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

TL;DR

This work addresses the problem of obtaining quantum dot energy levels in bilayer graphene from a computationally efficient yet accurate model. Starting from the four-component continuum Hamiltonian for BLG in a perpendicular magnetic field, the authors derive an approximate single-component equation for the QD states by eliminating select components and introducing a reduced wavefunction, leading to a Schrödinger-like equation with a radial potential $Q_+(r)$. They show that energies obtained from this approximate framework, including a harmonic-oscillator-like quantization around the maximum of $Q_+(r)$, agree remarkably well with exact four-component results across a broad range of parameters, including soft-wall and hard-wall wells and bias configurations. The approach captures key features such as anticrossings, linear dependence of energy on the dot depth $V_W$, and the emergence of QD states from Landau levels, while offering significant pedagogical and computational advantages. The work lays groundwork for efficient analytical and semiclassical explorations of QD physics in BLG and points to future extensions using WKB or SUSY methods to handle more complex regimes.

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 Schrödinger 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.

Figures (9)

• Figure 1: The function $Q_{+}$, defined in Eq. (\ref{['qfull']}), as a function of radial coordinate at different magnetic fields and potential wells for $s=2$, $V_{\rm b}=0$, and: (a) $m=2$, $E\approx 1$ meV, (b) $m=2$, $E\approx - 20$ meV, (c) $m=2$, $E\approx -25$ meV, and (d) $m=8$, $E\approx 15$ meV.
• Figure 2: Exact and approximate QD energy levels as a function of QD potential well for $s=2$, and $V_{\rm b}=0$. From lower to upper curve: $m=2$, 4, 7, 10, 13, 16, 20. Approximate QD levels are derived from Eq. (\ref{['oscil']}) for $n=0$.
• Figure 3: Exact and approximate QD energy levels as a function of QD potential well for $L_W=100$ nm, $B=4$ T, $s=2$, and $V_{\rm b}=0$. From lower to upper curve: $m=-4$, $-16$, $-25$, $-35$, $-45$. Approximate QD levels are derived from Eq. (\ref{['oscil']}) for different $n$ values.
• Figure 4: Exact and approximate QD energy levels as a function of QD potential width for $V_W\approx 48$ meV, $m=8$, $B=2$ T and different superscripts $s$ in Eq. (\ref{['potential']}). Approximate QD levels are derived from Eq. (\ref{['oscil']}) for $n=0$. For clarity the curves for $s=1$ and $s=2$ are shifted by 20 nm and 10 nm respectively along the horizontal axis.
• Figure 5: Exact and approximate QD energy levels as a function of QD potential well at different magnetic fields $B$ and angular momenta $m$ for $s=2$ and $V_{\rm b}=0$. Approximate QD levels are derived from the numerical solution of Eq. (\ref{['approx1']}).