Ising quantum Hall ferromagnetism in Landau levels $\left\vert N\right\vert \geq 1$ of bilayer graphene

TL;DR

This paper investigates the phase diagram of bilayer graphene's Landau levels with N≥1, revealing valley and spin ferromagnetic states and phase transitions induced by electric fields, using Hartree-Fock approximation.

Contribution

It provides the first detailed Hartree-Fock analysis of Ising quantum Hall ferromagnetism in higher Landau levels of bilayer graphene, highlighting phase transitions at integer fillings.

Findings

Identification of valley and spin polarized phases at odd and even fillings.

Discovery of first-order phase transitions induced by electric bias.

Discontinuous changes in transport properties at transition points.

Abstract

A magnetic field applied perpendicularly to the chiral two-dimensional electron gas (C2DEG)\ in a Bernal-stacked bilayer graphene quantizes the kinetic energy into a discrete set of Landau levels $N=0,\pm 1,\pm 2,...$ While Landau level $N=0$ is eighfold degenerate, higher Landau levels ($ \left\vert N\right\vert \geq 1$) are fourfold degenerate when counting spin and valley degrees of freedom. In this work, the Hartree-Fock approximation is used to study the phase diagram of the C2DEG at integer fillings $ \widetilde{\nu }=1,2,3$ of these higher Landau levels. At these filling factors, the C2DEG is a valley or spin Ising quantum Hall ferromagnet. At odd fillings, the C2DEG is spin polarized and has all its electrons in one valley or the other. There is no intervalley coherence in contrast with most of the the ground states in Landau level $N=0.$ At even filling, $\widetilde{\nu }=2,$ the C2DEG is either fully spin polarized with electrons occupying both valleys or spin unpolarized with electron occupying one of the two valleys. A finite electric field (or bias) applied perpendicularly to the plane of the C2DEG induces a series of first-order phase transitions between these different ground states. The transport gap or its slope is discontinuous at the bias where a transition occurs. Such discontinuity may result in a change in the transport properties of the C2DEG at that bias.