Construction and Accuracy of Electronic Continuum Models of Incommensurate Bilayer 2D Materials

Abstract

Single-particle continuum models such as the popular Bistritzer-MacDonald model have become powerful tools for predicting electronic phenomena of incommensurate 2D materials and the development of many-body models aimed to model unconventional superconductivity and correlated insulators. In this work, we introduce a procedure to construct continuum models of arbitrary accuracy relative to tight-binding models for moiré incommensurate bilayers. This is done by recognizing the continuum model as arising from Taylor expansions of a high accuracy momentum space approximation of the tight-binding model. We apply our procedure in full detail to two models of twisted bilayer graphene and demonstrate both admit the Bistritzer-MacDonald model as the leading order continuum model, while higher order expansions reveal qualitative spectral differences.

Figures (7)

• Figure 1.1: Left: atomic structure of stacked bilayer graphene with interlayer twist $5^\circ$, viewed from above. Atoms in the lower graphene sheet are colored orange and in the upper graphene sheet in blue. Moiré lattice vectors which generate the moiré lattice are shown in black. The continuum models we construct in the present work have coefficients which are periodic with respect to this lattice. Right: band structure of the Bistritzer-MacDonald continuum model Bistritzer2011 at the magic twist angle $1.1^\circ$, with parameter values taken from the ab initio tight-binding model proposed in Fang2016. For more details, see Figure \ref{['fig:bandstruct']}; this figure is the same as the middle right figure there. Many-body models of twisted bilayer graphene typically model electrons in the flat bands, the two bands nearest to zero energy which are approximately flat over most of the moiré Brillouin zone, interacting via the Coulomb force. Two of the key inputs to such models are the flat band eigenvalues and associated eigenfunctions. The goal of the present work is to construct moiré-scale single-particle continuum models of twisted bilayer graphene's electronic properties with essentially arbitrary accuracy relative to tight-binding models, in order to provide more accurate flat band eigenvalues and eigenfunctions for many-body models.
• Figure 2.2: The left picture shows the rotated monolayer graphene band structure, where the red regions correspond to ${B_{\Sigma+\eta}}$. The right picture shows the reciprocal unit cell of the rotated monolayer graphene, where the red regions are $\Gamma_1^*(B_{\Sigma+\eta})$, and by adding the blue regions around them we obtain $\Gamma_1^*(B_{\Sigma+\eta})+B_r(0)$. Here $K_1$ represents ${\rm mod}_1(K_1)$.
• Figure 3.3: Procedure for the construction of the continuum model.
• Figure 4.4: Magnitude of interlayer coupling in momentum space for the simplified TBG, AA and AB parts of the Wannierized TBG, respectively. The balls represent all $K_1+G_1$, where the black balls are hoppings corresponding to $\mathcal{B}_1$.
• Figure 4.5: Relative error ${\rm Err}(\Lambda,\Sigma)$ for the Wannierized TBG at $\theta=1.1^\circ$.

Theorems & Definitions (11)

• Definition 2.1
• Theorem 2.1
• Definition 2.2
• Theorem 2.2
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Example 3.1
• Theorem 3.1