Lattice Theory of Pseudospin Ferromagnetism in Bilayer Graphene: Competing Orders and Interaction Induced Quantum Hall States

In mean-field-theory bilayer graphene's massive Dirac fermion model has a family of broken inversion symmetry ground states with charge gaps and flavor dependent spontaneous inter layer charge transfers. We use a lattice Hartree-Fock model to explore some of the physics which controls whether or not this type of broken symmetry state, which can be viewed as a pseudospin ferromagnet, occurs in nature. We find that inversion symmetry is still broken in the lattice model and estimate that transferred areal densities are $\sim 10^{-5}$ electrons per carbon atom, that the associated energy gaps are $\sim 10^{-2} {\rm eV}$, that the ordering condensation energies are $\sim 10^{-7} eV$ per carbon atom, and that the energy differences between competing orders at the neutrality point are $\sim 10^{-9} eV$ per carbon atom. We explore the quantum phase transitions induced by external magnetic fields and by externally controlled electric potential differences between the layers. We find, in particular, that in an external magnetic field coupling to spontaneous orbital moments favors broken time-reversal-symmetry states that have spontaneous quantized anomalous Hall effects. Our theory predicts a non monotonic behavior of the band gap at neutrality as a function of interlayer potential difference in qualitative agreement with recent experiments.Comment: 12 pages, 3 tables, 7 figure

Electronic Highways in Bilayer Graphene

Bilayer graphene with an interlayer potential difference has an energy gap and, when the potential difference varies spatially, topologically protected one-dimensional states localized along the difference's zero-lines. When disorder is absent, electronic travel directions along zero-line trajectories are fixed by valley Hall properties. Using the Landauer-B\"uttiker formula and the non-equilibrium Green's function technique we demonstrate numerically that collisions between electrons traveling in opposite directions, due to either disorder or changes in path direction, are strongly suppressed. We find that extremely long mean free paths of the order of hundreds of microns can be expected in relatively clean samples. This finding suggests the possibility of designing low power nanoscale electronic devices in which transport paths are controlled by gates which alter the inter-layer potential landscape.Comment: 8 pages, 5 figure

Moir\'e pattern interlayer potentials in van der Waals materials from random-phase approximation calculations

Stacking-dependent interlayer interactions are important for understanding the structural and electronic properties in incommensurable two dimensional material assemblies where long-range moir\'e patterns arise due to small lattice constant mismatch or twist angles. Here, we study the stacking-dependent interlayer coupling energies between graphene (G) and hexagonal boron nitride (BN) homo- and hetero-structures using high-level random-phase approximation (RPA) ab initio calculations. Our results show that although total binding energies within LDA and RPA differ substantially between a factor of 200%-400%, the energy differences as a function of stacking configuration yield nearly constant values with variations smaller than 20% meaning that LDA estimates are quite reliable. We produce phenomenological fits to these energy differences, which allows us to calculate various properties of interest including interlayer spacing, sliding energetics, pressure gradients and elastic coefficients to high accuracy. The importance of long-range interactions (captured by RPA but not LDA) on various properties is also discussed. Parameterisations for all fits are provided.Comment: 10 pages, 6 figures, 2 table

Origin of band gaps in graphene on hexagonal boron nitride

Recent progress in preparing well controlled 2D van der Waals heterojunctions has opened up a new frontier in materials physics. In this paper we address the intriguing energy gaps that are sometimes observed when a graphene sheet is placed on a hexagonal boron nitride substrate, demonstrating that they are produced by an interesting interplay between structural and electronic properties, including electronic many-body exchange interactions. Our theory is able to explain the observed gap behavior by accounting first for the structural relaxation of graphene's carbon atoms when placed on a boron nitride substrate and then for the influence of the substrate on low-energy $\pi$-electrons located at relaxed carbon atom sites. The methods we employ can be applied to many other van der Waals heterojunctions.Comment: 16 pages 15 figures. This version corrects minor numerical error