Phonons in twisted bilayer graphene

We theoretically investigate phonon dispersion in AA-stacked, AB-stacked, and twisted bilayer graphene with various rotation angles. The calculations are performed using the Born–von Karman model for the intralayer atomic interactions and the Lennard-Jones potential for the interlayer interactions. It is found that the stacking order affects the out-of-plane acoustic phonon modes the most. The difference in the phonon densities of states in the twisted bilayer graphene and in AA- or AB-stacked bilayer graphene appears in the phonon frequency range 90–110 cm${}^{-1}$. Twisting bilayer graphene leads to the emergence of different phonon branches—termed hybrid folded phonons—which originate from the mixing of phonon modes from different high-symmetry directions in the Brillouin zone. The frequencies of the hybrid folded phonons depend strongly on the rotation angle and can be used for noncontact identification of the twist angles in graphene samples. The obtained results and the tabulated frequencies of phonons in twisted bilayer graphene are important for the interpretation of experimental Raman data and in determining the thermal conductivity of these material systems.

Figure 1

Schematic of the (a) lattice structure and (b) Brillouin zone in twisted bilayer graphene with a rotational angle $\theta =21.8^{\circ }$.

Figure 2

Four coordination spheres in single-layer graphene used in the theoretical description.

Figure 3

Phonon energy dispersions in (a) single-layer graphene, (b) AA-stacked bilayer graphene, and in the twisted bilayer graphene with (c) $\theta =21.8^{\circ }$ and (d) $\theta =13.2^{\circ }$. All dispersion relations are shown for the Γ-$K$ direction in the Brillouin zone of SLG, BLG, and T-BLG, correspondingly.

Figure 4

Phonon energy dispersions (a) near the Γ point and (b) near the $K$ point of the Brillouin zone in AA-stacked (red dashed curves) and AB-stacked (black solid curves) bilayer graphene.

Figure 5

Phonon energy dispersions in AA-stacked bilayer graphene shown for the (a) Γ${}_0$-$K_0$ and (b) Γ${}_1$-$K_1$ directions of the Brillouin zone of AA-BLG. (c) Brillouin zones in AA-BLG and T-BLG with $\theta =21.8^{\circ }$. Note that seven high-symmetry directions Γ${}_i$-K${}_i$ ($i=0,...,6$) in the Brillouin zone of AA-BLG are equivalent to the high-symmetry direction Γ${}_0$$-K_0$ in the Brillouin zone of T-BLG.

Figure 6

Zone-center phonon dispersions of the (a) out-of-plane and (b) in-plane acoustic modes in AA-BLG (black curves), T-BLG with $\theta =21.8^{\circ }$ (red curves), and T-BLG with $\theta =13.2^{\circ }$ (blue curves). Note how a change in the interlayer interaction due to twisting affects the TA${}_2$, LA${}_2$, and ZA${}_2$ modes. The region where anticrossing of LA${}_1$ and TA${}_2$ hybrid folded phonon branches occurs is shown by a dashed circle.

Figure 7

Two-dimensional phonon density of states as a function of the phonon frequency in AA-BLG (black curves), T-BLG with $\theta =21.8^{\circ }$ (red curves), and T-BLG with $\theta =13.2^{\circ }$ (blue curves). Phonon DOS are presented (a) for an extended frequency range and (b) for the frequency interval of 90–110 cm${}^{-1}$ where the maximum difference in the phonon DOS is observed.

Figure 8

Phonon average group velocity as a function of the phonon frequency in AA-BLG (black curves), T-BLG with $\theta =21.8^{\circ }$ (red curves), and T-BLG with $\theta =13.2^{\circ }$ (blue curves).