On the Entire Structure of the Energy Bands of 1D Moir\'{e} Superchain

TL;DR

This paper develops a comprehensive model for the energy band structure of one-dimensional moiré superchains, revealing how the entire spectrum and flat bands can be understood through a single recurrence relation and WKB analysis.

Contribution

It introduces a unified framework using a three-term recurrence relation to analyze the full energy spectrum of 1D moiré superchains with arbitrary dispersions and couplings.

Findings

Spectrum governed by a single three-term recurrence relation

Identification of flat bands via potential functions

Analysis of the chiral limit as a 1D analog of twisted bilayer graphene

Abstract

We consider a general model of two atomic chains forming a moir\'{e} pattern due to a small mismatch in their lattice spacings, given by $\theta = (a_{1} - a_{2})/a_{2}$. Assuming arbitrary single-band dispersion relations $\varepsilon_{1}(p)$ and $\varepsilon_{2}(q)$ for the chains, along with an arbitrary inter-chain coupling term $T(x)$, we show that the entire spectrum of such a one-dimensional moir\'{e} superchain is governed by a single three-term recurrence (TTR) relation. We analyze this TTR relation using the discrete WKB method and demonstrate how the entire structure of the spectrum as well as emergence of flat bands can be easily identified from a pair of upper and lower potential functions of the TTR relation. We also comment on the chiral limit of the moir\'{e} superchain, which can be viewed, in some sense, as a 1D analog of the chiral limit of Twisted Bilayer Graphene.