Electronic Raman Scattering in Twistronic Few-Layer Graphene

We study electronic contribution to the Raman scattering signals of two-, three- and four-layer graphene with layers at one of the interfaces twisted by a small angle with respect to each other. We find that the Raman spectra of these systems feature two peaks produced by van Hove singularities in moiré minibands of twistronic graphene, one related to direct hybridization of the Dirac states, and the other resulting from band folding caused by moiré superlattice. The positions of both peaks strongly depend on the twist angle, so that their detection can be used for noninvasive characterization of the twist, even in hBN-encapsulated structures.

Figure 1

(a) Sketch of ($1+N$) twistronic graphene. The red and blue balls correspond to the two different sublattice sites in each graphene layer and green ellipse represents a dimer bonding leading to direct interlayer coupling. (b) Brillouin zones of the $N$-layer stack (dashed black line) and the top graphene (dashed purple line) with corners $\mathit{K}$ (bottom layer) and ${\mathit{K}}^{\prime }$ (top), respectively, as well as the effective moiré Brillouin zone, shown both centred around the valleys (solid black line) as the preferred choice in this Letter, and with valleys in its corners (dashed gray), centered at $\mathit{\gamma }$. (c) Feynman diagrams for the scattering amplitudes $\mathcal{R}$ contributing to the electronic Raman features discussed in this Letter.

Figure 2

Left: Electronic miniband structures of twisted bilayer graphene for $\theta =1.8°$ and $\theta =1.1°$ across the corresponding mBZ (solid black hexagons). The dashed lines indicate the positions of the valleys $\mathit{K}$ (bottom layer) and ${\mathit{K}}^{\prime }$ (top) within the mBZ. The pink arrows identify the minibands and the black arrows point to saddle points which give rise to van Hove singularities (vHs). Right: ERS intensity map for $0.8°<\theta <2°$. The green solid lines show cuts for the angles $\theta =1.8°$ and $\theta =1.1°$, indicated with the green dashed lines. The gray and orange peaks mark spectral features corresponding to excitations indicated with vertical arrows of the same color in the miniband dispersions which involve dispersion saddle points. The hexagonal insets next to band structures show in yellow mBZ regions which contribute to the $1^{-}\to 1^{+}$ ERS peak for $\theta =1.8°$ and $2^{-}\to 2^{+}$ peak for $\theta =1.1°$; red dots indicate positions of dispersion saddle points on the valence side. The additional two insets for $\theta =1.1°$ depict real space distribution of the saddle points wave function in the top or bottom layers; black solid lines mark boundaries of the moiré supercell and the letters indicate local stacking.

Figure 3

Top: ERS intensity map for ($1+2$) graphene. The green dashed lines indicate twist angles $\theta =1.8°$ and $\theta =1.1°$ and the green solid lines show the ERS spectra for these angles. Bottom: Miniband structure for the ($1+2$) stack for $\theta =1.1°$. The top inset shows in yellow the mBZ regions which contribute to the $2^{-}\to 2^{+}$ ERS peak; red dots indicate positions of dispersion saddle points on the valence side. The saddle points wave function is predominantly located in the bottom layer and its real space distribution is presented in the bottom inset; black solid lines mark boundaries of the moiré supercell and the letters indicate local stacking (from the bottom layer to the top).

Figure 4

Comparison of ERS intensity maps for (a) $(1+3B)$, (b) $(1+3R)$, (c) $(2AB+2BA)$, and (d) $(2AB+2AB)$ tetralayers. Green solid lines show the Raman spectra for twist angles $\theta =1.8°$ and $\theta =1.1°$.