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

In bilayer graphene, the exact energy levels of quantum dots (QDs) can be derived from the four-component continuum Hamiltonian. Here, we study the QD 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 QD is defined by a continuous potential well in a uniform magnetic field. Using some simple arguments, we demonstrate realistic regimes where approximate QD equations can be derived. Interestingly, these approximate equations can be solved semianalytically, 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 QD model. We show that the approximate QD 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.