Density Dependent Exchange Contribution to @ =@n and Compressibility in Graphene

We calculate @ =@n (where chemical potential and n electron density), which is associated with the compressibility, in graphene as a function of n, within the Hartree-Fock approximation. The exchange-driven Dirac-point logarithmic singularity in the quasiparticle velocity of intrinsic graphene disappears in the extrinsic case. The calculated renormalized @ =@n in extrinsic graphene on SiO2 has the same n 1=2 density dependence but is 20% larger than the inverse bare density of states, a relatively weak effect compared to the corresponding parabolic-band case. We predict that the renormalization effect can be enhanced to about 50% by changing the graphene substrate.