Effect of lattice relaxation on electronic spectra of helically twisted trilayer graphene: Large-scale atomistic simulation approach

Twisted trilayer graphene hosts two moir\'e superlattices originating from two interfaces between graphene layers. However, the system is generally unstable to lattice relaxation at small twist angles and is expected to show a significantly modified electronic band structure. In particular, a helical trilayer graphene - whose two twisted angles have the same sign - provides an attractive platform with a flat band isolated by large energy gaps near the magic angle, but the interplay between the lattice and the electronic degrees of freedom is not well understood. Here, we performed a large-scale molecular dynamics simulation to study the lattice relaxation of helical trilayer graphenes and evaluated their electronic spectra with a tight-binding model calculation. The comparison of the electronic spectra both with and without the lattice relaxation reveals how the lattice relaxation significantly modifies the electronic spectra particularly near the charge neutrality point. We also investigated the local density of states to visualize the spatially-varying electronic spectra that accords with domain patterns of moir\'e lattice stackings. We propose these characteristic spectral features in the electronic degrees of freedom of a relaxed helical trilayer graphene to be confirmed by scanning probe techniques, such as scanning single-electron transistors and scanning tunneling microscopes.