Competition between excitonic gap generation and disorder scattering in graphene

We study the disorder effect on the excitonic gap generation caused by strong Coulomb interaction in graphene. By solving the self-consistently coupled equations of dynamical fermion gap $m$ and disorder scattering rate $\Gamma$, we found a critical line on the plane of interaction strength $\lambda$ and disorder strength $g$. The phase diagram is divided into two regions: in the region with large $\lambda$ and small $g$, $m \neq 0$ and $\Gamma = 0$; in the other region, $m = 0$ and $\Gamma \neq 0$ for nonzero $g$. In particular, there is no coexistence of finite fermion gap and finite scattering rate. These results imply a strong competition between excitonic gap generation and disorder scattering. This conclusion does not change when an additional contact four-fermion interaction is included. For sufficiently large $\lambda$, the growing disorder may drive a quantum phase transition from an excitonic insulator to a metal.