Reducible Fermi surface for multi-layer quantum graphs including stacked graphene

TL;DR

This paper constructs multi-layer quantum graphs, including stacked graphene, showing how their Fermi surfaces become reducible into multiple components, with implications for band structure and Dirac cone behavior.

Contribution

It introduces a method to analyze multi-layer quantum graphs, demonstrating the reducibility of their Fermi surfaces and the effects of stacking configurations on Dirac cones.

Findings

Fermi surface becomes reducible into several components.

Dirac cones break in multi-layer graphene except for AA-stacking.

A surgery-type calculus for dispersion functions is developed.

Abstract

We construct two types of multi-layer quantum graphs (Schr\"odinger operators on metric graphs) for which the dispersion function of wave vector and energy is proved to be a polynomial in the dispersion function of the single layer. This leads to the reducibility of the algebraic Fermi surface, at any energy, into several components. Each component contributes a set of bands to the spectrum of the graph operator. When the layers are graphene, AA-, AB-, and ABC-stacking are allowed within the same multi-layer structure. Conical singularities (Dirac cones) characteristic of single-layer graphene break when multiple layers are coupled, except for special AA-stacking. One of the tools we introduce is a surgery-type calculus for obtaining the dispersion function for a periodic quantum graph by gluing two graphs together.