectors defined in Eq. \ref{['eqn:G_vectors']}. (b) Average Drude weight $\overline D$ as a function of $P_f$. The solid line is Eq. \ref{['eqn:average_Drude']} evaluated numerically, and the dashed line is Eq. \ref{['eqn:D_avg']}. Inset: diagram showing allowed single-particle scattering directions.
• Figure 4: RPA and PNM spectra [Eq. \ref{['eqn:spectral_function']}] along high-symmetry lines for a short-range interaction. (a) RPA spectrum [from Eqs. \ref{['eqn:polarization_as_sum']}--\ref{['eqn:Psi2']}, \ref{['eqn:epsilon_RPA']}] for $P_f = 0.4$ and $K = 0.25$. The Fermi energy $\varepsilon_F$ is indicated in Fig. \ref{['fig:drude_weight']}(a). (b) Local PNM spectrum [from Eqs. \ref{['eqn:S']}, \ref{['eqn:T']}--\ref{['eqn:B']}] for the same parameters. The symbols label the frequencies $\omega_\triangle$ and $\omega_\Diamond$ defined in Eqs. \ref{['eqn:flat_band1']} and \ref{['eqn:flat_band2']}. (c) Enlarged views of the long-wavelength, low-frequency parts of the plots above. Note that at the $\mathrm{K}$ point $q L = 4\pi / 3$.
• Figure 5: RPA and PNM spectra along high-symmetry lines for an unscreened Coulomb interaction. Parameters: $\alpha N = 10$, $\ell = 0.01L$, $P_f$ and $\varepsilon_F$ are the same as in Fig. \ref{['fig:bands']}. (a) RPA spectrum [from Eqs. \ref{['eqn:polarization_as_sum']}--\ref{['eqn:Psi2']}, \ref{['eqn:epsilon_RPA']}]. (b) Non-local PNM spectrum [Eqs. \ref{['eqn:polarization_as_sum']}, \ref{['eqn:epsilon_RPA']}, \ref{['eqn:polarization3']}--\ref{['eqn:lambda_s_a']}]. (c) Enlarged views of the long-wavelength, low-frequency parts of the plots above.