Canted antiferromagnetism and excitonic order in gated double-layer graphene

We study the effects of the electron-electron interactions on the excitonic properties and charge-density modulations in the AB stacked double-layer (DL) graphene, placed in the external gate-potential $V$. The coexistence of the canted antiferromagnetic order and excitonic pairing gap has been studied with the help of the generalized Hubbard model. We calculate the chemical potential $\mu$, the average charge density difference between the layers $\delta{\bar{n}}$, the antiferromagnetic gap-function $\Delta_{\rm AFM}$ and the excitonic order parameters $\Delta_{\sigma}$ in the zero temperature limit. We found that the excitonic pairing order parameter has a larger energy scale than the canted antiferromagnetic gap-function. The charge neutrality, in the DL graphene system, occurs only in the absence of the external gate-potential $V$. Moreover, we have shown that the values of the antiferromagnetic gap-function $\Delta_{\rm AFM}$ and excitonic order parameter $\Delta_{\sigma}$ are always increasing at the large values of inter-layer Coulomb interaction, while they are decreasing for large values of the applied gate-potential $V$.