Moir\'e phonons and impact of electronic symmetry breaking in twisted trilayer graphene

TL;DR

This paper investigates phonon modes in twisted trilayer graphene, revealing how electronic symmetry breaking and lattice relaxation influence phonon gaps and modes, with implications for understanding superconductivity and resistivity.

Contribution

It introduces the classification of shear phonon modes in twisted trilayer graphene, highlighting the effects of mirror symmetry breaking and lattice relaxation on phonon properties.

Findings

Mirror-even shear modes are gapless and analogous to twisted bilayer graphene phasons.

Mirror-odd shear modes have a finite gap proportional to lattice relaxation.

Symmetry breaking can induce finite angular momentum in phonon branches.

Abstract

Twisted trilayer graphene is a particularly promising moir\'e superlattice system, due to its tunability, strong superconductivity, and complex electronic symmetry breaking. Motivated by these properties, we study lattice relaxation and the long-wavelength phonon modes of this system. We show that mirror-symmetric trilayer graphene hosts, aside from the conventional acoustic phonon modes, two classes of shear modes, which are even and odd under mirror reflection. The mirror-even modes are found to be gapless and equivalent to the "phason" modes of twisted bilayer graphene, with appropriately rescaled parameters. The modes odd under mirror symmetry have no analogue in twisted bilayer graphene and exhibit a finite gap, which we show is directly proportional to the degree of lattice relaxation. We also discuss the impact of mirror-symmetry breaking, which can be tuned by a displacement field or result from a stacking shift, and of rotational- as well as time-reversal-symmetry breaking, resulting from spontaneous electronic order. We demonstrate that this can induce finite angular momentum to the phonon branches. Our findings are important to the interpretation of recent experiments, concerning the origin of superconductivity and of linear-in-$T$ resistivity.