Landau Levels in Bilayer Graphene under Pressure

Bilayer graphene under an applied magnetic field is an appealing platform for many-body ordered states built out of Landau levels because its ground state is ordered in the spin, valley, and orbital degrees of freedom, and this order can be controlled with applied magnetic or electric fields or by applying uniaxial pressure to reduce the distance between the graphene layers.

In our study, we find and characterize the ground state as a function of magnetic field, electric field, and pressure, using a tight-binding model with interactions treated at the Hartree-Fock level. We show that pressure and magnetic field are direct experimental methods for tuning the energetic balance between the Landau level separation and Coulomb interaction. We predict that using pressure to tune this balance, five ordered states can be observed, two of which are only accessible with applied pressure. These states are both orbitally polarized. One is maximally orbitally polarized but unpolarized in valley and spin, while the other is partially polarized in orbital and spin but unpolarized in valley.

We have posted our work on the arXiv and published it in Physical Review B.

Main result:

Below are the phase diagrams depicting the five possible ground states in our model and magnetic field, electric field, and pressure which may be used to construct them. The legend at the bottom indicates which Landau levels are filled and hence in which degrees of freedom the state is ordered.

Phase diagrams of LLs in BLGMany-body ground states for LLs in BLGPhase diagrams for (a) zero, (b) intermediate, and (c) high pressures are given; (d) magnifies (c). Five Landau-level Slater determinant and no Landau-level coherent states appear. Notice that applied pressure literally compresses the phase diagram so that all transitions occur at progressively lower fields, as explained in the text, but that the overall topology remains unchanged. (e) A schematic of the dot-diagram depiction of states devised by Lambert and Cote, and the dot-diagram representation of the different states appearing in our phase diagrams.

Abstract:

Bilayer graphene in a magnetic field hosts a variety of ordered phases built from eight Landau levels close in energy to the neutrality point. These levels are characterized by orbital n=0,1, valley ξ=+,- and spin σ=↑,↓; their relative energies depend strongly on the Coulomb interaction, magnetic field, and interlayer bias. We treat interactions at the Hartree-Fock level, including the effects of metallic gates, layer separation, spatial extent of the pz; orbitals, all Slonczewski-Weiss-McClure tight-binding parameters, and pressure. We obtain the ground state as function of the applied magnetic field, bias, and pressure. The gates, layer separation and extent of the pz orbitals weaken the Coulomb interaction at different length scales; these effects distort the phase diagram but do not change its topology. However, previously-predicted continuous transitions become discontinuous when all tight-binding parameters are included nonperturbatively. We find that pressure increases the importance of the noninteracting scale with respect to the Coulomb energy, which drives phase transitions to occur at lower fields. This brings two orbitally polarized states not yet predicted or observed into the experimentally accessible region of the phase diagram, in addition to previously-identified valley-, spin-, and partially orbitally polarized states.