Pseudospin magnetism in graphene

We predict that neutral graphene bilayers are pseudospin magnets in which the charge density contribution from each valley and spin spontaneously shifts to one of the two layers. The band structure of this system is characterized by a momentum-space vortex, which is responsible for unusual competition between band and kinetic energies, leading to symmetry breaking in the vortex core. We discuss the possibility of realizing a pseudospin version of ferromagnetic metal spintronics in graphene bilayers based on hysteresis associated with this broken symmetry.

Figure 1

(Color online) Pseudospin orientation in a graphene bilayer broken symmetry state. In this figure the arrows represent both the magnitude and the direction of the $\widehat{x}\text{−}\widehat{y}$ projection of the pseudospin orientation $\widehat{n}$ as obtained from a mean-field-theory calculation for a neutral, unbiased bilayer with coupling constant $\alpha =1$. The arrows are shorter in the core of the momentum-space vortex because the pseudospins in the core have rotated spontaneously toward the $\widehat{z}$ or $-\widehat{z}$ direction.

Figure 2

(Color online) Phase diagrams of C2DESs with $J=2$ and 1. For the $J=2$ bilayer case we have taken $\overline{d}=0.2$. Pseudospin magnetism occurs at strong coupling $\alpha$ and weak doping $f$. ($1+f=n_{\uparrow }+n_{\downarrow }$ where the pseudospin density $n_{\sigma }={\sum }_{\mathit{k},i}⟨{\widehat{c}}_{\mathit{k},i,\sigma }^{†}{\widehat{c}}_{\mathbf{k},i,\sigma }⟩∕\mathcal{N}$ and $\mathcal{N}={\sum }_{\mathit{k},i}1$.) In the $J=2$ bilayer case, the Hartree potential favors smaller total polarization so that the initial normal (N) state instability (blue separatrix) is to antiferromagnetic (AF) states in which the pseudospin polarizations of different valley and spin components cancel. At larger $\alpha$, the normal state is unstable (green separatrix) to ferromagnetic (F) pseudospin states. In the $J=1$ monolayer case $\overline{d}=0$ so the phase boundaries (red separatrix) of F and AF broken-symmetry states coincide.

Figure 3

(Color online) Condensation energy per electron $\delta \epsilon$ [in units of ${\epsilon }_0\left(k_{\mathrm{c}}\right)$] as a function of $\alpha$ for an undoped $(f=0)$ $J=2$ C2DES. This figure shows results for both F (top) and AF (bottom) states.

Figure 4

(Color online) Metastable configurations of the pseudospin ferromagnet as a function of bias voltage $V_{\mathrm{g}}$ [in units of ${\epsilon }_0\left(k_{\mathrm{c}}\right)$] with $\alpha =1$ and $f=0$. We find self-consistent solutions of the gap equations (7, 8) in which the pseudospin polarization has the same sense in all four components (ferromagnetic), in three of the four components (ferrimagnetic), or in half of the four components (antiferromagnetic).