Optical response of alternating twisted trilayer graphene

TL;DR

This work analyzes the optical response of alternating-twist trilayer graphene using a unitary transformation that maps the trilayer Hamiltonian to a direct sum of an effective twisted bilayer and a single-layer system, enabling a Kubo-formalism treatment of layer-resolved conductivities. It shows that, under mirror symmetry, the in-plane magnetic response is exceedingly small and optical activity is absent, while a local magnetoelectric coupling arising from vertical gradients of the magnetic field and moment persists. The layer-resolved conductivity is expressed in terms of the known bilayer and monolayer responses plus a coupling term $oldsymbol{}_c$ that encodes bilayer–monolayer mixing, with explicit expressions provided for these quantities. A microscopic estimate using a localization ansatz for nearly flat bands demonstrates that the coupling-induced in-plane magnetic response is suppressed by the large moiré length, aligning with the observed absence of in-plane orbital magnetism; a local chiral coupling remains possible but does not produce bulk optical activity due to mirror symmetry. The framework connects the trilayer optical response to the microscopic parameters of the constituent subsystems and lays groundwork for extensions to symmetry-broken trilayers and tetralayer structures.

Abstract

We study the optical response of the alternating twisted trilayer graphene by making use of a unitary transformation for the trilayer Hamiltonian and the Kubo formulation of linear response theory. The layer-resolved optical conductivities are expressed in terms of contributions from effective twisted bilayer and single-layer systems along with their coupling. We show that the in-plane magnetic response is proportional to this coupling between the twisted bilayer and single-layer systems; and, due to the different energy scales, the in-plane magnetic response is negligibly small. We also formulate a local electro-magnetic response that involves the vertical gradients of the magnetic field and moment.

Figures (2)

• Figure 1: Schematic of the alternating-twist trilayer configuration. Three infinite flat graphene sheets labeled by $\ell=1,\,2,\,3$ are parallel to the $xy$-plane, and have twist angles $(-1)^{\ell}\theta/2$ and interlayer distance equal to $a/2$ ($0<\theta< \pi/2$). The layers are immersed in a homogeneous medium.
• Figure 2: Schematic view of the currents when only the even mode $\boldsymbol {\mathcal{\bm J}}_+^-$ is present; see Eqs. \ref{['eq:matrix-M']} and \ref{['eq:transformed-currents']}. The solid arrows define the in-plane current densities ${\bf J}_1$, ${\bf J}_2$ and ${\bf J}_3$ with ${\bf J}_1={\bf J}_3=-{\bf J}_2/2$. These currents give rise to the magnetic moments ${\bf m}_1$ and ${\bf m}_2$ with ${\bf m}_1=-{\bf m}_2$.