[Paper Review] Theory for the electronic structure of incommensurate twisted bilayer graphene

Doried Ghader, Dominik Szczęśniak+1 more|arXiv (Cornell University)|Jan 26, 2015

Graphene research and applications3 citations

TL;DR

This paper develops a tight-binding virtual crystal approximation (TB-VCA) theory to model the electronic structure of incommensurate twisted bilayer graphene (tBLG) for any twist angle θ, treating the disordered interface as an effective medium. The method yields quasi-Hermitian Hamiltonians that recover Hermitian physics via redefined inner products, revealing angle-dependent Van Hove singularities in the local density of states and confirming that the Fermi velocity remains unchanged from monolayer graphene, consistent with STM/STS experiments.

ABSTRACT

The experimental control over the twist angle in twisted bilayer graphene has not been reported and its realistic structure is most likely incommensurate. In this paper, we develop a tight-binding virtual crystal approximation theory to study the electronic properties in incommensurate twisted bilayer graphene. The theory yields the electronic band structure and the local density of states for any incommensurate twist angle θ between the graphene sheets. Angle dependent Van Hove singularities are observed in the numerically calculated local density of states. In accord with observations in scanning tunneling microscopy and spectroscopy, our theoretical calculation indicates that the rotation angle between graphene sheets does not result in a significant reduction in the Fermi velocity in comparison with monolayer graphene. The developed theory is quite general and can be applied to investigate the electronic properties in any incommensurate multilayer heterostructures.

Motivation & Objective

• To address the lack of a general theoretical framework for electronic structure in realistic incommensurate twisted bilayer graphene (tBLG), which is experimentally prevalent due to the vanishing probability of commensurate twist angles.
• To develop a systematic method capable of calculating electronic band structure and local density of states (LDOS) for any incoherent twist angle θ.
• To establish a theoretical foundation for incommensurate tBLG that captures disorder effects via an effective medium approximation.
• To validate the model against experimental observations, particularly the persistence of Dirac cone-like behavior and Fermi velocity in monolayer graphene.

Proposed method

• The TB-VCA method constructs an effective medium Hamiltonian by averaging over the quasi-infinite set of interlayer atomic configurations in incommensurate tBLG.
• The resulting Hamiltonian is quasi-Hermitian due to the incommensurate twist, but its physical equivalence to a Hermitian operator is restored by transposing the inner product in the Hilbert space.
• The theory uses a tight-binding model with onsite energies, in-plane hopping integrals, and interlayer hopping integrals derived from numerical averaging over atomic configurations.
• The effective Hamiltonian is solved to compute the electronic band structure and local density of states (LDOS) along high-symmetry paths in the first Brillouin zone.
• Interlayer hopping integrals are numerically averaged and reported in Table 1 for θ = 7°, 5°, 2°, and 1°, showing minimal variation with angle.
• The approach generalizes to multilayer and heterostructured incommensurate systems by extending the effective medium formalism.

Experimental results

Research questions

• RQ1 How can the electronic structure of incommensurate twisted bilayer graphene be systematically modeled despite its non-periodic, disordered nature?
• RQ2 What is the role of the twist angle θ in determining the position and strength of Van Hove singularities in the local density of states?
• RQ3 Does the Fermi velocity in incommensurate tBLG remain comparable to that of monolayer graphene, as suggested by recent STM/STS experiments?
• RQ4 Can a non-Hermitian but pseudo-Hermitian Hamiltonian formalism accurately describe the electronic properties of incommensurate tBLG systems?
• RQ5 To what extent can the virtual crystal approximation capture the essential physics of interlayer coupling in disordered tBLG?

Key findings

• The Fermi velocity in incommensurate tBLG is found to be identical to that of massless Dirac fermions in monolayer graphene, confirming experimental STM/STS observations.
• Angle-dependent Van Hove singularities appear in the local density of states (LDOS), with peak positions shifting to lower energies as the twist angle θ decreases.
• The numerical calculations show two overlapping Dirac cones centered at the K1 and K2 points of the respective layers, with saddle points emerging from band overlap.
• The averaged interlayer hopping integrals remain nearly constant across different twist angles: 0.2907 eV for A1(0)A2(0) and 0.1340–0.1348 eV for A1(0)A2(1), indicating weak angular dependence.
• The TB-VCA model successfully retrieves Hermitian physics via a pseudo-Hermitian representation, with the effective Hamiltonian being quasi-Hermitian but physically equivalent to a Hermitian operator through redefined inner products.
• The theory is generalizable to incommensurate multilayer and heterostructured graphene systems, offering a framework for studying electronic and optoelectronic properties in disordered 2D heterostructures.