Confinement in bilayer graphene via intra- and inter-layer interactions

TL;DR

This paper explores how Dirac fermions in bilayer graphene can be confined using inhomogeneous interactions and magnetic fields, providing analytical solutions for localized states in various scenarios.

Contribution

It introduces a method to find localized states in bilayer graphene through an effective Schrödinger equation with energy-dependent potential, considering multiple inhomogeneous interaction scenarios.

Findings

Localized states can be achieved via inhomogeneous interactions and magnetic fields.

Analytical solutions are provided for states localized by interaction fluctuations or periodic defects.

The framework simplifies the problem to two Dirac-type Hamiltonians, enabling explicit analysis.

Abstract

We consider confinement of Dirac fermions in $AB$-stacked bilayer graphene by inhomogeneous on-site interactions, (pseudo-)magnetic field or inter-layer interaction. Working within the framework of four-band approximation, we focus on the systems where the stationary equation is reducible into two stationary equations with $2\times2$ Dirac-type Hamiltonians and auxiliary interactions. We show that it is possible to find localized states by solving an effective Schr\"odinger equation with energy-dependent potential. We consider several scenarios where bilayer graphene is subject to inhomogneous (pseudo-)magnetic field, on-site interactions or inter-layer coupling. In explicit examples, we provide analytical solutions for the states localized by local fluctuations or periodicity defects of the interactions.