Quantum dot energy levels in bilayer graphene: Exact and approximate study

TL;DR

This paper compares exact and approximate methods for calculating quantum dot energy levels in bilayer graphene, demonstrating that simplified equations can accurately reproduce the exact results with minimal computational effort.

Contribution

It introduces a semi-analytical approximation for quantum dot energy levels in bilayer graphene and validates its accuracy against exact solutions.

Findings

Approximate equations closely match exact energy levels across various parameters.

The semi-analytical approach simplifies calculations with minimal loss of accuracy.

Realistic regimes identified where the approximation's error is negligible.

Abstract

In bilayer graphene the exact energy levels of quantum dots can be derived from the four-component continuum Hamiltonian. Here, we study the quantum dot energy levels with approximate equations and compare them with the exact levels. The starting point of our approach is the four-component continuum model and the quantum dot is defined by a continuous potential well in a uniform magnetic field. Using some simple arguments we demonstrate realistic regimes where approximate quantum dot equations can be derived. Interestingly these approximate equations can be solved semi-analytically, in the same context as a single-component Schr\"odinger equation. The approximate equations provide valuable insight into the physics with minimal numerical effort compared with the four-component quantum dot model. We show that the approximate quantum dot energy levels agree very well with the exact levels in a broad range of parameters and find realistic regimes where the relative error is vanishingly small.