Charged moiré phonons in twisted bilayer graphene

TL;DR

This work demonstrates that moiré phonons in twisted bilayer graphene can become infrared active upon doping, with several modes including the phason acquiring dipole moments through interband electron-phonon coupling. Using a continuum stacking-field model that includes relaxation, the authors show the phason carries a charge equal to the number of electrons per moiré cell added/removed, a result rooted in a sliding $Chern$-number. The optical response features phason-related resonances within the single-electron gap that can be described by a Drude-like conductivity with an effective mass set by the moiré geometry and relaxation, offering a tunable knob via adhesion and disorder. These findings provide a concrete path for THz spectroscopy to characterize EPC and disorder in moiré systems and generalize to other charged moiré materials at small twist angles.

Abstract

Moiré phonons describe collective vibrations of a moiré superlattice produced by long-wavelength relative displacements of the constituent layers. Despite coming from the backfolding of the acoustic phonons of the individual layers, many of these modes become infrared active when the system is doped. We illustrate this effect by a direct calculation of the optical absorption of twisted bilayer graphene (tBG) around different twist angles, including the magic angle. Several modes -- including the acoustic-like phason -- acquire a dipole moment via interband matrix elements of the electron-phonon coupling (EPC) when the flat band is filled or emptied, giving rise to new resonances in the optical conductivity within the single-electron gap that are strongly affected by relaxation. The phason in particular gains a charge that equals the amount of electrons per moiré cell added/removed to/from neutrality. Geometrically, this can be understood as the topological quantization of a sliding Chern number. The charged phason yields a Drude-like conductivity with an effective mass that increases with lattice relaxation. Our findings are testable via THz spectroscopy, and provide an experimental knob to characterize EPC strength and disorder in moiré materials at small twist angles.

Figures (7)

• Figure 1: Optical absorption of tBG at the magic angle in the absence of interlayer adhesion at two different fillings of the flat bands. Small green arrows highlight the moiré phonon peaks. The inset shows the electronic band structure, where color distinguishes valleys. The interband transitions giving rise to peaks at finite frequency in the electron-hole continuum are indicated by the blue and red arrows.
• Figure 2: (a) Optical absorption of tBG for the neutral (dashed lines) and for the doped system with four electrons per moiré cell (solid lines) at three different twist angles in the presence of interlayer adhesion forces. (b) Phason peaks in the optical absorption. For visualization, they are pinned to $0.01\omega_{m}$, where the characteristic phonon frequency imposed by folding scales as $\omega_{m}\propto\theta$SI. (c) Optical absorption at the magic angle in the range of frequencies of the lowest-energy infrared-active gapped modes.
• Figure 3: Diagrams for the calculation of the optical conductivity. Dashed lines represent phonons, solid lines represent electrons (with band/valley and momentum indices indicated in the figure), and curvy lines correspond to photons. (a) Electron-phonon (given in Eq. \ref{['eqn:g1']}) and electron-photon interaction vertices. (b) Optical conductivity. The first diagram represents the direct current-current response, Eq. \ref{['eqn:sigma(0)']}, while the second diagram involves mixed responses of current and stacking, Eq. \ref{['eqn:sigma(1)']}.
• Figure 4: Optical absorption of the phason mode with no pinning for the three twist angles of reference. The greater the relaxation (i.e. the smaller the twist angle), the more electronic interband transitions are needed in Eq. \ref{['eqn:sigma(1)']} in order to reproduce the analytical curve given by Eq. \ref{['eqn:sigmaph']}.
• Figure S1: Moiré phonon dispersion at different twist angles. The characteristic energy scale is $\omega_{m} = \sqrt{\frac{\mu}{3\varrho}}\frac{4\pi}{L_{m}}\propto\theta$, where $\mu = 9.57$ eV/Å$^{2}$ and $\varrho = 7.55 \cdot 10^{-27}$ kg/Å$^{2}$ are the graphene shear modulus and mass density, respectively. $L_{m}$ is the moiré periodicity. We indicate with green arrows the first three $E_{1}$-branches for $\theta=1.05^{\circ}$. For the other two angles the irreducible representations are ordered in the same way.