Low-Density Ferromagnetism in Biased Bilayer Graphene

We compute the phase diagram of a biased graphene bilayer. The existence of a ferromagnetic phase is discussed with respect to both carrier density and temperature. We find that the ferromagnetic transition is first-order, lowering the value of $U$ relatively to the usual Stoner criterion. We show that in the ferromagnetic phase the two planes have unequal magnetization and that the electronic density is holelike in one plane and electronlike in the other.

Figure 1

(a) Bilayer graphene DOS for $U=0$. Inset: Zoom near the gap region. (b) $U_c$ vs $\delta n$ in the low doping regime. Inset: The same as a function of $\tilde{\mu }$.

Figure 2

(a)–(c) show the $T=0$ solution for $m$, $\Delta m$, and $\Delta n$, respectively. (d) shows the $U$ vs $\delta n$ phase diagram at $T=0$: symbols are inferred from (a) and signal a first-order transition; lines stand for the second-order one given by Eq. (1). Labels: $P$, paramagnetic; $F$, ferromagnetic.

Figure 3

Hartree-Fock bands for $\uparrow$ (solid lines) and $\downarrow$ (dashed lines) spin polarizations.

Figure 4

(a)–(c) show the finite $T$ solution for $m$, $\Delta m$, and $\Delta n$, respectively, with $T$ measured in K. (d) shows the $U$ vs $T$ phase diagram.