Nonuniform electron-hole pairing in graphene bilayer

Electron-hole (e-h) pairing caused by Coulomb interaction in the system of independently gated graphene layers is considered. The influence of the mismatch of concentrations of e and h and trigonal warping of their spectrum on critical temperature is studied. We predict the appearance of the state with finite value of Cooper pair momentum-Larkin-Ovchinnikov-Fulde-Ferrell-like (LOFF) state at mismatch of the concentrations of e and above the critical value. Internal Josephson effect in LOFF-like state caused by interlayer tunneling of e and h is considered. We suggest a new phase-sensitive experiment based on the internal Josephson effect for probing spatial structure of order parameter in a LOFF-like state.