Ising quantum Hall ferromagnetism in Landau levels|N|≥1of bilayer graphene

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 $N=0,\ifmmode\pm\else\textpm\fi{}1,\ifmmode\pm\else\textpm\fi{}2,....$ While Landau level $N=0$ is eightfold degenerate, higher Landau levels $(|N|\ensuremath{\ge}1)$ are fourfold degenerate when spin and valley degrees of freedom are counted. In this work, the Hartree-Fock approximation is used to study the phase diagram of the C2DEG at integer fillings $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\nu}}=1,2,3$ 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 ground states in Landau level $N=0.$ At even filling $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\nu}}=2,$ the C2DEG is either fully spin polarized with electrons occupying both valleys or spin unpolarized with electrons 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.