Energetic stability and spatial inhomogeneity in the local electronic structure of relaxed twisted trilayer graphene

TL;DR

This study investigates the energetic stability and local electronic inhomogeneity in relaxed twisted trilayer graphene, revealing how slight variations in twist angles and structural relaxation influence the density of states near the Fermi level.

Contribution

It provides a detailed analysis of the structural relaxation effects and stacking-dependent electronic properties in twisted trilayer graphene with small twist angles.

Findings

Commensurate TTG with equal twist angles has a local energy minimum.

DOS near the Fermi level varies significantly with stacking configurations.

Structural relaxation critically affects the high DOS and its spatial inhomogeneity.

Abstract

We study the energetic stability and the local electronic structure of the general twisted trilayer graphene (TTG) with the top and bottom layers rotated with respect to the middle layer respectively by $\theta$ and $\theta'$. Approximate supercells of the moir\'{e}-of-moir\'{e} superlattices with $\theta$ and $\theta^{\prime}$ within $1^{\circ}\sim 2^{\circ}$ are established to describe the structural and electronic properties of relaxed TTG with the periodic boundary condition. Full relaxation demonstrates that the commensurate TTG with $\theta=\theta^{\prime}$ has the local minimum total energy ($E_{tol}$) at a fixed $\theta$, while $E_{tol}$ first reaches a local maximum and begins to drop with decreasing $\theta^{\prime}$ for $\theta^{\prime} < \theta$. Some regions exhibit enhanced in-plane relaxation in the top and bottom layers but suppressed relaxation in the middle layer and form a hexagonal network with the moir\'{e}-of-moir\'{e} length scale. The stacking configurations with the atoms in the three layers vertically aligned at the origin of the relaxed TTG supercells at $\theta$ around $1.6^{\circ}$ and $\theta^{\prime}$ around $1.4^{\circ}$ have a high density of states (DOS) near the Fermi level ($E_F$), which can reach that of the mirror symmetric TTG with equal twist angles of about $1.7^{\circ}$. In contrast, some other stackings can have rather low DOS around $E_F$. The significant stacking dependence of DOS for some TTG supercells demonstrates that the local electronic structure of TTG can exhibit strong spatial inhomogeneity when the twist angles are slightly away from those of the small supercells with large variations of DOS among different stackings. Moreover, the structural relaxation of TTG plays a crucial role in the high DOS and its strong stacking dependence.