Electron-electron interactions and the phase diagram of a graphene bilayer

We study the effects of long and short-range electron-electron interactions in a graphene bilayer. Using a variational wave function technique we show that in the presence of long-range Coulomb interactions the clean bilayer is always unstable to electron and hole pocket formation with a finite ferromagnetic polarization. Furthermore, we argue that short-range electron-electron interactions lead to a staggered orientation of the ordered ferromagnetic moment in each layer (that is, $c$-axis antiferromagnetism). We also comment on the effects of doping and trigonal distortions of the electronic bands.

Figure 1

(Color online) Lattice structure of the bilayer. The $A$ sublattices are indicated by the darker spheres.

Figure 2

Band dispersions near the $K$ points in the bilayer. Bands are labeled by the numbers 1–4 as in the text.

Figure 3

(Color online) Sketch of the trial states: (a) Half-filled noninteracting ground state, (b) trial state with particle-hole pockets built upon (a), (c) doped noninteracting ground state, and (d) trial state with particle-hole pockets built upon (c).

Figure 4

(Color online) Left: Phase diagram of the graphene bilayer as a function of electron density away from half-filling, $n$ (electrons per carbon), and coupling strength, $g=e^2∕\left({ϵ}_0{v}_F\right)$, with $t_{\perp }=0.05$. Inset: $\Delta E$ as a function of $x$ (as defined in the text) in the paramagnetic (A), critical (B), and ferromagnetic (C) regions of the phase diagram.